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The stability of current vortex sheets with transverse magnetic field

Binqiang Xie, Yueyang Feng, Ying Zhang

TL;DR

This work analyzes the stability of current vortex sheets in compressible MHD with a transverse magnetic field. By linearizing about a rectilinear baseline and reformulating on fixed domains, it reduces the front dynamics to a pseudo-differential equation and analyzes its symbol via Fourier–Laplace methods. The main results show that a transverse magnetic field does not universally stabilize the flow; rather, stability (well-posedness) is ensured for the transverse-field regime when the magneto-acoustic Mach number satisfies $M_B>\sqrt{2}$. The study provides detailed a priori estimates and an existence theory (Theorems 2.1 and 2.2) for the front under this stability condition, clarifying the role of $M_B$ and the front symbol in the linearized problem.

Abstract

Compared to the results in \cite{Shivamoggi}, using the normal mode method, we have rigorously confirmed that a transverse magnetic field reduces the stability of the system. Specifically, a larger velocity is required for stability in the presence of a magnetic field than in its absence. More precisely, when the magnitude of the magneto-acoustic Mach number $M_{B}:=\frac{\dot{v}_1^{+}}{\bar{C}_{B}}>\sqrt{2}$, we proved the well-posedness of the current vortex sheet problem for compressible MHD flows with a transverse magnetic field.

The stability of current vortex sheets with transverse magnetic field

TL;DR

This work analyzes the stability of current vortex sheets in compressible MHD with a transverse magnetic field. By linearizing about a rectilinear baseline and reformulating on fixed domains, it reduces the front dynamics to a pseudo-differential equation and analyzes its symbol via Fourier–Laplace methods. The main results show that a transverse magnetic field does not universally stabilize the flow; rather, stability (well-posedness) is ensured for the transverse-field regime when the magneto-acoustic Mach number satisfies . The study provides detailed a priori estimates and an existence theory (Theorems 2.1 and 2.2) for the front under this stability condition, clarifying the role of and the front symbol in the linearized problem.

Abstract

Compared to the results in \cite{Shivamoggi}, using the normal mode method, we have rigorously confirmed that a transverse magnetic field reduces the stability of the system. Specifically, a larger velocity is required for stability in the presence of a magnetic field than in its absence. More precisely, when the magnitude of the magneto-acoustic Mach number , we proved the well-posedness of the current vortex sheet problem for compressible MHD flows with a transverse magnetic field.

Paper Structure

This paper contains 12 sections, 10 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Rectilinear solution
  3. The new formulations
  4. The wave equation for the log-density $h$ and the magnetic $B$
  5. Linearized problems and main theorem
  6. Construction of special solution of the linearized system.
  7. Proof of Theorem 2.1
  8. The symbol of equation \ref{['3.155']} for the front
  9. Study of the roots $\mu^{\pm}$
  10. Study of the symbol $\Sigma$
  11. The stability case
  12. Proof of Theorem 2.2

Key Result

Lemma 1.1

Define the function $\Psi$ by Then $\Psi: (t,x)\mapsto (t,x_{1}, x_{2},x_{3}+ \psi(t,x))$ are diffeomorphism of $\Omega^{\pm}$ for all $t \in[0, T]$.

Theorems & Definitions (22)

  • Lemma 1.1
  • proof
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 4.1
  • Lemma 4.2
  • ...and 12 more