Large Scale Quasi-geostrophic Magnetohydrodynamics

TL;DR

This paper derives a new quasi-geostrophic MHD equation for shallow fluid layers on rotating planets, revealing unique inverse cascade behavior and energy accumulation in zonal flows, relevant to Earth's outer core dynamics.

Contribution

It introduces a novel large-scale MHD equation with unique cascade properties and an extra invariant, extending quasi-geostrophic theory to magnetized planetary fluids.

Findings

Inverse cascade leads to energy transfer to smaller scales.

Energy accumulates in larger scales and zonal flows.

The equation predicts a Kolmogorov-type spectrum for the inverse cascade.

Abstract

We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on a rapidly rotating planet or star. The presence of a background toroidal magnetic field is assumed, and the "shallow water" beta-plane approximation is used. We derive a single equation for the slow large length scale dynamics. The range of validity of this equation fits the MHD of the lighter fluid at the top of Earth's outer core. The form of this equation is similar to the quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the parameters are essentially different. Our equation also implies the inverse cascade; but contrary to the usual Q-G situation, the energy cascades to smaller length scales, while the enstrophy cascades to the larger scales. We find the Kolmogorov-type spectrum for the inverse cascade. The spectrum indicates the energy accumulation in larger scales. In addition to the energy and enstrophy, the obtained equation possesses an extra invariant. Its presence is shown to imply energy accumulation in the $30^\circ$ sector around zonal flow. With some special energy input, the extra invariant can lead to the accumulation of energy in zonal flow; this happens if the input of the extra invariant is small, while the energy input is considerable.