S Mathis

L A L V A N

F Remus

INTERNAL WAVES AND TIDES IN STAR-PLANET SYSTEMS

Internal waves and tidal interactions constitute powerful mechanisms for angular momentum exchanges in star-planet systems. In this lecture, we review the state of the art of their modelling and we discuss their impact on the evolution of stellar systems. In this context, we first show how internal waves are able to deeply modify the secular transport of angular momentum in stellar interiors in the whole Hertzsprung-Russelll diagram. Then, we show how tidal waves, as well as the related exchanges of angular momentum, strongly depend on the internal structure of stars and planets.

1 Internal gravity waves in stellar interiors

Motivation

 and asteroseismology (e.g.

radiative core (Charbonnel & Talon 2005), the light elements mixing in low-mass stars (Montalban & Schatzman 1996;Talon & Charbonnel 2005)t h ew e a kd i ff e rential rotation in subgiant and red giant stars (Eggenberger et al. 2012;Cellier et al. 2012;M a r q u e set al. 2013) and the transport of angular momentum necessary to explain mass-loss in active massive stars such as Be stars (Huat et al. 2009;L e e&S a i o1993;L e e2013).

Internal gravity waves thus constitute an essential transport and mixing mechanism in (differentially) rotating stellar radiation zones during stars' evolution. In the first part of this lecture, we will thus first describe the present state of the art for the treatment of the secular transport of angular momentum in stars (Sect. 1.2). Then, we will focus on the action of internal gravity waves and on the related challenges (Sect. 1.3). Finally, we will conclude (Sect. 1.4).

Secular angular momentum transport mechanisms in stars

1.2.1 General context Stars are dynamical turbulent rotating and magnetic objects. So, turbulence, rotation and magnetic fields modify their evolution as well as their interactions with their environment. For example, differential rotation induces "non-standard" mixing processes, which modify their lifetime, their late stages of evolution and their nucleosynthetic properties (see e.g. [START_REF] Maeder | Physics, Formation and Evolution of Rotating Stars[END_REF]). Therefore, they should be taken into account to get realistic and coherent stellar models. Moreover, since very strong constraints have been obtained on the Sun's internal rotation profile until 0.2 R ⊙ (e.g. Garcia et al. 2007;G a r c i aet al. 2011) using helioseismology (GOLF/SOHO) and on more evolved low-mass stars (e.g. Beck et al. 2012;Deheuvels et al. 2012;Mosser et al. 2012) using space asteroseismology (CoRoT, Kepler), it becomes necessary to build models computing MHD processes both on dynamical and secular time-scales. From now on in this review, we therefore adress the most recent progresses achieved for the modelling of secular exchanges of angular momentum in stellar interiors (see also the chapter by A. Palacios in this volume). Therefore, we focus on mechanisms acting in their radiation zones while the reader could refer to [START_REF] Brun | EAS Publications Series[END_REF] for a review on dynamical time-scales and on convective regions.

Differential rotation and associated large-scale meridional circulation and shear-induced turbulence

In addition to convective penetration (Zahn 1991;Z h a n g2013), the first hydrodynamical processes that should be understood are the differential rotation and the associated large-scale meridional circulation (e.g. Zahn 1992;M a e d e r&Z a h n 1998; Mathis & Zahn 2004) and shear-induced turbulence (e.g. Talon & Zahn 1997;M a e d e r2003; Mathis et al. 2004;Prat & Lignières 2013). A global understanding of their couplings has been obtained by Rieutord (2006) and Decressin et al. (2009) (see Fig. 1). First, viscous turbulent transport, stellar winds (and other applied external torques; see e.g. Sect. 1.3) and structural adjustments induce meridional currents. Then, they advect heat that leads to thermal relaxation inducing latitudinal gradients of temperature. Finally, the associated baroclinic torque builds a new differential rotation profile, which can be understood using the so-called thermal-wind equation. This transport loop is represented in Figure 7. However, these three mechanisms are unable to reproduce the angular velocity profile of the radiative core of the present Sun [START_REF] Pinsonneault | [END_REF]Turck-Chieze et al. 2010) and the rotational evolution of solar-type stars on the main sequence (e.g. Talon & Charb onnel 2005) and of low-mass subgiant and red giant stars (Palacios et al. 2006;Eggenberger et al. 2012;Cellier et al. 2012; Marques et al.M a r q u e set al. 2013). Therefore, other processes such as magnetic fields (e.g. Rudiger & Kitchatinov 1997;Gough & McIntyre 1998;Garaud & Garaud 2008;Wood et al. 2011) or internal waves excited by penetrative convection (e.g. Schatzman 1993b;Z a h net al. 1997) must be studied.

Fossil magnetic field

The main admitted paradigm for magnetic field origin in stellar radiation zones is the fossil field hypothesis [START_REF] Mestel | IAU Symposium, Stellar and Solar Magnetic Fields[END_REF]W a d eet al. 2011). Fossil fields are believed to originate from the trapping of the interstellar magnetic flux during star formation or to be a remnant of a field generated by a dynamo during the pre-main-sequence (e.g. Mathis et al. 2011 and references therein). Then, if the initial field is a small-scale turbulent field, we have to understand and predict the topology of the resulting field after the birth of the radiative region. This field geometry will then impact angular momentum transport properties. This problem is related to MHD relaxation processes, in which an initial turbulent magnetic field is converted into a large-scale configuration thanks to a selective decay of the ideal MHD invariants (Braithwaite & Spruit 2004;Braithwaite & Nordlund 2006;Braithwaite 2008;D u e z&M a t h i s2010). Then, resulting non force-free relaxed states minimize the total energy for given magnetic helicity and fluid invariant due to the stable stratification of stellar radiation zones. Roughly axisymmetric dipolar twisted configurations are obtained if the initial magnetic energy is confined near the center (Braithwaite & Nordlund 2006;D u e z&M a t h i s2010) while one gets complex non-axisymmetric fields in the case where the energy is distributed in the whole radiation zone (Braithwaite 2008). The field is then organised on large-scale, mixed (poloidal and toroidal), non force-free configurations, which are initially stable as demonstrated by Braithwaite (2009)a n dD u e zet al. ( 2010).

Once this initial (axi-or non-axisymmetric) non force-free magnetic relaxed configuration has been established, it interacts with differential rotation and Alfvén waves propagate along poloidal field lines to damp this latter (Mestel & Weiss 1987). Then, in the case of axisymmetric configurations, the angular velocity becomes frozen along the meridional field lines: this is the so-called Ferraro's state (e.g. Brun & Zahn 2006;Spada et al. 2010;Strugarek et al. 2011), which is not observed in the Sun. In the case of non-axisymmetric configurations, two regimes have been proposed by Spruit (1999). On one hand, in the case of a strong initial field, the rotation will become uniform in the whole radiation zone (e.g. the oblique rotator's case; see Moss 1992). On the other hand, in the case of a weak initial field, Spruit (1999) proposed that the magnetic configuration evolves in a first step to an axisymmetric state because of rotational smoothing processes, that leads finally to the Ferraro's state for the rotation profile. This scenario for an initial weak non-axisymmetric field has not been confirmed by the first 3-D numerical simulations by Strugarek et al.( 2011) but more complete studies must be achieved in a near future. In each case, meridional currents will be driven, until the star reaches a torque-free state, by the mean axisymmetric Lorentz torque and Maxwell stresses due to correlations of non-axisymmetric magnetic fluctuations in addition to external applied torques and structural adjustments. Then, the meridional circulation advection of angular momentum balances the residual magnetic torque [START_REF] Mestel | [END_REF]Mathis & Zahn 2005).

This picture could be modified by magnetic instabilities, if during the first phase of evolution, the residual differential rotation on each magnetic surface is able to generate a strong toroidal component of the field that becomes unstable (e.g. Tayler 1973) and if this instability is able to trigger a dynamo action through an α-effect in stellar radiation zones; this question is still strongly debated (Spruit 2002;Z a h net al. 2007;Rudiger et al. 2012).

Internal gravity waves

The transport of angular momentum by internal gravity waves

The last angular momentum transport mechanism that should be studied is the action of internal gravity waves (hereafter IGWs) propagating in stably stratified radiative cores of the Sun and low-mass stars and external envelopes of intermediate-mass and massive stars. IGWs are excited by turbulent convective flows at radiation/convection borders, by the κ-m e c h a n i s mi fs t a r sa r ei nt h er elated instability strip, and by tides if there is a close companion (see Sect. 1.3.2).

IGWs transport angular momentum (hereafter AM), which is deposited where they are dissipated by radiative and viscous diffusion processes (e.g. Zahn et al. 1997) or where they reach their critical layers where their frequency is proportional to the local angular velocity (see Goldreich & Nicholson 1989b;A l v a net al. 2011;A l v a net al. 2013, and Sect. 1.3.2). Prograde IGWs (for which m<0, where m is the azimuthal wave number) transport positive AM while retrograde IGWs (m>0) transport negative AM. Moreover, the mean vertical flux of AM transported by prograde waves has the same sign that the mean vertical energy flux injected in radiation zones by the excitation source while the mean vertical flux of AM transported by retrograde waves is directed in the opposite direction. Therefore, in the case of low-mass stars, prograde IGWs deposit AM into the radiative core while retrograde IGWs extract AM towards the surface. In the case of intermediate-mass and massive stars, the situation is inverted, and prograde waves deposit AM towards the exterior while retrograde waves transport AM towards the center of the star. Then, if the star is rotating uniformly, the total amount of AM transported by prograde IGWs cancels the one transported by retrograde ones if we assume a symmetric excitation. However, if differential rotation is initially present or sustained during the evolution of the star, a net flux of AM is transported. Indeed, the Doppler effect that affects the diffusive damping actings on IGWs scales as [σ (r)]

-4 (see the detailed equations in the chapter by A. Palacios in this volume), where σ (r)=σ exc + m Ω(r) -Ω exc , Ω and Ω exc being the horizontally averaged angular velocity of the examined layer and of the excitation region (at r = r exc in the star, e.g. at the boundaries convection/radiation) respectively while σ exc is the studied excited frequency. Then, if we assume a negative initial gradient of angular velocity in a low-mass star (i.e. the radiative core is rotating faster than the convective envelope), prograde waves are damped closer to the excitation region than retrograde waves, leading to a net extraction of AM and thus a spin-down of the core (e.g. Talon & Charbonnel 2005). This net extraction is then constituted by several AM fronts, which propagate from the center of the star towards its surface. That leads, combined with the strong horizontal shear-induced turbulence due to the stable stratification, to a quasi-uniform rotation profile (see Fig. 2). IGWs thus constitute a solid candidate to explain the observed quasi-uniform rotation rate until 0.2 R ⊙ of the solar radiative core as well as the mixing in low-mass stars (Charbonnel & Talon 2005;Talon & Charbonnel 2005). Moreover, during the propagation of AM extraction fronts, the meridional circulation is strongly affected and becomes multi-cellular with a possible large number of loops in the radial direction (see Fig. 2). Such modification of meridional flows topology, which is simplest without IGWs (see Fig. 1), is induced by the action of Reynolds stresses due to the correlations of waves' velocity components [START_REF] Alvan | [END_REF] above the position of the extraction front. In the same way, if such a negative gradient is assumed in an upper main-sequence star (i.e. the convective core is rotating faster than the radiative envelope), retrograde waves are damped closer to the convective core border than prograde waves leading to an acceleration (or spin-up) of the envelope (e.g. Rogers et al. 2012). Finally, IGWs will be excited during all phases of stellar evolution including the Pre-Main-Sequence and advanced stages (Talon & Charbonnel 2008;Charbonnel et al. 2013). Therefore, it is necessary to take them into account in all stellar models and to improve their modelling.

Towards a coherent modelling of internal gravity waves

Excitation mechanisms in single stars

The treatment of the excitation of IGWs in stellar interiors is the first key ingredient that should be improved to obtain a quantitative and realistic estimation of the transport of AM in stars by IGWs.

On one hand, we should get a coherent and complete modelling of the highly non-linear excitation of IGWs by turbulent convection at the border between convection and radiation zones because of penetrative convection (the surfacic excitation) and in the bulk of convection zones (the volumetric excitation). First theoretical models for such processes were built once the interest of transport and mixing in stars by IGWs was identified and with the development of solar and stellar seismology (Press 1981;Goldreich & Kumar 1990;Garcia Lopez & Spruit 1991). However, these models suffered from the rough description of turbulence in convection zones and at their borders. These weaknesses then motivated cartesian numerical simulations of the dynamics of a convection zone above a given stratified layer to characterise the amplitude and the frequency spectrum of IGWs [START_REF] Hubbard | The Interior of Saturn[END_REF]Kiraga et al. 2003Kiraga et al. , 2005;;Dintrans et al. 2005). Then, first groundbreaking progresses were obtained in 2D equatorial set-up by the use of realistic stellar stratification (Rogers & Glatzmaier 2005, 2006;Rogers et al. 2008;2012). Next, IGWs excitation was also simulated in 2D and 3D part of a star (e.g. Mocák et al. 2011). Finally, we are now reaching a very high level of realism by using 3D global numerical simulations of the internal dynamics of different types of stars that give direct predictions for excited IGWs amplitude and spectrum (Browning et al. 2004;B r u net al. 2011;A l v a net al. 2012 and Fig. 3). Moreover, these simulations give realistic turbulent spectrum and time-correlation functions for semi-analytical models of the volumetric excitation in the bulk of convection zones (Belkacem et al. 2009;Samadi et al. 2010), which are used in helio and asteroseismology. In this context, it is important to point that as emphasised by Lecoanet & Quataert (2013), a careful treatment of the convection/radiation transition should be adopted. Moreover, to guide us laboratory experiments can also be used to quantify the excitation rate (e.g. Ansong & Sutherland 2010;Perrard et al. 2013).

On the other hand, as for IGWs stochastic excitation described just above, a precise prediction of IGW amplitudes induced by the κ-mechanism due to opacity bumps is necessary. Then, if the prediction of the instability strip (and of the related linear growth-rate) is now more or less under control without and with rotation (see e.g. [START_REF] Unno | Nonradial Oscillations of Stars Wade[END_REF]Lee & Baraffe 1995 respectively), it is now necessary to understand non-linear saturation of this mechanism. This task has been undertaken using both theoretical approach (Lee 2012) and direct numerical simulations (Gastine & Dintrans 2008a,b).

The couplings between IGWs and shear-induced turbulence: The critical layers

IGWs and the vertical shear instability are intrinsically coupled [START_REF] Drazin | Hydrodynamic Stability[END_REF]. Moreover as emphasized by (Goldreich & Nicholson 1989b), corotation layers, also called critical layers, constitute the only place where IGWs can transport a net amount of AM without any dissipative mechanism. This is the reason why [START_REF] Alvan | Proceedings of the Annual Meeting of the French Society of Astronomy and Astrophysics[END_REF]; [START_REF] Alvan | [END_REF] have undertaken the detailed study of the interaction between IGWs and the shear of the mean flow that occurs at co-rotation layers in stably stratified stellar radiation zones. Then, taking advantage of the work realised in the literature about atmospheric and oceanic fluids (e.g. Booker & Bretherton 1967;Lindzen & Barker 1985), they highlighted the similarities with such stellar regions and derived a complete approach adapted to the related case of deep spherical shells. Then, the use of spherical coordinates brings differences in the equations and make their resolution more complicated but the final results are comparable. [START_REF] Alvan | [END_REF] unravelled the intrinsic couplings between IGWs and the shear-induced instabilities and turbulence that can thus not be added linearly Fig. 4. The two possible configurations for critical layers in a low-mass star with an external convective envelope. r c is the position of the studied CL. Left: the case of a stable CL where IGWs are damped by radiative damping and because of the corotation resonance. Right: the case of an unstable CL where IGWs can be over-reflected/transmitted or can tunnel. 2δ is the thickness of the shear-unstable layer. (Taken from Alvan et al. 2013, courtesy Astronomy & Astrophysics.) as done previously in stellar evolution literature and they have emphasised the existence of two regimes where the interactions between IGWs and the shear at critical layers are strongly different (see Fig. 4):

• in the first regime, the fluid is stable and IGW amplitude is overdamped by the critical layer compared to the classical case where only radiative (and viscous) dampings are taken into account (see the left panel of Fig. 4);

• in the second regime, the fluid is shear-unstable and turbulent and the critical layer can act as a secondary excitation region. Indeed, an overreflection/transmission mechanism can occur. That is to say that the reflection coefficient R and the transmission one T are greater than 1. In this case, energy is taken from the unstable shear that increases the amplitude of an incident IGW (see the right panel of Fig. 4). In the case of simple reflection and transmission (|R| < 1and|T | < 1), they demonstrate the existence of IGWs tunneling through unstable regions as identified by Sutherlan Yewchuk ( 2004) in laboratory experiments and by Brown & Sutherland (2007); Nault & Sutherland (2007) in Geophysics.

Finally, this study clearly demonstrates that IGWs Reynolds stresses must only be taken into account in stellar models in regions where they are propagative and not in unstable layers where AM transport is ensured by turbulent stresses and the meridional circulation (e.g. Sect. 1.2.2).

The modification by rotation and magnetic fields

IGWs are low-frequency waves in comparison with acoustic waves. Therefore, in the case of rapid rotators, such as young low-mass stars and massive stars, and of magnetic stars, the Coriolis acceleration and the Lorentz force can strongly modify their propagation, dissipation and induced AM transport. Indeed, IGWs become in these cases gravito-inertial and magneto-gravito-inertial waves because of the combined action of the buoyancy force, the Coriolis acceleration and the magnetic f o r c e( s e eF i g . 5). This is the reason why the modification of their structure and dissipation by the (differential) rotation has been studied with taking into account the Coriolis acceleration (Lee & Saio 1997;Dintrans & Rieutord 2000;Mathis et al. 2008;Mathis 2009;Ballot et al. 2010). Then, depending on the ratio between excited frequencies (σ exc ) and the inertial frequency (2Ω, where Ω is the angular velocity), waves are propagating at all latitudes in the super-inertial regime (σ exc > 2Ω) while they are trapped in an equatorial belt below a given critical latitude in the sub-inertial one (σ exc < 2Ω) (see Fig. 6). This modifies the geometrical transmission of the energy coming from turbulent convective flows to IGWs, which is reduced in the sub-inertial regime if we consider a given turbulent source that does not depend on rotation. Moreover, the thermal diffusion is enhanced and waves are thus damped closer to their excitation region. This leads, combined with the equatorial trapping, to a modification of the efficiency of the AM transport by gravito-inertial waves as soon as the ratio σ exc /2Ω diminishes (Pantillon et al. 2007;Mathis et al. 2008). Moreover, the bottom of convective envelopes in solar-type stars and the top of convective cores in intermediate-mass and massive stars are the seat of dynamo action. Indeed, the tachocline is believed to be the place of the large-scale toroidal magnetic field storage in the Sun (Browning et al. 2006). Moreover, as it has been discussed in Section 1.2.3, stellar radiation zones may host fossil magnetic fields. We thus have to also take into account the magnetic field's action on IGWs dynamics. They become magneto-gravito-inertial waves because of the three restoring forces: i.e. the buoyancy force, the Coriolis acceleration and the Lorentz force. The first studies have been devoted to the case of IGWs dynamics with an axisymmetric toroidal field (e.g. Schatzman 1993a; Kim & Mac Gregor 2003;Rogers & Mac Gregor 2011;M a t h i s&d eB r y e2011a). In this case, waves become vertically trapped as soon as 1 -m 2 ω 2 A /σ 2 exc < 0, where we have introduced the Alfvén frequency ω A . Moreover, IGWs are submitted to the same equatorial trapping phenomenon that in the purely hydrodynamical case of gravito-inertial waves. The main difference in the magnetic case is that this latitudinal trapping due to the combined action of the Coriolis acceleration and the Lorentz force is different for pro-and retrograde waves because of the magnetic force. Then, the horizontal trapping of prograde waves is stronger than those of retrograde waves. Furthermore, IGWs radiative damping becomes stronger when magnetic field amplitude increases. The efficiency of the induced tranport of AM is thus a function of the rotation and of the magnetic field amplitude (see Fig. 6), which is strongly modified as soon as vertical and horizontal trappings modify waves' dynamics, with a net bias in favor of retrograde waves in the case of purely azimuthal axisymmetric magnetic fields (Mathis & de Brye 2012).

Couplings with stellar environment

As this has been emphasised in previous sections, torques applied at stellar surfaces modify internal transport processes: for example they can drive the large-scale meridional circulation (e.g. Zahn 1992) and feed IGWs induced transport of AM. Therefore, we must get a coherent physical modelling of the interaction of stars with their environment when their rotational evolution is studied (e.g. Bouvier 2008). First, stellar winds must be carefully examined as a function of the stars' rotation rate and of their magnetic field topology and amplitude (e.g. For this latters, the equilibrium tide associated to the hydrostatic adjustment of the star to the tidal excitation leads to a net torque applied on the external convective envelope of solar-type stars (Zahn 1966b;R e m u set al. 2012a)t h a t modifies their rotational evolution (Zahn & Bouchet 1989).

Moreover, IGWs are also excited by the tidal potential at the border of convective regions and are able to transport AM in the same way that those excited by the turbulent penetrative convection: this is the dynamical tide (see e.g. Zahn 1975; Ogilvie & Lin 2007), which has now been detected in KOI 54 by Kepler (Welsh et al. 2011). Then, tidal IGWs dissipation by radiative damping in stellar radiation zones is a key actor to drive systems' circularisation and synchronisation and the damping of the stellar and planetary obliquities and of the orbital inclination. Moreover, as in single stars, the Coriolis acceleration action is determinant because of the so-called resonance locking states where strong tidal evolution occurs (Witte & Savonije 1999b;2002). Moreover, if large amplitude IGWs are excited by tides, they can lead to the creation of critical layers at the center of solar-type stars and below the surface of intermediate-mass and massive stars and to an efficient deposit of AM (Barker & Ogilvie 2010;B a r k e r2011; Rogers et al. 2012). Finally, parametric instabilities can take place that would modify tidal dissipation because of associated non-linear mechanisms (Weinberg et al. 2012). These processes will be detailed in Section 2.

Conclusions

In the first part of this lecture, we have recalled that a transport of AM by magnetic fields and/or IGWs is needed for low-mass main-sequence and evolved stars and for active massive stars. Then, we have presented the way in which transport of momentum is modified by IGWs leading to a net extraction of AM from low-mass stars' radiative core and to the possibility to accelerate the external radiative envelope of massive stars. Furthermore, we have summarised the different progresses which have been obtained on IGWs excitation and on their interaction with shear instabilities and turbulence, rotation and magnetic fields. IGWs action and interactions are summarised in Figure 7.

Therefore, we clearly show that it becomes necessary to take IGWs to get a precise and complete picture of the dynamical evolution of stars in the whole Hertzsprung-Russell diagram. To reach this objective, constrains coming from helioseismology and asteroseismology will be our best guide. For this reason we can cite the title of a paper published by E. Schatzman in 1996: "Do Not Forget Gravity Waves" (Schatzman 1996).

2 Tidal interactions and dissipation mechanisms in star-planet systems 2.1 The dissipation of tidal kinetic energy: The driver of systems evolution Tidal interactions can be broken down in two steps. First, if we adopt an ideal adiabatic view of the action of the tidal potential exerted by the secondary on the primary, this latter becomes elongated along the line of centers. However, dissipative processes that convert the tidal kinetic energy into heat, such as viscous friction and heat diffusion, have to be taken into account (see Fig. 8). Then, the response of the studied body to the tidal excitation presents a delay (the tidal lag), which translates into the tidal angle δ between the tidal bulge and the line of centers. Those tidal delay and angle are thus directly related to the dissipative mechanisms and the associated quality factor Q =1 / sin (2δ) (Goldreich & Soter 1966) and their dependence on the tidal frequency. A high value of Q corresponds to a weak tidal dissipation and a small angle δ and a weak value of Q corresponds to a strong dissipation and a larger δ. Therefore, to predict the fate of a binary system, one has to identify and to model the dissipative processes achieving the conversion of kinetic energy into thermal energy that take place in fluid and solid layers and at their interfaces, from which one can derive the characteristic times of circularisation, synchronisation and spins' alignment or of spiraling onto the central body depending on the properties of the system (Hut 1980).

Tidal dissipation in fluid bodies

While stars are fluid bodies, planets host gaseous and liquid layers: the deep envelopes of giant planets, as well as the internal core, the atmosphere and the possible ocean of telluric ones (see Figs. 9 & 10). Therefore, one has to obtain a complete understanding of flows that are tidally excited and dissipated in such regions in planetary systems.

Type of fluid tides

Two types of tides operate in stars and in fluid planetary layers: the equilibrium and dynamical tides. On one hand, the equilibrium tide designates the large-scale flow induced by the hydrostatic adjustment of studied fluid layers in response to the gravitational force exerted by the companion (Zahn 1966a,c). On the other hand, the dynamical tide corresponds to the fluid eigenmodes that are excited by the tidal potential. Let us now detail the different types of eigenmodes that should be studied in stellar and fluid planetary regions. First, if Figure 11 is considered, four characteristic frequencies are introduced: the Alfvén frequency (ω A = B √ μρr sin θ ,w h e r eB is the field amplitude and μ is the magnetic permeability), the inertial frequency (2Ω), the Brunt-Väisäla frequency (N ), and the Lamb frequency (f L ). These delimit the frequency domain of corresponding Alfvén waves (ω<ω A ), inertial waves (ω<2Ω), internal gravity waves (for which ω<N ; see Sect. 1), and acoustic waves (ω>f L ); these are respectively driven by the magnetic tension force, the Coriolis acceleration, the buoyancy force and the compressibility of the studied layers. If we now focus on low-frequency waves, inertial waves are propagating in convective regions while internal waves are propagating in stably stratified regions. For these latters, if considered frequencies are of the same order of magnitude that the Alvén and the inertial frequencies, these become gravito-inertial waves if we add the action of the Coriolis acceleration to the one of buoyancy and magneto-gravito-inertial waves if the magnetic field is taken into account. In this picture, tidal excitation is mostly efficient for low-frequency eigenmodes, thus for inertial and internal waves. Moreover, for acoustic waves, which are high-frequency waves, the action of tides is only a perturbation. Therefore, one has to focus on inertial and on gravito-inertial waves to study the dynamical tide respectively in convective and in stably stratified regions.

Next, dissipative processes that convert kinetic energy of tidally excited fluid velocities into heat have to be identified.

First, stellar and planetary convective layers host strong turbulent flows because of the high value of the Reynolds number in such celestial bodies. In such regions, the action of turbulence on the tidal flows (the equilibrium tide and the dynamical tide, i.e. the inertial waves excited by the tidal potential) can be modeled as a viscous force with a turbulent viscosity coefficient (see for example Zahn 1966c;Goldreich & Keeley 1977). This implicitely assumes that the respective length scales of tidal and convective flows allow to distinguish one from the other. Such rough modelling has now been confirmed using direct numerical simulations of the interaction between an highly turbulent convection and a tidal velocity (see for example Penev et al. 2009a,b note that in these works, the prescription given by Zahn 1966c where the turbulent viscosity scales linearly with the tidal period (and thus the inverse of the tidal frequency) has been confirmed). So, in convective regions, the kinetic energy of tidal flows is dissipated into heat because of the turbulent viscous friction.

Next, in stably stratified stellar and planetary regions, the dynamical tide (i.e. the gravito-inertial waves) is dissipated through viscous and thermal diffusions (see for example Goldreich & Nicholson 1989a,c). Then, the ratio between the viscous and the radiative damping is govern by the Prandtl number (Pr = ν/K,w h e r eν and K are respectively the viscosity and the thermal diffusivity). Moreover, Zahn (1966d) has demonstrated that the dissipation of the equilibrium tide in stellar radiation zones can be neglected.

Finally, planetary interiors host solid/fluid interfaces. There, viscous friction occurs that contributes to the dissipation of tides kinetic energy.

All of this underlines the importance of the internal structure of the studied bodies, because each stellar or planetary layer dissipates the kinetic energy in function of its fluid or solid nature and of its stability with respect to the convective instability for fluid regions. A summary of the dissipation mechanisms that contribute to the tidal friction in liquid and gaseous regions is given in Figure 12.

The fluid equilibrium tide

Let us first consider the equilibrium tide. As it has been explained in previous sections, the studied component adjusts in a quasi-hydrostatic way when it is submitted to the tidal potential exerted by the companion. Then, a large-scale Fig. 12. Dissipative processes acting in solar-type stars (left) and in massive stars and in Jupiter (and Saturn) -type giant planets (right). CE, RE, CC, RC are respectively convective and radiative envelopes and cores. Note that for giant planets, a rocky/icy core could exist Guillot (1999Guillot ( , 2005)), Baraffe (2005), Fortney & Nettelmann (2010).

flow in phase with the tidal potential is excited as a response to such structural adjustment with an amplitude scaling with the tidal frequency; this is the equilibrium tide (see for example Zahn 1966aZahn , 1977;;Ivanov & Papaloizou 2004;R e m u s et al. 2012a). Since the tidal force is derived from a potential, the density is constant on an isobar, which is also an equipotential of the total potential (the sum of the self-gravitation potential and of the tidal one). Then, by definition of the equilibrium tide, the tidal deformation and the related structural variables (the total gravific potential and the density) and the equilibrium tide velocity field are time-independent in a frame rotating with the studied Fourier component of the tidal potential; because of that property, the equilibrium tide velocity field is divergence-free (see Remus et al. 2012a) contrary to the claim of Scharlemann (1981); Eggleton et al.( 1998). It is also important to point that since this velocity field represented in Figure 13 verifies the momentum equation in this rotating frame, the Coriolis acceleration must be taken into account in its derivation (the equilibrium tide velocity field has thus both a poloidal and a toroidal components). Next, the view of the equilibrium tide we gave above is an adiabatic view, and we must now intoduce the associated friction mechanism. For the equilibrium tide this is the turbulent friction in convective regions that will act to convert its kinetic energy into heat (note that Zahn 1966d has demonstrated that its dissipation in stably stratified regions is negligible).

Then, the interaction between turbulent convective flows and the equilibrium tide velocity field has to be examined. From now on, two main assumptions are made: first, we consider that the respective scales of the turbulent convection and the equilibrium tide are different enough to be separated; next, we assume that the action of the turbulent convection on the tidal flow can be modeled as a viscous force, where the used viscosity is a "turbulent viscosity" which is enhanced compared to the molecular viscosity of the plasma. Since the adiabatic equilibrium tide has both poloidal and toroidal components, so does the related viscous force that sustains a secondary toroidal flow, called the convective or dissipative equilibrium tide, in quadrature with the tidal potential. By redistributing the density, this velocity field leads to a dissipative perturbation of the gravific potential that drives the secular evolution of the orbit and the spins of the system components.

To describe the dissipation of the kinetic energy of the adiabatic equilibrium tide, namely its amplitude and its dependence on the tidal frequency, the key physical ingredient is the assumed prescription for the turbulent viscosity coefficient. Then, two different regimes can be drawn: the "slow tide" and "fast tide" regimes. In the first one, the orbital period of the perturber is longer that the characteristic convective turn-over time. Then, the turbulent friction can be efficient to dissipate the kinetic energy into heat. Conversely, in the fast tide regime, the orbital period is shorter that the characteristic convective turn-over time and the turbulent friction losses part of its efficiency to convert the kinetic energy into heat, leading to a saturation of the associated energy dissipation. The way in which the dissipation becomes less efficient when the tidal period becomes shorter remains one of the unsolved question in the treatment of the equilibrium tide. Two main prescriptions have been given today in the litterature: those by Zahn (1966cZahn ( , 1989) ) and by Goldreich & Keeley (1977). In the first one the turbulent viscosity scales linearly with the tidal period while in the second one it scales as the squared tidal period. The most efficient way to probe such prescriptions on the action of turbulence is then to used three-dimensional numerical simulations of highly turbulent convective flows submitted to a periodic forcing, which is often modeled as a shear that oscillates with time. The most recent numerical simulations have been achieved with such set-up in cartesian coordinates (Penev et al. 2009a,b;Penev & Sasselov 2011) that tends to confirm the prescription by Zahn (1966c) (See also Ogilvie & Lesur 2012). In a near future, more simulations have to be computed in order to reach flows that are more turbulent and to take into account the spherical (ellipsoidal) geometry of the problem. For the moment, one can at least conclude that the viscous dissipation of the equilibrium tide varies as the product of the tidal frequency (on which depends the velocity field) with the frequency dependence of the turbulent viscosity. In the case of the linear prescription given by Zahn (1966c), we shall note that in the fast tide case the turbulent viscosity scales as the inverse of the tidal frequency, leading to a constant tidal dissipation, while in the slow tide regime the turbulent viscosity is constant, so that the dissipation scales here with the tidal frequency (see in Fig. 14).

To conclude this part on the equilibrium tide, we shall note that in future works both differential rotation and magnetic fields have to be taken into account since they modify at the same time the equilibrium tide velocity field, the convective flows and the associated turbulence properties.

The fluid dynamical tide: Inertial and gravito-inertial waves

Inertial waves

Once the equilibrium tide has been studied, it is then necessary to focus on the dynamical tide, i.e. the eigenmodes of the studied body, which are excited by the tidal potential. To achieve this aim, let us first consider convective zones in stellar

0.1 1 1 1 0.1 0.1 t k 2 Q t conv / P tide
Fig. 14. The two regimes of the turbulent dissipation of the equilibrium tide according to the prescription by Zahn (1966c). As long as the local convective turn-over time remains shorter than the tidal period (t conv <P tide ), the turbulent viscosity ν t (in black dashed line) is independent of the tidal frequency, and the inverse quality factor k 2 /Q (in red continuous line) varies proportionally to the tidal frequency (σ l ) (so does also the tidal lag angle). When t conv >P tide , ν t varies proportionally to the tidal period, whereas k 2 /Q does no longer depend on the tidal frequency. ν t and k 2 /Q have been scaled by the value they take respectively for t conv /P tide → 0 and →∞(taken from Remus et al. 2012a, courtesy Astronomy & Astrophysics). and planetary interiors. In those regions, if we neglect magnetic field, two type of waves are propagating: the acoustic and the inertial waves. However, as it has been explained above acoustic waves are high-frequency waves and are thus only weakly perturbed by the tidal potential. This is not the case of inertial waves for which the Coriolis acceleration is the restoring force. Then, inertial waves can be efficiently coupled with the tidal potential (see Ogilvie & Lin 2004;Papaloizou & Ivanov 2005;W u2005a,b;Ogilvie & Lin 2007;Ogilvie 2009;Ivanov & Papaloizou 2010;Papaloizou & Ivanov 2010;Rieutord & Valdettaro 2010;Ogilvie 2013). Such coupling can then lead to an important tidal dissipation in the convective envelopes of low-mass stars and giant planets and in the convective cores of stars and telluric planets. Because of inertial waves properties, the necessary condition to get such dissipation is that the tidal frequency is such that |σ T |∈[0, 2Ω], where from now on σ T is the tidal frequency.

Let us now examine the properties of tidally excited inertial waves. First, two configurations can occur in stellar and planetary interiors; first, the tidal potential can be coupled with inertial waves that propagate in a full sphere (case of an entirely convective star or fluid planet or of a stellar convective core); next, inertial waves can propagate between concentric spheres with stress-free or no-slip boundary conditions depending on if we are studying stellar or planetary interiors (case of the convective envelopes of low-mass stars and of giant planets if those have an heavy element rocky or icy core). Because of the cylindrical geometry related to the Coriolis accelaration, boundary conditions then strongly influence the excited flows and the related dissipation.

The "Full sphere" configuration

The first case of entirely convective sphere has been studied for example by Wu (2005a,b) (Fig. 15). In this work, the original solution for inertial waves in a full sphere with a constant density derived by Bryan (1889) has been generalised to the stratified case. This leads to global modes which then couple to the tidal potential as in the academic case of a forced oscillator. To study the dissipation of their kinetic energy, the same assumptions that in the case of the equilibrium tide (i.e. the spatial scales separation between the convective flows and the tidal inertial waves and the modelling of the friction using a viscous force with a turbulent viscosity) are assumed. Numerous resonances between eigenmodes and the tidal potential are obtained and identified. Then, the main important properties to study is the related amplitude of the dissipation and its dependance on the tidal frequency. First, the tidal dissipation can be increased by several order of magnitudes when a resonance is encountered compared to the case of the equilibrium tide. Next, the main difference with the equilibrium tide is the strong dependance on the tidal frequency of the tidal dissipation because of the resonances.

The cored configuration

The second case of inertial waves propagating in spherical shells has been studied in Ogilvie & Lin (2004) for the planetary case and in Ogilvie & Lin (2007)f o rt h e stellar case. Then, because of the conflict between the spherical geometry of the problem and the one related to the Coriolis acceleration, the propagation of inertial waves becomes more complex (Fig. 16). First, inertial waves propagate along characteristic rays that are inclined to the rotation axis at a certain angle, which depends on the wave frequency. This angle is necessary preserved in reflections of the waves from boundaries so that a beam is foucused of defocused in such a reflection. When propagating in such spherical annulus (in opposite to the first full sphere case), inertial waves are focused onto attractors where an intense dissipation occurs (Rieutord & Valdettaro 1997;R i e u t o r det al. 2001, 2002;Ogilvie 2009). Then, Ogilvie (2005) demonstrated mathematically that the mean dissipation rate associated with waves attractors (in a simplified wave equation) becomes independent of the viscosity in the limit of very small Ekman number (i.e. for very small viscosity). This constitutes a remarkable behaviour compared for example to the case of the equilibrium tide dissipation, which is directly proportional to the viscosity. In the complete global modelling by Ogilvie & Lin (2004), a similar behaviour is observed in the limit of small viscosity and obtained solutions indicate that the dissipation is typically concentrated along the rays that emanate from the critical latitude on the inner boundary (see also Rieutord & Valdettaro 2010). Goodman & Lackner (2009) have proposed a physical interpretation of such phenomena: using WKB methods, they demonstrate the production of short inertial waves by scattering of the equilibrium tide off the core at critical latitudes. The tidal dissipation rate associated with these waves scales as the fifth power of the core radius. They also find that even if the core of rock or ice is unlikely to be rigid, Ogilvie and Lin's mechanism should still operate if the core is substantially denser than its immediate surroundings.

As a partial conclusion, we must point that the viscous dissipation of inertial waves are one of the most important processes to take into account in stellar and in planetary interiors. Let us now draw some perspectives on what should be done to improve the modelling of such processes. First, as in the case of the equilibrium tide, the differential rotation and the magnetic field have to be taken into account. First, convective flows are those that establish the differential Fig. 16. A) high-resolution calculation (E =10 -9 ) of the tidal response of a uniformly rotating planet with an internal core. The rms velocity of the total tide (equilibrium and dynamical) is plotted in a meridional slice through the convective region (the velocity scale is linear, black representing zero). Attractors with associated inertial wave beams can be identified. The forcing tidal frequency is chosen to be near the peak of an inertial mode resonance. (Taken from Ogilvie & Lin 2004, courtesy The Astrophysical Journal.) B) production of short inertial waves by scattering of the equilibrium tide off the core at critical latitudes to explain results obtained by Ogilvie & Lin (2004) rotation that depends both on radius and on latitude in convective regions (Brun &T oomre2002). Next such regions host dynamo generated magnetic fields [START_REF] Brun | [END_REF]) that are generated because of the simultaneous action of differential rotation and of convective turbulence that are themselves modified by the magnetic field because of the Lorentz force feed-back in the momentum equation. Then, inertial waves propagation and dissipation will be modified both by differential rotation (Baruteau & Rieutord 2013) and magnetic fields. Indeed, the dissipation will be changed by corotation resonances between inertial waves and the sheared rotation (see also the case of gravito-inertial waves) and by the Ohmic heating that constitute supplementary dissipation sources.

Inertial waves instabilities

Finally, it is important to point the possibility of tidal inertial waves' related instabilities. In the case of inertial waves, main instabilities can come for the interaction with differential rotation (see also the case of gravito-inertial waves) and from the tidal elliptic instability. This latter corresponds to the astrophysical version of the generic elliptical instability, which affects all rotating fuids with elliptically-deformed streamlines (Kerswell 2002;Lacaze et al. 2004Lacaze et al. , 2005;;LeBars et al. 2007). In the astrophysical case, the origin of such elliptic instability is a resonnance between inertial waves in rotating stars and planets and the tidal wave i.e. the underlying strain field responsible for the elliptic deformation [START_REF] Waleffe | [END_REF]). This instability is able to generate and sustain large-scale flows (for example the so-called spin-over mode) that superpose to basic flows such that differential rotation or convection in planetary and in stellar interiors (see Fig. 17). Then, as in the case of inertial waves, viscous forces can act to dissipate the generated kinetic energy that leads to potential important evolution of the considered system. Recent studies of the tidal elliptic instability have been recently achieved in the context of binary stars (Le Bars et al. 2010) ). In the case of convective regions, the elliptic instability can thus develop with a growth-rate that diminishes with the intensity of the convection; thus, the flow generated with the tidal instability can superpose to the convective one. Next, such tidal flow may play an important role in the induction of a magnetic field leading to a "tidal dynamo" (Lacaze et al. 2006;Herreman et al. 2009) in planetary interiors and may be in stellar ones. This last point constitutes one of the most important question to examine in a near future to see a possible impact of tidal interactions on celestial bodies magnetic activity. Finally, we must point that the dissipation related to the elliptic instability depends on boundary conditions that are applied (see Sect. 3.4), that can lead to inportant differences between no-slip boundary conditions (in planetary cores of telluric planets and at the interface between a central core and a surrounding fluid envelope in giant planets) and stress-free conditions (as at giant planets' and stars' surfaces.)

Internal waves

Let us now consider the case of stably stratified zones in stellar and planetary interiors. As it has been described in Sect. 2.2.1, gravity (and gravito-inertial waves) are propagating in such regions. These are excited at the border with adjacent convective regions both by turbulent movements, and in the case where there is a close companion, by pressure fluctuations induced by tidally excited inertial waves (for example those of inertial waves attractors in the case of external convective envelopes). Then, in the case of binary or multiple systems, gravitoinertial waves will be forced in stellar radiation zones and in stably stratified planetary layers (for example in non-convective layers just below the surface of giant planets in our solar system or of giant extra-solar planets, where those layers can be created because of the heating of the surface by the close star, and in stably stratified regions in telluric planets). Then, as in the case of convective regions, the displacement is the sum of the equilibrium tide and of the dynamical tide (i.e. tidal gravito-inertial waves). Let us now consider the properties of the tidal dissipation related to gravito-inertial waves. We must first recall that the main dissipative mechanism acting on such waves is the thermal diffusion. Then, as in the case of inertial waves, the tidal dissipation can be increased by several orders of magnitude compared to the one of the equilibrium tide, in particular in resonances that occurs for gravity waves as this has been shown by Zahn (1975); Rocca (1982Rocca ( , 1987Rocca ( , 1989)); Savonije et al.( 1995); Savonije & Papaloizou (1997); Papaloizou & Savonije (1997); Ogilvie & Lin (2004, 2007). Moreover, because of such increase of the tidal dissipation during resonances, its behaviour is highly dependant on the tidal frequency (see Fig. 19).

Let us now discuss the modification of gravito-inertial waves propagation by rotation and magnetic field. First, for the rotation, gravito-inertial waves propagation will strongly depends on the value of the tidal frequency compared to the inertial frequency (2Ω) where we recall that Ω is the rotation of the studied body. First, in the super-inertial regime (σ T > 2Ω), gravito-inertial waves are propagating in the whole sphere. However, in the sub-inertial one (σ T < 2Ω), waves become trapped in an equatorial region, propagating only above a given so-called critical latitude (Mathis et al. 2008;Mathis 2009) (see Fig. 18 and Sect. 1.3.2). Such modification of gravito-inertial propagation is very important for their coupling with tidally-induced displacements in adjacent convective regions. Moreover, as this has been discovered by Witte & Savonije (1999a,c, 2001); Savonije & Witte (2002); Witte & Savonije (2002), the Coriolis acceleration can lead to the so-called "tapping in resonance" where retrograde and prograde waves exert respectively a Fig. 18. We consider here a solar-twin star with an external convective envelope and a radiative core. The rotation profile in this latter is flat (i.e. Ω=Ω m = 430 nHz) for r ∈ [0.2 R, 0.7 R], where R is the stellar radius. In the central region, Ω increases until Ω=5Ω m in the center. Colored regions correspond to those where gravito-inertial waves are propagative while black regions correspond to "dead" zones for waves propagation. We choose three frequencies (σ = {500, 750, 1000}nHz) that shows that for frequency below 2Ω m equatorial trapping phenomena appear. The white line corresponds to the critical surface (the critical latitude in the case of uniform rotation) at the level of which the wave propagation regime changes. The central region is always a non-propagative region because of the central rapid rotation.

negative and an positive torques that act to block the studied system in a resonant state where the tidal dissipation is very efficient. Then, the tidal dissipation is also dependent on the rotation rate and the associated Coriolis acceleration. Next, for example in the case of solar-type stars, gravito-inertial waves propagation is modified by the presence of magnetic field, for example at the bottom of the convective envelope. Then, waves becomes magneto-gravito-inertial waves where the Lorenz force has to be taken into account. First, waves excited with frequencies close to the Alvén frequency will be vertically trapped. Then, as in the gravito-inertial case, an equatorial trapping can occur depending both on ω A and 2Ω (see Sect. 1.3.2).

As in the case of inertial waves, we can thus conclude that the tidal dissipation will be strongly dependent on the tidal frequency, on the rotation and of potential magnetic fields.

Interactions with shear flows and instabilities

Finally, let us discuss the interaction between differential rotation and the dynamical tide in stably stratified layers. First, as this has been explained by Goldreich & Nicholson (1989a,c), gravity waves can transfer angular momentum only if these are damped by a dissipative mechanism (which is mostly the thermal diffusion) or if they reach corotation resonances during their propagation (i.e. if we consider Fig. 19. A) tidal dissipation by gravito-inertial waves in a solar-type star and the corresponding quality factor as a function of the tidal frequency computed by Ogilvie & Lin (2007) (courtesy the Astrophysical Journal). B) breaking of large-amplitude tidally excited internal waves at the center of a solar-type star computed by Barker & Ogilvie (2010). There, waves deposit their angular momentum that can accelerates the center of the star and makes evolve the orbit of the planetary companion. (Courtesy Monthly Notices of the Royal Astronomical Society.) a "shellular" rotation that depends only on r,r a d i u sw h e r eΩ( r c ) is proportional to the tidal frequency). Let us first examine the thermal diffusion effect: an important point is that the thermal damping depends on the prograde or retrograde behaviour of the wave because of the Doppler shift.

Then, in the case of a differentially rotating body, the synchronisation of each layer will progress from the surface to deeper regions (Goldreich & Nicholson 1989a). Let us now focus on corotation regions, i.e. on critical layers which have been described in Section 1.3.2. There, these are strong interactions between internal waves and the shear of the differential rotation: first, if the studied layer is stable with respect to the shear instabilities, waves deposit their angular momentum, the damping rate being dependant on the so-called Richardson number, which compares the strength of the stabilisation by the stratification and the destabilisation by the shear gradient; then, if the layer is already turbulent, internal waves can be reflected and transmitted by such layers with an amplitude greater than their initial one because waves take energy from the turbulent flows. In this context, this is important to study the possible instabilities that could affect internal waves dynamics. First, if waves are excited with a large amplitude, waves will break and then, these could overturn the stable stratification (see for example Barker & Ogilvie (2010, 2011) and Fig. 19 for dynamical tide dynamics at the center of solar-type star). Then, even for a weak amplitude, internal waves can undergo parametric instabilities where a "parent" wave give birth to "daughter" waves that could be then also dissipated (Weinberg et al. 2012). Thus, as in the case of inertial waves, the interaction with differential rotation as well as their own instabilities could strongly modify the value of the tidal dissipation.

Tidal dissipation in rocky or icy planetary regions

As this has been shown in previous sections, the tidal potential is able to excite several types of velocity fields in fluid stellar and planetary layers leading to a possibly high tidal dissipation, which is function of the tidal frequency both for the equilibrium and the dynamical tides. However, planets (and their associated natural satellites) are composed of both fluid and solid layers, and tidal dissipation in these latters and at the fluid-solid interfaces should also be treated.

In this sense, the treatment of what is often called "the bodily tide", in other word the solid tide, has been one of the first studies of tidal dissipation using continuum mechanics (see for example [START_REF] Love | Some Problems of Geodynamics[END_REF]. These studies were of course motivated by the Earth case where tidal interactions with both the Moon and the Sun have to be taken into account (Neron de Surgy & Laskar 1997). In solid layers, tidal physical mechanisms are similar to those occuring in fluids. First, a solid equilibrium tide is generated that consists on a permanent large-scale displacement (with a zero velocity field that constitutes a difference with the fluid case). In an adiabatic modelling, this displacement is allowed by the elasticity of the material and directed along the line of centers (see the right panel of Fig. 20). However, as in the fluid case, solid layers host dissipation because of their anelasticity and the tidal energy is dissipated into heat, leading at the same time to internal heating, to a net applied torque, and to a small delay between the tidal bulge and the line of centers. This anelasticity, which is often modeled as a viscous behaviour that adds to the elasticity as in the Maxwell's body model, depends on the intrisic properties of the considered material (for example silicates or ices), which are described by its rheology (e.g. Henning et al. 2009). This latter is given by the constitutive equation that links the strain tensor ( ) to the stress tensor (σ). Under small deformations (as tidal perturbations), it is customary to assume that the strain-stress relationship is linear, and materials that obey this law are called Hookean materials. If we also assume that they are isotropic and incompressible, the Hooke's law states that: σ = μ (σ T ) ,w h e r e μ is the complex shear modulus (also called rigidity, which measures the stiffness of the material) which depends on the tidal frequency (σ T ). Its real and imaginary parts represent respectively the energy storage and the energy losses of the system. Moreover, thanks to the correspondence principle (Biot 1954), one can calculate the tidal dissipation and the associated quality factor Q for any linear rheology.

Such type of computation has already been performed for the Moon (Peale & Cassen 1978), for rocky core of giant planets (Dermott 1979), for icy natural satellites (see for example Tobie et al. 2005), and for telluric extrasolar planets as Earth-like planets or Super-Earths (Henning et al. 2009

;E f r o i m s k y2012).
Such type of solid tide occurs for example in a two-layer planet with an internal rocky part and an external fluid envelope as studied by Dermott (1979) and Remus et al.( 2012b) (see Fig. 20). This corresponds to the cases of a telluric planet with an external ocean or atmosphere or a gaseous or icy giant planet with a potential rocky/icy core borned during the planetary formation (Guillot 1999(Guillot , 2005)). According to such models, the resulting solid tidal dissipation can reach values greater by several orders of magnitude than those due to fluid tidal velocities described in previous sections, for realistic values of viscoelastic parameters (see Figs. 21 & 22 for the solid cores of Jupiter-and Saturn-like planets). Observational measurements of tidal dissipation in our Solar system, from astrometry for example (see Lainey et al. 2009 for Jupiter, andLainey et al. 2012 for Saturn confirmed by realistic scenarii of natural satellites formation (Charnoz et al. 2011)), provide precious constraints to discriminate one tidal mechanism from an other. Indeed, each process has its own characteristics: for example, the study of the tidal frequency-dependence of solid dissipation shows a smooth behaviour (Fig. 22) compared to the case of inertial and gravito-inertial waves where the dissipation varies of several orders of magnitude (cf. Sect. 2.2). In the same time, some other behaviors can be shared by different mechanisms as the sensitivity to the size of the solid cores that have a similar influence on the core's anelastic tidal dissipation and on the one by tidal inertial waves. This shows how this becomes crucial to get constraints on the size of the rocky core of giant planets from observations and theoretical predictions. Moreover, the obtained large tidal dissipation and its dependence on the internal structure and rheology shows the strong need to go beyond phenomenological prescriptions (Efroimsky & Lainey 2007). 2012) for Saturn also needed by Charnoz et al.( 2011) to form its mid-sized satellites. We recall the values of R p = {10.97, 9.14} (in units of R ⊕ p , the Earth radius), M p = {317.8, 95.16} (in units of M ⊕ p , the Earth mass), and M c = {6.41, 18.65}×M ⊕ p (see respectively Guillot 1999;[START_REF] Hubbard | The Interior of Saturn[END_REF]. Moreover, we have followed Gavrilov & Zharkov (1977)

Boundary layers

As this has been shown previously, tidal interactions excite flows (i.e. the equilibrium tide, the dynamical tide, and fluid movements that result from their instabilities). Then, depending on the internal structure of the studied body, boundary conditions, and particularly at the surface and near solid/fluid interfaces (in planetary interiors) should be examined carefully. Indeed, strongly sheared boundary Note that the curves are represented with a logarithmic scale. The red and blue dotted lines correspond to the mean value of Qeff = {(3.56 ± 0.56) × 10 4 , (1.682 ± 0.540) × 10 3 } (for Jupiter and Saturn respectively) determined by Lainey et al.( 2009Lainey et al.( , 2012)). Their zone of uncertainty is also represented in the corresponding color. The blue dots correspond to the values (obtained by Lainey et al. 2012) of the dissipation induced in Saturn by Enceladus, Thetys, Dione and Rhea with their respective error bars. We recall the values of R p = {10.97, 9.14} (in units of R ⊕ p , the Earth radius), M p = {317.8, 95.16} (in units of M ⊕ p , the Earth mass), M c = {6.41, 18.65}×M ⊕ p ,a n d R c = {0.15, 0.26} R p . We take for the viscoelastic parameters G = {2.73, 6.51}× 10 10 (Pa), and η = {8.65 × 10 13 , 2.50 × 10 14 } (Pa • s) for Jupiter and Saturn respectively. (Adapted from Remus et al. 2012b;courtesy Astronomy & Astrophysics.) flows can develop there, that may lead to a strong dissipation. Let us first consider the case of the surface of stars or of planets. Tassoul & Tassoul (1992a,b, 1997) have proposed that Eckman boundary layers take place and lead to an important viscous dissipation. However, following Rieutord (1992); Rieutord & Zahn (1997), we must point that such layers constitute free surfaces with stress-free boundary conditions for the velocity field. Therefore, the dissipation related to associated boundary layers will be very weak. If we now consider solid/fluid interfaces, we are in the case of no-slip boundary conditions that correspond to the classical Eckman boundary layers (see Fig. 23) where a strong viscous dissipation may occur, particularly if the studied region is turbulent. As a partial conclusion on boundary flows, one has thus to remember that solid/fluid regions (at the top of rocky/icy cores of giant planets and at boundaries of liquid cores in telluric planets) may host strong viscous dissipation while this will not be the case below the surface of stars or planets with a fluid envelope. At solid/fluid boundaries, we must also point that couplings between the fluid dynamical tide (inertial or gravito-inertial) and the (an)-elastic tide in solid regions must be studied in a near future since these may modify the related tidal dissipation.

Hierarchy between dissipative physical processes and the associated obtained states

In this review, we have tried to give a complete picture of dissipative processes acting on tidally excited velocity fields in stellar and in planetary interiors. It is thus interesting at that point to draw a hierarchy between the intensity of the related dissipations. First, for fluid regions, dynamical tides (inertial and gravito-inertial waves) as well as related instabilities (the elliptic instability for inertial waves and the convective and parametric instabilities fo internal waves) lead to a stronger dissipation that can dominate by several order of magnitude the one associated to the equilibrium tide. Next, if we study planetary interiors, we have isolated that tidal dissipation in rocky/icy regions can dominate the one in fluid regions depending on their respective size. Remember also, that each type of tidal dissipation have a different dependence on the tidal frequency that can be constrained by observations to unravel the action of the different physical mechanisms. Finally, it is important to recall that once one have identified all the dissipation processes and derived the associated torques, equilibrium states for orbital and spin properties can be obtained (see for example Correia & Laskar 2003a,b;Correia et al. 2003;Correia et al. 2008 for telluric planets) and compared to observational constraints obtained in our Solar system and in exoplanetary systems.

Conclusions

In the second part of this lecture, we have tried to give the must complete review of tidal interactions in planetary systems. In the first section, we have pointed the crucial dependence of systems' evolution on dissipative mechanisms that convert the kinetic energy of tidal flows into heat. Next, we have described the important diversity of mechanisms that take place in stellar and planetary interiors. We have given their main properties, namely their relative amplitude and their dependence on the internal structure and on the tidal frequency that can be highly complex. This shows how it is now necessary to study tidal interactions with a good description of the bodies' internal structure and to go beyond the rough approximations that are often adopted to describe star-planet systems' evolution due to tidal interactions.

Fig. 1 .

 1 Fig. 1. Highly non-linear interactions between the differential rotation (upper rightpanel ), the meridional circulation (bottom left-panel where we represent the iso-contours of the stream meridional function (ξ); the meridional circulation is tangent to these lines), and the related thermal imbalance (δT )( bottom right-panel ) during the evolution on the main sequence of a 1.5 M ⊙ star with a solar metallicity and an initial equatorial velocity V ini = 100 km.s -1 . The rotation profile is represented for three positions in the Hertzsprung-Russelll diagram (upper left-corner ): X c = {0.675, 0.32, 0.0},w h e r eX c is the central fraction in mass in Hydrogen. The meridional currents driven by the torque of the wind, which slows down the surface, and the related temperature fluctuations are represented for X c =0.32. (Adapted from Decressin et al. 2009, courtesy Astronomy & Astrophysics.)

Fig. 2 .

 2 Fig. 2. Left: Secular extraction of angular momentum in the radiative core of a solartype star (taken from Alvan et al. 2013; see also Talon & Charbonnel 2005). Associated highly multi-cellular meridional circulation (Right top-panel ) and thermal imbalance (Right bottom-panel ). (Taken from Mathis et al. 2013.)

Fig. 3 .

 3 Fig. 3. 3D rendering of an ASH numerical simulation of the solar interior dynamics where the convective envelope and the radiative core are treated simultaneously. (Taken from Brun et al. 2011;A l v a net al. 2012.) An octant has been removed in order to visualise the structure within the meridional and the equatorial planes. IGWs are excited by turbulent convective plumes and propagate in the core.

Fig. 5 .

 5 Fig.5. Wave types in a rotating and magnetic stellar radiation zone and associated frequencies (N is the buoyancy frequency, 2Ω the inertial frequency (Ω being the angular velocity of the star), ω A the Alfvén frequency and f L the Lamb's frequency). Values for the Sun (⊙) at the bottom of the tachocline are given in brackets. (Taken from Mathis &d eB r y e2011a, courtesy Astronomy & Astrophysics.)

Fig. 6 .

 6 Fig. 6. Left: synopsis of the structure and the induced transport of AM in stars with an external and an internal convection zone. Convection and radiation zones are depicted in grey and orange. The short dashed blue lines represent the equatorial trapping due to the Coriolis acceleration and the Lorentz force and the purple long-dashed lines represent the vertical magnetic trapping. Finally, the vertical fluxes of energy and of angular momentum are represented by the thin dark grey arrow and the large green one. Right: net bias between energy geometrical transmition from convection to IGWs (here for an azimutal number |m| = 1) as a function of the wave Rossby (σ exc /2Ω) and Elsasser (ω 2 A /Ω σ exc ) numbers. Because of the strongest equatorial trapping of prograde waves, retrograde IGWs and their related net extraction of AM are favored. (Taken from Mathis &d eB r y e2012, courtesy Astronomy & Astrophysics.)

Fig. 7 .

 7 Fig. 7. Hydrodynamical transport loop in stellar radiation zones when taking into account IGWs. The different applied torques (winds & tides) and interactions with shear, rotation and magnetic fields are explicited. (Adapted from Mathis & de Brye 2012, courtesy Astronomy & Astrophysics.)

Fig. 8 .

 8 Fig.8. Top: the tidal energy dissipation: first, the gravific tidal potential energy generates tidal flows with a given kinetic energy. Next, the internal friction, related to the internal structure properties of the studied body, dissipates this kinetic energy into heat. Bottom: this process leads to the evolution of the system by modifying spins and orbital properties because of the net secular torque applied on the tidal bulge and change the internal structure of its components by tidal heating (e.g.Leconte et al. 2010).

Fig. 9 .

 9 Fig. 9. Internal structure of main-sequence stars as a function of their mass represented using Kippenhahn diagram: stellar masses are given in the horizontal axis while the fraction in mass and the position of convection zones (cloudy regions) and of radiation zones are given along the vertical axis (adapted from Kippenhahn & Weigert 1990, courtesy Springer-Verlag). Very low-mass stars are entirely convective. Next, as the stellar mass grows the radiative core becomes more and more important. Finally, a transition occurs because hydrogen is converted into helium through the CNO cycle and a convective core takes place with an external radiative envelope.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Top: internal structure of telluric planets in the Solar system and of the Moon. Bottom: internal structure of gaseous giant planets (Jupiter & Saturn) and of icy giant planets (Uranus & Neptune) in the Solar system (see Guillot 1999, 2005; Baraffe 2005; Fortney & Nettelmann 2010). (Courtesy NASA/JPL-Caltech.)

Fig. 13 .

 13 Fig. 13. Top-left: 3-D view of the total (poloidal and toroidal) adiabatic equilibrium tide velocity field (white arrows). The red and orange arrows indicate the direction of the primary's rotation axis and the line of centers respectively. Top-right: representation of this velocity field at the surface of the primary (black arrows); the color-scaled background represents the normalised tidal potential intensity (blue and red for the minimum and maximum values respectively). Bottom-left: view of the velocity field (white arrows) in its equatorial plane of symmetry; the color-scaled background represents the velocity value (black and orange for the minimum and maximum values respectively). Bottomright: view of the velocity field (white arrows) in its meridional plane of symmetry; the color-scaled background represents the velocity value (black and purple for the minimum and maximum values respectively (taken from Remus et al. 2012a, courtesy Astronomy & Astrophysics).

  Fig. 15. A: density perturbation (left) and radial and latitudinal components (right) of a tidally excited retrograde inertial mode with |m| = 2. The mode amplitude and the wavevector remain relatively uniform over much of the planet and rise sharply toward the surface; this rise is most striking near the critical latitude θ c =a r c o s( σ T /2Ω) marked by straight lines (Ω is the planet rotation). (Taken from Wu 2005a, courtesy The Astrophysical Journal.) B: viscous dissipation of such inertial waves excited by the tidal potential in a fully convective planet and the associated value of the tidal quality factor Q as a function of the tidal frequency for an Eckman number E = ν/ΩR 2 =10 -7 , where ν is the viscosity and R the radius of the studied planet. (Taken from Wu 2005b, courtesy The Astrophysical Journal.)

  Fig. 16. A) high-resolution calculation (E =10 -9 ) of the tidal response of a uniformly rotating planet with an internal core. The rms velocity of the total tide (equilibrium and dynamical) is plotted in a meridional slice through the convective region (the velocity scale is linear, black representing zero). Attractors with associated inertial wave beams can be identified. The forcing tidal frequency is chosen to be near the peak of an inertial mode resonance. (Taken from Ogilvie & Lin 2004, courtesy The Astrophysical Journal.) B) production of short inertial waves by scattering of the equilibrium tide off the core at critical latitudes to explain results obtained by Ogilvie & Lin (2004). (Taken from Goodman & Lackner 2009, courtesy The Astrophysical Journal.) C) viscous dissipation of such inertial waves excited by the tidal potential as a function of the tidal frequency for an Eckman number E =10 -7 ; the dotted line corresponds to Q =10 5 .( T a k e nf r o m Ogilvie & Lin 2004, courtesy The Astrophysical Journal.)

Fig. 17 .

 17 Fig. 17. Spin-over mode of the tidal elliptic instability observed in the laboratory (A) and computed in numerical simulations (B). (Adapted from the Ph.D. Thesis of D. Cébron and Cébron et al. 2010a, courtesy D. Cébron and Physics of the Earth and Planetary Interiors.)

  , planetary cores (Cébron et al. 2010b; Cebron et al. 2010a) and extra-solar planetary systems (Cebron et al. 2011). Then, interesting behaviour of this instability have been isolated. First, this can develop both in convective or stably stratified regions (where inertial waves become gravito-inertial because of the supplementary stabilising buoyancy force) (Le Bars & Le Dizès et al. 2006;G u i m b a r det al. 2010;C ébron et al. 2010b; Lavorel & Le Bars 2010

Fig. 20 .

 20 Fig. 20. Left: two-layer model of planetary internal structure (body A of mass M A rotating at angular velocity Ω) formed by an internal rocky core surrounded by a fluid envelope perturbed by a companion (body B of mass M B orbiting around A with mean motion n). R c and R p are respectively the core and the planetary radii, ρ o and ρ c the densities of the fluid envelope and of the core, and μ is the shear modulus of the core. Right: purely elastic tidal displacement in the solid core free of fluid envelope; the red and orange arrows indicate respectively the symmetry axis of the planetary core and the direction of the companion. (Taken from Remus et al. 2012b; courtesy Astronomy & Astrophysics.)

Fig. 21 .

 21 Fig. 21. Dissipation quality factor Q eff of the Maxwell model in function of the viscoelastic parameters G (the stiffness) and η (the viscosity). Top : for a Jupiter-like two-layer planet tidally perturbed at the Io's frequency σ T =2 .79 × 10 -4 rad • s -1 . Bottom: for a Saturn-like two-layer planet tidally perturbed at the Enceladus' frequency σ T =2 .25 × 10 -4 rad • s -1 . The red dashed lines correspond to the value of Q eff obtained by Lainey et al.( 2009) for Jupiter, and by Lainey eta l .( 2012) for Saturn also needed by Charnoz et al.( 2011) to form its mid-sized satellites. We recall the values of R p = {10.97, 9.14} (in units of R ⊕ p , the Earth radius), M p = {317.8, 95.16} (in units of M ⊕ p , the Earth mass), and M c = {6.41, 18.65}×M ⊕ p (see respectively Guillot 1999;[START_REF] Hubbard | The Interior of Saturn[END_REF]. Moreover, we have followedGavrilov & Zharkov (1977) for the values of the adiabatic Love numbers of Jupiter and Saturn (respectively k 2 = {0.379, 0.341}). The blue lines correspond to the lower and upper limits of the more realistic values taken by the viscoelastic parameters G and η at very high pressure for an unknown mix of ice and silicates. (Taken fromRemus et al. 2012b; courtesy Astronomy & Astrophysics.)

  Fig. 21. Dissipation quality factor Q eff of the Maxwell model in function of the viscoelastic parameters G (the stiffness) and η (the viscosity). Top : for a Jupiter-like two-layer planet tidally perturbed at the Io's frequency σ T =2 .79 × 10 -4 rad • s -1 . Bottom: for a Saturn-like two-layer planet tidally perturbed at the Enceladus' frequency σ T =2 .25 × 10 -4 rad • s -1 . The red dashed lines correspond to the value of Q eff obtained by Lainey et al.( 2009) for Jupiter, and by Lainey eta l .( 2012) for Saturn also needed by Charnoz et al.( 2011) to form its mid-sized satellites. We recall the values of R p = {10.97, 9.14} (in units of R ⊕ p , the Earth radius), M p = {317.8, 95.16} (in units of M ⊕ p , the Earth mass), and M c = {6.41, 18.65}×M ⊕ p (see respectively Guillot 1999;[START_REF] Hubbard | The Interior of Saturn[END_REF]. Moreover, we have followedGavrilov & Zharkov (1977) for the values of the adiabatic Love numbers of Jupiter and Saturn (respectively k 2 = {0.379, 0.341}). The blue lines correspond to the lower and upper limits of the more realistic values taken by the viscoelastic parameters G and η at very high pressure for an unknown mix of ice and silicates. (Taken fromRemus et al. 2012b; courtesy Astronomy & Astrophysics.)

Fig. 22 .

 22 Fig.22. Dependence to the perturbative strain pulsation ω of the tidal quality factor Q eff for Jupiter-like (red solid line) and Saturn-like (blue dashed line) giant planets. Note that the curves are represented with a logarithmic scale. The red and blue dotted lines correspond to the mean value of Qeff = {(3.56 ± 0.56) × 10 4 , (1.682 ± 0.540) × 10 3 } (for Jupiter and Saturn respectively) determined byLainey et al.( 2009Lainey et al.( , 2012)). Their zone of uncertainty is also represented in the corresponding color. The blue dots correspond to the values (obtained byLainey et al. 2012) of the dissipation induced in Saturn by Enceladus, Thetys, Dione and Rhea with their respective error bars. We recall the values of R p = {10.97, 9.14} (in units of R ⊕ p , the Earth radius), M p = {317.8, 95.16} (in units of M ⊕ p , the Earth mass), M c = {6.41, 18.65}×M ⊕ p ,a n d R c = {0.15, 0.26} R p . We take for the viscoelastic parameters G = {2.73, 6.51}× 10 10 (Pa), and η = {8.65 × 10 13 , 2.50 × 10 14 } (Pa • s) for Jupiter and Saturn respectively. (Adapted fromRemus et al. 2012b; courtesy Astronomy & Astrophysics.)

Fig. 23 .

 23 Fig. 23. Schematic view of Eckman boundary layers that can be important at boundaries of planetary cores. (Taken from Rieutord & Zahn 1997 in which details concerning boundary velocity fields b 1 and b 2 are given, courtesy the Astrophysical Journal.)

  Kawaler 1988;U d D o u l aet al. 2009;P i n t oet al. 2011;M a t tet al. 2012). Furthermore, if studied stars host a planetary system (Udry & Santos 2007; Perryman 2011), the coupling with the protoplanetary disk (Matt & Pudritz 2005; Zanni & Ferreira 2013) as well as tidal interactions should be taken into account.

This work was supported by the French Programmes Nationaux de Physique Stellaire (PNPS) et de Planétologie (PNP) of CNRS/INSU, by the CNES SOHO/GOLF & CoRoT-Kepler grants and asteroseismology support in CEA-Saclay, by the TOUPIES project funded by the French National Agency for Research (ANR), by the Campus Spatial de l'Université Paris-Diderot and by the Programme Physique Théorique et ses Interfaces (CNRS/INP). S. M. is grateful to the organisers of the Evry Schatzman School for the invitation to give this review. F. Rémus thanks the Fundation l'Oréal-Académie des Sciences-Unesco "Pour les femmes et la Science".