Axisymmetric perturbations of thin gaseous disks. I - Unstable convective modes and their consequences for the solar nebula

The axisymmetric perturbations of a thin, differentially rotating gas disk in which the vertical temperature stratification is superadiabatic are analyzed. The growth rate of adiabatic convective normal modes is calculated, departures from simple polytropic disk models are briefly discussed, and the detailed vertical structure of the eigenfunctions is analyzed. The stabilizing effect of radiative diffusion on convective modes is considered. It is found that rotation and compressibility tend to reduce the rate of growth of the disturbances, and that the growth rate increases without limit and is proportional to the square root of the radial wavenumber in polytropic equilibria. The maximum radial size of convective eddies scales like the square root of the degree of superadiabaticity times the size of the convective zone. There is significant convective penetration into radiatively stable layers only for the fundamental and low-order harmonic modes, whose vertical wavelength is comparable to the size of the convective layer.