Inertial waves in rotating bodies: a WKBJ formalism for inertial modes and a comparison with numerical results

TL;DR

This paper develops a WKBJ formalism and perturbation theory to analytically study inertial modes in uniformly rotating, spherically symmetric bodies, showing good agreement with numerical results and enabling better understanding of inertial waves in astrophysical objects.

Contribution

It introduces a new WKBJ formalism and first-order perturbation approach for inertial modes in rotating bodies, validated against numerical simulations.

Findings

Analytical eigenfrequencies match numerical results across models.

First-order perturbation theory accurately estimates frequency corrections.

The approach is applicable to various astrophysical rotating bodies.

Abstract

(abbreviated) In this paper we develop a consistent WKBJ formalism, together with a formal first order perturbation theory for calculating the properties of the inertial modes of a uniformly rotating coreless body (modelled as a polytrope and referred hereafter to as a planet) under the assumption of a spherically symmetric structure. The eigenfrequencies, spatial form of the associated eigenfunctions and other properties we obtained analytically using the WKBJ eigenfunctions are in good agreement with corresponding results obtained by numerical means for a variety of planet models even for global modes with a large scale distribution of perturbed quantities. This indicates that even though they are embedded in a dense spectrum, such modes can be identified and followed as model parameters changed and that first order perturbation theory can be applied. This is used to estimate corrections to the eigenfrequencies as a consequence of the anelastic approximation, which we argue here to be small when the rotation frequency is small. These are compared with simulation results in an accompanying paper with a good agreement between theoretical and numerical results. The results reported here may provide a basis of theoretical investigations of inertial waves in many astrophysical and other applications, where a rotating body can be modelled as a uniformly rotating barotropic object, for which the density has, close to its surface, an approximately power law dependence on distance from the surface.