Nonlinear tides in a homogeneous rotating planet or star: global modes and elliptical instability

Abstract

We revisit the global modes and instabilities of homogeneous rotating ellipsoidal fluid masses, which are the simplest global models of rotationally and tidally deformed gaseous planets or stars. The tidal flow in a short-period planet may be unstable to the elliptical instability, a hydrodynamic instability that can drive tidal evolution. We perform a global (and local WKB) analysis to study this instability using the elegant formalism of Lebovitz & Lifschitz. We survey the parameter space of global instabilities with harmonic orders $\ell \le 5$, for planets with spins that are purely aligned (prograde) or anti-aligned (retrograde) with their orbits. In general, the instability has a much larger growth rate if the planetary spin and orbit are anti-aligned rather than aligned. We have identified a violent instability for anti-aligned spins outside of the usual frequency range for the elliptical instability (when $\frac{n}{\Omega }\lesssim -1$, where $n$ and $\Omega$ are the orbital and spin angular frequencies, respectively) if the tidal amplitude is sufficiently large. We also explore the instability in a rigid ellipsoidal container, which is found to be quantitatively similar to that with a realistic free surface. Finally, we study the effect of rotation and tidal deformation on mode frequencies. We find that larger rotation rates and larger tidal deformations both decrease the frequencies of the prograde sectoral surface gravity modes. This increases the prospect of their tidal excitation, potentially enhancing the tidal response over expectations from linear theory. In a companion paper, we use our results to interpret global simulations of the elliptical instability.

arXiv comment16 pages, 10 figures, accepted for publication in MNRAS (2016 March 22)

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Cited by in corpus (16)

• Nonlinear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability
• Experimental study of the non-linear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence
• Turbulent viscosity acting on the equilibrium tidal flow in convective stars
• Magnetic fields driven by tidal mixing in radiative stars
• Efficiency of tidal dissipation in slowly rotating fully convective stars or planets
• On the interaction between fast tides and convection
• Fossil field decay due to nonlinear tides in massive binaries
• Dynamical tidal Love numbers of rapidly rotating planets and stars
• Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits
• Parametric instability and wave turbulence driven by tidal excitation of internal waves
• The effects of nonlinearities on tidal flows in the convective envelopes of rotating stars and planets in exoplanetary systems
• Tidal dissipation due to the elliptical instability and turbulent viscosity in convection zones in rotating giant planets and stars
• Rotational dynamics of planetary cores: instabilities driven by precession, libration and tides
• Tidal dissipation in stratified and semi-convective regions of giant planets
• The interactions of the elliptical instability and convection
• Rapid falling of an orbiting moon to its parent planet due to tidal-seismic resonance