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Earth’s temporal principal moments of inertia and variable rotation

Based on the gravity field models EGM96 and EIGEN-GL04C, the Earth’s time-dependent principal moments of inertia A, B, C are obtained, and the variable rotation of the Earth is determined. Numerical results show that A, B, and C have increasing tendencies; the tilt of the rotation axis increases 2.1×10−8 mas/yr; the third component of the rotational angular velocity, ω3, has a decrease of 1.0×10−22 rad/s2, which is around 23% of the present observed value. Studies show in detail that both θ and ω3 experience complex fluctuations at various time scales due to the variations of A, B and C.


Introduction
Concerning the study of the Earth's rotation, man has investigated the causes of the variations of the angular velocity and of the position of the Earth's rotation axis in space and on Earth (precession, nutations, polar motion, and polar wander). The variation of the Earth's angular velocity is directly related to the variation of the length-of-day (LOD). Polar motion corresponds to the displacement of the rotation axis in a frame tied to the Earth. It takes place within a square, 15 m long on each side during seasonal or sub-seasonal time scales. At the seasonal time scale, polar motion has two main components: the Chandler wobble with a period of about 435 days, and an annual component. The polar wander is the long-term component of the motion of the pole. The precession is the long-term component of the motion of the pole in space; nutations are the corresponding periodic components (Dehant & de Viron, 2002).
As we know, the mass redistribution within the Earth is an important factor that influences the rotation of the Earth but was poorly determined (Lambeck, 1980). However, at present, high-accuracy gravimetry has been developed and it is possible to obtain the variations of the inertia tensor since the inertia tensor is relevant up to the second-degree potential coefficients (Marchenko & Abrikosov, 2001). Some gravity models, such as EGM96 (1996) and EIGEN-GL04C (2006), provide the ratio of the variation of the low-order coefficients and thus enable us to model the variation of the inertia tensor caused by the mass redistribution.
In this paper we study the Earth's rotation based on the temporary principal moments of inertia of the Earth determined by the temporary gravity field.

The solutions of the Earth's free wobble
In the Tisserand axes system , the rotation of a nonrigid Earth could be described by the Liouville Equation (Lambeck, 1980;Goldstain et al, 2002): where L is the external torque; Ȧ is the angular velocity vector. The Tisserand axes are actually the mean axes of the principal axes A, B and C, respectively. We can obtain the time dependent longitude i λ and latitude i θ ( ) of axis i according to the variable second-degree potential coefficients. Then the orientations of the axes T x , T y and T z could be expressed as ( , ) , where X means the mean value of X. So the Tisserand axes system is not the Earth's principal axes system.
In the Earth's principal axes system, the inertia tensor could be expressed as: while in the Tisserand axes system: 11 And the Earth-fixed coordinate system can be transformed to the Tisserand axes system by 2 cos cos sin cos sin sin cos 0 cos sin sin sin cos Then the transformation metric, from the principal axes system to the Tisserand axes system, could be expressed as: According to the norm of metric transformation, one gets (Huang, et al, 2003) 1 T − = = I P I P P I P (7) On the other hand, I could be decomposed as: 11 12 13 Similarly, the angular velocity Ȧ could be expressed as: In this paper, we investigate the free rotation of the Earth. In this case, 0 = L , and Eq.(10) reduces to 33 33 3

The Earth's inertia moment tensor and its temporal variation
In order to determine the Earth's inertia moment tensor, we need to solve the Earth's principal axes and moments of inertia which correspond to its principal axes. The principal axes and moments of inertia are usually determined based on the harmonic analysis of the gravity field using the second-order harmonic coefficients. The solution of the principal moments of inertia can be expressed as (Marchenko and Abrikosov, 2001) 3 1 where 2 15 6 In Eqs. (12) Consequently, the included angles of the Earth's principal axes and the geocentric coordinate system's axes respectively are ( , , Thus, the Earth's principal axes and moments of inertia are determined. In fact, the harmonic coefficients of the gravity model are time-dependent (see Table 1 for relevant  Tables 2 and  Table 3)

Variations in rotation rate and polar motion
We use the time-dependent values, obtained from the last section, solving Eqs.(7)~(11), and we obtain the variations of 3 ω and polar motion (see Fig.1). ω have periods of 22, 14, and 8 years (known as the decadal variation), as well as other short-period variations ranging from a few days to two or three years. Thus we can conclude that the mass redistribution should be partly responsible for the decadal oscillation of the length of day (LOD).
Conventionally, the material slowing down of the Earth's rotation is attributed to tidal frictions. However, our results show that the increase of the Earth's inertial moment also contributes to the deceleration. As to the free rotation, the Earth's system should hold the conservation of the angular moment. Then the deceleration of the Earth's rotation is a result of the increasing inertial moment. From this point of view, our results seem to be reliable.
On the other hand, the amplitude of Chandler wobble, θ , also has complex fluctuations. In 20 years, θ seems to have an increasing tendency of 2.06×10 8 mas/yr (a fitting result). Here we still adapt the coif5 wavelet to decompose θ into 12 layers, and the results are shown in Fig.4. Investigating Fig.4, it can be concluded that increasing A, B, C would lead to the polar motions with periods ranging 24 (a 12 ), 14 (d 12 ), 8 (d 11 ), 3 (d 10 ) and 2 (d 9 ) years; and the annual and sub-annual polar motions also appear. These values are in good agreement with the observations (Lambeck, 1980).

Conclusions
Based on the gravity models EGM96 and EIGEN-GL04C, we first calculate the temporal principal moments of inertia of the Earth, and then obtain the numerical solutions to Liouville's equations. Calculations show that the Earth's principal moments of iner-tia are increasing, which might be a clue of the Earth's expansion ; both 3 ω and θ have short-period fluctuations but have secular increasing and decreasing tendency. Our results show that variations of 3 ω and θ in all scales are relevant to the increase of A, B and C. Thus we should not treat the Earth's principal moments of inertia as constants as man traditionally did, because they vary with time and could contribute to the observed variations in the Earth's rotation.