The scalar boundary conditions for the motion of the elastic earth to second order in ellipticity

The scalar equations of infinitesimal elastic gravitational motion for a ro tating, slightly elliptical Earth are always used to study the Earth's nuta tion and tides theoretically, while the determination of the integration of the equations depends, to a certain extent, on the choice of a set of appr opriate boundary conditions. In this paper, a continuity quantity related t o the displacement is first transformed from the elliptical reference bound ary to the corresponding effective spherical domain, and then converted fro m a vector (or tensor) form to a scalar form by generalized surface spheric al harmonics expansion. All the related components, including the displacem ent vector field (or the stress tensor field), are then decomposed into the poloidal and toroidal field using the symmetry restrictions on the normal mode eigenfunctions. After truncation, the boundary conditions are finally derived, in a scalar ordinary differential form. The process of the derivat ion is second order in ellipticity and in full detail. Moreover, the other boundary conditions are also presented as second order in ellipticity at th e end of this paper.