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Numerical simulations of fully nonlinear viscoresistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics is affected by shaping. It is shown that independent of the plasma shape, the quantity , comparing the hydrodynamic pressure to the magnetic pressure, decreases for increasing values of the Lundquist number, but the pressure gradient fluctuations remain roughly constant, when compared to the Lorentz force. Different elliptical shapes of the cross-section of the domain lead to the excitation of different toroidal (or axial) magnetic and dynamic modes. Furthermore, it is found that in a geometry with circular cross-section, a significant local poloidal angular momentum is observed, absent in the geometries with elliptical cross-section. Because the confinement is dominantly determined by plasma movement, and the dynamics of the velocity and magnetic field are modified by the modification of the geometry, shaping can thus affect the performance of RFP devices."> <meta name="citation_doi" content="10.1017/S0022377821001094"> <link rel="alternate" href="/core/journals/journal-of-plasma-physics/article/abs/effect-of-shaping-on-turbulent-dynamics-in-reversedfield-pinch-simulations/27954E6E18E0B6BF60FD6C952D1D89D5" hreflang="en" /> <link rel="icon" href="/core/cambridge-core/public/images/favicon.ico" type="image/x-icon"/> <link rel="shortcut icon" href="/core/cambridge-core/public/images/favicon.ico" type="image/x-icon"/> <link href='//fonts.googleapis.com/css?family=Noto+Sans:400,700,400italic,700italic' rel='stylesheet' type='text/css'>  <link rel="stylesheet" href="/core/cambridge-core/public/css/app.css?version=v7.337.1-hotfix"/> <link rel="dns-prefetch" href="https://usage.prod.aop.cambridge.org/v1/events"> <link 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app-link--accent" data-v-63dfaf6e data-v-3692cf84><span data-v-63dfaf6e>Volume 87 Issue 6</span> </a></li><li class="page-breadcrumbs__item" data-v-3692cf84><span aria-hidden="true" class="breadcrumbs-wrapper__arrow" data-v-3692cf84>></span><span data-v-3692cf84>Effect of shaping on turbulent dynamics in reversed-field...</span></li></ul></div></div> <div class="language" data-v-3692cf84><ul class="language-switch" data-v-6b1118dd data-v-3692cf84><li aria-label="English" data-v-6b1118dd><span class="language-option current divider" data-v-6b1118dd>English</span></li><li aria-label="Français" data-v-6b1118dd><span role="button" tabindex="0" href="#" lang="fr" class="language-option" data-v-6b1118dd> Français </span></li></ul></div></div></div></div> <div class="container container__modified" data-v-01274b1d>  <div class="row" data-v-01274b1d><div role="complementary" aria-label="table of content" class="column__left" data-v-01274b1d><div class="col journal-container row" data-v-146270e8 data-v-01274b1d><img src="https://static.cambridge.org/covers/PLA_0_0_0/journal-of-plasma-physics.jpg" alt="" class="journal__image" data-v-146270e8> <a href="/core/journals/journal-of-plasma-physics" class="app-link journal__title app-link__text app-link--underlined" data-v-63dfaf6e data-v-146270e8><span class="text" data-v-63dfaf6e>Journal of Plasma Physics </span> </a> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-146270e8></div>  <div id="toc" class="table-of-content" data-v-01274b1d><h2>Article contents</h2> <div id="toc-list-wrapper" class="table-of-content__wrapper"><ul id="toc-list" class="list"><li class="list__item"><a href="#sec0" class="list__item__link"><span class="toc-title">Abstract</span></a></li> <li class="list__item"><a href="#footnotes-list" class="list__item__link"><span class="toc-title">Footnotes</span></a></li> <li class="list__item"><a href="#references-list" class="list__item__link"><span class="toc-title">References</span></a></li></ul></div></div></div> <div class="column__main" data-v-01274b1d><div class="row" data-v-01274b1d><div class="column__main__left" data-v-01274b1d><div id="maincontent" class="col" data-v-862424e6 data-v-01274b1d> <hgroup data-v-862424e6><h1 data-v-862424e6>Effect of shaping on turbulent dynamics in reversed-field pinch simulations</h1> </hgroup>     <div class="row published-date" data-v-862424e6><p data-v-862424e6> Published online by Cambridge University Press: <strong data-v-862424e6>29 November 2021</strong></p></div>   <div class="contributors-details" data-v-99f6eb26 data-v-862424e6><div class="row contributors" data-v-99f6eb26><div class="col" data-v-99f6eb26><div class="row contributor-type" data-v-792406ce data-v-99f6eb26> <div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Robert%20Chahine&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Robert Chahine</span> </a>  <span data-v-792406ce>,</span></div><div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Kai%20Schneider&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Kai Schneider</span> </a> <a target="_blank" href="https://orcid.org/0000-0003-1243-6621" data-test-orcid="Kai Schneider" class="app-link contributor-type__contributor__orcid app-link__icon app-link--" data-v-63dfaf6e data-v-792406ce><img src="data:image/svg+xml;base64,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" alt="Open the ORCID record for Kai Schneider" class="app-icon icon orcid" data-v-d2c09870 data-v-63dfaf6e> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a> <span data-v-792406ce> and</span></div><div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Wouter%20J.T.%20Bos&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Wouter J.T. Bos</span> </a> <a target="_blank" href="https://orcid.org/0000-0003-3510-0362" data-test-orcid="Wouter J.T. Bos" class="app-link contributor-type__contributor__orcid app-link__icon app-link--" data-v-63dfaf6e data-v-792406ce><img src="data:image/svg+xml;base64,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" alt="Open the ORCID record for Wouter J.T. Bos" class="app-icon icon orcid" data-v-d2c09870 data-v-63dfaf6e> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a> <span data-v-792406ce></span></div> </div> </div> <div class="col-2 collapse-link" data-v-99f6eb26><a href="#authors-details" data-toggle="collapse" aria-expanded="false" aria-controls="authors-details" class="app-link collapsed app-link__text-icon app-link--secondary reverse" data-v-63dfaf6e data-v-99f6eb26><img src="data:image/svg+xml;base64,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" alt="" class="app-icon icon arrow-down" data-v-d2c09870 data-v-63dfaf6e><span class="text" data-v-63dfaf6e>Show author details </span> </a></div></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-99f6eb26> <dl id="authors-details" class="authors-details collapse" data-v-2edb8da6 data-v-99f6eb26><div data-test-author="Robert Chahine" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Robert Chahine</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Univ Claude Bernard Lyon 1, LMFA, UMR5509, 69340 Ecully, France</span> </span></div></dd></div><div data-test-author="Kai Schneider" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Kai Schneider</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Aix-Marseille Université, CNRS, Institut de Mathématiques de Marseille (I2M), UMR 7373, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France</span> </span></div></dd></div><div data-test-author="Wouter J.T. Bos" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Wouter J.T. Bos*</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Univ Claude Bernard Lyon 1, LMFA, UMR5509, 69340 Ecully, France</span> </span></div></dd></div> <div class="row" data-v-2edb8da6><dt class="col-sm-2 col-12 title" data-v-2edb8da6> * </dt> <dd class="col content" data-v-2edb8da6><div class="row" data-v-2edb8da6><div class="d-sm-flex d-inline flex-sm-1 flex-sm-wrap" data-v-2edb8da6><div class="d-inline" data-v-2edb8da6><span data-v-2edb8da6><div class="corresp"><span class="label">†</span>Email address for correspondence: <a href="mailto:wouter.bos@ec-lyon.fr">wouter.bos@ec-lyon.fr</a></div></span></div></div></div></dd></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-2edb8da6></dl></div></div> <div id="app-tabs" 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href="https://s100.copyright.com/AppDispatchServlet?publisherName=CUP&publication=PLA&title=Effect%20of%20shaping%20on%20turbulent%20dynamics%20in%20reversed-field%20pinch%20simulations&publicationDate=29%20November%202021&author=Robert%20Chahine%2C%20Kai%20Schneider%2C%20Wouter%20J.T.%20Bos©right=Copyright%20%C2%A9%20The%20Author(s)%2C%202021.%20Published%20by%20Cambridge%20University%20Press&contentID=10.1017%2FS0022377821001094&startPage=&endPage=&orderBeanReset=True&volumeNum=87&issueNum=6&oa=" data-test-id="buttonRightLink" class="app-link rights-link d-print-none app-link__text-icon app-link--accent" data-v-63dfaf6e data-v-92ee52a2 data-v-43a4d572><img src="/core/page-component/img/rights-icon.d4a677c.svg" alt="" class="app-icon icon rights" data-v-d2c09870 data-v-63dfaf6e><span class="text" data-v-63dfaf6e>Rights & Permissions </span> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a></div> <hr aria-hidden="true" class="separator default" data-v-7036083a></div> <div class="share-modal-overlay" style="display:none;" data-v-43a4d572></div> <div data-spy="scroll" data-target="#toc" class="scrollspy-content" data-v-43a4d572> <div id="sec0" class="col article-abstract sec" data-v-2fa8b348 data-v-43a4d572><div class="abstract-text-container" data-v-2fa8b348><div lang="en"><h2>Abstract</h2>  <div class="abstract-content"><div class="abstract" data-abstract-type="normal"><p>We study the influence of the shape of the plasma container on the dynamics of the reversed-field pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear viscoresistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics is affected by shaping. It is shown that independent of the plasma shape, the quantity <span class="inlineFormula"><span class="alternatives"><span class="mathjax-tex-wrapper" data-mathjax-type="texmath"><span class="tex-math mathjax-tex-math mathjax-on">$\beta$</span></span><img class="inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off" data-mimesubtype="png" data-type src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20211126154034391-0479:S0022377821001094:S0022377821001094_inline1.png" /></span></span>, comparing the hydrodynamic pressure to the magnetic pressure, decreases for increasing values of the Lundquist number, but the pressure gradient fluctuations remain roughly constant, when compared to the Lorentz force. Different elliptical shapes of the cross-section of the domain lead to the excitation of different toroidal (or axial) magnetic and dynamic modes. Furthermore, it is found that in a geometry with circular cross-section, a significant local poloidal angular momentum is observed, absent in the geometries with elliptical cross-section. Because the confinement is dominantly determined by plasma movement, and the dynamics of the velocity and magnetic field are modified by the modification of the geometry, shaping can thus affect the performance of RFP devices.</p> </div></div> <hr aria-hidden="true" class="abstract-divider separator default" data-v-7036083a></div></div>   </div>  <div class="keywords" data-v-86c27100 data-v-43a4d572><h2 data-v-86c27100>Keywords</h2> <div class="row keywords__pills" data-v-86c27100><a href="/core/search?filters[keywords]=MHD and Electrohydrodynamics" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>MHD and Electrohydrodynamics</span></a><a href="/core/search?filters[keywords]=MHD turbulence" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>MHD turbulence</span></a><a href="/core/search?filters[keywords]=plasmas" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>plasmas</span></a></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-86c27100></div>  <dl class="article-details" data-v-6e32a161 data-v-43a4d572><div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Type </dt> <dd class="col content" data-v-6e32a161>Research Article</dd></div> <div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Information </dt> <dd class="col content" data-v-6e32a161><div class="content__journal" data-v-6e32a161><a href="/core/journals/journal-of-plasma-physics" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Journal of Plasma Physics </span> </a> <span data-v-6e32a161> , <a href="/core/journals/journal-of-plasma-physics/volume/7FC091DA30C15E529273A7B57565D1FD" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Volume 87 </span> </a></span> <span data-v-6e32a161> , <a href="/core/journals/journal-of-plasma-physics/issue/0FF8E7AFBEDC5EFC2F4158E39CD846E4" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Issue 6 </span> </a></span> <span data-v-6e32a161>, December 2021</span>   <span data-v-6e32a161>, 905870608</span></div> <div class="doi-data" data-v-6e32a161><div data-v-6e32a161>DOI: <a target="_blank" href="https://doi.org/10.1017/S0022377821001094" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>https://doi.org/10.1017/S0022377821001094 </span> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a></div> <a data-target="crossmark" aria-label="Check for updates" href="#" class="crossmark-widget" data-v-6e32a161><img src="/core/page-component/img/crossmark-logo.61d5da3.svg" alt="Check for updates" data-v-6e32a161></a></div>  <div data-v-6e32a161><a target="_blank" href="https://ui.adsabs.harvard.edu/abs/10.1017/S0022377821001094" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>NASA ADS Abstract Service </span> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a></div></dd></div>   <div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Copyright </dt> <dd class="col content" data-v-6e32a161><div data-v-6e32a161> Copyright © The Author(s), 2021. Published by Cambridge University Press </div></dd></div></dl> <div id="access-block" class="d-print-none" data-v-43a4d572><div class="access-options"><h2 class="access-options__heading">Access options</h2> <span>Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)</span> <div class="access-options__content access-options__new"> </div></div></div> <div id="content-container" class="content-container" data-v-43a4d572> <hr aria-hidden="true" class="list-divider separator default" data-v-7036083a> <div id="footnotes-list" class="circle-list"><h2>Footnotes</h2> <div data-type="footnote" class="circle-list__item"><div class="circle-list__item__indicator"></div> <div class="circle-list__item__grouped"><div class="circle-list__item__grouped__content"><div class="fn"><span class="label">‡</span><p>Present address: Rio Tinto Aluminium, Voreppe, France.</p></div></div></div></div> </div> <hr aria-hidden="true" class="list-divider separator default" data-v-7036083a> <div id="references-list" class="circle-list"><h2>References</h2> <div id="reference-header-17kmk" aria-flowto="reference-1-content reference-1-button"><h4> REFERENCES </h4> <div class="circle-list__item__indicator"></div> <div aria-hidden="true" data-test-hidden="true" class="circle-list__item__number"> </div> <div class="circle-list__item__grouped"><div id="reference-1-content" class="circle-list__item__grouped__content"></div></div></div><div id="ref1" aria-flowto="reference-2-content reference-2-button" class="circle-list__item"> <div class="circle-list__item__indicator"></div> <div aria-hidden="true" data-test-hidden="true" class="circle-list__item__number"> </div> <div class="circle-list__item__grouped"><div id="reference-2-content" class="circle-list__item__grouped__content"><span class="string-name"><span class="surname">Almagri</span>, <span class="given-names">A.</span></span>, <span class="string-name"><span class="surname">Assadi</span>, <span class="given-names">S.</span></span>, <span class="string-name"><span class="surname">Dexter</span>, <span class="given-names">R.</span></span>, <span class="string-name"><span class="surname">Prager</span>, <span class="given-names">S.</span></span>, <span class="string-name"><span class="surname">Sarff</span>, <span class="given-names">J.</span></span> & <span class="string-name"><span class="surname">Sprott</span>, <span class="given-names">J.</span></span> <span class="year">1987</span> <span class="article-title">Studies of large, non-circular, reversed field pinch discharges</span>. <span class="source">Nucl. 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Bos",searchUrl:"\u002Fcore\u002Fsearch?filters%5BauthorTerms%5D=Wouter%20J.T.%20Bos&eventCode=SE-AU",orcidUrl:"https:\u002F\u002Forcid.org\u002F0000-0003-3510-0362"}]},translators:{contributors:[],label:"Translated by"}},collections:[],publishedDate:"29 November 2021",keywords:[{url:"\u002Fcore\u002Fsearch?filters[keywords]=MHD and Electrohydrodynamics",name:"MHD and Electrohydrodynamics"},{url:"\u002Fcore\u002Fsearch?filters[keywords]=MHD turbulence",name:"MHD turbulence"},{url:"\u002Fcore\u002Fsearch?filters[keywords]=plasmas",name:"plasmas"}],openPracticeBadges:[],doi:{url:"https:\u002F\u002Fdoi.org\u002F10.1017\u002FS0022377821001094",value:"10.1017\u002FS0022377821001094"},copyright:{statement:["Copyright © The Author(s), 2021. Published by Cambridge University Press"],holder:["The Author(s)"],year:[2021]},creativeCommons:e,acceptedManuscript:b,type:"research-article",typeDescription:"Research Article",eNumber:"905870608",commentsCount:r,topicsAndSubtopics:e},journal:{id:E,title:m,titleSlug:F,mnemonic:"PLA",titleHistory:[],isFirstView:b,journalSlug:F,isCompanion:b,parentCompanionJournalName:m,associatedParentCollection:e,paymentInfo:{prices:{"£":{price:26,sku:n,skuNew:l,currency:"£"},"€":{price:31,sku:n,skuNew:l,currency:"€"},US$:{price:36,sku:n,skuNew:l,currency:"US$"},AU$:{price:51,sku:n,skuNew:l,currency:"AU$"}}},url:G,firstViewUrl:"\u002Fcore\u002Fjournals\u002Fjournal-of-plasma-physics\u002Ffirstview",coverUrl:"https:\u002F\u002Fstatic.cambridge.org\u002Fcovers\u002FPLA_0_0_0\u002Fjournal-of-plasma-physics.jpg",submitMaterialsUrl:"\u002Fcore\u002Fjournals\u002Fjournal-of-plasma-physics\u002Finformation\u002Fauthor-instructions\u002Fsubmitting-your-materials",hasHistory:b,latestTitle:m,latestId:E,hasPastTitle:b,issue:{id:"0FF8E7AFBEDC5EFC2F4158E39CD846E4",number:"6",title:"Issue 6",publishedDate:"December 2021",printPublishTimestamp:1638316800000,url:H},volume:{id:"7FC091DA30C15E529273A7B57565D1FD",number:"87",title:"Volume 87",publishedDate:"October 2021",printPublishTimestamp:1633046400000,url:"\u002Fcore\u002Fjournals\u002Fjournal-of-plasma-physics\u002Fvolume\u002F7FC091DA30C15E529273A7B57565D1FD"}},abstract:{textAbstracts:[{title:"Abstract",content:"\u003Cdiv class=\"abstract\" data-abstract-type=\"normal\"\u003E\u003Cp\u003EWe study the influence of the shape of the plasma container on the dynamics of the reversed-field pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear viscoresistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics is affected by shaping. It is shown that independent of the plasma shape, the quantity \u003Cspan class=\"inlineFormula\"\u003E\u003Cspan class=\"alternatives\"\u003E\u003Cspan class=\"mathjax-tex-wrapper\" data-mathjax-type=\"texmath\"\u003E\u003Cspan class=\"tex-math mathjax-tex-math mathjax-on\"\u003E$\\beta$\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E\u003Cimg class=\"inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off\" data-mimesubtype=\"png\" data-type src=\"https:\u002F\u002Fstatic.cambridge.org\u002Fbinary\u002Fversion\u002Fid\u002Furn:cambridge.org:id:binary:20211126154034391-0479:S0022377821001094:S0022377821001094_inline1.png\" \u002F\u003E\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, comparing the hydrodynamic pressure to the magnetic pressure, decreases for increasing values of the Lundquist number, but the pressure gradient fluctuations remain roughly constant, when compared to the Lorentz force. Different elliptical shapes of the cross-section of the domain lead to the excitation of different toroidal (or axial) magnetic and dynamic modes. Furthermore, it is found that in a geometry with circular cross-section, a significant local poloidal angular momentum is observed, absent in the geometries with elliptical cross-section. Because the confinement is dominantly determined by plasma movement, and the dynamics of the velocity and magnetic field are modified by the modification of the geometry, shaping can thus affect the performance of RFP devices.\u003C\u002Fp\u003E\n\u003C\u002Fdiv\u003E",lang:I}]},content:{html:a,tableOfContent:[],footnotes:[{content:"\u003Cdiv class=\"fn\"\u003E\u003Cspan class=\"label\"\u003E‡\u003C\u002Fspan\u003E\u003Cp\u003EPresent address: Rio Tinto Aluminium, Voreppe, France.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E",targetId:"cor2"}],fulltextNotes:[],references:[{item:[],id:"reference-header-17kmk",title:"REFERENCES",description:a},{id:"ref1",displayNumber:a,existInContent:b,content:"\u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EAlmagri\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EA.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EAssadi\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003ES.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EDexter\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003ER.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EPrager\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003ES.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003ESarff\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EJ.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E & \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003ESprott\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EJ.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E \u003Cspan class=\"year\"\u003E1987\u003C\u002Fspan\u003E \u003Cspan class=\"article-title\"\u003EStudies of large, non-circular, reversed field pinch discharges\u003C\u002Fspan\u003E. \u003Cspan class=\"source\"\u003ENucl. 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\u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EAlmagri\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EA.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EAnderson\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EJ.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EBonomo\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EF.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EBrower\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003ED.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003ECombs\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003ES.K.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan 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Fusion\u003C\u002Fspan\u003E \u003Cspan class=\"volume\"\u003E49\u003C\u002Fspan\u003E (\u003Cspan class=\"issue\"\u003E1\u003C\u002Fspan\u003E), \u003Cspan class=\"fpage\"\u003E015003\u003C\u002Fspan\u003E.\u003Ca class='ref-link' target='_blank' aria-label='CrossRef link for Plasma behaviour at high $\\beta$ and high density in the Madison Symmetric Torus RFP' href=https:\u002F\u002Fdx.doi.org\u002F10.1088\u002F0029-5515\u002F49\u002F1\u002F015003\u003ECrossRef\u003C\u002Fa\u003E\u003Ca class='ref-link' target='_blank' aria-label='Google Scholar link for Plasma behaviour at high $\\beta$ and high density in the Madison Symmetric Torus RFP' 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Fusion","Phys. Rev. Lett","Phys. Plasmas","Plasma Phys. Control. Fusion","S0022377821001094","Journal of Plasma Physics","article_pla_ind_pur","2020","Phys. Fluids","western",0,"book","Nonlinear Magnetohydrodynamics","unknown","2013","2008","Spectral Methods in Fluid Dynamics","2000","2009","49","Tokamaks","Effect of shaping on turbulent dynamics in reversed-field pinch simulations","Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Univ Claude Bernard Lyon 1, LMFA, UMR5509, 69340 Ecully, France","F8F44ED0833DA6BE0A78F7639898FA08","journal-of-plasma-physics","\u002Fcore\u002Fjournals\u002Fjournal-of-plasma-physics","\u002Fcore\u002Fjournals\u002Fjournal-of-plasma-physics\u002Fissue\u002F0FF8E7AFBEDC5EFC2F4158E39CD846E4","en","Studies of large, non-circular, reversed field pinch discharges","27","1987","10.1088\u002F0029-5515\u002F27\u002F11\u002F005","10.1017\u002FCBO9780511599965","Experimental-like helical self-organization in reversed-field pinch modeling","10.1103\u002FPhysRevLett.111.085002","Fast ion transport in the quasi-single helical reversed-field pinch","26","2019","10.1063\u002F1.5084059","Rapid generation of angular momentum in bounded magnetized plasma","10.1103\u002FPhysRevLett.101.235003","Self-organization and symmetry-breaking in two-dimensional plasma turbulence","17","2010","10.1063\u002F1.3466030","Reconnection processes and scaling laws in reversed field pinch magnetohydrodynamics","10.1088\u002F0029-5515\u002F36\u002F5\u002FI05","Bifurcation in viscoresistive MHD: the Hartmann number and the reversed field pinch","85","10.1103\u002FPhysRevLett.85.3838","On the role and value of $\\beta$ in incompressible MHD simulations","25","2018","10.1063\u002F1.5018666","Quasi-single-helicity reversed-field-pinch plasmas","10.1103\u002FPhysRevLett.85.1662","Single and multiple helicity ohmic states in reversed-field pinches","Phys. Fluids B","10.1063\u002F1.860082","Improved particle confinement in transition from multiple-helicity to quasi-single-helicity regimes of a reversed-field pinch","10.1103\u002FPhysRevLett.97.175001","Dynamic equilibria and magnetohydrodynamic instabilities in toroidal plasmas with non-uniform transport coefficients","10.1063\u002F1.4919960","Role of resistivity and viscosity in the excitation of stable $m= 0$ modes during the RFP sawtooth crash","10.1063\u002F1.5054578","Does shaping bring an advantage for reversed field pinch plasmas?","53","10.1088\u002F0029-5515\u002F53\u002F11\u002F113035","Spontaneous angular momentum generation of two-dimensional fluid flow in an elliptic geometry","Phys. Rev. E","10.1103\u002FPhysRevE.78.036301","Effects of trapped electrons and impurity ions on ITG modes in reversed-field pinch plasmas","Europhys. Lett","10.1209\u002F0295-5075\u002F127\u002F45002","Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas","Nat. Phys","10.1038\u002Fnphys1308","Overview of RFX-mod results","10.1088\u002F0029-5515\u002F49\u002F10\u002F104019","Quasi-single helicity states in the reversed field pinch: beyond the standard paradigm","7","1984","10.1063\u002F1.874054","Overview of quasi-single helicity experiments in reversed field pinches","10.1088\u002F0029-5515\u002F43\u002F12\u002F028","Modeling of formation of helical structures in reversed-field pinch","Plasma Fusion Res","2403117","2012","10.1585\u002Fpfr.7.2403117","The phenomenology of reconnection events in the reversed field pinch","10.1088\u002F1741-4326\u002Fab7d4e","Intrinsic rotation of toroidally confined magnetohydrodynamics","10.1103\u002FPhysRevLett.109.175002","On the effect of toroidicity on reversed field pinch dynamics","2014","10.1088\u002F0741-3335\u002F56\u002F9\u002F095024","Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization","J. Comput. Phys","10.1016\u002Fj.jcp.2014.05.038","Breaking up-down symmetry with magnetic perturbations in tokamak plasmas: increase of axisymmetric steady-state velocities","10.1063\u002F5.0016566","Ideal toroidal stability beta limits and shaping effect for reversed field pinch configurations","10.1088\u002F0029-5515\u002F31\u002F10\u002F008","RFX-mod: a multi-configuration fusion facility for three-dimensional physics studies","10.1063\u002F1.4806765","Influence of external 3D magnetic fields on helical equilibrium and plasma flow in RFX-mod","2011","10.1088\u002F0741-3335\u002F53\u002F8\u002F084005","Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control","10.1088\u002F0029-5515\u002F49\u002F8\u002F085036","Stepped pressure equilibrium with relaxed flow and applications in reversed-field pinch plasmas","10.1088\u002F1361-6587\u002Fab7fc5","Minimum energy state of a toroidal discharge","10.1063\u002F1.862857","Taylor relaxation in a torus of arbitrary aspect ratio and cross section","10.1063\u002F1.863468","Control of ideal and resistive magnetohydrodynamic modes in reversed field pinches with a resistive wall","10.1063\u002F1.3506819","Magnetohydrodynamic stabilization through rotation","10.1103\u002FPhysRevLett.73.1624","Relaxation of toroidal plasma and generation of reverse magnetic fields","10.1103\u002FPhysRevLett.33.1139","Relaxation and magnetic reconnection in plasmas","Rev. Mod. Phys","10.1103\u002FRevModPhys.58.741","Enhanced confinement and quasi-single-helicity regimes induced by poloidal current drive","10.1103\u002FPhysRevLett.99.095001","MHD-limits to plasma confinement","10.1088\u002F0741-3335\u002F26\u002F1A\u002F319","Magnetohydrodynamics modelling successfully predicts new helical states in reversed-field pinch fusion plasmas","10.1088\u002F1741-4326\u002Faa7f46","Plasma behaviour at high $\\beta$ and high density in the Madison Symmetric Torus RFP","10.1088\u002F0029-5515\u002F49\u002F1\u002F015003"));</script><script src="/core/page-component/06cd607.js" defer></script><script src="/core/page-component/20147ec.js" defer></script><script src="/core/page-component/722ace3.js" defer></script><script src="/core/page-component/6f4302c.js" defer></script> </div> </div> <div id="article-new-home-productCitations" class="product-citations-modal reveal-modal medium" data-reveal role="dialog" aria-labelledby="article-new-home-citedByModalHeader"> <div 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If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. <a href="/core/help">Find out more about saving content to Google Drive</a>. </p> </div> </div> <div class="row book-title"> <div class="small-12 columns"> <div class="heading_08"><div class="title">Effect of shaping on turbulent dynamics in reversed-field pinch simulations</div></div> </div> </div> <div class="row book-details book-overview"> <div class="small-12 columns"> <ul> <li class="volume-issue">Volume 87, <a href="">Issue 6</a></li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=Robert Chahine&eventCode=SE-AU" class="more-by-this-author ">Robert Chahine</a> <sup data-affiliation-id="a1" class="">(a1)</sup><span class="separator ">, </span><a href="/core/search?filters%5BauthorTerms%5D=Kai Schneider&eventCode=SE-AU" class="more-by-this-author ">Kai Schneider</a><a target="_blank" class="orcid-logo " href="https://orcid.org/0000-0003-1243-6621" aria-label="Open the ORCID record for Kai Schneider in new tab/window"></a> <sup data-affiliation-id="a2" class="">(a2)</sup><span class="separator "> and </span><a href="/core/search?filters%5BauthorTerms%5D=Wouter J.T. 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fill="#0072CF" d="M20.4 25.3V15.2h-4.9v2.2h1.1v7.9h-1.1v2.2h6v-2.2h-1.1zm-1.9-11.4c1 0 1.9-1 1.9-2.2 0-1.2-.8-2.2-1.9-2.2s-1.9 1-1.9 2.2c0 1.2.9 2.2 1.9 2.2z"/> </svg> </a> </span> <div id="comment-email-tip_--x--" data-dropdown-content role="dialog" aria-modal="true" aria-hidden="true" tabindex="-1" class="f-dropdown content medium" data-remove-focus="true" aria-label="Email information"> <div class="close-container"> <a href="#" class="button small transparent-no-border radius tooltip-close-btn" id="tooltip-close-link-comment-email-tip_--x--"> <span class="custom-tooltip-button-remove"> <span class="sr-only">Close Email help</span> <svg width="20" height="20" viewBox="0 0 20 20" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" focusable="false"> <defs> <path d="M9.986 0C4.479 0 0 4.434 0 9.916c0 5.469 4.465 9.916 9.986 9.916 5.507 0 9.986-4.433 9.986-9.916C19.958 4.434 15.493 0 9.986 0z" id="a-comment-email-tip_--x--"/> <mask id="b-comment-email-tip_--x--" x="0" y="0" width="19.972" height="19.832" fill="#fff"><use xlink:href="#a-comment-email-tip_--x--"/></mask> </defs> <g fill="none" fill-rule="evenodd"> <use stroke="#436FCC" mask="url(#b-comment-email-tip_--x--)" stroke-width="2" xlink:href="#a-comment-email-tip_--x--"/> <path d="M14.778 13.325a.983.983 0 0 1 0 1.385.982.982 0 0 1-.704.28c-.254 0-.507-.098-.704-.28l-3.352-3.329-3.353 3.329a.973.973 0 0 1-.69.28 1.01 1.01 0 0 1-.69-.28.983.983 0 0 1 0-1.385l3.352-3.328-3.352-3.33a.983.983 0 0 1 0-1.384 1 1 0 0 1 1.395 0l3.352 3.329 3.352-3.329a1 1 0 0 1 1.394 0 .983.983 0 0 1 0 1.385l-3.352 3.329 3.352 3.328z" fill="#436FCC"/> </g> </svg> </span> </a> </div> <p id="info-comment-email-tip_--x--" class="text-left">Your email address will be used in order to notify you when your comment has been reviewed by the moderator and in case the author(s) of the article or the moderator need to contact you directly.</p> </div> <input name="emailAddress_--x--" 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