3 Dynamics: The torque

Evaluation of the torque  ${\bf\Gamma}^{(n,m)}_\omega$ on the Earth due to the potential  $V_{\omega}^{(n,m)}$ of Eq. (4) is done in Appendix A for general (n,m). Though the potential wave is strictly retrograde, the complex combination

$$\tilde\Gamma^{(n,m)}_\omega\equiv \left(\Gamma_{\omega,1}^{(n,m)}+{\rm i}\Gamma_{\omega,2}^{(n,m)}\right)$$ (23)

of the equatorial components of the torque ${\bf\Gamma}^{(n,m)}_\omega$ is seen to be composed of both prograde and retrograde parts:

$$\tilde\Gamma^{(n,m)}_\omega(t) = \tilde\Gamma^{(n,m)}_\omega(... ...ma_{\rm r})\ {\rm e}^{{\rm -i}(\Theta_\omega(t)-\zeta_{n,m})},$$ (24)

where the subscripts p and r denote "prograde'' and "retrograde'', respectively. The dependence of  $\Theta_\omega(t)$ and $-\Theta_\omega(t)$ on time is given by Eq. (8), and

$$\sigma_{\rm p}=\omega/\Omega_0, \qquad \sigma_{\rm r}=-\omega/\Omega_0=-\sigma_{\rm p}.$$ (25)

It follows now from Eq. (7) that for the torques due to nonzonal (m>0) potentials, and for the wobbles excited by them, $\sigma_{\rm p}$ and  $\sigma_{\rm r}$ lie in separate bands of width 1 cpsd each, centered at m cpsd and -m cpsd, respectively, while for m=0, the prograde and retrograde frequencies together constitute a single band of width 1 cpsd centered at 0 cpsd. The frequency bands of the nutations produced by  $V_{\omega}^{(n,m)}$ are shifted by +1 cpsd relative to those of the associated wobbles, in view of (14).

We present now explicit expressions for the amplitudes  $\tilde\Gamma^{(n,m)}_\omega(\sigma_{\rm p})$ and  $\tilde\Gamma^{(n,m)}_\omega(\sigma_{\rm r})$ (see Appendix A). The retrograde amplitude has distinct forms for the zonal (m=0) and nonzonal cases.

$$\tilde\Gamma^{(n,m)}_\omega(\sigma_{\rm p}) = ({\rm i}\Omega_0^2 ... ...}~ G_{n,m}^{(+)}~H^{(n,m)}_\omega\ (C_{n,m+1}+{\rm i}S_{n,m+1}), ~~ (0\le m<n),$$ (26a)

$$\tilde\Gamma^{(n,m)}_\omega(\sigma_{\rm r})=({\rm i}\Omega_0^2 \b... ...G_{n,m}^{(-)}~H^{(n,m)}_\omega\ (C_{n,m-1}-{\rm i}S_{n,m-1}), ~~ (1\le m\le n),$$ (26b)

$$\tilde\Gamma^{(n,0)}_\omega(\sigma_{\rm r}) -({\rm i}\Omega_0^2 \bar A) ~ G_{n,0}^{(+)}~H_\omega^{(n,0)}~ (C_{n,1}+{\rm i}S_{n,1})$$ =

$$\tilde\Gamma^{(n,0)}_\omega(\sigma_{\rm p}),$$ = (26c)

wherein $\bar A\equiv (A+B)/2$ stands for the mean equatorial moment of inertia, and G_n,m⁽⁺⁾ and G_n,m^(-) are given by

$$(n-m)(n+m+1)G_{n,m},\qquad G_{n,m}^{(-)}= {2\over (2-\delta_{m,1})}G_{n,m},$$                       G_n,m⁽⁺⁾ =

$$\left({2n+1\over 4\pi}{(n+m)!\over(n-m)!}\right)^{1/2}~ {g_{\rm e} M_{\rm E}\over 4\Omega_0^2 \bar A}\cdot$$ G_n,m = (27)

The expressions (26) show that the prograde part of the torque is due to the action of  $V_{\omega}^{(n,m)}$ on C_n,m+1 and S_n,m+1, and the retrograde part is, for all m>0, due to its action on C_n,m-1 and S_n,m-1. In the case of zonal potentials, both the prograde and the retrograde torques result from the action on C_n,1 and S_n,1.

In the special case of the constant term ($\omega=0$) present in the spectrum of any zonal potential of even order n, $\Theta_\omega(t)=0$, and the expression (24) for the torque reduces to

$$\tilde\Gamma^{(n,0)}_0(t)=-2 ~({\rm i}\Omega_0^2 \bar A) ~G_{... ...)} ~H_0^{(n,0)}~(C_{n1}+{\rm i}S_{n1}), \qquad (n\ {\rm even})$$ (28)

on using the relevant expressions from Eqs. (26). The nutation due to this term requires special consideration, as will be seen in Sect. 6.

It should be noted that torques due to the degree 2 sectorial (m=2) and zonal potentials are ignorable because C_2,1 and S_2,1 are. The origin of nutations and polar motions in the various frequency bands, as displayed in Table 1, is clear from the above considerations.

 

 

Table 1: Origin of nutations, wobbles, and polar motions in different frequency bands.

Nutations	Frequency	due to	by potentials	Wobbles &
	band (cpsd)	action	of type	Polar motions
Long period	(-0.5,+0.5)	Cn,0	(n,1)	Retro diurnal
Pro diurnal	(+0.5,+1.5)	(Cn,1,Sn,1)	(n,0)	Long period
Retro diurnal	(-1.5,-0.5)	(Cn,1,Sn,1)	(n,2)	Retro semidiurnal
Pro semidiurnal	(+1.5,+2.5)	(Cn,2,Sn,2)	(n,1)	Pro diurnal
Retro semidiurnal	(-2.5,-1.5)	(Cn,2,Sn,2)	(n,3)	Retro terdiurnal
Pro terdiurnal	(+2.5,+3.5)	(Cn,3,Sn,3)	(n,2)	Pro semidiurnal

To compute the nutations or polar motions in a particular band of frequencies, one starts by identifying from Table 1 the types of potentials and geopotential coefficients relevant to that band, then picking out the expressions for the torques which they produce, and finally, solving the dynamical equations with these as the driving torques. For prograde semidiurnal nutations (or prograde diurnal polar motions), for instance, the relevant potentials are of type (n,1), acting on C_n,2 and S_n,2. The values of the geopotential coefficients of relevance to this work are shown in Table 2. The first set (labeled IERS92) gives the values from McCarthy (1992), used by Bretagnon et al. (1997) and other workers in computing the high frequency nutations of the rigid Earth, and the second set lists the JGM3 (Joint Gravity Model 3) values transformed to the same normalization as IERS92. We use the latter (more recent) set for our computations for the nonrigid Earth.