Stellar Mixing: I. Formalism

In this paper we use the Reynolds stress models (RSM) to derive algebraic expressions for the following variables: a) heat fluxes; b) mu fluxes; and c) momentum fluxes. These relations, which are fully 3D, include: 1) stable and unstable stratification, represented by the Brunt-Vaisala frequency, N-2 = -gH(p)(-1)(del - del(ad))(1 - R-mu); 2) double diffusion, salt-fingers, and semi-convection, represented by the density ratio R-mu = del(mu)(del - del(ad))(-1); 3) shear (differential rotation), represented by the mean squared shear Sigma(2) or by the Richardson number, Ri = N-2 Sigma(-2); 4) radiative losses represented by a Peclet number, Pe; 5) a complete analytical solution of the 1D version of the model. In general, the model requires the solution of two differential equations for the eddy kinetic energy K and its rate of dissipation, e. In the local and stationary cases, when production equals dissipation, the model equations are all algebraic.