Nonlinear evolution of the interface of the sausage instability

In this work, I study the nonlinear evolution of the $m=0$ sausage instability, which is a well-known magnetohydrodynamic (MHD) instability occurring in Bennett-type pinches, that is, axially uniform, axisymmetric, cylindrical plasmas. In the proposed model, the plasma column is assumed incompressible and perfectly conducting. To describe the nonlinear dynamics of the plasma surface, I introduce a contour-dynamics formulation, where the interface of the plasma column is modeled as a series of interacting co-axial vortex rings. The radius, axial location, and vortex strength of each ring are allowed to dynamically vary, and the corresponding equations of motion are derived. For small initial sinusoidal perturbations, the calculated linear growth rate agrees with the well-known growth rate determined using simple potential-flow theory. In the nonlinear regime, initial sinusoidal perturbations grow into a "spindle" structure with broad minima in plasma radius and sharp disk-like maxima. Using this model, I numerically investigate the long-time behavior of the sausage instability and quantify its temporal evolution for various initial conditions. For code-validation purposes, I compare the results of this model to numerical simulations using the 2D radiation-MHD code HYDRA.