Layered semi-convection and tides in giant planet interiors

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 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 use a local Cartesian analytical model, taking into account the complete Coriolis acceleration at any latitude, thus generalizing previous works. We use 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-, latitudeand 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.

Wave propagation in semiconvective regions of giant planets

“…This transfer matrix correctly recovers the Cartesian geometry results in Belyaev et al (2015), Sutherland (2016) and André et al (2017), once we take the double limits l 1 and 1 n , and we identify…”

“…A layered density structure could play an important role in affecting the propagation of waves in planetary interiors. Previous work has analysed the free modes of a density staircase (Belyaev et al 2015;André et al 2017) and quantified the transmission of waves through such a structure (Sutherland 2016; André et al 2017). These previous calculations adopted a local Cartesian model to study a small patch of a density staircase.…”

“…This formalism allows us to determine the solution in the (m + 1)-th layer in terms of the solution in the 0-th layer by repeatedly applying the transfer matrix. Note that T n depends on the radius of the n-th interface, which complicates the following analysis compared with the Cartesian case (even with constant d and ∆ρ) in André et al (2017). But we may still define a 2 × 2 matrix such that,…”

“…The free modes were found to differ from those of a continuously stratified medium, with those waves with wavelengths that are comparable with a step-size being affected the most. The transmission of internal waves through a density staircase in a similar Cartesian model was studied by Sutherland (2016), who adopted the "traditional approximation" to incorporate rotation (this assumes that the buoyancy force dominates the Coriolis acceleration in the direction of stratification, thereby disallowing inertial waves), and subsequently André et al (2017) studied the free modes and transmission of internal and inertial waves in a local model that included the full Coriolis acceleration at any latitude in a planet. The density staircase was found to strongly affect the transmission of waves through such a region in a frequency and wavelength-dependent manner.…”

Internal wave attractors in 3D geometries : A dynamical systems approach

“…Indeed, it is usual to consider the interior of gaseous planets as a fluid, (at least partially) stratified radially, around a solid or very dense liquid core (Dintrans, Rieutord & Valdettaro 1999). Internal waves propagating within this spherical shell may indeed strongly influence the dynamics of the planet (André, Barker & Mathis 2017). With a view of modelling internal tides in a channel, their three-dimensional behaviour was also investigated numerically in a rotating, uniformly-stratified parabolic channel (Drijfhout & Maas 2007).…”