Planet formation and dynamics

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 time-scale $t_{\rm conv}$, 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 $\delta(r) \sim r \omega_{\rm osc} / \left( g(r) t_{\rm conv}(r) \right)$, 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 $\delta(r)$ 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.