Narrow Results

The $z$-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the $z$-direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such $z$-pinch is always unstable. We also establish the existence of a largest growing mode, which dominates the linear growth of the linear MHD system.

The level set method for compressible flows [13] is simple to implement, especially in the presence of topological changes. However, this method was shown to suffer from large spurious oscillations in [11]. In [4], a new Ghost Fluid Method (GFM) was shown to remove these spurious oscillations by minimizing the numerical smearing in the entropy field with the help of an Isobaric Fix [6] technique. The original GFM was designed for the inviscid Euler equations. In this paper, we extend the formulation of the GFM and apply the extended formulation to the viscous Navier-Stokes equations. The resulting numerical method is robust and easy to implement along the lines of [15].