The thermohaline, Richardson, Rayleigh-Taylor, Solberg-Hoiland, and GSF criteria in rotating stars

Aims. We examine the interactions of various instabilities in rotating stars, which usually are considered as independent. Methods. An analytical study of the problem is performed accounting for radiative losses, mu-gradients, and horizontal turbulence. Results. The diffusion coefficient for an ensemble of instabilities is not given by the sum of the specific coefficients for each instability, but by the solution of a general equation. We find that thermohaline mixing is possible in low-mass red giants only if the horizontal turbulence is very weak. In rotating stars the Rayleigh-Taylor and the shear instabilities need simultaneous treating. This leads to rotation laws of the form 1/r(alpha) being predicted to be unstable for alpha > 1.6568, while the usual Rayleigh criterion only predicts instability for alpha > 2. Also, the shear instabilities are somehow reduced in main sequence stars by the effect of the Rayleigh-Taylor criterion. Various instability criteria should be expressed differently in rotating stars than in simplified geometries.