Non-linear 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 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 n Ω −1, where n and Ω 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.

Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits

“…This phenomenon has already been noticed in previous global analyses performed at lower degrees n 7 (e.g. Kerswell 1993b; , 2013Vantieghem et al 2015;Barker et al 2016). Indeed, more resonances are expected when n increases.…”

“…The other monomials, denoted h i (r), contain z as factor. We index the set of these polynomials as (Lebovitz 1989;Barker et al 2016)…”

“…Their physical relevance remains elusive, since the fluid instabilities that can grow upon the basic state were not considered. The hydrodynamic stability of ellipsoidal fluid masses has been tackled by Lebovitz & Lifschitz (1996b,a) for isolated fluid masses and recently by Barker et al (2016); Barker (2016a) for circular orbital motions.…”

Rotational Dynamics of Planetary Cores: Instabilities Driven By Precession, Libration and Tides

“…The tidal force field is represented in figure 2 and bears two important symmetries: it is invariant by rotation around the planet-moon axis (OX ) and by reflection relative to the (Y OZ) plane. The deformation induced by the tidal potential (1) can be analytically determined (see for instance discussions in Barker (2016) and Barker et al (2016)). At the lowest order, it can be shown that the planet adopts an ellipsoidal shape; the combination of both equatorial -or rotational-and tidal bulges forces the three axes of this ellipsoid to have different lengths.…”

Rapid falling of an orbiting moon to its parent planet due to tidal-seismic resonance

“…However, in some cases can tidal force frequencies intrude into the frequency range of the planet normal modes. For example, the tidal force on a rapid rotating planet can excite normal modes of the planet (Braviner and Ogilvie, 2014a, b;Barker et al, 2016). Interaction between Saturn ring and Saturn can excite acoustic free oscillation of Saturn (Marley, 1991;Marley and Porco, 1993;Marley, 2014).…”