On the energetics of a tidally oscillating convective flow

TL;DR

This paper investigates the energy exchange mechanisms between oscillating tidal flows and convective motions in planetary interiors, deriving phase lag and dissipation factors consistent with observations of Jupiter and Saturn's moons.

Contribution

It introduces a theoretical framework linking oscillation-convection energy exchange to observable tidal dissipation factors, incorporating compressibility and rotation effects.

Findings

Energy exchange rate scales with convective velocity squared

Phase lag correlates with local oscillation frequency and convective timescale

Predicted dissipation factors match observed values for Jupiter and Saturn's moons

Abstract

This paper examines the energetics of a convective flow subject to an oscillation with a period $t_{\rm osc}$ much smaller than the convective timescale $t_{\rm conv}$, allowing for compressibility and uniform rotation. We show that the energy of the oscillation is exchanged with the kinetic energy of the convective flow at a rate $D_R$ that couples the Reynolds stress of the oscillation with the convective velocity gradient. For the equilibrium tide and inertial waves, this is the only energy exchange term, whereas for p modes there are also exchanges with the potential and internal energy of the convective flow. Locally, $\left| D_R \right| \sim u^{\prime 2} /t_{\rm conv}$, where $u^{\prime}$ is the oscillating velocity. If $t_{\rm conv} \ll t_{\rm osc}$ and assuming mixing length theory, $\left| D_R \right|$ is $\left( \lambda_{\rm conv}/ \lambda_{\rm osc} \right)^2$ smaller, where $\lambda_{\rm conv}$ and $\lambda_{\rm osc}$ are the characteristic scales of convection and the oscillation. Assuming local dissipation, we show that the equilibrium tide lags behind the tidal potential by a phase $\delta \left( r \right) \sim r \omega_{\rm osc}/ \left( g \left( r \right) t_{\rm conv} \left( r \right)\right)$, where $g$ is the gravitational acceleration. The equilibrium tide can be described locally as a harmonic oscillator with natural frequency $\left( g / r \right)^{1/2}$ and subject to a damping force $-u^{\prime}/t_{\rm conv}$. Although $\delta \left( r \right) $ varies by orders of magnitude through the flow, it is possible to define an average phase shift $\overline{\delta}$ which is in good agreement with observations for Jupiter and some of the moons of Saturn. Finally, $1 / \overline{\delta}$ is shown to be equal to the standard tidal dissipation factor.