Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability

Abstract

I present results from the first global hydrodynamical simulations of the elliptical instability in a tidally deformed gaseous planet (or star) with a free surface. The elliptical instability is potentially important for tidal evolution of the shortest-period hot Jupiters. I model the planet as a spin-orbit aligned or anti-aligned, and nonsynchronously rotating, tidally deformed, homogeneous fluid body. A companion paper presented an analysis of the global modes and instabilities of such a planet. Here I focus on the nonlinear evolution of the elliptical instability. This is observed to produce bursts of turbulence that drive the planet towards synchronism with its orbit in an erratic manner. If the planetary spin is initially anti-aligned, the elliptical instability also drives spin-orbit alignment on a similar timescale as the spin synchronisation. The instability generates differential rotation inside the planet in the form of zonal flows, which play an important role in the saturation of the instability, and in producing the observed burstiness. These results are broadly consistent with the picture obtained using a local Cartesian model (where columnar vortices played the role of zonal flows). I also simulate the instability in a container that is rigid (but stressfree) rather than free, finding broad quantitative agreement. The dissipation resulting from the elliptical instability could explain why the shortest-period hot Jupiters tend to have circular orbits inside about 2-3 days, and predicts spin-synchronisation (and spin-orbit alignment) out to about 10-15 days. However, other mechanisms must be invoked to explain tidal circularisation for longer orbital periods.

On turbulence driven by axial precession and tidal evolution of the spin–orbit angle of close-in giant planets

“…2 This can be compared with the timescale that that we have previously estimated for the related elliptical instability (Barker 2016). In that case, simulations suggest the alignment timescale to be comparable with the synchronisation timescale, so that…”

“…These columnar vortices are commonly found in turbulence subjected to rapid rotation, and would be expected to correspond with zonal flows in a global model (Barker 2016). In the presence of even a weak magnetic field, the properties of the flow are significantly altered, and the formation of these vortices is inhibited.…”

Turbulent Kinematic Dynamos in Ellipsoids Driven by Mechanical Forcing

“…Local methods based on finite elements or finite volumes are more common due to their inherent flexibility for the geometry (see, e.g., Ernst-Hullermann et al, 2013;Vantieghem et al, 2016;Wu & Roberts, 2009). Nek5000 has been used to study flows driven by tides (Barker, 2016a;Favier et al, 2014;Grannan et al, 2016) and libration (Favier et al, 2015). Spectral elements can be seen as an intermediate between fully spectral and local methods.…”

Inertial Wave Turbulence Driven by Elliptical Instability

“…First, columnar vortices are typically not observed in global simulations. Instead, geostrophic modes emerge as steady axisymmetric flows, which do not necessarily inhibit the instability mechanism [17] (but see [20]). Moreover, the inverse cascade shown in Fig.…”

“…The saturation of the instability can lead to either sustained flows [15][16][17] or cyclic behaviors between laminar and turbulent states [12,[18][19][20], reminiscent of the "resonant collapse" of inertial waves observed by McEwan [21]. The presence or absence of geostrophic modes appears to be important, but the diversity remains to be explained in detail.…”