Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

Abstract

A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u=(0,sin⁡x) in the infinite (x,y)-plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B0, in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with −1/2<ℓ≤ 1/2. In parallel, asymptotic approximations are developed, valid in the limits k→0, ℓ→0, using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B0 is increased from zero. However with increasing B0, further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number Pm<1, as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for Pm>1 in the presence of an additional imposed x-directed fluid flow U0. For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B0 increases. The above instabilities are present for Bloch wavenumber ℓ=0; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even 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. Detailed comparisons are given between theory for small k and ℓ, and numerical results.