Unstable transition properties of the driven magnetohydrodynamic sheet pinch

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Abstract
The unstable transition behaviour of a bounded, current-carrying, two-dimensional magnetofluid is explored, using the hydrodynamic theory developed for parallel shear flows as a guide. The time development of a perturbed driven magnetohydrodynamic sheet pinch is simulated numerically. The nonlinear partial differential equations of two-dimensional, incompressible, viscoresistive magnetohydrodynamics are advanced in time numerically by means of a semi-implicit, mixed Fourier collocation-finite difference algorithm. Nonlinear excitation of the higher wavenumbers results in the development of electric current sheets of finite extent, as well as the formation of ‘attraction currents’ centred at the magnetic O-points. A secondary instability mechanism, the dynamic tearing of the electric current sheet, is also observed. This dynamic tearing leads to sawtooth-like temporal oscillations in certain global quantities. The long time state of the system resembles a nonlinearly saturated state with significant excitation of many wavenumbers. Some features of this state can be understood by means of a Landau nonlinear stability theory based on certain assumptions about the perturbation energy balance.