4 The dynamical equations

Since the amplitudes of the HF nutations are expected to be no more than about 20 $\mu$as, the effects of the solid inner core are expected to be negligible at the level of 0.1 $\mu$as, which is the accuracy we aim for. So we treat the Earth as consisting of only two regions - the mantle and a fluid core.

The coupled rotational motions of of these two regions which are excited by potentials of any type $(n,m)\ne (2,1)$ are governed by dynamical equations of the Sasao et al. (1980) form for the wobble variables  $\tilde m(t)$ and  $\tilde m_{\rm f}(t)$. They are, in the time domain,

$$\left ({{\rm d}\over {\rm d}t} -{\rm i}e\Omega_0\right)\tilde... ...lde m_{\rm f}(t)\right]\!=\!{\tilde\Gamma(t)\over A\Omega_0},$$ (29a)

$$\left ({{\rm d}\over {\rm d}t} +{\rm i}\Omega_0~(1+e_{\rm f})... ...ilde m(t)+{\tilde c^{\rm f}_3(t)\over A_{\rm f}} \right) = 0.$$ (29b)

Here $\tilde\Gamma(t)$ stands for $\tilde\Gamma^{(n,m)}_\omega(t)$; $e,~ e_{\rm f}$ are the ellipticities of the whole Earth and of the fluid core, and $\tilde c_3(t),~ \tilde c^{\rm f}_3(t)$ are complex combinations,

$$\tilde c_3=c_{1,3}+{\rm i}c_{2,3}\quad {\rm and}\quad \tilde c^{\rm f}_3=c^{\rm f}_{1,3}+ {\rm i}c^{\rm f}_{2,3}$$ (30)

of the indicated elements of their respective inertia tensors. The mean equatorial moments of inertia $\bar A, \bar A_{\rm f}$, and $\bar A_{\rm m}$ of the Earth and of its core and mantle are written here simply as $A, A_{\rm f}, A_{\rm m}$; the overbars will be restored in Sect. 7 where triaxiality needs to be taken into account.

The quantity $\tilde c_3$ in (30) represents the deformation of type (2,1) of the whole Earth due to the direct action of the tidal potential, together with that due to incremental centrifugal potentials associated with the wobbles $\tilde m$ and  $\tilde m_{\rm f}$. The origins of  $\tilde c^{\rm f}_3$ are similar, but the relevant deformation is that of the core alone. For a spherically symmetric Earth, the deformation induced by a tidal potential of type (n,m) is strictly of the same type, and will therefore contribute nothing to  $\tilde c_3$ or  $\tilde c^{\rm f}_3$ unless (n,m)=(2,1). Although the ellipticity of the Earth would result in the presence of a small part (of the order of 1/300) of type (2,1) in the deformation due to (n,1) potentials with $n\ne 2$, its effect on the already small nutations produced by such potentials is entirely negligible. However, the incremental centrifugal potentials are necessarily of type (2,1) irrespective of the nature of the tidal potential, and so is its contribution to deformation. Consequently, the expressions of Sasao et al. (1980) for  $\tilde c_3$ and  $\tilde c^{\rm f}_3$ take the following modified form in the general case:

$$\tilde c_3 A~\left[~\kappa~ (\tilde m-\tilde\phi~ \delta_{n,2}\delta_{m,1}) +\xi\tilde m_{\rm f}~\right],$$ = (31a)

$$\tilde c^{\rm f}_3 A_{\rm f}~\left[~\gamma~ (\tilde m-\tilde\phi~ \delta_{n,2}\delta_{m,1}) +\beta\tilde m_{\rm f}~\right],$$ = (31b)

where $\tilde\phi$ is equivalent to the tidal potential but is expressed in suitable dimensionless units, and $\kappa, \xi, \gamma$, and $\beta$ are compliance (deformability) parameters (see Sasao et al. 1980; Mathews et al. 1991). The Kronecker delta function $\delta_{p,q}$ is unity if p=q, and is zero otherwise.

There is a caveat to be entered in regard to the above expressions, because the compliances have to be considered to be frequency dependent, in general, even within a particular tidal band. The reason is that the deformation due to ocean tides raised by the tidal potential, and the effect of mantle anelasticity on the deformations, are both reflected in the values of the compliances - and one or the other of these could be strongly frequency dependent, depending on the type (n,m) of the potential involved (see, for instance, Mathews et al. 2002). This dependence, to the extent that it is known, can be readily taken into account if the expressions (31) are taken in the frequency domain, i.e., as expressions for  $\tilde c_3(\sigma)$ and  $\tilde c^{\rm f}_3(\sigma)$ in terms of the spectral amplitudes $\tilde m(\sigma),~ \tilde m_{\rm f}(\sigma)$, and  $\tilde\phi(\sigma)$; this is what we do.

The frequency domain version of Eqs. (29) is now readily obtained by replacing ${\rm d}/{\rm d}t$ by ${\rm i}\Omega_0\sigma$ and introducing the expressions (31). One should keep in mind that $\tilde\Gamma$ here stands for  $\tilde\Gamma^{(n,m)}_\omega$, which has both prograde and retrograde parts. For any term belonging to the prograde part, $\sigma$ is to be taken as $\sigma_{\rm p}\equiv \omega/\Omega_0$, and $\tilde\Gamma(\sigma)$ is given by (26a), while for terms in the retrograde part, $\sigma$ is $\sigma_{\rm r}\equiv -\omega/\Omega_0$and $\tilde\Gamma(\sigma)$ is given by (26b) or (26c) according as $m\ne 0$ or m=0. With this understanding, we continue to use the generic symbol $\sigma$ in writing down the pair of frequency domain equations for the wobbles due to a tidal perturbation of general type $(n,m)\ne (2,1)$:

$$\left[(\sigma-e)+(1+\sigma)\kappa\right]~\tilde m(\sigma)+ (1... ... m_{\rm f}(\sigma)= \tilde\Gamma(\sigma)/({\rm i}A\Omega_0^2),$$ (32a)

$$(1+\gamma)~\sigma~\tilde m(\sigma)+ \left[1+{\rm e}_{\rm f}+(1+\beta)\sigma\right]~\tilde m_{\rm f}= 0.$$ (32b)

It is a pertinent to ask, at this point, whether the nonlinear terms in the dynamical equations (Mathews et al. 2002), which had been ignored so far, could contribute to $\tilde p(\sigma)\equiv \tilde m(\sigma)/(1+\sigma)$ at the 0.1 $\mu$as level. As may be seen from the Appendix A of that paper, the largest of the contributions to $\tilde m(t)$ from nonlinear terms are $(c_{1,1} - c_{3,3})^{(Z)}\tilde\phi/A$ and $(c_{1,1}+{\rm i}c_{1,2})^{(S)}\tilde\phi^*/A$, wherein all the quantities are in the time domain; $\tilde\phi$ is the tesseral tidal potential, and the inerita tensor elements involved arise from the action of the zonal or sectorial tides, as identified by the superscript Z or S. The (c_i,j/A) are of the order of  $\kappa\phi^{(Z)}$ and  $\kappa\tilde\phi^{(S)}$ in the two cases. Since $\kappa\approx 10^{-3}$ and the largest of the spectral components of the various potentials are of O(10^-5), it would appear that contributions to  $\tilde m(\sigma)$ of the order of 10^-13 radians, i.e., about 0.1 $\mu$as, could arise. The fact that this does happen in the case of the retrograde diurnal wobbles is not relevant, however, for wobbles in the other bands. The reason is that the frequency spectrum of  c_i,j^(Z) lies in the low frequency band and that of  c_i,j^(S) is in the retrograde semidiurnal band, while the spectra of  $\tilde\phi$ and  $\tilde\phi^*$ are in the retrograde and prograde diurnal bands respectively. It follows then then the spectra of the above-mentioned contributions to  $\tilde m(t)$ are only in the retrograde diurnal band and are of no consequence for the other bands with which this work is concerned. Therefore the use of the linear Eqs. (29) or (32) for our purposes is entirely justified.

 

 

Table 2: Values of geopotential coefficients.

	IERS92		JGM3
(k,l)	Ck,l	Sk,l	Ck,l	Sk,l
(2, 2)	1.574 410	-0.903 757	1.574 536	-0.903 868
(3, 1)	2.190 181	0.269 185	2.192 799	0.268 012
(3, 2)	0.308 936	-0.211 582	0.309 016	-0.211 402
(3, 3)	0.100 447	0.197 157	0.100 559	0.197 201
(4, 1)	-0.508 638	-0.449 141	-0.508 725	-0.449 460
(4, 2)	$\cdots$	$\cdots$	0.350 670	0.662 571