Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732286508"> <meta name="title" content="Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks"> <meta name="description" content="The shear flow instabilities under the presence of magnetic fields in the primordial disk can greatly facilitate the formation of density structures that serve as seeds prior to the onset of the gravitational Jeans instability. We evaluate the effects of the Parker, magnetorotational and kinematic dynamo instabilities by comparing the properties of these instabilities. We calculate the mass spectra of coagulated density structures by the above mechanism in the radial direction for an axisymmetric magnetohydrodynamic (MHD) torus equilibrium and power density profile models. Our local three-dimensional MHD simulation indicates that the coupling of the Parker and magnetorotational instability creates spiral arms and gas blobs in an accretion disk, reinforcing the theory and model. Such a mechanism for the early structure formation may be tested in a laboratory. The recent progress in experiments involving shear flows in rotating tokamak, field reversed configuration (FRC) and laser plasmas may become a key element to advance in nonlinear studies." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks"> <meta name="dc.creator" content="Koichi Noguchi"> <meta name="dc.creator" content="Toshiki Tajima"> <meta name="dc.creator" content="Wendell Horton"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 411"> <meta name="dc.date" content="2024-10-31"> <meta name ="dc.identifier" content="10.3390/universe10110411"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="The shear flow instabilities under the presence of magnetic fields in the primordial disk can greatly facilitate the formation of density structures that serve as seeds prior to the onset of the gravitational Jeans instability. We evaluate the effects of the Parker, magnetorotational and kinematic dynamo instabilities by comparing the properties of these instabilities. We calculate the mass spectra of coagulated density structures by the above mechanism in the radial direction for an axisymmetric magnetohydrodynamic (MHD) torus equilibrium and power density profile models. Our local three-dimensional MHD simulation indicates that the coupling of the Parker and magnetorotational instability creates spiral arms and gas blobs in an accretion disk, reinforcing the theory and model. Such a mechanism for the early structure formation may be tested in a laboratory. The recent progress in experiments involving shear flows in rotating tokamak, field reversed configuration (FRC) and laser plasmas may become a key element to advance in nonlinear studies." > <meta name="dc.subject" content="magnetorotational instability" > <meta name="dc.subject" content="structure formation" > <meta name="dc.subject" content="field reversed configuration" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-10-31"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="11"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="411" > <meta name="citation_issn" content="2218-1997"> <meta name="citation_journal_title" content="Universe"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks"> <meta name="citation_publication_date" content="2024/11"> <meta name="citation_online_date" content="2024/10/31"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="11"> <meta name="citation_firstpage" content="411"> <meta name="citation_author" content="Noguchi, Koichi"> <meta name="citation_author" content="Tajima, Toshiki"> <meta name="citation_author" content="Horton, Wendell"> <meta name="citation_doi" content="10.3390/universe10110411"> <meta name="citation_id" content="mdpi-universe10110411"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/11/411"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/11/411/pdf?version=1730372062"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/11/411/pdf?version=1730372062"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/11/411/pdf?version=1730372062"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/11/411/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/411/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/11/411/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/411/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/11/411/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/11/411/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/universe-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2218-1997/10/11/411" /> <meta property="og:title" content="Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks" /> <meta property="og:description" content="The shear flow instabilities under the presence of magnetic fields in the primordial disk can greatly facilitate the formation of density structures that serve as seeds prior to the onset of the gravitational Jeans instability. We evaluate the effects of the Parker, magnetorotational and kinematic dynamo instabilities by comparing the properties of these instabilities. We calculate the mass spectra of coagulated density structures by the above mechanism in the radial direction for an axisymmetric magnetohydrodynamic (MHD) torus equilibrium and power density profile models. Our local three-dimensional MHD simulation indicates that the coupling of the Parker and magnetorotational instability creates spiral arms and gas blobs in an accretion disk, reinforcing the theory and model. Such a mechanism for the early structure formation may be tested in a laboratory. The recent progress in experiments involving shear flows in rotating tokamak, field reversed configuration (FRC) and laser plasmas may become a key element to advance in nonlinear studies." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g001-550.jpg?1730372207" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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href="/cdn-cgi/l/email-protection#557a363b317836323c7a397a3038343c397825273a213036213c3a3b76656565656261646064306431646c646066616564646264316034646464656564"><sup><i class="fa fa-envelope-o"></i></sup></a> and </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13341835' data-options='is_hover:true, hover_timeout:5000'> Wendell Horton</div><div id="profile-card-drop13341835" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Wendell Horton</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a 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style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Optical Devices Division, Patent and Design Examination Department (Physics, Optics, Social Infrastructure and Design), Japan Patent Office, 3-4-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8915, Japan</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>3</sup></div> <div class="affiliation-name ">Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712, USA</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(11), 411; <a href="https://doi.org/10.3390/universe10110411">https://doi.org/10.3390/universe10110411</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 28 September 2024</span> / <span style="display: inline-block">Revised: 26 October 2024</span> / <span style="display: inline-block">Accepted: 28 October 2024</span> / <span style="display: inline-block">Published: 31 October 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/universe/special_issues/1C1KP05INH ">Multi-wavelength Properties of Active Galactic Nuclei</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1511246" aria-controls="drop-supplementary-1511246" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1511246" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2218-1997/10/11/411/pdf?version=1730372062" data-name="Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks" data-journal="universe">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2218-1997/10/11/411/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2218-1997/10/11/411/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2218-1997/10/11/411/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2218-1997/10/11/411/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g001.png?1730372207" title=" <strong>Figure 1</strong><br/> <p>A schematic view of MHD instabilities in early stage of the primordial evolution. Shear flow and magnetic field makes an accretion disk 1mstable to the magnetorotational instability. The disk is also unstable to the Parker instability because of gravity from the central star and magnetic field inside of the disk.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png?1730372208" title=" <strong>Figure 2</strong><br/> <p>Model picture of local simulation box (<b>right</b>) in accretion disk (<b>left</b>). Within the box, the Cartesian coordinate <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math> is along the radial cylindrical coordinate <math display="inline"><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> (with respect to the center of the accretion disk) and <math display="inline"><semantics> <mrow> <mi>y</mi> </mrow> </semantics></math> is along <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> </mrow> </semantics></math> is same in both the systems, where <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> are the equatorial plane and the height of the accretion disk, respectively. The planes A-A’ and B-B’ indicate the snapshot planes in Figures 5 and 6, respectively.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png?1730372209" title=" <strong>Figure 3</strong><br/> <p>Mass spectrum of an accretion disk in the outer region (<math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics></math> AU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in arbitrary units. We plot the spectrum for two different density profile, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> <mo>)</mo> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the profile given by Equation (22).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png?1730372209" title=" <strong>Figure 4</strong><br/> <p>Mass spectrum of an accretion disk in the inner region (r &lt; lAU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in units of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) shows the spectrum due to the self-gravity of gas, and (<b>b</b>) shows the spectrum due to the magnetic dynamo. We plot the spectrum for two different density profiles, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> </mrow> </mfenced> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the density profile given by Equation (22).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png?1730372212" title=" <strong>Figure 5</strong><br/> <p>Snapshots of the local disk structure in the plane A-A’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png?1730372215" title=" <strong>Figure 6</strong><br/> <p>Snapshots of the local disk structure in the plane B-B’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png?1730372217" title=" <strong>Figure 7</strong><br/> <p>A schematic view of a laser experiment. Electric current is applied in the direction of rotation to create magnetic field in the azimuthal direction. The directions of laser injection are slightly out of axis to create shear flow in the azimuthal direction. The injected laser compresses the target radially to make target Parker unstable.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png?1730372219" title=" <strong>Figure 8</strong><br/> <p>A typical FRC magnetic configurations and their possible beam injected shear appearance. (<b>a</b>) The axial magnetic field lines of an FRC overall outside of the separatrix, including the middle region (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>~</mo> <mn>0</mn> </mrow> </semantics></math>), the endplug regions (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>~</mo> <mn>10</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mo>~</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics></math>) and divertor regions (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>&gt;</mo> <mn>18</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo>&lt;</mo> <mo>−</mo> <mn>18</mn> </mrow> </semantics></math>). (after Ref. [<a href="#B13-universe-10-00411" class="html-bibr">13</a>]). (<b>b</b>) Meanwhile, magnetic fields within or near the separatrix are shown in (<b>b</b>), possibly with the beam driving sheared magnetic fields.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/11/411/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">The shear flow instabilities under the presence of magnetic fields in the primordial disk can greatly facilitate the formation of density structures that serve as seeds prior to the onset of the gravitational Jeans instability. We evaluate the effects of the Parker, magnetorotational and kinematic dynamo instabilities by comparing the properties of these instabilities. We calculate the mass spectra of coagulated density structures by the above mechanism in the radial direction for an axisymmetric magnetohydrodynamic (MHD) torus equilibrium and power density profile models. Our local three-dimensional MHD simulation indicates that the coupling of the Parker and magnetorotational instability creates spiral arms and gas blobs in an accretion disk, reinforcing the theory and model. Such a mechanism for the early structure formation may be tested in a laboratory. The recent progress in experiments involving shear flows in rotating tokamak, field reversed configuration (FRC) and laser plasmas may become a key element to advance in nonlinear studies.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=magnetorotational+instability">magnetorotational instability</a>; <a href="/search?q=structure+formation">structure formation</a>; <a href="/search?q=field+reversed+configuration">field reversed configuration</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-universe-10-00411' type='intro'><h2 data-nested='1'> 1. Introduction: Magnetohydrodynamic Processes of the Primordial Disk</h2><div class='html-p'>Magnetohydrodynamic (MHD) instabilities are important in the early stage of primordial evolution for the following reasons. (We show a schematic view of the system in <a href="#universe-10-00411-f001" class="html-fig">Figure 1</a>). In the inner region of the accretion disk, axial and azimuthal magnetic fields from the central object, which reciprocally convert back and forth to each other, can be enhanced exponentially in several disk rotations due to the kinematic dynamo. In the outer region, the magnetorotational instability renders the disk turbulent, which also converts axial and azimuthal components of magnetic field into each other and enhances their strength exponentially in time. As the magnetic pressure inside of the disk grows, the disk becomes unstable to the Parker instability because of the magnetic buoyancy. The Parker instability accumulates matter as blobs at the foot points of the magnetic loops on the disk, while it reduces magnetic fields away from the disk. Once these blobs develop, the Jeans instability sets in to dominate the disk evolution. All of these processes, except for the Jeans instability, do not need self-gravitation, accelerating the evolution of primordial disks in the early pre-self-gravitation stage.</div><div class='html-p'>The evolution of an accretion disk remains a challenging problem. The main problem arises from the treatment of the flow shear in the accretion disk [<a href="#B1-universe-10-00411" class="html-bibr">1</a>,<a href="#B2-universe-10-00411" class="html-bibr">2</a>,<a href="#B3-universe-10-00411" class="html-bibr">3</a>,<a href="#B4-universe-10-00411" class="html-bibr">4</a>,<a href="#B5-universe-10-00411" class="html-bibr">5</a>]. A local analysis of shear flows (i.e., the Fourier analysis) fails due to the radial nonuniformity of the accretion disk, which also prevents one from applying the WKB method to this issue. We adopt the nonlocal eigenvalue problem [<a href="#B5-universe-10-00411" class="html-bibr">5</a>].</div><div class='html-p'>The other problem is the creation and annihilation of magnetic fields. The magnetic field in an accretion disk is enhanced due to the magnetorotational instability [<a href="#B6-universe-10-00411" class="html-bibr">6</a>], leading to the buoyancy becoming strong enough to make the disk unstable to the Parker instability. Magnetic fields form loops emanating from the inside to outside of the disk due to the Parker instability. Since the magnetic field is incessantly created from both the magnetorotational instability and the kinematic dynamo, this dynamic process occurs recursively, or in another word, is self-sustained. Moreover, the disk remains turbulent due to the magnetorotational instability that is essential for sustaining the kinematic dynamo. The combination of these instabilities is a complex problem. In this paper, we compare these instabilities to estimate which instability is dominant under what conditions. These processes lead to highly nonlinear and mutually interacting plasma flows and magnetic fields under gravitational influence, which also manifest not only turbulent transport, but also structure formation (such as jets and relaxation oscillations).</div><div class='html-p'>Some of the processes associated with this structure formation of plasma with shear flows may be studied in laboratory plasmas such as tokamaks [<a href="#B7-universe-10-00411" class="html-bibr">7</a>,<a href="#B8-universe-10-00411" class="html-bibr">8</a>,<a href="#B9-universe-10-00411" class="html-bibr">9</a>], liquid metal Couette flow [<a href="#B10-universe-10-00411" class="html-bibr">10</a>,<a href="#B11-universe-10-00411" class="html-bibr">11</a>], FRC [<a href="#B12-universe-10-00411" class="html-bibr">12</a>,<a href="#B13-universe-10-00411" class="html-bibr">13</a>,<a href="#B14-universe-10-00411" class="html-bibr">14</a>] and in laser experiments [<a href="#B15-universe-10-00411" class="html-bibr">15</a>,<a href="#B16-universe-10-00411" class="html-bibr">16</a>]. Recent experiments show that toroidal shear flows in tokamak creates the radial momentum transport and the barrier inside of plasma that regulates momentum transfer to the wall. Though the strength of the magnetic field and density of plasma may be different from the accretion disk case, the configuration of the system and the instabilities that may occur in tokamak experiments can be similar to those of the accretion disk system. Shear flows and magnetic fields make plasma unstable to the magnetorotational instability, and the curvature of magnetic field in the toroidal direction makes plasma Rayleigh–Taylor (Parker) unstable. Thus, the unstable conditions may be experimentally examined in tokamak devices. In experiments trying to simulate supernovae, the surface of target suffers the Rayleigh–Taylor instability. If we apply sheared rotation and the magnetic field, essential for the magnetorotational instability, to the target, we may be able to simulate the accretion disk evolution by laser experiments. We estimate parameters for those cases and discuss if such experiments are plausible.</div></section><section id='sec2-universe-10-00411' type=''><h2 data-nested='1'> 2. The Parker, Magnetorotational and Kinematic Dynamo Instabilities</h2><div class='html-p'>The MHD shear flow instabilities, either the magnetorotational or the dynamo instabilities, play a role as an agent that keeps (or even enhances) the presence and strength of the disk magnetic field and its level of turbulence. These shear flow instabilities, by and large, determine the amount of mass that may be aggregated. The elevated level of magnetic fields in the disk through these instabilities gives rise to the activation of the Parker instability. The nonlinear evolution of the Parker instability is known [<a href="#B17-universe-10-00411" class="html-bibr">17</a>,<a href="#B18-universe-10-00411" class="html-bibr">18</a>,<a href="#B19-universe-10-00411" class="html-bibr">19</a>] to produce high density plasma blobs at the feet of the unstable buoyant loop. This nonlinear evolution can become a robust mechanism that yields a significant density structure within a matter of several multiples of the rotational period. Note that this mechanism can be operative even much before the self-gravitating Jeans instability becomes effective. The instabilities for plasma evolution require a weak, but finite initial poloidal or dipolar magnetic field.</div><div class='html-p'>For simplicity, we ignore the curvature effect and use the local Cartesian coordinates <math display='inline'><semantics> <mrow> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> </mrow> </mfenced> </mrow> </semantics></math> in the local co-rotating frame at <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> at frequency <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> </mrow> </semantics></math> are the radial, azimuthal and vertical direction, respectively) and assume the sheared angular velocity <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math> arising from <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mi mathvariant="sans-serif">Ω</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <mo>−</mo> <mn>2</mn> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> for <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mo>−</mo> <mo>(</mo> <mi>r</mi> <mi>d</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>/</mo> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">r</mi> <mo>)</mo> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> is the Oort’s constant (<a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>). Under a constant vertical gravity <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, the growth rate of the Parker instability <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> </semantics></math> is [<a href="#B20-universe-10-00411" class="html-bibr">20</a>] <div class='html-disp-formula-info' id='FD1-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> <mfenced open="[" close="]" separators="|"> <mrow> <msqrt> <mn>4</mn> <msup> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mrow> <mrow> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>−</mo> <mn>2</mn> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfenced> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </msqrt> <mo>−</mo> <mn>2</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <span class='html-italic'>H</span> is the vertical scale height of the disk, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mi>γ</mi> <mi>P</mi> <mo>/</mo> <mi>ρ</mi> </msqrt> </mrow> </semantics></math> is the sound speed, <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>8</mn> <mi>π</mi> <mi>P</mi> <mo>/</mo> <msup> <mrow> <mi>B</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math> is the wavenumber in the azimuthal direction. The height <math display='inline'><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> depends on <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> by <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>β</mi> </mrow> </mfenced> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>/</mo> <mi>H</mi> <mo>.</mo> </mrow> </semantics></math> We also assume <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mi>H</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math> for maximum growth rate, where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> is the radial component of the wavenumber. When <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mtext> </mtext> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>16</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> makes the growth rate maximum, <div class='html-disp-formula-info' id='FD2-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>S</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>8</mn> <msup> <mrow> <mi>β</mi> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mn>8</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> while <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mfenced separators="|"> <mrow> <mn>2</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>/</mo> <mi>β</mi> </msqrt> <mo>−</mo> <mn>3</mn> </mrow> </mfenced> <mo>/</mo> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> gives the maximum growth rate <div class='html-disp-formula-info' id='FD3-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>S</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>β</mi> <mo>−</mo> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mi>β</mi> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> </mrow> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> </mfrac> </mstyle> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div></div><div class='html-p'>When the magnetic field is purely toroidal, the maximum growth rate of the magnetorotational instability <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> can be estimated from the following local dispersion relation [<a href="#B5-universe-10-00411" class="html-bibr">5</a>]:<div class='html-disp-formula-info' id='FD4-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>A</mi> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>−</mo> <mi>A</mi> </mrow> </mfrac> </mstyle> <msubsup> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>∝</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>A</mi> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mfrac> </mstyle> <msubsup> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>k</mi> <mo>⋅</mo> <mi>B</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>/</mo> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </mrow> </semantics></math> is the Alfvén frequency. Since global analysis shows that the growth rate is maximum when <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>/</mo> <mi mathvariant="sans-serif">Ω</mi> <mo>~</mo> <mn>1</mn> </mrow> </semantics></math> [<a href="#B5-universe-10-00411" class="html-bibr">5</a>], the maximum grow rate for magnetorotational instability is <math display='inline'><semantics> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>~</mo> <mi>A</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> The growth time is revealed to be a dozen multiples of the rotation period of the disk.</div><div class='html-p'>We introduce the ratio <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> of the squared growth rates to assess the relative importance of these two processes as <div class='html-disp-formula-info' id='FD5-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>≃</mo> <mi>A</mi> <mi mathvariant="sans-serif">Ω</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>8</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> for <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math> and <div class='html-disp-formula-info' id='FD6-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>≃</mo> <mi>A</mi> <mi mathvariant="sans-serif">Ω</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div> for <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>. For the primordial disk, <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math> since the magnetic field in an accretion disk is weak to begin with. The vertical gravitational force of the central object can be expressed by the radius <span class='html-italic'>r</span> from the center and the height <span class='html-italic'>z</span> from the equator of the disk:<div class='html-disp-formula-info' id='FD7-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>z</mi> </mrow> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>G</mi> </mrow> </semantics></math> is the gravitational constant and <math display='inline'><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math> is the mass of the central object. When <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>≫</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> can be reduced to <math display='inline'><semantics> <mrow> <mi>G</mi> <mi>M</mi> <mi>z</mi> <mo>/</mo> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> Since <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msqrt> <mi>G</mi> <mi>M</mi> <mo>/</mo> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>3</mn> <mi mathvariant="sans-serif">Ω</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math> for the Keplerian disk, the ratio <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> is written as <div class='html-disp-formula-info' id='FD8-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mn>6</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>H</mi> </mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div></div><div class='html-p'>In this case, the number of Parker e-foldings in one rotation period is <div class='html-disp-formula-info' id='FD9-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>π</mi> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mstyle> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>β</mi> </mrow> </mfrac> </mstyle> </msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> showing the dominance of the Parker instability away from the equatorial plane of the disk. On the surface of the accretion disk (<math display='inline'><semantics> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> <mo>=</mo> <mi>H</mi> </mrow> </semantics></math>), <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> only depends on <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>. If a protostellar disk is weakly magnetized (<math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> = 10), <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> is the order of unity. Since the vertical gravity decreases towards the equatorial plane of the disk, <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> increases when <math display='inline'><semantics> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> </mrow> </semantics></math> decreases. However, if it is more strongly magnetized (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>), the ratio <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> is calculated from Equation (6), <div class='html-disp-formula-info' id='FD10-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>3</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>H</mi> </mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div></div><div class='html-p'>Though the magnetization of the disk rarely becomes this high, Equation (10) indicates the dominance of the Parker instability on the surface of the strongly magnetized disks (<math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>). Therefore, the magnetorotational instability is dominant in the early stage of the disk evolution (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math>), and the magnetization of the disk increases with time due to the magnetorotational instability. In the later stage (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>~</mo> <mn>1</mn> </mrow> </semantics></math>), the Parker instability sets in because of the enhanced magnetic fields, and the Parker instability dominates the evolution of the surface of the disk, whereas the magnetorotational instability still enhances magnetic fields near the equatorial plane <math display='inline'><semantics> <mrow> <mo>(</mo> <mfenced open="|" close="|" separators="|"> <mrow> <mi>z</mi> </mrow> </mfenced> <mo>≪</mo> <mi>H</mi> </mrow> </semantics></math>).</div><div class='html-p'>Next, we compare the growth rates of the kinematic dynamo and the Parker instability. We evaluate the <math display='inline'><semantics> <mrow> <mi>α</mi> <mi>ω</mi> </mrow> </semantics></math> dynamo, which includes the effect from helicity (<math display='inline'><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> effect) and the differential rotation effect (<math display='inline'><semantics> <mrow> <mi>ω</mi> </mrow> </semantics></math> effect). The governing equations of kinematic dynamo are [<a href="#B20-universe-10-00411" class="html-bibr">20</a>] <div class='html-disp-formula-info' id='FD11-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <mi>η</mi> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div><div class='html-disp-formula-info' id='FD12-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>−</mo> <mi>η</mi> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> is the mean helicity, <math display='inline'><semantics> <mrow> <mi>η</mi> </mrow> </semantics></math> is the effective magnetic diffusivity and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math> is the azimuthal component of the vector potential. The growth rate of the <math display='inline'><semantics> <mrow> <mi>α</mi> <mi>ω</mi> </mrow> </semantics></math> dynamo is given by [<a href="#B17-universe-10-00411" class="html-bibr">17</a>] <div class='html-disp-formula-info' id='FD13-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>η</mi> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mfenced open="[" close="]" separators="|"> <mrow> <mi>Re</mi> <mfenced separators="|"> <mrow> <msqrt> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msubsup> <msubsup> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <mi>η</mi> <msup> <mrow> <mi>k</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mn>2</mn> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </msqrt> </mrow> </mfenced> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msup> <mrow> <mi>k</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </semantics></math> is the square of the dynamo wavenumber, and <div class='html-disp-formula-info' id='FD14-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>α</mi> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>k</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div> is the dynamo number, respectively. The base level resistivity set by electron-neutral collisions is small, typically estimated as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>η</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>6</mn> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">c</mi> <msup> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> However, the turbulent fluid motions produce a large effective resistivity. We assume that the turbulent motion in the inner region produces a resistivity of <math display='inline'><semantics> <mrow> <mi>η</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>16</mn> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">c</mi> <msup> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> [<a href="#B21-universe-10-00411" class="html-bibr">21</a>] for a typical dynamo number for dipole, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>~</mo> <mn>600</mn> <mo>.</mo> </mrow> </semantics></math> Since <math display='inline'><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> can be estimated as <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mi>r</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <msup> <mrow> <mo>(</mo> <mi>r</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> depends on <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <msup> <mrow> <mo>(</mo> <mi>r</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> [<a href="#B21-universe-10-00411" class="html-bibr">21</a>]. We estimate <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> by assuming <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>≃</mo> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>≥</mo> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math> in Equation (14), which yields <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>≃</mo> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mfenced separators="|"> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </mfenced> <mo>.</mo> </mrow> </semantics></math> In this case, we cannot use the dispersion relation (1) since <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mi>H</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math> and we introduce another expression [<a href="#B22-universe-10-00411" class="html-bibr">22</a>] <div class='html-disp-formula-info' id='FD15-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mfenced open="[" close="]" separators="|"> <mrow> <msqrt> <msup> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mn>4</mn> <mi mathvariant="normal">k</mi> </mrow> <mrow> <mi mathvariant="normal">y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mrow> <mrow> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>−</mo> <mn>2</mn> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </msqrt> <mo>−</mo> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> </mfenced> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div></div><div class='html-p'>When <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math>, the maximum growth rate arises for <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mfenced separators="|"> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mo>/</mo> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and is <div class='html-disp-formula-info' id='FD16-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced separators="|"> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div></div><div class='html-p'>While for <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mfenced separators="|"> <mrow> <msqrt> <mn>2</mn> <mo>/</mo> <mi>β</mi> </msqrt> <mo>−</mo> <mn>3</mn> </mrow> </mfenced> <mo>/</mo> <mn>4</mn> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> makes the growth rate maximum, <div class='html-disp-formula-info' id='FD17-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msqrt> <mi>β</mi> </msqrt> <mi>g</mi> </mrow> <mrow> <mn>4</mn> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <mi>β</mi> </msqrt> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> <mi>H</mi> </mrow> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div></div><div class='html-p'>Now, we introduce another growth rate ratio <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> comparing the growth rate of the Parker instability to that of the <math display='inline'><semantics> <mrow> <mi>α</mi> <mi>ω</mi> </mrow> </semantics></math> dynamo with the same wavenumber <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> <mo>,</mo> </mrow> </semantics></math><div class='html-disp-formula-info' id='FD18-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo>√</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>d</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msubsup> <mi>H</mi> </mrow> <mrow> <msup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>z</mi> </mrow> </mfrac> </mstyle> <mfenced open="[" close="]" separators="|"> <mrow> <mi>Re</mi> <mfenced separators="|"> <mrow> <msqrt> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <msubsup> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>k</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </msqrt> <mo>+</mo> <mn>2</mn> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </mfenced> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div> for <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≫</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <div class='html-disp-formula-info' id='FD19-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msqrt> <mi>β</mi> <mo>/</mo> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>d</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msubsup> <mi>H</mi> </mrow> <mrow> <msup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>z</mi> </mrow> </mfrac> </mstyle> <mfenced open="[" close="]" separators="|"> <mrow> <mi>Re</mi> <mfenced separators="|"> <mrow> <msqrt> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <msubsup> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>k</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </msqrt> <mo>+</mo> <mn>2</mn> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </mfenced> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div> for <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>. For the small thick disk case (<math display='inline'><semantics> <mrow> <mi>r</mi> <mo>~</mo> <mn>1</mn> </mrow> </semantics></math> AU, <math display='inline'><semantics> <mrow> <mi>H</mi> <mo>/</mo> <mi>r</mi> <mo>~</mo> <mn>0.1</mn> <mo>)</mo> </mrow> </semantics></math> with a temperature of 100 K, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo>(</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>)</mo> </mrow> </semantics></math> with <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> with <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> on the surface of the disk. Thus, the dynamo effect is weak at the beginning of the protostellar evolution (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>), and magnetic fields created by the dynamo (<math display='inline'><semantics> <mrow> <mi>z</mi> <mo>≪</mo> <mi>H</mi> </mrow> </semantics></math>) cannot be maintained because of the Parker instability. However, in the later stage of the evolution (<math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>), the magnetic field is maintained by the dynamo effect and the Parker instability creates large structures in the inner region. Note that the <math display='inline'><semantics> <mrow> <mi>α</mi> <mi>ω</mi> </mrow> </semantics></math> dynamo grows faster than that of the Parker instability even when <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math> and makes the inner region turbulent.</div></section><section id='sec3-universe-10-00411' type=''><h2 data-nested='1'> 3. Radial Mass Spectrum of Accretion Disk</h2><div class='html-p'>The radial mass spectrum of aggregated matter that may be organized as a result of the nonlinear evolution of the MHD instabilities given in <a href="#sec2-universe-10-00411" class="html-sec">Section 2</a> for a primordial accretion disk is now estimated from the characteristics of the magnetorotational and kinematic dynamo instabilities. We choose two different types of density profiles and compare the mass spectra of these profiles. First, we assume the radial dependence of density is parameterized by [<a href="#B23-universe-10-00411" class="html-bibr">23</a>] <div class='html-disp-formula-info' id='FD20-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>r</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'>Then, the total mass <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics></math> between two Alfvén singular points in corotational coordinates, which is the typical scale length of the magnetorotational instability in the radial direction, can be given by <div class='html-disp-formula-info' id='FD21-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>ρ</mi> <mo>×</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <mi>A</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mstyle> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>π</mi> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mfrac> </mstyle> </msqrt> <mi>B</mi> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mi>ν</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(21)</label> </div> </div> for pure azimuthal perturbation in a Keplerian disk. Since the typical value of <math display='inline'><semantics> <mrow> <mi>ν</mi> </mrow> </semantics></math> is 2 in an early stage of accretion disk evolution [<a href="#B17-universe-10-00411" class="html-bibr">17</a>,<a href="#B23-universe-10-00411" class="html-bibr">23</a>,<a href="#B24-universe-10-00411" class="html-bibr">24</a>], <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> increases even though the density decreases with <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math>. <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> is the maximum seed mass available from the MHD evolution between the two Alfvén layers separated by <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi>r</mi> <mo>=</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>/</mo> <mi>A</mi> </mrow> </semantics></math>.</div><div class='html-p'>Another, more specific mass density model introduced by Okada et al. [<a href="#B25-universe-10-00411" class="html-bibr">25</a>] assumes that the magnetic field exists only in the azimuthal direction and that the density distribution is obtained by integrating the equations of motion into the potential form, <div class='html-disp-formula-info' id='FD22-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mrow> <mrow> <mi mathvariant="normal">max</mi> </mrow> <mo>⁡</mo> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>,</mo> <mn>0</mn> </mrow> </mfenced> </mrow> </mrow> </mrow> <mrow> <mi>K</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mfrac> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mfenced open="[" close="]" separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </msup> </mrow> </mfenced> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(22)</label> </div> </div> where <div class='html-disp-formula-info' id='FD23-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> <msubsup> <mrow> <mi>v</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <msubsup> <mrow> <mi>v</mi> </mrow> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(23)</label> </div> </div> is the value of potential on the surface of the torus, where the mass density is <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>K</mi> </mrow> </semantics></math> is a constant, <math display='inline'><semantics> <mrow> <mi>R</mi> <mo>=</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> <math display='inline'><semantics> <mrow> <mi>L</mi> </mrow> </semantics></math> is the angular momentum, <math display='inline'><semantics> <mrow> <mi>γ</mi> </mrow> </semantics></math> is the specific heat ratio and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>β</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>K</mi> <mo>/</mo> <mi>H</mi> </mrow> </semantics></math> is the plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> at <math display='inline'><semantics> <mrow> <mfenced separators="|"> <mrow> <mi>r</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> </mrow> </mfenced> <mo>=</mo> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> </mrow> </mfenced> <mo>.</mo> </mrow> </semantics></math> In the outer region <math display='inline'><semantics> <mrow> <mo>(</mo> <mi>r</mi> <mo>≫</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>ρ</mi> </mrow> </semantics></math> reduces to <div class='html-disp-formula-info' id='FD24-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>max</mi> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>r</mi> </mrow> </mfrac> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> </mfenced> </mrow> <mrow> <mi>K</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mfrac> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(24)</label> </div> </div> which yields the mass spectrum of the maximum coagulated matter between the Alfvén resonance, <div class='html-disp-formula-info' id='FD25-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mstyle> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>π</mi> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mfrac> </mstyle> </msqrt> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>ϕ</mi> </mrow> </msub> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>5</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </msup> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>max</mi> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>r</mi> </mrow> </mfrac> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> </mfenced> </mrow> <mrow> <mi>K</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mfrac> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mfenced separators="|"> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(25)</label> </div> </div></div><div class='html-p'>Thus, the radial dependence of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> is determined by the factors <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>γ</mi> </mrow> </semantics></math>. It vanishes if <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Φ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics></math>.</div><div class='html-p'>We show the <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> dependence of those two mass spectra in <a href="#universe-10-00411-f003" class="html-fig">Figure 3</a>. For the power law model in Equation (21), we choose <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">ν</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and the other parameters are chosen to match the initial condition of the simulation performed by Machida et al. [<a href="#B6-universe-10-00411" class="html-bibr">6</a>]. As we described above, the mass spectrum increases with r if the density profile obeys the power law. On the other hand, if the density profile is given by the potential model in Equation (22), it takes a maximum value at some radius.</div><div class='html-p'>In the inner region (r < 1 AU), the kinematic dynamo is predominant in determining the turbulence of plasma. In this case, structure formation is inhibited inside of the Roche limit <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mn>2.546</mn> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mo>⊙</mo> </mrow> </msub> <mo>/</mo> <mi>ρ</mi> </mrow> </mfenced> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mo>⊙</mo> </mrow> </msub> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mo>⊙</mo> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mo>⊙</mo> </mrow> </msub> </mrow> </semantics></math> are the density and the radius of the central object, respectively. If we only consider the tidal force and self-gravitation, the typical wavelength for exponentially growth of density waves in the radial direction is <div class='html-disp-formula-info' id='FD26-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> </mrow> <mrow> <mi>r</mi> <msup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(26)</label> </div> </div> which yields the mass spectrum <div class='html-disp-formula-info' id='FD27-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>ρ</mi> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <msup> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(27)</label> </div> </div></div><div class='html-p'>This mass spectrum can either increase or decrease with <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math>, depending on what kind of density profile we choose.</div><div class='html-p'>Another characteristic wavelength is the width of dynamo eigenmodes. Stepinski et al. [<a href="#B21-universe-10-00411" class="html-bibr">21</a>] showed the eigenmodes extend from the axis of the rotation to <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>5</mn> <mo>~</mo> <mn>10</mn> <mi>H</mi> </mrow> </semantics></math>. Since the height of the disk <math display='inline'><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> is given by <div class='html-disp-formula-info' id='FD28-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mi>γ</mi> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mrow> </mfenced> <msup> <mrow> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <mi>γ</mi> <mi>G</mi> <mi>M</mi> <mi>z</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(28)</label> </div> </div> the mass spectrum due to the dynamo effect can be estimated in the form <div class='html-disp-formula-info' id='FD29-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>10</mn> <mo>×</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>β</mi> </mrow> </mfrac> </mrow> </mfenced> <msup> <mrow> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <mi>γ</mi> <mi>G</mi> <mi>M</mi> <mi>z</mi> </mrow> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(29)</label> </div> </div></div><div class='html-p'>The mass spectra of those cases are shown in <a href="#universe-10-00411-f004" class="html-fig">Figure 4</a>. In <a href="#universe-10-00411-f004" class="html-fig">Figure 4</a>a, we calculate the mass spectrum due to the self-gravitation of gas. When the density profile is constant, the spectrum monotonically increases with <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math>. The peak near the rotational axis may be too close to the central object. This dense region creates a disk because of the tidal force. However, when the density profile obeys Equation (22), the spectrum rises from <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>~</mo> <mn>0.65</mn> <mi>H</mi> </mrow> </semantics></math>, has a peak around <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1.5</mn> <mi>H</mi> </mrow> </semantics></math> and then decreases.</div><div class='html-p'>We show the mass spectrum due to the magnetic dynamo in <a href="#universe-10-00411-f004" class="html-fig">Figure 4</a>b. We assume constant pressure (<math display='inline'><semantics> <mrow> <mi>T</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> <mo>=</mo> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mo>(</mo> <mi>r</mi> <mo>/</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>)</mo> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math>) for a power law profile, and for the potential model Equation (22), <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> is given by <div class='html-disp-formula-info' id='FD30-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>K</mi> <mi>γ</mi> <msup> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(30)</label> </div> </div></div><div class='html-p'>The mass spectrum increases monotonically with r in this case, but the power law profile increases less rapidly than in the torus case.</div></section><section id='sec4-universe-10-00411' type=''><h2 data-nested='1'> 4. Numerical Simulation of Coupled Instabilities</h2><div class='html-p'>We present the results of three-dimensional MHD simulations in a corotating reference frame. The basic equation we use are ideal MHD equations in the rotating frame where the gas is stationary at <math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>. In this frame, <math display='inline'><semantics> <mrow> <mn>2</mn> <mi mathvariant="sans-serif">Ω</mi> <mo>(</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> gives the Coriolis acceleration and <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mfenced separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>=</mo> <mi>A</mi> <mi>x</mi> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mi>r</mi> <mi>d</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>/</mo> <mi>d</mi> <mi>r</mi> </mrow> </semantics></math>. We assume that the curvature effect is negligible and use the local cartesian coordinate system <math display='inline'><semantics> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math>. A schematic of the simulation box is shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>. The bottom (<math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) and top (<math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mi>H</mi> </mrow> </semantics></math>) of the box corresponds to the equatorial plane and surface of the disk, and uniform gravity is applied in the <math display='inline'><semantics> <mrow> <mi>z</mi> </mrow> </semantics></math> direction. An initial shear flow is applied in the <math display='inline'><semantics> <mrow> <mi>y</mi> </mrow> </semantics></math> direction, which corresponds to the flow in the azimuthal direction in the accretion disk case, and magnetic field in the <math display='inline'><semantics> <mrow> <mi>y</mi> </mrow> </semantics></math> direction supports fluid against gravity. We assume the periodic boundary in the <math display='inline'><semantics> <mrow> <mi>x</mi> <mtext> </mtext> </mrow> </semantics></math>(radial) direction and a free boundary at <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mi>H</mi> </mrow> </semantics></math>. The boundary condition at <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is antisymmetric for the magnetic field and symmetric for other field variables. We adopt the shear boundary condition in the <math display='inline'><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math> direction [<a href="#B2-universe-10-00411" class="html-bibr">2</a>,<a href="#B5-universe-10-00411" class="html-bibr">5</a>], <div class='html-disp-formula-info' id='FD31-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> </mrow> </mfenced> <mo>=</mo> <mi>f</mi> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>+</mo> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>A</mi> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> </mrow> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> </mrow> </mfenced> <mo>=</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>x</mi> <mo>+</mo> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>A</mi> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> </mrow> </mfenced> <mo>+</mo> <mn>2</mn> <mi>A</mi> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(31)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> is the sizes of the simulation box in the <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>z</mi> </mrow> </semantics></math> directions, respectively.</div><div class='html-p'>We show the simulation results in <a href="#universe-10-00411-f005" class="html-fig">Figure 5</a> and <a href="#universe-10-00411-f006" class="html-fig">Figure 6</a>, where plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> is unity on the equatorial plane (<math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), and <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1.05</mn> </mrow> </semantics></math> for nearly isothermal dynamics. We put sinusoidal incompressible perturbation whose wavelength corresponds to <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>R</mi> </mrow> </msub> </mrow> </semantics></math> in <math display='inline'><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math> direction and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> </semantics></math> in y direction to the velocity field. At <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.05</mn> <msup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, the Parker instability creates magnetic loops in the azimuthal direction and plasma accretes to the foot points of loops. Plasma also accretes in the radial direction due to the magnetorotational instability. The combination of these two instabilities makes high density stripe regions in the equatorial plane.</div><div class='html-p'>At a later time (<math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.93</mn> <msup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>), the density at the bottom becomes larger and shows fewer dense blobs. We note that even though strong shear flows exist, blobs and spiral dense regions do not disappear due to phase mixing. Magnetic fields continuously escape through the top of the box because of the buoyancy, but the maximum strength of magnetic field increases because magnetic fields are continuously created by the magnetorotational instability.</div><div class='html-p'>Global simulations [<a href="#B6-universe-10-00411" class="html-bibr">6</a>] showed that the competition between the magnetorotational and the Parker instability become quasi-periodic after some rotations. We have stopped the simulation before it reaches the quasi-periodic state, but we can see all the major features of the global simulation appear in the local simulation.</div></section><section id='sec5-universe-10-00411' type=''><h2 data-nested='1'> 5. Shear Flow-Induced Structures in Laboratory Plasmas</h2><div class='html-p'>Next, we consider the “simulation” of the above processes of structure formation in laboratory plasmas. In recent tokamak plasmas, strong shear flows have been observed either due to the external drive, such as the neutral beam injection (NBI) [<a href="#B7-universe-10-00411" class="html-bibr">7</a>,<a href="#B26-universe-10-00411" class="html-bibr">26</a>], or due to the internal flow drive mechanism [<a href="#B8-universe-10-00411" class="html-bibr">8</a>]. In another magnetic configuration called FRC (field reversible configuration), there are end plugs at the end of the FRC where shear rotated plasmas may be formed [<a href="#B12-universe-10-00411" class="html-bibr">12</a>,<a href="#B13-universe-10-00411" class="html-bibr">13</a>,<a href="#B14-universe-10-00411" class="html-bibr">14</a>]. It may be also possible to generate shear flows in laser-irradiated targets [<a href="#B15-universe-10-00411" class="html-bibr">15</a>,<a href="#B16-universe-10-00411" class="html-bibr">16</a>].</div><div class='html-p'>For a plasma in tokamak, the gravitational force is replaced by the centrifugal force, which is created by the thermal velocity and/or flow in toroidal direction and is some substantial fraction <math display='inline'><semantics> <mrow> <mi>ϵ</mi> </mrow> </semantics></math> of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math>, <div class='html-disp-formula-info' id='FD33-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>ϵ</mi> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(32)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> is the major radius of the torus. Then, the growth rate ration in Equation (6) can be written in the form <div class='html-disp-formula-info' id='FD34-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>A</mi> <mi mathvariant="sans-serif">Ω</mi> <msubsup> <mrow> <mi>R</mi> </mrow> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <msup> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mi>β</mi> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> </mrow> </mfenced> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(33)</label> </div> </div> Since the toroidal flow <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> is equal to <math display='inline'><semantics> <mrow> <mi>δ</mi> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>, we can further reduce R to <div class='html-disp-formula-info' id='FD35-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>A</mi> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mi>β</mi> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(34)</label> </div> </div> The value of <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> in a typical experiment is about 0.01 at peak and is smaller near the wall. If we assume <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> becomes a function of the ratio <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </semantics></math><div class='html-disp-formula-info' id='FD36-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">R</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>A</mi> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mfrac> </mstyle> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(35)</label> </div> </div></div><div class='html-p'>The flow shear rate <math display='inline'><semantics> <mrow> <mi>A</mi> </mrow> </semantics></math> in the transport barrier region can be estimated from the toroidal flow profile. The typical value of <math display='inline'><semantics> <mrow> <mi>A</mi> </mrow> </semantics></math> is 100 kHz [<a href="#B27-universe-10-00411" class="html-bibr">27</a>]. The toroidal rotation of the plasma is driven by the neutral beam injection. In modern large tokamaks, the toroidal velocity reaches 200~300 km/s, which yields <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>~</mo> </mrow> </semantics></math>100 kHz. The estimated value of <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> is about <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> for the transport barrier region. Thus, for a typical experiment configuration, the magnetorotational instability grows faster to make plasma turbulent. The details of the turbulent transport from both rotational shear, magnetic shear and the thermal gradient are complex in tokamaks [<a href="#B27-universe-10-00411" class="html-bibr">27</a>].</div><div class='html-p'>Another problem is to maximize the growth rate of the magnetorotational instability by equalizing the Alfvén frequency <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> </semantics></math> to the rotational speed <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </semantics></math>. For typical tokamak devices, the poloidal magnetic field is ~0.1T and density <math display='inline'><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> is <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>13</mn> </mrow> </msup> <mo> </mo> <msup> <mrow> <mi>cm</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>, which yields a Alfvén velocity of the order of <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> km/s. Thus, the typical length of the magnetorotational instability in the poloidal direction [<a href="#B5-universe-10-00411" class="html-bibr">5</a>], <div class='html-disp-formula-info' id='FD37-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <mi>A</mi> </mrow> </mfrac> </mstyle> <mo>≃</mo> <mi>R</mi> <mi>t</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msqrt> <mi>α</mi> </msqrt> </mrow> <mrow> <mi>ϵ</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(36)</label> </div> </div> is the order of <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mtext> </mtext> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> cm. The typical wavelength of the Parker instability in the toroidal direction is <div class='html-disp-formula-info' id='FD38-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfenced> </mrow> <mrow> <msqrt> <mi>β</mi> </msqrt> <msup> <mrow> <mfenced separators="|"> <mrow> <mn>2</mn> <msqrt> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>β</mi> </msqrt> <mo>−</mo> <mn>3</mn> <mi>β</mi> </mrow> </mfenced> </mrow> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(37)</label> </div> </div> whose value is <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>6</mn> </mrow> </msup> <mtext> </mtext> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math> cm.</div><div class='html-p'>The magnetic (Maxwell stress) viscosity coefficient <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math> caused by the magnetorotational instability is given by [<a href="#B5-universe-10-00411" class="html-bibr">5</a>] <div class='html-disp-formula-info' id='FD39-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced open="⟨" close="⟩" separators="|"> <mrow> <mi>δ</mi> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mi>δ</mi> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> <mo><</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced open="⟨" close="⟩" separators="|"> <mrow> <mi>δ</mi> <msup> <mrow> <mi>B</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(38)</label> </div> </div></div><div class='html-p'>Substituting all the variables in Equation (38), the magnetic viscosity in tokamak device is estimated to be <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>≪</mo> <mn>0.01</mn> </mrow> </semantics></math>.</div><div class='html-p'>Next, we consider the possibility of laser plasma experiments. A schematic view of an experiment is shown in <a href="#universe-10-00411-f007" class="html-fig">Figure 7</a> [<a href="#B15-universe-10-00411" class="html-bibr">15</a>]. Though it may not be easy to apply a strong magnetic field to the target, we may apply an electric current to a target perpendicular to the direction of laser injections to create a magnetic field in the azimuthal direction. The injection directions of lasers are offset from the symmetry axis, so that the target obtains angular momentum from the laser beams. The target is nonuniform to create a sheared rotation in the horizontal plane. Since the target is Rayleigh–Taylor unstable due to the pressure of the laser and magnetorotationally unstable due to the sheared rotation and magnetic field, we should be able to observe how the global structure grows experimentally if we choose the variables properly.</div><div class='html-p'>A critical condition for laser–space plasma experiments is to make the equations of motion invariant under the scale transformation [<a href="#B16-universe-10-00411" class="html-bibr">16</a>]:<div class='html-disp-formula-info' id='FD40-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>,</mo> </mrow> <mspace linebreak="newline"/> <mrow> <msub> <mrow> <mi>p</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>p</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>,</mo> </mrow> <mspace linebreak="newline"/> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(39)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi>ρ</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>p</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>h</mi> </mrow> </semantics></math> are the characteristic values of density, pressure and spatial scale, respectively, and subscripts <math display='inline'><semantics> <mrow> <mi>d</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>e</mi> </mrow> </semantics></math> indicate that those values correspond to the accretion disk or experiment. The ratios <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </semantics></math> are determined from those parameters. Under the transformation, the characteristic values of time scale <math display='inline'><semantics> <mrow> <mi>τ</mi> </mrow> </semantics></math>, magnetic field <math display='inline'><semantics> <mrow> <mi>B</mi> </mrow> </semantics></math> and angular frequency <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </semantics></math> are transformed as <div class='html-disp-formula-info' id='FD43-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </msqrt> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>≡</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>,</mo> </mrow> <mspace linebreak="newline"/> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </msubsup> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>,</mo> </mrow> <mspace linebreak="newline"/> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mi>d</mi> <mo>=</mo> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(40)</label> </div> </div></div><div class='html-p'>First, we determine the ranges of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> from the density and magnetic field strength. The typical strengths of magnetic field are <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>~</mo> <mn>1</mn> </mrow> </semantics></math> G and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mi mathvariant="normal">G</mi> <mo>~</mo> <mn>1</mn> <mtext> </mtext> <mi mathvariant="normal">T</mi> </mrow> </semantics></math>, which yields <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>0</mn> </mrow> </msup> </mrow> </semantics></math>. Since the typical number density of a protostellar accretion disk is <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>9</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>12</mn> </mrow> </msup> <mi mathvariant="normal">c</mi> <msup> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>, whose content is mostly hydrogen (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <mo>~</mo> <mtext> </mtext> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">g</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> </mrow> <msup> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>), and that of target is <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>~</mo> <mn>10</mn> <msup> <mrow> <mrow> <mi mathvariant="normal">g</mi> <mo>/</mo> <mi>cm</mi> </mrow> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>, the ratio <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> is the order of <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>16</mn> </mrow> </msup> <mo>~</mo> <mtext> </mtext> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math>. The pressure of the disk is estimated by assuming its temperature <math display='inline'><semantics> <mrow> <mi>T</mi> <mo>~</mo> <mn>100</mn> <mtext> </mtext> <mi mathvariant="normal">K</mi> </mrow> </semantics></math>, which yields <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>p</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mi>ρ</mi> <mo>/</mo> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> Pa and an experiment <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>p</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>/</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math> Pa.</div><div class='html-p'>Next, we determine the range of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> again, with a typical scale length and time scale. The typical scale length of a primordial disk can be determined by the distance between the Alfvén singularities in corotational frame [<a href="#B5-universe-10-00411" class="html-bibr">5</a>], <div class='html-disp-formula-info' id='FD46-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>~</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <mi>A</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>B</mi> </mrow> <mrow> <msqrt> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </msqrt> <mi>A</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>9</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>13</mn> </mrow> </msup> <mi>cm</mi> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(41)</label> </div> </div> where we use <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>3</mn> <mi mathvariant="sans-serif">Ω</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>9</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, whose period is <math display='inline'><semantics> <mrow> <mn>10</mn> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> yr. For laser experiments, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>~</mo> <mtext> </mtext> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>, which yields <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>11</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>17</mn> </mrow> </msup> </mrow> </semantics></math>. The typical time scale of accretion disk evolution is estimated by the growth rate of the magnetorotational instability, <div class='html-disp-formula-info' id='FD47-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>~</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>7</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>9</mn> </mrow> </msup> <mi mathvariant="normal">s</mi> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(42)</label> </div> </div></div><div class='html-p'>The typical time scale of experiments is <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mtext> </mtext> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, which makes scaling factor <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> </mrow> </semantics></math> as <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>11</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>17</mn> </mrow> </msup> </mrow> </semantics></math>. Since <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> is given by <div class='html-disp-formula-info' id='FD48-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(43)</label> </div> </div> to make the equation of motion invariant under the transformation, the range of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> is <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>28</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>0</mn> </mrow> </msup> <mtext> </mtext> </mrow> </semantics></math>(due to the square in Equation (43)), which overlaps with the range of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> calculated from the magnetic field strength. Thus, we find that a laser experiment for an accretion disk may be achieved in the range <math display='inline'><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo><</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo><</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>0</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>The rotation frequency for experiments <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> </mrow> </semantics></math> is given by <div class='html-disp-formula-info' id='FD49-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </msqrt> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(44)</label> </div> </div></div><div class='html-p'>For realistic experiments, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <msup> <mrow> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, which corresponds to <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <mn>1</mn> </mrow> </semantics></math> G, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> Pa, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> cm, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> s.</div><div class='html-p'>We also apply this scale transformation to active galactic nuclei (AGNs). The typical ranges of variables for AGNs are <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>A</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>A</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>5</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>6</mn> </mrow> </msup> <mi mathvariant="normal">g</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <msup> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>A</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mtext> </mtext> <mi>K</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>A</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mo>⊙</mo> </mrow> </msub> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>A</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>~</mo> <mn>100</mn> <mtext> </mtext> <mi>AU</mi> </mrow> </semantics></math>. Substituting all the variables to Equations (39) and (40), we can estimate realistic experimental values as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> G, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> g/cm<sup>3</sup>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>9</mn> </mrow> </msup> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>11</mn> </mrow> </msup> <mi>Pa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> cm, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> s and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>10</mn> <mtext> </mtext> <msup> <mrow> <mi mathvariant="normal">s</mi> </mrow> <mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mrow> </msup> </mrow> </semantics></math>. The comparison of the estimated values is shown in <a href="#universe-10-00411-t001" class="html-table">Table 1</a>. In both cases, the ratios <math display='inline'><semantics> <mrow> <mi mathvariant="script">R</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> may be small. It is not easy to adjust the rotation frequency and the shear rate of the target to the ideal values. The Parker instability is relatively easy to control since the pressure of the laser directly makes the target Parker unstable. The resulting gas buoyancy may be too strong to couple either the magnetorotational instability or kinematic dynamo.</div><div class='html-p'>A similar physical situation of dynamically twisting magnetic field lines and consequent plasma dynamics may be seen in the FRC plasma [<a href="#B12-universe-10-00411" class="html-bibr">12</a>,<a href="#B13-universe-10-00411" class="html-bibr">13</a>,<a href="#B14-universe-10-00411" class="html-bibr">14</a>,<a href="#B28-universe-10-00411" class="html-bibr">28</a>]. The FRC plasma are often injected by rotating beams in the middle (See <a href="#universe-10-00411-f008" class="html-fig">Figure 8</a>). Such a rotation of one part of the FRC plasma in one part of the axial position may be different at the other part of the axial position, which could cause a shearing magnetic field that originally lacked an azimuthal component. This situation is not different from the situation with the astrophysical disk-induced magnetic shear [<a href="#B28-universe-10-00411" class="html-bibr">28</a>]. It would be of interest to see some evidence of various plasma instabilities, their nonlinear evolutions and a certain class of structure formation of plasmas seen in astrophysical plasmas [<a href="#B16-universe-10-00411" class="html-bibr">16</a>] in laboratories.</div></section><section id='sec6-universe-10-00411' type='discussion'><h2 data-nested='1'> 6. Discussion</h2><div class='html-p'>We have shown that the combined effects of the magnetorotational and Parker instabilities give rise to density blobs in a primordial disk. We have calculated the mass spectrum of such aggregations from the typical wavelengths of the magnetorotational instability, self-gravitation and magnetic dynamo. By transforming the equations of motion, we have found that this coupling instability may be observed experimentally in laboratory plasma. (The azimuthal dependence of the coupled instability remains to be an issue of investigation.)</div><div class='html-p'>The wavenumber dependence of the magnetorotational instability arises only from the ratio <math display='inline'><semantics> <mrow> <mi>q</mi> <mo>=</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>/</mo> <msubsup> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>θ</mi> </mrow> </msub> </mrow> </semantics></math> is the wavenumber in the azimuthal direction, and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> is in the direction of the rotation axis [<a href="#B5-universe-10-00411" class="html-bibr">5</a>]. By using the radius of the disk <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> and azimuthal mode number <math display='inline'><semantics> <mrow> <mi>m</mi> </mrow> </semantics></math>, we estimate <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>θ</mi> </mrow> </msub> </mrow> </semantics></math> as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo>/</mo> <mi>r</mi> </mrow> </semantics></math>. For the thick disk case, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> can be estimated by the reciprocal of the <math display='inline'><semantics> <mrow> <mi>π</mi> </mrow> </semantics></math> divided by the half height of the disk <math display='inline'><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> and a vertical mode number <math display='inline'><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>π</mi> <mo>/</mo> <mi>H</mi> </mrow> </semantics></math>. The ratio <math display='inline'><semantics> <mrow> <mi>q</mi> </mrow> </semantics></math> is then given by <div class='html-disp-formula-info' id='FD50-universe-10-00411'> <div class='f'> <math display='block'><semantics> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>m</mi> <mi>H</mi> </mrow> <mrow> <mi>π</mi> <mi>n</mi> <mi>r</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(45)</label> </div> </div></div><div class='html-p'>We have discussed the possibility of experiments of analogous phenomena in laboratory plasmas. First, the radiation pressure may be stronger in laser plasmas than primordial accretion disks (perhaps more fitting in disks with AGNs). Since the temperature of the accretion disk gas is typically low (<math display='inline'><semantics> <mrow> <mi>T</mi> <mo>~</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>K</mi> </mrow> </semantics></math>), we neglect the radiation pressure in our calculation and simulation. However, the temperature in the analog laser experiments is typically high (10~100 eV) so that the radiation pressure is not negligible. Second, the control of shear flows is challenging in laboratory plasmas for its role in the transport of angular momentum. In tokamak devices, shear flows are created by an unbalanced NBI and maintained by plasma itself. To create near-Keplerian shear flows in the device, the injection angle of the NBI, external magnetic field and plasma density should be carefully considered. In laser plasmas, shear flows may be created by injected laser from opposite direction. Rigid rotating flows may be possible to create. Inhomogeneites of the laser irradiation to a target also prevent shear flows from achieving uniformity. Despite these issues, analogous experiments may be important to observe and provide a database for nonlinear evolutions of accretion disks with large density and magnetic field ranges. Our estimations can be applied to not only to AGNs, but also to any accretion disks with strongly magnetized plasma, such as accretion disks around dwarf novae and black holes. Analog experiments of these disks are also possible.</div><div class='html-p'>As discussed above, the cooperation of magnetic field and plasma in the astrophysical objects (of a certain class) initiates and/or enhances the structure formation, provides efficient conversion of gravitational energies into flows and accelerates the structure formation and evolution of astrophysical objects [<a href="#B17-universe-10-00411" class="html-bibr">17</a>]. By combining the proposed experiments with theoretical considerations, we should be able to gain a deeper understanding of structure formation in the disk, and the star- and galaxy-scale structure formations that are linked to it.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, K.N., T.T. and W.H.; methodology, K.N. and T.T.; software, K.N.; validation, K.N. and T.T.; formal analysis, K.N., T.T. and W.H.; writing—original draft preparation K.N.; writing—review and editing, K.N., T.T. and W.H.; visualization, K.N.; supervision, T.T. and W.H. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This work is partially supported by NSF grant number ATM 98-15809, the US Department of Energy under contract number W-7405-ENG-48 (at UT Austin) and the Norman Rostoker Chair (at UCI).</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>We would like to thank Wen-Chien Chou, R. Matsumoto and K. Shibata for useful discussions. In addition, we would like to thank S. Inutsuka from Nagoya Univ. and T. Suzuki from Tokyo Univ. for their valuable comments. K.N. also thanks A. Imaura from JPO for the permission to publish this work. 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Shear flow and magnetic field makes an accretion disk 1mstable to the magnetorotational instability. The disk is also unstable to the Parker instability because of gravity from the central star and magnetic field inside of the disk.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f001"> <div class="html-caption"> <b>Figure 1.</b> A schematic view of MHD instabilities in early stage of the primordial evolution. Shear flow and magnetic field makes an accretion disk 1mstable to the magnetorotational instability. The disk is also unstable to the Parker instability because of gravity from the central star and magnetic field inside of the disk.</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g001.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g001.png" alt="Universe 10 00411 g001" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f002"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png" alt="Universe 10 00411 g002" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Model picture of local simulation box (<b>right</b>) in accretion disk (<b>left</b>). Within the box, the Cartesian coordinate <math display='inline'><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math> is along the radial cylindrical coordinate <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> (with respect to the center of the accretion disk) and <math display='inline'><semantics> <mrow> <mi>y</mi> </mrow> </semantics></math> is along <math display='inline'><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mi>z</mi> </mrow> </semantics></math> is same in both the systems, where <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> are the equatorial plane and the height of the accretion disk, respectively. The planes A-A’ and B-B’ indicate the snapshot planes in Figures 5 and 6, respectively.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f002"> <div class="html-caption"> <b>Figure 2.</b> Model picture of local simulation box (<b>right</b>) in accretion disk (<b>left</b>). Within the box, the Cartesian coordinate <math display='inline'><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math> is along the radial cylindrical coordinate <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> (with respect to the center of the accretion disk) and <math display='inline'><semantics> <mrow> <mi>y</mi> </mrow> </semantics></math> is along <math display='inline'><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mi>z</mi> </mrow> </semantics></math> is same in both the systems, where <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>H</mi> </mrow> </semantics></math> are the equatorial plane and the height of the accretion disk, respectively. The planes A-A’ and B-B’ indicate the snapshot planes in Figures 5 and 6, respectively.</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png" alt="Universe 10 00411 g002" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f003"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png" alt="Universe 10 00411 g003" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Mass spectrum of an accretion disk in the outer region (<math display='inline'><semantics> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics></math> AU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in arbitrary units. We plot the spectrum for two different density profile, <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> <mo>)</mo> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the profile given by Equation (22).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f003"> <div class="html-caption"> <b>Figure 3.</b> Mass spectrum of an accretion disk in the outer region (<math display='inline'><semantics> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics></math> AU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in arbitrary units. We plot the spectrum for two different density profile, <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> <mo>)</mo> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the profile given by Equation (22).</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png" alt="Universe 10 00411 g003" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f004"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png" alt="Universe 10 00411 g004" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> Mass spectrum of an accretion disk in the inner region (r < lAU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in units of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) shows the spectrum due to the self-gravity of gas, and (<b>b</b>) shows the spectrum due to the magnetic dynamo. We plot the spectrum for two different density profiles, <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> </mrow> </mfenced> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the density profile given by Equation (22).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f004"> <div class="html-caption"> <b>Figure 4.</b> Mass spectrum of an accretion disk in the inner region (r < lAU). The horizontal axis indicates the radial distance from the rotational axis of the disk, in units of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) shows the spectrum due to the self-gravity of gas, and (<b>b</b>) shows the spectrum due to the magnetic dynamo. We plot the spectrum for two different density profiles, <math display='inline'><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>r</mi> </mrow> </mfenced> </mrow> <mrow> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math> and the density profile given by Equation (22).</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png" alt="Universe 10 00411 g004" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g004.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f005"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png" alt="Universe 10 00411 g005" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Snapshots of the local disk structure in the plane A-A’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f005"> <div class="html-caption"> <b>Figure 5.</b> Snapshots of the local disk structure in the plane A-A’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png" alt="Universe 10 00411 g005" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g005.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f006"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png" alt="Universe 10 00411 g006" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Snapshots of the local disk structure in the plane B-B’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f006"> <div class="html-caption"> <b>Figure 6.</b> Snapshots of the local disk structure in the plane B-B’ shown in <a href="#universe-10-00411-f002" class="html-fig">Figure 2</a>, at t = 2.05 rotations (<b>a</b>,<b>c</b>,<b>e</b>) and 2.93 rotations (<b>b</b>,<b>d</b>,<b>f</b>). Density (<b>a</b>,<b>b</b>) and plasma <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>(<b>c</b>,<b>d</b>) contours, and velocity vectors (<b>e</b>,<b>f</b>) are shown. The solid lines correspond to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>), while the dotted lines corresponds to <math display='inline'><semantics> <mrow> <mi>δ</mi> <mi>ρ</mi> <mo>/</mo> <mi>ρ</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) <math display='inline'><semantics> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi>δ</mi> <mi>β</mi> <mo>/</mo> <mi>β</mi> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>).</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png" alt="Universe 10 00411 g006" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g006.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f007"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png" alt="Universe 10 00411 g007" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> A schematic view of a laser experiment. Electric current is applied in the direction of rotation to create magnetic field in the azimuthal direction. The directions of laser injection are slightly out of axis to create shear flow in the azimuthal direction. The injected laser compresses the target radially to make target Parker unstable.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f007"> <div class="html-caption"> <b>Figure 7.</b> A schematic view of a laser experiment. Electric current is applied in the direction of rotation to create magnetic field in the azimuthal direction. The directions of laser injection are slightly out of axis to create shear flow in the azimuthal direction. The injected laser compresses the target radially to make target Parker unstable.</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png" alt="Universe 10 00411 g007" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g007.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00411-f008"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f008"> <img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png" alt="Universe 10 00411 g008" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#fig_body_display_universe-10-00411-f008"></a> </div> </div> <div class="html-fig_description"> <b>Figure 8.</b> A typical FRC magnetic configurations and their possible beam injected shear appearance. (<b>a</b>) The axial magnetic field lines of an FRC overall outside of the separatrix, including the middle region (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>~</mo> <mn>0</mn> </mrow> </semantics></math>), the endplug regions (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>~</mo> <mn>10</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mo>~</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics></math>) and divertor regions (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>18</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo><</mo> <mo>−</mo> <mn>18</mn> </mrow> </semantics></math>). (after Ref. [<a href="#B13-universe-10-00411" class="html-bibr">13</a>]). (<b>b</b>) Meanwhile, magnetic fields within or near the separatrix are shown in (<b>b</b>), possibly with the beam driving sheared magnetic fields.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00411-f008"> <div class="html-caption"> <b>Figure 8.</b> A typical FRC magnetic configurations and their possible beam injected shear appearance. (<b>a</b>) The axial magnetic field lines of an FRC overall outside of the separatrix, including the middle region (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>~</mo> <mn>0</mn> </mrow> </semantics></math>), the endplug regions (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>~</mo> <mn>10</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mo>~</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics></math>) and divertor regions (<math display='inline'><semantics> <mrow> <mi mathvariant="normal">z</mi> <msub> <mrow> <mo>/</mo> <mi>R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>18</mn> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo><</mo> <mo>−</mo> <mn>18</mn> </mrow> </semantics></math>). (after Ref. [<a href="#B13-universe-10-00411" class="html-bibr">13</a>]). (<b>b</b>) Meanwhile, magnetic fields within or near the separatrix are shown in (<b>b</b>), possibly with the beam driving sheared magnetic fields.</div> <div class="html-img"><img data-large="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png" data-original="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png" alt="Universe 10 00411 g008" data-lsrc="/universe/universe-10-00411/article_deploy/html/images/universe-10-00411-g008.png" /></div> </div> <div class="html-table-wrap" id="universe-10-00411-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href='#table_body_display_universe-10-00411-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/411/display" href="#table_body_display_universe-10-00411-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Typical quantities of primordial disks and experimental plasmas. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00411-t001"> <div class="html-caption"><b>Table 1.</b> Typical quantities of primordial disks and experimental plasmas.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Quantities</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Primordial Disks</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Tokamak</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Laser</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >L (cm)</td><td align='center' valign='middle' class='html-align-center' >10<sup>13</sup>~10<sup>15</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>3</sup>~10<sup>7</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>−4</sup>~10<sup>−2</sup></td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>t</span> (s)</td><td align='center' valign='middle' class='html-align-center' >10<sup>6</sup>~10<sup>12</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>−5</sup>~10<sup>−4</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>−8</sup>~10<sup>−2</sup></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >P (Pa)</td><td align='center' valign='middle' class='html-align-center' >10<sup>9</sup>~10<sup>11</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>3</sup>~10<sup>4</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>9</sup>~10<sup>11</sup></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >ρ (g/cm<sup>3</sup>)</td><td align='center' valign='middle' class='html-align-center' >10<sup>5</sup>~10<sup>6</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>−11</sup>~10<sup>−10</sup></td><td align='center' valign='middle' class='html-align-center' >1~10</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >B (G)</td><td align='center' valign='middle' class='html-align-center' >10<sup>−3</sup>~10<sup>−1</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>3</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>−1</sup></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >T (eV)</td><td align='center' valign='middle' class='html-align-center' >10<sup>−1</sup>~1</td><td align='center' valign='middle' class='html-align-center' >10<sup>4</sup></td><td align='center' valign='middle' class='html-align-center' >1</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >a (cm/s<sup>2</sup>)</td><td align='center' valign='middle' class='html-align-center' >10<sup>−11</sup>~10<sup>−6</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>10</sup>~10<sup>13</sup></td><td align='center' valign='middle' class='html-align-center' >10<sup>11</sup>~10<sup>16</sup></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msup> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10<sup>6</sup>~10<sup>9</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10<sup>1</sup>~10<sup>2</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >≪1</td></tr></tbody> </table> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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Structure Formation Through Magnetohydrodynamical Instabilities in Primordial Disks