Geostrophic wind - Wikipedia

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class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equilibri_geostr%C3%B2fic" title="Equilibri geostròfic – Catalan" lang="ca" hreflang="ca" data-title="Equilibri geostròfic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Geostrofisk_balance" title="Geostrofisk balance – Danish" lang="da" hreflang="da" data-title="Geostrofisk balance" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geostrophischer_Wind" title="Geostrophischer Wind – German" lang="de" hreflang="de" data-title="Geostrophischer Wind" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geostroofiline_tuul" title="Geostroofiline tuul – Estonian" lang="et" hreflang="et" data-title="Geostroofiline tuul" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Viento_geostr%C3%B3fico" title="Viento geostrófico – Spanish" lang="es" hreflang="es" data-title="Viento geostrófico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vent_g%C3%A9ostrophique" title="Vent géostrophique – French" lang="fr" hreflang="fr" data-title="Vent géostrophique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gaoth_gheostr%C3%B3fach" title="Gaoth gheostrófach – Irish" lang="ga" hreflang="ga" data-title="Gaoth gheostrófach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%80%EA%B7%A0%ED%92%8D" title="지균풍 – Korean" lang="ko" hreflang="ko" data-title="지균풍" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Geostrofi%C4%8Dki_vjetar" title="Geostrofički vjetar – Croatian" lang="hr" hreflang="hr" data-title="Geostrofički vjetar" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Vento_geostrofico" title="Vento geostrofico – Italian" lang="it" hreflang="it" data-title="Vento geostrofico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%94%E1%83%9D%E1%83%A1%E1%83%A2%E1%83%A0%E1%83%9D%E1%83%A4%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A5%E1%83%90%E1%83%A0%E1%83%98" title="გეოსტროფული ქარი – Georgian" lang="ka" hreflang="ka" data-title="გეოსტროფული ქარი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Vent_geostr%C3%B2fich" title="Vent geostròfich – Lombard" lang="lmo" hreflang="lmo" data-title="Vent geostròfich" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geostrofische_wind" title="Geostrofische wind – Dutch" lang="nl" hreflang="nl" data-title="Geostrofische wind" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9C%B0%E8%A1%A1%E9%A2%A8" title="地衡風 – Japanese" lang="ja" hreflang="ja" data-title="地衡風" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geostrofisk_vind" title="Geostrofisk vind – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Geostrofisk vind" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geostrofisk_vind" title="Geostrofisk vind – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Geostrofisk vind" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geostrofik_shamol" title="Geostrofik shamol – Uzbek" lang="uz" hreflang="uz" data-title="Geostrofik shamol" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wiatr_geostroficzny" title="Wiatr geostroficzny – Polish" lang="pl" hreflang="pl" data-title="Wiatr geostroficzny" data-language-autonym="Polski" data-language-local-name="Polish" 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/Vento_geostr%C3%B3fico" title="Vento geostrófico – Portuguese" lang="pt" hreflang="pt" data-title="Vento geostrófico" data-language-autonym="Português" data-language-local-name="Portuguese" 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/V%C3%A2nt_geostrofic" title="Vânt geostrofic – Romanian" lang="ro" hreflang="ro" data-title="Vânt geostrofic" data-language-autonym="Română" data-language-local-name="Romanian" 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%93%D0%B5%D0%BE%D1%81%D1%82%D1%80%D0%BE%D1%84%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B2%D0%B5%D1%82%D0%B5%D1%80" title="Геострофический ветер – Russian" lang="ru" hreflang="ru" data-title="Геострофический ветер" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Geostrofinen_tuuli" title="Geostrofinen tuuli – Finnish" lang="fi" hreflang="fi" data-title="Geostrofinen tuuli" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Geostrofisk_vind" title="Geostrofisk vind – Swedish" lang="sv" hreflang="sv" data-title="Geostrofisk vind" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%81%D1%82%D1%80%D0%BE%D1%84%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%B2%D1%96%D1%82%D0%B5%D1%80" title="Геострофічний вітер – Ukrainian" lang="uk" hreflang="uk" data-title="Геострофічний вітер" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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/%E5%9C%B0%E8%BD%89%E9%A2%A8" title="地轉風 – Chinese" lang="zh" hreflang="zh" data-title="地轉風" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in atmospheric science</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Geostrophic flow" redirects here. For oceanic wind, see <a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Geostrophic_wind" title="Special:EditPage/Geostrophic wind">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Geostrophic+wind%22">"Geostrophic wind"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Geostrophic+wind%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Geostrophic+wind%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Geostrophic+wind%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Geostrophic+wind%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Geostrophic+wind%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Atmospheric_science" title="Atmospheric science">atmospheric science</a>, <b>geostrophic flow</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/dʒ/: 'j' in 'jam'">dʒ</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/ə/: 'a' in 'about'">ə</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'s' in 'sigh'">s</span><span title="'t' in 'tie'">t</span><span title="'r' in 'rye'">r</span><span title="/ɒ/: 'o' in 'body'">ɒ</span><span title="'f' in 'find'">f</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="'k' in 'kind'">k</span></span>,<span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/dʒ/: 'j' in 'jam'">dʒ</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-,<span class="wrap"> </span>-<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'s' in 'sigh'">s</span><span title="'t' in 'tie'">t</span><span title="'r' in 'rye'">r</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-/</a></span></span><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><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><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>) is the theoretical <a href="/wiki/Wind" title="Wind">wind</a> that would result from an exact balance between the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis force</a> and the <a href="/wiki/Pressure_gradient" title="Pressure gradient">pressure gradient</a> force. This condition is called <i><a href="/wiki/Geostrophic_equilibrium" class="mw-redirect" title="Geostrophic equilibrium">geostrophic equilibrium</a></i> or <i>geostrophic balance</i> (also known as <i>geostrophy</i>). The geostrophic wind is directed <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to <a href="/wiki/Isobar_(meteorology)" class="mw-redirect" title="Isobar (meteorology)">isobars</a> (lines of constant <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">pressure</a> at a given height). This balance seldom holds exactly in nature. The true wind almost always differs from the geostrophic wind due to other forces such as <a href="/wiki/Friction" title="Friction">friction</a> from the ground. Thus, the actual wind would equal the geostrophic wind only if there were no friction (e.g. above the <a href="/wiki/Atmospheric_boundary_layer" class="mw-redirect" title="Atmospheric boundary layer">atmospheric boundary layer</a>) and the isobars were perfectly straight. Despite this, much of the atmosphere outside the <a href="/wiki/Tropics" title="Tropics">tropics</a> is close to geostrophic flow much of the time and it is a valuable first approximation. Geostrophic flow in air or water is a zero-frequency <a href="/wiki/Inertial_waves" class="mw-redirect" title="Inertial waves">inertial wave</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin">Origin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=1" title="Edit section: Origin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A useful heuristic is to imagine <a href="/wiki/Air" class="mw-redirect" title="Air">air</a> starting from rest, experiencing a force directed from areas of high <a href="/wiki/Pressure" title="Pressure">pressure</a> toward areas of low pressure, called the <a href="/wiki/Pressure_gradient" title="Pressure gradient">pressure gradient</a> force. If the air began to move in response to that force, however, the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a> would deflect it, to the right of the motion in the <a href="/wiki/Northern_hemisphere" class="mw-redirect" title="Northern hemisphere">northern hemisphere</a> or to the left in the <a href="/wiki/Southern_hemisphere" class="mw-redirect" title="Southern hemisphere">southern hemisphere</a>. As the air accelerated, the deflection would increase until the Coriolis force's strength and direction balanced the pressure gradient force, a state called geostrophic balance. At this point, the flow is no longer moving from high to low pressure, but instead moves along <a href="/wiki/Isobar_(meteorology)" class="mw-redirect" title="Isobar (meteorology)">isobars</a>. Geostrophic balance helps to explain why, in the northern hemisphere, <a href="/wiki/Low-pressure_system" class="mw-redirect" title="Low-pressure system">low-pressure systems</a> (or <i><a href="/wiki/Cyclone" title="Cyclone">cyclones</a></i>) spin counterclockwise and <a href="/wiki/High-pressure_area" title="High-pressure area">high-pressure systems</a> (or <i><a href="/wiki/Anticyclone" title="Anticyclone">anticyclones</a></i>) spin clockwise, and the opposite in the southern hemisphere. </p> <div class="mw-heading mw-heading2"><h2 id="Geostrophic_currents">Geostrophic currents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=2" title="Edit section: Geostrophic currents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></div> <p>Flow of ocean water is also largely geostrophic. Just as multiple weather balloons that measure pressure as a function of height in the atmosphere are used to map the atmospheric pressure field and infer the geostrophic wind, measurements of density as a function of depth in the ocean are used to infer geostrophic currents. <a href="/wiki/Satellite_altimetry" class="mw-redirect" title="Satellite altimetry">Satellite altimeters</a> are also used to measure sea surface height anomaly, which permits a calculation of the geostrophic current at the surface. </p> <div class="mw-heading mw-heading2"><h2 id="Limitations_of_the_geostrophic_approximation">Limitations of the geostrophic approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=3" title="Edit section: Limitations of the geostrophic approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The effect of friction, between the air and the land, breaks the geostrophic balance. Friction slows the flow, lessening the effect of the Coriolis force. As a result, the pressure gradient force has a greater effect and the air still moves from high pressure to low pressure, though with great deflection. This explains why high-pressure system winds radiate out from the center of the system, while low-pressure systems have winds that spiral inwards. </p><p>The geostrophic wind neglects <a href="/wiki/Friction" title="Friction">frictional</a> effects, which is usually a good <a href="/wiki/Approximation" title="Approximation">approximation</a> for the <a href="/wiki/Synoptic_scale_meteorology" title="Synoptic scale meteorology">synoptic scale</a> instantaneous flow in the midlatitude mid-<a href="/wiki/Troposphere" title="Troposphere">troposphere</a>.<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> Although <a href="/wiki/Ageostrophic" class="mw-redirect" title="Ageostrophic">ageostrophic</a> terms are relatively small, they are essential for the time evolution of the flow and in particular are necessary for the growth and decay of storms. <a href="/wiki/Quasi-geostrophic_equations" title="Quasi-geostrophic equations">Quasigeostrophic</a> and semi geostrophic theory are used to model flows in the atmosphere more widely. These theories allow for a divergence to take place and for weather systems to then develop. </p> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=4" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Geostrophic_current#Formulation" title="Geostrophic current">Geostrophic current § Formulation</a></div> <p><a href="/wiki/Newton%27s_Second_Law" class="mw-redirect" title="Newton's Second Law">Newton's Second Law</a> can be written as follows if only the pressure gradient, gravity, and friction act on an air parcel, where bold symbols are vectors: </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 {D{\boldsymbol {U}} \over Dt}=-{1 \over \rho }\nabla p-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}+{\boldsymbol {g}}+{\boldsymbol {F}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">U</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">U</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">g</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {D{\boldsymbol {U}} \over Dt}=-{1 \over \rho }\nabla p-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}+{\boldsymbol {g}}+{\boldsymbol {F}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afbecd00c059e6ddb7596ebb1adbc34dc059d3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.184ex; height:5.843ex;" alt="{\displaystyle {D{\boldsymbol {U}} \over Dt}=-{1 \over \rho }\nabla p-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}+{\boldsymbol {g}}+{\boldsymbol {F}}_{r}}"></span></dd></dl> <p>Here <i><b>U</b></i> is the velocity field of the air, <b>Ω</b> is the angular velocity vector of the planet, <i>ρ</i> is the density of the air, P is the air pressure, <b>F</b><sub>r</sub> is the friction, <b>g</b> is the <a href="/wiki/Standard_gravity" title="Standard gravity">acceleration vector due to gravity</a> and <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">D</span><span class="sr-only">/</span><span class="den">D<i>t</i></span></span>⁠</span> is the <a href="/wiki/Material_derivative" title="Material derivative">material derivative</a>. </p><p>Locally this can be expanded in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, with a positive <i>u</i> representing an eastward direction and a positive <i>v</i> representing a northward direction. Neglecting friction and vertical motion, as justified by the <a href="/wiki/Taylor%E2%80%93Proudman_theorem" title="Taylor–Proudman theorem">Taylor–Proudman theorem</a>, we have: </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 {Du \over Dt}=-{1 \over \rho }{\partial P \over \partial x}+fv+0+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>u</mi> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>f</mi> <mi>v</mi> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Du \over Dt}=-{1 \over \rho }{\partial P \over \partial x}+fv+0+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86be403ad942f965d043fe6e272d4d700c391cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.187ex; height:5.843ex;" alt="{\displaystyle {Du \over Dt}=-{1 \over \rho }{\partial P \over \partial x}+fv+0+0}"></span></dd> <dd></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 {Dv \over Dt}=-{1 \over \rho }{\partial P \over \partial y}-fu+0+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>v</mi> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>f</mi> <mi>u</mi> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Dv \over Dt}=-{1 \over \rho }{\partial P \over \partial y}-fu+0+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ce33a437062a747bff5d70800dd27616d1b1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.187ex; height:6.009ex;" alt="{\displaystyle {Dv \over Dt}=-{1 \over \rho }{\partial P \over \partial y}-fu+0+0}"></span></dd> <dd></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 {Dw \over Dt}=-{1 \over \rho }{\partial P \over \partial z}+0-g+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>w</mi> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mi>g</mi> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Dw \over Dt}=-{1 \over \rho }{\partial P \over \partial z}+0-g+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7c83bf1fae25ff64f244da52b83a945ebed9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.231ex; height:5.843ex;" alt="{\displaystyle {Dw \over Dt}=-{1 \over \rho }{\partial P \over \partial z}+0-g+0}"></span></dd></dl> <p>With <span class="nowrap"><i>f</i> = 2Ω sin <i>φ</i></span> the <a href="/wiki/Coriolis_frequency" title="Coriolis frequency">Coriolis parameter</a> (approximately <span class="nowrap"><span data-sort-value="6996100000000000000♠"></span>10<sup>−4</sup> s<sup>−1</sup></span>, varying with latitude). </p><p>Assuming geostrophic balance, the system is stationary and the first two equations become: </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 fv={1 \over \rho }{\partial P \over \partial x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fv={1 \over \rho }{\partial P \over \partial x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc73214113c83b0559c2fb5b5930ab0aeee0f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.442ex; height:5.843ex;" alt="{\displaystyle fv={1 \over \rho }{\partial P \over \partial x}}"></span></dd> <dd></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 fu=-{1 \over \rho }{\partial P \over \partial y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>u</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fu=-{1 \over \rho }{\partial P \over \partial y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237e1c84f73d860ef5e9bf2563d7d4d484300ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.453ex; height:6.009ex;" alt="{\displaystyle fu=-{1 \over \rho }{\partial P \over \partial y}}"></span></dd> <dd></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 -g-{1 \over \rho }{\partial P \over \partial z}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>g</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -g-{1 \over \rho }{\partial P \over \partial z}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97cb503c16ab53c7b327eece58858c4c895dd6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.963ex; height:5.843ex;" alt="{\displaystyle -g-{1 \over \rho }{\partial P \over \partial z}=0}"></span></dd></dl> <p>By substituting using the third equation above, we have: </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}fv&={\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial x}}:={\frac {\;-g\;}{\;c\;}}a\\[5px]fu&=-{\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial y}}:=+{\frac {\;g\;}{\;c\;}}b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mo>−</mo> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mo>−</mo> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mo>−</mo> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>:=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial x}}:={\frac {\;-g\;}{\;c\;}}a\\[5px]fu&=-{\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial y}}:=+{\frac {\;g\;}{\;c\;}}b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74230973dd76c3d54ba86f4585c87c568477f2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:27.088ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial x}}:={\frac {\;-g\;}{\;c\;}}a\\[5px]fu&=-{\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial y}}:=+{\frac {\;g\;}{\;c\;}}b\end{aligned}}}"></span></dd></dl> <p>with <i>z</i> the <a href="/wiki/Geopotential_height" title="Geopotential height">geopotential height</a> of the <u>constant pressure</u> surface, satisfying </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 {\rm {d}}P={\partial P \over \partial x}{\rm {d}}x+{\partial P \over \partial y}{\rm {d}}y+{\partial P \over \partial z}{\rm {d}}z:=a{\rm {d}}x+b{\rm {d}}y+c{\rm {d}}z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> <mo>:=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {d}}P={\partial P \over \partial x}{\rm {d}}x+{\partial P \over \partial y}{\rm {d}}y+{\partial P \over \partial z}{\rm {d}}z:=a{\rm {d}}x+b{\rm {d}}y+c{\rm {d}}z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c5a628e63549fe42bd7dbed9f375e6def561c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.339ex; height:6.009ex;" alt="{\displaystyle {\rm {d}}P={\partial P \over \partial x}{\rm {d}}x+{\partial P \over \partial y}{\rm {d}}y+{\partial P \over \partial z}{\rm {d}}z:=a{\rm {d}}x+b{\rm {d}}y+c{\rm {d}}z=0}"></span></dd></dl> <p>Further simplify those formulae above: </p><p><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}fv&={\frac {\;-g\;}{\;c\;}}a=+g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}x}}{\biggr )}_{{\rm {d}}y=0}\\[5px]fu&=+{\frac {\;g\;}{\;c\;}}b=-g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}y}}{\biggr )}_{{\rm {d}}x=0}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mo>−</mo> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mi>a</mi> <mo>=</mo> <mo>+</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mi>g</mi> <mspace width="thickmathspace" /> </mrow> <mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;c\;}}a=+g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}x}}{\biggr )}_{{\rm {d}}y=0}\\[5px]fu&=+{\frac {\;g\;}{\;c\;}}b=-g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}y}}{\biggr )}_{{\rm {d}}x=0}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71223d1bd00ec0b06da1dc4aae4aeac95456565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.705ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;c\;}}a=+g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}x}}{\biggr )}_{{\rm {d}}y=0}\\[5px]fu&=+{\frac {\;g\;}{\;c\;}}b=-g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}y}}{\biggr )}_{{\rm {d}}x=0}\end{aligned}}}"></span> </p><p>This leads us to the following result for the geostrophic wind components: </p><p><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 v_{g}={g \over f}{{\rm {d}}z \over {\rm {d}}x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>f</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{g}={g \over f}{{\rm {d}}z \over {\rm {d}}x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc0e331a35831eff0792045a5cd680d69fe767a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.821ex; height:5.843ex;" alt="{\displaystyle v_{g}={g \over f}{{\rm {d}}z \over {\rm {d}}x}}"></span> </p><p><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 u_{g}=-{g \over f}{{\rm {d}}z \over {\rm {d}}y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>f</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{g}=-{g \over f}{{\rm {d}}z \over {\rm {d}}y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a4d02c9cf195e91032b7a500fd8da61959d09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.657ex; height:5.843ex;" alt="{\displaystyle u_{g}=-{g \over f}{{\rm {d}}z \over {\rm {d}}y}}"></span> </p><p>The validity of this approximation depends on the local <a href="/wiki/Rossby_number" title="Rossby number">Rossby number</a>. It is invalid at the equator, because <i>f</i> is equal to zero there, and therefore generally not used in the <a href="/wiki/Tropics" title="Tropics">tropics</a>. </p><p>Other variants of the equation are possible; for example, the geostrophic wind vector can be expressed in terms of the gradient of the <a href="/wiki/Geopotential_height" title="Geopotential height">geopotential</a> Φ on a surface of constant pressure: </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 \mathbf {V} _{\mathrm {g} }={\frac {\hat {\mathbf {k} }}{f}}\times \nabla _{p}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mi>f</mi> </mfrac> </mrow> <mo>×</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{\mathrm {g} }={\frac {\hat {\mathbf {k} }}{f}}\times \nabla _{p}\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e710e3f644f8c61d9662910c66e2e421fd8700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.933ex; height:6.343ex;" alt="{\displaystyle \mathbf {V} _{\mathrm {g} }={\frac {\hat {\mathbf {k} }}{f}}\times \nabla _{p}\Phi }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li> <li><a href="/wiki/Thermal_wind" title="Thermal wind">Thermal wind</a></li> <li><a href="/wiki/Gradient_wind" class="mw-redirect" title="Gradient wind">Gradient wind</a></li> <li><a href="/wiki/Prevailing_winds" title="Prevailing winds">Prevailing winds</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.dictionary.com/browse/geostrophic">"geostrophic"</a>. <i><a href="/wiki/Dictionary.com" title="Dictionary.com">Dictionary.com Unabridged</a></i> (Online). n.d<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-01-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=geostrophic&rft.btitle=Dictionary.com+Unabridged&rft_id=https%3A%2F%2Fwww.dictionary.com%2Fbrowse%2Fgeostrophic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeostrophic+wind" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20211223131345/https://www.lexico.com/definition/geostrophic">"geostrophic"</a>. <i><a href="/wiki/Lexico" title="Lexico">Lexico</a> UK English Dictionary</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. Archived from <a rel="nofollow" class="external text" href="http://www.lexico.com/definition/geostrophic">the original</a> on 2021-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=geostrophic&rft.btitle=Lexico+UK+English+Dictionary&rft.pub=Oxford+University+Press&rft_id=http%3A%2F%2Fwww.lexico.com%2Fdefinition%2Fgeostrophic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeostrophic+wind" class="Z3988"></span> </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/geostrophic">"geostrophic"</a>. <i><a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster.com Dictionary</a></i>. Merriam-Webster<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-01-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster.com+Dictionary&rft.atitle=geostrophic&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fgeostrophic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeostrophic+wind" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoltonHakim2012" class="citation book cs1">Holton, James R.; Hakim, Gregory J. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hcxcqQp7XOsC&pg=PA42">"2.4.1 Geostrophic Approximation and Geostrophic Wind"</a>. <i>An Introduction to Dynamic Meteorology</i>. International Geophysics. Vol. 88 (5th ed.). Academic Press. pp. 42–43. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-384867-3" title="Special:BookSources/978-0-12-384867-3"><bdi>978-0-12-384867-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.4.1+Geostrophic+Approximation+and+Geostrophic+Wind&rft.btitle=An+Introduction+to+Dynamic+Meteorology&rft.series=International+Geophysics&rft.pages=42-43&rft.edition=5th&rft.pub=Academic+Press&rft.date=2012&rft.isbn=978-0-12-384867-3&rft.aulast=Holton&rft.aufirst=James+R.&rft.au=Hakim%2C+Gregory+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhcxcqQp7XOsC%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeostrophic+wind" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geostrophic_wind&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://atmos.nmsu.edu/education_and_outreach/encyclopedia/geostrophic.htm">Geostrophic approximation</a></li> <li><a rel="nofollow" class="external text" href="http://nsidc.org/cryosphere/glossary/term/geostrophic-wind">Definition of geostrophic wind</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110726123313/http://atmo.tamu.edu/class/atmo203/tut/windpres/wind8.html">Geostrophic wind description</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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Geostrophic wind - Wikipedia