Forced nutations of the Earth: contributions from the effects of ellipticity and rotation on the elastic deformations

We determine the deformation produced by the lunisolar tidal potential in a rotating, spheroidal model Earth. We proceed by decomposing the equations of motion into separate, though coupled, equations for the nutational and deformational parts of the Earth's response. Using this scheme, we derive a simpler set of equations for the deformational displacements, where the driving forces include not only the tidal terms but also inertial forces and gravitational perturbations associated with the nutational motions. We show that the deformations are affected only to a very small extent by the Earth's asphericity and rotation. This fact is exploited to set up a perturbative procedure, whereby the equation governing the deformation is separated into equations of zeroth and first orders in the perturbation. In the initial calculation (the zeroth order), the influences of the Earth's asphericity and the inertial forces associated with the deformation are neglected, while the forces arising from the nutational motions are taken into account. The resulting calculation for the quasi-static deformation is equivalent to the so-called spherical approximation used by Sasao et al. (1980), although the solutions obtained here are physically more insightful. This zeroth-order calculation is used to determine the compliances defined in the work of Mathews et al. (1991a), which characterize the deformability of the Earth. In the second step of the calculation, the solutions obtained under the spherical approximation are used to determine corrections to the deformation for the omitted effects of ellipticity and inertia (including the Coriolis force). Corresponding corrections to the zeroth-order compliances used by Mathews et al. (1991b) are found to be nominally O(ε) smaller than the zeroth-order compliances, where ε is the geometric ellipticity (surface flattening) of the Earth. As a consequence of these corrections to the compliance parameters, changes in the nutation amplitudes as computed by Mathews et al. (1991b) are produced, which amount to -0.18, 0.46, and 0.26 milliarcseconds, in the prograde semiannual, and the retrograde annual and 18.6-year terms, respectively. Additional corrections are introduced if we require the theoretical value of the retrograde annual nutation to match the determination made using very long baseline interferometry. The procedure presented here to account for the effects of ellipticity and rotation could also be used to determine corrections to nutations for the effects of anelasticity in the mantle and inner core or for the effect; of lateral heterogeneity in the Earth's density and elastic properties