A new nutation series for a more realistic model earth

The frequency-dependent correction coefficients with respect to the forced nutations of a rigid earth are computed using the complex scalar gravitational-motion equations for an earth model with an anelastic mantle. Oceanic loads and tidal currents enter the model via outer boundary conditions. The ellipticity of the core-mantle boundary and the dynamical ellipticity are adjusted to observations. This requires the behaviour inside the model earth to be regarded as non-hydrostatic. Some relevant equations for the evaluation of boundary conditions and some terms in the equations of motion are expanded to second order in ellipticity. The computation of the equipotential-surface ellipticity profile is carried to second order as well. These second-order expansions lead to increased accuracy of the results in general. Moreover, one achieves a better reliability for the integration at frequencies close to a resonance. This allows the integration of the equations of motion at any relevant nutation period without the need for a normal-mode expansion. A complete new nutation series for a realistic model earth is presented.     

Internal waves in a rotating stratified spherical shell: asymptotic solutions

Summary. Small amplitude oscillations of a rotating, density-stratified fluid bounded by a spherical shell are examined. No restrictions are placed on the thickness of the shell. The internal mode spectrum is examined in the complete rotation-stratification parameter range including the regime that is appropriate for a plausible stratification distribution in the Earth's fluid core. A mathematical model is derived in terms of an eigenvalue PDE of mixed type. The existence of oscillatory solutions is exhibited in the limits of no rotation and no stratification. The frequency spectrum is extended asymptotically away from these limiting cases. A reduction in the complexity of the PDE for modes oscillating at the inertial frequency is exploited. A variational formulation is constructed in which the stratification parameter is treated as an eigenvalue of the system for fixed wave frequency. The spectral information is again extended asymptotically away from these 'accessible' points. Although the PDE reduces to Laplace's tidal equations (LTE) only under stringent parameter restrictions, it is observed that aspects of the behaviour of low frequency LTE modes are reproduced in the general model.     

The forced nutations of an elliptical, rotating, elastic and oceanless earth

Summary. We compute the luni-solar forced nutations of an elliptical, rotating, self-gravitating, elastic, hydrostatically prestressed and oceanless earth. Several recent structural models are considered, each possessing a fluid outer core and solid inner core. Complete results are given for the nutation of the 'axis of figure for the Tisserand mean surface' which best represents the observational effects of the Earth's nutational motion. Differences between results for different structural models are observationally insignificant. Differences between our results and Molodensky's are as large as ∼ 0.002 arcsec at six month and at 18.6 yr.     

Seismic rays in an ellipsoidal earth model: theoretical analysis and estimation of deviations due to the ellipticity

Summary. A first-order form of the Euler's equations for rays in an ellipsoidal model of the Earth is obtained. The conditions affecting the velocity law for a monotonic increase, with respect to the arc length, in the angular distance to the epicentre, and in the angle of incidence, are the same in the ellipsoidal and spherical models. It is therefore possible to trace rays and to compute travel times directly in an ellipsoidal earth as in the spherical model. Thus comparison with the rays of the same coordinates in a spherical earth provides an estimate of the various deviations of these rays due to the Earth's flattening, and the corresponding travel-time differences, for mantle P -waves and for shallow earthquakes. All these deviations are functions both of the latitude and of the epicentral distance. The difference in the distance to the Earth's centre at points with the same geocentric latitude on rays in the ellipsoidal and in the spherical model may reach several kilometres. Directly related to the deformation of the isovelocity surfaces, this difference is the only cause of significant perturbation in travel times. Other differences, such as that corresponding to the ray torsion, are of the first order in ellipticity, and may exceed 1 km. They induce only small differences in travel time, less than 0.01s. Thus, we show that the ellipticity correction obtained by Jeffreys (1935) and Bullen (1937) by a perturbational method can be recovered by a direct evaluation of the travel times in an ellipsoidal model of the Earth. Moreover, as stated by Dziewonski & Gilbert (1976), we verify the non-dependence of this correction on the choice of the velocity law.