The Effective Tidal Viscosity in Close Solar-Type Binaries

A major problem confronting the understanding of tidal evolution of close solar-type binaries is the inefficiency of the turbulent convection. The value of the effective viscosity estimated, in the framework of the mixing length theory (MLT), implies circularization timescales which are almost two orders of magnitude longer than observed. Moreover, the reduction of the effective viscosity due to the fast time-variation of the tidal shear in short period binaries, increases the discrepancy to about three orders of magnitude. This state of affairs has motivated suggestions that tidal orbital evolution, notably circularization occurs mainly during the pre-main-sequence phase. However, observational data accumulated over the recent decades imply that circularization does occur during the the main-sequence phase (Mazeh 2008). In this work, we examine the possibility that the apparent inefficiency of turbulent convection is merely a shortcoming of MLT approach. Indeed, a recent 3D numerical simulation (Penev et al. 2007), suggests that the true convective viscosity is probably larger than the MLT value and that the reduction due to the time-variation of the shear is not drastic. We employ a model for stellar turbulent convection (Canuto, Goldman & Mazzitelli 1996) to evaluate the effective viscosity both for a steady for and time dependent tidal shear. The model is physically based, self-consistent, and accounts for the full spectrum of the turbulent eddies. It has been found advantageous, compared to the MLT, in many applications. We use an analytic approximation to the turbulent spectrum to obtain the reduction of the efficiency due to the time-variation of the tide. The results are: (i) an enhanced effective viscosity ( by a factor of ∼ 4.5) and more importantly (ii) only a mild reduction due to the time-variation of the tidal shear. Overall, for binaries with orbital period of 15 days the discrepancy is ”only” a factor of ∼ 30 down from a factor of ∼ 1000. These encouraging results should motivate an investigation of rigorous non-analytic solutions.