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

TL;DR

This paper investigates two-dimensional ideal MHD waves on a rotating sphere with non-Malkus magnetic fields, revealing the emergence of a continuous spectrum and wave absorption at critical latitudes, advancing understanding of Earth's core and solar tachocline dynamics.

Contribution

It demonstrates that non-Malkus magnetic fields lead to a continuous spectrum of eigenmodes and wave absorption phenomena, extending the classical discrete mode analysis.

Findings

Continuous spectrum replaces discrete modes in non-Malkus fields.

Wave packets are absorbed at critical latitudes due to Alfvén resonance.

Eigenfunctions show propagation patterns consistent with theoretical predictions.

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 Alfv\'en waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfv\'en 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 Alfv\'en and slow MR discrete modes. The critical latitudes attributed to the Alfv\'en 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 Alfv\'en 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.