6 Low frequency polar motions and prograde diurnal nutations

We consider now the motions excited by torques of low frequencies ( $\vert\sigma\vert<1/2$). The potentials responsible for these torques are of type (n,0). The nutations produced are below our cut-off level of 0.05 $\mu$asamplitude for all n>4.

6.1 The Chandler resonance

The Chandler resonance plays a major role in the wobble response to forcing in this band; so the value of $\kappa$ which appears in the Chandler frequency assumes significance. The effects of mantle anelasticity and ocean tides on nutations can be dealt with by taking into account the complex increments that they produce to the values of $\kappa$ and other compliances, a fact that was exploited by Mathews et al. (2002) in their treatment of low frequency nutations and retrograde diurnal wobbles. In that context, the anelasticity contribution was practically independent of frequency while the ocean tide admittances were strongly frequency dependent, not only due to the FCN resonance, but also because of other aspects of ocean dynamics. In the low frequency tidal band that we are concerned with now, the ocean tides are believed to be essentially equibrium tides, with a constant admittance; but the anelasticity effect varies strongly with frequency across the band, making the anelasticity contribution to $\kappa$ strongly dependent on the forcing frequency. So the apparent frequency of the Chandler resonance, $\sigma_1=(A/A_{\rm m}) ~(e-\kappa)$ cpsd, is itself a function of the excitation frequency $\sigma$. More precisely, the polar motion response (37) to forcing at $\sigma$ cpsd is as if there is a resonance at $\sigma=\sigma_1(\sigma)$. The Chandler eigenfrequency $\sigma_{\rm CW}$ (i.e. the frequency of the free Chandler wobble mode) is not variable, of course. It is given by the value of  $\sigma_1(\sigma)$ for the specific excitation frequency $\sigma$ at which Re $\sigma_1(\sigma)=\sigma$. Another aspect that cannot be ignored is that the imaginary part of the anelasticity contribution to $\kappa$ has to be taken with a sign opposite to that of the forcing frequency: positive for retrograde wobbles and negative for prograde ones. This is required for ensuring that the tidal deformation lags behind the tidal forcing; see Mathews et al. (2002), Appendix C, for details.

6.2 Anelasticity model and the resonance frequency

The anelasticity model adopted in this work is the one employed by Mathews et al. (2002). It belongs to the class of models that Wahr & Bergen (1986) refer to as the $Q_\mu$ model of Sailor & Dziewonski (1978). The essential feature of these models is that the value of a deformational response parameter (e.g., $\kappa$) of the anelastic Earth to harmonic excitation at some frequency  $\omega_{\rm e}$ differs from the elastic-Earth value at a reference frequency  $\omega_{\rm m}$ by an amount proportional to

$$F(\omega_{\rm e};\omega_{\rm m},\alpha)=\cot\left({\alpha\pi\... ...t({\omega_{\rm m}\over\vert\omega_{\rm e}\vert}\right)^\alpha,$$ (39)

where $s_{\rm e}=\omega_{\rm e}/\vert\omega_{\rm e}\vert$. The reference frequency in the seismic frequency range was taken to be $\omega_{\rm m}=(2\pi/T_{\rm m})$ with $T_{\rm m}=200$ s. Compliances of the anelastic Earth in the retrograde diurnal frequency range, computed with $\alpha=0.15$, were found to give optimal results for the low frequency nutations. The same model played a crucial role in the work of Mathews et al. (2002) (see their Appendix D) in demonstrating that their estimate from nutation data for the period of the Chandler resonance (about 383 days) in the Earth's response to forcing at retrograde diurnal frequencies was entirely compatible with the observed Chandler eigenperiod of about 430 days: starting from the former, and taking account of the difference between the ocean tide contributions at the diurnal period of about 1 day and at the observed Chandler period, together with the difference in the anelasticity contributions  $\Delta\kappa_{\rm D}$ and  $\Delta\kappa_{\rm CW}$ to $\kappa$ at these periods as computed from (39), they arrived at the value $\sigma_{\rm CW} = (2.3175+0.0131~{\rm i}) \times 10^{-3}$ for the Chandler eigenferquency. The corresponding period and Q are 430.3 days and 88 respectively. These are quite compatible with estimates from observations of long period polar motions. We felt safe, therefore, in employing this model to compute the anelasticity contribution  $\Delta\kappa^{\rm AE}$ to $\kappa$ as a function of frequency down to the lowest of the wobble frequencies ($\approx$1/7000 cpsd) relevant to the diurnal nutations listed in the various rigid Earth series cited in the Introduction. If we denote $(\omega_{\rm m}/\omega_{\rm e})^\alpha$ by $f(\sigma)$ for  $\omega_{\rm e}=\Omega_0~\sigma$ and $f(\sigma_{\rm CW})$ for  $\omega_{\rm e}=\Omega_0~({\rm Re}\ \sigma_{\rm CW})$, one can see from (39) that $\Delta\kappa^{\rm AE}(\sigma)$, the part of  $\Delta\kappa(\sigma)$ that is due to anelasticity, may be readily obtained from $\Delta\kappa^{\rm AE}(\sigma_{\rm CW})$ for which the value $(4.381-1.205~{\rm i}) \times 10^{-5}$ has been found in the above-cited work: ${\rm Re}\ \Delta\kappa^{\rm AE}(\sigma)=[~(1-f(\sigma))/ (1-f(\sigma_{\rm CW}))~]\ {\rm Re}\ \Delta\kappa^{\rm AE}(\sigma_{\rm CW})$, and ${\rm Im}\ \Delta\kappa^{\rm AE}(\sigma) = [~s_\sigma f(\sigma)/f(\sigma_{\rm CW})~]\ {\rm Im}\ \Delta\kappa^{\rm AE}(\sigma_{\rm CW})$, where $s_\sigma$ is the sign of $\sigma$. The effective Chandler resonance frequency in the response to excitation at any frequency $\sigma$ in the low frequency band may then be computed as $\sigma_1(\sigma)=\sigma_{\rm CW} +(A/A_{\rm m}) [~\Delta\kappa^{\rm AE}(\sigma_{\rm CW})-\Delta\kappa^{\rm AE}(\sigma)~]$, in view of the expression for $\sigma_1$ in (35).

6.3 Excitation by time independent potential

It is evident, however, that the model (39) cannot remain valid down to zero frequency: F would become infinite at $\omega=0$, leading to an infinite anelasticity contribution. The zero frequency term present in the (4, 0) tidal potential presents, therefore, an exception that cannot be handled by the above procedure. The deformational responses to an incessantly acting potential should actually be characterized by the so-called secular or fluid Love numbers. Now, the compliance $\kappa$ is known to have a simple relation to the k Love number (see, for example, Sasao et al. 1980):

$$\kappa = \left(\Omega_0^2 a^5/ 3 GA\right)~k.$$ (40a)

On using for k its fluid (secular) value

$$k_{\rm f}=\left(3GA/ \Omega_0^2 a^5\right)~e$$ (40b)

which characterizes the response to a time independent degree 2 tidal potential, it follows that $\kappa=e$for $\omega=0$.

This finding has interesting consequences. One sees trivially that with $\sigma=0$ and $(e-\kappa)=0$, the frequency domain Eqs. (32) lead to the unphysical result that  $\tilde m(0)$ is infinite. One has to go back therefore to the time domain Eqs. (29), with  $\tilde c_3(t)$ and  $\tilde c^{\rm f}_3(t)$ replaced by the expressions (31), noting that the $\tilde\phi$ terms drop out since m=0 the present case. One sees immediately that the terms proportional to  $\tilde m(t)$ in Eq. (29a) cancel out as a consequence of the vanishing of $\kappa-e$, and that in the entirely adequate approximation wherein terms of  $O(\epsilon)$ are neglected, the two equations then take the forms ${\rm d}\tilde m/{\rm d}t +(A_{\rm f}/A)Y=\tilde\Gamma/(A\Omega_0)$ and ${\rm d}\tilde m/{\rm d}t+Y=0$, with $Y={\rm d}\tilde m_{\rm f}/{\rm d}t+{\rm i}\Omega_0 \tilde m_{\rm f}$. Subtraction of one from the other yields $Y=-\tilde\Gamma/(A_{\rm m} \Omega_0)$, which is a constant in the present case since  $\tilde\Gamma$ is. The solution for  $\tilde m(t)$ is then immediate:

$$\tilde m(t)= {\rm i}\Omega_0~K t+\tilde m_0, \qquad K={\tilde\Gamma(0)\over {\rm i}\Omega_0^2 A_{\rm m}}\cdot$$ (41)

The initial value $\tilde m_0$ of $\tilde m(t)$ is arbitrary. The value of K in the case of the (4, 0) potential may be obtained from Eqs. (28) and (27):

$$K= -{15\over (4\pi)^{1/2}}~{g_{\rm e}M_{\rm E}\over \Omega_0^2 A_{\rm m}}~H^{(4,0)}_0~ (C_{4,1}+{\rm i}S_{4,1}).$$ (42)

The solution (41) describes a secular motion, relative to the pole of maximum moment of inertia, of the Earth's instantaneous rotation pole. The associated nutation  $\tilde\eta(t)$, obtained by integrating Eq. (11) after introducing (41), is

$$\tilde\eta(t)= (P-{\rm i}K\Omega_0t)~ {\rm e}^{{\rm i}~\Omega_0(t-t_0)}+Q,$$ (43)

where P and Q depend on the values $\tilde m_0$ and $\tilde\eta_0$ of $\tilde m$ and  $\tilde\eta$ at t=0:

$$P = K+\tilde m_0/{\rm i}\Omega_0,\qquad Q=\tilde\eta_0-P~{\rm e}^{{\rm -i}\Omega_0t_0}.$$ (44)

The "polar motion'' $\tilde p(t)$ can then be easily seen to be

$$\tilde p(t)= -\tilde\eta(t){\rm e}^{{\rm -i}\Omega_0(t-t_0)} ={\r... ...-(\tilde m_0/{\rm i}\Omega_0)\left(1-{\rm e}^{{\rm -i}\Omega_0t}\right)\right].$$ (45)

The first term in (45), linear in t, being of a type not encountered before, is of considerable interest. It represents a steady drift of the celestial pole in relation to the TRF. The next term, also proportional to the forcing, is of such small magnitude as to be entirely negligible. The remaining terms involve the arbitrary initial values $\tilde m_0$ and $\tilde p_0$. Retaining just the linear term, we find that

$$x_{\rm p}(t)=-K_{\rm I} \Omega_0t,\qquad y_{\rm p}(t)=-K_{\rm R}\Omega_0t,$$ (46)

where the subscripts R and I represent the real and imaginary parts. $K_{\rm R}$ and $K_{\rm I}$ are proportional to C₄₁ and S₄₁, respectively.