Parametric instability and wave turbulence driven by tidal excitation of internal waves

We investigate the stability of stratified fluid layers undergoing homogeneous and periodic tidal deformation. We first introduce a local model which allows to study velocity and buoyancy fluctuations in a Lagrangian domain periodically stretched and sheared by the tidal base flow. While keeping the key physical ingredients only, such a model is efficient to simulate planetary regimes where tidal amplitudes and dissipation are small. With this model, we prove that tidal flows are able to drive parametric subharmonic resonances of internal waves, in a way reminiscent of the elliptical instability in rotating fluids. The growth rates computed via Direct Numerical Simulations (DNS) are in very good agreement with WKB analysis and Floquet theory. We also investigate the turbulence driven by this instability mechanism. With spatio-temporal analysis, we show that it is a weak internal wave turbulence occurring at small Froude and buoyancy Reynolds numbers. When the gap between the excitation and the Brunt-V\"ais\"al\"a frequencies is increased, the frequency spectrum of this wave turbulence displays a -2 power law reminiscent of the high-frequency branch of the Garett and Munk spectrum (Garrett & Munk 1979) which has been measured in the oceans. In addition, we find that the mixing efficiency is altered compared to what is computed in the context of DNS of stratified turbulence excited at small Froude and large buoyancy Reynolds numbers and is consistent with a superposition of waves.Comment: Accepted for publication in Journal of Fluid Mechanics, 27 pages, 21 figure

High-Rayleigh-number convection in porous-fluid layers

We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-B\'enard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection in both the porous and the unconfined fluid regions, while the porous flow still remains strongly confined and governed by Darcy's law. We demonstrate that the heat flux and average thermal structure of the system can be predicted using previous results of convection in individual fluid or porous layers. We revisit a controversy about the role of subcritical "penetrative convection" in the porous medium, and confirm that such induced flow does not contribute to the heat flux through the system. Lastly, we briefly study the temporal coupling between the two layers and find that the turbulent fluid convection above acts as a low-pass filter on the longer-timescale variability of convection in the porous layer.Comment: Accepted for publication in Journal of Fluid Mechanics, 25 pages, 13 figure

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

International audienceIn this paper, we present an experimental investigation of the turbulent saturation of the flow driven by the parametric resonance of inertial waves in a rotating fluid. In our setup , a half-metre wide ellipsoid filled with water is brought to solid-body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly nonlinear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (Phys. Rev. Lett., vol. 119 (3), 2017, 034502) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed

Internally heated porous convection: an idealised model for Enceladus' hydrothermal activity

Recent planetary data and geophysical modelling suggest that hydrothermal activity is ongoing under the ice crust of Enceladus, one of Saturn's moons. According to these models, hydrothermal flow in the porous, rocky core of the satellite is driven by tidal deformation that induces dissipation and volumetric internal heating. Despite the effort in the modelling of Enceladus' interior, systematic understanding---and even basic scaling laws---of internally-heated porous convection and hydrothermal activity are still lacking. In this article, using an idealised model of an internally-heated porous medium, we explore numerically and theoretically the flows that develop close and far from the onset of convection. In particular, we quantify heat-transport efficiency by convective flows as well as the typical extent and intensity of heat-flux anomalies created at the top of the porous layer. With our idealised model, we derive simple and general laws governing the temperature and hydrothermal velocity that can be driven in the oceans of icy moons. In the future, these laws could help better constraining models of the interior of Enceladus and other icy satellites.Comment: 23 pages, 13 figure

Evidence of the Zakharov-Kolmogorov spectrum in numerical simulations of inertial wave turbulence

Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived analytically by Galtier (Phys. Rev. E, 68, 2003), thus offering numerical proof of the relevance of wave turbulence theory for three-dimensional, anisotropic waves. Lastly, we show that the same forcing leads to either geostrophic or wave turbulence depending on the initial condition. Our results thus bring further evidence for bi-stability in rotating turbulent flows.Comment: 7 pages, 4 figure

Inertial wave turbulence driven by elliptical instability

The combination of elliptical deformation of streamlines and vorticity can lead to the destabilisation of any rotating flow via the elliptical instability. Such a mechanism has been invoked as a possible source of turbulence in planetary cores subject to tidal deformations. The saturation of the elliptical instability has been shown to generate turbulence composed of non-linearly interacting waves and strong columnar vortices with varying respective amplitudes, depending on the control parameters and geometry. In this paper, we present a suite of numerical simulations to investigate the saturation and the transition from vortex-dominated to wave-dominated regimes. This is achieved by simulating the growth and saturation of the elliptical instability in an idealised triply periodic domain, adding a frictional damping to the geostrophic component only, to mimic its interaction with boundaries. We reproduce several experimental observations within one idealised local model and complement them by reaching more extreme flow parameters. In particular, a wave-dominated regime that exhibits many signatures of inertial wave turbulence is characterised for the first time. This regime is expected in planetary interiors

Régimes asymptotiques des écoulements en rotation excités par forçage mécanique dans les noyaux planétaires: saturation turbulente et organisation à grande échelle

Many terrestrial bodies, including the Earth, are surrounded by a magnetic field protecting them from high energy stellar particles. It originates in the turbulent motion of the liquid, conducting iron core of these planets and moons. The complex motion of liquid iron in planetary cores is often thought to be driven by thermal and solutal convection, but such a model is sometimes hard to conciliate with the heat budget of terrestrial planets, especially the smaller ones. To explain the existence of magnetic fields surrounding small moons such as Ganymede and Io, mechanical forcing induced by tides has been proposed as an alternative source of turbulence in planetary cores. Tidal interaction between a terrestrial body and a companion results in a distortion of the shape of the body, a deformation that remains mostly directed towards the companion and may rotate at a different rate compared to the planet or the moon spinning rate. This is the case for instance of the Earth-Moon system: the Earths tidal bulge rotates at the Moons orbiting rate (in 27 days) whereas the Earths spinning rate is much larger (1 day). Another effect of tidal interaction is to force periodic variations of the length of the day, an oscillation called libration. These two effects (differential rotation and libration) have been shown to excite parametric resonance of inertial waves, the latter being spontaneous oscillations of rotating fluids interiors induced by the restoring action of the Coriolis force. This resonance is called the elliptical instability. The inertial waves grow exponentially and eventually collapse into turbulence. Although the saturation of the instability is the most important state for dynamo action and orbital evolution of planets, it remains poorly understood. The work presented throughout this dissertation aims at better characterising the turbulence resulting from the elliptical instability, in particular in regimes that are relevant to geo- and astrophysics when both the tidal forcing amplitude and the viscous dissipation are weak. This investigation of the non-linear saturation of the parametric resonance is carried out with experiments and idealised numerical simulations, complemented by theoretical investigations. In the experiment, we reveal that two regimes exist in the saturation of the instability. The first one, which is classical of turbulence in rotating fluids, is dominated by strong vortices invariant along the rotation axis, or geostrophic. Additionally, we exhibit a new regime which is dominated by inertial waves in non-linear resonant interactions, a state called inertial wave turbulence. To extend our understanding of these two states and to fully characterise the inertial waves interactions, we proceed to idealised numerical simulations in a local cartesian model of tidal flows. It allows producing the two regimes of saturation and exploring the weak forcing and dissipation regime. With this ideal model, we show that the transition between the two regimes mentioned earlier is caused by an instability that vanishes below a finite forcing amplitude. We also explore the possibility for direct forcing of strong geostrophic motion by the resonant waves directly, but our simulations suggest that they should not dominated in the geophysical limit. We therefore conclude that the superposition of inertial waves type of saturation is the relevant one for planetary cores. We finally investigate the stability of stably stratified planetary cores undergoing tidal distortion. Similarly to the elliptical instability, we exhibit a resonance of internal waves, which are oscillations caused by the stable density stratification. We show with idealised numerical simulations that the resonant waves give rise to internal wave turbulence in the non-linear saturation of the instability.De nombreux corps telluriques, dont la Terre, sont entourés d'un champ magnétique les protégeant des particules à haute énergie émises par les étoiles. Celui-ci trouve son origine dans les mouvements turbulents du noyau de ces corps, constitué de fer conducteur liquide. Les mouvements complexes de ce fluide sont souvent attribués à la convection thermique et solutale, mais ce modèle est parfois difficile à concilier avec le bilan thermique des corps telluriques, particulièrement les plus petits. Pour expliquer l'existence des champs magnétiques entourant par exemple Ganymède ou Io, les forçages mécaniques provoqués par les marées ont été proposés comme une source alternative de mouvements turbulents dans les noyaux planétaires. L'interaction de marée entre un corps tellurique et un astre compagnon se traduit par la déformation du premier donnant un bourrelet qui suit le mouvement de l'astre compagnon, celui-ci orbitant à une vitesse différente de la rotation du corps considéré. C'est le cas par exemple du système Terre-Lune : le bourrelet de marée terrestre suit l'orbite de la Lune et accomplit une rotation en 27 jours, tandis que la période de rotation de la Terre sur elle-même est de 1 jour. L'interaction de marée cause également des variations périodiques de la durée du jour appelées "libration". Des études précédentes ont montré que ces deux effets (la rotation différentielle et la libration) excitent des résonances paramétriques d'ondes inertielles, ces dernières étant des oscillations spontanées causée par la force de Coriolis. Ce mécanisme de résonance est appelé "instabilité elliptique". Les ondes inertielles croissent exponentiellement pour finalement s'effondrer en turbulence. Bien que la saturation de l'instabilité soit la phase la plus importante pour comprendre la génération de champ magnétique et l'évolution orbitale des planètes, elle reste mal comprise. Les travaux présentés dans cette thèse visent à mieux caractériser la turbulence résultant de la saturation de l'instabilité elliptique, qui plus est en s'approchant des régimes pertinent pour la géo- et l'astrophysique lorsque le forçage de marée et la dissipation visqueuse sont faibles. Cette étude de la saturation de l'instabilité elliptique est menée à travers des expériences et des simulations numériques idéalisées, complétée par des développements théoriques. Les expériences révèlent que deux régimes existent dans la saturation de l'instabilité. Le premier est dominé par des vortex invariants suivant l'axe de rotation, qualifiés de "géostrophiques". Le second est dominé par des ondes inertielles en interaction non-linéaire, un état appelé "turbulence d'ondes". Pour mieux comprendre ces deux états et mieux caractériser les interactions entre ondes, nous complétons les expériences par des simulations numériques dans un modèle local cartésien d'un noyau soumis aux forces de marées. Celui-ci nous permet de produire les deux types de saturation et de pousser davantage l'exploration des régimes de faibles forçages et dissipation. Nous montrons de plus que la transition entre les deux régimes peut être attribuée à une instabilité qui disparaît au-dessous d'une amplitude de forçage non-nul. Nous explorons également la possibilité pour que les écoulements géostrophiques soient directement forcés par les ondes résonantes, mais nous montrons que ce mécanisme n'est pas dominant dans les régimes géophysiques. Nous concluons par conséquent que le régime de turbulence d'onde est attendu dans les noyaux planétaires. Enfin, nous étudions la stabilité des couches fluides stablement stratifiées subissant des déformations de marées. Nous montrons l'existence de résonances paramétriques d'ondes internes, ces dernières étant dues à une compétition entre inertie et gravité, analogues à l'instabilité elliptique. Nous montrons par des simulations numériques idéalisées que les ondes résonantes donnent lieu à une turbulence d'ondes internes lorsque l'instabilité sature

Asymptotic regimes of flows excited by harmonic forcing in planetary cores : turbulent saturation and large-scale organisation

Cette thèse explore les écoulements turbulents induits par les effets de marées dans les noyaux planétaires. Plus particulièrement, nous nous intéressons à la saturation des instabilités elliptiques dues à l'effet conjoint de la rotation planétaire et de la déformation de marée. De telles instabilités se produisent notamment lorsque la rotation de la déformation diffère de la rotation planétaire, conduisant à la croissance exponentielle d'ondes inertielles. De telles résonances sont également excitées lorsque les interactions de marées produisent des variations de la vitesse de rotation planétaires appelées "libration". Dans un premier temps, la saturation de l'instabilité elliptique est étudiée expérimentalement en forçant la libration d'un ellipsoïde rempli d'eau. Nous mettons en évidence deux régimes de saturation : aux faibles amplitudes de libration, l'écoulement saturé se constitue d'ondes inertielles en interactions triadiques résonnantes, un état appelé "turbulence d'ondes". À plus forte amplitude, la saturation est dominée par un écoulement moyen géostrophique fort. Par une étude numérique et théorique, cette transition est attribuée à une instabilité affectant les ondes inertielles d'amplitude finie. La saturation de l'instabilité elliptique est également étudiée numériquement à l'aide d'un modèle local cartésien. Celui-ci permet de retrouver les deux régimes de saturation observés expérimentalement, d'en étudier finement les propriétés et d'en explorer le comportement dans la limite géophysique. Enfin, une dernière étude est consacrée à l'instabilité elliptique dans les milieux stratifiés.The present thesis explores the turbulent flows induced by tidal interaction in planetary cores. In particular, we focus on the saturation of the elliptical instabilities induced by the effect of planetary rotation and tidal deformation. Such instabilities arise for instance when the rotation rate of the tidal bulge and the planet differ and lead to the exponential growth of inertial waves. Similar resonances are also excited when tidal interactions induce harmonic modulation of the planetary rotation rate called "libration." The saturation of the elliptical instability is first investigated experimentally via the forcing the libration of an ellipsoid filled with water. We find two saturation regimes: at low forcing amplitude, the saturation flow is a superposition of inertial waves in non-linear, triadic resonant interaction, a state called "wave turbulence." At larger forcing amplitude, the saturation flow is dominated by a strong geostrophic mean flow. With a numerical and theoretical study, we find that the transition between these two regimes is due to an instability of finite amplitude inertial waves. Furthermore, we explore the saturation of the elliptical instability with an idealised, local cartesian model. We retrieve the two saturation regimes found in the experiment and give a detail account of their spatial and temporal content. The local model also allows exploring the behaviour of these two saturation regimes in the geophysical limit. Lastly, a last study is targeted at the elliptical instability in stratified planetary cores. We find that internal waves are excited by tidal interaction and also saturate into inertial wave turbulence

Internal wave turbulence driven by tides

Poster presented at Euromech-Ercoftac conference <i>Turbulent Cascades II.</i> <br