Atmospheric tide - Wikipedia

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href="#Solar_atmospheric_tides"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Solar atmospheric tides</span> </div> </a> <button aria-controls="toc-Solar_atmospheric_tides-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Solar atmospheric tides subsection</span> </button> <ul id="toc-Solar_atmospheric_tides-sublist" class="vector-toc-list"> <li id="toc-Migrating_solar_tides" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Migrating_solar_tides"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Migrating solar tides</span> </div> </a> <ul id="toc-Migrating_solar_tides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-migrating_solar_tides" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-migrating_solar_tides"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Non-migrating solar tides</span> </div> </a> <ul id="toc-Non-migrating_solar_tides-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lunar_atmospheric_tides" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lunar_atmospheric_tides"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lunar atmospheric tides</span> </div> </a> <ul id="toc-Lunar_atmospheric_tides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_tidal_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classical_tidal_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Classical tidal theory</span> </div> </a> <button aria-controls="toc-Classical_tidal_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classical tidal theory subsection</span> </button> <ul id="toc-Classical_tidal_theory-sublist" class="vector-toc-list"> <li id="toc-Basic_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Basic equations</span> </div> </a> <ul id="toc-Basic_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separation_of_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separation_of_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Separation of variables</span> </div> </a> <ul id="toc-Separation_of_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laplace's_tidal_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace's_tidal_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Laplace's tidal equation</span> </div> </a> <ul id="toc-Laplace's_tidal_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_solution_of_Laplace's_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_solution_of_Laplace's_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>General solution of Laplace's equation</span> </div> </a> <ul id="toc-General_solution_of_Laplace's_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vertical_structure_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vertical_structure_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Vertical structure equation</span> </div> </a> <ul id="toc-Vertical_structure_equation-sublist" 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tide</span> </div> </a> <ul id="toc-Effects_of_atmospheric_tide-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes and references</span> </div> </a> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Atmospheric tide</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="Go to an article in another language. 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href="https://no.wikipedia.org/wiki/Atmosf%C3%A6risk_tidevann" title="Atmosfærisk tidevann – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Atmosfærisk tidevann" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Atmosf%C3%A6risk_tidvatn" title="Atmosfærisk tidvatn – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Atmosfærisk tidvatn" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A7%E6%B0%A3%E6%BD%AE" title="大氣潮 – Chinese" lang="zh" hreflang="zh" data-title="大氣潮" data-language-autonym="中文" data-language-local-name="Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Global-scale periodic oscillations of the atmosphere</div> <p><b>Atmospheric tides</b> are global-scale periodic oscillations of the <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a>. In many ways they are analogous to <a href="/wiki/Ocean_tides" class="mw-redirect" title="Ocean tides">ocean tides</a>. They can be excited by: </p> <ul><li>The regular <a href="/wiki/Day" title="Day">day</a>-<a href="/wiki/Night" title="Night">night</a> cycle in the Sun's heating of the atmosphere (<a href="/wiki/Insolation" class="mw-redirect" title="Insolation">insolation</a>)</li> <li>The <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> pull of the <a href="/wiki/Moon" title="Moon">Moon</a></li> <li><a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">Non-linear</a> interactions between tides and <a href="/wiki/Planetary_wave" class="mw-redirect" title="Planetary wave">planetary waves</a></li> <li>Large-scale <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a> release due to <a href="/wiki/Deep_convection" class="mw-redirect" title="Deep convection">deep convection</a> in the <a href="/wiki/Tropics" title="Tropics">tropics</a></li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="General_characteristics">General characteristics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=1" title="Edit section: General characteristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The largest-amplitude atmospheric tides are mostly generated in the <a href="/wiki/Troposphere" title="Troposphere">troposphere</a> and <a href="/wiki/Stratosphere" title="Stratosphere">stratosphere</a> when the <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a> is periodically heated, as <a href="/wiki/Water_vapor" title="Water vapor">water vapor</a> and <a href="/wiki/Ozone" title="Ozone">ozone</a> absorb <a href="/wiki/Solar_radiation" class="mw-redirect" title="Solar radiation">solar radiation</a> during the day. These tides propagate away from the source regions and ascend into the <a href="/wiki/Mesosphere" title="Mesosphere">mesosphere</a> and <a href="/wiki/Thermosphere" title="Thermosphere">thermosphere</a>. Atmospheric tides can be measured as regular fluctuations in <a href="/wiki/Wind" title="Wind">wind</a>, <a href="/wiki/Temperature" title="Temperature">temperature</a>, <a href="/wiki/Density" title="Density">density</a> and <a href="/wiki/Pressure" title="Pressure">pressure</a>. Although atmospheric tides share much in common with ocean tides they have two key distinguishing features: </p> <ol><li>Atmospheric tides are primarily excited by the <a href="/wiki/Sun" title="Sun">Sun</a>'s heating of the atmosphere whereas ocean tides are excited by the <a href="/wiki/Moon" title="Moon">Moon</a>'s gravitational pull and to a lesser extent by the Sun's <a href="/wiki/Gravity" title="Gravity">gravity</a>. This means that most atmospheric tides have periods of <a href="/wiki/Oscillation" title="Oscillation">oscillation</a> related to the 24-hour length of the <a href="/wiki/Solar_day" class="mw-redirect" title="Solar day">solar day</a> whereas ocean tides have periods of oscillation related both to the <a href="/wiki/Solar_day" class="mw-redirect" title="Solar day">solar day</a> as well as to the longer <a href="/wiki/Lunar_day#Alternate_usage" title="Lunar day">tidal lunar day</a> (time between successive lunar transits) of about 24 <a href="/wiki/Hour" title="Hour">hours</a> 51 <a href="/wiki/Minute" title="Minute">minutes</a>.</li> <li>Atmospheric tides propagate in an atmosphere where density varies significantly with <a href="/wiki/Height" title="Height">height</a>. A consequence of this is that their <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a> naturally <a href="/wiki/Increase_exponentially" class="mw-redirect" title="Increase exponentially">increase exponentially</a> as the tide ascends into progressively more rarefied regions of the atmosphere (for an explanation of this phenomenon, see below). In contrast, the <a href="/wiki/Density" title="Density">density</a> of the oceans varies only slightly with depth and so there the tides do not necessarily vary in amplitude with depth.</li></ol> <p>At ground level, atmospheric tides can be detected as regular but small oscillations in surface pressure with periods of 24 and 12 hours. However, at greater heights, the amplitudes of the tides can become very large. In the mesosphere (heights of about 50–100 km (30–60 mi; 200,000–300,000 ft)) atmospheric tides can reach amplitudes of more than 50 m/s and are often the most significant part of the motion of the atmosphere. </p><p>The reason for this dramatic growth in amplitude from tiny fluctuations near the ground to oscillations that dominate the motion of the mesosphere lies in the fact that the density of the atmosphere decreases with increasing height. As tides or waves propagate upwards, they move into regions of lower and lower density. If the tide or wave is not dissipating, then its <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> density must be conserved. Since the density is decreasing, the amplitude of the tide or wave increases correspondingly so that <a href="/wiki/Energy_is_conserved" class="mw-redirect" title="Energy is conserved">energy is conserved</a>. </p><p>Following this growth with height atmospheric tides have much larger amplitudes in the middle and upper atmosphere than they do at ground level. </p> <div class="mw-heading mw-heading2"><h2 id="Solar_atmospheric_tides">Solar atmospheric tides</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=2" title="Edit section: Solar atmospheric tides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The largest amplitude atmospheric tides are generated by the periodic heating of the atmosphere by the Sun – the atmosphere is heated during the day and not heated at night. This regular diurnal (daily) cycle in heating generates <b>thermal tides</b> that have periods related to the solar day. It might initially be expected that this diurnal heating would give rise to tides with a period of 24 hours, corresponding to the heating's periodicity. However, observations reveal that large amplitude tides are generated with periods of 24 and 12 hours. Tides have also been observed with periods of 8 and 6 hours, although these latter tides generally have smaller amplitudes. This set of periods occurs because the solar heating of the atmosphere occurs in an approximate <a href="/wiki/Square_wave" title="Square wave">square wave</a> profile and so is rich in harmonics. When this pattern is decomposed into separate frequency components using a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, as well as the mean and daily (24-hour) variation, significant oscillations with periods of 12, 8 and 6 hours are produced. Tides generated by the gravitational effect of the Sun are very much smaller than those generated by solar heating. Solar tides will refer to only thermal solar tides from this point. </p><p>Solar energy is absorbed throughout the atmosphere some of the most significant in this context are<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="ungrammaticality (March 2022)">clarification needed</span></a></i>]</sup> <a href="/wiki/Water_vapor" title="Water vapor">water vapor</a> at about 0–15 km in the <a href="/wiki/Troposphere" title="Troposphere">troposphere</a>, <a href="/wiki/Ozone" title="Ozone">ozone</a> at about 30–60 km in the <a href="/wiki/Stratosphere" title="Stratosphere">stratosphere</a> and molecular oxygen and molecular nitrogen at about 120–170 km) in the <a href="/wiki/Thermosphere" title="Thermosphere">thermosphere</a>. Variations in the global distribution and density of these species result in changes in the amplitude of the solar tides. The tides are also affected by the environment through which they travel. </p><p>Solar tides can be separated into two components: <b>migrating</b> and <b>non-migrating</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Migrating_solar_tides">Migrating solar tides</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=3" title="Edit section: Migrating solar tides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Timed100kmsabertidisep2005.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Timed100kmsabertidisep2005.gif/220px-Timed100kmsabertidisep2005.gif" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Timed100kmsabertidisep2005.gif/330px-Timed100kmsabertidisep2005.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Timed100kmsabertidisep2005.gif/440px-Timed100kmsabertidisep2005.gif 2x" data-file-width="1000" data-file-height="625" /></a><figcaption>Figure 1. Tidal temperature and wind perturbations at 100 km altitude for September 2005 as a function of universal time. The animation is based upon observations from the SABER and TIDI instruments on board the <a href="/wiki/TIMED" title="TIMED">TIMED</a> satellite. It shows the superposition of the most important diurnal and semidiurnal tidal components (migrating and nonmigrating).</figcaption></figure> <p>Migrating tides are Sun synchronous – from the point of view of a stationary observer on the ground they propagate westwards with the apparent motion of the Sun. As the migrating tides stay fixed relative to the Sun a pattern of excitation is formed that is also fixed relative to the Sun. Changes in the tide observed from a stationary viewpoint on the <a href="/wiki/Earth%27s_surface" class="mw-redirect" title="Earth's surface">Earth's surface</a> are caused by the rotation of the Earth with respect to this fixed pattern. Seasonal variations of the tides also occur as the Earth tilts relative to the Sun and so relative to the pattern of excitation.<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>The migrating solar tides have been extensively studied both through observations and mechanistic models.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-migrating_solar_tides">Non-migrating solar tides</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=4" title="Edit section: Non-migrating solar tides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Non-migrating tides can be thought of as global-scale waves with the same periods as the migrating tides. However, non-migrating tides do not follow the apparent motion of the Sun. Either they do not propagate horizontally, they propagate eastwards or they propagate westwards at a different speed to the Sun. These non-migrating tides may be generated by differences in <a href="/wiki/Topography" title="Topography">topography</a> with longitude, land-sea contrast, and surface interactions. An important source is <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a> release due to <a href="/wiki/Deep_convection" class="mw-redirect" title="Deep convection">deep convection</a> in the <a href="/wiki/Tropics" title="Tropics">tropics</a>. </p><p>The primary source for the 24-hr tide is in the lower atmosphere where surface effects are important. This is reflected in a relatively large non-migrating component seen in longitudinal differences in tidal amplitudes. Largest amplitudes have been observed over <a href="/wiki/South_America" title="South America">South America</a>, <a href="/wiki/Africa" title="Africa">Africa</a> and <a href="/wiki/Australia" title="Australia">Australia</a>.<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-heading2"><h2 id="Lunar_atmospheric_tides">Lunar atmospheric tides</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=5" title="Edit section: Lunar atmospheric tides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atmospheric tides are also produced through the gravitational effects of the Moon.<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> Lunar (gravitational) tides are much weaker than solar thermal tides and are generated by the motion of the Earth's oceans (caused by the Moon) and to a lesser extent the effect of the Moon's gravitational attraction on the atmosphere. </p> <div class="mw-heading mw-heading2"><h2 id="Classical_tidal_theory">Classical tidal theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=6" title="Edit section: Classical tidal theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic characteristics of the atmospheric tides are described by the <i>classical tidal theory</i>.<sup id="cite_ref-ChapmanLindzen_5-0" class="reference"><a href="#cite_note-ChapmanLindzen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> By neglecting <a href="/w/index.php?title=Mechanical_forcing&action=edit&redlink=1" class="new" title="Mechanical forcing (page does not exist)">mechanical forcing</a> and <a href="/wiki/Dissipation" title="Dissipation">dissipation</a>, the classical tidal theory assumes that atmospheric wave motions can be considered as linear perturbations of an initially motionless zonal mean state that is horizontally <a href="/wiki/Atmospheric_stratification" class="mw-redirect" title="Atmospheric stratification">stratified</a> and <a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a>. The two major results of the classical theory are </p> <ul><li>atmospheric tides are <a href="/wiki/Eigenmodes" class="mw-redirect" title="Eigenmodes">eigenmodes</a> of the atmosphere described by <a href="/wiki/Hough_functions" class="mw-redirect" title="Hough functions">Hough functions</a></li> <li>amplitudes grow exponentially with height.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Basic_equations">Basic equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=7" title="Edit section: Basic equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Primitive_equations" title="Primitive equations">primitive equations</a> lead to the linearized equations for perturbations (primed variables) in a spherical isothermal atmosphere:<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> <div><ul><li>horizontal momentum equations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rrl}{\frac {\partial u'}{\partial t}}\,-\,2\Omega \sin \varphi \,v'\,+&{\frac {1}{a\,\cos \varphi }}\,{\frac {\partial \Phi '}{\partial \lambda }}&=0\\{\frac {\partial v'}{\partial t}}\,+\,2\Omega \sin \varphi \,u'\,+&{\frac {1}{a}}\,{\frac {\partial \Phi '}{\partial \varphi }}&=0\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <msup> <mi>v</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>+</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <msup> <mi>u</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>+</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rrl}{\frac {\partial u'}{\partial t}}\,-\,2\Omega \sin \varphi \,v'\,+&{\frac {1}{a\,\cos \varphi }}\,{\frac {\partial \Phi '}{\partial \lambda }}&=0\\{\frac {\partial v'}{\partial t}}\,+\,2\Omega \sin \varphi \,u'\,+&{\frac {1}{a}}\,{\frac {\partial \Phi '}{\partial \varphi }}&=0\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ece44bfa9a9179966beaec28d28a1f97391b83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.594ex; margin-bottom: -0.244ex; width:38.326ex; height:8.843ex;" alt="{\displaystyle {\begin{array}{rrl}{\frac {\partial u'}{\partial t}}\,-\,2\Omega \sin \varphi \,v'\,+&{\frac {1}{a\,\cos \varphi }}\,{\frac {\partial \Phi '}{\partial \lambda }}&=0\\{\frac {\partial v'}{\partial t}}\,+\,2\Omega \sin \varphi \,u'\,+&{\frac {1}{a}}\,{\frac {\partial \Phi '}{\partial \varphi }}&=0\end{array}}}"></span></li><li>energy equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}}{\partial t\partial z}}\Phi '\,+\,N^{2}w'={\frac {\kappa J'}{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <msup> <mi>J</mi> <mo>′</mo> </msup> </mrow> <mi>H</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}}{\partial t\partial z}}\Phi '\,+\,N^{2}w'={\frac {\kappa J'}{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec437bf55abec6f2629f802b81a39ce4f9676b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.388ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial ^{2}}{\partial t\partial z}}\Phi '\,+\,N^{2}w'={\frac {\kappa J'}{H}}}"></span></li><li>continuity equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a\,\cos \varphi }}\,\left({\frac {\partial u'}{\partial \lambda }}\,+\,{\frac {\partial }{\partial \varphi }}(v'\,\cos \varphi )\right)\,+\,{\frac {1}{\varrho _{o}}}\,{\frac {\partial }{\partial z}}(\varrho _{o}w')=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>v</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a\,\cos \varphi }}\,\left({\frac {\partial u'}{\partial \lambda }}\,+\,{\frac {\partial }{\partial \varphi }}(v'\,\cos \varphi )\right)\,+\,{\frac {1}{\varrho _{o}}}\,{\frac {\partial }{\partial z}}(\varrho _{o}w')=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf753817cbfc619593fa6f4255ae474b073b53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.888ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{a\,\cos \varphi }}\,\left({\frac {\partial u'}{\partial \lambda }}\,+\,{\frac {\partial }{\partial \varphi }}(v'\,\cos \varphi )\right)\,+\,{\frac {1}{\varrho _{o}}}\,{\frac {\partial }{\partial z}}(\varrho _{o}w')=0}"></span></li></ul></div> <p>with the definitions </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> eastward zonal wind</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> northward meridional wind</li> <li><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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> upward vertical wind</li> <li><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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" 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 \Phi }"></span> geopotential, <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 \int g(z,\varphi )\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int g(z,\varphi )\,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea45dd52fb6a40b934228377be05a515e202b528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.839ex; height:5.676ex;" alt="{\displaystyle \int g(z,\varphi )\,dz}"></span></li> <li><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 N^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe131b76af8a2bc86e01b14a7ba843db69c1a164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.177ex; height:2.676ex;" alt="{\displaystyle N^{2}}"></span> square of Brunt-Vaisala (buoyancy) frequency</li> <li><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> angular velocity of the Earth</li> <li><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 \varrho _{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varrho _{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1265d740ec3f0bcd43abf5b39512cd01f7ad6ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.232ex; height:2.009ex;" alt="{\displaystyle \varrho _{o}}"></span> density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto \exp(-z/H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∝</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \propto \exp(-z/H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4697f4ed9409ec7f230b778bafc898a15934e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.937ex; height:2.843ex;" alt="{\displaystyle \propto \exp(-z/H)}"></span></li> <li><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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> altitude</li> <li><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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> geographic longitude</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> geographic latitude</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> heating rate per unit mass</li> <li><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> radius of the Earth</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> gravity acceleration</li> <li><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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> constant scale height</li> <li><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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> time</li></ul> <div class="mw-heading mw-heading3"><h3 id="Separation_of_variables">Separation of variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=8" title="Edit section: Separation of variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of equations can be solved for <i>atmospheric tides</i>, i.e., longitudinally propagating waves of zonal wavenumber <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> and frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>. Zonal wavenumber <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is a positive integer so that positive values for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> correspond to eastward propagating tides and negative values to westward propagating tides. A separation approach of the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Phi '(\varphi ,\lambda ,z,t)&={\hat {\Phi }}(\varphi ,z)\,e^{i(s\lambda -\sigma t)}\\{\hat {\Phi }}(\varphi ,z)&=\sum _{n}\Theta _{n}(\varphi )\,G_{n}(z)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>λ</mi> <mo>−</mo> <mi>σ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Phi '(\varphi ,\lambda ,z,t)&={\hat {\Phi }}(\varphi ,z)\,e^{i(s\lambda -\sigma t)}\\{\hat {\Phi }}(\varphi ,z)&=\sum _{n}\Theta _{n}(\varphi )\,G_{n}(z)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b913b5be6c26c76c884e071ff9bef0913cfb6574" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.628ex; margin-bottom: -0.21ex; width:32.355ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\Phi '(\varphi ,\lambda ,z,t)&={\hat {\Phi }}(\varphi ,z)\,e^{i(s\lambda -\sigma t)}\\{\hat {\Phi }}(\varphi ,z)&=\sum _{n}\Theta _{n}(\varphi )\,G_{n}(z)\end{aligned}}}"></span> </p><p>and doing some manipulations<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> yields expressions for the latitudinal and vertical structure of the tides. </p> <div class="mw-heading mw-heading3"><h3 id="Laplace's_tidal_equation"><span id="Laplace.27s_tidal_equation"></span>Laplace's tidal equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=9" title="Edit section: Laplace's tidal equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The latitudinal structure of the tides is described by the <i>horizontal structure equation</i> which is also called <i>Laplace's tidal equation</i>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {L}{\Theta }_{n}+\varepsilon _{n}{\Theta }_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {L}{\Theta }_{n}+\varepsilon _{n}{\Theta }_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e74680451cbcb862af54dee23c71cb8730ee7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.039ex; height:2.509ex;" alt="{\displaystyle {L}{\Theta }_{n}+\varepsilon _{n}{\Theta }_{n}=0}"></span> </p><p>with <i>Laplace operator</i> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {L}={\frac {\partial }{\partial \mu }}\left[{\frac {(1-\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}\,{\frac {\partial }{\partial \mu }}\right]-{\frac {1}{\eta ^{2}-\mu ^{2}}}\,\left[-{\frac {s}{\eta }}\,{\frac {(\eta ^{2}+\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}+{\frac {s^{2}}{1-\mu ^{2}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>η</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {L}={\frac {\partial }{\partial \mu }}\left[{\frac {(1-\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}\,{\frac {\partial }{\partial \mu }}\right]-{\frac {1}{\eta ^{2}-\mu ^{2}}}\,\left[-{\frac {s}{\eta }}\,{\frac {(\eta ^{2}+\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}+{\frac {s^{2}}{1-\mu ^{2}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf631260c9cd2011a2e5035f96aaa5f8d4dd45ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.639ex; height:7.509ex;" alt="{\displaystyle {L}={\frac {\partial }{\partial \mu }}\left[{\frac {(1-\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}\,{\frac {\partial }{\partial \mu }}\right]-{\frac {1}{\eta ^{2}-\mu ^{2}}}\,\left[-{\frac {s}{\eta }}\,{\frac {(\eta ^{2}+\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}+{\frac {s^{2}}{1-\mu ^{2}}}\right]}"></span> </p><p>using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c6a715464349da91b71a33dd68ee5c6b30b961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.263ex; height:2.676ex;" alt="{\displaystyle \mu =\sin \varphi }"></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 \eta =\sigma /(2\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\sigma /(2\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e2a6a36b986041987c2995f1584ee0fe67ef28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.41ex; height:2.843ex;" alt="{\displaystyle \eta =\sigma /(2\Omega )}"></span> and <i>eigenvalue</i> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{n}=(2\Omega a)^{2}/gh_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{n}=(2\Omega a)^{2}/gh_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec8c3cfb3e4c3bac74cf3c26aad3848fb1a6e95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.817ex; height:3.176ex;" alt="{\displaystyle \varepsilon _{n}=(2\Omega a)^{2}/gh_{n}.}"></span> </p><p>Hence, atmospheric tides are eigenoscillations (<a href="/wiki/Eigenmodes" class="mw-redirect" title="Eigenmodes">eigenmodes</a>)of Earth's atmosphere with <a href="/wiki/Eigenfunctions" class="mw-redirect" title="Eigenfunctions">eigenfunctions</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 \Theta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200ad4d14e0c5f55ce756c7ccba038e66d50533b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle \Theta _{n}}"></span>, called <a href="/wiki/Hough_functions" class="mw-redirect" title="Hough functions">Hough functions</a>, and <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</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 \varepsilon _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfa578876505b4b9541f182a3f2213060a0e093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.302ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{n}}"></span>. The latter define the <i>equivalent depth</i> <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 h_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc869ca124a4bbb264113f1b3b40d78e09c055be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.557ex; height:2.509ex;" alt="{\displaystyle h_{n}}"></span> which couples the latitudinal structure of the tides with their vertical structure. </p> <div class="mw-heading mw-heading3"><h3 id="General_solution_of_Laplace's_equation"><span id="General_solution_of_Laplace.27s_equation"></span>General solution of Laplace's equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=10" title="Edit section: General solution of Laplace's equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eigenvalue.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Eigenvalue.jpg/480px-Eigenvalue.jpg" decoding="async" width="480" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Eigenvalue.jpg/720px-Eigenvalue.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Eigenvalue.jpg/960px-Eigenvalue.jpg 2x" data-file-width="983" data-file-height="600" /></a><figcaption>Figure 2. Eigenvalue <span class="texhtml mvar" style="font-style:italic;">ε</span> of wave modes of zonal wave number <span class="texhtml"><var style="padding-right: 1px;">s</var> = 1</span> vs. normalized frequency <span class="texhtml"><var style="padding-right: 1px;">ν</var> = <var style="padding-right: 1px;">ω</var>/Ω</span> where <span class="texhtml">Ω = <span class="nowrap"><span data-sort-value="6995727000000000000♠"></span>7.27<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup> s<sup>−1</sup></span></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of one <a href="/wiki/Solar_day" class="mw-redirect" title="Solar day">solar day</a>. Waves with positive (negative) frequencies propagate to the east (west). The horizontal dashed line is at <span class="texhtml"><var style="padding-right: 1px;">ε</var><sub>c</sub> ≃ 11</span> and indicates the transition from internal to external waves. Meaning of the symbols: 'RH' Rossby-Haurwitz waves (<span class="texhtml"><var style="padding-right: 1px;">ε</var> = 0</span>); 'Y' Yanai waves; 'K' Kelvin waves; 'R' Rossby waves; 'DT' Diurnal tides (<span class="texhtml"><var style="padding-right: 1px;">ν</var> = −1</span>); 'NM' Normal modes (<span class="texhtml"><var style="padding-right: 1px;">ε</var> ≃ <var style="padding-right: 1px;">ε</var><sub>c</sub></span>)</figcaption></figure> <p>Longuet-Higgins<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> has completely solved Laplace's equations and has discovered tidal modes with negative eigenvalues <span class="texhtml"><span style="white-space: nowrap;"><span style="font-style: italic;"><var style="padding-right: 1px;">ε</var></span><span style="font-size: 40%;"> </span><span style="font-size: 70%;"><span style="display:inline-block; vertical-align: -0.4em; line-height:1.1em;"><var style="padding-right: 1px;">s</var><br /><var style="padding-right: 1px;">n</var></span><span style="font-size: 40%;"> </span></span></span></span> (Figure 2). There exist two kinds of waves: class 1 waves, (sometimes called gravity waves), labelled by positive n, and class 2 waves (sometimes called rotational waves), labelled by negative n. Class 2 waves owe their existence to the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis</a> force and can only exist for periods greater than 12 hours (or <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><var style="padding-right: 1px;">ν</var></span>| ≤ 2</span>). Tidal waves can be either internal (travelling waves) with positive eigenvalues (or equivalent depth) which have finite vertical wavelengths and can transport wave energy upward, or external (evanescent waves) with negative eigenvalues and infinitely large vertical wavelengths meaning that their phases remain constant with altitude. These external wave modes cannot transport wave energy, and their amplitudes decrease exponentially with height outside their source regions. Even numbers of n correspond to waves symmetric with respect to the equator, and odd numbers corresponding to antisymmetric waves. The transition from internal to external waves appears at <span class="texhtml"><var style="padding-right: 1px;">ε</var> ≃ <var style="padding-right: 1px;">ε</var><sub>c</sub></span>, or at the vertical wavenumber <span class="texhtml"><var style="padding-right: 1px;">k</var><sub><var style="padding-right: 1px;">z</var></sub> = 0</span>, and <span class="texhtml"><var style="padding-right: 1px;">λ</var><sub><var style="padding-right: 1px;">z</var></sub> ⇒ ∞</span>, respectively. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Meridionalstr.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Meridionalstr.jpg/480px-Meridionalstr.jpg" decoding="async" width="480" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Meridionalstr.jpg/720px-Meridionalstr.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Meridionalstr.jpg/960px-Meridionalstr.jpg 2x" data-file-width="1000" data-file-height="500" /></a><figcaption>Figure 3. Pressure amplitudes vs. latitude of the <a href="/wiki/Hough_functions" class="mw-redirect" title="Hough functions">Hough functions</a> of the diurnal tide (<span class="texhtml"><var style="padding-right: 1px;">s</var> = 1</span>; <span class="texhtml"><var style="padding-right: 1px;">ν</var> = −1</span>) (left) and of the semidiurnal tides (<span class="texhtml"><var style="padding-right: 1px;">s</var> = 2</span>; <span class="texhtml"><var style="padding-right: 1px;">ν</var> = −2</span>) (right) on the northern hemisphere. Solid curves: symmetric waves; dashed curves: antisymmetric waves</figcaption></figure> <p>The fundamental solar diurnal tidal mode which optimally matches the solar heat input configuration and thus is most strongly excited is the <a href="/wiki/Hough_function" title="Hough function">Hough</a> mode (1, −2) (Figure 3). It depends on <a href="/wiki/Time_zone" title="Time zone">local time</a> and travels westward with the Sun. It is an external mode of class 2 and has the eigenvalue of <span class="texhtml"><span style="white-space: nowrap;"><span style="font-style: italic;"><var style="padding-right: 1px;">ε</var></span><span style="font-size: 40%;"> </span><span style="font-size: 70%;"><span style="display:inline-block; vertical-align: -0.4em; line-height:1.1em;">1<br />−2</span><span style="font-size: 40%;"> </span></span></span> = −12.56</span>. Its maximum pressure amplitude on the ground is about 60 Pa.<sup id="cite_ref-ChapmanLindzen_5-1" class="reference"><a href="#cite_note-ChapmanLindzen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The largest solar semidiurnal wave is mode (2, 2) with maximum pressure amplitudes at the ground of 120 Pa. It is an internal class 1 wave. Its amplitude increases exponentially with altitude. Although its solar excitation is half of that of mode (1, −2), its amplitude on the ground is larger by a factor of two. This indicates the effect of suppression of external waves, in this case by a factor of four.<sup id="cite_ref-VollandAT_9-0" class="reference"><a href="#cite_note-VollandAT-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Vertical_structure_equation">Vertical structure equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=11" title="Edit section: Vertical structure equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For bounded solutions and at altitudes above the forcing region, the <i>vertical structure equation</i> in its canonical form is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}G_{n}^{\star }}{\partial x^{2}}}\,+\,\alpha _{n}^{2}\,G_{n}^{\star }=F_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}G_{n}^{\star }}{\partial x^{2}}}\,+\,\alpha _{n}^{2}\,G_{n}^{\star }=F_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852b797e83abfdded2047b6b4389677722e632f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.982ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}G_{n}^{\star }}{\partial x^{2}}}\,+\,\alpha _{n}^{2}\,G_{n}^{\star }=F_{n}(x)}"></span> </p><p>with solution </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}^{\star }(x)\sim {\begin{cases}e^{-|\alpha _{n}|x}&{\text{:}}\,\alpha _{n}^{2}<0,\,{\text{ evanescent or trapped}}\\e^{i\alpha _{n}x}&{\text{:}}\,\alpha _{n}^{2}>0,\,{\text{ propagating}}\\e^{\left(\kappa -{\frac {1}{2}}\right)x}&{\text{:}}\,h_{n}=H/(1-\kappa ),F_{n}(x)=0\,\forall x,\,{\text{ Lamb waves (free solutions)}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>:</mtext> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo><</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> evanescent or trapped</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>:</mtext> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> propagating</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>κ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>:</mtext> </mrow> <mspace width="thinmathspace" /> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>κ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> Lamb waves (free solutions)</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}^{\star }(x)\sim {\begin{cases}e^{-|\alpha _{n}|x}&{\text{:}}\,\alpha _{n}^{2}<0,\,{\text{ evanescent or trapped}}\\e^{i\alpha _{n}x}&{\text{:}}\,\alpha _{n}^{2}>0,\,{\text{ propagating}}\\e^{\left(\kappa -{\frac {1}{2}}\right)x}&{\text{:}}\,h_{n}=H/(1-\kappa ),F_{n}(x)=0\,\forall x,\,{\text{ Lamb waves (free solutions)}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75222d34201b640db099a0cae18ca45db37abc0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:83.636ex; height:10.176ex;" alt="{\displaystyle G_{n}^{\star }(x)\sim {\begin{cases}e^{-|\alpha _{n}|x}&{\text{:}}\,\alpha _{n}^{2}<0,\,{\text{ evanescent or trapped}}\\e^{i\alpha _{n}x}&{\text{:}}\,\alpha _{n}^{2}>0,\,{\text{ propagating}}\\e^{\left(\kappa -{\frac {1}{2}}\right)x}&{\text{:}}\,h_{n}=H/(1-\kappa ),F_{n}(x)=0\,\forall x,\,{\text{ Lamb waves (free solutions)}}\end{cases}}}"></span> </p><p>using the definitions </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha _{n}^{2}&={\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}\\x&={\frac {z}{H}}\\G_{n}^{\star }&=G_{n}\,\varrho _{o}^{\frac {1}{2}}\,N^{-1}\\F_{n}(x)&=-{\frac {\varrho _{o}^{-{\frac {1}{2}}}}{i\sigma N}}\,{\frac {\partial }{\partial x}}(\varrho _{o}J_{n}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <mi>H</mi> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>H</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msubsup> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msubsup> <mrow> <mi>i</mi> <mi>σ</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ϱ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha _{n}^{2}&={\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}\\x&={\frac {z}{H}}\\G_{n}^{\star }&=G_{n}\,\varrho _{o}^{\frac {1}{2}}\,N^{-1}\\F_{n}(x)&=-{\frac {\varrho _{o}^{-{\frac {1}{2}}}}{i\sigma N}}\,{\frac {\partial }{\partial x}}(\varrho _{o}J_{n}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cf9ac9f90e51fef9baf94e48082a22cfab3939" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.604ex; margin-bottom: -0.234ex; width:27.63ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}\alpha _{n}^{2}&={\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}\\x&={\frac {z}{H}}\\G_{n}^{\star }&=G_{n}\,\varrho _{o}^{\frac {1}{2}}\,N^{-1}\\F_{n}(x)&=-{\frac {\varrho _{o}^{-{\frac {1}{2}}}}{i\sigma N}}\,{\frac {\partial }{\partial x}}(\varrho _{o}J_{n}).\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Propagating_solutions">Propagating solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=12" title="Edit section: Propagating solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Therefore, each wavenumber/frequency pair (a tidal <i>component</i>) is a superposition of associated <a href="/wiki/Hough_functions" class="mw-redirect" title="Hough functions">Hough functions</a> (often called tidal <i>modes</i> in the literature) of index <i>n</i>. The nomenclature is such that a negative value of <i>n</i> refers to evanescent modes (no vertical propagation) and a positive value to propagating modes. The equivalent depth <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 h_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc869ca124a4bbb264113f1b3b40d78e09c055be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.557ex; height:2.509ex;" alt="{\displaystyle h_{n}}"></span> is linked to the vertical wavelength <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 \lambda _{z,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{z,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a910f9ee96fc4a2a7a3b15c9c23448d4d0b6d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.801ex; height:2.843ex;" alt="{\displaystyle \lambda _{z,n}}"></span>, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}/H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}/H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267057c8db274aa92eae44601b2dde7bab69acf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:2.843ex;" alt="{\displaystyle \alpha _{n}/H}"></span> is the vertical wavenumber: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{z,n}={\frac {2\pi \,H}{\alpha _{n}}}={\frac {2\pi \,H}{\sqrt {{\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>H</mi> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>H</mi> </mrow> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <mi>H</mi> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{z,n}={\frac {2\pi \,H}{\alpha _{n}}}={\frac {2\pi \,H}{\sqrt {{\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51acb5d443aa7a50f92b4e33e62d6845ac695f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:27.326ex; height:8.009ex;" alt="{\displaystyle \lambda _{z,n}={\frac {2\pi \,H}{\alpha _{n}}}={\frac {2\pi \,H}{\sqrt {{\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}}}}.}"></span> </p><p>For propagating solutions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha _{n}^{2}>0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha _{n}^{2}>0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44549515aa59e9e319b691d99d9b60f3ffd83b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.776ex; height:3.009ex;" alt="{\displaystyle (\alpha _{n}^{2}>0)}"></span>, the vertical group velocity </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{gz,n}=H{\frac {\partial \sigma }{\partial \alpha _{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>z</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>σ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{gz,n}=H{\frac {\partial \sigma }{\partial \alpha _{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ef74478503f8f3244bd1ed8d51ba31334918ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.264ex; height:5.843ex;" alt="{\displaystyle c_{gz,n}=H{\frac {\partial \sigma }{\partial \alpha _{n}}}}"></span> </p><p>becomes positive (upward energy propagation) only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37d804e6f0270a340a7bbcbf43ab851d1ed8680c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.967ex; height:2.509ex;" alt="{\displaystyle \alpha _{n}>0}"></span> for westward <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sigma <0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo><</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma <0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8054ccc4b9c9da13b75301272e89b63b04f9a51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.4ex; height:2.843ex;" alt="{\displaystyle (\sigma <0)}"></span> or if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf9d901f34d3c96e838d9b7519e2a95d391286c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.967ex; height:2.509ex;" alt="{\displaystyle \alpha _{n}<0}"></span> for eastward <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sigma >0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma >0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9491c10fcbb473062bd9c73d9c50dbfb3fef656a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.4ex; height:2.843ex;" alt="{\displaystyle (\sigma >0)}"></span> propagating waves. At a given height <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 x=z/H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=z/H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ed620f512e16e516f4fa755b0d36a1e1fa1894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.742ex; height:2.843ex;" alt="{\displaystyle x=z/H}"></span>, the wave maximizes for </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}=s\lambda +\alpha _{n}x-\sigma t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mi>λ</mi> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>x</mi> <mo>−</mo> <mi>σ</mi> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}=s\lambda +\alpha _{n}x-\sigma t=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e04f6d9a33228157f886bf9e91d0018aeca64b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.529ex; height:2.509ex;" alt="{\displaystyle K_{n}=s\lambda +\alpha _{n}x-\sigma t=0.}"></span> </p><p>For a fixed longitude <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, this in turn always results in downward phase progression as time progresses, independent of the propagation direction. This is an important result for the interpretation of observations: downward phase progression in time means an upward propagation of energy and therefore a tidal forcing lower in the atmosphere. Amplitude increases with height <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 \propto e^{z/2H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \propto e^{z/2H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efba6ff62c04a0ebcaf746394087b034d2626847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.642ex; height:2.843ex;" alt="{\displaystyle \propto e^{z/2H}}"></span>, as density decreases. </p> <div class="mw-heading mw-heading2"><h2 id="Dissipation">Dissipation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=13" title="Edit section: Dissipation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">Damping</a> of the tides occurs primarily in the lower thermosphere region, and may be caused by <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> from breaking <a href="/wiki/Gravity_wave" title="Gravity wave">gravity waves</a>. A similar phenomenon to ocean waves breaking on a <a href="/wiki/Beach" title="Beach">beach</a>, the <a href="/wiki/Energy" title="Energy">energy</a> dissipates into the background atmosphere. Molecular <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> also becomes increasingly important at higher levels in the lower thermosphere as the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> increases in the rarefied atmosphere.<sup id="cite_ref-:0_10-0" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="Originally cited with only author, journal title, volume, year and page number - no article title or issue number. This journal doesn't provide page numbers in the citations on its website so this couldn't identify an individual article (February 2022)">verification needed</span></a></i>]</sup> </p><p>At thermospheric heights, <a href="/wiki/Attenuation" title="Attenuation">attenuation</a> of atmospheric waves, mainly due to collisions between the neutral gas and the ionospheric plasma, becomes significant so that at above about 150 km altitude, all wave modes gradually become external waves, and the <a href="/wiki/Hough_function" title="Hough function">Hough functions</a> degenerate to <a href="/wiki/Zonal_spherical_function" title="Zonal spherical function">spherical functions</a>; e.g., mode (1, −2) develops to the spherical function <span class="texhtml"><span style="white-space: nowrap;"><span style="font-style: italic;">P</span><span style="font-size: 40%;"> </span><span style="font-size: 70%;"><span style="display:inline-block; vertical-align: -0.4em; line-height:1.1em;">1<br />1</span><span style="font-size: 40%;"> </span></span></span>(<var style="padding-right: 1px;">θ</var>)</span>, mode (2, 2) becomes <span class="texhtml"><span style="white-space: nowrap;"><span style="font-style: italic;">P</span><span style="font-size: 40%;"> </span><span style="font-size: 70%;"><span style="display:inline-block; vertical-align: -0.4em; line-height:1.1em;">2<br />2</span><span style="font-size: 40%;"> </span></span></span>(<var style="padding-right: 1px;">θ</var>)</span>, with <span class="texhtml mvar" style="font-style:italic;">θ</span> the co-latitude, etc.<sup id="cite_ref-VollandAT_9-1" class="reference"><a href="#cite_note-VollandAT-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Within the <a href="/wiki/Thermosphere" title="Thermosphere">thermosphere</a>, mode (1, −2) is the predominant mode reaching diurnal temperature amplitudes at the <a href="/wiki/Exosphere" title="Exosphere">exosphere</a> of at least 140 K and horizontal winds of the order of 100 m/s and more increasing with geomagnetic activity.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It is responsible for the electric Sq currents within the <a href="/wiki/Ionospheric_dynamo_region" title="Ionospheric dynamo region">ionospheric dynamo region</a> between about 100 and 200 km altitude.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Both diurnal and semidiurnal tides can be observed across the <a href="/wiki/Ionospheric_dynamo_region" title="Ionospheric dynamo region">ionospheric dynamo region</a> with <a href="/wiki/Incoherent_scatter" title="Incoherent scatter">incoherent scatter</a> radars by tracking the tidal motion of ionospheric plasma. <sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effects_of_atmospheric_tide">Effects of atmospheric tide</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=14" title="Edit section: Effects of atmospheric tide"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The tides form an important mechanism for transporting energy from the lower atmosphere into the upper atmosphere,<sup id="cite_ref-:0_10-1" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> while dominating the dynamics of the mesosphere and lower thermosphere. Therefore, understanding the atmospheric tides is essential in understanding the atmosphere as a whole. Modeling and observations of atmospheric tides are needed in order to monitor and predict changes in the Earth's atmosphere.<sup id="cite_ref-VollandAT_9-2" class="reference"><a href="#cite_note-VollandAT-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Atmospheric_wave" title="Atmospheric wave">Atmospheric wave</a></li> <li><a href="/wiki/Tide" title="Tide">Tide</a></li> <li><a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a></li> <li><a href="/wiki/Mesosphere" title="Mesosphere">Mesosphere</a></li> <li><a href="/wiki/Thermosphere" title="Thermosphere">Thermosphere</a></li> <li><a href="/wiki/Ionospheric_dynamo_region" title="Ionospheric dynamo region">Ionospheric dynamo region</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atmospheric_tide&action=edit&section=16" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.hao.ucar.edu/modeling/gswm/gswm.html">Global Scale Wave Model</a> <a href="/wiki/University_Corporation_for_Atmospheric_Research" title="University Corporation for Atmospheric Research">UCAR</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.hao.ucar.edu/modeling/gswm/refs.html">GSWM References</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHaganForbesRichmond2003" class="citation encyclopaedia cs1">Hagan, M. E.; Forbes, J. M.; Richmond, A. (2003). "Atmospheric Tides". <i>Encyclopedia of Atmospheric Sciences</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Atmospheric+Tides&rft.btitle=Encyclopedia+of+Atmospheric+Sciences&rft.date=2003&rft.aulast=Hagan&rft.aufirst=M.+E.&rft.au=Forbes%2C+J.+M.&rft.au=Richmond%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200129015554/https://news.google.com/newspapers?id=X30QAAAAIBAJ&sjid=wZMDAAAAIBAJ&pg=3035,1211515&dq=lyle-medal&hl=en">"Tides found in atmosphere"</a>. <i><a href="/wiki/Sydney_Morning_Herald" class="mw-redirect" title="Sydney Morning Herald">Sydney Morning Herald</a></i>. September 9, 1947. p. 17. Archived from <a rel="nofollow" class="external text" href="https://news.google.com/newspapers?id=X30QAAAAIBAJ&sjid=wZMDAAAAIBAJ&pg=3035,1211515&dq=lyle-medal&hl=en">the original</a> on January 29, 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sydney+Morning+Herald&rft.atitle=Tides+found+in+atmosphere&rft.pages=17&rft.date=1947-09-09&rft_id=https%3A%2F%2Fnews.google.com%2Fnewspapers%3Fid%3DX30QAAAAIBAJ%26sjid%3DwZMDAAAAIBAJ%26pg%3D3035%2C1211515%26dq%3Dlyle-medal%26hl%3Den&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-ChapmanLindzen-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-ChapmanLindzen_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ChapmanLindzen_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChapmanLindzen1970" class="citation book cs1">Chapman, S.; Lindzen, R. S. (1970). <i>Atmospheric Tides</i>. Norwell, Massachusetts: D. Reidel.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atmospheric+Tides&rft.place=Norwell%2C+Massachusetts&rft.pub=D.+Reidel&rft.date=1970&rft.aulast=Chapman&rft.aufirst=S.&rft.au=Lindzen%2C+R.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolton1975" class="citation journal cs1">Holton, J. R. (1975). "The Dynamic Meteorology of the Stratosphere and Mesosphere". <i>Meteorological Monographs</i>. <b>15</b> (37). Massachusetts: American Meteorological Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Meteorological+Monographs&rft.atitle=The+Dynamic+Meteorology+of+the+Stratosphere+and+Mesosphere&rft.volume=15&rft.issue=37&rft.date=1975&rft.aulast=Holton&rft.aufirst=J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">J. Oberheide (2007). <a rel="nofollow" class="external text" href="http://deposit.d-nb.de/cgi-bin/dokserv?idn=985343273&dok_var=d1&dok_ext=pdf&filename=985343273.pdf"><i>On large-scale wave coupling across the stratopause</i></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110722191107/http://deposit.d-nb.de/cgi-bin/dokserv?idn=985343273&dok_var=d1&dok_ext=pdf&filename=985343273.pdf">Archived</a> July 22, 2011, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Appendix A2, pp. 113–117. University of Wuppertal.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1968" class="citation journal cs1">Longuet-Higgins, Michael Selwyn (1968). <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1968.0003">"The eigenfunctions of Laplace's tidal equation over a sphere"</a>. <i>Philosophical Transactions. Royal Society of London</i>. <b>A262</b>: <span class="nowrap">511–</span>607. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1968.0003">10.1098/rsta.1968.0003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions.+Royal+Society+of+London&rft.atitle=The+eigenfunctions+of+Laplace%27s+tidal+equation+over+a+sphere&rft.volume=A262&rft.pages=%3Cspan+class%3D%22nowrap%22%3E511-%3C%2Fspan%3E607&rft.date=1968&rft_id=info%3Adoi%2F10.1098%2Frsta.1968.0003&rft.aulast=Longuet-Higgins&rft.aufirst=Michael+Selwyn&rft_id=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2Fpdf%2F10.1098%2Frsta.1968.0003&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-VollandAT-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-VollandAT_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-VollandAT_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-VollandAT_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVolland1988" class="citation book cs1">Volland, Hans (1988). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-94-009-2861-9"><i>Atmospheric Tidal and Planetary Waves</i></a>. Dordrecht, Netherlands: Kluwer Academic Publisher.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atmospheric+Tidal+and+Planetary+Waves&rft.place=Dordrecht%2C+Netherlands&rft.pub=Kluwer+Academic+Publisher&rft.date=1988&rft.aulast=Volland&rft.aufirst=Hans&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-94-009-2861-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-:0-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFForbesZhangPaloRussell2008" class="citation journal cs1">Forbes, J. M.; Zhang, X.; Palo, S.; Russell, J.; Mertens, C. J.; Mlynczak, M. (22 February 2008). <a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2007JA012737">"Tidal variability in the ionospheric dynamo region"</a>. <i>Journal of Geophysical Research: Space Physics</i>. <b>113</b> (A2). <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JGRA..113.2310F">2008JGRA..113.2310F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2007JA012737">10.1029/2007JA012737</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geophysical+Research%3A+Space+Physics&rft.atitle=Tidal+variability+in+the+ionospheric+dynamo+region&rft.volume=113&rft.issue=A2&rft.date=2008-02-22&rft_id=info%3Adoi%2F10.1029%2F2007JA012737&rft_id=info%3Abibcode%2F2008JGRA..113.2310F&rft.aulast=Forbes&rft.aufirst=J.+M.&rft.au=Zhang%2C+X.&rft.au=Palo%2C+S.&rft.au=Russell%2C+J.&rft.au=Mertens%2C+C.+J.&rft.au=Mlynczak%2C+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1029%252F2007JA012737&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKohlKing1967" class="citation journal cs1">Kohl, H.; King, J. W. (1967). "Atmospheric winds between 100 and 700 km and their effects on the ionosphere". <i>Journal of Atmospheric and Terrestrial Physics</i>. <b>29</b> (9): <span class="nowrap">1045–</span>1062. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967JATP...29.1045K">1967JATP...29.1045K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9169%2867%2990139-0">10.1016/0021-9169(67)90139-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Atmospheric+and+Terrestrial+Physics&rft.atitle=Atmospheric+winds+between+100+and+700+km+and+their+effects+on+the+ionosphere&rft.volume=29&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1045-%3C%2Fspan%3E1062&rft.date=1967&rft_id=info%3Adoi%2F10.1016%2F0021-9169%2867%2990139-0&rft_id=info%3Abibcode%2F1967JATP...29.1045K&rft.aulast=Kohl&rft.aufirst=H.&rft.au=King%2C+J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKato1966" class="citation journal cs1">Kato, S. (1 July 1966). "Diurnal atmospheric oscillation: 2. Thermal excitation in the upper atmosphere". <i>Journal of Geophysical Research</i>. <b>71</b> (13): <span class="nowrap">3211–</span>3214. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966JGR....71.3211K">1966JGR....71.3211K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2FJZ071i013p03211">10.1029/JZ071i013p03211</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geophysical+Research&rft.atitle=Diurnal+atmospheric+oscillation%3A+2.+Thermal+excitation+in+the+upper+atmosphere&rft.volume=71&rft.issue=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3211-%3C%2Fspan%3E3214&rft.date=1966-07-01&rft_id=info%3Adoi%2F10.1029%2FJZ071i013p03211&rft_id=info%3Abibcode%2F1966JGR....71.3211K&rft.aulast=Kato&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGünzkoferPokhotelovStober2022" class="citation journal cs1">Günzkofer, F.; Pokhotelov, D.; Stober, G.; et al. (2022-09-25). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?gldocs-11858/10460">"Determining the Origin of Tidal Oscillations in the Ionospheric Transition Region With EISCAT Radar and Global Simulation Data"</a>. <i>Journal of Geophysical Research: Space Physics</i>. <b>127</b> (10). <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022JGRA..12730861G">2022JGRA..12730861G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2022JA030861">10.1029/2022JA030861</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:252535576">252535576</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geophysical+Research%3A+Space+Physics&rft.atitle=Determining+the+Origin+of+Tidal+Oscillations+in+the+Ionospheric+Transition+Region+With+EISCAT+Radar+and+Global+Simulation+Data&rft.volume=127&rft.issue=10&rft.date=2022-09-25&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A252535576%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1029%2F2022JA030861&rft_id=info%3Abibcode%2F2022JGRA..12730861G&rft.aulast=G%C3%BCnzkofer&rft.aufirst=F.&rft.au=Pokhotelov%2C+D.&rft.au=Stober%2C+G.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3Fgldocs-11858%2F10460&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+tide" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Atmospheric_tide&oldid=1265969131">https://en.wikipedia.org/w/index.php?title=Atmospheric_tide&oldid=1265969131</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Atmospheric_dynamics" title="Category:Atmospheric dynamics">Atmospheric dynamics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_March_2022" title="Category:Wikipedia articles needing clarification from March 2022">Wikipedia articles needing clarification from March 2022</a></li><li><a href="/wiki/Category:All_pages_needing_factual_verification" title="Category:All pages needing factual verification">All pages needing factual verification</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_factual_verification_from_February_2022" title="Category:Wikipedia articles needing factual verification from February 2022">Wikipedia articles needing factual verification from February 2022</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 December 2024, at 13:25<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Atmospheric tide - Wikipedia