Oceanic Tides from Earth-Like to Ocean Planets P

A&A 615, A23 (2018) Astronomy https://doi.org/10.1051/0004-6361/201732249 & © ESO 2018 Astrophysics

Oceanic tides from Earth-like to ocean planets P. Auclair-Desrotour1,2,4, S. Mathis2,3, J. Laskar1, and J. Leconte4

1 IMCCE, Observatoire de Paris, CNRS UMR 8028, PSL, 77 Avenue Denfert-Rochereau, 75014 Paris, France e-mail: [email protected] 2 Laboratoire AIM Paris-Saclay, CEA/DRF - CNRS - Université Paris Diderot, IRFU/DAp Centre de Saclay, 91191 Gif-sur-Yvette, France 3 LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France e-mail: [email protected] 4 Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France e-mail: [email protected]

Received 7 November 2017 / Accepted 24 January 2018

ABSTRACT

Context. Oceanic tides are a major source of tidal dissipation. They drive the evolution of planetary systems and the rotational dynamics of planets. However, two-dimensional (2D) models commonly used for the Earth cannot be applied to extrasolar telluric planets hosting potentially deep oceans because they ignore the three-dimensional (3D) effects related to the ocean’s vertical structure. Aims. Our goal is to investigate, in a consistant way, the importance of the contribution of internal gravity waves in the oceanic tidal response and to propose a modelling that allows one to treat a wide range of cases from shallow to deep oceans. Methods. A 3D ab initio model is developed to study the dynamics of a global planetary ocean. This model takes into account compressibility, stratification, and sphericity terms, which are usually ignored in 2D approaches. An analytic solution is computed and used to study the dependence of the tidal response on the tidal frequency and on the ocean depth and stratification. Results. In the 2D asymptotic limit, we recover the frequency-resonant behaviour due to surface inertial-gravity waves identified by early studies. As the ocean depth and Brunt–Väisälä frequency increase, the contribution of internal gravity waves grows in importance and the tidal response becomes 3D. In the case of deep oceans, the stable stratification induces resonances that can increase the tidal dissipation rate by several orders of magnitude. It is thus able to significantly affect the evolution time scale of the planetary rotation. Key words. hydrodynamics – planet-star interactions – planets and satellites: oceans – planets and satellites: terrestrial planets

1. Introduction same order of magnitude and even greater than that associated with the solid tide, as is seen on Earth. Observed as an exoplanet, Even though the number of detected extrasolar planets located the Earth appears as a very dry rocky planet with an oceanic 4 in the habitable zone of their host stars has kept growing layer of depth approximatively equal to 6 10− R (Eakins & continuously for the past two decades, two major discoveries Sharman 2010). Yet, the Earth’s ocean is today× the main⊕ contrib- recently aroused excitement within the community of exoplanet utor to the total energy tidally dissipated by the Earth-Moon-Sun research. Firstly, in Summer, 2016, a telluric extrasolar planet system. It accounts for roughly 95% of the planetary dissipa- was found orbiting the Sun’s closest stellar neighbour, the red tion rate of 2.54 TW generated by the Lunar semidiurnal tide dwarf Proxima Centauri (Anglada-Escudé et al. 2016; Ribas (see Lambeck 1977; Ray et al. 2001; Egbert & Ray 2001, and et al. 2016). This planet, Proxima b, has a minimum mass of references therein). Moreover, the oceanic dissipation rate can 1.3 M and an equilibrium temperature that is within the range vary significantly owing to the frequency-resonant behaviour of where⊕ water could be liquid on its surface, which makes it the fluid layer (e.g. Webb 1980), thus leading to the present-day resemble a potential twin sister of the Earth. A few months high transfer of angular momentum from the spin of the Earth later, another star was in the spotlight, the ultra-cool dwarf to the orbit of the Moon (Lambeck 1980; Bills & Ray 1999). star TRAPPIST-1 (Gillon et al. 2017). The system hosted by More generally, oceanic tides have an impact on the history of TRAPPIST-1 drew attention due to its remarkable architecture, the spin rotation and states of equilibrium of the Earth (Neron de composed of eight telluric planets of masses between 0.7 M Surgy & Laskar 1997) and terrestrial exoplanets (Correia et al. and 1.2 M and radii between 0.1 R and 1.5 R (Gillon et al.⊕ 2008; Leconte et al. 2015; Auclair-Desrotour et al. 2017b). There- 2017; Wang⊕ et al. 2017). Most of these⊕ planets (i.e.⊕ b, c, d, e, f, g, fore, the oceanic tidal dissipation rate needs to be quantified in h) exhibit small densities suggesting that water stands for a large the case of extrasolar planets to study the history of observed fraction of their masses (Wang et al. 2017). Hence, Proxima b planetary systems and constrain their physical properties. and several of the telluric planets orbiting TRAPPIST-1 could Since Laplace’s pioneering works (Laplace 1798), the host deep oceans of liquid water. Earth’s oceanic tides have been examined through different Oceans play a crucial role in the evolution of planetary sys- approaches. The dissipation rate they induce is now well con- tems. Similarly to rocky cores, they are distorted by the tidal strained thanks to the fit of the altimetric data provided by gravitational forcings of perturbers such as stars or satellites. The the TOPEX/Poseidon satellite with numerical simulations made energy dissipated by the resulting oceanic tides can be of the using general circulation models (GCM; Egbert & Ray 2001,

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2003). Moreover, numerous studies have characterized their dependence on the planet parameters by developing linear ab initio global models, such as the hemispherical ocean model (Proudman & Doodson 1936; Doodson 1938; Longuet-Higgins & Pond 1970; Webb 1980). This approach has recently been used to estimate the tidal dissipation rate in the oceans of icy satellites orbiting Jupiter and Saturn (Tyler 2011, 2014; Chen et al. 2014; Matsuyama 2014). Simplified ab initio models are very convenient to explore the domain of parameters because they involve a small number of control parameters and require a relatively small computational cost compared to numerical models taking into account local features (e.g. topography and mean flows). They are commonly based on a two dimensional (2D) spherical geometry and the thin shell hypothesis. In this approach, the tidal response of the ocean is controlled by the Laplace tidal equations (LTE), Fig. 1. Quadrupolar semidiurnal oceanic tide in a rotating terrestrial which describe its barotropic response, that is the component planet gravitationally forced by a perturber (star of satellite). Refer- associated with the propagation of surface-gravity waves. This is ence frame and system of coordinates used in the modelling. The spin typically the tidal response that the ocean will have if it is unstrat- frequency of the planet and the orbital frequency of the perturber ified (uniform density). Nevertheless, solutions to the barotropic are denoted Ω and norb, respectively. The notation δtide designates the angular lag of the tidal bulge. equations can be applied to the case of stably stratified oceans to describe the contribution of internal gravity waves, restored by the Archimedean force. As was shown a long time ago (e.g. tidal potential of a perturber. The considered system is a Chapman & Lindzen 1970, in the case of the atmosphere), the spherical telluric planet of external radius R uniformly covered tidal response of a stratified fluid layer in the classical tidal the- by an oceanic layer of depth H (Fig.1). The radius of the solid ory is characterized by a frequency-dependent equivalent depth, part is designated by Rc = R H. We assume that the ocean − which is analogous to the barotropic ocean depth (see e.g. Tyler rotates uniformly with the rocky part at the angular velocity Ω, 2011). Results obtained in the barotropic approximation can thus the corresponding spin vector being denoted Ω. This allows us be reinterpreted by replacing the ocean depth by the equivalent to write the fluid dynamics in the natural equatorial reference depth associated with the studied mode. However, the com- frame rotating with the planet, RE;T : O, XE, YE, ZE , where O is the centre of the planet, and { define the equatorial} plete solution controlling the tidal response of a deep stratified XE YE ocean, including its vertical component, has never been explicitly plane and Z = Ω/ Ω . We use the spherical basis er, eθ, eϕ E | | resolved nor analysed to our knowledge. and coordinates (r, θ, ϕ) where r is the radial coordinate, θ the Therefore, following along the line of Webb(1980) and Tyler colatitude and ϕ the longitude (the position vector is r = r er). (2011), we develop here a linear global model of oceanic tides Finally, the time is denoted t. including three-dimensional (3D) effects related to the ocean sphericity and vertical structure. This model, by taking into 2.1. Forced dynamics equations account the vertical stratification of the ocean with respect to convection (i.e. its convective stability), allows us to quantify in The planet generates a gravity field, denoted g, oriented radially. a consistent way the contribution of internal gravity waves to the The centrifugal acceleration due to rotation tends to make the oceanic tidal response through explicit solutions derived in sim- effective acceleration depend on latitude. We ignore this depen- p plified cases. The obtained results can be used as a tool to (1) dence by considering that Ω Ωc, where Ωc = g/r represents characterize the tidal behaviour of deep global oceans, (2) con- the Keplerian critical rotation velocity (the rotation velocity of strain the structure and history of observed telluric planets from the planet is far slower than the critical velocity for which the the architecture of their host planetary system, and (3) provide a distortion due to the centrifugal acceleration destroys the oceanic simplified diagnosis of the oceanic tidal dissipation rate. layer). We assume then that the ocean is stratified radially in In Sect.2, we establish the equations governing the dynam- pressure p and density ρ. These quantities are written ics of the 3D oceanic tidal response, identify the possible tidal regimes, and express the tidal Love numbers and torque exerted p (r, t) = p0 (r) + δp (r, t) , ρ (r, t) = ρ0 (r) + δρ (r, t) , (1) on the planet in the general case. In Sect.3, we simplify the dynamics by assuming a uniform stratification as a first step and where the superscript 0 refers to the background distribution and derive an analytic solution for the tidal response and the resulting δ to a small fluctuation at the vicinity of the equilibrium. Sim- ilarly, the velocity of the flows generated by the perturbation is dissipation rate. This solution is then applied to idealized Earth and TRAPPIST-1 f planets in Sect.4 in order to illustrate the denoted V (r, t) = Vr, Vθ, Vϕ and the associated displacement difference between shallow and deep oceans. We also use this ξ (such that V = ∂t ξ, Unno et al. 1989). We note that, because solution in Sect.5 to explore the dependence of the dissipation of the solid rotation assumption, there is no meridional or zonal rate on the ocean depth and stratification. Finally, we discuss the background circulation. We consider in the following that the simplifications assumed in the modelling in Sect.6 and give our tidal perturbation can be approximated by a linear model and conclusions in Sect.7. therefore ignore terms of order greater than 1 with respect to ξ, V, δp and δρ. Moreover, we assume the Cowling approxi- 2. Dynamics of a gravitationally forced thick ocean mation (Cowling 1941), that is, the self-gravitational effect of the variation of mass distribution is not taken into account. We establish in this section the equations that govern the dynam- Finally, following early works (e.g. Webb 1980; Egbert & Ray ics of tides in a thick oceanic shell submitted to the gravitational 2001, 2003; Tyler 2011), we introduce dissipation by using a A23, page 2 of 15 P. Auclair-Desrotour et al.: Oceanic tides from Earth-like to ocean planets

Rayleigh friction, σRV, where σR is a constant effective fre- super-inertial waves (e.g. Mathis 2009; Auclair-Desrotour et al. quency associated with the damping (we note that Egbert & Ray 2018). 2001, 2003, consider two different models to describe friction, Finally, the equations of mass conservation (3) and of the the Rayleigh friction model and a quadratic one depending on buoyancy (4) can be written respectively the total velocity of the flow and including all tidal components). As demonstrated by Ogilvie(2009), the results obtained with 1 2 ρ0 h i ∂ δρ + ∂ r ρ V = ∂θ (sin θVθ) + ∂ϕVϕ , (7) such a simplified approach are a reasonably good approximation t r2 r 0 r −r sin θ of those obtained by taking into account the complete viscous force. It also seems to be a reasonable approach for situations and such as exoplanets, where the solid-fluid coupling is naturally 1 ρ N2 unknown. ∂ δρ = ∂ δ + 0 . t 2 t p Vr (8) Under these assumptions, the Navier-Stokes equation cs g describing the distortion of the ocean forced by the tidal poten- The tidal perturbation is periodic in time and longitude. tial U can be expressed in the frame rotating with the planet as Therefore, introducing the longitudinal wavenumber m and the (Gerkema & Zimmerman 2008) tidal frequency σ, we expand the perturbed quantities in Fourier 1 g series of t and ϕ. Any quantity f is written ∂tV + 2Ω V = ∇δp δρer + ∇U σ V. (2) ∧ −ρ − ρ − R X 0 0 f = f m,σ (r, θ) ei(σt+mϕ), (9) It is completed by the equation of mass conservation m,σ where the f m,σ are the spatial distributions of f associated with dρ0 ∂tδρ + Vr + ρ0∇ V = 0, (3) the doublet (m, σ). In the case of a perfect fluid (σR = 0), the dr · latitudinal structure of the perturbation is fully characterized by and the equation of buoyancy m and the real parameter ν = 2Ω/σ, which is the so-called spin parameter (e.g. Lee & Saio 1997). Viscous friction slightly mod- ! dρ 1 dp ifies the latitudinal structure by introducing a dependence on σR. ∂ δρ + 0 = ∂ δ + 0 , t Vr 2 t p Vr (4) Thus, we define the complex tidal frequency and spin parameter dr cs dr as where cs (r) = Γ (p0, ρ0, S 0) designates the sound velocity, Γ 2Ω σ˜ = σ iσ and ν˜ = , (10) being any state function and S 0 the salinity of the fluid. It shall − R σ˜ be noted here that we only consider isohaline and isentropic processes. Diabatic processes such as the thermal expansion of the and the latitudinal operators ocean are ignored. The radial stratification of the oceanic layer is characterized by the Brunt–Väisälä frequency (e.g. Gerkema & m,ν˜ 1 = [∂θ + mν˜ cot θ] , and Zimmerman 2008), Lθ 1 ν˜2 cos2 θ (11) − 1 m " # m,ν˜ = ν θ∂ + , g d ln ρ gH ϕ 2 2 ˜ cos θ N2 = 0 + , (5) L 1 ν˜ cos θ sin θ − H dx c2 − s such that ! where we have introduced the non-dimensional reduced alti- δ m,σ = + m,σ i m,ν˜ p m,σ tude x, defined by r Rc Hx. The three components of the Vθ = θ U , and momentum equation given by Eq. (2) are strongly coupled by σ˜ r L ρ0 − m,σ ! (12) the Coriolis acceleration. To make the problem tractable analyt- m,σ 1 m,ν˜ δp m,σ Vϕ = ϕ U . ically, we assume the commonly used traditional approximation −σ˜ r L ρ0 − (e.g. Eckart 1960; Mathis et al. 2008; Auclair-Desrotour et al. 2017a). This simplification consists in ignoring the latitudinal As mentioned above, the traditional approximation allows us component of the rotation vector in the Coriolis acceleration to separate the coordinates r and θ in the Fourier coefficients of (i.e. 2Ω sin θVr and 2Ω sin θVϕ along eϕ and er respectively), Eq. (9). Therefore, we write these functions as giving X X ! ξm,σ = ξm,σ (r) Θm,ν˜ (θ) , Vm,σ = Vm,σ (r) Θm,ν˜ (θ) , 1 δp r r;n n r r;n n ∂ Vθ 2Ω cos θVϕ = ∂θ U σ Vθ, n n t R m,σ X m,σ m,ν˜ m,σ X m,σ m,ν˜ − − r ρ0 − − ξ = ξ (r) Θ (θ) , V = V (r) Θ (θ) , ! θ θ;n θ;n θ θ;n θ;n 1 δp Xn Xn ∂tVϕ + 2Ω cos θVθ = ∂ϕ U σRVϕ, (6) m,σ m,σ m,ν˜ m,σ m,σ m,ν˜ −r sin θ ρ0 − − ξϕ = ξϕ;n (r) Θϕ;n (θ) , Vϕ = Vϕ;n (r) Θϕ;n (θ) , 1 g nX Xn ∂ V = ∂ δp δρ + ∂ U σ V , m,σ m,σ m,ν˜ m,σ m,σ m,ν˜ t r r r R r δp = δpn (r) Θn (θ) , δρ = δρn (r) Θn (θ) , −ρ0 − ρ0 − Xn n m,σ m,σ m,ν˜ thus making it possible to separate θ and r coordinates in solu- U = Un (r) Θn (θ) , (13) tions, as shown hereafter. As discussed in Sect.6, the traditional n approximation is usually considered to be satisfactory in stably where n designates the latitudinal wavenumber of a component 2 m,ν˜ stratified layers (2Ω N ) and for tidal frequencies satisfying (n N if ν 1 ; n Z if ν > 1), the Θn are the so- the condition 2Ω < σ N2, which corresponds to the regime of called∈ Hough| | functions≤ ∈ (Hough| 1898| ), defined on the interval A23, page 3 of 15 A&A 615, A23 (2018)

Fig. 2. Real (top) and imaginary (bottom) parts of even Hough functions (n even) associated with a quadrupolar tidal perturbation (m = 2) and deﬁned by ν˜ = ν/ (1 iγ) with ν = 1 (ordinary/gravity modes) and various positive values of γ = σR/σ. From left to right, γ = 0 (non-frictional regime), γ = 0.1 (weakly− frictional regime), γ = 1 and γ = 10 (strongly frictional regime). Hough functions are plotted as functions of the latitude δ (degrees).

h i θ [0, π], and the Θm,ν˜ (θ) = m,ν˜ Θm,ν˜ (θ) and Θm,ν˜ (θ) = The Hough functions associated with gravity modes degen- ∈ θ;n Lθ n ϕ;n 1 m m,ν˜ h m,ν˜ i erate in the associated Legendre polynomials Pl (with ϕ Θn (θ) stand for the latitudinal functions associated l = m + n) when ν 0. L → with the latitudinal and longitudinal velocities and displace- A strongly frictional regime ( σ . σR), where friction sig- ments, respectively. Hough functions are the solutions of the • nificantly affects the horizontal| | structure of tidal waves. In eigenfunctions-eigenvalues problem defined by the Laplace’s this regime, the hierarchy of eigenvalues can change, as tidal equation (Laplace 1798), demonstrated by Volland(1974a) who notes that the corresponding critical points appear for σ σR. For σR & m,ν˜Θ = ΛΘ, (14) Ω , inertial modes converge towards gravity| | ∼ modes. Grav- L − ity| | and Rossby Hough functions merge asymptotically when m,ν˜ where Θ is a function, Λ C and the operator σR + , and converge to the associated Legendre poly- ∈ L nomials.→ ∞ Thus, a strong friction makes the Coriolis effects ! negligible, as if the planet were not rotating. In this asymp- m,ν˜ 1 d sin θ d = totic regime, the angular lag between the tidal bulge and the L sin θ dθ 1 ν˜2 cos2 θ dθ − ! (15) direction of the perturber is given by the imaginary part of 1 1 + ν˜2 cos2 θ m2 mν˜ + . the vertical profiles of perturbed quantities. The behaviour of −1 ν˜2 cos2 θ 1 ν˜2 cos2 θ sin2 θ tidal modes in the dissipative regime is discussed thoroughly − − by Volland & Mayr(1972) and Volland(1974a). m,ν˜ m,ν˜ m,ν˜ m,ν˜ m,ν˜ Hence, for a given n, Θn is such that Θn = Λn Θn , Hough functions associated with symmetric modes of degree Λm,ν˜ L − the parameter n being the associated eigenvalue. We note m = 2 are plotted in Fig.2 for various values of the ratio σR/σ. here that the only difference with the usual case where fric- The equations describing the vertical structure are obtained tion is not taken into account (e.g. Lee & Saio 1997) is the by substituting the expansions given by Eqs. (9) and (13) in nature of ν˜ which is a complex number and not a real one (see Eqs. (6–8). In order to lighten expressions, the superscripts m,ν˜ m,ν˜ Eq. (10)). It follows that the Θn and Λn are complex func- (m, σ) is omitted up to the end of Sect. 2.1, where no con- tions and parameters in the general case (e.g. Volland 1974a,b). fusion will arise. Let us introduce the notation yn = δpn/ρ0 Looking at the expression of ν˜, one can identify two dissipation and assume the ocean to be at the hydrostatic equilibrium regimes: (dp /dx = Hgρ ), where x is the reduced altitude introduced 0 − 0 A weakly frictional regime ( σ σR), where Hough func- in Eq. (5). After some manipulations, one obtains • | | m,ν tions and eigenvalues are real (they are denoted Θn and m,ν Λn ). This regime corresponds to the case treated usually " # " # " # in the theory of tides (Chapman & Lindzen 1970; Lee & dY A1 (x) B1 (x) C1 (x) yn = Y + , with Y = 2 , (16) Saio 1997; Auclair-Desrotour et al. 2014). Tidal modes can dx A2 (x) B2 (x) C2 (x) r ξr;n be divided into two families: the so-called gravity modes (n 0), which are defined both in the regime of super- ≥ 1 The Pm represent here the normalized associated Legendre polynomi- inertial waves ( ν 1) and in the regime of sub-inertial l h i m/2 l waves ( ν > 1), and| | ≤ the inertial modes (n < 0), defined only als, expressed as Pm = ( 1)m / 2ll! 1 x2 dl+m/dxl+m x2 1 l − − − in the regime| | of sub-inertial waves (see Lee & Saio 1997). (Abramowitz & Stegun 1972).

A23, page 4 of 15 P. Auclair-Desrotour et al.: Oceanic tides from Earth-like to ocean planets and the coefficients The vertical profiles of the other quantities are deduced from m,σ ,ν ! Ψn straightforwardly thanks to polarization relations. We get, N2H Λm ˜ r2 A = , A = H n , for the velocity field 1 g 2 σσ˜ − c2 s " ! # H gH iσ dΨ dU = σσ 2 , = , (17) Vm,σ = Φ n + Ψ n , (26) B1 2 ˜ N B2 2 r;n 2 n n n r − cs − H N σσ˜ dx A − dx m,ν˜ − dUn HΛn m,σ i C = , C = U . V = (ΦnΨn Un) , (27) 1 dx 2 − σσ˜ n θ;n σ˜ r − m,σ 1 This system is then written as a single second-order ordinary Vϕ;n = (ΦnΨn Un) , (28) differential equation, −σ˜ r − for the displacement d2y dy n + n + y = , 2 A (x) B (x) n C (x) (18) " ! # dx dx m,σ 1 dΨn dUn ξr;n = Φn + nΨn , (29) with the coefficients − H N2 σσ˜ dx A − dx − d ln ρ0 m,σ 1 A (x) = K , (19) ξθ;n = (ΦnΨn Un) , (30) dx − ◦ σσ˜ r − m,σ i ! ξ = (ΦnΨn Un) , (31) H2 N2 ϕ;n σσ˜ r − = Λm,ν˜ ε B (x) 2 n 1 1 s;n (20) r σσ˜ − − and for scalar quantities ! ! d gH N2H K , δ m,σ = ρ Φ Ψ , − dx − c2 − ◦ g pn 0 n n (32) s " ! # ρ N2 dΨ dU ! ! δρm,σ = 0 Φ n + Ψ n , 2 2 2 n 2 n n n (33) H m,ν˜ N gH dUn d Un −gH N σσ˜ dx B − dx C (x) = Λn 1 Un + K + . (21) − r2 σσ˜ − − c2 ◦ dx dx2 s with the frequency-dependent factors Here, we have introduced the frequency-dependent, dimension- " # less acoustic, and sphericity parameters 1 gH N2H n (x) = + K , (34) ! A 2 c2 − g ◦ r2σσ˜ r2 d N2 σσ˜ s εs;n = ,ν and K = − . (22) m ˜ 2 ◦ 2 σσ x 2 " 2 # Λn cs N ˜ d r 1 Hg σσ˜ HN − = + . n (x) 2 2 2 1 K (35) 2 B 2 cs N − − g ◦ The acoustic parameter can be written εs;n = σσ/σ˜ s;n, m,ν˜1/2 where σs;n = Λn cs/r is the cut-off frequency of acous- Through equations describing the tidal dynamics, we iden- tic waves (also called the Lamb frequency) for the degree-n tify the families of waves involved in the tidal response. First, m,ν˜ + because of rotation, inertial waves are generated in the frequency mode when Λn R . The regime of acoustic waves thus corre- ∈ range σ < 2Ω. Then, we identify surface-gravity waves, which sponds to εs;n & 1. Typically, for an Earth-like planet of radius are restored| | by gravity and are characterized by the cut-off fre- R = 6371 km and angular velocity Ω = 7.25 10 5 s 1 (NASA − − quency σ = √gH/r. If the ocean is stably stratified (N > 0), fact⊕ sheets), with a water oceanic⊕ layer characterized× by g internal gravity waves can propagate in the frequency range c 1.545 km.s 1 (Gerkema & Zimmerman 2008), the gravest s − σ < N. These waves are restored by the Archimedean force. acoustic≈ modes of the quadrupolar semidiurnal tide (m = 2, Finally,| | in the high-frequency range delimited by the acous- = Λ2,ν˜ Ω . Ω n 0, 0 11) appear for & 5 5 . The other parame- tic cut-off frequency σ = c /r, the tidal response is partly ter, K , corresponds∼ to the effect of the spherical⊕ geometry of s s ◦ composed of horizontally propagating acoustic Lamb modes, the layer on the structure of tidal waves. It is usually ignored restored by compressibility. The possible regimes of oceanic in 2D modellings where r = R. We see in the following section tides are determined by the hierarchy of the characteristic fre- that K H/R when H/R 1. Eventually, using the change of ◦ ∝ m,σ m,σ m,σ quencies of the system, namely σ, σR, 2Ω, N, σg and σs. variables yn = Φn Ψn , where Φn is the radial function The frequency spectrum of waves composing the oceanic tidal " Z x # response is summarized by Fig.3. m,σ 1 Φn (x) = exp A x0 dx0 , (23) −2 0 2.2. Tidal potential, Love numbers, and tidal torque we write the vertical structure equation (Eq. (18)) as the Schrödinger-like equation The variation of mass distribution due to the tidal distortion modifies the self-gravitational potential of the planet. The tidal 2 m,σ d Ψn 2 m,σ m,σ 1 potential of the excitation is usually expanded in Fourier series + kˆ Ψ = Φ − C, (24) dr2 n n n and spherical harmonics (see for instance the Kaula’s multipolar expansion; Kaula 1962). It is thus convenient to write the fluctu- in which kˆn represents the vertical wavenumber of the mode, ations of the self-gravitational potential, denoted , in the same defined as form, U 2 ! 1 dA A X ,σ kˆ2 = B + . (25) = m (x) Pm (cos θ) ei(σt+mϕ). (36) n U Ul l − 2 dx 2 σ,l,m A23, page 5 of 15 A&A 615, A23 (2018)

m,ν˜ m,ν˜ where the An,l and Bk,n designate the mutual projection coefﬁ- cients of the Hough functions on normalized associated Legen- dre polynomials, deﬁned respectively by the expansions X m,ν˜ m,ν˜ m Θn (θ) = An,l Pl (cos θ) , (43) l m X≥ m m,ν˜ m,ν˜ Pk (cos θ) = Bk,n Θn (θ) . (44) n

m,ν˜ The Cl,n,k coefficients quantify the coupling induced by the Coriolis effects in the dynamics of the tidal response. They are real in the non-frictional case (σR = 0 and ν˜ = ν), where m,ν m m,ν m,ν m,ν m m,ν An,l = Pl , Θn and Bk,n = Θn , Pk = An,k , the notation h i h i m m,ν , standing for the scalar product defined for any Pl and Θn ash· ·i Z π m m,ν m m,ν Pl , Θn = Pl (cos θ) Θn (θ) sin θ dθ. (45) Fig. 3. Frequency spectrum of waves likely to be excited by the tidal h i 0 perturbation and of the corresponding tidal regimes. The characteristic σ In the case of a non-rotating planet (Ω = 0 and ν = 0), there is no frequencies of the system are, from left to right, the drag frequency R, m,0 m,0 the inertia frequency 2Ω, the surface-gravity waves cut-off frequency coupling. Therefore, C = 1 if l = n + m = k, else C = 0. l,n,k l,n,k σg = √gH/R, the Brunt–Väisälä frequency N, and the acoustic cut- The notations ξm,σ = ξm,σ Um,σ and δρm,σ = δρm,σ Um,σ in off frequency σ = c /r. Their hierarchy can vary as a function of r;n,k r;n k n,k n k s s Eqs. (40) and (41) stand for the vertical displacement and density the physical and dynamical properties of the planet. In the case of m 5 1 4 1 variations due to the P -component of the forcing projected on the Earth’s ocean, σR 10− s− (Webb 1980), 2Ω = 1.46 10− s− , k 5 1 ≈ 3 1 × σg = 3.1 10− s− , N 2.2 10− s− (Gerkema & Zimmerman 2008), the (n, m, ν˜)-Hough function. The expressions of the components × 3 1≈ × and σs = 6.3 10− s− . of the tidal potential (Eqs. (40) and (41)) allow us to compute × complex Love numbers, which are commonly used to quantify tidal dissipation in celestial bodies (Tobie et al. 2005; Remus The radial profiles of the (m, σ)-components are themselves et al. 2012; Ogilvie 2014). Love numbers are defined as the ratio written between the gravitational potential due to the distortion and the tidal gravitational potential taken at the external surface of the m,σ = m,σ + m,σ, (37) body (r = R). Thus, oceanic Love numbers associated with the Ul Uξ;l Uρ;l m,σ triplet (l, m, σ), denoted kl , are defined as where we have introduced the contribution of the surface dis- m,σ m,σ m,σ m,σ l placement and of the internal density variations . In k = U . (46) Uξ;l Uρ;l l Um,σ the following, we use the subscripts ξ and ρ to make the distinc- l x=1 tion between the two contributions in a systematic way. At the n o n o Here, km,σ and km,σ stand for the tidal response of the planet surface (x = 1), the two components of the potential are < l = l expressed as ocean to the (l, m)-component of the excitating tidal potential. m The Ul are provided by the Kaula’s multipolar expansion of the tidal gravitational potential and are expressed as functions of the m,σ 4πG R m,σ (1) = ρ0 (1) ξ (1) , (38) Keplerian elements of the planet-perturber system (Kaula 1966; Uξ;l + r;l 2l 1 Mathis & Le Poncin-Lafitte 2009). "Z 1 l+2 # m,σ 4πG R H m,σ Because of internal dissipation, the variation of mass distri- ρ;l (1) = H 1 + x0 δρl dx0 , (39) U 2l + 1 0 R bution due to the tidal perturbation generates a tidal torque. The torque exerted by the perturber on the oceanic layer with respect to the spin axis of the planet, denoted , is given by with G representing the universal gravity constant. Expanded in T Hough functions, they write X = m,σ + m,σ . (47) T Tξ Tρ m,σ m,σ 4πG R X X m,ν˜ m,σ (1) = ρ0 (1) C ξ (1) , (40) Uξ;l 2l + 1 l,n,k r;n,k m,σ m,σ n k m The components and ρ of the (m, σ)-mode are defined ≥ Tξ T X X Z 1 l+2 as m,σ 4πG R m,ν˜ H m,σ ρ;n (1) = H Cl,n,k 1+ x0 δρn,k dx0. ( Z m,σ ) U 2l+1 0 R m,σ 1 ∂U m,σ n k m = ρ ξ (1) ∗ dS , (48) ≥ ξ s r (41) T < 2 ∂V0 ∂ϕ r=R

( Z m,σ ) In these expressions, the complex weighting coefficients m,σ 1 ∂U m,σ = (δρ )∗ dV , (49) m,ν˜ Tρ < ∂ϕ Cl,n,k are expressed as 2 V0 where stands for the conjugate of a complex number and its m,ν˜ m,ν˜ m,ν˜ ∗ < Cl,n,k = Bk,n An,l , (42) real part ( will be used here for the imaginary part), ρ = ρ (1) = s 0 A23, page 6 of 15 P. Auclair-Desrotour et al.: Oceanic tides from Earth-like to ocean planets denotes the density at the surface ocean, V0 the spatial domain Hence, we express the vertical structure equation (Eq. (24)) filled by the oceanic shell at rest, ∂V0 its upper boundary, and dS 2 m,σ 2 2 ! and dV infinitesimal surface and volume parcels, respectively. d Ψn 2 m,σ H m,ν˜ N δx + kˆ Ψ = Λ 1 U e− . (56) Similarly to the tidal potential due to the distortion, these two dx2 n n R2 n σσ˜ − n components can be expanded on Hough functions and associated Legendre polynomials. For l m and k m, one thus obtains, ˆ2 ≥ ≥ The vertical wavenumber to square kn, given by Eq. (25) in the X n h i o general case, is now the constant m,σ 2 m,ν˜ ∗ σ,m m,σ ∗ ξ = mπR ρs Cl,n,k Ul (1) ξr;n,k (1) , (50) T − = 2 2 ! ! 2 l,n,k H ,ν N gH N H kˆ2 = Λm ˜ 1 1 ε + + K δ2. (57) ( Z 1 ) n 2 n s;n 2 X h i R σσ˜ − − cs ◦ g − m,σ = mπR2H Cm,ν˜ ∗ Um,σ δρm,σ ∗ dx . (51) Tρ − = l,n,k l n,k l,n,k 0 In this expression, the first term is the vertical wavenumber of internal gravity waves; it dominates when σσ˜ N2, that is, for a stable stratification. Other terms correspond| | to acous- 3. Tides in a thin stably stratified ocean tic and sphericity terms usually ignored in anelastic (cs = + ) and thin-shell (H/R 0) approximations. Here, we will only∞ In some observed cases, the oceanic layer of terrestrial planets → and satellites is thin compared to the radius of the body (for ignore sphericity terms,that is, we assume that K = 0. We will 4 ◦ instance, the Earth’s ocean has an aspect ratio of 6 10− ; Eakins keep acoustic terms because they do not bring supplementary & Sharman 2010). Therefore, in this section, we apply× our gen- mathematical complexities into the analytic treatment and can, eral modelling to the simplified case of an oceanic layer of small furthermore, be comparable to those associated to stratification. depth H R, and compute an analytic solution of the oceanic For instance, in the case of the Earth’s ocean, density increases 3 3 tidal response, Love numbers, and tidal torque. from 1022 kg m− at the surface to 1070 kg m− at 10 km depth (Gerkema & Zimmerman 2008), which gives the mean gra- 2 2 2 dient dρ0/dz = 0.0048 kg m− . As ρ0g/cs 0.0043 kg m− , 3.1. Tidal waves dynamics the two terms of− Eq. (5) are comparable and≈ must be retained. The thin shell hypothesis allows us to simplify the physical setup. To solve the vertical structure equation, two boundary condi- Given that the relative difference of gravity between the lower tions are required. At x = 0, we set the impenetrable rigid-wall and the upper boundaries is proportional to the ratio H/R, g condition ξr = 0. At the upper boundary, we apply the usual free- is supposed to be constant in the oceanic layer. Similarly, the surface condition (e.g. Unno et al. 1989), δp = gρ0ξr. It follows radial variations of the tidal gravitational potential with x can that 2 2 be neglected (dUn/dx 0 and d Un/dx 0). Furthermore, (0) ≈ ≈ Ψn n δx for simplicity, we assume uniform stratification and compress- Ψn (x) = ne− ibility, that is, that the Brunt–Väisälä frequency (N) and the n D D δ h i sound velocity (cs) are constants. We note that the case of the + ( n δ) e− n sin kˆn x kˆn cos kˆn x N C − A − (58) Earth’s ocean does not well follow this assumption since decays over several orders of magnitude from the pycnocline + ( n δ) kˆn cos kˆn (1 x) (the upper region of the ocean where the density is changing A − − 1 o most rapidly) where its typical values are about 0.01 s− , to the + ( n δ) n sin kˆn (1 x) , 1 abyssal ocean, where they fall to 0.001 s− (e.g. Gerkema & A − C − Zimmerman 2008). Although a more realistic model would be where Ψ(0) is the constant given by more appropriate in this case, the uniform stratification appears n as a useful first step to solve the vertical structure equation and 2 m,ν˜ 2 ! (0) H Λn N derive explicit solutions controlling the tidal response of a strati- Ψn = 1 Un, (59) 2 2 ˆ2 2 σσ˜ − fied ocean. As shown by Eq. (5), setting N to a constant implies R kn + δ a density decreasing exponentially with altitude, and n and n the dimensionless coefficients expressed as τ(1 x) C D ρ0 (x) = ρse − , (52) H 2 n = n + N σσ˜ , (60) where τ designates the decreasing rate given by C A g − ˆ ˆ ˆ2 ˆ ! n = kn ( n n) cos kn n n + kn sin kn . (61) N2 g D C − A − A C τ = + . H 2 (53) g cs 3.2. Second-order tidal Love number and tidal torque The acoustic and sphericity terms (Eq. (22)) are simplified into By substituting Eq. (58) in Eqs. (29) and (33) we obtain the com- R2σ2 H ponents of the variation of mass distribution intervening in the εs;n and K 2 . (54) Love numbers and tidal torque (see Eqs. (50) and (51)) as explicit ≈ Λm,ν˜ 2 ◦ ≈ − R n cs functions of the internal structure parameters, Φ It follows that n (see Eq. (22)) writes Z 1 m,σ m,σ m,σ m,σ m,σ m,σ ξr;n,k (1) = H ξ;n Uk and δρn,k (x) dx = ρs ρ;n Uk , δx 1 Q 0 Q Φn (x) = e with δ = (τ + K ) . (55) 2 ◦ (62)

A23, page 7 of 15 A&A 615, A23 (2018)

h i ˆ2 ˆ2 n = ( n δ) γ n n + kn + kn ( n n) (69) K C − Ah B A − B i eτ ( δ) γ + kˆ2 + kˆ2 ( ) . − An − BnCn n n Cn − Bn m,σ Hence, ξ;n stands for the intrinsic response of the planet Q m,σ due to the variations of the oceanic surface level, while ρ;n corresponds to the contribution of internal inertial-gravityQ waves. This will later be equal to zero in the case of an incompressible and neutrally stratiﬁed ocean (τ = 0). The contribution of internal gravity waves to the tidal response with respect to that

of surface-gravity waves is weighted by the ratio ρ;n/ ξ;n . By using the expressions of the solution, we retrieveQ theQ weighting factor given in the literature (see e.g. Hendershott 1981, 2 Eq. 10.40), that is, ρ;n/ ξ;n N H/g. The oceanic second- Q2 Q ∼ order Love number k2 is then deduced from Eqs. (40), (41) and (46) straightforwardly. In the quadrupolar approximation, where terms of degrees l > 2 are neglected, it writes Fig. 4. Vertical displacement due to the quadrupolar semidiurnal G Moc X ,ν ,σ tide (σ = 2 (Ω n ), n being the dynamical frequency) obtained k2 = C2 ˜ 2 + 2,σ , (70) − orb orb 2 5R 2,n,2 Qξ;n Qρ;n using the analytic solution established for the thin layer approxima- n Z tion (Eq. (58)). The displacement, normalized by the quadrupolar tidal ∈ 2 potential, is plotted in the equatorial plane of the planet as a func- where Moc = 4πR Hρs designates the total mass of the ocean tion of longitude (horizontal axis) and normalized altitude (vertical in the weak stratification approximation (τ 1). Similarly, the axis). The position ϕ = 0 corresponds to the sub-perturber point; the n o tidal quality factor Q = k2/ k2 (Mathis & Le Poncin-Lafitte oceanic floor and surface are located at x = 0 and x = 1, respectively. 2 = 2 For this computation, the rotation period of the planet Pspin = 2π/Ω 2009) is expressed as and orbital period of the perturber Porb = 2π/norb are set to Pspin = X ,ν ,σ 6.0 d and Porb = 365.25 d. The used values of parameters are R = R , C2 ˜ 2 + 2,σ 2 3 1 1 ⊕ 2,n,2 ξ;n ρ;n H = 100 km, g = 9.81 m s− , N = 10− s− , cs = 1545 m s− and Q Q 5 1 n σR = 10− s− . Q = X n o , (71) C2,ν˜ 2,σ + 2,σ = 2,n,2 Qξ;n Qρ;n n where the frequency-dependent parameters m,σ and m,σ are Qξ;n Qρ;n and the quadrupolar tidal torque, which corresponds to the l = expressed as m = 2 component, as

,ν Λm ˜ n h io 1 2 X n o m,σ n 1 ˆ 2,σ = M U2,σ C2,ν˜ 2,σ + 2,σ . (72) ξ = n δ + n− n + n sin kn , oc 2 2,n,2 ξ;n ρ;n Q ;n − 2 ˆ2 2 A − D E F T 2 = Q Q R σσ˜ kn + δ n

(63) 2,σ where the quadrupolar component of the tidal potential U2 is 2 m,ν˜ given by m,σ N HΛn n 1 τ ρ;n = τ− ( n δ)(e 1) (64) r Q − 2 2 2 B − − 2 gR σσ˜ kˆ + δ 3 R G M? n 2,σ = .  U2 (73) + cos kˆ + sin kˆ  5 a a Gn Hn n Kn n  +  , γ2 + kˆ2  Expressed as a function of the degree-two tidal Love number Dn n given by Eq. (70), the tidal torque is written with γ = δ τ and the dimensionless coefﬁcients 5 − 2,σ 3 2 R n 2o = G M? k2 , (74) =kˆ ( δ)( ) eδ, (65) T 2 a6 = En n An − An − Cn which is the well-known expression of the tidal torque associated with the quadrupolar semidiurnal tide (e.g. Efroimsky & = ( δ) 2 + kˆ2 , (66) Fn Cn − An n Williams 2009; Makarov 2012; Correia et al. 2014).

n h i 3.3. Case of the neutrally stratified ocean (N = 0) ˆ δ ˆ2 n =kn ( n δ) e γ ( n n) + n n + kn (67) G A − h B − C B C io In analytic solutions describing oceanic tides, the stratification is γ ˆ2 + ( n δ) e− γ ( n n) + n n + kn , usually not taken into account, the layer being 2D and the fluid C − B − A A B assumed to be incompressible (e.g. Webb 1980; Tyler 2011, 2014; Chen et al. 2014; Matsuyama 2014). This reduces the oceanic n h i ˆ τ ˆ2 n = kn e ( n δ) γ ( n n) + n n + kn (68) tidal dynamics to horizontal flows. We show here that we recover H − Ah − B − C B C io the results given by this approach with our modelling, in which + ( δ) γ ( ) + + kˆ2 , Cn − Bn − An AnBn n the shallow-water case is a particular case. A23, page 8 of 15 P. Auclair-Desrotour et al.: Oceanic tides from Earth-like to ocean planets

Following the early works mentioned above, let us set N = 0 Table 1. Values of parameters used in Sect.4. (neutral stratification) and cs = + (incompressible fluid). The vertical wavenumber thus becomes∞ q Parameters Units Earth TRAPPIST-1 f H m,ν˜ kˆn = i Λn . (75) MM 1 0.36 R ⊕ RR 1 1.045 As shown by Eq. (75), kˆ is now directly proportional to the g ⊕ 2 . . n q m s− 9 81 3 23 ˆ m,ν˜ 2 H km 4.0 1000 horizontal wavenumber k ;n = Λn /R . In the weak-friction 3 ⊥ ρs kg m− 1022 1022 approximation (σR σ ) and the regime of super-inertial waves 1 | | cs m s− 1545 1545 ( ν˜ 1), it does not depend on the tidal frequency. Indeed, 1 3 3 | | ≤ ,ν , N s− 10− 10− in this case, the associated eigenvalues are Λm ˜ Λm 0 = n n M kg 7.346 1022 1.59 1029 (m + n)(m + n + 1) with n N (the relation between the≈ degree pert P days 27.32× 9.20 × l of associated Legendre polynomials∈ and the degree n has been orb m,ν˜ defined as l = m + n). The frequency dependence of Λn can- Notes. For the Earth, M , R , g, Porb and Mpert are given by NASA not be ignored any more in the regime of sub-inertial waves. fact sheets. The oceanic⊕ parameters⊕ H, ρ , c and N come from s s Because of the thin-layer approximation, kˆn 1. It follows Gerkema & Zimmerman(2008). For TRAPPIST-1 f, we use the values given by Wang et al.(2017) for M, R, M and P . The surface that sin kˆn x kˆn x and cos kˆn x 1. Hence, in the case of the pert orb ≈ ≈ gravity g is estimated using the definition g = G M/R and we use the neutrally stratified incompressible ocean, the analytic solution of oceanic parameters of the Earth. The ocean depth is arbitrarily set to Eq. (58) simply reduces to H = 1000 km. ˆ2 H ,σ gHk ;n R σσ˜ x Ψm = U ⊥ − , (76) n n σ σ σ σ+ 4.1. The Earth − n− − n + where σn− and σn are the frequencies of resonance of large-scale The Sun and the Moon exert on the Earth gravitational forcings of comparable intensities. The resulting tidal perturbation ( kˆ ;nR 1) damped surface inertial-gravity waves (see Fig.3) associated⊥ with the gravity mode of degree n. These frequen- is the sum of two contributions, the solid and oceanic tides, cies characterize the tidal response of the 2D incompressible which drive the rotational evolution of the planet as well as the ocean, where only surface-gravity waves can propagate (e.g. orbital evolution of the Earth-Moon system. We focus here on Webb 1980, Eq. (2.12)). They are expressed as the Lunar quadrupolar tide, which corresponds to the tidal fre- r quency σ = 2 (Ω norb), where Ω and norb stand for the spin σ σ 2 − R ˆ2 R frequency of the Earth and orbital frequency of the Moon. We σn± = i gHk ;n . (77) 2 ± ⊥ − 2 introduce the corresponding rotation (Prot = 2π/Ω) and orbital (P = 2π/n ) periods. Following Webb(1980), we use for We recognize in this expression the dispersion relation of orb orb the uniform oceanic depth the mean depth of the Earth’s ocean, large-scale surface-gravity waves, σ2 = gHkˆ2 (see e.g. Vallis ⊥ that is H = 4 km (Webb 1980). The radius of the planet is 2006, in the adiabatic limit). Like the vertical wavenumber, σn− = = + m,ν˜ set to R R km (with R 6378 km), the surface grav- and σ formally depend on σ through Λ but this depen- ⊕ ⊕ 2 3 n n ity and density to g = 9.81 m s− and ρs = 1022 kg m− . The 4 2 1 dence can be neglected in the regime of super-inertial waves if Earth’s ocean is characterized by N 10− 10− s− and the weak-friction approximation is assumed. The above approx- 1 ∼ − cs 1545 m s− (Gerkema & Zimmerman 2008), which means imations imply τ = 0 and m,σ = 0. Thus, the Love numbers ≈ Qρ;n that τ 1. Thus, the effects of stratification are negligible in and tidal torque (Eqs. (70) and (72)) are determined by the this case. We set the Brunt–Väisälä frequency to the intermedi- 3 1 contribution of the surface displacement solely, which reads ate value N = 10− s− . All these quantities are summarized in Λm,ν˜ 1 Table1. m,σ = n . ξ;n 2 + (78) The drag frequency characterizing the effective Rayleigh Q − R σ σn− σ σn − − friction (σR) is more difficult to specify because it models the m,σ The expression of Qξ,n shows that, in the absence of stable strat- effects of several mechanisms, such as turbulent friction, vis- ification, the coupling induced by the Coriolis effect will lead to cous friction, friction with topography, and breaking of tidal a set of resonances (one per Hough mode) enhancing the tidal waves (Garrett & Munk 1979; Garrett & Kunze 2007). Thus, we dissipation and tidal torque at frequencies σn±. begin by studying the dependence of the oceanic tidal response on σR. The frequency spectra of the imaginary part of the tidal Love number (Eq. (70)) and of the associated tidal quality fac- 4. Application to illustrative cases tor (Eq. (71)) are plotted in Fig.5 (left column) as functions of the normalized frequency ω = (Ω norb) /Ω (Ω stands for In this section, we apply the results established above to repre- − ⊕ ⊕ sentative telluric planets by using the thin-shell approximation of the today Earth rotation rate) for various orders of magnitude of Sect.3. We first treat the case of the Earth, which illustrates the σR. We observe on these plots the resonances associated with thin-layer asymptotic behaviour. However, we shall bear in mind surface-gravity modes modified by rotation. They correspond to the eigenfrequencies σ given by Eq. (77). As σ decreases, the that the global ocean approximation is a rough hypothesis in the n o n± R case of the Earth, where the interaction of the tidal perturbation variability of k2 increases. Particularly, the level of the non- = 2 with continents plays a central role (Webb 1980; Egbert & Ray resonant background decreases proportionally to σR, while the 1 2001, 2003). We then apply the model to TRAPPIST-1 f, which peaks heights increase as σR− , which is in good agreement with could be covered with a deep global ocean of liquid water owing the scaling laws derived in Auclair Desrotour et al.(2015). In to its small density (Mp 0.36 M and R 1.045 R , see Wang the asymptotic regime of strong friction, the behaviour of the et al. 2017). ≈ ⊕ ≈ ⊕ ocean is regular. The imaginary part of the tidal Love number

A23, page 9 of 15 A&A proofs:A&Amanuscript 615, A23 no. (2018) auclair-desrotour

2 2 γ = −78 γ = −8 ------+ + + + + + + - + γ = −8 1 6 5 4 3 2 1 0 0 1 2 3 4 5 6 γ = −7 1 0 0 γ = −7 6 γ = −7 γ = −6 γ = −6 0 0 γ = −6 γ = −55 γ = −5 γ = −5

)|] −1 γ = −4 )|] −1 γ = −4 2 2 γ = −432 2 γγ = = − −34 −2 −2 γ = −3 3 [Q] [|Im(k −3 [|Im(k −3 10 10 10 −4 2 −4 log log log −5 1 −5

−6 0 −6

−7 −1 −7 −4 −2 0 2 4 −4 −2 0 2 4 7 7 ω γ =− −28 ω γ = −8 −4 −2 0 2 4 6 γ = −7 6 γ = −7 γ = −6 ω γ = −6 5 5 γ = −5 γ = −5 4 γ = −4 4 γ = −4 γ = −3 γ = −3 3 3 [Q] [Q] 10 10 2 2 log log 1 1

0 0

−1 −1

−2 −2 −4 −2 0 2 4 −4 −2 0 2 4 ω ω Fig. 5. SemidiurnalSemidiurnal oceanic oceanic tide tide of of the the Earth Earth (left (left column) column) and and Trappist-1 TRAPPIST-1 f (right f (right column). column The). logarithms The logarithms of the of imaginary the imaginary part of part the of tidal the Love tidal Lovenumber number (left) and (left tidal) and quality tidal quality factor (right) factor are (right plotted) are as plotted functions as functions of the normalized of the normalized forcing frequency forcing frequencyω = (Ω nωorb=) /(Ω (wherenorb) /ΩΩ (wheredesignatesΩ − ⊕− ⊕⊕ ⊕ designatesthe today rotation the today rate rotation of therate Earth) of forthe various Earth) for orders various of magnitude orders ofmagnitude of the drag of parameter the dragγ parameter= log (σRγ).= Theselog (σ frequencyR). These spectra frequency are spectra computed are computedusing the expressions using the expressions given by Eqs. given (70 by) and Eqs. (71 (70).) In and each (71 case,). In each the parameter case, the parameternorb is assumednorb is to assumed be constant to be and constant the rotation and the rate rotation of the rate planet of thevaries planet with varies the tidal with frequency the tidal following frequency the following formula theΩ = formulanorb + σ/Ω2. = Thenorb resonances+ σ/2. The associated resonances with associated surface inertial-gravity with surface inertial-gravity modes are designated modes areby black designated dashed by lines black in dashed the top lines panels in and the numbers top panels indicate and numbers the degree indicaten of the the corresponding degree n of the Hough corresponding modes and Hough the sign modes of eigenfrequencies and the sign of eigenfrequenciesgiven by Eq. (77).The given values by Eq. of (77 parameters).The values used of for parameters this evaluation used for are this summarized evaluation in are Table summarized1. in Table1. orbital evolutionn o of the Earth-Moon system. We focus here on Table 1. Values of parameters used in Sect.4. For the Earth, M , R , 2 of rocky planets (for instance, the mean density of the Earth⊕ ⊕ is scales as k σ in the zero-frequency limit (σ 0) and as g, P and M are given by NASA fact sheets. The oceanic param- then Lunaro = quadrupolar2 ∝ tide, which corresponds to the→ tidal fre- ρ¯ =orb 5514 kgpert m 3; see NASA fact sheets). This means that 2 3 eters H, ρ , c and−N come from Gerkema & Zimmerman(2008). For quencyk σσ=− 2at(Ωσ norb+),. where Ω and norb stand for the spin ⊕ s s = 2 ∝ | −| → ∞ TRAPPIST-1water could f,stand we use for the an values important given fraction by Wang of et the al.(2017 planet) for mass.M, frequencyThe observed of the Earth receding and orbitalof the frequencyMoon is estimated of the Moon. to 3.82 We Typically, the water fraction is estimated to be between 25 and introduce the corresponding rotation (P = 2π/Ω) and orbital R, Mpert and Porb. The surface gravity g is estimated using the definition cm per year (Dickey et al. 1994; Billsrot & Ray 1999). This 100%= of/ the planet mass. Moreover, with a greenhouse effect, (P = 2π/n )n periods.o Following Webb(1980), we use for g G M R and we use the oceanic parameters of the Earth. The ocean correspondsorb toorb k2 2.56 10 2 for the tidal Love number depthTRAPPIST-1 is arbitrarily f is set sufficiently to H = 1000 irradiated km. by its host star to have the uniform oceanic= 2 depth≈ the× mean− depth of the Earth’s ocean, 16 a surface temperature compatible with liquid water (its black and =4.50 10 N m for the tidal torque, which are the that isTH ≈ 4 km× (Webb 1980). The radius of the planet is set to body equilibriumarameters temperaturenits isarth estimated to 219rappist K, seef Wang valuesR = R obtainedkm (with byR setting= 6378 the km), Rayleigh the surface drag gravity frequency and den- to P U E T -1 ⊕ 5 1 ⊕ et al. 2017). All these features argue for the possible. presence σsityR = to10g −= s9−.81in m. ours 2 and modelρ = (Fig.10225, kg right.m 3 panel).. The Earth’s We hence ocean MM 1 0 36 − s − of a deep oceanic layer⊕ on TRAPPIST-1 f. In. such a case, if retrieveis characterized the 30 h by frictionN 10 time4 scale10 2 estimateds 1 and c by Webb1545(1980 m.s )1 RR 1 1 045 ∼ − − − s − the layer is stably stratified,⊕ 2 the effect of stratification cannot with a similar approach.∼ This value− of σR is used≈τ in Sect.5 to g m.s− 9.81 3.23 (Gerkema & Zimmerman 2008), which means that 1. Thus, be ignored any more because the variation rate of the density explorethe effects the of domain stratification of parameters. are negligible in this case. We set the H km 4.0 1000 3 1 profileρ (τ defined by Eq.. (353)) is not necessary very small with Brunt-Väisälä frequency to the intermediate value N = 10− s− . s kg m− 1022 1022 respect to 1. Thus, as in the1 case of the Earth, the tidal response is 4.2.All these TRAPPIST-1 quantities f are summarized in Table1. cs m.s− 1545 1545 mainlyN composed ofs surface-gravity1 10 3 waves modified10 3 by rotation. To illustrateThe drag the frequency behaviour characterizing of a deep, stably the stratified effective ocean, Rayleigh we − − − But itM is also composedkg of internal 7.346 gravity1022 waves,1.59 restored1029 by considerfriction (σ the) is case more of di TRAPPIST-1fficult to specify f. TRAPPIST-1 because it models f is one the ef- of pert R the ArchimedeanP forcedays and inducing 27.32× internal 9density.20 × variations thefects eight of several telluric mechanisms, planets recently such discovered as turbulent in friction, the vicinity viscous of (Fig.3).orb thefriction, star TRAPPIST-1 friction with ( topography,Gillon et al. and2017 breaking; Wang et of al. tidal 2017 waves). Its We investigate this complex behaviour by considering an radius(Garrett is &very Munk close 1979 to that; Garrett of the &Earth Kunze (R = 20071.045). Thus,M ) whereas we be- idealized TRAPPIST-1 f planet with a global ocean of depth itsgin mass by studying is only M the= 0 dependence.36 M . As of a consequence, the oceanic tidal its⊕ mean response den- torH = (Eq.1000 71)km are and plotted a stratification in Fig.5 (left similar column) to that as of functions the Earth’s of ⊕ 3 3 1 sityon σ isR. equal The frequency to ρ¯ = 1740 spectra kg m− of, which the imaginary is far smaller part of than the those tidal theocean, normalized that is N frequency= 10− s− ω. Assuming= (Ω norb the) / orbitalΩ (Ω periodstands of forthe Love number (Eq. 70) and of the associated tidal quality fac- the today Earth rotation rate) for various− orders⊕ of⊕ magnitude A23, page 10 of 15 Article number, page 10 of 15 P. Auclair-Desrotour et al.: Oceanic tides from Earth-like to ocean planets planet (norb) to be constant, we study the frequency dependence inertial-gravity waves tend to be mixed with acoustic waves of the quadrupolar component of the semidiurnal stellar tide (cf. while Prot 0. In the sub-synchronous regime (area located Table1) by letting Ω vary with σ. Similarly to the previous above the diagonal→ line), the forcing frequency is greater than n o case, the corresponding spectra of k2 and Q are plotted on the inertia frequency (super-inertial regime) and the only mode = 2 Fig.5 (right column) as a function of the forcing frequency. In coupled with the forcing is the gravity mode of degree n = 0. the high-frequency range we retrieve the resonances associated As a consequence, only one resonance appears in the super- 2,σ 2 with surface-gravity modes. The two peaks appearing around inertial regime. As the tidal potential scales as U2 norb (see 4 ∝ ω 4.5 correspond to the Hough mode of degree n = 0, which Eq. (73)), the tidal torque scales as Porb− (see Eq. (72)), which cor- is≈ the ± surface-gravity mode of lowest eigenfrequency. Given responds to the horizontal colour gradient of the maps plotted in that σn± √gH in the weakly frictional regime (see Eq. (77)), Fig.7. the resonances∝ are translated towards the high-frequency range By moving to deeper oceans, we observe that the resonances with respect to the case of the Earth (left column). In addi- are translated towards the small-period range, following the scal- 3 1 tion to surface-gravity modes, we observe resonances caused by ing law σn± √gH identified above. For N = 10− s− , the the propagation of internal gravity waves in the low-frequency stratification is∝ sufficiently strong to allow internal gravity waves range. These modes can exist for σ . N (see Fig.3), which to propagate. Therefore, new resonances appear in the frequency corresponds to ω . 5. | | range σ . N (see right panels of Figs.6 and7). This effect is | | enhanced| | for H = 1000 km. As the eigenfrequencies of internal gravity waves do not depend on Ω like those of surface gravity 5. From shallow to deep oceans waves, their wings patterns are symmetrical with respect to the diagonal axis. Moreover, given that the contribution of internal In the previous sections, we identified the depth of the ocean waves is weighted by the dimensionless number N2H/g, the non- and the stability of its stratification as the key structure parame- resonant background is amplified in the frequency range σ . N. ters characterizing the oceanic tidal response. The depth deter- The effects of the resonances associated with internal| gravity| mines the eigenvalues of resonant surface-gravity modes and waves are attenuated by the Rayleigh drag. When σR 0, reso- the Brunt–Väisälä frequency, the frequency range of internal nances tend to increase the dependence of the tidal torque→ and gravity modes. Further, the contribution of these modes is Love number on the forcing frequency (as expected from the 2 weighted by the dimensionless number N H/g, which compares general scaling laws derived in Auclair Desrotour et al. 2015). the Archimedean force to the gravity force. Therefore, we examine in this section the sensitivity of tidal dissipation to H and N. We consider the case of idealized Earth sisters hosting a 6. Discussion global ocean and submitted to the semidiurnal gravitational forcing of any perturber orbiting in their equatorial plane. These In this work, we have opted for a simplified linear analysis allow- planets are characterized by various orders of magnitude of ing us to widely explore the domain of parameters at a reasonable ocean depths (H = 10, 100, and 1000 km) and Brunt–Väisälä computational cost. This approach is convenient to examine the 4 3 1 frequency (N = 10− , 10− s− ), all the other parameters being frequency-resonant behaviour of a global ocean and highlight the unchanged (see Table1). Following the discussion of the pre- key parameters of the problem, which can be explained in more vious section, the frequency associated with the Rayleigh drag detail in a second phase. Particularly, we identify and quantify 5 1 is set to σR = 10− s− in a first case (friction time scale of in a consistant way the contribution of internal gravity waves 6 1 the Earth’s ocean). In a second case, it is set to σR = 10− s− , restored by the stable stratification. This contribution can be which corresponds to a weaker friction, to broaden the explored important in the case of deep oceans, where the combined effects parameter space. Hence, for each planet, we plot in Fig.6 the of tides and stratification induce important density fluctuations. imaginary part of the tidal Love number as a function of the However, we shall discuss here the main simplifications assumed orbital period of the perturber and the rotation period of the in the model: planet. Then, assuming that the perturber is a Moon-like satel- Global ocean of uniform depth – The large scale topo- 22 • lite (Mpert = 7.346 10 kg), we plot the corresponding tidal graphical features of the oceanic floor and continents are torque exerted on planets× in Fig.7. not taken into account. Because of this spherical symme- As it may be noted, the three maps plotted for H = 10 km try, the obtained oceanic tidal response is very regular and (bottom panels of Figs.6 and7) have the same aspect what- exhibits resonances corresponding to global spherical modes ever the value of N. This can be interpreted as the fact that (see e.g. Fig.5). The case of a hemispherical ocean centred the tidal response generated by the gravitational forcing exerted at the equator was treated in early studies in the shallow- on the ocean is mainly composed of surface-gravity waves. In water approximation (Proudman & Doodson 1936; Doodson other words, the tidal response is dominated by its barotropic 1938; Longuet-Higgins & Pond 1970; Webb 1980). Meridian component, which causes the role played by the stratification to continental shelves prevent large-scale gravity waves from appear negligible in this case, as mentioned before. We identify propagating. They thus couple the global gravitational forc- the resonances due to surface inertial-gravity modes observed ing to modes determined by the scale of the basin. This in the case of the Earth in the previous section (Fig.5, left significantly modifies the aspect of the frequency spectrum column). They form a typical wings pattern, which is proper of the tidal response (see e.g. Webb 1980). to these waves (see e.g. Fuller & Lai 2014, Fig. 7, in the Rayleigh friction – All of the dissipative mechanisms damp- case of white dwarves). However, we note that this pattern is • ing the tidal response are reduced to a single parameter, not symmetrical with respect to the diagonal line designating the effective Rayleigh coefficient σR. This approximation synchronization. This is a consequence of the effects of the Cori- is common in the literature (e.g. Webb 1980; Egbert & olis acceleration. In the super-synchronous regime (area located Ray 2001, 2003; Ogilvie 2009; Tyler 2011) and assumed below the diagonal line), they induce a distortion that cou- for convenience. In addition to the traditional approxima- ples the quadrupolar forcing to several Hough modes. These tion, discussed in the following paragraph, it allows us to

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n 2o n 2o Fig. 6. Imaginary part of the tidal Love number k2 associated with the quadrupolar oceanic tide. The logarithm of k2 is plotted as a = = 6 1 function of the orbital (horizontal axis) and rotation (vertical axis) periods in logarithmic scale by using Eq. (70) for σR = 10− s− (bottom) and 5 1 σR = 10− s− (top), and various values of H and N. Horizontally, H = 10 km (left), H = 100 km (middle) and H = 1000 km (right). Vertically, 4 1 5 1 N = 10− s− (bottom) and N = 10− s− (top). Colours correspond to logarithmic decades (colour-bars on the right). The diagonal black dashed line corresponds to spin-orbit synchronization.

separate the x and θ coordinates in the dynamics. Rayleigh waves on continental shelves although this effect stands for friction can be reasonably used to describe the drag caused the major part of the tidal energy in the case of the Earth. by small-scale topographical features homogeneously dis- Traditional approximation – This simpliﬁcation is com- tributed around the planet, as in the numerical model of • monly used in the literature to solve the latitudinal and Egbert & Ray(2001). It can also be considered a simpliﬁed vertical structure of the tidal response separately (e.g. Unno approximation of viscous and turbulent frictions if a length- et al. 1989). It consists in neglecting the latitudinal com- scale L such that ∆V V/L2 is introduced. However, it does ponent of the rotation vector in the Coriolis acceleration. not at all model the∼ local dissipation due the breaking of This means that we ignore both the radial component of the

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Fig. 7. Tidal torque associated with the Lunar quadrupolar oceanic tide. The logarithm of is plotted as a function of the orbital (horizontal axis) 6 1T 5 1 and rotation (vertical axis) periods in logarithmic scale by using Eq. (72) for σR = 10− s− (bottom) and σR = 10− s− (top), and various values of 4 1 3 1 H and N. Horizontally, H = 10 km (left), H = 100 km (middle) and H = 1000 km (right). Vertically, N = 10− s− (bottom) and N = 10− s− (top). Colours correspond to logarithmic decades (colour-bars on the right). The diagonal black dashed line corresponds to spin-orbit synchronization.

Coriolis force and the horizontal component of the Corio- stratified ocean (N 0). This requires one to use 3D (2D if lis force associated with radial motions. Consequently, this solutions eimϕ are≈ assumed) numerical or semi-analytical approximation is appropriate to 2D models, where vertical methods (see∝ e.g. Ogilvie & Lin 2004). The Archimedean motions are negligible relative to horizontal ones. In the case force acts as a restoring force on fluid particles in the verti- of deeper oceans, its domain of applicability depends on the cal direction. Therefore, if this restoring force is sufficiently hierarchy of the inertia (2Ω), Brunt–Väisälä (N), and forc- strong, it prevents vertical motions from being comparable in ing (σ) frequencies. In the super-inertial asymptotic regime amplitude to horizontal ones. This condition is expressed as ( 2Ω σ ), the forcing time scale is small compared to the 2Ω N (e.g. Auclair-Desrotour et al. 2017a). The validity rotation| | period,| | which makes the traditional approximation |of the| traditional approximation has been examined in vari- appropriate. However, when this condition is not satisfied, ous fluid layers, from planetary oceans and atmospheres (e.g. motions are fully tridimensional in the case of a neutrally Gerkema & Shrira 2005; Gerkema & Zimmerman 2008) to

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stellar interiors (e.g. Mathis 2009; Tort & Dubos 2014; Prat The tidal response is composed of resonant surface inertial- et al. 2017). gravity modes due to the conjugated effects of gravity and Solid rotation – As we consider in our model that the oceanic rotation. In this case, the role played by stratification is negli- • layer rotates uniformly with the planet at the angular veloc- gible. The fluid compressibility can affect the tidal response in ity Ω, we ignore the complex interplay existing between the high-frequency range, where horizontally propagating Lamb tidal waves and mean flows (e.g. Favier et al. 2014; Guenel modes can be excited. The importance of the role played by the et al. 2016). By studying the propagation of internal gravity stratification grows with the ocean depth. The stable stratification waves in a shear flow, Booker & Bretherton(1967) showed allows internal gravity waves to propagate. It induces inter- that the amplitude of oscillations is attenuated by a fac- nal density fluctuations characterized by a frequency-resonant tor depending on the local Richardson number of the fluid, behaviour, similar to surface gravity waves. The contribution 2 2 Ri = N dV0/dr − (where V0 stands for the velocity vec- of this component is not negligible in the case of deep oceans tor of the| shear| flow), as waves pass through a critical and sensitively increases the evolution timescales of the planet- level at which their horizontal phase speed is equal to the perturber system. zonal velocity of the flow. By this mechanism, tidal waves Although the linear analysis developed in this work is can transfer horizontal momentum to mean flows, which simplified with respect to numerical models, it provides a modifies the tidal response. very convenient way to explore the broad domain of plane- Free-surface boundary condition – In this work, a free- tary parameters and unravel the complex dynamics of oceanic • surface boundary condition is used to derive analytic solu- tides with a reasonable computational cost and few physical tions. This condition corresponds well to the case of external control parameters. These analytical results can be used in oceanic layers, where the upper surface can move freely. the future as benchmark validations for fully 3D GCM tidal However, it is not appropriate to subsurface oceans such as studies. A next step should be to introduce meridian bound- those presumedly hosted by Europa-like icy satellites (e.g. aries in the model in order to study the effect of blocking Khurana et al. 1998; Kivelson et al. 2002). In order to study the progression of the tidal response and thus better character- these objects, results obtained in this work can easily be ize the role played by continental shelves in the oceanic tidal applied to the case of a rigid lid by replacing for instance the dissipation. standard free-surface condition by a rigid-wall condition. Coupling with the solid part – In order to simplify the anal- Acknowledgements. The authors acknowledge funding by the European Research • Council through ERC grants SPIRE 647383 and WHIPLASH 679030, the Pro- ysis, the solid part of the planet has been considered as a gramme National de Planétologie (INSU/CNRS) and the CNRS PLATO grants sphere of infinite rigidity. In reality, the solid part behaves at CEA/IRFU/DAp. They wish to thank the referee, Robert Tyler, for his helpful as a visco-elastic body (e.g. Efroimsky 2012) forced both suggestions and remarks. by the tidal potential and the loading induced by the tidal distortion of the ocean. This solid-fluid coupling affects the frequencies given by Eq. (77) and generates an additional References coupling between tidal modes (e.g. Matsuyama 2014). We Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical Functions, simply ignored this coupling so that we did not need to 1046 consider the parameters of the solid response. Anglada-Escudé, G., Amado, P. J., Barnes, J., et al. 2016, Nature, 536, 437 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 Auclair-Desrotour, P., Laskar, J., & Mathis, S. 2017a, A&A, 603, A107 7. Conclusion Auclair-Desrotour, P., Laskar, J., Mathis, S., & Correia, A. C. 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