Interaction between tides and convection in stars and giant planets

The aim of the project is to develop a new formalism for the dissipation of rapidly varying oscillations in the convective envelopes of solar-type stars and giant planets. This work builds on my recent findings, which have shown that the theory used over the past six decades is incorrect. I have proposed new ideas for an alternative formalism, which will be explored throughout the project.

The project involves both analytical and numerical approaches and lies at the intersection of astrophysics and fluid dynamics. It addresses a critical problem that has been a central focus of theoretical astrophysics for nearly 60-years.

Background: Dissipation of tidal oscillations in stars shapes the orbits of close binary systems, whereas in giant planets it controls the orbital evolution of close moons. In late--type stars and giant planets, dissipation occurs predominantly in convective envelopes. The standard theory describes the tidal oscillation as a mean flow which interacts with fluctuating convective eddies (Zahn 1966).

It is assumed that energy is transferred from the tides to the convective flow, and mixing length theory is used to model convection as providing a dissipative `turbulent viscosity'. In many cases of interest, the tidal periods are significantly smaller than the convective turnover timescale, such that convective eddies cannot transport and exchange momentum with their environment during a tidal period, and dissipation is suppressed.

This has been modelled by incorporating a period-dependent term in the expression for the turbulent viscosity (Zahn 1966, Goldreich Nicholson 1977). However, tidal dissipation calculated this way is orders of magnitude too small to account for either the circularization period of late-type binaries, or the tidal dissipation factor of Jupiter and Saturn inferred from the orbital motion of their satellites.

A new formulation: I have recently revisited this theory (Terquem 2021, 2023) and proposed a new formulation. By writing the energy conservation equations for a flow which is the superposition of a rapidly varying tidal oscillation and slow convective motions, I have shown that traditional roles are in fact reversed: the tidal oscillation is the fluctuation, whereas convection is the mean flow.

This implies that mixing length theory is simply not applicable. In this context, I have obtained a new expression for the rate ''DR'' at which energy is exchanged between the tides and convection. It involves a coupling between the Reynolds stress associated with the tidal velocity and the convective shear flow, and not the other way round, as assumed in the standard theory.

Crucially, the scale of this new term DR is much larger than the standard value calculated using the convective turbulent viscosity. The sign of DR is not known a priori, but observations indicate that tidal oscillations are dissipated when interacting with a convective flow: this is evidenced by the circularization of late-type binaries and the orbital evolution of the satellites of Jupiter and Saturn.

This corresponds to the integral of DR over the convective envelope being positive. I have accordingly assumed that DR is locally positive, and investigated whether such a coupling yields results in agreement with observations. I have calculated the tidal dissipation Q factor for Jupiter and Saturn, the circularization periods of late-type binaries and evolution timescales for hot Jupiters.

All these results are in good quantitative agreement with observations, except for one satellite of Saturn. Terquem & Martin (2021) calculated the circularization timescale as a function of orbital period for late-type binaries, comparing observations and the calculations based on the new formalism. The theoretical curve gives the correct circularization timescale for the oldest binaries in the halo, which no other theory has been able to explain.

Main objectives of the project: The formal derivation of the rate DR at which energy