Do nonlinear effects disrupt tidal dissipation predictions in convective envelopes?

Most prior works studying tidal interactions in tight star/planet or star/star binary systems have employed linear theory of a viscous fluid in a uniformly-rotating two-dimensional spherical shell. However, compact systems may have sufficiently large tidal amplitudes for nonlinear effects to be important. We compute tidal flows subject to nonlinear effects in a 3D, thin (solar-like) convective shell, spanning the entire frequency range of inertial waves. Tidal frequency-averaged dissipation predictions of linear theory with solid body rotation are approximately reproduced in our nonlinear simulations (though we find it to be reduced by a factor of a few), but we find significant differences, potentially by orders of magnitude, at a fixed tidal frequency corresponding to a specific two-body system at a given epoch. This is largely due to tidal generation of differential rotation (zonal flows) and their effects on the waves.Comment: 2 pages, 1 figure, proceeding of the Annual meeting of the French Society of Astronomy and Astrophysics (SF2A 2023

Tidally-excited inertial waves in stars and planets: exploring the frequency-dependent and averaged dissipation with nonlinear simulations

We simulate the nonlinear hydrodynamical evolution of tidally-excited inertial waves in convective envelopes of rotating stars and giant planets modelled as spherical shells containing incompressible, viscous and adiabatically-stratified fluid. This model is relevant for studying tidal interactions between close-in planets and their stars, as well as close low-mass star binaries. We explore in detail the frequency-dependent tidal dissipation rates obtained from an extensive suite of numerical simulations, which we compare with linear theory, including with the widely-employed frequency-averaged formalism to represent inertial wave dissipation. We demonstrate that the frequency-averaged predictions appear to be quite robust and is approximately reproduced in our nonlinear simulations spanning the frequency range of inertial waves as we vary the convective envelope thickness, tidal amplitude, and Ekman number. Yet, we find nonlinear simulations can produce significant differences with linear theory for a given tidal frequency (potentially by orders of magnitude), largely due to tidal generation of differential rotation and its effects on the waves. Since the dissipation in a given system can be very different both in linear and nonlinear simulations, the frequency-averaged formalism should be used with caution. Despite its robustness, it is also unclear how accurately it represents tidal evolution in real (frequency-dependent) systems.Comment: 14 pages, 7 figures, 2 tables, to be published in ApJ

How do tidal waves interact with convective vortices in rapidly-rotating planets and stars?

The dissipation of tidal inertial waves in planetary and stellar convective regions is one of the key mechanisms that drive the evolution of star-planet/planet-moon systems. In this context, the interaction between tidal inertial waves and turbulent convective flows must be modelled in a realistic and robust way. In the state-of-the-art simulations, the friction applied by convection on tidal waves is modelled most of the time by an effective eddy-viscosity. This approach may be valid when the characteristic length scales of convective eddies are smaller than those of tidal waves. However, it becomes highly questionable in the case where tidal waves interact with potentially stable large-scale vortices, as those observed at the pole of Jupiter and Saturn. They are potentially triggered by convection in rapidly-rotating bodies in which the Coriolis acceleration forms the flow in columnar vortical structures along the direction of the rotation axis. In this paper, we investigate the complex interactions between a tidal inertial wave and a columnar convective vortex. We use a quasi-geostrophic semi-analytical model of a convective columnar vortex. We perform linear stability analysis to identify the unstable regime and conduct linear numerical simulations for the interactions between the convective vortex and an incoming tidal inertial wave. We verify that in the unstable regime, an incoming tidal inertial wave triggers the most unstable mode of the vortex leading to turbulent dissipation. For stable vortices, the wave-vortex interaction leads to the momentum mixing while it creates a low-velocity region around the vortex core and a new wave-like perturbation in the form of a progressive wave radiating in the far field. The emission of this secondary wave is the strongest when the wavelength of the incoming wave is close to the characteristic size of the vortex.Comment: 20 pages, 15 figures, accepted in Astronomy & Astrophysic

The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regions:I. Free waves

Quantifying tidal interactions in close-in two-body systems is of prime interest since they have a crucial impact on the architecture and on the rotational history of the bodies. Various studies have shown that the dissipation of tides in either body is very sensitive to its structure and to its dynamics, like differential rotation which exists in the outer convective enveloppe of solar-like stars and giant gaseous planets. In particular, tidal waves may strongly interact with zonal flows at the so-called corotation resonances, where the wave's Doppler-shifted frequency cancels out. We aim to provide a deep physical understanding of the dynamics of tidal inertial waves at corotation resonances, in the presence of differential rotation profiles typical of low-mass stars and giant planets. By developping an inclined shearing box, we investigate the propagation and the transmission of free inertial waves at corotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model. We find that inertial waves can be either fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux to the fluid when damped at critical levels, or they can extract action flux to the fluid when amplified at critical levels. Both situations could lead to significant angular momentum exchange between the tidally interacting bodies.Comment: 25 pages, 12 figures, 4 tables, accepted for publication in Astronomy & Astrophysic

Tidal dissipation in rotating and evolving giant planets with application to exoplanet systems

We study tidal dissipation in models of rotating giant planets with masses in the range $0.1 - 10 M_\mathrm{J}$ throughout their evolution. Our models incorporate a frequency-dependent turbulent effective viscosity acting on equilibrium tides (including its modification by rapid rotation consistent with hydrodynamical simulations) and inertial waves in convection zones, and internal gravity waves in the thin radiative atmospheres. We consider a range of planetary evolutionary models for various masses and strengths of stellar instellation. Dissipation of inertial waves is computed using a frequency-averaged formalism fully accounting for planetary structures. Dissipation of gravity waves in the radiation zone is computed assuming these waves are launched adiabatically and are subsequently fully damped (by wave breaking/radiative damping). We compute modified tidal quality factors $Q'$ and evolutionary timescales for these planets as a function of their ages. We find inertial waves to be the dominant mechanism of tidal dissipation in giant planets whenever they are excited. Their excitation requires the tidal period ($P_\mathrm{tide}$) to be longer than half the planetary rotation ($P_\mathrm{rot}/2$), and we predict inertial waves to provide a typical $Q'\sim 10^3 (P_\mathrm{rot}/1 \mathrm{d})^2$, with values between $10^5$ and $10^6$ for a 10-day period. We show correlations of observed exoplanet eccentricities with tidal circularisation timescale predictions, highlighting the key role of planetary tides. A major uncertainty in planetary models is the role of stably-stratified layers resulting from compositional gradients, which we do not account for here, but which could modify predictions for tidal dissipation rates.Comment: Accepted by MNRAS. 12 pages, 6 figure

Tidally Excited Inertial Waves in Stars and Planets: Exploring the Frequency-dependent and Averaged Dissipation with Nonlinear Simulations

We simulate the nonlinear hydrodynamical evolution of tidally excited inertial waves in convective envelopes of rotating stars and giant planets modeled as spherical shells containing incompressible, viscous, and adiabatically stratified fluid. This model is relevant for studying tidal interactions between close-in planets and their stars, as well as close low-mass star binaries. We explore in detail the frequency-dependent tidal dissipation rates obtained from an extensive suite of numerical simulations, which we compare with linear theory, including with the widely employed frequency-averaged formalism to represent inertial wave dissipation. We demonstrate that the frequency-averaged predictions appear to be quite robust and are approximately reproduced in our nonlinear simulations spanning the frequency range of inertial waves as we vary the convective envelope thickness, tidal amplitude, and Ekman number. Yet, we find nonlinear simulations can produce significant differences with linear theory for a given tidal frequency (potentially by orders of magnitude), largely due to tidal generation of differential rotation and its effects on the waves. Since the dissipation in a given system can be very different both in linear and nonlinear simulations, the frequency-averaged formalism should be used with caution. Despite its robustness, it is also unclear how accurately it represents tidal evolution in real (frequency-dependent) systems

Tidal dissipation modelling in gas giant planets

Gas giant planets are turbulent rotating magnetic objects that have strong and complex interactions with their environment. In such systems, the dissipation of tidal waves excited by tidal forces shape the orbital architecture and the rotational dynamics of the planets. During the last decade, a revolution has occurred for our understanding of tides in these systems and for our knowledge of the interiors of giant planets thanks to the space mission JUNO and the grand finale of the CASSINI mission. Our objective is thus to predict tidal dissipation using internal structure models, which agree with these latest observational constrains. To accomplish that, we build a new ab-initio model of tidal dissipation in giant planets that coherently takes into account the interactions of tidal waves with their complex structure. This model is a semi-global model in the planetary equatorial plane. We study the linear excitation of tidal (magneto-)gravito-inertial progressive waves and standing modes. We present here the general formalism and the potential regimes of parameters that should be explored. This will pave the way for full 3D numerical simulations that will take into account complex internal structure and dynamics of gas giant (exo-)planets

Tidal dissipation modelling in gaseous giant planets at the time of space missions

International audienceGaseous giant planets (Jupiter and Saturn in our solar system and hot Jupiters around other stars) are turbulent rotating magnetic objects that have strong and complex interactions with their environment (their moons in the case of Jupiter and Saturn and their host stars in the case of hot Jupiters/Saturns). In such systems, the dissipation of tidal waves excited by tidal forces shape the orbital architecture and the rotational dynamics of the planets.During the last decade, a revolution has occurred for our understanding of tides in these systems. First, Lainey et al. (2009, 2012, 2017) have measured tidal dissipation stronger by one order of magnitude than expected in Jupiter and Saturn. Second, unexplained broad diversity of orbital architectures and large radius of some hot Jupiters are observed in exoplanetary systems. Finally, new constraints obtained thanks to Kepler/K2 and TESS indicate that tidal dissipation in gaseous giant exoplanets is weaker than in Jupiter and in Saturn (Ogilvie 2014, Van Eylen et al. 2018, Huber et al. 2019).Furthermore, the space mission JUNO and the grand finale of the CASSINI mission have revolutionized our knowledge of the interiors of giant planets. We now know, for example, that Jupiter is a very complex planet: it is a stratified planet with, from the surface to the core, a differentially rotating convective envelope, a first mixing zone (with stratified convection), a uniformly rotating magnetised convective zone, a second magnetized mixing zone (the diluted core, potentially in stratified convection) and a solid core (Debras & Chabrier 2019). So far, tides in these planets have been studied by assuming a simplified internal structure with a stable rocky and icy core (Remus et al. 2012, 2015) and a deep convective envelope surrounded by a thin stable atmosphere (Ogilvie & Lin 2004) where mixing processes, differential rotation and magnetic field were completely neglected.Our objective is thus to predict tidal dissipation using internal structure models, which agree with these last observational constrains. In this work, we build a new ab-initio model of tidal dissipation in giant planets that coherently takes into account the interactions of tidal waves with their complex stratification induced by the mixing of heavy elements, their zonal winds, and (dynamo) magnetic fields. This model is a semi-global model in the planetary equatorial plane. We study the linear excitation of tidal magneto-gravito-inertial progressive waves and standing modes. We take into account the buoyancy, the compressibility, the Coriolis acceleration (including differential rotation), and the Lorentz force. The tidal waves are submitted to the different potential dissipative processes: Ohmic, thermal, molecular diffusivities, and viscosity. We here present the general formalism and the potential regimes of parameters that should be explored. The quantities of interest such as tidal torque, dissipation, and heating are derived. This will pave the way for full 3D numerical simulations that will take into account complex internal structure and dynamics of gaseous giant (exo-)planets in spherical/spheroidal geometry

Effect of differential rotation on tidal waves in the convective envelope of low-mass stars and giant planets

International audienceOver 4000 exoplanets have been discovered in the past 25 years, most of which orbit around low-mass stars. For the shortest period exoplanets, like Hot-Jupiter systems, star-planet tidal interactions are likely to have played a major role in the orbital architecture and rotational evolution of the planets and their host star, through stellar and planetary tidal dissipation (see e.g. Ogilvie 2014, Mathis 2019, and references therein). Moreover, stellar tidal dissipation is known to vary considerably with the mass, age, rotation, and metallicity of the star (Mathis 2015, Gallet et al. 2017, Bolmont et al. 2017). To draw a comprehensive picture of tidal dissipation in stars and giant gaseous planets, one key physical mechanisms need to be further explored: differential rotation. Recent improvements in asterosismology and spectropolarimetry have helped to establish that the convective envelope of low-mass stars is differentially rotating like in the Sun (e.g. Barnes et al. 2017, Benomar et al. 2018). Similarly, data furnished by Cassini Grand Finale and the probe Juno, showed that cylindrical differential rotation extends into the convective envelope of Saturn and Jupiter below their surface (Galanti et al. 2019, Kaspi et al. 2017, respectively), which augurs similar rotation profiles for giant gaseous exoplanets. Nevertheless, differential rotation is rarely taken into account to compute tidal dissipation. Yet, Baruteau & Rieutord (2013) and Guenel et al. (2016a,2016b) have shown, using rotation profiles inspired by those encountered in solar-like stars and giant gaseous planets, that differential rotation can strongly affect the dynamics and dissipation properties of linear tidal waves. In particular, they pointed out that tidal waves deeply interact with zonal flows at the so-called corotation resonances, where the Doppler-shifted wave frequency vanishes, with the possibility that this interaction becomes unstable. This phenomenon that gives rise to an intense and localised energy dissipation is still poorly understood. In that context, we have developed a new local shear box that models a small patch of the convective zone of a low-mass star or a giant planet, in order to understand the complex interplay between tidal waves and zonal flows. The inclination of the box with respect to the rotation axis allows us to study the impact of cylindrical rotation profile when the box is at the pole (similar to Jupiter, Saturn and rapidly rotating stars rotation profiles, e.g. Gastine et al. 2013), and conical differential rotation profile when the box is tilted (analogous to that of the Sun and solar-type stars, e.g. Brun et al. 2017). We investigate the transmission of free linear inertial waves, within a fluid in thermal-wind balance, at corotation resonances and more broadly at critical levels that are singularities in the differential equation governing wave propagation. By the use of an invariant of energy flux, we diagnose each critical level according to the used rotation profile. Indeed, we find different regimes in which waves can be either fully transmitted, damped (as shown in the figures below) or even amplified with strong consequences for the tidal angular momentum exchanges. These different regimes of wave transmission are found both with conical and cylindrical rotation profiles, and depend on the critical level encountered, on the wave properties (direction, wavenumbers) and on the profile of the mean flow (first and second derivative). In particular, we demonstrate that a criterion analogous to the Miles-Howard theorem for the stability of gravity waves with vertical shear, is expected when using a cylindrical profile. Alongside this analytical study, we have numerically solved the leading equations for free inertial waves in a dissipative medium, where turbulent friction is taken into account thanks to a frictional damping force. We observed a good consistency between the analytical and numerical results, though some discrepancies are reported when using a non-linear mean flow profile at corotation. Based on these results, we discussed possible applications to the interiors of giant-gaseous planets and solar-like stars. Thanks to values of shear contrast given by asterosismology observations or 3D numerical simulations for solar-like stars, and to an analytical equatorial model for giant gaseous planets, we determine which regime of wave transmission is expected at each critical level in the convective envelope of these objects. It turns out that for K and G-type stars along their lifetime, and for Jupiter and Saturn at the present time, a regime where inertial waves are strongly damped is largely preferred in the convective envelope of these objects

Hydrodynamic modelling of dynamical tides dissipation in Jupiter's interior as revealed by Juno

The Juno spacecraft has acquired exceptionally precise data on Jupiter's gravity field, offering invaluable insights into Jupiter's tidal response, interior structure, and dynamics, establishing crucial constraints. We develop a new model for calculating Jupiter's tidal response based on its latest interior model, while also examining the significance of different dissipation processes for the evolution of its system. We study the dissipation of dynamical tides in Jupiter by thermal, viscous and molecular diffusivities acting on gravito-inertial waves in stably stratified zones and inertial waves in convection ones. We solve the linearised equations for the equilibrium tide. Next, we compute the dynamical tides using linear hydrodynamical simulations based on a spectral method. The Coriolis force is fully taken into account, but the centrifugal effect is neglected. We study the dynamical tides occurring in Jupiter using internal structure models that respect Juno's constraints. We study specifically the dominant quadrupolar tidal components and our focus is on the frequency range that corresponds to the tidal frequencies associated with Jupiter's Galilean satellites. By incorporating the different dissipation mechanisms, we calculate the total dissipation and determine the imaginary part of the tidal Love number. We find a significant frequency dependence in dissipation spectra, indicating a strong relationship between dissipation and forcing frequency. Furthermore, our analysis reveals that, in the chosen parameter regime in which kinematic viscosity, thermal and molecular diffusivities are equal, the dominant mechanism contributing to dissipation is viscosity, exceeding in magnitude both thermal and chemical dissipation. We find that the presence of stably stratified zones plays an important role in explaining the high dissipation observed in Jupiter