The subseismic approximation in core dynamics - II. Love numbers and surface gravity

Abstract

This paper extends our earlier examinations of the utility of various approximations for treating the dynamics of the Earth's liquid core on time-scales of the order of 104 to 108 s. We discuss the effects of representing the response of the mantle and inner core by static (versus dynamic) Love numbers, and of invoking the subseismic approximation for treating core flow, used either only in the interior of the liquid core (SSA-1) or also at the boundaries (SSA-2). The success of each approximation (or combinations thereof) is measured by comparing the resulting surface gravity effects (computed for a given earthquake excitation), and (for the Slichter mode) the distribution of translational momentum, with reference calculations in which none of these approximations is made. We conclude that for calculations of the Slichter triplet, none of the approximations is satisfactory, i.e. a full solution (using dynamic Love numbers at elastic boundaries and no core flow approximation) is required in order to avoid spurious eigenfrequencies and to yield correct eigenfunctions (e.g. conserving translational momentum) and surface gravity. For core undertones, the use of static Love numbers at rigid boundaries is acceptable, along with SSA-1 (i.e. provided the subseismic approximation is not invoked at the core boundaries). Although the calculations presented here are for a non-rotating earth model, we argue that the principal conclusions should be applicable to the rotating Earth. Shortcomings of the subseismic approximation appear to arise because both SSA-1 and SSA-2 lower the order of the governing system of differential equations (giving rise to a singular perturbation problem), and because SSA-2 overdetermines the boundary conditions (making it impossible for solutions to satisfy all continuity requirements at core boundaries).