Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

Abstract

This paper concerns the classic problem of stability of the Kolmogorov flow u = (0, sin x) in the infinite (x, y)-plane. A mean magnetic field of strength B 0 is introduced and the MHD linear stability problem is studied for modes with a wave-number k in the y-direction, and sometimes a Bloch wavenumber in the x-direction. The key parameters governing the problem are the Reynolds number ν −1 , magnetic Prandtl number P , and dimensionless magnetic field strength B 0 corresponding to an inverse magnetic Mach number. The mean magnetic field can also be taken to have an arbitrary direction in the (x, y)-plane and a mean x-directed flow U 0 can be incorporated, in the most general formulation.First the paper considers Kolmogorov flow with a y-directed mean magnetic field, which for convenience we refer to as 'vertical'. Taking = 0, the suppression of the classic hydrodynamic instability is observed with increasing field strength B 0 . A branch of strong-field instabilities occurs when the magnetic Prandtl number P is less than unity, as found recently by A.E. Fraser, I.G. Cresser and P. Garaud (J. Fluid Mech. 949, A43, 2022). Analytical results based on eigenvalue perturbation theory in the limit k → 0 support the numerics for both weak-field and strong-field instabilities, and show their origin in the coupling of large-scale weakly damped modes with x-wavenumber n = 0, to smaller-scale modes having n = ±1.The paper then considers the case of 'horizontal' or x-directed mean magnetic field. Here the unperturbed state consists of steady, wavey magnetic field lines distorted by the underlying Kolmogorov flow, and with the driving body force balancing both viscosity and the Lorentz force. As the magnetic field is increased from zero, the purely hydrodynamic instability is suppressed again, but for stronger fields a new branch of instabilities appears. Allowing a non-zero Bloch wavenumber allows further instability, and in some circumstances when the system is hydrodynamically stable, arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Numerical results are presented together with eigenvalue perturbation theory in the limits k, → 0. The theory gives analytical approximations for growth rates and thresholds that are in good agreement with those computed.