Wirbelstärke – Wikipedia

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vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Inhaltsverzeichnis</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">In die Seitenleiste verschieben</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">Verbergen</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Anfang)</div> </a> </li> <li id="toc-Formale_Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formale_Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Formale Notation</span> </div> </a> <ul id="toc-Formale_Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hydrodynamik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hydrodynamik"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Hydrodynamik</span> </div> </a> <ul id="toc-Hydrodynamik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Meteorologie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Meteorologie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Meteorologie</span> </div> </a> <button aria-controls="toc-Meteorologie-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>Unterabschnitt Meteorologie umschalten</span> </button> <ul id="toc-Meteorologie-sublist" class="vector-toc-list"> <li id="toc-Potentielle_Vortizität" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Potentielle_Vortizität"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Potentielle Vortizität</span> </div> </a> <ul id="toc-Potentielle_Vortizität-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Anmerkungen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Anmerkungen"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Anmerkungen</span> </div> </a> <ul id="toc-Anmerkungen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Literatur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Literatur"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Literatur</span> </div> </a> <ul id="toc-Literatur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weblinks" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Weblinks"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Weblinks</span> </div> </a> <ul id="toc-Weblinks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einzelnachweise" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Einzelnachweise"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Einzelnachweise</span> </div> </a> <ul id="toc-Einzelnachweise-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="Inhaltsverzeichnis" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Inhaltsverzeichnis" > <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="Inhaltsverzeichnis umschalten" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Inhaltsverzeichnis umschalten</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Wirbelstärke</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Zu einem Artikel in einer anderen Sprache gehen. Verfügbar in 24 Sprachen" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-24" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">24 Sprachen</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D9%85%D9%8A%D8%A9" title="دوامية – Arabisch" lang="ar" hreflang="ar" data-title="دوامية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%B2%D1%96%D1%85%D1%80%D0%B0%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Завіхранасць – Belarussisch" lang="be" hreflang="be" data-title="Завіхранасць" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Vorticitat" title="Vorticitat – Katalanisch" lang="ca" hreflang="ca" data-title="Vorticitat" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vorticita" title="Vorticita – Tschechisch" lang="cs" hreflang="cs" data-title="Vorticita" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Vorticity" title="Vorticity – Englisch" lang="en" hreflang="en" data-title="Vorticity" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vorticidad" title="Vorticidad – Spanisch" lang="es" hreflang="es" data-title="Vorticidad" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bortizitate" title="Bortizitate – Baskisch" lang="eu" hreflang="eu" data-title="Bortizitate" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%B1%D8%AA%DB%8C%D8%B3%DB%8C%D8%AA%D9%87" title="ورتیسیته – Persisch" lang="fa" hreflang="fa" data-title="ورتیسیته" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A8%D7%91%D7%95%D7%9C%D7%99%D7%95%D7%AA" title="ערבוליות – Hebräisch" lang="he" hreflang="he" data-title="ערבוליות" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AD%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BF%E0%A4%B2%E0%A4%A4%E0%A4%BE" title="भ्रमिलता – Hindi" lang="hi" hreflang="hi" data-title="भ्रमिलता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Vorticit%C3%A0" title="Vorticità – Italienisch" lang="it" hreflang="it" data-title="Vorticità" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B8%A6%E5%BA%A6" title="渦度 – Japanisch" lang="ja" hreflang="ja" data-title="渦度" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EC%9A%A9%EB%8F%8C%EC%9D%B4%EB%8F%84" title="소용돌이도 – Koreanisch" lang="ko" hreflang="ko" data-title="소용돌이도" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vorticiteit" title="Vorticiteit – Niederländisch" lang="nl" hreflang="nl" data-title="Vorticiteit" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvervling" title="Kvervling – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Kvervling" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wirowo%C5%9B%C4%87" title="Wirowość – Polnisch" lang="pl" hreflang="pl" data-title="Wirowość" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Vorticidade" title="Vorticidade – Portugiesisch" lang="pt" hreflang="pt" data-title="Vorticidade" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vorticitate" title="Vorticitate – Rumänisch" lang="ro" hreflang="ro" data-title="Vorticitate" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%B2%D0%B8%D1%85%D1%80%D0%B5%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Завихренность – Russisch" lang="ru" hreflang="ru" data-title="Завихренность" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vorticity" title="Vorticity – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Vorticity" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vorticitet" title="Vorticitet – Schwedisch" lang="sv" hreflang="sv" data-title="Vorticitet" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%81%E0%AE%B4%E0%AE%BF%E0%AE%AE%E0%AF%88" title="சுழிமை – Tamil" lang="ta" hreflang="ta" data-title="சுழிமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%B2%D0%B8%D1%85%D1%80%D0%B5%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Завихреність – Ukrainisch" lang="uk" hreflang="uk" data-title="Завихреність" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B6%A1%E9%87%8F" title="涡量 – Chinesisch" lang="zh" hreflang="zh" data-title="涡量" data-language-autonym="中文" data-language-local-name="Chinesisch" 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 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Weitergeleitet von <a href="/w/index.php?title=Vorticity&redirect=no" class="mw-redirect" title="Vorticity">Vorticity</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Wirbelstärke</b> oder <b>Wirbeldichte</b><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> <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 {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}" /></span> bzw. <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 {\vec {\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dacdf0eaa3c2bde6ba1723200bb63252acaaad38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.27ex; height:2.843ex;" alt="{\displaystyle {\vec {\eta }}}" /></span> bzw. <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 {\vec {\zeta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\zeta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc55f29d1759ae452f91abad0b49b59bf5bf1cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.338ex; height:3.343ex;" alt="{\displaystyle {\vec {\zeta }}}" /></span> beziffert eine zentrale Größe der <a href="/wiki/Str%C3%B6mungsmechanik" title="Strömungsmechanik">Strömungsmechanik</a> und der <a href="/wiki/Meteorologie" title="Meteorologie">Meteorologie</a>, indem sie dem <a href="/wiki/Strudel_(Physik)" title="Strudel (Physik)">Strudel</a> und den kreis- oder spiralförmigen Strömungen ein <a href="/wiki/Geschwindigkeitsfeld" title="Geschwindigkeitsfeld">Feld von Geschwindigkeiten</a> zuordnet. Die gleichwertige Bezeichnung <b>Vortizität</b> von lateinisch vortex = „Wirbel, Strudel“, <span style="font-style:normal;font-weight:normal"><a href="/wiki/Englische_Sprache" title="Englische Sprache">englisch</a></span> <span lang="en-Latn" style="font-style:italic">Vorticity</span>, wird mit <i>Wirbelhaftigkeit</i> übersetzt. </p><p>In der Strömungsmechanik werden kleine Unterschiede in Geschwindigkeit und Richtung von <a href="/wiki/Gas" title="Gas">Gasen</a> und <a href="/wiki/Fl%C3%BCssigkeit" title="Flüssigkeit">Flüssigkeiten</a> als <a href="/wiki/Scherung_(Mechanik)#Scherung_von_Fluiden" title="Scherung (Mechanik)">Scherung</a> bezeichnet. Die <a href="/wiki/Stromlinien" class="mw-redirect" title="Stromlinien">Stromlinien</a> sind anderseits geometrische Hilfsmittel zur anschaulichen Beschreibung einer Strömung als gerichtete Bewegung von Teilchen. Schließlich ist die <a href="/wiki/Viskosit%C3%A4t" title="Viskosität">Viskosität</a> die Zähflüssigkeit oder Zähigkeit von <a href="/wiki/Fluid" title="Fluid">Fluiden</a>, also der Widerstand des Fluids gegenüber Scherung. </p><p>Anschaulich entspricht die Wirbelstärke der Tendenz eines Fluidelements zur Eigen<a href="/wiki/Rotation_(Physik)" title="Rotation (Physik)">drehung</a> um eine Achse, aus der eine <a href="/wiki/Zirkulation_(Feldtheorie)" title="Zirkulation (Feldtheorie)">Zirkulation</a> von fließenden oder strömenden Medien in einem geschlossenen Gebiet entsteht. Weiter wird das <a href="/wiki/Quadratisches_Mittel" title="Quadratisches Mittel">Mittel der quadratischen</a> Wirbelstärke über einer bestimmten Fläche als <a href="/wiki/Enstrophie" title="Enstrophie">Enstrophie</a> bezeichnet, welche z. B. das Strömungsverhalten von Glas-<a href="/wiki/Doppelfassade" title="Doppelfassade">Doppelfassaden</a> beschreibt. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formale_Notation">Formale Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=1" title="Abschnitt bearbeiten: Formale Notation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Formale Notation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Wirbelstärke <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 {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}" /></span>, in der Meteorologie angelehnt an die Zirkulation mit <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 {\vec {\zeta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\zeta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc55f29d1759ae452f91abad0b49b59bf5bf1cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.338ex; height:3.343ex;" alt="{\displaystyle {\vec {\zeta }}}" /></span> bezeichnet, ist definiert als die <a href="/wiki/Rotation_(Mathematik)" class="mw-redirect" title="Rotation (Mathematik)">Rotation</a> der Geschwindigkeit <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" /></span> eines <a href="/wiki/Vektorfeld" title="Vektorfeld">Vektorfelds</a>: </p> <dl><dd><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 {\vec {\omega }}:=\operatorname {rot} {\vec {v}}={\vec {\nabla }}\times {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}:=\operatorname {rot} {\vec {v}}={\vec {\nabla }}\times {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641a604a4371b605dbd159864b2941184e45bbef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.782ex; height:2.843ex;" alt="{\displaystyle {\vec {\omega }}:=\operatorname {rot} {\vec {v}}={\vec {\nabla }}\times {\vec {v}}}" /></span></dd></dl> <p>Sie hat die <a href="/wiki/SI-Einheit" class="mw-redirect" title="SI-Einheit">SI-Einheit</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 {\tfrac {1}{\mathrm {s} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\mathrm {s} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f824b8b6a7cb425dfb823d82dea835f003498d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.658ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{\mathrm {s} }}}" /></span> und ist wie jede Rotation eines Vektorfelds ein <a href="/wiki/Pseudovektor" title="Pseudovektor">Pseudovektor</a>feld. </p><p>Weil sich in einem <a href="/wiki/Abgeschlossenes_System" title="Abgeschlossenes System">abgeschlossenen System</a> die <a href="/wiki/Erhaltungsgr%C3%B6%C3%9Fe" class="mw-redirect" title="Erhaltungsgröße">Erhaltungsgrößen</a> nicht ändern, ist die Wirbelstärke gleich der flächenbezogenen <a href="/wiki/Zirkulation_(Feldtheorie)" title="Zirkulation (Feldtheorie)">Zirkulations</a>rate <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 \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span>:<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> </p> <dl><dd><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 \Gamma :=\oint _{\partial A}{\vec {v}}\cdot \mathrm {d} {\vec {r}}=\int _{A}\;\operatorname {rot} ({\vec {v}})\cdot \mathrm {d} {\vec {A}}=\int _{A}{\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>:=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mspace width="thickmathspace"></mspace> <mi>rot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma :=\oint _{\partial A}{\vec {v}}\cdot \mathrm {d} {\vec {r}}=\int _{A}\;\operatorname {rot} ({\vec {v}})\cdot \mathrm {d} {\vec {A}}=\int _{A}{\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad7efa02ef82cdd930bc4691dd7935390044189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.614ex; height:5.676ex;" alt="{\displaystyle \Gamma :=\oint _{\partial A}{\vec {v}}\cdot \mathrm {d} {\vec {r}}=\int _{A}\;\operatorname {rot} ({\vec {v}})\cdot \mathrm {d} {\vec {A}}=\int _{A}{\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}" /></span></dd> <dd><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 \Rightarrow {\vec {\omega }}\cdot {\vec {n}}={\frac {\mathrm {d} \Gamma }{\mathrm {d} A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi mathvariant="normal">Γ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>A</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\vec {\omega }}\cdot {\vec {n}}={\frac {\mathrm {d} \Gamma }{\mathrm {d} A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01259d415cf7f12d5c3ff557c8e86f90c4e901de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.459ex; height:5.509ex;" alt="{\displaystyle \Rightarrow {\vec {\omega }}\cdot {\vec {n}}={\frac {\mathrm {d} \Gamma }{\mathrm {d} A}}}" /></span></dd></dl> <p>mit der <a href="/wiki/Normale" class="mw-redirect" title="Normale">Normalen</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 {\vec {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" /></span>. </p><p>In der Meteorologie liegen – außer bei echt dreidimensionalen <a href="/wiki/Wirbel_(Str%C3%B6mungslehre)" title="Wirbel (Strömungslehre)">Wirbeln</a> wie <a href="/wiki/Tornado" title="Tornado">Tornados</a> – oft zweidimensionale <a href="/wiki/Geschwindigkeitsfeld" title="Geschwindigkeitsfeld">Geschwindigkeitsfelder</a> vor. Die entsprechende Vortizität zeigt in <i>z</i>-Richtung und lautet </p> <dl><dd><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 {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}=\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right){\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mi>rot</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}=\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right){\vec {e}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f13f6340d6732d62fc4b5693271cd3f8b0f8ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.797ex; height:6.343ex;" alt="{\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}=\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right){\vec {e}}_{z}}" /></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Hydrodynamik">Hydrodynamik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=2" title="Abschnitt bearbeiten: Hydrodynamik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Hydrodynamik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Hydrodynamik" class="mw-redirect" title="Hydrodynamik">Hydrodynamik</a> ist die Vortizität oder Wirbeldichte die Rotation der Fluidgeschwindigkeit, die in Richtung der Drehachse bzw. bei zweidimensionalen <a href="/wiki/Fluss_(Physik)" title="Fluss (Physik)">Flüssen</a> senkrecht zur Flussebene orientiert ist. Für Fluide mit einer festen Rotation <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 {\vec {\omega }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c1134146f17ac8fa94b01ee366ead3d4104db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle {\vec {\omega }}_{0}}" /></span> um eine Achse (z. B. einen rotierenden Zylinder) ist die Geschwindigkeit <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" /></span> eines Teilchens in <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 {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" /></span> identisch mit<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <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 {\vec {v}}={\vec {\omega }}_{0}\times {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\vec {\omega }}_{0}\times {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74f03e5985d14f556ec24d11011508625d398ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.837ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}={\vec {\omega }}_{0}\times {\vec {r}}}" /></span>. Damit ist die Wirbelstärke <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 {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}" /></span> gleich der doppelten <a href="/wiki/Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</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 {\vec {\omega }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c1134146f17ac8fa94b01ee366ead3d4104db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle {\vec {\omega }}_{0}}" /></span> des Fluidelements<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>: </p> <dl><dd><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 {\vec {\omega }}=2{\vec {\omega }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=2{\vec {\omega }}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afecdb9fa12744275fff55d882d3b41716d260ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.207ex; height:2.676ex;" alt="{\displaystyle {\vec {\omega }}=2{\vec {\omega }}_{0}}" /></span></dd> <dd><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 {\vec {\omega }}={\vec {\nabla }}\times {\vec {v}}={\vec {\nabla }}\times \left({\vec {\omega }}_{0}\times {\vec {r}}\right)={\vec {\omega }}_{0}\left({\vec {\nabla }}\cdot {\vec {r}}\right)-\left({\vec {\omega }}_{0}\cdot {\vec {\nabla }}\right){\vec {r}}=3{\vec {\omega }}_{0}-{\vec {\omega }}_{0}=2{\vec {\omega }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>3</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}={\vec {\nabla }}\times {\vec {v}}={\vec {\nabla }}\times \left({\vec {\omega }}_{0}\times {\vec {r}}\right)={\vec {\omega }}_{0}\left({\vec {\nabla }}\cdot {\vec {r}}\right)-\left({\vec {\omega }}_{0}\cdot {\vec {\nabla }}\right){\vec {r}}=3{\vec {\omega }}_{0}-{\vec {\omega }}_{0}=2{\vec {\omega }}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3998a29514fe1c2e48995121f85266598c31a84d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.546ex; height:4.843ex;" alt="{\displaystyle {\vec {\omega }}={\vec {\nabla }}\times {\vec {v}}={\vec {\nabla }}\times \left({\vec {\omega }}_{0}\times {\vec {r}}\right)={\vec {\omega }}_{0}\left({\vec {\nabla }}\cdot {\vec {r}}\right)-\left({\vec {\omega }}_{0}\cdot {\vec {\nabla }}\right){\vec {r}}=3{\vec {\omega }}_{0}-{\vec {\omega }}_{0}=2{\vec {\omega }}_{0}}" /></span></dd></dl> <p>Fluide ohne Wirbelstärke heißen rotations- oder <a href="/wiki/Wirbelfrei" class="mw-redirect" title="Wirbelfrei">wirbelfrei</a> mit <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 {\vec {\omega }}=0\;{\text{bzw.}}\;\zeta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>bzw.</mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi>ζ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=0\;{\text{bzw.}}\;\zeta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb79a6eb5e0cf59bde54fe91cf6e47c1445bee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.003ex; height:2.676ex;" alt="{\displaystyle {\vec {\omega }}=0\;{\text{bzw.}}\;\zeta =0}" /></span>. Allerdings können auch die Fluidelemente eines solchen rotationsfreien Fluids eine Winkelgeschwindigkeit <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 {\vec {\omega }}_{0}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}_{0}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbd702b1e66d15109343df22acf9400f453a45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.761ex; height:2.843ex;" alt="{\displaystyle {\vec {\omega }}_{0}\neq 0}" /></span> besitzen, d. h. sich auf gekrümmten Bahnen bewegen, vgl. die folgende Abbildung, wobei der Buchstabe <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"> <mi>ω</mi> </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/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span> im Text für die Wirbelstärke und in der Abbildung für die Winkelgeschwindigkeit steht: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Vortizit%C3%A4t.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Vortizit%C3%A4t.svg/260px-Vortizit%C3%A4t.svg.png" decoding="async" width="260" height="331" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Vortizit%C3%A4t.svg/390px-Vortizit%C3%A4t.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Vortizit%C3%A4t.svg/520px-Vortizit%C3%A4t.svg.png 2x" data-file-width="550" data-file-height="700" /></a><figcaption>Vortizität und Winkelgeschwindigkeit</figcaption></figure> <p>Man betrachtet ein <a href="/wiki/Infinitesimal" class="mw-redirect" title="Infinitesimal">infinitesimal</a> kleines, quadratisches Gebiet einer Flüssigkeit. Wenn dieses Gebiet rotiert, ist die Wirbelstärke der Strömung ungleich null. Die Wirbelstärke bezieht sich auf <a href="/wiki/Wirbel_(Str%C3%B6mungslehre)#Erzwungener_Wirbel" title="Wirbel (Strömungslehre)">erzwungene Wirbel</a> mit <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 {\vec {\omega }}=\operatorname {rot} {\vec {v}}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=\operatorname {rot} {\vec {v}}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ad8e20bd2cb69ce20086e3588903dfed1a0877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.346ex; height:2.843ex;" alt="{\displaystyle {\vec {\omega }}=\operatorname {rot} {\vec {v}}\neq 0}" /></span>. </p><p>Die Vortizität ist ein geeignetes Mittel für Flüssigkeiten mit kleiner <a href="/wiki/Viskosit%C3%A4t" title="Viskosität">Viskosität</a>. Dann kann die Vortizität an fast allen Orten der Strömung als gleich null angesehen werden. Dies ist offensichtlich für zweidimensionale Strömungen, in denen der Fluss auf der <a href="/wiki/Komplexe_Ebene" class="mw-redirect" title="Komplexe Ebene">komplexen Ebene</a> dargestellt werden kann. Derartige Probleme können meist analytisch gelöst werden. </p><p>Durch Anwendung der Rotation <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 {\vec {\nabla }}\times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f618f7c6314f26d01727fe5dd66353adb6a8698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}\times }" /></span> auf die <a href="/wiki/Navier-Stokes-Gleichungen" title="Navier-Stokes-Gleichungen">Navier-Stokes-Gleichungen</a> für die Geschwindigkeit <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" /></span> lässt sich eine Transportgleichung für die Wirbelstärke gewinnen. Für <a href="/wiki/Inkompressibel" class="mw-redirect" title="Inkompressibel">inkompressible</a>, nichtviskose Flüssigkeiten lautet diese:<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> </p> <dl><dd><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 {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8265fcb8a99e7b1d753962c462e88e09862fa6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.782ex; height:5.343ex;" alt="{\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}}" /></span></dd></dl> <p>Auch für reale Strömungen (dreidimensional, endliche <a href="/wiki/Reynoldszahl" class="mw-redirect" title="Reynoldszahl">Reynoldszahl</a>, d. h. Viskosität ungleich Null) ist die Betrachtung des Flusses über die Wirbelstärke mit Einschränkungen nutzbar, wenn man annimmt, dass das Vortizitätsfeld als eine Anordnung einzelner Wirbel darstellbar ist. Die <a href="/wiki/Diffusion" title="Diffusion">Diffusion</a> dieser Wirbel durch die Strömung wird durch die Wirbeltransportgleichung beschrieben<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> <dl><dd><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 {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}+{\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mi>ρ</mi> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}+{\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14b19aee63487e8c188c79e397431d0b0e72889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.585ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }})\ {\vec {v}}+{\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}" /></span></dd></dl> <p>wobei <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 \nabla ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}" /></span> den <a href="/wiki/Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> bezeichnet.<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> Hier wurde die Wirbeldichtegleichung durch den Diffusionsterm <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 {\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mi>ρ</mi> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be199dc37e708310874265b7aacb75c1481aa96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.962ex; height:5.343ex;" alt="{\displaystyle {\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}" /></span> ergänzt. </p><p>Für hochviskose Strömungen, beispielsweise <a href="/wiki/Taylor-Couette-Str%C3%B6mung" title="Taylor-Couette-Strömung">Taylor-Couette-Strömungen</a>, kann es sinnvoller sein, direkt das Geschwindigkeitsfeld des Fluids anstelle der Wirbelstärke zu betrachten, da die hohe Viskosität zu einer sehr starken Diffusion der Wirbel führt. </p><p>Die <a href="/wiki/Wirbellinie" title="Wirbellinie">Wirbellinie</a> hängt direkt mit der Wirbelstärke zusammen, indem Wirbellinien Tangenten an die Wirbelstärke sind. Die Gesamtheit der durch ein Flächenelement <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 dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1d35422f3d3b3cfcfd9b9aebe9176599fed24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.959ex; height:2.176ex;" alt="{\displaystyle dA}" /></span> gehenden Wirbellinien wird als <i>Wirbelfaden</i> bezeichnet. Die <a href="/wiki/Helmholtzsche_Wirbels%C3%A4tze" title="Helmholtzsche Wirbelsätze">Helmholtzschen Wirbelsätze</a> sagen aus, dass der Wirbelfluss <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 \iint {\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint {\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f60acb02abfcba19fd0fcb4ba4a8a72ea28ae034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.968ex; height:5.676ex;" alt="{\displaystyle \iint {\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}" /></span> sowohl zeitlich als auch räumlich konstant ist. </p> <div class="mw-heading mw-heading2"><h2 id="Meteorologie">Meteorologie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=3" title="Abschnitt bearbeiten: Meteorologie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Meteorologie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der Meteorologie wird mit der Vortizität hauptsächlich die Rotation von Luft um eine Achse beschrieben. </p><p>Die absolute Vortizität <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 _{\mathrm {abs} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {abs} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a759a9592dc9f4583ac8867ef4cd80cc9ed76c0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.062ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {abs} }}" /></span> eines Volumenelements oder eines Körpers in der Meteorologie setzt sich zusammen aus zwei Summanden, der planetaren Vortizität <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 f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></span> und der relativen Vortizität <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 _{\mathrm {rel} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {rel} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06c202c1e733f2ae254495863fdcfd2c2d5fd55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.51ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {rel} }}" /></span> bzw. <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 \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" /></span>: </p> <dl><dd><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 {\begin{aligned}\omega _{\mathrm {abs} }=\colon \eta &=f_{\mathrm {c} }+\omega _{\mathrm {rel} }\\&=f_{\mathrm {c} }+\zeta \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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>:</mo> <mi>η</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>ζ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega _{\mathrm {abs} }=\colon \eta &=f_{\mathrm {c} }+\omega _{\mathrm {rel} }\\&=f_{\mathrm {c} }+\zeta \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9bb5389ae5f22daff1c64603667f90148a5f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.02ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\omega _{\mathrm {abs} }=\colon \eta &=f_{\mathrm {c} }+\omega _{\mathrm {rel} }\\&=f_{\mathrm {c} }+\zeta \end{aligned}}}" /></span></dd></dl> <p>Aufgrund der <a href="/wiki/Erddrehung" class="mw-redirect" title="Erddrehung">Erddrehung</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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}" /></span> erfährt jeder Körper in Erdnähe eine Rotation um die <a href="/wiki/Erdachse" title="Erdachse">Erdachse</a> und besitzt somit eine feste Vortizität. Diese wird bestimmt durch den <a href="/wiki/Corioliskraft" title="Corioliskraft">Coriolisfaktor</a> </p> <dl><dd><dl><dd><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 f_{\mathrm {c} }=2\omega _{0}\sin \varphi \approx 10^{-4}\,{\frac {1}{\mathrm {s} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>≈</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }=2\omega _{0}\sin \varphi \approx 10^{-4}\,{\frac {1}{\mathrm {s} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50665ca3b32289c6b762964aa27be510eccb74c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.154ex; height:5.176ex;" alt="{\displaystyle f_{\mathrm {c} }=2\omega _{0}\sin \varphi \approx 10^{-4}\,{\frac {1}{\mathrm {s} }}}" /></span>,</dd></dl></dd></dl> <p>der vom <a href="/wiki/Breitengrad" class="mw-redirect" title="Breitengrad">Breitengrad</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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> abhängt, und als <i>planetare Vortizität</i> bezeichnet. </p><p>Die <i>relative Vortizität</i> ist die mit der Eigendrehung des Körpers zusammenhängende Größe. Da in der Meteorologie meist zweidimensionale Strömungsfelder auftreten, wird sie oft durch die Rotation in zwei Dimensionen ausgedrückt: </p> <dl><dd><dl><dd><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 {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mi>rot</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca5fbd35f50cf68d29b4a32529a8233984ba51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.039ex; height:3.343ex;" alt="{\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {v}}_{2D}}" /></span></dd></dl></dd></dl> <p>Die Richtung des Wirbelstärke-Vektors lässt sich mit der <a href="/wiki/Korkenzieherregel" title="Korkenzieherregel">Korkenzieherregel</a> bestimmen: Dreht sich das Fluid <i>gegen</i> den <a href="/wiki/Uhrzeigersinn" class="mw-redirect" title="Uhrzeigersinn">Uhrzeigersinn</a>, so zeigt die Wirbelstärke nach oben und ist positiv. Auf der <a href="/wiki/Nordhalbkugel" title="Nordhalbkugel">Nordhalbkugel</a> wird Rotation gegen den Uhrzeigersinn, also mit positivem <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span>, als <a href="/wiki/Zyklonale_Rotation" title="Zyklonale Rotation">zyklonale Rotation</a> bezeichnet und mit negativem <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span> als <a href="/wiki/Antizyklonale_Rotation" title="Antizyklonale Rotation">antizyklonale Rotation</a>. Auf der <a href="/wiki/S%C3%BCdhalbkugel" title="Südhalbkugel">Südhalbkugel</a> gilt dies jeweils entsprechend umgekehrt. </p><p>In <a href="/w/index.php?title=Nat%C3%BCrliche_Koordinaten&action=edit&redlink=1" class="new" title="Natürliche Koordinaten (Seite nicht vorhanden)">natürlichen Koordinaten</a> ergibt sich: </p> <dl><dd><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 {\begin{alignedat}{2}\zeta &=\zeta _{C}&&+\zeta _{S}\\&=v\cdot K_{s}&&-{\frac {\partial v}{\partial n}}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>ζ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>+</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mo>⋅</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}\zeta &=\zeta _{C}&&+\zeta _{S}\\&=v\cdot K_{s}&&-{\frac {\partial v}{\partial n}}\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2cc5fac53455928e0ab77c02f4e0dda78bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.433ex; margin-bottom: -0.239ex; width:17.117ex; height:8.509ex;" alt="{\displaystyle {\begin{alignedat}{2}\zeta &=\zeta _{C}&&+\zeta _{S}\\&=v\cdot K_{s}&&-{\frac {\partial v}{\partial n}}\end{alignedat}}}" /></span></dd></dl> <p>mit<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> </p> <ul><li>der Krümmungsvortizität <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 \zeta _{C}:=v\cdot K_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>:=</mo> <mi>v</mi> <mo>⋅</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{C}:=v\cdot K_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/005b4a1fa613220a56aed726a64301e3176b2d39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.028ex; height:2.509ex;" alt="{\displaystyle \zeta _{C}:=v\cdot K_{s}}" /></span> <ul><li>der <a href="/wiki/Kr%C3%BCmmung" title="Krümmung">Krümmung</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 K_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44d82e5e2fdd046a55645a8b3f967cfa7fd0a8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.976ex; height:2.509ex;" alt="{\displaystyle K_{s}}" /></span> der <a href="/wiki/Stromlinie" title="Stromlinie">Stromlinien</a></li></ul></li> <li>der Scherungsvortizität <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 \zeta _{S}:=-{\frac {\partial v}{\partial n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>:=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{S}:=-{\frac {\partial v}{\partial n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce9e7b2679ad26dac64b8b65d25c003125bbbd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.413ex; height:5.509ex;" alt="{\displaystyle \zeta _{S}:=-{\frac {\partial v}{\partial n}}}" /></span> <ul><li>den Komponenten <i>n</i> und <i>s</i> des Koordinatensystems.</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Potentielle_Vortizität"><span id="Potentielle_Vortizit.C3.A4t"></span>Potentielle Vortizität</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=4" title="Abschnitt bearbeiten: Potentielle Vortizität" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Potentielle Vortizität"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <a href="/wiki/Helmholtzsche_Wirbels%C3%A4tze" title="Helmholtzsche Wirbelsätze">Helmholtzschen Erhaltungssätze für den Wirbelfluss</a> führen zur <a href="/wiki/Potentielle_Vortizit%C3%A4t" title="Potentielle Vortizität">potentiellen Vortizität</a> PV:<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> <dl><dd><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 PV={\frac {\eta }{\Delta p}}={\frac {f_{c}+\zeta }{\Delta p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>+</mo> <mi>ζ</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV={\frac {\eta }{\Delta p}}={\frac {f_{c}+\zeta }{\Delta p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6d1848fcc613a82f5ddaa7d969da1efea5c902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.526ex; height:5.843ex;" alt="{\displaystyle PV={\frac {\eta }{\Delta p}}={\frac {f_{c}+\zeta }{\Delta p}}}" /></span></dd></dl> <p>Durch Kombination der Wirbeldichtegleichung mit der <a href="/wiki/Kontinuit%C3%A4tsgleichung" title="Kontinuitätsgleichung">Kontinuitätsgleichung</a> kann man zeigen, dass die potentielle Vortizität zeitlich erhalten ist: </p> <dl><dd><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 {\frac {\mathrm {d} PV}{\mathrm {d} t}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mi>V</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} PV}{\mathrm {d} t}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/522a54f8853f606c1e8ef8acd15ebb17db279240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.922ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} PV}{\mathrm {d} t}}=0}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=5" title="Abschnitt bearbeiten: Anmerkungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Anmerkungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Literatur enthält auch die Definition<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><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> </p> <dl><dd><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 {\vec {\omega }}={\frac {1}{2}}\operatorname {rot} {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}={\frac {1}{2}}\operatorname {rot} {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1a18524afa004050ad3bd0b3bc07e8052b376e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.471ex; height:5.176ex;" alt="{\displaystyle {\vec {\omega }}={\frac {1}{2}}\operatorname {rot} {\vec {v}}}" /></span></dd></dl> <p>Die Begriffe <i>Wirbelstärke</i>, <i>Wirbeldichte</i>, <i>Wirbelhaftigkeit</i>, <i>Wirbeligkeit</i>, <i>Wirbelung</i>, <i>Vortizität</i>, <i>Wirbelfaden</i> sowie die Benennung der <i>Wirbeldichte-</i> und <i>Wirbeltransportgleichung</i> sind nicht klar definiert und somit schwer gegeneinander abgrenzbar. In der Literatur finden sich teilweise widersprüchliche Angaben und Definitionen. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=6" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hans Stephani, Gerhard Kluge: <i>Theoretische Mechanik</i>. Spektrum Akademischer Verlag, Heidelberg 1995, <a href="/wiki/Spezial:ISBN-Suche/3860252844" class="internal mw-magiclink-isbn">ISBN 3-86025-284-4</a></li> <li>Ludwig Bergmann, Clemens Schaefer: <i>Lehrbuch der Experimentalphysik</i>. Band 1: <i>Mechanik, Relativität, Wärme</i>. de Gruyter, Berlin 1998. <a href="/wiki/Spezial:ISBN-Suche/3110128705" class="internal mw-magiclink-isbn">ISBN 3-11-012870-5</a></li> <li>Lew D. Landau, Jewgeni M. Lifschitz: <i>Lehrbuch der theoretischen Physik</i>. Band 6: <i>Hydrodynamik</i>. Verlag Harri Deutsch, Frankfurt am Main 2007. <a href="/wiki/Spezial:ISBN-Suche/9783817113316" class="internal mw-magiclink-isbn">ISBN 978-3-8171-1331-6</a></li> <li>Koji Ohkitani: <i>Elementary Account Of Vorticity And Related Equations</i>. Cambridge University Press, 2005. <a href="/wiki/Spezial:ISBN-Suche/0521819849" class="internal mw-magiclink-isbn">ISBN 0-521-81984-9</a></li> <li>Andrew J. Majda, Andrea L. Bertozzi: <i>Vorticity and Incompressible Flow</i>. Cambridge University Press, 2002. <a href="/wiki/Spezial:ISBN-Suche/0521639484" class="internal mw-magiclink-isbn">ISBN 0-521-63948-4</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=7" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div><b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Vorticity?uselang=de"><span lang="en">Commons</span>: Wirbelstärke</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</div> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wirbelst%C3%A4rke&veaction=edit&section=8" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wirbelst%C3%A4rke&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Etienne Guyon, Jean-Pierre Hulin, Luc Petit: <cite style="font-style:italic">Hydrodynamik -</cite>. 3. Auflage. Vieweg + Teubner, Wiesbaden 1997, <a href="/wiki/Spezial:ISBN-Suche/9783528072766" class="internal mw-magiclink-isbn">ISBN 978-3-528-07276-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>105</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Wirbelst%C3%A4rke&rft.au=Etienne+Guyon%2C+Jean-Pierre+Hulin%2C+Luc+Petit&rft.btitle=Hydrodynamik+-&rft.date=1997&rft.edition=3.&rft.genre=book&rft.isbn=9783528072766&rft.pages=105&rft.place=Wiesbaden&rft.pub=Vieweg+%2B+Teubner" style="display:none"> </span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Ludwig Bergmann, Clemens Schaefer: <i>Lehrbuch der Experimentalphysik, Band 1: Mechanik, Relativität, Wärme</i>, S. 564. de Gruyter, Berlin 1998. <a href="/wiki/Spezial:ISBN-Suche/3110128705" class="internal mw-magiclink-isbn">ISBN 3-11-012870-5</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, <a href="/wiki/Spezial:ISBN-Suche/3871443743" class="internal mw-magiclink-isbn">ISBN 3-87144-374-3</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>104</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Wirbelst%C3%A4rke&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=104&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">I.N. Bronstein, K. A. Semendjajew: <cite style="font-style:italic">Taschenbuch der Mathematik</cite>. 19. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1980, <a href="/wiki/Spezial:ISBN-Suche/3871444928" class="internal mw-magiclink-isbn">ISBN 3-87144-492-8</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>626</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Wirbelst%C3%A4rke&rft.au=I.N.+Bronstein%2C+K.+A.+Semendjajew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=1980&rft.edition=19.&rft.genre=book&rft.isbn=3871444928&rft.pages=626&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none"> </span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="cite">Roland Netz: <a rel="nofollow" class="external text" href="https://einrichtungen.ph.tum.de/T37/WS1011/KontMech/SkriptKontinuumsmech.pdf"><i>Mechanik der Kontinua.</i></a> (PDF; 671 kB)<span class="Abrufdatum"> Abgerufen am 25. Mai 2011</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWirbelst%C3%A4rke&rft.title=Mechanik+der+Kontinua&rft.description=Mechanik+der+Kontinua&rft.identifier=http%3A%2F%2Feinrichtungen.ph.tum.de%2FT37%2FWS1011%2FKontMech%2FSkriptKontinuumsmech.pdf&rft.creator=Roland+Netz"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Etienne Guyon, Jean-Pierre Hulin, Luc Petit: <cite style="font-style:italic">Hydrodynamik -</cite>. 3. Auflage. Vieweg + Teubner, Wiesbaden 1997, <a href="/wiki/Spezial:ISBN-Suche/9783528072766" class="internal mw-magiclink-isbn">ISBN 978-3-528-07276-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>282</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Wirbelst%C3%A4rke&rft.au=Etienne+Guyon%2C+Jean-Pierre+Hulin%2C+Luc+Petit&rft.btitle=Hydrodynamik+-&rft.date=1997&rft.edition=3.&rft.genre=book&rft.isbn=9783528072766&rft.pages=282&rft.place=Wiesbaden&rft.pub=Vieweg+%2B+Teubner" style="display:none"> </span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080921183217/http://www.ihr.uni-stuttgart.de/fileadmin/user_upload/autoren/mh/structure/node24.html"><i>Wirbeltransportgleichungen.</i></a> Archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">.mw-parser-output .dewiki-iconexternal>a{background-position:center right!important;background-repeat:no-repeat!important}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/a/a4/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px!important;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/3/30/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/9/96/Link-external-small-ltr-progressive.svg")!important;background-size:0.857em!important;padding-right:1em!important}</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.ihr.uni-stuttgart.de%2Ffileadmin%2Fuser_upload%2Fautoren%2Fmh%2Fstructure%2Fnode24.html">Original</a></span> (nicht mehr online verfügbar) am <span style="white-space:nowrap;">21. September 2008</span><span>;</span><span class="Abrufdatum"> abgerufen am 25. Mai 2011</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWirbelst%C3%A4rke&rft.title=Wirbeltransportgleichungen&rft.description=Wirbeltransportgleichungen&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20080921183217%2Fhttp%3A%2F%2Fwww.ihr.uni-stuttgart.de%2Ffileadmin%2Fuser_upload%2Fautoren%2Fmh%2Fstructure%2Fnode24.html&rft.source=http://www.ihr.uni-stuttgart.de/fileadmin/user_upload/autoren/mh/structure/node24.html"> </span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="http://www.wetter3.de/vorticity.html"><i>Vorticity.</i></a><span class="Abrufdatum"> Abgerufen am 25. Mai 2011</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWirbelst%C3%A4rke&rft.title=Vorticity&rft.description=Vorticity&rft.identifier=http%3A%2F%2Fwww.wetter3.de%2Fvorticity.html"> </span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150218233406/http://www.iapmw.unibe.ch/teaching/vorlesungen/atmosphaerenphysik/FS_2011/AT-phys_FS11_Kapitel6b.pdf"><i>Atmosphärenphysik.</i></a> (PDF; 337 kB) Archiviert vom <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r250917974" /><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.iapmw.unibe.ch%2Fteaching%2Fvorlesungen%2Fatmosphaerenphysik%2FFS_2011%2FAT-phys_FS11_Kapitel6b.pdf">Original</a></span> (nicht mehr online verfügbar) am <span style="white-space:nowrap;">18. Februar 2015</span><span>;</span><span class="Abrufdatum"> abgerufen am 25. Mai 2011</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWirbelst%C3%A4rke&rft.title=Atmosph%C3%A4renphysik&rft.description=Atmosph%C3%A4renphysik&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20150218233406%2Fhttp%3A%2F%2Fwww.iapmw.unibe.ch%2Fteaching%2Fvorlesungen%2Fatmosphaerenphysik%2FFS_2011%2FAT-phys_FS11_Kapitel6b.pdf&rft.source=http://www.iapmw.unibe.ch/teaching/vorlesungen/atmosphaerenphysik/FS_2011/AT-phys_FS11_Kapitel6b.pdf"> </span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/Vorticity.html"><i>Vorticity -- from Eric Weisstein's World of Physics.</i></a> In: <i>scienceworld.wolfram.com.</i><span class="Abrufdatum"> Abgerufen am 25. Mai 2011</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWirbelst%C3%A4rke&rft.title=Vorticity+--+from+Eric+Weisstein%27s+World+of+Physics&rft.description=Vorticity+--+from+Eric+Weisstein%27s+World+of+Physics&rft.identifier=http%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FVorticity.html"> </span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Hans Stephani, Gerhard Kluge: <i>Theoretische Mechanik</i>. Spektrum Akademischer Verlag, Heidelberg 1995, <a href="/wiki/Spezial:ISBN-Suche/3860252844" class="internal mw-magiclink-isbn">ISBN 3-86025-284-4</a>, S. 273.</span> </li> </ol></div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Wirbelstärke&oldid=252546015">https://de.wikipedia.org/w/index.php?title=Wirbelstärke&oldid=252546015</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Str%C3%B6mungsmechanik" title="Kategorie:Strömungsmechanik">Strömungsmechanik</a></li><li><a href="/wiki/Kategorie:Meteorologische_Gr%C3%B6%C3%9Fe" title="Kategorie:Meteorologische Größe">Meteorologische Größe</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 23. 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class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Wirbelstärke</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>24 Sprachen</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Abschnitt hinzufügen</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> 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Wirbelstärke – Wikipedia