Non-linear evolution of the tidal elliptical instability in gaseous planets and stars

Abstract

Tidally distorted rotating stars and gaseous planets are subject to a well-known linear fluid instability-the elliptical instability. It has been proposed that this instability might drive enough energy dissipation to solve the long-standing problem of the origin of tidal dissipation in stars and planets. But the nonlinear outcome of the elliptical instability has yet to be investigated in the parameter regime of interest, and the resulting turbulent energy dissipation has not yet been quantified. We do so by performing three dimensional hydrodynamical simulations of a small patch of a tidally deformed fluid planet or star subject to the elliptical instability. We show that when the tidal deformation is weak, the nonlinear outcome of the instability leads to the formation of long-lived columnar vortices aligned with the axis of rotation. These vortices shut off the elliptical instability, and the net result is insufficient energy dissipation to account for tidal dissipation. However, further work is required to account for effects neglected here, including magnetic fields, turbulent convection, and realistic boundary conditions.

Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

“…This paper has primarily focused on the longest-wavelength global modes. This is because, if these are excited, they might be expected to dominate the instability-driven turbulence and its resulting tidal dissipation (as indicated by the previous local simulations of Barker & Lithwick 2013. In addition, the non-negligible tidal amplitude for some hot Jupiters increases the prospect of global modes being excited inside these planets (since these modes can be excited out of exact resonance).…”

“…4 and plotted in Fig. 1 of Barker & Lithwick 2013). This explains why the growth rate is smaller for positive n (aligned) compared with negative n (anti-aligned) for the same departure from synchronism γ.…”

“…We also aim to study the modes that we might obtain in our global simulations that we present in the companion paper (Barker 2016). We will map out the parameter space for the global instabilities with ℓ 5, since these modes might be thought to be the most important for controlling the tidal dissipation (by providing an "outer scale" for the turbulence, as evidenced by the local numerical simulations of Barker & Lithwick 2013).…”

“…This is a nonlinear mechanism of tidal dissipation that requires a finite tidal deformation. The outcome of this instability has been studied in laboratory experiments (Lacaze et al 2004;Le Bars et al 2007, as well as idealised local (Barker & Lithwick 2013 and global numerical simulations (Cébron et al 2010(Cébron et al , 2013. These works suggest that the elliptical instability could contribute to tidal dissipation at sufficiently short orbital periods.…”

“…Nonlinear tides are particularly difficult to study because they tend to require numerical simulations to quantify, so that some progress towards understanding them has only been made very recently (e.g. Goodman & Lackner 2009;Barker c 2016RAS & Ogilvie 2010Weinberg et al 2012;Barker & Lithwick 2013Favier et al 2014). Given the complexity of the tidal response in the fluid layers of rotating planets and stars, it is instructive to consider simplified models that capture the most important elements, before attempting to understand a realistic model.…”