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

TL;DR

This paper analyzes the global modes and elliptical instability in rotating homogeneous ellipsoidal planets or stars, revealing how spin orientation and tidal deformation influence instability growth rates and mode frequencies, with implications for tidal evolution.

Contribution

It provides a comprehensive global and local analysis of elliptical instability in rotating ellipsoids, including new insights into the effects of spin alignment and large tidal amplitudes.

Findings

Anti-aligned spins lead to larger instability growth rates.

Violent instability occurs outside the usual frequency range for large tidal amplitudes.

Rotation and tidal deformation decrease mode frequencies, enhancing tidal excitation.

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 $\ell\leq 5$, 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.