Two-dimensional ideal magnetohydrodynamic waves on a rotating sphere under a non-Malkus field: I. Continuous spectrum and its ray-theoretical interpretation

This is a Preprint and has not been peer reviewed. The published version of this Preprint is available: https://doi.org/10.1080/03091929.2024.2384388. This is version 2 of this Preprint.

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Authors

Ryosuke Nakashima , Shigeo Yoshida

Abstract

Two-dimensional ideal incompressible magnetohydrodynamic (MHD) linear waves at the surface of a rotating sphere are studied as a model to imitate the outermost layer of the Earth's core or the solar tachocline. This thin conducting layer is permeated by a toroidal magnetic field the magnitude of which depends only on the latitude. The Malkus background field, which is proportional to the sine of the colatitude, provides two well-known groups of branches; on one branch, retrograde Alfvén waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfvén waves undergo a gradual transition into slow MR waves. In the case of non-Malkus fields, we demonstrate that the associated eigenvalue problems can yield a continuous spectrum instead of Alfvén and slow MR discrete modes. The critical latitudes attributed to the Alfvén resonance eliminate these discrete eigenvalues and produce an infinite number of singular eigenmodes. The theory of slowly varying wave trains in an inhomogeneous magnetic field shows that a wave packet related to this continuous spectrum propagates toward a critical latitude corresponding to the wave and is eventually absorbed there. The expected behaviour whereby the retrograde propagating packets pertaining to the continuous spectrum approach the latitudes from the equatorial side and the prograde ones approach from the polar side is consistent with the profiles of their eigenfunctions derived using our numerical calculations. Further in-depth discussions of the Alfvén continuum would develop the theory of the ``wave–mean field interaction'' in the MHD system and the understanding of the dynamics in such thin layers.

DOI

https://doi.org/10.31223/X5Z67T

Subjects

Fluid Dynamics

Keywords

Magnetic Rossby waves, Continuous modes, Stably stratified layer, Geomagnetic variations

Dates

Published: 2023-10-05 03:35

Last Updated: 2024-08-18 16:55

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License

CC-BY Attribution-NonCommercial-ShareAlike 4.0 International

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Conflict of interest statement:
None