Normal mode analysis of fluid discontinuities: numerical method and application to magnetohydrodynamics

TL;DR

This paper introduces a numerical framework for analyzing fluid discontinuities using normal mode analysis, enabling accurate stability studies of shock fronts and vortex sheets in magnetohydrodynamics, including nonideal effects.

Contribution

It provides a systematic numerical method to study the coupling of normal modes across fluid discontinuities, applicable to complex conservation laws and plasma conditions.

Findings

Successfully applied to magnetohydrodynamic shocks and shear layers.

Exactly recovers linear stability properties of studied discontinuities.

Allows inclusion of nonideal effects like dispersion and dissipation.

Abstract

Fluid discontinuities, such as shock fronts and vortex sheets, can reflect waves and become unstable to corrugation. Analytical calculations of these phenomena are tractable in the simplest cases only, while their numerical simulations are biased by truncation errors inherent to discretization schemes. The author lays down a computational framework to study the coupling of normal modes (plane linear waves) through discontinuities satisfying arbitrary conservation laws, as is relevant to a variety of fluid mechanical problems. A systematic method is provided to solve these problems numerically, along with a series of validation cases. As a demonstration, it is applied to magnetohydrodynamic shocks and shear layers to exactly recover their linear stability properties. The straightforward inclusion of nonideal (dispersive, dissipative) effects notably opens a route to investigate how these phenomena are altered in weakly ionized plasmas.