Double-diffusive mixing-length theory, semiconvection and massive star evolution

Double-diffusive convection refers to mixing where the effects of thermal and composition gradients compete to determine the stability of a fluid. In addition to the familiar fast convective instability, such fluids exhibit the slow, direct salt finger instability and the slow, overstable semiconvective instability. Previous approaches to this subject usually have been based on linear stability analyses. We develop here the non-linear mixing-length theory (MLT) of double-diffusive convection, in analogy to the more familiar MLT for a fluid of homogeneous composition. We present approximate solutions for the mixing rate in the various regimes, and show that the familiar Schwarzschild and Ledoux stability criteria are good approximations to the precise criteria in stellar interiors. We have implemented the self-consistent computation of the temperature gradient and turbulent mixing rate in a stellar evolution code, and solved a diffusion equation to mix the composition at the appropriate rate. We have evolved 15- and 30-M⊙ stars from the zero-age main sequence to the end of core He-burning. Semiconvective mixing is fast enough to alter stellar composition profiles on relevant time-scales, but not so fast that instantaneous readjustment is appropriate.