Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.