On the energetics of a tidally oscillating convective flow

This paper examines the energetics of a convective flow subject to an oscillation with a period $t_{\rm osc}$ much smaller than the convective timescale $t_{\rm conv}$, allowing for compressibility and uniform rotation. We show that the energy of the oscillation is exchanged with the kinetic energy of the convective flow at a rate $D_R$ that couples the Reynolds stress of the oscillation with the convective velocity gradient. For the equilibrium tide and inertial waves, this is the only energy exchange term, whereas for p modes there are also exchanges with the potential and internal energy of the convective flow. Locally, $\left| D_R \right| \sim u^{\prime 2} /t_{\rm conv}$, where $u^{\prime}$ is the oscillating velocity. If $t_{\rm conv} \ll t_{\rm osc}$ and assuming mixing length theory, $\left| D_R \right|$ is $\left( \lambda_{\rm conv}/ \lambda_{\rm osc} \right)^2$ smaller, where $\lambda_{\rm conv}$ and $\lambda_{\rm osc}$ are the characteristic scales of convection and the oscillation. Assuming local dissipation, we show that the equilibrium tide lags behind the tidal potential by a phase $\delta \left( r \right) \sim r \omega_{\rm osc}/ \left( g \left( r \right) t_{\rm conv} \left( r \right)\right)$, where $g$ is the gravitational acceleration. The equilibrium tide can be described locally as a harmonic oscillator with natural frequency $\left( g / r \right)^{1/2}$ and subject to a damping force $-u^{\prime}/t_{\rm conv}$. Although $\delta \left( r \right) $ varies by orders of magnitude through the flow, it is possible to define an average phase shift $\overline{\delta}$ which is in good agreement with observations for Jupiter and some of the moons of Saturn. Finally, $1 / \overline{\delta}$ is shown to be equal to the standard tidal dissipation factor.

The fact that observations can be accounted for by assuming that energy is systematically transferred from the tides to convection, both when the tides act as the fluctuations or as the mean flow, is compelling. However, it is worth noting that in either case there is no proof so far that tidal energy is al w ays dissipated.
The analysis in  was restricted to incompressible and non-rotating flows. The present paper aims at generalizing this analysis by including compressibility and uniform rotation. The addition of compressibility will enable the application of the formalism to the case where the oscillations are pressure modes, which are associated with time-scales much shorter than the conv ectiv e time-scales in stars. It is believed that they are damped through their interaction with the conv ectiv e flow, and this is usually modelled using mixing length theory. Ho we ver, this has been found not to be in good agreement with numerical simulations (Basu 2016 ), which is not surprising in the context of the new formalism which applies when t osc t conv . The addition of uniform rotation will also enable the application of the formalism to inertial waves. This is important because, if the tidal frequency is smaller than twice the rotation frequency, the response of a rotating star or giant planet to a tidal perturbation is the superimposition of an equilibrium tide and propagating inertial waves. It has been assumed in previous studies that these waves are damped through their interaction with a turbulent conv ectiv e viscosity. Howev er, like for the equilibrium tide, damping of inertial waves cannot be described by mixing length theory when t osc t conv . This paper examines in full generality in which form energy is transferred between an oscillation and convection. We take into account all the energy stores and examine all the routes that the energy of the oscillation could potentially travel through.
We then focus on tidal oscillations and, assuming local dissipation, calculate the phase by which the tide lags behind the perturbing potential. We also explore the damped harmonic oscillator model for the equilibrium tide.
The plan of the paper is as follows. In Section 2 , we write the energy conservation equations for the flow as a whole, without separating the oscillation from the conv ectiv e flow. Assuming t osc t conv , we then perform a Reynolds decomposition and write the kinetic, potential, and internal energy conservation equations for the mean conv ectiv e flow and for the oscillation separately in Section 3 . These equations are av eraged first o v er the time-scale t osc and then o v er a time-scale long compared to t conv . We show that the energy of the oscillation is exchanged with the kinetic energy of the conv ectiv e flow at a rate D R per unit mass that couples the Reynolds stress of the oscillation with the conv ectiv e velocity gradient. If the oscillation is the equilibrium tide or an inertial wave, D R is the only term that exchanges energy between the oscillation and the conv ectiv e flow, and it is al w ays balanced by the w ork done by the tidal force, whether this is positiv e or ne gativ e. If the oscillation is a p mode, there is an additional exchange with the potential and internal energy of the conv ectiv e flow, because of compressibility. In Section 4 , we discuss how to express D R in terms of the flow velocities. We re vie w the standard cases of viscous and turbulent shear flows, and examine oscillations with t osc either small or large compared to t conv . This makes it clear that many of the questions that arise about the direction of energy transfer when the tides are fast are also rele v ant when the tides are slow and play the role of the mean flow. The two cases should therefore be examined together. When t conv t osc , | D R | ∼ u 2 /t conv , where u is the oscillating velocity. If t conv t osc and assuming mixing length theory, | D R | is ( λ conv / λ osc ) 2 smaller, where λ conv is the mixing length and λ osc is the spatial scale on which the oscillation varies. This applies to p modes, the equilibrium tide and inertial waves. In Section 5 , we focus on tidal oscillations and assume local dissipation. We show that the phase shift between the oscillation and the tidal potential varies by orders of magnitude through the flow. For the equilibrium tide, which we model as a harmonic oscillator, it is ho we ver possible to define a mean phase shift for the flow which can be compared to the v alues deri ved from observ ations for Jupiter and Saturn. The inverse of this mean phase shift is equal to the standard tidal dissipation factor Q = 2 πE p / E , where E p is the potential energy in the tide and E is the total energy dissipated during a tidal period. Finally, we summarize and discuss our results in Section 6 .

T H E D I F F E R E N T F O R M S O F E N E R G Y
We consider a body (solar-type star or giant planet) in uniform rotation which has an envelope in which energy is transported by convection. We note Ω the angular velocity vector of the body, u the velocity of the gas in the rotating frame, ρ its density and p its pressure. The body is subject to a tidal force per unit mass f t = −∇ Ψ t , where Ψ t is the tidal potential in the rotating frame.

Conser v ation of energy
To help the discussion, we first recall the equations expressing conservation of energy for the flow in the conv ectiv e env elope without separating tidal oscillations and conv ectiv e motions. The flow satisfies the mass conservation equation and Navier-Stokes equation, which i -component in Cartesian coordinates is ρ ∂u i ∂t + ρu j ∂u i ∂x j = − ∂p ∂x i + ρg i + ρf c,i + ρf t,i + ∂σ ij ∂x j , where g i is the (ne gativ e) acceleration due to the gravity of the body itself in the i -direction, f c = −2 Ω×u is the Coriolis force per unit mass, σ ij is the viscous stress tensor and repeated indices are summed o v er. In the cases of interest, the centrifugal force is very small compared MNRAS 525, 508-526 (2023) to the gravitational force, into which it can be subsumed. F or e xample, in the Sun, the centrifugal force is about 10 5 times smaller than the gravitational force (Miesch 2005 ). For this reason, it is neglected here. In a solar-type star, almost all of the mass is in the radiative core, so that g −GM / r 2 , where M is the total mass of the star. In principle, to calculate g in the envelope of a giant planet, self-gravity has to be taken into account. Ho we ver , in the parts of the en v elope where tides hav e a significant amplitude, g is also well approximated by −GM / r 2 , where M is the total mass of the planet. This approximation will therefore be used thereafter. We note Ψ 0 the associated gravitational potential, defined through g = −∇ Ψ 0 (i.e. in spherical coordinates, Ψ 0 = −GM /r ). Multiplying equation ( 2 ) by u i , summing up o v er i and using equation ( 1 ) yields the equation for kinetic energy conservation ∂ ∂t An equation for the gravitational potential energy is obtained by multiplying equation ( 1 ) by the potential energy per unit mass Ψ = Ψ 0 + Ψ t .
Assuming Ψ 0 to be independent of time, this yields Equations ( 3 ) and ( 4 ) show that potential and kinetic energies are exchanged through the work ρ ( g + f t ) · u done by the total gravitational force, which includes both the tidal force and the self-gravitational force. The term ρ ( ∂ Ψ t /∂ t ) represents the excess or deficit of potential energy due to the time dependence of the tidal potential. Although it does not appear to involve work done, we will see in Section 3.7 that, when integrated over the envelope, its average over a tidal period is actually equal to the average work done by the tidal force. An equation for the internal energy is obtained from the first law of thermodynamics where e int is the internal energy per unit volume and q is the radiative flux of thermal energy. Conv ectiv e and radiativ e transport of energy are contained in the first and last terms on the right-hand side, respectively. Equations ( 3 ) and ( 5 ) show that kinetic and internal energies are exchanged through viscous dissipation and through the work p ∇ · u done by the pressure force. By adding equations ( 3 ), ( 4 ), and ( 5 ), we obtain a conservation equation for the total energy per unit volume e tot = ρu 2 / 2 + ρΨ + e int The work done by the tidal force does not appear as a source term for the total mechanical energy, no more than the work done by the buoyancy force does, because the tidal potential is included in the potential energy.

Energy dissipation
We now integrate equation ( 6 ) over the whole volume of the convective envelope. The divergence term on the right-hand side becomes an inte gral o v er the surf aces. The contribution from the outer surf ace is ne gligible as ρ and p are v ery small there. At the inner surface, the conv ectiv e v elocity vanishes, the v elocity of the equilibrium tide is v ery small, and the radial v elocity of inertial wav es has to vanish to satisfy the boundary condition with either a solid core or a radiative layer. The contribution from the inner surface is therefore also negligible, and the surface integral can be ignored.
In the absence of tides, transport of energy throughout the envelope adjusts itself so that a steady state is maintained (i.e. e tot integrated o v er the envelope is constant) over time-scales long compared to that of the flow and small compared to the time-scale on which the internal structure of the body evolves. When tides are present, the total energy has contribution from the tidal oscillation. Averaged over a tidal period, this contribution is constant o v er a time-scale short compared to the tidal evolution time-scale of the binary. Therefore, o v er such time-scales, the integral of the term on the left-hand side of equation ( 6 ) is zero and we then obtain where V is the volume of the conv ectiv e env elope, F rad is the radiativ e flux through the outer surface S, and E core is the thermal energy entering through the inner surface (due to nuclear energy production in a star and other mechanisms in a giant planet). The brackets denote an average o v er the tidal period.
The equation abo v e shows that, if the tidal potential results in an excess of potential energy (i.e. the integral on the left-hand side is positive), it has to be ultimately radiated away along with the energy E core produced in the core of the body. As potential energy cannot be converted directly into internal energy, which is the only form of energy which can be radiated away, it first has to be converted into kinetic energy. Equation ( 7 ) can also be written as  (2023) where the term on the right-hand side is the perturbation to the net surface flux due to the tidal oscillation, which has contribution from the perturbation to the flux itself, but also from the perturbation to the surface normal and to the surface area. An explicit calculation shows that this term is proportional to the Lagrangian variation of the radial flux at the surface, which itself depends on the radial component of the tidal displacement there (Dziembowski 1977 ;Bunting & Terquem 2021 ). Numerical simulations aiming at calculating tidal dissipation should therefore allow the outer surface of the flow to oscillate under tidal forcing. Enforcing rigid boundaries may prevent the energy to be released and create spurious results.

E Q UAT I O N S F O R T H E O S C I L L AT I O N A N D T H E M E A N F L OW
The response of a rotating conv ectiv e flow to tidal forcing includes a non-w avelik e part (equilibrium tide) to which are superimposed propagating inertial waves when the tidal frequency is smaller than twice the rotation frequency. In addition to tidally forced oscillations, a conv ectiv e flow can also support propagating pressure waves.
In the analysis presented below, we consider a forced oscillation in a compressible conv ectiv e flow in uniform rotation. The oscillation can therefore be identified with either an equilibrium tide, the sum of an equilibrium tide and a propagating inertial wave if the rotation frequency is large enough, or a propagating pressure wave (for which f t has to be thought of as the force that excites the wave rather than a tidal force).
We note t osc and ω osc the period and frequenc y, respectiv ely, of the oscillation in the rotating frame. F or tidal oscillations, we hav e ω osc = 2 | ω orb − Ω| , where ω orb is the orbital frequency of the binary (we only consider circular orbits). In this section, we assume t osc t conv , where t conv is the characteristic time-scale associated with conv ectiv e motions. We will however also discuss the case where the oscillation time-scale is the longest one in the following section.

Reynolds decomposition
We use the Reynolds decomposition in which the total velocity in the rotating frame is written as u ( r , t ) = V ( r , t ) + u ( r , t ) , where u is the velocity associated with the periodic oscillation in the rotating frame. In other words, if we note . . . a time-average over t osc , then u ( r , t ) = 0 . This defines V ( r , t ) ≡ u ( r , t ) as the mean velocity, and this is the conv ectiv e v elocity in the rotating frame. As evidenced by the Sun, convection in the presence of rotation may induce differential rotation. This is not al w ays the case, ho we ver, as Jupiter for example is mostly in rigid body rotation. When differential rotation is present, then it can be thought of as being included in V (e.g. Durney & Spruit 1979 ).
A Reynolds decomposition can also be made for the pressure p ( r , t ) = p 0 ( r ) + δp ( r , t ) + p ( r , t ) and the mass density ρ ( r , t ) = ρ 0 ( r ) + δρ ( r , t ) + ρ ( r , t ) , with p ( r , t ) = ρ ( r , t ) = 0. In other words, ρ and p are the zero-mean density and pressure perturbations associated with the oscillation, δρ and δp are the fluctuations due to convection, and ρ 0 and p 0 are the density and pressure in the fluid at hydrostatic equilibrium (i.e. the values the density and pressure would have in the absence of convection and oscillation).
We will also assume that the time deri v ati ve of the oscillating quantities u , ρ , and p average to zero over a time t osc , as is the case for periodic oscillations.
The Reynolds decomposition above allows the oscillation to couple to convection through the non-linear term in Navier-Stokes equation, but does not allow mode-mode coupling for p modes or inertial waves. This could only be captured by having a sum of different u , each for a different wave, in the decomposition for u . Since p modes are excited by the turbulent conv ectiv e flow itself, modes with different frequencies co-exist and interact with each other. However, p mode damping through mode-mode coupling is believed to be negligible in the Sun due to the small amplitude of the modes (Kumar & Goldreich 1989 ;Weinberg, Arras & Pramanik 2021 ), and therefore this is neglected here. Similarly, a tidally excited inertial wave could in principle interact with free inertial waves if these were present in the flo w. Ho we ver, numerical simulations of Jupiter have found that free inertial waves could not be maintained, probably because of their interaction with convection and gravity waves (Glatzmaier 2018 ). Although the question of whether these waves are present or not is still open, we will neglect the possibility of them coupling with tidally excited modes. Our analysis therefore does not include possible resonances. Note that we also neglect possible interactions of tidal oscillations or p modes with magnetic waves. If these mechanisms are a source of damping, the corresponding terms can be added to the terms we include in the analysis presented below.

Equations for the kinetic energy averaged over the oscillation time-scale
As will be discussed in Section 3.9 , neglecting density perturbations in Navier-Stokes equation before performing time averages to derive equations for the oscillation and the mean flow yields inconsistencies. Therefore, we first derive the equations for the conservation of kinetic energy without making any approximations. The small density perturbation limit will then be examined in the next subsection.
MNRAS 525, 508-526 (2023) Substituting the Reynolds decomposition in Navier-Stokes equation ( 2 ) and averaging over t osc yields where | i denotes the i component. To derive the abo v e equation, we have assumed that g and Ω are not affected by the perturbations. We have also interchanged time averages with space deri v ati ves. Note that p 0 and ρ 0 satisfy the hydrostatic equilibrium equation −∂ p 0 / ∂ x i + ρ 0 g i = 0. This could be subtracted off, but we keep these terms as it makes the discussion about energies more clear. Averaging the mass conservation equation ( 1 ) over t osc yields and subtracting from equation ( 1 ) gives We obtain an equation for E k by multiplying equation ( 10 ) by V i and summing o v er i . Using equations ( 11 ) and ( 12 ) then yields where we have defined Similarly, we obtain an equation for e k by multiplying equation ( 2 ) by u i , summing o v er i , using equation ( 1 ) and averaging over t osc . This yields where No approximations have been made to obtain the equations above, i.e all the terms have been retained. As pointed out above, these equations apply whether the oscillation is an equilibrium tide, the sum of an equilibrium tide and a propagating inertial wave, or a propagating pressure wave (in which case f t has to be thought of as the force that excites the wave). They also apply even if the oscillations are not periodic, in which case the average has to be taken o v er a time long compared to t osc and short compared to t conv . Therefore, these equations can be used when the role of the tidal oscillations and that of convection are reversed, i.e. when the tides are the mean flow and conv ectiv e motions are the rapid fluctuations, which is appropriate when t conv t osc (note, ho we ver, that in that case the fluctuating velocity does not average to zero, but to a value which is second order in the fluctuations, e.g. Nordlund, Stein & Asplund 2009 ).

The case of small perturbations
We now approximate the energy conservation equations ( 13 ) and ( 16 ) using ρ ρ 0 and u | V | , which is al w ays satisfied for tidal oscillations or pressure modes in a conv ectiv e flow.
The mass conservation equation ( 12 ) can then be approximated as We note λ conv the characteristic spatial scale of the conv ectiv e eddies (mixing length), such that V ∼ λ conv / t conv (if V includes significant contribution from differential rotation, then t conv and λ conv are themselves affected by rotation). We further note λ osc the characteristic spatial scale o v er which the oscillating quantities vary. For the equilibrium tide in Jupiter, Saturn, or the Sun, and in the parts of the envelope where tides are significant, λ osc / λ conv 1 near the surface and decreases to reach values on the order of unity deeper in the envelope. Since inertial wav es are driv en by the Coriolis force acting on the equilibrium tide (Ogilvie 2013 ), their λ osc is comparable to that of the equilibrium tide. For p modes, λ osc is larger than the pressure scale height H p . Since λ conv ∼ 2 H p , it follows that λ conv ࣠ λ osc in all cases.
MNRAS 525, 508-526 (2023) Therefore, ∂ ρ V j /∂x j ∼ ρ V /λ conv ∼ ρ /t conv . Since t osc t conv , this is very small compared to ∂ρ /∂t ∼ ρ /t osc , so that equation ( 18 ) can be further approximated as: This yields ρ /t osc ∼ ∂ ρ 0 u j /∂x j ∼ ρ 0 u /λ osc or ∼ ρ 0 u /r, depending on whether the oscillation is compressible or not. This implies Therefore, in equation ( 9 ), ρ u · V / e k ∼ 1. The kinetic energy averaged over the tidal period is then e k E k + e k , which is the sum of the kinetic energy of the mean flow and that of the oscillation.
Furthermore, having 1 and ρ ρ 0 yields D R ρ 0 D R (where we have also used | δρ| ρ 0 ), with which is the parameter that was first introduced in Terquem ( 2021 ).

Av eraging o v er the conv ecti v e time-scale
As we are interested in the exchange of energy between convection and the oscillation over a time-scale long compared to the convective time-scale, we now average equations ( 13 ) and ( 16 ) over such a time-scale. The exact same procedure is followed when the mixing length approximation is used, which may be appropriate when t conv t osc , and which corresponds to the tidal oscillation being the mean flow and convection being the rapid fluctuations. In that case, the first time averaging is done o v er the conv ectiv e time-scale, and the second o v er the oscillation period. Mixing length theory assumes that D R , which involv es a coupling between the conv ectiv e Re ynolds stress and the gradient of the tidal velocity, is al w ays ne gativ e, corresponding to local dissipation of the tides (mean flow). Therefore, D R does not average to zero o v er an oscillation period, even though it is linear in the gradient of the tidal velocity (see Appendix A for a more detailed discussion).
Similarly, here, where t conv t osc and D R involves a coupling between the Reynolds stress of the oscillation and the gradient of the conv ectiv e v elocity, we allow for the possibility that the oscillation is locally dissipated. This corresponds to D R > 0 and implies that D R = 0, where the o v erline denotes an average over a time long compared to the convective time-scale.
Locally, | D R | given by equation ( 21 ) is of order u 2 V /λ conv ∼ u 2 /t conv . If the oscillation is locally dissipated, then D R is positive everywhere and at all times, and D R = | D R | . Ho we ver, if the oscillation is not locally dissipated, D R may be much smaller than | D R | , in which case the rate of energy dissipation is too low to explain the circularization of solar-type binaries (Terquem & Martin 2021 ). We will neglect in equations ( 13 ) and ( 16 ) the terms which are small compared to ρ 0 | D R | when av eraged o v er a long time-scale, while retaining ρ 0 D R to allow for the possibility of local dissipation. If D R is actually small compared to | D R | , then the exchange of energy between the oscillation and conv ectiv e eddies with long time-scale is negligible and cannot account for the observations, in which case the terms that we neglect are not important anyway.
We therefore neglect ρ u i ( ∂ V i /∂ t ) in equation ( 16 ), as it is times smaller than ρ 0 | D R | . We also note that, in equation ( 13 ), ∂ ρ u i /∂t = 0 because ρ u i is the sum of constant and periodic terms.
We now compare the Coriolis term, which redistributes kinetic energy among the different components of the velocities, to ρ 0 | D R | . We have ρ Ω×u · V ∼ ρ u ΩV , as ρ is almost in phase with u ϕ (they would be exactly in phase if there were no exchange of energy between the oscillation and the conv ectiv e flow). The ratio of this quantity to ρ 0 | D R | is ( λ conv /λ osc ) Ωt osc . If Ω is large, then Ωt osc π , so that this ratio is of order unity. Ho we ver, mass conserv ation implies that Nordlund et al. 2009 ). Therefore the Coriolis term averaged over a long time-scale is very small compared to ρ 0 | D R | , and will therefore be neglected.
The work done by the tidal force on the conv ectiv e flow is given by the last term on the right-hand side of equation ( 13 ). Av eraged o v er a long time-scale, this is ρ f t · V ∼ ρ f t V . The work done by the tidal force on the oscillation is given by the last term on the right-hand side of equation ( 16 ), and is ρ f t · u ∼ ρ 0 f t u δ, where δ is the (small) phase shift between the tidal displacement and the tidal force which results from the exchange of energy between the oscillation and convection. The ratio of these two terms is therefore η ≡ ρ V / ρ 0 u δ = V / ( V δ) . The work done by the buoyancy force g | δρ| over ∼λ conv is equal to the kinetic energy per unit volume of conv ectiv e motions (Schwarzschild 1958 ), so that | δρ| /ρ 0 ∼ V 2 / ( gλ conv ) ∼ V / ( gt conv ) , and this is equal to V /V . Therefore, η ∼ V /( gt conv δ). We will show in Section 5.1 that η 1 for tidal oscillations, so that the last term on the right-hand side of equation ( 13 ) can be neglected: most of the tidal work is done on the oscillating velocities, not on the mean flow. This implies that the mean flow 'feels' the effect of the tidal forcing through exchanging energy with the oscillation, rather than directly. (For p modes, there is no work done by the forcing on the conv ectiv e flow, as the forcing comes from convection itself).
Finally, we note that V V also implies that the term ∂ e k V j /∂x j averaged over a long time-scale is very small compared to ρ 0 | D R | (we are assuming that the gradient of the conv ectiv e v elocity, not the v elocity itself, may couple to the tidal Re ynolds stress).
Using the small perturbation approximations described in Section 3.3 and averaging equations ( 13 ) and ( 16 ) where the average over the long time-scale is now taken as read. Equations ( 22 ) and ( 23 ) show that kinetic energy is exchanged between the mean flow and the oscillation at a rate ρ 0 D R per unit volume, which extends the result that was established by  in the incompressible and non-rotating case.

Equations for the potential and internal energy
We now write the equations for the potential and internal energy of the oscillation and mean flow to identify the terms which exchange the kinetic energy of the oscillation with other forms of energy. The potential energy per unit volume is e p = ρ 0 + δρ + ρ ( Ψ 0 + Ψ t ) . Av eraged o v er t osc , we get e p = E p + e p with E p = ( ρ 0 + δρ) Ψ 0 being the potential energy of the mean flow and e p = ρ Ψ t being the mean potential energy of the oscillation.
We obtain an equation for E p by multiplying equation ( 11 ) by Ψ 0 , which yields: where we have used the fact that Ψ 0 and ρ 0 are independent of time. A conservation equation for e p is obtained by multiplying equation ( 1 ) by Ψ t and averaging over t osc : where we have used η 1. For a perfect gas, the internal energy per unit volume is e int = p /( γ − 1), where γ is the ratio of the heat capacity at constant pressure to that at constant volume. Assuming this parameter not to be affected by the perturbation, we have p / ( γ − 1 ) = 0 and e int = ( p 0 + δp ) / ( γ − 1 ) is then the internal energy of the mean flow: on av erage o v er a tidal period, there is no internal energy in the tidal oscillation. A conservation equation for e int is obtained by averaging equation ( 5 )

Identifying all the terms responsible for energy transfer between oscillation and mean flow
As can be seen from equation ( 22 ) to ( 26 ), to leading order, the work done by the tidal force enters through the kinetic and potential energies of the oscillation only. The potential energy of the oscillation cannot be exchanged with the mean flo w. Ho we ver, the kinetic energy of the oscillation can be exchanged with the kinetic energy of the conv ectiv e flow through the term ρ 0 D R , with the potential energy of the conv ectiv e flow through the buoyancy term ρ g · u , and with the internal energy of the convecti ve flo w through the pressure term p ∇ · u and viscous term D v . The compressibility associated with low-frequency tidal oscillations is negligible. This can be seen by comparing t osc with the time t s = λ osc / c s it takes a sound wave to cross the characteristic spatial scale of the oscillation, where c s is the sound speed. We have We have checked that this is satisfied for the tidal periods of interest in Jupiter, Saturn, and the Sun, although it is only marginally satisfied very close to the surface in Jupiter and Saturn. Therefore, the effect of compressibility on tidal oscillations can be neglected (Lighthill 1978 ), which means that terms involving ∇ · u (and therefore also ∇ · ξ , where ξ is the Lagrangian tidal displacement) can be discarded.
Therefore, both the pressure and buoyancy terms can be neglected for tidal oscillations. Note that they were also neglected compared to the term involving the Reynolds stress in the study of gravity waves or f -modes interacting with convection in Press ( 1981 ), Goldreich & Kumar ( 1990 ), Lecoanet & Quataert ( 2013 ).
For p modes ho we ver, t s / t osc becomes of order unity near the surface of the conv ectiv e zone of the Sun, and compressibility therefore plays a role for the damping of these modes. This yields a phase shift between ρ and p which results in net work done by the pressure force (Samadi, Belkacem & Sonoi 2015 , see also Goldreich & Kumar 1990 for a comparison of these terms with the term involving the Reynolds stress).
The viscous flux and D v in equation ( 23 ) can also be neglected, because microscopic viscosity has a negligible effect on oscillations, except possibly near the surfaces of the envelope. In general, viscous dissipation is negligible away from the surfaces of the conv ectiv e env elope, Downloaded from https://academic.oup.com/mnras/article/525/1/508/7227355 by guest on 13 August 2023 MNRAS 525, 508-526 (2023) both in the Sun (Miesch 2005 ) and in Jupiter (Guillot et al. 2004 ). In the bulk of the conv ectiv e zone, the energy contained in the large scale eddies is transported towards the surface mostly by the enthalpy flux, with comparatively very little dissipation due to viscous stresses acting on the smaller scales. Near the surf ace, the enthalp y flux f alls off and energy is transported by the radiative flux (e.g. Featherstone & Miesch 2015 ).

Tidal work and dissipation
We now consider the specific case of a tidal oscillation, for which the terms related to buoyancy, compressibility and viscosity can be neglected in equation ( 23 ). The divergence term in this equation can also be neglected, because p u is associated with compressibility, which we have just shown is not playing any role.
Equation ( 23 ) can therefore be written as Equations ( 25 ) and ( 27 ) ( 25 ) and ( 27 ) become Integrating equation ( 28 ) over the volume V of the convective envelope yields where we have used the fact that ρ 0 is very small on the outer surface and u is very small on the inner surface (as in Section 2.2 ), so that the divergence term does not contribute. Comparing this equation with equation ( 8 ) shows that the energy which is ultimately radiated away comes from the work done by the tidal force (when this is positive). We now discuss the implications of equation ( 29 ). When there is no exchange of energy between the tidal flow and convection, i.e. D R = 0, the tidal displacement ξ is in phase with the tidal force f t , so that u = ∂ ξ /∂t is in quadrature and f t · u = 0. This means that f t · u is positive during half a tidal period and ne gativ e during the other half. When it is positive, the kinetic energy of the oscillation increases while its potential energy decreases by the same amount. When it is ne gativ e, the kinetic energy decreases and the potential energy increases.
When there is dissipation, i.e. D R > 0, ξ lags behind f t by a (positive) phase shift δ, which yields f t · u ∝ sin δ with f t · u > 0. In other words, when the tidal flow transfers energy to convection, work is done by the tidal force, which supplies the energy being transferred. This ultimately comes from the orbital motion of the binary.
If D R < 0 instead, the tidal flow extracts kinetic energy from convection. In that case, ξ leads f t and the situation is as above but with δ < 0. The work done by the tidal force on the flow is now ne gativ e. In other words, the energy extracted from convection by the tidal oscillation is remo v ed from the flow by the work done by the tidal force, and is ultimately fed to the orbital motion of the binary.
The analysis abo v e shows that tidal energy cannot be dissipated by the oscillation itself, as it cannot be converted into thermal energy of the oscillation. It can only be dissipated by being transferred to the kinetic energy of the mean conv ectiv e flow first, where it becomes part of the conv ectiv e ener gy budget and is transformed into thermal ener gy in the standard way.

Equilibrium tide and inertial waves
As already mentioned abo v e, when the tidal frequency in the rotating frame is less than twice the rotational frequency, propagating inertial waves are excited. This corresponds to t orb ≥ t rot /2, where t rot is the rotational period.
When this condition is satisfied, the oscillation is the sum of a non-wave like part and an inertial wave. The non-wave like part is the equilibrium tide, which corresponds to the flow being al w ays instantaneously at hydrostatic equilibrium in the perturbed potential, and is therefore a solution of the equations without the Coriolis force. The Coriolis force acting on the equilibrium tide is then a forcing term that drives inertial waves (see Ogilvie 2013 for a thorough discussion). Denoting u e and u w the velocities associated with the equilibrium tide and inertial wav es, respectiv ely, we then have u e / u w ∼ | ω osc | / ( 2 Ω) ≤ 1. When inertial waves are present, D R can be written as D R , e + D R , w + D R , ew , with MNRAS 525, 508-526 (2023) This shows that the exchange of energy between the conv ectiv e flow and the tidal oscillation can be written in the same way whether the perturbation is dominated by the equilibrium tide, by inertial waves, or by both. Therefore, those inertial waves which have a period small compared to the conv ectiv e time-scale can only be dissipated by interaction with convection if D R , w > 0.

Comments on Barker & Astoul ( 2021 )
Barker & Astoul ( 2021 ) have claimed that the term exchanging kinetic energy between convection and tidal oscillations is ρ 0 D R − t 1 with t 1 ≡ V i u i ∂ρ /∂t (using their notations), instead of ρ 0 D R , as found abo v e. Their result is based on identifying the term exchanging kinetic energy as being ρ 0 V · u · ∇ u only, i.e. neglecting the contribution from the local time deri v ati ve ρ ( ∂ u /∂t ) and also other terms in ρ ( u · ∇ ) u in Navier-Stokes equation. Ho we v er, as the y bring ∂ρ /∂t back from the mass conservation equation, their analysis is not self-consistent and does not yield the correct energy conservation equation. Including the local time deri v ati ve would indeed add the term V · ρ ∂ u /∂t which, added to t 1 , gives V i ∂ ρ u i /∂t , which is zero (see Section 3.4 ). When all the terms are taken into account in a self-consistent manner, the correct exchange term is therefore ρ 0 D R , as shown in the analysis presented abo v e.
We also note that, contrary to what is argued by Barker & Astoul ( 2021 ), the term ρ 0 V i V j ∂u i /∂x j , which has been regarded as the term exchanging energy between tides and convection in previous studies (as will be discussed in the next section), is not an alternative to ρ 0 D R when t osc t conv . Indeed, ρ 0 V i V j ∂u i /∂x j averages to zero over the shortest time-scale t osc . This is discussed in more details in Appendix A .

V I S C O U S A N D T U R B U L E N T S H E A R F L OW S , S L OW A N D FA S T T I D E S
We now discuss how to express D R in terms of the flow velocities. Before considering tidal oscillations in conv ectiv e flo ws, we first re vie w the classical treatment of viscous and turbulent shear flows, as this is instructive and this has been used to approximate tidal dissipation in the standard approach, when the tides are the mean flow. We then discuss both the standard approach and the case of fast tides to which the analysis of the previous section applies. Although, in this section, we refer to the oscillation as a tide, the discussion also applies to p modes. For those modes ho we ver, D R is only part of the exchange energy rate with the convective flow.

Viscous and turbulent shear flows
In a viscous flow with mean velocity U , momentum is transported in the direction of decreasing momentum by the fluctuating (thermal) part c of the particle v elocities. After trav elling through a mean free path λ, particles collide with each other and redistribute momentum. For an incompressible Newtonian fluid, the viscous stress is given by where ν ∼ cλ/ 3 is the kinematic viscosity. This corresponds to a rate of change of energy per unit mass for the mean flow given by As D R , visc < 0, kinetic energy is irreversibly lost by the mean flow. In this case, the correlations < c i c j > between the components of the velocity fluctuations are a result of the shear, and are given the sign required for kinetic energy to be transformed into thermal energy.
In the classical case of a turbulent shear flow, as described, e.g., in Tennekes & Lumley ( 1972 ), the Reynolds stress −ρ < c i c j > is associated with the velocity c of the turbulent fluctuations. The rate of change of energy per unit mass for the mean shear flow, which has velocity U , is still D R , turb = < c i c j > ( ∂ U i / ∂ x j ). Because the length scale of the turbulent eddies is small compared to the scale of the shear flow, eddies are stretched by the shear flow, and conservation of angular momentum then produces a correlation of the components of the turbulent velocity yielding D R , turb < 0. Therefore, here again, < c i c j > is determined by the shear. This corresponds to a transfer of energy from the mean flow to the largest turbulent eddies and the subsequent cascade results in a small scale viscous dissipation of the free energy present in the shear flow. This is consistent with the fact that the turbulence is due to instabilities of the mean shear flow itself, so that the energy of the turbulent eddies comes from the shear.
We now discuss energy transfer in conv ectiv e flows subject to tidal oscillations.

Slow tides t conv t osc and mixing length theory
In the analysis done in the previous section, it was assumed that the tidal period was the shortest time-scale. Ho we ver, the same analysis could be done for the case where the conv ectiv e time-scale is the shortest time-scale. Convective motions would then be the rapid fluctuations, whereas the tidal oscillation would be the mean flow . This is the standard picture which has been considered by previous authors. This yields a rate of change of kinetic energy which is still given by Downloaded from https://academic.oup.com/mnras/article/525/1/508/7227355 by guest on 13 August 2023 MNRAS 525, 508-526 (2023) but with V being the tidal velocity and u being the convective velocity , and the average is o v er a time long compared to the conv ectiv e time-scale and short compared to the tidal period (see Appendix A ). Since the mean flow is now the tidal oscillation, transfer of energy from the tides to convection requires D R , slow < 0. In previous studies, starting with Zahn ( 1966 ), the mixing length approximation has been used to write D R , slow in terms of a turbulent conv ectiv e viscosity. This relies on assuming that conv ectiv e eddies behav e like particles in a fluid, and exchange momentum with their environment when the y dissolv e after having travelled over a mixing length λ conv , which is (at most) twice the pressure scale height. The Reynolds stress, −ρ u i u j , is then expressed by analogy with the viscous stress ( 32 ) as where ν t ∼ u λ conv is the turbulent conv ectiv e viscosity. The rate of energy dissipation is then given by equation ( 33 ) with ν being replaced by ν t , which yields where we have used | ∂ V i / ∂ x j | ∼ V / λ osc and u ∼ λ conv /t conv . Therefore, D R , slow is linear in the conv ectiv e v elocity u and quadratic in the tidal velocity V . The mixing length approximation assumes that conv ectiv e eddies al w ays extract energy from the background shear flow, i.e. transport the momentum associated with the tides from regions where it is high to regions where it is lower. In other words, in this picture, the tidal oscillation dictates the direction in which conv ectiv e eddies trav el, yielding the correlations u i u j between the components of the conv ectiv e velocity to have the sign required for tides to be dissipated. Note that, as the tidal velocity changes sign periodically, u i u j has to change sign on the same time-scale to keep D R , slow < 0.
There are two issues with this picture. First, it is difficult to envision how tidal velocities, which are orders of magnitude smaller than conv ectiv e v elocities in the flows of interest, could influence the conv ectiv e v elocities. Second, ev en if the tides could affect conv ectiv e motions, it is not clear this would result in the correlations u i u j having the required amplitude and the required sign. Indeed, although convection does correlate velocity and density fluctuations very ef fecti vely (yielding positive work from the buoyancy force), it does not necessarily produce correlations of the components of the flow velocity (Tennekes & Lumley 1972 , Section 3.4 ). And when such correlations are produced, they do not necessarily have the sign required for convection to act as a viscosity (Starr 1968 ). The fact that differential rotation in the Sun is produced by the conv ectiv e Re ynolds stress is a clear e xample of conv ection acting as a ne gativ e viscosity.
Finally, we comment on the fact that the mixing length theory is based on a diffusion approximation which is not formally valid when the length scale o v er which the fluctuations vary is comparable to that o v er which the mean flow varies, which is the case when considering tides in a conv ectiv e flo w. This is e ven more of a problem in the presence of rotation, which affects the motion of the conv ectiv e eddies as they mo v e o v er a mixing length. In such a case, the motion of conv ectiv e eddies cannot be treated in a similar way as the motion of molecules, for which rotation is irrele v ant as they move over a mean free path.
There are therefore unjustified assumptions behind the model of convection acting as a turbulent viscosity. Ho we ver, it gi ves theoretical dissipation rates which are in agreement with observed circularization periods for binaries for which t conv t osc (Verbunt & Phinney 1995 ).

Fast tides t conv t osc
Here, D R is given by equation ( 21 ), where u is the tidal velocity (fluctuations) and V is the convective velocity (mean flow). To leading order, u r and u θ are in phase (any phase shift would be a result of energy exchange between the oscillation and the conv ectiv e flow), so that u r u θ ∼ u 2 . We also have u 2 where we have used | ∂ V i / ∂ x j | ∼ V / λ conv ∼ 1/ t conv . Therefore, here again, D R is linear in the conv ectiv e v elocity V and quadratic in the tidal velocity u . This is consistent with equation ( 29 ), as the amplitude of the tidal velocity is proportional to that of the tidal force, so that the dependence on u is quadratic. The dependence on the conv ectiv e v elocity comes from the phase shift between the tidal force and the tidal displacement, which is due to the interaction between the tides and convection.
In previous studies, it has been argued that mixing-length theory is applicable to both the case of slow and fast tides, but that in the regime of fast tides the turbulent viscosity has to be reduced (Zahn 1966 ;Goldreich & Nicholson 1977 ). The results abo v e show that mixing-length theory does not apply to fast tides, but none the less the tidal dissipation rate has the same form as that given by this theory, except without the reduction factor . The dissipation rate is actually larger than for slow tides, as | D R , slow / D R | ∼ ( λ conv / λ osc ) 2 (remembering that the roles of the tidal and conv ectiv e v elocities hav e been e xchanged in the e xpression for D R , slow ). This is much smaller than unity in the parts of the conv ectiv e envelopes of the Sun, Jupiter, and Saturn which contribute to tidal dissipation, which yields enhanced exchange of energy when t conv t osc . In general, at a given location in a convecti ve flo w, there is a range of conv ectiv e time-scales associated with eddies of different sizes. Therefore, even when t osc is small compared to the longest conv ectiv e time-scale, it is still likely to be large compared to the shortest one. This implies that values of t osc / t conv smaller and larger than unity simultaneously contribute to exchange of energy. Ho we ver, since | D R , slow | is MNRAS 525, 508-526 (2023) significantly smaller than | D R | , the rate of exchange of energy is dominated by the interaction with the eddies which have the longest convective time-scale. Note, ho we ver, that the analysis presented here does not apply when t conv ∼ t osc , so there is still a possibility that resonant interaction between tides and convection plays a significant role.
In the case of fast tides, since the tidal oscillations are the rapid fluctuations, D R > 0 corresponds to a transfer of energy from the tidal flow to convection, whereas D R < 0 correspond to the tidal flow extracting kinetic energy from convection. If the tide is approximated by its equilibrium value, u i u j > 0 and therefore tidal dissipation requires the gradient of the conv ectiv e v elocity to couple to this Reynolds stress in a particular way. As in the case of slow tides, it is difficult to envision how the tidal oscillation could induce such a coupling.
Ho we ver, like for the case of slow tides, Terquem & Martin ( 2021 ) showed that assuming D R > 0 in the regime t conv t osc yields dissipation rates which account for the circularization of solar-type binaries, which has been a longstanding theoretical puzzle.

P H A S E L AG A N D T I DA L D I S S I PAT I O N FAC T O R
The tidal dissipation factor is only rele v ant when the tide is locally dissipated by interaction with convection, i.e. when D R > 0 for fast tides or D R , slow < 0 for slow tides. Therefore, in this section, we assume that the tide is indeed locally dissipated and derive an expression for the phase lag and tidal dissipation factor using the formalism presented abo v e.

Phase lag
As already mentioned, when there is dissipation, the Lagrangian displacement ξ lags behind the tidal force f t by a phase δ. Therefore, if the tidal potential is proportional to cos (2 ϕ − ω osc t ) in the rotating frame, then u r and u θ are proportional to sin (2 ϕ − ω osc t − δ) while u ϕ is proportional to cos (2 ϕ − ω osc t − δ). (We are assuming here that the phase shift is the same for all the components of ξ , which may not actually be the case; e.g. Bunting, Papaloizou & Terquem 2019 ). This yields f t · u ∼ f t ( r ) u ( r ) sin δ where f t ( r ) and u ( r) denote positive characteristic values of f t and u at r and where we assume averages over θ. From equation ( 29 ), we then have δ ∼ D R / f t ( r ) u ( r ) , where we have used sin δ δ as dissipation is weak. Equation ( 37 ) gives D R ∼ u 2 ( r ) /t conv ( r ), where again we assume av erages o v er θ. This then yields For the equilibrium tide, the characteristic value of the tidal displacement is ξ ( r) ∼ Ψ t ( r) /g( r) (Ogilvie 2014 ). Since u ( r) = ω osc ξ ( r), we then get Note that, here, we rely on first-order perturbation theory to compute the phase shift, i.e. we use the velocity calculated while ignoring dissipation to derive the phase shift that results from dissipation. This approach, which was also used in Terquem et al. ( 1998 ), is valid because the energy dissipated during a tidal cycle is small compared to the energy contained in the tides. If the mass M of the star or planet is centrally condensed, then g ( r ) GM / r 2 . As already mentioned abo v e, this is a very good approximation in the conv ectiv e env elope of the Sun. It is also a reasonably good approximation in the outer parts of the envelope of Jupiter and Saturn, where the tides are significant. Therefore, the phase shift can be approximated as This is exactly the same expression as that obtained by Darwin ( 1879 ) for a viscous sphere, except that in Darwin's expression r is the radius of the sphere and t conv is a coefficient which is assumed to be uniform. Zahn ( 1977 ) later used Darwin's formula by identifying this coefficient with what he called the 'friction time', which is the time it takes for convection, assumed to act as a turbulent viscosity, to transport energy throughout the conv ectiv e env elope of the star. In Section 3.4 , we calculated that the ratio of the work done by the tidal force on the conv ectiv e flow to that done on the oscillation is η = V /( gt conv δ), where is given by equation ( 20 ). Using equation ( 38 ), this yields η ∼ λ conv t osc /( rt conv ) 1, which confirms that most of the work is done on the oscillation.
If the amplitude of inertial waves dominates over that of the equilibrium tide, then u ( r) ∼ 2 Ωξ ( r), where ξ is the displacement corresponding to the equilibrium tide (see Section 3.8 ). Therefore, equations ( 38 ) and ( 39 ) still apply but with ω osc being replaced by 2 Ω, with the caveat that possible strong latitudinal dependence of inertial waves are not captured by the averaging over θ.
The time lag t is defined through δ( r ) = ω osc t ( r ), and therefore equation ( 38 ) yields a time lag independent of frequency for the equilibrium tide.
The e xpression abo v e has been deriv ed assuming fast tides. F or slow tides, D R is ( λ conv / λ osc ) 2 smaller, so that the right-hand side of equations ( 38 ) and ( 39 ) has to be multiplied by this factor.
In previous studies, it has been assumed that tidal dissipation in a giant planet or in a star could be quantified by a single uniform phase lag. Ho we ver, equation ( 38 ) shows that δ is strongly dependent on r , increasing sharply towards the surface. The phase lag is uniform when the response of the body is viscoelastic, but not in the presence of a fluid.

Phase lags associated with different satellites
Let us consider the case where there are two moons, contributing tidal forces per unit mass f t, 1 and f t, 2 with frequencies in the rotating frame ω osc, 1 and ω osc, 2 , respectively. Then u = u 1 + u 2 , where u i (with i = 1, 2) oscillates with frequency ω osc, i . Averaging over a time long compared to both 2 π / ω osc, 1 and 2 π / ω osc, 2 remo v es the terms involving a product of perturbations with different frequencies (even if the frequencies are commensurate). Equation ( 29 ) then becomes with D R, 1 = u 1 ,i u 1 ,j ∂ V i /∂ x j and similarly for D R , 2 . Because, on average, f t, 1 does not work on u 2 , and f t, 2 does not work on u 1 , the dissipation rate D R , i can only be due to the work done by f t,i on u i . The equation abo v e then implies that f t,i · u i = D R,i for both i = 1 and i = 2, so that equation ( 38 ) is satisfied for each of the moons.

Harmonic oscillator and tidal dissipation factor
We now focus on the equilibrium tide. Ever since Goldreich ( 1963 ), Kaula ( 1964 ), andMacDonald ( 1964 ), the tidal dissipation factor Q has been e xtensiv ely used to quantify the amount of energy dissipation in moons, planets and stars.
In Appendix B , we give a brief review of the calculation of the phase lag δ and Q factor for a driven and damped harmonic oscillator. Comparing the expressions obtained for sin δ in both the case of the equilibrium tide (equation 38 ) and the harmonic oscillator (equation B4 ), we see that we can model the equilibrium tide as a harmonic oscillator with natural frequency ω 0 = ( g / r ) 1/2 ( GM / r 3 ) 1/2 , driving frequency ω = ω osc and damping coefficient γ = 1/ t conv , providing ω 2 0 ω 2 osc γ ω osc . For both Saturn and Jupiter interacting with their closest moons, these inequalities are well satisfied, although in the outer parts of the envelope ω 2 0 is only about 5 times ω 2 osc . For Saturn interacting with Rhea and Titan, this approximation is only marginally satisfied at the surface of the envelope, where ω 2 0 /ω 2 osc ∼ 2. We can therefore approximate the equilibrium tide at some location r as a one dimensional harmonic oscillator which equation of motion is where ξ and u are characteristic values at r of (any component of) the tidal displacement and v elocity, respectiv ely, and f t is a characteristic value at r of the amplitude of the tidal force per unit mass. In this equation, −ω 2 0 ξ and −γ u are the restoring and friction forces per unit mass, respectively. The restoring force is the gravitational force since ω 2 0 ξ ≡ gξ/r = gu / ( ω osc r ) ∼ ρ g/ρ 0 , where the mass conservation ( 19 ) has been used. For a harmonic oscillator, the rate of work done by the friction force, γ u 2 , is equal to the rate of work done on av erage o v er a period by the driving force. As γ u 2 = D R , identifying γ u with the friction force is consistent with equation ( 29 ).
The equilibrium tide assumes that the tidal displacement adjusts itself so that the flow is al w ays at equilibrium in the perturbing potential. This is only satisfied if d 2 ξ/ d t 2 ω 2 0 ξ , i.e. ω 2 osc ω 2 0 , which is consistent with the condition required for sin δ to be approximated by equation ( B5 ). In that case, the energy of the oscillator is dominated by the potential energy, which is e p ( r ) = ρ 0 ω 2 0 ξ 2 / 2 ∼ ρ gξ ∼ ρ Ψ t per unit volume, so that the peak energy E in the expression ( B8 ) of the Q factor is the peak potential energy, not the peak kinetic energy.
It is clear from the discussion abo v e that the equilibrium tide can only be described locally as a harmonic oscillator. Fig. 1 shows the time lag t ( r ) = δ( r )/ ω osc for both Jupiter and Saturn and Q ( r ) ≡ 1/sin δ( r ) for Jupiter interacting with Io and for Saturn interacting with Enceladus as a function of position in the planet. The models for Jupiter and Saturn have been provided by I. Baraffe (Baraffe, Chabrier & Barman 2008 ). The model for Saturn has a ratio of the mixing length to pressure scale height α = 0.5, instead of the value of 2 used in stars, as it has been argued that this may be better suited for planetary interiors (see Terquem 2021 for a more detailed discussion about these models). As can be seen on this figure, δ, and therefore Q , vary by orders of magnitude throughout the envelope.

Comparison with obser v ations for Jupiter and Saturn
Observational constraints on tidal dissipation in a planet are obtained by calculating the tidal deformation of the planet due to a moon, and the gravitational potential V ext that this deformation produces at the position of the moon. It is assumed that, because of tidal dissipation in the planet, the tidal bulge (equilibrium tide) lags behind the line joining the centres of the planet and moon, which corresponds to a phase lag δ. Therefore, V ext results in the planet e x erting a torque onto the moon. The component Γ z of this torque along the direction of the orbital angular momentum yields a secular acceleration of the moon. Since Γ z ∝ k 2 sin δ, where k 2 is the Lo v e number of the planet, comparing the solutions of the equations of orbital evolution with the observations enables constraints to be put on the phase shift ( k 2 being independently constrained). The tidal dissipation factor Q is calculated through Q = 1/tan δ. More details can be found in Goldreich & Soter ( 1966 ), Mignard ( 1980 ), Murray & Dermott ( 1999 ), and Lainey, Dehant & P ätzold ( 2007 ).
When more than one moon is present, each moon i is assumed to be associated with a distinct phase shift δ i . This corresponds to a potential V ext, i produced by the planet at the position of all the moons, but this potential is only associated with a torque on moon i when an average is taken o v er a time-scale long compared to all the tidal periods. Constraints are derived by taking into account the interactions between all the Time lag t = δ/ ω osc for Jupiter (red curve) and Saturn (orange curve) and tidal dissipation factor Q ≡ 1/sin δ for the tides raised on Jupiter by Io (blue curve) and the tides raised on Saturn by Enceladus (cyan curve), in logarithmic scale, versus r / R planet , where R planet is either the radius of Jupiter or that of Saturn. t is independent of ω osc whereas Q ∝ 1/ ω osc . The phase lag δ varies by orders of magnitude through the envelope of the planets. The observations give Q = 3.56 × 10 4 for Jupiter interacting with Io, and Q = 2.45 × 10 3 for Saturn interacting with Enceladus. objects in the system, and the secular acceleration of the different moons has contribution from dissipation in the planet but also in the moons themselves.
Although these studies were initially developed for rocky planets, for which the response is viscoelastic and therefore well described by a uniform phase shift, they have also been used to quantify tidal dissipation in giant planets (Lainey et al. 2009(Lainey et al. , 2012Jacobson 2022 ).
As shown abo v e, δ for a gaseous planet is not uniform, and varies by orders of magnitude throughout the planet. In order to relate the analysis done in this paper to observational constraints, we now show that we can define an average phase shift for the planet. The torque Γ z e x erted by the planet on the moon is equal and opposite to that e x erted by the moon on the planet. Therefore where V is the volume of the envelope of the planet. For a circular orbit, the tidal potential is given by Ψ t ( r, θ, ϕ, t ) = 3 f r 2 sin 2 θ cos ( 2 ϕ − ω osc t ) with f = −GM p /(4 a 3 ), where M p is the mass of the companion which excites the tides and a is the binary separation. We have shown in Section 3.6 that ρ g · u was negligible. This implies that the phase shift between ρ and ξ r is very small compared to the phase shift δ between ξ r and the tidal potential (otherwise ρ g · u would be comparable to ρ 0 f t · u , and therefore to ρ 0 D R ). Therefore, ρ lags behind Ψ t by δ and we can write ρ ( r, θ, ϕ, t ) = ρ ( r) h ( θ ) cos ( 2 ϕ − ω osc t − δ) , with h a function of θ. We have assumed here that the variables are separable, which is not the case when there is rotation. Ho we ver, for uniform rotation (applicable for most of the interior of Jupiter and Saturn), the tidal displacement is well approximated by that corresponding to a non-rotating body (Ioannou & Lindzen 1993 ). Using these expressions of Ψ t and ρ in equation ( 42 ) then yields where R i and R p are the inner and outer radii of the env elope, respectiv ely. We can write Γ z as being proportional to an average sin δ for the planet by defining this average as This yields an average time lag t = δ/ω osc where δ sin δ.
Since sin δ( r) ∼ D R / f t ( r) u ( r) ∼ rD R / Ψ t ( r) u ( r) and ρ ( r) ∼ ρ 0 u ( r) / ( rω osc ) , equation ( 44 ) can also be written as The numerator is the total energy dissipated per unit time, whereas the integral in the denominator is the total potential energy E p in the tide. Therefore, we have sin δ ∼ 1 /Q with: where E is the total energy dissipated during one oscillation period.
Downloaded from https://academic.oup.com/mnras/article/525/1/508/7227355 by guest on 13 August 2023 Figure 2. Dissipation of the tides raised in Jupiter (left-hand panel) and Saturn (right-hand panel) by Io and Enceladus, respectively. The blue and cyan curves show r R i ρ ( r ) r 4 sin δ ( r ) d r normalized to unity versus r / R planet . The parts of the envelope which contribute most to sin δ are abo v e the value of r at which this quantity becomes non negligible. Therefore, the curves show that most of the dissipation occurs in the ∼15 outer per cent of the envelopes. The red and orange curves show 1/ t conv normalized to unity. This is a measure of the Brunt-V äis äl ä frequency, and therefore of non-adiabaticity.  incorrectly took twice the kinetic energy instead of the potential energy to compute Q , arguing equipartition between kinetic and potential energy. Although those energies are comparable in the outer parts of Jupiter's env elope, the y are ho we ver not the same because ω 0 is a few times ω osc . This also led an incorrect dependence on the tidal frequency.
To compute the average of sin δ, equation ( 44 ) shows that it has to be weighed by ρ , and not by ρ 0 . This is because sin δ, which does not depend on the perturbation, is only rele v ant in the parts of the flow where the tidal perturbation is significant. The exact form of ρ ( r) is therefore not important when computing sin δ, as long as it tracks the tidal perturbation. To get numerical values of sin δ, we then use ρ corresponding to the standard equilibrium tide, which is incompressible, i.e. ρ ( r) = − ( d ρ 0 / d r ) ξ r ( r), with ξ r ( r ) = −3 fr 2 / g ( r ), even though in the regime t osc < t conv it is not the correct form of the equilibrium tide (Terquem et al. 1998 ;Goodman & Dickson 1998 ). We have checked that using ξ r to weigh sin δ in equation ( 44 ), instead of ρ , does not make a difference, which pro v es that using an approximate form of ρ is sufficient. To calculate Q directly from E p though, as given by equation ( 46 ), the correct form of ρ has to be used. We have indeed checked that the standard equilibrium tide gives a value of Q from equation ( 46 ) which is about an order or magnitude larger than 1 / sin δ calculated from equation ( 44 ).
For Mimas, Enceladus, Tethys, and Dione interacting with Saturn, Laine y et al. ( 2020 ), and Jacobson ( 2022 ) deriv ed a value of t roughly between 0.3 and 3.7 from observations, which is in reasonable agreement with our value for t . For Titan, Jacobson ( 2022 ) also has a value within that range, whereas Lainey et al. ( 2020 ) derived a time lag 10 times larger. For Rhea, both studies report values of t close to 10.
For both Jupiter and Saturn, Fig. 2 shows the regions which contribute most to sin δ, and therefore to tidal dissipation. In principle, we could also calculate directly the orbital evolution time-scale t a = a /(d a /d t ) = | E orb | /(d E /d t ), where E orb = −GMM p /(2 a ) is the orbital energy (with M being the planet's mass and M p the satellite's mass) and d E /d t is the energy dissipated per unit time.
where the integral is o v er the volume of the conv ectiv e env elope, we obtain t −1 a = 0 . 3 × 10 −10 yr −1 for Io. Ho we ver, this cannot be compared meaningfully to the observations, which give t −1 a = 0 . 09 × 10 −10 yr −1 (Lainey et al. 2009 ), because our calculation does not include the contribution from the dissipation of tidal energy in Io itself, nor the effect of Europa and Ganymede, which is important because of the Laplace resonance the satellites are in. The fact that Io is moving towards Jupiter, instead of away from it as would be the case if only tidal dissipation in the planet were important, shows that the motion of Io is dominated by these other contributions.

Summary
The work presented in this paper shows that the energy of a tidal oscillation in a conv ectiv e flow can only be exchanged with the convective flow by changing the kinetic energy of this flow, not its internal nor potential energy. The analysis has been done for t osc t conv , and in this case the rate D R of energy exchange couples the Reynolds stress associated with the oscillation to the gradient of the conv ectiv e v elocity. This result is valid even when the flows are compressible and in the presence of uniform rotation, and applies whether the oscillation is the equilibrium tide or a superposition of the equilibrium tide and a propagating inertial wave. If the oscillation is a p mode, the rate at which the kinetic energy of the oscillation is exchanged with the kinetic energy of the conv ectiv e flow is still given by D R . However, in that case, MNRAS 525, 508-526 (2023) and because of compressibility, there is also an exchange between the kinetic energy of the oscillation and the potential and internal energy of convection.
The analysis would still apply when t osc t conv , and the rate of energy exchange per unit mass would still be D R , but with this term now coupling the Reynolds stress associated with the convective velocities to the gradient of the velocity of the oscillation.
In the case t osc t conv , | D R | ∼ u 2 V /λ conv , where u and V are the velocities of the oscillation and conv ection, respectiv ely. In the case t osc t conv , and assuming that mixing length theory applies in this regime, | D R | has the same form but is ( λ conv / λ osc ) 2 smaller. Therefore, not only is the energy exchange not suppressed for fast tides, contrary to what has been assumed in previous studies, it is actually much larger than for slow tides! Local dissipation of the oscillation requires D R > 0 when t osc t conv and D R < 0 when t osc t conv . This means that whichever flo w is v arying faster has to transport the momentum associated with the slo wly v arying flo w from regions where it is high to regions where it is lower. It is not clear how, or even if, that happens.
Focusing on tidal oscillations, and assuming local dissipation of the tides, we have calculated the phase lag δ( r ) between the oscillation and the tidal potential. We have shown that this is simply given by r ω osc /( gt conv ), where the gravitational acceleration g and t conv have to be e v aluated locally. The equilibrium tide can be described locally as a harmonic oscillator with natural frequency ( g / r ) 1/2 and subject to a damping force −u /t conv .
Although δ( r ) varies by orders of magnitude through the conv ectiv e env elope of a planet, it is possible to define an av erage phase shift δ which can be compared to the phase shift derived from observations. For the equilibrium tide, we have found that 1 / δ is equal to the standard tidal dissipation factor Q = 2 πE p / E. As δ ∝ ω osc , the time lag associated with this phase shift does not depend on frequency (i.e. it is uniquely defined for a planet, independently of the moon that raises the tides), and Q ∝ 1/ ω osc .

Discussion
It has been proposed that the dissipation of inertial waves could explain the circularization of solar type binaries (Barker 2020(Barker , 2022. In these studies, no specific dissipation mechanism is being proposed, it is just assumed that dissipation takes place. As the analysis we have presented abo v e applies to propagating inertial waves as well as to the equilibrium tide, it shows that inertial waves cannot in general be an alternative to the equilibrium tide to explain dissipation: if they are dissipated, the equilibrium tide is dissipated as well. Of course, inertial waves could in principle provide more dissipation. Ho we ver, this is not borne out by the results of Barker ( 2022 ) for solar-type stars. Their fig. 3 shows that solar-type stars can only reach the circularization periods observed for 10 Gyr clusters if they circularize up to about 9 d on the PMS, which is significantly abo v e the value of 7 d derived from observations. Therefore, circularization on the PMS is clearly o v erestimated in this calculation (this is achieved by starting the tidal interaction when the stars are only 0.15 Myr). If circularization is actually only achieved up to orbital periods of 7 d on the PMS, as suggested by observations, significant tidal dissipation is needed after the MS, when the star rotates much more slowly. For the Sun in its current state, inertial waves would only be tidally excited in binaries with orbital periods larger than 13 d. They could therefore not explain the increase of the circularization period from 7 d on the PMS to 10-12 d at the beginning of the RGB. Finally, we note that, in these studies, the energy dissipated by inertial waves is calculated using an av erage o v er all frequencies. This formalism was initially proposed by Ogilvie ( 2013 ) to calculate the dissipation of energy when the forcing is impulsive. It is appropriate if tides are raised during a brief encounter or in very eccentric orbits, but not in a circular binary when only one dominant frequency contributes to the tides. The reason invoked by Barker ( 2020 ) for using this averaging is that the dissipation of energy of inertial waves varies by orders of magnitude depending on the frequency. Ho we ver, this is by no means a justification for using an averaging over all possible frequencies.
If D R has the sign required for local dissipation of the oscillation to occur, then dissipation of the equilibrium tide alone explains the circularization periods of solar-type stars derived from observations, and the interaction does not need to be started before about 0.4 Myr for circularization up to periods of 7 d to be achieved on the PMS (Terquem & Martin 2021 ). For Jupiter and Saturn, the results presented in this paper show that the phase shift due to tidal dissipation of the equilibrium tide is also consistent with observations, except for the tides raised by Rhea in Saturn, and maybe also for the tides raised by Titan in this planet. Ho we ver, for these moons, the equilibrium tide approximation may not apply in the outer parts of the envelope, which contribute most to dissipation, as the natural frequency ω 0 is comparable to the oscillation frequency ω osc there, as pointed in Section 5.3 . Resonance locking with inertial waves has been proposed as a possible mechanism driving the evolution of the moons of Saturn (Fuller, Luan & Quataert 2016 ;Lainey et al. 2020 ), and it explains the phase shift of Rhea and Titan. But again, this can only occur if these inertial waves are dissipated by interaction with convection. Resonance locking also requires the tidal oscillation to resonate with a free inertial wave in the planet. Whether such free modes can be maintained is still an open question.
In this context, it is important to note that mode-mode coupling for inertial waves, whether it is a tidally driven oscillation resonating with a free mode, or free modes parametrically interacting with each other, cannot be modelled in the presence of a turbulent viscosity arising from convection. As demonstrated in this paper , con vection does not act as a turbulent viscosity. Instead, damping of an inertial wave which interacts with a conv ectiv e flow is itself a result of mode-mode coupling between the inertial wave and the unstable gravity waves which characterize convection.
The very important question that remains to be answered is whether D R has the sign needed for the tidal oscillation to be damped. It has al w ays been assumed to be the case for slow tides, when mixing length theory is used, and it is indeed what numerical simulations show (Ogilvie & Lesur 2012 ;Duguid, Barker & Jones 2020 ;Vidal & Barker 2020 ). For fast tides, there is some suggestion in the simulations performed by Barker & Astoul ( 2021 ) that D R integrated over the flow domain is positive, which corresponds to tidal dissipation. The total rate MNRAS 525, 508-526 (2023) of energy dissipation is found to be significantly smaller than what we obtain by assuming local dissipation, but the results may be affected by the use of rigid boundary conditions, as discussed in Section 2.2 .
Further work, and in particular numerical simulations, are of course needed to investigate the dissipation of tidal oscillations when t osc t conv .
As compressiblility has been taken into account, the formalism presented in this paper applies to p modes. The oscillation period of p modes is much smaller than the conv ectiv e time-scale of the slowest eddies in a large part of the conv ectiv e zone, and therefore mixing length theory does not apply to describe the interaction of the modes with these eddies. It has been proposed that the damping of p modes is dominated by resonant interactions with convection, i.e. by interactions with eddies which have a conv ectiv e time-scale comparable to that of the oscillation, and this interaction has been studied using the mixing length approximation (Goldreich & Keeley 1977 ). However, the analysis presented here shows that, if this approximation applies, it is only in the regime t conv t osc . Numerical simulations actually confirm that mixing length theory is not a good approximation to model the damping of p modes (Basu 2016 ). Resonant interaction is not captured by the analysis presented in this paper, which relies on a separation of time-scales. If important, resonant interaction needs to be investigated using a different approach. It would still be interesting to study the interaction of p modes with the slowest eddies, to obtain some estimate of the energy damping rate to which they contribute. Existing theories are indeed not fully successful at reproducing the mode linewidths (Houdek & Dupret 2015 ), and it has been argued that a no v el approach is needed (Belkacem et al. 2019 ).

AC K N OW L E D G E M E N T S
I thank Steven Balbus for his support, his patience in answering questions and very insightful comments that have been invaluable in shaping the work presented in this paper. I am grateful to John Papaloizou for commenting on various parts of this work, and sharing his vast knowledge of the topic. I also thank David Marshall and Michael McIntyre for giving me some interesting perspectives from oceanography and atmospheric sciences, and Isabelle Baraffe, Sacha Brun, Antoine Strugarek, and Dimitar Vlaykov for enlightening discussions about convection in the Sun. I have also benefited greatly from very stimulating discussions with all the participants of the Tidal Evolution Research Review for Astrophysics (TERRA) w orkshop, which w as held in Leiden in January 2023, and particularly with Dong Lai, Valery Lainey, Gordon Oglvie and Jack Wisdom. Finally, I thank Scott Tremaine and K évin Belkacem for comments on an earlier draft of this paper, which led to some important corrections, and an anonymous referee whose very thoughtful re vie ws resulted in very significant improvements.

DATA AVA I L A B I L I T Y
No new data were generated or analysed in support of this research.
MNRAS 525, 508-526 (2023) whereas, if ω 2 ω 2 0 γ 2 ω 2 , then: When ω 2 0 ω 2 γ 2 ω 2 , E is equal to the maximum of the potential energy, which is reached at t = T /4 for our choice of initial conditions, and can be calculated while neglecting dissipation. Therefore Using | sin δ| | cos δ| together with equations ( B8 ) and ( B10 ) then yields This paper has been typeset from a T E X/L A T E X file prepared by the author.