On a variational formulation of the weakly nonlinear magnetic Rayleigh--Taylor instability

The magnetic-Rayleigh--Taylor (MRT) instability is a ubiquitous phenomenon that occurs in magnetically-driven Z-pinch implosions. It is important to understand this instability since it can decrease the performance of such implosions. In this work, I present a theoretical model for the weakly nonlinear MRT instability. I obtain such model by asymptotically expanding an action principle, whose Lagrangian leads to the fully nonlinear MRT equations. After introducing a suitable choice of coordinates, I show that the theory can be cast as a Hamiltonian system, whose Hamiltonian is calculated up to sixth order in a perturbation parameter. The resulting theory captures the harmonic generation of MRT modes. In particular, it is shown that the saturation amplitude of the linear MRT instability grows as the stabilization effect of the magnetic-field tension increases. Overall, the theory provides an intuitive interpretation of the weakly nonlinear MRT instability and provides a systematic approach for studying this instability in more complex settings.