7 Prograde diurnal polar motions and semidiurnal nutations

7.1 Triaxiality contribution to angular momentum

The prograde semidiurnal nutations arise primarily from the direct action of the (2,1) tidal potential on the Earth's triaxial structure. Triaxiality, i.e., inequality of the two principal equatorial moments of inertia A and B, is characterized by the parameter e':

$$e'\equiv {(B-A)\over 2\bar A},\quad {\rm where}\quad \bar A \equiv {A+B\over 2},$$ (47a)

$$A=\bar A (1-e'),\qquad B=\bar A (1+e').$$ (47b)

The geopotential coefficients C_2,2 and S_2,2 are a reflection of triaxiality and are related to e' through

$$\bar A e' \cos 2\alpha= 2 M_{\rm E} a_{\rm E}^2~C_{2,2},$$ (48a)

$$\bar A e' \sin 2\alpha= 2 M_{\rm E} a_{\rm E}^2~S_{2,2},$$ (48b)

where $\alpha$ is the longitude of the direction of the axis of maximum equatorial moment of inertia. The numbers given by Bretagnon et al. (1997) lead to

$$e'=1.10157\times 10^{-5}, \qquad \alpha=14\hbox{$.\!\!^\circ$ }9285\ {\rm West}.$$ (49)

We define, for later convenience, the complex parameter

$$Z=e'{\rm e}^{2{\rm i}\alpha}= (2M_{\rm E} a_{\rm e}^2/\bar A)~(C_{2,2}+{\rm i}S_{2,2}).$$ (50)

Contributions to the prograde semidiurnal nutations arise from the appearance of triaxiality through $A, B, A_{\rm f}$, and $B_{\rm f}$ in the equatorial components of the angular momentum vectors ${\vec H}$ and  ${\vec H}_{\rm f}$. In a coordinate system with its axes along the principal axes of the inertia tensor, the equatorial components of ${\vec H}$ are, for the triaxial Earth,

$$H_1' = A m'_1=\bar A(1-e') m'_1, \quad H'_2 = B m'_2=\bar A(1+e') m'_2,$$ (51a)

$$\tilde H'(t) = H_1'(t)+{\rm i}H_2'(t)=\bar A~[~\tilde m'(t) -e'\tilde m'^*(t)].$$ (51b)

In reality, the equatorial principal axes make an angle $\alpha$ to the axes of the generally used terrestrial reference frame, and so $\tilde m(t)' = {\rm e}^{{\rm -i}\alpha} \tilde m(t)$, $\tilde m'^*(t) = {\rm e}^{{\rm i}\alpha} \tilde m^*(t)$, and $\tilde H'(t)={\rm e}^{{\rm -i}\alpha}\tilde H(t)$, as may be readily seen. It follows then that

$$\tilde H(t)= \bar A~[\tilde m(t)-Z~\tilde m^*(t)~ ].$$ (52)

If the core is triaxial, $\tilde H_{\rm f}(t)$ also contains triaxiality terms:

$$\tilde H_{\rm f}(t) H_{f1}(t)+{\rm i}H_{f2}(t)$$ =

$$\bar A_{\rm f}~\left[(\tilde m(t)+\tilde m_{\rm f}(t)) -Z_{\rm f}~(\tilde m^*(t)+\tilde m_{\rm f}^*(t))\right],$$ =

$$Z_{\rm f} e_{\rm f}'{\rm e}^{2{\rm i}\alpha}.$$ = (53)

The meaning of the symbols used here should be obvious. In using the same angle $\alpha$ here as in the case of $\tilde H(t)$, we are assuming that the principal axes of the core have the same directions as those of the whole Earth, which seems the most likely scenario on physical grounds.

7.2 Wobble equations including triaxiality

The triaxiality terms appearing through the expressions (52) and (53) had not been included in the dynamical equations in the earlier sections, as the triaxiality contributions to the nutations considered there were far below the 0.1 $\mu$asbecause of the weakness of the driving potentials of degree >2. With the inclusion of these terms here, the dynamical Eqs. (29) get modified to

$$\left({{\rm d}\over {\rm d}t} -{\rm i}e\Omega_0\right)\tilde m(t)+ {1\over \bar A}\left({{\rm d}\over {\rm d}t}+ {\rm i}\Omega_0\right)$$

$$\times\left[\tilde c_3(t)+\bar A_{\rm f}\tilde m_{\rm f}(t)-\bar ... ... Z_{\rm f}\tilde m_{\rm f}^*(t)\right] = {\tilde\Gamma(t)\over \bar A\Omega_0},$$ (54a)

$${{\rm d}\over {\rm d}t}\left(\tilde m\!-\!Z_{\rm f}\tilde m^*\!+\... ...m f})\right)\tilde m_{\rm f}\!-\!Z_{\rm f}{{\rm d}m_{\rm f}^*\over {\rm d}t}=0,$$ (54b)

where $\tilde\Gamma(t)$ is the full torque due to the (2,1) potential, containing both prograde and retrograde diurnal components.

Consider now the frequency domain version of the above equations, corresponding to a spectral component having the time dependence  ${\rm e}^{{\rm i}\sigma\Omega_0t}$. Noting that the amplitude of the term in  $\tilde m^*(t)$ which has this time dependence is  $\tilde m^*(-\sigma)$, and similarly for $\tilde m_{\rm f}^*(t)$, one sees that the equations are:

$$(\sigma-e)\tilde m(\sigma)+(1/\bar A)(1+\sigma)\big[\tilde c_3(\sigma)+\bar A_{\rm f} \tilde m_{\rm f}(\sigma)-\bar A Z~\tilde m^*(-\sigma)$$

$$-\bar A_{\rm f} Z_{\rm f}~\tilde m_{\rm f}^*(-\sigma)\big] = \tilde\Gamma(\sigma)/({\rm i}\bar A\Omega_0^2),$$ (55a)

$$\sigma~[~\tilde m(\sigma)+\tilde c^{\rm f}_3(\sigma)/\bar A_{\rm f}~]-\sigma Z_{\rm f}~\tilde m^*(-\sigma)$$

$$+(\sigma+1+e_{\rm f})~\tilde m_{\rm f}(\sigma) -\sigma~Z_{\rm f}~\tilde m_{\rm f}^*(-\sigma)=0.$$ (55b)

It is evident that these equations couple the amplitude of any prograde wobble ($\sigma>0$) to that of the corresponding retrograde wobble with frequency $-\sigma$.

7.3 Free and forced wobbles

It is to be expected that the triaxiality terms in Eqs. (55) will lead to small increments to the frequencies of the free wobbles. These increments are of the second order in triaxiality, as may be shown very generally from the structure of these equations. For an elementary illustration, consider the rigid Earth, whose free wobbles are governed by the equation $(\sigma-e)\tilde m(\sigma)- (1+\sigma)~ Z~\tilde m^*(-\sigma)= 0$, which is the relevant special case of (55a). On taking its complex conjugate and making the replacement $\sigma\to -\sigma$, one obtains a second equation relating  $\tilde m(\sigma)$ and  $\tilde m^*(-\sigma)$. This pair of equations yields the secular equation $\sigma^2=(e^2-e'^2)$, where e'=|Z|. Thus the change in the Chandler eigenfrequency due to triaxiality is from e to e(1-e'²/2e²), to the lowest order in the triaxiality. Given the magnitudes of e and e', the fractional change is seen to be of the order of 10^-5, which is much too small be of interest. For the Earth with a fluid core, a similar procedure starting with both the Eqs. (55) and their complex conjugates with the sign of $\sigma$ reversed, leads to a similar result also for the second eigenfrequency associated with the free core nutation.

Moving on now to forced prograde diurnal wobbles due to the (2,1) potential, it is advantageous, in view of their coupling to the retrograde diurnals, to express the potential in the form employed by Mathews et al. (1991) in their treatment of the retrograde diurnal wobbles:

$$\phi^{(2,1)}({\vec r};t) = -\Omega_0^2\left[~\phi_1(t) xz+\phi_2(t) yz~\right].$$ (56)

To obtain the torque due to $\phi^{(2,1)}$, it is simplest to start with (56) reexpressed in the principal axis frame before evaluation of the relevant integrals. The components of the torque can then be transformed back to the original frame. The result, written down for $\tilde\Gamma= {\bf\Gamma}_1+{\rm i}{\bf\Gamma}_2$, is

$$\tilde\Gamma= -{\rm i}\bar A\Omega_0^2\ \left[~e\tilde\phi(t)+Z\tilde\phi^*(t)~\right].$$ (57)

Higher order terms which are irrelevant in the present context have been dropped.

For a spectral component of the potential (56) with angular frequency $\omega$, $\phi^{2,1}$ is proportional to $\sin(\Theta_\omega(t)+\lambda)$, $\lambda$ being the longitude, according to the Cartwright-Tayler convention. This is ensured by taking the spectral components of $\phi_1(t)$ and $\phi_2(t)$ to be

$$\phi_1(t)= \phi(\sigma_{\rm r})~\cos (\Theta_\omega-\pi/2),~ \phi_2(t)=-\phi(\sigma_{\rm r})~\sin (\Theta_\omega-\pi/2),$$ (58)

with $\phi(\sigma_{\rm r})$ real and $-3/2<\sigma_{\rm r}<-1/2$. The complex quantities  $\tilde\phi(t)$ and  $\tilde\phi^*(t)$ are now

$$% \tilde\phi(t) \equiv \phi_1(t)+{\rm i}\phi_2(t)=\phi(\sigma_{\rm r}) {\rm e}^{{\rm -i}(\Theta_\omega-\pi/2)} \quad\ {\rm and}$$

$$\tilde\phi^*(t) \phi(\sigma_{\rm r}) {\rm e}^{{\rm i}(\Theta_\omega-\pi/2)}.$$ = (59)

The torque (57) becomes now

$$\tilde\Gamma_\omega= \tilde\Gamma(\sigma_{\rm r})~{\rm e}^{{\... ...Gamma(\sigma_{\rm p})~ {\rm e}^{{\rm i}(\Theta_\omega-\pi/2)},$$ (60)

$$\tilde\Gamma(\sigma_{\rm r})= -{\rm i}\bar A\Omega_0^2~ \phi(... ...ma_{\rm p})= -{\rm i}\bar A\Omega_0^2~ \phi(\sigma_{\rm r})~Z.$$ (61)

On using (60), the solution of Eqs. (54) is seen to have the form

$$\tilde m(t)= \tilde m(\sigma_{\rm r}){\rm e}^{{\rm -i}(\Theta... ...tilde m(\sigma_{\rm p}) {\rm e}^{{\rm i}(\Theta_\omega-\pi/2)}$$ (62)

together with a similar expression for $\tilde m_{\rm f}(t)$.

We can now specialize Eqs. (55) to the case of prograde wobbles by setting $\sigma=\sigma_{\rm p}$ and $\tilde\Gamma(\sigma)=\tilde\Gamma(\sigma_{\rm p})$ and introducing from Eqs. (31) the expressions

$$\tilde c_3(\sigma_{\rm p}) \bar A~[\kappa \tilde m(\sigma_{\rm p}) +\xi\tilde m_{\rm f}(\sigma_{\rm p})],$$ =

$$\tilde c^{\rm f}_3(\sigma_{\rm p}) \bar A_{\rm f}~[\gamma \tilde m(\sigma_{\rm p}) +\beta\tilde m_{\rm f}(\sigma_{\rm p})],$$ = (63)

noting that $\tilde\phi(\sigma_{\rm p})=0$ since the potential is strictly retrograde. We thus obtain the wobble equations for $\sigma=\sigma_{\rm p}$ in explicit form as

$$\left[~(\sigma_{\rm p}-e)+(1+\sigma_{\rm p})\kappa~\right]~\tilde... ...+(1+\sigma_{\rm p})(\bar A_{\rm f}/\bar A+\xi)~\tilde m_{\rm f}(\sigma_{\rm p})$$

$$=-~Z\phi(\sigma_{\rm r})+(1+\sigma_{\rm p})~ \left[Z\tilde m^*(\s... ...r})+(\bar A_{\rm f}/\bar A)~Z_{\rm f}\tilde m_{\rm f}^*(\sigma_{\rm r})\right],$$ (64a)

$$(1+\gamma)\sigma_{\rm p}~\tilde m(\sigma_{\rm p})+ [1+(1+\beta)\sigma_{\rm p}+e_{\rm f}]~\tilde m_{\rm f}(\sigma_{\rm p})$$

$$=Z_{\rm f}\sigma_{\rm p}~\tilde m^*(\sigma_{\rm r})+Z_{\rm f}\sigma_{\rm p}~\tilde m_{\rm f}^*(\sigma_{\rm r}).$$ (64b)

Now, $\tilde m^*(\sigma_{\rm r})$ and $\tilde m_{\rm f}^*(\sigma_{\rm r})$ relate to the retrograde diurnal wobbles. The triaxiality parts of the angular momenta of the Earth and its core, which are the source of the terms containing these quantities in the above equations, are thus seen to couple the prograde diurnal wobbles to the retrograde ones. Thus the mechanism through which the resonances in the latter get to affect the prograde semidiurnal nutations becomes transparent.

In the coefficients of $\tilde m(\sigma_{\rm p})$ and $\tilde m_{\rm f}(\sigma_{\rm p})$ in Eqs. (64), $\sigma_{\rm p}$ is close to 1 while all other quantities except  $A_{\rm f}/A$ are of  $O(\epsilon)$. Substituting for $(1+\sigma_{\rm p})\tilde m_{\rm f}(\sigma_{\rm p})$ in Eqs. (64a) from (64b), one finds, with the neglect of small quantities of $O(\epsilon)$, that

$$(\bar A_{\rm m}/\bar A)\sigma_{\rm p}\tilde m(\sigma_{\rm p})=-Z ... ...~\tilde m_{\rm f}^*(\sigma_{\rm r})-\sigma_{\rm p}\tilde m^*(\sigma_{\rm r})~].$$ (65)

This result enables us to compute the prograde semidiurnal nutations from the amplitudes of the LF nutations and of the associated wobbles of the core, since both $\tilde\phi(\sigma_{\rm r})$ and $\tilde m(\sigma_{\rm r})$ may be expressed in terms of the amplitude of the associated nutation with frequency $\nu=1+\sigma_{\rm r}$.

Before proceeding futher on these lines, it is useful to get an idea of the relative magnitudes of the various terms on the right hand side of (65). The dominant term is the first one representing the external torque acting on the triaxiality. Both $\tilde m^*(\sigma_{\rm r})$ and  $\tilde m_{\rm f}^*(\sigma_{\rm r})$ in the remaining terms are affected by the resonance at $\sigma_{\rm r}\approx -1.002319$ associated with the FCN. Nevertheless, the factor $\tilde m^*(\sigma_{\rm r})/\tilde\phi(\sigma_{\rm r})$ stays close to e at the frequencies of interest here, deviating from it only by about 30% even at the $\psi_1$ frequency ( $\sigma_{\rm r}=-1.002730$) responsible for the retrograde annual nutation. Therefore the contribution from the second term on the right in (65) relative to that of the first term is $\approx$ $2e\approx 1/150$, since  $\sigma_{\rm p}$ is close to unity; hence the contribution of this term to any of the semidiurnal polar motion coefficients is at most about 0.1 $\mu$as, the largest of the coefficients being about 15 $\mu$as. If the core is triaxial ( $Z_{\rm f}\ne 0$), the huge resonance in $\tilde m_{\rm f}^*(\sigma_{\rm r})$ enters the picture: $\vert\tilde m_{\rm f}^*(\sigma_{\rm r})/\tilde m^*(\sigma_{\rm r})\vert$ is as large as 800 for the $\psi_1$ frequency, and about 200 at the frequency ($\sigma= -1$) of the K₁ tide (which causes the precession) and at the nearby frequency with which the 18.6 year retrograde nutation is associated. Thus, if the triaxiality of the core were to have the same magnitude as that of the whole Earth ( $Z_{\rm f}=Z$), the magnitude of the contribution of the $\tilde m_{\rm f}^*(\sigma_{\rm r})$ term in (65) relative to that of the external torque would become $\vert(A_{\rm f}/A)~(\tilde m_{\rm f}^*(\sigma_{\rm r})/\tilde m^*(\sigma_{\rm r... ...\rm r})/\tilde\phi(\sigma_{\rm r}))\vert \approx (1/9)(200)(1/300)\approx 0.075$ for $\sigma_{\rm p}=1$ (the 0.99727 day polar motion driven by the K₁ tide); the corresponding number is large, about 0.39, for the PM due to the $\psi_1$ tide close to the resonance, but only 0.0022 for that due to the O₁ tide far from the resonance. The magnitude of the actual contribution to the larger of the PM coefficients for the 0.99727 day polar motion turns out to be about 1 $\mu$as, which is not insignificant. Further discussion of the numerical results on the effects of core triaxiality will be deferred to Sect. 8, where a comparison will be made with the results of Escapa et al. (2002) who have already drawn attention to the role of the FCN resonance in the context of semidiurnal nutations (equivalent to the prograde diurnal polar motions).

Returning now to the development of the theoretical expressions, we note that no generally accepted quantitative estimates are avaliable for the triaxiality of the core, and so we ignore its effects hereafter. We have then, to an approximation which neglects terms of $O(\epsilon)$,

$$\tilde m(\sigma_{\rm p}) = -(Z/e) (\bar A/ \bar A_{\rm m} ) ~\tilde m_{\rm R}(\sigma_{\rm r}).$$ (66)

In this approximation, the transfer function from the rigid to the nonrigid Earth is simply the constant  $A/A_{\rm m}$.

Now, in view of Eq. (62), the prograde part of the wobble in the time domain is

$$\tilde m(t)=\tilde m(\sigma_{\rm p})~{\rm e}^{{\rm i}(\Theta_... ... i}\tilde m(\sigma_{\rm p})~{\rm e}^{{\rm i}\Theta_\omega(t)}.$$ (67)

The corresponding nutation is obtained using Eqs. (13), (15), and (16) with the upper sign, remembering that m=1 and ${\rm e}^{{\rm -i}\zeta_{n,m}} = -{\rm i}$ in the present case:

$$\tilde\eta(t)= {{\rm i}\tilde m(\sigma_{\rm p})\over (\sigma_... ...ega(t)+GMST)}={\rm i}\tilde\eta(\nu){\rm e}^{{\rm i}\Xi_\nu},$$ (68)

where $\tilde\eta(\nu)=-\tilde m(\sigma_{\rm p})/(1+\sigma_{\rm p})$ as in (17), and $\nu=\nu_+\equiv 1+\sigma_{\rm p}$. As for $\tilde p(t)$, one finds from (19) on using (68) that

$$\tilde p(t)={-{\rm i}\tilde m(\sigma_{\rm p})\over (\sigma_{\rm p}+1)}~{\rm e}^{{\rm i}\Theta_\omega(t)}.$$ (69)

If we now use the approximation (66) for $\tilde m(\sigma_{\rm p})$ and then replace $\tilde m_{\rm R}(\sigma_{\rm r})$ by $-(1+\sigma_{\rm r})\tilde\eta(\nu_-)$ where $\nu_-=1-\sigma_{\rm p}$, and use the expression (50) for Z, we obtain

$$\tilde p(t) =-{\rm i}~ {2M_{\rm E}a_{\rm e}^2\over \bar A_{\r... ...-)~(C_{2,2}+{\rm i}S_{2,2})~{\rm e}^{{\rm i}\Theta_\omega(t)}.$$ (70)

This expression may be evaluated by taking the amplitudes  $\tilde\eta_{\rm R}(\nu_-)$ of LF nutations from the appropriate rigid Earth nutation series. For exact results, one needs to compute the solution for  $\tilde m(\sigma_{\rm p})$ from Eqs. (64), which calls for a knowledge of $\tilde m(\sigma_{\rm r})$ (and  $\tilde m_{\rm f}(\sigma_{\rm r})$ too if $Z_{\rm f}$ is set to a nonzero value). These wobble amplitudes may be obtained, for instance, by solving the standard Sasao et al. (1980) equations. It turns out that the largest of the errors caused by the use of (70) is at the 0.1 $\mu$as level if triaxiality of the core is ignored. The results shown in the tables in the next section are the exact ones, of course.