Journal Article

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

P. B. Ivanov and J. C. B. Papaloizou

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 407, issue 3, pages 1609-1630
Published in print September 2010 | ISSN: 0035-8711
Published online September 2010 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1111/j.1365-2966.2010.17009.x
Inertial waves in rotating bodies: a WKBJ formalism for inertial modes and a comparison with numerical results

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Inertial waves governed by Coriolis forces may play an important role in several astrophysical settings, such as tidal interactions, which may occur in extrasolar planetary systems and close binary systems, or in rotating compact objects emitting gravitational waves. Additionally, they are of interest in other research fields, e.g. in geophysics.

However, their analysis is complicated by the fact that in the inviscid case the normal mode spectrum is either everywhere dense or continuous in any frequency interval contained within the inertial range. Moreover, the equations governing the corresponding eigenproblem are, in general, non-separable.

In this paper we develop a consistent WKBJ formalism, together with a formal first-order perturbation theory for calculating the properties of the normal 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.

Keywords: hydrodynamics; planet; star interactions; binaries: general; stars: oscillations; stars: rotation

Journal Article.  16088 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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