Weakly nonlinear surface waves in magnetohydrodynamics

TL;DR

This paper develops a systematic method to construct highly accurate weakly nonlinear surface wave solutions in incompressible magnetohydrodynamics, advancing understanding of high-frequency oscillations and their governing equations.

Contribution

It introduces a novel approach for constructing arbitrarily many correctors to approximate solutions of free boundary problems in MHD with high-frequency oscillations.

Findings

Constructed approximate solutions with arbitrary accuracy.

Derived solvability conditions using algebra and combinatorics.

Showed the non-occurrence of the rectification phenomenon in this context.

Abstract

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the incompressible magnetohydrodynamics equations. Current vortex sheets are piecewise smooth solutions to the incompressible magnetohydrodynamics equations that satisfy suitable jump conditions for the velocity and magnetic field on the (free) discontinuity surface. In this work, we complete an earlier work by Ali and Hunter and construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to `surface waves' on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobitype equation known as the `HIZ equation' (standing for Hamilton-Il'insky-Zabolotskaya) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. Based on a suitable duality formula, we exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. Theverification of these solvability conditions is based on a combination of mere algebra and arguments of combinatorial analysis. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the free boundary problem. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves does not arise in the same way for the current vortex sheet problem.