Onda di Rossby - Wikipedia

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<a class="vector-toc-link" href="#Trattazione_semplificata"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Trattazione semplificata</span> </div> </a> <button aria-controls="toc-Trattazione_semplificata-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>Attiva/disattiva la sottosezione Trattazione semplificata</span> </button> <ul id="toc-Trattazione_semplificata-sublist" class="vector-toc-list"> <li id="toc-Effetto_beta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effetto_beta"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Effetto beta</span> </div> </a> <ul id="toc-Effetto_beta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modello_semplificato" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modello_semplificato"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Modello semplificato</span> </div> </a> <ul id="toc-Modello_semplificato-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relazione_di_dispersione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relazione_di_dispersione"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relazione di dispersione</span> </div> </a> <button aria-controls="toc-Relazione_di_dispersione-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>Attiva/disattiva la sottosezione Relazione di dispersione</span> </button> <ul id="toc-Relazione_di_dispersione-sublist" class="vector-toc-list"> <li id="toc-Onde_corte" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Onde_corte"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Onde corte</span> </div> </a> <ul id="toc-Onde_corte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Onde_lunghe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Onde_lunghe"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Onde lunghe</span> </div> </a> <ul id="toc-Onde_lunghe-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Onda di Rossby</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="Vai a una voce in un'altra lingua. Disponibile in 25 lingue" > <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-25" 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">25 lingue</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%A3%D9%85%D9%88%D8%A7%D8%AC_%D8%B1%D9%88%D8%B3%D8%A8%D9%8A" title="أمواج روسبي - arabo" lang="ar" hreflang="ar" data-title="أمواج روسبي" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rossbyho_vlna" title="Rossbyho vlna - ceco" lang="cs" hreflang="cs" data-title="Rossbyho vlna" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rossby-Welle" title="Rossby-Welle - tedesco" lang="de" hreflang="de" data-title="Rossby-Welle" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Rossby_wave" title="Rossby wave - inglese" lang="en" hreflang="en" data-title="Rossby wave" data-language-autonym="English" data-language-local-name="inglese" 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/Onda_de_Rossby" title="Onda de Rossby - spagnolo" lang="es" hreflang="es" data-title="Onda de Rossby" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Rossby_lained" title="Rossby lained - estone" lang="et" hreflang="et" data-title="Rossby lained" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC_%D8%B1%D8%A7%D8%B3%D8%A8%DB%8C" title="موج راسبی - persiano" lang="fa" hreflang="fa" data-title="موج راسبی" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rossby-aalto" title="Rossby-aalto - finlandese" lang="fi" hreflang="fi" data-title="Rossby-aalto" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Onde_de_Rossby" title="Onde de Rossby - francese" lang="fr" hreflang="fr" data-title="Onde de Rossby" data-language-autonym="Français" data-language-local-name="francese" 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/Tonnta_Rossby" title="Tonnta Rossby - irlandese" lang="ga" hreflang="ga" data-title="Tonnta Rossby" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%9C_%D7%A8%D7%95%D7%A1%D7%91%D7%99" title="גל רוסבי - ebraico" lang="he" hreflang="he" data-title="גל רוסבי" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gelombang_Rossby" title="Gelombang Rossby - indonesiano" lang="id" hreflang="id" data-title="Gelombang Rossby" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AD%E3%82%B9%E3%83%93%E3%83%BC%E6%B3%A2" title="ロスビー波 - giapponese" lang="ja" hreflang="ja" data-title="ロスビー波" data-language-autonym="日本語" data-language-local-name="giapponese" 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/%EB%A1%9C%EC%8A%A4%EB%B9%84_%ED%8C%8C%EB%8F%99" title="로스비 파동 - coreano" lang="ko" hreflang="ko" data-title="로스비 파동" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Ond_del_Rossby" title="Ond del Rossby - lombardo" lang="lmo" hreflang="lmo" data-title="Ond del Rossby" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rosbio_bangos" title="Rosbio bangos - lituano" lang="lt" hreflang="lt" data-title="Rosbio bangos" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rossby-golf" title="Rossby-golf - olandese" lang="nl" hreflang="nl" data-title="Rossby-golf" data-language-autonym="Nederlands" data-language-local-name="olandese" 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/Rossbyb%C3%B8lgje" title="Rossbybølgje - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Rossbybølgje" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rossbyb%C3%B8lge" title="Rossbybølge - norvegese bokmål" lang="nb" hreflang="nb" data-title="Rossbybølge" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Fale_Rossby%E2%80%99ego" title="Fale Rossby’ego - polacco" lang="pl" hreflang="pl" data-title="Fale Rossby’ego" data-language-autonym="Polski" data-language-local-name="polacco" 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/Onda_de_Rossby" title="Onda de Rossby - portoghese" lang="pt" hreflang="pt" data-title="Onda de Rossby" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BE%D0%BB%D0%BD%D1%8B_%D0%A0%D0%BE%D1%81%D1%81%D0%B1%D0%B8" title="Волны Россби - russo" lang="ru" hreflang="ru" data-title="Волны Россби" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rossbyv%C3%A5g" title="Rossbyvåg - svedese" lang="sv" hreflang="sv" data-title="Rossbyvåg" data-language-autonym="Svenska" data-language-local-name="svedese" 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%A5%D0%B2%D0%B8%D0%BB%D1%96_%D0%A0%D0%BE%D1%81%D1%81%D0%B1%D1%96" title="Хвилі Россбі - ucraino" lang="uk" hreflang="uk" data-title="Хвилі Россбі" data-language-autonym="Українська" data-language-local-name="ucraino" 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/%E7%BD%97%E6%96%AF%E8%B4%9D%E6%B3%A2" title="罗斯贝波 - cinese" lang="zh" hreflang="zh" data-title="罗斯贝波" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1053589#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Onda_di_Rossby" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" 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indichino puntualmente la provenienza delle informazioni. Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Onda_di_Rossby&action=edit">migliorare questa voce</a> <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">citando le fonti</a> più precisamente. Segui i suggerimenti del <a href="/wiki/Progetto:Fisica" title="Progetto:Fisica">progetto di riferimento</a>.</div> </div> </div> </div> </div><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:N_Jetstream_Rossby_Waves_N.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/N_Jetstream_Rossby_Waves_N.gif/440px-N_Jetstream_Rossby_Waves_N.gif" decoding="async" width="440" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/N_Jetstream_Rossby_Waves_N.gif/660px-N_Jetstream_Rossby_Waves_N.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/N_Jetstream_Rossby_Waves_N.gif/880px-N_Jetstream_Rossby_Waves_N.gif 2x" data-file-width="1042" data-file-height="253" /></a><figcaption>Meandri della <a href="/wiki/Corrente_a_getto" title="Corrente a getto">corrente a getto</a> nell'emisfero Nord, il loro sviluppo (a, b), e infine il distacco di una "goccia" di aria fredda (c). Arancio: masse di aria più calde; rosa: <a href="/wiki/Corrente_a_getto" title="Corrente a getto">corrente a getto</a>.</figcaption></figure> <p>In <a href="/wiki/Fisica_dell%27atmosfera" title="Fisica dell'atmosfera">fisica dell'atmosfera</a> e <a href="/wiki/Oceanografia" title="Oceanografia">oceanografia</a> le <b>onde di Rossby</b>, dette anche <b>onde planetarie</b>, sono strutture che caratterizzano i moti dei <a href="/wiki/Fluido" title="Fluido">fluidi</a> geofisici a <a href="/wiki/Scala_sinottica" class="mw-redirect" title="Scala sinottica">scala sinottica</a> e <a href="/wiki/Scala_planetaria" class="mw-redirect" title="Scala planetaria">planetaria</a>. Esse si possono osservare sia in <a href="/wiki/Atmosfera_terrestre" title="Atmosfera terrestre">atmosfera</a>, ad esempio nei <a href="/wiki/Meandri" class="mw-redirect" title="Meandri">meandri</a> compiuti dalla <a href="/wiki/Corrente_a_getto" title="Corrente a getto">corrente a getto</a> sub-polare alle medie latitudini, sia nell'<a href="/wiki/Oceano" title="Oceano">oceano</a>, nell'evoluzione delle perturbazioni del <a href="/wiki/Termoclino" title="Termoclino">termoclino</a> e assumono notevole importanza in <a href="/wiki/Meteorologia" title="Meteorologia">meteorologia</a> e <a href="/wiki/Climatologia" title="Climatologia">climatologia</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Descrizione">Descrizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=1" title="Modifica la sezione Descrizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=1" title="Edit section's source code: Descrizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_G._A._Rossby_LCCN2016875745_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg/220px-Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg" decoding="async" width="220" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg/330px-Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg/440px-Carl_G._A._Rossby_LCCN2016875745_%28cropped%29.jpg 2x" data-file-width="7469" data-file-height="8583" /></a><figcaption><a href="/wiki/Carl-Gustav_Arvid_Rossby" title="Carl-Gustav Arvid Rossby">Carl-Gustav Arvid Rossby</a></figcaption></figure> <p>Sono dovute all'aumento del <a href="/wiki/Parametro_di_Coriolis" class="mw-redirect" title="Parametro di Coriolis">parametro di Coriolis</a> con la <a href="/wiki/Latitudine" title="Latitudine">latitudine</a>, unitamente alla condizione di conservazione del <a href="/wiki/Momento_angolare" title="Momento angolare">momento angolare</a>, presentandosi come oscillazioni inerziali attorno al "<a href="/wiki/Punto_di_equilibrio" title="Punto di equilibrio">punto di equilibrio</a>" dato dal perfetto <a href="/wiki/Bilancio_geostrofico" title="Bilancio geostrofico">bilancio geostrofico</a>, situazione in cui le forze di <a href="/wiki/Pressione" title="Pressione">pressione</a> sono esattamente bilanciate dalla <a href="/wiki/Forza_di_Coriolis" title="Forza di Coriolis">forza di Coriolis</a>. </p><p>Devono il loro nome al <a href="/wiki/Fisico" title="Fisico">fisico</a> svedese <a href="/wiki/Carl-Gustav_Arvid_Rossby" title="Carl-Gustav Arvid Rossby">Carl-Gustav Arvid Rossby</a>, che per primo le identificò nell'atmosfera e ne diede la spiegazione nel <a href="/wiki/1939" title="1939">1939</a>. Dal punto vista meteorologico all'onda di Rossby e ai suoi meandri si associano, attraverso divergenza e convergenza d'aria, rispettivamente zone anticicloniche o di <a href="/wiki/Alta_pressione" class="mw-redirect" title="Alta pressione">alta pressione</a> e zone cicloniche o di <a href="/wiki/Bassa_pressione" class="mw-redirect" title="Bassa pressione">bassa pressione</a> (come ad es. <a href="/wiki/Saccatura" title="Saccatura">saccature</a> fino all'eventuale <i><a href="/wiki/Cut_off_(meteorologia)" title="Cut off (meteorologia)">cut off</a></i> sotto forma di gocce fredde) determinando <a href="/wiki/Avvezione" title="Avvezione">avvezioni</a> di <a href="/wiki/Massa_d%27aria" title="Massa d'aria">masse d'aria</a> con caratteristiche fisiche diverse. Tali onde, ed i fenomeni atmosferici a esse associati, rappresentano dunque uno dei mezzi fondamentali attraverso i quali la <a href="/wiki/Circolazione_atmosferica" title="Circolazione atmosferica">circolazione atmosferica</a> terrestre scambia <a href="/wiki/Calore" title="Calore">calore</a> dall'<a href="/wiki/Equatore" title="Equatore">equatore</a> ai <a href="/wiki/Regioni_polari" title="Regioni polari">poli</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Trattazione_semplificata">Trattazione semplificata</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=2" title="Modifica la sezione Trattazione semplificata" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=2" title="Edit section's source code: Trattazione semplificata"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le onde di Rossby sono originate dal fatto che i <a href="/wiki/Venti_occidentali" title="Venti occidentali">flussi zonali occidentali</a>, cioè i flussi diretti da Ovest a Est, possono oscillare in direzione meridionale (Nord-Sud) intorno alla posizione di equilibrio, per effetto della conservazione della <a href="/wiki/Vorticit%C3%A0_potenziale" title="Vorticità potenziale">vorticità potenziale</a>, dato che la <a href="/wiki/Vorticit%C3%A0_planetaria" title="Vorticità planetaria">vorticità planetaria</a> aumenta con la latitudine. Infatti se un flusso occidentale ha una traiettoria curva che lo porta verso Nord, ha inizialmente <a href="/wiki/Vorticit%C3%A0_relativa" class="mw-redirect" title="Vorticità relativa">vorticità relativa</a> positiva. Spostandosi verso Nord la <a href="/wiki/Vorticit%C3%A0_planetaria" title="Vorticità planetaria">vorticità planetaria</a> aumenta, e conseguentemente diminuisce la vorticità relativa. Quando la vorticità relativa raggiunge un valore negativo il flusso curva verso Sud. Nel movimento verso Sud si verifica un meccanismo analogo, a causa del quale la vorticità relativa torna a crescere e il flusso torna a rivolgersi verso Nord. Insomma, il flusso mediamente è sempre rivolto verso Est, e compie delle oscillazioni intorno a questa direzione<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>. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Potential_vorticity_westerly_flux.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c4/Potential_vorticity_westerly_flux.png" decoding="async" width="419" height="101" class="mw-file-element" data-file-width="419" data-file-height="101" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Effetto_beta">Effetto beta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=3" title="Modifica la sezione Effetto beta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=3" title="Edit section's source code: Effetto beta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per semplificare la trattazione delle onde di Rossby si linearizza la variazione del <a href="/wiki/Parametro_di_Coriolis" class="mw-redirect" title="Parametro di Coriolis">parametro di Coriolis</a> intorno al punto di equilibrio: </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 f=f_{0}+\beta \ \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mtext> </mtext> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=f_{0}+\beta \ \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ee371f196c4d7cceb34281e30dbc971d6debd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.676ex;" alt="{\displaystyle f=f_{0}+\beta \ \delta y}"></span></dd></dl> <p>dove <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 \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc454b7f0e03259e9c5e8568fcef1c8e6b78230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.204ex; height:2.676ex;" alt="{\displaystyle \delta y}"></span> rappresenta lo scostamento meridionale dal punto di equilibrio <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 y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}"></span>, <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> è la derivata del <a href="/wiki/Parametro_di_Coriolis" class="mw-redirect" title="Parametro di Coriolis">parametro di Coriolis</a> nel punto di equilibrio. </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 \beta ={\frac {df}{dy}}(y_{0})=2{\frac {\Omega }{R}}\cos(y_{0}/R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Ω</mi> <mi>R</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {df}{dy}}(y_{0})=2{\frac {\Omega }{R}}\cos(y_{0}/R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1f96fb3de6f7f5f176c4bdc779de930ec4c17d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.052ex; height:5.843ex;" alt="{\displaystyle \beta ={\frac {df}{dy}}(y_{0})=2{\frac {\Omega }{R}}\cos(y_{0}/R)}"></span></dd></dl> <p>dove <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 mathvariant="normal">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/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> è la <a href="/wiki/Velocit%C3%A0_angolare" title="Velocità angolare">velocità angolare</a> della <a href="/wiki/Terra" title="Terra">Terra</a> su se stessa, R è il suo raggio. </p><p>In questa approssimazione, detta di piano beta<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>, la variazione del parametro di Coriolis con la latitudine viene chiamato effetto beta <i>(beta-effect)</i><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>. </p> <div class="mw-heading mw-heading3"><h3 id="Modello_semplificato">Modello semplificato</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=4" title="Modifica la sezione Modello semplificato" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=4" title="Edit section's source code: Modello semplificato"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se si considera l'atmosfera come un sottile strato uniforme di spessore costante, la conservazione della <a href="/wiki/Vorticit%C3%A0_potenziale" title="Vorticità potenziale">vorticità potenziale</a> implica la conservazione della vorticità. Quindi, se il fluido ha vorticità relativa nulla nel punto di latitudine y, si ha: </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 \zeta (y+\delta y)=f(y)-f(y+\delta y)=-\beta \ \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>δ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>δ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mtext> </mtext> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (y+\delta y)=f(y)-f(y+\delta y)=-\beta \ \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b99cebf8fca3b53fa8d1ce3b33d27d84ed10fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.598ex; height:2.843ex;" alt="{\displaystyle \zeta (y+\delta y)=f(y)-f(y+\delta y)=-\beta \ \delta y}"></span></dd></dl> <p>Dove <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> è la <a href="/wiki/Vorticit%C3%A0_relativa" class="mw-redirect" title="Vorticità relativa">vorticità relativa</a>, f è la <a href="/wiki/Vorticit%C3%A0_planetaria" title="Vorticità planetaria">vorticità planetaria</a>. </p><p>Se consideriamo costante la componente zonale u della velocità, la vorticità è determinata dalla sola componente meridionale v: </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 \zeta (y)={\frac {\partial v}{\partial x}}={\frac {\partial (D\delta y/Dt)}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </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> <mo stretchy="false">(</mo> <mi>D</mi> <mi>δ</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (y)={\frac {\partial v}{\partial x}}={\frac {\partial (D\delta y/Dt)}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21571328b6ba547c4a92a05d596ad803acf00e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.759ex; height:5.843ex;" alt="{\displaystyle \zeta (y)={\frac {\partial v}{\partial x}}={\frac {\partial (D\delta y/Dt)}{\partial x}}}"></span></dd></dl> <p>Eguagliando le due espressioni sopra si ottiene: </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 {\partial (D\delta y/Dt)}{\partial x}}=-\beta \ \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mi>δ</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mtext> </mtext> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (D\delta y/Dt)}{\partial x}}=-\beta \ \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d3dac0e892a4dd235c37bb2c4e71615aa282d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.042ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (D\delta y/Dt)}{\partial x}}=-\beta \ \delta y}"></span></dd></dl> <p>Se sostituiamo 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 \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc454b7f0e03259e9c5e8568fcef1c8e6b78230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.204ex; height:2.676ex;" alt="{\displaystyle \delta y}"></span> un'espressione "ondosa" come </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 \delta y=a\operatorname {sen}(k(x-ct))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta y=a\operatorname {sen}(k(x-ct))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcea25610f3e810b5683548224414a56bd7b1231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.007ex; height:2.843ex;" alt="{\displaystyle \delta y=a\operatorname {sen}(k(x-ct))}"></span></dd></dl> <p>si ottiene </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 c=-\beta /k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-\beta /k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3090c3a31efda24e07cbb2061f509a5523128cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.673ex; height:3.176ex;" alt="{\displaystyle c=-\beta /k^{2}}"></span></dd></dl> <p>che è una versione semplificata della <a href="/wiki/Relazione_di_dispersione" title="Relazione di dispersione">relazione di dispersione</a> delle onde di Rossby in atmosfera alle <a href="/wiki/Medie_latitudini" class="mw-redirect" title="Medie latitudini">medie latitudini</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>. Si noti che la <a href="/wiki/Velocit%C3%A0_di_fase" title="Velocità di fase">velocità di fase</a> dell'onda così ottenuta è sempre negativa, cioè rivolta verso Ovest. Quindi l'onda si propaga verso Ovest <i>rispetto al flusso principale</i>, che è diretto verso Est. </p> <div class="mw-heading mw-heading2"><h2 id="Relazione_di_dispersione">Relazione di dispersione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=5" title="Modifica la sezione Relazione di dispersione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=5" title="Edit section's source code: Relazione di dispersione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dato che le onde di Rossby si osservano nei moti su larga scala, la relazione di dispersione può essere dedotta con ragionevole precisione utilizzando l'<a href="/wiki/Approssimazione_shallow_homogeneous_layer" class="mw-redirect" title="Approssimazione shallow homogeneous layer">approssimazione <i>shallow homogeneous layer</i></a>, considerando cioè l'atmosfera (o l'oceano) come uno strato di fluido sottile e omogeneo, in cui la componente verticale del moto è trascurabile<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>. Si parte dall'<a href="/wiki/Equazione_della_vorticit%C3%A0" title="Equazione della vorticità">equazione della vorticità</a> in questa approssimazione, che alle medie latitudini, dove <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>>\zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>>></mo> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f>>\zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a445c8abb7370b84a630c7d5ed378c11c2421c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.28ex; height:2.509ex;" alt="{\displaystyle f>>\zeta }"></span>, è data da: </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 {Df+\zeta }{Dt}}+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>f</mi> <mo>+</mo> <mi>ζ</mi> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </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> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Df+\zeta }{Dt}}+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c8f1b2166ebbdf15725a9d0c51cb1f5c9e5de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.797ex; height:6.176ex;" alt="{\displaystyle {\frac {Df+\zeta }{Dt}}+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}"></span></dd></dl> <p>dove f è la <a href="/wiki/Vorticit%C3%A0_planetaria" title="Vorticità planetaria">vorticità planetaria</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 \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> è la <a href="/wiki/Vorticit%C3%A0_relativa" class="mw-redirect" title="Vorticità relativa">vorticità relativa</a>. </p><p>Sostituendo in questa equazione l'espressione discussa precedentemente per l'effetto beta <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=f_{0}+\beta \ \delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mtext> </mtext> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=f_{0}+\beta \ \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ee371f196c4d7cceb34281e30dbc971d6debd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.676ex;" alt="{\displaystyle f=f_{0}+\beta \ \delta y}"></span>, rimaneggiando e trascurando i termini di ordine superiore a uno si ottiene: </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 {\partial \zeta }{\partial t}}+\beta v+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ζ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>β</mi> <mi>v</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </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> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \zeta }{\partial t}}+\beta v+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de6a67677fcd26c45bea56064232ec35b3c70ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.371ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \zeta }{\partial t}}+\beta v+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}"></span></dd></dl> <p>Dato che il fluido viene considerato omogeneo e incomprimibile l'<a href="/wiki/Equazione_di_continuit%C3%A0" title="Equazione di continuità">equazione di continuità</a> della massa è: </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 {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </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> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</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 {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0843cb698528bbaf6e3eeaf08246df3edabab447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.554ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}"></span></dd></dl> <p>dove (u, v, w) sono le tre componenti della velocità. Integrando questa equazione sulla profondità H si ottiene: </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 {\partial Hu}{\partial x}}+{\frac {\partial Hv}{\partial y}}=-\int _{z=0}{H}{\frac {\partial w}{\partial z}}dz=-w(H)=-{\frac {\partial \eta }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> <mi>u</mi> </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> <mi>H</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial Hu}{\partial x}}+{\frac {\partial Hv}{\partial y}}=-\int _{z=0}{H}{\frac {\partial w}{\partial z}}dz=-w(H)=-{\frac {\partial \eta }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c44474171e095efd48ea3fea0491b9bbf614a52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.669ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial Hu}{\partial x}}+{\frac {\partial Hv}{\partial y}}=-\int _{z=0}{H}{\frac {\partial w}{\partial z}}dz=-w(H)=-{\frac {\partial \eta }{\partial t}}}"></span></dd></dl> <p>dove si sono considerate le componenti orizzontali della velocità indipendenti dall'altezza, e dove <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> rappresenta l'innalzamento della colonna di fluido rispetto alla profondità media H: si ipotizza che valga l'approssimazione <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 <<H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo><<</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta <<H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8408db07cce81844487be92c1592c4bbc94ff317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.14ex; height:2.676ex;" alt="{\displaystyle \eta <<H}"></span>. </p><p>Riarrangiando, risulta: </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 {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=-{\frac {1}{H}}{\frac {\partial \eta }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </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> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>H</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=-{\frac {1}{H}}{\frac {\partial \eta }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5778b4e075c21070966da036a438242eb537f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.764ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=-{\frac {1}{H}}{\frac {\partial \eta }{\partial t}}}"></span></dd></dl> <p>Sostituendo questa espressione nell'equazione della vorticità si ottiene: </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 {\partial }{\partial t}}\left(\zeta -f{\frac {\eta }{H}}\right)+\beta v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ζ</mi> <mo>−</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mi>H</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>β</mi> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\left(\zeta -f{\frac {\eta }{H}}\right)+\beta v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fe09fe80a45c32c2d4ebcd7e881f10c4b49d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.831ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\left(\zeta -f{\frac {\eta }{H}}\right)+\beta v=0}"></span></dd></dl> <p>Questa relazione è chiamata equazione della vorticità potenziale ed è una relazione d'importanza centrale nella descrizione dei moti quasi-geostrofici<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>. </p><p>Ipotizzando che localmente i moti siano approssimativamente <a href="/wiki/Bilancio_geostrofico" title="Bilancio geostrofico">geostrofici</a> valgono le seguenti relazioni: </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 fu=g{\frac {\partial \eta }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fu=g{\frac {\partial \eta }{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c77810d919273e330d4292246f58721f2e5799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.307ex; height:5.676ex;" alt="{\displaystyle fu=g{\frac {\partial \eta }{\partial x}}}"></span></dd></dl> <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=-g{\frac {\partial \eta }{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>v</mi> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fv=-g{\frac {\partial \eta }{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3cc9607b8002d0622ab04dec9d25eb7bc0f23f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.752ex; height:6.176ex;" alt="{\displaystyle fv=-g{\frac {\partial \eta }{\partial y}}}"></span></dd></dl> <p>Sostituendo queste espressioni nell'equazione sopra, e ricordando che per moti quasi piani la vorticità ha la sola componente verticale non nulla, che è data da <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 =\partial v/\partial x\ -\ \partial u/\partial y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =\partial v/\partial x\ -\ \partial u/\partial y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd352e512d46e0bbd7b816c95eb2d36e61b61c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.735ex; height:2.843ex;" alt="{\displaystyle \zeta =\partial v/\partial x\ -\ \partial u/\partial y}"></span> si ottiene: </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 {\partial }{\partial t}}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}-{\frac {f^{2}}{gH}}\eta \right)+\beta {\frac {\partial \eta }{\partial x}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>g</mi> <mi>H</mi> </mrow> </mfrac> </mrow> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}-{\frac {f^{2}}{gH}}\eta \right)+\beta {\frac {\partial \eta }{\partial x}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf2826026137983b57d2940b6baaa18342af743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.526ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}-{\frac {f^{2}}{gH}}\eta \right)+\beta {\frac {\partial \eta }{\partial x}}=0}"></span></dd></dl> <p>Imponendo 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 \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> la seguente forma: </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 \eta =\eta _{0}\operatorname {sen}(kx+ly-\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mi>l</mi> <mi>y</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\eta _{0}\operatorname {sen}(kx+ly-\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50dc72d811f74e22662cc01a04c3cf457045ff5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.271ex; height:2.843ex;" alt="{\displaystyle \eta =\eta _{0}\operatorname {sen}(kx+ly-\omega t)}"></span></dd></dl> <p>si ottiene la relazione di dispersione delle onde di Rossby: </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 \omega ={\frac {-\beta k}{k^{2}+l^{2}+f^{2}/gH}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>β</mi> <mi>k</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> <mi>H</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}+f^{2}/gH}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901a5a988b783d4d31a878743e7271be3aa4163a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.791ex; height:6.176ex;" alt="{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}+f^{2}/gH}}}"></span></dd></dl> <p>Infine, sostituendo l'espressione per il <a href="/wiki/Raggio_di_Rossby" class="mw-redirect" title="Raggio di Rossby">raggio di Rossby</a>, dato da <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 a={\sqrt {gH}}/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>g</mi> <mi>H</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {gH}}/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473509295213f489bf783ff693af7626bf7e1630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.273ex; height:3.509ex;" alt="{\displaystyle a={\sqrt {gH}}/f}"></span> si ottiene: </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 \omega ={\frac {-\beta k}{k^{2}+l^{2}+1/a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>β</mi> <mi>k</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}+1/a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4285e0fbe51505011c6c60591ced43ae751aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.683ex; height:6.176ex;" alt="{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}+1/a^{2}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Onde_corte">Onde corte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=6" title="Modifica la sezione Onde corte" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=6" title="Edit section's source code: Onde corte"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se la <a href="/wiki/Lunghezza_d%27onda" title="Lunghezza d'onda">lunghezza d'onda</a> è piccola rispetto al raggio di Rossby si parla di onde di Rossby corte. In questo caso il termine <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 1/a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d9f42ce02ca87684dc2baa62a7db7eea0198ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.609ex; height:3.176ex;" alt="{\displaystyle 1/a^{2}}"></span> è piccolo, e la relazione di dispersione diventa: </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 \omega ={\frac {-\beta k}{k^{2}+l^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>β</mi> <mi>k</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8824a42d3931f5b207603cf4584d98f0535e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.234ex; height:5.843ex;" alt="{\displaystyle \omega ={\frac {-\beta k}{k^{2}+l^{2}}}}"></span></dd></dl> <p>Si noti che questa espressione è molto simile a quella ottenuta nel paragrafo sul modello semplificato. Essa infatti può essere ricavata facendo un ragionamento analogo ma considerando anche le oscillazioni della componente zonale della velocità, dunque la componente <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> del <a href="/wiki/Vettore_d%27onda" title="Vettore d'onda">vettore d'onda</a>. Confrontando l'equazione della vorticità usata nel modello semplificato con l'equazione della vorticità potenziale ricavata nella derivazione della relazione di dispersione, si nota che la differenza tra le due è data dalla mancanza nella prima del termine <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\eta /H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\eta /H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650a4503837042c1e03aad3ea1908250249f74ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle f\eta /H}"></span>. Quindi si può comprendere che nelle onde di Rossby corte l'innalzamento <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> della colonna di fluido è trascurabile, l'<a href="/wiki/Energia_potenziale_gravitazionale" title="Energia potenziale gravitazionale">energia potenziale gravitazionale</a> associata ai moti è piccola rispetto all'<a href="/wiki/Energia_cinetica" title="Energia cinetica">energia cinetica</a>, e l'onda è determinata dal solo <i>effetto beta</i> che si esercita sul fluido in movimento<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>. Le onde di Rossby corte si osservano in atmosfera, dove il raggio di Rossby è nell'ordine dei 1000 km<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>, e nei pressi dell'<a href="/wiki/Equatore" title="Equatore">equatore</a> dove il raggio di Rossby tende a infinito. </p><p>La <a href="/wiki/Velocit%C3%A0_di_fase" title="Velocità di fase">velocità di fase</a> delle <a href="/wiki/Onde_corte" title="Onde corte">onde corte</a> è data da: </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 c={\frac {\omega }{k}}=-{\frac {\beta }{k^{2}+l^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {\omega }{k}}=-{\frac {\beta }{k^{2}+l^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eb564acd36aada2410947e0e2bb220765bc963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.983ex; height:5.843ex;" alt="{\displaystyle c={\frac {\omega }{k}}=-{\frac {\beta }{k^{2}+l^{2}}}}"></span></dd></dl> <p>è sempre negativa, cioè rivolta verso Ovest. </p><p>La <a href="/wiki/Velocit%C3%A0_di_gruppo" title="Velocità di gruppo">velocità di gruppo</a> è data da: </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 {\partial \omega }{\partial k}}=\beta {\frac {k^{2}-l^{2}}{(k^{2}+l^{2})^{2}}}\sim {\frac {\beta }{k^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \omega }{\partial k}}=\beta {\frac {k^{2}-l^{2}}{(k^{2}+l^{2})^{2}}}\sim {\frac {\beta }{k^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c76bd5d217669646bc139985b02635dac8357a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.784ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial \omega }{\partial k}}=\beta {\frac {k^{2}-l^{2}}{(k^{2}+l^{2})^{2}}}\sim {\frac {\beta }{k^{2}}}}"></span></dd></dl> <p>dato che <i>di solito</i> l << k. Quindi le onde di Rossby corte <i>di solito</i> si propagano verso Est rispetto al flusso principale. Come regola generale le onde atmosferiche di lunghezza inferiore ai 3000 km si muovono verso Est rispetto alla superficie terrestre.<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><p>L'analisi che Rossby effettuò nel 1939 riguarda le sole onde corte. </p> <div class="mw-heading mw-heading3"><h3 id="Onde_lunghe">Onde lunghe</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=7" title="Modifica la sezione Onde lunghe" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=7" title="Edit section's source code: Onde lunghe"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se la <a href="/wiki/Lunghezza_d%27onda" title="Lunghezza d'onda">lunghezza d'onda</a> è grande rispetto al raggio di Rossby si parla di onde di Rossby lunghe. In questo caso la relazione di dispersione diventa: </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 \omega =-\beta ka^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mi>k</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =-\beta ka^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8e8c5b8e925ecb66cb584ff0352e5176026695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.18ex; height:3.009ex;" alt="{\displaystyle \omega =-\beta ka^{2}}"></span></dd></dl> <p>Le onde Rossby lunghe sono <a href="/w/index.php?title=Onde_non_dispersive&action=edit&redlink=1" class="new" title="Onde non dispersive (la pagina non esiste)">onde non dispersive</a>, cioè la <a href="/wiki/Velocit%C3%A0_di_fase" title="Velocità di fase">velocità di fase</a> è indipendente dalla <a href="/wiki/Lunghezza_d%27onda" title="Lunghezza d'onda">lunghezza d'onda</a> ed equivale alla <a href="/wiki/Velocit%C3%A0_di_gruppo" title="Velocità di gruppo">velocità di gruppo</a>. Essa è data da: </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 c={\frac {\omega }{k}}=-\beta a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>β</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {\omega }{k}}=-\beta a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350dcbb082bfba6f4cf3c70f74d4c1743a5c475f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.91ex; height:4.843ex;" alt="{\displaystyle c={\frac {\omega }{k}}=-\beta a^{2}}"></span></dd></dl> <p>È sempre negativa, quindi le onde di Rossby lunghe si propagano verso Ovest. </p><p>Per le onde di Rossby lunghe il termine <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\eta /H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\eta /H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650a4503837042c1e03aad3ea1908250249f74ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle f\eta /H}"></span>, nell'<i>equazione della vorticità potenziale</i>, è più importante della <a href="/wiki/Vorticit%C3%A0_relativa" class="mw-redirect" title="Vorticità relativa">vorticità relativa</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 \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> e i moti sono approssimativamente in <a href="/wiki/Bilancio_geostrofico" title="Bilancio geostrofico">bilancio geostrofico</a>: si parla infatti di moti quasi geostrofici. Le onde di Rossby lunghe si possono osservare nell'oceano, nell'evoluzione delle perturbazioni del <a href="/wiki/Termoclino" title="Termoclino">termoclino</a>, dato che il raggio di Rossby per questi moti è di soli 10-30 km<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>. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=8" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=8" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFHolton_(2004)">Holton (2004)</a>, p. 98</cite>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFGill_(1982)">Gill (1982)</a>, p. 495</cite>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFHolton_(2004)">Holton (2004)</a>, p. 214</cite>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFHolton_(2004)">Holton (2004)</a>, p. 215</cite>.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFGill_(1982)">Gill (1982)</a>, p. 444</cite>.</span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFGill_(1982)">Gill (1982)</a>, p. 447</cite>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFGill_(1982)">Gill (1982)</a>, p. 503</cite>.</span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFHolton_(2004)">Holton (2004)</a>, p. 259</cite>.</span> </li> <li id="cite_note-9"><a href="#cite_ref-9"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFHolton_(2004)">Holton (2004)</a>, p. 154</cite>.</span> </li> <li id="cite_note-10"><a href="#cite_ref-10"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFGill_(1982)">Gill (1982)</a>, p. 207</cite>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=9" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=9" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFHolton_(2004)" class="citation libro" style="font-style:normal"> James R. Holton, <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/54400282"><span style="font-style:italic;">An introduction to dynamic meteorology</span></a>, Fourth edition, Elsevier Academic Press, 2004, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-12-354015-1" title="Speciale:RicercaISBN/0-12-354015-1">0-12-354015-1</a>, <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/54400282">54400282</a>.</cite></li> <li><cite id="CITEREFGill_(1982)" class="citation libro" style="font-style:normal"> Adrian E. Gill, <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/277664880"><span style="font-style:italic;">Atmosphere-ocean dynamics</span></a>, Academic Press, 1982, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-08-057052-5" title="Speciale:RicercaISBN/978-0-08-057052-5">978-0-08-057052-5</a>, <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/277664880">277664880</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=10" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=10" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Meteorologia_dinamica" title="Meteorologia dinamica">Meteorologia dinamica</a></li> <li><a href="/wiki/Geofluidodinamica" title="Geofluidodinamica">Geofluidodinamica</a></li> <li><a href="/wiki/Vortice_polare" title="Vortice polare">Vortice polare</a></li> <li><a href="/wiki/Corrente_a_getto" title="Corrente a getto">Corrente a getto</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=11" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=11" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Rossby waves"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Rossby_waves?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Rossby_waves?uselang=it">Onda di Rossby</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onda_di_Rossby&veaction=edit&section=12" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onda_di_Rossby&action=edit&section=12" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/science/Rossby-wave"><span style="font-style:italic;">Rossby wave</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q1053589#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85115467">sh85115467</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4216707-3">4216707-3</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007546163105171">987007546163105171</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; 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