On the fast rotation asymptotics of a non-homogeneous incompressible MHD system

TL;DR

This paper investigates the asymptotic behavior of a 2D incompressible MHD system with density variations under rapid rotation, identifying different limit dynamics in quasi-homogeneous and fully non-homogeneous regimes using advanced mathematical techniques.

Contribution

It provides a rigorous analysis of the fast rotation limit for non-homogeneous incompressible MHD systems, including new limit equations and the handling of ill-prepared initial data.

Findings

Limit dynamics in quasi-homogeneous regime identified as homogeneous MHD with density transport.

In fully non-homogeneous regime, magnetic field equations couple with a linear relation involving density and velocity.

The analysis accommodates density-dependent viscosity and resistivity coefficients.

Abstract

This paper is devoted to the analysis of a singular perturbation problem for a $2$-D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non-constant profile. In this case, in the limit, the equation for the magnetic field combines with an underdetermined linear equation, which links the limit density variation function with the limit velocity field. The proof is based on a compensated compactness argument, which enables us to consider general ill-prepared initial data. An application of Di Perna-Lions theory for transport equations allows to treat the case of density-dependent viscosity and resistivity coefficients.