Semiconvection as the occasional breaking of weakly amplified internal waves

Abstract

An analysis is made of the semiconvective zones which arise in the evolution of many stars. Despite the presence of a strongly stabilizing solute gradient, growing overstable modes are possible for a slightly superadiabatic temperature gradient, because heat diffusion greatly exceeds solute diffusion or molecular viscosity. These modes are essentially identical to weakly amplified internal waves (or hydromagnetic waves if rotation and magnetic field are present). Their amplification is balanced by a cascade to higher wavevectors (caused in part by subharmonic parametric instabilities) leading to infrequent wavebreaking 'events' which provide the required redistribution of solute. Detailed quantification of this model is impractical, but a simplified analysis indicates that the ratio of the superadiabaticity to the solute gradient is at most of order (D_e/K)^(1/2) where D_e is the solute 'eddy' diffusivity and K is the thermal diffusivity. Evolutionary models require D_e << K, so the Schwarzschild-Härm criterion for semiconvection is essentially correct. A field in excess of about 10^4 gauss modifies the model somewhat, but does not invalidate it. Propagation of the waves out into the radiative envelope of the star is unimportant. The related phenomenon to semiconvection which occurs in differentiating black dwarfs (e.g. Jupiter, Saturn) is also briefly discussed.

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