NOTES AND CORRESPONDENCE A note on the annual wobble excitation due to the seasonal atmospheric loading on continents

Northern Eurasian continent has been regarded to contribute as the major source area, of which seasonal atmospheric pressure loading and unloading cycle leads to the annual wobble of the Earth. In the early days when reliable data of global coverage were not accessible, this dominance has remained as only a hypothesis. Nowadays, however, European Centre for Medium-Range Weather Forecasts and National Centers for Environmental Prediction produce reliable datasets, therefore, such unique feature has become clearer with quantitative evidences. Having both the Earth’s polar motion and the global atmospheric state known with unprecedented accuracy, we hereby identify and scrutinize whether Siberia and Manchuria dominate the annual polar motion of the Earth. Article history: Received 23 February 2018


INTRODUCTION
Although the Earth rotation is fairly uniform and consistent in times of days or thousand years, at close observation aided by space geodetic measurements, the Earth's spin rotation axis undergoes wobbling which is rather small (usually its amplitude is a few meters on the Earth's surface) but quite complicated in nature. Change in Earth rotation can be categorized into two kinds; (1) change in its angular spin rate or length of day (LOD) and (2) change in the orientation of the Earth's spin axis, that is, precession-nutation and wobble. LOD variation associated with the seasonal change in the global atmospheric wind pattern had been suspected and was discovered in the middle of the last century. Other interesting topics are involved with secular and decadal variations of LOD. Precession and nutation are forced motions of the Earth's rotational axis in the celestial sphere due mainly to lunisolar tidal force on the equatorial bulge of the Earth, and therefore, they are not associated with the processes in the Earth. In modern and formal definition, wobble is the movement of the Earth's rotational pole in the terrestrial reference system of which frequency is either greater than -0.5 or smaller than -1.5 cycle per sidereal day [IERS Conventions 2010 (Petit andLuzum 2010); here the minus sign refers to 'retrograde', i.e., the reverse of the Earth's spin]. In Fig. 1, recent polar motion since 1981 is illustrated based on the dataset of IERS EOP C04. EOP C04 is basically daily basis dataset composed of pole position coordinates and their uncertainties with changes in LOD and UT1, and it is a combined solution from different space geodetic observations; VLBI, GPS, SLR, and DORIS (Bizouard and Gambis 2009). Annual wobble is one of the two main components of Earth's wobbling motion. While Chandler wobble excitation mechanism has been controversial for a long time, it is evident that annual wobble is driven by geophysical phenomena having periodic Terr. Atmos. Ocean. Sci., Vol. 29, No. 6, 721-729, December 2018 seasonal variations (see for example, Munk and MacDonald 1960;Lambeck 1980;Gross 2009). And it has been presumed since early times that Siberia does a noticeable role in the annual wobble excitation, i.e., periodic loading/unloading of air mass over the wide Siberian land may result in major contribution on the Earth's annual wobble (Munk and MacDonald 1960). Although the torque associated with the seasonal changes in the global atmospheric circulation pattern result in the excitation of the same order of magnitude, however, such contribution on each continents are generally regarded as smaller than that induced by mass changes.
Information of worldwide weather are gathered and assimilated in European Centre for Medium-Range Weather Forecasts (ECMWF) and National Centers for Environmental Prediction (NCEP) for the purpose of weather forecasting as well as to provide datasets necessary for scientific investigations. The 6-hr period global data coverage by ECMWF on wind velocity and barometric pressure can be used to infer the atmospheric excitation of Earth rotation.
Monthly average values of surface atmospheric pressure in the year of 2016 are illustrated in Fig. 2. The pressure values in this figure were not adjusted down to sea-level but taken as on-site raw values so that they are used for evaluation of excitation integral. However pressure deviation from local average value should be used unto the integral unless whole globe is simultaneously considered. Monthly average values of differential surface atmospheric pressure from local yearly average in the year of 2016 are illustrated in Fig. 3. On the northern part of Eurasian continent, i.e., Siberia and Manchuria, high atmospheric pressure exists from October to March, while low atmospheric pressure prevails from May to August. For closer comparison between winter and summer seasons January and July average pressure as well as the deviatoric pressure from the local year average are il-lustrated together in Fig. 4. Although this seasonal loading/ unloading cycle has been thought as the main input to annual wobble since the beginning of last century (Munk and MacDonald 1960), reliable estimations have been deferred until the advent of accurate source of information, such as ECMWF dataset or comparable sort.
In this study, we compare (1) the observed geodetic excitation inferred from the recent polar motion data and (2) the calculated atmospheric excitation due to seasonal atmospheric loading/unloading on Northern Eurasia as well as four other continents, namely, North America, South America, Australia, and Africa. We used IERS EOP C04 and ECMWF atmospheric pressure datasets each.

For complex polar motion
, one can define their Fourier transform pairs as ( ) P~ and ( ) X~. Then a simple relation holds between ( ) P~ and ( ) X~ in the frequency domain as follows (Na 2013).
m with the Earth's rotation angular velocity 0 and the Chandler wobble quality factor Q (taken as 100). Therefore, from a dataset of polar motion, one can derive its excitation function by Fourier transform and successive manipulation according to Eq. (1) and then inverse Fourier transform. Usually the excitation function derived from polar motion time series through this procedure is called 'geodetic excitation'.
With known perturbation due to certain geophysical   processes on the Earth's surface or its interior, polar motion excitation function can be expressed again as follows (Gross 2009).
( ) where C and A are the two principal moments of inertia of the Earth along z-axis and x-axis, and h i and ΔI i3 are the imposed perturbing angular momentum and the perturbation in the Earth's inertia tensor. In case of considering barometric pressure loading/unloading only, the polar motion excitation function can be expressed as follows (Eubanks 1993;Gross 2009).
( , )cos sin cos C A g a P d d 1 100 ) .
( , )cos sin sin C A g a P d d 1 100 where ΔP is the excessive barometric pressure, and a and g are the radius and surface gravity of the Earth.
To extract seasonal variation (1-year period sinusoidal oscillation) from given time series of polar motion, we used least square fitting based on the following model (Chung and Na 2016).
where 1 X and 2 X are the frequencies of Chandler and annual wobbles. A k and B k are the amplitudes of each components. Similarly, least square error fittings have been done on the observed or calculated excitation functions as follows.

RESULTS
The geodetic (observed) excitation function was acquired from the polar motion dataset, IERS EOP C04. The 1 | component of the observed excitation function after 2010 is illustrated in Fig. 5. Its five graphs are (1) geodetic excitation acquired from polar motion data, (2) atmospheric excitation (wind + pressure: global), (3) atmospheric excitation (wind only: global), (4) atmospheric excitation (pressure only: global), and (5) atmospheric excitation (pressure: Siberia and Manchria). Similarly the excitation function 2 | after 2010 is illustrated in Fig. 6. It can be found that atmospheric pressure excitation on 2 | is roughly about half of the observed excitation. Figure 7 shows the geodetic excitation function and together with its annual oscillation component acquired by least square fitting. It is noted that fitting itself was done on seventeen-year dataset, while only the latter five-year data    and fit are shown on the figure. Although not shown here, the annual wobble component is acquired similarly on the polar motion. Figure 8 shows two elliptical loci of (1) the annual wobble component and (2) the annual excitation.
We repeated the computation for atmospheric pressure excitation on other continents, North America, South America, Australia, and Africa. The amplitudes of the geodetic excitation and calculated excitation due to atmospheric pressure on the five continents are provided in Table 1. And, in different forms, the total-observed and calculated excitations are again illustrated together in Fig. 9 for their comparison. It has been found that the atmospheric pressure loading/unloading annual cycle on Siberia and Manchuria gave rise to 15.61 milliarcsec excitation in 2 | component, which is 54% of total excitation observed, while the same area gave rise to much smaller 1 | excitation. The atmospheric loading/unloading on Australia gave rise to rather large excitation both on 1 | and 2 | .

DISCUSSION AND CONCLUSION
Atmospheric seasonal loading/unloading cycle on Siberia and Manchuria results in more than fifty percentage (54%) of the observed excitation. And this holds not only for 2 | but also for total amplitude including 1 | component. To see more clearly the seasonal atmospheric loading cycle on Siberia, which is shown in the monthly average barometric pressure distribution in Figs. 2 and 3, another kinds of maps are drawn as Figs. 10 and 11, which show the January and July monthly average barometric pressure on the northern hemisphere using equal area projection. From the maps of Figs. 10 and 11, one can readily see the atmospheric loading cycle on the northern Eurasia -Siberia and Manchuria. Also one can see why 1 | excitation associated with Siberia and Manchuria area is much smaller than 2 | excitation because the center of the area (Siberia + Manchuria) is quite close to y-axis (longitude 90c m = + ). It should be noted that high/low barometric pressure on the vast area of oceans has almost no contribution to polar motion excitation. This is due to the fact that ocean water would re-distribute under atmospheric pressure (often called 'inverted barometer') so that ocean bottom pressure exerts virtually no seasonal perturbation on the Earth's rotation.
Although the amplitude of 2 | component of the observed excitation is much larger than that of 1 | , the annual polar motion has been more or less circular (Fig. 8). Other feature is that high frequency components exist in large amounts in both 1 | and 2 | components of the observed excitation (Figs. 5,6,and 7), in other words, the polar motion time series show much smoother variation than its excitation time series. Actually, this smaller amount of high frequency content is due the fact that polar motion is, in a sense, a time integral of the polar motion excitation function.
Atmospheric pressure excitation functions calculated   notable that North American continent has a small fraction of contribution to the polar motion excitation. Africa, being centered near the equator, may be overlooked as to exert small amount of loading excitation. However, it does affect in considerable amount as shown in Fig. 9 and Table 1. Total atmospheric excitation 2 | acquired from ECM-WF data exceeds the observed excitation during the whole time span as observed in Fig. 6. Had we have data for other possible causes, such as, ground water level change or glacier melting rate variation, that inequality might be reasonably explained.
From the comparison of observed geodetic excitation inferred from IERS C04 polar motion dataset and calculated excitation function via integration of ECMWF pressure data, we hereby confirm that the atmospheric seasonal loading cycle in the area of Siberia and Manchuria with their neighbors in the Northern Eurasian continent, plays the major role in the annual wobble of the Earth's spin rotation axis in the Earth's reference system.