Efficiency of tidal dissipation in slowly rotating fully convective stars or planets

Turbulent convection is thought to act as an effective viscosity in damping equilibrium tidal flows, driving spin and orbital evolution in close convective binary systems. Compared to mixing-length predictions, this viscosity ought to be reduced when the tidal frequency $|\omega_t|$ exceeds the turnover frequency $\omega_{c\nu}$ of the dominant convective eddies, but the efficiency of this reduction has been disputed. We reexamine this long-standing controversy using direct numerical simulations of an idealized global model. We simulate thermal convection in a full sphere, and externally forced by the equilibrium tidal flow, to measure the effective viscosity $\nu_E$ acting on the tidal flow when $|\omega_t|/\omega_{c\nu} \gtrsim 1$. We demonstrate that the frequency reduction of $\nu_E$ is correlated with the frequency spectrum of the (unperturbed) convection. For intermediate frequencies below those in the turbulent cascade ($|\omega_t|/\omega_{c\nu} \sim 1-5$), the frequency spectrum displays an anomalous $1/\omega^\alpha$ power law that is responsible for the frequency-reduction $\nu_E \propto 1/|\omega_t|^{\alpha}$, where $\alpha < 1$ depends on the model parameters. We then get $|\nu_E| \propto 1/|\omega_t|^{\delta}$ with $\delta > 1$ for higher frequencies, and $\delta=2$ is obtained for a Kolmogorov turbulent cascade. A generic $|\nu_E| \propto 1/|\omega_t|^{2}$ suppression is next found for higher frequencies within the dissipation range of the convection (but with negative values). Our results indicate that a better knowledge of the frequency spectrum of convection is necessary to accurately predict the efficiency of tidal dissipation in stars and planets resulting from this mechanism.