The aim of these simulations is to examine under what conditions the radial shear of differential rotation operating in the thin solar tachocline layer can amplify a strong enough large scale toroidal magnetic field that undergoes magnetic buoyancy instabilities and develops buoyantly rising structures. The numerical simulations together with a subsequent analytical study (Vasil and Brummell, 2009) show that magnetic buoyancy instabilities can indeed develop in the shear-generated magnetic layer (Figure 11) if the forcing that drives the shear flow is sufficiently large. The needed forcing is such that, in the absence of the magnetic field, it imposes a hydrodynamically unstable shear. It is found that the imposed shear needs to have a Richardson number being less than 1, where measures the relative importance of the stabilizing effect of the stratification over the strength of the shear to overturn the fluid (Vasil and Brummell, 2009; Silvers et al., 2009). This result is not surprising because in order for the magnetic layer to be buoyantly unstable, the imposed shear flow needs to transfer enough energy to the magnetic field for it to overcome the stable background stratification (Silvers et al., 2009). It is not clear whether such strong forcing of the shear exists in the solar tachocline. For the observed shear in the solar tachocline, the Richardson number is estimated to be much greater than 1, (Gough, 2007). However, the observed shear in the tachocline may not correspond to the forcing shear, but is the end steady-state reached when the forcing is balanced by the built-up magnetic stress and turbulent transport.
Silvers et al. (2009) further extend the studies of Vasil and Brummell (2008) and Vasil and Brummell (2009) by considering the fact that the ratio of the magnetic diffusivity () over the thermal diffusivity () in the solar tachocline is very small: . Under such conditions the double-diffusive magnetic buoyancy instabilities can develop at a much less steep magnetic pressure gradient for the magnetic layer compared to that required for magnetic buoyancy instabilities under the assumption of adiabatic evolution (see end of Section 4.2). The stabilizing effect of the subadiabatic stratification is significantly reduced by the thermal diffusion. Simulations by Silvers et al. (2009) verify that double-diffusive magnetic buoyancy instabilities indeed can develop for a magnetic layer generated by a weak forcing shear that is hydrodynamically stable (with ).
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