Turbulence in Stars. III. Unified Treatment of Diffusion, Convection, Semiconvection, Salt Fingers, and Differential Rotation

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Abstract
The goal of this paper is to propose a unified treatment of diffusion, convection, semiconvection, salt fingers, overshooting, and rotational mixing. The detection of SN 1987A has served, among other things, to highlight the incompleteness of our understanding of such phenomena. Moreover, the variety of solutions proposed thus far to deal with each phenomenon separately, the uncertainty about the Ledoux-Schwarzschild criteria, the extent of overshooting, the effect of a μ gradient, the role of differential rotational mixing, etc., have added further urgency to the need of a unified, rather than a case-by-case, treatment of these processes. Since at the root of these difficulties lies the fact that we are dealing with a highly nonlinear, turbulent regime under the action of three gradients T (temperature), C (concentration), and (mean flow), it is not surprising that such difficulties have arisen. In this paper we propose a unified treatment based on a turbulence model. A key difference with previous models is that we do not employ heuristic arguments to determine the five basic timescales that enter the problem and that entail a corresponding number of adjustable constants. These timescales are computed using renormalization group (RNG) techniques. The model comes in three flavors: (a) all the turbulent variables are treated nonlocally; (b) the turbulent kinetic energy K and its rate of dissipation are nonlocal, while the remaining turbulence variables (fluxes, Reynolds stresses, etc.) are treated locally; and (c) all turbulence variables are local. In the latter case, one must specify a mixing length. Some of the results are as follows: 1. The local model entails the solution of two algebraic equations, one being the flux conservation law. By solving them, we obtain the desired - ad versus μ relations for semiconvection and salt fingers. We also derive other variables of interest, turbulent diffusivities, Peclet number, turbulent velocity, etc. 2. Schwarzschild and Ledoux criteria for instability are replaced by a new criterion that is physically equivalent to the requirement that turbulent mixing can exist only so long as the turbulent kinetic energy is positive. In addition to , ad, and μ, the new criterion depends on the turbulent diffusivities for temperature and concentration that only a turbulence model can provide. 3. We derive the dynamic equations necessary to quantify the extent of overshooting OV in the presence of a μ barrier. 4. We prove that OV(μ) < OV(μ = const.). Although this result is physically understandable, no direct proof has been available as yet. 5. We derive the turbulent diffusivity for a passive scalar, one that does not affect a preexisting turbulence, e.g., a sedimentation of He. We show that it differs from that of an active scalar, e.g., a μ field causing semiconvection and/or salt fingers. Such diffusivity is a function of the temperature gradient (stable/unstable) and shear (rotational mixing). 6. We show that the turbulent diffusivities of momentum (entering the angular momentum equation), of heat (entering the model of convection), and concentration (entering the diffusion equation and/or semiconvection and salt fingers) are different from one another and should not be taken to be the same, as has been done thus far. 7. We consider the effect of shear. We solve the local turbulence problem analytically and derive the turbulent diffusivities for momentum, heat, and concentration in terms of the three gradients of the mean fields, T, C, and . Since shear is itself a source of turbulent mixing, one could expect it to enhance the diffusivities. However, its interaction with salt fingers and semiconvection is a subtle one, and the opposite may occur, a phenomenon for which we offer a physical interpretation and a validation with laboratory data. 8. A comparison is made with previous models.

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