Semiconvection: theory

TL;DR

This paper develops a theoretical model for heat and solute transport in semiconvective layers under astrophysical conditions, focusing on layer formation, thickness, and stability criteria.

Contribution

It introduces a model that predicts layer behavior and effective diffusivities, assuming rapid layer formation and weak dependence on layer thickness.

Findings

Effective diffusivities weakly depend on layer thickness when energy flux is specified.

The model predicts a maximum density ratio for stable layered states.

Layer thickness evolution can be estimated over time in semiconvective zones.

Abstract

A model is developed for the transport of heat and solute in a system of double-diffusive layers under astrophysical conditions (viscosity and solute diffusivity low compared with the thermal diffusivity). The process of formation of the layers is not part of the model but, as observed in geophysical and laboratory settings, is assumed to be fast compared to the life time of the semiconvective zone. The thickness of the layers is \tbf{a} free parameter of the model. When the energy flux of the star is specified, the effective semiconvective diffusivities are only weakly dependent on this parameter. An estimate is given of the evolution of layer thickness with time in a semiconvective zone. The model predicts that the density ratio has a maximum for which a stationary layered state can exist, $R_\rho\la \mr{Le}^{-1/2}$. Comparison of the model predictions with a grid of numerical simulations is presented in a companion paper.