Polar motion of the Earth is the motion of the Earth's rotational axis
relative to its crust.:1 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.
3 Basic principles
6.1 Annual component
6.2 Chandler wobble
7 See also
Polar motion is defined relative to a conventionally defined reference
axis, the CIO (Conventional International Origin), being the pole's
average location over the year 1900. It consists of three major
components: a free oscillation called
Chandler wobble with a period of
about 435 days, an annual oscillation, and an irregular drift in the
direction of the 80th meridian west, which has lately been shifted
toward the east.:1
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.[dubious – discuss] The latter are determined
based on the International Terrestrial Reference System, which follows
the polar motion.
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
Greenland ice sheet melts, and to isostatic rebound, i.e.
the slow rise of land that was formerly burdened with ice sheets or
glaciers.:2 The drift is roughly along the 80th meridian west.
However, since about year 2000, the pole has found new direction of
drift, which is roughly along the central meridian. This dramatic
eastward shift in drift direction is attributed to the global scale
mass transport between the oceans and the continents.:2
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.
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
Euler period of free nutation is
(1) τE = 1/νE = A/(C − A) sidereal days ≈ 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.
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.
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
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. 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
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. 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.
Polar motion is observed routinely by very-long-baseline
interferometry, lunar laser ranging and satellite laser
ranging. 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.
Chandler wobble is usually considered a resonance phenomenon, a
free nutation that is excited by a source and then dies away with a
time constant τD of the order of 100 years. It is a measure of
the elastic reaction of the Earth. 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. This reciprocal behavior
between amplitude and frequency has been described by the empirical
(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. Atmospheric excitation seems to be the most
likely candidate. Others propose a combination of atmospheric
and oceanic processes, with the dominant excitation mechanism being
ocean‐bottom pressure fluctuations.
Current and historic polar motion data is available from the
International Earth Rotation and Reference Systems Service
International Earth Rotation and Reference Systems Service Earth
Orientation products. 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.
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)
There is now general agreement that the annual component of polar
motion is a forced motion excited predominantly by atmospheric
dynamics. 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:
(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/m2 =νC
where νC is the Chandler resonance frequency. The result is in good
agreement with the observations. From Figure 2 together with
eq.(4), one obtains νC = 0.83, corresponding to a Chandler resonance
(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%.
It is improbable that the internal parameters of the Earth responsible
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, and a regional
14-month signal in regional sea surface temperature has been
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:
(6) m = 14.5 po νC/[(ν - νC)2 + νD2]1/2 (for ν <
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 ≫
The maximum amplitude of m at ν = νC becomes
(8) mmax = 14.5 po νC/νD
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.
Pole shift hypothesis
True polar wander
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