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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

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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

f tidal dissipation in star-planet systems
P. Auclair-Desrotour1,2, S. Mathis2,3, C. Le Poncin-Lafitte4

Talk given at OHP-2015 Colloquium

1IMCCE, CNRS UMR 8028, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France

(pierre.auclair-desrotour@obspm.fr)

2Laboratoire AIM Paris-Saclay, CEA/DSM - CNRS - Universit´

e Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette Cedex, France

3LESIA, 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

4SYRTE, 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 affected 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 diffusion, 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 effects of tidal perturbations have to be characterized and quantified. Moreover, observational constraints are now

btained 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 differ in nature from what is observed for solids and simplified fluid equilibrium tide (Efroimsky & Lainey 2007; Auclair-Desrotour et al. 2014). Because

f its complexity, the tidal response of planetary and stellar fluid layers has motivated numerous theoretical studies

92

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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 diffusion, 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 diffu- 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 diffusion 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 diffusion. This allows us to identify the possible regimes of tidal dissi- pation as a function of the characteristic frequencies, viscosity and thermal diffusion. 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 RO : {O, XE, YE, ZE} the reference frame rotating with the body at the spin frequency Ω with respect

to ZE.The spin vector Ω is thus given by Ω = ΩZE. The point M is defined by the spherical coordinates (r, θ, ϕ) and the corresponding spherical basis is denoted

er, eθ, eϕ
. We also define the local Cartesian coordinates x = (x, y, z)

and associated reference frame R :

M, ex, ey, ez
, which is such that ez = er, ex = eϕ and ey = −eθ. In this frame,

the local gravity acceleration, assumed to be constant, is aligned with the vertical direction, i.e. g = −gez, and the spin vector is decomposed as follows: Ω = Ω

cos θez + sin θey
, where θ is the colatitude.

−8 −6 −4 −2 2 4 5 10 15 20

log10 [J.kg1]

the

Figure 1: Left: Local Cartesian model, frame, and coordinates. Right: Energy dissipated (ζ) and its viscous and thermal components, ζvisc and ζtherm respectively, as functions of the reduced tidal frequency (ω) for θ = 0, A = 102, E = 10−4 and K = 10−2, which gives Pr = 10−2 (see Sect. 2 for the definition of these quantities).

93

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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 The fluid is Newtonian and locally homogeneous, of kinematic viscosity ν and thermal diffusivity κ. To com- plete the set of parameters, we introduce the Brunt-V¨ ais¨ al¨ a frequency N given by N2 = −g d log ρ dz − 1 γ d log P dz

(1) where γ = (∂ ln P/∂ ln ρ)S is the adiabatic exponent (S being the specific macroscopic entropy), and P and ρ are the radial distributions of pressure and density of the background, respectively. These distributions are assumed to be rather smooth to consider P and ρ constant in the box. The regions studied are stably stratified (N2 > 0) or convective (N2 ≈ 0 or N2 < 0). Finally, we suppose that the fluid is in solid rotation.

2.2 Analytic expressions of dissipated energies

The fluid is perturbed by a tidal force F =

Fx, Fy, Fz
, periodic in time (denoted t) and space, at the frequency χ. Its

tidal response takes the form of local variations of pressure p′, density ρ′, velocity field u = (u, v, w) and buoyancy B, which is defined as follows: B = Bez = −gρ

′ (x, t)

ρ ez . (2) Introducing the dimensionless time and space coordinates, tidal frequency, normalized buoyancy, and force per unit mass T = 2Ωt, X = x L, Y = y L, Z = z L, ω = χ 2Ω, b = B 2Ω, f = F 2Ω, (3) and using the Navier-Stokes, continuity and heat transport equations, we compute a solution of the tidally forced waves and perturbation, denoted s = {p′, ρ′, u, b, f}, of the form s = ℜ

smnei2π(mX+nZ)e−iωT

, where ℜ stands for the real part of a complex number. In this expression, m and n are the longitudinal and vertical degrees of Fourier modes and smn the associated amplitude coefficient. We now introduce the control parameters of the system, A, E (the Ekman number) and K, given by A = N 2Ω 2 , E = 2π2ν ΩL2 , and K = 2π2κ ΩL2 . (4) Hence, introducing the complex frequencies ˜ ω = ω+iE

m2 + n2

and ˆ ω = ω+iK

m2 + n2

, we get for the velocity field                                                      umn = n i ˜ ω (nfmn − mhmn) − n cos θgmn m2 + n2 ˜ ω2 − n2 cos2 θ − Am2 ˜ ω ˆ ω , vmn = n cos θ (nfmn − mhmn) + i

m2 + n2

˜ ω − Am2 ˆ ω

m2 + n2 ˜ ω2 − n2 cos2 θ − Am2 ˜ ω ˆ ω , wmn = −m i ˜ ω (nfmn − mhmn) − n cos θgmn m2 + n2 ˜ ω2 − n2 cos2 θ − Am2 ˜ ω ˆ ω , (5) and for the buoyancy perturbation bmn = iAm ω i ˜ ω (nfmn − mhmn) − n cos θgmn m2 + n2 ˜ ω2 − n2 cos2 θ − Am2 ˜ ω ˆ ω , (6) where fmn, gmn and hmn are the coefficients of the Fourier expansion in longitude and radius. From this, we derive the expressions of the energies dissipated per unit mass over a rotation period by viscous friction and thermal diffusion:

94

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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 ζvisc = 2πE

(m,n)∈Z∗2

m2 + n2
u2

ζtherm = 2πKA−2

(m,n)∈Z∗2

m2 + n2

|bmn|2 . (7)

Inertial waves Convective Zone Dissipation controlled by viscosity Dissipation controlled by thermal diffusion Gravito-inertial waves Stably stratified Zone

a" b" c" d"

Viscous'fric*on' Thermal'diffusion'

Figure 2: Map of the asymptotic behaviours of the tidal response. The horizontal (vertical) axis measures the parameter A (Pr = ν/κ) in logarithmic scales. Regions on the left correspond to inertial waves (a and c) and to gravito-inertial waves on the right (b and d). The fluid viscosity (thermal diffusivity) drives the behaviour of the fluid in regions a and b (c and d). The pink (grey) zone corresponds to the regime of parameters where ζtherm (ζvisc) predominates in the tidal dissipation.

3 Asymptotic regimes and scaling laws

To make easy the comparison of the obtained results with those of Ogilvie & Lin (2004), we use the same forcing as this early work: fmn = −i/

4 |m| n2

, gmn = 0 and hmn = 0. Plotting ζvisc and ζtherm (Eq. 7) as functions of the tidal frequency (ω), we get the graph of Fig. 1 (right panel). This graph reveals the highly resonant behaviour

f tidal dissipation. As demonstrated by Fig. 4 in Appendix A, the properties of the frequency spectra strongly

depend on the control parameters identified above (the Ekman number on the Figure). Therefore, tidal dissipation can correspond to the four different asymptotic regimes that we identify (Fig. 2). These regimes are defined by the frequency ratio A and the Prandtl number of the system, which is defined by Pr = ν/κ. Each of them corresponds to a colored region on the map:

1. A ≪ Amn and Pr ≫ Pr;mn: inertial waves controlled by viscous diffusion (blue);
2. A ≫ Amn and Pr ≫ Pr;mn: gravity waves controlled by viscous diffusion (red);
3. A ≪ Amn and Pr ≪ Pr;mn: inertial waves controlled by thermal diffusion (purple);
4. A ≫ Amn and Pr ≪ Pr;mn: gravity waves controlled by thermal diffusion (orange),

where Amn and Pr;mn are the vertical and horizontal transition parameters associated with the mode (m, n). In addition, we identify the regions where the fluid response is mainly damped by thermal diffusion (pink), for which ξtherm ≫ ξvisc, or by viscous friction (grey), for which ξtherm ≪ ξvisc. The transition, materialized by the pink line, corresponds to Pr = Pdiss

. The model allows us to compute for all regimes analytical formulae quantifying properties of the dissipation spectrum such as the number Nkc, positions ωmn, width lmn and height Hmn of the

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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 resonant peaks, the height of the non-resonant background Hbg, which corresponds to the equilibrium tide, and the sharpness ratio Ξ = H11/Hbg. Some of these formulae are given in Fig. 3. We finally deduce from these analytic solutions the scaling laws characterizing the dissipation regimes of Fig. 2, summarized in Table 1.

−4 −2 2 4 0.5 1 1.5 2

log10 [J.kg1]

E = 104

− − − − − − − −

Ξ = 1

2 cos2 +A

A + cos2 θ 3 AK + 2 cos2 +A E2 h C1

in cos2 θ + C1 gravA

− − − − − − − −

− − − − − − −

Nkc ∼

8 > > > < > > > : 1 2

2 cos2 θ + A

A + cos2 θ 3 AK + 2 cos2 +A E2 h C1

in cos2 θ + C1 gravA

i 9 > > > = > > > ; 1 4 .

− − − − − − − −

Hbg = 4πF2E

gravA + C1 in cos2 θ

A + cos2 θ2

− − − − − − − − − − − − − −

− − − − − − − − − − − − −

− − − − − − − − − − − − − − −

Hmn =

8πF2E m2n2 m2 + n22

2n2 cos2 θ + Am2

n2 cos2 θ + Am2 Am2K + 2n2 cos2 θ + Am2 E2 , (49)

lmn =
m2 + n2 Am2K +
2n2 cos2 θ + Am2

E n2 cos2 θ + Am2 .

Figure 3: Energy dissipated by viscous friction as a function of the reduced tidal frequency (ω) and the formulae giving the properties of the spectrum as functions of the colatitude and control parameters of the box (A, E and K). Table 1: Scaling laws for the properties of the energy dissipated in the different asymptotic regimes. The transition Prandtl number Pdiss

r;11 indicates the transition zone between a dissipation led by viscous friction and a dissipation

led by heat diffusion. The parameter A11 indicates the transition between tidal inertial and gravity waves. The parameter Preg

r;11 is defined as Preg r;11 = max

Pr;11, Pdiss

r;11

Domain A ≪ A11 A ≫ A11 Pr ≫ Preg

r;11

lmn ∝ E ωmn ∝ n cos θ √ m2 + n2 lmn ∝ E ωmn ∝ m √ A √ m2 + n2 Hmn ∝ E−1 Nkc ∝ E− 1

Hmn ∝ E−1 Nkc ∝ A

1 4 E− 1 2

Hbg ∝ E Ξ ∝ E−2 Hbg ∝ A−1E Ξ ∝ AE−2 Pr ≪ Preg

r;11

Pr ≫ Pr;11 lmn ∝ E ωmn ∝ n cos θ √ m2 + n2 Pr ≫ Pdiss

r;11

lmn ∝ EP−1

ωmn ∝ m √ A √ m2 + n2 Hmn ∝ E−1P−1

Nkc ∝ E− 1

Hmn ∝ E−1P2

Nkc ∝ A

1 4 E− 1 2 P 1 2

Hbg ∝ EP−1

Ξ ∝ E−2 Hbg ∝ A−1E Ξ ∝ AE−2P2

Pr ≪ Pr;11 lmn ∝ AEP−1

ωmn ∝ n cos θ √ m2 + n2 Pr ≪ Pdiss

r;11

lmn ∝ EP−1

ωmn ∝ m √ A √ m2 + n2 Hmn ∝ A−2E−1Pr Nkc ∝ A− 1

2 E− 1 2 P 1 2

Hmn ∝ A−1E−1Pr Nkc ∝ A

1 4 E− 1 2 P 1 2

Hbg ∝ EP−1

Ξ ∝ A−2E Hbg ∝ A−2EP−1

Ξ ∝ AE−2P2

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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

4 Conclusions

This work describes a general method which can be used to understand the complex mechanisms responsible for tidal dissipation in stars and fluid planetary layers. By focusing on the local behaviour of fluid body submitted to an academic tidal forcing, we establish the relation between the dissipated energy and the parameters of the internal structure and dynamics. This framework allows us to highlight the four possible regimes of tidal dissipation as func- tion of the Prandtl number, Ekman number and the inertial, Brunt-V¨ ais¨ al¨ a and tidal frequency. We note its strong dependence on tidal frequency, the frequency spectrum of tidal waves being determined by the control parameters

f the system. In this formalism, we are able to compute analytically the scaling laws associated with the identified

regimes for various representative properties of the spectrum: the positions, widths, heights, number of resonant peaks, the level of the non-resonant background as well as the sharpness ratio, which indicates the sensitivity of tidal dissipation to the tidal frequency. The method will be generalized in future works to magnetized fluid bod- ies, for which we expect additional regimes resulting from the propagation of Alv´ en waves and the corresponding Ohmic diffusion. Acknowledgments: This work was supported by the French Programme National de Plan´ etologie (CNRS/INSU), the CNES-CoRoT grant at CEA-Saclay, the ”Axe f´ ed´ erateur Etoile” of Paris Observatory Scientific Council, and the International Space Institute (ISSI; team ENCELADE 2.0). P. Auclair-Desrotour and S. Mathis acknowledge funding by the European Research Council through ERC grant SPIRE 647383.

References

Auclair-Desrotour, P., Le Poncin-Lafitte, C., & Mathis, S. 2014, A&A, 561, L7 Auclair Desrotour, P., Mathis, S., & Le Poncin-Lafitte, C. 2015, A&A, 581, A118 C´ ebron, D., Bars, M. L., Gal, P. L., et al. 2013, Icarus, 226, 1642 C´ ebron, D., Le Bars, M., Moutou, C., & Le Gal, P. 2012, A&A, 539, A78 Efroimsky, M. & Lainey, V. 2007, Journal of Geophysical Research (Planets), 112, 12003 Gerkema, T. & Shrira, V. I. 2005, Journal of Fluid Mechanics, 529, 195 Lainey, V., Arlot, J.-E., Karatekin, ¨ O., & van Hoolst, T. 2009, Nature, 459, 957 Lainey, V., Jacobson, R. A., Tajeddine, R., et al. 2015, ArXiv e-prints Lainey, V., Karatekin, ¨ O., Desmars, J., et al. 2012, ApJ, 752, 14 Ogilvie, G. I. 2014, Annual Review of Astronomy and Astrophysics, 52, 171 Ogilvie, G. I. & Lin, D. N. C. 2004, ApJ, 610, 477 Ogilvie, G. I. & Lin, D. N. C. 2007, ApJ, 661, 1180 Remus, F., Mathis, S., & Zahn, J.-P. 2012, A&A, 544, A132 Wu, Y. 2005, ApJ, 635, 688 Zahn, J. P. 1966a, Annales d’Astrophysique, 29, 313 Zahn, J. P. 1966b, Annales d’Astrophysique, 29, 489 Zahn, J. P. 1966c, Annales d’Astrophysique, 29, 565 Zahn, J.-P. 1975, A&A, 41, 329 Zahn, J.-P. 1977, A&A, 57, 383 Zahn, J.-P. 1989, A&A, 220, 112

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A Dependence of the frequency spectra of tidal dissipation on the fluid control parameters

The expressions given by Eq. (7) allow us to perform parametric studies easily. For example, we draw in Fig. 4 the frequency spectrum of the energy dissipated by viscous friction for various values of the Ekman number. We thus retrieve the results presented by Ogilvie & Lin (2004), which give an idea of the crucial role played by the internal structure and rotation. When E decreases, the number of peaks and their heights increase while their width and the level of the non-resonant background decay. Convective layers in gaseous giant planets and stars are characterized by low Ekman numbers and therefore highly resonant frequency spectra for viscous friction.

−4 −2 2 4 0.5 1 1.5 2

log10 visc [J.kg1]

E = 102

−4 −2 2 4 0.5 1 1.5 2

log10 visc [J.kg1]

E = 103

−4 −2 2 4 0.5 1 1.5 2

log10 visc [J.kg1]

E = 104

−4 −2 2 4 0.5 1 1.5 2

log10 visc [J.kg1]

E = 105

Figure 4: Frequency spectra of the energy per unit mass locally dissipated by viscous friction ζvisc for inertial waves (A = 0) at the position θ = 0 for K = 0 and different Ekman numbers. In abscissa, the normalized frequency ω = χ/2Ω. Top left: E = 10−2. Top right: E = 10−3. Bottom left: E = 10−4. Bottom right: E = 10−5. The results obtained by Ogilvie & Lin (2004) are recovered. Note that the non-vanishing viscous dissipation at ω = 0 is the one of the geostrophic equilibrium.