[Paper Review] Nonlinear Stability and Instablity in Rayleight--Taylor Problem of Stratisfied Compressible MHD Fluids

Fei Jiang, Song Jiang|arXiv (Cornell University)|Feb 24, 2017

Navier-Stokes equation solutions4 references3 citations

TL;DR

This paper establishes nonlinear stability and instability criteria for the stratified compressible magnetohydrodynamic (MHD) Rayleigh–Taylor problem in Lagrangian coordinates. It proves that a sufficiently strong vertical magnetic field stabilizes the flow under $Ψ < 1$, while instability occurs when $Ψ > 1$, demonstrating that compressibility weakens magnetic stabilization but pressure can counteract this effect. The results are extended to viscoelastic fluids, showing elasticity provides stronger stabilization than magnetic fields.

ABSTRACT

We establish criteria of stability and instability for the stratified compressible magnetic Rayleigh--Taylor (RT) problem. More precisely, if under the stability condition $Ξ&lt;1$, we show the existence of unique solution with algebraic decay in time for the (compressible) magnetic RT problem with proper initial data in Lagrangian coordinates. The stability result presents that sufficiently large vertical (base) magnetic field can inhibit the development of RT instability. On the other hand, if $Ξ&gt;1$, there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This shows that the RT instability still occurs when the strength of base magnetic field is small or the base magnetic field is horizontal with proper large horizontal period cell. Moreover, by analyzing the stability condition in magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in vertical direction can be inhibited by the stabilizing effect of pressure, which also plays an important role in the mathematical proof for stability of the magnetic RT problem. In addition, we will extend the results in magnetic RT problem to the (compressible) viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than the one of magnetic fields.

Motivation & Objective

• To establish rigorous nonlinear stability and instability criteria for the stratified compressible MHD Rayleigh–Taylor problem.
• To investigate how compressibility affects the stabilizing role of magnetic fields in the presence of gravity.
• To determine whether the magnetic inhibition effect observed in incompressible MHD fluids persists in compressible regimes.
• To compare the stabilizing effects of magnetic fields and elasticity in stratified compressible fluids.
• To extend the stability framework to the viscoelastic RT problem and assess relative stabilization strength.

Proposed method

• Formulating the compressible MHD equations in Lagrangian coordinates with a vertical gravitational field and a base magnetic field.
• Defining a stability condition $Ψ < 1$ based on energy estimates and functional inequalities involving density, pressure, and magnetic field strength.
• Constructing a Lyapunov-type energy functional $χ\mathcal{E}$ and proving its decay via the dissipation term $χ\mathcal{D}$, leading to algebraic time decay of solutions.
• Using integration by parts and Poincaré-type inequalities to bound the magnetic and elastic energy terms in the energy functional.
• Proving instability in the Hadamard sense by constructing initial data that lead to exponential growth of velocity components over time.
• Extending results to the viscoelastic RT problem by analyzing the elasticity coefficient $κ$ and comparing its stabilizing effect to that of magnetic fields.

Experimental results

Research questions

• RQ1 Under what conditions does a vertical magnetic field stabilize the compressible MHD Rayleigh–Taylor instability?
• RQ2 How does compressibility affect the stabilizing influence of magnetic fields in the RT problem?
• RQ3 Can the instability still occur when the magnetic field is weak or horizontal, and what determines this behavior?
• RQ4 How does the stabilizing effect of elasticity in viscoelastic fluids compare to that of magnetic fields in MHD fluids?
• RQ5 What role does pressure play in counteracting the destabilizing effect of compressibility on magnetic stabilization?

Key findings

• When the stability condition $\Xi < 1$ holds, a unique solution exists with algebraic decay in time, proving nonlinear stability under sufficiently strong vertical magnetic fields.
• For $\Xi > 1$, the system exhibits nonlinear instability in the Hadamard sense, with solutions showing exponential growth of velocity components over time.
• Compressibility weakens the stabilizing effect of vertical magnetic fields, but this destabilizing influence is counteracted by the stabilizing role of pressure.
• The instability condition $\Xi > 1$ is satisfied for sufficiently small elasticity coefficient $\kappa$, confirming that weak elasticity fails to stabilize the system.
• In the viscoelastic RT problem, elasticity provides stronger stabilization than magnetic fields, as evidenced by the dominance of $\kappa$ in the stability criterion.
• The stability condition $\Xi < 1$ is verified to hold under the condition $\kappa_{-}, \kappa_{+} > 0$ and appropriate bounds on density and pressure gradients.