The Influence of a Magnetic Field on Schwarzschild's Criterion for Convective Instability in an Ideally Conducting Fluid

The influence of a magnetic field on the onset of adiabatic thermal convection in a compressible ideally conducting fluid is studied. In the absence of a magnetic field Schwarzschild's criterion for the onset of convection is a purely local criterion; a magnetic field connects fluid at different levels and a local criterion which is necessary and sufficient for stability against convection can no longer be obtained. However, it is shown that, for certain simple magnetic field configurations which do not introduce hydromagnetic instabilities, there are simple local criteria which are sufficient for the stability of the system. It is pointed out that, in a fluid of finite electrical resistivity, a magnetic field cannot be expected to lead to complete stabilization of convective motions. In such a case, if the sufficient condition is satisfied, there should probably be a substantial reduction in the growth rate of the instability and in the energy carried by fully developed convection. One of the sufficient conditions for stability is applied to a sunspot model recently constructed by Chitre and it is found that it is satisfied throughout most of the model. Although the theory is not general enough to apply to this case, it is suggestive that a high degree of suppression of convection by a magnetic field should occur in such a spot.