Tidal dissipation in star-planet systems can occur through various mechanisms, among which is the elliptical instability. This acts on elliptically deformed equilibrium tidal flows in rotating fluid planets and stars, and excites inertial waves in convective regions if the dimensionless tidal amplitude (ε) is sufficiently large. We study its interaction with turbulent convection, and attempt to constrain the contributions of both elliptical instability and convection to tidal dissipation. For this, we perform an extensive suite of Cartesian hydrodynamical simulations of rotating Rayleigh-Bénard convection in a small patch of a planet. We find that tidal dissipation resulting from the elliptical instability, when it operates, is consistent with ε3, as in prior simulations without convection. Convective motions also act as an effective viscosity on large-scale tidal flows, resulting in continuous tidal dissipation (scaling as ε2). We derive scaling laws for the effective viscosity using (rotating) mixing-length theory, and find that they predict the turbulent quantities found in our simulations very well. In addition, we examine the reduction of the effective viscosity for fast tides, which we observe to scale with tidal frequency (ω) as ω−2. We evaluate our scaling laws using interior models of Hot Jupiters computed with MESA. We conclude that rotation reduces convective length scales, velocities and effective viscosities (though not in the fast tides regime). We estimate that elliptical instability is efficient for the shortest-period Hot Jupiters, and that effective viscosity of turbulent convection is negligible in giant planets compared with inertial waves.
The elliptical instability is an instability of elliptical streamlines, which can be excited by large-scale tidal flows in rotating fluid bodies, and excites inertial waves if the dimensionless tidal amplitude (ε) is sufficiently large. It operates in convection zones but its interactions with turbulent convection has not been studied in this context. We perform an extensive suite of Cartesian hydrodynamical simulations in wide boxes to explore the interactions of the elliptical instability and Rayleigh-Bénard convection. We find that geostrophic vortices generated by the elliptical instability dominate the flow, with energies far exceeding those of the inertial waves. Furthermore, we find that the elliptical instability can operate with convection, but it is suppressed for sufficiently strong convection, primarily by convectively-driven large-scale vortices. We examine the flow in Fourier space, allowing us to determine the energetically dominant frequencies and wave numbers. We find that power primarily concentrates in geostrophic vortices, in wave numbers that are convectively unstable, and along the inertial wave dispersion relation, even in non-elliptically deformed convective flows. Examining linear growth rates on a convective background, we find that convective large-scale vortices suppress the elliptical instability in the same way as the geostrophic vortices created by the elliptical instability itself. Finally, convective motions act as an effective viscosity on large-scale tidal flows, providing a sustained energy transfer (scaling as ε2). Furthermore, we find that the energy transfer resulting from bursts of elliptical instability, when it operates, is consistent with the ε3 scaling found in prior work.
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