CONSTRAINTS ON THE MAGNETIC BUOYANCY INSTABILITIES OF A SHEAR-GENERATED MAGNETIC LAYER

Abstract

In a recent paper addressing the solar Ω-effect, we found that the action of forced vertical (radial) shear on the vertical component of poloidal magnetic field could induce magnetic buoyancy that produced rising, undulating tubular magnetic structures, as often envisaged arising from the solar tachocline. However, such dynamics were only found to exist under extreme circumstances (extremely large forcing). Here, we analytically examine the reasons underpinning the difficulties in obtaining magnetic buoyancy in this type of system. Specifically, under various assumptions about the maintenance of the shear, we derive some analytic limits on the ability of localized radial shear to produce a toroidal magnetic field from a poloidal field. First, we consider a local time-dependent context, where an unmaintained shear builds a toroidal magnetic layer over time by shearing the poloidal field. Second, we consider the possibility that maintenance is necessary to obtain stationary buoyant dynamics, and examine a local time-independent context, where the shear is maintained by a weak forcing. In both situations, we derive estimates or mathematical bounds for the toroidal magnetic energy that can be realized and its gradients, and thus evaluate the likelihood of magnetic buoyancy instabilities. We find that the results can be expressed as conditions on the imposed shear-flow Richardson number, Ri, and the magnetic Prandtl number, σM, requiring either low Ri or high σM for instability. We found the former in earlier work by Vasil & Brummell, and perform a new simulation here confirming the latter. These results suggest that, for the case of the solar tachocline where σM is small and Ri is large, the assumptions of our models must be violated, and a more comprehensive model is likely required.