Turbulent transport and equilibrium profiles in two-dimensional magnetohydrodynamics with background shear

Turbulent transport and equilibrium profile are studied in two-dimensional magnetohydrodynamics (2D MHD) in the presence of a background shear flow and a large-scale magnetic field; the latter quantities are assumed to be parallel and to vary in the perpendicular direction. The nonuniformity of the background is incorporated, to first order, by using the Gabor transform. The magnetic vector potential and momentum fluxes (or total stress) are calculated in both kinematic and dynamic cases in the case of unit magnetic Prandtl number, which then determines turbulent diffusivity and viscosity and equilibrium profile of the mean shear flow. The turbulent diffusion is found to be suppressed for a strong (large-scale) magnetic field. The Lorentz force changes the sign of the total stress resulting in the turbulent viscosity with an opposite sign compared to that in the hydrodynamical case. The former reduces the amplitude of the total stress for a fixed shear due to the cancellation between Reynolds and Maxwell stresses, therefore leading to the reduction in momentum transport. Since the divergence of momentum flux acts as an effective force on the background shear, the presence of the magnetic field can lead to an equilibrium shear profile which is different from that of the pure hydrodynamic case. In particular, the Lorentz force is shown to laminarize the mean shear flow.