Emergence of Unstable Modes for Shock Waves in Ideal MHD

TL;DR

This paper investigates the stability of classical MHD shock waves in ideal plasmas, identifying a critical parameter manifold where unstable modes emerge, with detailed analysis of symmetric and nearly parallel shocks.

Contribution

It provides an explicit algebraic description of the critical manifold and numerical insights into the emergence of unstable modes in ideal MHD shocks.

Findings

Unstable transverse modes emerge at the critical manifold.

Explicit algebraic form of the critical manifold for parallel shocks.

Unstable mode pairs originate from non-zero imaginary spectral values in nearly parallel shocks.

Abstract

This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies a critical manifold in parameter space, across which slow classical MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. In the reflectionally symmetric case of parallel shocks, this emergence happens at the spectral value 0, and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerical treatment show that for only almost parallel shocks the unstable mode pair emerges from a pair of non-zero imaginary spectral values.