Nonlinear evolution of the resistive tearing mode

The nonlinear behavior of the tearing instability is investigated with numerical solutions of the resistive, incompressible, magnetohydrodynamic equations. Simulations have been completed for values of the Lundquist number S from 102 to 106 and wavelength parameter α=2πa/λ from 0.042 to 0.5. The initial state for the nonlinear computations is provided by the linear instability. In all cases, the nonlinear mode initially evolves at the linear rate, followed by a period of considerably reduced growth. At high S and low α, secondary-flow vortices, opposite in direction to the linear vortices, generate a new magnetic island centered at the initial X point. The nonlinear growth rate of the one constant-ψ (in the linear regime) solution considered is approximately an order of magnitude less than that of a comparable nonconstant-ψ solution over the same time period. The nonconstant-ψ computations indicate a reduction of from 8% to 27% of the initial shear-layer magnetic energy, with the larger reductions occurring for the longer wavelength disturbances. The constant-ψ simulation shows a reduction two orders of magnitude smaller. The island width of the nonconstant-ψ solution becomes larger than twice the scale of the initial shear layer. For all cases, the electric field parallel to the magnetic field grows, at the end of the run, to about 10−3 times the Driecer field.