Hamilton's principle and normal mode coupling in an aspherical planet with a fluid core

Abstract

We apply Hamilton's principle to obtain the exact equations of motion for an elastic planet that is rotating, self-gravitating and comprises both fluid and solid regions. This variational problem is complicated by the occurrence of tangential slip at fluid-solid boundaries, but we show how this can be accommodated both directly and using the method of Lagrange multipliers. A novelty of our approach is that the planet's motion is described relative to an arbitrary reference configuration, with this generality offering advantages for numerical calculations. In particular, aspherical topography on the free surface or internal boundaries of the planet's equilibrium configuration can be converted exactly into effective volumetric heterogeneities within a geometrically spherical reference body by applying a suitable particle relabelling transformation. The theory is then specialized to consider the linearized motion of a planet about a steadily rotating equilibrium configuration, with these results having applications to normal mode coupling calculations used within studies of long-period seismology, tidal deformation and related fields. In particular, we explain how our new theory will, for the first time, allow aspherical boundary topography to be incorporated exactly within such coupling calculations.