Tidal flows with convection: frequency-dependence of the effective viscosity and evidence for anti-dissipation

TL;DR

This study investigates how tidal flows interact with convection in stars and planets, revealing frequency-dependent effective viscosities and evidence for anti-dissipation, which impacts orbital evolution models.

Contribution

The paper provides the first detailed numerical and analytical analysis of the frequency dependence of effective viscosity in convective zones under tidal forcing, including the possibility of negative viscosities.

Findings

Effective viscosity is frequency-independent at low frequencies.

Viscosity scales inversely with the square of tidal frequency at high frequencies.

Negative effective viscosities suggest potential anti-dissipation effects.

Abstract

Tidal interactions are important in driving spin and orbital evolution in planetary and stellar binary systems, but the fluid dynamical mechanisms responsible remain incompletely understood. One key mechanism is the interaction between tidal flows and convection. Turbulent convection is thought to act as an effective viscosity in damping large-scale tidal flows, but there is a long-standing controversy over the efficiency of this mechanism when the tidal frequency exceeds the turnover frequency of the dominant convective eddies. This high frequency regime is relevant for many applications, such as for tides in stars hosting hot Jupiters. We explore the interaction between tidal flows and convection using hydrodynamical simulations within a local Cartesian model of a small patch of a convection zone of a star or planet. We adopt the Boussinesq approximation and simulate Rayleigh-B\'enard convection, modelling the tidal flow as a background oscillatory shear flow. We demonstrate that the effective viscosity of both laminar and turbulent convection is approximately frequency-independent for low frequencies. When the forcing frequency exceeds the dominant convective frequency, the effective viscosity scales inversely with the square of the tidal frequency. We also show that negative effective viscosities are possible, particularly for high frequency tidal forcing, suggesting the surprising possibility of tidal anti-dissipation. These results are supported by a complementary high frequency asymptotic analysis that extends prior work by Ogilvie & Lesur. We discuss the implications of these results for interpreting the orbital decay of hot Jupiters, and for several other astrophysical problems.