Transport and instability in driven two-dimensional magnetohydrodynamic flows
Journal of Fluid Mechanics
Cambridge University Press (CUP)
This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.
Reason for embargo
This paper concerns the generation of large scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a beta-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed ‘zonostrophic instability’ and studied in a variety of settings, both numerically and analytically. The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given, random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress. The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear, and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full MHD system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and beta-effect on zonostrophic instability and magnetically driven instabilities
We are grateful to the EPSRC for funding SD via a DTG research studentship.
Vol. 799, July 2016, pp. 541- 578