Magnetic Buoyancy Instability in the Solar Tachocline

Magnetic buoyancy has been suggested as a probable mechanism for the rise of flux tubes through the solar convection zone to emerge as the structures we observe at the surface. The large scale of these structures, however, implies that rising flux interacts with the effects of the small-scale, turbulent convection in the region through which they pass in such a way as to preserve the large scale variation. With this motivation, we consider the linear stability of a horizontal layer to magnetic buoyancy, as a model for the escape of field from the solar tachocline. We assume a turbulent region in the upper part of the layer and a non-turbulent region below. The effects of turbulent convective motion are captured via the turbulent pumping and turbulent diffusion effects implied by mean field dynamo theory. We produce a self-consistent equilibrium state given these effects, and solve for linear perturbations to this state. We consider the effects of parameter changes and of the vertical profiles of the turbulent effects on the growth rate, horizontal scale, and vertical variation of perturbations. We find that for stronger turbulent effects in the upper part of the layer, 2D interchange modes are preferred over 3D modes. We also apply the turbulent pumping and turbulent diffusion preferentially to larger horizontal scales, in light of the assumption of mean field theory. However, we find that the primary effect of the turbulent pumping and diffusion on stability for our parameters is via their influence on the initial equilibrium field gradient, as opposed to their action directly on the perturbations. In addition, following the asymptotic approach of Gilman (1970), we consider the non-diffusive case for modes with small spatial scale, to derive an analytic expression for the growth rate, given the effect of mean field turbulent pumping. In the small-scale, non-diffusive limit we find that, when the turbulent pumping is included, the stability is no longer determined by an effective vertically dependent dispersion relation but instead by a second order ODE for 3D modes, and first order for interchange. We focus on the interchange case and compare with the more general non-diffusive case, with no small-scale assumption, and find a third order eigenvalue problem for interchange modes. We consider two third order model problems in relation to this system, which we solve asymptotically in the limit of small turbulent pumping. We then consider a local approximation to the non-diffusive linear system and derive dispersion relations for the cases of first an isothermal and then an adiabatic system.