Turbulent damping of fast tidal oscillations by three-dimensional Rayleigh-B\'enard convection with a radiating free surface

TL;DR

This study uses 3D simulations to analyze how turbulent Rayleigh-Bénard convection with a radiating surface affects the damping of fast tidal oscillations, validating a formalism for tidal dissipation.

Contribution

It demonstrates that oscillatory tidal forcing transfers kinetic energy to mean flows in convective envelopes, supporting a formalism for fast tide dissipation in stars and planets.

Findings

Oscillations transfer energy to mean flow when Reynolds number exceeds threshold.

Energy transfer rate matches observed tidal circularization timescales.

Free surface effects significantly influence the energy transfer correlations.

Abstract

We present three-dimensional Dedalus simulations of Rayleigh-B\'enard convection with a blackbody-radiating free upper surface, subject to a low-amplitude oscillatory forcing that mimics tidal perturbations in convective envelopes of stars and planets. The forcing period is 10-100 times shorter than the convective timescale, $t_{\rm conv}$. Using a Reynolds decomposition of the velocity field averaged over one oscillation period, in which the tidal oscillations naturally constitute the fluctuating field and convection the mean flow, we elucidate the kinetic energy exchange between the two. Provided the oscillatory Reynolds number exceeds a modest threshold, we find that the oscillations systematically transfer kinetic energy to the mean flow at a volume-averaged rate $D_R \sim u'^2 t_{\rm conv}^{-1}$, where $u'$ is the rms fluctuation velocity. This reflects strong, order-unity correlations between the fluctuation velocities and the mean flow. These arise because the oscillatory forcing displaces fluid elements that are then redirected by buoyancy and incompressibility in the same manner as the mean flow. The transfer is dominated by correlations involving vertical velocity fluctuations and vertical gradients of the mean flow. The resulting energy transfer rate is consistent, within the equilibrium-tide framework, with the observed tidal circularisation of solar-type binaries and with the orbital evolution of moons of Jupiter and Saturn. This validates the formalism proposed by Terquem (2021) for the dissipation of fast tides, a longstanding problem. Replacing the free surface with a rigid upper boundary significantly and artificially modifies the correlations.