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 $|ω_t|$ exceeds the turnover frequency $ω_{cν}$ 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 $ν_E$ acting on the tidal flow when $|ω_t|/ω_{cν} \gtrsim 1$. We demonstrate that the frequency reduction of $ν_E$ is correlated with the frequency spectrum of the (unperturbed) convection. For intermediate frequencies below those in the turbulent cascade ($|ω_t|/ω_{cν} \sim 1-5$), the frequency spectrum displays an anomalous $1/ω^α$ power law that is responsible for the frequency-reduction $ν_E \propto 1/|ω_t|^α$, where $α< 1$ depends on the model parameters. We then get $|ν_E| \propto 1/|ω_t|^δ$ with $δ> 1$ for higher frequencies, and $δ=2$ is obtained for a Kolmogorov turbulent cascade. A generic $|ν_E| \propto 1/|ω_t|^{2}$ 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.

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