Asymptotic and numerical solutions of the initial value problem in rotating planetary fluid cores

P>An initial state of fluid motion in planetary cores or atmospheres, excited, for example, by the giant impact of an asteroid or an earthquake and then damped by viscous dissipation, decays towards the state of rigid-body rotation. The process of how the initial state approaches the final state, the initial value problem, is investigated both analytically and numerically for rotating fluid spheres. We derive an explicit asymptotic expression for the time-dependent solution of the initial value problem valid for an asymptotically small Ekman number E. We also perform a fully numerical analysis to simulate time-dependent solutions of the initial value problem for a small value of E. Comparison between the asymptotic solution and the corresponding numerical simulation shows a satisfactory quantitative agreement. For the purpose of illustrating why spherical geometry represents an intricate and exceptional case, we also briefly discuss the initial value problem in a rotating fluid channel. Geophysical and planetary physical implications of the result are also discussed