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 B 0 , 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 B 0 is increased from zero. However with increasing B 0 , 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 P m < 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 1 $ ]]> P m > 1 in the presence of an additional imposed x-directed fluid flow U 0 . For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B 0 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.