Splitting of seismic-free oscillations and of the Slichter triplet using the normal mode theory of a rotating, ellipsoidal earth

The normal mode theory of a rotating, ellipsoidal earth offers the possibility to treat in a unified way the seismic normal modes, whose frequencies are much greater than the earth’s rotation speed, and the five known rotational modes, the periods of which are diurnal, nearly-diurnal or longer than a few hundred days. However, except for the Slichter mode, the seismic modes have not been studied using this method. Instead, the splitting of them by rotation and ellipticity is usually computed in the frame of a perturbation method applied to a non-rotating, spherical earth. In this paper, we apply the normal mode theory to the study of the splitting of the seismic modes by rotation and ellipticity. This approach enables us to treat isolated modes as well as modes that have close frequencies on a non-rotating, spherical model and that are strongly coupled by rotation and ellipticity. We find that the computed eigenfrequencies agree to within the observational uncertainties with the eigenfrequencies obtained by a perturbation technique. The plotted eigenfunctions show the difficulty of unambiguously identifying the dominant spheroidal or toroidal characteristic of singlets stemming from coupled multiplets. Finally, we determine the split frequencies of the Slichter modes for the PREM and the 1066A models. We find that the ellipticity and the centrifugal force are responsible for a shift of at most 0.3% relative to the eigenfrequencies obtained by considering the Coriolis force as the single perturbation to the non-rotating, spherical case.