Accretion disk material differs from normal stellar gas by the existence of the small-scale ‘‘α-turbulence'’ which enables the disk to form. This background turbulence transports both angular momentum in the radial direction and energy in the vertical direction, and at the same time produces heat by friction through its own eddy viscosity. The questions are whether in such a medium a convective instability with moderate scale-length would develop, and what form the instability (‘‘Schwarzschild'') criterion for such a convective motion would take. We find that the interaction of rotation and turbulence is an effective generator of (axisymmetric) instabilities even for the temperature gradients which normally, according to the traditional Schwarzschild criterion, guarantee stability. For different reasons, however, the new instabilities seem to be of minor relevance for modelling the thin accretion disks that in reality exist. Only in one case, in which the turbulent energy transport influences the fluctuations, does a convection develop for parameter values which are interesting in accretion disk theory.