On the stability of mean-field models of the solar convection zone

The stability of the solutions of the mean-field theories of turbulent media is questioned. It is done here for the model equations for the solar convection zone which have been used, in particular, to explain the differential rotation. We present an approximation valid for axisymmetric, short-wave disturbances. A critical local Rayleigh number can be defined - involving eddy diffusivities - above which the stratification becomes unstable. For mixing-length models of the solar convection zone we always find sub-critical Rayleigh numbers. One must be careful, however, with other theoretical models. Those we have considered do not reach sufficiently high surface pressure values so that there the associated Rayleigh numbers exceed their critical limits. In the outermost layers in such models, therefore, the solutions could really be unstable.