Stellar mixing II. Double diffusion processes

In this paper, salt-fingers (also called thermohaline convection) and semi-convection are treated under the name of double-diffusion (DD). We present and discuss the solutions of the RSM(Reynolds stress models) equations that provide the momentum, heat, mu fluxes, and their corresponding diffusivities denoted by K-m,K-h,K-mu. Such fluxes are given by a set of linear, algebraic equations that depend on the following variables: mean velocity gradient (differential rotation), temperature gradients (for both stable and unstable regimes), and mu-gradients (DD). Some key results are as follows. Salt-fingers. When shear is strong and DD is inefficient, heat and mu diffusivities are identical. Second, when shear is weak K-mu > K-h and the difference can be sizeable O(10) meaning that heat and mu diffusivities must therefore be treated as different. Third, for strong-to-moderate shears and for R-mu less than 0.8, both heat and mu diffusivities are practically independent of R-mu. Fourth, the latter result favors parameterizations of the type K-h,K-mu similar to CR mu 0 suggested by some authors. Our results, however, show that C is not a constant but a linear function of the Reynolds number Re = epsilon(nu N-2)(-1) defined in terms of the kinematic viscosity nu, the Brunt-Vaisala frequency N, and the rate of energy input into the system, epsilon. Fifth, we suggest that epsilon is an essential ingredient that has been missing in all diffusivity models, but which ought to be present because without a source of energy, turbulence dies out and so does the turbulent mixing (for example, the turbulent kinetic energy is proportional to the power 2/3 of epsilon). Moreover, since different stellar environments have different epsilon, its presence is necessary for differentiating mixing regimes in different stars. Semi-convection. In this case the destabilizing effect is the T-gradient, and when shear is weak, K-h > K-mu. Since the model is symmetric under the change R-mu to R-mu(-1), most of the results obtained in the previous case can be translated to this case.