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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Polar motion</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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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"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about motion relative to the crust. For motion relative to an astronomical framework, see <a href="/wiki/Axial_precession" title="Axial precession">axial precession</a> and <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">astronomical nutation</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Motion of Earth's rotational axis relative to its crust</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PolarMotion.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/PolarMotion.png/330px-PolarMotion.png" decoding="async" width="330" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/PolarMotion.png/495px-PolarMotion.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/PolarMotion.png/660px-PolarMotion.png 2x" data-file-width="684" data-file-height="547" /></a><figcaption>Polar motion in <a href="/wiki/Arc-seconds" class="mw-redirect" title="Arc-seconds">arc-seconds</a> as function of time in days (0.1 arcsec ≈ 3 meters).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><b>Polar motion</b> of the Earth is the motion of the <a href="/wiki/Earth%27s_rotation" title="Earth's rotation">Earth's rotational axis</a> relative to its <a href="/wiki/Earth%27s_crust" title="Earth's crust">crust</a>.<sup id="cite_ref-Lambeck_2-0" class="reference"><a href="#cite_note-Lambeck-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1">: 1 </span></sup> This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called <i>Earth-centered, Earth-fixed</i> or <a href="/wiki/ECEF" class="mw-redirect" title="ECEF">ECEF</a> reference frame). This variation is a few meters on the surface of the Earth. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=1" title="Edit section: Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polar motion is defined relative to a conventionally defined reference axis, the CIO (<a href="/wiki/Conventional_International_Origin" title="Conventional International Origin">Conventional International Origin</a>), being the pole's average location over the year 1900. It consists of three major components: a free oscillation called <a href="/wiki/Chandler_wobble" title="Chandler wobble">Chandler wobble</a> with a period of about 435 days, an annual oscillation, and an irregular drift in the direction of the 80th <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridian</a> west,<sup id="cite_ref-IERS_3-0" class="reference"><a href="#cite_note-IERS-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> which has lately been less extremely west.<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><sup id="cite_ref-Adhikari_5-0" class="reference"><a href="#cite_note-Adhikari-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1">: 1 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Causes">Causes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=2" title="Edit section: Causes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The slow drift, about 20 m since 1900, is partly due to motions in the Earth's core and mantle, and partly to the redistribution of water mass as the <a href="/wiki/Greenland_ice_sheet" title="Greenland ice sheet">Greenland ice sheet</a> melts, and to <a href="/wiki/Isostatic_rebound" class="mw-redirect" title="Isostatic rebound">isostatic rebound</a>, i.e. the slow rise of land that was formerly burdened with ice sheets or glaciers.<sup id="cite_ref-Lambeck_2-1" class="reference"><a href="#cite_note-Lambeck-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 2">: 2 </span></sup> The drift is roughly along the <a href="/wiki/80th_meridian_west" title="80th meridian west">80th meridian west</a>. Since about 2000, the pole has found a less extreme drift, which is roughly along the central meridian. This less dramatically westward drift of motion is attributed to the global scale mass transport between the oceans and the continents.<sup id="cite_ref-Adhikari_5-1" class="reference"><a href="#cite_note-Adhikari-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 2">: 2 </span></sup> </p><p>Major <a href="/wiki/Earthquakes" class="mw-redirect" title="Earthquakes">earthquakes</a> cause abrupt polar motion by altering the volume distribution of the Earth's solid mass. These shifts are quite small in magnitude relative to the long-term core/mantle and isostatic rebound components of polar motion.<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 class="mw-heading mw-heading2"><h2 id="Principle">Principle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=3" title="Edit section: Principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the absence of external torques, the vector of the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> <b>M</b> of a rotating system remains constant and is directed toward a fixed point in space. If the earth were perfectly symmetrical and rigid, <b>M</b> would remain aligned with its axis of symmetry, which would also be its <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a>. In the case of the Earth, it is almost identical with its axis of rotation, with the discrepancy due to shifts of mass on the planet's surface. The vector of the <a href="/wiki/Figure_axis" class="mw-redirect" title="Figure axis">figure axis</a> <b>F</b> of the system (or maximum principal axis, the axis which yields the largest value of moment of inertia) wobbles around <b>M</b>. This motion is called <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>'s <a href="/wiki/Free_nutation" class="mw-redirect" title="Free nutation">free nutation</a>. For a rigid Earth which is an oblate <a href="/wiki/Spheroid" title="Spheroid">spheroid</a> to a good approximation, the figure axis <b>F</b> would be its geometric axis defined by the geographic north and south pole, and identical with the axis of its polar moment of inertia. The Euler period of free nutation is </p><p>(1) <span style="padding-left:4em;"> </span> τ<sub>E</sub> = 1/ν<sub>E</sub> = A/(C − A) sidereal days ≈ 307 sidereal days ≈ 0.84 sidereal years </p><p><span class="nowrap">ν<sub>E</sub> = 1.19</span> is the normalized Euler frequency (in units of reciprocal years), <span class="nowrap">C = 8.04 × 10<sup>37</sup> kg m<sup>2</sup></span> is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and <span class="nowrap">C − A = 2.61 × 10<sup>35</sup> kg m<sup>2</sup></span>.<sup id="cite_ref-Lambeck_2-2" class="reference"><a href="#cite_note-Lambeck-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The observed angle between the figure axis of the Earth <b>F</b> and its angular momentum <b>M</b> is a few hundred <a href="/wiki/Milliarcseconds" class="mw-redirect" title="Milliarcseconds">milliarcseconds</a> (mas). This rotation can be interpreted as a linear <a href="/wiki/Displacement_(physics)" class="mw-redirect" title="Displacement (physics)">displacement</a> of either <a href="/wiki/Geographical_pole" title="Geographical pole">geographical pole</a> amounting to several meters on the surface of the Earth: 100 mas <a href="/wiki/Subtended_angle" title="Subtended angle">subtends</a> an <a href="/wiki/Arc_length" title="Arc length">arc length</a> of 3.082 m, when converted to radians and multiplied by the <a href="/wiki/Earth%27s_figure" class="mw-redirect" title="Earth's figure">Earth's</a> <a href="/wiki/Polar_radius" class="mw-redirect" title="Polar radius">polar radius</a> (6,356,752.3 m). Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing the apparent motion of the rotation axis about the geometric axis of the Earth. This is the so-called polar motion.<sup id="cite_ref-MoritzMueller_8-0" class="reference"><a href="#cite_note-MoritzMueller-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Observations show that the figure axis exhibits an annual wobble forced by surface mass displacement via atmospheric and/or ocean dynamics, while the free nutation is much larger than the Euler period and of the order of 435 to 445 sidereal days. This observed free nutation is called <a href="/wiki/Chandler_wobble" title="Chandler wobble">Chandler wobble</a>. There exist, in addition, polar motions with smaller periods of the order of decades.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Finally, a secular polar drift of about 0.10<span class="nowrap"> </span>m per year in the direction of 80° west has been observed which is due to mass redistribution within the Earth's interior by continental drift, and/or slow motions within mantle and core which gives rise to changes of the moment of inertia.<sup id="cite_ref-MoritzMueller_8-1" class="reference"><a href="#cite_note-MoritzMueller-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of the variation of the latitude of stars, while <a href="/wiki/Seth_Carlo_Chandler" title="Seth Carlo Chandler">S.C. Chandler</a> found the free nutation in 1891.<sup id="cite_ref-MoritzMueller_8-2" class="reference"><a href="#cite_note-MoritzMueller-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Both periods superpose, giving rise to a <a href="/wiki/Beat_frequency" class="mw-redirect" title="Beat frequency">beat frequency</a> with a period of about 5 to 8 years (see Figure 1). </p><p>This polar motion should not be confused with the changing direction of the <a href="/wiki/Earth%27s_rotation_axis" class="mw-redirect" title="Earth's rotation axis">Earth's rotation axis</a> relative to the stars with different periods, caused mostly by the torques on the <a href="/wiki/Geoid" title="Geoid">Geoid</a> due to the gravitational attraction of the Moon and Sun. They are also called <a href="/wiki/Nutation" title="Nutation">nutations</a>, except for the slowest, which is the <a href="/wiki/Precession_of_the_equinoxes" class="mw-redirect" title="Precession of the equinoxes">precession of the equinoxes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Observations">Observations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=4" title="Edit section: Observations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polar motion is observed routinely by <a href="/wiki/Space_geodesy" title="Space geodesy">space geodesy</a> methods such as <a href="/wiki/Very-long-baseline_interferometry" title="Very-long-baseline interferometry">very-long-baseline interferometry</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Lunar_Laser_Ranging_experiment" class="mw-redirect" title="Lunar Laser Ranging experiment">lunar laser ranging</a> and <a href="/wiki/Satellite_laser_ranging" title="Satellite laser ranging">satellite laser ranging</a>.<sup id="cite_ref-Eubanks_11-0" class="reference"><a href="#cite_note-Eubanks-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The annual component is rather constant in amplitude, and its frequency varies by not more than 1 to 2%. The amplitude of the Chandler wobble, however, varies by a factor of three, and its frequency by up to 7%. Its maximum amplitude during the last 100 years never exceeded 230 mas. </p><p>The <a href="/wiki/Chandler_wobble" title="Chandler wobble">Chandler wobble</a> is usually considered a resonance phenomenon, a free <a href="/wiki/Nutation" title="Nutation">nutation</a> that is excited by a source and then dies away with a time constant τ<sub>D</sub> of the order of 100 years. It is a measure of the elastic reaction of the Earth.<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> It is also the explanation for the deviation of the Chandler period from the Euler period. However, rather than dying away, the Chandler wobble, continuously observed for more than 100 years, varies in amplitude and shows a sometimes rapid frequency shift within a few years.<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> This reciprocal behavior between amplitude and frequency has been described by the empirical formula:<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>(2) <span style="padding-left:4em;"> </span> m = 3.7/(ν − 0.816) <span style="padding-left:2em;"> </span> (for 0.83 < ν < 0.9) </p><p>with m the observed amplitude (in units of mas), and ν the frequency (in units of reciprocal sidereal years) of the Chandler wobble. In order to generate the Chandler wobble, recurring excitation is necessary. Seismic activity, groundwater movement, snow load, or atmospheric interannual dynamics have been suggested as such recurring forces, e.g.<sup id="cite_ref-Eubanks_11-1" class="reference"><a href="#cite_note-Eubanks-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Atmospheric excitation seems to be the most likely candidate.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-VollandAR_17-0" class="reference"><a href="#cite_note-VollandAR-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Others propose a combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Current and historic polar motion data is available from the <a href="/wiki/International_Earth_Rotation_and_Reference_Systems_Service" title="International Earth Rotation and Reference Systems Service">International Earth Rotation and Reference Systems Service</a>'s <a href="/wiki/Earth_orientation_parameters" title="Earth orientation parameters">Earth orientation parameters</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Note in using this data that the convention is to define <span class="texhtml"><i>p</i><sub><i>x</i></sub></span> to be positive along 0° longitude and <span class="texhtml"><i>p</i><sub><i>y</i></sub></span> to be positive along 90°E longitude.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=5" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Annual_component">Annual component</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=6" title="Edit section: Annual component"><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:Annualmotion1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Annualmotion1.jpg/310px-Annualmotion1.jpg" decoding="async" width="310" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Annualmotion1.jpg/465px-Annualmotion1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Annualmotion1.jpg/620px-Annualmotion1.jpg 2x" data-file-width="700" data-file-height="640" /></a><figcaption>Figure 2. <a href="/wiki/Rotation_vector" class="mw-redirect" title="Rotation vector">Rotation vector</a> <b>m</b> of the annual component of polar motion as function of year. Numbers and tick marks indicate the beginning of each calendar month. The dash-dotted line is in the direction of the major axis. The line in the direction of the minor axis is the location of the excitation function vs. time of year. (<span class="nowrap">100 mas (milliarcseconds) = 3.082 m</span> on the Earth's surface at the poles)</figcaption></figure> <p>There is now general agreement that the annual component of polar motion is a forced motion excited predominantly by atmospheric dynamics.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> There exist two external forces to excite polar motion: atmospheric winds, and pressure loading. The main component is pressure forcing, which is a standing wave of the form:<sup id="cite_ref-VollandAR_17-1" class="reference"><a href="#cite_note-VollandAR-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>(3) <span style="padding-left:4em;"> </span> p = p<sub>0</sub>Θ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">−3</sub></span></span>(θ) cos[2πν<sub>A</sub>(t − t<sub>0</sub>)] cos(λ − λ<sub>0</sub>) </p><p>with p<sub>0</sub> a pressure amplitude, Θ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">−3</sub></span></span> a <a href="/wiki/Hough_function" title="Hough function">Hough function</a> describing the latitude distribution of the atmospheric pressure on the ground, θ the geographic co-latitude, t the time of year, t<sub>0</sub> a time delay, <span class="nowrap">ν<sub>A</sub> = 1.003</span> the normalized frequency of one solar year, λ the longitude, and λ<sub>0</sub> the longitude of maximum pressure. The Hough function in a first approximation is proportional to sin θ cos θ. Such standing wave represents the seasonally varying spatial difference of the Earth's surface pressure. In northern winter, there is a pressure high over the North Atlantic Ocean and a pressure low over Siberia with temperature differences of the order of 50°, and vice versa in summer, thus an unbalanced mass distribution on the surface of the Earth. The position of the vector <b>m</b> of the annual component describes an ellipse (Figure 2). The calculated ratio between major and minor axis of the ellipse is </p><p>(4) <span style="padding-left:4em;"> </span> m<sub>1</sub>/m<sub>2</sub> = ν<sub>C</sub> </p><p>where ν<sub>C</sub> is the Chandler resonance frequency. The result is in good agreement with the observations.<sup id="cite_ref-Lambeck_2-3" class="reference"><a href="#cite_note-Lambeck-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>From Figure 2 together with eq.(4), one obtains <span class="nowrap">ν<sub>C</sub> = 0.83</span>, corresponding to a Chandler resonance period of </p><p>(5) <span style="padding-left:4em;"> </span> τ<sub>C</sub> = 441 sidereal days = 1.20 sidereal years </p><p><span class="nowrap">p<sub>0</sub> = 2.2 hPa</span>, <span class="nowrap">λ<sub>0</sub> = −170°</span> the latitude of maximum pressure, and <span class="nowrap">t<sub>0</sub> = −0.07 years = −25 days</span>. </p><p>It is difficult to estimate the effect of the ocean, which may slightly increase the value of maximum ground pressure necessary to generate the annual wobble. This ocean effect has been estimated to be of the order of 5–10%.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Chandler_wobble">Chandler wobble</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=7" title="Edit section: Chandler wobble"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chandler_wobble" title="Chandler wobble">Chandler wobble</a></div> <p>It is improbable that the internal parameters of the Earth responsible for the Chandler wobble would be time dependent on such short time intervals. Moreover, the observed stability of the annual component argues against any hypothesis of a variable Chandler resonance frequency. One possible explanation for the observed frequency-amplitude behavior would be a forced, but slowly changing quasi-periodic excitation by interannually varying atmospheric dynamics. Indeed, a quasi-14 month period has been found in coupled ocean-atmosphere general circulation models,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and a regional 14-month signal in regional <a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">sea surface temperature</a> has been observed.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>To describe such behavior theoretically, one starts with the Euler equation with pressure loading as in eq.(3), however now with a slowly changing frequency ν, and replaces the frequency ν by a complex frequency <span class="nowrap">ν + iν<sub>D</sub></span>, where ν<sub>D</sub> simulates dissipation due to the elastic reaction of the Earth's interior. As in Figure 2, the result is the sum of a prograde and a retrograde circular polarized wave. For frequencies ν < 0.9 the retrograde wave can be neglected, and there remains the circular propagating prograde wave where the vector of polar motion moves on a circle in anti-clockwise direction. The magnitude of <b>m</b> becomes:<sup id="cite_ref-VollandAR_17-2" class="reference"><a href="#cite_note-VollandAR-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>(6) <span style="padding-left:4em;"> </span> m = 14.5 p<sub>0</sub> ν<sub>C</sub>/[(ν − ν<sub>C</sub>)<sup>2</sup> + ν<sub>D</sub><sup>2</sup>]<sup><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span></sup> <span style="padding-left:5em;"> </span> (for ν < 0.9) </p><p>It is a resonance curve which can be approximated at its flanks by </p><p>(7) <span style="padding-left:4em;"> </span> m ≈ 14.5 p<sub>0</sub> ν<sub>C</sub>/|ν − ν<sub>C</sub>| <span style="padding-left:5em;"> </span> (for (ν − ν<sub>C</sub>)<sup>2</sup> ≫ ν<sub>D</sub><sup>2</sup>) </p><p>The maximum amplitude of m at <span class="nowrap">ν = ν<sub>C</sub></span> becomes </p><p>(8) <span style="padding-left:4em;"> </span> m<sub>max</sub> = 14.5 p<sub>0</sub> ν<sub>C</sub>/ν<sub>D</sub> </p><p>In the range of validity of the empirical formula eq.(2), there is reasonable agreement with eq.(7). From eqs.(2) and (7), one finds the number <span class="nowrap">p<sub>0</sub> ∼ 0.2 hPa</span>. The observed maximum value of m yields <span class="nowrap">m<sub>max</sub> ≥ 230 mas</span>. Together with eq.(8), one obtains </p><p>(9) <span style="padding-left:4em;"> </span> τ<sub>D</sub> = 1/ν<sub>D</sub> ≥ 100 years </p><p>The number of the maximum pressure amplitude is tiny, indeed. It clearly indicates the resonance amplification of Chandler wobble in the environment of the Chandler resonance frequency. </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=Polar_motion&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/International_Polar_Motion_Service" class="mw-redirect" title="International Polar Motion Service">International Polar Motion Service</a></li> <li><a href="/wiki/Pole_shift_hypothesis" class="mw-redirect" title="Pole shift hypothesis">Pole shift hypothesis</a></li> <li><a href="/wiki/Pole_tide" class="mw-redirect" title="Pole tide">Pole tide</a></li> <li><a href="/wiki/True_polar_wander" title="True polar wander">True polar wander</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap 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"><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="CITEREFFolgueira2005" class="citation journal cs1">Folgueira, M. 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Lett</i>. <b>16</b> (3): 247. <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/1989GeoRL..16..247H">1989GeoRL..16..247H</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%2Fgl016i003p00247">10.1029/gl016i003p00247</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geophys.+Res.+Lett.&rft.atitle=Simulation+of+the+14-month+Chandler+wobble+in+a+global+climatic+model&rft.volume=16&rft.issue=3&rft.pages=247&rft.date=1989&rft_id=info%3Adoi%2F10.1029%2Fgl016i003p00247&rft_id=info%3Abibcode%2F1989GeoRL..16..247H&rft.aulast=Hameed&rft.aufirst=S.&rft.au=Currie%2C+R.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+motion" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Kikuchi, I., and I. Naito 1982 Sea surface temperature analysis near the Chandler period, Proceedings of the International Latitude Observatory of Mizusawa, <b>21 K</b>, 64</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_motion&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCarthy2000" class="citation journal cs1">McCarthy, Dennis D. (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0252921100061364">"Polar Motion — an Overview"</a>. <i>International Astronomical Union Colloquium</i>. <b>178</b>. Cambridge University Press (CUP): 221–236. <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/2000ASPC..208..223M">2000ASPC..208..223M</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/10.1017%2Fs0252921100061364">10.1017/s0252921100061364</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0252-9211">0252-9211</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Astronomical+Union+Colloquium&rft.atitle=Polar+Motion+%E2%80%94+an+Overview&rft.volume=178&rft.pages=221-236&rft.date=2000&rft.issn=0252-9211&rft_id=info%3Adoi%2F10.1017%2Fs0252921100061364&rft_id=info%3Abibcode%2F2000ASPC..208..223M&rft.aulast=McCarthy&rft.aufirst=Dennis+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252Fs0252921100061364&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+motion" class="Z3988"></span></li> <li>Fisher, Rick (1996). "<a rel="nofollow" class="external text" href="http://www.cv.nrao.edu/~rfisher/Ephemerides/earth_rot.html">Earth Rotation and Equatorial Coordinates</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110818022004/http://www.cv.nrao.edu/~rfisher/Ephemerides/earth_rot.html">Archived</a> 2011-08-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>". Retrieved Jun. 5, 2005.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunk2002" class="citation journal cs1"><a href="/wiki/Walter_Munk" title="Walter Munk">Munk, Walter</a> (14 May 2002). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC124440">"Twentieth century sea level: An enigma"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. <b>99</b> (10): 6550–6555. <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/2002PNAS...99.6550M">2002PNAS...99.6550M</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/10.1073%2Fpnas.092704599">10.1073/pnas.092704599</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC124440">124440</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12011419">12011419</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=Twentieth+century+sea+level%3A+An+enigma&rft.volume=99&rft.issue=10&rft.pages=6550-6555&rft.date=2002-05-14&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC124440%23id-name%3DPMC&rft_id=info%3Apmid%2F12011419&rft_id=info%3Adoi%2F10.1073%2Fpnas.092704599&rft_id=info%3Abibcode%2F2002PNAS...99.6550M&rft.aulast=Munk&rft.aufirst=Walter&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC124440&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+motion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDick2000" class="citation journal cs1">Dick, Steven J. (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0252921100061170">"Polar Motion: A Historical Overview on the Occasion of the Centennial of the International Latitude Service"</a>. <i>International Astronomical Union Colloquium</i>. <b>178</b>. Cambridge University Press (CUP): 1–24. <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/2000ASPC..208....3D">2000ASPC..208....3D</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/10.1017%2Fs0252921100061170">10.1017/s0252921100061170</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0252-9211">0252-9211</a>.</cite><span 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