On viscous decay factors for spherical inertial modes in rotating planetary fluid cores: Comparison between asymptotic and numerical analysis

The initial value problem of how an initial state of fluid motion, excited by earthquake or tide and then damped by viscous dissipation, decays toward the state of rigid-body rotation is considered for rapidly rotating fluid spheres like planetary fluid cores. An essential element in an asymptotic time-dependent solution for the initial value problem is the viscous decay factors for spherical inertial modes. We derive an analytical expression for the viscous decay factors valid for a broad range of the inertial modes that are required for an asymptotic solution of the initial value problem at an arbitrarily small but fixed Ekman number. We also perform fully numerical analysis to compute the viscous decay factors for several selected inertial modes, showing a quantitative agreement between the asymptotic and numerical analysis. It is argued that the correct viscous decay factors cannot be derived using an asymptotic expansion based on the half powers of a small Ekman number.