Twenty years of giant exoplanets - Proceedings of the Haute Provence Observatory Colloquium, 5-9 October 2015 Edited by I. Boisse, O. Demangeon, F. Bouchy & L. Arnold Hydrodynamical scaling laws to explore the physics of tidal dissipation in star-planet systems P. Auclair-Desrotour 1 , 2 , S. Mathis 2 , 3 , C. Le Poncin-Lafitte 4 Talk given at OHP-2015 Colloquium 1 IMCCE, CNRS UMR 8028, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France ( pierre.auclair-desrotour@obspm.fr ) 2 Laboratoire AIM Paris-Saclay, CEA / DSM - CNRS - Universit´ e Paris Diderot, IRFU / SAp Centre de Saclay, 91191 Gif-sur-Yvette Cedex, France 3 LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit´ es, UPMC Univ. Paris 6, Univ. Paris Diderot, Sorbonne Paris Cit´ e, 5 place Jules Janssen, 92195 Meudon, France 4 SYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit´ es, UPMC Universit´ e Paris 6, LNE, 61 avenue de l’Observatoire, 75014 Paris, France Abstract Fluid celestial bodies can be strongly a ff ected by tidal perturbations, which drive the evolution of close planetary systems over long timescales. While the tidal response of solid bodies varies smoothly with the tidal frequency, fluid bodies present a highly frequency-resonant tidal dissipation resulting from the complex hydrodynamical response. In these bodies, tides have the form of a combination of inertial waves restored by the Coriolis acceleration and gravity waves in the case of stably stratified layers, which are restored by the Archimedean force. Because of processes such as viscous friction and thermal di ff usion, the energy given by the tidal forcing is dissipated. This directly impact the architecture of planetary systems. In this study, we detail a local analytical model which makes us able to characterize the internal dissipation of fluid bodies as functions of identified control parameters such as the inertial, Brunt-V¨ ais¨ al¨ a and tidal frequencies, and the Ekman and Prandtl numbers. 1 Introduction Since they result from mutual interactions between celestial bodies, tides are intrinsic to planetary systems. Owing to their impact on the architecture of these systems, as well as the physical properties of the bodies themselves, the e ff ects of tidal perturbations have to be characterized and quantified. Moreover, observational constraints are now obtained on tidal dissipation inside giant gaseous planets in our Solar system (Lainey et al. 2009, 2012, 2015) and exoplanetary systems (see e.g. Ogilvie 2014, and references therein). Like planetary solids, fluid layers in stars and planets are submitted to gravitational tidal potentials. But they are also the place of thermal forcings induced by the insolation flux of irradiating host stars. Hence the tidal response of fluid bodies results both from the complex coupling between the properties of their internal structure and these forcings. As proved by recent works (e.g. Ogilvie & Lin 2004, 2007; Gerkema & Shrira 2005), this response is characterized by tidal waves which strongly depend on the frequencies of the forcings and belong to well-identified families: • inertial waves, which result from the spin rotation of the body and are restored by the Coriolis acceleration, • gravity waves, which can propagate in stably stratified fluids and are restored by the Archimedean force, • Alfv´ en waves, which can propagate in magnetized fluids and are restored by magnetic forces. The energy tidally dissipated by these waves can vary with tidal frequency over several orders of magnitude, which leads to a potentially erratic evolution of the planetary systems dynamics that di ff er in nature from what is observed for solids and simplified fluid equilibrium tide (Efroimsky & Lainey 2007; Auclair-Desrotour et al. 2014). Because of its complexity, the tidal response of planetary and stellar fluid layers has motivated numerous theoretical studies 92
Twenty years of giant exoplanets - Proceedings of the Haute Provence Observatory Colloquium, 5-9 October 2015 Edited by I. Boisse, O. Demangeon, F. Bouchy & L. Arnold since the middle of the twentieth century (see e.g. Zahn 1966a,b,c, 1975, 1977, 1989; Ogilvie & Lin 2004; Wu 2005; Ogilvie & Lin 2007; Remus et al. 2012; C´ ebron et al. 2012, 2013). All these works highlighted the crucial role played by the properties of the internal structure (stratification, viscous friction, thermal di ff usion, etc.) and dynamics (e.g. spin rotation). As they make possible to explore the whole domain of parameters, asymptotic analytic models appear as an interesting approach to unravel the physics of tidal dissipation. For this reason, we follow Ogilvie & Lin (2004), who detailed in Appendix A of their paper a robust simplified set-up providing expressions of the energy dissipated by viscous friction as explicit functions of the tidal frequency and fluid parameters. Generalizing this early work, we focus on a local cross-section belonging to a rotating celestial fluid body, which can be either a star or a planet. The fluid is supposed to be possibly stably stratified and rotating with the body. We take into account two dissipative processes: viscous friction which characterizes turbulent convective zones in planets and stars, and thermal di ff u- sion which predominates in stellar radiative zones. Magnetism, which will be introduced in forthcoming works, is not taken into account in the present one. Therefore, we study the tidal dissipation induced by viscous friction and thermal di ff usion acting on gravito-inertial waves. The local fluid section is submitted to an academic harmonic tidal forcing with periodic boundary conditions. From the dynamics, we establish the expression of the energy dissipated by viscous friction and thermal di ff usion. This allows us to identify the possible regimes of tidal dissi- pation as a function of the characteristic frequencies, viscosity and thermal di ff usion. Then, we compute scaling laws characterizing the properties of the frequency dissipation spectrum, such as the positions of resonant peaks, their widths, heights and number, the level of the non-resonant background, as functions of the tidal frequency and fluid control parameters. For a detailed development, the reader will refer to Auclair Desrotour et al. (2015). In Sect. 2, we present the layout. In Sect. 3, we summarize the results and finally, in Sect. 4, give our conclusions and prospects. 2 Physical set-up 2.1 Local model Our local model is a Cartesian fluid box of side length L centered on a point M of a planetary fluid layer, or star (see Fig. 1). Let be R O : { O , X E , Y E , Z E } the reference frame rotating with the body at the spin frequency Ω with respect to Z E .The spin vector Ω is thus given by Ω = Ω Z E . The point M is defined by the spherical coordinates ( r , θ, ϕ ) and � � the corresponding spherical basis is denoted e r , e θ , e ϕ . We also define the local Cartesian coordinates x = ( x , y , z ) � � and associated reference frame R : M , e x , e y , e z , which is such that e z = e r , e x = e ϕ and e y = − e θ . In this frame, the local gravity acceleration, assumed to be constant, is aligned with the vertical direction, i.e. g = − g e z , and the � � spin vector is decomposed as follows: Ω = Ω cos θ e z + sin θ e y , where θ is the colatitude. 4 � 2 � vis log 10 � [J.kg � 1 ] � the 0 − 2 − 4 − 6 − 8 0 5 10 15 20 � Figure 1: Left: Local Cartesian model, frame, and coordinates. Right: Energy dissipated ( ζ ) and its viscous and thermal components, ζ visc and ζ therm respectively, as functions of the reduced tidal frequency ( ω ) for θ = 0, A = 10 2 , E = 10 − 4 and K = 10 − 2 , which gives P r = 10 − 2 (see Sect. 2 for the definition of these quantities). 93
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