Mean flow instabilities of two-dimensional convection in strong magnetic fields

The interaction of magnetic fields with convection is of great importance in astrophysics. Two well-known aspects of the interaction are the tendency of convection cells to become narrow in the perpendicular direction when the imposed field is strong, and the occurrence of streaming instabilities involving horizontal shears. Previous studies have found that the latter instability mechanism operates only when the cells are narrow, and so we investigate the occurrence of the streaming instability for large imposed fields, when the cells are naturally narrow near onset. The basic cellular solution can be treated in the asymptotic limit as a nonlinear eigenvalue problem. In the limit of large imposed field, the instability occurs for asymptotically small Prandtl number. The determination of the stability boundary turns out to be surprisingly complicated. At leading order, the linear stability problem is the linearisation of the same nonlinear eigenvalue problem, and as a result, it is necessary to go to higher order to obtain a stability criterion. We establish that the flow can only be unstable to a horizontal mean flow if the Prandtl number is smaller than order , where B0 is the imposed magnetic field, and that the mean flow is concentrated in a horizontal jet of width in the middle of the layer. The result applies to stress-free or no-slip boundary conditions at the top and bottom of the layer

Magnetic buoyancy instabilities in the presence of magnetic flux pumping at the base of the solar convection zone

We perform idealized numerical simulations of magnetic buoyancy instabilities in three dimensions, solving the equations of compressible magnetohydrodynamics in a model of the solar tachocline. In particular, we study the effects of including a highly simplified model of magnetic flux pumping in an upper layer (‘the convection zone’) on magnetic buoyancy instabilities in a lower layer (‘the upper parts of the radiative interior – including the tachocline’), to study these competing flux transport mechanisms at the base of the convection zone. The results of the inclusion of this effect in numerical simulations of the buoyancy instability of both a preconceived magnetic slab and a shear-generated magnetic layer are presented. In the former, we find that if we are in the regime that the downward pumping velocity is comparable with the Alfvén speed of the magnetic layer, magnetic flux pumping is able to hold back the bulk of the magnetic field, with only small pockets of strong field able to rise into the upper layer. In simulations in which the magnetic layer is generated by shear, we find that the shear velocity is not necessarily required to exceed that of the pumping (therefore the kinetic energy of the shear is not required to exceed that of the overlying convection) for strong localized pockets of magnetic field to be produced which can rise into the upper layer. This is because magnetic flux pumping acts to store the field below the interface, allowing it to be amplified both by the shear and by vortical fluid motions, until pockets of field can achieve sufficient strength to rise into the upper layer. In addition, we find that the interface between the two layers is a natural location for the production of strong vertical gradients in the magnetic field. If these gradients are sufficiently strong to allow the development of magnetic buoyancy instabilities, strong shear is not necessarily required to drive them (cf. previous work by Vasil & Brummell). We find that the addition of magnetic flux pumping appears to be able to assist shear-driven magnetic buoyancy in producing strong flux concentrations that can rise up into the convection zone from the radiative interior