Layered semi-convection and tides in giant planet interiors

I. Propagation of internal waves

¹ Laboratoire AIM Paris-Saclay, CEA/DRF, CNRS, Univ. Paris-Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France
e-mail: quentin.andre@cea.fr
² Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
³ Département de Physique, ENS Paris-Saclay, Université Paris-Saclay, 61 Avenue du Président Wilson, 94230 Cachan, France
⁴ Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
e-mail: a.j.barker@leeds.ac.uk
⁵ LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris-Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
e-mail: stephane.mathis@cea.fr

Received: 10 March 2017
Accepted: 27 April 2017

Abstract

Context. Layered semi-convection is a possible candidate to explain Saturn’s luminosity excess and the abnormally large radius of some hot Jupiters. In giant planet interiors, it could lead to the creation of density staircases, which are convective layers separated by thin stably stratified interfaces. These are also observed on Earth in some lakes and in the Arctic Ocean.

Aims. We aim to study the propagation of internal waves in a region of layered semi-convection, with the aim to predict energy transport by internal waves incident upon a density staircase. The goal is then to understand the resulting tidal dissipation when these waves are excited by other bodies such as moons in giant planets systems.

Methods. We used a local Cartesian analytical model, taking into account the complete Coriolis acceleration at any latitude, thus generalising previous works. We used a model in which stably stratified interfaces are infinitesimally thin, before relaxing this assumption with a second model that assumes a piecewise linear stratification.

Results. We find transmission of incident internal waves to be strongly affected by the presence of a density staircase, even if these waves are initially pure inertial waves (which are restored by the Coriolis acceleration). In particular, low-frequency waves of all wavelengths are perfectly transmitted near the critical latitude, defined by θ_c = sin^-1(ω/ 2Ω), where ω is the wave’s frequency and Ω is the rotation rate of the planet. Otherwise, short-wavelength waves are only efficiently transmitted if they are resonant with a free mode (interfacial gravity wave or short-wavelength inertial mode) of the staircase. In all other cases, waves are primarily reflected unless their wavelengths are longer than the vertical extent of the entire staircase (not just a single step).

Conclusions. We expect incident internal waves to be strongly affected by the presence of a density staircase in a frequency-, latitude- and wavelength-dependent manner. First, this could lead to new criteria to probe the interior of giant planets by seismology; and second, this may have important consequences for tidal dissipation and our understanding of the evolution of giant planet systems.