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Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids | PNAS
<!DOCTYPE html> <html lang="en" dir="ltr" xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <head profile="http://www.w3.org/1999/xhtml/vocab"><script type="text/javascript" src="/_static/js/bundle-playback.js?v=HxkREWBo" charset="utf-8"></script> <script type="text/javascript" src="/_static/js/wombat.js?v=txqj7nKC" charset="utf-8"></script> <script>window.RufflePlayer=window.RufflePlayer||{};window.RufflePlayer.config={"autoplay":"on","unmuteOverlay":"hidden"};</script> <script type="text/javascript" src="/_static/js/ruffle/ruffle.js"></script> <script type="text/javascript"> __wm.init("https://web.archive.org/web"); __wm.wombat("http://www.pnas.org:80/content/114/9/2119","20180714115531","https://web.archive.org/","web","/_static/", "1531569331"); </script> <link rel="stylesheet" type="text/css" href="/_static/css/banner-styles.css?v=S1zqJCYt" /> <link rel="stylesheet" type="text/css" href="/_static/css/iconochive.css?v=3PDvdIFv" /> <!-- End Wayback Rewrite JS Include --> <!--[if IE]><![endif]--> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/> <link rel="dns-prefetch" href="//web.archive.org/web/20180714115531/http://pnas-movie.glencoesoftware.com/"/> <link rel="dns-prefetch" href="//web.archive.org/web/20180714115531/http://scholar.google.com/"/> <link rel="dns-prefetch" href="//web.archive.org/web/20180714115531/http://stats.g.doubleclick.net/"/> <link rel="dns-prefetch" href="//web.archive.org/web/20180714115531/http://cdnjs.cloudflare.com/"/> <link rel="dns-prefetch" href="//web.archive.org/web/20180714115531/http://www.google-analytics.com/"/> <meta name="viewport" content="initial-scale=1, maximum-scale=1, width=device-width, user-scalable=yes"/> <link rel="shortcut icon" href="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/sites/default/files/images/favicon.ico" type="image/vnd.microsoft.icon"/> <meta name="citation_funding_source" content="citation_funder_id=100000879;citation_grant_number=FG-2015‐65783;"/> <meta name="citation_funding_source" content="citation_funder_id=100000913;citation_grant_number=JSMF Grant no. 22020474;"/> <meta name="issue_cover_image" content="http://www.pnas.org/sites/default/files/highwire/pnas/114/9.cover-source.jpg"/> <meta name="type" content="article"/> <meta name="category" content="research-article"/> <meta name="HW.identifier" content="/pnas/114/9/2119.atom"/> <meta name="HW.pisa" content="pnas;114/9/2119"/> <meta name="DC.Format" content="text/html"/> <meta name="DC.Language" content="en"/> <meta name="DC.Title" content="Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids"/> <meta name="DC.Identifier" content="10.1073/pnas.1614721114"/> <meta name="DC.Date" content="2017-02-28"/> <meta name="DC.Publisher" content="National Academy of Sciences"/> <meta name="DC.Rights" content="© . Freely available online through the PNAS open access option."/> <meta name="DC.AccessRights" content="open-access"/> <meta name="DC.Relation" content="10.1073/pnas.1703403114"/> <meta name="DC.Relation" content="10.1073/pnas.1703403114"/> <meta name="DC.Relation" content="10.1073/pnas.1703403114"/> <meta name="DC.Relation" content="10.1073/pnas.1703403114"/> <meta name="DC.Description" content="Turbulence provides an important mechanism for energy redistribution and mixing in interstellar gases, planetary atmospheres, and the oceans. Classical turbulence theory suggests for ordinary 3D fluids or gases, such as water or air, that larger vortices can transform into smaller ones but not vice versa, thus limiting energy transfer from smaller to larger scales. Our calculations predict that bacterial suspensions and other pattern-forming active fluids can deviate from this paradigm by creating turbulent flow structures that spontaneously break mirror symmetry. These results imply that the collective dynamics of swimming microorganisms can enhance fluid mixing more strongly than previously thought."/> <meta name="DC.Contributor" content="Jonasz Słomka"/> <meta name="DC.Contributor" content="Jörn Dunkel"/> <meta name="og:title" property="og:title" content="Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids"/> <meta name="og:url" property="og:url" content="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/114/9/2119"/> <meta name="og:site_name" property="og:site_name" content="PNAS"/> <meta name="og:image" property="og:image" content="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/sites/default/files/highwire/pnas/114/9.cover-source.jpg"/> <meta name="og:description" property="og:description" content="Turbulence provides an important mechanism for energy redistribution and mixing in interstellar gases, planetary atmospheres, and the oceans. Classical turbulence theory suggests for ordinary 3D fluids or gases, such as water or air, that larger vortices can transform into smaller ones but not vice versa, thus limiting energy transfer from smaller to larger scales. Our calculations predict that bacterial suspensions and other pattern-forming active fluids can deviate from this paradigm by creating turbulent flow structures that spontaneously break mirror symmetry. These results imply that the collective dynamics of swimming microorganisms can enhance fluid mixing more strongly than previously thought."/> <meta name="og:type" property="og:type" content="article"/> <meta name="article:published_time" content="2017-02-28"/> <meta name="article:section" content="Physical Sciences"/> <meta name="citation_title" content="Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids"/> <meta name="citation_abstract" lang="en" content="<p>Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicity-driven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.</p>"/> <meta name="citation_abstract" lang="en" scheme="executive-summary" content="<h3>Significance</h3> <p>Turbulence provides an important mechanism for energy redistribution and mixing in interstellar gases, planetary atmospheres, and the oceans. Classical turbulence theory suggests for ordinary 3D fluids or gases, such as water or air, that larger vortices can transform into smaller ones but not vice versa, thus limiting energy transfer from smaller to larger scales. Our calculations predict that bacterial suspensions and other pattern-forming active fluids can deviate from this paradigm by creating turbulent flow structures that spontaneously break mirror symmetry. These results imply that the collective dynamics of swimming microorganisms can enhance fluid mixing more strongly than previously thought.</p>"/> <meta name="citation_journal_title" content="Proceedings of the National Academy of Sciences"/> <meta name="citation_publisher" content="National Academy of Sciences"/> <meta name="citation_publication_date" content="2017/02/28"/> <meta name="citation_mjid" content="pnas;114/9/2119"/> <meta name="citation_id" content="114/9/2119"/> <meta name="citation_public_url" content="http://www.pnas.org/content/114/9/2119"/> <meta name="citation_abstract_html_url" content="http://www.pnas.org/content/114/9/2119.abstract"/> <meta name="citation_full_html_url" content="http://www.pnas.org/content/114/9/2119.full"/> <meta name="citation_pdf_url" content="http://www.pnas.org/content/114/9/2119.full.pdf"/> <meta name="citation_issn" content="0027-8424"/> <meta name="citation_issn" content="1091-6490"/> <meta name="citation_journal_abbrev" content="PNAS"/> <meta name="citation_doi" content="10.1073/pnas.1614721114"/> <meta name="citation_pmid" content="28193853"/> <meta name="citation_volume" content="114"/> <meta name="citation_issue" content="9"/> <meta name="citation_num_pages" content="6"/> <meta name="citation_article_type" content="Research Article"/> <meta name="citation_section" content="Physical Sciences"/> <meta name="citation_firstpage" content="2119"/> <meta name="citation_lastpage" content="2124"/> <meta name="citation_access" content="all"/> <meta name="citation_author" content="Jonasz Słomka"/> <meta name="citation_author_institution" content="Massachusetts Institute of Technology"/> <meta name="citation_author" content="Jörn Dunkel"/> <meta name="citation_author_institution" content="Massachusetts Institute of Technology"/> <meta name="citation_author_email" content="dunkel@math.mit.edu"/> <meta name="citation_reference" content="citation_title=Turbulence;citation_year=2004"/> <meta name="citation_reference" content="citation_title=The Physics of Fluid Turbulence;citation_year=1990"/> <meta name="citation_reference" content="citation_journal_title=Astrophys J;citation_author=JC. 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class="main-content-wrapper col-narrow-15"> <div class="panel-panel panel-region-content"> <div class="inside"><div class="panel-pane pane-highwire-article-crossmark"> <div class="pane-content"> <!-- Start CrossMark Snippet v2.0 --> <a data-target="crossmark" class="crossmark-square"><img src="https://web.archive.org/web/20180714115531im_/https://crossmark-cdn.crossref.org/widget/v2.0/logos/CROSSMARK_Color_square.svg" alt=""></a> <div id="crossmark-dialog" style="display: none;" title=""> <!-- the external CrossMark data is loaded inside this iframe --> <iframe id="crossmark-dialog-frame"></iframe> </div> <!-- End CrossMark Snippet --> </div> </div> <div class="panel-separator"></div><div class="panel-pane pane-highwire-article-citation"> <div class="pane-content"> <div class="highwire-article-citation highwire-citation-type-highwire-article" data-node-nid="2011" id="node2011" data-pisa="pnas;114/9/2119" data-pisa-master="pnas;1614721114" data-apath="/pnas/114/9/2119.atom" data-hw-author-tooltip-instance="highwire_author_tooltip"><div class="highwire-cite highwire-cite-highwire-article highwire-citation-pnas-article-title-complete clearfix has-author-tooltip highwire-citation-highwire-article-top-a"> <h1 class="highwire-cite-title" id="page-title">Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids</h1> <div class="highwire-cite-authors"><span class="highwire-citation-authors"><span class="highwire-citation-author first" data-delta="0">Jonasz Słomka</span> and <span class="highwire-citation-author" data-delta="1">Jörn Dunkel</span></span></div> <div class="highwire-cite-metadata"><span class="highwire-cite-metadata-journal highwire-cite-metadata">PNAS </span><span class="highwire-cite-metadata-date highwire-cite-metadata">February 28, 2017. </span><span class="highwire-cite-metadata-volume highwire-cite-metadata">114 </span><span class="highwire-cite-metadata-issue highwire-cite-metadata">(9) </span><span class="highwire-cite-metadata-pages highwire-cite-metadata">2119-2124; </span><span class="highwire-cite-metadata-papdate highwire-cite-metadata">published ahead of print February 13, 2017. </span><span class="highwire-cite-metadata-doi highwire-cite-metadata"><a href="https://web.archive.org/web/20180714115531/https://doi.org/10.1073/pnas.1614721114">https://doi.org/10.1073/pnas.1614721114</a> </span></div> <div class="highwire-cite-extras"><span class="highwire-foxycart-add-to-cart-ahah highwire-foxycart-add-to-cart-ahah" data-text="Add to Cart (%short-price)" data-apath="/pnas/114/9/2119.atom" data-type="link" data-font-icon="" data-parent-id="1458"></span></div> </div> <div id="hw-article-author-popups-node2011" style="display: none;"><div class="author-tooltip-0"><div class="author-tooltip-name">Jonasz Słomka</div><div class="author-tooltip-affiliation"><span class="author-tooltip-text"><span class="nlm-aff"><span class="nlm-sup">a</span>Department of Mathematics, <span class="nlm-institution" data-hwp-id="institution-1">Massachusetts Institute of Technology</span>, Cambridge, <span class="nlm-addr-line">MA</span> 02139-4307</span></span></div><ul class="author-tooltip-find-more"><li class="author-tooltip-gs-link first"><a href="/web/20180714115531/http://www.pnas.org/lookup/google-scholar?link_type=googlescholar&gs_type=author&author%5B0%5D=J.%2BS%C5%82omka" target="_blank" class="" data-icon-position="" data-hide-link-title="0">Find this author on Google Scholar</a></li> <li class="author-tooltip-pubmed-link"><a href="/web/20180714115531/http://www.pnas.org/lookup/external-ref?access_num=S%C5%82omka%20J&link_type=AUTHORSEARCH" target="_blank" class="" data-icon-position="" data-hide-link-title="0">Find this author on PubMed</a></li> <li class="author-site-search-link last"><a href="/web/20180714115531/http://www.pnas.org/search/author1%3AJonasz%2BS%25C5%2582omka" rel="nofollow" class="" data-icon-position="" data-hide-link-title="0">Search for this author on this site</a></li> </ul></div><div class="author-tooltip-1"><div class="author-tooltip-name">Jörn Dunkel</div><div class="author-tooltip-affiliation"><span class="author-tooltip-text"><span class="nlm-aff"><span class="nlm-sup">a</span>Department of Mathematics, <span class="nlm-institution" data-hwp-id="institution-1">Massachusetts Institute of Technology</span>, Cambridge, <span class="nlm-addr-line">MA</span> 02139-4307</span></span></div><ul class="author-tooltip-find-more"><li class="author-tooltip-gs-link first"><a href="/web/20180714115531/http://www.pnas.org/lookup/google-scholar?link_type=googlescholar&gs_type=author&author%5B0%5D=J.%2BDunkel" target="_blank" class="" data-icon-position="" data-hide-link-title="0">Find this author on Google Scholar</a></li> <li class="author-tooltip-pubmed-link"><a href="/web/20180714115531/http://www.pnas.org/lookup/external-ref?access_num=Dunkel%20J&link_type=AUTHORSEARCH" target="_blank" class="" data-icon-position="" data-hide-link-title="0">Find this author on PubMed</a></li> <li class="author-site-search-link"><a href="/web/20180714115531/http://www.pnas.org/search/author1%3AJ%25C3%25B6rn%2BDunkel" rel="nofollow" class="" data-icon-position="" data-hide-link-title="0">Search for this author on this site</a></li> <li class="author-corresp-email-link last"><span>For correspondence: <a href="https://web.archive.org/web/20180714115531/mailto:dunkel@math.mit.edu" class="" data-icon-position="" data-hide-link-title="0">dunkel@math.mit.edu</a></span></li> </ul></div></div></div> </div> </div> <div class="panel-separator"></div><div class="panel-pane pane-highwire-markup article-citation-contributors"> <div class="pane-content"> <div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" id="content-block-markup" xmlns:xhtml="http://www.w3.org/1999/xhtml"><ol xmlns:xhtml="http://www.w3.org/1999/xhtml" class="fn-track"><li class="fn-edited-by" id="fn-1"><p id="p-1">Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 4, 2017 (received for review September 2, 2016)</p></li></ol></div></div> </div> </div> <div class="panel-separator"></div><div class="panel-pane pane-highwire-corrections"> <div class="pane-content"> <div class="messages highwire-corrections highwire-correction-type-correction highwire-correction-type-correction"><h3 class="highwire-corrections-title">This article has a correction. Please see:</h3><ul class="highwire-corrections-list"><li class="relation-2496 first last"><a href="/web/20180714115531/http://www.pnas.org/content/114/15/E3159" class="" data-icon-position="" data-hide-link-title="0">Correction for Słomka and Dunkel, Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids</a></li> </ul></div> </div> </div> <div class="panel-separator"></div><div class="panel-pane pane-highwire-panel-tabs pane-panels-ajax-tab-tabs"> <div class="pane-content"> <div class="item-list"><ul class="tabs inline panels-ajax-tab"><li class="first"><a href="/web/20180714115531/http://www.pnas.org/content/114/9/2119" class="panels-ajax-tab-tab" data-panel-name="jnl_pnas_tab_art" data-target-id="highwire_article_tabs" data-entity-context="node:2011" data-trigger="" data-url-enabled="1">Article</a><a href="/web/20180714115531/http://www.pnas.org/panels_ajax_tab/jnl_pnas_tab_art/node:2011/1" rel="nofollow" style="display:none" class="js-crawler-link"></a></li> <li><a href="/web/20180714115531/http://www.pnas.org/content/114/9/2119/tab-figures-data" class="panels-ajax-tab-tab" data-panel-name="jnl_pnas_tab_data" data-target-id="highwire_article_tabs" data-entity-context="node:2011" data-trigger="tab-figures-data" data-url-enabled="1">Figures & SI</a><a href="/web/20180714115531/http://www.pnas.org/panels_ajax_tab/jnl_pnas_tab_data/node:2011/1" rel="nofollow" style="display:none" class="js-crawler-link"></a></li> <li><a href="/web/20180714115531/http://www.pnas.org/content/114/9/2119/tab-article-info" class="panels-ajax-tab-tab" data-panel-name="jnl_pnas_tab_info" data-target-id="highwire_article_tabs" data-entity-context="node:2011" data-trigger="tab-article-info" data-url-enabled="1">Authors & Info</a><a href="/web/20180714115531/http://www.pnas.org/panels_ajax_tab/jnl_pnas_tab_info/node:2011/1" rel="nofollow" style="display:none" class="js-crawler-link"></a></li> <li class="last"><a href="/web/20180714115531/http://www.pnas.org/content/114/9/2119.full.pdf" class="panels-ajax-tab-tab" data-panel-name="jnl_pnas_tab_pdf" data-target-id="highwire_article_tabs" data-entity-context="node:2011" data-trigger="tab-pdf" data-url-enabled="1"><i class="icon-file-alt"></i> PDF</a><a href="/web/20180714115531/http://www.pnas.org/panels_ajax_tab/jnl_pnas_tab_pdf/node:2011/1" rel="nofollow" style="display:none" class="js-crawler-link"></a></li> </ul></div> </div> </div> <div class="panel-separator"></div><div class="panel-pane pane-highwire-panel-tabs-container"> <div class="pane-content"> <div data-panels-ajax-tab-preloaded="jnl_pnas_tab_art" id="panels-ajax-tab-container-highwire_article_tabs" class="panels-ajax-tab-container"><div class="panels-ajax-tab-loading" style="display:none"><img class="loading" src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/sites/all/modules/contrib/panels_ajax_tab/images/loading.gif" alt="Loading" title="Loading"/></div><div class="panels-ajax-tab-wrap-jnl_pnas_tab_art"><div class="panel-display panel-1col clearfix"> <div class="panel-panel panel-col"> <div><div class="panel-pane pane-highwire-markup"> <div class="pane-content"> <div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" id="content-block-markup" data-highwire-cite-ref-tooltip-instance="highwire_reflinks_tooltip" xmlns:xhtml="http://www.w3.org/1999/xhtml"><div class="article fulltext-view "><span class="highwire-journal-article-marker-start"></span><div class="executive-summary"><h2 class="">Significance</h2><p id="p-5">Turbulence provides an important mechanism for energy redistribution and mixing in interstellar gases, planetary atmospheres, and the oceans. Classical turbulence theory suggests for ordinary 3D fluids or gases, such as water or air, that larger vortices can transform into smaller ones but not vice versa, thus limiting energy transfer from smaller to larger scales. Our calculations predict that bacterial suspensions and other pattern-forming active fluids can deviate from this paradigm by creating turbulent flow structures that spontaneously break mirror symmetry. These results imply that the collective dynamics of swimming microorganisms can enhance fluid mixing more strongly than previously thought.</p></div><div class="section abstract" id="abstract-2"><h2>Abstract</h2><p id="p-6">Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicity-driven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.</p></div><ul class="kwd-group"><li class="kwd"><a href="/web/20180714115531/http://www.pnas.org/keyword/active-turbulence" class="hw-term hw-article-keyword hw-article-keyword-active-turbulence" rel="nofollow">active turbulence</a></li><li class="kwd"><a href="/web/20180714115531/http://www.pnas.org/keyword/inverse-cascade" class="hw-term hw-article-keyword hw-article-keyword-inverse-cascade" rel="nofollow">inverse cascade</a></li><li class="kwd"><a href="/web/20180714115531/http://www.pnas.org/keyword/beltrami-flows" class="hw-term hw-article-keyword hw-article-keyword-beltrami-flows" rel="nofollow">Beltrami flows</a></li><li class="kwd"><a href="/web/20180714115531/http://www.pnas.org/keyword/pattern-formation" class="hw-term hw-article-keyword hw-article-keyword-pattern-formation" rel="nofollow">pattern formation</a></li></ul><p id="p-7" class="flushleft">Turbulence, the chaotic motion of liquids and gases, remains one of the most widely studied phenomena in classical physics (<a id="xref-ref-1-1" class="xref-bibr" href="#ref-1">1</a>, <a id="xref-ref-2-1" class="xref-bibr" href="#ref-2">2</a>). Turbulent flows determine energy transfer and material mixing over a vast range of scales, from the interstellar medium (<a id="xref-ref-3-1" class="xref-bibr" href="#ref-3">3</a>, <a id="xref-ref-4-1" class="xref-bibr" href="#ref-4">4</a>) and solar winds (<a id="xref-ref-5-1" class="xref-bibr" href="#ref-5">5</a>) to the Earth’s atmosphere (<a id="xref-ref-6-1" class="xref-bibr" href="#ref-6">6</a>, <a id="xref-ref-7-1" class="xref-bibr" href="#ref-7">7</a>), ocean currents (<a id="xref-ref-8-1" class="xref-bibr" href="#ref-8">8</a>), and our morning cup of coffee. Of particular recent interest is the interplay of turbulence and active biological matter (<a id="xref-ref-9-1" class="xref-bibr" href="#ref-9">9</a>), owing to its relevance for carbon fixation and nutrient transport in marine ecosystems (<a id="xref-ref-10-1" class="xref-bibr" href="#ref-10">10</a>). Although much has been learned about the statistical and spectral properties of turbulent flows both experimentally (<a id="xref-ref-11-1" class="xref-bibr" href="#ref-11">11</a><a id="xref-ref-12-1" class="xref-down-link" href="#ref-12"><span>⇓</span></a>–<a id="xref-ref-13-1" class="xref-bibr" href="#ref-13">13</a>) and theoretically (<a id="xref-ref-14-1" class="xref-bibr" href="#ref-14">14</a><a id="xref-ref-15-1" class="xref-down-link" href="#ref-15"><span>⇓</span></a><a id="xref-ref-16-1" class="xref-down-link" href="#ref-16"><span>⇓</span></a><a id="xref-ref-17-1" class="xref-down-link" href="#ref-17"><span>⇓</span></a><a id="xref-ref-18-1" class="xref-down-link" href="#ref-18"><span>⇓</span></a><a id="xref-ref-19-1" class="xref-down-link" href="#ref-19"><span>⇓</span></a><a id="xref-ref-20-1" class="xref-down-link" href="#ref-20"><span>⇓</span></a>–<a id="xref-ref-21-1" class="xref-bibr" href="#ref-21">21</a>) over the last 75 years, several fundamental physical and mathematical (<a id="xref-ref-22-1" class="xref-bibr" href="#ref-22">22</a>) questions still await their answer. One of the most important among them, with profound implications for the limits of hydrodynamic mixing, concerns whether 3D turbulent flows can develop an inverse cascade that transports energy from smaller to larger scales (<a id="xref-ref-19-2" class="xref-bibr" href="#ref-19">19</a>, <a id="xref-ref-23-1" class="xref-bibr" href="#ref-23">23</a>, <a id="xref-ref-24-1" class="xref-bibr" href="#ref-24">24</a>).</p><p id="p-8">Kolmogorov’s 1941 theory of turbulence (<a id="xref-ref-14-2" class="xref-bibr" href="#ref-14">14</a>) assumes that turbulent energy transport in 3D proceeds primarily from larger to smaller scales through the decay of vortices. This forward (Richardson) cascade is a consequence of the fact that the 3D inviscid Euler equations conserve energy (<a id="xref-ref-1-2" class="xref-bibr" href="#ref-1">1</a>). In 1967, Kraichnan (<a id="xref-ref-17-2" class="xref-bibr" href="#ref-17">17</a>) realized that the presence of a second conserved quantity, enstrophy, in 2D turbulent flows implies the existence of two dual cascades (<a id="xref-ref-25-1" class="xref-bibr" href="#ref-25">25</a>): a vorticity-induced cascade to smaller scales and an inverse energy cascade to larger scales (<a id="xref-ref-20-2" class="xref-bibr" href="#ref-20">20</a>, <a id="xref-ref-26-1" class="xref-bibr" href="#ref-26">26</a>). Two years later, Moffatt (<a id="xref-ref-27-1" class="xref-bibr" href="#ref-27">27</a>) discovered a new invariant of the 3D Euler equations, which he termed helicity. Could helicity conservation generate an inverse turbulent cascade in 3D? Building on thermodynamic considerations, Kraichnan (<a id="xref-ref-23-2" class="xref-bibr" href="#ref-23">23</a>) argued in 1973 that this should not be possible, but he also conceded that turbulent flows do not necessarily follow equilibrium statistics. Since then, insightful theoretical studies (<a id="xref-ref-19-3" class="xref-bibr" href="#ref-19">19</a>, <a id="xref-ref-24-2" class="xref-bibr" href="#ref-24">24</a>) have elucidated other important conditions for the emergence of helicity-driven inverse cascades in 3D fluids, in particular identifying mirror-symmetry breaking as a key mechanism (<a id="xref-ref-24-3" class="xref-bibr" href="#ref-24">24</a>). However, no natural or artificially engineered fluid system exhibiting this phenomenon has been identified to date.</p><p id="p-9">Here, we predict that fluid flows in active nonequilibrium liquids, such as bacterial suspensions, can spontaneously break mirror symmetry, resulting in a 3D inverse cascade. Broken mirror symmetry plays an important role in nature, exemplified by the parity-violating weak interactions (<a id="xref-ref-28-1" class="xref-bibr" href="#ref-28">28</a>) in the standard model of particle physics, by the helical structure of DNA (<a id="xref-ref-29-1" class="xref-bibr" href="#ref-29">29</a>) or, at the macroscale, by chiral seed pods (<a id="xref-ref-30-1" class="xref-bibr" href="#ref-30">30</a>). Another, fluid-based realization (<a id="xref-ref-31-1" class="xref-bibr" href="#ref-31">31</a>) of a spontaneously broken chiral symmetry was recently observed in confined bacterial suspensions (<a id="xref-ref-32-1" class="xref-bibr" href="#ref-32">32</a>, <a id="xref-ref-33-1" class="xref-bibr" href="#ref-33">33</a>), which form stable vortices of well-defined circulation when the container dimensions match the correlation scale <span class="inline-formula" id="inline-formula-1"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>70</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m of the collective cell motion in bulk (<a id="xref-ref-34-1" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-1" class="xref-bibr" href="#ref-35">35</a>). Motivated by these observations, we investigate a generalized Navier–Stokes model (<a id="xref-ref-36-1" class="xref-bibr" href="#ref-36">36</a>, <a id="xref-ref-37-1" class="xref-bibr" href="#ref-37">37</a>) for pattern-forming nonequilibrium fluids that are driven by an active component, which could be swimming bacteria (<a id="xref-ref-34-2" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-2" class="xref-bibr" href="#ref-35">35</a>) or ATP-driven microtubules (<a id="xref-ref-38-1" class="xref-bibr" href="#ref-38">38</a>, <a id="xref-ref-39-1" class="xref-bibr" href="#ref-39">39</a>) or artificial microswimmers (<a id="xref-ref-40-1" class="xref-bibr" href="#ref-40">40</a><a id="xref-ref-41-1" class="xref-down-link" href="#ref-41"><span>⇓</span></a>–<a id="xref-ref-42-1" class="xref-bibr" href="#ref-42">42</a>). The theory uses only generic assumptions about the symmetries and long-wavelength structure of active stress tensors and captures the experimentally observed bulk vortex dynamics in 3D bacterial suspensions (<a id="xref-ref-34-3" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-3" class="xref-bibr" href="#ref-35">35</a>) and in flows driven by isotropic active microtubule networks (<a id="xref-ref-38-2" class="xref-bibr" href="#ref-38">38</a>) (<a id="xref-fig-1-1" class="xref-fig" href="#F1">Fig. S1</a>).</p><div id="F1" class="fig pos-float type-supplementary-material odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F1.large.jpg?width=800&height=600&carousel=1" title="Fit results for the generalized Navier–Stokes model defined in Eq. 1 of the main text compared with recent experimental data for (A–C) bacterial suspensions and (D) microtubule networks. (A) PDFs of the Cartesian in-plane velocity components, normalized by their mean values and standard deviations. The black curve labeled “theory” represents a five-parameter continuum model for the bacterial dynamics described in ref. 35. The blue curve is obtained for the generalized three-parameter Navier–Stokes model of the solvent flow defined in Eq. 1 of the main text, using the fit parameters listed in the description of C. Note that only the Navier–Stokes model correctly captures the tails of the velocity distribution. (B) The equal-time VCFs indicate the characteristic pattern formation scale. The black curve labeled “theory” again represents the continuum model for the bacterial dynamics described in ref. 35. The blue curve is obtained for the Navier–Stokes model for the solvent flow defined in Eq. 1 of the main text, using the fit parameters listed in the description of C. (C) VACFs obtained for three different values of the fit parameters corresponding to three different activity levels of the bacteria: Γ0= 4.77μm2/s, Γ2/Γ0=−1.15×103μm2, Γ4/Γ0= 9.80×104μm4 (magenta line), Γ0= 6.82μm2/s, Γ2/Γ0=−1.15×103μm2, Γ4/Γ0= 9.80×104μm4 (blue lines), Γ0= 1.59×101μm2/s, Γ2/Γ0=−1.15×103μm2, and Γ4/Γ0= 9.80×104μm4 (green line). In terms of the characteristic vortex size Λ, growth time τ, speed U, and bandwidth κ, these parameters correspond to Λ= 41μm, τ= 15s, U= 17.2μm/s, and κ= 73mm−1 (magenta); Λ= 41μm, τ= 10.5s, U= 24.5μm/s, and κ= 73mm−1 (blue); and Λ= 41μm, τ= 4.5s, U= 57.2μm/s, and κ= 73mm−1 (green). (D) Equal-time VCF for an active fluid driven by a microtubule network. The black line is obtained for the Navier–Stokes model for the solvent flow defined in Eq. 1 of the main text, using fit parameters Γ0= 9.05×μm2/s, Γ2/Γ0=−8.61×103μm2, Γ4/Γ0= 7.37×106μm4 corresponding to Λ= 130μm, τ= 125s, U= 6.5μm/s, and κ= 21mm−1. All simulations were performed on a large 3D domain of size L= 32Λ to exclude finite size effects. A–C were reprinted with permission from ref. 35. D was reprinted from ref. 38 by permission from Macmillan Publishers Ltd: Nature, copyright (2012)." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Fit results for the generalized Navier–Stokes model defined in Eq. <strong>1</strong> of the main text compared with recent experimental data for (<em>A–C</em>) bacterial suspensions and (<em>D</em>) microtubule networks. (<em>A</em>) PDFs of the Cartesian in-plane velocity components, normalized by their mean values and standard deviations. The black curve labeled “theory” represents a five-parameter continuum model for the bacterial dynamics described in ref. 35. The blue curve is obtained for the generalized three-parameter Navier–Stokes model of the solvent flow defined in Eq. <strong>1</strong> of the main text, using the fit parameters listed in the description of <em>C</em>. Note that only the Navier–Stokes model correctly captures the tails of the velocity distribution. (<em>B</em>) The equal-time VCFs indicate the characteristic pattern formation scale. The black curve labeled “theory” again represents the continuum model for the bacterial dynamics described in ref. 35. The blue curve is obtained for the Navier–Stokes model for the solvent flow defined in Eq. <strong>1</strong> of the main text, using the fit parameters listed in the description of <em>C</em>. (<em>C</em>) VACFs obtained for three different values of the fit parameters corresponding to three different activity levels of the bacteria: <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 4.77</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span> (magenta line), <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 6.82</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span> (blue lines), <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn> 1.59</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>1</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span>, and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span> (green line). In terms of the characteristic vortex size <span class="mathjax mml-math"><mml:math><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span>, growth time <span class="mathjax mml-math"><mml:math><mml:mi>τ</mml:mi></mml:math></span>, speed <span class="mathjax mml-math"><mml:math><mml:mi>U</mml:mi></mml:math></span>, and bandwidth <span class="mathjax mml-math"><mml:math><mml:mi>κ</mml:mi></mml:math></span>, these parameters correspond to <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 15</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 17.2</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span> (magenta); <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 10.5</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 24.5</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span> (blue); and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 4.5</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 57.2</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span> (green). (<em>D</em>) Equal-time VCF for an active fluid driven by a microtubule network. The black line is obtained for the Navier–Stokes model for the solvent flow defined in Eq. <strong>1</strong> of the main text, using fit parameters <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn> 9.05</mml:mn><mml:mo>×</mml:mo><mml:mi>μ</mml:mi></mml:mrow><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>8.61</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 7.37</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span> corresponding to <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 130</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 125</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 6.5</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span>, and <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 21</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span>. All simulations were performed on a large 3D domain of size <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span> to exclude finite size effects. <em>A–C</em> were reprinted with permission from ref. 35. <em>D</em> was reprinted from ref. 38 by permission from Macmillan Publishers Ltd: Nature, copyright (2012).</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. S1." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F1.medium.gif" width="440" height="95"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. S1." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F1.medium.gif" width="440" height="95"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F1.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. S1." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F1.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9855" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption" xmlns:xhtml="http://www.w3.org/1999/xhtml"><span class="fig-label">Fig. S1.</span> <p id="p-10" class="first-child">Fit results for the generalized Navier–Stokes model defined in <a id="xref-disp-formula-1-1" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text compared with recent experimental data for (<em>A–C</em>) bacterial suspensions and (<em>D</em>) microtubule networks. (<em>A</em>) PDFs of the Cartesian in-plane velocity components, normalized by their mean values and standard deviations. The black curve labeled “theory” represents a five-parameter continuum model for the bacterial dynamics described in ref. <a id="xref-ref-35-4" class="xref-bibr" href="#ref-35">35</a>. The blue curve is obtained for the generalized three-parameter Navier–Stokes model of the solvent flow defined in <a id="xref-disp-formula-1-2" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text, using the fit parameters listed in the description of <em>C</em>. Note that only the Navier–Stokes model correctly captures the tails of the velocity distribution. (<em>B</em>) The equal-time VCFs indicate the characteristic pattern formation scale. The black curve labeled “theory” again represents the continuum model for the bacterial dynamics described in ref. <a id="xref-ref-35-5" class="xref-bibr" href="#ref-35">35</a>. The blue curve is obtained for the Navier–Stokes model for the solvent flow defined in <a id="xref-disp-formula-1-3" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text, using the fit parameters listed in the description of <em>C</em>. (<em>C</em>) VACFs obtained for three different values of the fit parameters corresponding to three different activity levels of the bacteria: <span class="inline-formula" id="inline-formula-2"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 4.77</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-3"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-4"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (magenta line), <span class="inline-formula" id="inline-formula-5"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 6.82</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-6"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-7"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (blue lines), <span class="inline-formula" id="inline-formula-8"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn> 1.59</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>1</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-9"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>1.15</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-10"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 9.80</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (green line). In terms of the characteristic vortex size <span class="inline-formula" id="inline-formula-11"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span>, growth time <span class="inline-formula" id="inline-formula-12"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math></span></span>, speed <span class="inline-formula" id="inline-formula-13"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math></span></span>, and bandwidth <span class="inline-formula" id="inline-formula-14"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span>, these parameters correspond to <span class="inline-formula" id="inline-formula-15"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-16"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 15</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-17"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 17.2</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-18"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (magenta); <span class="inline-formula" id="inline-formula-19"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-20"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 10.5</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-21"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 24.5</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-22"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (blue); and <span class="inline-formula" id="inline-formula-23"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-24"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 4.5</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-25"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 57.2</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-26"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 73</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> (green). (<em>D</em>) Equal-time VCF for an active fluid driven by a microtubule network. The black line is obtained for the Navier–Stokes model for the solvent flow defined in <a id="xref-disp-formula-1-4" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text, using fit parameters <span class="inline-formula" id="inline-formula-27"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn> 9.05</mml:mn><mml:mo>×</mml:mo><mml:mi>μ</mml:mi></mml:mrow><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-28"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:mn>8.61</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-29"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 7.37</mml:mn><mml:mo>×</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> corresponding to <span class="inline-formula" id="inline-formula-30"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 130</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-31"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 125</mml:mn></mml:mpadded><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-32"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 6.5</mml:mn></mml:mpadded><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-33"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 21</mml:mn></mml:mpadded><mml:msup><mml:mi>mm</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:math></span></span>. All simulations were performed on a large 3D domain of size <span class="inline-formula" id="inline-formula-34"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> to exclude finite size effects. <em>A–C</em> were reprinted with permission from ref. <a id="xref-ref-35-6" class="xref-bibr" href="#ref-35">35</a>. <em>D</em> was reprinted from ref. <a id="xref-ref-38-3" class="xref-bibr" href="#ref-38">38</a> by permission from Macmillan Publishers Ltd: Nature, copyright (2012).</p><div class="sb-div caption-clear"></div></div></div><p id="p-11">To demonstrate the existence of a helicity-driven inverse cascade in 3D active bulk fluids, we first verify analytically the existence of exact parity-violating Beltrami-flow (<a id="xref-ref-43-1" class="xref-bibr" href="#ref-43">43</a><a id="xref-ref-44-1" class="xref-down-link" href="#ref-44"><span>⇓</span></a>–<a id="xref-ref-45-1" class="xref-bibr" href="#ref-45">45</a>) solutions. We then confirm numerically that active bulk flows starting from random initial conditions approach attractors that spontaneously break mirror symmetry and are statistically close to Beltrami-vector fields. Finally, we demonstrate that the broken mirror symmetry leads to an inverse cascade with triad interactions as predicted by Waleffe (<a id="xref-ref-19-4" class="xref-bibr" href="#ref-19">19</a>) about 25 years ago.</p><div class="section results" id="sec-1"><h2 class="">Results</h2><div id="sec-2" class="subsection"><h3>Theory.</h3><p id="p-12">We consider pattern-forming nonequilibrium fluids consisting of a passive solvent component, such as water, and a stress-generating active component, which could be bacteria (<a id="xref-ref-34-4" class="xref-bibr" href="#ref-34">34</a>), ATP-driven microtubules (<a id="xref-ref-38-4" class="xref-bibr" href="#ref-38">38</a>), or Janus particles (<a id="xref-ref-46-1" class="xref-bibr" href="#ref-46">46</a>, <a id="xref-ref-47-1" class="xref-bibr" href="#ref-47">47</a>). In contrast to earlier studies, which analyzed the velocity field of the active matter component (<a id="xref-ref-35-7" class="xref-bibr" href="#ref-35">35</a>, <a id="xref-ref-48-1" class="xref-bibr" href="#ref-48">48</a>, <a id="xref-ref-49-1" class="xref-bibr" href="#ref-49">49</a>), we focus here on the incompressible solvent velocity field <span class="inline-formula" id="inline-formula-35"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> described by<span class="disp-formula" id="disp-formula-1"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo>∇</mml:mo><mml:mo>⋅</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[1a]</span></span><span class="disp-formula" id="disp-formula-2"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>⋅</mml:mo><mml:mi>𝝈</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[1b]</span></span>where <span class="inline-formula" id="inline-formula-36"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> is the local pressure. The effective stress tensor <span class="inline-formula" id="inline-formula-37"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> comprises passive contributions from the intrinsic solvent fluid viscosity and active contributions representing the stresses exerted by the microswimmers on the fluid (<a id="xref-ref-50-1" class="xref-bibr" href="#ref-50">50</a><a id="xref-ref-51-1" class="xref-down-link" href="#ref-51"><span>⇓</span></a><a id="xref-ref-52-1" class="xref-down-link" href="#ref-52"><span>⇓</span></a>–<a id="xref-ref-53-1" class="xref-bibr" href="#ref-53">53</a>). Experiments (<a id="xref-ref-32-2" class="xref-bibr" href="#ref-32">32</a>, <a id="xref-ref-34-5" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-8" class="xref-bibr" href="#ref-35">35</a>, <a id="xref-ref-38-5" class="xref-bibr" href="#ref-38">38</a>, <a id="xref-ref-54-1" class="xref-bibr" href="#ref-54">54</a>, <a id="xref-ref-55-1" class="xref-bibr" href="#ref-55">55</a>) show that active stresses typically lead to vortex scale selection in the ambient solvent fluid. This mesoscale pattern formation stands in contrast to the scale-free vortex structures in externally driven classical turbulence and can be described phenomenologically through the stress tensor (<a id="xref-ref-36-2" class="xref-bibr" href="#ref-36">36</a>, <a id="xref-ref-37-2" class="xref-bibr" href="#ref-37">37</a>)<span class="disp-formula" id="disp-formula-3"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>𝝈</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>⊤</mml:mo></mml:msup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[1c]</span></span>where the higher-order derivatives <span class="inline-formula" id="inline-formula-38"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msup><mml:mo>∇</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mpadded><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-39"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>n</mml:mi></mml:mpadded><mml:mo>≥</mml:mo><mml:mn> 2</mml:mn></mml:mrow></mml:math></span></span> account for non-Newtonian effects (<a id="xref-ref-56-1" class="xref-bibr" href="#ref-56">56</a>) (<a id="xref-sec-16-1" class="xref-sec" href="#sec-16"><em>Model Justification</em></a>). Such higher-order stresses arise naturally from diagrammatic expansions (<a id="xref-ref-57-1" class="xref-bibr" href="#ref-57">57</a>). Similar 1D and 2D models have been studied in the context of soft-mode turbulence and seismic waves (<a id="xref-ref-58-1" class="xref-bibr" href="#ref-58">58</a><a id="xref-ref-59-1" class="xref-down-link" href="#ref-59"><span>⇓</span></a>–<a id="xref-ref-60-1" class="xref-bibr" href="#ref-60">60</a>).</p><p id="p-13">The parameters <span class="inline-formula" id="inline-formula-40"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span> encode microscopic interactions, thermal and athermal fluctuations, and other nonequilibrium processes. For <span class="inline-formula" id="inline-formula-41"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span>, <a id="xref-disp-formula-1-5" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> reduce to the standard Navier–Stokes equations with kinematic viscosity <span class="inline-formula" id="inline-formula-42"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>></mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span>. For <span class="inline-formula" id="inline-formula-43"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>></mml:mo><mml:mn> 0</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mpadded><mml:mo>></mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-44"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span>, <a id="xref-disp-formula-3-1" class="xref-disp-formula" href="#disp-formula-3">Eq. <strong>1c</strong></a> defines the simplest ansatz for an active stress tensor that is isotropic, selects flow patterns of a characteristic scale (<a id="xref-fig-2-1" class="xref-fig" href="#F2">Fig. 1</a>), and yields a stable theory at small and large wavenumbers (<a id="xref-ref-36-3" class="xref-bibr" href="#ref-36">36</a>, <a id="xref-ref-37-3" class="xref-bibr" href="#ref-37">37</a>). The active-to-passive phase transition corresponds to a sign change from <span class="inline-formula" id="inline-formula-45"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo><</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></span></span> to <span class="inline-formula" id="inline-formula-46"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></span></span>, which can be realized experimentally through ATP or nutrient depletion. The nonnegativity of <span class="inline-formula" id="inline-formula-47"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></span></span> and <span class="inline-formula" id="inline-formula-48"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></span></span> follows from stability considerations. <span class="inline-formula" id="inline-formula-49"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></span></span> describes the damping of long-wavelength perturbations on scales much larger than the correlation length of the coherent flow structures, whereas <span class="inline-formula" id="inline-formula-50"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></span></span> and <span class="inline-formula" id="inline-formula-51"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></span></span> account for the growth and damping of modes at intermediate and small scales (<a id="xref-fig-2-2" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>). The resulting nonequilibrium flow structures can be characterized in terms of the typical vortex size <span class="inline-formula" id="inline-formula-52"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>π</mml:mi><mml:msqrt><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:mrow></mml:math></span></span>, growth time scale (<a id="xref-ref-37-4" class="xref-bibr" href="#ref-37">37</a>)<span class="disp-formula" id="disp-formula-4"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:msubsup><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span>circulation speed <span class="inline-formula" id="inline-formula-53"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, and spectral bandwidth (<a id="xref-fig-2-3" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>)<span class="disp-formula" id="disp-formula-5"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo maxsize="210%" minsize="210%">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mfrac><mml:mo>−</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:msqrt><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mfrac></mml:msqrt></mml:mrow></mml:mrow><mml:mo maxsize="210%" minsize="210%">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span></span></p><div id="F2" class="fig pos-float type-featured odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F2.large.jpg?width=800&height=600&carousel=1" title="Exact Beltrami-flow solutions and spontaneous mirror-symmetry breaking in 3D simulations. (A) Linear stability analysis of Eq. 1 distinguishes three different regions in Fourier space: Domains I and III are dissipative, whereas domain II represents active modes. The radius of the active shell II corresponds approximately to the inverse of characteristic pattern formation scale Λ. The bandwidth κ measures the ability of the active fluid component to concentrate power input in Fourier space. (B) Two examples of exact stationary bulk solutions of Eq. 1 realizing Beltrami vector fields of opposite helicity, obtained from Eq. 5 by combining modes of the same helicity located on one of the marginally stable gray surfaces in A. (C) Simulations with random initial condition spontaneously select one of two helicity branches. The histogram represents an average over 150 runs with random initial conditions, sampled over the statistically stationary state starting at time t=20τ (dashed line). Simulation parameters: Λ=75μm, U=72μm/s, κI=0.9/Λ, L=8Λ (see also Fig. 2 and Supporting Information for larger simulations)." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Exact Beltrami-flow solutions and spontaneous mirror-symmetry breaking in 3D simulations. (<em>A</em>) Linear stability analysis of Eq. <strong>1</strong> distinguishes three different regions in Fourier space: Domains I and III are dissipative, whereas domain II represents active modes. The radius of the active shell II corresponds approximately to the inverse of characteristic pattern formation scale <span class="mathjax mml-math"><mml:math><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span>. The bandwidth <span class="mathjax mml-math"><mml:math><mml:mi>κ</mml:mi></mml:math></span> measures the ability of the active fluid component to concentrate power input in Fourier space. (<em>B</em>) Two examples of exact stationary bulk solutions of Eq. <strong>1</strong> realizing Beltrami vector fields of opposite helicity, obtained from Eq. <strong>5</strong> by combining modes of the same helicity located on one of the marginally stable gray surfaces in <em>A</em>. (<em>C</em>) Simulations with random initial condition spontaneously select one of two helicity branches. The histogram represents an average over <span class="mathjax mml-math"><mml:math><mml:mn>150</mml:mn></mml:math></span> runs with random initial conditions, sampled over the statistically stationary state starting at time <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>20</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span> (dashed line). Simulation parameters: <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>75</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span>m, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>72</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span>m/s, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mn>0.9</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>8</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span> (see also Fig. 2 and <em>Supporting Information</em> for larger simulations).</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. 1." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F2.medium.gif" width="440" height="117"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. 1." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F2.medium.gif" width="440" height="117"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F2.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. 1." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F2.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9858" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. 1.</span> <p id="p-14" class="first-child">Exact Beltrami-flow solutions and spontaneous mirror-symmetry breaking in 3D simulations. (<em>A</em>) Linear stability analysis of <a id="xref-disp-formula-1-6" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> distinguishes three different regions in Fourier space: Domains I and III are dissipative, whereas domain II represents active modes. The radius of the active shell II corresponds approximately to the inverse of characteristic pattern formation scale <span class="inline-formula" id="inline-formula-54"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span>. The bandwidth <span class="inline-formula" id="inline-formula-55"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span> measures the ability of the active fluid component to concentrate power input in Fourier space. (<em>B</em>) Two examples of exact stationary bulk solutions of <a id="xref-disp-formula-1-7" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> realizing Beltrami vector fields of opposite helicity, obtained from Eq. <a id="xref-disp-formula-9-1" class="xref-disp-formula" href="#disp-formula-9"><strong>5</strong></a> by combining modes of the same helicity located on one of the marginally stable gray surfaces in <em>A</em>. (<em>C</em>) Simulations with random initial condition spontaneously select one of two helicity branches. The histogram represents an average over <span class="inline-formula" id="inline-formula-56"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>150</mml:mn></mml:math></span></span> runs with random initial conditions, sampled over the statistically stationary state starting at time <span class="inline-formula" id="inline-formula-57"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>20</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> (dashed line). Simulation parameters: <span class="inline-formula" id="inline-formula-58"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>75</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, <span class="inline-formula" id="inline-formula-59"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>72</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m/s, <span class="inline-formula" id="inline-formula-60"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mn>0.9</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-61"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>8</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> (see also <a id="xref-fig-3-1" class="xref-fig" href="#F3">Fig. 2</a> and <a id="xref-sec-13-1" class="xref-sec" href="#sec-13"><em>Supporting Information</em></a> for larger simulations).</p><div class="sb-div caption-clear"></div></div></div><p id="p-15">Specifically, we find <span class="inline-formula" id="inline-formula-62"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 41</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, <span class="inline-formula" id="inline-formula-63"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 57</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m/s, and <span class="inline-formula" id="inline-formula-64"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn>73</mml:mn></mml:mrow></mml:math></span></span> mm<sup>−1</sup> for flows measured in <em>Bacillus subtilis</em> suspensions (<a id="xref-ref-34-6" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-9" class="xref-bibr" href="#ref-35">35</a>) and <span class="inline-formula" id="inline-formula-65"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 130</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, <span class="inline-formula" id="inline-formula-66"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 6.5</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m/s, and <span class="inline-formula" id="inline-formula-67"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 21</mml:mn></mml:mrow></mml:math></span></span> mm<sup>−1</sup> for ATP-driven microtubule–network suspensions (<a id="xref-ref-38-6" class="xref-bibr" href="#ref-38">38</a>) (<a id="xref-sec-13-2" class="xref-sec" href="#sec-13"><em>Comparison with Experiments</em></a>). We emphasize, however, that truncated polynomial stress tensors of the form <a id="xref-disp-formula-3-2" class="xref-disp-formula" href="#disp-formula-3"><strong>1c</strong></a> can provide useful long-wavelength approximations for a broad class of pattern-forming liquids, including magnetically (<a id="xref-ref-61-1" class="xref-bibr" href="#ref-61">61</a>), electrically (<a id="xref-ref-62-1" class="xref-bibr" href="#ref-62">62</a>), thermally (<a id="xref-ref-46-2" class="xref-bibr" href="#ref-46">46</a>, <a id="xref-ref-63-1" class="xref-bibr" href="#ref-63">63</a>, <a id="xref-ref-64-1" class="xref-bibr" href="#ref-64">64</a>), or chemically (<a id="xref-ref-47-2" class="xref-bibr" href="#ref-47">47</a>, <a id="xref-ref-65-1" class="xref-bibr" href="#ref-65">65</a>) driven flows.</p></div><div id="sec-3" class="subsection"><h3>Exact Beltrami-Flow Solutions and Broken-Mirror Symmetry.</h3><p id="p-16">The higher-order Navier–Stokes equations defined by <a id="xref-disp-formula-1-8" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> are invariant under the parity transformation <span class="inline-formula" id="inline-formula-68"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒙</mml:mi><mml:mo>→</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝒙</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>. Their solutions, however, can spontaneously break this mirror symmetry. To demonstrate this explicitly, we construct a family of exact nontrivial stationary solutions in free space by decomposing the Fourier series <span class="inline-formula" id="inline-formula-69"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> of the divergence-free velocity field <span class="inline-formula" id="inline-formula-70"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> into helical modes (<a id="xref-ref-19-5" class="xref-bibr" href="#ref-19">19</a>, <a id="xref-ref-24-4" class="xref-bibr" href="#ref-24">24</a>)<span class="disp-formula" id="disp-formula-6"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[2]</span></span>where <span class="inline-formula" id="inline-formula-71"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:math></span></span> are the eigenvectors of the curl operator, <span class="inline-formula" id="inline-formula-72"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>𝒌</mml:mi></mml:mrow><mml:mo>∧</mml:mo><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>±</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> with <span class="inline-formula" id="inline-formula-73"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>. Projecting <a id="xref-disp-formula-2-1" class="xref-disp-formula" href="#disp-formula-2">Eq. <strong>1b</strong></a> onto helicity eigenstates (<a id="xref-ref-19-6" class="xref-bibr" href="#ref-19">19</a>) yields the evolution equation for the mode amplitudes <span class="inline-formula" id="inline-formula-74"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:math></span></span>,<span class="disp-formula" id="disp-formula-7"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒒</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[3]</span></span>where <span class="inline-formula" id="inline-formula-75"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> is the active stress contribution, and the nonlinear advection is represented by all triadic interactions (<a id="xref-ref-19-7" class="xref-bibr" href="#ref-19">19</a>, <a id="xref-ref-24-5" class="xref-bibr" href="#ref-24">24</a>)<span class="disp-formula" id="disp-formula-8"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>,</mml:mo><mml:mtext>𝒒</mml:mtext><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>p</mml:mi></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mi>q</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle displaystyle="true"><mml:mover accent="true"><mml:mi>𝒉</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mstyle></mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:msup><mml:mo>∧</mml:mo><mml:msup><mml:mrow><mml:mstyle displaystyle="true"><mml:mover accent="true"><mml:mi>𝒉</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mstyle></mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mstyle displaystyle="true"><mml:mover accent="true"><mml:mi>𝒉</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mstyle></mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mstyle displaystyle="true"><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mstyle></mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:msup><mml:msup><mml:mrow><mml:mstyle displaystyle="true"><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mstyle></mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span><span class="disp-formula-label">[4]</span></span>between helical <strong><em>k</em></strong> modes and <span class="inline-formula" id="inline-formula-76"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒑</mml:mi><mml:mo>,</mml:mo><mml:mtext>𝒒</mml:mtext></mml:mrow></mml:math></span></span> modes, where <span class="inline-formula" id="inline-formula-77"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mo>±</mml:mo><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> are the corresponding helicity indexes [overbars denote complex conjugates of <span class="inline-formula" id="inline-formula-78"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msup><mml:mi>𝒉</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:msup></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>𝒉</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒑</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, etc. (<a id="xref-ref-19-8" class="xref-bibr" href="#ref-19">19</a>)]. There are 2 degrees of freedom per wavevector and hence eight types of interactions for every triple <span class="inline-formula" id="inline-formula-79"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>,</mml:mo><mml:mtext>𝒒</mml:mtext><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span>. As evident from <a id="xref-disp-formula-8-1" class="xref-disp-formula" href="#disp-formula-8">Eq. <strong>4</strong></a>, arbitrary superpositions of modes with identical wavenumber <span class="inline-formula" id="inline-formula-80"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>p</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mpadded width="+1.7pt"><mml:mi>q</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:math></span></span> and same helicity index annihilate the advection term, because <span class="inline-formula" id="inline-formula-81"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>p</mml:mi></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mpadded width="+1.7pt"><mml:mi>q</mml:mi></mml:mpadded></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span> in this case. Therefore, by choosing <span class="inline-formula" id="inline-formula-82"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></span></span> to be a root of the polynomial <span class="inline-formula" id="inline-formula-83"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>, corresponding to the gray surfaces in <a id="xref-fig-2-4" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>, we obtain exact stationary solutions<span class="disp-formula" id="disp-formula-9"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>𝒉</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>𝒌</mml:mi></mml:mrow><mml:mo>⋅</mml:mo><mml:mi>𝒙</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[5]</span></span>where <span class="inline-formula" id="inline-formula-84"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝒌</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> ensures real-valued flow fields. In particular, these solutions (<a id="xref-disp-formula-9-2" class="xref-disp-formula" href="#disp-formula-9">Eq. <strong>5</strong></a>) correspond to Beltrami flows (<a id="xref-ref-43-2" class="xref-bibr" href="#ref-43">43</a><a id="xref-ref-44-2" class="xref-down-link" href="#ref-44"><span>⇓</span></a>–<a id="xref-ref-45-2" class="xref-bibr" href="#ref-45">45</a>), obeying <span class="inline-formula" id="inline-formula-85"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>±</mml:mo><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>. Applying the parity operator to any right-handed solution <span class="inline-formula" id="inline-formula-86"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> generates the corresponding left-handed solution <span class="inline-formula" id="inline-formula-87"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> and vice versa (<a id="xref-fig-2-5" class="xref-fig" href="#F2">Fig. 1<em>B</em></a>).</p><p id="p-17">Although the exact solutions <span class="inline-formula" id="inline-formula-88"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> describe stationary Beltrami fields (<a id="xref-ref-43-3" class="xref-bibr" href="#ref-43">43</a><a id="xref-ref-44-3" class="xref-down-link" href="#ref-44"><span>⇓</span></a>–<a id="xref-ref-45-3" class="xref-bibr" href="#ref-45">45</a>) of fixed total helicity <span class="inline-formula" id="inline-formula-89"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msup><mml:mi mathvariant="script">H</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>x</mml:mi></mml:mpadded><mml:msup><mml:mi>𝒗</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow><mml:mo>⋅</mml:mo><mml:msup><mml:mi>𝝎</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, where <span class="inline-formula" id="inline-formula-90"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝎</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> is the vorticity, it is not yet clear whether parity violation is a generic feature of arbitrary time-dependent solutions of <a id="xref-disp-formula-1-9" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a>. As we demonstrate next, simulations with random initial conditions do indeed converge to statistically stationary flow states that spontaneously break mirror symmetry and are close to Beltrami flows.</p></div><div id="sec-4" class="subsection"><h3>Spontaneous Mirror-Symmetry Breaking in Time-Dependent Solutions.</h3><p id="p-18">We simulate the full nonlinear <a id="xref-disp-formula-1-10" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> on a periodic cubic domain (size <span class="inline-formula" id="inline-formula-91"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math></span></span>) using a spectral algorithm (<a id="xref-sec-20-1" class="xref-sec" href="#sec-20"><em>Numerical Methods</em></a>). Simulations are performed for typical bacterial parameters <span class="inline-formula" id="inline-formula-92"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span>, keeping the vortex scale <span class="inline-formula" id="inline-formula-93"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 75</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m and circulation speed <span class="inline-formula" id="inline-formula-94"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 72</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m/s fixed (<a id="xref-ref-34-7" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-10" class="xref-bibr" href="#ref-35">35</a>) and comparing three spectral bandwidths <span class="inline-formula" id="inline-formula-95"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 0.63</mml:mn><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mn>8.4</mml:mn></mml:mrow></mml:math></span></span> mm<sup>−1</sup>, <span class="inline-formula" id="inline-formula-96"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 0.90</mml:mn><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mpadded width="+1.7pt"><mml:mn> 12</mml:mn></mml:mpadded></mml:mrow></mml:math></span></span> mm<sup>−1</sup>, and <span class="inline-formula" id="inline-formula-97"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 2.11</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>28.1</mml:mn></mml:mrow></mml:math></span></span> mm<sup>−1</sup>, corresponding to active fluids with a small (S), an intermediate (I), and a wide (W) range of energy injection scales. A small bandwidth means that the active stresses inject energy into a narrow shell in Fourier space, whereas a wide bandwidth means energy is pumped into a wide range of Fourier modes (<a id="xref-fig-2-6" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>). All simulations are initiated with weak incompressible random flow fields. For all three values of <span class="inline-formula" id="inline-formula-98"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span>, we observe spontaneous mirror-symmetry breaking indicated by the time evolution of the mean helicity <span class="inline-formula" id="inline-formula-99"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>H</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>x</mml:mi></mml:mpadded><mml:mi>h</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, where <span class="inline-formula" id="inline-formula-100"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>h</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> is the local helicity. During the initial relaxation phase, the flow dynamics are attracted to states of well-defined total helicity and remain in such a statistically stationary configuration for the rest of the simulation. As an illustration, <a id="xref-fig-2-7" class="xref-fig" href="#F2">Fig. 1<em>C</em></a> shows results from 150 runs for <span class="inline-formula" id="inline-formula-101"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-102"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 8</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, with flow settling into a positive (negative) mean helicity state 72 (78) times. This spontaneous mirror-symmetry breaking is robust against variations of the bandwidth and simulation box size, as evident from the local vorticity and helicity fields for <span class="inline-formula" id="inline-formula-103"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>κ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-104"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> in <a id="xref-fig-3-2" class="xref-fig" href="#F3">Fig. 2 <em>A</em> and <em>B</em></a>.</p><div id="F3" class="fig pos-float type-figure odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F3.large.jpg?width=800&height=600&carousel=1" title="Active fluids spontaneously break mirror symmetry by realizing Beltrami-type flows. (A) Snapshot of a representative vorticity component field ωx (Movie S1) for an active fluid with small bandwidth κS=0.63/Λ, as defined in Fig. 1A. (B) The corresponding helicity field signals parity-symmetry breaking, leading to a positive-helicity flow in this example. (C) Histograms of the angles between velocity v and vorticity 𝝎 quantify the alignment between the two fields for different active bandwidths κS" class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Active fluids spontaneously break mirror symmetry by realizing Beltrami-type flows. (<em>A</em>) Snapshot of a representative vorticity component field <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:math></span> (Movie S1) for an active fluid with small bandwidth <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mn>0.63</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>, as defined in Fig. 1<em>A</em>. (<em>B</em>) The corresponding helicity field signals parity-symmetry breaking, leading to a positive-helicity flow in this example. (<em>C</em>) Histograms of the angles between velocity <strong><em>v</em></strong> and vorticity <span class="mathjax mml-math"><mml:math><mml:mi>𝝎</mml:mi></mml:math></span> quantify the alignment between the two fields for different active bandwidths <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:mrow></mml:math></span>: The smaller the bandwidth is, the stronger the alignment between <strong><em>v</em></strong> and <span class="mathjax mml-math"><mml:math><mml:mi>𝝎</mml:mi></mml:math></span>. (<em>D</em>) Numerically estimated distributions of the Beltrami measure, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒗</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span>, shown on a log scale. An ideal Beltrami flow with <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span> produces a delta peak centered at <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 1</mml:mn></mml:mrow></mml:math></span>. Identifying <span class="mathjax mml-math"><mml:math><mml:mi>λ</mml:mi></mml:math></span> with the midpoint of the active shell (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>λ</mml:mi><mml:mo>≈</mml:mo><mml:mrow><mml:mi>π</mml:mi><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>), which approximately corresponds to the most unstable wavenumber and the characteristic pattern formation scale, we observe that a smaller active bandwidth leads to a sharper peak and hence more Beltrami-like flows. Data were taken at a single representative time point long after the characteristic relaxation time. Simulation parameters: <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 75</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span>m, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 72</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span>m/s, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>.</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. 2." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F3.medium.gif" width="440" height="106"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. 2." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F3.medium.gif" width="440" height="106"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F3.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. 2." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F3.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/790814" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. 2.</span> <p id="p-19" class="first-child">Active fluids spontaneously break mirror symmetry by realizing Beltrami-type flows. (<em>A</em>) Snapshot of a representative vorticity component field <span class="inline-formula" id="inline-formula-105"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ω</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:math></span></span> (<a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a>) for an active fluid with small bandwidth <span class="inline-formula" id="inline-formula-106"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mn>0.63</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>, as defined in <a id="xref-fig-2-8" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>. (<em>B</em>) The corresponding helicity field signals parity-symmetry breaking, leading to a positive-helicity flow in this example. (<em>C</em>) Histograms of the angles between velocity <strong><em>v</em></strong> and vorticity <span class="inline-formula" id="inline-formula-107"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span> quantify the alignment between the two fields for different active bandwidths <span class="inline-formula" id="inline-formula-108"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:mrow></mml:math></span></span>: The smaller the bandwidth is, the stronger the alignment between <strong><em>v</em></strong> and <span class="inline-formula" id="inline-formula-109"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span>. (<em>D</em>) Numerically estimated distributions of the Beltrami measure, <span class="inline-formula" id="inline-formula-110"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒗</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, shown on a log scale. An ideal Beltrami flow with <span class="inline-formula" id="inline-formula-111"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> produces a delta peak centered at <span class="inline-formula" id="inline-formula-112"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 1</mml:mn></mml:mrow></mml:math></span></span>. Identifying <span class="inline-formula" id="inline-formula-113"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math></span></span> with the midpoint of the active shell (<span class="inline-formula" id="inline-formula-114"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>≈</mml:mo><mml:mrow><mml:mi>π</mml:mi><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>), which approximately corresponds to the most unstable wavenumber and the characteristic pattern formation scale, we observe that a smaller active bandwidth leads to a sharper peak and hence more Beltrami-like flows. Data were taken at a single representative time point long after the characteristic relaxation time. Simulation parameters: <span class="inline-formula" id="inline-formula-115"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 75</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, <span class="inline-formula" id="inline-formula-116"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn> 72</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m/s, <span class="inline-formula" id="inline-formula-117"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>.</p><div class="sb-div caption-clear"></div></div></div></div><div id="sec-5" class="subsection"><h3>Beltrami-Flow Attractors.</h3><p id="p-20">Having confirmed spontaneous parity violation for the time-dependent solutions of <a id="xref-disp-formula-1-11" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a>, we next characterize the chaotic flow attractors. To this end, we measure and compare the histograms of the angles between the local velocity field <span class="inline-formula" id="inline-formula-118"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> and vorticity field <span class="inline-formula" id="inline-formula-119"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝝎</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> for the three bandwidths <span class="inline-formula" id="inline-formula-120"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:mrow></mml:math></span></span>. Our numerical results reveal that a smaller active bandwidth, corresponding to a more sharply defined scale selection, causes a stronger alignment of the two fields (<a id="xref-fig-3-3" class="xref-fig" href="#F3">Fig. 2<em>C</em></a>). Recalling that perfect alignment, described by <span class="inline-formula" id="inline-formula-121"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝎</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> with eigenvalue <span class="inline-formula" id="inline-formula-122"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math></span></span>, is the defining feature of Beltrami flows (<a id="xref-ref-43-4" class="xref-bibr" href="#ref-43">43</a><a id="xref-ref-44-4" class="xref-down-link" href="#ref-44"><span>⇓</span></a>–<a id="xref-ref-45-4" class="xref-bibr" href="#ref-45">45</a>), we introduce the Beltrami measure <span class="inline-formula" id="inline-formula-123"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒗</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>. For ideal Beltrami fields, the distribution of <span class="inline-formula" id="inline-formula-124"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math></span></span> becomes a delta peak centered at <span class="inline-formula" id="inline-formula-125"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 1</mml:mn></mml:mrow></mml:math></span></span>. Identifying <span class="inline-formula" id="inline-formula-126"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math></span></span> with the midpoint of the active shell (<span class="inline-formula" id="inline-formula-127"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>λ</mml:mi></mml:mpadded><mml:mo>≈</mml:mo><mml:mrow><mml:mi>π</mml:mi><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>), which approximately corresponds to the most dominant pattern formation scale in <a id="xref-disp-formula-1-12" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a>, we indeed find that the numerically computed flow fields exhibit <span class="inline-formula" id="inline-formula-128"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math></span></span> distributions that are sharply peaked at <span class="inline-formula" id="inline-formula-129"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>β</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 1</mml:mn></mml:mrow></mml:math></span></span> (<a id="xref-fig-3-4" class="xref-fig" href="#F3">Fig. 2<em>D</em></a>). Keeping <span class="inline-formula" id="inline-formula-130"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-131"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math></span></span> constant, the sharpness of the peak increases with decreasing active bandwidth <span class="inline-formula" id="inline-formula-132"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span>. These results imply that active fluids with well-defined intrinsic scale selection realize flow structures that are statistically close to Beltrami fields, as suggested by the particular analytical solutions derived earlier.</p></div></div><div class="section discussion" id="sec-6"><h2 class="">Discussion</h2><div id="sec-7" class="subsection"><h3>Spontaneous Parity Breaking vs. Surgical Mode Removal.</h3><p id="p-21">Important previous studies identified bifurcation mechanisms (<a id="xref-ref-66-1" class="xref-bibr" href="#ref-66">66</a><a id="xref-ref-67-1" class="xref-down-link" href="#ref-67"><span>⇓</span></a>–<a id="xref-ref-68-1" class="xref-bibr" href="#ref-68">68</a>) leading to parity violation in 1D and 2D (<a id="xref-ref-69-1" class="xref-bibr" href="#ref-69">69</a>) continuum models of pattern-forming nonequilibrium systems (<a id="xref-ref-70-1" class="xref-bibr" href="#ref-70">70</a>, <a id="xref-ref-71-1" class="xref-bibr" href="#ref-71">71</a>). The above analytical and numerical results generalize these ideas to 3D fluid flows, by showing that an active scale selection mechanism can induce spontaneous helical mirror-symmetry breaking. Such self-organized parity violation can profoundly affect energy transport and mixing in 3D active fluids, which do not satisfy the premises of Kraichnan’s thermodynamic no-go argument (<a id="xref-ref-23-3" class="xref-bibr" href="#ref-23">23</a>). An insightful recent study (<a id="xref-ref-24-6" class="xref-bibr" href="#ref-24">24</a>), based on the classical Navier–Stokes equation, found that an ad hoc projection of solutions to positive or negative helicity subspaces can result in an inverse cascade but it has remained an open question whether such a surgical mode removal can be realized experimentally in passive fluids. By contrast, active fluids spontaneously achieve helical parity breaking (<a id="xref-fig-2-9" class="xref-fig" href="#F2">Fig. 1<em>C</em></a>) by approaching Beltrami-flow states (<a id="xref-fig-3-5" class="xref-fig" href="#F3">Fig. 2 <em>C</em> and <em>D</em></a>), suggesting the possibility of a self-organized inverse energy cascade even in 3D. Before testing this hypothesis we recall that the model defined by <a id="xref-disp-formula-1-13" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> merely assumes the existence of linear active stresses to account for pattern scale selection as observed in a wide range of microbial suspensions (<a id="xref-ref-35-11" class="xref-bibr" href="#ref-35">35</a>, <a id="xref-ref-38-7" class="xref-bibr" href="#ref-38">38</a>, <a id="xref-ref-54-2" class="xref-bibr" href="#ref-54">54</a>, <a id="xref-ref-72-1" class="xref-bibr" href="#ref-72">72</a>), but does not introduce nonlinearities beyond those already present in the classical Navier–Stokes equations. That is, energy redistribution in the solvent fluid is governed by the advective nonlinearities as in conventional passive liquids.</p></div><div id="sec-8" class="subsection"><h3>Inverse Cascade in 3D Active Fluids.</h3><p id="p-22">To quantify how pattern scale selection controls parity breaking and energy transport in active fluids, we analyzed large-scale simulations (<span class="inline-formula" id="inline-formula-133"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>; <a id="xref-fig-3-6" class="xref-fig" href="#F3">Fig. 2 <em>A</em> and <em>B</em></a>) for different values of the activity bandwidth <span class="inline-formula" id="inline-formula-134"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span> (<a id="xref-fig-2-10" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>) while keeping the pattern scale <span class="inline-formula" id="inline-formula-135"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span> and the circulation speed <span class="inline-formula" id="inline-formula-136"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math></span></span> fixed. The active shell (red domain II in <a id="xref-fig-2-11" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>) corresponds to the energy injection range in Fourier space and provides a natural separation between large flow scales (blue domain I) and small flow scales (blue domain III). Consequently, the forward cascade corresponds to a net energy flux from domain II to domain III, whereas an inverse cascade transports energy from domain II to I. We calculate energy spectra <span class="inline-formula" id="inline-formula-137"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and energy fluxes <span class="inline-formula" id="inline-formula-138"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> directly from our simulation data, by decomposing the velocity field into helical modes as in <a id="xref-disp-formula-6-1" class="xref-disp-formula" href="#disp-formula-6">Eq. <strong>2</strong></a>, which yields a natural splitting into cumulative energy and flux contributions <span class="inline-formula" id="inline-formula-139"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-140"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> from helical modes <span class="inline-formula" id="inline-formula-141"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> lying on the wavenumber shell <span class="inline-formula" id="inline-formula-142"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></span></span> (<a id="xref-sec-20-2" class="xref-sec" href="#sec-20"><em>Numerical Methods</em></a>). Time-averaged spectra and fluxes are computed for each simulation run after the system has relaxed to a statistically stationary state (<a id="xref-fig-5-1" class="xref-fig" href="#F5">Fig. S2</a>). For a small injection bandwidth <span class="inline-formula" id="inline-formula-143"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span>, the energy spectra <span class="inline-formula" id="inline-formula-144"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> reflect the broken mirror symmetry, with most of the energy being stored in either the positive-helicity or the negative-helicity modes (<a id="xref-fig-4-1" class="xref-fig" href="#F4">Fig. 3<em>A</em></a>), depending on the initial conditions. Moreover, in addition to the expected 3D forward transfer, the simulation data for <span class="inline-formula" id="inline-formula-145"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span> also show a significant inverse transfer, signaled by the negative values of the total flux <span class="inline-formula" id="inline-formula-146"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> (yellow curve in <a id="xref-fig-4-2" class="xref-fig" href="#F4">Fig. 3<em>B</em></a>) in domain I. As evident from the blue curves in <a id="xref-fig-4-3" class="xref-fig" href="#F4">Fig. 3 <em>A</em> and <em>B</em></a>, this inverse cascade is facilitated by the helical modes that carry most of the energy. For a large injection bandwidth <span class="inline-formula" id="inline-formula-147"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:mpadded><mml:mo>≫</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mrow></mml:math></span></span>, the energy spectra continue to show signatures of helical symmetry breaking (<a id="xref-fig-4-4" class="xref-fig" href="#F4">Fig. 3<em>E</em></a>), but the energy transported to larger scales becomes negligible relative to the forward cascade, as contributions from opposite-helicity modes approximately cancel in the long-wavelength domain I (<a id="xref-fig-4-5" class="xref-fig" href="#F4">Fig. 3<em>F</em></a>). Results for the intermediate case <span class="inline-formula" id="inline-formula-148"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">M</mml:mi></mml:msub></mml:math></span></span> still show a significant inverse transfer (<a id="xref-fig-6-1" class="xref-fig" href="#F6">Figs. S3</a> and <a id="xref-fig-7-1" class="xref-fig" href="#F7">S4<em>E</em></a>), demonstrating how the activity bandwidth—or, equivalently, the pattern selection range—controls both parity violation and inverse cascade formation in an active fluid. The upward transfer is noninertial at intermediate scales, as indicated by the wavenumber dependence of the energy flux (<a id="xref-fig-4-6" class="xref-fig" href="#F4">Fig. 3<em>B</em></a>). At very large scales <span class="inline-formula" id="inline-formula-149"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≫</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math></span></span>, however, the flux approaches an inertial plateau (<a id="xref-sec-27-1" class="xref-sec" href="#sec-27"><em>Cascade Characteristics</em></a>). In contrast to the energy-mediated 2D inverse cascade in passive fluids, the helicity-driven 3D inverse cascade in active fluids is linked to the formation of extended vortex chain complexes that move collectively through the fluid (<a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a> and <a id="xref-sec-27-2" class="xref-sec" href="#sec-27"><em>Cascade Characteristics</em></a>).</p><div id="F4" class="fig pos-float type-figure odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F4.large.jpg?width=800&height=600&carousel=1" title="Scale selection controls mirror-symmetry breaking and induces an inverse energy cascade. We demonstrate these effects for active fluids with (A–D) a small active bandwidth κS and (E–H) a wide bandwidth κW (Fig. 1A). The intermediate case κI is presented in Fig. S3. (A) Energy spectra e±(k) of the helical velocity-field modes show strong symmetry breaking for small bandwidth parameter κS. In this example, the system spontaneously selects positive helicity modes, such that e+(k)>e−(k) at all dominant wavenumbers. Dashed vertical lines indicate the boundaries of the energy injection domain II. (B) The resulting energy fluxes Π±(k) combine into the total flux Π(k), which is negative in region I and positive in region III, signaling inverse and forward energy transfers, respectively. (C) Contributions to the energy flow ⟨TKPQsKsPsQ⟩ between the three spectral domains I, II, and III (18 possibilities, columns) from the eight types of triad interactions (rows). In reflection-invariant turbulence, this table remains unchanged under upside-down flipping (+↔−). Instead, we observe a strong asymmetry, with two cumulative triads (D) dominating the energy transfer. Red and blue arrows represent transfer toward large and small scales, respectively, and thickness represents magnitude of energy flow. Green arrows (H) represent transfer within the same spectral domain. The direction of the energy flow is in agreement with the instability assumption of Waleffe (19). In this case, 18.2% of the injected energy is transferred from region II to region I and 81.8% is transferred from region II to III. (E–H) The same plots for an active fluid with wide active bandwidth κW. (E and F) Energy spectra show weaker parity breaking (E) and suppression of the inverse energy cascade (F). (G) The energy flow table partially recovers the upside-down (+↔−) symmetry. (H) The most active triads now favor the forward cascade, so that only 1.1% of the injected energy flows into region I, whereas 98.9% is transferred into region III. Data represent averages over single runs (Fig. S2). Simulation parameters are identical to those in Fig. 2." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Scale selection controls mirror-symmetry breaking and induces an inverse energy cascade. We demonstrate these effects for active fluids with (<em>A–D</em>) a small active bandwidth <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span> and (<em>E–H</em>) a wide bandwidth <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span> (Fig. 1<em>A</em>). The intermediate case <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:math></span> is presented in Fig. S3. (<em>A</em>) Energy spectra <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span> of the helical velocity-field modes show strong symmetry breaking for small bandwidth parameter <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span>. In this example, the system spontaneously selects positive helicity modes, such that <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span> at all dominant wavenumbers. Dashed vertical lines indicate the boundaries of the energy injection domain II. (<em>B</em>) The resulting energy fluxes <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span> combine into the total flux <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span>, which is negative in region I and positive in region III, signaling inverse and forward energy transfers, respectively. (<em>C</em>) Contributions to the energy flow <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:math></span> between the three spectral domains I, II, and III (18 possibilities, columns) from the eight types of triad interactions (rows). In reflection-invariant turbulence, this table remains unchanged under upside-down flipping (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>↔</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span>). Instead, we observe a strong asymmetry, with two cumulative triads (<em>D</em>) dominating the energy transfer. Red and blue arrows represent transfer toward large and small scales, respectively, and thickness represents magnitude of energy flow. Green arrows (<em>H</em>) represent transfer within the same spectral domain. The direction of the energy flow is in agreement with the instability assumption of Waleffe (19). In this case, 18.2% of the injected energy is transferred from region II to region I and 81.8% is transferred from region II to III. (<em>E–H</em>) The same plots for an active fluid with wide active bandwidth <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span>. (<em>E</em> and <em>F</em>) Energy spectra show weaker parity breaking (<em>E</em>) and suppression of the inverse energy cascade (<em>F</em>). (<em>G</em>) The energy flow table partially recovers the upside-down (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>↔</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span>) symmetry. (<em>H</em>) The most active triads now favor the forward cascade, so that only 1.1% of the injected energy flows into region I, whereas 98.9% is transferred into region III. Data represent averages over single runs (Fig. S2). Simulation parameters are identical to those in Fig. 2.</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. 3." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F4.medium.gif" width="440" height="278"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. 3." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F4.medium.gif" width="440" height="278"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F4.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. 3." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F4.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9861" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. 3.</span> <p id="p-23" class="first-child">Scale selection controls mirror-symmetry breaking and induces an inverse energy cascade. We demonstrate these effects for active fluids with (<em>A–D</em>) a small active bandwidth <span class="inline-formula" id="inline-formula-150"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span> and (<em>E–H</em>) a wide bandwidth <span class="inline-formula" id="inline-formula-151"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span></span> (<a id="xref-fig-2-12" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>). The intermediate case <span class="inline-formula" id="inline-formula-152"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:math></span></span> is presented in <a id="xref-fig-6-2" class="xref-fig" href="#F6">Fig. S3</a>. (<em>A</em>) Energy spectra <span class="inline-formula" id="inline-formula-153"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> of the helical velocity-field modes show strong symmetry breaking for small bandwidth parameter <span class="inline-formula" id="inline-formula-154"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span>. In this example, the system spontaneously selects positive helicity modes, such that <span class="inline-formula" id="inline-formula-155"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> at all dominant wavenumbers. Dashed vertical lines indicate the boundaries of the energy injection domain II. (<em>B</em>) The resulting energy fluxes <span class="inline-formula" id="inline-formula-156"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> combine into the total flux <span class="inline-formula" id="inline-formula-157"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>, which is negative in region I and positive in region III, signaling inverse and forward energy transfers, respectively. (<em>C</em>) Contributions to the energy flow <span class="inline-formula" id="inline-formula-158"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:math></span></span> between the three spectral domains I, II, and III (18 possibilities, columns) from the eight types of triad interactions (rows). In reflection-invariant turbulence, this table remains unchanged under upside-down flipping (<span class="inline-formula" id="inline-formula-159"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>↔</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span></span>). Instead, we observe a strong asymmetry, with two cumulative triads (<em>D</em>) dominating the energy transfer. Red and blue arrows represent transfer toward large and small scales, respectively, and thickness represents magnitude of energy flow. Green arrows (<em>H</em>) represent transfer within the same spectral domain. The direction of the energy flow is in agreement with the instability assumption of Waleffe (<a id="xref-ref-19-9" class="xref-bibr" href="#ref-19">19</a>). In this case, 18.2% of the injected energy is transferred from region II to region I and 81.8% is transferred from region II to III. (<em>E–H</em>) The same plots for an active fluid with wide active bandwidth <span class="inline-formula" id="inline-formula-160"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span></span>. (<em>E</em> and <em>F</em>) Energy spectra show weaker parity breaking (<em>E</em>) and suppression of the inverse energy cascade (<em>F</em>). (<em>G</em>) The energy flow table partially recovers the upside-down (<span class="inline-formula" id="inline-formula-161"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>↔</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span></span>) symmetry. (<em>H</em>) The most active triads now favor the forward cascade, so that only 1.1% of the injected energy flows into region I, whereas 98.9% is transferred into region III. Data represent averages over single runs (<a id="xref-fig-5-2" class="xref-fig" href="#F5">Fig. S2</a>). Simulation parameters are identical to those in <a id="xref-fig-3-7" class="xref-fig" href="#F3">Fig. 2</a>.</p><div class="sb-div caption-clear"></div></div></div><div id="F5" class="fig pos-float type-supplementary-material odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F5.large.jpg?width=800&height=600&carousel=1" title="Numerical estimation of the stationary energy spectra for the narrow bandwidth in Fig. 3A of the main text. (A and B) Kinetic energy (A) and helicity time series (B) are used to determine the relaxation time R to stationary state (vertical dashed line, R= 31.7τ in this case). (C) Convergence of the energy spectra estimates using |⟨ϵ±⟩R,Δ−⟨ϵ±⟩R,Δmax|2/|⟨ϵ±⟩R,Δmax|2 (Eq. S35), as a function of the averaging interval Δ. (D) Relative difference (l2 norm) between the momentary energy spectra and their (stationary) mean as a function of time." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Numerical estimation of the stationary energy spectra for the narrow bandwidth in Fig. 3<em>A</em> of the main text. (<em>A</em> and <em>B</em>) Kinetic energy (<em>A</em>) and helicity time series (<em>B</em>) are used to determine the relaxation time <span class="mathjax mml-math"><mml:math><mml:mi mathvariant="script">R</mml:mi></mml:math></span> to stationary state (vertical dashed line, <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="script">R</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 31.7</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span> in this case). (<em>C</em>) Convergence of the energy spectra estimates using <span class="mathjax mml-math"><mml:math><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></span> (Eq. <strong>S35</strong>), as a function of the averaging interval <span class="mathjax mml-math"><mml:math><mml:mi mathvariant="normal">Δ</mml:mi></mml:math></span>. (<em>D</em>) Relative difference (<span class="mathjax mml-math"><mml:math><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></span> norm) between the momentary energy spectra and their (stationary) mean as a function of time.</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. S2." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F5.medium.gif" width="437" height="440"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. S2." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F5.medium.gif" width="437" height="440"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F5.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. S2." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F5.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9863" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. S2.</span> <p id="p-24" class="first-child">Numerical estimation of the stationary energy spectra for the narrow bandwidth in <a id="xref-fig-4-7" class="xref-fig" href="#F4">Fig. 3<em>A</em></a> of the main text. (<em>A</em> and <em>B</em>) Kinetic energy (<em>A</em>) and helicity time series (<em>B</em>) are used to determine the relaxation time <span class="inline-formula" id="inline-formula-162"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">R</mml:mi></mml:math></span></span> to stationary state (vertical dashed line, <span class="inline-formula" id="inline-formula-163"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="script">R</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 31.7</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> in this case). (<em>C</em>) Convergence of the energy spectra estimates using <span class="inline-formula" id="inline-formula-164"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></span></span> (<a id="xref-disp-formula-62-1" class="xref-disp-formula" href="#disp-formula-62">Eq. <strong>S35</strong></a>), as a function of the averaging interval <span class="inline-formula" id="inline-formula-165"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Δ</mml:mi></mml:math></span></span>. (<em>D</em>) Relative difference (<span class="inline-formula" id="inline-formula-166"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></span></span> norm) between the momentary energy spectra and their (stationary) mean as a function of time.</p><div class="sb-div caption-clear"></div></div></div><div id="F6" class="fig pos-float type-supplementary-material odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F6.large.jpg?width=800&height=600&carousel=1" title="Mirror-symmetry breaking and inverse energy cascade for an active fluid with the intermediate bandwidth κI, showing the same quantities as in Fig. 3 of the main text. Overall, 15.4% of the injected energy flows into region I, whereas 84.6% flows into region III." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup">Mirror-symmetry breaking and inverse energy cascade for an active fluid with the intermediate bandwidth <span xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:math></span>, showing the same quantities as in Fig. 3 of the main text. Overall, 15.4% of the injected energy flows into region I, whereas 84.6% flows into region III.</div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. S3." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F6.medium.gif" width="339" height="440"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. S3." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F6.medium.gif" width="339" height="440"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F6.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. S3." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F6.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9865" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. S3.</span> <p id="p-25" class="first-child">Mirror-symmetry breaking and inverse energy cascade for an active fluid with the intermediate bandwidth <span class="inline-formula" id="inline-formula-167"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:msub></mml:math></span></span>, showing the same quantities as in <a id="xref-fig-4-8" class="xref-fig" href="#F4">Fig. 3</a> of the main text. Overall, 15.4% of the injected energy flows into region I, whereas 84.6% flows into region III.</p><div class="sb-div caption-clear"></div></div></div><div id="F7" class="fig pos-float type-supplementary-material odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F7.large.jpg?width=800&height=600&carousel=1" title="Characterization of the inverse energy cascade. (A and B) Two horizontal cuts through the 3D simulation domain for a small bandwidth κ𝐒, showing that the inverse cascade is not characterized by vortex mergers, but rather by chain-like complexes, visible as “dark” structures in Movie S1, Left. (C and D) Same flow-field snapshots as in A and B but now represented through the local helicity field. The chain-like large-scale structures carry most of the helicity. They do not merge, but rather form extended filaments and clusters that move throughout the simulation domain (Movie S1). Domain size L= 32Λ. (E) The proportion of the energy injected by the active component that is transported to region I (corresponding to large scales, compare with Fig. 1A of the main text) as a function of the active bandwidth κ. (F) Absolute value of the energy flux for an active fluid with small bandwidth κ𝐒 for different simulation domain sizes. In region I, corresponding to large scales, the upward transfer is noninertial at intermediate wavenumbers with the flux exhibiting k3 scaling. For k→0, however, the flux approaches a constant plateau value, indicating that inertial effects start dominate at very large scales ≫Λ." class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">Characterization of the inverse energy cascade. (<em>A</em> and <em>B</em>) Two horizontal cuts through the 3D simulation domain for a small bandwidth <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi>𝐒</mml:mi></mml:msub></mml:math></span>, showing that the inverse cascade is not characterized by vortex mergers, but rather by chain-like complexes, visible as “dark” structures in Movie S1, <em>Left</em>. (<em>C</em> and <em>D</em>) Same flow-field snapshots as in <em>A</em> and <em>B</em> but now represented through the local helicity field. The chain-like large-scale structures carry most of the helicity. They do not merge, but rather form extended filaments and clusters that move throughout the simulation domain (Movie S1). Domain size <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>. (<em>E</em>) The proportion of the energy injected by the active component that is transported to region I (corresponding to large scales, compare with Fig. 1<em>A</em> of the main text) as a function of the active bandwidth <span class="mathjax mml-math"><mml:math><mml:mi>κ</mml:mi></mml:math></span>. (<em>F</em>) Absolute value of the energy flux for an active fluid with small bandwidth <span class="mathjax mml-math"><mml:math><mml:msub><mml:mi>κ</mml:mi><mml:mi>𝐒</mml:mi></mml:msub></mml:math></span> for different simulation domain sizes. In region I, corresponding to large scales, the upward transfer is noninertial at intermediate wavenumbers with the flux exhibiting <span class="mathjax mml-math"><mml:math><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></span> scaling. For <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></span>, however, the flux approaches a constant plateau value, indicating that inertial effects start dominate at very large scales <span class="mathjax mml-math"><mml:math><mml:mrow><mml:mo>≫</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math></span>.</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. S4." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F7.medium.gif" width="440" height="300"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. S4." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F7.medium.gif" width="440" height="300"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F7.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. S4." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F7.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/790815" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. S4.</span> <p id="p-26" class="first-child">Characterization of the inverse energy cascade. (<em>A</em> and <em>B</em>) Two horizontal cuts through the 3D simulation domain for a small bandwidth <span class="inline-formula" id="inline-formula-168"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi>𝐒</mml:mi></mml:msub></mml:math></span></span>, showing that the inverse cascade is not characterized by vortex mergers, but rather by chain-like complexes, visible as “dark” structures in <a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a>, <em>Left</em>. (<em>C</em> and <em>D</em>) Same flow-field snapshots as in <em>A</em> and <em>B</em> but now represented through the local helicity field. The chain-like large-scale structures carry most of the helicity. They do not merge, but rather form extended filaments and clusters that move throughout the simulation domain (<a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a>). Domain size <span class="inline-formula" id="inline-formula-169"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>. (<em>E</em>) The proportion of the energy injected by the active component that is transported to region I (corresponding to large scales, compare with <a id="xref-fig-2-13" class="xref-fig" href="#F2">Fig. 1<em>A</em></a> of the main text) as a function of the active bandwidth <span class="inline-formula" id="inline-formula-170"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math></span></span>. (<em>F</em>) Absolute value of the energy flux for an active fluid with small bandwidth <span class="inline-formula" id="inline-formula-171"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi>𝐒</mml:mi></mml:msub></mml:math></span></span> for different simulation domain sizes. In region I, corresponding to large scales, the upward transfer is noninertial at intermediate wavenumbers with the flux exhibiting <span class="inline-formula" id="inline-formula-172"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></span></span> scaling. For <span class="inline-formula" id="inline-formula-173"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></span></span>, however, the flux approaches a constant plateau value, indicating that inertial effects start dominate at very large scales <span class="inline-formula" id="inline-formula-174"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≫</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math></span></span>.</p><div class="sb-div caption-clear"></div></div></div></div><div id="sec-9" class="subsection"><h3>Triad Interactions.</h3><p id="p-27">Our numerical flux measurements confirm directly the existence of a self-sustained 3D inverse cascade induced by spontaneous parity violation, consistent with earlier projection-based arguments for the classical Navier–Stokes equations (<a id="xref-ref-24-7" class="xref-bibr" href="#ref-24">24</a>). An inverse energy cascade can exist in 3D active fluids because mirror-symmetry breaking favors only a subclass of all possible triad interactions, which describe advective energy transfer in Fourier space between velocity modes <span class="inline-formula" id="inline-formula-175"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒑</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mtext>𝒒</mml:mtext><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math></span></span> with <span class="inline-formula" id="inline-formula-176"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>+</mml:mo><mml:mpadded width="+1.7pt"><mml:mtext>𝒒</mml:mtext></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span> (compare with <a id="xref-disp-formula-8-2" class="xref-disp-formula" href="#disp-formula-8">Eq. <strong>4</strong></a>). To analyze in detail which triads are spontaneously activated in a pattern-forming nonequilibrium fluid, we consider combinations <span class="inline-formula" id="inline-formula-177"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi mathvariant="normal">I</mml:mi><mml:mo>,</mml:mo><mml:mi>II</mml:mi><mml:mo>,</mml:mo><mml:mi>III</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> of the spectral domains in <a id="xref-fig-2-14" class="xref-fig" href="#F2">Fig. 1<em>A</em></a> and distinguish modes by their helicity index <span class="inline-formula" id="inline-formula-178"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mo>±</mml:mo><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>. The helicity-resolved integrated energy flow into the region <span class="inline-formula" id="inline-formula-179"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span> due to interaction with regions <span class="inline-formula" id="inline-formula-180"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-181"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span> is given by (<a id="xref-sec-20-3" class="xref-sec" href="#sec-20"><em>Numerical Methods</em></a>)<span class="disp-formula" id="disp-formula-10"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>Q</mml:mi><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[6]</span></span>where the unsymmetrized flows are defined by<span class="disp-formula" id="disp-formula-11"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>x</mml:mi><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:msubsup></mml:mrow><mml:mo>⋅</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>P</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:msubsup><mml:mo>⋅</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>Q</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[7]</span></span>with <span class="inline-formula" id="inline-formula-182"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> denoting the helical Littlewood–Paley velocity components, obtained by projecting on modes of a given helicity index <span class="inline-formula" id="inline-formula-183"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mo>±</mml:mo><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> restricted to the Fourier domain <span class="inline-formula" id="inline-formula-184"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math></span></span>. Entries of <span class="inline-formula" id="inline-formula-185"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">T</mml:mi></mml:math></span></span> are large when the corresponding triads are dominant.</p><p id="p-28">For active fluids, Fourier space is naturally partitioned into three regions (<a id="xref-fig-2-15" class="xref-fig" href="#F2">Fig. 1<em>A</em></a>) and there are <span class="inline-formula" id="inline-formula-186"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math></span></span> helicity index combinations. The triad tensor <span class="inline-formula" id="inline-formula-187"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">T</mml:mi></mml:math></span></span> is symmetric in the last two indexes, so that <span class="inline-formula" id="inline-formula-188"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">T</mml:mi></mml:math></span></span> has <span class="inline-formula" id="inline-formula-189"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>8</mml:mn><mml:mo>×</mml:mo><mml:mn>18</mml:mn></mml:mrow></mml:math></span></span> independent components encoding the fine structure of the advective energy transport. Stationary time averages for <span class="inline-formula" id="inline-formula-190"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:math></span></span>, measured from our simulations (<a id="xref-sec-20-4" class="xref-sec" href="#sec-20"><em>Numerical Methods</em></a>) for small (<span class="inline-formula" id="inline-formula-191"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span>) and wide (<span class="inline-formula" id="inline-formula-192"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span></span>) energy injection bandwidths, are shown in <a id="xref-fig-4-9" class="xref-fig" href="#F4">Fig. 3 <em>C</em> and <em>G</em></a>. For reflection-symmetric turbulent flows, these two tables would remain unchanged under an upside-down flip (<span class="inline-formula" id="inline-formula-193"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>↔</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span></span>). By contrast, we find a strong asymmetry for a narrow bandwidth <span class="inline-formula" id="inline-formula-194"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span> (<a id="xref-fig-4-10" class="xref-fig" href="#F4">Fig. 3<em>C</em></a>), which persists in weakened form for <span class="inline-formula" id="inline-formula-195"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span></span> (<a id="xref-fig-4-11" class="xref-fig" href="#F4">Fig. 3<em>G</em></a>). Specifically, we observe for <span class="inline-formula" id="inline-formula-196"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span> two dominant cumulative triads with energy flowing out of the active spectral range II into the two passive domains I and III (<a id="xref-fig-4-12" class="xref-fig" href="#F4">Fig. 3<em>D</em></a>). These cumulative triads visualize dominant entries of the tables in <a id="xref-fig-4-13" class="xref-fig" href="#F4">Fig. 3 <em>C</em> and <em>G</em></a> and represent the total contributions from all triadic interactions between modes with given helicity indexes and with “legs” lying in the specified spectral domain. The observed energy transfer directions, with energy flowing out of the intermediate domain II when the small-scale modes carry the same helicity index, are in agreement with a turbulent instability mechanism proposed by Waleffe (<a id="xref-ref-19-10" class="xref-bibr" href="#ref-19">19</a>). Interestingly, however, our numerical results show that both “R”-interaction channels <span class="inline-formula" id="inline-formula-197"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-198"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:math></span></span> contribute substantially even in the case of strong parity breaking (<span class="inline-formula" id="inline-formula-199"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span>; <a id="xref-fig-4-14" class="xref-fig" href="#F4">Fig. 3<em>D</em></a>); when one surgically projects the full dynamics onto states with fixed parity, only the +++ channel remains (<a id="xref-ref-24-8" class="xref-bibr" href="#ref-24">24</a>). By contrast, for a wide bandwidth <span class="inline-formula" id="inline-formula-200"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:msub></mml:math></span></span>, the dominating triad interactions (<a id="xref-fig-4-15" class="xref-fig" href="#F4">Fig. 3<em>H</em></a>) favor the forward cascade. Hence, the inverse energy cascade in 3D active fluids is possible because only a subset of triadic interactions is active in the presence of strong mirror-symmetry breaking. This phenomenon is controlled by the spectral bandwidth of the scale selection mechanism.</p></div><div id="sec-10" class="subsection"><h3>Enhanced Mixing.</h3><p id="p-29"><a id="xref-disp-formula-1-14" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> describe a 3D isotropic fluid capable of transporting energy from smaller to larger scales. Previously, self-organized inverse cascades were demonstrated only in effectively 2D flows (<a id="xref-ref-6-2" class="xref-bibr" href="#ref-6">6</a>, <a id="xref-ref-18-2" class="xref-bibr" href="#ref-18">18</a>, <a id="xref-ref-20-3" class="xref-bibr" href="#ref-20">20</a>, <a id="xref-ref-25-2" class="xref-bibr" href="#ref-25">25</a>, <a id="xref-ref-73-1" class="xref-bibr" href="#ref-73">73</a><a id="xref-ref-74-1" class="xref-down-link" href="#ref-74"><span>⇓</span></a><a id="xref-ref-75-1" class="xref-down-link" href="#ref-75"><span>⇓</span></a><a id="xref-ref-76-1" class="xref-down-link" href="#ref-76"><span>⇓</span></a><a id="xref-ref-77-1" class="xref-down-link" href="#ref-77"><span>⇓</span></a><a id="xref-ref-78-1" class="xref-down-link" href="#ref-78"><span>⇓</span></a>–<a id="xref-ref-79-1" class="xref-bibr" href="#ref-79">79</a>). The 2D inverse cascade has been intensely studied in meteorology (<a id="xref-ref-6-3" class="xref-bibr" href="#ref-6">6</a>, <a id="xref-ref-7-2" class="xref-bibr" href="#ref-7">7</a>), a prominent example being Jovian atmospheric dynamics (<a id="xref-ref-80-1" class="xref-bibr" href="#ref-80">80</a>), because of its importance for the mixing of thin fluid layers (<a id="xref-ref-81-1" class="xref-bibr" href="#ref-81">81</a><a id="xref-ref-82-1" class="xref-down-link" href="#ref-82"><span>⇓</span></a>–<a id="xref-ref-83-1" class="xref-bibr" href="#ref-83">83</a>). Analogously, the 3D inverse cascade and the underlying Beltrami-flow structure are expected to enhance mixing and transport in active fluids. Arnold (<a id="xref-ref-43-5" class="xref-bibr" href="#ref-43">43</a>) showed that steady solutions of the incompressible Euler equations include Beltrami-type ABC flows (<a id="xref-ref-44-5" class="xref-bibr" href="#ref-44">44</a>) characterized by chaotic streamlines. Similarly, the Beltrami structure of the active-flow attractors of <a id="xref-disp-formula-1-15" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> implies enhanced local mixing. Combined with the presence of an inverse cascade, which facilitates additional large-scale mixing through the excitation of long-wavelength modes, these results suggest that active biological fluids, such as microbial suspensions (<a id="xref-ref-35-12" class="xref-bibr" href="#ref-35">35</a>, <a id="xref-ref-54-3" class="xref-bibr" href="#ref-54">54</a>, <a id="xref-ref-72-2" class="xref-bibr" href="#ref-72">72</a>), can be more efficient at stirring fluids and transporting nutrients than previously thought.</p></div></div><div class="section conclusions" id="sec-11"><h2 class="">Conclusions</h2><p id="p-30" class="flushleft">To detect Beltrami flows in biological or engineered active fluids, one has to construct histograms and spectra as shown in <a id="xref-fig-3-8" class="xref-fig" href="#F3">Figs. 2 <em>C</em> and <em>D</em></a> and <a id="xref-fig-4-16" class="xref-fig" href="#F4">3 <em>A</em> and <em>E</em></a> from experimental fluid velocity and helicity data, which is possible with current fluorescence imaging techniques (<a id="xref-ref-13-2" class="xref-bibr" href="#ref-13">13</a>, <a id="xref-ref-35-13" class="xref-bibr" href="#ref-35">35</a>). Moreover, helical tracer particles (<a id="xref-ref-84-1" class="xref-bibr" href="#ref-84">84</a>) can help distinguish left-handed and right-handed flows. The above analysis predicts that Beltrami-flow structures, mirror-symmetry breaking, and the inverse 3D cascade appear more pronounced when the pattern selection is focused in a narrow spectral range. Our simulations further suggest that the relaxation time required for completion of the mirror-symmetry breaking process depends on the domain size (<a id="xref-fig-8-1" class="xref-fig" href="#F8">Fig. S5</a>). For small systems, the relaxation is exponentially fast, whereas for large domains relaxation proceeds in two stages, first exponentially and then linearly. In practice, it may therefore be advisable to accelerate relaxation by starting experiments from rotating initial conditions.</p><div id="F8" class="fig pos-float type-supplementary-material odd"><div class="highwire-figure"><div class="fig-inline-img-wrapper"><div class="fig-inline-img"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F8.large.jpg?width=800&height=600&carousel=1" title="(A) Relaxation time for spontaneous symmetry breaking depends on the domain size. (B and C) Kinetic energy (B) and helicity (C) as a function of time for a very large domain (L= 48Λ). The relaxation proceeds in two stages, the initial stage characterized by a rapid exponential growth rate (t" class="highwire-fragment fragment-images colorbox-load" rel="gallery-fragment-images-92291554" data-figure-caption="<div class="highwire-markup"><div xmlns="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML">(<em>A</em>) Relaxation time for spontaneous symmetry breaking depends on the domain size. (<em>B</em> and <em>C</em>) Kinetic energy (<em>B</em>) and helicity (<em>C</em>) as a function of time for a very large domain (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 48</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span>). The relaxation proceeds in two stages, the initial stage characterized by a rapid exponential growth rate (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo><</mml:mo><mml:mrow><mml:mn>20</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span>), followed by a slower linear growth until full relaxation (<span class="mathjax mml-math"><mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>t</mml:mi></mml:mpadded><mml:mo>≈</mml:mo><mml:mrow><mml:mn> 100</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span>). (<em>D–F</em>) Energy spectra at various stages of the relaxation process (compare with dashed lines in <em>B</em> and <em>C</em>) show how the system realizes a state with broken mirror symmetry.</div></div>" data-icon-position="" data-hide-link-title="0"><span class="hw-responsive-img"><img class="highwire-fragment fragment-image lazyload" alt="Fig. S5." src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-src="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F8.medium.gif" width="440" height="274"/><noscript><img class="highwire-fragment fragment-image" alt="Fig. S5." src="https://web.archive.org/web/20180714115531im_/http://www.pnas.org/content/pnas/114/9/2119/F8.medium.gif" width="440" height="274"/></noscript></span></a></div></div><ul class="highwire-figure-links inline"><li class="download-fig first"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F8.large.jpg?download=true" class="highwire-figure-link highwire-figure-link-download" title="Download Fig. S5." data-icon-position="" data-hide-link-title="0">Download figure</a></li> <li class="new-tab"><a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/content/pnas/114/9/2119/F8.large.jpg" class="highwire-figure-link highwire-figure-link-newtab" target="_blank" data-icon-position="" data-hide-link-title="0">Open in new tab</a></li> <li class="download-ppt last"><a href="/web/20180714115531/http://www.pnas.org/highwire/powerpoint/9869" class="highwire-figure-link highwire-figure-link-ppt" data-icon-position="" data-hide-link-title="0">Download powerpoint</a></li> </ul></div><div class="fig-caption"><span class="fig-label">Fig. S5.</span> <p id="p-31" class="first-child">(<em>A</em>) Relaxation time for spontaneous symmetry breaking depends on the domain size. (<em>B</em> and <em>C</em>) Kinetic energy (<em>B</em>) and helicity (<em>C</em>) as a function of time for a very large domain (<span class="inline-formula" id="inline-formula-201"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 48</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>). The relaxation proceeds in two stages, the initial stage characterized by a rapid exponential growth rate (<span class="inline-formula" id="inline-formula-202"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo><</mml:mo><mml:mrow><mml:mn>20</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>), followed by a slower linear growth until full relaxation (<span class="inline-formula" id="inline-formula-203"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>t</mml:mi></mml:mpadded><mml:mo>≈</mml:mo><mml:mrow><mml:mn> 100</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>). (<em>D–F</em>) Energy spectra at various stages of the relaxation process (compare with dashed lines in <em>B</em> and <em>C</em>) show how the system realizes a state with broken mirror symmetry.</p><div class="sb-div caption-clear"></div></div></div></div><div class="section methods" id="sec-12"><h2 class="">Methods</h2><p id="p-32" class="flushleft"><a id="xref-disp-formula-1-16" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> was solved numerically in the vorticity-vector potential form with periodic boundary conditions using a spectral code with 3/2 anti-aliasing (<a id="xref-sec-20-5" class="xref-sec" href="#sec-20"><em>Numerical Methods</em></a>). Tables in <a id="xref-fig-4-17" class="xref-fig" href="#F4">Fig. 3</a> were calculated using the Littlewood–Paley decomposition and collocation.</p></div><div class="section supplementary-material" id="sec-13"><h2 class="">Comparison with Experiments</h2><p id="p-33" class="flushleft">The generalized Navier–Stokes equations defined in <a id="xref-disp-formula-1-17" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text aim to provide an effective three-parameter description of solvent flows driven by an active component. Although the flow structures seen in the simulations look visually similar to those observed in experiments on bacterial and other active suspensions, a quantitative comparison with experimental data is needed to evaluate the practical applicability of the theory.<a id="xref-fn-5-1" class="xref-fn" href="#fn-5"><sup>*</sup></a> To contribute toward closing the gap between theory and experiments, we performed systematic parameter scans, comparing fluid flow statistics measured in our simulations with recently reported experimental data for two different classes of active fluids: (<em>i</em>) concentrated quasi-3D suspensions of swimming <em>B. subtilis</em> bacteria (<a id="xref-ref-35-14" class="xref-bibr" href="#ref-35">35</a>) and (<em>ii</em>) ATP-driven microtubule networks (<a id="xref-ref-38-8" class="xref-bibr" href="#ref-38">38</a>). This analysis identified specific parameter values <span class="inline-formula" id="inline-formula-204"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span> <span class="inline-formula" id="inline-formula-205"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-206"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></span></span> for these two paradigmatic experimental systems, demonstrating in both cases good agreement between theory and available experimental data for velocity distributions and correlation functions (<a id="xref-fig-1-2" class="xref-fig" href="#F1">Fig. S1</a>).</p><div id="sec-14" class="subsection"><h3>Bacterial Suspensions.</h3><p id="p-34">The experiments reported in ref. <a id="xref-ref-35-15" class="xref-bibr" href="#ref-35">35</a> studied dense suspensions of rod-like <em>B. subtilis</em> bacteria swimming in a quasi-3D microfluidic channel (height <span class="inline-formula" id="inline-formula-207"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mrow><mml:mpadded width="+3.3pt"><mml:mn>80</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, radius <span class="inline-formula" id="inline-formula-208"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mrow><mml:mpadded width="+3.3pt"><mml:mn>750</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m). The bacterial velocity field was reconstructed from bright-field micoscopy videos using particle imaging velocimetry (PIV), and the solvent flow dynamics were measured by particle tracking velocimetry (PTV) using micrometer-sized fluorescent tracer beads. The experimental setup allowed the observation of 2D slices through the 3D velocity field, yielding data for the in-plane velocity components. From these 2D data, velocity distributions and correlation functions for bacteria and passive tracer particles were reconstructed, showing close correlations between bacterial dynamics and solvent flows. To compare the experimental measurements in ref. <a id="xref-ref-35-16" class="xref-bibr" href="#ref-35">35</a> with our 3D simulations, we mimicked the experimental setup by selecting arbitrary 2D planes in our 3D simulation volume. We then measured the in-plane velocity components and compared the numerically calculated velocity statistics with the corresponding experimental data (<a id="xref-fig-1-3" class="xref-fig" href="#F1">Fig. S1 <em>A–C</em></a>).</p><p id="p-35"><a id="xref-fig-1-4" class="xref-fig" href="#F1">Fig. S1<em>A</em></a> compares the experimentally measured velocity distribution for bacteria (open circles) and solvent tracer particles (solid circles) with the statistics of the five-parameter model for the bacterial velocity field considered in ref. <a id="xref-ref-35-17" class="xref-bibr" href="#ref-35">35</a> (black line labeled “theory”) and our generalized Navier–Stokes model (blue line). As discussed by the authors of ref. <a id="xref-ref-35-18" class="xref-bibr" href="#ref-35">35</a>, their model for the bacterial dynamics fails to capture the tails of the velocity distributions as it includes an effective fourth-order velocity potential (representing steric alignment interactions) that dominates the tails of velocity distributions in their simulations. By contrast, our generalized Navier–Stokes model accurately captures the experimentally measured Gaussian velocity probability distribution functions (PDFs) over the whole range of the available experimental data (see <a id="xref-fig-1-5" class="xref-fig" href="#F1">Fig. S1<em>C</em></a> legend for a summary of fit parameters).</p><p id="p-36"><a id="xref-fig-1-6" class="xref-fig" href="#F1">Fig. S1 <em>B</em></a> compares the equal-time (in-plane) velocity correlation functions (VCFs) for the bacteria, tracer particles, and the theories. As mentioned in ref. <a id="xref-ref-35-19" class="xref-bibr" href="#ref-35">35</a>, the VCFs for tracer particles become unreliable at large distances <span class="inline-formula" id="inline-formula-209"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>r</mml:mi><mml:mo>></mml:mo><mml:mrow><mml:mn>50</mml:mn><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>m, due to the deliberately low seeding densities of the tracer particles in these experiments. Low tracer densities were required to minimize feedback from passive tracers on the active suspension dynamics. This meant, however, that tracer particle pairs with large spatial separation <span class="inline-formula" id="inline-formula-210"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math></span></span> are significantly less frequently observed. Notwithstanding such experimental limitations, we find that the two complementary continuum models for bacterial and solvent flow yield qualitatively and quantitatively similar VCFs, correctly reflecting the typical vortex size <span class="inline-formula" id="inline-formula-211"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mrow><mml:mn>50</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mpadded width="+3.3pt"><mml:mn>70</mml:mn></mml:mpadded><mml:mi mathvariant="normal">μ</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>m in the negative part of the VCFs.</p><p id="p-37"><a id="xref-fig-1-7" class="xref-fig" href="#F1">Fig. S1 <em>C</em></a> compares the simulation results for the velocity autocorrelation functions (VACFs) with the corresponding experimental results at different bacterial activities (<a id="xref-ref-35-20" class="xref-bibr" href="#ref-35">35</a>) due to oxygen depletion. PTV-based VACFs were not given in ref. <a id="xref-ref-35-21" class="xref-bibr" href="#ref-35">35</a> as a specific tracer particle typically spends only a short time in the 2D field of view of the microscope before diffusing out of view. As evident from <a id="xref-fig-1-8" class="xref-fig" href="#F1">Fig. S1 <em>C</em></a>, our generalized Navier–Stokes model can correctly reproduce the functional form of the PIV-based VACFs at high (green), intermediate (blue), and low (magenta) activities. With regard to a future quantitative characterization and classification of active fluids, we find it encouraging that a three-parameter model can account for the key velocity statistics reported in ref. <a id="xref-ref-35-22" class="xref-bibr" href="#ref-35">35</a>.</p><p id="p-38">Another interesting experimental observation reported but not rationalized in ref. <a id="xref-ref-35-23" class="xref-bibr" href="#ref-35">35</a> is the linear scaling of kinetic energy and enstrophy (figure 2d in ref. <a id="xref-ref-35-24" class="xref-bibr" href="#ref-35">35</a>). We note that such a linear scaling is consistent with the Beltrami-like flows found in our simulations, which satisfy <span class="inline-formula" id="inline-formula-212"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>∝</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:math></span></span> (<a id="xref-fig-3-9" class="xref-fig" href="#F3">Fig. 2 <em>C</em> and <em>D</em></a> of the main text). Generally, we hope that the good agreement between the generalized Navier–Stokes model defined in <a id="xref-disp-formula-1-18" class="xref-disp-formula" href="#disp-formula-1">Eqs. <strong>1</strong></a> of the main text and the experimental data for <em>B. subtilis</em> will stimulate additional 3D measurements on bacterial suspension in the near future, to test the Beltrami flow prediction directly and to explore the possibility of spontaneous mirror-symmetry breaking in detail.</p></div><div id="sec-15" class="subsection"><h3>ATP-Driven Microtubule Networks.</h3><p id="p-39">The generalized Navier–Stokes equations defined in <a id="xref-disp-formula-1-19" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text merely assume that active stresses in an otherwise passive fluid lead to scale selection. They should therefore also apply to other types of active fluids, including ATP-driven microtubule suspensions. To test this hypothesis, we performed additional simulations to compare our model with experimental data published recently in ref. <a id="xref-ref-38-9" class="xref-bibr" href="#ref-38">38</a>. The authors of this study report VCF data for tracer particles diffusing in fluid flows driven by predominantly extensile microtubule–kinesin bundles that form complex, approximately isotropic networks. The flows created by these active networks exhibit turbulent vortices on scales larger than the typical bundle-bending radii, suggesting that these flows are generated by the collective extensile dynamics of the bundles. <a id="xref-fig-1-9" class="xref-fig" href="#F1">Fig. S1 <em>D</em></a> shows the experimental VCF data reported in ref. <a id="xref-ref-38-10" class="xref-bibr" href="#ref-38">38</a> (colored circles and lines) and a fit (black solid line) obtained from simulations of our generalized Navier–Stokes model, using the parameters specified in the legend.<a id="xref-fn-6-1" class="xref-fn" href="#fn-6"><sup>†</sup></a> Different ATP-controlled activity levels can be reproduced in our model through a trivial adjustment of the velocity scale <span class="inline-formula" id="inline-formula-213"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math></span></span> . Strikingly, changing the activity does not significantly alter the shape of the VCF after rescaling by kinetic energy for both bacterial and active microtubule suspensions (<a id="xref-fig-1-10" class="xref-fig" href="#F1">Fig. S1 <em>B</em> and <em>D</em></a>), corroborating the idea that active suspensions can be robustly described by the leading-order terms of stress tensor expansions. More generally, the good agreement between the generalized Navier–Stokes model and two microscopically distinct active fluids supports the view that the main results and predictions of our study apply to a broad range of pattern-forming nonequilibrium fluids.</p></div></div><div class="section supplementary-material" id="sec-16"><h2 class="">Model Justification</h2><p id="p-40" class="flushleft">The generalized Navier–Stokes model defined in <a id="xref-disp-formula-1-20" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text describes the solvent flows in active suspensions through effective higher-order stresses that account phenomenologically for the experimentally observed flow-pattern scale selection (<a id="xref-ref-35-25" class="xref-bibr" href="#ref-35">35</a>, <a id="xref-ref-38-11" class="xref-bibr" href="#ref-38">38</a>). By contrast, derivations of effective higher-order continuum models (88) often focus on the complementary problem of obtaining a higher-order equation for the orientational order-parameter fields of the active component by ignoring nonlinear inertial effects in the fluid. The generalized Navier–Stokes <a id="xref-disp-formula-1-21" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> in the main text avoid the latter oversimplification and assume a linear response between orientational order-parameter fields and ambient fluid (<a id="xref-ref-72-3" class="xref-bibr" href="#ref-72">72</a>).</p><div id="sec-17" class="subsection"><h3>Inertial Effects.</h3><p id="p-41">The standard argument for neglecting inertial terms in the Navier–Stokes equations for dilute microbial suspensions is based on the typical length scale and swimming speed of a single bacterium and the viscosity of water (<a id="xref-ref-89-1" class="xref-bibr" href="#ref-89">89</a>). This argument is certainly correct for very low bacterial volume fractions when collective dynamical effects are negligible. The argument becomes invalid, however, at sufficiently high concentrations when collective effects dominate the suspension dynamics. There are three reasons for this: First, collective locomotion speeds of bacteria at moderate-to-high concentrations ( <span class="inline-formula" id="inline-formula-214"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> volume fraction) can be more than an order of magnitude larger than the self-propulsion speed of an individual bacterium (<a id="xref-ref-72-4" class="xref-bibr" href="#ref-72">72</a>). Second, the typical scale of a vortex is one or two magnitudes larger than the length of an individual cell (<a id="xref-ref-34-8" class="xref-bibr" href="#ref-34">34</a>, <a id="xref-ref-35-26" class="xref-bibr" href="#ref-35">35</a>). Third, recent studies (<a id="xref-ref-90-1" class="xref-bibr" href="#ref-90">90</a>, <a id="xref-ref-91-1" class="xref-bibr" href="#ref-91">91</a>) show that the collective dynamics can reduce the effective viscosity of a bacterial suspension by an order of magnitude. The combination of these three effects means that, in the collective swimming regime, the effective Reynolds number can approach 1 and, hence, inertial terms cannot a priori be neglected.</p></div><div id="sec-18" class="subsection"><h3>Active Stresses.</h3><p id="p-42">At the fundamental continuum level, the fluid dynamics of a passive solvent are described by the Navier–Stokes equations. The effect of the active components on the fluid can be written as a collection of point forces entering on the right-hand side (rhs) of the Navier–Stokes equations. An important feature of intrinsically driven active suspensions (in contrast to externally forced colloidal suspensions) is given by the experimentally confirmed fact that bacteria and other microbes achieve locomotion through shape changes that require zero net force (<a id="xref-ref-89-2" class="xref-bibr" href="#ref-89">89</a>, <a id="xref-ref-92-1" class="xref-bibr" href="#ref-92">92</a><a id="xref-ref-93-1" class="xref-down-link" href="#ref-93"><span>⇓</span></a>–<a id="xref-ref-94-1" class="xref-bibr" href="#ref-94">94</a>). Considering, for instance, the simplest force dipole model, this means that forces can be paired, so that monopole contributions cancel and the leading-order contributions entering the Navier–Stokes equations take the form of divergences of stress tensors (<a id="xref-ref-92-2" class="xref-bibr" href="#ref-92">92</a>).</p></div><div id="sec-19" class="subsection"><h3>Slaving and Linear Response.</h3><p id="p-43">In sufficiently dense suspensions, the net effective stress tensor depends on the collective dynamics, which are typically characterized through orientational order-parameter fields. If the fluid flows generated by the collective action of the active constituents dominate over their individual swimming dynamics, then one can assume that tensorial order parameters become “slaved” to the solvent dynamics. In this case, assuming a linear and isotropic response, one arrives at closure conditions for polar and nematic order parameters <em><strong>p</strong></em> and <em><strong>Q</strong></em> of the form <span class="disp-formula" id="disp-formula-12"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>𝒑</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi mathvariant="normal">∞</mml:mi></mml:munderover></mml:mstyle><mml:msub><mml:mi>p</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>𝒗</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi mathvariant="bold-italic">Q</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi mathvariant="normal">∞</mml:mi></mml:munderover></mml:mstyle><mml:msub><mml:mi>q</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo mathvariant="bold">∇</mml:mo><mml:mi>𝒗</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></span><span class="disp-formula-label">[S1]</span></span> where the superscript “ <span class="inline-formula" id="inline-formula-215"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>+</mml:mo></mml:math></span></span> ” denotes the traceless symmetric tensor part. Further assuming that the conventional viscous stresses and the active stresses are additive, that the leading active stresses are linear in the order parameters <em><strong>p</strong></em> and <em><strong>Q</strong></em>, and by truncating at order <span class="inline-formula" id="inline-formula-216"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>n</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 3</mml:mn></mml:mrow></mml:math></span></span>, one is led to a sixth-order stress tensor as in <a id="xref-disp-formula-3-3" class="xref-disp-formula" href="#disp-formula-3">Eq. <strong>1c</strong></a> of the main text. This reasoning can be formalized systematically through diagrammatic expansion techniques (<a id="xref-ref-57-2" class="xref-bibr" href="#ref-57">57</a>).<a id="xref-fn-7-1" class="xref-fn" href="#fn-7"><sup>‡</sup></a> The successful comparison with the experiments above suggests that such truncated stress tensors can capture essential aspects of the collective dynamics.</p></div></div><div class="section supplementary-material" id="sec-20"><h2 class="">Numerical Methods</h2><p id="p-44" class="flushleft">Numerical simulations were performed using a Fourier spectral method with a 3/2 rule to avoid aliasing when calculating the advection term through collocation (<a id="xref-ref-85-1" class="xref-bibr" href="#ref-85">85</a>). We typically used grids of size <span class="inline-formula" id="inline-formula-217"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>243</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:math></span></span> . Larger resolutions are not necessary, because the highest-order term in <a id="xref-disp-formula-1-22" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> provides strong damping <span class="inline-formula" id="inline-formula-218"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math></span></span> at large wavenumbers <span class="inline-formula" id="inline-formula-219"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span> . We find solutions to <a id="xref-disp-formula-1-23" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> by using the Hodge decomposition (<a id="xref-ref-86-1" class="xref-bibr" href="#ref-86">86</a>) and solving the corresponding vorticity-vector potential problem <span class="disp-formula" id="disp-formula-13"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo>∧</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>6</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S2a]</span></span><span class="disp-formula" id="disp-formula-14"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S2b]</span></span> where <span class="inline-formula" id="inline-formula-220"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝎</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> is the vorticity and <span class="inline-formula" id="inline-formula-221"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝍</mml:mi></mml:math></span></span> is the divergence-free vector potential related to the velocity field through <span class="inline-formula" id="inline-formula-222"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝒗</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> . <a id="xref-disp-formula-13-1" class="xref-disp-formula" href="#disp-formula-13">Eq. <strong>S2</strong></a> are evolved in time, using a third-order semiimplicit backward differentiation time-stepping scheme (<a id="xref-ref-86-2" class="xref-bibr" href="#ref-86">86</a>), calculating the nonlinear advection term explicitly while inverting the linear part implicitly. The discretized <a id="xref-disp-formula-13-2" class="xref-disp-formula" href="#disp-formula-13">Eq. <strong>S2</strong></a> maintain <span class="inline-formula" id="inline-formula-223"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-224"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝍</mml:mi></mml:math></span></span> divergence-free in exact arithmetic. To avoid slow buildup of nonzero divergence when working in double-precision arithmetic, numerical solutions are projected onto the divergence-free manifold during the time stepping. To calculate the energy transfer tables in <a id="xref-fig-4-18" class="xref-fig" href="#F4">Fig. 3 <em>C</em> and <em>G</em></a> efficiently, we decompose the velocity field into Littlewood–Paley components and use collocation.</p><div id="sec-21" class="subsection"><h3>Vorticity-Vector Potential Formulation.</h3><p id="p-45">We find the vorticity-vector potential formulation of the system <a id="xref-disp-formula-1-24" class="xref-disp-formula" href="#disp-formula-1">1</a> of the main text on the three-torus, <span class="inline-formula" id="inline-formula-225"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msup><mml:mi>𝕋</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></span></span> . This is a manifold without boundary, so the usual Hodge decomposition applies (<a id="xref-ref-86-3" class="xref-bibr" href="#ref-86">86</a>). For a vector field <em><strong>v</strong></em>, the decomposition takes the form <span class="disp-formula" id="disp-formula-15"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>∇</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo>∧</mml:mo><mml:mi>𝝍</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mi mathvariant="bold-italic">H</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S3]</span></span>where <em><strong>H</strong></em> is an element of the 3D space of harmonic vector fields, which implies on a torus that <span class="inline-formula" id="inline-formula-226"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="bold-italic">H</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>𝒚</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="normal">z</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> for some constants <span class="inline-formula" id="inline-formula-227"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:mrow></mml:math></span></span> . For divergence-free flows, we have <span class="inline-formula" id="inline-formula-228"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>ϕ</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span> and hence <span class="inline-formula" id="inline-formula-229"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>ϕ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span> because <span class="inline-formula" id="inline-formula-230"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>𝕋</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></span></span> is compact and without boundary. In this case, we interpret <em><strong>H</strong></em> as the fluid center of mass motion. By working in the center of mass frame, we are left with <span class="disp-formula" id="disp-formula-16"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S4]</span></span> Taking the curl of <a id="xref-disp-formula-1-25" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text gives <span class="disp-formula" id="disp-formula-17"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S5]</span></span> where <span class="inline-formula" id="inline-formula-231"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝎</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> and <span class="disp-formula" id="disp-formula-18"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S6]</span></span></p><p id="p-46">We can simplify the advection term by using the following standard identity <span class="disp-formula" id="disp-formula-19"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mi>𝝎</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>⋅</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>⋅</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S7]</span></span> because both fields are divergence-free. This identity speeds up the computational cost of evaluating the advection term, as it requires fewer applications of the fast Fourier transform. Because the divergence of <span class="inline-formula" id="inline-formula-232"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝍</mml:mi></mml:math></span></span> does not affect the decomposition S3, we fix the gauge and work with a divergence-free vector potential. In this case, taking the curl of <a id="xref-disp-formula-16-1" class="xref-disp-formula" href="#disp-formula-16">Eq. <strong>S4</strong></a> gives <span class="inline-formula" id="inline-formula-233"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>𝝍</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> . In summary, in the vorticity-vector potential formulation, the equations of motion read <span class="disp-formula" id="disp-formula-20"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo>∧</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S8a]</span></span><span class="disp-formula" id="disp-formula-21"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S8b]</span></span></p></div><div id="sec-22" class="subsection"><h3>Characteristic Scales.</h3><p id="p-47">The linear growth rate associated with the operator <span class="inline-formula" id="inline-formula-234"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">L</mml:mi></mml:math></span></span> is <span class="inline-formula" id="inline-formula-235"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> . For <span class="inline-formula" id="inline-formula-236"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span>, the most unstable mode <span class="inline-formula" id="inline-formula-237"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>argmax</mml:mi><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> is well approximated by the maximum <span class="inline-formula" id="inline-formula-238"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></span></span> of the function <span class="inline-formula" id="inline-formula-239"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mpadded width="+1.7pt"><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, yielding <span class="disp-formula" id="disp-formula-22"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S9]</span></span> We prefer to express characteristic scales in terms of <span class="inline-formula" id="inline-formula-240"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></span></span> (instead of <span class="inline-formula" id="inline-formula-241"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></span></span>) as this simplifies subsequent formulas, and <span class="inline-formula" id="inline-formula-242"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></span></span> is generally close to <span class="inline-formula" id="inline-formula-243"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></span></span> for sufficiently small injection bandwidths. The associated wavelength is <span class="inline-formula" id="inline-formula-244"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>λ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:math></span></span> . This wavelength represents two vortices, one with positive and one with negative vorticity, each of characteristic diameter <span class="disp-formula" id="disp-formula-23"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>λ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mn>2</mml:mn></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mi>π</mml:mi><mml:msqrt><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:msqrt></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S10]</span></span> The corresponding growth rate is <span class="disp-formula" id="disp-formula-24"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:msubsup><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span> which defines the time scale <span class="disp-formula" id="disp-formula-25"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S11]</span></span> <span class="inline-formula" id="inline-formula-245"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-246"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math></span></span> can be used to define a characteristic circulation speed <span class="inline-formula" id="inline-formula-247"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>U</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 2</mml:mn><mml:mi>π</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> .</p></div><div id="sec-23" class="subsection"><h3>Nondimensionalization in Numerical Simulations.</h3><p id="p-48">For simulation purposes, we rescale time and space as <span class="inline-formula" id="inline-formula-248"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>t</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-249"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝒙</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>L</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>𝒙</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, where <span class="inline-formula" id="inline-formula-250"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math></span></span> is the domain size. We further introduce <span class="inline-formula" id="inline-formula-251"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝍</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>𝝍</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-252"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝝎</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>𝝎</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> . <a id="xref-disp-formula-20-1" class="xref-disp-formula" href="#disp-formula-20">Eqs. <strong>S8</strong></a> then become (after dropping the tildes) <span class="disp-formula" id="disp-formula-26"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>T</mml:mi></mml:mfrac><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi>ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>∇</mml:mo></mml:mrow></mml:mrow><mml:mo>∧</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>6</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S12a]</span></span><span class="disp-formula" id="disp-formula-27"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mfrac><mml:msub><mml:mi>ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S12b]</span></span></p><p id="p-49">Setting <span class="inline-formula" id="inline-formula-253"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>T</mml:mi></mml:mfrac></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-254"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>T</mml:mi></mml:mfrac></mml:mrow></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-255"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac></mml:mrow></mml:math></span></span>, and defining <span class="inline-formula" id="inline-formula-256"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-257"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>γ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>L</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math></span></span>, we obtain the nondimensionalized equations <span class="disp-formula" id="disp-formula-28"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo>∧</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>6</mml:mn></mml:msup><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S13a]</span></span><span class="disp-formula" id="disp-formula-29"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S13b]</span></span></p></div><div id="sec-24" class="subsection"><h3>Time Discretization.</h3><p id="p-50">For the time stepping, we use the third-order semiimplicit backward differentiation scheme introduced by Ascher et al. (<a id="xref-ref-87-1" class="xref-bibr" href="#ref-87">87</a>), <span class="disp-formula" id="disp-formula-30"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>11</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>t</mml:mi><mml:mi mathvariant="script">L</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>𝝎</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S14a]</span></span> where <span class="disp-formula" id="disp-formula-31"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mo>∇</mml:mo><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S14b]</span></span><span class="disp-formula" id="disp-formula-32"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝝍</mml:mi><mml:mo>,</mml:mo><mml:mi>𝝎</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>∇</mml:mo></mml:mrow><mml:mo>∧</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>∧</mml:mo><mml:mi>𝒗</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S14c]</span></span> recalling that <span class="inline-formula" id="inline-formula-258"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝒗</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>∧</mml:mo><mml:mi>𝝍</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> . We then solve for the vector potential <span class="disp-formula" id="disp-formula-33"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>𝝍</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S15]</span></span></p></div><div id="sec-25" class="subsection"><h3>Space Discretization.</h3><p id="p-51">We work with a Fourier spectral method. If we denote the rhs of <a id="xref-disp-formula-30-1" class="xref-disp-formula" href="#disp-formula-30">Eq. <strong>S14a</strong></a> by <span class="inline-formula" id="inline-formula-259"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="normal">b</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></span></span>, then the update formula for the Fourier coefficients reads <span class="disp-formula" id="disp-formula-34"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:msup><mml:mi mathvariant="normal">b</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S16a]</span></span><span class="disp-formula" id="disp-formula-35"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mi>𝝍</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:msup><mml:mi>𝝎</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S16b]</span></span> where <span class="inline-formula" id="inline-formula-260"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn> 11</mml:mn><mml:mo>/</mml:mo><mml:mn>6</mml:mn></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> . In addition, we always have <span class="inline-formula" id="inline-formula-261"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝝎</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mpadded width="+1.7pt"><mml:mi>𝒌</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span>, because the vorticity is defined by taking the curl of <em><strong>v</strong></em>, and we can set <span class="inline-formula" id="inline-formula-262"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝝍</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mpadded width="+1.7pt"><mml:mi>𝒌</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn> 0</mml:mn></mml:mrow></mml:math></span></span> by gauge freedom. Because both <span class="inline-formula" id="inline-formula-263"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-264"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝍</mml:mi></mml:math></span></span> are divergence-free, we have to impose <span class="disp-formula" id="disp-formula-36"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>⋅</mml:mo><mml:mi>𝝍</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:mrow></mml:mrow></mml:math></span><span class="disp-formula-label">[S17]</span></span></p><p id="p-52">If we initiate the simulations with divergence-free fields, then the update rule S16 preserves this property in exact arithmetic. Nevertheless, numerical errors will always build up after several iterations in double-precision arithmetic. We project back onto the divergence-free manifold every several steps by mimicking gauge transformation. Suppose <span class="inline-formula" id="inline-formula-265"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span> has small divergence that we want to remove. We set <span class="inline-formula" id="inline-formula-266"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>f</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mo>⋅</mml:mo><mml:mi>𝝎</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> . We then solve the Poisson equation, <span class="disp-formula" id="disp-formula-37"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mi>λ</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S18]</span></span> and subsequently remove the divergence from <span class="inline-formula" id="inline-formula-267"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝝎</mml:mi></mml:math></span></span> according to <span class="disp-formula" id="disp-formula-38"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>→</mml:mo><mml:mrow><mml:mi>𝝎</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S19]</span></span></p></div><div id="sec-26" class="subsection"><h3>Calculation of Shell Interactions.</h3><p id="p-53">We next explain how the energy spectra, fluxes, and energy flow tables are calculated numerically (<a id="xref-fig-4-19" class="xref-fig" href="#F4">Fig. 3</a> of the main text and <a id="xref-fig-6-3" class="xref-fig" href="#F6">Fig. S3</a>). To establish notation, we first recall the derivation of the energy balance equation as given in Waleffe (<a id="xref-ref-19-11" class="xref-bibr" href="#ref-19">19</a>). Expanding the velocity and pressure fields in Fourier series, <a id="xref-disp-formula-1-26" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text give <span class="disp-formula" id="disp-formula-39"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>⋅</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span><span class="disp-formula" id="disp-formula-40"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒒</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒑</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>𝒌</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span> where <span class="inline-formula" id="inline-formula-268"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>k</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> . By projecting on helical modes one finds <a id="xref-disp-formula-7-1" class="xref-disp-formula" href="#disp-formula-7">Eq. <strong>3</strong></a> of the main text. To find the equation for the energy in mode <em><strong>k</strong></em> we relabel <span class="inline-formula" id="inline-formula-269"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>𝒌</mml:mi></mml:mpadded><mml:mo>→</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝒌</mml:mi></mml:mrow></mml:mrow></mml:math></span></span> in the second equation, multiply by <span class="inline-formula" id="inline-formula-270"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>, sum over <span class="inline-formula" id="inline-formula-271"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi></mml:math></span></span>, and use the incompressibility condition to get <span class="disp-formula" id="disp-formula-41"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝒌</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒒</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span> where we dropped the explicit time dependence for ease of notation. We now add the above equation to its complex conjugate and use <span class="inline-formula" id="inline-formula-272"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>𝒌</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mover accent="true"><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span>, <span class="disp-formula" id="disp-formula-42"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒑</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒒</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mtext mathvariant="normal">c.c.</mml:mtext></mml:mrow></mml:mrow></mml:math></span></span></p><p id="p-54">The energy in shell <span class="inline-formula" id="inline-formula-273"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></span></span> is defined as <span class="disp-formula" id="disp-formula-43"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>𝒗</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S20]</span></span></p><p id="p-55">The corresponding evolution equation is <span class="disp-formula" id="disp-formula-44"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mi>p</mml:mi></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mi>q</mml:mi></mml:munder></mml:mstyle><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span></span>where <span class="disp-formula" id="disp-formula-45"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mfrac><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mstyle scriptlevel="+1"><mml:mtable columnspacing="0.4em" rowspacing="0.2ex"><mml:mtr><mml:mtd><mml:mtext mathvariant="normal">shells</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mstyle></mml:munder></mml:mstyle><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒉</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒒</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mn>𝟎</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mtext mathvariant="normal">c.c.</mml:mtext></mml:mrow></mml:mrow></mml:math></span></span><span class="disp-formula" id="disp-formula-46"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mstyle scriptlevel="+1"><mml:mtable columnspacing="0.4em" rowspacing="0.2ex"><mml:mtr><mml:mtd><mml:mtext mathvariant="normal">shells</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mstyle></mml:munder></mml:mstyle><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi>𝒌</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒉</mml:mi><mml:mo>+</mml:mo><mml:mi>𝒒</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mn>𝟎</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S21]</span></span></p><p id="p-56">We used the fact that the sum over all modes can be split into radial and shell parts <span class="inline-formula" id="inline-formula-274"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mi>𝒌</mml:mi></mml:msub><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and we defined <span class="disp-formula" id="disp-formula-47"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mstyle scriptlevel="+1"><mml:mtable columnspacing="0.4em" rowspacing="0.2ex"><mml:mtr><mml:mtd><mml:mtext mathvariant="normal">shells</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mstyle></mml:munder></mml:mstyle><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒉</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒉</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒒</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo>,</mml:mo><mml:mi>𝒒</mml:mi><mml:mo>′</mml:mo><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">p</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span></span></p><p id="p-57">Symmetrizing as <span class="inline-formula" id="inline-formula-275"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> gives the usual energy balance equation (<a id="xref-ref-1-3" class="xref-bibr" href="#ref-1">1</a>, <a id="xref-ref-19-12" class="xref-bibr" href="#ref-19">19</a>) <span class="disp-formula" id="disp-formula-48"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S22a]</span></span> where <span class="disp-formula" id="disp-formula-49"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mi>p</mml:mi></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mi>q</mml:mi></mml:munder></mml:mstyle><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S22b]</span></span></p><p id="p-58">The quantity <span class="inline-formula" id="inline-formula-276"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> is the energy transfer into the shell <span class="inline-formula" id="inline-formula-277"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span> due to all triad interactions with shells <span class="inline-formula" id="inline-formula-278"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-279"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math></span></span> at time <span class="inline-formula" id="inline-formula-280"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math></span></span>, and <span class="inline-formula" id="inline-formula-281"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> is the energy transfer into the shell <span class="inline-formula" id="inline-formula-282"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span> due to all triad interactions. The energy flux across <span class="inline-formula" id="inline-formula-283"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span> is defined as <span class="disp-formula" id="disp-formula-50"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>></mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span><span class="disp-formula-label">[S23]</span></span> and represents energy flow from wavenumbers below <span class="inline-formula" id="inline-formula-284"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span> to those above it at time <span class="inline-formula" id="inline-formula-285"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math></span></span> .</p><p id="p-59">Projecting the velocity field onto the helical modes reveals additional substructure (<a id="xref-ref-19-13" class="xref-bibr" href="#ref-19">19</a>). The energy spectrum splits into two helical components <span class="disp-formula" id="disp-formula-51"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S24]</span></span></p><p id="p-60">The energy flow and energy flux split into eight components, one for each possible assignment of the helicity index over the triads <span class="disp-formula" id="disp-formula-52"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:munderover><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:munderover><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:munderover><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S25]</span></span> where we follow the binary ordering as in ref. <a id="xref-ref-19-14" class="xref-bibr" href="#ref-19">19</a> ( <span class="inline-formula" id="inline-formula-286"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 1</mml:mn></mml:mrow></mml:math></span></span> corresponds to <span class="inline-formula" id="inline-formula-287"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:math></span></span>, <span class="inline-formula" id="inline-formula-288"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mn> 2</mml:mn></mml:mrow></mml:math></span></span> to <span class="inline-formula" id="inline-formula-289"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>+</mml:mo><mml:mo></mml:mo><mml:mo>−</mml:mo></mml:mrow></mml:math></span></span>, etc.). The energy conservation for helical shells becomes <span class="disp-formula" id="disp-formula-53"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S26]</span></span> where <span class="inline-formula" id="inline-formula-290"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>4</mml:mn></mml:msubsup><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-291"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:msubsup><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> .</p><p id="p-61">We now consider time averages. For a quantity <span class="inline-formula" id="inline-formula-292"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi></mml:math></span></span> we define <span class="disp-formula" id="disp-formula-54"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo movablelimits="false">lim</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mo>→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi></mml:mfrac><mml:mrow><mml:msubsup><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mi mathvariant="script">R</mml:mi><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>d</mml:mi><mml:mpadded width="+1.7pt"><mml:mi>t</mml:mi></mml:mpadded><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S27]</span></span> In practice, <span class="inline-formula" id="inline-formula-293"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">R</mml:mi></mml:math></span></span> is the relaxation time for the system, and <span class="inline-formula" id="inline-formula-294"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Δ</mml:mi></mml:math></span></span> is the length of the averaging interval. In the stationary regime, the averages become time independent, which we denote by <span class="disp-formula" id="disp-formula-55"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span><span class="disp-formula-label">[S28]</span></span> and <span class="disp-formula" id="disp-formula-56"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:mrow></mml:math></span><span class="disp-formula-label">[S29]</span></span> Taking averages reduces <a id="xref-disp-formula-48-1" class="xref-disp-formula" href="#disp-formula-48">Eq. <strong>S22a</strong></a> to <span class="disp-formula" id="disp-formula-57"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S30]</span></span> Thus, in the stationary regime, the energy flux can be derived from the spectrum according to <span class="disp-formula" id="disp-formula-58"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>></mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S31]</span></span> Similarly, for the helical projections we get <span class="disp-formula" id="disp-formula-59"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S32]</span></span> and <span class="disp-formula" id="disp-formula-60"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>></mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S33]</span></span></p><p id="p-62">We numerically estimate the discrete stationary spectra <span class="inline-formula" id="inline-formula-295"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:math></span></span> as follows. At each time step <span class="inline-formula" id="inline-formula-296"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math></span></span>, we calculate <span class="disp-formula" id="disp-formula-61"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:msubsup><mml:mi>u</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒌</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S34]</span></span> We then apply the discrete version of the formula S27, <span class="disp-formula" id="disp-formula-62"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi mathvariant="normal">Δ</mml:mi></mml:mfrac><mml:mrow><mml:munderover><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S35]</span></span> where we choose <span class="inline-formula" id="inline-formula-297"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">R</mml:mi></mml:math></span></span> to be the relaxation time of the energy and helicity time series and the averaging interval <span class="inline-formula" id="inline-formula-298"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Δ</mml:mi></mml:math></span></span> is taken long enough to ensure convergence of statistical observables (<a id="xref-fig-5-3" class="xref-fig" href="#F5">Fig. S2</a>). We recover the helical flux contributions using formula <a id="xref-disp-formula-59-1" class="xref-disp-formula" href="#disp-formula-59"><strong>S32</strong></a> and the total flux from <span class="inline-formula" id="inline-formula-299"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="normal">Π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> .</p><p id="p-63">For plotting purposes, we connect the discrete energy spectra to their continuous definitions. The mean kinetic energy in the system of size <span class="inline-formula" id="inline-formula-300"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math></span></span> is <span class="disp-formula" id="disp-formula-63"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mi>k</mml:mi></mml:munder></mml:mstyle><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S36]</span></span> where <span class="inline-formula" id="inline-formula-301"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>n</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 1</mml:mn><mml:mo>,</mml:mo><mml:mn>2,3</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></span></span> and <span class="inline-formula" id="inline-formula-302"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:math></span></span> . In the limit <span class="inline-formula" id="inline-formula-303"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:math></span></span>, we recover the continuous definition of the energy spectrum <span class="disp-formula" id="disp-formula-64"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mn>0</mml:mn><mml:mi mathvariant="normal">∞</mml:mi></mml:msubsup><mml:mrow><mml:mi>e</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S37]</span></span><span class="disp-formula" id="disp-formula-65"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo movablelimits="false">lim</mml:mo><mml:mrow><mml:mi>L</mml:mi><mml:mo>→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mfrac><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S38]</span></span> When plotting energy spectra, we thus use the discrete (finite box size) approximation of the continuous definition <span class="disp-formula" id="disp-formula-66"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>k</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">⟩</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S39]</span></span></p><p id="p-64">The spectral domains I, II, and III in <a id="xref-fig-2-16" class="xref-fig" href="#F2">Fig. 1 <em>A</em></a> of the main text have finite thickness. To calculate the energy flow between the regions, we have to sum over shells contained in a given region. For example, <span class="disp-formula" id="disp-formula-67"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:msub><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi mathvariant="normal">I</mml:mi><mml:mo>,</mml:mo><mml:mi>II</mml:mi><mml:mo>,</mml:mo><mml:mi>III</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mtext mathvariant="normal">region I</mml:mtext></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mtext mathvariant="normal">region II</mml:mtext></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mtext mathvariant="normal">region III</mml:mtext></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></span><span class="disp-formula-label">[S40]</span></span> is the energy flow into region I due to all triad interactions with modes supported on regions II and III. To calculate <span class="inline-formula" id="inline-formula-304"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub></mml:math></span></span>, where <span class="inline-formula" id="inline-formula-305"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi mathvariant="normal">I</mml:mi><mml:mo>,</mml:mo><mml:mtext mathvariant="normal">II</mml:mtext><mml:mo>,</mml:mo><mml:mtext mathvariant="normal">III</mml:mtext><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>, consider the following integral, <span class="disp-formula" id="disp-formula-68"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>x</mml:mi></mml:mpadded><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mrow><mml:mo>⋅</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo>⋅</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S41]</span></span> where <span class="inline-formula" id="inline-formula-306"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></span></span> is the Littlewood–Paley component corresponding to region <span class="inline-formula" id="inline-formula-307"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math></span></span> and similarly for <span class="inline-formula" id="inline-formula-308"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:math></span></span> and <span class="inline-formula" id="inline-formula-309"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:math></span></span> . Specifically, <span class="inline-formula" id="inline-formula-310"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></span></span> is obtained from <em><strong>v</strong></em> by keeping only the Fourier amplitudes supported on the region <span class="inline-formula" id="inline-formula-311"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math></span></span>, etc. In terms of Fourier series, we find that <span class="disp-formula" id="disp-formula-69"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mtext mathvariant="normal">region</mml:mtext><mml:mi>K</mml:mi></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mtext mathvariant="normal">region</mml:mtext><mml:mi>P</mml:mi></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo largeop="true" movablelimits="false" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mtext mathvariant="normal">region</mml:mtext><mml:mi>Q</mml:mi></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S42]</span></span> where <span class="inline-formula" id="inline-formula-312"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>t</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></span></span> is given by <a id="xref-disp-formula-46-1" class="xref-disp-formula" href="#disp-formula-46">Eq. <strong>S21</strong></a>. We symmetrize in the last to indexes, by defining <span class="disp-formula" id="disp-formula-70"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>Q</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S43]</span></span></p><p id="p-65">To split <span class="inline-formula" id="inline-formula-313"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo largeop="true" symmetric="true">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:msubsup><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mrow></mml:math></span></span> into the contributions from the eight types of helical triad interactions, it is convenient to consider equivalent integral representations of the form <span class="disp-formula" id="disp-formula-71"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo largeop="true" symmetric="true">∫</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mpadded width="+1.7pt"><mml:mi>x</mml:mi></mml:mpadded><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:msubsup></mml:mrow><mml:mo>⋅</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>P</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:msubsup><mml:mo>⋅</mml:mo><mml:mo>∇</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>Q</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S44]</span></span> where <span class="inline-formula" id="inline-formula-314"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:msubsup></mml:math></span></span> is constructed from <span class="inline-formula" id="inline-formula-315"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒗</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></span></span> by projection onto modes with helicity index <span class="inline-formula" id="inline-formula-316"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mo>±</mml:mo><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math></span></span>, etc. The symmetrization <span class="disp-formula" id="disp-formula-72"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="script">T</mml:mi><mml:mo stretchy="false">∼</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi><mml:mi>Q</mml:mi><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></span><span class="disp-formula-label">[S45]</span></span> represents the energy flow into modes with helicity index <span class="inline-formula" id="inline-formula-317"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></span></span> lying in region <span class="inline-formula" id="inline-formula-318"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math></span></span>, due to interactions with modes with helicity indexes <span class="inline-formula" id="inline-formula-319"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:math></span></span> and <span class="inline-formula" id="inline-formula-320"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:math></span></span> in regions <span class="inline-formula" id="inline-formula-321"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math></span></span> and <span class="inline-formula" id="inline-formula-322"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math></span></span>, respectively. Expressions of the form S44 are calculated, at a given time step, by collocation: Evaluate the three projections in the physical domain on an equally spaced grid, perform the point-wise multiplication, go back to Fourier space, and integrate by reading off the value of the zeroth Fourier mode. All these operations are done efficiently using the fast Fourier transform. To calculate the stationary energy flows <span class="inline-formula" id="inline-formula-323"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">⟨</mml:mo><mml:msubsup><mml:mi mathvariant="script">T</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mi>P</mml:mi><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:msub><mml:mi>s</mml:mi><mml:mi>Q</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo stretchy="false">⟩</mml:mo></mml:mrow></mml:math></span></span> shown in the energy flow tables (<a id="xref-fig-4-20" class="xref-fig" href="#F4">Fig. 3 <em>C</em> and <em>G</em></a> in the main text and <a id="xref-fig-6-4" class="xref-fig" href="#F6">Fig. S3 <em>C</em></a>), we adopt a procedure analogous to that used to estimate the energy spectra.</p></div></div><div class="section supplementary-material" id="sec-27"><h2 class="">Cascade Characteristics</h2><p id="p-66" class="flushleft">The phenomenology of the inverse cascade in passive 2D turbulent flows is often characterized in terms of vortex mergers. By contrast, in active fluids with a well-defined vortex scale <span class="inline-formula" id="inline-formula-324"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></span></span> and a small injection bandwidth <span class="inline-formula" id="inline-formula-325"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:math></span></span>, vortex mergers are suppressed by the dominant pattern-scale selection processes. This raises the question how the inverse cascade, which can transport a considerable fraction of energy to larger scales (<a id="xref-fig-7-2" class="xref-fig" href="#F7">Fig. S4 <em>E</em></a>), manifests itself in the flow-field structure of a 3D active fluid. Our simulations demonstrate that pattern-forming nonequilibrium fluids can achieve energy transport to larger scales by forming chain-like vortex complexes that propagate through the fluid (<a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a>). To illustrate this phenomenon in more detail, <a id="xref-fig-7-3" class="xref-fig" href="#F7">Fig. S4 <em>A–D</em></a> shows two horizontal 2D <span class="inline-formula" id="inline-formula-326"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></span></span> slices of a large 3D simulation domain (size <span class="inline-formula" id="inline-formula-327"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>L</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mn> 32</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mrow></mml:math></span></span>) at a fixed representative time for an active fluid with small active bandwidth <span class="inline-formula" id="inline-formula-328"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mtext>𝐒</mml:mtext></mml:msub></mml:math></span></span> (using the same parameters as in the main text). In <a id="xref-fig-7-4" class="xref-fig" href="#F7">Fig. S4 <em>A</em> and <em>B</em></a>, the flow field is visualized through the perpendicular <span class="inline-formula" id="inline-formula-329"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi></mml:math></span></span> component of the vorticity, <span class="inline-formula" id="inline-formula-330"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ω</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:math></span></span>, and in <a id="xref-fig-7-5" class="xref-fig" href="#F7">Fig. S4 <em>C</em> and <em>D</em></a> through the local helicity <span class="inline-formula" id="inline-formula-331"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math></span></span> . The thin black lines in <a id="xref-fig-7-6" class="xref-fig" href="#F7">Fig. S4 <em>A</em> and <em>B</em></a> indicate in-plane portions of filaments consisting of alternating vortices that correspond to 3D filamentous clusters of high helicity in <a id="xref-fig-7-7" class="xref-fig" href="#F7">Fig. S4 <em>C</em> and <em>D</em></a>. The kinetic energy transported to large scales manifests itself as the formation and motion of such vortex chains (<a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental/pnas.1614721114.sm01.mov" class="in-nw">Movie S1</a>). These results illustrate that the helicity-driven 3D inverse cascade in active fluids is distinctly different from the energy-driven 2D inverse cascade in passive fluids.</p><p id="p-67">A detailed spectral characterization of this helicity-driven 3D active turbulence can be obtained by analyzing the upward energy transfer into region I in Fourier space (defined in <a id="xref-fig-2-17" class="xref-fig" href="#F2">Fig. 1 <em>A</em></a> of the main text). <a id="xref-fig-7-8" class="xref-fig" href="#F7">Fig. S4 <em>E</em></a> shows the absolute value of the energy flux for an active fluid with small bandwidth <span class="inline-formula" id="inline-formula-332"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>κ</mml:mi><mml:mtext>𝐒</mml:mtext></mml:msub></mml:math></span></span> for three different simulation domain sizes. In the case of an inertial energy cascade, one expects the flux to be independent of the wavenumber <span class="inline-formula" id="inline-formula-333"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math></span></span>, at least over some range. In such inertial ranges there is no dissipation of energy, just purely nonlinear redistribution. In our simulations, we see that the energy flux, upon entering the spectral region I from above (i.e., coming from region II), is at first noninertial with an approximate <span class="inline-formula" id="inline-formula-334"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></span></span> scaling, implying that the transfer is assisted by strong dissipation effects. At very large scales <span class="inline-formula" id="inline-formula-335"><span class="mathjax mml-math"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≫</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math></span></span>, however, the flux develops a plateau, indicating that the transfer becomes mostly inertial. Increasing the simulation domain size broadens the plateau and increases the magnitude of the flux that reaches the plateau. Interestingly, at these very large scales, the model defined in <a id="xref-disp-formula-1-27" class="xref-disp-formula" href="#disp-formula-1">Eq. <strong>1</strong></a> of the main text effectively reduces to the classical Navier–Stokes equations.</p></div><div class="section ack" id="ack-1"><h2>Acknowledgments</h2><p id="p-68">We thank Aden Forrow, Francis Woodhouse, Luca Biferale, Moritz Linkmann, and Michael Tribelsky for helpful discussions.</p></div><div class="section fn-group" id="fn-group-1"><h2>Footnotes</h2><ul class="pnas-footnotes"><li class="corresp" id="corresp-1"><a class="rev-xref" href="#xref-corresp-1-1">↵</a><sup>1</sup>To whom correspondence should be addressed. Email: <span class="em-link"><span class="em-addr">dunkel{at}math.mit.edu</span></span>.</li></ul><ul><li class="fn-other" id="fn-2"><p id="p-2">Author contributions: J.S. and J.D. designed research; J.S. performed research; and J.S. and J.D. wrote the paper.</p></li><li class="fn-conflict" id="fn-3"><p id="p-69">The authors declare no conflict of interest.</p></li><li class="fn-other" id="fn-4"><p id="p-70">This article is a PNAS Direct Submission.</p></li><li class="fn-other" id="fn-5"><p id="p-71"><a class="rev-xref" href="#xref-fn-5-1">↵</a>*We thank an anonymous reviewer for insisting on a detailed comparison with experiments. The comparison presented here benefited from the fact that one of us (J.D.) was involved in the original analysis of the experimental data in ref. <a id="xref-ref-35-27" class="xref-bibr" href="#ref-35">35</a>.</p></li><li class="fn-other" id="fn-6"><p id="p-72"><a class="rev-xref" href="#xref-fn-6-1">↵</a><sup>†</sup>These parameters agree well with the typical velocity, length, and time scales expected from microbial suspensions.</p></li><li class="fn-other" id="fn-7"><p id="p-73"><a class="rev-xref" href="#xref-fn-7-1">↵</a><sup>‡</sup>Structurally similar sixth-order hydrodynamic equations are obtained by systematically reducing magneto-hydrodynamic models in the vicinity of flow bifurcations (Geoffrey Vasil, personal communication).</p></li><li class="fn-supplementary-material" id="fn-8"><p id="p-74">This article contains supporting information online at <a href="https://web.archive.org/web/20180714115531/http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental" class="in-nw">www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental</a>.</p></li></ul></div><div class="license" id="license-1"><p id="p-3">Freely available online through the PNAS open access option.</p><p id="p-4"></p></div><div class="section ref-list" id="ref-list-1"><h2>References</h2><ol class="cit-list"><li><a class="rev-xref-ref" href="#xref-ref-1-1" title="View reference 1 in text" id="ref-1">↵</a><div class="cit ref-cit ref-book" id="cit-114.9.2119.1"><div class="cit-metadata"><ol class="cit-auth-list"><li><span class="cit-auth"><span class="cit-name-surname">Frisch</span> <span class="cit-name-given-names">U</span></span></li></ol><cite> (<span class="cit-pub-date">2004</span>) <span class="cit-source">Turbulence</span> (<span class="cit-publ-name">Cambridge Univ Press</span>, <span class="cit-publ-loc">Cambridge, UK</span>).</cite></div>.<div class="cit-extra"></div></div></li><li><a class="rev-xref-ref" href="#xref-ref-2-1" title="View reference 2 in text" id="ref-2">↵</a><div class="cit ref-cit ref-book" id="cit-114.9.2119.2"><div class="cit-metadata"><ol class="cit-auth-list"><li><span class="cit-auth"><span class="cit-name-surname">McComb</span> <span class="cit-name-given-names">WD</span></span></li></ol><cite> (<span class="cit-pub-date">1990</span>) <span class="cit-source">The Physics of Fluid Turbulence</span> (<span class="cit-publ-name">Oxford Science Publications</span>, <span class="cit-publ-loc">New York</span>).</cite></div>.<div class="cit-extra"></div></div></li><li><a class="rev-xref-ref" href="#xref-ref-3-1" title="View reference 3 in text" id="ref-3">↵</a><div class="cit ref-cit ref-journal" id="cit-114.9.2119.3"><div class="cit-metadata"><ol class="cit-auth-list"><li><span class="cit-auth"><span class="cit-name-surname">Higdon</span> <span class="cit-name-given-names">JC</span></span></li></ol><cite> (<span class="cit-pub-date">1984</span>) <span class="cit-article-title">Density fluctuations in the interstellar medium: Evidence for anisotropic magnetogasdynamic turbulence. 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