Tidal dissipation and spin-orbit alignment due to the precessional instability in convection zones in rotating giant planets and stars

Abstract

Tidal dissipation in star-planet systems occurs through various mechanisms, including the precessional instability. This is an instability of laminar flows (``Poincaré flows") forced by axial precession of a rotating, oblate, spin-orbit misaligned fluid planet or star, which excites inertial waves in convective regions if the dimensionless precession rate (``Poincaré number" $\mathrm{Po}$) is sufficiently large. We constrain the contribution of the precessional instability to tidal dissipation and heat transport, using Cartesian hydrodynamical simulations in a small patch of a planet, and study its interaction with turbulent convection, modelled as rotating Rayleigh-Bénard convection. The precessional instability without convection results in laminar flow at low values and turbulent flow at sufficiently high values of $\mathrm{Po}$. The associated tidal dissipation rate scales as $\mathrm{Po}^2$ and $\mathrm{Po}^3$ in each regime, respectively. With convection, the Poincaré number at which turbulent flow is achieved shifts to lower values for stronger convective driving. Convective motions also act on large-scale tidal flows like an effective viscosity, resulting in continuous tidal dissipation (scaling as $\mathrm{Po}^2$), which obfuscates or suppresses tidal dissipation due to precessional instability. The effective viscosities obtained agree with scaling laws previously derived using (rotating) mixing-length theory. By evaluating our scaling laws using interior models of Hot Jupiters, we find that the precessional instability is significantly more efficient than the effective viscosity of convection. The former drives alignment in 1 Gyr for a Jupiter-like planet orbiting within 23 days. Linearly excited inertial waves can be even more effective for wider orbits, aligning spins for orbits within 53-142 days.

Figures (14)

• Figure 1: The location of the local box in the convection zone of a Hot Jupiter. We indicate the rotation vector $\bm{\Omega}$ in black, the precession vector $\bm{\Omega}_p$ in dark grey and the background flow $\bm{U}_0$ in green, all in the boundary frame and not to scale relative to each other. The background flow takes the form of a shear in $z$, the direction of which rotates as a function of time. The precession vector also rotates as a function of time, always lying in the $x,y$-plane. The time dependence of the direction of both $\bm{\Omega}_p$ and $\bm{U}_{0}$ is illustrated by dark and green dashed circles respectively. The local temperature gradient is represented by the red (hot) and blue (cold) sides of the box.
• Figure 2: Benchmark simulations of the linear growth rate of the precessional instability and convective instability, each in isolation at $\mathrm{Ek}=5\cdot 10^{-5}$ in a 1-by-1-by-1 box using both nek5000 (black) and dedalus (red). Left: growth rate of the precessional instability. The most unstable mode pair, after taking the viscous reduction into account, is $n_A=1,\ n_B=2$. The theoretical prediction for small $\mathrm{Po}$ for this mode is given in blue. The simulation data obtained using both codes match each other and the theoretical prediction well in the interval $\mathrm{Po}\in[0.1,0.15]$, and accurately predict the critical value of $\mathrm{Po}$ for inhibition of the instability by viscosity. Deviation from the predicted maximum growth at higher $\mathrm{Po}$ partly arises from effects that are of higher order in $\mathrm{Po}$ not captured by the linear analysis. In addition, the most unstable horizontal wavenumber changes as $\mathrm{Po}$ is increased, see Fig. 3a in masonkerswell, and therefore the mode becomes increasingly more de-tuned. Right: Benchmark simulations of the linear growth rate of convection in isolation at $\mathrm{Ek}=5\cdot 10^{-5}$ in a 1-by-1-by-1 box. The growth rates in the simulations match each other and the theoretical growth rate of convection well.
• Figure 3: Snapshots of the vertical vorticity $\omega_z$ of the flow in simulations executed using nek5000 with $\mathrm{Ek}=2.5\cdot10^{-5}$. The top left and top right panels, both obtained from the same simulation, show snapshots taken during a burst of the energy injection and during the period of large-scale flow after the burst, respectively. The middle panels, both taken from a simulation with stronger precessional driving than the top panels, show vortices in the flow instead. A rapid "secondary transition" appears to occur in this simulation, indicated by a large increase in vorticity and kinetic energy from the panel on the left to the one on the right. The bottom left panel shows a simulation with weak convective driving; the bursty behaviour of the precessional instability appears absent and instead the convection appears to dominate this flow. In the bottom right panel a snapshot of the simulation with stronger convective driving is shown. This flow strongly resembles that of the middle right panel, but it is unclear whether the vortex is driven by convection or precession.
• Figure 4: Time series of the precessional instability and convection with $\mathrm{Ek}=2.5\cdot10^{-5}$, executed using nek5000. Vertical dotted-black lines correspond to the times at which the snapshots in Fig. \ref{['fig:Nek_snapshots']} are taken. The top left and right panels show the precessional instability in isolation. Note the different $x$-axis ranges between the two panels. The simulation in the top left panel produces the expected bursty behaviour, while the top right panel with stronger precessional driving features a secondary transition to a continuously turbulent state. The introduction of convection inhibits the bursty behaviour in the figure on the middle left, displaying a continuous energy injection instead. The simulation on the middle right with stronger precessional driving than the one on the middle left shows similar behaviour to the panel on the top right. The simulation on the bottom left with strong convective driving shows evidence for a secondary transition that is more gradual than in the top right panel, so convection might allow the precessional instability to become continuously turbulent at lower values of the Poincaré number. Finally, the simulation on the bottom right also shows the secondary transition, but stronger convective driving appears to allow the transition to occur sooner.
• Figure 5: Horizontal energy spectra taken at $z=0.5$ of the simulation with $\mathrm{Po}=0.1$, $\mathrm{Ra}=0$ from $t=0.106$ to $t=0.140$, corresponding to the interval between the vertical dotted lines in Fig. \ref{['fig:Ded_Po0.1']}. The simulation goes through a burst and decay period in this interval. In the spectra this is indicated by the increase and subsequent decrease in the main energy injection spike in the wavenumber bin with $k_\perp=6\pi$, while the 2D energy, here located in the smallest wavenumber bin, lags slightly behind this spike. The spectra clearly do not match the Kolmogorov scaling as $k_\perp^{-5/3}$ in dashed-black and therefore this case is not turbulent.