On the energetics of a tidally oscillating convective flow

Abstract

This paper examines the energetics of a convective flow subject to an oscillation with a period $t_{\mathrm{osc}}$ much smaller than the convective timescale $t_{\mathrm{conv}}$, 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, $\left|D_R\right|\sim u^{\prime 2}/t_{\mathrm{conv}}$, where $u^{\prime }$ is the oscillating velocity. If $t_{\mathrm{conv}}\ll t_{\mathrm{osc}}$ and assuming mixing length theory, $\left|D_R\right|$ is ${\left({\lambda }_{\mathrm{conv}}/{\lambda }_{\mathrm{osc}}\right)}^2$ smaller, where ${\lambda }_{\mathrm{conv}}$ and ${\lambda }_{\mathrm{osc}}$ 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 $\delta \left(r\right)\sim r{\omega }_{\mathrm{osc}}/\left(g\left(r\right)t_{\mathrm{conv}}\left(r\right)\right)$, 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 $-u^{\prime }/t_{\mathrm{conv}}$. 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.

arXiv commentPublished in MNRAS (2023, vol. 525, p. 508-526)

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