On the viscous damping of inertial oscillation in planetary fluid interiors

Abstract

The effect of viscous dissipations on inertial oscillations in a rapidly rotating fluid sphere like the Earth’s core and planetary interiors is investigated analytically for an asymptotically small Ekman number. Two different cases are studied. In the first case, we employ the stress-free velocity boundary condition which is appropriate for giant fluid planets such as Jupiter. In this case, an explicit general analytical expression in closed form for the decay rates of all inertial oscillation modes due to viscous dampings is obtained. We show that the smallest non-zero decay rate is associated with an inertial oscillation mode that is a purely toroidal, westward travelling wave with the azimuthal wavenumber, m=2 and with the maximum azimuthal flow at middle latitudes, θ=±45°. In the second case, we employ the non-slip velocity boundary condition which is appropriate for the Earth’s fluid core or laboratory experiments. An explicit general analytical expression for the decay rates of all inertial oscillation modes is also derived. However, except for a few special modes such as the spin-over mode, numerical integration is generally required to evaluate a one-dimensional integral in the analytical expression. It is found that the polar modes which have complex spatial structure decay slower than that of the spin-over mode that is purely toroidal and has the simplest spatial structure.