Astronomy and Astrophysics
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, μ fluxes, and their corresponding diffusivities denoted by K m,h,μ . Such fluxes are given by a set of linear, algebraic equations that depend on the following variables: mean velocity gradient (differential rotation), temperature gradients
... ature gradients (for both stable and unstable regimes), and μ-gradients (DD). Some key results are as follows. Salt−fingers. When shear is strong and DD is inefficient, heat and μ diffusivities are identical. Second, when shear is weak K μ > K h and the difference can be sizeable O(10) meaning that heat and μ diffusivities must therefore be treated as different. Third, for strong-to-moderate shears and for R μ less than 0.8, both heat and μ diffusivities are practically independent of R μ . Fourth, the latter result favors parameterizations of the type K h,μ ∼ CR 0 μ suggested by some authors. Our results, however, show that C is not a constant but a linear function of the Reynolds number Re = ε(νN 2 ) −1 defined in terms of the kinematic viscosity ν, the Brunt-Väisälä frequency N, and the rate of energy input into the system, ε. Fifth, we suggest that ε 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 ε). Moreover, since different stellar environments have different ε, 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 μ . Since the model is symmetric under the change R μ to R −1 μ , most of the results obtained in the previous case can be translated to this case.