Overstable Alfvén waves in periodic shear flows at low magnetic Prandtl number

Periodic shear flows are known to exhibit negative eddy viscosity effects (Dubrulle & Frisch 1991). Here, we show that this effect can lead to a destabilization of shear Alfvén waves. In particular, 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 system is known to be unstable to the Kelvin-Helmholtz instability in the hydrodynamic case, and in ideal MHD, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. We demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we call the Alfvénic Dubrulle-Frisch mode due to its connection to shear Alfvén waves, exists for any nonzero magnetic field strength as long as the magnetic Prandtl number Pm<1. We present a reduced model for this instability that reveals its connection to the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (1991). We also demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.