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This paper concerns the stability of Kolmogorov flow u = (0, sin x) in theinfinite (x,y)-plane. A mean magnetic field of strength B0 is introduced andthe MHD linear stability problem studied for modes with wave-number k in they-direction, and Bloch wavenumber l in the x-direction. The parametersgoverning the problem are Reynolds number 1/nu, magnetic Prandtl number P, anddimensionless magnetic field strength B0. The mean magnetic field can be takento have an arbitrary direction in the (x,y)-plane and a mean x-directed flow U0can be incorporated.  First the paper considers Kolmogorov flow with y-directed mean magneticfield, referred to as vertical. Taking l=0, the suppression of the purehydrodynamic instability is observed with increasing field strength B0. Abranch of strong-field instabilities occurs for magnetic Prandtl number P lessthan unity, as found by A.E. Fraser, I.G. Cresser and P. Garaud (J. Fluid Mech.949, A43, 2022). Analytical results using eigenvalue perturbation theory in thelimit k->0 support the numerics for both weak- and strong-field instabilities,and originate in the coupling of large-scale modes with x-wavenumber n=0, tosmaller-scale modes.  The paper considers the case of horizontal or x-directed mean magnetic field.The unperturbed state consists of steady, wavey magnetic field lines. As themagnetic field is increased, the purely hydrodynamic instability is suppressedagain, but for stronger fields a new branch of instabilities appears. Allowinga non-zero Bloch wavenumber l allows further instability, and in somecircumstances when the system is hydrodynamically stable, arbitrarily weakmagnetic fields can give growing modes. Numerical results are presentedtogether with eigenvalue perturbation theory in the limits k,l->0. The theorygives analytical approximations for growth rates and thresholds in goodagreement with those computed.

Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow