[Paper Review] Nonlinear Rayleigh-Taylor Instability for Nonhomogeneous Incompressible Viscous Magnetohydrodynamic Flows

Fei Jiang, Song Jiang+1 more|arXiv (Cornell University)|Apr 20, 2013

Navier-Stokes equation solutions22 references3 citations

TL;DR

This paper establishes the nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic (MHD) flows with zero resistivity in a 3D horizontally periodic domain. By constructing growing solutions to the linearized problem in Sobolev spaces and leveraging local well-posedness of the nonlinear system, the authors prove instability in density, velocity, and, when the magnetic field is small and vertical, in the magnetic field itself—confirming the physical mechanism where velocity instability induces magnetic field instability via the induction equation.

ABSTRACT

We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.

Motivation & Objective

• To investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution in 3D nonhomogeneous incompressible viscous MHD flows with zero resistivity.
• To analyze the linearized equations around the steady-state solution and construct unstable solutions that grow in time in Sobolev spaces.
• To extend linear instability results to the full nonlinear problem using local well-posedness of smooth solutions.
• To verify the physical mechanism where velocity instability induces magnetic field instability through the induction equation, particularly when the magnetic field is small and vertical.

Proposed method

• Linearize the 3D nonhomogeneous incompressible MHD equations with zero resistivity around a steady-state solution involving a density profile with increasing density in height.
• Construct solutions to the linearized problem that grow in time in the $H^k$ Sobolev space, establishing linear instability.
• Use a local well-posedness result for the original nonlinear problem to propagate the instability from the linear to the nonlinear regime.
• Apply Gronwall's inequality and energy estimates to control the evolution of perturbations in density, velocity, and magnetic field.
• Analyze the induction equation to show that instability in velocity leads to instability in the magnetic field when the steady magnetic field is small and vertical.
• Verify that the constructed unstable solutions satisfy the full nonlinear system and the divergence-free constraints for velocity and magnetic field.

Experimental results

Research questions

• RQ1 Does the Rayleigh-Taylor instability persist in the nonlinear regime for nonhomogeneous incompressible viscous MHD flows with zero resistivity?
• RQ2 Can unstable solutions of the linearized MHD system be used to construct unstable solutions for the full nonlinear problem?
• RQ3 How does the presence of a steady magnetic field affect the instability of the velocity and magnetic field components?
• RQ4 What is the role of the induction equation in transferring instability from velocity to magnetic field?
• RQ5 Under what conditions does the magnetic field become unstable due to velocity-driven instability?

Key findings

• The paper proves the existence of solutions to the linearized MHD equations that grow in time in the $H^k$ Sobolev space, confirming linear instability for the Rayleigh-Taylor problem.
• Nonlinear instability is established for the density, horizontal and vertical velocities in the full nonlinear MHD system, using the constructed unstable linear solutions and local well-posedness.
• When the steady magnetic field is small and vertical, the magnetic field also becomes unstable, verifying the physical mechanism of instability propagation via the induction equation.
• The instability of the velocity field directly induces instability in the magnetic field, as predicted by the induction equation, even in the absence of magnetic diffusivity.
• The analysis is conducted in a 3D horizontally periodic domain with a smooth, non-constant density profile satisfying $\bar{\rho}'(x_3^0) > 0$ at some point, enabling the classical RT instability.