On PSI Interactions in Internal Gravity Wave Fields and the Decay of Baroclinic Tides

Barotropic tidal oscillations over seafloor topography generate baroclinic tides that may be damped in turn via nonlinear triad interactions with internal gravity waves, fueling the ambient wave field. We derive the kinetic equations for this tidal damping and the energy transfer to the ambient wave field and compute damping times and energy transfer rates for the M2 tide and a Garrett–Munk-like ambient wave field. We show that parametric subharmonic instability (PSI) interactions are important, where the tide interacts resonantly with two background waves, each of half the tidal frequency. PSI is restricted to the latitude belt 28.8°N/S and yields under typical conditions damping times of about 20 days for tides with low vertical wavenumber. Damping times decrease with equivalent mode number j roughly as 1/j2. Outside the critical latitudes PSI is not possible, and damping times are from one to two orders of magnitude larger. The energy transfer to the ambient wave field is concentrated at half the tidal frequency ω at all latitudes within the critical latitude belt. Outside, the transfer is much smaller and peaks at ω + f and N. An estimation of the tidal spectral transfer on the global scale is hampered by insufficient knowledge of the baroclinic energy distribution over the vertical modes. Using results from a numerical circulation model with tidal forcing, we find an energy transfer from the tide to the ambient wave field of typically 0.3 TW, about half of what is currently proposed for the conversion of barotropic to baroclinic energy.

Tidal flows with convection: frequency-dependence of the effective viscosity and evidence for anti-dissipation

Tidal interactions are important in driving spin and orbital evolution in planetary and stellar binary systems, but the fluid dynamical mechanisms responsible remain incompletely understood. One key mechanism is the interaction between tidal flows and convection. Turbulent convection is thought to act as an effective viscosity in damping large-scale tidal flows, but there is a long-standing controversy over the efficiency of this mechanism when the tidal frequency exceeds the turnover frequency of the dominant convective eddies. This high frequency regime is relevant for many applications, such as for tides in stars hosting hot Jupiters. We explore the interaction between tidal flows and convection using hydrodynamical simulations within a local Cartesian model of a small patch of a convection zone of a star or planet. We adopt the Boussinesq approximation and simulate Rayleigh-Bénard convection, modelling the tidal flow as a background oscillatory shear flow. We demonstrate that the effective viscosity of both laminar and turbulent convection is approximately frequency-independent for low frequencies. When the forcing frequency exceeds the dominant convective frequency, the effective viscosity scales inversely with the square of the tidal frequency. We also show that negative effective viscosities are possible, particularly for high frequency tidal forcing, suggesting the surprising possibility of tidal anti-dissipation. These results are supported by a complementary high-frequency asymptotic analysis that extends prior work by Ogilvie & Lesur. We discuss the implications of these results for interpreting the orbital decay of hot Jupiters, and for several other astrophysical problems.

Semidiurnal thermal tides in asynchronously rotating hot Jupiters

Context. Thermal tides can torque the atmosphere of hot Jupiters into asynchronous rotation, while these planets are usually assumed to be locked into spin-orbit synchronization with their host star. Aims. In this work, our goal is to characterize the tidal response of a rotating hot Jupiter to the tidal semidiurnal thermal forcing of its host star by identifying the structure of tidal waves responsible for variation of mass distribution, their dependence on the tidal frequency, and their ability to generate strong zonal flows. Methods. We develop an ab initio global modelling that generalizes the early approach of Arras & Socrates (2010, ApJ, 714, 1) to rotating and non-adiabatic planets. We analytically derive the torque exerted on the body and the associated timescales of evolution, as well as the equilibrium tidal response of the atmosphere in the zero-frequency limit. Finally, we numerically integrate the equations of thermal tides for three cases, including dissipation and rotation step by step. Results. The resonances associated with tidally generated gravito-inertial waves significantly amplify the resulting tidal torque in the range 1–30 days. This torque can globally drive the atmosphere into asynchronous rotation, as its sign depends on the tidal frequency. The resonant behaviour of the tidal response is enhanced by rotation, which couples the forcing to several Hough modes in the general case, while the radiative cooling tends to regularize it and diminish its amplitude.

Atmospheric tides in Earth-like planets

Context. Atmospheric tides can strongly affect the rotational dynamics of planets. In the family of Earth-like planets, which includes Venus, this physical mechanism coupled with solid tides makes the angular velocity evolve over long timescales and determines the equilibrium configurations of their spin. Aims. Unlike the solid core, the atmosphere of a planet is subject to both tidal gravitational potential and insolation flux coming from the star. The complex response of the gas is intrinsically linked to its physical properties. This dependence has to be characterized and quantified for application to the wide variety of extrasolar planetary systems. Methods. We develop a theoretical global model where radiative losses, which are predominant in slowly rotating atmospheres, are taken into account. We analytically compute the perturbation of pressure, density, temperature, and velocity field caused by a thermogravitational tidal perturbation. From these quantities, we deduce the expressions of atmospheric Love numbers and tidal torque exerted on the fluid shell by the star. The equations are written for the general case of a thick envelope and the simplified one of a thin isothermal atmosphere. Results. The dynamics of atmospheric tides depends on the frequency regime of the tidal perturbation: the thermal regime near synchronization and the dynamical regime characterizing fast-rotating planets. Gravitational and thermal perturbations imply different responses of the fluid, i.e. gravitational tides and thermal tides, which are clearly identified. The dependence of the torque on the tidal frequency is quantified using the analytic expressions of the model for Earth-like and Venus-like exoplanets and is in good agreement with the results given by global climate models (GCM) simulations.Introducing dissipative processes such as radiation regularizes the tidal response of the atmosphere, otherwise it is singular at synchronization. Conclusions. We demonstrate the important role played by the physical and dynamical properties of a super-Earth atmosphere (e.g. Coriolis, stratification, basic pressure, density, temperature, radiative emission) in its response to a tidal perturbation. We point out the key parameters defining tidal regimes (e.g. inertia, Brunt-Väisälä, radiative frequencies, tidal frequency) and characterize the behaviour of the fluid shell in the dissipative regime, which cannot be studied without considering the radiative losses.

The tidally induced bottom boundary layer in a rotating frame: similarity of turbulence

To investigate turbulent properties of the tidally induced bottom boundary layer (TBBL) in a rotating frame, we performed three-dimensional numerical experiments under unstratified conditions, varying the temporal Rossby number Rot = |σ*/f*|, where σ* and f* are the tidal frequency and the Coriolis parameter, respectively. The vertical profiles of the time-averaged currents and stresses showed good similarity and coincided well with the turbulent Ekman layer, when they were normalized by the modified ‘outer’ scales, the frictional velocity u*τ, T* = 1/|f* + σ*| and δ* = u*τ/|f* + σ*| for the velocity, time and length scales (σ* is positive when the tidal ellipse rotates anticlockwise). This means that the similarity in turbulent statistics is universally applicable to the TBBL in the world's ocean except near the equator. Although strong inertial waves contaminated the perturbation field when Rot ~ 1 and masked the similarity, the apparent diffusivity κ*ap estimated by tracer experiments again showed similarity, since the inertial waves did not affect the mixing process in the present experiments. Thus, κ*ap can be represented in terms of the three external parameters: the latitude (f*), the tidal frequency (σ*) and the tidal amplitude (u*τ). The obtained scaling of u*τ δ* = u*τ2/|f*+σ*| for κ*ap suggests that effective mixing may occur when Rot ~ 1, i.e. near the critical latitude.

Interharmonics in internal gravity waves generated by tide-topography interaction

The dynamics and spectrum of internal gravity waves generated in a linearly stratified fluid by tidal flow over a flat-topped ridge are investigated at five different latitudes using an inviscid two-dimensional numerical model. The resulting wave field includes progressive freely propagating waves which satisfy the dispersion relation, and forced waves which are trapped non-propagating oscillations with frequencies outside the internal wave band. The flow is largely stable with respect to shear instabilities, and, throughout the runs, there is a negligibly small amount of overturning which is confined to the highly nonlinear regions along the sloping topography and where tidal beams reflect from the boundaries. The wave spectrum exhibits a self-similar structure with prominent peaks at tidal harmonics and interharmonics, whose magnitudes decay exponentially with frequency. Two strong subharmonics are generated by an instability of tidal beams which is particularly strong for near-critical latitudes where the Coriolis frequency is half the tidal frequency. When both subharmonics are within the free internal wave range (as in cases 0°–20° N), they form a resonant triad with the tidal harmonic. When at least one of the two subharmonics is outside of the range (as in cases 30°–40° N) the observed instability is no longer a resonant triad interaction. We argue that the two subharmonics are generated by parametric subharmonic instability that can produce both progressive and forced waves. Other interharmonics are produced through wave–wave interactions and are not an artefact of Doppler shifting as assumed by previous authors. As the two subharmonics are, in general, not proper fractions of the tidal frequency, the wave–wave interactions are capable of transferring energy to a continuum of frequencies.