ON A NEW, ONE-DIMENSIONAL, TIME-DEPENDENT MODEL FOR TURBULENCE AND CONVECTION .1. A BASIC DISCUSSION OF THE MATHEMATICAL-MODEL

A model for turbulence and convection in regimes of very large Reynolds numbers is introduced by coupling an equation for turbulent energy to the equations of radiation hydrodynamics. These in turn are modified by nonlinear terms due to interactions with turbulence. All appearing Reynolds correlations are consistently modeled using a diffusion approximation. The equations of the model are discussed for spherical symmetry. The related equations for stellar structure are derived. In the hydrostatic regime, the free parameters are determined in the limit of a fully developed convection zone by comparison with the usual mixing length formulations. The general properties of the turbulence equation are characterized. Its consistency within the framework of the equations of radiation hydrodynamics is highlighted. It is shown that familiar notions like "semi-convection", "overshooting", and the criterion for convective instability find a natural interpretation in the model.