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.

Recent observations of Jupiter and Saturn suggest that heavy elements may be diluted in the gaseous envelope, providing a compositional gradient that could stabilise ordinary convection and produce a stably-stratified layer near the core of these planets. This region could consist of semi-convective layers with a staircase-like density profile, which have multiple convective zones separated by thin stably-stratified interfaces, as a result of double-diffusive convection. These layers could have important effects on wave propagation and tidal dissipation that have not been fully explored. We analyse the effects of these layers on the propagation and transmission of internal waves within giant planets, extending prior work in a local Cartesian model. We adopt a simplified global Boussinesq planetary model in which we explore the internal waves in a non-rotating spherical body. We begin by studying the free modes of a region containing semi-convective layers. We then analyse the transmission of internal waves through such a region. The free modes depend strongly on the staircase properties, and consist of modes with both internal and interfacial gravity wave-like behaviour. We determine the frequency shifts of these waves as a function of the number of steps to explore their potential to probe planetary internal structures. We also find that wave transmission is strongly affected by the presence of a staircase. Very large-wavelength waves are transmitted efficiently, but small-scale waves are only transmitted if they are resonant with one of the free modes. The effective size of the core is therefore larger for non-resonant modes.