Y. Rogister
EOST, Strasbourg, France
Yves.Rogister@eost.u-strasbg.fr
The eigenfrequencies $\omega$ of a non-rotating spherical Earth model are degenerate, the frequency of a normal mode of harmonic degree $\ell$ being $2 \ell + 1$ times degenerate. The slow and nearly steady rotation of the Earth acts in two ways to split each multiplet in $2 \ell +1$ singlets and to shift the mean frequency of the multiplet. First, if the Earth is supposed to be a rotating body initially in hydrostatic equilibrium, its shape is slightly ellipsoidal. The flattening of the ellipsoidal strata ranges from about 1/415 at the center to approximately 1/300 at the surface. The departure from spherical symmetry is responsible for the splitting and the shift of the degenerate eigenfrequencies. Second, the oscillations of the ellipsoidal model are computed in a frame of reference rotating steadily at the mean angular speed $\Omega$ of the Earth. Therefore, apparent inertia forces, {\em i.e.}the Coriolis force and the centrifugal force, must be taken into account. The effect of the Coriolis force is similar to the Zeeman effect in quantum mechanics. In this case, the degenerate energy levels of an atom of hydrogen are split owing to the presence of a magnetic field. The centrifugal force also splits and shifts the degenerate eigenfrequencies. It has to be noticed that the first observation of normal mode splitting goes back to 1961 after the 1960 Chilean earthquake. The computation of the frequencies of the seismic modes of a rotating, ellipsoidal Earth model is generally based on a perturbation technique. In this scheme, the eigenfrequencies and eigenfunctions of the corresponding non-rotating spherical model are first computed. The ellipticity and the apparent inertia forces are then considered to be small perturbations. The Coriolis force is the dominant perturbation since it is of the order of $\frac{\Omega}{\omega}$. The effects of the centrifugal force and of the ellipticity are of the order of $\left( \frac{\Omega}{\omega} \right)^2$. In order that this approach be coherent, the effect of the Coriolis force has to be calculated up to the second order in $\left( \frac{\Omega}{\omega} \right)^2$. In this paper, we investigate another method for computing the eigenfrequencies of a rotating ellipsoidal model. It consists in a direct integration of the equations of motion of a rotating ellipsoidal model. Until now, this method has mainly been applied to the study of rotational modes, namely the Chandler wobble, the Free Core Nutation or the Free Inner Core Nutation. But it is general enough to allow for the study of the seismic modes. We focus on the modes which are significantly influenced by gravity. These modes have a frequency below 1 mHz. One of them is the translational mode of the inner core, also named the Slichter mode. We find slight discrepancies with the results obtained using the pertubation technique mentioned above. By considering the separate effects of the Coriolis force, the centrifugal force and the ellipticity, we explain the discrepancies.