Dissipation efficiency of turbulent convective zones

The turbulent dissipation in convective zones is important for a wide range of astrophysical problems. Usually, this dissipation is parametrized by an effective viscosity coefficient, which is estimated by assuming Kolmogorov cascade for the turbulent flow, and applying some heuristic (Zahn, 1966, 1989; Goldreich & Nicholson, 1977; Goldreich & Keeley, 1977) or perturbative (Goodman & Oh, 1997) argument. Because Kolmogorov turbulence is isotropic all such prescriptions predict isotropic viscosity, and the only parameter the dissipation efficiency is allowed to depend on is the period of the external shear. The assumption of Kolmogorov turbulence is justified, only as long as there are no external length scales to the problem. However, for many astrophysical problems, the timescales of interest correspond to the turnover times of eddies with sizes comparable to or larger than the pressure scale height of the convective zone, especially near the surface of the star. For such cases, no analytical prescriptions exist for the properties of the turbulent flow, and one has to rely on numerical simulations. This work begins by adapting the Goodman & Oh (1997) perturbative formalism for estimating the effective viscosity to the output of numerical models of a stratified convective layer. The calculation is performed for simulations of small patches of the surface convective zones of three low mass stars, and the non-Kolmogorov turbulence is seen to lead to much more efficient dissipation at low periods, and to significantly anisotropic turbulent viscosity. Next, a specialized numerical model of a convective layer is introduced, based on the anelastic approximation, which includes external forcing directly and not as a perturbation. A number of tests are performed in order to verify the model and show it is suitable for studying turbulent dissipation in convective zones. Two methods are presented for deriving the effective viscosity from the output of the above model, and a large number of simulations are performed in order to study its anisotropy, period and amplitude dependence. The effective viscosities from the perturbative and direct methods are combined to construct a complete viscosity tensor which is then applied to several astrophysical systems, which give constraints for the dissipation efficiency in stellar convective zones.