Editing Polar motion (section)

==Theory==

===Annual component===
[[File:annualmotion1.jpg|thumb|upright=1.4|Figure 2. [[Rotation vector]] '''m''' 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. ({{nowrap|100 mas (milliarcseconds) {{=}} 3.082 m}} on the Earth's surface at the poles)]]
There is now general agreement that the annual component of polar motion is a forced motion excited predominantly by atmospheric dynamics.<ref>{{cite journal | last1 = Wahr | first1 = J.M. | year = 1988 | title = The Earth's Rotation | journal = Annu. Rev. Earth Planet. Sci. | volume = 16 | page = 231 | doi=10.1146/annurev.ea.16.050188.001311| bibcode = 1988AREPS..16..231W| s2cid = 54540284 }}</ref> 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:<ref name=VollandAR/>

(3) {{pad|4em}} p = p<sub>0</sub>Θ{{sup sub|1|−3}}(θ) cos[2πν<sub>A</sub>(t − t<sub>0</sub>)] cos(λ − λ<sub>0</sub>)

with p<sub>0</sub> a pressure amplitude, Θ{{sup sub|1|−3}} a [[Hough function]] 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, {{nowrap|ν<sub>A</sub> {{=}} 1.003}} 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&nbsp;θ cos&nbsp;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 '''m''' of the annual component describes an ellipse (Figure 2). The calculated ratio between major and minor axis of the ellipse is

(4) {{pad|4em}} m<sub>1</sub>/m<sub>2</sub> = ν<sub>C</sub>

where ν<sub>C</sub> is the Chandler resonance frequency. The result is in good agreement with the observations.<ref name=Lambeck/><ref>Jochmann, H., The Earth rotation as a cyclic process and as an indicator within the Earth's interior, Z. geol. Wiss., '''12''', 197, 1984</ref>

From Figure 2 together with eq.(4), one obtains {{nowrap|ν<sub>C</sub> {{=}} 0.83}}, corresponding to a Chandler resonance period of

(5) {{pad|4em}} τ<sub>C</sub> = 441 sidereal days = 1.20 sidereal years

{{nowrap|p<sub>0</sub> {{=}} 2.2 hPa}}, {{nowrap|λ<sub>0</sub> {{=}} −170°}} the latitude of maximum pressure, and {{nowrap|t<sub>0</sub> {{=}} −0.07 years {{=}} −25 days}}.

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%.<ref>[[John M. Wahr|Wahr, J.M.]], The effects of the atmosphere and oceans on the Earth's wobble — I. Theory, Geophys. Res. J. R. Astr. Soc., '''70''', 349, 1982 {{doi|10.1111/j.1365-246X.1982.tb04972.x}}</ref>

===Chandler wobble===
{{main|Chandler wobble}}

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,<ref>{{cite journal | last1 = Hameed | first1 = S. | last2 = Currie | first2 = R.G. | year = 1989 | title = Simulation of the 14-month Chandler wobble in a global climatic model | journal = Geophys. Res. Lett. | volume = 16 | issue = 3| page = 247 | doi=10.1029/gl016i003p00247 | bibcode=1989GeoRL..16..247H}}</ref> and a regional 14-month signal in regional [[sea surface temperature]] has been observed.<ref>Kikuchi, I., and I. Naito 1982 Sea surface temperature analysis near the Chandler period, Proceedings of the International Latitude Observatory of Mizusawa, '''21 K''', 64</ref>

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 {{nowrap|ν + iν<sub>D</sub>}}, 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 '''m''' becomes:<ref name=VollandAR/>

(6) {{pad|4em}} 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>{{frac|1|2}}</sup> {{pad|5em}} (for ν < 0.9)

It is a resonance curve which can be approximated at its flanks by

(7) {{pad|4em}} m ≈ 14.5 p<sub>0</sub> ν<sub>C</sub>/|ν − ν<sub>C</sub>| {{pad|5em}} (for (ν − ν<sub>C</sub>)<sup>2</sup> ≫ ν<sub>D</sub><sup>2</sup>)

The maximum amplitude of m at {{nowrap|ν {{=}} ν<sub>C</sub>}} becomes

(8) {{pad|4em}} m<sub>max</sub> = 14.5 p<sub>0</sub> ν<sub>C</sub>/ν<sub>D</sub>

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 {{nowrap|p<sub>0</sub> ∼ 0.2 hPa}}. The observed maximum value of m yields {{nowrap|m<sub>max</sub> ≥ 230 mas}}. Together with eq.(8), one obtains

(9) {{pad|4em}} τ<sub>D</sub> = 1/ν<sub>D</sub> ≥ 100 years

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.