Appendix A: Torque due to the potential of type (n, m)

We start with the expression (4), written slightly more explicitly:

$$V_{\omega}^{(n,m)}({\vec r};t) {1\over 2} \left({r\over a_{\rm e}}\right)^n \sum_\omega g_{\rm e... ...ega \big[~ Y_n^m(\theta,\lambda){\rm e}^{{\rm i}(\Theta_\omega(t)-\zeta_{n,m})}$$ =

$$+ {Y_n^m}^*(\theta,\lambda){\rm e}^{{\rm -i}(\Theta_\omega(t)-\zeta_{n,m})}\big].$$ (A.1)

The surface spherical harmonics Y_n^m, ( $0\le m\le n)$, are defined here by

$$Y_n^m(\theta,\lambda) = N_n^m P_n^m(\cos\theta){\rm e}^{{\rm i}m\lambda},$$

$$N_n^m=(-1)^m\left({2n+1\over 4\pi}{(n-m)!\over (n+m)!}\right)^{1/2},$$ (A.2a)

$$Y_n^{-m} =(-1)^m {Y_n^m}^*,~{\rm and}~ \int {Y_n^m}^*~Y_q^p \sin\theta~ {\rm d}\theta~ {\rm d}\lambda= \delta_{n,q}\delta_{m,p}.$$ (A.2b)

The torque on the Earth due to the potential $V_{\omega}^{(n,m)}$ is

$${\bf\Gamma}^{nm}_\omega= \int \rho({\vec r})~({\vec r}\times\nabla)~V_{\omega}^{(n,m)}({\vec r};t){\rm d}^3r,$$ (A.3)

where $\rho({\vec r})$ is the density at the postion ${\vec r}$, and the integral is over the volume of the Earth. The operator  $({\vec r}\times\nabla)$ is proportional to the angular momentum operator ${\vec L}$ of quantum mechanics. Knowing the action of ${\vec L}$ on Y_n^m (see, for instance, Mathews & Venkatesan 1976), one finds immediately that

$$({\vec r}\times\nabla)_+~Y_n^m {\rm i}c_n^m~Y_n^{m+1},$$ = (A.4a)

$$({\vec r}\times\nabla)_-~Y_n^m {\rm i}c_n^{-m}~Y_n^{m-1},$$ = (A.4b)

where $({\vec r}\times\nabla)_\pm=({\vec r}\times\nabla)_1\pm {\rm i}({\vec r}\times\nabla)_2$ and

c_n^m = [ (n-m)(n+m+1) ]^1/2. (A.5)

We can now carry out the angular part of the integration in (A3) with the help of a spherical harmonic expansion of  $\rho({\vec r})$:

$$\rho({\vec r}) =\sum_{l=0}^\infty\sum_{k=-l}^l\rho_{lk}(r) ~{Y_l^k}^*(\theta,\lambda),$$ (A.6)

$$\rho_{nm}(r)=\int \rho({\vec r})~{Y_n^m}(\theta,\lambda)~\sin\theta~ {\rm d}\theta~ {\rm d}\lambda= (-1)^m\rho_{n,-m}^*(r).$$ (A.7)

On introducing (A6) together with (A1) into (A3) and making use of Eqs. (A2) and (A4), we obtain the following expressions for the familiar complex combination  $\tilde\Gamma^{(n,m)}_\omega$ of the first two components of  ${\bf\Gamma}^{(nm)}_\omega$:

$$\tilde\Gamma^{(n,m)}_\omega= {\rm i}{g_{\rm e}\over 2a_{\rm e}^n}... ... -c_n^{-m}~ {Q_{n,m-1}}^*~{\rm e}^{{\rm -i}(\Theta_\omega(t)-\zeta_{n,m})}\Big]$$ (A.8)

for m> 0, and

$$\tilde\Gamma^{(n0)}_\omega={\rm i}{g_{\rm e}\over 2a_{\rm e}^... ...})}+ ~{\rm e}^{{\rm -i}(\Theta_\omega(t)-\zeta_{n,0})}~\right]$$ (A.9)

for m=0, where

$$Q_{nm} = \int \rho_{nm}(r)r^{n+2} {\rm d}r=(-1)^m Q_{n,-m}^*.$$ (A.10)

The Q_nm are related, of course, to the geopotential coefficients (C_nm,S_nm). The relation is

$$(2-\delta_{m0})4\pi N_{nm} Q_{nm}= (2n+1)M_{\rm E} a_{\rm e}^n (C_{nm}+{\rm i}S_{nm}).$$ (A.11)

The expressions given in Eqs. (26) of the text are obtained on substituting for Q_nm in (A8) and (A9) from (A11). The entries in Table 1 regarding the origins of the wobbles and nutations in various frequency bands are based on Eqs. (A8) and (A9). The prograde/retrograde nature of the two terms in each of these equations is evident from the fact that the contribution to  $(\Gamma_x^{(n,m)},\Gamma_y^{(n,m)})$ from the first term is proportional to  $(\cos\Theta_\omega(t), \sin\Theta_\omega(t))$which rotates in the prograde sense, while the contribution from the second term, proportional to  $(\cos\Theta_\omega(t),-\sin\Theta_\omega(t))$, is retrograde.