Stellar mixing

II. Double diffusion processes^⋆

¹ NASA, Goddard Institute for Space Studies, New York, NY 10025, USA
² Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA
e-mail: vmc13@columbia.edu

Received: 17 March 2010
Accepted: 25 August 2010

Abstract

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 (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 suggested by some authors. Our results, however, show that C is not a constant but a linear function of the Reynolds number Re = ε(νN²)^-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 , most of the results obtained in the previous case can be translated to this case.