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, account is given to radiative losses, mu-gradients and horizontal tur
bulence. 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 has for consequence that rotation laws of the form 1/r^(alpha) are predicted to be unstable for alpha > 1.6568, while the usual Rayleigh criterion predicts instability only 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.
The aim of this paper is to examine the effects of the horizontal turbulence in differentially rotating stars on the GSF instability and apply our results to pre-supernova models. For this purpose we derive the expression for the GSF instability with
account of the thermal transport and smoothing of the mu-gradient by the horizontal turbulence. We apply the new expressions in numerical models of a 20 solar mass star. We show that if N^2_{Omega} < 0 the Rayleigh-Taylor instability cannot be killed by the stabilizing thermal and mu-gradients, so that the GSF instability is always there and we derive the corresponding diffusion coefficient. The GSF instability grows towards the very latest stages of stellar evolution. Close to the deep convective zones in pre-supernova stages, the transport coefficient of elements and angular momentum by the GSF instability can very locally be larger than the shear instability and even as large as the thermal diffusivity. However the zones over which the GSF instability is acting are extremely narrow and there is not enough time left before the supernova explosion for a significant mixing to occur. Thus, even when the inhibiting effects of the mu-gradient are reduced by the horizontal turbulence, the GSF instability remains insignificant for the evolution. We conclude that the GSF instability in pre-supernova stages cannot be held responsible for the relatively low rotation rate of pulsars compared to the predictions of rotating star models.
