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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <button aria-controls="toc-Formal_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Formal definition subsection</span> </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Tidal_heating_supporting_a_dynamo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tidal_heating_supporting_a_dynamo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Tidal heating supporting a dynamo</span> </div> </a> <ul id="toc-Tidal_heating_supporting_a_dynamo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kinematic_dynamo_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kinematic_dynamo_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Kinematic dynamo theory</span> </div> </a> <button aria-controls="toc-Kinematic_dynamo_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Kinematic dynamo theory subsection</span> </button> <ul id="toc-Kinematic_dynamo_theory-sublist" class="vector-toc-list"> <li id="toc-Practical_measure_of_possible_dynamos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Practical_measure_of_possible_dynamos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Practical measure of possible dynamos</span> </div> </a> <ul id="toc-Practical_measure_of_possible_dynamos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spontaneous_breakdown_of_a_topological_supersymmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spontaneous_breakdown_of_a_topological_supersymmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Spontaneous breakdown of a topological supersymmetry</span> </div> </a> <ul id="toc-Spontaneous_breakdown_of_a_topological_supersymmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nonlinear_dynamo_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nonlinear_dynamo_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Nonlinear dynamo theory</span> </div> </a> <button aria-controls="toc-Nonlinear_dynamo_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Nonlinear dynamo theory subsection</span> </button> <ul id="toc-Nonlinear_dynamo_theory-sublist" class="vector-toc-list"> <li id="toc-Energy_conversion_between_magnetic_and_kinematic_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy_conversion_between_magnetic_and_kinematic_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Energy conversion between magnetic and kinematic energy</span> </div> </a> <ul id="toc-Energy_conversion_between_magnetic_and_kinematic_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Order_of_magnitude_of_the_magnetic_field_created_by_Earth's_dynamo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Order_of_magnitude_of_the_magnetic_field_created_by_Earth's_dynamo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Order of magnitude of the magnetic field created by Earth's dynamo</span> </div> </a> <ul id="toc-Order_of_magnitude_of_the_magnetic_field_created_by_Earth's_dynamo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numerical_models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numerical_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Numerical models</span> </div> </a> <button aria-controls="toc-Numerical_models-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Numerical models subsection</span> </button> <ul id="toc-Numerical_models-sublist" class="vector-toc-list"> <li id="toc-Observations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Observations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Observations</span> </div> </a> <ul id="toc-Observations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_modelling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_modelling"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Modern modelling</span> </div> </a> <ul id="toc-Modern_modelling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notable_people" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notable_people"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notable people</span> </div> </a> <ul id="toc-Notable_people-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button 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href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%AF%D9%8A%D9%86%D8%A7%D9%85%D9%88" title="نظرية الدينامو – Arabic" lang="ar" hreflang="ar" data-title="نظرية الدينامو" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BE%E0%A6%AF%E0%A6%BC%E0%A6%A8%E0%A6%BE%E0%A6%AE%E0%A7%8B_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="ডায়নামো তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="ডায়নামো তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Hip%C3%B2tesi_de_la_dinamo" title="Hipòtesi de la dinamo – Catalan" lang="ca" hreflang="ca" data-title="Hipòtesi de la dinamo" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dynamotheorie" title="Dynamotheorie – German" lang="de" hreflang="de" data-title="Dynamotheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Hip%C3%B3tesis_de_la_d%C3%ADnamo" title="Hipótesis de la dínamo – Spanish" lang="es" hreflang="es" data-title="Hipótesis de la dínamo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%AF%DB%8C%D9%86%D8%A7%D9%85%D9%88" title="نظریه دینامو – Persian" lang="fa" hreflang="fa" data-title="نظریه دینامو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%9D%B4%EB%84%88%EB%AA%A8_%EC%9D%B4%EB%A1%A0" title="다이너모 이론 – Korean" lang="ko" hreflang="ko" data-title="다이너모 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_dinamo" title="Teori dinamo – Indonesian" lang="id" hreflang="id" data-title="Teori dinamo" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%90%D7%95%D7%A8%D7%99%D7%99%D7%AA_%D7%94%D7%93%D7%99%D7%A0%D7%9E%D7%95" title="תאוריית הדינמו – Hebrew" lang="he" hreflang="he" data-title="תאוריית הדינמו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eodinamo" title="Ģeodinamo – Latvian" lang="lv" hreflang="lv" data-title="Ģeodinamo" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_dinamo" title="Teori dinamo – Malay" lang="ms" hreflang="ms" data-title="Teori dinamo" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dynamotheorie" title="Dynamotheorie – Dutch" lang="nl" hreflang="nl" data-title="Dynamotheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%80%E3%82%A4%E3%83%8A%E3%83%A2%E7%90%86%E8%AB%96" title="ダイナモ理論 – Japanese" lang="ja" hreflang="ja" data-title="ダイナモ理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Dynamoteori" title="Dynamoteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Dynamoteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_do_d%C3%ADnamo" title="Teoria do dínamo – Portuguese" lang="pt" hreflang="pt" data-title="Teoria do dínamo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%BE" title="Теория динамо – Russian" lang="ru" hreflang="ru" data-title="Теория динамо" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Dynamoteoria" title="Dynamoteoria – Finnish" lang="fi" hreflang="fi" data-title="Dynamoteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B9%84%E0%B8%94%E0%B8%99%E0%B8%B2%E0%B9%82%E0%B8%A1" title="ทฤษฎีไดนาโม – Thai" lang="th" hreflang="th" data-title="ทฤษฎีไดนาโม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dinamo_teorisi" title="Dinamo teorisi – Turkish" lang="tr" hreflang="tr" data-title="Dinamo teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Thuy%E1%BA%BFt_dynamo" title="Thuyết dynamo – Vietnamese" lang="vi" hreflang="vi" data-title="Thuyết dynamo" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%91%E7%94%B5%E6%9C%BA%E7%90%86%E8%AE%BA" title="发电机理论 – Chinese" lang="zh" hreflang="zh" data-title="发电机理论" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1269129#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about a proposed theory for the source of a planet's magnetic field. For an explanation of the operation of a mechanical dynamo, see <a href="/wiki/Dynamo" title="Dynamo">Dynamo</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg/330px-Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg/495px-Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg/660px-Dynamo_Theory_-_Outer_core_convection_and_magnetic_field_generation.svg.png 2x" data-file-width="3000" data-file-height="3000" /></a><figcaption>Illustration of the dynamo mechanism that generates the Earth's magnetic field: <a href="/wiki/Convection" title="Convection">convection</a> currents of fluid metal in the Earth's <a href="/wiki/Internal_structure_of_Earth" title="Internal structure of Earth">outer core</a>, driven by heat flow from the inner core, organized into rolls by the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a>, generate circulating electric currents, which supports the magnetic field.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>dynamo theory</b> proposes a mechanism by which a celestial body such as <a href="/wiki/Earth" title="Earth">Earth</a> or a <a href="/wiki/Star" title="Star">star</a> generates a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. The dynamo theory describes the process through which a rotating, <a href="/wiki/Convection" title="Convection">convecting</a>, and <a href="/wiki/Electric" class="mw-redirect" title="Electric">electrically</a> conducting fluid can maintain a magnetic field over <a href="/wiki/Astronomical" class="mw-redirect" title="Astronomical">astronomical</a> time scales. A dynamo is thought to be the source of the <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a> and the magnetic fields of Mercury and the <a href="/wiki/Jovian_planets" class="mw-redirect" title="Jovian planets">Jovian planets</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_of_theory">History of theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=1" title="Edit section: History of theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When <a href="/wiki/William_Gilbert_(astronomer)" class="mw-redirect" title="William Gilbert (astronomer)">William Gilbert</a> published <i><a href="/wiki/De_Magnete" title="De Magnete">de Magnete</a></i> in 1600, he concluded that the Earth is magnetic and proposed the first hypothesis for the origin of this magnetism: permanent magnetism such as that found in <a href="/wiki/Lodestone" title="Lodestone">lodestone</a>. In 1822, <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">André-Marie Ampère</a> proposed that internal currents are responsible for Earth's magnetism.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In 1919, <a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Joseph Larmor</a> proposed that a <a href="/wiki/Dynamo" title="Dynamo">dynamo</a> might be generating the field.<sup id="cite_ref-Larmor1919_3-0" class="reference"><a href="#cite_note-Larmor1919-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> However, even after he advanced his hypothesis, some prominent scientists advanced alternative explanations. The <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a> winner <a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Patrick Blackett</a> did a series of experiments looking for a fundamental relation between <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moment</a>, but found none.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Walter_M._Elsasser" title="Walter M. Elsasser">Walter M. Elsasser</a>, considered a "father" of the presently accepted dynamo theory as an explanation of the Earth's magnetism, proposed that this magnetic field resulted from electric currents induced in the fluid outer core of the Earth. He revealed the history of the Earth's magnetic field through pioneering the study of the magnetic orientation of minerals in rocks. </p><p>In order to maintain the magnetic field against <a href="/wiki/Ohm" title="Ohm">ohmic</a> decay (which would occur for the dipole field in 20,000 years), the outer core must be convecting. The <a href="/wiki/Convection" title="Convection">convection</a> is likely some combination of thermal and compositional convection. The mantle controls the rate at which heat is extracted from the core. Heat sources include gravitational energy released by the compression of the core, gravitational energy released by the rejection of light elements (probably <a href="/wiki/Sulfur" title="Sulfur">sulfur</a>, <a href="/wiki/Oxygen" title="Oxygen">oxygen</a>, or <a href="/wiki/Silicon" title="Silicon">silicon</a>) at the inner core boundary as it grows, latent heat of crystallization at the inner core boundary, and radioactivity of <a href="/wiki/Potassium" title="Potassium">potassium</a>, <a href="/wiki/Uranium" title="Uranium">uranium</a> and <a href="/wiki/Thorium" title="Thorium">thorium</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>At the dawn of the 21st century, numerical modeling of the Earth's magnetic field has not been successfully demonstrated. Initial models are focused on field generation by convection in the planet's fluid outer core. It was possible to show the generation of a strong, Earth-like field when the model assumed a uniform core-surface temperature and exceptionally high viscosities for the core fluid. Computations which incorporated more realistic parameter values yielded magnetic fields that were less Earth-like, but indicated that model refinements <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (February 2022)">which?</span></a></i>]</sup> may ultimately lead to an accurate analytic model. Slight variations in the core-surface temperature, in the range of a few millikelvins, result in significant increases in convective flow and produce more realistic magnetic fields.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid acts to maintain a magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. The conductive fluid in the geodynamo is liquid iron in the outer core, and in the <a href="/wiki/Solar_dynamo" title="Solar dynamo">solar dynamo</a> is ionized gas at the <a href="/wiki/Tachocline" title="Tachocline">tachocline</a>. Dynamo theory of astrophysical bodies uses <a href="/wiki/Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamic</a> equations to investigate how the fluid can continuously regenerate the magnetic field.<sup id="cite_ref-hydromagnetic_dynamo_10-0" class="reference"><a href="#cite_note-hydromagnetic_dynamo-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>It was once believed that the <a href="/wiki/Dipole" title="Dipole">dipole</a>, which comprises much of the <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a> and is misaligned along the rotation axis by 11.3 degrees, was caused by permanent magnetization of the materials in the earth. This means that dynamo theory was originally used to explain the Sun's magnetic field in its relationship with that of the Earth. However, this hypothesis, which was initially proposed by <a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Joseph Larmor</a> in 1919, has been modified due to extensive studies of magnetic <a href="/wiki/Secular_variation" title="Secular variation">secular variation</a>, <a href="/wiki/Paleomagnetism" title="Paleomagnetism">paleomagnetism</a> (including <a href="/wiki/Geomagnetic_reversal" title="Geomagnetic reversal">polarity reversals</a>), seismology, and the solar system's abundance of elements. Also, the application of the theories of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> to magnetic observations showed that Earth's magnetic field had an internal, rather than external, origin. </p><p>There are three requisites for a dynamo to operate: </p> <ul><li>An electrically conductive fluid medium</li> <li>Kinetic energy provided by planetary rotation</li> <li>An internal energy source to drive convective motions within the fluid.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <p>In the case of the Earth, the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effect</a> caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns (also see <a href="/wiki/Taylor_column" title="Taylor column">Taylor columns</a>) aligned with the rotation axis. Induction or generation of magnetic field is described by the <a href="/wiki/Induction_equation" title="Induction equation">induction equation</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {u} \times \mathbf {B} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>η</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {u} \times \mathbf {B} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053cb874e129517d28552c9d1288f580c133c76e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.867ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {u} \times \mathbf {B} )}"></span> where <span class="texhtml"><b>u</b></span> is velocity, <span class="texhtml"><b>B</b></span> is magnetic field, <span class="texhtml mvar" style="font-style:italic;">t</span> is time, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =1/(\sigma \mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1/(\sigma \mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ef8dfacd87c18d48d95d396bd7f016b171e012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.133ex; height:2.843ex;" alt="{\displaystyle \eta =1/(\sigma \mu )}"></span> is the <a href="/wiki/Magnetic_diffusivity" title="Magnetic diffusivity">magnetic diffusivity</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> electrical conductivity and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a>. The ratio of the second term on the right hand side to the first term gives the <a href="/wiki/Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic Reynolds number</a>, a dimensionless ratio of advection of magnetic field to diffusion. </p> <div class="mw-heading mw-heading3"><h3 id="Tidal_heating_supporting_a_dynamo">Tidal heating supporting a dynamo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=3" title="Edit section: Tidal heating supporting a dynamo"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tidal forces between celestial orbiting bodies cause friction that heats up their interiors. This is known as tidal heating, and it helps keep the interior in a liquid state. A liquid interior that can conduct electricity is required to produce a dynamo. Saturn's Enceladus and Jupiter's Io have enough tidal heating to liquify their inner cores, but they may not create a dynamo because they cannot conduct electricity.<sup id="cite_ref-Enceladus_12-0" class="reference"><a href="#cite_note-Enceladus-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Io_geologic_13-0" class="reference"><a href="#cite_note-Io_geologic-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Mercury, despite its small size, has a magnetic field, because it has a conductive liquid core created by its iron composition and friction resulting from its highly elliptical orbit.<sup id="cite_ref-mercury_core_14-0" class="reference"><a href="#cite_note-mercury_core-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It is theorized that the Moon once had a magnetic field, based on evidence from magnetized lunar rocks, due to its short-lived closer distance to Earth creating tidal heating.<sup id="cite_ref-lunar_dynamo_15-0" class="reference"><a href="#cite_note-lunar_dynamo-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> An orbit and rotation of a planet helps provide a liquid core, and supplements kinetic energy that supports a dynamo action. </p> <div class="mw-heading mw-heading2"><h2 id="Kinematic_dynamo_theory">Kinematic dynamo theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=4" title="Edit section: Kinematic dynamo theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In kinematic dynamo theory the velocity field is <i>prescribed</i>, instead of being a dynamic variable: The model makes no provision for the flow distorting in response to the magnetic field. This method cannot provide the time variable behaviour of a fully nonlinear chaotic dynamo, but can be used to study how magnetic field strength varies with the flow structure and speed. </p><p>Using <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> simultaneously with the curl of <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a>, one can derive what is basically a linear eigenvalue equation for magnetic fields (<b><span class="texhtml">B</span></b>), which can be done when assuming that the magnetic field is independent from the velocity field. One arrives at a critical <i><a href="/wiki/Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic Reynolds number</a></i>, above which the flow strength is sufficient to amplify the imposed magnetic field, and below which the magnetic field dissipates. </p> <div class="mw-heading mw-heading3"><h3 id="Practical_measure_of_possible_dynamos">Practical measure of possible dynamos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=5" title="Edit section: Practical measure of possible dynamos"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most functional feature of kinematic dynamo theory is that it can be used to test whether a velocity field is or is not capable of dynamo action. By experimentally applying a certain velocity field to a small magnetic field, one can observe whether the magnetic field tends to grow (or not) in response to the applied flow. If the magnetic field does grow, then the system is either capable of dynamo action or is a dynamo, but if the magnetic field does not grow, then it is simply referred to as “not a dynamo”. </p><p>An analogous method called the <i><a href="/wiki/Membrane_paradigm" title="Membrane paradigm">membrane paradigm</a></i> is a way of looking at <a href="/wiki/Black_hole" title="Black hole">black holes</a> that allows for the material near their surfaces to be expressed in the language of dynamo theory. </p> <div class="mw-heading mw-heading3"><h3 id="Spontaneous_breakdown_of_a_topological_supersymmetry">Spontaneous breakdown of a topological supersymmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=6" title="Edit section: Spontaneous breakdown of a topological supersymmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kinematic dynamo can be also viewed as the phenomenon of the spontaneous breakdown of the topological supersymmetry of the associated stochastic differential equation related to the flow of the background matter.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Within <a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">stochastic supersymmetric theory</a>, this supersymmetry is an intrinsic property of <i>all</i> <a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equations</a>, its interpretation is that the model's phase space preserves continuity via continuous time flows. When the continuity of that flow spontaneously breaks down, the system is in the stochastic state of <a href="/wiki/Chaos_theory" title="Chaos theory"><i>deterministic chaos</i></a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In other words, kinematic dynamo arises because of chaotic flow in the underlying background matter. </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinear_dynamo_theory">Nonlinear dynamo theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=7" title="Edit section: Nonlinear dynamo theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The kinematic approximation becomes invalid when the magnetic field becomes strong enough to affect the fluid motions. In that case the velocity field becomes affected by the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a>, and so the induction equation is no longer linear in the magnetic field. In most cases this leads to a quenching of the amplitude of the dynamo. Such dynamos are sometimes also referred to as <i>hydromagnetic dynamos</i>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos. </p><p>The main idea of the theory is that any small magnetic field existing in the outer core creates currents in the moving fluid there due to Lorentz force. These currents create further magnetic field due to <a href="/wiki/Ampere%27s_law" class="mw-redirect" title="Ampere's law">Ampere's law</a>. With the fluid motion, the currents are carried in a way that the magnetic field gets stronger (as long as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4448b391a4ababddc98d6179f1785cbdfa6451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.386ex; height:2.843ex;" alt="{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"></span> is negative<sup id="cite_ref-Kono2002_19-0" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>). Thus a "seed" magnetic field can get stronger and stronger until it reaches some value that is related to existing non-magnetic forces. </p><p>Numerical models are used to simulate fully nonlinear dynamos. The following equations are used: </p> <ul><li>The induction equation, presented above.</li> <li>Maxwell's equations for negligible electric field: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\nabla \cdot \mathbf {B} =0\\[1ex]&\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\nabla \cdot \mathbf {B} =0\\[1ex]&\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb7f07611a6797a341d97c2b5938b50097c1e74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.364ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}&\nabla \cdot \mathbf {B} =0\\[1ex]&\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} \end{aligned}}}"></span></li> <li>The <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> for <a href="/wiki/Conservation_of_mass" title="Conservation of mass">conservation of mass</a>, for which the <a href="/wiki/Boussinesq_approximation_(buoyancy)" title="Boussinesq approximation (buoyancy)">Boussinesq approximation</a> is often used: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {u} =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {u} =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e12812c8a9d3ffb93c86f1042bef079d7d61cebe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.008ex; height:2.509ex;" alt="{\displaystyle \nabla \cdot \mathbf {u} =0,}"></span></li> <li>The <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier-Stokes equation</a> for conservation of <a href="/wiki/Momentum" title="Momentum">momentum</a>, again in the same approximation, with the magnetic force and gravitation force as the external forces: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho _{0}}}\nabla p+\nu \nabla ^{2}\mathbf {u} +\rho '\mathbf {g} +2{\boldsymbol {\Omega }}\times \mathbf {u} +{\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}\times \mathbf {R} +{\frac {1}{\rho _{0}}}\mathbf {J} \times \mathbf {B} ~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>ν</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <msup> <mi>ρ</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho _{0}}}\nabla p+\nu \nabla ^{2}\mathbf {u} +\rho '\mathbf {g} +2{\boldsymbol {\Omega }}\times \mathbf {u} +{\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}\times \mathbf {R} +{\frac {1}{\rho _{0}}}\mathbf {J} \times \mathbf {B} ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf47af41fe77c19e6ddc9d6d44ce76222dbe1cce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.891ex; height:5.676ex;" alt="{\displaystyle {\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho _{0}}}\nabla p+\nu \nabla ^{2}\mathbf {u} +\rho '\mathbf {g} +2{\boldsymbol {\Omega }}\times \mathbf {u} +{\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}\times \mathbf {R} +{\frac {1}{\rho _{0}}}\mathbf {J} \times \mathbf {B} ~,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\nu \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>ν</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\nu \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587412a8f75acc6cb64d4553cfc26c1945182b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.006ex; height:1.676ex;" alt="{\displaystyle \,\nu \,}"></span> is the kinematic <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\rho _{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\rho _{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3f54757e7b7816da5a271e40309b3607bd6b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.03ex; height:2.176ex;" alt="{\displaystyle \,\rho _{0}\,}"></span>is the mean density and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2400eb83af71f78fcb442e55bf294cebdfb9803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.887ex; height:3.009ex;" alt="{\displaystyle \rho '}"></span> is the relative density perturbation that provides buoyancy (for thermal convection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\rho '=\alpha \Delta T\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msup> <mi>ρ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>α</mi> <mi mathvariant="normal">Δ</mi> <mi>T</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\rho '=\alpha \Delta T\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4703b17f83621a820131d3f8915ed5c50caf9a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.335ex; height:3.009ex;" alt="{\displaystyle \;\rho '=\alpha \Delta T\;}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\alpha \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>α</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\alpha \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec11bf0bf0c4713df31060433e8ed7db766a6daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.262ex; height:1.676ex;" alt="{\displaystyle \,\alpha \,}"></span> is <a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">coefficient of thermal expansion</a>), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\Omega \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\Omega \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46711322bdaf8a80e055f8f087007247fd489dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.452ex; height:2.176ex;" alt="{\displaystyle \,\Omega \,}"></span> is the <a href="/wiki/Earth%27s_rotation" title="Earth's rotation">rotation rate of the Earth</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\mathbf {J} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mathbf {J} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a5c09b0f1bb79f063f35f463a2aa3d3d588834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.155ex; height:2.176ex;" alt="{\displaystyle \,\mathbf {J} \,}"></span> is the electric current density.</li> <li>A transport equation, usually of heat (sometimes of light element concentration): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\,\partial T\,}{\partial t}}=\kappa \nabla ^{2}T+\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>κ</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>T</mi> <mo>+</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\,\partial T\,}{\partial t}}=\kappa \nabla ^{2}T+\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393e70d6fe4cd2e5b39431780cc2d30505da1448" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.552ex; height:5.509ex;" alt="{\displaystyle {\frac {\,\partial T\,}{\partial t}}=\kappa \nabla ^{2}T+\varepsilon }"></span> where <span class="texhtml mvar" style="font-style:italic;">T</span> is temperature, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\kappa =k/\rho c_{p}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>κ</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\kappa =k/\rho c_{p}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99da5fc78f94f3b83a4cec00f97b8a45d9e2810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.369ex; height:3.009ex;" alt="{\displaystyle \;\kappa =k/\rho c_{p}\;}"></span> is the thermal diffusivity with <span class="texhtml mvar" style="font-style:italic;">k</span> thermal conductivity, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,c_{p}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,c_{p}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b02573b5aac7f7f093ba3c29f11c2c0b201ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.84ex; height:2.343ex;" alt="{\displaystyle \,c_{p}\,}"></span> heat capacity, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> density, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> is an optional heat source. Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed.</li></ul> <p>These equations are then non-dimensionalized, introducing the non-dimensional parameters, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathsf {a}}={\frac {\,g\alpha TD^{3}\,}{\nu \kappa }}\;,\quad E={\frac {\nu }{\,\Omega D^{2}\,}}\;,\quad P_{\mathsf {r}}={\frac {\,\nu \,}{\kappa }}\;,\quad P_{\mathsf {m}}={\frac {\,\nu \,}{\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>g</mi> <mi>α</mi> <mi>T</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mrow> <mi>ν</mi> <mi>κ</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>ν</mi> <mspace width="thinmathspace" /> </mrow> <mi>κ</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>ν</mi> <mspace width="thinmathspace" /> </mrow> <mi>η</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathsf {a}}={\frac {\,g\alpha TD^{3}\,}{\nu \kappa }}\;,\quad E={\frac {\nu }{\,\Omega D^{2}\,}}\;,\quad P_{\mathsf {r}}={\frac {\,\nu \,}{\kappa }}\;,\quad P_{\mathsf {m}}={\frac {\,\nu \,}{\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb76982191858a1828c6812da2072223b30f1b57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.058ex; height:6.176ex;" alt="{\displaystyle R_{\mathsf {a}}={\frac {\,g\alpha TD^{3}\,}{\nu \kappa }}\;,\quad E={\frac {\nu }{\,\Omega D^{2}\,}}\;,\quad P_{\mathsf {r}}={\frac {\,\nu \,}{\kappa }}\;,\quad P_{\mathsf {m}}={\frac {\,\nu \,}{\eta }}}"></span> where <span class="texhtml mvar" style="font-style:italic;">R</span><sub>a</sub> is the <a href="/wiki/Rayleigh_number" title="Rayleigh number">Rayleigh number</a>, <span class="texhtml mvar" style="font-style:italic;">E</span> the <a href="/wiki/Ekman_number" title="Ekman number">Ekman number</a>, <span class="texhtml mvar" style="font-style:italic;">P</span><sub>r</sub> and <span class="texhtml mvar" style="font-style:italic;">P</span><sub>m</sub> the <a href="/wiki/Prandtl_number" title="Prandtl number">Prandtl</a> and <a href="/wiki/Magnetic_Prandtl_number" title="Magnetic Prandtl number">magnetic Prandtl number</a>. Magnetic field scaling is often in <a href="/wiki/Elsasser_number" title="Elsasser number">Elsasser number</a> units <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(\rho \Omega /\sigma )^{1/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>σ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(\rho \Omega /\sigma )^{1/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302918aae2e229d92ba4f38223953026be5b9304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.389ex; height:3.343ex;" alt="{\displaystyle B=(\rho \Omega /\sigma )^{1/2}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Energy_conversion_between_magnetic_and_kinematic_energy">Energy conversion between magnetic and kinematic energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=8" title="Edit section: Energy conversion between magnetic and kinematic energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The scalar product of the above form of Navier-Stokes equation with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\rho _{0}\mathbf {u} \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\rho _{0}\mathbf {u} \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d9f134820b90549fef4b9a1d7c15304f5e92c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.032ex; height:2.176ex;" alt="{\displaystyle \;\rho _{0}\mathbf {u} \;}"></span> gives the rate of increase of kinetic energy density, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\tfrac {1}{2}}\rho _{0}u^{2}c\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\tfrac {1}{2}}\rho _{0}u^{2}c\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a502483cce6f13d3dcab030fc24100a6d84b295d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.595ex; height:3.509ex;" alt="{\displaystyle \;{\tfrac {1}{2}}\rho _{0}u^{2}c\;}"></span>, on the left-hand side. The last term on the right-hand side is then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4448b391a4ababddc98d6179f1785cbdfa6451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.386ex; height:2.843ex;" alt="{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"></span>, the local contribution to the kinetic energy due to <a href="/wiki/Lorentz_force#Continuous_charge_distribution" title="Lorentz force">Lorentz force</a>. </p><p>The scalar product of the induction equation with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/630fb3d0b9dc777f6c1e28e708a3add5670cebda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.56ex; height:3.843ex;" alt="{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} }"></span> gives the rate of increase of the magnetic energy density, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\tfrac {1}{2}}\mu _{0}B^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\tfrac {1}{2}}\mu _{0}B^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e841085974a05fb9d6e35cc5ff56844ac54c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.223ex; height:3.509ex;" alt="{\displaystyle \;{\tfrac {1}{2}}\mu _{0}B^{2}\;}"></span>, on the left-hand side. The last term on the right-hand side is then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} \cdot \left(\nabla \times \left(\mathbf {u} \times \mathbf {B} \right)\right)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} \cdot \left(\nabla \times \left(\mathbf {u} \times \mathbf {B} \right)\right)\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08674a8fd34aebd1a73c509c1a84ea434293072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:22.54ex; height:3.843ex;" alt="{\textstyle {\tfrac {1}{\mu _{0}}}\mathbf {B} \cdot \left(\nabla \times \left(\mathbf {u} \times \mathbf {B} \right)\right)\;.}"></span> Since the equation is volume-integrated, this term is <a href="/wiki/Integration_by_parts" title="Integration by parts">equivalent up to a boundary term</a> (and with the double use of the <a href="/wiki/Triple_product#Scalar_triple_product" title="Triple product">scalar triple product</a> identity) to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;-\mathbf {u} \cdot \left({\tfrac {1}{\mu _{0}}}\left(\nabla \times \mathbf {B} \right)\times \mathbf {B} \right)=-\mathbf {u} \cdot \left(\mathbf {J} \times \mathbf {B} \right)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thickmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \;-\mathbf {u} \cdot \left({\tfrac {1}{\mu _{0}}}\left(\nabla \times \mathbf {B} \right)\times \mathbf {B} \right)=-\mathbf {u} \cdot \left(\mathbf {J} \times \mathbf {B} \right)~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1554e6eb14444493489e9ba073ea5691ed9680ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.637ex; height:4.843ex;" alt="{\textstyle \;-\mathbf {u} \cdot \left({\tfrac {1}{\mu _{0}}}\left(\nabla \times \mathbf {B} \right)\times \mathbf {B} \right)=-\mathbf {u} \cdot \left(\mathbf {J} \times \mathbf {B} \right)~}"></span> (where one of Maxwell's equations was used). This is the local contribution to the magnetic energy due to fluid motion. </p><p>Thus the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;-\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;-\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a09d335630396ed594f9bd9705af759fe7849e86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.194ex; height:2.843ex;" alt="{\displaystyle \;-\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}"></span> is the rate of transformation of kinetic energy to magnetic energy. This has to be non-negative at least in part of the volume, for the dynamo to produce magnetic field.<sup id="cite_ref-Kono2002_19-1" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>From the diagram above, it is not clear why this term should be positive. A simple argument can be based on consideration of net effects. To create the magnetic field, the net electric current must wrap around the axis of rotation of the planet. In that case, for the term to be positive, the net flow of conducting matter must be towards the axis of rotation. The diagram only shows a net flow from the poles to the equator. However mass conservation requires an additional flow from the equator toward the poles. If that flow was along the axis of rotation, that implies the circulation would be completed by a flow from the ones shown towards the axis of rotation, producing the desired effect. </p> <div class="mw-heading mw-heading3"><h3 id="Order_of_magnitude_of_the_magnetic_field_created_by_Earth's_dynamo"><span id="Order_of_magnitude_of_the_magnetic_field_created_by_Earth.27s_dynamo"></span>Order of magnitude of the magnetic field created by Earth's dynamo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=9" title="Edit section: Order of magnitude of the magnetic field created by Earth's dynamo"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above formula for the rate of conversion of kinetic energy to magnetic energy, is equivalent to a rate of work done by a force of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathbf {J} \times \mathbf {B} \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathbf {J} \times \mathbf {B} \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfc48245b841e4f839706d513b73413a9b42c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.413ex; height:2.176ex;" alt="{\displaystyle \;\mathbf {J} \times \mathbf {B} \;}"></span> on the outer core matter, whose velocity is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span>. This work is the result of non-magnetic forces acting on the fluid. </p><p>Of those, the gravitational force and the <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a> are <a href="/wiki/Conservative_vector_field" title="Conservative vector field">conservative</a> and therefore have no overall contribution to fluid moving in closed loops. Ekman number (defined above), which is the ratio between the two remaining forces, namely the viscosity and Coriolis force, is very low inside Earth's outer core, because its viscosity is low (1.2–1.5 ×10<sup>−2</sup> <a href="/wiki/Pascal-second" class="mw-redirect" title="Pascal-second">pascal-second</a><sup id="cite_ref-Wijs1998_20-0" class="reference"><a href="#cite_note-Wijs1998-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>) due to its liquidity. </p><p>Thus the main time-averaged contribution to the work is from Coriolis force, whose size is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;-2\rho \,\mathbf {\Omega } \times \mathbf {u} \;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo>−</mo> <mn>2</mn> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;-2\rho \,\mathbf {\Omega } \times \mathbf {u} \;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071fcb054615f045ee15412f0266c3f092e7b77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.754ex; height:2.676ex;" alt="{\displaystyle \;-2\rho \,\mathbf {\Omega } \times \mathbf {u} \;,}"></span> though this quantity and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83cd51a8b32d38f1fbea2710eb97404763e75701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.122ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} \times \mathbf {B} }"></span> are related only indirectly and are not in general equal locally (thus they affect each other but not in the same place and time). </p><p>The current density <span class="texhtml mvar" style="font-style:italic;">J</span> is itself the result of the magnetic field according to <a href="/wiki/Ohm%27s_law#Magnetic_effects" title="Ohm's law">Ohm's law</a>. Again, due to matter motion and current flow, this is not necessarily the field at the same place and time. However these relations can still be used to deduce orders of magnitude of the quantities in question. </p><p>In terms of order of magnitude, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;J\,B\sim \rho \,\Omega \,u\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>J</mi> <mspace width="thinmathspace" /> <mi>B</mi> <mo>∼</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> <mi>u</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;J\,B\sim \rho \,\Omega \,u\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3897f8d7b06d1761fe50de4c66b9d1cf06170d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.995ex; height:2.676ex;" alt="{\displaystyle \;J\,B\sim \rho \,\Omega \,u\;}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;J\sim \sigma uB\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>J</mi> <mo>∼</mo> <mi>σ</mi> <mi>u</mi> <mi>B</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;J\sim \sigma uB\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48cafeb9ef6ec3dd36d9de04ad026d68bc9be3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.284ex; height:2.176ex;" alt="{\displaystyle \;J\sim \sigma uB\;}"></span>, giving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sigma \,u\,B^{2}\sim \rho \,\Omega \,u\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>σ</mi> <mspace width="thinmathspace" /> <mi>u</mi> <mspace width="thinmathspace" /> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> <mi>u</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sigma \,u\,B^{2}\sim \rho \,\Omega \,u\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0178af70d0758d827d97f87fa3cc943e3522e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.271ex; height:3.176ex;" alt="{\displaystyle \;\sigma \,u\,B^{2}\sim \rho \,\Omega \,u\;,}"></span> or: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\sim {\sqrt {{\frac {\,\rho \,\Omega \,}{\sigma }}\;}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> </mrow> <mi>σ</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\sim {\sqrt {{\frac {\,\rho \,\Omega \,}{\sigma }}\;}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0eb9694bf60977ce9890e953bb2a04b31f7f8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.709ex; height:6.176ex;" alt="{\displaystyle B\sim {\sqrt {{\frac {\,\rho \,\Omega \,}{\sigma }}\;}}}"></span> </p><p>The exact ratio between both sides is the square root of <a href="/wiki/Elsasser_number" title="Elsasser number">Elsasser number</a>. </p><p>Note that the magnetic field direction cannot be inferred from this approximation (at least not its sign) as it appears squared, and is, indeed, sometimes <a href="/wiki/Earth%27s_magnetic_field#Magnetic_field_reversals" title="Earth's magnetic field">reversed</a>, though in general it lies on a similar axis to that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Omega } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Omega } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bbc5c501b899d658ddaa37ae734fe62c91f6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Omega } }"></span>. </p><p>For earth outer core, <span class="texhtml mvar" style="font-style:italic;">ρ</span> is approximately 10<sup>4</sup> kg/m<sup>3</sup>,<sup id="cite_ref-Wijs1998_20-1" class="reference"><a href="#cite_note-Wijs1998-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>   <span class="texhtml">Ω</span> = 2<span class="texhtml mvar" style="font-style:italic;">π</span>/day = 7.3×10<sup>−5</sup>/second   and   <span class="texhtml mvar" style="font-style:italic;">σ</span>   is approximately   10<sup>7</sup>Ω<sup>−1</sup>m<sup>−1</sup> .<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This gives   2.7×10<sup>−4</sup> <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">Tesla</a>. </p><p>The magnetic field of a <a href="/wiki/Magnetic_dipole" title="Magnetic dipole">magnetic dipole</a> has an inverse cubic dependence in distance, so its order of magnitude at the earth surface can be approximated by multiplying the above result with <span class="nowrap"><span style="font-size:120%">(</span><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num"><span class="texhtml mvar" style="font-style:italic;">R</span><sub>outer core</sub></span>⁄<span class="den"><span class="texhtml mvar" style="font-style:italic;">R</span><sub>Earth</sub> </span></span><span style="font-size:120%">)</span><sup>3</sup> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">2890</span>⁄<span class="den">6370</span></span>)<sup>3</sup> = 0.093 ,</span> giving   2.5×10<sup>−5</sup> Tesla, not far from the measured value of 3×10<sup>−5</sup> Tesla at the <a href="/wiki/Equator" title="Equator">equator</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_models">Numerical models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=10" title="Edit section: Numerical models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geodynamo_Before_Reversal.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Geodynamo_Before_Reversal.gif/220px-Geodynamo_Before_Reversal.gif" decoding="async" width="220" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Geodynamo_Before_Reversal.gif/330px-Geodynamo_Before_Reversal.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Geodynamo_Before_Reversal.gif/440px-Geodynamo_Before_Reversal.gif 2x" data-file-width="717" data-file-height="787" /></a><figcaption>A visual representation of the Glatzmaier model before dipole reversal</figcaption></figure> <p>Broadly, models of the geodynamo attempt to produce magnetic fields consistent with observed data given certain conditions and equations as mentioned in the sections above. Implementing the <a href="/wiki/Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamic</a> equations successfully was of particular significance because they pushed dynamo models to self-consistency. Though geodynamo models are especially prevalent, dynamo models are not necessarily restricted to the geodynamo; solar and general dynamo models are also of interest. Studying dynamo models has utility in the field of geophysics as doing so can identify how various mechanisms form magnetic fields like those produced by astrophysical bodies like Earth and how they cause magnetic fields to exhibit certain features, such as pole reversals. </p><p>The equations used in numerical models of dynamo are highly complex. For decades, theorists were confined to two dimensional <i>kinematic dynamo</i> models described above, in which the fluid motion is chosen in advance and the effect on the magnetic field calculated. The progression from linear to nonlinear, three dimensional models of dynamo was largely hindered by the search for solutions to magnetohydrodynamic equations, which eliminate the need for many of the assumptions made in kinematic models and allow self-consistency. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geodynamo_In_Reversal.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Geodynamo_In_Reversal.gif/220px-Geodynamo_In_Reversal.gif" decoding="async" width="220" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Geodynamo_In_Reversal.gif/330px-Geodynamo_In_Reversal.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Geodynamo_In_Reversal.gif/440px-Geodynamo_In_Reversal.gif 2x" data-file-width="717" data-file-height="787" /></a><figcaption>A visual representation of the Glatzmaier model during dipole reversal</figcaption></figure> <p>The first <i>self-consistent</i> dynamo models, ones that determine both the fluid motions and the magnetic field, were developed by two groups in 1995, one in Japan<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and one in the United States.<sup id="cite_ref-selfconsistent_23-0" class="reference"><a href="#cite_note-selfconsistent-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The latter was made as a model with regards to the geodynamo and received significant attention because it successfully reproduced some of the characteristics of the Earth's field.<sup id="cite_ref-Kono2002_19-2" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Following this breakthrough, there was a large swell in development of reasonable, three dimensional dynamo models.<sup id="cite_ref-Kono2002_19-3" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Though many self-consistent models now exist, there are significant differences among the models, both in the results they produce and the way they were developed.<sup id="cite_ref-Kono2002_19-4" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Given the complexity of developing a geodynamo model, there are many places where discrepancies can occur such as when making assumptions involving the mechanisms that provide energy for the dynamo, when choosing values for parameters used in equations, or when normalizing equations. In spite of the many differences that may occur, most models have shared features like clear axial dipoles. In many of these models, phenomena like <a href="/wiki/Secular_variation" title="Secular variation">secular variation</a> and <a href="/wiki/Geomagnetic_reversal" title="Geomagnetic reversal">geomagnetic polarity reversals</a> have also been successfully recreated.<sup id="cite_ref-Kono2002_19-5" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Observations">Observations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=11" title="Edit section: Observations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geodynamo_After_Reversal.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Geodynamo_After_Reversal.gif/220px-Geodynamo_After_Reversal.gif" decoding="async" width="220" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Geodynamo_After_Reversal.gif/330px-Geodynamo_After_Reversal.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Geodynamo_After_Reversal.gif/440px-Geodynamo_After_Reversal.gif 2x" data-file-width="717" data-file-height="787" /></a><figcaption>A visual representation of the Glatzmaier model after dipole reversal</figcaption></figure> <p>Many observations can be made from dynamo models. Models can be used to estimate how magnetic fields vary with time and can be compared to observed <a href="/wiki/Paleomagnetism" title="Paleomagnetism">paleomagnetic</a> data to find similarities between the model and the Earth. Due to the uncertainty of paleomagnetic observations, however, comparisons may not be entirely valid or useful.<sup id="cite_ref-Kono2002_19-6" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Simplified geodynamo models have shown relationships between the dynamo number (determined by <a href="/wiki/Differential_rotation" title="Differential rotation">variance in rotational rates</a> in the outer core and mirror-asymmetric convection (e.g. when convection favors one direction in the north and the other in the south)) and magnetic pole reversals as well as found similarities between the geodynamo and the Sun's dynamo.<sup id="cite_ref-Kono2002_19-7" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In many models, it appears that magnetic fields have somewhat random magnitudes that follow a normal trend that average to zero.<sup id="cite_ref-Kono2002_19-8" class="reference"><a href="#cite_note-Kono2002-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In addition to these observations, general observations about the mechanisms powering the geodynamo can be made based on how accurately the model reflects actual data collected from Earth. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_modelling">Modern modelling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=12" title="Edit section: Modern modelling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The complexity of dynamo modelling is so great that models of the geodynamo are limited by the current power of <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a>, particularly because calculating the <a href="/wiki/Ekman_number" title="Ekman number">Ekman</a> and <a href="/wiki/Rayleigh_number" title="Rayleigh number">Rayleigh</a> number of the outer core is extremely difficult and requires a vast number of computations. </p><p>Many improvements have been proposed in dynamo modelling since the self-consistent breakthrough in 1995. One suggestion in studying the complex magnetic field changes is applying <a href="/wiki/Spectral_method" title="Spectral method">spectral methods</a> to simplify computations.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Ultimately, until considerable improvements in computer power are made, the methods for computing realistic dynamo models will have to be made more efficient, so making improvements in methods for computing the model is of high importance for the advancement of numerical dynamo modelling. </p> <div class="mw-heading mw-heading2"><h2 id="Notable_people">Notable people</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=13" title="Edit section: Notable people"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stanislav_I._Braginsky" class="mw-redirect" title="Stanislav I. Braginsky">Stanislav I. Braginsky</a>, research geophysicist</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Antidynamo_theorem" title="Antidynamo theorem">Antidynamo theorem</a></li> <li><a href="/wiki/Rotating_magnetic_field" title="Rotating magnetic field">Rotating magnetic field</a></li> <li><a href="/wiki/Secular_variation" title="Secular variation">Secular variation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamo_theory&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150118213104/http://www.usgs.gov/faq/?q=categories%2F9782%2F2738">"How does the Earth's core generate a magnetic field?"</a>. <i>USGS FAQs</i>. United States Geological Survey. Archived from <a rel="nofollow" class="external text" href="https://www.usgs.gov/faq/?q=categories/9782/2738">the original</a> on 18 January 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">21 October</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=USGS+FAQs&rft.atitle=How+does+the+Earth%27s+core+generate+a+magnetic+field%3F&rft_id=http%3A%2F%2Fwww.usgs.gov%2Ffaq%2F%3Fq%3Dcategories%2F9782%2F2738&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmpère1822" class="citation book cs1 cs1-prop-foreign-lang-source">Ampère, André-Marie (1822). <a rel="nofollow" class="external text" href="http://catalogue.bnf.fr/ark:/12148/cb37284089w"><i>Recueil d'observations électro-dynamiques : contenant divers mémoires, notices, extraits de lettres ou d'ouvrages périodiques sur les sciences relatifs à l'action mutuelle de deux courants électriques, à celle qui existe entre un courant électrique et un aimant ou le globe terrestre, et à celle de deux aimants l'un sur l'autre</i></a> (in French). Paris: Crochard.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recueil+d%27observations+%C3%A9lectro-dynamiques+%3A+contenant+divers+m%C3%A9moires%2C+notices%2C+extraits+de+lettres+ou+d%27ouvrages+p%C3%A9riodiques+sur+les+sciences+relatifs+%C3%A0+l%27action+mutuelle+de+deux+courants+%C3%A9lectriques%2C+%C3%A0+celle+qui+existe+entre+un+courant+%C3%A9lectrique+et+un+aimant+ou+le+globe+terrestre%2C+et+%C3%A0+celle+de+deux+aimants+l%27un+sur+l%27autre&rft.place=Paris&rft.pub=Crochard&rft.date=1822&rft.aulast=Amp%C3%A8re&rft.aufirst=Andr%C3%A9-Marie&rft_id=http%3A%2F%2Fcatalogue.bnf.fr%2Fark%3A%2F12148%2Fcb37284089w&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span>[ISBN unspecified]</span> </li> <li id="cite_note-Larmor1919-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Larmor1919_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarmor1919" class="citation journal cs1">Larmor, J. (1919). "How could a rotating body such as the Sun become a magnet?". <i>Reports of the British Association</i>. <b>87</b>: 159–160.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reports+of+the+British+Association&rft.atitle=How+could+a+rotating+body+such+as+the+Sun+become+a+magnet%3F&rft.volume=87&rft.pages=159-160&rft.date=1919&rft.aulast=Larmor&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarmor1919" class="citation journal cs1">Larmor, J. (1919). "Possible rotational origin of magnetic fields of sun and earth". <i>Electrical Review</i>. <b>85</b>: 412ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electrical+Review&rft.atitle=Possible+rotational+origin+of+magnetic+fields+of+sun+and+earth&rft.volume=85&rft.pages=412ff&rft.date=1919&rft.aulast=Larmor&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span> Reprinted in <i>Engineering</i>, vol. 108, pages 461ff (3 October 1919).</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNye1999" class="citation journal cs1">Nye, Mary Jo (1 March 1999). "Temptations of theory, strategies of evidence: P. M. S. Blackett and the earth's magnetism, 1947–52". <i>The British Journal for the History of Science</i>. <b>32</b> (1): 69–92. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0007087498003495">10.1017/S0007087498003495</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143344977">143344977</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+British+Journal+for+the+History+of+Science&rft.atitle=Temptations+of+theory%2C+strategies+of+evidence%3A+P.+M.+S.+Blackett+and+the+earth%27s+magnetism%2C+1947%E2%80%9352&rft.volume=32&rft.issue=1&rft.pages=69-92&rft.date=1999-03-01&rft_id=info%3Adoi%2F10.1017%2FS0007087498003495&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143344977%23id-name%3DS2CID&rft.aulast=Nye&rft.aufirst=Mary+Jo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFMerrillMcElhinnyMcFadden1996">Merrill, McElhinny & McFadden 1996</a>, page 17 claim that in 1905, shortly after composing his <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> paper, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> described the origin of the <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a> as being one of the great unsolved problems facing modern <a href="/wiki/Physicist" title="Physicist">physicists</a>. However, they do not provide details on where he made this statement.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanders2003" class="citation news cs1">Sanders, Robert (2003-12-10). <a rel="nofollow" class="external text" href="http://www.berkeley.edu/news/media/releases/2003/12/10_heat.shtml">"Radioactive potassium may be major heat source in Earth's core"</a>. UC Berkeley News<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Radioactive+potassium+may+be+major+heat+source+in+Earth%27s+core&rft.date=2003-12-10&rft.aulast=Sanders&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.berkeley.edu%2Fnews%2Fmedia%2Freleases%2F2003%2F12%2F10_heat.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakurabaPaul_H._Roberts2009" class="citation journal cs1">Sakuraba, Ataru; Paul H. Roberts (4 October 2009). "Generation of a strong magnetic field using uniform heat flux at the surface of the core". <i>Nature Geoscience</i>. <b>2</b> (11): 802–805. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NatGe...2..802S">2009NatGe...2..802S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fngeo643">10.1038/ngeo643</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Geoscience&rft.atitle=Generation+of+a+strong+magnetic+field+using+uniform+heat+flux+at+the+surface+of+the+core&rft.volume=2&rft.issue=11&rft.pages=802-805&rft.date=2009-10-04&rft_id=info%3Adoi%2F10.1038%2Fngeo643&rft_id=info%3Abibcode%2F2009NatGe...2..802S&rft.aulast=Sakuraba&rft.aufirst=Ataru&rft.au=Paul+H.+Roberts&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuffett2009" class="citation journal cs1">Buffett, Bruce (2009). "Geodynamo: A matter of boundaries". <i>Nature Geoscience</i>. <b>2</b> (11): 741–742. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NatGe...2..741B">2009NatGe...2..741B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fngeo673">10.1038/ngeo673</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Geoscience&rft.atitle=Geodynamo%3A+A+matter+of+boundaries&rft.volume=2&rft.issue=11&rft.pages=741-742&rft.date=2009&rft_id=info%3Adoi%2F10.1038%2Fngeo673&rft_id=info%3Abibcode%2F2009NatGe...2..741B&rft.aulast=Buffett&rft.aufirst=Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-hydromagnetic_dynamo-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-hydromagnetic_dynamo_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrandenburg2007" class="citation journal cs1">Brandenburg, Axel (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.2309">"Hydromagnetic dynamo theory"</a>. <i>Scholarpedia</i>. <b>2</b> (3): 2309. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007SchpJ...2.2309B">2007SchpJ...2.2309B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.2309">10.4249/scholarpedia.2309</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Hydromagnetic+dynamo+theory&rft.volume=2&rft.issue=3&rft.pages=2309&rft.date=2007&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.2309&rft_id=info%3Abibcode%2F2007SchpJ...2.2309B&rft.aulast=Brandenburg&rft.aufirst=Axel&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.2309&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE._Pallé2010" class="citation book cs1">E. Pallé (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qLuVCJtRTV0C&pg=PA316"><i>The Earth as a Distant Planet: A Rosetta Stone for the Search of Earth-Like Worlds (Astronomy and Astrophysics Library)</i></a>. Berlin: Springer. pp. 316–317. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-1683-9" title="Special:BookSources/978-1-4419-1683-9"><bdi>978-1-4419-1683-9</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 July</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Earth+as+a+Distant+Planet%3A+A+Rosetta+Stone+for+the+Search+of+Earth-Like+Worlds+%28Astronomy+and+Astrophysics+Library%29&rft.place=Berlin&rft.pages=316-317&rft.pub=Springer&rft.date=2010&rft.isbn=978-1-4419-1683-9&rft.au=E.+Pall%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqLuVCJtRTV0C%26pg%3DPA316&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-Enceladus-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Enceladus_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteigerwald,_Bill2010" class="citation web cs1">Steigerwald, Bill (October 6, 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150324222212/http://www.nasa.gov/mission_pages/cassini/whycassini/cassini20100708-b.html">"Saturn's Icy Moon May Keep Oceans Liquid with Wobble"</a>. NASA. Archived from <a rel="nofollow" class="external text" href="https://www.nasa.gov/mission_pages/cassini/whycassini/cassini20100708-b.html">the original</a> on March 24, 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">August 14,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Saturn%27s+Icy+Moon+May+Keep+Oceans+Liquid+with+Wobble&rft.pub=NASA&rft.date=2010-10-06&rft.au=Steigerwald%2C+Bill&rft_id=http%3A%2F%2Fwww.nasa.gov%2Fmission_pages%2Fcassini%2Fwhycassini%2Fcassini20100708-b.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-Io_geologic-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Io_geologic_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCassis,_Nikki2012" class="citation web cs1">Cassis, Nikki (March 19, 2012). <a rel="nofollow" class="external text" href="https://clas.asu.edu/node/12161">"Geologic map of Jupiter's moon Io details an otherworldly volcanic surface"</a>. Astrogeology Science Center<span class="reference-accessdate">. Retrieved <span class="nowrap">August 14,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Geologic+map+of+Jupiter%27s+moon+Io+details+an+otherworldly+volcanic+surface&rft.pub=Astrogeology+Science+Center&rft.date=2012-03-19&rft.au=Cassis%2C+Nikki&rft_id=https%3A%2F%2Fclas.asu.edu%2Fnode%2F12161&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged September 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-mercury_core-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-mercury_core_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150118213837/http://carnegiescience.edu/news/mercury%E2%80%99s_surprising_core_and_landscape_curiosities">"Mercury's Surprising Core and Landscape Curiosities"</a>. <i>MESSENGER</i>. Carnegie Institution for Science. March 21, 2012. Archived from <a rel="nofollow" class="external text" href="http://carnegiescience.edu/news/mercury%E2%80%99s_surprising_core_and_landscape_curiosities">the original</a> on January 18, 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">August 14,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MESSENGER&rft.atitle=Mercury%27s+Surprising+Core+and+Landscape+Curiosities&rft.date=2012-03-21&rft_id=http%3A%2F%2Fcarnegiescience.edu%2Fnews%2Fmercury%25E2%2580%2599s_surprising_core_and_landscape_curiosities&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-lunar_dynamo-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-lunar_dynamo_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStevens,_Tim2011" class="citation web cs1">Stevens, Tim (November 9, 2011). <a rel="nofollow" class="external text" href="http://news.ucsc.edu/2011/11/lunar-dynamo.html">"Ancient lunar dynamo may explain magnetized moon rocks"</a>. University of California<span class="reference-accessdate">. Retrieved <span class="nowrap">August 14,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ancient+lunar+dynamo+may+explain+magnetized+moon+rocks&rft.pub=University+of+California&rft.date=2011-11-09&rft.au=Stevens%2C+Tim&rft_id=http%3A%2F%2Fnews.ucsc.edu%2F2011%2F11%2Flunar-dynamo.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOvchinnikov,_I.V.Ensslin,_T.A.2016" class="citation journal cs1">Ovchinnikov, I.V.; Ensslin, T.A. (April 2016). "Kinematic dynamo, supersymmetry breaking, and chaos". <i>Physical Review D</i>. <b>93</b> (8): 085023. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1512.01651">1512.01651</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhRvD..93h5023O">2016PhRvD..93h5023O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.93.085023">10.1103/PhysRevD.93.085023</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:59367815">59367815</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Kinematic+dynamo%2C+supersymmetry+breaking%2C+and+chaos&rft.volume=93&rft.issue=8&rft.pages=085023&rft.date=2016-04&rft_id=info%3Aarxiv%2F1512.01651&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59367815%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.93.085023&rft_id=info%3Abibcode%2F2016PhRvD..93h5023O&rft.au=Ovchinnikov%2C+I.V.&rft.au=Ensslin%2C+T.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOvchinnikov,_I.V.2016" class="citation journal cs1">Ovchinnikov, I.V. (March 2016). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe18040108">"Introduction to Supersymmetric Theory of Stochastics"</a>. <i>Entropy</i>. <b>18</b> (4): 108. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1511.03393">1511.03393</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016Entrp..18..108O">2016Entrp..18..108O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe18040108">10.3390/e18040108</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2388285">2388285</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Entropy&rft.atitle=Introduction+to+Supersymmetric+Theory+of+Stochastics&rft.volume=18&rft.issue=4&rft.pages=108&rft.date=2016-03&rft_id=info%3Aarxiv%2F1511.03393&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2388285%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3390%2Fe18040108&rft_id=info%3Abibcode%2F2016Entrp..18..108O&rft.au=Ovchinnikov%2C+I.V.&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fe18040108&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParker1955" class="citation journal cs1">Parker, Eugene N. (September 1955). "Hydromagnetic Dynamo Models". <i>The Astrophysical Journal</i>. <b>122</b>: 293–314. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1955ApJ...122..293P">1955ApJ...122..293P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F146087">10.1086/146087</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Hydromagnetic+Dynamo+Models&rft.volume=122&rft.pages=293-314&rft.date=1955-09&rft_id=info%3Adoi%2F10.1086%2F146087&rft_id=info%3Abibcode%2F1955ApJ...122..293P&rft.aulast=Parker&rft.aufirst=Eugene+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-Kono2002-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kono2002_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kono2002_19-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Kono2002_19-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Kono2002_19-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Kono2002_19-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Kono2002_19-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Kono2002_19-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Kono2002_19-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Kono2002_19-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKonoRoberts2002" class="citation journal cs1">Kono, Masaru; Roberts, Paul H. (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2000RG000102">"Recent geodynamo simulations and observations of the geomagnetic field"</a>. <i><a href="/wiki/Reviews_of_Geophysics" title="Reviews of Geophysics">Reviews of Geophysics</a></i>. <b>40</b> (4): 1–53. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002RvGeo..40.1013K">2002RvGeo..40.1013K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2000RG000102">10.1029/2000RG000102</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Geophysics&rft.atitle=Recent+geodynamo+simulations+and+observations+of+the+geomagnetic+field&rft.volume=40&rft.issue=4&rft.pages=1-53&rft.date=2002&rft_id=info%3Adoi%2F10.1029%2F2000RG000102&rft_id=info%3Abibcode%2F2002RvGeo..40.1013K&rft.aulast=Kono&rft.aufirst=Masaru&rft.au=Roberts%2C+Paul+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1029%252F2000RG000102&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-Wijs1998-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wijs1998_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wijs1998_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">de Wijs, G. A., Kresse, G., Vočadlo, L., Dobson, D., Alfe, D., Gillan, M. J., & Price, G. D. (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180420202850/https://pdfs.semanticscholar.org/8072/57a2b0ecc863b2f7ae143b9f54ae4b6d10cd.pdf">The viscosity of liquid iron at the physical conditions of the Earth's core.</a> Nature, 392(6678), 805.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Ohta, K., Kuwayama, Y., Hirose, K., Shimizu, K., & Ohishi, Y. (2016). Experimental determination of the electrical resistivity of iron at Earth’s core conditions. Nature, 534(7605), 95. Link to a <a rel="nofollow" class="external text" href="http://www.spring8.or.jp/pdf/en/res_fro/16/094_095.pdf">summary</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKageyamaSato1995" class="citation journal cs1">Kageyama, Akira; Sato, Tetsuya (1 January 1995). "Computer simulation of a magnetohydrodynamic dynamo. II". <i>Physics of Plasmas</i>. <b>2</b> (5): 1421–1431. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PhPl....2.1421K">1995PhPl....2.1421K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.871485">10.1063/1.871485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Plasmas&rft.atitle=Computer+simulation+of+a+magnetohydrodynamic+dynamo.+II&rft.volume=2&rft.issue=5&rft.pages=1421-1431&rft.date=1995-01-01&rft_id=info%3Adoi%2F10.1063%2F1.871485&rft_id=info%3Abibcode%2F1995PhPl....2.1421K&rft.aulast=Kageyama&rft.aufirst=Akira&rft.au=Sato%2C+Tetsuya&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-selfconsistent-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-selfconsistent_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlatzmaierRoberts1995" class="citation journal cs1">Glatzmaier, Gary A.; Roberts, Paul H. (1995). "A three-dimensional self-consistent computer simulation of a geomagnetic field reversal". <i>Nature</i>. <b>377</b> (6546): 203–209. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995Natur.377..203G">1995Natur.377..203G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F377203a0">10.1038/377203a0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4265765">4265765</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+three-dimensional+self-consistent+computer+simulation+of+a+geomagnetic+field+reversal&rft.volume=377&rft.issue=6546&rft.pages=203-209&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4265765%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F377203a0&rft_id=info%3Abibcode%2F1995Natur.377..203G&rft.aulast=Glatzmaier&rft.aufirst=Gary+A.&rft.au=Roberts%2C+Paul+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlatzmaierRoberts1995" class="citation journal cs1">Glatzmaier, G; Roberts, Paul H. (1995). "A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle". <i>Physics of the Earth and Planetary Interiors</i>. <b>91</b> (1–3): 63–75. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PEPI...91...63G">1995PEPI...91...63G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0031-9201%2895%2903049-3">10.1016/0031-9201(95)03049-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+the+Earth+and+Planetary+Interiors&rft.atitle=A+three-dimensional+convective+dynamo+solution+with+rotating+and+finitely+conducting+inner+core+and+mantle&rft.volume=91&rft.issue=1%E2%80%933&rft.pages=63-75&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0031-9201%2895%2903049-3&rft_id=info%3Abibcode%2F1995PEPI...91...63G&rft.aulast=Glatzmaier&rft.aufirst=G&rft.au=Roberts%2C+Paul+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAveryConstableDaviesGubbins2019" class="citation journal cs1">Avery, Margaret S.; <a href="/wiki/Catherine_Constable" title="Catherine Constable">Constable, Catherine G.</a>; Davies, Christopher J.; Gubbins, David (2019-01-01). <a rel="nofollow" class="external text" href="http://eprints.whiterose.ac.uk/140318/1/energy_balance%28freq%29_revised_v2.pdf">"Spectral methods for analyzing energy balances in geodynamo simulations"</a> <span class="cs1-format">(PDF)</span>. <i>Physics of the Earth and Planetary Interiors</i>. <b>286</b>: 127–137. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019PEPI..286..127A">2019PEPI..286..127A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.pepi.2018.10.002">10.1016/j.pepi.2018.10.002</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9201">0031-9201</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+the+Earth+and+Planetary+Interiors&rft.atitle=Spectral+methods+for+analyzing+energy+balances+in+geodynamo+simulations&rft.volume=286&rft.pages=127-137&rft.date=2019-01-01&rft.issn=0031-9201&rft_id=info%3Adoi%2F10.1016%2Fj.pepi.2018.10.002&rft_id=info%3Abibcode%2F2019PEPI..286..127A&rft.aulast=Avery&rft.aufirst=Margaret+S.&rft.au=Constable%2C+Catherine+G.&rft.au=Davies%2C+Christopher+J.&rft.au=Gubbins%2C+David&rft_id=http%3A%2F%2Feprints.whiterose.ac.uk%2F140318%2F1%2Fenergy_balance%2528freq%2529_revised_v2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemorest2001" class="citation web cs1">Demorest, Paul (21 May 2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070221094040/http://setiathome.berkeley.edu/~pauld/etc/210BPaper.pdf">"Dynamo Theory and Earth's magnetic Field (term paper)"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://setiathome.berkeley.edu/~pauld/etc/210BPaper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 21 February 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">14 October</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dynamo+Theory+and+Earth%27s+magnetic+Field+%28term+paper%29&rft.date=2001-05-21&rft.aulast=Demorest&rft.aufirst=Paul&rft_id=http%3A%2F%2Fsetiathome.berkeley.edu%2F~pauld%2Fetc%2F210BPaper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFitzpatrick2002" class="citation web cs1">Fitzpatrick, Richard (18 May 2002). <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/plasma/lectures/node70.html">"MHD Dynamo Theory"</a>. <i>Plasma Physics</i>. <a href="/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a><span class="reference-accessdate">. Retrieved <span class="nowrap">14 October</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Plasma+Physics&rft.atitle=MHD+Dynamo+Theory&rft.date=2002-05-18&rft.aulast=Fitzpatrick&rft.aufirst=Richard&rft_id=http%3A%2F%2Ffarside.ph.utexas.edu%2Fteaching%2Fplasma%2Flectures%2Fnode70.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerrillMcElhinnyMcFadden1996" class="citation book cs1">Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). <i>The magnetic field of the earth: Paleomagnetism, the core, and the deep mantle</i>. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-491246-5" title="Special:BookSources/978-0-12-491246-5"><bdi>978-0-12-491246-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+magnetic+field+of+the+earth%3A+Paleomagnetism%2C+the+core%2C+and+the+deep+mantle&rft.pub=Academic+Press&rft.date=1996&rft.isbn=978-0-12-491246-5&rft.aulast=Merrill&rft.aufirst=Ronald+T.&rft.au=McElhinny%2C+Michael+W.&rft.au=McFadden%2C+Phillip+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStern" class="citation web cs1">Stern, David P. <a rel="nofollow" class="external text" href="http://www.phy6.org/earthmag/dynamos.htm">"Chapter 12: The dynamo process"</a>. <i>The Great Magnet, the Earth</i><span class="reference-accessdate">. Retrieved <span class="nowrap">14 October</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Great+Magnet%2C+the+Earth&rft.atitle=Chapter+12%3A+The+dynamo+process&rft.aulast=Stern&rft.aufirst=David+P.&rft_id=http%3A%2F%2Fwww.phy6.org%2Fearthmag%2Fdynamos.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStern" class="citation web cs1">Stern, David P. <a rel="nofollow" class="external text" href="http://www.phy6.org/earthmag/dynamos2.htm">"Chapter 13: Dynamo in the Earth's Core"</a>. <i>The Great Magnet, the Earth</i><span class="reference-accessdate">. Retrieved <span class="nowrap">14 October</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Great+Magnet%2C+the+Earth&rft.atitle=Chapter+13%3A+Dynamo+in+the+Earth%27s+Core&rft.aulast=Stern&rft.aufirst=David+P.&rft_id=http%3A%2F%2Fwww.phy6.org%2Fearthmag%2Fdynamos2.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamo+theory" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol 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