  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.
