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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Plasma Turbulence — Astrophysical Plasmas – Solar wind" lang="en" name="keywords"/> <base href="/html/2502.08883v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S1" title="In Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2" title="In Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Methods</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS1" title="In 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>MMS Observations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS2" title="In 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Numerical Simulation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px1" title="In 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title">MHD plasma model</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px2" title="In 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title">Turbulent driving</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px3" title="In 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title">Initial conditions & steady-state magnetic field</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px4" title="In 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title">The inhomogeneous mass density field</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S3" title="In Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S4" title="In Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Discussion & Conclusions</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#A1" title="In Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Maximum Likelihood Fits</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_multiline"> <h1 class="ltx_title ltx_title_document">Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id20.id1">in-situ</span> space observations and high-Reynolds number simulations</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0002-6962-0959" title="">Riddhi Bandyopadhyay<sup class="ltx_sup" id="id23.2.2.id1"><span class="ltx_text ltx_font_italic" id="id23.2.2.id1.1">†</span></sup></a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Astrophysical Sciences, Princeton, NJ 08544, USA </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0001-9199-7771" title="">James R. Beattie<sup class="ltx_sup" id="id24.2.2.id1"><span class="ltx_text ltx_font_italic" id="id24.2.2.id1.1">‡</span></sup></a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Astrophysical Sciences, Princeton, NJ 08544, USA </span> <span class="ltx_contact ltx_role_affiliation">Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, M5S3H8, ON, Canada </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0001-6411-0178" title="">Amitava Bhattacharjee</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Astrophysical Sciences, Princeton, NJ 08544, USA </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id19.12">Understanding the nature of compressible fluctuations in a broad range of turbulent plasmas, from the intracluster medium to the solar wind, has been an active field of research in the past decades. Theoretical frameworks for weakly compressible MHD turbulence in an inhomogeneous background magnetic field predict a linear scaling of the normalized mass density fluctuation (<math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="id8.1.m1.1"><semantics id="id8.1.m1.1a"><mrow id="id8.1.m1.1.1" xref="id8.1.m1.1.1.cmml"><mrow id="id8.1.m1.1.1.2" xref="id8.1.m1.1.1.2.cmml"><mi id="id8.1.m1.1.1.2.2" xref="id8.1.m1.1.1.2.2.cmml">δ</mi><mo id="id8.1.m1.1.1.2.1" xref="id8.1.m1.1.1.2.1.cmml">⁢</mo><mi id="id8.1.m1.1.1.2.3" xref="id8.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="id8.1.m1.1.1.1" xref="id8.1.m1.1.1.1.cmml">/</mo><msub id="id8.1.m1.1.1.3" xref="id8.1.m1.1.1.3.cmml"><mi id="id8.1.m1.1.1.3.2" xref="id8.1.m1.1.1.3.2.cmml">ρ</mi><mn id="id8.1.m1.1.1.3.3" xref="id8.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="id8.1.m1.1b"><apply id="id8.1.m1.1.1.cmml" xref="id8.1.m1.1.1"><divide id="id8.1.m1.1.1.1.cmml" xref="id8.1.m1.1.1.1"></divide><apply id="id8.1.m1.1.1.2.cmml" xref="id8.1.m1.1.1.2"><times id="id8.1.m1.1.1.2.1.cmml" xref="id8.1.m1.1.1.2.1"></times><ci id="id8.1.m1.1.1.2.2.cmml" xref="id8.1.m1.1.1.2.2">𝛿</ci><ci id="id8.1.m1.1.1.2.3.cmml" xref="id8.1.m1.1.1.2.3">𝜌</ci></apply><apply id="id8.1.m1.1.1.3.cmml" xref="id8.1.m1.1.1.3"><csymbol cd="ambiguous" id="id8.1.m1.1.1.3.1.cmml" xref="id8.1.m1.1.1.3">subscript</csymbol><ci id="id8.1.m1.1.1.3.2.cmml" xref="id8.1.m1.1.1.3.2">𝜌</ci><cn id="id8.1.m1.1.1.3.3.cmml" type="integer" xref="id8.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.1.m1.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="id8.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>), as a function of the turbulent Mach number (<math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="id9.2.m2.1"><semantics id="id9.2.m2.1a"><msub id="id9.2.m2.1.1" xref="id9.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="id9.2.m2.1.1.2" xref="id9.2.m2.1.1.2.cmml">ℳ</mi><mi id="id9.2.m2.1.1.3" xref="id9.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="id9.2.m2.1b"><apply id="id9.2.m2.1.1.cmml" xref="id9.2.m2.1.1"><csymbol cd="ambiguous" id="id9.2.m2.1.1.1.cmml" xref="id9.2.m2.1.1">subscript</csymbol><ci id="id9.2.m2.1.1.2.cmml" xref="id9.2.m2.1.1.2">ℳ</ci><ci id="id9.2.m2.1.1.3.cmml" xref="id9.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.2.m2.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="id9.2.m2.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>), <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="id10.3.m3.1"><semantics id="id10.3.m3.1a"><mrow id="id10.3.m3.1.1" xref="id10.3.m3.1.1.cmml"><mrow id="id10.3.m3.1.1.2" xref="id10.3.m3.1.1.2.cmml"><mrow id="id10.3.m3.1.1.2.2" xref="id10.3.m3.1.1.2.2.cmml"><mi id="id10.3.m3.1.1.2.2.2" xref="id10.3.m3.1.1.2.2.2.cmml">δ</mi><mo id="id10.3.m3.1.1.2.2.1" xref="id10.3.m3.1.1.2.2.1.cmml">⁢</mo><mi id="id10.3.m3.1.1.2.2.3" xref="id10.3.m3.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="id10.3.m3.1.1.2.1" xref="id10.3.m3.1.1.2.1.cmml">/</mo><msub id="id10.3.m3.1.1.2.3" xref="id10.3.m3.1.1.2.3.cmml"><mi id="id10.3.m3.1.1.2.3.2" xref="id10.3.m3.1.1.2.3.2.cmml">ρ</mi><mn id="id10.3.m3.1.1.2.3.3" xref="id10.3.m3.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="id10.3.m3.1.1.1" xref="id10.3.m3.1.1.1.cmml">∝</mo><msub id="id10.3.m3.1.1.3" xref="id10.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id10.3.m3.1.1.3.2" xref="id10.3.m3.1.1.3.2.cmml">ℳ</mi><mi id="id10.3.m3.1.1.3.3" xref="id10.3.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="id10.3.m3.1b"><apply id="id10.3.m3.1.1.cmml" xref="id10.3.m3.1.1"><csymbol cd="latexml" id="id10.3.m3.1.1.1.cmml" xref="id10.3.m3.1.1.1">proportional-to</csymbol><apply id="id10.3.m3.1.1.2.cmml" xref="id10.3.m3.1.1.2"><divide id="id10.3.m3.1.1.2.1.cmml" xref="id10.3.m3.1.1.2.1"></divide><apply id="id10.3.m3.1.1.2.2.cmml" xref="id10.3.m3.1.1.2.2"><times id="id10.3.m3.1.1.2.2.1.cmml" xref="id10.3.m3.1.1.2.2.1"></times><ci id="id10.3.m3.1.1.2.2.2.cmml" xref="id10.3.m3.1.1.2.2.2">𝛿</ci><ci id="id10.3.m3.1.1.2.2.3.cmml" xref="id10.3.m3.1.1.2.2.3">𝜌</ci></apply><apply id="id10.3.m3.1.1.2.3.cmml" xref="id10.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="id10.3.m3.1.1.2.3.1.cmml" xref="id10.3.m3.1.1.2.3">subscript</csymbol><ci id="id10.3.m3.1.1.2.3.2.cmml" xref="id10.3.m3.1.1.2.3.2">𝜌</ci><cn id="id10.3.m3.1.1.2.3.3.cmml" type="integer" xref="id10.3.m3.1.1.2.3.3">0</cn></apply></apply><apply id="id10.3.m3.1.1.3.cmml" xref="id10.3.m3.1.1.3"><csymbol cd="ambiguous" id="id10.3.m3.1.1.3.1.cmml" xref="id10.3.m3.1.1.3">subscript</csymbol><ci id="id10.3.m3.1.1.3.2.cmml" xref="id10.3.m3.1.1.3.2">ℳ</ci><ci id="id10.3.m3.1.1.3.3.cmml" xref="id10.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id10.3.m3.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="id10.3.m3.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. However, so far the scaling relation has been tested only using moderate to low plasma beta (<math alttext="\beta\lesssim 1" class="ltx_Math" display="inline" id="id11.4.m4.1"><semantics id="id11.4.m4.1a"><mrow id="id11.4.m4.1.1" xref="id11.4.m4.1.1.cmml"><mi id="id11.4.m4.1.1.2" xref="id11.4.m4.1.1.2.cmml">β</mi><mo id="id11.4.m4.1.1.1" xref="id11.4.m4.1.1.1.cmml">≲</mo><mn id="id11.4.m4.1.1.3" xref="id11.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id11.4.m4.1b"><apply id="id11.4.m4.1.1.cmml" xref="id11.4.m4.1.1"><csymbol cd="latexml" id="id11.4.m4.1.1.1.cmml" xref="id11.4.m4.1.1.1">less-than-or-similar-to</csymbol><ci id="id11.4.m4.1.1.2.cmml" xref="id11.4.m4.1.1.2">𝛽</ci><cn id="id11.4.m4.1.1.3.cmml" type="integer" xref="id11.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.4.m4.1c">\beta\lesssim 1</annotation><annotation encoding="application/x-llamapun" id="id11.4.m4.1d">italic_β ≲ 1</annotation></semantics></math>) solar wind observational data where the compressibility is weak <math alttext="\delta\rho/\rho_{0}\sim 0.1" class="ltx_Math" display="inline" id="id12.5.m5.1"><semantics id="id12.5.m5.1a"><mrow id="id12.5.m5.1.1" xref="id12.5.m5.1.1.cmml"><mrow id="id12.5.m5.1.1.2" xref="id12.5.m5.1.1.2.cmml"><mrow id="id12.5.m5.1.1.2.2" xref="id12.5.m5.1.1.2.2.cmml"><mi id="id12.5.m5.1.1.2.2.2" xref="id12.5.m5.1.1.2.2.2.cmml">δ</mi><mo id="id12.5.m5.1.1.2.2.1" xref="id12.5.m5.1.1.2.2.1.cmml">⁢</mo><mi id="id12.5.m5.1.1.2.2.3" xref="id12.5.m5.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="id12.5.m5.1.1.2.1" xref="id12.5.m5.1.1.2.1.cmml">/</mo><msub id="id12.5.m5.1.1.2.3" xref="id12.5.m5.1.1.2.3.cmml"><mi id="id12.5.m5.1.1.2.3.2" xref="id12.5.m5.1.1.2.3.2.cmml">ρ</mi><mn id="id12.5.m5.1.1.2.3.3" xref="id12.5.m5.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="id12.5.m5.1.1.1" xref="id12.5.m5.1.1.1.cmml">∼</mo><mn id="id12.5.m5.1.1.3" xref="id12.5.m5.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="id12.5.m5.1b"><apply id="id12.5.m5.1.1.cmml" xref="id12.5.m5.1.1"><csymbol cd="latexml" id="id12.5.m5.1.1.1.cmml" xref="id12.5.m5.1.1.1">similar-to</csymbol><apply id="id12.5.m5.1.1.2.cmml" xref="id12.5.m5.1.1.2"><divide id="id12.5.m5.1.1.2.1.cmml" xref="id12.5.m5.1.1.2.1"></divide><apply id="id12.5.m5.1.1.2.2.cmml" xref="id12.5.m5.1.1.2.2"><times id="id12.5.m5.1.1.2.2.1.cmml" xref="id12.5.m5.1.1.2.2.1"></times><ci id="id12.5.m5.1.1.2.2.2.cmml" xref="id12.5.m5.1.1.2.2.2">𝛿</ci><ci id="id12.5.m5.1.1.2.2.3.cmml" xref="id12.5.m5.1.1.2.2.3">𝜌</ci></apply><apply id="id12.5.m5.1.1.2.3.cmml" xref="id12.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="id12.5.m5.1.1.2.3.1.cmml" xref="id12.5.m5.1.1.2.3">subscript</csymbol><ci id="id12.5.m5.1.1.2.3.2.cmml" xref="id12.5.m5.1.1.2.3.2">𝜌</ci><cn id="id12.5.m5.1.1.2.3.3.cmml" type="integer" xref="id12.5.m5.1.1.2.3.3">0</cn></apply></apply><cn id="id12.5.m5.1.1.3.cmml" type="float" xref="id12.5.m5.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id12.5.m5.1c">\delta\rho/\rho_{0}\sim 0.1</annotation><annotation encoding="application/x-llamapun" id="id12.5.m5.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∼ 0.1</annotation></semantics></math>. Here, we combine NASA’s Magnetospheric Multiscale Mission data in Earth’s magnetosheath, where <math alttext="\beta\sim 10" class="ltx_Math" display="inline" id="id13.6.m6.1"><semantics id="id13.6.m6.1a"><mrow id="id13.6.m6.1.1" xref="id13.6.m6.1.1.cmml"><mi id="id13.6.m6.1.1.2" xref="id13.6.m6.1.1.2.cmml">β</mi><mo id="id13.6.m6.1.1.1" xref="id13.6.m6.1.1.1.cmml">∼</mo><mn id="id13.6.m6.1.1.3" xref="id13.6.m6.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="id13.6.m6.1b"><apply id="id13.6.m6.1.1.cmml" xref="id13.6.m6.1.1"><csymbol cd="latexml" id="id13.6.m6.1.1.1.cmml" xref="id13.6.m6.1.1.1">similar-to</csymbol><ci id="id13.6.m6.1.1.2.cmml" xref="id13.6.m6.1.1.2">𝛽</ci><cn id="id13.6.m6.1.1.3.cmml" type="integer" xref="id13.6.m6.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id13.6.m6.1c">\beta\sim 10</annotation><annotation encoding="application/x-llamapun" id="id13.6.m6.1d">italic_β ∼ 10</annotation></semantics></math> is high, and <math alttext="\beta\sim 1" class="ltx_Math" display="inline" id="id14.7.m7.1"><semantics id="id14.7.m7.1a"><mrow id="id14.7.m7.1.1" xref="id14.7.m7.1.1.cmml"><mi id="id14.7.m7.1.1.2" xref="id14.7.m7.1.1.2.cmml">β</mi><mo id="id14.7.m7.1.1.1" xref="id14.7.m7.1.1.1.cmml">∼</mo><mn id="id14.7.m7.1.1.3" xref="id14.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id14.7.m7.1b"><apply id="id14.7.m7.1.1.cmml" xref="id14.7.m7.1.1"><csymbol cd="latexml" id="id14.7.m7.1.1.1.cmml" xref="id14.7.m7.1.1.1">similar-to</csymbol><ci id="id14.7.m7.1.1.2.cmml" xref="id14.7.m7.1.1.2">𝛽</ci><cn id="id14.7.m7.1.1.3.cmml" type="integer" xref="id14.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id14.7.m7.1c">\beta\sim 1</annotation><annotation encoding="application/x-llamapun" id="id14.7.m7.1d">italic_β ∼ 1</annotation></semantics></math> highly-compressible magnetohydrodynamic turbulence simulations at unprecedented resolutions. Both show that <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="id15.8.m8.1"><semantics id="id15.8.m8.1a"><mrow id="id15.8.m8.1.1" xref="id15.8.m8.1.1.cmml"><mrow id="id15.8.m8.1.1.2" xref="id15.8.m8.1.1.2.cmml"><mrow id="id15.8.m8.1.1.2.2" xref="id15.8.m8.1.1.2.2.cmml"><mi id="id15.8.m8.1.1.2.2.2" xref="id15.8.m8.1.1.2.2.2.cmml">δ</mi><mo id="id15.8.m8.1.1.2.2.1" xref="id15.8.m8.1.1.2.2.1.cmml">⁢</mo><mi id="id15.8.m8.1.1.2.2.3" xref="id15.8.m8.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="id15.8.m8.1.1.2.1" xref="id15.8.m8.1.1.2.1.cmml">/</mo><msub id="id15.8.m8.1.1.2.3" xref="id15.8.m8.1.1.2.3.cmml"><mi id="id15.8.m8.1.1.2.3.2" xref="id15.8.m8.1.1.2.3.2.cmml">ρ</mi><mn id="id15.8.m8.1.1.2.3.3" xref="id15.8.m8.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="id15.8.m8.1.1.1" xref="id15.8.m8.1.1.1.cmml">∝</mo><msub id="id15.8.m8.1.1.3" xref="id15.8.m8.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id15.8.m8.1.1.3.2" xref="id15.8.m8.1.1.3.2.cmml">ℳ</mi><mi id="id15.8.m8.1.1.3.3" xref="id15.8.m8.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="id15.8.m8.1b"><apply id="id15.8.m8.1.1.cmml" xref="id15.8.m8.1.1"><csymbol cd="latexml" id="id15.8.m8.1.1.1.cmml" xref="id15.8.m8.1.1.1">proportional-to</csymbol><apply id="id15.8.m8.1.1.2.cmml" xref="id15.8.m8.1.1.2"><divide id="id15.8.m8.1.1.2.1.cmml" xref="id15.8.m8.1.1.2.1"></divide><apply id="id15.8.m8.1.1.2.2.cmml" xref="id15.8.m8.1.1.2.2"><times id="id15.8.m8.1.1.2.2.1.cmml" xref="id15.8.m8.1.1.2.2.1"></times><ci id="id15.8.m8.1.1.2.2.2.cmml" xref="id15.8.m8.1.1.2.2.2">𝛿</ci><ci id="id15.8.m8.1.1.2.2.3.cmml" xref="id15.8.m8.1.1.2.2.3">𝜌</ci></apply><apply id="id15.8.m8.1.1.2.3.cmml" xref="id15.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="id15.8.m8.1.1.2.3.1.cmml" xref="id15.8.m8.1.1.2.3">subscript</csymbol><ci id="id15.8.m8.1.1.2.3.2.cmml" xref="id15.8.m8.1.1.2.3.2">𝜌</ci><cn id="id15.8.m8.1.1.2.3.3.cmml" type="integer" xref="id15.8.m8.1.1.2.3.3">0</cn></apply></apply><apply id="id15.8.m8.1.1.3.cmml" xref="id15.8.m8.1.1.3"><csymbol cd="ambiguous" id="id15.8.m8.1.1.3.1.cmml" xref="id15.8.m8.1.1.3">subscript</csymbol><ci id="id15.8.m8.1.1.3.2.cmml" xref="id15.8.m8.1.1.3.2">ℳ</ci><ci id="id15.8.m8.1.1.3.3.cmml" xref="id15.8.m8.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id15.8.m8.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="id15.8.m8.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> holds across a broad range of <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="id16.9.m9.1"><semantics id="id16.9.m9.1a"><mrow id="id16.9.m9.1.1" xref="id16.9.m9.1.1.cmml"><mrow id="id16.9.m9.1.1.2" xref="id16.9.m9.1.1.2.cmml"><mi id="id16.9.m9.1.1.2.2" xref="id16.9.m9.1.1.2.2.cmml">δ</mi><mo id="id16.9.m9.1.1.2.1" xref="id16.9.m9.1.1.2.1.cmml">⁢</mo><mi id="id16.9.m9.1.1.2.3" xref="id16.9.m9.1.1.2.3.cmml">ρ</mi></mrow><mo id="id16.9.m9.1.1.1" xref="id16.9.m9.1.1.1.cmml">/</mo><msub id="id16.9.m9.1.1.3" xref="id16.9.m9.1.1.3.cmml"><mi id="id16.9.m9.1.1.3.2" xref="id16.9.m9.1.1.3.2.cmml">ρ</mi><mn id="id16.9.m9.1.1.3.3" xref="id16.9.m9.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="id16.9.m9.1b"><apply id="id16.9.m9.1.1.cmml" xref="id16.9.m9.1.1"><divide id="id16.9.m9.1.1.1.cmml" xref="id16.9.m9.1.1.1"></divide><apply id="id16.9.m9.1.1.2.cmml" xref="id16.9.m9.1.1.2"><times id="id16.9.m9.1.1.2.1.cmml" xref="id16.9.m9.1.1.2.1"></times><ci id="id16.9.m9.1.1.2.2.cmml" xref="id16.9.m9.1.1.2.2">𝛿</ci><ci id="id16.9.m9.1.1.2.3.cmml" xref="id16.9.m9.1.1.2.3">𝜌</ci></apply><apply id="id16.9.m9.1.1.3.cmml" xref="id16.9.m9.1.1.3"><csymbol cd="ambiguous" id="id16.9.m9.1.1.3.1.cmml" xref="id16.9.m9.1.1.3">subscript</csymbol><ci id="id16.9.m9.1.1.3.2.cmml" xref="id16.9.m9.1.1.3.2">𝜌</ci><cn id="id16.9.m9.1.1.3.3.cmml" type="integer" xref="id16.9.m9.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id16.9.m9.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="id16.9.m9.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="id17.10.m10.1"><semantics id="id17.10.m10.1a"><msub id="id17.10.m10.1.1" xref="id17.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="id17.10.m10.1.1.2" xref="id17.10.m10.1.1.2.cmml">ℳ</mi><mi id="id17.10.m10.1.1.3" xref="id17.10.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="id17.10.m10.1b"><apply id="id17.10.m10.1.1.cmml" xref="id17.10.m10.1.1"><csymbol cd="ambiguous" id="id17.10.m10.1.1.1.cmml" xref="id17.10.m10.1.1">subscript</csymbol><ci id="id17.10.m10.1.1.2.cmml" xref="id17.10.m10.1.1.2">ℳ</ci><ci id="id17.10.m10.1.1.3.cmml" xref="id17.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id17.10.m10.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="id17.10.m10.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\beta" class="ltx_Math" display="inline" id="id18.11.m11.1"><semantics id="id18.11.m11.1a"><mi id="id18.11.m11.1.1" xref="id18.11.m11.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="id18.11.m11.1b"><ci id="id18.11.m11.1.1.cmml" xref="id18.11.m11.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="id18.11.m11.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="id18.11.m11.1d">italic_β</annotation></semantics></math>, demonstrating that <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="id19.12.m12.1"><semantics id="id19.12.m12.1a"><mrow id="id19.12.m12.1.1" xref="id19.12.m12.1.1.cmml"><mrow id="id19.12.m12.1.1.2" xref="id19.12.m12.1.1.2.cmml"><mrow id="id19.12.m12.1.1.2.2" xref="id19.12.m12.1.1.2.2.cmml"><mi id="id19.12.m12.1.1.2.2.2" xref="id19.12.m12.1.1.2.2.2.cmml">δ</mi><mo id="id19.12.m12.1.1.2.2.1" xref="id19.12.m12.1.1.2.2.1.cmml">⁢</mo><mi id="id19.12.m12.1.1.2.2.3" xref="id19.12.m12.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="id19.12.m12.1.1.2.1" xref="id19.12.m12.1.1.2.1.cmml">/</mo><msub id="id19.12.m12.1.1.2.3" xref="id19.12.m12.1.1.2.3.cmml"><mi id="id19.12.m12.1.1.2.3.2" xref="id19.12.m12.1.1.2.3.2.cmml">ρ</mi><mn id="id19.12.m12.1.1.2.3.3" xref="id19.12.m12.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="id19.12.m12.1.1.1" xref="id19.12.m12.1.1.1.cmml">∝</mo><msub id="id19.12.m12.1.1.3" xref="id19.12.m12.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id19.12.m12.1.1.3.2" xref="id19.12.m12.1.1.3.2.cmml">ℳ</mi><mi id="id19.12.m12.1.1.3.3" xref="id19.12.m12.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="id19.12.m12.1b"><apply id="id19.12.m12.1.1.cmml" xref="id19.12.m12.1.1"><csymbol cd="latexml" id="id19.12.m12.1.1.1.cmml" xref="id19.12.m12.1.1.1">proportional-to</csymbol><apply id="id19.12.m12.1.1.2.cmml" xref="id19.12.m12.1.1.2"><divide id="id19.12.m12.1.1.2.1.cmml" xref="id19.12.m12.1.1.2.1"></divide><apply id="id19.12.m12.1.1.2.2.cmml" xref="id19.12.m12.1.1.2.2"><times id="id19.12.m12.1.1.2.2.1.cmml" xref="id19.12.m12.1.1.2.2.1"></times><ci id="id19.12.m12.1.1.2.2.2.cmml" xref="id19.12.m12.1.1.2.2.2">𝛿</ci><ci id="id19.12.m12.1.1.2.2.3.cmml" xref="id19.12.m12.1.1.2.2.3">𝜌</ci></apply><apply id="id19.12.m12.1.1.2.3.cmml" xref="id19.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="id19.12.m12.1.1.2.3.1.cmml" xref="id19.12.m12.1.1.2.3">subscript</csymbol><ci id="id19.12.m12.1.1.2.3.2.cmml" xref="id19.12.m12.1.1.2.3.2">𝜌</ci><cn id="id19.12.m12.1.1.2.3.3.cmml" type="integer" xref="id19.12.m12.1.1.2.3.3">0</cn></apply></apply><apply id="id19.12.m12.1.1.3.cmml" xref="id19.12.m12.1.1.3"><csymbol cd="ambiguous" id="id19.12.m12.1.1.3.1.cmml" xref="id19.12.m12.1.1.3">subscript</csymbol><ci id="id19.12.m12.1.1.3.2.cmml" xref="id19.12.m12.1.1.3.2">ℳ</ci><ci id="id19.12.m12.1.1.3.3.cmml" xref="id19.12.m12.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id19.12.m12.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="id19.12.m12.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is a robust compressible turbulence relation, going beyond the asymptotics of the weakly compressible theory. We discuss the findings in the context of understanding the nature of strongly compressible turbulent fluctuations and the driving parameter in astrophysical and space plasmas.</p> </div> <div class="ltx_keywords">Plasma Turbulence — Astrophysical Plasmas – Solar wind </div> <span class="ltx_note ltx_note_frontmatter ltx_role_software" id="id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">software: </span>Data analysis and visualization software used in this study: <span class="ltx_text ltx_font_smallcaps" id="id1.1">C++</span> <cite class="ltx_cite ltx_citemacro_citep">(Stroustrup, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib73" title="">2013</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.2">numpy</span> <cite class="ltx_cite ltx_citemacro_citep">(Oliphant, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib57" title="">2006</a>; Harris et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib35" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.3">numba</span>, <cite class="ltx_cite ltx_citemacro_citep">(Lam et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib46" title="">2015</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.4">matplotlib</span> <cite class="ltx_cite ltx_citemacro_citep">(Hunter, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib38" title="">2007</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.5">cython</span> <cite class="ltx_cite ltx_citemacro_citep">(Behnel et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib8" title="">2011</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.6">visit</span> <cite class="ltx_cite ltx_citemacro_citep">(Childs et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib16" title="">2012</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.7">scipy</span> <cite class="ltx_cite ltx_citemacro_citep">(Virtanen et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib80" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.8">scikit-image</span> <cite class="ltx_cite ltx_citemacro_citep">(van der Walt et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib78" title="">2014</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.9">cmasher</span> <cite class="ltx_cite ltx_citemacro_citep">(van der Velden, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib77" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.10">pandas</span> <cite class="ltx_cite ltx_citemacro_citep">(pandas development team, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib60" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.11">joblib</span> <cite class="ltx_cite ltx_citemacro_citep">(Joblib Development Team, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib39" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.12">emcee</span> <cite class="ltx_cite ltx_citemacro_citep">(Foreman-Mackey et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib30" title="">2013</a>)</cite>, <span class="ltx_text ltx_font_smallcaps" id="id1.13">corner<cite class="ltx_cite ltx_citemacro_citep"><span class="ltx_text ltx_font_upright" id="id1.13.1.1">(</span>Foreman-Mackey<span class="ltx_text ltx_font_upright" id="id1.13.2.2.1.1">, </span><a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib29" title="">2016</a><span class="ltx_text ltx_font_upright" id="id1.13.3.3">)</span></cite></span></span></span></span> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Most of the visible Universe is made of ionized plasmas, and most naturally occurring plasmas exist in a turbulent state <cite class="ltx_cite ltx_citemacro_citep">(Matthaeus & Velli, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib50" title="">2011</a>)</cite>, ranging from the interstellar medium <cite class="ltx_cite ltx_citemacro_citep">(Elmegreen & Scalo, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib21" title="">2004</a>; Brandenburg & Lazarian, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib12" title="">2013</a>; Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite>, intracluster medium <cite class="ltx_cite ltx_citemacro_citep">(Mohapatra & Sharma, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib54" title="">2019</a>; Kunz et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib45" title="">2022</a>)</cite> to solar wind and planetary magnetospheres <cite class="ltx_cite ltx_citemacro_citep">(Verscharen et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib79" title="">2019</a>)</cite>. Turbulent fluctuations in plasmas have important effects on heating, transport, and overall evolution and structure in these systems <cite class="ltx_cite ltx_citemacro_citep">(Usmanov et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib76" title="">2014</a>; Adhikari et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib1" title="">2023</a>)</cite>. An important parameter controlling the dynamics in turbulent plasmas is the plasma beta <math alttext="\beta" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_β</annotation></semantics></math>, defined as</p> <table class="ltx_equation ltx_eqn_table" id="S1.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\beta=\frac{n\,k_{B}\,T}{B^{2}/\left(2\,\mu_{0}\right)}," class="ltx_Math" display="block" id="S1.E1.m1.2"><semantics id="S1.E1.m1.2a"><mrow id="S1.E1.m1.2.2.1" xref="S1.E1.m1.2.2.1.1.cmml"><mrow id="S1.E1.m1.2.2.1.1" xref="S1.E1.m1.2.2.1.1.cmml"><mi id="S1.E1.m1.2.2.1.1.2" xref="S1.E1.m1.2.2.1.1.2.cmml">β</mi><mo id="S1.E1.m1.2.2.1.1.1" xref="S1.E1.m1.2.2.1.1.1.cmml">=</mo><mfrac id="S1.E1.m1.1.1" xref="S1.E1.m1.1.1.cmml"><mrow id="S1.E1.m1.1.1.3" xref="S1.E1.m1.1.1.3.cmml"><mi id="S1.E1.m1.1.1.3.2" xref="S1.E1.m1.1.1.3.2.cmml">n</mi><mo id="S1.E1.m1.1.1.3.1" lspace="0.170em" xref="S1.E1.m1.1.1.3.1.cmml">⁢</mo><msub id="S1.E1.m1.1.1.3.3" xref="S1.E1.m1.1.1.3.3.cmml"><mi id="S1.E1.m1.1.1.3.3.2" xref="S1.E1.m1.1.1.3.3.2.cmml">k</mi><mi id="S1.E1.m1.1.1.3.3.3" xref="S1.E1.m1.1.1.3.3.3.cmml">B</mi></msub><mo id="S1.E1.m1.1.1.3.1a" xref="S1.E1.m1.1.1.3.1.cmml">⁢</mo><mi id="S1.E1.m1.1.1.3.4" xref="S1.E1.m1.1.1.3.4.cmml">T</mi></mrow><mrow id="S1.E1.m1.1.1.1" xref="S1.E1.m1.1.1.1.cmml"><msup id="S1.E1.m1.1.1.1.3" xref="S1.E1.m1.1.1.1.3.cmml"><mi id="S1.E1.m1.1.1.1.3.2" xref="S1.E1.m1.1.1.1.3.2.cmml">B</mi><mn id="S1.E1.m1.1.1.1.3.3" xref="S1.E1.m1.1.1.1.3.3.cmml">2</mn></msup><mo id="S1.E1.m1.1.1.1.2" xref="S1.E1.m1.1.1.1.2.cmml">/</mo><mrow id="S1.E1.m1.1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.1.1.cmml"><mo id="S1.E1.m1.1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.E1.m1.1.1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.1.1.cmml"><mn id="S1.E1.m1.1.1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.1.1.2.cmml">2</mn><mo id="S1.E1.m1.1.1.1.1.1.1.1" lspace="0.170em" xref="S1.E1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S1.E1.m1.1.1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.1.1.3.cmml"><mi id="S1.E1.m1.1.1.1.1.1.1.3.2" xref="S1.E1.m1.1.1.1.1.1.1.3.2.cmml">μ</mi><mn id="S1.E1.m1.1.1.1.1.1.1.3.3" xref="S1.E1.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S1.E1.m1.1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mrow><mo id="S1.E1.m1.2.2.1.2" xref="S1.E1.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E1.m1.2b"><apply id="S1.E1.m1.2.2.1.1.cmml" xref="S1.E1.m1.2.2.1"><eq id="S1.E1.m1.2.2.1.1.1.cmml" xref="S1.E1.m1.2.2.1.1.1"></eq><ci id="S1.E1.m1.2.2.1.1.2.cmml" xref="S1.E1.m1.2.2.1.1.2">𝛽</ci><apply id="S1.E1.m1.1.1.cmml" xref="S1.E1.m1.1.1"><divide id="S1.E1.m1.1.1.2.cmml" xref="S1.E1.m1.1.1"></divide><apply id="S1.E1.m1.1.1.3.cmml" xref="S1.E1.m1.1.1.3"><times id="S1.E1.m1.1.1.3.1.cmml" xref="S1.E1.m1.1.1.3.1"></times><ci id="S1.E1.m1.1.1.3.2.cmml" xref="S1.E1.m1.1.1.3.2">𝑛</ci><apply id="S1.E1.m1.1.1.3.3.cmml" xref="S1.E1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.E1.m1.1.1.3.3.1.cmml" xref="S1.E1.m1.1.1.3.3">subscript</csymbol><ci id="S1.E1.m1.1.1.3.3.2.cmml" xref="S1.E1.m1.1.1.3.3.2">𝑘</ci><ci id="S1.E1.m1.1.1.3.3.3.cmml" xref="S1.E1.m1.1.1.3.3.3">𝐵</ci></apply><ci id="S1.E1.m1.1.1.3.4.cmml" xref="S1.E1.m1.1.1.3.4">𝑇</ci></apply><apply id="S1.E1.m1.1.1.1.cmml" xref="S1.E1.m1.1.1.1"><divide id="S1.E1.m1.1.1.1.2.cmml" xref="S1.E1.m1.1.1.1.2"></divide><apply id="S1.E1.m1.1.1.1.3.cmml" xref="S1.E1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E1.m1.1.1.1.3.1.cmml" xref="S1.E1.m1.1.1.1.3">superscript</csymbol><ci id="S1.E1.m1.1.1.1.3.2.cmml" xref="S1.E1.m1.1.1.1.3.2">𝐵</ci><cn id="S1.E1.m1.1.1.1.3.3.cmml" type="integer" xref="S1.E1.m1.1.1.1.3.3">2</cn></apply><apply id="S1.E1.m1.1.1.1.1.1.1.cmml" xref="S1.E1.m1.1.1.1.1.1"><times id="S1.E1.m1.1.1.1.1.1.1.1.cmml" xref="S1.E1.m1.1.1.1.1.1.1.1"></times><cn id="S1.E1.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="S1.E1.m1.1.1.1.1.1.1.2">2</cn><apply id="S1.E1.m1.1.1.1.1.1.1.3.cmml" xref="S1.E1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E1.m1.1.1.1.1.1.1.3.1.cmml" xref="S1.E1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S1.E1.m1.1.1.1.1.1.1.3.2.cmml" xref="S1.E1.m1.1.1.1.1.1.1.3.2">𝜇</ci><cn id="S1.E1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.E1.m1.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.2c">\beta=\frac{n\,k_{B}\,T}{B^{2}/\left(2\,\mu_{0}\right)},</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.2d">italic_β = divide start_ARG italic_n italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG start_ARG italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p1.11">where <math alttext="n" class="ltx_Math" display="inline" id="S1.p1.2.m1.1"><semantics id="S1.p1.2.m1.1a"><mi id="S1.p1.2.m1.1.1" xref="S1.p1.2.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m1.1b"><ci id="S1.p1.2.m1.1.1.cmml" xref="S1.p1.2.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m1.1d">italic_n</annotation></semantics></math> is the plasma number density, <math alttext="k_{B}" class="ltx_Math" display="inline" id="S1.p1.3.m2.1"><semantics id="S1.p1.3.m2.1a"><msub id="S1.p1.3.m2.1.1" xref="S1.p1.3.m2.1.1.cmml"><mi id="S1.p1.3.m2.1.1.2" xref="S1.p1.3.m2.1.1.2.cmml">k</mi><mi id="S1.p1.3.m2.1.1.3" xref="S1.p1.3.m2.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p1.3.m2.1b"><apply id="S1.p1.3.m2.1.1.cmml" xref="S1.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S1.p1.3.m2.1.1.1.cmml" xref="S1.p1.3.m2.1.1">subscript</csymbol><ci id="S1.p1.3.m2.1.1.2.cmml" xref="S1.p1.3.m2.1.1.2">𝑘</ci><ci id="S1.p1.3.m2.1.1.3.cmml" xref="S1.p1.3.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m2.1c">k_{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m2.1d">italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> the Boltzmann constant, <math alttext="T" class="ltx_Math" display="inline" id="S1.p1.4.m3.1"><semantics id="S1.p1.4.m3.1a"><mi id="S1.p1.4.m3.1.1" xref="S1.p1.4.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m3.1b"><ci id="S1.p1.4.m3.1.1.cmml" xref="S1.p1.4.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m3.1d">italic_T</annotation></semantics></math> is the temperature, <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.5.m4.1"><semantics id="S1.p1.5.m4.1a"><mi id="S1.p1.5.m4.1.1" xref="S1.p1.5.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.5.m4.1b"><ci id="S1.p1.5.m4.1.1.cmml" xref="S1.p1.5.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m4.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m4.1d">italic_B</annotation></semantics></math> is the magnetic field, and <math alttext="\mu_{0}" class="ltx_Math" display="inline" id="S1.p1.6.m5.1"><semantics id="S1.p1.6.m5.1a"><msub id="S1.p1.6.m5.1.1" xref="S1.p1.6.m5.1.1.cmml"><mi id="S1.p1.6.m5.1.1.2" xref="S1.p1.6.m5.1.1.2.cmml">μ</mi><mn id="S1.p1.6.m5.1.1.3" xref="S1.p1.6.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p1.6.m5.1b"><apply id="S1.p1.6.m5.1.1.cmml" xref="S1.p1.6.m5.1.1"><csymbol cd="ambiguous" id="S1.p1.6.m5.1.1.1.cmml" xref="S1.p1.6.m5.1.1">subscript</csymbol><ci id="S1.p1.6.m5.1.1.2.cmml" xref="S1.p1.6.m5.1.1.2">𝜇</ci><cn id="S1.p1.6.m5.1.1.3.cmml" type="integer" xref="S1.p1.6.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m5.1c">\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m5.1d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> the vacuum permeability. Physically, <math alttext="\beta" class="ltx_Math" display="inline" id="S1.p1.7.m6.1"><semantics id="S1.p1.7.m6.1a"><mi id="S1.p1.7.m6.1.1" xref="S1.p1.7.m6.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p1.7.m6.1b"><ci id="S1.p1.7.m6.1.1.cmml" xref="S1.p1.7.m6.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m6.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m6.1d">italic_β</annotation></semantics></math> is the ratio of the thermal pressure, to the magnetic pressure <math alttext="P_{\rm mag}" class="ltx_Math" display="inline" id="S1.p1.8.m7.1"><semantics id="S1.p1.8.m7.1a"><msub id="S1.p1.8.m7.1.1" xref="S1.p1.8.m7.1.1.cmml"><mi id="S1.p1.8.m7.1.1.2" xref="S1.p1.8.m7.1.1.2.cmml">P</mi><mi id="S1.p1.8.m7.1.1.3" xref="S1.p1.8.m7.1.1.3.cmml">mag</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p1.8.m7.1b"><apply id="S1.p1.8.m7.1.1.cmml" xref="S1.p1.8.m7.1.1"><csymbol cd="ambiguous" id="S1.p1.8.m7.1.1.1.cmml" xref="S1.p1.8.m7.1.1">subscript</csymbol><ci id="S1.p1.8.m7.1.1.2.cmml" xref="S1.p1.8.m7.1.1.2">𝑃</ci><ci id="S1.p1.8.m7.1.1.3.cmml" xref="S1.p1.8.m7.1.1.3">mag</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m7.1c">P_{\rm mag}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m7.1d">italic_P start_POSTSUBSCRIPT roman_mag end_POSTSUBSCRIPT</annotation></semantics></math>. Although the interplanetary solar wind is mostly characterized by <math alttext="\beta\sim 1" class="ltx_Math" display="inline" id="S1.p1.9.m8.1"><semantics id="S1.p1.9.m8.1a"><mrow id="S1.p1.9.m8.1.1" xref="S1.p1.9.m8.1.1.cmml"><mi id="S1.p1.9.m8.1.1.2" xref="S1.p1.9.m8.1.1.2.cmml">β</mi><mo id="S1.p1.9.m8.1.1.1" xref="S1.p1.9.m8.1.1.1.cmml">∼</mo><mn id="S1.p1.9.m8.1.1.3" xref="S1.p1.9.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.9.m8.1b"><apply id="S1.p1.9.m8.1.1.cmml" xref="S1.p1.9.m8.1.1"><csymbol cd="latexml" id="S1.p1.9.m8.1.1.1.cmml" xref="S1.p1.9.m8.1.1.1">similar-to</csymbol><ci id="S1.p1.9.m8.1.1.2.cmml" xref="S1.p1.9.m8.1.1.2">𝛽</ci><cn id="S1.p1.9.m8.1.1.3.cmml" type="integer" xref="S1.p1.9.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m8.1c">\beta\sim 1</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m8.1d">italic_β ∼ 1</annotation></semantics></math>, high-beta plasmas <math alttext="(\beta\gg 1)" class="ltx_Math" display="inline" id="S1.p1.10.m9.1"><semantics id="S1.p1.10.m9.1a"><mrow id="S1.p1.10.m9.1.1.1" xref="S1.p1.10.m9.1.1.1.1.cmml"><mo id="S1.p1.10.m9.1.1.1.2" stretchy="false" xref="S1.p1.10.m9.1.1.1.1.cmml">(</mo><mrow id="S1.p1.10.m9.1.1.1.1" xref="S1.p1.10.m9.1.1.1.1.cmml"><mi id="S1.p1.10.m9.1.1.1.1.2" xref="S1.p1.10.m9.1.1.1.1.2.cmml">β</mi><mo id="S1.p1.10.m9.1.1.1.1.1" xref="S1.p1.10.m9.1.1.1.1.1.cmml">≫</mo><mn id="S1.p1.10.m9.1.1.1.1.3" xref="S1.p1.10.m9.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.p1.10.m9.1.1.1.3" stretchy="false" xref="S1.p1.10.m9.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.10.m9.1b"><apply id="S1.p1.10.m9.1.1.1.1.cmml" xref="S1.p1.10.m9.1.1.1"><csymbol cd="latexml" id="S1.p1.10.m9.1.1.1.1.1.cmml" xref="S1.p1.10.m9.1.1.1.1.1">much-greater-than</csymbol><ci id="S1.p1.10.m9.1.1.1.1.2.cmml" xref="S1.p1.10.m9.1.1.1.1.2">𝛽</ci><cn id="S1.p1.10.m9.1.1.1.1.3.cmml" type="integer" xref="S1.p1.10.m9.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.10.m9.1c">(\beta\gg 1)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m9.1d">( italic_β ≫ 1 )</annotation></semantics></math> are prevalent in many astrophysical environments, e.g., in the warmer phases of the interstellar medium <cite class="ltx_cite ltx_citemacro_citep">(Ferrière, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib28" title="">2020</a>)</cite> and the intracluster medium <cite class="ltx_cite ltx_citemacro_citep">(Kunz et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib45" title="">2022</a>)</cite>, where <math alttext="\beta\gg 1" class="ltx_Math" display="inline" id="S1.p1.11.m10.1"><semantics id="S1.p1.11.m10.1a"><mrow id="S1.p1.11.m10.1.1" xref="S1.p1.11.m10.1.1.cmml"><mi id="S1.p1.11.m10.1.1.2" xref="S1.p1.11.m10.1.1.2.cmml">β</mi><mo id="S1.p1.11.m10.1.1.1" xref="S1.p1.11.m10.1.1.1.cmml">≫</mo><mn id="S1.p1.11.m10.1.1.3" xref="S1.p1.11.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.11.m10.1b"><apply id="S1.p1.11.m10.1.1.cmml" xref="S1.p1.11.m10.1.1"><csymbol cd="latexml" id="S1.p1.11.m10.1.1.1.cmml" xref="S1.p1.11.m10.1.1.1">much-greater-than</csymbol><ci id="S1.p1.11.m10.1.1.2.cmml" xref="S1.p1.11.m10.1.1.2">𝛽</ci><cn id="S1.p1.11.m10.1.1.3.cmml" type="integer" xref="S1.p1.11.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.11.m10.1c">\beta\gg 1</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m10.1d">italic_β ≫ 1</annotation></semantics></math> plays an essential role in facilitating the conditions for mirror and firehose instabilities.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.7">For an isothermal plasma, <math alttext="\beta=2c_{s}^{2}/v_{A}^{2}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><mi id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml">β</mi><mo id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.cmml">=</mo><mrow id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml"><mrow id="S1.p2.1.m1.1.1.3.2" xref="S1.p2.1.m1.1.1.3.2.cmml"><mn id="S1.p2.1.m1.1.1.3.2.2" xref="S1.p2.1.m1.1.1.3.2.2.cmml">2</mn><mo id="S1.p2.1.m1.1.1.3.2.1" xref="S1.p2.1.m1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S1.p2.1.m1.1.1.3.2.3" xref="S1.p2.1.m1.1.1.3.2.3.cmml"><mi id="S1.p2.1.m1.1.1.3.2.3.2.2" xref="S1.p2.1.m1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S1.p2.1.m1.1.1.3.2.3.2.3" xref="S1.p2.1.m1.1.1.3.2.3.2.3.cmml">s</mi><mn id="S1.p2.1.m1.1.1.3.2.3.3" xref="S1.p2.1.m1.1.1.3.2.3.3.cmml">2</mn></msubsup></mrow><mo id="S1.p2.1.m1.1.1.3.1" xref="S1.p2.1.m1.1.1.3.1.cmml">/</mo><msubsup id="S1.p2.1.m1.1.1.3.3" xref="S1.p2.1.m1.1.1.3.3.cmml"><mi id="S1.p2.1.m1.1.1.3.3.2.2" xref="S1.p2.1.m1.1.1.3.3.2.2.cmml">v</mi><mi id="S1.p2.1.m1.1.1.3.3.2.3" xref="S1.p2.1.m1.1.1.3.3.2.3.cmml">A</mi><mn id="S1.p2.1.m1.1.1.3.3.3" xref="S1.p2.1.m1.1.1.3.3.3.cmml">2</mn></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><eq id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"></eq><ci id="S1.p2.1.m1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.2">𝛽</ci><apply id="S1.p2.1.m1.1.1.3.cmml" xref="S1.p2.1.m1.1.1.3"><divide id="S1.p2.1.m1.1.1.3.1.cmml" xref="S1.p2.1.m1.1.1.3.1"></divide><apply id="S1.p2.1.m1.1.1.3.2.cmml" xref="S1.p2.1.m1.1.1.3.2"><times id="S1.p2.1.m1.1.1.3.2.1.cmml" xref="S1.p2.1.m1.1.1.3.2.1"></times><cn id="S1.p2.1.m1.1.1.3.2.2.cmml" type="integer" xref="S1.p2.1.m1.1.1.3.2.2">2</cn><apply id="S1.p2.1.m1.1.1.3.2.3.cmml" xref="S1.p2.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.3.2.3.1.cmml" xref="S1.p2.1.m1.1.1.3.2.3">superscript</csymbol><apply id="S1.p2.1.m1.1.1.3.2.3.2.cmml" xref="S1.p2.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.3.2.3.2.1.cmml" xref="S1.p2.1.m1.1.1.3.2.3">subscript</csymbol><ci id="S1.p2.1.m1.1.1.3.2.3.2.2.cmml" xref="S1.p2.1.m1.1.1.3.2.3.2.2">𝑐</ci><ci id="S1.p2.1.m1.1.1.3.2.3.2.3.cmml" xref="S1.p2.1.m1.1.1.3.2.3.2.3">𝑠</ci></apply><cn id="S1.p2.1.m1.1.1.3.2.3.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.3.2.3.3">2</cn></apply></apply><apply id="S1.p2.1.m1.1.1.3.3.cmml" xref="S1.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.3.3.1.cmml" xref="S1.p2.1.m1.1.1.3.3">superscript</csymbol><apply id="S1.p2.1.m1.1.1.3.3.2.cmml" xref="S1.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.3.3.2.1.cmml" xref="S1.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="S1.p2.1.m1.1.1.3.3.2.2.cmml" xref="S1.p2.1.m1.1.1.3.3.2.2">𝑣</ci><ci id="S1.p2.1.m1.1.1.3.3.2.3.cmml" xref="S1.p2.1.m1.1.1.3.3.2.3">𝐴</ci></apply><cn id="S1.p2.1.m1.1.1.3.3.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\beta=2c_{s}^{2}/v_{A}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_β = 2 italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_v start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> where <math alttext="c_{s}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><msub id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">c</mi><mi id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1">subscript</csymbol><ci id="S1.p2.2.m2.1.1.2.cmml" xref="S1.p2.2.m2.1.1.2">𝑐</ci><ci id="S1.p2.2.m2.1.1.3.cmml" xref="S1.p2.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">c_{s}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is the sound speed, and <math alttext="v_{A}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><msub id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">v</mi><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1">subscript</csymbol><ci id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2">𝑣</ci><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">v_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_v start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> the Alfvén speed. Hence, in some ways, <math alttext="\beta" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_β</annotation></semantics></math>, which controls the speed of the fast wave relative to the Alfvén speed, characterizing how compressible a plasma is. The properties and origins of compressible fluctuations in turbulent space plasmas are still not well understood <cite class="ltx_cite ltx_citemacro_citep">(Klainerman & Majda, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib40" title="">1981</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib41" title="">1982</a>; Shebalin & Montgomery, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib70" title="">1988</a>; Du et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib19" title="">2023</a>)</cite>. Density fluctuation <math alttext="(\delta\rho=\left\langle(\rho-\left\langle\rho\right\rangle)^{2}\right\rangle^% {1/2})" class="ltx_Math" display="inline" id="S1.p2.5.m5.2"><semantics id="S1.p2.5.m5.2a"><mrow id="S1.p2.5.m5.2.2.1" xref="S1.p2.5.m5.2.2.1.1.cmml"><mo id="S1.p2.5.m5.2.2.1.2" stretchy="false" xref="S1.p2.5.m5.2.2.1.1.cmml">(</mo><mrow id="S1.p2.5.m5.2.2.1.1" xref="S1.p2.5.m5.2.2.1.1.cmml"><mrow id="S1.p2.5.m5.2.2.1.1.3" xref="S1.p2.5.m5.2.2.1.1.3.cmml"><mi id="S1.p2.5.m5.2.2.1.1.3.2" xref="S1.p2.5.m5.2.2.1.1.3.2.cmml">δ</mi><mo id="S1.p2.5.m5.2.2.1.1.3.1" xref="S1.p2.5.m5.2.2.1.1.3.1.cmml">⁢</mo><mi id="S1.p2.5.m5.2.2.1.1.3.3" xref="S1.p2.5.m5.2.2.1.1.3.3.cmml">ρ</mi></mrow><mo id="S1.p2.5.m5.2.2.1.1.2" xref="S1.p2.5.m5.2.2.1.1.2.cmml">=</mo><msup id="S1.p2.5.m5.2.2.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.cmml"><mrow id="S1.p2.5.m5.2.2.1.1.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.1.2.cmml"><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.2" xref="S1.p2.5.m5.2.2.1.1.1.1.2.1.cmml">⟨</mo><msup id="S1.p2.5.m5.2.2.1.1.1.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.cmml"><mrow id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.2" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.2.cmml">ρ</mi><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.1" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.2" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.2.1" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.1.1.cmml">⟨</mo><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">ρ</mi><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.2.2" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S1.p2.5.m5.2.2.1.1.1.1.1.1.3" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S1.p2.5.m5.2.2.1.1.1.1.1.3" xref="S1.p2.5.m5.2.2.1.1.1.1.2.1.cmml">⟩</mo></mrow><mrow id="S1.p2.5.m5.2.2.1.1.1.3" xref="S1.p2.5.m5.2.2.1.1.1.3.cmml"><mn id="S1.p2.5.m5.2.2.1.1.1.3.2" xref="S1.p2.5.m5.2.2.1.1.1.3.2.cmml">1</mn><mo id="S1.p2.5.m5.2.2.1.1.1.3.1" xref="S1.p2.5.m5.2.2.1.1.1.3.1.cmml">/</mo><mn id="S1.p2.5.m5.2.2.1.1.1.3.3" xref="S1.p2.5.m5.2.2.1.1.1.3.3.cmml">2</mn></mrow></msup></mrow><mo id="S1.p2.5.m5.2.2.1.3" stretchy="false" xref="S1.p2.5.m5.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.2b"><apply id="S1.p2.5.m5.2.2.1.1.cmml" xref="S1.p2.5.m5.2.2.1"><eq id="S1.p2.5.m5.2.2.1.1.2.cmml" xref="S1.p2.5.m5.2.2.1.1.2"></eq><apply id="S1.p2.5.m5.2.2.1.1.3.cmml" xref="S1.p2.5.m5.2.2.1.1.3"><times id="S1.p2.5.m5.2.2.1.1.3.1.cmml" xref="S1.p2.5.m5.2.2.1.1.3.1"></times><ci id="S1.p2.5.m5.2.2.1.1.3.2.cmml" xref="S1.p2.5.m5.2.2.1.1.3.2">𝛿</ci><ci id="S1.p2.5.m5.2.2.1.1.3.3.cmml" xref="S1.p2.5.m5.2.2.1.1.3.3">𝜌</ci></apply><apply id="S1.p2.5.m5.2.2.1.1.1.cmml" xref="S1.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p2.5.m5.2.2.1.1.1.2.cmml" xref="S1.p2.5.m5.2.2.1.1.1">superscript</csymbol><apply id="S1.p2.5.m5.2.2.1.1.1.1.2.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1"><csymbol cd="latexml" id="S1.p2.5.m5.2.2.1.1.1.1.2.1.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S1.p2.5.m5.2.2.1.1.1.1.1.1.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.5.m5.2.2.1.1.1.1.1.1.2.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1">superscript</csymbol><apply id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1"><minus id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S1.p2.5.m5.2.2.1.1.1.1.1.1.1.1.1.2">𝜌</ci><apply 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encoding="application/x-llamapun" id="S1.p2.5.m5.2d">( italic_δ italic_ρ = ⟨ ( italic_ρ - ⟨ italic_ρ ⟩ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> in the solar wind is typically small compared to background density <math alttext="(\rho_{0}=\langle\rho\rangle)" class="ltx_Math" display="inline" id="S1.p2.6.m6.2"><semantics id="S1.p2.6.m6.2a"><mrow id="S1.p2.6.m6.2.2.1" xref="S1.p2.6.m6.2.2.1.1.cmml"><mo id="S1.p2.6.m6.2.2.1.2" stretchy="false" xref="S1.p2.6.m6.2.2.1.1.cmml">(</mo><mrow id="S1.p2.6.m6.2.2.1.1" xref="S1.p2.6.m6.2.2.1.1.cmml"><msub id="S1.p2.6.m6.2.2.1.1.2" xref="S1.p2.6.m6.2.2.1.1.2.cmml"><mi id="S1.p2.6.m6.2.2.1.1.2.2" xref="S1.p2.6.m6.2.2.1.1.2.2.cmml">ρ</mi><mn id="S1.p2.6.m6.2.2.1.1.2.3" xref="S1.p2.6.m6.2.2.1.1.2.3.cmml">0</mn></msub><mo id="S1.p2.6.m6.2.2.1.1.1" xref="S1.p2.6.m6.2.2.1.1.1.cmml">=</mo><mrow id="S1.p2.6.m6.2.2.1.1.3.2" xref="S1.p2.6.m6.2.2.1.1.3.1.cmml"><mo id="S1.p2.6.m6.2.2.1.1.3.2.1" stretchy="false" xref="S1.p2.6.m6.2.2.1.1.3.1.1.cmml">⟨</mo><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">ρ</mi><mo id="S1.p2.6.m6.2.2.1.1.3.2.2" stretchy="false" xref="S1.p2.6.m6.2.2.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S1.p2.6.m6.2.2.1.3" stretchy="false" xref="S1.p2.6.m6.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.2b"><apply id="S1.p2.6.m6.2.2.1.1.cmml" xref="S1.p2.6.m6.2.2.1"><eq id="S1.p2.6.m6.2.2.1.1.1.cmml" xref="S1.p2.6.m6.2.2.1.1.1"></eq><apply id="S1.p2.6.m6.2.2.1.1.2.cmml" xref="S1.p2.6.m6.2.2.1.1.2"><csymbol cd="ambiguous" id="S1.p2.6.m6.2.2.1.1.2.1.cmml" xref="S1.p2.6.m6.2.2.1.1.2">subscript</csymbol><ci id="S1.p2.6.m6.2.2.1.1.2.2.cmml" xref="S1.p2.6.m6.2.2.1.1.2.2">𝜌</ci><cn id="S1.p2.6.m6.2.2.1.1.2.3.cmml" type="integer" xref="S1.p2.6.m6.2.2.1.1.2.3">0</cn></apply><apply id="S1.p2.6.m6.2.2.1.1.3.1.cmml" xref="S1.p2.6.m6.2.2.1.1.3.2"><csymbol cd="latexml" id="S1.p2.6.m6.2.2.1.1.3.1.1.cmml" xref="S1.p2.6.m6.2.2.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.2c">(\rho_{0}=\langle\rho\rangle)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.2d">( italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ⟨ italic_ρ ⟩ )</annotation></semantics></math>. Consequently, theoretical frameworks <cite class="ltx_cite ltx_citemacro_citep">(Matthaeus & Brown, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib48" title="">1988</a>; Matthaeus et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib49" title="">1991</a>; Zank & Matthaeus, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib82" title="">1993</a>)</cite> have been developed for small-amplitude compressive fluctuations <math alttext="(\delta\rho/\rho_{0}\ll 1)" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><mrow id="S1.p2.7.m7.1.1.1" xref="S1.p2.7.m7.1.1.1.1.cmml"><mo id="S1.p2.7.m7.1.1.1.2" stretchy="false" xref="S1.p2.7.m7.1.1.1.1.cmml">(</mo><mrow id="S1.p2.7.m7.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.cmml"><mrow id="S1.p2.7.m7.1.1.1.1.2" xref="S1.p2.7.m7.1.1.1.1.2.cmml"><mrow id="S1.p2.7.m7.1.1.1.1.2.2" xref="S1.p2.7.m7.1.1.1.1.2.2.cmml"><mi id="S1.p2.7.m7.1.1.1.1.2.2.2" xref="S1.p2.7.m7.1.1.1.1.2.2.2.cmml">δ</mi><mo id="S1.p2.7.m7.1.1.1.1.2.2.1" xref="S1.p2.7.m7.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S1.p2.7.m7.1.1.1.1.2.2.3" xref="S1.p2.7.m7.1.1.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.p2.7.m7.1.1.1.1.2.1" xref="S1.p2.7.m7.1.1.1.1.2.1.cmml">/</mo><msub id="S1.p2.7.m7.1.1.1.1.2.3" xref="S1.p2.7.m7.1.1.1.1.2.3.cmml"><mi id="S1.p2.7.m7.1.1.1.1.2.3.2" xref="S1.p2.7.m7.1.1.1.1.2.3.2.cmml">ρ</mi><mn id="S1.p2.7.m7.1.1.1.1.2.3.3" xref="S1.p2.7.m7.1.1.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.p2.7.m7.1.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.1.cmml">≪</mo><mn id="S1.p2.7.m7.1.1.1.1.3" xref="S1.p2.7.m7.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.p2.7.m7.1.1.1.3" stretchy="false" xref="S1.p2.7.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1"><csymbol cd="latexml" id="S1.p2.7.m7.1.1.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1.1.1">much-less-than</csymbol><apply 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id="S1.p2.7.m7.1c">(\delta\rho/\rho_{0}\ll 1)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">( italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ 1 )</annotation></semantics></math>, assuming a small turbulent Mach number, which is defined</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A1.EGx1"> <tbody id="S1.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{M}_{t}=\frac{\left\langle(u-\left\langle u\right\rangle)% ^{2}\right\rangle^{1/2}}{\left\langle c_{s}\right\rangle}=\frac{\delta u}{c_{s% }}," class="ltx_Math" display="inline" id="S1.E2.m1.4"><semantics id="S1.E2.m1.4a"><mrow id="S1.E2.m1.4.4.1" xref="S1.E2.m1.4.4.1.1.cmml"><mrow id="S1.E2.m1.4.4.1.1" xref="S1.E2.m1.4.4.1.1.cmml"><msub id="S1.E2.m1.4.4.1.1.2" xref="S1.E2.m1.4.4.1.1.2.cmml"><mi 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id="S1.E2.m1.4c">\displaystyle\mathcal{M}_{t}=\frac{\left\langle(u-\left\langle u\right\rangle)% ^{2}\right\rangle^{1/2}}{\left\langle c_{s}\right\rangle}=\frac{\delta u}{c_{s% }},</annotation><annotation encoding="application/x-llamapun" id="S1.E2.m1.4d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = divide start_ARG ⟨ ( italic_u - ⟨ italic_u ⟩ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT end_ARG start_ARG ⟨ italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⟩ end_ARG = divide start_ARG italic_δ italic_u end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.10">with <math alttext="\delta u" 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id="S1.p2.9.m2.1a"><msub id="S1.p2.9.m2.1.1" xref="S1.p2.9.m2.1.1.cmml"><mi id="S1.p2.9.m2.1.1.2" xref="S1.p2.9.m2.1.1.2.cmml">c</mi><mi id="S1.p2.9.m2.1.1.3" xref="S1.p2.9.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.9.m2.1b"><apply id="S1.p2.9.m2.1.1.cmml" xref="S1.p2.9.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.9.m2.1.1.1.cmml" xref="S1.p2.9.m2.1.1">subscript</csymbol><ci id="S1.p2.9.m2.1.1.2.cmml" xref="S1.p2.9.m2.1.1.2">𝑐</ci><ci id="S1.p2.9.m2.1.1.3.cmml" xref="S1.p2.9.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m2.1c">c_{s}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m2.1d">italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> as the mean sound speed. For a homogeneous background field, the nearly-incompressible MHD (NI-MHD) theories predict that the normalized density fluctuations scale with the square of <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p2.10.m3.1"><semantics id="S1.p2.10.m3.1a"><msub id="S1.p2.10.m3.1.1" xref="S1.p2.10.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.10.m3.1.1.2" xref="S1.p2.10.m3.1.1.2.cmml">ℳ</mi><mi id="S1.p2.10.m3.1.1.3" xref="S1.p2.10.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.10.m3.1b"><apply id="S1.p2.10.m3.1.1.cmml" xref="S1.p2.10.m3.1.1"><csymbol cd="ambiguous" id="S1.p2.10.m3.1.1.1.cmml" xref="S1.p2.10.m3.1.1">subscript</csymbol><ci id="S1.p2.10.m3.1.1.2.cmml" xref="S1.p2.10.m3.1.1.2">ℳ</ci><ci id="S1.p2.10.m3.1.1.3.cmml" xref="S1.p2.10.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m3.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m3.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A1.EGx2"> <tbody id="S1.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta\rho/\rho_{0}\propto\mathcal{M}_{t}^{2}." class="ltx_Math" display="inline" id="S1.E3.m1.1"><semantics id="S1.E3.m1.1a"><mrow id="S1.E3.m1.1.1.1" xref="S1.E3.m1.1.1.1.1.cmml"><mrow id="S1.E3.m1.1.1.1.1" xref="S1.E3.m1.1.1.1.1.cmml"><mrow id="S1.E3.m1.1.1.1.1.2" xref="S1.E3.m1.1.1.1.1.2.cmml"><mrow id="S1.E3.m1.1.1.1.1.2.2" xref="S1.E3.m1.1.1.1.1.2.2.cmml"><mi id="S1.E3.m1.1.1.1.1.2.2.2" xref="S1.E3.m1.1.1.1.1.2.2.2.cmml">δ</mi><mo id="S1.E3.m1.1.1.1.1.2.2.1" xref="S1.E3.m1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S1.E3.m1.1.1.1.1.2.2.3" xref="S1.E3.m1.1.1.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.E3.m1.1.1.1.1.2.1" xref="S1.E3.m1.1.1.1.1.2.1.cmml">/</mo><msub id="S1.E3.m1.1.1.1.1.2.3" xref="S1.E3.m1.1.1.1.1.2.3.cmml"><mi id="S1.E3.m1.1.1.1.1.2.3.2" xref="S1.E3.m1.1.1.1.1.2.3.2.cmml">ρ</mi><mn id="S1.E3.m1.1.1.1.1.2.3.3" xref="S1.E3.m1.1.1.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.E3.m1.1.1.1.1.1" xref="S1.E3.m1.1.1.1.1.1.cmml">∝</mo><msubsup id="S1.E3.m1.1.1.1.1.3" xref="S1.E3.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E3.m1.1.1.1.1.3.2.2" xref="S1.E3.m1.1.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S1.E3.m1.1.1.1.1.3.2.3" xref="S1.E3.m1.1.1.1.1.3.2.3.cmml">t</mi><mn id="S1.E3.m1.1.1.1.1.3.3" xref="S1.E3.m1.1.1.1.1.3.3.cmml">2</mn></msubsup></mrow><mo id="S1.E3.m1.1.1.1.2" lspace="0em" xref="S1.E3.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E3.m1.1b"><apply id="S1.E3.m1.1.1.1.1.cmml" xref="S1.E3.m1.1.1.1"><csymbol cd="latexml" id="S1.E3.m1.1.1.1.1.1.cmml" xref="S1.E3.m1.1.1.1.1.1">proportional-to</csymbol><apply id="S1.E3.m1.1.1.1.1.2.cmml" xref="S1.E3.m1.1.1.1.1.2"><divide id="S1.E3.m1.1.1.1.1.2.1.cmml" xref="S1.E3.m1.1.1.1.1.2.1"></divide><apply id="S1.E3.m1.1.1.1.1.2.2.cmml" xref="S1.E3.m1.1.1.1.1.2.2"><times id="S1.E3.m1.1.1.1.1.2.2.1.cmml" xref="S1.E3.m1.1.1.1.1.2.2.1"></times><ci id="S1.E3.m1.1.1.1.1.2.2.2.cmml" xref="S1.E3.m1.1.1.1.1.2.2.2">𝛿</ci><ci id="S1.E3.m1.1.1.1.1.2.2.3.cmml" xref="S1.E3.m1.1.1.1.1.2.2.3">𝜌</ci></apply><apply id="S1.E3.m1.1.1.1.1.2.3.cmml" xref="S1.E3.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.E3.m1.1.1.1.1.2.3.1.cmml" xref="S1.E3.m1.1.1.1.1.2.3">subscript</csymbol><ci id="S1.E3.m1.1.1.1.1.2.3.2.cmml" xref="S1.E3.m1.1.1.1.1.2.3.2">𝜌</ci><cn id="S1.E3.m1.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.E3.m1.1.1.1.1.2.3.3">0</cn></apply></apply><apply id="S1.E3.m1.1.1.1.1.3.cmml" xref="S1.E3.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E3.m1.1.1.1.1.3.1.cmml" xref="S1.E3.m1.1.1.1.1.3">superscript</csymbol><apply id="S1.E3.m1.1.1.1.1.3.2.cmml" xref="S1.E3.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E3.m1.1.1.1.1.3.2.1.cmml" xref="S1.E3.m1.1.1.1.1.3">subscript</csymbol><ci id="S1.E3.m1.1.1.1.1.3.2.2.cmml" xref="S1.E3.m1.1.1.1.1.3.2.2">ℳ</ci><ci id="S1.E3.m1.1.1.1.1.3.2.3.cmml" xref="S1.E3.m1.1.1.1.1.3.2.3">𝑡</ci></apply><cn id="S1.E3.m1.1.1.1.1.3.3.cmml" type="integer" xref="S1.E3.m1.1.1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E3.m1.1c">\displaystyle\delta\rho/\rho_{0}\propto\mathcal{M}_{t}^{2}.</annotation><annotation encoding="application/x-llamapun" id="S1.E3.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.11">However, a further generalization of NI-MHD theory, referred to here as the weakly compressible MHD theory for the case of an inhomogeneous background field <cite class="ltx_cite ltx_citemacro_citep">(Bhattacharjee et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib9" title="">1999</a>; Hunana & Zank, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib37" title="">2010</a>)</cite> predicts a linear relationship of normalized density fluctuations scaling with <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p2.11.m1.1"><semantics id="S1.p2.11.m1.1a"><msub id="S1.p2.11.m1.1.1" xref="S1.p2.11.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.11.m1.1.1.2" xref="S1.p2.11.m1.1.1.2.cmml">ℳ</mi><mi id="S1.p2.11.m1.1.1.3" xref="S1.p2.11.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.11.m1.1b"><apply id="S1.p2.11.m1.1.1.cmml" xref="S1.p2.11.m1.1.1"><csymbol cd="ambiguous" id="S1.p2.11.m1.1.1.1.cmml" xref="S1.p2.11.m1.1.1">subscript</csymbol><ci id="S1.p2.11.m1.1.1.2.cmml" xref="S1.p2.11.m1.1.1.2">ℳ</ci><ci id="S1.p2.11.m1.1.1.3.cmml" xref="S1.p2.11.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.11.m1.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.11.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A1.EGx3"> <tbody id="S1.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta\rho/\rho_{0}\propto\mathcal{M}_{t}." class="ltx_Math" display="inline" id="S1.E4.m1.1"><semantics id="S1.E4.m1.1a"><mrow id="S1.E4.m1.1.1.1" xref="S1.E4.m1.1.1.1.1.cmml"><mrow id="S1.E4.m1.1.1.1.1" xref="S1.E4.m1.1.1.1.1.cmml"><mrow id="S1.E4.m1.1.1.1.1.2" xref="S1.E4.m1.1.1.1.1.2.cmml"><mrow id="S1.E4.m1.1.1.1.1.2.2" xref="S1.E4.m1.1.1.1.1.2.2.cmml"><mi id="S1.E4.m1.1.1.1.1.2.2.2" xref="S1.E4.m1.1.1.1.1.2.2.2.cmml">δ</mi><mo id="S1.E4.m1.1.1.1.1.2.2.1" xref="S1.E4.m1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S1.E4.m1.1.1.1.1.2.2.3" xref="S1.E4.m1.1.1.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.E4.m1.1.1.1.1.2.1" xref="S1.E4.m1.1.1.1.1.2.1.cmml">/</mo><msub id="S1.E4.m1.1.1.1.1.2.3" xref="S1.E4.m1.1.1.1.1.2.3.cmml"><mi id="S1.E4.m1.1.1.1.1.2.3.2" xref="S1.E4.m1.1.1.1.1.2.3.2.cmml">ρ</mi><mn id="S1.E4.m1.1.1.1.1.2.3.3" xref="S1.E4.m1.1.1.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.E4.m1.1.1.1.1.1" xref="S1.E4.m1.1.1.1.1.1.cmml">∝</mo><msub id="S1.E4.m1.1.1.1.1.3" xref="S1.E4.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E4.m1.1.1.1.1.3.2" xref="S1.E4.m1.1.1.1.1.3.2.cmml">ℳ</mi><mi id="S1.E4.m1.1.1.1.1.3.3" xref="S1.E4.m1.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo id="S1.E4.m1.1.1.1.2" lspace="0em" xref="S1.E4.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E4.m1.1b"><apply id="S1.E4.m1.1.1.1.1.cmml" xref="S1.E4.m1.1.1.1"><csymbol cd="latexml" id="S1.E4.m1.1.1.1.1.1.cmml" xref="S1.E4.m1.1.1.1.1.1">proportional-to</csymbol><apply id="S1.E4.m1.1.1.1.1.2.cmml" xref="S1.E4.m1.1.1.1.1.2"><divide id="S1.E4.m1.1.1.1.1.2.1.cmml" xref="S1.E4.m1.1.1.1.1.2.1"></divide><apply id="S1.E4.m1.1.1.1.1.2.2.cmml" xref="S1.E4.m1.1.1.1.1.2.2"><times id="S1.E4.m1.1.1.1.1.2.2.1.cmml" xref="S1.E4.m1.1.1.1.1.2.2.1"></times><ci id="S1.E4.m1.1.1.1.1.2.2.2.cmml" xref="S1.E4.m1.1.1.1.1.2.2.2">𝛿</ci><ci id="S1.E4.m1.1.1.1.1.2.2.3.cmml" xref="S1.E4.m1.1.1.1.1.2.2.3">𝜌</ci></apply><apply id="S1.E4.m1.1.1.1.1.2.3.cmml" xref="S1.E4.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.E4.m1.1.1.1.1.2.3.1.cmml" xref="S1.E4.m1.1.1.1.1.2.3">subscript</csymbol><ci id="S1.E4.m1.1.1.1.1.2.3.2.cmml" xref="S1.E4.m1.1.1.1.1.2.3.2">𝜌</ci><cn id="S1.E4.m1.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.E4.m1.1.1.1.1.2.3.3">0</cn></apply></apply><apply id="S1.E4.m1.1.1.1.1.3.cmml" xref="S1.E4.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E4.m1.1.1.1.1.3.1.cmml" xref="S1.E4.m1.1.1.1.1.3">subscript</csymbol><ci id="S1.E4.m1.1.1.1.1.3.2.cmml" xref="S1.E4.m1.1.1.1.1.3.2">ℳ</ci><ci id="S1.E4.m1.1.1.1.1.3.3.cmml" xref="S1.E4.m1.1.1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E4.m1.1c">\displaystyle\delta\rho/\rho_{0}\propto\mathcal{M}_{t}.</annotation><annotation encoding="application/x-llamapun" id="S1.E4.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.13">For supersonic turbulence, <math alttext="\mathcal{M}_{t}\gg 1" class="ltx_Math" display="inline" id="S1.p2.12.m1.1"><semantics id="S1.p2.12.m1.1a"><mrow id="S1.p2.12.m1.1.1" xref="S1.p2.12.m1.1.1.cmml"><msub id="S1.p2.12.m1.1.1.2" xref="S1.p2.12.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.12.m1.1.1.2.2" xref="S1.p2.12.m1.1.1.2.2.cmml">ℳ</mi><mi id="S1.p2.12.m1.1.1.2.3" xref="S1.p2.12.m1.1.1.2.3.cmml">t</mi></msub><mo id="S1.p2.12.m1.1.1.1" xref="S1.p2.12.m1.1.1.1.cmml">≫</mo><mn id="S1.p2.12.m1.1.1.3" xref="S1.p2.12.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.12.m1.1b"><apply id="S1.p2.12.m1.1.1.cmml" xref="S1.p2.12.m1.1.1"><csymbol cd="latexml" id="S1.p2.12.m1.1.1.1.cmml" xref="S1.p2.12.m1.1.1.1">much-greater-than</csymbol><apply id="S1.p2.12.m1.1.1.2.cmml" xref="S1.p2.12.m1.1.1.2"><csymbol cd="ambiguous" id="S1.p2.12.m1.1.1.2.1.cmml" xref="S1.p2.12.m1.1.1.2">subscript</csymbol><ci id="S1.p2.12.m1.1.1.2.2.cmml" xref="S1.p2.12.m1.1.1.2.2">ℳ</ci><ci id="S1.p2.12.m1.1.1.2.3.cmml" xref="S1.p2.12.m1.1.1.2.3">𝑡</ci></apply><cn id="S1.p2.12.m1.1.1.3.cmml" type="integer" xref="S1.p2.12.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m1.1c">\mathcal{M}_{t}\gg 1</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≫ 1</annotation></semantics></math>, ubiquitous in the star-forming, colder phases of the interstellar medium <cite class="ltx_cite ltx_citemacro_citep">(Krumholz, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib44" title="">2015</a>)</cite>, with large magnetic field <cite class="ltx_cite ltx_citemacro_citep">(Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib5" title="">2020</a>)</cite> and mass density inhomogeneities <cite class="ltx_cite ltx_citemacro_citep">(Federrath, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib23" title="">2013</a>; Beattie & Federrath, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib3" title="">2020</a>)</cite>, a further model has been proposed, based on the Rankine-Hugoniot shock-jump conditions. For isothermal, hydrodynamic shocks <cite class="ltx_cite ltx_citemacro_citet">Padoan et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib58" title="">1997</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Passot & Vázquez-Semadeni (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib61" title="">1998</a>)</cite> showed that the density variance can be related to <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p2.13.m2.1"><semantics id="S1.p2.13.m2.1a"><msub id="S1.p2.13.m2.1.1" xref="S1.p2.13.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.13.m2.1.1.2" xref="S1.p2.13.m2.1.1.2.cmml">ℳ</mi><mi id="S1.p2.13.m2.1.1.3" xref="S1.p2.13.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.13.m2.1b"><apply id="S1.p2.13.m2.1.1.cmml" xref="S1.p2.13.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.13.m2.1.1.1.cmml" xref="S1.p2.13.m2.1.1">subscript</csymbol><ci id="S1.p2.13.m2.1.1.2.cmml" xref="S1.p2.13.m2.1.1.2">ℳ</ci><ci id="S1.p2.13.m2.1.1.3.cmml" xref="S1.p2.13.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.13.m2.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.13.m2.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx4"> <tbody id="S1.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta\rho/\rho_{0}=b\mathcal{M}_{t}," class="ltx_Math" display="inline" id="S1.E5.m1.1"><semantics id="S1.E5.m1.1a"><mrow id="S1.E5.m1.1.1.1" xref="S1.E5.m1.1.1.1.1.cmml"><mrow id="S1.E5.m1.1.1.1.1" xref="S1.E5.m1.1.1.1.1.cmml"><mrow id="S1.E5.m1.1.1.1.1.2" xref="S1.E5.m1.1.1.1.1.2.cmml"><mrow id="S1.E5.m1.1.1.1.1.2.2" xref="S1.E5.m1.1.1.1.1.2.2.cmml"><mi id="S1.E5.m1.1.1.1.1.2.2.2" xref="S1.E5.m1.1.1.1.1.2.2.2.cmml">δ</mi><mo id="S1.E5.m1.1.1.1.1.2.2.1" xref="S1.E5.m1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S1.E5.m1.1.1.1.1.2.2.3" xref="S1.E5.m1.1.1.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.E5.m1.1.1.1.1.2.1" xref="S1.E5.m1.1.1.1.1.2.1.cmml">/</mo><msub id="S1.E5.m1.1.1.1.1.2.3" xref="S1.E5.m1.1.1.1.1.2.3.cmml"><mi id="S1.E5.m1.1.1.1.1.2.3.2" xref="S1.E5.m1.1.1.1.1.2.3.2.cmml">ρ</mi><mn id="S1.E5.m1.1.1.1.1.2.3.3" xref="S1.E5.m1.1.1.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.E5.m1.1.1.1.1.1" xref="S1.E5.m1.1.1.1.1.1.cmml">=</mo><mrow id="S1.E5.m1.1.1.1.1.3" xref="S1.E5.m1.1.1.1.1.3.cmml"><mi id="S1.E5.m1.1.1.1.1.3.2" xref="S1.E5.m1.1.1.1.1.3.2.cmml">b</mi><mo id="S1.E5.m1.1.1.1.1.3.1" xref="S1.E5.m1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S1.E5.m1.1.1.1.1.3.3" xref="S1.E5.m1.1.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E5.m1.1.1.1.1.3.3.2" xref="S1.E5.m1.1.1.1.1.3.3.2.cmml">ℳ</mi><mi id="S1.E5.m1.1.1.1.1.3.3.3" xref="S1.E5.m1.1.1.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><mo id="S1.E5.m1.1.1.1.2" xref="S1.E5.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E5.m1.1b"><apply id="S1.E5.m1.1.1.1.1.cmml" xref="S1.E5.m1.1.1.1"><eq id="S1.E5.m1.1.1.1.1.1.cmml" xref="S1.E5.m1.1.1.1.1.1"></eq><apply id="S1.E5.m1.1.1.1.1.2.cmml" xref="S1.E5.m1.1.1.1.1.2"><divide id="S1.E5.m1.1.1.1.1.2.1.cmml" xref="S1.E5.m1.1.1.1.1.2.1"></divide><apply id="S1.E5.m1.1.1.1.1.2.2.cmml" xref="S1.E5.m1.1.1.1.1.2.2"><times id="S1.E5.m1.1.1.1.1.2.2.1.cmml" xref="S1.E5.m1.1.1.1.1.2.2.1"></times><ci id="S1.E5.m1.1.1.1.1.2.2.2.cmml" xref="S1.E5.m1.1.1.1.1.2.2.2">𝛿</ci><ci id="S1.E5.m1.1.1.1.1.2.2.3.cmml" xref="S1.E5.m1.1.1.1.1.2.2.3">𝜌</ci></apply><apply id="S1.E5.m1.1.1.1.1.2.3.cmml" xref="S1.E5.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.E5.m1.1.1.1.1.2.3.1.cmml" xref="S1.E5.m1.1.1.1.1.2.3">subscript</csymbol><ci id="S1.E5.m1.1.1.1.1.2.3.2.cmml" xref="S1.E5.m1.1.1.1.1.2.3.2">𝜌</ci><cn id="S1.E5.m1.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.E5.m1.1.1.1.1.2.3.3">0</cn></apply></apply><apply id="S1.E5.m1.1.1.1.1.3.cmml" xref="S1.E5.m1.1.1.1.1.3"><times id="S1.E5.m1.1.1.1.1.3.1.cmml" xref="S1.E5.m1.1.1.1.1.3.1"></times><ci id="S1.E5.m1.1.1.1.1.3.2.cmml" xref="S1.E5.m1.1.1.1.1.3.2">𝑏</ci><apply id="S1.E5.m1.1.1.1.1.3.3.cmml" xref="S1.E5.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.E5.m1.1.1.1.1.3.3.1.cmml" xref="S1.E5.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S1.E5.m1.1.1.1.1.3.3.2.cmml" xref="S1.E5.m1.1.1.1.1.3.3.2">ℳ</ci><ci id="S1.E5.m1.1.1.1.1.3.3.3.cmml" xref="S1.E5.m1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E5.m1.1c">\displaystyle\delta\rho/\rho_{0}=b\mathcal{M}_{t},</annotation><annotation encoding="application/x-llamapun" id="S1.E5.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_b caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.20">where <math alttext="b" class="ltx_Math" display="inline" id="S1.p2.14.m1.1"><semantics id="S1.p2.14.m1.1a"><mi id="S1.p2.14.m1.1.1" xref="S1.p2.14.m1.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S1.p2.14.m1.1b"><ci id="S1.p2.14.m1.1.1.cmml" xref="S1.p2.14.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.14.m1.1c">b</annotation><annotation encoding="application/x-llamapun" id="S1.p2.14.m1.1d">italic_b</annotation></semantics></math> is the so-called ‘driving parameter,’ which depends nonlinearly on how the turbulence is driven, i.e., if energy is injected with compressible modes (<math alttext="b=1" class="ltx_Math" display="inline" id="S1.p2.15.m2.1"><semantics id="S1.p2.15.m2.1a"><mrow id="S1.p2.15.m2.1.1" xref="S1.p2.15.m2.1.1.cmml"><mi id="S1.p2.15.m2.1.1.2" xref="S1.p2.15.m2.1.1.2.cmml">b</mi><mo id="S1.p2.15.m2.1.1.1" xref="S1.p2.15.m2.1.1.1.cmml">=</mo><mn id="S1.p2.15.m2.1.1.3" xref="S1.p2.15.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.15.m2.1b"><apply id="S1.p2.15.m2.1.1.cmml" xref="S1.p2.15.m2.1.1"><eq id="S1.p2.15.m2.1.1.1.cmml" xref="S1.p2.15.m2.1.1.1"></eq><ci id="S1.p2.15.m2.1.1.2.cmml" xref="S1.p2.15.m2.1.1.2">𝑏</ci><cn id="S1.p2.15.m2.1.1.3.cmml" type="integer" xref="S1.p2.15.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.15.m2.1c">b=1</annotation><annotation encoding="application/x-llamapun" id="S1.p2.15.m2.1d">italic_b = 1</annotation></semantics></math> for purely <math alttext="|\bm{\nabla}\times\bm{u}|=0" class="ltx_Math" display="inline" id="S1.p2.16.m3.1"><semantics id="S1.p2.16.m3.1a"><mrow id="S1.p2.16.m3.1.1" xref="S1.p2.16.m3.1.1.cmml"><mrow id="S1.p2.16.m3.1.1.1.1" xref="S1.p2.16.m3.1.1.1.2.cmml"><mo id="S1.p2.16.m3.1.1.1.1.2" stretchy="false" xref="S1.p2.16.m3.1.1.1.2.1.cmml">|</mo><mrow id="S1.p2.16.m3.1.1.1.1.1" xref="S1.p2.16.m3.1.1.1.1.1.cmml"><mo class="ltx_mathvariant_bold" id="S1.p2.16.m3.1.1.1.1.1.2" mathvariant="bold" xref="S1.p2.16.m3.1.1.1.1.1.2.cmml">∇</mo><mo id="S1.p2.16.m3.1.1.1.1.1.1" lspace="0em" rspace="0.222em" xref="S1.p2.16.m3.1.1.1.1.1.1.cmml">×</mo><mi id="S1.p2.16.m3.1.1.1.1.1.3" xref="S1.p2.16.m3.1.1.1.1.1.3.cmml">𝒖</mi></mrow><mo id="S1.p2.16.m3.1.1.1.1.3" stretchy="false" xref="S1.p2.16.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S1.p2.16.m3.1.1.2" xref="S1.p2.16.m3.1.1.2.cmml">=</mo><mn id="S1.p2.16.m3.1.1.3" xref="S1.p2.16.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.16.m3.1b"><apply id="S1.p2.16.m3.1.1.cmml" xref="S1.p2.16.m3.1.1"><eq id="S1.p2.16.m3.1.1.2.cmml" xref="S1.p2.16.m3.1.1.2"></eq><apply id="S1.p2.16.m3.1.1.1.2.cmml" xref="S1.p2.16.m3.1.1.1.1"><abs id="S1.p2.16.m3.1.1.1.2.1.cmml" xref="S1.p2.16.m3.1.1.1.1.2"></abs><apply id="S1.p2.16.m3.1.1.1.1.1.cmml" xref="S1.p2.16.m3.1.1.1.1.1"><times id="S1.p2.16.m3.1.1.1.1.1.1.cmml" xref="S1.p2.16.m3.1.1.1.1.1.1"></times><ci id="S1.p2.16.m3.1.1.1.1.1.2.cmml" xref="S1.p2.16.m3.1.1.1.1.1.2">bold-∇</ci><ci id="S1.p2.16.m3.1.1.1.1.1.3.cmml" xref="S1.p2.16.m3.1.1.1.1.1.3">𝒖</ci></apply></apply><cn id="S1.p2.16.m3.1.1.3.cmml" type="integer" xref="S1.p2.16.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.16.m3.1c">|\bm{\nabla}\times\bm{u}|=0</annotation><annotation encoding="application/x-llamapun" id="S1.p2.16.m3.1d">| bold_∇ × bold_italic_u | = 0</annotation></semantics></math> modes) or incompressible modes (<math alttext="b=1/3" class="ltx_Math" display="inline" id="S1.p2.17.m4.1"><semantics id="S1.p2.17.m4.1a"><mrow id="S1.p2.17.m4.1.1" xref="S1.p2.17.m4.1.1.cmml"><mi id="S1.p2.17.m4.1.1.2" xref="S1.p2.17.m4.1.1.2.cmml">b</mi><mo id="S1.p2.17.m4.1.1.1" xref="S1.p2.17.m4.1.1.1.cmml">=</mo><mrow id="S1.p2.17.m4.1.1.3" xref="S1.p2.17.m4.1.1.3.cmml"><mn id="S1.p2.17.m4.1.1.3.2" xref="S1.p2.17.m4.1.1.3.2.cmml">1</mn><mo id="S1.p2.17.m4.1.1.3.1" xref="S1.p2.17.m4.1.1.3.1.cmml">/</mo><mn id="S1.p2.17.m4.1.1.3.3" xref="S1.p2.17.m4.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.17.m4.1b"><apply id="S1.p2.17.m4.1.1.cmml" xref="S1.p2.17.m4.1.1"><eq id="S1.p2.17.m4.1.1.1.cmml" xref="S1.p2.17.m4.1.1.1"></eq><ci id="S1.p2.17.m4.1.1.2.cmml" xref="S1.p2.17.m4.1.1.2">𝑏</ci><apply id="S1.p2.17.m4.1.1.3.cmml" xref="S1.p2.17.m4.1.1.3"><divide id="S1.p2.17.m4.1.1.3.1.cmml" xref="S1.p2.17.m4.1.1.3.1"></divide><cn id="S1.p2.17.m4.1.1.3.2.cmml" type="integer" xref="S1.p2.17.m4.1.1.3.2">1</cn><cn id="S1.p2.17.m4.1.1.3.3.cmml" type="integer" xref="S1.p2.17.m4.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.17.m4.1c">b=1/3</annotation><annotation encoding="application/x-llamapun" id="S1.p2.17.m4.1d">italic_b = 1 / 3</annotation></semantics></math> for purely <math alttext="\bm{\nabla}\cdot\bm{u}=0" class="ltx_Math" display="inline" id="S1.p2.18.m5.1"><semantics id="S1.p2.18.m5.1a"><mrow id="S1.p2.18.m5.1.1" xref="S1.p2.18.m5.1.1.cmml"><mrow id="S1.p2.18.m5.1.1.2" xref="S1.p2.18.m5.1.1.2.cmml"><mo class="ltx_mathvariant_bold" id="S1.p2.18.m5.1.1.2.2" mathvariant="bold" xref="S1.p2.18.m5.1.1.2.2.cmml">∇</mo><mo id="S1.p2.18.m5.1.1.2.1" lspace="0em" rspace="0.222em" xref="S1.p2.18.m5.1.1.2.1.cmml">⋅</mo><mi id="S1.p2.18.m5.1.1.2.3" xref="S1.p2.18.m5.1.1.2.3.cmml">𝒖</mi></mrow><mo id="S1.p2.18.m5.1.1.1" xref="S1.p2.18.m5.1.1.1.cmml">=</mo><mn id="S1.p2.18.m5.1.1.3" xref="S1.p2.18.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.18.m5.1b"><apply id="S1.p2.18.m5.1.1.cmml" xref="S1.p2.18.m5.1.1"><eq id="S1.p2.18.m5.1.1.1.cmml" xref="S1.p2.18.m5.1.1.1"></eq><apply id="S1.p2.18.m5.1.1.2.cmml" xref="S1.p2.18.m5.1.1.2"><ci id="S1.p2.18.m5.1.1.2.1.cmml" xref="S1.p2.18.m5.1.1.2.1">⋅</ci><ci id="S1.p2.18.m5.1.1.2.2.cmml" xref="S1.p2.18.m5.1.1.2.2">bold-∇</ci><ci id="S1.p2.18.m5.1.1.2.3.cmml" xref="S1.p2.18.m5.1.1.2.3">𝒖</ci></apply><cn id="S1.p2.18.m5.1.1.3.cmml" type="integer" xref="S1.p2.18.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.18.m5.1c">\bm{\nabla}\cdot\bm{u}=0</annotation><annotation encoding="application/x-llamapun" id="S1.p2.18.m5.1d">bold_∇ ⋅ bold_italic_u = 0</annotation></semantics></math> modes) <cite class="ltx_cite ltx_citemacro_citep">(Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib25" title="">2010</a>)</cite>. This was later generalized to high-<math alttext="\beta" class="ltx_Math" display="inline" id="S1.p2.19.m6.1"><semantics id="S1.p2.19.m6.1a"><mi id="S1.p2.19.m6.1.1" xref="S1.p2.19.m6.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.19.m6.1b"><ci id="S1.p2.19.m6.1.1.cmml" xref="S1.p2.19.m6.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.19.m6.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.19.m6.1d">italic_β</annotation></semantics></math> shocks in <cite class="ltx_cite ltx_citemacro_citet">Molina et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib55" title="">2012</a>)</cite>, non-isothermal shocks in <cite class="ltx_cite ltx_citemacro_citet">Nolan et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib56" title="">2015</a>)</cite> and low-<math alttext="\beta" class="ltx_Math" display="inline" id="S1.p2.20.m7.1"><semantics id="S1.p2.20.m7.1a"><mi id="S1.p2.20.m7.1.1" xref="S1.p2.20.m7.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.20.m7.1b"><ci id="S1.p2.20.m7.1.1.cmml" xref="S1.p2.20.m7.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.20.m7.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.20.m7.1d">italic_β</annotation></semantics></math> shocks in <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib6" title="">2021</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.9">Numerous studies have been conducted to verify the relation of density fluctuation and turbulent Mach number, using both observational space and astrophysical data <cite class="ltx_cite ltx_citemacro_citep">(Matthaeus et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib49" title="">1991</a>; Tu & Marsch, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib75" title="">1994</a>; Bavassano & Bruno, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib2" title="">1995</a>; Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib27" title="">2016</a>; Menon et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib51" title="">2021</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib52" title="">2020</a>; Sharda et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib69" title="">2021</a>; Gerrard et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib33" title="">2023</a>; Cuesta et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib17" title="">2023</a>; Zhao et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib83" title="">2025</a>)</cite> and numerical simulations <cite class="ltx_cite ltx_citemacro_citep">(Padoan et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib58" title="">1997</a>; Passot & Vázquez-Semadeni, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib61" title="">1998</a>; Kowal et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib42" title="">2007</a>; Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib25" title="">2010</a>; Price et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib63" title="">2011</a>; Burkhart & Lazarian, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib15" title="">2012</a>; Nolan et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib56" title="">2015</a>; Pan et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib59" title="">2019</a>; Mohapatra et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib53" title="">2021</a>; Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib6" title="">2021</a>; Dhawalikar et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib18" title="">2022</a>; Du et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib19" title="">2023</a>)</cite>, and mostly support the linear relation (<a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.E4" title="4 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 4</span></a> and <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.E5" title="5 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 5</span></a>). Observationally in space plasmas, to the best of our knowledge, the <math alttext="\delta\rho-\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mrow id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml"><mi id="S1.p3.1.m1.1.1.2.2" xref="S1.p3.1.m1.1.1.2.2.cmml">δ</mi><mo id="S1.p3.1.m1.1.1.2.1" xref="S1.p3.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S1.p3.1.m1.1.1.2.3" xref="S1.p3.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">−</mo><msub id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml">ℳ</mi><mi id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><minus id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1.1"></minus><apply id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2"><times id="S1.p3.1.m1.1.1.2.1.cmml" xref="S1.p3.1.m1.1.1.2.1"></times><ci id="S1.p3.1.m1.1.1.2.2.cmml" xref="S1.p3.1.m1.1.1.2.2">𝛿</ci><ci id="S1.p3.1.m1.1.1.2.3.cmml" xref="S1.p3.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.3.1.cmml" xref="S1.p3.1.m1.1.1.3">subscript</csymbol><ci id="S1.p3.1.m1.1.1.3.2.cmml" xref="S1.p3.1.m1.1.1.3.2">ℳ</ci><ci id="S1.p3.1.m1.1.1.3.3.cmml" xref="S1.p3.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\delta\rho-\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_δ italic_ρ - caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> scaling relations have not been tested for high-<math alttext="\beta" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_β</annotation></semantics></math> regime, and nor have they been tested on a scale-by-scale basis in simulations, far away from the modes that drive the turbulence. Our goal in this work is to then further examine the scaling relation between <math alttext="\delta\rho-\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><mrow id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml"><mi id="S1.p3.3.m3.1.1.2.2" xref="S1.p3.3.m3.1.1.2.2.cmml">δ</mi><mo id="S1.p3.3.m3.1.1.2.1" xref="S1.p3.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S1.p3.3.m3.1.1.2.3" xref="S1.p3.3.m3.1.1.2.3.cmml">ρ</mi></mrow><mo id="S1.p3.3.m3.1.1.1" xref="S1.p3.3.m3.1.1.1.cmml">−</mo><msub id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.1.1.3.2" xref="S1.p3.3.m3.1.1.3.2.cmml">ℳ</mi><mi id="S1.p3.3.m3.1.1.3.3" xref="S1.p3.3.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><apply id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1"><minus id="S1.p3.3.m3.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1"></minus><apply id="S1.p3.3.m3.1.1.2.cmml" xref="S1.p3.3.m3.1.1.2"><times id="S1.p3.3.m3.1.1.2.1.cmml" xref="S1.p3.3.m3.1.1.2.1"></times><ci id="S1.p3.3.m3.1.1.2.2.cmml" xref="S1.p3.3.m3.1.1.2.2">𝛿</ci><ci id="S1.p3.3.m3.1.1.2.3.cmml" xref="S1.p3.3.m3.1.1.2.3">𝜌</ci></apply><apply id="S1.p3.3.m3.1.1.3.cmml" xref="S1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.3.1.cmml" xref="S1.p3.3.m3.1.1.3">subscript</csymbol><ci id="S1.p3.3.m3.1.1.3.2.cmml" xref="S1.p3.3.m3.1.1.3.2">ℳ</ci><ci id="S1.p3.3.m3.1.1.3.3.cmml" xref="S1.p3.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">\delta\rho-\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_δ italic_ρ - caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> for high-beta plasma. To test this, we utilize <span class="ltx_text ltx_font_italic" id="S1.p3.9.1">in-situ</span> data in Earth’s magnetosheath, where the effect of spatial inhomogeneities is strong and the typical <math alttext="\beta\sim 10" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mrow id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mi id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml">β</mi><mo id="S1.p3.4.m4.1.1.1" xref="S1.p3.4.m4.1.1.1.cmml">∼</mo><mn id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><csymbol cd="latexml" id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1.1">similar-to</csymbol><ci id="S1.p3.4.m4.1.1.2.cmml" xref="S1.p3.4.m4.1.1.2">𝛽</ci><cn id="S1.p3.4.m4.1.1.3.cmml" type="integer" xref="S1.p3.4.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">\beta\sim 10</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_β ∼ 10</annotation></semantics></math>. Along with the <span class="ltx_text ltx_font_italic" id="S1.p3.9.2">in-situ</span> data, we make use of an unprecedentedly high-resolution MHD simulation at Reynolds number <math alttext="\rm{Re}\gtrsim 10^{6}" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mrow id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><mi id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml">Re</mi><mo id="S1.p3.5.m5.1.1.1" xref="S1.p3.5.m5.1.1.1.cmml">≳</mo><msup id="S1.p3.5.m5.1.1.3" xref="S1.p3.5.m5.1.1.3.cmml"><mn id="S1.p3.5.m5.1.1.3.2" xref="S1.p3.5.m5.1.1.3.2.cmml">10</mn><mn id="S1.p3.5.m5.1.1.3.3" xref="S1.p3.5.m5.1.1.3.3.cmml">6</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><apply id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1"><csymbol cd="latexml" id="S1.p3.5.m5.1.1.1.cmml" xref="S1.p3.5.m5.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S1.p3.5.m5.1.1.2.cmml" xref="S1.p3.5.m5.1.1.2">Re</ci><apply id="S1.p3.5.m5.1.1.3.cmml" xref="S1.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.p3.5.m5.1.1.3.1.cmml" xref="S1.p3.5.m5.1.1.3">superscript</csymbol><cn id="S1.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.3.2">10</cn><cn id="S1.p3.5.m5.1.1.3.3.cmml" type="integer" xref="S1.p3.5.m5.1.1.3.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">\rm{Re}\gtrsim 10^{6}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">roman_Re ≳ 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\beta\sim 1" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mrow id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml"><mi id="S1.p3.6.m6.1.1.2" xref="S1.p3.6.m6.1.1.2.cmml">β</mi><mo id="S1.p3.6.m6.1.1.1" xref="S1.p3.6.m6.1.1.1.cmml">∼</mo><mn id="S1.p3.6.m6.1.1.3" xref="S1.p3.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><apply id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1"><csymbol cd="latexml" id="S1.p3.6.m6.1.1.1.cmml" xref="S1.p3.6.m6.1.1.1">similar-to</csymbol><ci id="S1.p3.6.m6.1.1.2.cmml" xref="S1.p3.6.m6.1.1.2">𝛽</ci><cn id="S1.p3.6.m6.1.1.3.cmml" type="integer" xref="S1.p3.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">\beta\sim 1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_β ∼ 1</annotation></semantics></math>. Regardless of <math alttext="\beta" class="ltx_Math" display="inline" id="S1.p3.7.m7.1"><semantics id="S1.p3.7.m7.1a"><mi id="S1.p3.7.m7.1.1" xref="S1.p3.7.m7.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p3.7.m7.1b"><ci id="S1.p3.7.m7.1.1.cmml" xref="S1.p3.7.m7.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m7.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m7.1d">italic_β</annotation></semantics></math>, within <math alttext="1\sigma" class="ltx_Math" display="inline" id="S1.p3.8.m8.1"><semantics id="S1.p3.8.m8.1a"><mrow id="S1.p3.8.m8.1.1" xref="S1.p3.8.m8.1.1.cmml"><mn id="S1.p3.8.m8.1.1.2" xref="S1.p3.8.m8.1.1.2.cmml">1</mn><mo id="S1.p3.8.m8.1.1.1" xref="S1.p3.8.m8.1.1.1.cmml">⁢</mo><mi id="S1.p3.8.m8.1.1.3" xref="S1.p3.8.m8.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.8.m8.1b"><apply id="S1.p3.8.m8.1.1.cmml" xref="S1.p3.8.m8.1.1"><times id="S1.p3.8.m8.1.1.1.cmml" xref="S1.p3.8.m8.1.1.1"></times><cn id="S1.p3.8.m8.1.1.2.cmml" type="integer" xref="S1.p3.8.m8.1.1.2">1</cn><ci id="S1.p3.8.m8.1.1.3.cmml" xref="S1.p3.8.m8.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m8.1c">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m8.1d">1 italic_σ</annotation></semantics></math> uncertainties, both data support a linear scaling <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p3.9.m9.1"><semantics id="S1.p3.9.m9.1a"><mrow id="S1.p3.9.m9.1.1" xref="S1.p3.9.m9.1.1.cmml"><mrow id="S1.p3.9.m9.1.1.2" xref="S1.p3.9.m9.1.1.2.cmml"><mrow id="S1.p3.9.m9.1.1.2.2" xref="S1.p3.9.m9.1.1.2.2.cmml"><mi id="S1.p3.9.m9.1.1.2.2.2" xref="S1.p3.9.m9.1.1.2.2.2.cmml">δ</mi><mo id="S1.p3.9.m9.1.1.2.2.1" xref="S1.p3.9.m9.1.1.2.2.1.cmml">⁢</mo><mi id="S1.p3.9.m9.1.1.2.2.3" xref="S1.p3.9.m9.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.p3.9.m9.1.1.2.1" xref="S1.p3.9.m9.1.1.2.1.cmml">/</mo><msub id="S1.p3.9.m9.1.1.2.3" xref="S1.p3.9.m9.1.1.2.3.cmml"><mi id="S1.p3.9.m9.1.1.2.3.2" xref="S1.p3.9.m9.1.1.2.3.2.cmml">ρ</mi><mn id="S1.p3.9.m9.1.1.2.3.3" xref="S1.p3.9.m9.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.p3.9.m9.1.1.1" xref="S1.p3.9.m9.1.1.1.cmml">∝</mo><msub id="S1.p3.9.m9.1.1.3" xref="S1.p3.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.9.m9.1.1.3.2" xref="S1.p3.9.m9.1.1.3.2.cmml">ℳ</mi><mi id="S1.p3.9.m9.1.1.3.3" xref="S1.p3.9.m9.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.9.m9.1b"><apply id="S1.p3.9.m9.1.1.cmml" xref="S1.p3.9.m9.1.1"><csymbol cd="latexml" id="S1.p3.9.m9.1.1.1.cmml" xref="S1.p3.9.m9.1.1.1">proportional-to</csymbol><apply id="S1.p3.9.m9.1.1.2.cmml" xref="S1.p3.9.m9.1.1.2"><divide id="S1.p3.9.m9.1.1.2.1.cmml" xref="S1.p3.9.m9.1.1.2.1"></divide><apply id="S1.p3.9.m9.1.1.2.2.cmml" xref="S1.p3.9.m9.1.1.2.2"><times id="S1.p3.9.m9.1.1.2.2.1.cmml" xref="S1.p3.9.m9.1.1.2.2.1"></times><ci id="S1.p3.9.m9.1.1.2.2.2.cmml" xref="S1.p3.9.m9.1.1.2.2.2">𝛿</ci><ci id="S1.p3.9.m9.1.1.2.2.3.cmml" xref="S1.p3.9.m9.1.1.2.2.3">𝜌</ci></apply><apply id="S1.p3.9.m9.1.1.2.3.cmml" xref="S1.p3.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S1.p3.9.m9.1.1.2.3.1.cmml" xref="S1.p3.9.m9.1.1.2.3">subscript</csymbol><ci id="S1.p3.9.m9.1.1.2.3.2.cmml" xref="S1.p3.9.m9.1.1.2.3.2">𝜌</ci><cn id="S1.p3.9.m9.1.1.2.3.3.cmml" type="integer" xref="S1.p3.9.m9.1.1.2.3.3">0</cn></apply></apply><apply id="S1.p3.9.m9.1.1.3.cmml" xref="S1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.p3.9.m9.1.1.3.1.cmml" xref="S1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S1.p3.9.m9.1.1.3.2.cmml" xref="S1.p3.9.m9.1.1.3.2">ℳ</ci><ci id="S1.p3.9.m9.1.1.3.3.cmml" xref="S1.p3.9.m9.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m9.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m9.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, even sharing (statistically) the same proportionality constants. This potentially demonstrates the importance of inhomogeneous fluctuations in compressible, turbulent plasmas, and supports the weakly compressible MHD turbulence theory in <cite class="ltx_cite ltx_citemacro_citet">Bhattacharjee et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.6">This paper is organized as follows. In <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S2" title="2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">section 2</span></a> we discuss the observational and simulation data that we use in this study. In <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3" title="3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">section 3</span></a> we calculate scale-dependent <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mrow id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml"><mi id="S1.p4.1.m1.1.1.2.2" xref="S1.p4.1.m1.1.1.2.2.cmml">δ</mi><mo id="S1.p4.1.m1.1.1.2.1" xref="S1.p4.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S1.p4.1.m1.1.1.2.3" xref="S1.p4.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">/</mo><msub id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml"><mi id="S1.p4.1.m1.1.1.3.2" xref="S1.p4.1.m1.1.1.3.2.cmml">ρ</mi><mn id="S1.p4.1.m1.1.1.3.3" xref="S1.p4.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><divide id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1"></divide><apply id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2"><times id="S1.p4.1.m1.1.1.2.1.cmml" xref="S1.p4.1.m1.1.1.2.1"></times><ci id="S1.p4.1.m1.1.1.2.2.cmml" xref="S1.p4.1.m1.1.1.2.2">𝛿</ci><ci id="S1.p4.1.m1.1.1.2.3.cmml" xref="S1.p4.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S1.p4.1.m1.1.1.3.cmml" xref="S1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p4.1.m1.1.1.3.1.cmml" xref="S1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S1.p4.1.m1.1.1.3.2.cmml" xref="S1.p4.1.m1.1.1.3.2">𝜌</ci><cn id="S1.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S1.p4.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><msub id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml">ℳ</mi><mi id="S1.p4.2.m2.1.1.3" xref="S1.p4.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1">subscript</csymbol><ci id="S1.p4.2.m2.1.1.2.cmml" xref="S1.p4.2.m2.1.1.2">ℳ</ci><ci id="S1.p4.2.m2.1.1.3.cmml" xref="S1.p4.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> curves from the simulation and combine them with the observational data to reveal that <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mrow id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml"><mrow id="S1.p4.3.m3.1.1.2" xref="S1.p4.3.m3.1.1.2.cmml"><mrow id="S1.p4.3.m3.1.1.2.2" xref="S1.p4.3.m3.1.1.2.2.cmml"><mi id="S1.p4.3.m3.1.1.2.2.2" xref="S1.p4.3.m3.1.1.2.2.2.cmml">δ</mi><mo id="S1.p4.3.m3.1.1.2.2.1" xref="S1.p4.3.m3.1.1.2.2.1.cmml">⁢</mo><mi id="S1.p4.3.m3.1.1.2.2.3" xref="S1.p4.3.m3.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S1.p4.3.m3.1.1.2.1" xref="S1.p4.3.m3.1.1.2.1.cmml">/</mo><msub id="S1.p4.3.m3.1.1.2.3" xref="S1.p4.3.m3.1.1.2.3.cmml"><mi id="S1.p4.3.m3.1.1.2.3.2" xref="S1.p4.3.m3.1.1.2.3.2.cmml">ρ</mi><mn id="S1.p4.3.m3.1.1.2.3.3" xref="S1.p4.3.m3.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S1.p4.3.m3.1.1.1" xref="S1.p4.3.m3.1.1.1.cmml">∝</mo><msub id="S1.p4.3.m3.1.1.3" xref="S1.p4.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.3.m3.1.1.3.2" xref="S1.p4.3.m3.1.1.3.2.cmml">ℳ</mi><mi id="S1.p4.3.m3.1.1.3.3" xref="S1.p4.3.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><apply id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1"><csymbol cd="latexml" id="S1.p4.3.m3.1.1.1.cmml" xref="S1.p4.3.m3.1.1.1">proportional-to</csymbol><apply id="S1.p4.3.m3.1.1.2.cmml" xref="S1.p4.3.m3.1.1.2"><divide id="S1.p4.3.m3.1.1.2.1.cmml" xref="S1.p4.3.m3.1.1.2.1"></divide><apply id="S1.p4.3.m3.1.1.2.2.cmml" xref="S1.p4.3.m3.1.1.2.2"><times id="S1.p4.3.m3.1.1.2.2.1.cmml" xref="S1.p4.3.m3.1.1.2.2.1"></times><ci id="S1.p4.3.m3.1.1.2.2.2.cmml" xref="S1.p4.3.m3.1.1.2.2.2">𝛿</ci><ci id="S1.p4.3.m3.1.1.2.2.3.cmml" xref="S1.p4.3.m3.1.1.2.2.3">𝜌</ci></apply><apply id="S1.p4.3.m3.1.1.2.3.cmml" xref="S1.p4.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S1.p4.3.m3.1.1.2.3.1.cmml" xref="S1.p4.3.m3.1.1.2.3">subscript</csymbol><ci id="S1.p4.3.m3.1.1.2.3.2.cmml" xref="S1.p4.3.m3.1.1.2.3.2">𝜌</ci><cn id="S1.p4.3.m3.1.1.2.3.3.cmml" type="integer" xref="S1.p4.3.m3.1.1.2.3.3">0</cn></apply></apply><apply id="S1.p4.3.m3.1.1.3.cmml" xref="S1.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p4.3.m3.1.1.3.1.cmml" xref="S1.p4.3.m3.1.1.3">subscript</csymbol><ci id="S1.p4.3.m3.1.1.3.2.cmml" xref="S1.p4.3.m3.1.1.3.2">ℳ</ci><ci id="S1.p4.3.m3.1.1.3.3.cmml" xref="S1.p4.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> over a broad range of <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mrow id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml"><mrow id="S1.p4.4.m4.1.1.2" xref="S1.p4.4.m4.1.1.2.cmml"><mi id="S1.p4.4.m4.1.1.2.2" xref="S1.p4.4.m4.1.1.2.2.cmml">δ</mi><mo id="S1.p4.4.m4.1.1.2.1" xref="S1.p4.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S1.p4.4.m4.1.1.2.3" xref="S1.p4.4.m4.1.1.2.3.cmml">ρ</mi></mrow><mo id="S1.p4.4.m4.1.1.1" xref="S1.p4.4.m4.1.1.1.cmml">/</mo><msub id="S1.p4.4.m4.1.1.3" xref="S1.p4.4.m4.1.1.3.cmml"><mi id="S1.p4.4.m4.1.1.3.2" xref="S1.p4.4.m4.1.1.3.2.cmml">ρ</mi><mn id="S1.p4.4.m4.1.1.3.3" xref="S1.p4.4.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><apply id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1"><divide id="S1.p4.4.m4.1.1.1.cmml" xref="S1.p4.4.m4.1.1.1"></divide><apply id="S1.p4.4.m4.1.1.2.cmml" xref="S1.p4.4.m4.1.1.2"><times id="S1.p4.4.m4.1.1.2.1.cmml" xref="S1.p4.4.m4.1.1.2.1"></times><ci id="S1.p4.4.m4.1.1.2.2.cmml" xref="S1.p4.4.m4.1.1.2.2">𝛿</ci><ci id="S1.p4.4.m4.1.1.2.3.cmml" xref="S1.p4.4.m4.1.1.2.3">𝜌</ci></apply><apply id="S1.p4.4.m4.1.1.3.cmml" xref="S1.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.p4.4.m4.1.1.3.1.cmml" xref="S1.p4.4.m4.1.1.3">subscript</csymbol><ci id="S1.p4.4.m4.1.1.3.2.cmml" xref="S1.p4.4.m4.1.1.3.2">𝜌</ci><cn id="S1.p4.4.m4.1.1.3.3.cmml" type="integer" xref="S1.p4.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><msub id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.5.m5.1.1.2" xref="S1.p4.5.m5.1.1.2.cmml">ℳ</mi><mi id="S1.p4.5.m5.1.1.3" xref="S1.p4.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><apply id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S1.p4.5.m5.1.1.1.cmml" xref="S1.p4.5.m5.1.1">subscript</csymbol><ci id="S1.p4.5.m5.1.1.2.cmml" xref="S1.p4.5.m5.1.1.2">ℳ</ci><ci id="S1.p4.5.m5.1.1.3.cmml" xref="S1.p4.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, regardless of <math alttext="\beta" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mi id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">italic_β</annotation></semantics></math>. Finally, in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S4" title="4 Discussion & Conclusions ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">section 4</span></a> we discuss the implications our results have for weakly compressible MHD theory and space and astrophysical plasmas, and in particular driving parameter measurements that may be contaminated by cascade effects.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="456" id="S1.F1.g1" src="x1.png" width="789"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>An example of the MMS data in Earth’s turbulent magnetosheath. The data shown are from the FGM and FPI instruments on-board the MMS1 spacecraft. The top panel shows the magnetic field measurements in GSE coordinates; the second panel shows the electron density; third panel shows the ion temperature; and the bottom panel shows the ion velocity in GSE coordinates. Each panel shows significant temporal turbulent fluctuations, which, under the assumption of Taylor’s frozen-in hypothesis, we can equate to sampling the spatial turbulent fluctuations.</figcaption> </figure> <figure class="ltx_figure" id="S1.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="896" id="S1.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>A two-dimensional slice of the logarithmic mass density fluctuations, <math alttext="\ln(\rho/\rho_{0})" class="ltx_Math" display="inline" id="S1.F2.8.m1.2"><semantics id="S1.F2.8.m1.2b"><mrow id="S1.F2.8.m1.2.2.1" xref="S1.F2.8.m1.2.2.2.cmml"><mi id="S1.F2.8.m1.1.1" xref="S1.F2.8.m1.1.1.cmml">ln</mi><mo id="S1.F2.8.m1.2.2.1b" xref="S1.F2.8.m1.2.2.2.cmml">⁡</mo><mrow id="S1.F2.8.m1.2.2.1.1" xref="S1.F2.8.m1.2.2.2.cmml"><mo id="S1.F2.8.m1.2.2.1.1.2" stretchy="false" xref="S1.F2.8.m1.2.2.2.cmml">(</mo><mrow id="S1.F2.8.m1.2.2.1.1.1" xref="S1.F2.8.m1.2.2.1.1.1.cmml"><mi id="S1.F2.8.m1.2.2.1.1.1.2" xref="S1.F2.8.m1.2.2.1.1.1.2.cmml">ρ</mi><mo id="S1.F2.8.m1.2.2.1.1.1.1" xref="S1.F2.8.m1.2.2.1.1.1.1.cmml">/</mo><msub id="S1.F2.8.m1.2.2.1.1.1.3" xref="S1.F2.8.m1.2.2.1.1.1.3.cmml"><mi id="S1.F2.8.m1.2.2.1.1.1.3.2" xref="S1.F2.8.m1.2.2.1.1.1.3.2.cmml">ρ</mi><mn id="S1.F2.8.m1.2.2.1.1.1.3.3" xref="S1.F2.8.m1.2.2.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S1.F2.8.m1.2.2.1.1.3" stretchy="false" xref="S1.F2.8.m1.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.8.m1.2c"><apply id="S1.F2.8.m1.2.2.2.cmml" xref="S1.F2.8.m1.2.2.1"><ln id="S1.F2.8.m1.1.1.cmml" xref="S1.F2.8.m1.1.1"></ln><apply id="S1.F2.8.m1.2.2.1.1.1.cmml" xref="S1.F2.8.m1.2.2.1.1.1"><divide id="S1.F2.8.m1.2.2.1.1.1.1.cmml" xref="S1.F2.8.m1.2.2.1.1.1.1"></divide><ci id="S1.F2.8.m1.2.2.1.1.1.2.cmml" xref="S1.F2.8.m1.2.2.1.1.1.2">𝜌</ci><apply id="S1.F2.8.m1.2.2.1.1.1.3.cmml" xref="S1.F2.8.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S1.F2.8.m1.2.2.1.1.1.3.1.cmml" xref="S1.F2.8.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S1.F2.8.m1.2.2.1.1.1.3.2.cmml" xref="S1.F2.8.m1.2.2.1.1.1.3.2">𝜌</ci><cn id="S1.F2.8.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S1.F2.8.m1.2.2.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.8.m1.2d">\ln(\rho/\rho_{0})</annotation><annotation encoding="application/x-llamapun" id="S1.F2.8.m1.2e">roman_ln ( italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>, where <math alttext="\rho_{0}" class="ltx_Math" display="inline" id="S1.F2.9.m2.1"><semantics id="S1.F2.9.m2.1b"><msub id="S1.F2.9.m2.1.1" xref="S1.F2.9.m2.1.1.cmml"><mi id="S1.F2.9.m2.1.1.2" xref="S1.F2.9.m2.1.1.2.cmml">ρ</mi><mn id="S1.F2.9.m2.1.1.3" xref="S1.F2.9.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S1.F2.9.m2.1c"><apply id="S1.F2.9.m2.1.1.cmml" xref="S1.F2.9.m2.1.1"><csymbol cd="ambiguous" id="S1.F2.9.m2.1.1.1.cmml" xref="S1.F2.9.m2.1.1">subscript</csymbol><ci id="S1.F2.9.m2.1.1.2.cmml" xref="S1.F2.9.m2.1.1.2">𝜌</ci><cn id="S1.F2.9.m2.1.1.3.cmml" type="integer" xref="S1.F2.9.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.9.m2.1d">\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.9.m2.1e">italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the volume-average, overlaid with in-plane magnetic field streamlines shown in white. The mean density <math alttext="\rho=\rho_{0}" class="ltx_Math" display="inline" id="S1.F2.10.m3.1"><semantics id="S1.F2.10.m3.1b"><mrow id="S1.F2.10.m3.1.1" xref="S1.F2.10.m3.1.1.cmml"><mi id="S1.F2.10.m3.1.1.2" xref="S1.F2.10.m3.1.1.2.cmml">ρ</mi><mo id="S1.F2.10.m3.1.1.1" xref="S1.F2.10.m3.1.1.1.cmml">=</mo><msub id="S1.F2.10.m3.1.1.3" xref="S1.F2.10.m3.1.1.3.cmml"><mi id="S1.F2.10.m3.1.1.3.2" xref="S1.F2.10.m3.1.1.3.2.cmml">ρ</mi><mn id="S1.F2.10.m3.1.1.3.3" xref="S1.F2.10.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.10.m3.1c"><apply id="S1.F2.10.m3.1.1.cmml" xref="S1.F2.10.m3.1.1"><eq id="S1.F2.10.m3.1.1.1.cmml" xref="S1.F2.10.m3.1.1.1"></eq><ci id="S1.F2.10.m3.1.1.2.cmml" xref="S1.F2.10.m3.1.1.2">𝜌</ci><apply id="S1.F2.10.m3.1.1.3.cmml" xref="S1.F2.10.m3.1.1.3"><csymbol cd="ambiguous" id="S1.F2.10.m3.1.1.3.1.cmml" xref="S1.F2.10.m3.1.1.3">subscript</csymbol><ci id="S1.F2.10.m3.1.1.3.2.cmml" xref="S1.F2.10.m3.1.1.3.2">𝜌</ci><cn id="S1.F2.10.m3.1.1.3.3.cmml" type="integer" xref="S1.F2.10.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.10.m3.1d">\rho=\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.10.m3.1e">italic_ρ = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is shown in black, over-densities <math alttext="\rho>\rho_{0}" class="ltx_Math" display="inline" id="S1.F2.11.m4.1"><semantics id="S1.F2.11.m4.1b"><mrow id="S1.F2.11.m4.1.1" xref="S1.F2.11.m4.1.1.cmml"><mi id="S1.F2.11.m4.1.1.2" xref="S1.F2.11.m4.1.1.2.cmml">ρ</mi><mo id="S1.F2.11.m4.1.1.1" xref="S1.F2.11.m4.1.1.1.cmml">></mo><msub id="S1.F2.11.m4.1.1.3" xref="S1.F2.11.m4.1.1.3.cmml"><mi id="S1.F2.11.m4.1.1.3.2" xref="S1.F2.11.m4.1.1.3.2.cmml">ρ</mi><mn id="S1.F2.11.m4.1.1.3.3" xref="S1.F2.11.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.11.m4.1c"><apply id="S1.F2.11.m4.1.1.cmml" xref="S1.F2.11.m4.1.1"><gt id="S1.F2.11.m4.1.1.1.cmml" xref="S1.F2.11.m4.1.1.1"></gt><ci id="S1.F2.11.m4.1.1.2.cmml" xref="S1.F2.11.m4.1.1.2">𝜌</ci><apply id="S1.F2.11.m4.1.1.3.cmml" xref="S1.F2.11.m4.1.1.3"><csymbol cd="ambiguous" id="S1.F2.11.m4.1.1.3.1.cmml" xref="S1.F2.11.m4.1.1.3">subscript</csymbol><ci id="S1.F2.11.m4.1.1.3.2.cmml" xref="S1.F2.11.m4.1.1.3.2">𝜌</ci><cn id="S1.F2.11.m4.1.1.3.3.cmml" type="integer" xref="S1.F2.11.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.11.m4.1d">\rho>\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.11.m4.1e">italic_ρ > italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in yellow and under-densities <math alttext="\rho<\rho_{0}" class="ltx_Math" display="inline" id="S1.F2.12.m5.1"><semantics id="S1.F2.12.m5.1b"><mrow id="S1.F2.12.m5.1.1" xref="S1.F2.12.m5.1.1.cmml"><mi id="S1.F2.12.m5.1.1.2" xref="S1.F2.12.m5.1.1.2.cmml">ρ</mi><mo id="S1.F2.12.m5.1.1.1" xref="S1.F2.12.m5.1.1.1.cmml"><</mo><msub id="S1.F2.12.m5.1.1.3" xref="S1.F2.12.m5.1.1.3.cmml"><mi id="S1.F2.12.m5.1.1.3.2" xref="S1.F2.12.m5.1.1.3.2.cmml">ρ</mi><mn id="S1.F2.12.m5.1.1.3.3" xref="S1.F2.12.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.12.m5.1c"><apply id="S1.F2.12.m5.1.1.cmml" xref="S1.F2.12.m5.1.1"><lt id="S1.F2.12.m5.1.1.1.cmml" xref="S1.F2.12.m5.1.1.1"></lt><ci id="S1.F2.12.m5.1.1.2.cmml" xref="S1.F2.12.m5.1.1.2">𝜌</ci><apply id="S1.F2.12.m5.1.1.3.cmml" xref="S1.F2.12.m5.1.1.3"><csymbol cd="ambiguous" id="S1.F2.12.m5.1.1.3.1.cmml" xref="S1.F2.12.m5.1.1.3">subscript</csymbol><ci id="S1.F2.12.m5.1.1.3.2.cmml" xref="S1.F2.12.m5.1.1.3.2">𝜌</ci><cn id="S1.F2.12.m5.1.1.3.3.cmml" type="integer" xref="S1.F2.12.m5.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.12.m5.1d">\rho<\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.12.m5.1e">italic_ρ < italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in green. Due to the strong <math alttext="\rho/\rho_{0}" class="ltx_Math" display="inline" id="S1.F2.13.m6.1"><semantics id="S1.F2.13.m6.1b"><mrow id="S1.F2.13.m6.1.1" xref="S1.F2.13.m6.1.1.cmml"><mi id="S1.F2.13.m6.1.1.2" xref="S1.F2.13.m6.1.1.2.cmml">ρ</mi><mo id="S1.F2.13.m6.1.1.1" xref="S1.F2.13.m6.1.1.1.cmml">/</mo><msub id="S1.F2.13.m6.1.1.3" xref="S1.F2.13.m6.1.1.3.cmml"><mi id="S1.F2.13.m6.1.1.3.2" xref="S1.F2.13.m6.1.1.3.2.cmml">ρ</mi><mn id="S1.F2.13.m6.1.1.3.3" xref="S1.F2.13.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.13.m6.1c"><apply id="S1.F2.13.m6.1.1.cmml" xref="S1.F2.13.m6.1.1"><divide id="S1.F2.13.m6.1.1.1.cmml" xref="S1.F2.13.m6.1.1.1"></divide><ci id="S1.F2.13.m6.1.1.2.cmml" xref="S1.F2.13.m6.1.1.2">𝜌</ci><apply id="S1.F2.13.m6.1.1.3.cmml" xref="S1.F2.13.m6.1.1.3"><csymbol cd="ambiguous" id="S1.F2.13.m6.1.1.3.1.cmml" xref="S1.F2.13.m6.1.1.3">subscript</csymbol><ci id="S1.F2.13.m6.1.1.3.2.cmml" xref="S1.F2.13.m6.1.1.3.2">𝜌</ci><cn id="S1.F2.13.m6.1.1.3.3.cmml" type="integer" xref="S1.F2.13.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.13.m6.1d">\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.13.m6.1e">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> contrasts and the coherent, over-dense filamentary structures and deep under-dense voids the mass-density (and magnetic field) is highly-inhomogeneous. More details of the <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S1.F2.14.m7.2"><semantics id="S1.F2.14.m7.2b"><mrow id="S1.F2.14.m7.2.2.1" xref="S1.F2.14.m7.2.2.2.cmml"><mn id="S1.F2.14.m7.1.1" xref="S1.F2.14.m7.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S1.F2.14.m7.2.2.1.2" xref="S1.F2.14.m7.2.2.2.cmml">,</mo></mpadded><msup id="S1.F2.14.m7.2.2.1.1" xref="S1.F2.14.m7.2.2.1.1.cmml"><mn id="S1.F2.14.m7.2.2.1.1.2" xref="S1.F2.14.m7.2.2.1.1.2.cmml">080</mn><mn id="S1.F2.14.m7.2.2.1.1.3" xref="S1.F2.14.m7.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.14.m7.2c"><list id="S1.F2.14.m7.2.2.2.cmml" xref="S1.F2.14.m7.2.2.1"><cn id="S1.F2.14.m7.1.1.cmml" type="integer" xref="S1.F2.14.m7.1.1">10</cn><apply id="S1.F2.14.m7.2.2.1.1.cmml" xref="S1.F2.14.m7.2.2.1.1"><csymbol cd="ambiguous" id="S1.F2.14.m7.2.2.1.1.1.cmml" xref="S1.F2.14.m7.2.2.1.1">superscript</csymbol><cn id="S1.F2.14.m7.2.2.1.1.2.cmml" type="integer" xref="S1.F2.14.m7.2.2.1.1.2">080</cn><cn id="S1.F2.14.m7.2.2.1.1.3.cmml" type="integer" xref="S1.F2.14.m7.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.14.m7.2d">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.14.m7.2e">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> simulation are shown in <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite>.</figcaption> </figure> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Methods</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>MMS Observations</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.4">We use <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.4.1">in-situ</span> data collected by NASA’s Magnetospheric Multiscale (MMS) mission <cite class="ltx_cite ltx_citemacro_citep">(Burch et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib13" title="">2016</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib14" title="">2016</a>)</cite> in Earth’s magnetosheath. The magnetosheath consists of the shocked solar wind plasma downstream of the bow shock. Unlike the interplanetary solar wind, the magnetosheath plasma has a high <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_β</annotation></semantics></math> and is relatively compressible. The MMS consists of four identical spacecraft, but here we do not utilize any multi-spacecraft technique. For each interval, we average over the values obtained from four spacecraft. We make use of the ion and electron plasma moments from the Fast Plasma Instrument <cite class="ltx_cite ltx_citemacro_citep">(Pollock et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib62" title="">2016</a>)</cite>. We use ion velocity and temperature data to evaluate <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><msub id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">ℳ</mi><mi id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">ℳ</ci><ci id="S2.SS1.p1.2.m2.1.1.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. But, whenever available, we use the electron number density to measure background density and density fluctuation. This is because the number density of electrons and ions are almost equal due to the quasi-neutrality of the plasma, and due to a larger thermal speed the electron density measurements are usually more accurate <cite class="ltx_cite ltx_citemacro_citep">(Gershman et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib34" title="">2018</a>)</cite>. Magnetic field data are obtained from the fluxgate magnetometer (FGM) <cite class="ltx_cite ltx_citemacro_citep">(Russell et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib65" title="">2016</a>; Torbert et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib74" title="">2016</a>)</cite> and combined with the plasma moments to obtain <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mi id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><ci id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_β</annotation></semantics></math> values. For example, we show the burst resolution MMS data (MMS1) obtained in the turbulent magnetosheath on 2017 January 18 from 00:45:53 to 00:47:43 UTC in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 1</span></a>. The interval shows strong turbulent fluctuations with considerable density variations throughout the interval. The interval has <math alttext="\beta\approx 19" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">β</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">≈</mo><mn id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">19</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><approx id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></approx><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝛽</ci><cn id="S2.SS1.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p1.4.m4.1.1.3">19</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">\beta\approx 19</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_β ≈ 19</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.5">We eliminate intervals with very low average number density <math alttext="n_{e}<5" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><msub id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml"><mi id="S2.SS1.p2.1.m1.1.1.2.2" xref="S2.SS1.p2.1.m1.1.1.2.2.cmml">n</mi><mi id="S2.SS1.p2.1.m1.1.1.2.3" xref="S2.SS1.p2.1.m1.1.1.2.3.cmml">e</mi></msub><mo id="S2.SS1.p2.1.m1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.cmml"><</mo><mn id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><lt id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1"></lt><apply id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.2.1.cmml" xref="S2.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.2.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2.2">𝑛</ci><ci id="S2.SS1.p2.1.m1.1.1.2.3.cmml" xref="S2.SS1.p2.1.m1.1.1.2.3">𝑒</ci></apply><cn id="S2.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">n_{e}<5</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT < 5</annotation></semantics></math>cm<sup class="ltx_sup" id="S2.SS1.p2.5.1"><span class="ltx_text ltx_font_italic" id="S2.SS1.p2.5.1.1">-3</span></sup>, or very high average density <math alttext="n_{e}>50" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mrow id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><msub id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2.2" xref="S2.SS1.p2.3.m3.1.1.2.2.cmml">n</mi><mi id="S2.SS1.p2.3.m3.1.1.2.3" xref="S2.SS1.p2.3.m3.1.1.2.3.cmml">e</mi></msub><mo id="S2.SS1.p2.3.m3.1.1.1" xref="S2.SS1.p2.3.m3.1.1.1.cmml">></mo><mn id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">50</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><gt id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1.1"></gt><apply id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.2.1.cmml" xref="S2.SS1.p2.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2.2">𝑛</ci><ci id="S2.SS1.p2.3.m3.1.1.2.3.cmml" xref="S2.SS1.p2.3.m3.1.1.2.3">𝑒</ci></apply><cn id="S2.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p2.3.m3.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">n_{e}>50</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT > 50</annotation></semantics></math>cm<sup class="ltx_sup" id="S2.SS1.p2.5.2"><span class="ltx_text ltx_font_italic" id="S2.SS1.p2.5.2.1">-3</span></sup> because of larger uncertainties in such intervals. The length of every interval spans many times that of the typical correlation length (equivalently time) of the magnetosheath, which is about <math alttext="\tau_{c}\sim 100" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mrow id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml"><msub id="S2.SS1.p2.5.m5.1.1.2" xref="S2.SS1.p2.5.m5.1.1.2.cmml"><mi id="S2.SS1.p2.5.m5.1.1.2.2" xref="S2.SS1.p2.5.m5.1.1.2.2.cmml">τ</mi><mi id="S2.SS1.p2.5.m5.1.1.2.3" xref="S2.SS1.p2.5.m5.1.1.2.3.cmml">c</mi></msub><mo id="S2.SS1.p2.5.m5.1.1.1" xref="S2.SS1.p2.5.m5.1.1.1.cmml">∼</mo><mn id="S2.SS1.p2.5.m5.1.1.3" xref="S2.SS1.p2.5.m5.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><apply id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1"><csymbol cd="latexml" id="S2.SS1.p2.5.m5.1.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1.1">similar-to</csymbol><apply id="S2.SS1.p2.5.m5.1.1.2.cmml" xref="S2.SS1.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.1.1.2.1.cmml" xref="S2.SS1.p2.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.5.m5.1.1.2.2.cmml" xref="S2.SS1.p2.5.m5.1.1.2.2">𝜏</ci><ci id="S2.SS1.p2.5.m5.1.1.2.3.cmml" xref="S2.SS1.p2.5.m5.1.1.2.3">𝑐</ci></apply><cn id="S2.SS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p2.5.m5.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">\tau_{c}\sim 100</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_τ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∼ 100</annotation></semantics></math>s <cite class="ltx_cite ltx_citemacro_citep">(e.g., Stawarz et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib72" title="">2019</a>)</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Numerical Simulation</h3> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">MHD plasma model</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p1.1">Along with the MMS observational data, we use a heavily modified version of the magnetohydrodynamical (MHD) code <span class="ltx_text ltx_font_smallcaps" id="S2.SS2.SSS0.Px1.p1.1.1">flash</span> <cite class="ltx_cite ltx_citemacro_citep">(Fryxell et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib32" title="">2000</a>; Dubey et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib20" title="">2008</a>)</cite> that has recently been run on <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.1.m1.2"><semantics id="S2.SS2.SSS0.Px1.p1.1.m1.2a"><mrow id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.2.cmml"><mn id="S2.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.2.cmml">,</mo></mpadded><msup id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.cmml"><mn id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.2.cmml">080</mn><mn id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.3" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.1.m1.2b"><list id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1"><cn id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1">10</cn><apply id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1">superscript</csymbol><cn id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.2">080</cn><cn id="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.1.m1.2c">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.1.m1.2d">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> grid scales <cite class="ltx_cite ltx_citemacro_citep">(Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite>. Our code uses a highly-optimized, hybrid-precision <cite class="ltx_cite ltx_citemacro_citep">(Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib24" title="">2021</a>)</cite>, positivity-preserving, second-order MUSCL-Hancock HLL5R Riemann scheme <cite class="ltx_cite ltx_citemacro_citep">(Bouchut et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib11" title="">2010</a>; Waagan et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib81" title="">2011</a>)</cite> to solve the compressible, ideal, MHD fluid equations in three dimensions,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx5"> <tbody id="S2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\partial_{t}\rho+\nabla\cdot\left(\rho\bm{u}\right)=0," class="ltx_Math" display="inline" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mrow id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.3.cmml"><msub id="S2.E6.m1.1.1.1.1.1.3.1" xref="S2.E6.m1.1.1.1.1.1.3.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.3.1.2" xref="S2.E6.m1.1.1.1.1.1.3.1.2.cmml">∂</mo><mi id="S2.E6.m1.1.1.1.1.1.3.1.3" xref="S2.E6.m1.1.1.1.1.1.3.1.3.cmml">t</mi></msub><mi id="S2.E6.m1.1.1.1.1.1.3.2" xref="S2.E6.m1.1.1.1.1.1.3.2.cmml">ρ</mi></mrow><mo id="S2.E6.m1.1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.2.cmml">+</mo><mrow id="S2.E6.m1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.1.3.cmml">∇</mo><mo id="S2.E6.m1.1.1.1.1.1.1.2" lspace="0em" rspace="0.222em" xref="S2.E6.m1.1.1.1.1.1.1.2.cmml">⋅</mo><mrow id="S2.E6.m1.1.1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.1.1.1.cmml"><mo 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xref="S2.E6.m1.1.1.1.1.1.2"></plus><apply id="S2.E6.m1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.3"><apply id="S2.E6.m1.1.1.1.1.1.3.1.cmml" xref="S2.E6.m1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.1.3.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.3.1">subscript</csymbol><partialdiff id="S2.E6.m1.1.1.1.1.1.3.1.2.cmml" xref="S2.E6.m1.1.1.1.1.1.3.1.2"></partialdiff><ci id="S2.E6.m1.1.1.1.1.1.3.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.3.1.3">𝑡</ci></apply><ci id="S2.E6.m1.1.1.1.1.1.3.2.cmml" xref="S2.E6.m1.1.1.1.1.1.3.2">𝜌</ci></apply><apply id="S2.E6.m1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1"><ci id="S2.E6.m1.1.1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.1.1.2">⋅</ci><ci id="S2.E6.m1.1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.1.3">∇</ci><apply id="S2.E6.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1.1.1"><times id="S2.E6.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1.1.1.1.1"></times><ci id="S2.E6.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.1.1.1.1.1.2">𝜌</ci><ci id="S2.E6.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.1.1.1.1.3">𝒖</ci></apply></apply></apply><cn id="S2.E6.m1.1.1.1.1.3.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">\displaystyle\partial_{t}\rho+\nabla\cdot\left(\rho\bm{u}\right)=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_ρ + ∇ ⋅ ( italic_ρ bold_italic_u ) = 0 ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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bold_italic_u ⊗ bold_italic_u + italic_p blackboard_I - divide start_ARG 1 end_ARG start_ARG italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG bold_italic_b ⊗ bold_italic_b ) = italic_ρ bold_italic_f ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> <tbody id="S2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\partial_{t}\bm{b}+\nabla\cdot(\bm{u}\otimes\bm{b}-\bm{b}\otimes% \bm{u})=0," class="ltx_Math" display="inline" id="S2.E8.m1.1"><semantics id="S2.E8.m1.1a"><mrow id="S2.E8.m1.1.1.1" xref="S2.E8.m1.1.1.1.1.cmml"><mrow id="S2.E8.m1.1.1.1.1" xref="S2.E8.m1.1.1.1.1.cmml"><mrow 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xref="S2.E8.m1.1.1.1.1.1.3.2">𝒃</ci></apply><apply id="S2.E8.m1.1.1.1.1.1.1.cmml" xref="S2.E8.m1.1.1.1.1.1.1"><ci id="S2.E8.m1.1.1.1.1.1.1.2.cmml" xref="S2.E8.m1.1.1.1.1.1.1.2">⋅</ci><ci id="S2.E8.m1.1.1.1.1.1.1.3.cmml" xref="S2.E8.m1.1.1.1.1.1.1.3">∇</ci><apply id="S2.E8.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1"><minus id="S2.E8.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.1"></minus><apply id="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.2"><csymbol cd="latexml" id="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.1">tensor-product</csymbol><ci id="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.2">𝒖</ci><ci id="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.2.3">𝒃</ci></apply><apply id="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.1">tensor-product</csymbol><ci id="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.2">𝒃</ci><ci id="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E8.m1.1.1.1.1.1.1.1.1.1.3.3">𝒖</ci></apply></apply></apply></apply><cn id="S2.E8.m1.1.1.1.1.3.cmml" type="integer" xref="S2.E8.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.1c">\displaystyle\partial_{t}\bm{b}+\nabla\cdot(\bm{u}\otimes\bm{b}-\bm{b}\otimes% \bm{u})=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.1d">∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_b + ∇ ⋅ ( bold_italic_u ⊗ bold_italic_b - bold_italic_b ⊗ bold_italic_u ) = 0 ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\nabla\cdot\bm{b}=0," class="ltx_Math" display="inline" id="S2.E9.m1.1"><semantics id="S2.E9.m1.1a"><mrow id="S2.E9.m1.1.1.1" xref="S2.E9.m1.1.1.1.1.cmml"><mrow id="S2.E9.m1.1.1.1.1" xref="S2.E9.m1.1.1.1.1.cmml"><mrow id="S2.E9.m1.1.1.1.1.2" xref="S2.E9.m1.1.1.1.1.2.cmml"><mo id="S2.E9.m1.1.1.1.1.2.2" xref="S2.E9.m1.1.1.1.1.2.2.cmml">∇</mo><mo id="S2.E9.m1.1.1.1.1.2.1" lspace="0em" rspace="0.222em" xref="S2.E9.m1.1.1.1.1.2.1.cmml">⋅</mo><mi id="S2.E9.m1.1.1.1.1.2.3" xref="S2.E9.m1.1.1.1.1.2.3.cmml">𝒃</mi></mrow><mo id="S2.E9.m1.1.1.1.1.1" xref="S2.E9.m1.1.1.1.1.1.cmml">=</mo><mn id="S2.E9.m1.1.1.1.1.3" xref="S2.E9.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S2.E9.m1.1.1.1.2" xref="S2.E9.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.1b"><apply id="S2.E9.m1.1.1.1.1.cmml" xref="S2.E9.m1.1.1.1"><eq id="S2.E9.m1.1.1.1.1.1.cmml" xref="S2.E9.m1.1.1.1.1.1"></eq><apply id="S2.E9.m1.1.1.1.1.2.cmml" xref="S2.E9.m1.1.1.1.1.2"><ci id="S2.E9.m1.1.1.1.1.2.1.cmml" xref="S2.E9.m1.1.1.1.1.2.1">⋅</ci><ci id="S2.E9.m1.1.1.1.1.2.2.cmml" xref="S2.E9.m1.1.1.1.1.2.2">∇</ci><ci id="S2.E9.m1.1.1.1.1.2.3.cmml" xref="S2.E9.m1.1.1.1.1.2.3">𝒃</ci></apply><cn id="S2.E9.m1.1.1.1.1.3.cmml" type="integer" xref="S2.E9.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.1c">\displaystyle\nabla\cdot\bm{b}=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.1d">∇ ⋅ bold_italic_b = 0 ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> <tbody id="S2.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p=c_{s}^{2}\rho+\frac{1}{2\mu_{0}}\bm{b}\cdot\bm{b}," class="ltx_Math" display="inline" id="S2.E10.m1.1"><semantics id="S2.E10.m1.1a"><mrow id="S2.E10.m1.1.1.1" xref="S2.E10.m1.1.1.1.1.cmml"><mrow id="S2.E10.m1.1.1.1.1" xref="S2.E10.m1.1.1.1.1.cmml"><mi id="S2.E10.m1.1.1.1.1.2" xref="S2.E10.m1.1.1.1.1.2.cmml">p</mi><mo id="S2.E10.m1.1.1.1.1.1" xref="S2.E10.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E10.m1.1.1.1.1.3" xref="S2.E10.m1.1.1.1.1.3.cmml"><mrow id="S2.E10.m1.1.1.1.1.3.2" xref="S2.E10.m1.1.1.1.1.3.2.cmml"><msubsup id="S2.E10.m1.1.1.1.1.3.2.2" xref="S2.E10.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E10.m1.1.1.1.1.3.2.2.2.2" xref="S2.E10.m1.1.1.1.1.3.2.2.2.2.cmml">c</mi><mi id="S2.E10.m1.1.1.1.1.3.2.2.2.3" xref="S2.E10.m1.1.1.1.1.3.2.2.2.3.cmml">s</mi><mn id="S2.E10.m1.1.1.1.1.3.2.2.3" xref="S2.E10.m1.1.1.1.1.3.2.2.3.cmml">2</mn></msubsup><mo id="S2.E10.m1.1.1.1.1.3.2.1" xref="S2.E10.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.E10.m1.1.1.1.1.3.2.3" xref="S2.E10.m1.1.1.1.1.3.2.3.cmml">ρ</mi></mrow><mo id="S2.E10.m1.1.1.1.1.3.1" xref="S2.E10.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S2.E10.m1.1.1.1.1.3.3" xref="S2.E10.m1.1.1.1.1.3.3.cmml"><mrow id="S2.E10.m1.1.1.1.1.3.3.2" xref="S2.E10.m1.1.1.1.1.3.3.2.cmml"><mstyle displaystyle="true" id="S2.E10.m1.1.1.1.1.3.3.2.2" xref="S2.E10.m1.1.1.1.1.3.3.2.2.cmml"><mfrac id="S2.E10.m1.1.1.1.1.3.3.2.2a" xref="S2.E10.m1.1.1.1.1.3.3.2.2.cmml"><mn id="S2.E10.m1.1.1.1.1.3.3.2.2.2" xref="S2.E10.m1.1.1.1.1.3.3.2.2.2.cmml">1</mn><mrow id="S2.E10.m1.1.1.1.1.3.3.2.2.3" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.cmml"><mn id="S2.E10.m1.1.1.1.1.3.3.2.2.3.2" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.2.cmml">2</mn><mo id="S2.E10.m1.1.1.1.1.3.3.2.2.3.1" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.1.cmml">⁢</mo><msub id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.cmml"><mi id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.2" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.2.cmml">μ</mi><mn id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.3" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.3.cmml">0</mn></msub></mrow></mfrac></mstyle><mo id="S2.E10.m1.1.1.1.1.3.3.2.1" xref="S2.E10.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><mi id="S2.E10.m1.1.1.1.1.3.3.2.3" xref="S2.E10.m1.1.1.1.1.3.3.2.3.cmml">𝒃</mi></mrow><mo id="S2.E10.m1.1.1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E10.m1.1.1.1.1.3.3.1.cmml">⋅</mo><mi id="S2.E10.m1.1.1.1.1.3.3.3" xref="S2.E10.m1.1.1.1.1.3.3.3.cmml">𝒃</mi></mrow></mrow></mrow><mo id="S2.E10.m1.1.1.1.2" xref="S2.E10.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m1.1b"><apply id="S2.E10.m1.1.1.1.1.cmml" xref="S2.E10.m1.1.1.1"><eq id="S2.E10.m1.1.1.1.1.1.cmml" xref="S2.E10.m1.1.1.1.1.1"></eq><ci id="S2.E10.m1.1.1.1.1.2.cmml" xref="S2.E10.m1.1.1.1.1.2">𝑝</ci><apply id="S2.E10.m1.1.1.1.1.3.cmml" xref="S2.E10.m1.1.1.1.1.3"><plus id="S2.E10.m1.1.1.1.1.3.1.cmml" xref="S2.E10.m1.1.1.1.1.3.1"></plus><apply id="S2.E10.m1.1.1.1.1.3.2.cmml" xref="S2.E10.m1.1.1.1.1.3.2"><times id="S2.E10.m1.1.1.1.1.3.2.1.cmml" xref="S2.E10.m1.1.1.1.1.3.2.1"></times><apply id="S2.E10.m1.1.1.1.1.3.2.2.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.1.3.2.2.1.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2">superscript</csymbol><apply id="S2.E10.m1.1.1.1.1.3.2.2.2.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.1.3.2.2.2.1.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S2.E10.m1.1.1.1.1.3.2.2.2.2.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2.2.2">𝑐</ci><ci id="S2.E10.m1.1.1.1.1.3.2.2.2.3.cmml" xref="S2.E10.m1.1.1.1.1.3.2.2.2.3">𝑠</ci></apply><cn id="S2.E10.m1.1.1.1.1.3.2.2.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.3.2.2.3">2</cn></apply><ci id="S2.E10.m1.1.1.1.1.3.2.3.cmml" xref="S2.E10.m1.1.1.1.1.3.2.3">𝜌</ci></apply><apply id="S2.E10.m1.1.1.1.1.3.3.cmml" xref="S2.E10.m1.1.1.1.1.3.3"><ci id="S2.E10.m1.1.1.1.1.3.3.1.cmml" xref="S2.E10.m1.1.1.1.1.3.3.1">⋅</ci><apply id="S2.E10.m1.1.1.1.1.3.3.2.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2"><times id="S2.E10.m1.1.1.1.1.3.3.2.1.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.1"></times><apply id="S2.E10.m1.1.1.1.1.3.3.2.2.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2"><divide id="S2.E10.m1.1.1.1.1.3.3.2.2.1.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2"></divide><cn id="S2.E10.m1.1.1.1.1.3.3.2.2.2.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.3.3.2.2.2">1</cn><apply id="S2.E10.m1.1.1.1.1.3.3.2.2.3.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3"><times id="S2.E10.m1.1.1.1.1.3.3.2.2.3.1.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.1"></times><cn id="S2.E10.m1.1.1.1.1.3.3.2.2.3.2.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.2">2</cn><apply id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.1.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3">subscript</csymbol><ci id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.2.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.2">𝜇</ci><cn id="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.3.3.2.2.3.3.3">0</cn></apply></apply></apply><ci id="S2.E10.m1.1.1.1.1.3.3.2.3.cmml" xref="S2.E10.m1.1.1.1.1.3.3.2.3">𝒃</ci></apply><ci id="S2.E10.m1.1.1.1.1.3.3.3.cmml" xref="S2.E10.m1.1.1.1.1.3.3.3">𝒃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.1c">\displaystyle p=c_{s}^{2}\rho+\frac{1}{2\mu_{0}}\bm{b}\cdot\bm{b},</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.1d">italic_p = italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ + divide start_ARG 1 end_ARG start_ARG 2 italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG bold_italic_b ⋅ bold_italic_b ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p1.15">where <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.2.m1.1"><semantics id="S2.SS2.SSS0.Px1.p1.2.m1.1a"><mi id="S2.SS2.SSS0.Px1.p1.2.m1.1.1" xref="S2.SS2.SSS0.Px1.p1.2.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.2.m1.1b"><ci id="S2.SS2.SSS0.Px1.p1.2.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.2.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.2.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.2.m1.1d">italic_ρ</annotation></semantics></math>, <math alttext="\bm{u}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.3.m2.1"><semantics id="S2.SS2.SSS0.Px1.p1.3.m2.1a"><mi id="S2.SS2.SSS0.Px1.p1.3.m2.1.1" xref="S2.SS2.SSS0.Px1.p1.3.m2.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.3.m2.1b"><ci id="S2.SS2.SSS0.Px1.p1.3.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.3.m2.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.3.m2.1c">\bm{u}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.3.m2.1d">bold_italic_u</annotation></semantics></math>, <math alttext="\bm{b}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.4.m3.1"><semantics id="S2.SS2.SSS0.Px1.p1.4.m3.1a"><mi id="S2.SS2.SSS0.Px1.p1.4.m3.1.1" xref="S2.SS2.SSS0.Px1.p1.4.m3.1.1.cmml">𝒃</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.4.m3.1b"><ci id="S2.SS2.SSS0.Px1.p1.4.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.4.m3.1.1">𝒃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.4.m3.1c">\bm{b}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.4.m3.1d">bold_italic_b</annotation></semantics></math> and <math alttext="\mu_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.5.m4.1"><semantics id="S2.SS2.SSS0.Px1.p1.5.m4.1a"><msub id="S2.SS2.SSS0.Px1.p1.5.m4.1.1" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.2" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1.2.cmml">μ</mi><mn id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.3" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.5.m4.1b"><apply id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1.2">𝜇</ci><cn id="S2.SS2.SSS0.Px1.p1.5.m4.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.5.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.5.m4.1c">\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.5.m4.1d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are the gas density, the velocity and magnetic fields, and the magnetic permittivity, respectively. Equation <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.E10" title="In MHD plasma model ‣ 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">10</span></a> relates the scalar pressure <math alttext="p" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.6.m5.1"><semantics id="S2.SS2.SSS0.Px1.p1.6.m5.1a"><mi id="S2.SS2.SSS0.Px1.p1.6.m5.1.1" xref="S2.SS2.SSS0.Px1.p1.6.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.6.m5.1b"><ci id="S2.SS2.SSS0.Px1.p1.6.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.6.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.6.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.6.m5.1d">italic_p</annotation></semantics></math> to <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.7.m6.1"><semantics id="S2.SS2.SSS0.Px1.p1.7.m6.1a"><mi id="S2.SS2.SSS0.Px1.p1.7.m6.1.1" xref="S2.SS2.SSS0.Px1.p1.7.m6.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.7.m6.1b"><ci id="S2.SS2.SSS0.Px1.p1.7.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.7.m6.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.7.m6.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.7.m6.1d">italic_ρ</annotation></semantics></math> via the isothermal equation of state with constant sound speed <math alttext="c_{s}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.8.m7.1"><semantics id="S2.SS2.SSS0.Px1.p1.8.m7.1a"><msub id="S2.SS2.SSS0.Px1.p1.8.m7.1.1" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.2" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1.2.cmml">c</mi><mi id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.3" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.8.m7.1b"><apply id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p1.8.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.8.m7.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.8.m7.1c">c_{s}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.8.m7.1d">italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, as well as the pressure contribution from the magnetic field. We work in units <math alttext="c_{s}=\rho_{0}=\mu_{0}=L=1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.9.m8.1"><semantics id="S2.SS2.SSS0.Px1.p1.9.m8.1a"><mrow id="S2.SS2.SSS0.Px1.p1.9.m8.1.1" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.cmml"><msub id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.2" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.2.cmml">c</mi><mi id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.3" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.3.cmml">s</mi></msub><mo id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.3" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.3.cmml">=</mo><msub id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.cmml"><mi id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.2" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.3" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.5" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.5.cmml">=</mo><msub id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.cmml"><mi id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.2" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.2.cmml">μ</mi><mn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.3" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.7" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.7.cmml">=</mo><mi id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.8" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.8.cmml">L</mi><mo id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.9" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.9.cmml">=</mo><mn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.10" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.10.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.9.m8.1b"><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"><and id="S2.SS2.SSS0.Px1.p1.9.m8.1.1a.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"></and><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1b.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"><eq id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.3"></eq><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.2">𝑐</ci><ci id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.2.3">𝑠</ci></apply><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.2">𝜌</ci><cn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.3">0</cn></apply></apply><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1c.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"><eq id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.5.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.5"></eq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px1.p1.9.m8.1.1.4.cmml" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1d.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"></share><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.1.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.2.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.2">𝜇</ci><cn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.3">0</cn></apply></apply><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1e.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"><eq id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.7.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.7"></eq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px1.p1.9.m8.1.1.6.cmml" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1f.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"></share><ci id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.8.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.8">𝐿</ci></apply><apply id="S2.SS2.SSS0.Px1.p1.9.m8.1.1g.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"><eq id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.9.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.9"></eq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px1.p1.9.m8.1.1.8.cmml" id="S2.SS2.SSS0.Px1.p1.9.m8.1.1h.cmml" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1"></share><cn id="S2.SS2.SSS0.Px1.p1.9.m8.1.1.10.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.9.m8.1.1.10">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.9.m8.1c">c_{s}=\rho_{0}=\mu_{0}=L=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.9.m8.1d">italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_L = 1</annotation></semantics></math>, where <math alttext="\rho_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.10.m9.1"><semantics id="S2.SS2.SSS0.Px1.p1.10.m9.1a"><msub id="S2.SS2.SSS0.Px1.p1.10.m9.1.1" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.2" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.3" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.10.m9.1b"><apply id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1.2">𝜌</ci><cn id="S2.SS2.SSS0.Px1.p1.10.m9.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.10.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.10.m9.1c">\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.10.m9.1d">italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the mean gas density and <math alttext="L" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.11.m10.1"><semantics id="S2.SS2.SSS0.Px1.p1.11.m10.1a"><mi id="S2.SS2.SSS0.Px1.p1.11.m10.1.1" xref="S2.SS2.SSS0.Px1.p1.11.m10.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.11.m10.1b"><ci id="S2.SS2.SSS0.Px1.p1.11.m10.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.11.m10.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.11.m10.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.11.m10.1d">italic_L</annotation></semantics></math> is the characteristic length scale of the system, such that <math alttext="L^{3}=\mathcal{V}=1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.12.m11.1"><semantics id="S2.SS2.SSS0.Px1.p1.12.m11.1a"><mrow 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xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1"></and><apply id="S2.SS2.SSS0.Px1.p1.12.m11.1.1b.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1"><eq id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.3"></eq><apply id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.2">𝐿</ci><cn id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.2.3">3</cn></apply><ci id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.4.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.4">𝒱</ci></apply><apply id="S2.SS2.SSS0.Px1.p1.12.m11.1.1c.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1"><eq id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.5.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.5"></eq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px1.p1.12.m11.1.1.4.cmml" id="S2.SS2.SSS0.Px1.p1.12.m11.1.1d.cmml" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1"></share><cn id="S2.SS2.SSS0.Px1.p1.12.m11.1.1.6.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.12.m11.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.12.m11.1c">L^{3}=\mathcal{V}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.12.m11.1d">italic_L start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = caligraphic_V = 1</annotation></semantics></math> is the volume. We discretize the equations over a triply periodic domain of <math alttext="[-L/2,L/2]" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.13.m12.2"><semantics id="S2.SS2.SSS0.Px1.p1.13.m12.2a"><mrow id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.3.cmml"><mo id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.3.cmml">[</mo><mrow id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1a" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.cmml">−</mo><mrow id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.2" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.2.cmml">L</mi><mo id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.1" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.3" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.3.cmml">2</mn></mrow></mrow><mo id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.4" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.3.cmml">,</mo><mrow id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.2" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.2.cmml">L</mi><mo id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.1" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.3" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.3.cmml">2</mn></mrow><mo id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.5" stretchy="false" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.13.m12.2b"><interval closure="closed" id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2"><apply id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1"><minus id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1"></minus><apply id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2"><divide id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.1"></divide><ci id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.2">𝐿</ci><cn id="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.13.m12.1.1.1.1.2.3">2</cn></apply></apply><apply id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2"><divide id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.1"></divide><ci id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.2">𝐿</ci><cn id="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.13.m12.2.2.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.13.m12.2c">[-L/2,L/2]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.13.m12.2d">[ - italic_L / 2 , italic_L / 2 ]</annotation></semantics></math> in each dimension, with grid resolution <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.14.m13.2"><semantics id="S2.SS2.SSS0.Px1.p1.14.m13.2a"><mrow id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.2.cmml"><mn id="S2.SS2.SSS0.Px1.p1.14.m13.1.1" xref="S2.SS2.SSS0.Px1.p1.14.m13.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.2" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.2.cmml">,</mo></mpadded><msup id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.cmml"><mn id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.2" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.2.cmml">080</mn><mn id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.3" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.14.m13.2b"><list id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1"><cn id="S2.SS2.SSS0.Px1.p1.14.m13.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.14.m13.1.1">10</cn><apply id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1">superscript</csymbol><cn id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.2">080</cn><cn id="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p1.14.m13.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.14.m13.2c">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.14.m13.2d">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> – the largest grid in the world for simulations of this fluid turbulence regime. In order to drive turbulence, a turbulent forcing term <math alttext="\bm{f}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.15.m14.1"><semantics id="S2.SS2.SSS0.Px1.p1.15.m14.1a"><mi id="S2.SS2.SSS0.Px1.p1.15.m14.1.1" xref="S2.SS2.SSS0.Px1.p1.15.m14.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.15.m14.1b"><ci id="S2.SS2.SSS0.Px1.p1.15.m14.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.15.m14.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.15.m14.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.15.m14.1d">bold_italic_f</annotation></semantics></math> is applied in the momentum equation (details below).</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p2.3">Our numerical model is an implicit large eddy simulation (ILES), which relies upon the spatial discretisation to supply the numerical viscosity and resistivity as a fluid closure model. <cite class="ltx_cite ltx_citemacro_citet">Malvadi Shivakumar & Federrath (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib47" title="">2023</a>)</cite> provide a detailed characterization of the numerical viscous and resistive properties of this solver by comparing the ILES model with direct numerical simulations (DNS), which have explicit viscous and resistive operators. They derived empirical models for transforming grid resolution. 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xref="S2.SS2.SSS0.Px1.p2.1.m1.2.2.cmml">080</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.1.m1.2b"><apply id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3"><eq id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.1"></eq><apply id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.2">𝑁</ci><ci id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.2.3">grid</ci></apply><list id="S2.SS2.SSS0.Px1.p2.1.m1.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.2.3.3.2"><cn id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1">10</cn><cn id="S2.SS2.SSS0.Px1.p2.1.m1.2.2.cmml" type="integer" 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xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1"></share><cn id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.8.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.8">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.1.m1.1c">u_{0}/c_{s}=\mathcal{M}_{t}=4.32\pm 0.18\approx 4</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.1.m1.1d">italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 4.32 ± 0.18 ≈ 4</annotation></semantics></math>. This enables us to obtain scaling results in regimes where the simulation overlaps with both NI-MHD and weakly compressible MHD models, but also covers regimes in which the asymptotic approximations tend to break down. We apply a non-helical stochastic forcing term <math alttext="\bm{f}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p1.2.m2.1a"><mi id="S2.SS2.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.2.m2.1b"><ci id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.2.m2.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.2.m2.1d">bold_italic_f</annotation></semantics></math> in Equation <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#S2.E7" title="In MHD plasma model ‣ 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">7</span></a>, following an Ornstein-Uhlenbeck stochastic process <cite class="ltx_cite ltx_citemacro_citep">(Eswaran & Pope, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib22" title="">1988</a>; Schmidt et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib68" title="">2009</a>; Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib25" title="">2010</a>)</cite>, using the <span class="ltx_text ltx_font_smallcaps" id="S2.SS2.SSS0.Px2.p1.7.1">TurbGen</span> turbulent forcing module <cite class="ltx_cite ltx_citemacro_citep">(Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib25" title="">2010</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib26" title="">2022</a>)</cite>. 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type="integer" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.4.2.3">2</cn></apply><ci id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.4.3.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.4.3">𝜋</ci></apply></apply><apply id="S2.SS2.SSS0.Px2.p1.3.m3.1.1c.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1"><eq id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.5.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px2.p1.3.m3.1.1.4.cmml" id="S2.SS2.SSS0.Px2.p1.3.m3.1.1d.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1"></share><cn id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.6.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.3.m3.1c">\ell_{0}^{-1}=k_{0}L/2\pi=2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_L / 2 italic_π = 2</annotation></semantics></math> and tending to zero parabolically in the interval <math alttext="1\leq kL/2\pi\leq 3" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.4.m4.1"><semantics id="S2.SS2.SSS0.Px2.p1.4.m4.1a"><mrow id="S2.SS2.SSS0.Px2.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml"><mn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.2" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.3" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.3.cmml">≤</mo><mrow id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.cmml"><mrow id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.cmml"><mrow id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.cmml"><mi id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.2" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.2.cmml">k</mi><mo id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.3" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.2.3.cmml">L</mi></mrow><mo id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.3" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.3.cmml">2</mn></mrow><mo id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.3" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.3.cmml">π</mi></mrow><mo id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.5" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.5.cmml">≤</mo><mn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.6" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.6.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.4.m4.1b"><apply id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1"><and id="S2.SS2.SSS0.Px2.p1.4.m4.1.1a.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1"></and><apply id="S2.SS2.SSS0.Px2.p1.4.m4.1.1b.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1"><leq id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.3.cmml" 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xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.2.3">2</cn></apply><ci id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.3.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.3">𝜋</ci></apply></apply><apply id="S2.SS2.SSS0.Px2.p1.4.m4.1.1c.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1"><leq id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.5.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.5"></leq><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px2.p1.4.m4.1.1.4.cmml" id="S2.SS2.SSS0.Px2.p1.4.m4.1.1d.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1"></share><cn id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.6.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.6">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.4.m4.1c">1\leq kL/2\pi\leq 3</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.4.m4.1d">1 ≤ italic_k italic_L / 2 italic_π ≤ 3</annotation></semantics></math>. To replenish the large-scale compressible modes and shocks, we decompose <math alttext="\bm{f}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.5.m5.1"><semantics id="S2.SS2.SSS0.Px2.p1.5.m5.1a"><mi id="S2.SS2.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.5.m5.1b"><ci id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.5.m5.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.5.m5.1d">bold_italic_f</annotation></semantics></math> into its incompressible (<math alttext="\nabla\cdot\bm{f}=0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.6.m6.1"><semantics id="S2.SS2.SSS0.Px2.p1.6.m6.1a"><mrow id="S2.SS2.SSS0.Px2.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.cmml"><mrow id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.2" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.2.cmml">∇</mo><mo id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.1" lspace="0em" rspace="0.222em" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.1.cmml">⋅</mo><mi id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.3" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.3.cmml">𝒇</mi></mrow><mo id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.1" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.1.cmml">=</mo><mn id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.3" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.6.m6.1b"><apply id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1"><eq id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.1"></eq><apply id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2"><ci id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.1">⋅</ci><ci id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.2">∇</ci><ci id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.2.3">𝒇</ci></apply><cn id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.6.m6.1c">\nabla\cdot\bm{f}=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.6.m6.1d">∇ ⋅ bold_italic_f = 0</annotation></semantics></math>) and compressible (<math alttext="|\nabla\times\bm{f}|=0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p1.7.m7.1"><semantics id="S2.SS2.SSS0.Px2.p1.7.m7.1a"><mrow id="S2.SS2.SSS0.Px2.p1.7.m7.1.1" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.cmml"><mrow id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.2.1.cmml">|</mo><mrow id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.2.cmml">∇</mo><mo id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.1" lspace="0em" rspace="0.222em" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.1.cmml">×</mo><mi id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.3.cmml">𝒇</mi></mrow><mo id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.2" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.2.cmml">=</mo><mn id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.3" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.7.m7.1b"><apply id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1"><eq id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.2"></eq><apply id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1"><abs id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.2"></abs><apply id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1"><times id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.1"></times><ci id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.2">∇</ci><ci id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.1.1.1.3">𝒇</ci></apply></apply><cn id="S2.SS2.SSS0.Px2.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.7.m7.1c">|\nabla\times\bm{f}|=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p1.7.m7.1d">| ∇ × bold_italic_f | = 0</annotation></semantics></math>) modes <cite class="ltx_cite ltx_citemacro_citep">(Federrath et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib25" title="">2010</a>)</cite>, and drive the turbulence with equal amounts of energy in each of the modes.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p2.5">We set the correlation time of <math alttext="\bm{f}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p2.1.m1.1a"><mi id="S2.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.1.m1.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.1.m1.1d">bold_italic_f</annotation></semantics></math> to <math alttext="t_{0}=\ell_{0}/u_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.2.m2.1"><semantics id="S2.SS2.SSS0.Px2.p2.2.m2.1a"><mrow id="S2.SS2.SSS0.Px2.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.cmml"><msub id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.2" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.2.cmml">t</mi><mn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml"><msub id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.2" mathvariant="normal" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.2.cmml">ℓ</mi><mn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.1" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.1.cmml">/</mo><msub id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.2" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.2.cmml">u</mi><mn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.3" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.3.cmml">0</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.2.m2.1b"><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1"><eq id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.1"></eq><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.2">𝑡</ci><cn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3"><divide id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.1"></divide><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.2">ℓ</ci><cn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.3">0</cn></apply><apply id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.2">𝑢</ci><cn id="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.2.m2.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.2.m2.1c">t_{0}=\ell_{0}/u_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.2.m2.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = roman_ℓ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, such that the correlation time and turnover time of the largest eddy are equal. We drive the simulation into a statistically steady state, such that all of the first moments of underlying field variables no longer vary on average. We sample the turbulence 20 times across an <math alttext="2t_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.3.m3.1"><semantics id="S2.SS2.SSS0.Px2.p2.3.m3.1a"><mrow id="S2.SS2.SSS0.Px2.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.cmml"><mn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2.cmml">2</mn><mo id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.1" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.1.cmml">⁢</mo><msub id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.2" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.2.cmml">t</mi><mn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.3" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.3.m3.1b"><apply id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1"><times id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.1"></times><cn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.2">2</cn><apply id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.2">𝑡</ci><cn id="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.3.m3.1c">2t_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.3.m3.1d">2 italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> interval, averaging all statistics (e.g., <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px2.p2.4.m4.1a"><msub id="S2.SS2.SSS0.Px2.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2.cmml">ℳ</mi><mi id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.4.m4.1b"><apply id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.2">ℳ</ci><ci id="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.4.m4.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.4.m4.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.5.m5.1"><semantics id="S2.SS2.SSS0.Px2.p2.5.m5.1a"><mrow id="S2.SS2.SSS0.Px2.p2.5.m5.1.1" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.cmml"><mrow id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.2.cmml">δ</mi><mo id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.1" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.3.cmml">ρ</mi></mrow><mo id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.cmml">/</mo><msub id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.2" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.3" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.5.m5.1b"><apply id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1"><divide id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.1"></divide><apply id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2"><times id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.1"></times><ci id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.2">𝛿</ci><ci id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.2.3">𝜌</ci></apply><apply id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.2">𝜌</ci><cn id="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px2.p2.5.m5.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.5.m5.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.5.m5.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> spectra, as discussed in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3" title="3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">section 3</span></a>) to ensure that our results are not sensitive to any intermittent events. We refer to <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite> for further information about the integral quantities and the time-evolution of the simulation.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Initial conditions & steady-state magnetic field</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px3.p1.8">We initialize <math alttext="\rho(x,y,z)=\rho_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.1.m1.3"><semantics id="S2.SS2.SSS0.Px3.p1.1.m1.3a"><mrow id="S2.SS2.SSS0.Px3.p1.1.m1.3.4" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.cmml"><mrow id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml">x</mi><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p1.1.m1.2.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.2.2.cmml">y</mi><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2.3" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml">,</mo><mi id="S2.SS2.SSS0.Px3.p1.1.m1.3.3" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.3.cmml">z</mi><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2.4" stretchy="false" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.1.cmml">=</mo><msub id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.cmml"><mi id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.3" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.1.m1.3b"><apply id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4"><eq id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.1"></eq><apply id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2"><times id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.1"></times><ci id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.2">𝜌</ci><vector id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.2.3.2"><ci id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1">𝑥</ci><ci id="S2.SS2.SSS0.Px3.p1.1.m1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.2.2">𝑦</ci><ci id="S2.SS2.SSS0.Px3.p1.1.m1.3.3.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.3">𝑧</ci></vector></apply><apply id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.2.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.2">𝜌</ci><cn id="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.1.m1.3.4.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.1.m1.3c">\rho(x,y,z)=\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.1.m1.3d">italic_ρ ( italic_x , italic_y , italic_z ) = italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{u}=\bm{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px3.p1.2.m2.1a"><mrow id="S2.SS2.SSS0.Px3.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.2" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.2.cmml">𝒖</mi><mo id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.1" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.3" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.2.m2.1b"><apply id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1"><eq id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.1"></eq><ci id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.2">𝒖</ci><cn id="S2.SS2.SSS0.Px3.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.2.m2.1c">\bm{u}=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.2.m2.1d">bold_italic_u = bold_0</annotation></semantics></math>. For our simulations, <math alttext="B_{0}=0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px3.p1.3.m3.1a"><mrow id="S2.SS2.SSS0.Px3.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.2" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.2.cmml">B</mi><mn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.3" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.3" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.3.m3.1b"><apply id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1"><eq id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.1"></eq><apply id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.2">𝐵</ci><cn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.2.3">0</cn></apply><cn id="S2.SS2.SSS0.Px3.p1.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.3.m3.1c">B_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.3.m3.1d">italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math>, and only the turbulent <math alttext="\delta\bm{b}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.4.m4.1"><semantics id="S2.SS2.SSS0.Px3.p1.4.m4.1a"><mrow id="S2.SS2.SSS0.Px3.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2.cmml">δ</mi><mo id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.1" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml">𝒃</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.4.m4.1b"><apply id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1"><times id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.1"></times><ci id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.2">𝛿</ci><ci id="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.4.m4.1.1.3">𝒃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.4.m4.1c">\delta\bm{b}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.4.m4.1d">italic_δ bold_italic_b</annotation></semantics></math> remains. <math alttext="\delta\bm{b}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.5.m5.1"><semantics id="S2.SS2.SSS0.Px3.p1.5.m5.1a"><mrow id="S2.SS2.SSS0.Px3.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.2" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.2.cmml">δ</mi><mo id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.3" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.3.cmml">𝒃</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.5.m5.1b"><apply id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1"><times id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.1"></times><ci id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.2">𝛿</ci><ci id="S2.SS2.SSS0.Px3.p1.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.5.m5.1.1.3">𝒃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.5.m5.1c">\delta\bm{b}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.5.m5.1d">italic_δ bold_italic_b</annotation></semantics></math> is maintained in a statistically stationary state via the turbulent dynamo <cite class="ltx_cite ltx_citemacro_citep">(Schekochihin et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib67" title="">2004</a>; Rincon, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib64" title="">2019</a>; Kriel et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib43" title="">2022</a>)</cite>. The saturated state of our magnetic fields results in an Alfvén Mach number, <math alttext="\mathcal{M}_{\rm A}=u_{0}/\left\langle{v_{A}^{2}}\right\rangle_{\mathcal{V}}^{% 1/2}=2.03\pm 0.04\approx 2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.6.m6.1"><semantics id="S2.SS2.SSS0.Px3.p1.6.m6.1a"><mrow id="S2.SS2.SSS0.Px3.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3.2" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3.2.cmml">ℳ</mi><mi id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3.3" mathvariant="normal" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.3.3.cmml">A</mi></msub><mo id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.4" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.4.cmml">=</mo><mrow id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.cmml"><msub id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.1.3.cmml"><mi 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xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.7"></approx><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px3.p1.6.m6.1.1.6.cmml" id="S2.SS2.SSS0.Px3.p1.6.m6.1.1f.cmml" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1"></share><cn id="S2.SS2.SSS0.Px3.p1.6.m6.1.1.8.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.6.m6.1.1.8">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.6.m6.1c">\mathcal{M}_{\rm A}=u_{0}/\left\langle{v_{A}^{2}}\right\rangle_{\mathcal{V}}^{% 1/2}=2.03\pm 0.04\approx 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.6.m6.1d">caligraphic_M start_POSTSUBSCRIPT roman_A end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / ⟨ italic_v start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT caligraphic_V end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT = 2.03 ± 0.04 ≈ 2</annotation></semantics></math>, where <math 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xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.3">2</cn></apply></apply><ci id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.1.3">𝒱</ci></apply><apply id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3"><divide id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.1"></divide><cn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.2">1</cn><cn id="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.7.m7.1c">\left\langle{v_{A}^{2}}\right\rangle_{\mathcal{V}}^{1/2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.7.m7.1d">⟨ italic_v start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT caligraphic_V end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is the rms Alfvén velocity. On volume-average, this provides <math alttext="\beta\sim 1/8\sim 1" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px3.p1.8.m8.1"><semantics id="S2.SS2.SSS0.Px3.p1.8.m8.1a"><mrow id="S2.SS2.SSS0.Px3.p1.8.m8.1.1" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.2.cmml">β</mi><mo id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.3.cmml">∼</mo><mrow id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.cmml"><mn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.2" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.1" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.3" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.3.cmml">8</mn></mrow><mo id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.5" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.5.cmml">∼</mo><mn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.6" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.6.cmml">1</mn></mrow><annotation-xml 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xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1"><csymbol cd="latexml" id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.5.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.5">similar-to</csymbol><share href="https://arxiv.org/html/2502.08883v1#S2.SS2.SSS0.Px3.p1.8.m8.1.1.4.cmml" id="S2.SS2.SSS0.Px3.p1.8.m8.1.1d.cmml" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1"></share><cn id="S2.SS2.SSS0.Px3.p1.8.m8.1.1.6.cmml" type="integer" xref="S2.SS2.SSS0.Px3.p1.8.m8.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.8.m8.1c">\beta\sim 1/8\sim 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px3.p1.8.m8.1d">italic_β ∼ 1 / 8 ∼ 1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">The inhomogeneous mass density field</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px4.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px4.p1.3"><a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.F2" title="Figure 2 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 2</span></a> shows a two-dimensional slice of the logarithmic<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Natural logarithmic being the most appropriate transformation of <math alttext="\rho/\rho_{0}" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mi id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml">ρ</mi><mo id="footnote1.m1.1.1.1" xref="footnote1.m1.1.1.1.cmml">/</mo><msub id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml"><mi id="footnote1.m1.1.1.3.2" xref="footnote1.m1.1.1.3.2.cmml">ρ</mi><mn id="footnote1.m1.1.1.3.3" xref="footnote1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><divide id="footnote1.m1.1.1.1.cmml" xref="footnote1.m1.1.1.1"></divide><ci id="footnote1.m1.1.1.2.cmml" xref="footnote1.m1.1.1.2">𝜌</ci><apply id="footnote1.m1.1.1.3.cmml" xref="footnote1.m1.1.1.3"><csymbol cd="ambiguous" id="footnote1.m1.1.1.3.1.cmml" xref="footnote1.m1.1.1.3">subscript</csymbol><ci id="footnote1.m1.1.1.3.2.cmml" xref="footnote1.m1.1.1.3.2">𝜌</ci><cn id="footnote1.m1.1.1.3.3.cmml" type="integer" xref="footnote1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, since we know that trans-to-supersonic turbulence gives rise to roughly lognormal distribution functions in mass density fluctuations <cite class="ltx_cite ltx_citemacro_citep">(Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib7" title="">2022</a>)</cite>, with small correction to the higher-order moments based on the void statistics <cite class="ltx_cite ltx_citemacro_citep">(Hopkins, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib36" title="">2013</a>; Squire & Hopkins, <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib71" title="">2017</a>; Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib7" title="">2022</a>)</cite>.</span></span></span>, mean-normalized mass-density field, <math alttext="\ln(\rho/\rho_{0})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px4.p1.1.m1.2"><semantics id="S2.SS2.SSS0.Px4.p1.1.m1.2a"><mrow 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xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.3" stretchy="false" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px4.p1.1.m1.2b"><apply id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.2.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1"><ln id="S2.SS2.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.1.1"></ln><apply id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1"><divide id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.1"></divide><ci id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.2">𝜌</ci><apply id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.2">𝜌</ci><cn id="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px4.p1.1.m1.2.2.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px4.p1.1.m1.2c">\ln(\rho/\rho_{0})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px4.p1.1.m1.2d">roman_ln ( italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>, with in-plane magnetic field lines illustrated in white. The slice is taken from within the statistically stationary state of the turbulence. The strong inhomogeneities in the mass density can be observed from both the numerous coherent structures (e.g., high-<math alttext="\rho/\rho_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px4.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px4.p1.2.m2.1a"><mrow id="S2.SS2.SSS0.Px4.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.2" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.1" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.1.cmml">/</mo><msub id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.2" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.3" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px4.p1.2.m2.1b"><apply id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1"><divide id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.1"></divide><ci id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.2">𝜌</ci><apply id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.2">𝜌</ci><cn id="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px4.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px4.p1.2.m2.1c">\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px4.p1.2.m2.1d">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> density filaments, shown in yellow and deep mass density voids, shown in green; both ubiquitous in supersonic MHD turbulence <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib6" title="">2021</a></cite>) presented in the simulation domain, and the roughly three orders of magnitude in <math alttext="\rho/\rho_{0}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px4.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px4.p1.3.m3.1a"><mrow id="S2.SS2.SSS0.Px4.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.2" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.2.cmml">ρ</mi><mo id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.1" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.1.cmml">/</mo><msub id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.2" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.2.cmml">ρ</mi><mn id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.3" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px4.p1.3.m3.1b"><apply id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1"><divide id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.1"></divide><ci id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.2">𝜌</ci><apply id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.2">𝜌</ci><cn id="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px4.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px4.p1.3.m3.1c">\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px4.p1.3.m3.1d">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> that are resolved.</p> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Results</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.5">For each magnetosheath interval, we calculate the relative density fluctuation and turbulent Mach number, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.E2" title="2 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 2</span></a>. We approximately use 1200 magnetosheath intervals, creating a joint probability distribution function between <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mrow id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml"><mi id="S3.p1.1.m1.1.1.2.2" xref="S3.p1.1.m1.1.1.2.2.cmml">δ</mi><mo id="S3.p1.1.m1.1.1.2.1" xref="S3.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.p1.1.m1.1.1.2.3" xref="S3.p1.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S3.p1.1.m1.1.1.1" xref="S3.p1.1.m1.1.1.1.cmml">/</mo><msub id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml"><mi id="S3.p1.1.m1.1.1.3.2" xref="S3.p1.1.m1.1.1.3.2.cmml">ρ</mi><mn id="S3.p1.1.m1.1.1.3.3" xref="S3.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><divide id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1.1"></divide><apply 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xref="S3.p1.3.m3.2.2.2.2.2.1.cmml">/</mo><msub id="S3.p1.3.m3.2.2.2.2.2.3" xref="S3.p1.3.m3.2.2.2.2.2.3.cmml"><mi id="S3.p1.3.m3.2.2.2.2.2.3.2" xref="S3.p1.3.m3.2.2.2.2.2.3.2.cmml">ρ</mi><mn id="S3.p1.3.m3.2.2.2.2.2.3.3" xref="S3.p1.3.m3.2.2.2.2.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S3.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2"><times id="S3.p1.3.m3.2.2.3.cmml" xref="S3.p1.3.m3.2.2.3"></times><ci id="S3.p1.3.m3.2.2.4.cmml" xref="S3.p1.3.m3.2.2.4">𝑝</ci><interval closure="open" id="S3.p1.3.m3.2.2.2.3.cmml" xref="S3.p1.3.m3.2.2.2.2"><apply id="S3.p1.3.m3.1.1.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.p1.3.m3.1.1.1.1.1.2">ℳ</ci><ci id="S3.p1.3.m3.1.1.1.1.1.3.cmml" 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id="S3.p1.3.m3.2c">p(\mathcal{M}_{t},\delta\rho/\rho_{0})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.2d">italic_p ( caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. In <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.F3" title="Figure 3 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 3</span></a> (blue) we show the contours of a Gaussian kernel density estimate for <math alttext="p(\mathcal{M}_{t},\delta\rho/\rho_{0})" class="ltx_Math" display="inline" id="S3.p1.4.m4.2"><semantics id="S3.p1.4.m4.2a"><mrow id="S3.p1.4.m4.2.2" xref="S3.p1.4.m4.2.2.cmml"><mi id="S3.p1.4.m4.2.2.4" xref="S3.p1.4.m4.2.2.4.cmml">p</mi><mo id="S3.p1.4.m4.2.2.3" xref="S3.p1.4.m4.2.2.3.cmml">⁢</mo><mrow id="S3.p1.4.m4.2.2.2.2" xref="S3.p1.4.m4.2.2.2.3.cmml"><mo id="S3.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S3.p1.4.m4.2.2.2.3.cmml">(</mo><msub id="S3.p1.4.m4.1.1.1.1.1" xref="S3.p1.4.m4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.4.m4.1.1.1.1.1.2" xref="S3.p1.4.m4.1.1.1.1.1.2.cmml">ℳ</mi><mi id="S3.p1.4.m4.1.1.1.1.1.3" xref="S3.p1.4.m4.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S3.p1.4.m4.2.2.2.2.4" xref="S3.p1.4.m4.2.2.2.3.cmml">,</mo><mrow id="S3.p1.4.m4.2.2.2.2.2" xref="S3.p1.4.m4.2.2.2.2.2.cmml"><mrow id="S3.p1.4.m4.2.2.2.2.2.2" xref="S3.p1.4.m4.2.2.2.2.2.2.cmml"><mi id="S3.p1.4.m4.2.2.2.2.2.2.2" xref="S3.p1.4.m4.2.2.2.2.2.2.2.cmml">δ</mi><mo id="S3.p1.4.m4.2.2.2.2.2.2.1" xref="S3.p1.4.m4.2.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.p1.4.m4.2.2.2.2.2.2.3" xref="S3.p1.4.m4.2.2.2.2.2.2.3.cmml">ρ</mi></mrow><mo id="S3.p1.4.m4.2.2.2.2.2.1" xref="S3.p1.4.m4.2.2.2.2.2.1.cmml">/</mo><msub id="S3.p1.4.m4.2.2.2.2.2.3" xref="S3.p1.4.m4.2.2.2.2.2.3.cmml"><mi id="S3.p1.4.m4.2.2.2.2.2.3.2" xref="S3.p1.4.m4.2.2.2.2.2.3.2.cmml">ρ</mi><mn id="S3.p1.4.m4.2.2.2.2.2.3.3" xref="S3.p1.4.m4.2.2.2.2.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S3.p1.4.m4.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.2b"><apply id="S3.p1.4.m4.2.2.cmml" xref="S3.p1.4.m4.2.2"><times id="S3.p1.4.m4.2.2.3.cmml" xref="S3.p1.4.m4.2.2.3"></times><ci id="S3.p1.4.m4.2.2.4.cmml" xref="S3.p1.4.m4.2.2.4">𝑝</ci><interval closure="open" id="S3.p1.4.m4.2.2.2.3.cmml" xref="S3.p1.4.m4.2.2.2.2"><apply id="S3.p1.4.m4.1.1.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.p1.4.m4.1.1.1.1.1.2.cmml" xref="S3.p1.4.m4.1.1.1.1.1.2">ℳ</ci><ci id="S3.p1.4.m4.1.1.1.1.1.3.cmml" xref="S3.p1.4.m4.1.1.1.1.1.3">𝑡</ci></apply><apply id="S3.p1.4.m4.2.2.2.2.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2"><divide id="S3.p1.4.m4.2.2.2.2.2.1.cmml" xref="S3.p1.4.m4.2.2.2.2.2.1"></divide><apply id="S3.p1.4.m4.2.2.2.2.2.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2.2"><times id="S3.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S3.p1.4.m4.2.2.2.2.2.2.1"></times><ci id="S3.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2.2.2">𝛿</ci><ci id="S3.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S3.p1.4.m4.2.2.2.2.2.2.3">𝜌</ci></apply><apply id="S3.p1.4.m4.2.2.2.2.2.3.cmml" xref="S3.p1.4.m4.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.p1.4.m4.2.2.2.2.2.3.1.cmml" xref="S3.p1.4.m4.2.2.2.2.2.3">subscript</csymbol><ci id="S3.p1.4.m4.2.2.2.2.2.3.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2.3.2">𝜌</ci><cn id="S3.p1.4.m4.2.2.2.2.2.3.3.cmml" type="integer" xref="S3.p1.4.m4.2.2.2.2.2.3.3">0</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.2c">p(\mathcal{M}_{t},\delta\rho/\rho_{0})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.2d">italic_p ( caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. The shade of the contours represents fixed values of the probability density for the underlying distribution. The contour plot qualitatively illustrates that the MMS data follow a linear relation closely between the two variables. We perform a maximum likelihood fit of the function <math alttext="\delta\rho/\rho_{0}=\theta_{1}\mathcal{M}_{t}^{\theta_{0}}" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mrow id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mrow id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml"><mrow id="S3.p1.5.m5.1.1.2.2" xref="S3.p1.5.m5.1.1.2.2.cmml"><mi id="S3.p1.5.m5.1.1.2.2.2" xref="S3.p1.5.m5.1.1.2.2.2.cmml">δ</mi><mo id="S3.p1.5.m5.1.1.2.2.1" xref="S3.p1.5.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S3.p1.5.m5.1.1.2.2.3" xref="S3.p1.5.m5.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S3.p1.5.m5.1.1.2.1" xref="S3.p1.5.m5.1.1.2.1.cmml">/</mo><msub id="S3.p1.5.m5.1.1.2.3" xref="S3.p1.5.m5.1.1.2.3.cmml"><mi id="S3.p1.5.m5.1.1.2.3.2" xref="S3.p1.5.m5.1.1.2.3.2.cmml">ρ</mi><mn id="S3.p1.5.m5.1.1.2.3.3" xref="S3.p1.5.m5.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.p1.5.m5.1.1.1" xref="S3.p1.5.m5.1.1.1.cmml">=</mo><mrow id="S3.p1.5.m5.1.1.3" xref="S3.p1.5.m5.1.1.3.cmml"><msub id="S3.p1.5.m5.1.1.3.2" xref="S3.p1.5.m5.1.1.3.2.cmml"><mi id="S3.p1.5.m5.1.1.3.2.2" xref="S3.p1.5.m5.1.1.3.2.2.cmml">θ</mi><mn id="S3.p1.5.m5.1.1.3.2.3" xref="S3.p1.5.m5.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.p1.5.m5.1.1.3.1" xref="S3.p1.5.m5.1.1.3.1.cmml">⁢</mo><msubsup id="S3.p1.5.m5.1.1.3.3" xref="S3.p1.5.m5.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.5.m5.1.1.3.3.2.2" xref="S3.p1.5.m5.1.1.3.3.2.2.cmml">ℳ</mi><mi id="S3.p1.5.m5.1.1.3.3.2.3" xref="S3.p1.5.m5.1.1.3.3.2.3.cmml">t</mi><msub id="S3.p1.5.m5.1.1.3.3.3" xref="S3.p1.5.m5.1.1.3.3.3.cmml"><mi id="S3.p1.5.m5.1.1.3.3.3.2" xref="S3.p1.5.m5.1.1.3.3.3.2.cmml">θ</mi><mn id="S3.p1.5.m5.1.1.3.3.3.3" xref="S3.p1.5.m5.1.1.3.3.3.3.cmml">0</mn></msub></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1"><eq id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1"></eq><apply id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2"><divide id="S3.p1.5.m5.1.1.2.1.cmml" xref="S3.p1.5.m5.1.1.2.1"></divide><apply id="S3.p1.5.m5.1.1.2.2.cmml" xref="S3.p1.5.m5.1.1.2.2"><times id="S3.p1.5.m5.1.1.2.2.1.cmml" xref="S3.p1.5.m5.1.1.2.2.1"></times><ci id="S3.p1.5.m5.1.1.2.2.2.cmml" xref="S3.p1.5.m5.1.1.2.2.2">𝛿</ci><ci id="S3.p1.5.m5.1.1.2.2.3.cmml" xref="S3.p1.5.m5.1.1.2.2.3">𝜌</ci></apply><apply id="S3.p1.5.m5.1.1.2.3.cmml" xref="S3.p1.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.2.3.1.cmml" xref="S3.p1.5.m5.1.1.2.3">subscript</csymbol><ci id="S3.p1.5.m5.1.1.2.3.2.cmml" xref="S3.p1.5.m5.1.1.2.3.2">𝜌</ci><cn id="S3.p1.5.m5.1.1.2.3.3.cmml" type="integer" xref="S3.p1.5.m5.1.1.2.3.3">0</cn></apply></apply><apply id="S3.p1.5.m5.1.1.3.cmml" xref="S3.p1.5.m5.1.1.3"><times id="S3.p1.5.m5.1.1.3.1.cmml" xref="S3.p1.5.m5.1.1.3.1"></times><apply id="S3.p1.5.m5.1.1.3.2.cmml" xref="S3.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.3.2.1.cmml" xref="S3.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="S3.p1.5.m5.1.1.3.2.2.cmml" xref="S3.p1.5.m5.1.1.3.2.2">𝜃</ci><cn id="S3.p1.5.m5.1.1.3.2.3.cmml" type="integer" xref="S3.p1.5.m5.1.1.3.2.3">1</cn></apply><apply id="S3.p1.5.m5.1.1.3.3.cmml" xref="S3.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.3.3.1.cmml" xref="S3.p1.5.m5.1.1.3.3">superscript</csymbol><apply id="S3.p1.5.m5.1.1.3.3.2.cmml" xref="S3.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.3.3.2.1.cmml" xref="S3.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.p1.5.m5.1.1.3.3.2.2.cmml" xref="S3.p1.5.m5.1.1.3.3.2.2">ℳ</ci><ci id="S3.p1.5.m5.1.1.3.3.2.3.cmml" xref="S3.p1.5.m5.1.1.3.3.2.3">𝑡</ci></apply><apply id="S3.p1.5.m5.1.1.3.3.3.cmml" xref="S3.p1.5.m5.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.3.3.3.1.cmml" xref="S3.p1.5.m5.1.1.3.3.3">subscript</csymbol><ci id="S3.p1.5.m5.1.1.3.3.3.2.cmml" xref="S3.p1.5.m5.1.1.3.3.3.2">𝜃</ci><cn id="S3.p1.5.m5.1.1.3.3.3.3.cmml" type="integer" xref="S3.p1.5.m5.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\delta\rho/\rho_{0}=\theta_{1}\mathcal{M}_{t}^{\theta_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> to the data and determine that the best fit parameters yield</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx6"> <tbody id="S3.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta\rho/\rho_{0}=(0.83^{+0.54}_{-0.58})\mathcal{M}_{t}^{0.92^{% 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xref="S3.E11.m1.1.1.1.1.1.cmml"><mrow id="S3.E11.m1.1.1.1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E11.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E11.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.E11.m1.1.1.1.1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.1.1.1.1.cmml"><mn id="S3.E11.m1.1.1.1.1.1.1.1.1.2.2" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.2.cmml">0.83</mn><mrow id="S3.E11.m1.1.1.1.1.1.1.1.1.3" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S3.E11.m1.1.1.1.1.1.1.1.1.3a" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.cmml">−</mo><mn id="S3.E11.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.2.cmml">0.58</mn></mrow><mrow id="S3.E11.m1.1.1.1.1.1.1.1.1.2.3" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.cmml"><mo id="S3.E11.m1.1.1.1.1.1.1.1.1.2.3a" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.cmml">+</mo><mn id="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.2" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.2.cmml">0.54</mn></mrow></msubsup><mo id="S3.E11.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E11.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E11.m1.1.1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S3.E11.m1.1.1.1.1.1.3" xref="S3.E11.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E11.m1.1.1.1.1.1.3.2.2" xref="S3.E11.m1.1.1.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S3.E11.m1.1.1.1.1.1.3.2.3" xref="S3.E11.m1.1.1.1.1.1.3.2.3.cmml">t</mi><msubsup id="S3.E11.m1.1.1.1.1.1.3.3" xref="S3.E11.m1.1.1.1.1.1.3.3.cmml"><mn id="S3.E11.m1.1.1.1.1.1.3.3.2.2" xref="S3.E11.m1.1.1.1.1.1.3.3.2.2.cmml">0.92</mn><mrow id="S3.E11.m1.1.1.1.1.1.3.3.3" xref="S3.E11.m1.1.1.1.1.1.3.3.3.cmml"><mo id="S3.E11.m1.1.1.1.1.1.3.3.3a" xref="S3.E11.m1.1.1.1.1.1.3.3.3.cmml">−</mo><mn id="S3.E11.m1.1.1.1.1.1.3.3.3.2" xref="S3.E11.m1.1.1.1.1.1.3.3.3.2.cmml">0.29</mn></mrow><mrow id="S3.E11.m1.1.1.1.1.1.3.3.2.3" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3.cmml"><mo id="S3.E11.m1.1.1.1.1.1.3.3.2.3a" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3.cmml">+</mo><mn id="S3.E11.m1.1.1.1.1.1.3.3.2.3.2" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3.2.cmml">0.30</mn></mrow></msubsup></msubsup></mrow></mrow><mo id="S3.E11.m1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E11.m1.1b"><apply id="S3.E11.m1.1.1.1.1.cmml" xref="S3.E11.m1.1.1.1"><eq id="S3.E11.m1.1.1.1.1.2.cmml" xref="S3.E11.m1.1.1.1.1.2"></eq><apply id="S3.E11.m1.1.1.1.1.3.cmml" xref="S3.E11.m1.1.1.1.1.3"><divide id="S3.E11.m1.1.1.1.1.3.1.cmml" xref="S3.E11.m1.1.1.1.1.3.1"></divide><apply id="S3.E11.m1.1.1.1.1.3.2.cmml" xref="S3.E11.m1.1.1.1.1.3.2"><times id="S3.E11.m1.1.1.1.1.3.2.1.cmml" xref="S3.E11.m1.1.1.1.1.3.2.1"></times><ci id="S3.E11.m1.1.1.1.1.3.2.2.cmml" xref="S3.E11.m1.1.1.1.1.3.2.2">𝛿</ci><ci id="S3.E11.m1.1.1.1.1.3.2.3.cmml" xref="S3.E11.m1.1.1.1.1.3.2.3">𝜌</ci></apply><apply id="S3.E11.m1.1.1.1.1.3.3.cmml" xref="S3.E11.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.3.3.1.cmml" xref="S3.E11.m1.1.1.1.1.3.3">subscript</csymbol><ci 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id="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.2.cmml" type="float" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.3.2">0.54</cn></apply></apply><apply id="S3.E11.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3"><minus id="S3.E11.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3"></minus><cn id="S3.E11.m1.1.1.1.1.1.1.1.1.3.2.cmml" type="float" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.2">0.58</cn></apply></apply><apply id="S3.E11.m1.1.1.1.1.1.3.cmml" xref="S3.E11.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3">superscript</csymbol><apply id="S3.E11.m1.1.1.1.1.1.3.2.cmml" xref="S3.E11.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.3.2.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.E11.m1.1.1.1.1.1.3.2.2.cmml" xref="S3.E11.m1.1.1.1.1.1.3.2.2">ℳ</ci><ci id="S3.E11.m1.1.1.1.1.1.3.2.3.cmml" xref="S3.E11.m1.1.1.1.1.1.3.2.3">𝑡</ci></apply><apply id="S3.E11.m1.1.1.1.1.1.3.3.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.3.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3">subscript</csymbol><apply id="S3.E11.m1.1.1.1.1.1.3.3.2.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.3.3.2.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3">superscript</csymbol><cn id="S3.E11.m1.1.1.1.1.1.3.3.2.2.cmml" type="float" xref="S3.E11.m1.1.1.1.1.1.3.3.2.2">0.92</cn><apply id="S3.E11.m1.1.1.1.1.1.3.3.2.3.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3"><plus id="S3.E11.m1.1.1.1.1.1.3.3.2.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3"></plus><cn id="S3.E11.m1.1.1.1.1.1.3.3.2.3.2.cmml" type="float" xref="S3.E11.m1.1.1.1.1.1.3.3.2.3.2">0.30</cn></apply></apply><apply id="S3.E11.m1.1.1.1.1.1.3.3.3.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3.3"><minus id="S3.E11.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.3.3.3"></minus><cn id="S3.E11.m1.1.1.1.1.1.3.3.3.2.cmml" type="float" xref="S3.E11.m1.1.1.1.1.1.3.3.3.2">0.29</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.1c">\displaystyle\delta\rho/\rho_{0}=(0.83^{+0.54}_{-0.58})\mathcal{M}_{t}^{0.92^{% +0.30}_{-0.29}},</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 0.83 start_POSTSUPERSCRIPT + 0.54 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.58 end_POSTSUBSCRIPT ) caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0.92 start_POSTSUPERSCRIPT + 0.30 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.29 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(\text{MMS data)}" class="ltx_math_unparsed" display="inline" id="S3.E11.m2.1"><semantics id="S3.E11.m2.1a"><mrow id="S3.E11.m2.1b"><mo id="S3.E11.m2.1.1" stretchy="false">(</mo><mtext id="S3.E11.m2.1.2">MMS data)</mtext></mrow><annotation encoding="application/x-tex" id="S3.E11.m2.1c">\displaystyle(\text{MMS data)}</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m2.1d">( MMS data)</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.8">consistent within <math alttext="1\sigma" class="ltx_Math" display="inline" id="S3.p1.6.m1.1"><semantics id="S3.p1.6.m1.1a"><mrow id="S3.p1.6.m1.1.1" xref="S3.p1.6.m1.1.1.cmml"><mn id="S3.p1.6.m1.1.1.2" xref="S3.p1.6.m1.1.1.2.cmml">1</mn><mo id="S3.p1.6.m1.1.1.1" xref="S3.p1.6.m1.1.1.1.cmml">⁢</mo><mi id="S3.p1.6.m1.1.1.3" xref="S3.p1.6.m1.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.6.m1.1b"><apply id="S3.p1.6.m1.1.1.cmml" xref="S3.p1.6.m1.1.1"><times id="S3.p1.6.m1.1.1.1.cmml" xref="S3.p1.6.m1.1.1.1"></times><cn id="S3.p1.6.m1.1.1.2.cmml" type="integer" xref="S3.p1.6.m1.1.1.2">1</cn><ci id="S3.p1.6.m1.1.1.3.cmml" xref="S3.p1.6.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m1.1c">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m1.1d">1 italic_σ</annotation></semantics></math> to the <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p1.7.m2.1"><semantics id="S3.p1.7.m2.1a"><mrow id="S3.p1.7.m2.1.1" xref="S3.p1.7.m2.1.1.cmml"><mrow id="S3.p1.7.m2.1.1.2" xref="S3.p1.7.m2.1.1.2.cmml"><mrow id="S3.p1.7.m2.1.1.2.2" xref="S3.p1.7.m2.1.1.2.2.cmml"><mi id="S3.p1.7.m2.1.1.2.2.2" xref="S3.p1.7.m2.1.1.2.2.2.cmml">δ</mi><mo id="S3.p1.7.m2.1.1.2.2.1" xref="S3.p1.7.m2.1.1.2.2.1.cmml">⁢</mo><mi id="S3.p1.7.m2.1.1.2.2.3" xref="S3.p1.7.m2.1.1.2.2.3.cmml">ρ</mi></mrow><mo 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encoding="application/x-llamapun" id="S3.p1.7.m2.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> relation predicted by the weakly compressible MHD turbulence model <cite class="ltx_cite ltx_citemacro_citep">(Bhattacharjee et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>, even in regimes where the asymptotic expansion in powers of the <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p1.8.m3.1"><semantics id="S3.p1.8.m3.1a"><msub id="S3.p1.8.m3.1.1" xref="S3.p1.8.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.8.m3.1.1.2" xref="S3.p1.8.m3.1.1.2.cmml">ℳ</mi><mi id="S3.p1.8.m3.1.1.3" xref="S3.p1.8.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.8.m3.1b"><apply id="S3.p1.8.m3.1.1.cmml" xref="S3.p1.8.m3.1.1"><csymbol cd="ambiguous" 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We show the corner plots for the fit in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#A1" title="Appendix A Maximum Likelihood Fits ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Appendix A</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="623" id="S3.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>The variation of relative density fluctuations, <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S3.F3.11.m1.1"><semantics id="S3.F3.11.m1.1b"><mrow id="S3.F3.11.m1.1.1" xref="S3.F3.11.m1.1.1.cmml"><mrow id="S3.F3.11.m1.1.1.2" xref="S3.F3.11.m1.1.1.2.cmml"><mi id="S3.F3.11.m1.1.1.2.2" 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xref="S3.F3.15.m5.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.15.m5.2c"><apply id="S3.F3.15.m5.2.2.cmml" xref="S3.F3.15.m5.2.2"><times id="S3.F3.15.m5.2.2.3.cmml" xref="S3.F3.15.m5.2.2.3"></times><ci id="S3.F3.15.m5.2.2.4.cmml" xref="S3.F3.15.m5.2.2.4">𝑝</ci><interval closure="open" id="S3.F3.15.m5.2.2.2.3.cmml" xref="S3.F3.15.m5.2.2.2.2"><apply id="S3.F3.15.m5.1.1.1.1.1.cmml" xref="S3.F3.15.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F3.15.m5.1.1.1.1.1.1.cmml" xref="S3.F3.15.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.F3.15.m5.1.1.1.1.1.2.cmml" xref="S3.F3.15.m5.1.1.1.1.1.2">ℳ</ci><ci id="S3.F3.15.m5.1.1.1.1.1.3.cmml" xref="S3.F3.15.m5.1.1.1.1.1.3">𝑡</ci></apply><apply id="S3.F3.15.m5.2.2.2.2.2.cmml" xref="S3.F3.15.m5.2.2.2.2.2"><divide id="S3.F3.15.m5.2.2.2.2.2.1.cmml" xref="S3.F3.15.m5.2.2.2.2.2.1"></divide><apply id="S3.F3.15.m5.2.2.2.2.2.2.cmml" xref="S3.F3.15.m5.2.2.2.2.2.2"><times id="S3.F3.15.m5.2.2.2.2.2.2.1.cmml" xref="S3.F3.15.m5.2.2.2.2.2.2.1"></times><ci id="S3.F3.15.m5.2.2.2.2.2.2.2.cmml" xref="S3.F3.15.m5.2.2.2.2.2.2.2">𝛿</ci><ci id="S3.F3.15.m5.2.2.2.2.2.2.3.cmml" xref="S3.F3.15.m5.2.2.2.2.2.2.3">𝜌</ci></apply><apply id="S3.F3.15.m5.2.2.2.2.2.3.cmml" xref="S3.F3.15.m5.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.F3.15.m5.2.2.2.2.2.3.1.cmml" xref="S3.F3.15.m5.2.2.2.2.2.3">subscript</csymbol><ci id="S3.F3.15.m5.2.2.2.2.2.3.2.cmml" xref="S3.F3.15.m5.2.2.2.2.2.3.2">𝜌</ci><cn id="S3.F3.15.m5.2.2.2.2.2.3.3.cmml" type="integer" xref="S3.F3.15.m5.2.2.2.2.2.3.3">0</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.15.m5.2d">p(\mathcal{M}_{t},\delta\rho/\rho_{0})</annotation><annotation encoding="application/x-llamapun" id="S3.F3.15.m5.2e">italic_p ( caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>) and from a <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S3.F3.16.m6.2"><semantics id="S3.F3.16.m6.2b"><mrow id="S3.F3.16.m6.2.2.1" xref="S3.F3.16.m6.2.2.2.cmml"><mn id="S3.F3.16.m6.1.1" xref="S3.F3.16.m6.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S3.F3.16.m6.2.2.1.2" xref="S3.F3.16.m6.2.2.2.cmml">,</mo></mpadded><msup id="S3.F3.16.m6.2.2.1.1" xref="S3.F3.16.m6.2.2.1.1.cmml"><mn id="S3.F3.16.m6.2.2.1.1.2" xref="S3.F3.16.m6.2.2.1.1.2.cmml">080</mn><mn id="S3.F3.16.m6.2.2.1.1.3" xref="S3.F3.16.m6.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.16.m6.2c"><list id="S3.F3.16.m6.2.2.2.cmml" xref="S3.F3.16.m6.2.2.1"><cn id="S3.F3.16.m6.1.1.cmml" type="integer" xref="S3.F3.16.m6.1.1">10</cn><apply id="S3.F3.16.m6.2.2.1.1.cmml" xref="S3.F3.16.m6.2.2.1.1"><csymbol cd="ambiguous" id="S3.F3.16.m6.2.2.1.1.1.cmml" xref="S3.F3.16.m6.2.2.1.1">superscript</csymbol><cn id="S3.F3.16.m6.2.2.1.1.2.cmml" type="integer" xref="S3.F3.16.m6.2.2.1.1.2">080</cn><cn id="S3.F3.16.m6.2.2.1.1.3.cmml" type="integer" xref="S3.F3.16.m6.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.16.m6.2d">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.16.m6.2e">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, highly-compressible MHD simulation (red dots; by combining <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E14" title="14 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 14</span></a> and <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E16" title="16 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 16</span></a>). A best fit to the MMS data gives <math alttext="\delta\rho/\rho_{0}=(0.83^{+0.54}_{-0.58})\mathcal{M}_{t}^{0.92^{+0.30}_{-0.29}}" class="ltx_Math" display="inline" id="S3.F3.17.m7.1"><semantics id="S3.F3.17.m7.1b"><mrow id="S3.F3.17.m7.1.1" xref="S3.F3.17.m7.1.1.cmml"><mrow id="S3.F3.17.m7.1.1.3" xref="S3.F3.17.m7.1.1.3.cmml"><mrow id="S3.F3.17.m7.1.1.3.2" xref="S3.F3.17.m7.1.1.3.2.cmml"><mi id="S3.F3.17.m7.1.1.3.2.2" xref="S3.F3.17.m7.1.1.3.2.2.cmml">δ</mi><mo id="S3.F3.17.m7.1.1.3.2.1" xref="S3.F3.17.m7.1.1.3.2.1.cmml">⁢</mo><mi id="S3.F3.17.m7.1.1.3.2.3" xref="S3.F3.17.m7.1.1.3.2.3.cmml">ρ</mi></mrow><mo id="S3.F3.17.m7.1.1.3.1" xref="S3.F3.17.m7.1.1.3.1.cmml">/</mo><msub id="S3.F3.17.m7.1.1.3.3" xref="S3.F3.17.m7.1.1.3.3.cmml"><mi id="S3.F3.17.m7.1.1.3.3.2" xref="S3.F3.17.m7.1.1.3.3.2.cmml">ρ</mi><mn id="S3.F3.17.m7.1.1.3.3.3" xref="S3.F3.17.m7.1.1.3.3.3.cmml">0</mn></msub></mrow><mo id="S3.F3.17.m7.1.1.2" xref="S3.F3.17.m7.1.1.2.cmml">=</mo><mrow id="S3.F3.17.m7.1.1.1" xref="S3.F3.17.m7.1.1.1.cmml"><mrow id="S3.F3.17.m7.1.1.1.1.1" xref="S3.F3.17.m7.1.1.1.1.1.1.cmml"><mo id="S3.F3.17.m7.1.1.1.1.1.2" stretchy="false" xref="S3.F3.17.m7.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.F3.17.m7.1.1.1.1.1.1" xref="S3.F3.17.m7.1.1.1.1.1.1.cmml"><mn id="S3.F3.17.m7.1.1.1.1.1.1.2.2" xref="S3.F3.17.m7.1.1.1.1.1.1.2.2.cmml">0.83</mn><mrow id="S3.F3.17.m7.1.1.1.1.1.1.3" xref="S3.F3.17.m7.1.1.1.1.1.1.3.cmml"><mo id="S3.F3.17.m7.1.1.1.1.1.1.3b" xref="S3.F3.17.m7.1.1.1.1.1.1.3.cmml">−</mo><mn id="S3.F3.17.m7.1.1.1.1.1.1.3.2" xref="S3.F3.17.m7.1.1.1.1.1.1.3.2.cmml">0.58</mn></mrow><mrow id="S3.F3.17.m7.1.1.1.1.1.1.2.3" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3.cmml"><mo id="S3.F3.17.m7.1.1.1.1.1.1.2.3b" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3.cmml">+</mo><mn id="S3.F3.17.m7.1.1.1.1.1.1.2.3.2" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3.2.cmml">0.54</mn></mrow></msubsup><mo id="S3.F3.17.m7.1.1.1.1.1.3" stretchy="false" xref="S3.F3.17.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.F3.17.m7.1.1.1.2" xref="S3.F3.17.m7.1.1.1.2.cmml">⁢</mo><msubsup id="S3.F3.17.m7.1.1.1.3" xref="S3.F3.17.m7.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F3.17.m7.1.1.1.3.2.2" xref="S3.F3.17.m7.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S3.F3.17.m7.1.1.1.3.2.3" xref="S3.F3.17.m7.1.1.1.3.2.3.cmml">t</mi><msubsup id="S3.F3.17.m7.1.1.1.3.3" xref="S3.F3.17.m7.1.1.1.3.3.cmml"><mn id="S3.F3.17.m7.1.1.1.3.3.2.2" xref="S3.F3.17.m7.1.1.1.3.3.2.2.cmml">0.92</mn><mrow id="S3.F3.17.m7.1.1.1.3.3.3" xref="S3.F3.17.m7.1.1.1.3.3.3.cmml"><mo id="S3.F3.17.m7.1.1.1.3.3.3b" xref="S3.F3.17.m7.1.1.1.3.3.3.cmml">−</mo><mn id="S3.F3.17.m7.1.1.1.3.3.3.2" xref="S3.F3.17.m7.1.1.1.3.3.3.2.cmml">0.29</mn></mrow><mrow id="S3.F3.17.m7.1.1.1.3.3.2.3" xref="S3.F3.17.m7.1.1.1.3.3.2.3.cmml"><mo id="S3.F3.17.m7.1.1.1.3.3.2.3b" xref="S3.F3.17.m7.1.1.1.3.3.2.3.cmml">+</mo><mn id="S3.F3.17.m7.1.1.1.3.3.2.3.2" xref="S3.F3.17.m7.1.1.1.3.3.2.3.2.cmml">0.30</mn></mrow></msubsup></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.17.m7.1c"><apply id="S3.F3.17.m7.1.1.cmml" xref="S3.F3.17.m7.1.1"><eq id="S3.F3.17.m7.1.1.2.cmml" xref="S3.F3.17.m7.1.1.2"></eq><apply id="S3.F3.17.m7.1.1.3.cmml" xref="S3.F3.17.m7.1.1.3"><divide id="S3.F3.17.m7.1.1.3.1.cmml" xref="S3.F3.17.m7.1.1.3.1"></divide><apply id="S3.F3.17.m7.1.1.3.2.cmml" xref="S3.F3.17.m7.1.1.3.2"><times id="S3.F3.17.m7.1.1.3.2.1.cmml" xref="S3.F3.17.m7.1.1.3.2.1"></times><ci id="S3.F3.17.m7.1.1.3.2.2.cmml" xref="S3.F3.17.m7.1.1.3.2.2">𝛿</ci><ci id="S3.F3.17.m7.1.1.3.2.3.cmml" xref="S3.F3.17.m7.1.1.3.2.3">𝜌</ci></apply><apply id="S3.F3.17.m7.1.1.3.3.cmml" xref="S3.F3.17.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.3.3.1.cmml" xref="S3.F3.17.m7.1.1.3.3">subscript</csymbol><ci id="S3.F3.17.m7.1.1.3.3.2.cmml" xref="S3.F3.17.m7.1.1.3.3.2">𝜌</ci><cn id="S3.F3.17.m7.1.1.3.3.3.cmml" type="integer" xref="S3.F3.17.m7.1.1.3.3.3">0</cn></apply></apply><apply id="S3.F3.17.m7.1.1.1.cmml" xref="S3.F3.17.m7.1.1.1"><times id="S3.F3.17.m7.1.1.1.2.cmml" xref="S3.F3.17.m7.1.1.1.2"></times><apply id="S3.F3.17.m7.1.1.1.1.1.1.cmml" xref="S3.F3.17.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.1.1.1.1.cmml" xref="S3.F3.17.m7.1.1.1.1.1">subscript</csymbol><apply id="S3.F3.17.m7.1.1.1.1.1.1.2.cmml" xref="S3.F3.17.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.1.1.1.2.1.cmml" xref="S3.F3.17.m7.1.1.1.1.1">superscript</csymbol><cn id="S3.F3.17.m7.1.1.1.1.1.1.2.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.1.1.1.2.2">0.83</cn><apply id="S3.F3.17.m7.1.1.1.1.1.1.2.3.cmml" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3"><plus id="S3.F3.17.m7.1.1.1.1.1.1.2.3.1.cmml" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3"></plus><cn id="S3.F3.17.m7.1.1.1.1.1.1.2.3.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.1.1.1.2.3.2">0.54</cn></apply></apply><apply id="S3.F3.17.m7.1.1.1.1.1.1.3.cmml" xref="S3.F3.17.m7.1.1.1.1.1.1.3"><minus id="S3.F3.17.m7.1.1.1.1.1.1.3.1.cmml" xref="S3.F3.17.m7.1.1.1.1.1.1.3"></minus><cn id="S3.F3.17.m7.1.1.1.1.1.1.3.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.1.1.1.3.2">0.58</cn></apply></apply><apply id="S3.F3.17.m7.1.1.1.3.cmml" xref="S3.F3.17.m7.1.1.1.3"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.3.1.cmml" xref="S3.F3.17.m7.1.1.1.3">superscript</csymbol><apply id="S3.F3.17.m7.1.1.1.3.2.cmml" xref="S3.F3.17.m7.1.1.1.3"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.3.2.1.cmml" xref="S3.F3.17.m7.1.1.1.3">subscript</csymbol><ci id="S3.F3.17.m7.1.1.1.3.2.2.cmml" xref="S3.F3.17.m7.1.1.1.3.2.2">ℳ</ci><ci id="S3.F3.17.m7.1.1.1.3.2.3.cmml" xref="S3.F3.17.m7.1.1.1.3.2.3">𝑡</ci></apply><apply id="S3.F3.17.m7.1.1.1.3.3.cmml" xref="S3.F3.17.m7.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.3.3.1.cmml" xref="S3.F3.17.m7.1.1.1.3.3">subscript</csymbol><apply id="S3.F3.17.m7.1.1.1.3.3.2.cmml" xref="S3.F3.17.m7.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.17.m7.1.1.1.3.3.2.1.cmml" xref="S3.F3.17.m7.1.1.1.3.3">superscript</csymbol><cn id="S3.F3.17.m7.1.1.1.3.3.2.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.3.3.2.2">0.92</cn><apply id="S3.F3.17.m7.1.1.1.3.3.2.3.cmml" xref="S3.F3.17.m7.1.1.1.3.3.2.3"><plus id="S3.F3.17.m7.1.1.1.3.3.2.3.1.cmml" xref="S3.F3.17.m7.1.1.1.3.3.2.3"></plus><cn id="S3.F3.17.m7.1.1.1.3.3.2.3.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.3.3.2.3.2">0.30</cn></apply></apply><apply id="S3.F3.17.m7.1.1.1.3.3.3.cmml" xref="S3.F3.17.m7.1.1.1.3.3.3"><minus id="S3.F3.17.m7.1.1.1.3.3.3.1.cmml" xref="S3.F3.17.m7.1.1.1.3.3.3"></minus><cn id="S3.F3.17.m7.1.1.1.3.3.3.2.cmml" type="float" xref="S3.F3.17.m7.1.1.1.3.3.3.2">0.29</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.17.m7.1d">\delta\rho/\rho_{0}=(0.83^{+0.54}_{-0.58})\mathcal{M}_{t}^{0.92^{+0.30}_{-0.29}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.17.m7.1e">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 0.83 start_POSTSUPERSCRIPT + 0.54 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.58 end_POSTSUBSCRIPT ) caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0.92 start_POSTSUPERSCRIPT + 0.30 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.29 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and the simulation data yields <math alttext="\delta\rho/\rho_{0}=(0.97^{+0.23}_{-0.23})\mathcal{M}_{t}^{0.94^{+0.15}_{-0.15}}" class="ltx_Math" display="inline" id="S3.F3.18.m8.1"><semantics id="S3.F3.18.m8.1b"><mrow id="S3.F3.18.m8.1.1" xref="S3.F3.18.m8.1.1.cmml"><mrow id="S3.F3.18.m8.1.1.3" xref="S3.F3.18.m8.1.1.3.cmml"><mrow id="S3.F3.18.m8.1.1.3.2" xref="S3.F3.18.m8.1.1.3.2.cmml"><mi id="S3.F3.18.m8.1.1.3.2.2" xref="S3.F3.18.m8.1.1.3.2.2.cmml">δ</mi><mo id="S3.F3.18.m8.1.1.3.2.1" xref="S3.F3.18.m8.1.1.3.2.1.cmml">⁢</mo><mi id="S3.F3.18.m8.1.1.3.2.3" xref="S3.F3.18.m8.1.1.3.2.3.cmml">ρ</mi></mrow><mo id="S3.F3.18.m8.1.1.3.1" xref="S3.F3.18.m8.1.1.3.1.cmml">/</mo><msub id="S3.F3.18.m8.1.1.3.3" xref="S3.F3.18.m8.1.1.3.3.cmml"><mi id="S3.F3.18.m8.1.1.3.3.2" xref="S3.F3.18.m8.1.1.3.3.2.cmml">ρ</mi><mn id="S3.F3.18.m8.1.1.3.3.3" xref="S3.F3.18.m8.1.1.3.3.3.cmml">0</mn></msub></mrow><mo id="S3.F3.18.m8.1.1.2" xref="S3.F3.18.m8.1.1.2.cmml">=</mo><mrow id="S3.F3.18.m8.1.1.1" xref="S3.F3.18.m8.1.1.1.cmml"><mrow id="S3.F3.18.m8.1.1.1.1.1" xref="S3.F3.18.m8.1.1.1.1.1.1.cmml"><mo id="S3.F3.18.m8.1.1.1.1.1.2" stretchy="false" xref="S3.F3.18.m8.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.F3.18.m8.1.1.1.1.1.1" xref="S3.F3.18.m8.1.1.1.1.1.1.cmml"><mn id="S3.F3.18.m8.1.1.1.1.1.1.2.2" xref="S3.F3.18.m8.1.1.1.1.1.1.2.2.cmml">0.97</mn><mrow id="S3.F3.18.m8.1.1.1.1.1.1.3" xref="S3.F3.18.m8.1.1.1.1.1.1.3.cmml"><mo id="S3.F3.18.m8.1.1.1.1.1.1.3b" xref="S3.F3.18.m8.1.1.1.1.1.1.3.cmml">−</mo><mn id="S3.F3.18.m8.1.1.1.1.1.1.3.2" xref="S3.F3.18.m8.1.1.1.1.1.1.3.2.cmml">0.23</mn></mrow><mrow id="S3.F3.18.m8.1.1.1.1.1.1.2.3" xref="S3.F3.18.m8.1.1.1.1.1.1.2.3.cmml"><mo id="S3.F3.18.m8.1.1.1.1.1.1.2.3b" xref="S3.F3.18.m8.1.1.1.1.1.1.2.3.cmml">+</mo><mn id="S3.F3.18.m8.1.1.1.1.1.1.2.3.2" xref="S3.F3.18.m8.1.1.1.1.1.1.2.3.2.cmml">0.23</mn></mrow></msubsup><mo id="S3.F3.18.m8.1.1.1.1.1.3" stretchy="false" xref="S3.F3.18.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.F3.18.m8.1.1.1.2" xref="S3.F3.18.m8.1.1.1.2.cmml">⁢</mo><msubsup id="S3.F3.18.m8.1.1.1.3" xref="S3.F3.18.m8.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F3.18.m8.1.1.1.3.2.2" xref="S3.F3.18.m8.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S3.F3.18.m8.1.1.1.3.2.3" xref="S3.F3.18.m8.1.1.1.3.2.3.cmml">t</mi><msubsup id="S3.F3.18.m8.1.1.1.3.3" xref="S3.F3.18.m8.1.1.1.3.3.cmml"><mn id="S3.F3.18.m8.1.1.1.3.3.2.2" xref="S3.F3.18.m8.1.1.1.3.3.2.2.cmml">0.94</mn><mrow id="S3.F3.18.m8.1.1.1.3.3.3" xref="S3.F3.18.m8.1.1.1.3.3.3.cmml"><mo id="S3.F3.18.m8.1.1.1.3.3.3b" xref="S3.F3.18.m8.1.1.1.3.3.3.cmml">−</mo><mn id="S3.F3.18.m8.1.1.1.3.3.3.2" xref="S3.F3.18.m8.1.1.1.3.3.3.2.cmml">0.15</mn></mrow><mrow id="S3.F3.18.m8.1.1.1.3.3.2.3" xref="S3.F3.18.m8.1.1.1.3.3.2.3.cmml"><mo id="S3.F3.18.m8.1.1.1.3.3.2.3b" xref="S3.F3.18.m8.1.1.1.3.3.2.3.cmml">+</mo><mn id="S3.F3.18.m8.1.1.1.3.3.2.3.2" xref="S3.F3.18.m8.1.1.1.3.3.2.3.2.cmml">0.15</mn></mrow></msubsup></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.18.m8.1c"><apply id="S3.F3.18.m8.1.1.cmml" xref="S3.F3.18.m8.1.1"><eq id="S3.F3.18.m8.1.1.2.cmml" xref="S3.F3.18.m8.1.1.2"></eq><apply id="S3.F3.18.m8.1.1.3.cmml" xref="S3.F3.18.m8.1.1.3"><divide id="S3.F3.18.m8.1.1.3.1.cmml" xref="S3.F3.18.m8.1.1.3.1"></divide><apply id="S3.F3.18.m8.1.1.3.2.cmml" xref="S3.F3.18.m8.1.1.3.2"><times id="S3.F3.18.m8.1.1.3.2.1.cmml" xref="S3.F3.18.m8.1.1.3.2.1"></times><ci id="S3.F3.18.m8.1.1.3.2.2.cmml" xref="S3.F3.18.m8.1.1.3.2.2">𝛿</ci><ci id="S3.F3.18.m8.1.1.3.2.3.cmml" xref="S3.F3.18.m8.1.1.3.2.3">𝜌</ci></apply><apply id="S3.F3.18.m8.1.1.3.3.cmml" xref="S3.F3.18.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.18.m8.1.1.3.3.1.cmml" xref="S3.F3.18.m8.1.1.3.3">subscript</csymbol><ci id="S3.F3.18.m8.1.1.3.3.2.cmml" xref="S3.F3.18.m8.1.1.3.3.2">𝜌</ci><cn id="S3.F3.18.m8.1.1.3.3.3.cmml" type="integer" xref="S3.F3.18.m8.1.1.3.3.3">0</cn></apply></apply><apply id="S3.F3.18.m8.1.1.1.cmml" xref="S3.F3.18.m8.1.1.1"><times id="S3.F3.18.m8.1.1.1.2.cmml" xref="S3.F3.18.m8.1.1.1.2"></times><apply id="S3.F3.18.m8.1.1.1.1.1.1.cmml" xref="S3.F3.18.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F3.18.m8.1.1.1.1.1.1.1.cmml" xref="S3.F3.18.m8.1.1.1.1.1">subscript</csymbol><apply id="S3.F3.18.m8.1.1.1.1.1.1.2.cmml" xref="S3.F3.18.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F3.18.m8.1.1.1.1.1.1.2.1.cmml" xref="S3.F3.18.m8.1.1.1.1.1">superscript</csymbol><cn id="S3.F3.18.m8.1.1.1.1.1.1.2.2.cmml" type="float" xref="S3.F3.18.m8.1.1.1.1.1.1.2.2">0.97</cn><apply id="S3.F3.18.m8.1.1.1.1.1.1.2.3.cmml" xref="S3.F3.18.m8.1.1.1.1.1.1.2.3"><plus id="S3.F3.18.m8.1.1.1.1.1.1.2.3.1.cmml" xref="S3.F3.18.m8.1.1.1.1.1.1.2.3"></plus><cn 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id="S3.F3.18.m8.1.1.1.3.3.1.cmml" xref="S3.F3.18.m8.1.1.1.3.3">subscript</csymbol><apply id="S3.F3.18.m8.1.1.1.3.3.2.cmml" xref="S3.F3.18.m8.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.18.m8.1.1.1.3.3.2.1.cmml" xref="S3.F3.18.m8.1.1.1.3.3">superscript</csymbol><cn id="S3.F3.18.m8.1.1.1.3.3.2.2.cmml" type="float" xref="S3.F3.18.m8.1.1.1.3.3.2.2">0.94</cn><apply id="S3.F3.18.m8.1.1.1.3.3.2.3.cmml" xref="S3.F3.18.m8.1.1.1.3.3.2.3"><plus id="S3.F3.18.m8.1.1.1.3.3.2.3.1.cmml" xref="S3.F3.18.m8.1.1.1.3.3.2.3"></plus><cn id="S3.F3.18.m8.1.1.1.3.3.2.3.2.cmml" type="float" xref="S3.F3.18.m8.1.1.1.3.3.2.3.2">0.15</cn></apply></apply><apply id="S3.F3.18.m8.1.1.1.3.3.3.cmml" xref="S3.F3.18.m8.1.1.1.3.3.3"><minus id="S3.F3.18.m8.1.1.1.3.3.3.1.cmml" xref="S3.F3.18.m8.1.1.1.3.3.3"></minus><cn id="S3.F3.18.m8.1.1.1.3.3.3.2.cmml" type="float" xref="S3.F3.18.m8.1.1.1.3.3.3.2">0.15</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.18.m8.1d">\delta\rho/\rho_{0}=(0.97^{+0.23}_{-0.23})\mathcal{M}_{t}^{0.94^{+0.15}_{-0.15}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.18.m8.1e">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 0.97 start_POSTSUPERSCRIPT + 0.23 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.23 end_POSTSUBSCRIPT ) caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0.94 start_POSTSUPERSCRIPT + 0.15 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.15 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. The two dashed lines of slope unity are shown for proportionality constants <math alttext="b=1/3" class="ltx_Math" display="inline" id="S3.F3.19.m9.1"><semantics id="S3.F3.19.m9.1b"><mrow id="S3.F3.19.m9.1.1" xref="S3.F3.19.m9.1.1.cmml"><mi id="S3.F3.19.m9.1.1.2" xref="S3.F3.19.m9.1.1.2.cmml">b</mi><mo id="S3.F3.19.m9.1.1.1" xref="S3.F3.19.m9.1.1.1.cmml">=</mo><mrow id="S3.F3.19.m9.1.1.3" xref="S3.F3.19.m9.1.1.3.cmml"><mn id="S3.F3.19.m9.1.1.3.2" xref="S3.F3.19.m9.1.1.3.2.cmml">1</mn><mo id="S3.F3.19.m9.1.1.3.1" xref="S3.F3.19.m9.1.1.3.1.cmml">/</mo><mn id="S3.F3.19.m9.1.1.3.3" xref="S3.F3.19.m9.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.19.m9.1c"><apply id="S3.F3.19.m9.1.1.cmml" xref="S3.F3.19.m9.1.1"><eq id="S3.F3.19.m9.1.1.1.cmml" xref="S3.F3.19.m9.1.1.1"></eq><ci id="S3.F3.19.m9.1.1.2.cmml" xref="S3.F3.19.m9.1.1.2">𝑏</ci><apply id="S3.F3.19.m9.1.1.3.cmml" xref="S3.F3.19.m9.1.1.3"><divide id="S3.F3.19.m9.1.1.3.1.cmml" xref="S3.F3.19.m9.1.1.3.1"></divide><cn id="S3.F3.19.m9.1.1.3.2.cmml" type="integer" xref="S3.F3.19.m9.1.1.3.2">1</cn><cn id="S3.F3.19.m9.1.1.3.3.cmml" type="integer" xref="S3.F3.19.m9.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.19.m9.1d">b=1/3</annotation><annotation encoding="application/x-llamapun" id="S3.F3.19.m9.1e">italic_b = 1 / 3</annotation></semantics></math> and <math alttext="b=1" class="ltx_Math" display="inline" id="S3.F3.20.m10.1"><semantics id="S3.F3.20.m10.1b"><mrow id="S3.F3.20.m10.1.1" xref="S3.F3.20.m10.1.1.cmml"><mi id="S3.F3.20.m10.1.1.2" xref="S3.F3.20.m10.1.1.2.cmml">b</mi><mo id="S3.F3.20.m10.1.1.1" xref="S3.F3.20.m10.1.1.1.cmml">=</mo><mn id="S3.F3.20.m10.1.1.3" xref="S3.F3.20.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.20.m10.1c"><apply id="S3.F3.20.m10.1.1.cmml" xref="S3.F3.20.m10.1.1"><eq id="S3.F3.20.m10.1.1.1.cmml" xref="S3.F3.20.m10.1.1.1"></eq><ci id="S3.F3.20.m10.1.1.2.cmml" xref="S3.F3.20.m10.1.1.2">𝑏</ci><cn id="S3.F3.20.m10.1.1.3.cmml" type="integer" xref="S3.F3.20.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.20.m10.1d">b=1</annotation><annotation encoding="application/x-llamapun" id="S3.F3.20.m10.1e">italic_b = 1</annotation></semantics></math> (see <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.E5" title="5 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 5</span></a>).</figcaption> </figure> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.5">For the simulation, we calculate the scale-dependent <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">ℳ</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">ℳ</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\rho/\rho_{0}" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><mi id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml">ρ</mi><mo id="S3.p2.2.m2.1.1.1" xref="S3.p2.2.m2.1.1.1.cmml">/</mo><msub id="S3.p2.2.m2.1.1.3" xref="S3.p2.2.m2.1.1.3.cmml"><mi id="S3.p2.2.m2.1.1.3.2" xref="S3.p2.2.m2.1.1.3.2.cmml">ρ</mi><mn id="S3.p2.2.m2.1.1.3.3" xref="S3.p2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><divide id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"></divide><ci id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2">𝜌</ci><apply id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.3.1.cmml" xref="S3.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.p2.2.m2.1.1.3.2.cmml" xref="S3.p2.2.m2.1.1.3.2">𝜌</ci><cn id="S3.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.p2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> directly from the one-dimensional power spectrum of <math alttext="\bm{u}" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mi id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><ci id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">\bm{u}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">bold_italic_u</annotation></semantics></math> and <math alttext="\rho/\rho_{0}-1" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><mrow id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><mrow id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml"><mi id="S3.p2.4.m4.1.1.2.2" xref="S3.p2.4.m4.1.1.2.2.cmml">ρ</mi><mo id="S3.p2.4.m4.1.1.2.1" xref="S3.p2.4.m4.1.1.2.1.cmml">/</mo><msub id="S3.p2.4.m4.1.1.2.3" xref="S3.p2.4.m4.1.1.2.3.cmml"><mi id="S3.p2.4.m4.1.1.2.3.2" xref="S3.p2.4.m4.1.1.2.3.2.cmml">ρ</mi><mn id="S3.p2.4.m4.1.1.2.3.3" xref="S3.p2.4.m4.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.p2.4.m4.1.1.1" xref="S3.p2.4.m4.1.1.1.cmml">−</mo><mn id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1"><minus id="S3.p2.4.m4.1.1.1.cmml" xref="S3.p2.4.m4.1.1.1"></minus><apply id="S3.p2.4.m4.1.1.2.cmml" xref="S3.p2.4.m4.1.1.2"><divide id="S3.p2.4.m4.1.1.2.1.cmml" xref="S3.p2.4.m4.1.1.2.1"></divide><ci id="S3.p2.4.m4.1.1.2.2.cmml" xref="S3.p2.4.m4.1.1.2.2">𝜌</ci><apply id="S3.p2.4.m4.1.1.2.3.cmml" xref="S3.p2.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.2.3.1.cmml" xref="S3.p2.4.m4.1.1.2.3">subscript</csymbol><ci id="S3.p2.4.m4.1.1.2.3.2.cmml" xref="S3.p2.4.m4.1.1.2.3.2">𝜌</ci><cn id="S3.p2.4.m4.1.1.2.3.3.cmml" type="integer" xref="S3.p2.4.m4.1.1.2.3.3">0</cn></apply></apply><cn id="S3.p2.4.m4.1.1.3.cmml" type="integer" xref="S3.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">\rho/\rho_{0}-1</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1</annotation></semantics></math>. For example, the <math alttext="\bm{u}" class="ltx_Math" display="inline" id="S3.p2.5.m5.1"><semantics id="S3.p2.5.m5.1a"><mi id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml">𝒖</mi><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><ci id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1">𝒖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">\bm{u}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.1d">bold_italic_u</annotation></semantics></math> spectrum is,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx7"> <tbody id="S3.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{P}_{u}(k)=\int\operatorname{d}\!{\Omega_{k}}\;4\pi k^{2}% \bm{u}(\bm{k})\bm{u}^{\dagger}(\bm{k})," class="ltx_Math" display="inline" id="S3.E12.m1.4"><semantics id="S3.E12.m1.4a"><mrow id="S3.E12.m1.4.4.1" xref="S3.E12.m1.4.4.1.1.cmml"><mrow id="S3.E12.m1.4.4.1.1" xref="S3.E12.m1.4.4.1.1.cmml"><mrow id="S3.E12.m1.4.4.1.1.2" xref="S3.E12.m1.4.4.1.1.2.cmml"><msub id="S3.E12.m1.4.4.1.1.2.2" xref="S3.E12.m1.4.4.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E12.m1.4.4.1.1.2.2.2" xref="S3.E12.m1.4.4.1.1.2.2.2.cmml">𝒫</mi><mi id="S3.E12.m1.4.4.1.1.2.2.3" xref="S3.E12.m1.4.4.1.1.2.2.3.cmml">u</mi></msub><mo id="S3.E12.m1.4.4.1.1.2.1" xref="S3.E12.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S3.E12.m1.4.4.1.1.2.3.2" xref="S3.E12.m1.4.4.1.1.2.cmml"><mo id="S3.E12.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S3.E12.m1.4.4.1.1.2.cmml">(</mo><mi id="S3.E12.m1.1.1" xref="S3.E12.m1.1.1.cmml">k</mi><mo id="S3.E12.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S3.E12.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E12.m1.4.4.1.1.1" xref="S3.E12.m1.4.4.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E12.m1.4.4.1.1.3" xref="S3.E12.m1.4.4.1.1.3.cmml"><mrow id="S3.E12.m1.4.4.1.1.3a" xref="S3.E12.m1.4.4.1.1.3.cmml"><mo id="S3.E12.m1.4.4.1.1.3.1" xref="S3.E12.m1.4.4.1.1.3.1.cmml">∫</mo><mrow id="S3.E12.m1.4.4.1.1.3.2" xref="S3.E12.m1.4.4.1.1.3.2.cmml"><mrow id="S3.E12.m1.4.4.1.1.3.2.2" xref="S3.E12.m1.4.4.1.1.3.2.2.cmml"><mi id="S3.E12.m1.4.4.1.1.3.2.2.1" mathvariant="normal" xref="S3.E12.m1.4.4.1.1.3.2.2.1.cmml">d</mi><mo id="S3.E12.m1.4.4.1.1.3.2.2a" xref="S3.E12.m1.4.4.1.1.3.2.2.cmml">⁡</mo><mrow id="S3.E12.m1.4.4.1.1.3.2.2.2" xref="S3.E12.m1.4.4.1.1.3.2.2.2.cmml"><msub id="S3.E12.m1.4.4.1.1.3.2.2.2.2" xref="S3.E12.m1.4.4.1.1.3.2.2.2.2.cmml"><mi id="S3.E12.m1.4.4.1.1.3.2.2.2.2.2" mathvariant="normal" xref="S3.E12.m1.4.4.1.1.3.2.2.2.2.2.cmml">Ω</mi><mi id="S3.E12.m1.4.4.1.1.3.2.2.2.2.3" xref="S3.E12.m1.4.4.1.1.3.2.2.2.2.3.cmml">k</mi></msub><mo id="S3.E12.m1.4.4.1.1.3.2.2.2.1" xref="S3.E12.m1.4.4.1.1.3.2.2.2.1.cmml">⁢</mo><mn id="S3.E12.m1.4.4.1.1.3.2.2.2.3" xref="S3.E12.m1.4.4.1.1.3.2.2.2.3.cmml"> 4</mn><mo id="S3.E12.m1.4.4.1.1.3.2.2.2.1a" 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stretchy="false" xref="S3.E12.m1.4.4.1.1.3.2.cmml">)</mo></mrow><mo id="S3.E12.m1.4.4.1.1.3.2.1a" xref="S3.E12.m1.4.4.1.1.3.2.1.cmml">⁢</mo><msup id="S3.E12.m1.4.4.1.1.3.2.4" xref="S3.E12.m1.4.4.1.1.3.2.4.cmml"><mi id="S3.E12.m1.4.4.1.1.3.2.4.2" xref="S3.E12.m1.4.4.1.1.3.2.4.2.cmml">𝒖</mi><mo id="S3.E12.m1.4.4.1.1.3.2.4.3" xref="S3.E12.m1.4.4.1.1.3.2.4.3.cmml">†</mo></msup><mo id="S3.E12.m1.4.4.1.1.3.2.1b" xref="S3.E12.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E12.m1.4.4.1.1.3.2.5.2" xref="S3.E12.m1.4.4.1.1.3.2.cmml"><mo id="S3.E12.m1.4.4.1.1.3.2.5.2.1" stretchy="false" xref="S3.E12.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S3.E12.m1.3.3" xref="S3.E12.m1.3.3.cmml">𝒌</mi><mo id="S3.E12.m1.4.4.1.1.3.2.5.2.2" stretchy="false" xref="S3.E12.m1.4.4.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mstyle></mrow><mo id="S3.E12.m1.4.4.1.2" xref="S3.E12.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E12.m1.4b"><apply id="S3.E12.m1.4.4.1.1.cmml" xref="S3.E12.m1.4.4.1"><eq 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encoding="application/x-llamapun" id="S3.E12.m1.4d">caligraphic_P start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_k ) = ∫ roman_d roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 4 italic_π italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_u ( bold_italic_k ) bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( bold_italic_k ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.7">where <math alttext="\Omega_{k}" class="ltx_Math" display="inline" id="S3.p2.6.m1.1"><semantics id="S3.p2.6.m1.1a"><msub id="S3.p2.6.m1.1.1" xref="S3.p2.6.m1.1.1.cmml"><mi id="S3.p2.6.m1.1.1.2" mathvariant="normal" xref="S3.p2.6.m1.1.1.2.cmml">Ω</mi><mi id="S3.p2.6.m1.1.1.3" xref="S3.p2.6.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.6.m1.1b"><apply id="S3.p2.6.m1.1.1.cmml" xref="S3.p2.6.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.6.m1.1.1.1.cmml" xref="S3.p2.6.m1.1.1">subscript</csymbol><ci id="S3.p2.6.m1.1.1.2.cmml" xref="S3.p2.6.m1.1.1.2">Ω</ci><ci id="S3.p2.6.m1.1.1.3.cmml" xref="S3.p2.6.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m1.1c">\Omega_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m1.1d">roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is the integral over solid angle (see <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a></cite> for more information on <math alttext="\mathcal{P}_{u}(k)" class="ltx_Math" display="inline" id="S3.p2.7.m2.1"><semantics id="S3.p2.7.m2.1a"><mrow id="S3.p2.7.m2.1.2" xref="S3.p2.7.m2.1.2.cmml"><msub id="S3.p2.7.m2.1.2.2" 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xref="S3.p2.7.m2.1.2.2.3">𝑢</ci></apply><ci id="S3.p2.7.m2.1.1.cmml" xref="S3.p2.7.m2.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m2.1c">\mathcal{P}_{u}(k)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m2.1d">caligraphic_P start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_k )</annotation></semantics></math>), such that the one-dimensional spectrum has a normalization,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx8"> <tbody id="S3.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\langle{u^{2}}\right\rangle_{\mathcal{V}}=\int\operatorname{% d}\!{k}\;\mathcal{P}_{u}(k)," class="ltx_Math" display="inline" id="S3.E13.m1.2"><semantics id="S3.E13.m1.2a"><mrow id="S3.E13.m1.2.2.1" xref="S3.E13.m1.2.2.1.1.cmml"><mrow id="S3.E13.m1.2.2.1.1" 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xref="S3.E13.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="S3.E13.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2">𝑢</ci><cn id="S3.E13.m1.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3">2</cn></apply></apply><ci id="S3.E13.m1.2.2.1.1.1.3.cmml" xref="S3.E13.m1.2.2.1.1.1.3">𝒱</ci></apply><apply id="S3.E13.m1.2.2.1.1.3.cmml" xref="S3.E13.m1.2.2.1.1.3"><int id="S3.E13.m1.2.2.1.1.3.1.cmml" xref="S3.E13.m1.2.2.1.1.3.1"></int><apply id="S3.E13.m1.2.2.1.1.3.2.cmml" xref="S3.E13.m1.2.2.1.1.3.2"><times id="S3.E13.m1.2.2.1.1.3.2.1.cmml" xref="S3.E13.m1.2.2.1.1.3.2.1"></times><apply id="S3.E13.m1.2.2.1.1.3.2.2.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2"><ci id="S3.E13.m1.2.2.1.1.3.2.2.1.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.1">d</ci><apply id="S3.E13.m1.2.2.1.1.3.2.2.2.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2"><times id="S3.E13.m1.2.2.1.1.3.2.2.2.1.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.1"></times><ci id="S3.E13.m1.2.2.1.1.3.2.2.2.2.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.2">𝑘</ci><apply id="S3.E13.m1.2.2.1.1.3.2.2.2.3.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.3"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.3.2.2.2.3.1.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.3">subscript</csymbol><ci id="S3.E13.m1.2.2.1.1.3.2.2.2.3.2.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.3.2">𝒫</ci><ci id="S3.E13.m1.2.2.1.1.3.2.2.2.3.3.cmml" xref="S3.E13.m1.2.2.1.1.3.2.2.2.3.3">𝑢</ci></apply></apply></apply><ci id="S3.E13.m1.1.1.cmml" xref="S3.E13.m1.1.1">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.2c">\displaystyle\left\langle{u^{2}}\right\rangle_{\mathcal{V}}=\int\operatorname{% d}\!{k}\;\mathcal{P}_{u}(k),</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.2d">⟨ italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT caligraphic_V end_POSTSUBSCRIPT = ∫ roman_d italic_k caligraphic_P start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_k ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.8">from Parseval’s theorem. Hence, the scale-dependent <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p2.8.m1.1"><semantics id="S3.p2.8.m1.1a"><msub id="S3.p2.8.m1.1.1" xref="S3.p2.8.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.8.m1.1.1.2" xref="S3.p2.8.m1.1.1.2.cmml">ℳ</mi><mi id="S3.p2.8.m1.1.1.3" xref="S3.p2.8.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.8.m1.1b"><apply id="S3.p2.8.m1.1.1.cmml" xref="S3.p2.8.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.8.m1.1.1.1.cmml" xref="S3.p2.8.m1.1.1">subscript</csymbol><ci id="S3.p2.8.m1.1.1.2.cmml" xref="S3.p2.8.m1.1.1.2">ℳ</ci><ci id="S3.p2.8.m1.1.1.3.cmml" xref="S3.p2.8.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m1.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is simply,</p> <table 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end_POSTSUPERSCRIPT roman_d italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT caligraphic_P start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.9">where <math alttext="k_{u}" class="ltx_Math" display="inline" id="S3.p2.9.m1.1"><semantics id="S3.p2.9.m1.1a"><msub id="S3.p2.9.m1.1.1" xref="S3.p2.9.m1.1.1.cmml"><mi id="S3.p2.9.m1.1.1.2" xref="S3.p2.9.m1.1.1.2.cmml">k</mi><mi id="S3.p2.9.m1.1.1.3" xref="S3.p2.9.m1.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.9.m1.1b"><apply id="S3.p2.9.m1.1.1.cmml" xref="S3.p2.9.m1.1.1"><csymbol cd="ambiguous" 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caligraphic_P start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_k ) ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.21">such that the integral is not contaminated with modes in the viscous dissipation range. We similarly construct the scale-dependent rms for the mass density,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx11"> <tbody id="S3.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{\delta\rho(\ell/L)}{\rho_{0}}=\left\langle\left(\frac{\rho(% \ell/L)}{\rho_{0}}\right)^{2}\right\rangle=\left(\int_{k}^{k_{u}}\operatorname% {d}\!{k^{\prime}}\;\mathcal{P}_{\rho/\rho_{0}-1}(k^{\prime})\right)^{1/2}." class="ltx_Math" display="inline" id="S3.E16.m1.3"><semantics id="S3.E16.m1.3a"><mrow id="S3.E16.m1.3.3.1" xref="S3.E16.m1.3.3.1.1.cmml"><mrow id="S3.E16.m1.3.3.1.1" xref="S3.E16.m1.3.3.1.1.cmml"><mstyle displaystyle="true" id="S3.E16.m1.1.1" xref="S3.E16.m1.1.1.cmml"><mfrac id="S3.E16.m1.1.1a" xref="S3.E16.m1.1.1.cmml"><mrow id="S3.E16.m1.1.1.1" xref="S3.E16.m1.1.1.1.cmml"><mi id="S3.E16.m1.1.1.1.3" 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\ell/L)}{\rho_{0}}\right)^{2}\right\rangle=\left(\int_{k}^{k_{u}}\operatorname% {d}\!{k^{\prime}}\;\mathcal{P}_{\rho/\rho_{0}-1}(k^{\prime})\right)^{1/2}.</annotation><annotation encoding="application/x-llamapun" id="S3.E16.m1.3d">divide start_ARG italic_δ italic_ρ ( roman_ℓ / italic_L ) end_ARG start_ARG italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = ⟨ ( divide start_ARG italic_ρ ( roman_ℓ / italic_L ) end_ARG start_ARG italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = ( ∫ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_d italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT caligraphic_P start_POSTSUBSCRIPT italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.20">Plotting <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E16" title="16 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 16</span></a> as a function of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E14" title="14 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 14</span></a> allows us to directly compute the rms mass density as a function of <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p2.10.m1.1"><semantics id="S3.p2.10.m1.1a"><msub id="S3.p2.10.m1.1.1" xref="S3.p2.10.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.10.m1.1.1.2" xref="S3.p2.10.m1.1.1.2.cmml">ℳ</mi><mi id="S3.p2.10.m1.1.1.3" xref="S3.p2.10.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.10.m1.1b"><apply id="S3.p2.10.m1.1.1.cmml" xref="S3.p2.10.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.10.m1.1.1.1.cmml" xref="S3.p2.10.m1.1.1">subscript</csymbol><ci id="S3.p2.10.m1.1.1.2.cmml" xref="S3.p2.10.m1.1.1.2">ℳ</ci><ci id="S3.p2.10.m1.1.1.3.cmml" xref="S3.p2.10.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m1.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="\ell/L" class="ltx_Math" display="inline" id="S3.p2.11.m2.1"><semantics id="S3.p2.11.m2.1a"><mrow id="S3.p2.11.m2.1.1" xref="S3.p2.11.m2.1.1.cmml"><mi id="S3.p2.11.m2.1.1.2" mathvariant="normal" xref="S3.p2.11.m2.1.1.2.cmml">ℓ</mi><mo id="S3.p2.11.m2.1.1.1" xref="S3.p2.11.m2.1.1.1.cmml">/</mo><mi id="S3.p2.11.m2.1.1.3" xref="S3.p2.11.m2.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.11.m2.1b"><apply id="S3.p2.11.m2.1.1.cmml" xref="S3.p2.11.m2.1.1"><divide id="S3.p2.11.m2.1.1.1.cmml" xref="S3.p2.11.m2.1.1.1"></divide><ci id="S3.p2.11.m2.1.1.2.cmml" xref="S3.p2.11.m2.1.1.2">ℓ</ci><ci id="S3.p2.11.m2.1.1.3.cmml" xref="S3.p2.11.m2.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.11.m2.1c">\ell/L</annotation><annotation encoding="application/x-llamapun" id="S3.p2.11.m2.1d">roman_ℓ / italic_L</annotation></semantics></math> in the simulation. This process is equivalent to splitting the domain with volume <math alttext="\mathcal{V}" class="ltx_Math" display="inline" id="S3.p2.12.m3.1"><semantics id="S3.p2.12.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p2.12.m3.1.1" xref="S3.p2.12.m3.1.1.cmml">𝒱</mi><annotation-xml encoding="MathML-Content" id="S3.p2.12.m3.1b"><ci id="S3.p2.12.m3.1.1.cmml" xref="S3.p2.12.m3.1.1">𝒱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.12.m3.1c">\mathcal{V}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.12.m3.1d">caligraphic_V</annotation></semantics></math> into sub-domains <math alttext="\mathcal{V}=\bigcup_{n}\mathcal{V}_{n}" class="ltx_Math" display="inline" id="S3.p2.13.m4.1"><semantics id="S3.p2.13.m4.1a"><mrow id="S3.p2.13.m4.1.1" xref="S3.p2.13.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.13.m4.1.1.2" xref="S3.p2.13.m4.1.1.2.cmml">𝒱</mi><mo id="S3.p2.13.m4.1.1.1" rspace="0.111em" xref="S3.p2.13.m4.1.1.1.cmml">=</mo><mrow id="S3.p2.13.m4.1.1.3" xref="S3.p2.13.m4.1.1.3.cmml"><msub id="S3.p2.13.m4.1.1.3.1" xref="S3.p2.13.m4.1.1.3.1.cmml"><mo id="S3.p2.13.m4.1.1.3.1.2" xref="S3.p2.13.m4.1.1.3.1.2.cmml">⋃</mo><mi id="S3.p2.13.m4.1.1.3.1.3" xref="S3.p2.13.m4.1.1.3.1.3.cmml">n</mi></msub><msub id="S3.p2.13.m4.1.1.3.2" xref="S3.p2.13.m4.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.13.m4.1.1.3.2.2" xref="S3.p2.13.m4.1.1.3.2.2.cmml">𝒱</mi><mi id="S3.p2.13.m4.1.1.3.2.3" xref="S3.p2.13.m4.1.1.3.2.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.13.m4.1b"><apply id="S3.p2.13.m4.1.1.cmml" xref="S3.p2.13.m4.1.1"><eq id="S3.p2.13.m4.1.1.1.cmml" xref="S3.p2.13.m4.1.1.1"></eq><ci id="S3.p2.13.m4.1.1.2.cmml" xref="S3.p2.13.m4.1.1.2">𝒱</ci><apply id="S3.p2.13.m4.1.1.3.cmml" xref="S3.p2.13.m4.1.1.3"><apply id="S3.p2.13.m4.1.1.3.1.cmml" xref="S3.p2.13.m4.1.1.3.1"><csymbol cd="ambiguous" id="S3.p2.13.m4.1.1.3.1.1.cmml" xref="S3.p2.13.m4.1.1.3.1">subscript</csymbol><union id="S3.p2.13.m4.1.1.3.1.2.cmml" xref="S3.p2.13.m4.1.1.3.1.2"></union><ci id="S3.p2.13.m4.1.1.3.1.3.cmml" xref="S3.p2.13.m4.1.1.3.1.3">𝑛</ci></apply><apply id="S3.p2.13.m4.1.1.3.2.cmml" xref="S3.p2.13.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.13.m4.1.1.3.2.1.cmml" xref="S3.p2.13.m4.1.1.3.2">subscript</csymbol><ci id="S3.p2.13.m4.1.1.3.2.2.cmml" xref="S3.p2.13.m4.1.1.3.2.2">𝒱</ci><ci id="S3.p2.13.m4.1.1.3.2.3.cmml" xref="S3.p2.13.m4.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.13.m4.1c">\mathcal{V}=\bigcup_{n}\mathcal{V}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.13.m4.1d">caligraphic_V = ⋃ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and computing the rms for each collection of <math alttext="\mathcal{V}_{n}" class="ltx_Math" display="inline" id="S3.p2.14.m5.1"><semantics id="S3.p2.14.m5.1a"><msub id="S3.p2.14.m5.1.1" xref="S3.p2.14.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.14.m5.1.1.2" xref="S3.p2.14.m5.1.1.2.cmml">𝒱</mi><mi id="S3.p2.14.m5.1.1.3" xref="S3.p2.14.m5.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.14.m5.1b"><apply id="S3.p2.14.m5.1.1.cmml" xref="S3.p2.14.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.14.m5.1.1.1.cmml" xref="S3.p2.14.m5.1.1">subscript</csymbol><ci id="S3.p2.14.m5.1.1.2.cmml" xref="S3.p2.14.m5.1.1.2">𝒱</ci><ci id="S3.p2.14.m5.1.1.3.cmml" xref="S3.p2.14.m5.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.14.m5.1c">\mathcal{V}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.14.m5.1d">caligraphic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Performing this analysis for all <math alttext="\mathcal{V}_{n}" class="ltx_Math" display="inline" id="S3.p2.15.m6.1"><semantics id="S3.p2.15.m6.1a"><msub id="S3.p2.15.m6.1.1" xref="S3.p2.15.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.15.m6.1.1.2" xref="S3.p2.15.m6.1.1.2.cmml">𝒱</mi><mi id="S3.p2.15.m6.1.1.3" xref="S3.p2.15.m6.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.15.m6.1b"><apply id="S3.p2.15.m6.1.1.cmml" xref="S3.p2.15.m6.1.1"><csymbol cd="ambiguous" id="S3.p2.15.m6.1.1.1.cmml" xref="S3.p2.15.m6.1.1">subscript</csymbol><ci id="S3.p2.15.m6.1.1.2.cmml" xref="S3.p2.15.m6.1.1.2">𝒱</ci><ci id="S3.p2.15.m6.1.1.3.cmml" xref="S3.p2.15.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.15.m6.1c">\mathcal{V}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.15.m6.1d">caligraphic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> means that instead of running multiple simulations with different plasma parameters (e.g., different <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.p2.16.m7.1"><semantics id="S3.p2.16.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p2.16.m7.1.1" xref="S3.p2.16.m7.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.p2.16.m7.1b"><ci id="S3.p2.16.m7.1.1.cmml" xref="S3.p2.16.m7.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.16.m7.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.16.m7.1d">caligraphic_M</annotation></semantics></math>, as in <cite class="ltx_cite ltx_citemacro_citet">Beattie & Federrath <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib3" title="">2020</a>; Beattie et al. <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib7" title="">2022</a></cite>), we can use our single highly-resolved simulation to study multiple combinations of <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S3.p2.17.m8.1"><semantics id="S3.p2.17.m8.1a"><mrow id="S3.p2.17.m8.1.1" xref="S3.p2.17.m8.1.1.cmml"><mrow id="S3.p2.17.m8.1.1.2" xref="S3.p2.17.m8.1.1.2.cmml"><mi id="S3.p2.17.m8.1.1.2.2" xref="S3.p2.17.m8.1.1.2.2.cmml">δ</mi><mo id="S3.p2.17.m8.1.1.2.1" xref="S3.p2.17.m8.1.1.2.1.cmml">⁢</mo><mi id="S3.p2.17.m8.1.1.2.3" xref="S3.p2.17.m8.1.1.2.3.cmml">ρ</mi></mrow><mo id="S3.p2.17.m8.1.1.1" xref="S3.p2.17.m8.1.1.1.cmml">/</mo><msub id="S3.p2.17.m8.1.1.3" xref="S3.p2.17.m8.1.1.3.cmml"><mi id="S3.p2.17.m8.1.1.3.2" xref="S3.p2.17.m8.1.1.3.2.cmml">ρ</mi><mn id="S3.p2.17.m8.1.1.3.3" xref="S3.p2.17.m8.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.17.m8.1b"><apply id="S3.p2.17.m8.1.1.cmml" xref="S3.p2.17.m8.1.1"><divide id="S3.p2.17.m8.1.1.1.cmml" xref="S3.p2.17.m8.1.1.1"></divide><apply id="S3.p2.17.m8.1.1.2.cmml" xref="S3.p2.17.m8.1.1.2"><times id="S3.p2.17.m8.1.1.2.1.cmml" xref="S3.p2.17.m8.1.1.2.1"></times><ci id="S3.p2.17.m8.1.1.2.2.cmml" xref="S3.p2.17.m8.1.1.2.2">𝛿</ci><ci id="S3.p2.17.m8.1.1.2.3.cmml" xref="S3.p2.17.m8.1.1.2.3">𝜌</ci></apply><apply id="S3.p2.17.m8.1.1.3.cmml" xref="S3.p2.17.m8.1.1.3"><csymbol cd="ambiguous" id="S3.p2.17.m8.1.1.3.1.cmml" xref="S3.p2.17.m8.1.1.3">subscript</csymbol><ci id="S3.p2.17.m8.1.1.3.2.cmml" xref="S3.p2.17.m8.1.1.3.2">𝜌</ci><cn id="S3.p2.17.m8.1.1.3.3.cmml" type="integer" xref="S3.p2.17.m8.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.17.m8.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.17.m8.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p2.18.m9.1"><semantics id="S3.p2.18.m9.1a"><msub id="S3.p2.18.m9.1.1" xref="S3.p2.18.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.18.m9.1.1.2" xref="S3.p2.18.m9.1.1.2.cmml">ℳ</mi><mi id="S3.p2.18.m9.1.1.3" xref="S3.p2.18.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.18.m9.1b"><apply id="S3.p2.18.m9.1.1.cmml" xref="S3.p2.18.m9.1.1"><csymbol cd="ambiguous" id="S3.p2.18.m9.1.1.1.cmml" xref="S3.p2.18.m9.1.1">subscript</csymbol><ci id="S3.p2.18.m9.1.1.2.cmml" xref="S3.p2.18.m9.1.1.2">ℳ</ci><ci id="S3.p2.18.m9.1.1.3.cmml" xref="S3.p2.18.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.18.m9.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.18.m9.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. This is only possible with very high-resolution simulation data, with large amounts of dynamical range within the turbulence cascade, and allows us to test how robust <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p2.19.m10.1"><semantics id="S3.p2.19.m10.1a"><mrow id="S3.p2.19.m10.1.1" xref="S3.p2.19.m10.1.1.cmml"><mrow id="S3.p2.19.m10.1.1.2" xref="S3.p2.19.m10.1.1.2.cmml"><mrow id="S3.p2.19.m10.1.1.2.2" xref="S3.p2.19.m10.1.1.2.2.cmml"><mi id="S3.p2.19.m10.1.1.2.2.2" xref="S3.p2.19.m10.1.1.2.2.2.cmml">δ</mi><mo id="S3.p2.19.m10.1.1.2.2.1" xref="S3.p2.19.m10.1.1.2.2.1.cmml">⁢</mo><mi id="S3.p2.19.m10.1.1.2.2.3" xref="S3.p2.19.m10.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S3.p2.19.m10.1.1.2.1" xref="S3.p2.19.m10.1.1.2.1.cmml">/</mo><msub id="S3.p2.19.m10.1.1.2.3" xref="S3.p2.19.m10.1.1.2.3.cmml"><mi id="S3.p2.19.m10.1.1.2.3.2" xref="S3.p2.19.m10.1.1.2.3.2.cmml">ρ</mi><mn id="S3.p2.19.m10.1.1.2.3.3" xref="S3.p2.19.m10.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.p2.19.m10.1.1.1" xref="S3.p2.19.m10.1.1.1.cmml">∝</mo><msub id="S3.p2.19.m10.1.1.3" xref="S3.p2.19.m10.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.19.m10.1.1.3.2" xref="S3.p2.19.m10.1.1.3.2.cmml">ℳ</mi><mi id="S3.p2.19.m10.1.1.3.3" xref="S3.p2.19.m10.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.19.m10.1b"><apply id="S3.p2.19.m10.1.1.cmml" xref="S3.p2.19.m10.1.1"><csymbol cd="latexml" id="S3.p2.19.m10.1.1.1.cmml" xref="S3.p2.19.m10.1.1.1">proportional-to</csymbol><apply id="S3.p2.19.m10.1.1.2.cmml" xref="S3.p2.19.m10.1.1.2"><divide id="S3.p2.19.m10.1.1.2.1.cmml" xref="S3.p2.19.m10.1.1.2.1"></divide><apply id="S3.p2.19.m10.1.1.2.2.cmml" xref="S3.p2.19.m10.1.1.2.2"><times id="S3.p2.19.m10.1.1.2.2.1.cmml" xref="S3.p2.19.m10.1.1.2.2.1"></times><ci id="S3.p2.19.m10.1.1.2.2.2.cmml" xref="S3.p2.19.m10.1.1.2.2.2">𝛿</ci><ci id="S3.p2.19.m10.1.1.2.2.3.cmml" xref="S3.p2.19.m10.1.1.2.2.3">𝜌</ci></apply><apply id="S3.p2.19.m10.1.1.2.3.cmml" xref="S3.p2.19.m10.1.1.2.3"><csymbol cd="ambiguous" id="S3.p2.19.m10.1.1.2.3.1.cmml" xref="S3.p2.19.m10.1.1.2.3">subscript</csymbol><ci id="S3.p2.19.m10.1.1.2.3.2.cmml" xref="S3.p2.19.m10.1.1.2.3.2">𝜌</ci><cn id="S3.p2.19.m10.1.1.2.3.3.cmml" type="integer" xref="S3.p2.19.m10.1.1.2.3.3">0</cn></apply></apply><apply id="S3.p2.19.m10.1.1.3.cmml" xref="S3.p2.19.m10.1.1.3"><csymbol cd="ambiguous" id="S3.p2.19.m10.1.1.3.1.cmml" xref="S3.p2.19.m10.1.1.3">subscript</csymbol><ci id="S3.p2.19.m10.1.1.3.2.cmml" xref="S3.p2.19.m10.1.1.3.2">ℳ</ci><ci id="S3.p2.19.m10.1.1.3.3.cmml" xref="S3.p2.19.m10.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.19.m10.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.19.m10.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is, far away from the <math alttext="k" class="ltx_Math" display="inline" id="S3.p2.20.m11.1"><semantics id="S3.p2.20.m11.1a"><mi id="S3.p2.20.m11.1.1" xref="S3.p2.20.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.p2.20.m11.1b"><ci id="S3.p2.20.m11.1.1.cmml" xref="S3.p2.20.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.20.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.p2.20.m11.1d">italic_k</annotation></semantics></math> modes influenced by the driving mechanism.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.2">The computed <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mrow id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mrow id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml"><mi id="S3.p3.1.m1.1.1.2.2" xref="S3.p3.1.m1.1.1.2.2.cmml">δ</mi><mo id="S3.p3.1.m1.1.1.2.1" xref="S3.p3.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.p3.1.m1.1.1.2.3" xref="S3.p3.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S3.p3.1.m1.1.1.1" xref="S3.p3.1.m1.1.1.1.cmml">/</mo><msub id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml"><mi id="S3.p3.1.m1.1.1.3.2" xref="S3.p3.1.m1.1.1.3.2.cmml">ρ</mi><mn id="S3.p3.1.m1.1.1.3.3" xref="S3.p3.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><divide id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1.1"></divide><apply id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2"><times id="S3.p3.1.m1.1.1.2.1.cmml" xref="S3.p3.1.m1.1.1.2.1"></times><ci id="S3.p3.1.m1.1.1.2.2.cmml" xref="S3.p3.1.m1.1.1.2.2">𝛿</ci><ci id="S3.p3.1.m1.1.1.2.3.cmml" xref="S3.p3.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S3.p3.1.m1.1.1.3.cmml" xref="S3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.3.1.cmml" xref="S3.p3.1.m1.1.1.3">subscript</csymbol><ci id="S3.p3.1.m1.1.1.3.2.cmml" xref="S3.p3.1.m1.1.1.3.2">𝜌</ci><cn id="S3.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><msub id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">ℳ</mi><mi id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p3.2.m2.1.1.1.cmml" xref="S3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2">ℳ</ci><ci id="S3.p3.2.m2.1.1.3.cmml" xref="S3.p3.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> values are plotted with red, open markers in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.F3" title="Figure 3 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 3</span></a>. Using the same maximum likelihood fitting process and model as in the MMS data, we find</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx12"> <tbody id="S3.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta\rho/\rho_{0}=(0.97^{+0.23}_{-0.23})\mathcal{M}_{t}^{0.94^{% +0.15}_{-0.15}}," class="ltx_Math" display="inline" id="S3.E17.m1.1"><semantics id="S3.E17.m1.1a"><mrow id="S3.E17.m1.1.1.1" xref="S3.E17.m1.1.1.1.1.cmml"><mrow id="S3.E17.m1.1.1.1.1" xref="S3.E17.m1.1.1.1.1.cmml"><mrow id="S3.E17.m1.1.1.1.1.3" xref="S3.E17.m1.1.1.1.1.3.cmml"><mrow id="S3.E17.m1.1.1.1.1.3.2" xref="S3.E17.m1.1.1.1.1.3.2.cmml"><mi id="S3.E17.m1.1.1.1.1.3.2.2" xref="S3.E17.m1.1.1.1.1.3.2.2.cmml">δ</mi><mo id="S3.E17.m1.1.1.1.1.3.2.1" xref="S3.E17.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.E17.m1.1.1.1.1.3.2.3" xref="S3.E17.m1.1.1.1.1.3.2.3.cmml">ρ</mi></mrow><mo id="S3.E17.m1.1.1.1.1.3.1" xref="S3.E17.m1.1.1.1.1.3.1.cmml">/</mo><msub id="S3.E17.m1.1.1.1.1.3.3" xref="S3.E17.m1.1.1.1.1.3.3.cmml"><mi id="S3.E17.m1.1.1.1.1.3.3.2" xref="S3.E17.m1.1.1.1.1.3.3.2.cmml">ρ</mi><mn id="S3.E17.m1.1.1.1.1.3.3.3" xref="S3.E17.m1.1.1.1.1.3.3.3.cmml">0</mn></msub></mrow><mo id="S3.E17.m1.1.1.1.1.2" xref="S3.E17.m1.1.1.1.1.2.cmml">=</mo><mrow id="S3.E17.m1.1.1.1.1.1" xref="S3.E17.m1.1.1.1.1.1.cmml"><mrow id="S3.E17.m1.1.1.1.1.1.1.1" xref="S3.E17.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E17.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E17.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.E17.m1.1.1.1.1.1.1.1.1" xref="S3.E17.m1.1.1.1.1.1.1.1.1.cmml"><mn id="S3.E17.m1.1.1.1.1.1.1.1.1.2.2" xref="S3.E17.m1.1.1.1.1.1.1.1.1.2.2.cmml">0.97</mn><mrow id="S3.E17.m1.1.1.1.1.1.1.1.1.3" xref="S3.E17.m1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S3.E17.m1.1.1.1.1.1.1.1.1.3a" xref="S3.E17.m1.1.1.1.1.1.1.1.1.3.cmml">−</mo><mn id="S3.E17.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.E17.m1.1.1.1.1.1.1.1.1.3.2.cmml">0.23</mn></mrow><mrow id="S3.E17.m1.1.1.1.1.1.1.1.1.2.3" xref="S3.E17.m1.1.1.1.1.1.1.1.1.2.3.cmml"><mo id="S3.E17.m1.1.1.1.1.1.1.1.1.2.3a" xref="S3.E17.m1.1.1.1.1.1.1.1.1.2.3.cmml">+</mo><mn id="S3.E17.m1.1.1.1.1.1.1.1.1.2.3.2" xref="S3.E17.m1.1.1.1.1.1.1.1.1.2.3.2.cmml">0.23</mn></mrow></msubsup><mo id="S3.E17.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E17.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E17.m1.1.1.1.1.1.2" xref="S3.E17.m1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S3.E17.m1.1.1.1.1.1.3" xref="S3.E17.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E17.m1.1.1.1.1.1.3.2.2" xref="S3.E17.m1.1.1.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S3.E17.m1.1.1.1.1.1.3.2.3" xref="S3.E17.m1.1.1.1.1.1.3.2.3.cmml">t</mi><msubsup id="S3.E17.m1.1.1.1.1.1.3.3" xref="S3.E17.m1.1.1.1.1.1.3.3.cmml"><mn id="S3.E17.m1.1.1.1.1.1.3.3.2.2" xref="S3.E17.m1.1.1.1.1.1.3.3.2.2.cmml">0.94</mn><mrow 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xref="S3.E17.m1.1.1.1.1.1.3.3.2.3"></plus><cn id="S3.E17.m1.1.1.1.1.1.3.3.2.3.2.cmml" type="float" xref="S3.E17.m1.1.1.1.1.1.3.3.2.3.2">0.15</cn></apply></apply><apply id="S3.E17.m1.1.1.1.1.1.3.3.3.cmml" xref="S3.E17.m1.1.1.1.1.1.3.3.3"><minus id="S3.E17.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S3.E17.m1.1.1.1.1.1.3.3.3"></minus><cn id="S3.E17.m1.1.1.1.1.1.3.3.3.2.cmml" type="float" xref="S3.E17.m1.1.1.1.1.1.3.3.3.2">0.15</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E17.m1.1c">\displaystyle\delta\rho/\rho_{0}=(0.97^{+0.23}_{-0.23})\mathcal{M}_{t}^{0.94^{% +0.15}_{-0.15}},</annotation><annotation encoding="application/x-llamapun" id="S3.E17.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 0.97 start_POSTSUPERSCRIPT + 0.23 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.23 end_POSTSUBSCRIPT ) caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0.94 start_POSTSUPERSCRIPT + 0.15 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.15 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(\text{simulation data)}" class="ltx_math_unparsed" display="inline" id="S3.E17.m2.1"><semantics id="S3.E17.m2.1a"><mrow id="S3.E17.m2.1b"><mo id="S3.E17.m2.1.1" stretchy="false">(</mo><mtext id="S3.E17.m2.1.2">simulation data)</mtext></mrow><annotation encoding="application/x-tex" id="S3.E17.m2.1c">\displaystyle(\text{simulation data)}</annotation><annotation encoding="application/x-llamapun" id="S3.E17.m2.1d">( simulation data)</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.3">again showing that the simulation data is 1<math alttext="\sigma" class="ltx_Math" display="inline" id="S3.p3.3.m1.1"><semantics id="S3.p3.3.m1.1a"><mi id="S3.p3.3.m1.1.1" xref="S3.p3.3.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.p3.3.m1.1b"><ci id="S3.p3.3.m1.1.1.cmml" xref="S3.p3.3.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m1.1d">italic_σ</annotation></semantics></math> consistent with the weakly compressible model, with a constant that is very close to unity. We discuss the proportionality constant further in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S4" title="4 Discussion & Conclusions ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">section 4</span></a>, and show the corner plots for the fits in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#A1" title="Appendix A Maximum Likelihood Fits ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Appendix A</span></a>.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.4">Both observation and simulation results shown here follow closely the scaling prediction from weakly compressible MHD theory with an inhomogeneous background field. Note that this is independent of the underlying <math alttext="\beta" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_β</annotation></semantics></math> (<math alttext="\beta\sim 1" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mrow id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><mi id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">β</mi><mo id="S3.p4.2.m2.1.1.1" xref="S3.p4.2.m2.1.1.1.cmml">∼</mo><mn id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><csymbol cd="latexml" id="S3.p4.2.m2.1.1.1.cmml" xref="S3.p4.2.m2.1.1.1">similar-to</csymbol><ci id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2">𝛽</ci><cn id="S3.p4.2.m2.1.1.3.cmml" type="integer" xref="S3.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">\beta\sim 1</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_β ∼ 1</annotation></semantics></math> for the simulation and <math alttext="\beta\sim 10" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mrow id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml"><mi id="S3.p4.3.m3.1.1.2" xref="S3.p4.3.m3.1.1.2.cmml">β</mi><mo id="S3.p4.3.m3.1.1.1" xref="S3.p4.3.m3.1.1.1.cmml">∼</mo><mn id="S3.p4.3.m3.1.1.3" xref="S3.p4.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1"><csymbol cd="latexml" 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id="S3.p4.4.m4.1d">italic_β</annotation></semantics></math></p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Discussion & Conclusions</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.5">Due to the weak <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml"><mrow id="S4.p1.1.m1.1.1.2" xref="S4.p1.1.m1.1.1.2.cmml"><mi id="S4.p1.1.m1.1.1.2.2" xref="S4.p1.1.m1.1.1.2.2.cmml">δ</mi><mo id="S4.p1.1.m1.1.1.2.1" xref="S4.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.p1.1.m1.1.1.2.3" xref="S4.p1.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S4.p1.1.m1.1.1.1" xref="S4.p1.1.m1.1.1.1.cmml">/</mo><msub id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.3.cmml"><mi id="S4.p1.1.m1.1.1.3.2" xref="S4.p1.1.m1.1.1.3.2.cmml">ρ</mi><mn id="S4.p1.1.m1.1.1.3.3" xref="S4.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1"><divide id="S4.p1.1.m1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1"></divide><apply id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.2"><times id="S4.p1.1.m1.1.1.2.1.cmml" xref="S4.p1.1.m1.1.1.2.1"></times><ci id="S4.p1.1.m1.1.1.2.2.cmml" xref="S4.p1.1.m1.1.1.2.2">𝛿</ci><ci id="S4.p1.1.m1.1.1.2.3.cmml" xref="S4.p1.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S4.p1.1.m1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.3.1.cmml" xref="S4.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.p1.1.m1.1.1.3.2.cmml" xref="S4.p1.1.m1.1.1.3.2">𝜌</ci><cn id="S4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, the interplanetary solar wind behaves very close to an incompressible fluid. Indeed, until the recent advent of Parker Solar Probe <cite class="ltx_cite ltx_citemacro_citep">(PSP; Fox et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib31" title="">2016</a>)</cite> data in the near-Sun solar wind, the relative amplitude of <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mrow id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml"><mrow id="S4.p1.2.m2.1.1.2" xref="S4.p1.2.m2.1.1.2.cmml"><mi id="S4.p1.2.m2.1.1.2.2" xref="S4.p1.2.m2.1.1.2.2.cmml">δ</mi><mo id="S4.p1.2.m2.1.1.2.1" xref="S4.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.p1.2.m2.1.1.2.3" xref="S4.p1.2.m2.1.1.2.3.cmml">ρ</mi></mrow><mo id="S4.p1.2.m2.1.1.1" xref="S4.p1.2.m2.1.1.1.cmml">/</mo><msub id="S4.p1.2.m2.1.1.3" xref="S4.p1.2.m2.1.1.3.cmml"><mi id="S4.p1.2.m2.1.1.3.2" xref="S4.p1.2.m2.1.1.3.2.cmml">ρ</mi><mn id="S4.p1.2.m2.1.1.3.3" xref="S4.p1.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1"><divide id="S4.p1.2.m2.1.1.1.cmml" xref="S4.p1.2.m2.1.1.1"></divide><apply id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.2"><times id="S4.p1.2.m2.1.1.2.1.cmml" xref="S4.p1.2.m2.1.1.2.1"></times><ci id="S4.p1.2.m2.1.1.2.2.cmml" xref="S4.p1.2.m2.1.1.2.2">𝛿</ci><ci id="S4.p1.2.m2.1.1.2.3.cmml" xref="S4.p1.2.m2.1.1.2.3">𝜌</ci></apply><apply id="S4.p1.2.m2.1.1.3.cmml" xref="S4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.p1.2.m2.1.1.3.1.cmml" xref="S4.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.p1.2.m2.1.1.3.2.cmml" xref="S4.p1.2.m2.1.1.3.2">𝜌</ci><cn id="S4.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S4.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> has, for the majority of cases, remained much smaller than unity <cite class="ltx_cite ltx_citemacro_citep">(e.g., Matthaeus et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib49" title="">1991</a>)</cite>. Therefore, weakly compressible MHD is usually adequate to describe the interplanetary solar wind fluctuations. However, in several astrophysical settings, where <span class="ltx_text ltx_font_italic" id="S4.p1.5.1">in-situ</span> measurements are not yet feasible, the compressive fluctuations can be significant, e.g., <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S4.p1.3.m3.1"><semantics id="S4.p1.3.m3.1a"><mrow id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml"><mrow id="S4.p1.3.m3.1.1.2" xref="S4.p1.3.m3.1.1.2.cmml"><mi id="S4.p1.3.m3.1.1.2.2" xref="S4.p1.3.m3.1.1.2.2.cmml">δ</mi><mo id="S4.p1.3.m3.1.1.2.1" xref="S4.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S4.p1.3.m3.1.1.2.3" xref="S4.p1.3.m3.1.1.2.3.cmml">ρ</mi></mrow><mo id="S4.p1.3.m3.1.1.1" xref="S4.p1.3.m3.1.1.1.cmml">/</mo><msub id="S4.p1.3.m3.1.1.3" xref="S4.p1.3.m3.1.1.3.cmml"><mi id="S4.p1.3.m3.1.1.3.2" xref="S4.p1.3.m3.1.1.3.2.cmml">ρ</mi><mn id="S4.p1.3.m3.1.1.3.3" xref="S4.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.1b"><apply id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1"><divide id="S4.p1.3.m3.1.1.1.cmml" xref="S4.p1.3.m3.1.1.1"></divide><apply id="S4.p1.3.m3.1.1.2.cmml" xref="S4.p1.3.m3.1.1.2"><times id="S4.p1.3.m3.1.1.2.1.cmml" xref="S4.p1.3.m3.1.1.2.1"></times><ci id="S4.p1.3.m3.1.1.2.2.cmml" xref="S4.p1.3.m3.1.1.2.2">𝛿</ci><ci id="S4.p1.3.m3.1.1.2.3.cmml" xref="S4.p1.3.m3.1.1.2.3">𝜌</ci></apply><apply id="S4.p1.3.m3.1.1.3.cmml" xref="S4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.p1.3.m3.1.1.3.1.cmml" xref="S4.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.p1.3.m3.1.1.3.2.cmml" xref="S4.p1.3.m3.1.1.3.2">𝜌</ci><cn id="S4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S4.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> varies by many orders of magnitude, as is the case in our simulations. Furthermore, <math alttext="\beta" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><mi id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><ci id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">italic_β</annotation></semantics></math> is another parameter which is limited to a rather narrow band of values in the solar wind, but could be large in many astrophysical plasmas. Earth’s magnetosheath provides an excellent natural laboratory to test high <math alttext="\beta" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mi id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><ci id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_β</annotation></semantics></math>, strongly compressive plasma environment using <span class="ltx_text ltx_font_italic" id="S4.p1.5.2">in-situ</span> data <cite class="ltx_cite ltx_citemacro_citep">(Sahraoui et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib66" title="">2020</a>)</cite>.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.15">The proportionality constant, <math alttext="b" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_b</annotation></semantics></math>, from <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S1.E5" title="5 ‣ 1 Introduction ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 5</span></a>, has been measured in a number of galactic <cite class="ltx_cite ltx_citemacro_citep">(e.g., Menon et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib52" title="">2020</a>)</cite> and even extra-galactic sources <cite class="ltx_cite ltx_citemacro_citep">(e.g., Sharda et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib69" title="">2021</a>; Gerrard et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib33" title="">2023</a>)</cite>. In <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.F3" title="Figure 3 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 3</span></a> we show a very interesting effect. By measuring <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mrow id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mrow id="S4.p2.2.m2.1.1.2" xref="S4.p2.2.m2.1.1.2.cmml"><mrow id="S4.p2.2.m2.1.1.2.2" xref="S4.p2.2.m2.1.1.2.2.cmml"><mi id="S4.p2.2.m2.1.1.2.2.2" xref="S4.p2.2.m2.1.1.2.2.2.cmml">δ</mi><mo id="S4.p2.2.m2.1.1.2.2.1" xref="S4.p2.2.m2.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p2.2.m2.1.1.2.2.3" xref="S4.p2.2.m2.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.2.m2.1.1.2.1" xref="S4.p2.2.m2.1.1.2.1.cmml">/</mo><msub id="S4.p2.2.m2.1.1.2.3" xref="S4.p2.2.m2.1.1.2.3.cmml"><mi id="S4.p2.2.m2.1.1.2.3.2" xref="S4.p2.2.m2.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p2.2.m2.1.1.2.3.3" xref="S4.p2.2.m2.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p2.2.m2.1.1.1" xref="S4.p2.2.m2.1.1.1.cmml">∝</mo><msub id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.2.m2.1.1.3.2" xref="S4.p2.2.m2.1.1.3.2.cmml">ℳ</mi><mi id="S4.p2.2.m2.1.1.3.3" xref="S4.p2.2.m2.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="latexml" id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1.1">proportional-to</csymbol><apply id="S4.p2.2.m2.1.1.2.cmml" xref="S4.p2.2.m2.1.1.2"><divide id="S4.p2.2.m2.1.1.2.1.cmml" xref="S4.p2.2.m2.1.1.2.1"></divide><apply id="S4.p2.2.m2.1.1.2.2.cmml" xref="S4.p2.2.m2.1.1.2.2"><times id="S4.p2.2.m2.1.1.2.2.1.cmml" xref="S4.p2.2.m2.1.1.2.2.1"></times><ci id="S4.p2.2.m2.1.1.2.2.2.cmml" xref="S4.p2.2.m2.1.1.2.2.2">𝛿</ci><ci id="S4.p2.2.m2.1.1.2.2.3.cmml" xref="S4.p2.2.m2.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p2.2.m2.1.1.2.3.cmml" xref="S4.p2.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.2.3.1.cmml" xref="S4.p2.2.m2.1.1.2.3">subscript</csymbol><ci id="S4.p2.2.m2.1.1.2.3.2.cmml" xref="S4.p2.2.m2.1.1.2.3.2">𝜌</ci><cn id="S4.p2.2.m2.1.1.2.3.3.cmml" type="integer" xref="S4.p2.2.m2.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.3.1.cmml" xref="S4.p2.2.m2.1.1.3">subscript</csymbol><ci id="S4.p2.2.m2.1.1.3.2.cmml" xref="S4.p2.2.m2.1.1.3.2">ℳ</ci><ci id="S4.p2.2.m2.1.1.3.3.cmml" xref="S4.p2.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> across a broad range of scales in the turbulence (red markers) we find <math alttext="\delta\rho/\rho_{0}\approx 1/3\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mrow id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml"><mrow id="S4.p2.3.m3.1.1.2" xref="S4.p2.3.m3.1.1.2.cmml"><mrow id="S4.p2.3.m3.1.1.2.2" xref="S4.p2.3.m3.1.1.2.2.cmml"><mi id="S4.p2.3.m3.1.1.2.2.2" xref="S4.p2.3.m3.1.1.2.2.2.cmml">δ</mi><mo id="S4.p2.3.m3.1.1.2.2.1" xref="S4.p2.3.m3.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p2.3.m3.1.1.2.2.3" xref="S4.p2.3.m3.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.3.m3.1.1.2.1" xref="S4.p2.3.m3.1.1.2.1.cmml">/</mo><msub id="S4.p2.3.m3.1.1.2.3" xref="S4.p2.3.m3.1.1.2.3.cmml"><mi id="S4.p2.3.m3.1.1.2.3.2" xref="S4.p2.3.m3.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p2.3.m3.1.1.2.3.3" xref="S4.p2.3.m3.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p2.3.m3.1.1.1" xref="S4.p2.3.m3.1.1.1.cmml">≈</mo><mrow id="S4.p2.3.m3.1.1.3" xref="S4.p2.3.m3.1.1.3.cmml"><mrow id="S4.p2.3.m3.1.1.3.2" xref="S4.p2.3.m3.1.1.3.2.cmml"><mn id="S4.p2.3.m3.1.1.3.2.2" xref="S4.p2.3.m3.1.1.3.2.2.cmml">1</mn><mo id="S4.p2.3.m3.1.1.3.2.1" xref="S4.p2.3.m3.1.1.3.2.1.cmml">/</mo><mn id="S4.p2.3.m3.1.1.3.2.3" xref="S4.p2.3.m3.1.1.3.2.3.cmml">3</mn></mrow><mo id="S4.p2.3.m3.1.1.3.1" xref="S4.p2.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S4.p2.3.m3.1.1.3.3" xref="S4.p2.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.3.m3.1.1.3.3.2" xref="S4.p2.3.m3.1.1.3.3.2.cmml">ℳ</mi><mi id="S4.p2.3.m3.1.1.3.3.3" xref="S4.p2.3.m3.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><apply id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1"><approx id="S4.p2.3.m3.1.1.1.cmml" xref="S4.p2.3.m3.1.1.1"></approx><apply id="S4.p2.3.m3.1.1.2.cmml" xref="S4.p2.3.m3.1.1.2"><divide id="S4.p2.3.m3.1.1.2.1.cmml" xref="S4.p2.3.m3.1.1.2.1"></divide><apply id="S4.p2.3.m3.1.1.2.2.cmml" xref="S4.p2.3.m3.1.1.2.2"><times id="S4.p2.3.m3.1.1.2.2.1.cmml" xref="S4.p2.3.m3.1.1.2.2.1"></times><ci id="S4.p2.3.m3.1.1.2.2.2.cmml" xref="S4.p2.3.m3.1.1.2.2.2">𝛿</ci><ci id="S4.p2.3.m3.1.1.2.2.3.cmml" xref="S4.p2.3.m3.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p2.3.m3.1.1.2.3.cmml" xref="S4.p2.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S4.p2.3.m3.1.1.2.3.1.cmml" xref="S4.p2.3.m3.1.1.2.3">subscript</csymbol><ci id="S4.p2.3.m3.1.1.2.3.2.cmml" xref="S4.p2.3.m3.1.1.2.3.2">𝜌</ci><cn id="S4.p2.3.m3.1.1.2.3.3.cmml" type="integer" xref="S4.p2.3.m3.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p2.3.m3.1.1.3.cmml" xref="S4.p2.3.m3.1.1.3"><times id="S4.p2.3.m3.1.1.3.1.cmml" xref="S4.p2.3.m3.1.1.3.1"></times><apply id="S4.p2.3.m3.1.1.3.2.cmml" xref="S4.p2.3.m3.1.1.3.2"><divide id="S4.p2.3.m3.1.1.3.2.1.cmml" xref="S4.p2.3.m3.1.1.3.2.1"></divide><cn id="S4.p2.3.m3.1.1.3.2.2.cmml" type="integer" xref="S4.p2.3.m3.1.1.3.2.2">1</cn><cn id="S4.p2.3.m3.1.1.3.2.3.cmml" type="integer" xref="S4.p2.3.m3.1.1.3.2.3">3</cn></apply><apply id="S4.p2.3.m3.1.1.3.3.cmml" xref="S4.p2.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.p2.3.m3.1.1.3.3.1.cmml" xref="S4.p2.3.m3.1.1.3.3">subscript</csymbol><ci id="S4.p2.3.m3.1.1.3.3.2.cmml" xref="S4.p2.3.m3.1.1.3.3.2">ℳ</ci><ci id="S4.p2.3.m3.1.1.3.3.3.cmml" xref="S4.p2.3.m3.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">\delta\rho/\rho_{0}\approx 1/3\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≈ 1 / 3 caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on the outer scale (<math alttext="b=1/3)" class="ltx_math_unparsed" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><mrow id="S4.p2.4.m4.1b"><mi id="S4.p2.4.m4.1.1">b</mi><mo id="S4.p2.4.m4.1.2">=</mo><mn id="S4.p2.4.m4.1.3">1</mn><mo id="S4.p2.4.m4.1.4">/</mo><mn id="S4.p2.4.m4.1.5">3</mn><mo id="S4.p2.4.m4.1.6" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">b=1/3)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_b = 1 / 3 )</annotation></semantics></math>, indicating incompressible driving, and then it changes to <math alttext="\delta\rho/\rho_{0}\approx\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mrow id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml"><mrow id="S4.p2.5.m5.1.1.2" xref="S4.p2.5.m5.1.1.2.cmml"><mrow id="S4.p2.5.m5.1.1.2.2" xref="S4.p2.5.m5.1.1.2.2.cmml"><mi id="S4.p2.5.m5.1.1.2.2.2" xref="S4.p2.5.m5.1.1.2.2.2.cmml">δ</mi><mo id="S4.p2.5.m5.1.1.2.2.1" xref="S4.p2.5.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p2.5.m5.1.1.2.2.3" xref="S4.p2.5.m5.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.5.m5.1.1.2.1" xref="S4.p2.5.m5.1.1.2.1.cmml">/</mo><msub id="S4.p2.5.m5.1.1.2.3" xref="S4.p2.5.m5.1.1.2.3.cmml"><mi id="S4.p2.5.m5.1.1.2.3.2" xref="S4.p2.5.m5.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p2.5.m5.1.1.2.3.3" xref="S4.p2.5.m5.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p2.5.m5.1.1.1" xref="S4.p2.5.m5.1.1.1.cmml">≈</mo><msub id="S4.p2.5.m5.1.1.3" xref="S4.p2.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.5.m5.1.1.3.2" xref="S4.p2.5.m5.1.1.3.2.cmml">ℳ</mi><mi id="S4.p2.5.m5.1.1.3.3" xref="S4.p2.5.m5.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><apply id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1"><approx id="S4.p2.5.m5.1.1.1.cmml" xref="S4.p2.5.m5.1.1.1"></approx><apply id="S4.p2.5.m5.1.1.2.cmml" xref="S4.p2.5.m5.1.1.2"><divide id="S4.p2.5.m5.1.1.2.1.cmml" xref="S4.p2.5.m5.1.1.2.1"></divide><apply id="S4.p2.5.m5.1.1.2.2.cmml" xref="S4.p2.5.m5.1.1.2.2"><times id="S4.p2.5.m5.1.1.2.2.1.cmml" xref="S4.p2.5.m5.1.1.2.2.1"></times><ci id="S4.p2.5.m5.1.1.2.2.2.cmml" xref="S4.p2.5.m5.1.1.2.2.2">𝛿</ci><ci id="S4.p2.5.m5.1.1.2.2.3.cmml" xref="S4.p2.5.m5.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p2.5.m5.1.1.2.3.cmml" xref="S4.p2.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S4.p2.5.m5.1.1.2.3.1.cmml" xref="S4.p2.5.m5.1.1.2.3">subscript</csymbol><ci id="S4.p2.5.m5.1.1.2.3.2.cmml" xref="S4.p2.5.m5.1.1.2.3.2">𝜌</ci><cn id="S4.p2.5.m5.1.1.2.3.3.cmml" type="integer" xref="S4.p2.5.m5.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p2.5.m5.1.1.3.cmml" xref="S4.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.p2.5.m5.1.1.3.1.cmml" xref="S4.p2.5.m5.1.1.3">subscript</csymbol><ci id="S4.p2.5.m5.1.1.3.2.cmml" xref="S4.p2.5.m5.1.1.3.2">ℳ</ci><ci id="S4.p2.5.m5.1.1.3.3.cmml" xref="S4.p2.5.m5.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">\delta\rho/\rho_{0}\approx\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≈ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="b=1)" class="ltx_math_unparsed" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><mrow id="S4.p2.6.m6.1b"><mi id="S4.p2.6.m6.1.1">b</mi><mo id="S4.p2.6.m6.1.2">=</mo><mn id="S4.p2.6.m6.1.3">1</mn><mo id="S4.p2.6.m6.1.4" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">b=1)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">italic_b = 1 )</annotation></semantics></math> on smaller scales (smaller <math alttext="\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.7.m7.1"><semantics id="S4.p2.7.m7.1a"><msub id="S4.p2.7.m7.1.1" xref="S4.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.7.m7.1.1.2" xref="S4.p2.7.m7.1.1.2.cmml">ℳ</mi><mi id="S4.p2.7.m7.1.1.3" xref="S4.p2.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.7.m7.1b"><apply id="S4.p2.7.m7.1.1.cmml" xref="S4.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.p2.7.m7.1.1.1.cmml" xref="S4.p2.7.m7.1.1">subscript</csymbol><ci id="S4.p2.7.m7.1.1.2.cmml" xref="S4.p2.7.m7.1.1.2">ℳ</ci><ci id="S4.p2.7.m7.1.1.3.cmml" xref="S4.p2.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m7.1c">\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m7.1d">caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S4.p2.8.m8.1"><semantics id="S4.p2.8.m8.1a"><mrow id="S4.p2.8.m8.1.1" xref="S4.p2.8.m8.1.1.cmml"><mrow id="S4.p2.8.m8.1.1.2" xref="S4.p2.8.m8.1.1.2.cmml"><mi id="S4.p2.8.m8.1.1.2.2" xref="S4.p2.8.m8.1.1.2.2.cmml">δ</mi><mo id="S4.p2.8.m8.1.1.2.1" xref="S4.p2.8.m8.1.1.2.1.cmml">⁢</mo><mi id="S4.p2.8.m8.1.1.2.3" xref="S4.p2.8.m8.1.1.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.8.m8.1.1.1" xref="S4.p2.8.m8.1.1.1.cmml">/</mo><msub id="S4.p2.8.m8.1.1.3" xref="S4.p2.8.m8.1.1.3.cmml"><mi id="S4.p2.8.m8.1.1.3.2" xref="S4.p2.8.m8.1.1.3.2.cmml">ρ</mi><mn id="S4.p2.8.m8.1.1.3.3" xref="S4.p2.8.m8.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.8.m8.1b"><apply id="S4.p2.8.m8.1.1.cmml" xref="S4.p2.8.m8.1.1"><divide id="S4.p2.8.m8.1.1.1.cmml" xref="S4.p2.8.m8.1.1.1"></divide><apply id="S4.p2.8.m8.1.1.2.cmml" xref="S4.p2.8.m8.1.1.2"><times id="S4.p2.8.m8.1.1.2.1.cmml" xref="S4.p2.8.m8.1.1.2.1"></times><ci id="S4.p2.8.m8.1.1.2.2.cmml" xref="S4.p2.8.m8.1.1.2.2">𝛿</ci><ci id="S4.p2.8.m8.1.1.2.3.cmml" xref="S4.p2.8.m8.1.1.2.3">𝜌</ci></apply><apply id="S4.p2.8.m8.1.1.3.cmml" xref="S4.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.p2.8.m8.1.1.3.1.cmml" xref="S4.p2.8.m8.1.1.3">subscript</csymbol><ci id="S4.p2.8.m8.1.1.3.2.cmml" xref="S4.p2.8.m8.1.1.3.2">𝜌</ci><cn id="S4.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S4.p2.8.m8.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m8.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m8.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>). This value of <math alttext="b" class="ltx_Math" display="inline" id="S4.p2.9.m9.1"><semantics id="S4.p2.9.m9.1a"><mi id="S4.p2.9.m9.1.1" xref="S4.p2.9.m9.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.p2.9.m9.1b"><ci id="S4.p2.9.m9.1.1.cmml" xref="S4.p2.9.m9.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.9.m9.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.p2.9.m9.1d">italic_b</annotation></semantics></math> is theorized to be a signature of when the driving mechanism (<math alttext="\bm{f}" class="ltx_Math" display="inline" id="S4.p2.10.m10.1"><semantics id="S4.p2.10.m10.1a"><mi id="S4.p2.10.m10.1.1" xref="S4.p2.10.m10.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S4.p2.10.m10.1b"><ci id="S4.p2.10.m10.1.1.cmml" xref="S4.p2.10.m10.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m10.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m10.1d">bold_italic_f</annotation></semantics></math> in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S2.E7" title="7 ‣ MHD plasma model ‣ 2.2 Numerical Simulation ‣ 2 Methods ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 7</span></a>) for the turbulent fluctuations is compressive. This means that deep within the cascade <math alttext="\delta\rho/\rho_{0}=\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.11.m11.1"><semantics id="S4.p2.11.m11.1a"><mrow id="S4.p2.11.m11.1.1" xref="S4.p2.11.m11.1.1.cmml"><mrow id="S4.p2.11.m11.1.1.2" xref="S4.p2.11.m11.1.1.2.cmml"><mrow id="S4.p2.11.m11.1.1.2.2" xref="S4.p2.11.m11.1.1.2.2.cmml"><mi id="S4.p2.11.m11.1.1.2.2.2" xref="S4.p2.11.m11.1.1.2.2.2.cmml">δ</mi><mo id="S4.p2.11.m11.1.1.2.2.1" xref="S4.p2.11.m11.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p2.11.m11.1.1.2.2.3" xref="S4.p2.11.m11.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.11.m11.1.1.2.1" xref="S4.p2.11.m11.1.1.2.1.cmml">/</mo><msub id="S4.p2.11.m11.1.1.2.3" xref="S4.p2.11.m11.1.1.2.3.cmml"><mi id="S4.p2.11.m11.1.1.2.3.2" xref="S4.p2.11.m11.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p2.11.m11.1.1.2.3.3" xref="S4.p2.11.m11.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p2.11.m11.1.1.1" xref="S4.p2.11.m11.1.1.1.cmml">=</mo><msub id="S4.p2.11.m11.1.1.3" xref="S4.p2.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.11.m11.1.1.3.2" xref="S4.p2.11.m11.1.1.3.2.cmml">ℳ</mi><mi id="S4.p2.11.m11.1.1.3.3" xref="S4.p2.11.m11.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.11.m11.1b"><apply id="S4.p2.11.m11.1.1.cmml" xref="S4.p2.11.m11.1.1"><eq id="S4.p2.11.m11.1.1.1.cmml" xref="S4.p2.11.m11.1.1.1"></eq><apply id="S4.p2.11.m11.1.1.2.cmml" xref="S4.p2.11.m11.1.1.2"><divide id="S4.p2.11.m11.1.1.2.1.cmml" xref="S4.p2.11.m11.1.1.2.1"></divide><apply id="S4.p2.11.m11.1.1.2.2.cmml" xref="S4.p2.11.m11.1.1.2.2"><times id="S4.p2.11.m11.1.1.2.2.1.cmml" xref="S4.p2.11.m11.1.1.2.2.1"></times><ci id="S4.p2.11.m11.1.1.2.2.2.cmml" xref="S4.p2.11.m11.1.1.2.2.2">𝛿</ci><ci id="S4.p2.11.m11.1.1.2.2.3.cmml" xref="S4.p2.11.m11.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p2.11.m11.1.1.2.3.cmml" xref="S4.p2.11.m11.1.1.2.3"><csymbol cd="ambiguous" id="S4.p2.11.m11.1.1.2.3.1.cmml" xref="S4.p2.11.m11.1.1.2.3">subscript</csymbol><ci id="S4.p2.11.m11.1.1.2.3.2.cmml" xref="S4.p2.11.m11.1.1.2.3.2">𝜌</ci><cn id="S4.p2.11.m11.1.1.2.3.3.cmml" type="integer" xref="S4.p2.11.m11.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p2.11.m11.1.1.3.cmml" xref="S4.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.p2.11.m11.1.1.3.1.cmml" xref="S4.p2.11.m11.1.1.3">subscript</csymbol><ci id="S4.p2.11.m11.1.1.3.2.cmml" xref="S4.p2.11.m11.1.1.3.2">ℳ</ci><ci id="S4.p2.11.m11.1.1.3.3.cmml" xref="S4.p2.11.m11.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m11.1c">\delta\rho/\rho_{0}=\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m11.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, regardless of the nature of <math alttext="\bm{f}" class="ltx_Math" display="inline" id="S4.p2.12.m12.1"><semantics id="S4.p2.12.m12.1a"><mi id="S4.p2.12.m12.1.1" xref="S4.p2.12.m12.1.1.cmml">𝒇</mi><annotation-xml encoding="MathML-Content" id="S4.p2.12.m12.1b"><ci id="S4.p2.12.m12.1.1.cmml" xref="S4.p2.12.m12.1.1">𝒇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m12.1c">\bm{f}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m12.1d">bold_italic_f</annotation></semantics></math> (a mix of compressible and incompressible modes in our simulation). Based on the compressible / incompressible mode decomposition for the simulation in <cite class="ltx_cite ltx_citemacro_citet">Beattie et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite>, on the scales that <math alttext="\delta\rho/\rho_{0}=\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.13.m13.1"><semantics id="S4.p2.13.m13.1a"><mrow id="S4.p2.13.m13.1.1" xref="S4.p2.13.m13.1.1.cmml"><mrow id="S4.p2.13.m13.1.1.2" xref="S4.p2.13.m13.1.1.2.cmml"><mrow id="S4.p2.13.m13.1.1.2.2" xref="S4.p2.13.m13.1.1.2.2.cmml"><mi id="S4.p2.13.m13.1.1.2.2.2" xref="S4.p2.13.m13.1.1.2.2.2.cmml">δ</mi><mo id="S4.p2.13.m13.1.1.2.2.1" xref="S4.p2.13.m13.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p2.13.m13.1.1.2.2.3" xref="S4.p2.13.m13.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p2.13.m13.1.1.2.1" xref="S4.p2.13.m13.1.1.2.1.cmml">/</mo><msub id="S4.p2.13.m13.1.1.2.3" xref="S4.p2.13.m13.1.1.2.3.cmml"><mi id="S4.p2.13.m13.1.1.2.3.2" xref="S4.p2.13.m13.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p2.13.m13.1.1.2.3.3" xref="S4.p2.13.m13.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p2.13.m13.1.1.1" 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demonstrates that even if the momentum modes are mostly incompressible, if one measures <math alttext="b" class="ltx_Math" display="inline" id="S4.p2.14.m14.1"><semantics id="S4.p2.14.m14.1a"><mi id="S4.p2.14.m14.1.1" xref="S4.p2.14.m14.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.p2.14.m14.1b"><ci id="S4.p2.14.m14.1.1.cmml" xref="S4.p2.14.m14.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.14.m14.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.p2.14.m14.1d">italic_b</annotation></semantics></math> deep in the cascade, where there is no driving source, one gets the compressible-driving relation <math alttext="\delta\rho/\rho_{0}=\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p2.15.m15.1"><semantics id="S4.p2.15.m15.1a"><mrow id="S4.p2.15.m15.1.1" xref="S4.p2.15.m15.1.1.cmml"><mrow id="S4.p2.15.m15.1.1.2" xref="S4.p2.15.m15.1.1.2.cmml"><mrow id="S4.p2.15.m15.1.1.2.2" xref="S4.p2.15.m15.1.1.2.2.cmml"><mi 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xref="S4.p2.15.m15.1.1.1"></eq><apply id="S4.p2.15.m15.1.1.2.cmml" xref="S4.p2.15.m15.1.1.2"><divide id="S4.p2.15.m15.1.1.2.1.cmml" xref="S4.p2.15.m15.1.1.2.1"></divide><apply id="S4.p2.15.m15.1.1.2.2.cmml" xref="S4.p2.15.m15.1.1.2.2"><times id="S4.p2.15.m15.1.1.2.2.1.cmml" xref="S4.p2.15.m15.1.1.2.2.1"></times><ci id="S4.p2.15.m15.1.1.2.2.2.cmml" xref="S4.p2.15.m15.1.1.2.2.2">𝛿</ci><ci id="S4.p2.15.m15.1.1.2.2.3.cmml" xref="S4.p2.15.m15.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p2.15.m15.1.1.2.3.cmml" xref="S4.p2.15.m15.1.1.2.3"><csymbol cd="ambiguous" id="S4.p2.15.m15.1.1.2.3.1.cmml" xref="S4.p2.15.m15.1.1.2.3">subscript</csymbol><ci id="S4.p2.15.m15.1.1.2.3.2.cmml" xref="S4.p2.15.m15.1.1.2.3.2">𝜌</ci><cn id="S4.p2.15.m15.1.1.2.3.3.cmml" type="integer" xref="S4.p2.15.m15.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p2.15.m15.1.1.3.cmml" xref="S4.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="S4.p2.15.m15.1.1.3.1.cmml" xref="S4.p2.15.m15.1.1.3">subscript</csymbol><ci id="S4.p2.15.m15.1.1.3.2.cmml" xref="S4.p2.15.m15.1.1.3.2">ℳ</ci><ci id="S4.p2.15.m15.1.1.3.3.cmml" xref="S4.p2.15.m15.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.15.m15.1c">\delta\rho/\rho_{0}=\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.15.m15.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.9">In this paper, we investigate the scaling of <math alttext="\delta\rho/\rho_{0}" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mrow id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml"><mi id="S4.p3.1.m1.1.1.2.2" xref="S4.p3.1.m1.1.1.2.2.cmml">δ</mi><mo id="S4.p3.1.m1.1.1.2.1" xref="S4.p3.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.p3.1.m1.1.1.2.3" xref="S4.p3.1.m1.1.1.2.3.cmml">ρ</mi></mrow><mo id="S4.p3.1.m1.1.1.1" xref="S4.p3.1.m1.1.1.1.cmml">/</mo><msub id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml"><mi id="S4.p3.1.m1.1.1.3.2" xref="S4.p3.1.m1.1.1.3.2.cmml">ρ</mi><mn id="S4.p3.1.m1.1.1.3.3" xref="S4.p3.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><divide id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1.1"></divide><apply id="S4.p3.1.m1.1.1.2.cmml" xref="S4.p3.1.m1.1.1.2"><times id="S4.p3.1.m1.1.1.2.1.cmml" xref="S4.p3.1.m1.1.1.2.1"></times><ci id="S4.p3.1.m1.1.1.2.2.cmml" xref="S4.p3.1.m1.1.1.2.2">𝛿</ci><ci id="S4.p3.1.m1.1.1.2.3.cmml" xref="S4.p3.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S4.p3.1.m1.1.1.3.cmml" xref="S4.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p3.1.m1.1.1.3.1.cmml" xref="S4.p3.1.m1.1.1.3">subscript</csymbol><ci id="S4.p3.1.m1.1.1.3.2.cmml" xref="S4.p3.1.m1.1.1.3.2">𝜌</ci><cn id="S4.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S4.p3.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">\delta\rho/\rho_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with the turbulent Mach number in high <math alttext="\beta" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mi id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">italic_β</annotation></semantics></math>, highly-compressible regime. This has been accomplished using a large number of MMS measurements (over 1200 intervals) in Earth’s magnetosheath and an unprecedentedly high-resolution <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S4.p3.3.m3.2"><semantics id="S4.p3.3.m3.2a"><mrow id="S4.p3.3.m3.2.2.1" xref="S4.p3.3.m3.2.2.2.cmml"><mn id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S4.p3.3.m3.2.2.1.2" xref="S4.p3.3.m3.2.2.2.cmml">,</mo></mpadded><msup id="S4.p3.3.m3.2.2.1.1" xref="S4.p3.3.m3.2.2.1.1.cmml"><mn id="S4.p3.3.m3.2.2.1.1.2" xref="S4.p3.3.m3.2.2.1.1.2.cmml">080</mn><mn id="S4.p3.3.m3.2.2.1.1.3" xref="S4.p3.3.m3.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.2b"><list id="S4.p3.3.m3.2.2.2.cmml" xref="S4.p3.3.m3.2.2.1"><cn id="S4.p3.3.m3.1.1.cmml" type="integer" xref="S4.p3.3.m3.1.1">10</cn><apply id="S4.p3.3.m3.2.2.1.1.cmml" xref="S4.p3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.p3.3.m3.2.2.1.1.1.cmml" xref="S4.p3.3.m3.2.2.1.1">superscript</csymbol><cn id="S4.p3.3.m3.2.2.1.1.2.cmml" type="integer" xref="S4.p3.3.m3.2.2.1.1.2">080</cn><cn id="S4.p3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.p3.3.m3.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.2c">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.2d">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, highly-compressible MHD simulation <cite class="ltx_cite ltx_citemacro_citep">(Beattie et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib4" title="">2024</a>)</cite>. Not only do both the <span class="ltx_text ltx_font_italic" id="S4.p3.9.1">in-situ</span> data as well as the simulation support the scaling <math alttext="\delta\rho/\rho_{0}\propto\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p3.4.m4.1"><semantics id="S4.p3.4.m4.1a"><mrow id="S4.p3.4.m4.1.1" xref="S4.p3.4.m4.1.1.cmml"><mrow id="S4.p3.4.m4.1.1.2" xref="S4.p3.4.m4.1.1.2.cmml"><mrow id="S4.p3.4.m4.1.1.2.2" xref="S4.p3.4.m4.1.1.2.2.cmml"><mi id="S4.p3.4.m4.1.1.2.2.2" xref="S4.p3.4.m4.1.1.2.2.2.cmml">δ</mi><mo id="S4.p3.4.m4.1.1.2.2.1" xref="S4.p3.4.m4.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p3.4.m4.1.1.2.2.3" xref="S4.p3.4.m4.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="S4.p3.4.m4.1.1.2.1" xref="S4.p3.4.m4.1.1.2.1.cmml">/</mo><msub id="S4.p3.4.m4.1.1.2.3" xref="S4.p3.4.m4.1.1.2.3.cmml"><mi id="S4.p3.4.m4.1.1.2.3.2" xref="S4.p3.4.m4.1.1.2.3.2.cmml">ρ</mi><mn id="S4.p3.4.m4.1.1.2.3.3" xref="S4.p3.4.m4.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p3.4.m4.1.1.1" xref="S4.p3.4.m4.1.1.1.cmml">∝</mo><msub id="S4.p3.4.m4.1.1.3" xref="S4.p3.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p3.4.m4.1.1.3.2" xref="S4.p3.4.m4.1.1.3.2.cmml">ℳ</mi><mi id="S4.p3.4.m4.1.1.3.3" xref="S4.p3.4.m4.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.4.m4.1b"><apply id="S4.p3.4.m4.1.1.cmml" xref="S4.p3.4.m4.1.1"><csymbol cd="latexml" id="S4.p3.4.m4.1.1.1.cmml" xref="S4.p3.4.m4.1.1.1">proportional-to</csymbol><apply id="S4.p3.4.m4.1.1.2.cmml" xref="S4.p3.4.m4.1.1.2"><divide id="S4.p3.4.m4.1.1.2.1.cmml" xref="S4.p3.4.m4.1.1.2.1"></divide><apply id="S4.p3.4.m4.1.1.2.2.cmml" xref="S4.p3.4.m4.1.1.2.2"><times id="S4.p3.4.m4.1.1.2.2.1.cmml" xref="S4.p3.4.m4.1.1.2.2.1"></times><ci id="S4.p3.4.m4.1.1.2.2.2.cmml" xref="S4.p3.4.m4.1.1.2.2.2">𝛿</ci><ci id="S4.p3.4.m4.1.1.2.2.3.cmml" xref="S4.p3.4.m4.1.1.2.2.3">𝜌</ci></apply><apply id="S4.p3.4.m4.1.1.2.3.cmml" xref="S4.p3.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S4.p3.4.m4.1.1.2.3.1.cmml" xref="S4.p3.4.m4.1.1.2.3">subscript</csymbol><ci id="S4.p3.4.m4.1.1.2.3.2.cmml" xref="S4.p3.4.m4.1.1.2.3.2">𝜌</ci><cn id="S4.p3.4.m4.1.1.2.3.3.cmml" type="integer" xref="S4.p3.4.m4.1.1.2.3.3">0</cn></apply></apply><apply id="S4.p3.4.m4.1.1.3.cmml" xref="S4.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.p3.4.m4.1.1.3.1.cmml" xref="S4.p3.4.m4.1.1.3">subscript</csymbol><ci id="S4.p3.4.m4.1.1.3.2.cmml" xref="S4.p3.4.m4.1.1.3.2">ℳ</ci><ci id="S4.p3.4.m4.1.1.3.3.cmml" xref="S4.p3.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m4.1c">\delta\rho/\rho_{0}\propto\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m4.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∝ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> predicted by the weakly incompressible theory in the presence of background inhomogeneities <cite class="ltx_cite ltx_citemacro_citep">(Bhattacharjee et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>, but they also agree within 1<math alttext="\sigma" class="ltx_Math" display="inline" id="S4.p3.5.m5.1"><semantics id="S4.p3.5.m5.1a"><mi id="S4.p3.5.m5.1.1" xref="S4.p3.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.5.m5.1b"><ci id="S4.p3.5.m5.1.1.cmml" xref="S4.p3.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m5.1d">italic_σ</annotation></semantics></math> with one another, even in proportionality constants. This might be due to the fact that both the magnetosheath plasma and the simulation used here possess abundant inhomogeneities, e.g., mass density filaments, voids, current sheets, sheared flows, and other coherent structures. The structures and strong variations in the plasma variables render the plasma sufficiently inhomogeneous such that linear <math alttext="\sim\mathcal{M}_{t}" class="ltx_Math" display="inline" id="S4.p3.6.m6.1"><semantics id="S4.p3.6.m6.1a"><mrow id="S4.p3.6.m6.1.1" xref="S4.p3.6.m6.1.1.cmml"><mi id="S4.p3.6.m6.1.1.2" xref="S4.p3.6.m6.1.1.2.cmml"></mi><mo id="S4.p3.6.m6.1.1.1" xref="S4.p3.6.m6.1.1.1.cmml">∼</mo><msub id="S4.p3.6.m6.1.1.3" xref="S4.p3.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p3.6.m6.1.1.3.2" xref="S4.p3.6.m6.1.1.3.2.cmml">ℳ</mi><mi id="S4.p3.6.m6.1.1.3.3" xref="S4.p3.6.m6.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.6.m6.1b"><apply id="S4.p3.6.m6.1.1.cmml" xref="S4.p3.6.m6.1.1"><csymbol cd="latexml" id="S4.p3.6.m6.1.1.1.cmml" xref="S4.p3.6.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.p3.6.m6.1.1.2.cmml" xref="S4.p3.6.m6.1.1.2">absent</csymbol><apply id="S4.p3.6.m6.1.1.3.cmml" xref="S4.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.p3.6.m6.1.1.3.1.cmml" xref="S4.p3.6.m6.1.1.3">subscript</csymbol><ci id="S4.p3.6.m6.1.1.3.2.cmml" xref="S4.p3.6.m6.1.1.3.2">ℳ</ci><ci id="S4.p3.6.m6.1.1.3.3.cmml" xref="S4.p3.6.m6.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m6.1c">\sim\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m6.1d">∼ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> scaling becomes valid instead of quadratic <math alttext="\sim\mathcal{M}_{t}^{2}" class="ltx_Math" display="inline" id="S4.p3.7.m7.1"><semantics id="S4.p3.7.m7.1a"><mrow id="S4.p3.7.m7.1.1" xref="S4.p3.7.m7.1.1.cmml"><mi id="S4.p3.7.m7.1.1.2" xref="S4.p3.7.m7.1.1.2.cmml"></mi><mo id="S4.p3.7.m7.1.1.1" xref="S4.p3.7.m7.1.1.1.cmml">∼</mo><msubsup id="S4.p3.7.m7.1.1.3" xref="S4.p3.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p3.7.m7.1.1.3.2.2" xref="S4.p3.7.m7.1.1.3.2.2.cmml">ℳ</mi><mi id="S4.p3.7.m7.1.1.3.2.3" xref="S4.p3.7.m7.1.1.3.2.3.cmml">t</mi><mn id="S4.p3.7.m7.1.1.3.3" xref="S4.p3.7.m7.1.1.3.3.cmml">2</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.7.m7.1b"><apply id="S4.p3.7.m7.1.1.cmml" xref="S4.p3.7.m7.1.1"><csymbol cd="latexml" id="S4.p3.7.m7.1.1.1.cmml" xref="S4.p3.7.m7.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.p3.7.m7.1.1.2.cmml" xref="S4.p3.7.m7.1.1.2">absent</csymbol><apply id="S4.p3.7.m7.1.1.3.cmml" xref="S4.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.p3.7.m7.1.1.3.1.cmml" xref="S4.p3.7.m7.1.1.3">superscript</csymbol><apply id="S4.p3.7.m7.1.1.3.2.cmml" xref="S4.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.p3.7.m7.1.1.3.2.1.cmml" xref="S4.p3.7.m7.1.1.3">subscript</csymbol><ci id="S4.p3.7.m7.1.1.3.2.2.cmml" xref="S4.p3.7.m7.1.1.3.2.2">ℳ</ci><ci id="S4.p3.7.m7.1.1.3.2.3.cmml" xref="S4.p3.7.m7.1.1.3.2.3">𝑡</ci></apply><cn id="S4.p3.7.m7.1.1.3.3.cmml" type="integer" xref="S4.p3.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.7.m7.1c">\sim\mathcal{M}_{t}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.7.m7.1d">∼ caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> scaling, as we show in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.F3" title="Figure 3 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 3</span></a>. This demonstrates both the robustness of the <cite class="ltx_cite ltx_citemacro_citet">Bhattacharjee et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite> model for a variety of <math alttext="\beta" class="ltx_Math" display="inline" id="S4.p3.8.m8.1"><semantics id="S4.p3.8.m8.1a"><mi id="S4.p3.8.m8.1.1" xref="S4.p3.8.m8.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.p3.8.m8.1b"><ci id="S4.p3.8.m8.1.1.cmml" xref="S4.p3.8.m8.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.8.m8.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.p3.8.m8.1d">italic_β</annotation></semantics></math> plasmas, and the broad applicability of the <math alttext="10,\!080^{3}" class="ltx_Math" display="inline" id="S4.p3.9.m9.2"><semantics id="S4.p3.9.m9.2a"><mrow id="S4.p3.9.m9.2.2.1" xref="S4.p3.9.m9.2.2.2.cmml"><mn id="S4.p3.9.m9.1.1" xref="S4.p3.9.m9.1.1.cmml">10</mn><mpadded width="0.275em"><mo id="S4.p3.9.m9.2.2.1.2" xref="S4.p3.9.m9.2.2.2.cmml">,</mo></mpadded><msup id="S4.p3.9.m9.2.2.1.1" xref="S4.p3.9.m9.2.2.1.1.cmml"><mn id="S4.p3.9.m9.2.2.1.1.2" xref="S4.p3.9.m9.2.2.1.1.2.cmml">080</mn><mn id="S4.p3.9.m9.2.2.1.1.3" xref="S4.p3.9.m9.2.2.1.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.9.m9.2b"><list id="S4.p3.9.m9.2.2.2.cmml" xref="S4.p3.9.m9.2.2.1"><cn id="S4.p3.9.m9.1.1.cmml" type="integer" xref="S4.p3.9.m9.1.1">10</cn><apply id="S4.p3.9.m9.2.2.1.1.cmml" xref="S4.p3.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S4.p3.9.m9.2.2.1.1.1.cmml" xref="S4.p3.9.m9.2.2.1.1">superscript</csymbol><cn id="S4.p3.9.m9.2.2.1.1.2.cmml" type="integer" xref="S4.p3.9.m9.2.2.1.1.2">080</cn><cn id="S4.p3.9.m9.2.2.1.1.3.cmml" type="integer" xref="S4.p3.9.m9.2.2.1.1.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.9.m9.2c">10,\!080^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.9.m9.2d">10 , 080 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> MHD dataset to both space and astrophysical plasmas.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_font_bold ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">The data used in this analysis are Level 2 FIELDS and FPI data products, in cooperation with the instrument teams and in accordance with their guidelines. All MMS data are available at <a class="ltx_ref ltx_href" href="https://lasp.colorado.edu/mms/sdc/" title="">https://lasp.colorado.edu/mms/sdc/</a>. This research was supported in part by the NASA Heliospheric GI Grant No. 80NSSC21K0739 and NASA Grant No. 80NSSC21K1458. J. R. B. acknowledges the high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Center for Supercomputing grant pn76gi pr73fi and pn76ga, and Compute Ontario and the Digital Research Alliance of Canada (alliancecan.ca) compute allocation rrg-ripperda. J. .R. B. and A. B further acknowledge the support from NSF Award 2206756.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Adhikari et al. (2023)</span> <span class="ltx_bibblock"> Adhikari, L., Zank, G. P., Wang, B., et al. 2023, The Astrophysical Journal, 953, 44, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/1538-4357/acde57" title=""><span class="ltx_ref ltx_nolink">10.3847/1538-4357/acde57</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bavassano & Bruno (1995)</span> <span class="ltx_bibblock"> Bavassano, B., & Bruno, R. 1995, Journal of Geophysical Research: Space Physics, 100, 9475, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1029/94JA03048" title=""><span class="ltx_ref ltx_nolink">10.1029/94JA03048</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beattie & Federrath (2020)</span> <span class="ltx_bibblock"> Beattie, J. R., & Federrath, C. 2020, The Monthly Notices of The Royal Astronomical Society, 492, 668, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stz3377" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stz3377</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beattie et al. (2024)</span> <span class="ltx_bibblock"> Beattie, J. R., Federrath, C., Klessen, R. S., Cielo, S., & Bhattacharjee, A. 2024, arXiv e-prints, arXiv:2405.16626, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.48550/arXiv.2405.16626" title=""><span class="ltx_ref ltx_nolink">10.48550/arXiv.2405.16626</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beattie et al. (2020)</span> <span class="ltx_bibblock"> Beattie, J. R., Federrath, C., & Seta, A. 2020, The Monthly Notices of The Royal Astronomical Society, 498, 1593, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/staa2257" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/staa2257</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beattie et al. (2021)</span> <span class="ltx_bibblock"> Beattie, J. R., Mocz, P., Federrath, C., & Klessen, R. S. 2021, The Monthly Notices of The Royal Astronomical Society, 504, 4354, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stab1037" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stab1037</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Beattie et al. (2022)</span> <span class="ltx_bibblock"> —. 2022, The Monthly Notices of The Royal Astronomical Society, 517, 5003, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stac3005" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stac3005</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Behnel et al. (2011)</span> <span class="ltx_bibblock"> Behnel, S., Bradshaw, R., Citro, C., et al. 2011, Computing in Science & Engineering, 13, 31 </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bhattacharjee et al. (1999)</span> <span class="ltx_bibblock"> Bhattacharjee, A., Ng, C. S., Ghosh, S., & Goldstein, M. L. 1999, Journal of Geophysical Research: Space Physics, 104, 24835, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1029/1999JA900327" title=""><span class="ltx_ref ltx_nolink">10.1029/1999JA900327</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bhattacharjee et al. (1998)</span> <span class="ltx_bibblock"> Bhattacharjee, A., Ng, C. S., & Spangler, S. R. 1998, The Astrophysical Journal, 494, 409, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1086/305184" title=""><span class="ltx_ref ltx_nolink">10.1086/305184</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bouchut et al. (2010)</span> <span class="ltx_bibblock"> Bouchut, F., Klingenberg, C., & Waagan, K. 2010, Numerische Mathematik, 115, 647, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s00211-010-0289-4" title=""><span class="ltx_ref ltx_nolink">10.1007/s00211-010-0289-4</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Brandenburg & Lazarian (2013)</span> <span class="ltx_bibblock"> Brandenburg, A., & Lazarian, A. 2013, Space Science Reviews, 178, 163, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-013-0009-3" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-013-0009-3</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Burch et al. (2016)</span> <span class="ltx_bibblock"> Burch, J. L., Moore, T. E., Torbert, R. B., & Giles, B. L. 2016, Space Science Reviews, 199, 5, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-015-0164-9" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-015-0164-9</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Burch et al. (2016)</span> <span class="ltx_bibblock"> Burch, J. L., Torbert, R. B., Phan, T. D., et al. 2016, Science, 352, aaf2939, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1126/science.aaf2939" title=""><span class="ltx_ref ltx_nolink">10.1126/science.aaf2939</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Burkhart & Lazarian (2012)</span> <span class="ltx_bibblock"> Burkhart, B., & Lazarian, A. 2012, The Astrophysical Journal, 755, L19, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1088/2041-8205/755/1/l19" title=""><span class="ltx_ref ltx_nolink">10.1088/2041-8205/755/1/l19</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Childs et al. (2012)</span> <span class="ltx_bibblock"> Childs, H., Brugger, E., Whitlock, B., et al. 2012, in High Performance Visualization–Enabling Extreme-Scale Scientific Insight (Taylor & Francis), 357–372 </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Cuesta et al. (2023)</span> <span class="ltx_bibblock"> Cuesta, M. E., Chhiber, R., Fu, X., et al. 2023, The Astrophysical Journal Letters, 949, L19, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/2041-8213/acd4c2" title=""><span class="ltx_ref ltx_nolink">10.3847/2041-8213/acd4c2</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Dhawalikar et al. (2022)</span> <span class="ltx_bibblock"> Dhawalikar, S., Federrath, C., Davidovits, S., et al. 2022, MNRAS, 514, 1782, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stac1480" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stac1480</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Du et al. (2023)</span> <span class="ltx_bibblock"> Du, S., Li, H., Gan, Z., & Fu, X. 2023, The Astrophysical Journal, 946, 74, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/1538-4357/acc10b" title=""><span class="ltx_ref ltx_nolink">10.3847/1538-4357/acc10b</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Dubey et al. (2008)</span> <span class="ltx_bibblock"> Dubey, A., Fisher, R., Graziani, C., et al. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 385, Numerical Modeling of Space Plasma Flows, ed. N. V. Pogorelov, E. Audit, & G. P. Zank, 145 </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Elmegreen & Scalo (2004)</span> <span class="ltx_bibblock"> Elmegreen, B. G., & Scalo, J. 2004, ARA&A, 42, 211, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1146/annurev.astro.41.011802.094859" title=""><span class="ltx_ref ltx_nolink">10.1146/annurev.astro.41.011802.094859</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Eswaran & Pope (1988)</span> <span class="ltx_bibblock"> Eswaran, V., & Pope, S. B. 1988, Computers and Fluids, 16, 257 </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Federrath (2013)</span> <span class="ltx_bibblock"> Federrath, C. 2013, The Monthly Notices of The Royal Astronomical Society, 436, 1245, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stt1644" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stt1644</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Federrath et al. (2021)</span> <span class="ltx_bibblock"> Federrath, C., Klessen, R. S., Iapichino, L., & Beattie, J. R. 2021, Nature Astronomy, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1038/s41550-020-01282-z" title=""><span class="ltx_ref ltx_nolink">10.1038/s41550-020-01282-z</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Federrath et al. (2010)</span> <span class="ltx_bibblock"> Federrath, C., Roman-Duval, J., Klessen, R., Schmidt, W., & Mac Low, M. M. 2010, Astronomy and Astrophysics, 512, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1051/0004-6361/200912437" title=""><span class="ltx_ref ltx_nolink">10.1051/0004-6361/200912437</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Federrath et al. (2022)</span> <span class="ltx_bibblock"> Federrath, C., Roman-Duval, J., Klessen, R. S., Schmidt, W., & Mac Low, M. M. 2022, TG: Turbulence Generator, Astrophysics Source Code Library, record ascl:2204.001. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_href" href="http://ascl.net/2204.001" title=""><span class="ltx_ref ltx_nolink">http://ascl.net/2204.001</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Federrath et al. (2016)</span> <span class="ltx_bibblock"> Federrath, C., Rathborne, J. M., Longmore, S. N., et al. 2016, The Astrophysical Journal, 832, 143, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/0004-637X/832/2/143" title=""><span class="ltx_ref ltx_nolink">10.3847/0004-637X/832/2/143</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Ferrière (2020)</span> <span class="ltx_bibblock"> Ferrière, K. 2020, Plasma Physics and Controlled Fusion, 62, 014014, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1088/1361-6587/ab49eb" title=""><span class="ltx_ref ltx_nolink">10.1088/1361-6587/ab49eb</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Foreman-Mackey (2016)</span> <span class="ltx_bibblock"> Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.21105/joss.00024" title=""><span class="ltx_ref ltx_nolink">10.21105/joss.00024</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Foreman-Mackey et al. (2013)</span> <span class="ltx_bibblock"> Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1086/670067" title=""><span class="ltx_ref ltx_nolink">10.1086/670067</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Fox et al. (2016)</span> <span class="ltx_bibblock"> Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Science Reviews, 204, 7, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-015-0211-6" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-015-0211-6</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Fryxell et al. (2000)</span> <span class="ltx_bibblock"> Fryxell, B., Olson, K., Ricker, P., et al. 2000, The Astrophysical Journal Supplement, 131, 273, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1086/317361" title=""><span class="ltx_ref ltx_nolink">10.1086/317361</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gerrard et al. (2023)</span> <span class="ltx_bibblock"> Gerrard, I. A., Federrath, C., Pingel, N. M., et al. 2023, MNRAS, 526, 982, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stad2718" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stad2718</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib34"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gershman et al. (2018)</span> <span class="ltx_bibblock"> Gershman, D. J., Vinas, A. F., Dorelli, J. C., et al. 2018, Phys. Plasmas, 25, 022303, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1063/1.5009158" title=""><span class="ltx_ref ltx_nolink">10.1063/1.5009158</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib35"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Harris et al. (2020)</span> <span class="ltx_bibblock"> Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1038/s41586-020-2649-2" title=""><span class="ltx_ref ltx_nolink">10.1038/s41586-020-2649-2</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib36"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hopkins (2013)</span> <span class="ltx_bibblock"> Hopkins, P. F. 2013, The Monthly Notices of The Royal Astronomical Society, 430, 1880, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stt010" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stt010</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib37"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hunana & Zank (2010)</span> <span class="ltx_bibblock"> Hunana, P., & Zank, G. P. 2010, The Astrophysical Journal, 718, 148, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1088/0004-637X/718/1/148" title=""><span class="ltx_ref ltx_nolink">10.1088/0004-637X/718/1/148</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib38"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Hunter (2007)</span> <span class="ltx_bibblock"> Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1109/MCSE.2007.55" title=""><span class="ltx_ref ltx_nolink">10.1109/MCSE.2007.55</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib39"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Joblib Development Team (2020)</span> <span class="ltx_bibblock"> Joblib Development Team. 2020, Joblib: running Python functions as pipeline jobs. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_url ltx_font_typewriter" href="https://joblib.readthedocs.io/" title="">https://joblib.readthedocs.io/</a> </span> </li> <li class="ltx_bibitem" id="bib.bib40"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Klainerman & Majda (1981)</span> <span class="ltx_bibblock"> Klainerman, S., & Majda, A. 1981, Communications on Pure and Applied Mathematics, 34, 481, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1002/cpa.3160340405" title=""><span class="ltx_ref ltx_nolink">10.1002/cpa.3160340405</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib41"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Klainerman & Majda (1982)</span> <span class="ltx_bibblock"> —. 1982, Communications on Pure and Applied Mathematics, 35, 629, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1002/cpa.3160350503" title=""><span class="ltx_ref ltx_nolink">10.1002/cpa.3160350503</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib42"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kowal et al. (2007)</span> <span class="ltx_bibblock"> Kowal, G., Lazarian, A., & Beresnyak, A. 2007, The Astrophysical Journal, 658, 423, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1086/511515" title=""><span class="ltx_ref ltx_nolink">10.1086/511515</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib43"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kriel et al. (2022)</span> <span class="ltx_bibblock"> Kriel, N., Beattie, J. R., Seta, A., & Federrath, C. 2022, The Monthly Notices of The Royal Astronomical Society, 513, 2457, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stac969" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stac969</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib44"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Krumholz (2015)</span> <span class="ltx_bibblock"> Krumholz, M. R. 2015, ArXiv e-prints, arXiv:1511.03457. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_href" href="https://arxiv.org/abs/1511.03457" title=""><span class="ltx_ref ltx_nolink">https://arxiv.org/abs/1511.03457</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib45"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kunz et al. (2022)</span> <span class="ltx_bibblock"> Kunz, M. W., Jones, T. W., & Zhuravleva, I. 2022, in Handbook of X-ray and Gamma-ray Astrophysics, ed. C. Bambi & A. Sangangelo, 56, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/978-981-16-4544-0_125-1" title=""><span class="ltx_ref ltx_nolink">10.1007/978-981-16-4544-0_125-1</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib46"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Lam et al. (2015)</span> <span class="ltx_bibblock"> Lam, S. K., Pitrou, A., & Seibert, S. 2015, in Proc. Second Workshop on the LLVM Compiler Infrastructure in HPC, 1–6, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1145/2833157.2833162" title=""><span class="ltx_ref ltx_nolink">10.1145/2833157.2833162</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib47"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Malvadi Shivakumar & Federrath (2023)</span> <span class="ltx_bibblock"> Malvadi Shivakumar, L., & Federrath, C. 2023, arXiv e-prints, arXiv:2311.10350, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.48550/arXiv.2311.10350" title=""><span class="ltx_ref ltx_nolink">10.48550/arXiv.2311.10350</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib48"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Matthaeus & Brown (1988)</span> <span class="ltx_bibblock"> Matthaeus, W. H., & Brown, M. R. 1988, The Physics of Fluids, 31, 3634, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1063/1.866880" title=""><span class="ltx_ref ltx_nolink">10.1063/1.866880</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib49"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Matthaeus et al. (1991)</span> <span class="ltx_bibblock"> Matthaeus, W. H., Klein, L. W., Ghosh, S., & Brown, M. R. 1991, Journal of Geophysical Research: Space Physics, 96, 5421, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1029/90JA02609" title=""><span class="ltx_ref ltx_nolink">10.1029/90JA02609</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib50"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Matthaeus & Velli (2011)</span> <span class="ltx_bibblock"> Matthaeus, W. H., & Velli, M. 2011, Space Science Reviews, 160, 145, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-011-9793-9" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-011-9793-9</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib51"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Menon et al. (2021)</span> <span class="ltx_bibblock"> Menon, S. H., Federrath, C., Klaassen, P., Kuiper, R., & Reiter, M. 2021, The Monthly Notices of The Royal Astronomical Society, 500, 1721, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/staa3271" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/staa3271</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib52"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Menon et al. (2020)</span> <span class="ltx_bibblock"> Menon, S. H., Federrath, C., & Kuiper, R. 2020, The Monthly Notices of The Royal Astronomical Society, 493, 4643, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/staa580" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/staa580</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib53"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Mohapatra et al. (2021)</span> <span class="ltx_bibblock"> Mohapatra, R., Federrath, C., & Sharma, P. 2021, The Monthly Notices of The Royal Astronomical Society, 500, 5072, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/staa3564" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/staa3564</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib54"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Mohapatra & Sharma (2019)</span> <span class="ltx_bibblock"> Mohapatra, R., & Sharma, P. 2019, MNRAS, 484, 4881, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stz328" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stz328</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib55"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Molina et al. (2012)</span> <span class="ltx_bibblock"> Molina, F. Z., Glover, S. C. O., Federrath, C., & Klessen, R. S. 2012, The Monthly Notices of The Royal Astronomical Society, 423, 2680, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1111/j.1365-2966.2012.21075.x" title=""><span class="ltx_ref ltx_nolink">10.1111/j.1365-2966.2012.21075.x</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib56"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Nolan et al. (2015)</span> <span class="ltx_bibblock"> Nolan, C. A., Federrath, C., & Sutherland, R. S. 2015, The Monthly Notices of The Royal Astronomical Society, 451, 1380, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stv1030" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stv1030</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib57"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Oliphant (2006)</span> <span class="ltx_bibblock"> Oliphant, T. 2006, NumPy: A guide to NumPy, USA: Trelgol Publishing. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_url ltx_font_typewriter" href="http://www.numpy.org/" title="">http://www.numpy.org/</a> </span> </li> <li class="ltx_bibitem" id="bib.bib58"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Padoan et al. (1997)</span> <span class="ltx_bibblock"> Padoan, P., Nordlund, P., & Jones, B. J. T. 1997, Commmunications of the Konkoly Observatory Hungary, 100, 341 </span> </li> <li class="ltx_bibitem" id="bib.bib59"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Pan et al. (2019)</span> <span class="ltx_bibblock"> Pan, L., Padoan, P., & Nordlund, Å. 2019, The Astrophysical Journal, 881, 155, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/1538-4357/ab2ed6" title=""><span class="ltx_ref ltx_nolink">10.3847/1538-4357/ab2ed6</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib60"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">pandas development team (2020)</span> <span class="ltx_bibblock"> pandas development team, T. 2020, pandas-dev/pandas: Pandas, latest, Zenodo, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.5281/zenodo.3509134" title=""><span class="ltx_ref ltx_nolink">10.5281/zenodo.3509134</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib61"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Passot & Vázquez-Semadeni (1998)</span> <span class="ltx_bibblock"> Passot, T., & Vázquez-Semadeni, E. 1998, Phys. Rev. E, 58, 4501, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1103/PhysRevE.58.4501" title=""><span class="ltx_ref ltx_nolink">10.1103/PhysRevE.58.4501</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib62"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Pollock et al. (2016)</span> <span class="ltx_bibblock"> Pollock, C., Moore, T., Jacques, A., et al. 2016, Space Science Reviews, 199, 331, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-016-0245-4" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-016-0245-4</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib63"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Price et al. (2011)</span> <span class="ltx_bibblock"> Price, D. J., Federrath, C., & Brunt, C. M. 2011, The Astrophysical Journal Letters, 727, 1380, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1088/2041-8205/727/1/L21" title=""><span class="ltx_ref ltx_nolink">10.1088/2041-8205/727/1/L21</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib64"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Rincon (2019)</span> <span class="ltx_bibblock"> Rincon, F. 2019, Journal of Plasma Physics, 85, 205850401, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1017/S0022377819000539" title=""><span class="ltx_ref ltx_nolink">10.1017/S0022377819000539</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib65"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Russell et al. (2016)</span> <span class="ltx_bibblock"> Russell, C. T., Anderson, B. J., Baumjohann, W., et al. 2016, Space Science Reviews, 199, 189, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-014-0057-3" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-014-0057-3</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib66"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Sahraoui et al. (2020)</span> <span class="ltx_bibblock"> Sahraoui, F., Hadid, L., & Huang, S. 2020, Reviews of Modern Plasma Physics, 4, 4, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s41614-020-0040-2" title=""><span class="ltx_ref ltx_nolink">10.1007/s41614-020-0040-2</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib67"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Schekochihin et al. (2004)</span> <span class="ltx_bibblock"> Schekochihin, A. A., Cowley, S. C., Taylor, S. F., Maron, J. L., & McWilliams, J. C. 2004, The Astrophysical Journal, 612, 276, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1086/422547" title=""><span class="ltx_ref ltx_nolink">10.1086/422547</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib68"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Schmidt et al. (2009)</span> <span class="ltx_bibblock"> Schmidt, W., Federrath, C., Hupp, M., Kern, S., & Niemeyer, J. C. 2009, Astronomy and Astrophysics, 494, 127, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1051/0004-6361:200809967" title=""><span class="ltx_ref ltx_nolink">10.1051/0004-6361:200809967</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib69"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Sharda et al. (2021)</span> <span class="ltx_bibblock"> Sharda, P., Menon, S. H., Federrath, C., et al. 2021, The Monthly Notices of The Royal Astronomical Society, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stab3048" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stab3048</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib70"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Shebalin & Montgomery (1988)</span> <span class="ltx_bibblock"> Shebalin, J. V., & Montgomery, D. 1988, Journal of Plasma Physics, 39, 339–367, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1017/S0022377800013076" title=""><span class="ltx_ref ltx_nolink">10.1017/S0022377800013076</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib71"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Squire & Hopkins (2017)</span> <span class="ltx_bibblock"> Squire, J., & Hopkins, P. F. 2017, The Monthly Notices of The Royal Astronomical Society, 471, 3753, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1093/mnras/stx1817" title=""><span class="ltx_ref ltx_nolink">10.1093/mnras/stx1817</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib72"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Stawarz et al. (2019)</span> <span class="ltx_bibblock"> Stawarz, J. E., Eastwood, J. P., Phan, T. D., et al. 2019, The Astrophysical Journal Letters, 877, L37, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/2041-8213/ab21c8" title=""><span class="ltx_ref ltx_nolink">10.3847/2041-8213/ab21c8</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib73"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Stroustrup (2013)</span> <span class="ltx_bibblock"> Stroustrup, B. 2013, The C++ Programming Language, 4th edn. (Addison-Wesley Professional) </span> </li> <li class="ltx_bibitem" id="bib.bib74"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Torbert et al. (2016)</span> <span class="ltx_bibblock"> Torbert, R. B., Russell, C. T., Magnes, W., et al. 2016, Space Science Reviews, 199, 105, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s11214-014-0109-8" title=""><span class="ltx_ref ltx_nolink">10.1007/s11214-014-0109-8</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib75"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Tu & Marsch (1994)</span> <span class="ltx_bibblock"> Tu, C. Y., & Marsch, E. 1994, Journal of Geophysical Research: Space Physics, 99, 21481, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1029/94JA00843" title=""><span class="ltx_ref ltx_nolink">10.1029/94JA00843</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib76"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Usmanov et al. (2014)</span> <span class="ltx_bibblock"> Usmanov, A. V., Goldstein, M. L., & Matthaeus, W. H. 2014, Astrophys. J., 788, 43. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_url ltx_font_typewriter" href="http://stacks.iop.org/0004-637X/788/i=1/a=43" title="">http://stacks.iop.org/0004-637X/788/i=1/a=43</a> </span> </li> <li class="ltx_bibitem" id="bib.bib77"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">van der Velden (2020)</span> <span class="ltx_bibblock"> van der Velden, E. 2020, The Journal of Open Source Software, 5, 2004, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.21105/joss.02004" title=""><span class="ltx_ref ltx_nolink">10.21105/joss.02004</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib78"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">van der Walt et al. (2014)</span> <span class="ltx_bibblock"> van der Walt, S., Schönberger, J. L., Nunez-Iglesias, J., et al. 2014, PeerJ, 2, e453, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.7717/peerj.453" title=""><span class="ltx_ref ltx_nolink">10.7717/peerj.453</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib79"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Verscharen et al. (2019)</span> <span class="ltx_bibblock"> Verscharen, D., Klein, K. G., & Maruca, B. A. 2019, Living Reviews in Solar Physics, 16, 5, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1007/s41116-019-0021-0" title=""><span class="ltx_ref ltx_nolink">10.1007/s41116-019-0021-0</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib80"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Virtanen et al. (2020)</span> <span class="ltx_bibblock"> Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: <a class="ltx_ref ltx_href" href="http://doi.org/https://doi.org/10.1038/s41592-019-0686-2" title=""><span class="ltx_ref ltx_nolink">https://doi.org/10.1038/s41592-019-0686-2</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib81"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Waagan et al. (2011)</span> <span class="ltx_bibblock"> Waagan, K., Federrath, C., & Klingenberg, C. 2011, Journal of Computational Physics, 230, 3331, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1016/j.jcp.2011.01.026" title=""><span class="ltx_ref ltx_nolink">10.1016/j.jcp.2011.01.026</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib82"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zank & Matthaeus (1993)</span> <span class="ltx_bibblock"> Zank, G. P., & Matthaeus, W. H. 1993, Physics of Fluids A: Fluid Dynamics, 5, 257, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.1063/1.858780" title=""><span class="ltx_ref ltx_nolink">10.1063/1.858780</span></a> </span> </li> <li class="ltx_bibitem" id="bib.bib83"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zhao et al. (2025)</span> <span class="ltx_bibblock"> Zhao, L.-L., Silwal, A., Zhu, X., Li, H., & Zank, G. P. 2025, The Astrophysical Journal Letters, 979, L4, doi: <a class="ltx_ref ltx_href" href="http://doi.org/10.3847/2041-8213/ada3d8" title=""><span class="ltx_ref ltx_nolink">10.3847/2041-8213/ada3d8</span></a> </span> </li> </ul> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Maximum Likelihood Fits</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.5">We fit the simple model <math alttext="\delta\rho/\rho_{0}=\theta_{1}\mathcal{M}_{t}^{\theta_{0}}\iff\log_{10}\delta% \rho/\rho_{0}=\theta_{0}\log_{10}\mathcal{M}_{t}+\log_{10}\theta_{1}" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><mrow id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mrow id="A1.p1.1.m1.1.1.2" xref="A1.p1.1.m1.1.1.2.cmml"><mrow id="A1.p1.1.m1.1.1.2.2" xref="A1.p1.1.m1.1.1.2.2.cmml"><mrow id="A1.p1.1.m1.1.1.2.2.2" xref="A1.p1.1.m1.1.1.2.2.2.cmml"><mi id="A1.p1.1.m1.1.1.2.2.2.2" xref="A1.p1.1.m1.1.1.2.2.2.2.cmml">δ</mi><mo id="A1.p1.1.m1.1.1.2.2.2.1" xref="A1.p1.1.m1.1.1.2.2.2.1.cmml">⁢</mo><mi id="A1.p1.1.m1.1.1.2.2.2.3" xref="A1.p1.1.m1.1.1.2.2.2.3.cmml">ρ</mi></mrow><mo id="A1.p1.1.m1.1.1.2.2.1" xref="A1.p1.1.m1.1.1.2.2.1.cmml">/</mo><msub id="A1.p1.1.m1.1.1.2.2.3" xref="A1.p1.1.m1.1.1.2.2.3.cmml"><mi id="A1.p1.1.m1.1.1.2.2.3.2" xref="A1.p1.1.m1.1.1.2.2.3.2.cmml">ρ</mi><mn id="A1.p1.1.m1.1.1.2.2.3.3" xref="A1.p1.1.m1.1.1.2.2.3.3.cmml">0</mn></msub></mrow><mo id="A1.p1.1.m1.1.1.2.1" xref="A1.p1.1.m1.1.1.2.1.cmml">=</mo><mrow id="A1.p1.1.m1.1.1.2.3" xref="A1.p1.1.m1.1.1.2.3.cmml"><msub id="A1.p1.1.m1.1.1.2.3.2" xref="A1.p1.1.m1.1.1.2.3.2.cmml"><mi id="A1.p1.1.m1.1.1.2.3.2.2" xref="A1.p1.1.m1.1.1.2.3.2.2.cmml">θ</mi><mn id="A1.p1.1.m1.1.1.2.3.2.3" xref="A1.p1.1.m1.1.1.2.3.2.3.cmml">1</mn></msub><mo id="A1.p1.1.m1.1.1.2.3.1" xref="A1.p1.1.m1.1.1.2.3.1.cmml">⁢</mo><msubsup id="A1.p1.1.m1.1.1.2.3.3" xref="A1.p1.1.m1.1.1.2.3.3.cmml"><mi 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id="A1.p1.1.m1.1.1.3.3.3.cmml" xref="A1.p1.1.m1.1.1.3.3.3"><apply id="A1.p1.1.m1.1.1.3.3.3.1.cmml" xref="A1.p1.1.m1.1.1.3.3.3.1"><csymbol cd="ambiguous" id="A1.p1.1.m1.1.1.3.3.3.1.1.cmml" xref="A1.p1.1.m1.1.1.3.3.3.1">subscript</csymbol><log id="A1.p1.1.m1.1.1.3.3.3.1.2.cmml" xref="A1.p1.1.m1.1.1.3.3.3.1.2"></log><cn id="A1.p1.1.m1.1.1.3.3.3.1.3.cmml" type="integer" xref="A1.p1.1.m1.1.1.3.3.3.1.3">10</cn></apply><apply id="A1.p1.1.m1.1.1.3.3.3.2.cmml" xref="A1.p1.1.m1.1.1.3.3.3.2"><csymbol cd="ambiguous" id="A1.p1.1.m1.1.1.3.3.3.2.1.cmml" xref="A1.p1.1.m1.1.1.3.3.3.2">subscript</csymbol><ci id="A1.p1.1.m1.1.1.3.3.3.2.2.cmml" xref="A1.p1.1.m1.1.1.3.3.3.2.2">𝜃</ci><cn id="A1.p1.1.m1.1.1.3.3.3.2.3.cmml" type="integer" xref="A1.p1.1.m1.1.1.3.3.3.2.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">\delta\rho/\rho_{0}=\theta_{1}\mathcal{M}_{t}^{\theta_{0}}\iff\log_{10}\delta% \rho/\rho_{0}=\theta_{0}\log_{10}\mathcal{M}_{t}+\log_{10}\theta_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⇔ roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> to both the MMS and simulation data, to primarily test if there is statistical agreement between the data and the weakly incompressible theory from <cite class="ltx_cite ltx_citemacro_citet">Bhattacharjee et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>. Naturally, we are able to also probe the similarities and differences between the MMS data and the simulation, too. We use a maximum likelihood approach, utilizing the <cite class="ltx_cite ltx_citemacro_citet">Foreman-Mackey et al. (<a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib30" title="">2013</a>)</cite> sampler. We have very weakly constrained priors, with <math alttext="-4\leq\theta_{0}\leq 4" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><mrow id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml"><mrow id="A1.p1.2.m2.1.1.2" xref="A1.p1.2.m2.1.1.2.cmml"><mo id="A1.p1.2.m2.1.1.2a" xref="A1.p1.2.m2.1.1.2.cmml">−</mo><mn id="A1.p1.2.m2.1.1.2.2" xref="A1.p1.2.m2.1.1.2.2.cmml">4</mn></mrow><mo id="A1.p1.2.m2.1.1.3" xref="A1.p1.2.m2.1.1.3.cmml">≤</mo><msub id="A1.p1.2.m2.1.1.4" xref="A1.p1.2.m2.1.1.4.cmml"><mi id="A1.p1.2.m2.1.1.4.2" xref="A1.p1.2.m2.1.1.4.2.cmml">θ</mi><mn id="A1.p1.2.m2.1.1.4.3" xref="A1.p1.2.m2.1.1.4.3.cmml">0</mn></msub><mo id="A1.p1.2.m2.1.1.5" xref="A1.p1.2.m2.1.1.5.cmml">≤</mo><mn id="A1.p1.2.m2.1.1.6" xref="A1.p1.2.m2.1.1.6.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><apply id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1"><and id="A1.p1.2.m2.1.1a.cmml" xref="A1.p1.2.m2.1.1"></and><apply id="A1.p1.2.m2.1.1b.cmml" 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encoding="application/x-tex" id="A1.p1.2.m2.1c">-4\leq\theta_{0}\leq 4</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">- 4 ≤ italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ 4</annotation></semantics></math> and <math alttext="-2\leq\log_{10}\theta_{1}\leq 1" class="ltx_Math" display="inline" id="A1.p1.3.m3.1"><semantics id="A1.p1.3.m3.1a"><mrow id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml"><mrow id="A1.p1.3.m3.1.1.2" xref="A1.p1.3.m3.1.1.2.cmml"><mo id="A1.p1.3.m3.1.1.2a" xref="A1.p1.3.m3.1.1.2.cmml">−</mo><mn id="A1.p1.3.m3.1.1.2.2" xref="A1.p1.3.m3.1.1.2.2.cmml">2</mn></mrow><mo id="A1.p1.3.m3.1.1.3" xref="A1.p1.3.m3.1.1.3.cmml">≤</mo><mrow id="A1.p1.3.m3.1.1.4" xref="A1.p1.3.m3.1.1.4.cmml"><msub id="A1.p1.3.m3.1.1.4.1" xref="A1.p1.3.m3.1.1.4.1.cmml"><mi id="A1.p1.3.m3.1.1.4.1.2" xref="A1.p1.3.m3.1.1.4.1.2.cmml">log</mi><mn id="A1.p1.3.m3.1.1.4.1.3" xref="A1.p1.3.m3.1.1.4.1.3.cmml">10</mn></msub><mo id="A1.p1.3.m3.1.1.4a" lspace="0.167em" 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encoding="application/x-tex" id="A1.p1.3.m3.1c">-2\leq\log_{10}\theta_{1}\leq 1</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.1d">- 2 ≤ roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ 1</annotation></semantics></math>, ensuring that we do not over-constrain the fits. We use a regular likelihood function, and we sample the posterior with 32 walkers, for 5000 steps, getting rid of the first 500 steps as a burn-in stage. We show the posterior for the MMS (a) and simulation (b) panels in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#A1.F4" title="Figure 4 ‣ Appendix A Maximum Likelihood Fits ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Figure 4</span></a>, which indicates that the posterior is well-sampled, not over-constrained, and the parameters have a strong covariance (as expected). As we discussed in the main text, the fits show two key aspects. Firstly, that (within 1<math alttext="\sigma" class="ltx_Math" display="inline" id="A1.p1.4.m4.1"><semantics id="A1.p1.4.m4.1a"><mi id="A1.p1.4.m4.1.1" xref="A1.p1.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A1.p1.4.m4.1b"><ci id="A1.p1.4.m4.1.1.cmml" xref="A1.p1.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A1.p1.4.m4.1d">italic_σ</annotation></semantics></math>) <math alttext="\delta\rho/\rho_{0}=\mathcal{M}_{t}" class="ltx_Math" display="inline" id="A1.p1.5.m5.1"><semantics id="A1.p1.5.m5.1a"><mrow id="A1.p1.5.m5.1.1" xref="A1.p1.5.m5.1.1.cmml"><mrow id="A1.p1.5.m5.1.1.2" xref="A1.p1.5.m5.1.1.2.cmml"><mrow id="A1.p1.5.m5.1.1.2.2" xref="A1.p1.5.m5.1.1.2.2.cmml"><mi id="A1.p1.5.m5.1.1.2.2.2" xref="A1.p1.5.m5.1.1.2.2.2.cmml">δ</mi><mo id="A1.p1.5.m5.1.1.2.2.1" xref="A1.p1.5.m5.1.1.2.2.1.cmml">⁢</mo><mi id="A1.p1.5.m5.1.1.2.2.3" xref="A1.p1.5.m5.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="A1.p1.5.m5.1.1.2.1" xref="A1.p1.5.m5.1.1.2.1.cmml">/</mo><msub id="A1.p1.5.m5.1.1.2.3" xref="A1.p1.5.m5.1.1.2.3.cmml"><mi id="A1.p1.5.m5.1.1.2.3.2" xref="A1.p1.5.m5.1.1.2.3.2.cmml">ρ</mi><mn id="A1.p1.5.m5.1.1.2.3.3" xref="A1.p1.5.m5.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="A1.p1.5.m5.1.1.1" xref="A1.p1.5.m5.1.1.1.cmml">=</mo><msub id="A1.p1.5.m5.1.1.3" xref="A1.p1.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p1.5.m5.1.1.3.2" xref="A1.p1.5.m5.1.1.3.2.cmml">ℳ</mi><mi id="A1.p1.5.m5.1.1.3.3" xref="A1.p1.5.m5.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.5.m5.1b"><apply id="A1.p1.5.m5.1.1.cmml" xref="A1.p1.5.m5.1.1"><eq id="A1.p1.5.m5.1.1.1.cmml" xref="A1.p1.5.m5.1.1.1"></eq><apply id="A1.p1.5.m5.1.1.2.cmml" xref="A1.p1.5.m5.1.1.2"><divide id="A1.p1.5.m5.1.1.2.1.cmml" xref="A1.p1.5.m5.1.1.2.1"></divide><apply id="A1.p1.5.m5.1.1.2.2.cmml" xref="A1.p1.5.m5.1.1.2.2"><times 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encoding="application/x-llamapun" id="A1.p1.5.m5.1d">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, as predicted from weakly compressible MHD theory <cite class="ltx_cite ltx_citemacro_citep">(Bhattacharjee et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>, and secondly, both the MMS and simulation data agree with one another.</p> </div> <figure class="ltx_figure" id="A1.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="364" id="A1.F4.g1" src="x4.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>The corner plots for the two <math alttext="\log_{10}\delta\rho/\rho_{0}=\theta_{0}\log_{10}\mathcal{M}_{t}+\log_{10}% \theta_{1}" class="ltx_Math" display="inline" id="A1.F4.5.m1.1"><semantics 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id="A1.F4.5.m1.1.1.3.3.1.3.cmml" type="integer" xref="A1.F4.5.m1.1.1.3.3.1.3">10</cn></apply><apply id="A1.F4.5.m1.1.1.3.3.2.cmml" xref="A1.F4.5.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="A1.F4.5.m1.1.1.3.3.2.1.cmml" xref="A1.F4.5.m1.1.1.3.3.2">subscript</csymbol><ci id="A1.F4.5.m1.1.1.3.3.2.2.cmml" xref="A1.F4.5.m1.1.1.3.3.2.2">𝜃</ci><cn id="A1.F4.5.m1.1.1.3.3.2.3.cmml" type="integer" xref="A1.F4.5.m1.1.1.3.3.2.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.5.m1.1d">\log_{10}\delta\rho/\rho_{0}=\theta_{0}\log_{10}\mathcal{M}_{t}+\log_{10}% \theta_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.F4.5.m1.1e">roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> fits (shown in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E11" title="11 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 11</span></a> and <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.08883v1#S3.E17" title="17 ‣ 3 Results ‣ Density fluctuation - Mach number scaling in compressible, high plasma beta turbulence: in-situ space observations and high-Reynolds number simulations"><span class="ltx_text ltx_ref_tag">Equation 17</span></a>) to the MMS (a) and simulation data (b), showing that (1) within 1<math alttext="\sigma" class="ltx_Math" display="inline" id="A1.F4.6.m2.1"><semantics id="A1.F4.6.m2.1b"><mi id="A1.F4.6.m2.1.1" xref="A1.F4.6.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A1.F4.6.m2.1c"><ci id="A1.F4.6.m2.1.1.cmml" xref="A1.F4.6.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.6.m2.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="A1.F4.6.m2.1e">italic_σ</annotation></semantics></math>, <math alttext="\delta\rho/\rho_{0}=\mathcal{M}_{t}" class="ltx_Math" display="inline" id="A1.F4.7.m3.1"><semantics id="A1.F4.7.m3.1b"><mrow id="A1.F4.7.m3.1.1" xref="A1.F4.7.m3.1.1.cmml"><mrow id="A1.F4.7.m3.1.1.2" xref="A1.F4.7.m3.1.1.2.cmml"><mrow id="A1.F4.7.m3.1.1.2.2" xref="A1.F4.7.m3.1.1.2.2.cmml"><mi id="A1.F4.7.m3.1.1.2.2.2" xref="A1.F4.7.m3.1.1.2.2.2.cmml">δ</mi><mo id="A1.F4.7.m3.1.1.2.2.1" xref="A1.F4.7.m3.1.1.2.2.1.cmml">⁢</mo><mi id="A1.F4.7.m3.1.1.2.2.3" xref="A1.F4.7.m3.1.1.2.2.3.cmml">ρ</mi></mrow><mo id="A1.F4.7.m3.1.1.2.1" xref="A1.F4.7.m3.1.1.2.1.cmml">/</mo><msub id="A1.F4.7.m3.1.1.2.3" xref="A1.F4.7.m3.1.1.2.3.cmml"><mi id="A1.F4.7.m3.1.1.2.3.2" xref="A1.F4.7.m3.1.1.2.3.2.cmml">ρ</mi><mn id="A1.F4.7.m3.1.1.2.3.3" xref="A1.F4.7.m3.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="A1.F4.7.m3.1.1.1" xref="A1.F4.7.m3.1.1.1.cmml">=</mo><msub id="A1.F4.7.m3.1.1.3" xref="A1.F4.7.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.F4.7.m3.1.1.3.2" xref="A1.F4.7.m3.1.1.3.2.cmml">ℳ</mi><mi id="A1.F4.7.m3.1.1.3.3" xref="A1.F4.7.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.F4.7.m3.1c"><apply id="A1.F4.7.m3.1.1.cmml" xref="A1.F4.7.m3.1.1"><eq id="A1.F4.7.m3.1.1.1.cmml" xref="A1.F4.7.m3.1.1.1"></eq><apply id="A1.F4.7.m3.1.1.2.cmml" xref="A1.F4.7.m3.1.1.2"><divide id="A1.F4.7.m3.1.1.2.1.cmml" xref="A1.F4.7.m3.1.1.2.1"></divide><apply id="A1.F4.7.m3.1.1.2.2.cmml" xref="A1.F4.7.m3.1.1.2.2"><times id="A1.F4.7.m3.1.1.2.2.1.cmml" xref="A1.F4.7.m3.1.1.2.2.1"></times><ci id="A1.F4.7.m3.1.1.2.2.2.cmml" xref="A1.F4.7.m3.1.1.2.2.2">𝛿</ci><ci id="A1.F4.7.m3.1.1.2.2.3.cmml" xref="A1.F4.7.m3.1.1.2.2.3">𝜌</ci></apply><apply id="A1.F4.7.m3.1.1.2.3.cmml" xref="A1.F4.7.m3.1.1.2.3"><csymbol cd="ambiguous" id="A1.F4.7.m3.1.1.2.3.1.cmml" xref="A1.F4.7.m3.1.1.2.3">subscript</csymbol><ci id="A1.F4.7.m3.1.1.2.3.2.cmml" xref="A1.F4.7.m3.1.1.2.3.2">𝜌</ci><cn id="A1.F4.7.m3.1.1.2.3.3.cmml" type="integer" xref="A1.F4.7.m3.1.1.2.3.3">0</cn></apply></apply><apply id="A1.F4.7.m3.1.1.3.cmml" xref="A1.F4.7.m3.1.1.3"><csymbol cd="ambiguous" id="A1.F4.7.m3.1.1.3.1.cmml" xref="A1.F4.7.m3.1.1.3">subscript</csymbol><ci id="A1.F4.7.m3.1.1.3.2.cmml" xref="A1.F4.7.m3.1.1.3.2">ℳ</ci><ci id="A1.F4.7.m3.1.1.3.3.cmml" xref="A1.F4.7.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.7.m3.1d">\delta\rho/\rho_{0}=\mathcal{M}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.F4.7.m3.1e">italic_δ italic_ρ / italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, as predicted from weakly compressible turbulence theory <cite class="ltx_cite ltx_citemacro_citep">(Bhattacharjee et al., <a class="ltx_ref" href="https://arxiv.org/html/2502.08883v1#bib.bib10" title="">1998</a>)</cite>, and (2) that with 1<math alttext="\sigma" class="ltx_Math" display="inline" id="A1.F4.8.m4.1"><semantics id="A1.F4.8.m4.1b"><mi id="A1.F4.8.m4.1.1" xref="A1.F4.8.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A1.F4.8.m4.1c"><ci id="A1.F4.8.m4.1.1.cmml" xref="A1.F4.8.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.8.m4.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="A1.F4.8.m4.1e">italic_σ</annotation></semantics></math> both the MMS and simulation data agree with one another.</figcaption> </figure> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Feb 13 01:38:48 2025 by <a 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