Main Oscillatory Polar Motion of the Earth and Its Excitation: Investigation in Both the Frequency and Time Domains

The Fourier spectrum of the polar motion time series shown in Fig. 1, is illustrated in Fig. 2. On the Fig. 2(a), the amplitude spectrum |P(ω)| is drawn in logarithmic scale with distinction of the sense of rotation: prograde and retrograde. On the Fig. 2(b), the power spectrum |P(ω)|² of the polar motion time series is illustrated for a period range of 330–480 days. Although the dominance of two components – Chandler and annual – is quite certain, there exist numerous minor components as shown in Fig. 2(a). It is noted that certain procedures; elimination of linear trend, proper tapering at both ends, and zero paddings were employed on the data time series before calculation of the Fourier spectrum through FFT.

Fig. 2. Polar motion spectrum: (a) Amplitude spectrum in log scale with separation of prograde and retrograde components, (b) Power spectrum near the range of two dominant periodicities. Two half power widths are indicated by arrows.

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The period range of Fig. 2(b) is narrowed for close inspection. The central periods of the two peaks are T = 1.185 yr (432.7 days) and T = 1 yr (almost exactly 365.24 days) for Chandler and annual wobbles. And the corresponding two half-widths are found as ΔT = 8.86 days and ΔT = 4.85 days respectively. For harmonic oscillator response under driving force of uniform amplitude, the quality factor is given as Q = T / ΔT (Fowles 1977). Under the same assumption we inferred two quality factors of Chandler and annual wobbles as Q = 49 and 75 respectively.

Since polar motion is free from external torques, the early formulation of polar motion and its excitation had been made by Munk & MacDonald (1960) starting from the law of angular momentum conservation. The current theory after developments continued was thoroughly documented by Gross (2009). A convenient relation between the polar motion and its excitation function can be written in the frequency domain as the following formula (Na et al. 2016).

where P(ω) and X(ω) are each Fourier transforms of complex polar motion p_c = x_p – iy_p and complex excitation function χ_c = χ₁ + iχ₂ respectively. In fact, this straightforward relation of Eq. (1) is the crucial connection between polar motion and excitation as will be shown below. The Chandler frequency is defined as Ω = ω₀ (1 + 0.5i / Q) / T, where ω₀ is the nominal value of the Earth’s spin rotational angular velocity, and T and Q are the period and quality factor of Chandler wobble. Two components of polar motion excitation function are defined as follows.

where C and A are the two principal moments of inertia of the Earth, and h_i and ΔI_i3 are the perturbing angular momentum and small changes in the Earth inertia tensor (Gross 2009). A brief derivation of Eqs. (1) and (2) is given in Appendix 1. Explicit expressions for the excitation due to the atmospheric pressure and wind are shown in Appendix 2, which can be readily applied to evaluation of the excitation due to the ocean or the continental hydrosphere.

The perturbing roles of atmosphere and ocean on the Earth’s rotation were studied with early data in the 1980s and 1990s. Among those foremost groups, are cited here as; Barnes et al. (1983), Salstein & Rosen (1989), Chao (1993), and Nastula & Ponte (1999). Since 2000, IERS has been publishing the three kinds of effective angular momentum due each to the atmosphere, ocean, and continental hydrosphere (Dobslaw et al. 2010). As a matter of fact, these datasets were provided by GFZ, and ESMGFZ datasets exist quite early from year 1976 and afterwards. In Fig. 3, two components χ₁1 and χ₂ of the combined time series of three excitations together [atmo + ocean + hydro(con)] are shown. Two sets of each separate contributions from different fluid spheres are also shown in Appendix 2.

Fig. 3. Two components χ₁ and χ₂ of the combined polar motion excitation. These are the direct sum of the atmospheric, oceanic, and continental hydrosphere excitations. Data from IERS (2025).

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The polar motion excitation can also be deduced from polar motion data by Eq. (1) via forward/inverse Fourier transformations. And this excitation function has been called ‘geodetic excitation’ to make a distinction from the excitation function estimated from the information on the Earth’s fluid spheres. In fact, after comparison of two kinds of power spectra; fluid spheres excitation and geodetic excitation, Gross early addressed that ‘there seems to be enough power to maintain Chandler wobble in the ocean and atmosphere together’ (Plag et al. 2005). Yet such comparison has not been frequently made in the time domain except two cases (Xu et al. 2024; Yamaguchi & Furuya 2024). From comparison of the geodetic excitation (acquired for selected values of the two parameters T and Q of Chandler wobble) with fluid spheres excitation, one may determine optimal values of T and Q. Here the residual sum is defined as follows.

where summation is carried for the frequency range of Chandler wobble. This sum of Eq. (3) has been repeatedly calculated in the ranges of 428 < T < 438 days and 0 < Q < 250. As a result, the optimal set: T = 433.2 days and Q = 62.8 has been found in this study and are illustrated in Fig. 4. In fact, we repeated calculations with different time windows for the optimal values. Our 1σ-interval estimates of period and quality factor are T = 433.2 ± 1.4 days and Q = 63 ± 17. Former estimates made by similar approach have varied as 429 < T < 432 days and 83 < Q < 107 (Gross 2009). It is noted here that we took care in the sum (3): (i) first separate the time series of daily basis into four ones of 4 day-interval (t=4k, 4k+1, 4k+2, 4k+3), (ii) perform Fourier transform for each separate time series, and (iii) add up the four each spectrum for each T and Q. In this way, the spectral resolution was enhanced four times. The Nyquist frequency is lowered accordingly, but that does not lead to any difficulty. This method was applied to both the fluid excitation and polar motion.

Fig. 4. Determination of Chandler wobble period and quality factor from minimum residual between fluid spheres excitation and geodetic excitation. Unit of the residual: [10^-6 arcsec²].

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For many decades earthquakes were regarded as possible energy sources of polar motion, particularly Chandler wobble. However, compared with damages at the Earth’s surface or well-detected seismic waves, the perturbation of Earth’s spin rotational state due to earthquakes is not quite noticeable. Dahlen derived the formula for the excitation in terms of seismic moment (Dahlen 1971, 1973). Lambeck compiled studies of Dahlen with other related ones, and noted the importance of aseismic excitation (Lambeck 1980). Gross and Chao calculated the seismic excitation on the polar motion (Gross 1986; Chao & Gross 1987; Gross & Chao 2006). Brzezinsky compiled and extended the formulation (Brzezinsky 2005). Smylie early addressed that large earthquakes should affect the pole position in discernable amount and later claimed that prediction of large earthquake can be made from the polar motion monitoring (Smylie & Mansinha 1968; Smylie & Zuberi 2009). However, seismic pole shifts are not large enough to be detected from polar motion observation, because the coseismic jump of the principal axis does not lead to discontinuity in the polar motion (Chung & Na 2016). Cambiotti et al. did extensive study to model polar motion due to earthquakes by using a new and elaborate scheme (Cambiotti et al. 2016). Xu and Chao renewed the estimate of seismic excitation and addressed that the cumulative seismic excitations should be accounted in the long-term polar motion (Xu & Chao 2019). By using an empirical relation, Na and Kyung approximately attained the effect of post seismic deformation unto the polar motion excitation by earthquakes (Hearn 2003; Na & Kyung 2016).

The procedure to calculate the coseismically excited polar motion for a single earthquake is followings: (i) first, evaluate seismic moment tensor M_ij for the earthquake, (ii) calculate the corresponding change in the Earth’s rotational inertia tensor components ΔI₁₃ and ΔI₂₃, (iii) calculate the excitation components (χ₁, χ₂) using Eq. (2), (iv) calculate the resultant coseismic polar motion using Eq. (1) (Fig. 5).

Fig. 5. Excited Chandler wobble due to largest earthquakes since 1980: (a) Two calculated components χ₁ and χ₂ of the polar motion excitation function by 37 largest earthquakes, (b) Two components of seismically excited Chandler wobble for assuming τ = 125 days, (c) 3-dimensional illustration of (b).

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To evaluate polar motion excitation with assuming postseismic deformation is quite difficult task. One possible way is to adopt a homogeneous relaxation model and then apply it to assess polar motion excitation associated with it. Na & Kyung (2016) attempted this approach in their calculation of polar motion excitation due to earthquake including postseismic deformation. Their simple model for postseismic deformation was taken as follows,

in which formula, the postseismic deformation is described as exponential decay with assigned characteristic time τ and the ratio between u₀ and u₁. The constants here were adopted as u₁ = 0.655u₀ for τ = 125 days. In fact, our choice of time constant here is the logarithmic mean value of two limits: τ = 80 days and τ = 200 days after Hearn (2003). The value u₁ = 0.655u₀ is determined by interpolation of two limiting values from fitting the observed crust relaxation after 2010 Tohoku Earthquake (Na & Kyung 2016). More theoretical details with parts of the calculated excitations are given in Appendix 3.

37 major earthquakes with magnitudes greater than 8 have occurred on the globe since 1981 until Feb 2025. Using the information of earthquakes from the USGS website, two components (χ₁, χ₂) of polar motion excitation function due to those earthquakes were assessed and are illustrated in Fig. 5(a). The illustrated three cases of excitations for assuming the post seismic deformation were calculated for the three characteristic relaxation time: 80, 125 and 200 days. The amounts of excitation increased by 52, 66 and 82 percents accordingly from the coseismic excitation. The greatest coseismic excitation of polar motion was (–2.064, 2.356) milliarcsec for the 2011 Tohokou Earthquake in Japan. And the corresponding coseismic pole shift was 9.7 cm at the Earth’s surface. The second largest coseismic pole shift was (–0.623, 0.561) milliarcsec for the 2004 Sumatra Earthquake. The seismically excited Chandler wobble due to those large earthquakes after 1980 is calculated and illustrated in Fig. 5(b) and 5(c). The polar motion excitation due to each earthquake is smaller than the combined excitation of atmospheric, oceanic, and hydrologic excitations by two orders of magnitude. The total effect of earthquakes with assuming post seismic relaxation is still smaller than the fluid spheres excitation by one order of magnitude.

It seems plausible that certain motion deep in the Earth’s core should affect the rotation of the Earth. Geomagnetic jerk has been suspected as an energy source of Chandler wobble (Gibert & Le Mouël 2008; An & Ding 2022; and several references cited in them). We tried to confirm their findings that Chandler phase variation has been triggered by geomagnetic jerks. However, unlike their claims, the phase variation of Chandler wobble does not show close causal relationship with the jerks (Fig. 6).

Fig. 6. Phase variation of Chandler wobble shown together with dotted lines at epochs of the known geomagnetic jerks.

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According to Aubert & Finlay (2019), the amount of typical torque and time duration of geomagnetic jerk are approximately 1 × 10¹⁶ [Nm] and about 1 yr, which corresponds to angular momentum of 3.2 × 10²³ [kg m² s^–1]. Since the spin rotational angular momentum of the Earth is 5.86 × 10³³ [kg m² s^–1], two limiting cases can be assumed as follows: (i) torque exerts to deflect the Earth rotation axis, (ii) torque exerts to accelerate (decelerate) the earth rotation angular velocity. For the first case, the deflection angle will be 5.4 × 10^–11 [rad], which corresponds to 0.34 mm at the surface of the Earth. With the second case Δl.o.d = 4.6 × 10^–6 [sec]. Since these two simple calculations are for the maximum possible exertion of geomagnetic torque in each case, the truth would be smaller than these. Therefore, we do not agree with the former investigations addressing the possibility that Chandler wobble is significantly affected or driven by the geomagnetic jerks. The related details are given in Appendix 4.

From the known values of years-long polar motion excitation, one can deduce the corresponding polar motion by using the simple relation of the two Fourier transform pair in the frequency domain; Eq. (1). We then separately calculated the transient polar motion and added it unto the acquired polar motion after Eq. (1). Fig. 7 is a composite of two panels: (a) the constructed polar motion from the combination of atmospheric, ocean, and hydrologic excitations, (b) observed polar motion. The seismically excited polar motion is mostly of Chandler wobble period. For comparison with Fig. 7(a), the earthquake excited portion and secular movement have been deducted from the polar motion of Fig. 7(b). It is noted here that we initially acquired the fluid excitation time series data from IERS and calculated for only after year 2000. We learned the existence of original and longer dataset in ESMGFZ starting from 1976. The polar motion before 1981 is again omitted (constructed polar motion of these several year time span is much smaller and our seismic effect calculation starts in 1981).

Fig. 7. Comparison of constructed and observed polar motion time series sets: (a) polar motion due to fluid spheres excitation (atmo + ocean + hydro), (b) observed polar motion devoid of follows; (i) slow trend of Earth’s principal axis movement and (ii) earthquake excitation. Units of polar motion: [arcsec].

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The cause of polar motion as known as the combination of atmospheric, oceanic, and hydrologic effects is explicitly shown in Fig. 7; where the constructed polar motion is compared with the observed polar motion. Not only their amplitudes but also the waveforms of the two sets of time series (Fig. 7(a) and 7(b)) are comparable and in fair match to each other. However, the root mean square value of the difference between the constructed time series and the observed one is found 0.062 arcsec. Therefore, there still remains substantial misfit, which should be ascribed to the limited accuracy in the dataset of the fluid spheres states.

As noted above, atmospheric perturbation was early claimed as the cause of polar motion, and later oceanic and hydrologic effects were recognized. Dozens of studies then followed. However, for more than three decades, almost all those analyses were carried in the frequency domain only. Quite recently, there were two reports in which construction of polar motion is attempted (Xu et al. 2024; Yamaguchi & Furuya 2024). This was due to the shortage of dataset of fluid spheres excitation before 2000s, and also the difficulty in integral evaluation of polar motion in the time domain (Barnes et al. 1983; Chao 1985).

We constructed polar motion for four more cases of different combination of parameters: period T and Q-value of Chandler wobble: (T, Q) = (437, 120), (429, 120), (429, 25), and (437, 25). The four pairs of these additionally constructed polar motions are illustrated in Fig. 8. We found that the corresponding misfit is increased in the r.m.s. value as 0.135, 0.087, 0.097 and 0.110 arcsec respectively.

Fig. 8. More comparison between the observed polar motion and the constructed polar motion. Two components x_p and y_p are separately shown (a), (b). In both (a) and (b), from the top, observed polar motion and five constructed ones. ‘con (i, ii, iii, iv)’ stand for constructed with (T, Q) = (437, 120), (429, 120), (429, 25), and (437, 25). Duplicate observed polar motion in magenta (Unit: 0.2 arcsec).

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