A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfven and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.

/pdf/nonlinear-dynamics-of-non-uniform-current-vortex-sheets-in-1q2scbklpj.pdf

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagate along the plasma-vacuum interface, when it is linearly weakly stable. 
Following the approach of Ali and Hunter, we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.

/pdf/nonlinear-surface-waves-on-the-plasma-vacuum-interface-51i5964k74.pdf

Nonlinear surface waves on the plasma-vacuum interface

We describe examples of hyperbolic surface waves and discuss their connection with initial boundary value and discontinuity problems for hyperbolic systems of PDEs that are weakly but not uniformly stable.

https://www.math.ucdavis.edu/~hunter/papers/surf_corrected.pdf

Nonlinear Hyperbolic Surface Waves

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well as the quadratic potential V (x), ∂tU +( U − Ωx ⊥ ) · ∇xU = −ΩU ⊥ −∇xV , with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross-Pitaevskii equation for Bose-Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection. We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations. Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is quantified in terms of the initial spectral gap associated with the 2 × 2 initial velocity gradient, λ2(0) − λ1(0) ,λ j (0) = λj (∇xU0) as well as the initial divergence, divx(U0). We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the deformed flow map.

https://orion.math.iastate.edu/hliu/Pub/2006/Liu-ZAMP06.pdf

Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations

We derive a quadratically nonlinear equation that describes the motion of a tangential discontinuity in incompressible magnetohydrodynamics near the onset of a Kelvin–Helmholtz instability. We show that nonlinear effects enhance the instability and give a criterion for it to occur in terms of a longitudinal strain of the fluid motion along the discontinuity.

/pdf/on-the-weakly-nonlinear-kelvin-helmholtz-instability-of-3zts53bk4p.pdf

On the weakly nonlinear kelvin-helmholtz instability of tangential discontinuities in mhd

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

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On the weakly nonlinear kelvin-helmholtz instability of tangential discontinuities in mhd

Nonlinear hyperbolic wave propagation in a one-dimensional random medium