Resistive tearing mode instability with shear flow and viscosity

The linear theory of the resistive tearing mode instability in slab geometry, has been recently extended by introducing the effect of equilibrium shear flow and viscosity [ Phys. Fluids 29, 2563 (1986); Phys. Fluids B 1,2224 (1989); ibid. 2,495 (1990); ibid. 2,2575 (1990) 1. In the present analysis, numerical solutions of the time-dependent resistive equations are generalized to this problem and growth rate scaling is obtained. The results of the computations are compared to previous work, and the computed growth rate scalings agree with analytical predictions. Namely, the “constant-$” growth rate scales as S i’2 and the “nonconstant+” growth rate scales as S “3, where S is the magnetic Reynolds number. The Furth-KilleenRosenbluth (FKR) scaling of S 3’5 is reproduced for small values of shear flow. The presence of flow introduces a new peak in the eigenfunction, which is outside of the peak that occurs in the case without flow. The introduction of viscosity and small shear alters the growth rate scaling to S “3(S,/S) IF6 where S is the ratio of the viscous time to the Alfvitn time. When the shear flow is large, the grow& rate behaves in a more complex way, and KelvinHelmholtz instability effects are present.