Analysis of singular inertial modes in a spherical shell: the slender toroidal shell model

Abstract

We derive the asymptotic spectrum (as the Ekman number E → 0) of axisymmetric inertial modes when the problem is restricted to two dimensions. We show that the damping rate of such modes scales with the square root of the Ekman number and that the width of the shear layers of the eigenfunctions scales with E^1/4. The eigenfunctions obey a Schrödinger equation with a quadratic potential; we provide the analytical expression for eigenvalues (frequency and damping rate). These results validate the picture that attractors act like a potential well, trapping inertial waves which resist confinement owing to viscosity. Using three-dimensional numerical solutions, we show that the results can be applied to equatorially trapped modes in a thin spherical shell; in fact, these two-dimensional solutions give the first step (the zeroth order) of a perturbative approach to three-dimensional solutions in a spherical shell. Our method is applicable in a straightforward way to any other container where bi-dimensionality dominates.