# Polar motion

Polar motion as function of time. z-axis in day units, x&y in arcsec (0.1 arcsec ≈ 3 meters)

Polar motion of the Earth is the movement of the point where Earth's rotational axis intersects its surface. This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called Earth-centered, Earth-fixed or ECEF reference frame). This variation is only a few meters.

## Analysis

It consists of two quasi-periodic components and a gradual drift, mostly in the direction of the 80th meridian west, of the Earth's instantaneous rotational axis or North pole, from a conventionally defined reference axis, the CIO (Conventional International Origin), being the pole's average location over the year 1900.

The two periodic parts are a more or less circular motion called Chandler wobble with a period of about 435 days, and a yearly circular motion. There is also a longer term variation which is less well understood. These motions are illustrated on the Earth Orientation Center of the International Earth Rotation and Reference Systems Service (mas = milliarcseconds corresponds to about 3 cm):

The mean displacement far exceeds the magnitude of the wobbles. This can lead to errors in software for Earth observing spacecraft, since analysts may read off a 5 meter circular motion and ignore it, while a 20 meter offset exists, fouling the accuracy of the calculated latitude and longitude. The latter are determined based on the International Terrestrial Reference System, which follows the polar motion.

## Causes

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 Greenland ice sheet melts, and to isostatic rebound, i.e. the slow rise of land that was formerly burdened with ice sheets or glaciers.[1] The drift is roughly along the 80th meridian west.

Major earthquakes cause abrupt polar motion by altering the volume distribution of the Earth's solid mass. These shifts, however, are quite small in magnitude relative to the long-term core/mantle and isostatic rebound components of polar motion.[2]

## Basic Principles

In the absence of external torques, the vector of the angular momentum M of a rotating system remains constant and is directed toward a fixed point in space. In the case of the Earth, it is almost identical with its axis of rotation. The vector of the figure axis F of the system wobbles around M. This motion is called Euler's free nutation. For a rigid Earth which is an oblate spheroid to a good approximation, the figure axis F is its geometric axis defined by the geographic north and south pole. It is identical with the axis of its polar moment of inertia. The Euler period of free nutation is

(1)   τE = 1/νE = A/(C - A) = 307 sidereal days = 0.84 sidereal years

νE = 1.19 is the normalized Euler frequency (in units of reciprocal years), C = 8.04 × 1037  kg m2 is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and C - A = 2.61 × 1035 kg m2.[3][4]

The observed angle between M and F is a few hundred milliarcseconds (mas) which gives rise to a surface displacement of several meters (100 mas corresponds to 3.09 m) between the figure axis of the Earth and its angular momentum. 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.[5]

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 Chandler wobble. There exist, in addition, polar motions with smaller periods of the order of decades.[6] Finally, a secular polar drift of about 0.10 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.[5]

The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of the variation of the latitude of stars, while S.C. Chandler found the free nutation in 1891.[5] Both periods superpose, giving rise to a beat frequency with a period of about 5 to 8 years (see Figure 1).

This polar motion should not be confused with the changing direction of the Earth's spin axis relative to the stars with different periods, caused mostly by the torques on the Geoid due to the gravitational attraction of the Moon and Sun. They are also called nutations, except for the slowest, which is the precession of the equinoxes.

## Observations

Polar motion is observed routinely by Very Long Base Line Interferometry,[7] by Lunar Laser Ranging, and by Satellite Laser Ranging.[8] 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.

The Chandler wobble is usually considered a resonance phenomenon, a free nutation, which is excited by some source and then dying away with a time constant τD of the order of 100 years. This term is a measure of the elastic reaction of the Earth.[9] This 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.[10] This reciprocal behavior between amplitude and frequency has been described by the empirical formula:[11]

(2)   m = 3.7/(ν - 0.816)   (for 0.83 < ν < 0.9)

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.[8][12] Atmospheric excitation seems to be the most likely candidate.[13][14] Others propose a combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations.[15]

## Data

Current and historic polar motion data is available from the International Earth Rotation and Reference Systems Service Earth Orientation products.[16] Note in using this data that the convention is to define px to be positive along 0° longitude and py to be positive along 90°W longitude.[17]

## Theory

### Annual Component

Figure 2. Displacement 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. (100 mas (milliarcseconds) = 3.09 m on the Earth's surface)

It is now general agreement that the annual component of polar motion is a forced motion excited predominantly by atmospheric dynamics.[18] 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:[14]

(3)   p = poΘ-31(θ) cos[(2πνA (t - to)] cos(λ - λo)

with po a pressure amplitude, Θ-31 a Hough function describing the latitude distribution of the atmospheric pressure on the ground, θ the geographic co-latitude, t the time of year, to a time delay, νA = 1.003 the normalized frequency of one solar year, λ the longitude, and λo the longitude of maximum pressure. The Hough function in a first approximation is proportional to sinθ cosθ. 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)   m1/m2C

where νC is the Chandler resonance frequency. The result is in good agreement with the observations.[4][19] From Figure 2 together with eq.(4), one obtains νC = 0.83, corresponding to a Chandler resonance period of

(5)  τC = 441 sidereal days = 1.20 sidereal years

po = 2.2 hPa, λo = - 170° the latitude of maximum pressure, and to = - 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%.[20]

### 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,[21] and a regional 14-month signal in regional sea surface temperature has been observed.[22]

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 ν + iνD, where νD 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:[14]

(6)   m = 14.5 po νC/[(ν - νC)2 + νD2]1/2   (for ν < 0.9)

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

(7)   m ≈ 14.5 po νC/|ν - νC|   (for (ν - νC)2 ≫ νD2)

The maximum amplitude of m at ν = νC becomes

(8)   mmax = 14.5 po νCD

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 po ∼ 0.2 hPa. The observed maximum value of m yields mmax ≥ 230 mas. Together with eq.(8), one obtains

(9)   τD = 1/νD ≥ 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.

## References

1. ^ Munk, Walter (2002).
2. ^ [1]
3. ^ Munk W. H. and MacDonald, G. J. F., The Rotation of the Earth, Cambridge University Press, London, 1960
4. ^ a b Lambeck, K., The Earth's Variable Rotation: Geophysical Causes and Consequences, Cambridge University Press, Cambridge, 1980
5. ^ a b c Moritz, H., and I. Mueller, "Earth Rotation", Ungarn Publ. Comp., New York, 1987
6. ^ Gross, R.S., and U.J. Lindqwister, Atmospheric excitation of polar motion during the GIG 91 measurement campaign, Geophys. Res. Lett., 19, 849, 1992
7. ^ Schuh, H., Earth's rotation measured by VLBI, in Brosche, O., and J. Sündermann (eds.): "Earth's Rotation from Eons to Days", Springer, Heidelberg, 1990
8. ^ a b Eubanks, T.M., Variations in the orientation of the earth, in Smith, D.E., and D.L. Turcotte (eds.): "Contribution of Space Geodynamics", Geodynamic Series, 24, AGU, Washington, D.C., 1993
9. ^ Dickey, J.O., and Eubanks, T.M., Earth Rotation and Polar Motion: Measurements and Implications, IEEE Transact. Geosci. Remote Sensing, GE-23, 373, 1985
10. ^ Guinot, B., The Chandrerian wobble from 1900 to 1970, Astron. Astrophys., 19, 07, 1992
11. ^ Vondrak, J., Long-periodic behaviour of polar motion between 1900 and 1980, A. Geophys., 3, 351, 1985
12. ^ Runcorn, S.K., et al., The excitation of the Chandler wobble, Surv. Geophys., 9, 419, 1988
13. ^ Hide, Rotation of the atmosphere of the earth and planets, Phil. Trans. R. Soc., A313, 107, 1984
14. ^ a b c Volland, H., Atmosphere and Earth' Rotation, Surv. Geophys., 17, 101, 1996
15. ^ Gross, R., The excitation of the Chandler Wobble, Geophys. Res. Letters, 27, 2329, 2001
16. ^ http://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html
17. ^ "IERS Conventions 2010: Chapter 8". p. §8.3.
18. ^ Wahr, J.M., The Earth's Rotation, Ann. Rev. Earth Planet. Sci., 16, 231, 1988
19. ^ Jochmann, H., The Earth rotation as a cyclic process and as an indicator within the Earth's interior, Z. geol. Wiss., 12, 197, 1984
20. ^ 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
21. ^ Hameed, S., and R.G. Currie, Simulation of the 14-month Chandler wobble in a global climatic model, Geophys. Res. Lett., 16, 247, 1989
22. ^ Kikuchi, I., and I. Naito, Sea surface temperature analysis near the Chandler period, Proceedings of the International Latitude Observatory of Mizusawa, 21 K, 64, 1982