* Granot et al. (2018) Granot J., Gill R., Guetta D., De Colle F., 2018, MNRAS, 481, 1597
  * Guillochon et al. (2018) Guillochon J., Nicholl M., Villar V. A., Mockler B., Narayan G., Mandel K. S., Berger E., Williams P. K. G., 2018, ApJS, 236, 6
  * Henkel et al. (2022) Henkel A., Foucart F., Raaijmakers G., Nissanke S., 2022, arXiv e-prints, p. arXiv:2207.07658
  * Hosseinzadeh et al. (2019) Hosseinzadeh G., et al., 2019, ApJ, 880, L4
  * Hotokezaka et al. (2019) Hotokezaka K., Nakar E., Gottlieb O., Nissanke S., Masuda K., Hallinan G., Mooley K. P., Deller A. T., 2019, Nature Astronomy, 3, 940
  * Howlett & Davis (2020) Howlett C., Davis T. M., 2020, MNRAS, 492, 3803
  * Hu et al. (2017) Hu L., et al., 2017, Science Bulletin, 62, 1433
  * Hurley et al. (2000) Hurley J. R., Pols O. R., Tout C. A., 2000, MNRAS, 315, 543
  * Ioka & Nakamura (2018) Ioka K., Nakamura T., 2018, Progress of Theoretical and Experimental Physics, 2018, 043E02
  * Ishizaki et al. (2021a) Ishizaki W., Ioka K., Kiuchi K., 2021a, ApJ, 916, L13
  * Ishizaki et al. (2021b) Ishizaki W., Kiuchi K., Ioka K., Wanajo S., 2021b, ApJ, 922, 185
  * Jin et al. (2018) Jin Z.-P., et al., 2018, ApJ, 857, 128
  * Jürgen Krüger & Foucart (2020) Jürgen Krüger C., Foucart F., 2020, arXiv e-prints, p. arXiv:2002.07728
  * Kasen et al. (2017) Kasen D., Metzger B., Barnes J., Quataert E., Ramirez-Ruiz E., 2017, Nature, 551, 80
  * Kasliwal et al. (2017) Kasliwal M. M., et al., 2017, Science, 358, 1559
  * Kasliwal et al. (2020) Kasliwal M. M., et al., 2020, ApJ, 905, 145
  * Kawaguchi et al. (2020) Kawaguchi K., Shibata M., Tanaka M., 2020, ApJ, 889, 171
  * Kawaguchi et al. (2022) Kawaguchi K., Fujibayashi S., Hotokezaka K., Shibata M., Wanajo S., 2022, arXiv e-prints, p. arXiv:2202.13149
  * Kim et al. (2017) Kim S., et al., 2017, ApJ, 850, L21
  * Kiuchi et al. (2019) Kiuchi K., Kyutoku K., Shibata M., Taniguchi K., 2019, ApJ, 876, L31
  * Korobkin et al. (2021) Korobkin O., et al., 2021, ApJ, 910, 116
  * Lattimer (2012) Lattimer J. M., 2012, Annual Review of Nuclear and Particle Science, 62, 485
  * Li & Paczyński (1998) Li L.-X., Paczyński B., 1998, ApJ, 507, L59
  * Lipunov et al. (2017) Lipunov V. M., et al., 2017, ApJ, 850, L1
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, ARA&A, 52, 415
  * Maggiore et al. (2020) Maggiore M., et al., 2020, J. Cosmology Astropart. Phys., 2020, 050
  * Mandel (2018) Mandel I., 2018, ApJ, 853, L12
  * Margalit & Metzger (2017) Margalit B., Metzger B. D., 2017, ApJ, 850, L19
  * Martin et al. (2015) Martin D., Perego A., Arcones A., Thielemann F. K., Korobkin O., Rosswog S., 2015, ApJ, 813, 2
  * Metzger (2019) Metzger B. D., 2019, Living Reviews in Relativity, 23, 1
  * Metzger & Fernández (2021) Metzger B. D., Fernández R., 2021, ApJ, 916, L3
  * Metzger et al. (2010) Metzger B. D., et al., 2010, MNRAS, 406, 2650
  * Miller et al. (2021) Miller M. C., et al., 2021, ApJ, 918, L28
  * Most et al. (2020) Most E. R., Papenfort L. J., Weih L. R., Rezzolla L., 2020, MNRAS, 499, L82
  * Nakar & Piran (2021) Nakar E., Piran T., 2021, ApJ, 909, 114
  * Nakar et al. (2018) Nakar E., Gottlieb O., Piran T., Kasliwal M. M., Hallinan G., 2018, ApJ, 867, 18
  * Nedora et al. (2021) Nedora V., et al., 2021, ApJ, 906, 98
  * Nicholl et al. (2017) Nicholl M., Guillochon J., Berger E., 2017, ApJ, 850, 55
  * Pang et al. (2022) Pang P. T. H., et al., 2022, arXiv e-prints, p. arXiv:2205.08513
  * Perego et al. (2017) Perego A., Radice D., Bernuzzi S., 2017, ApJ, 850, L37
  * Petrov et al. (2022) Petrov P., et al., 2022, ApJ, 924, 54
  * Pian et al. (2017) Pian E., et al., 2017, Nature, 551, 67
  * Pozanenko et al. (2018) Pozanenko A. S., et al., 2018, ApJ, 852, L30
  * Raaijmakers et al. (2021) Raaijmakers G., et al., 2021, ApJ, 922, 269
  * Radice et al. (2018a) Radice D., Perego A., Bernuzzi S., Zhang B., 2018a, MNRAS, 481, 3670
  * Radice et al. (2018b) Radice D., Perego A., Hotokezaka K., Fromm S. A., Bernuzzi S., Roberts L. F., 2018b, ApJ, 869, 130
  * Reitze et al. (2019) Reitze D., et al., 2019, in Bulletin of the American Astronomical Society. p. 35 (arXiv:1907.04833), https://ui.adsabs.harvard.edu/abs/2019BAAS...51g..35R
  * Romani et al. (2022) Romani R. W., Kandel D., Filippenko A. V., Brink T. G., Zheng W., 2022, ApJ, 934, L17
  * Rossi et al. (2020) Rossi A., et al., 2020, MNRAS, 493, 3379
  * Savchenko et al. (2017) Savchenko V., et al., 2017, ApJ, 848, L15
  * Schutz (2011) Schutz B. F., 2011, Classical and Quantum Gravity, 28, 125023
  * Scolnic et al. (2018) Scolnic D., et al., 2018, ApJ, 852, L3
  * Setzer et al. (2019) Setzer C. N., Biswas R., Peiris H. V., Rosswog S., Korobkin O., Wollaeger R. T., Collaboration L. D. E. S., 2019, MNRAS, 485, 4260
  * Setzer et al. (2022) Setzer C. N., Peiris H. V., Korobkin O., Rosswog S., 2022, arXiv e-prints, p. arXiv:2205.12286
  * Shappee et al. (2017) Shappee B. J., et al., 2017, Science, 358, 1574
  * Siegel (2019) Siegel D. M., 2019, European Physical Journal A, 55, 203
  * Smartt et al. (2017) Smartt S. J., et al., 2017, Nature, 551, 75
  * Suwa et al. (2018) Suwa Y., Yoshida T., Shibata M., Umeda H., Takahashi K., 2018, MNRAS, 481, 3305
  * Tanvir et al. (2017) Tanvir N. R., et al., 2017, ApJ, 848, L27
  * Tauris et al. (2017) Tauris T. M., et al., 2017, ApJ, 846, 170
  * The LIGO Scientific Collaboration et al. (2021) The LIGO Scientific Collaboration et al., 2021, arXiv e-prints, p. arXiv:2111.03634
  * Tohuvavohu et al. (2019) Tohuvavohu A., et al., 2019, GRB Coordinates Network, 24353, 1
  * Troja et al. (2017) Troja E., et al., 2017, Nature, 551, 71
  * Troja et al. (2019) Troja E., et al., 2019, MNRAS, 489, 2104
  * Typel et al. (2010) Typel S., Röpke G., Klähn T., Blaschke D., Wolter H. H., 2010, Phys. Rev. C, 81, 015803
  * Utsumi et al. (2017) Utsumi Y., et al., 2017, PASJ, 69, 101
  * Valenti et al. (2017) Valenti S., et al., 2017, ApJ, 848, L24
  * Villar et al. (2017) Villar V. A., et al., 2017, ApJ, 851, L21
  * Wanajo (2018) Wanajo S., 2018, ApJ, 868, 65
  * Wollaeger et al. (2019) Wollaeger R. T., et al., 2019, ApJ, 880, 22
  * Wollaeger et al. (2021) Wollaeger R. T., et al., 2021, ApJ, 918, 10
  * Yagi & Yunes (2017) Yagi K., Yunes N., 2017, Phys. Rep., 681, 1
  * Yu et al. (2018) Yu Y.-W., Liu L.-D., Dai Z.-G., 2018, ApJ, 861, 114
  * Zhu et al. (2020) Zhu J.-P., Yang Y.-P., Liu L.-D., Huang Y., Zhang B., Li Z., Yu Y.-W., Gao H., 2020, ApJ, 897, 20
  * Zhu et al. (2021) Zhu Y. L., Lund K. A., Barnes J., Sprouse T. M., Vassh N., McLaughlin G. C., Mumpower M. R., Surman R., 2021, ApJ, 906, 94

##  Appendix A The Posterior Distributions of the Model Parameters Fitted By
the AT2017gfo Light Curves

Figure 14:  The corner plot for the posterior distributions of 9 of the $24$
model parameters (the opacities and the temperatures of the blue, purple and
the red components, and the parameters of the free neutron skin of the
dynamical ejecta) fitted with the LCs of AT2017gfo, and the best fits to the
LCs are shown in Fig. 1 and the best fit parameters are listed in Table 1.
Figure 15:  Similar to Fig. 14, but only showing the posterior distributions
11 of the $24$ model parameters (the accretion rate, the total mass, the
opacity and the temperature of the fallback heating photosphere, and the
related angles and the velocities parameters of disc wind).  Figure 16:
Similar to Fig. 14, but only showing the posterior distributions for $m_{\rm
disc}$, $m^{\rm d}_{\rm ej}$, $v_{\rm dyn}$, $\theta_{\rm dyn}$, $\theta_{\rm
view}$, respectively, for clarity. These five model parameters are set
according to NR simulations for general BNS mergers.


# Comment on ‘Heating of Enceladus due to the dissipation of ocean tides’ by
R. Tyler

Mikael Beuthe
Royal Observatory of Belgium,
Avenue Circulaire 3, 1180 Brussels, Belgium
E-mail<EMAIL_ADDRESS>
(Published in Icarus 350 (2020) 113934. doi:10.1016/j.icarus.2020.113934)

###### Abstract

Dissipation of ocean potential energy is proposed by Tyler [2020] as a new
mechanism leading to possible high-power states for Enceladus. I show here
that this process actually results from viscoelastic dissipation within the
crust. For plausible values of Enceladus’s ocean thickness, crustal
dissipation can be computed with the standard approach of static deformations
of solid layers by equilibrium tides.

##  1 Introduction

I will comment on several claims made in ‘Heating of Enceladus due to the
dissipation of ocean tides’ [Tyler, 2020], which I summarize with the
following extracts:

  1. 1.

‘The ability to associate dissipation with the ocean tidal potential energy
extends the previous work (…) to form two basic families under which generic
tidal solutions can be explored.’

  2. 2.

‘Tyler follows the precedent in terrestrial oceanography and considers “ocean
tidal dissipation” to refer to the dissipation of ocean tidal energy –
regardless of the ultimate location where this energy is deposited or
converted to heat.’

  3. 3.

‘Even if the dissipation takes place mostly in the overlying ice layer, it is
predominantly or solely the motion of the ocean driving that of the ice’

  4. 4.

‘in the viscoelastic membrane ice model, for example, the ice is massless and
therefore it is not forced directly by the tidal forces’

  5. 5.

‘The results show plausible states whereby the ocean tides (even if Enceladus
is covered with thick ice) generate the observed heat flux.’

I start by giving general counterarguments to the above points, before delving
into the technical justification.

  1. 1.

Dissipation of ocean potential energy is not a new dissipation mechanism but
is in fact equal to crustal dissipation in a viscoelastic shell. The most
obvious argument is that this kind of dissipation directly depends on the
rheology of the viscoelastic crust (through the imaginary part of the
effective shear modulus of the crust), and thus vanishes if the crust is
elastic. This argument is supported by a complete computation of crustal
dissipation as the product of stress and strain rate in the shell. Since
stress and strain are each proportional to the radial displacement of the
bottom of the shell, crustal dissipation is proportional to the squared radial
displacement of the top of the ocean. This explains why crustal dissipation
looks a bit like dissipation of ocean potential energy.

  2. 2.

In the planetology literature, with the exception of Tyler’s papers, ocean
dissipation always refers to dissipation associated with fluid motion in the
ocean. It is important to note that dissipation of ocean potential energy
still occurs in the static limit of equilibrium tides, for which only
dissipation within solid layers is significant. Not only is it confusing to
call ‘ocean dissipation’ heat that is dissipated inside solid layers, but it
gives the impression that one should compute it with a dynamical approach (for
example the Laplace Tidal Equations), whereas it can be computed much more
simply from equilibrium tides if the ocean is deep enough. Moreover,
dissipation of ocean potential energy cannot be physically interpreted as heat
dissipated per unit volume of the ocean, as is done in Figure 10 of Tyler’s
paper. By contrast, it makes good sense to understand it as the integrated
volumetric heating rate within the crust. Finally, the ultimate location where
the energy is converted to heat matters a lot because the amount of
dissipation depends sensitively on the microscopic dissipation processes.

  3. 3.

The ocean can be called the driver of tidal dissipation in two cases. First,
if the shell has a much smaller mass than the ocean and is soft enough that
the tidal amplitude is nearly the one of an ice-free ocean (as on the snowball
Earth and maybe Europa). Second, if there are large dynamical effects in the
ocean; but even so the elastic shell can have a strong controlling effect on
the resonances by shifting them to much smaller depths. For Enceladus, a
plausible model is 20-25 km-thick shell above a 35-40 km-thick ocean, much
deeper than the resonance range. The mass of the shell is thus comparable (in
order of magnitude) to the mass of the ocean, and both are strongly coupled by
the tidal potential. Moreover, Enceladus’s shell is quite rigid and strongly
reduces the tidal amplitude of the ice-free ocean. Mechanically, the most
important role of a deep ocean is to provide a free-slip condition at the
bottom of the shell, thus enhancing the tidal flexing of the shell. If the
ocean were always the driver of the shell, the fluid core of the Earth would
also be the driver of mantle dissipation. In fact, the concept of ‘dominant
driving force’ is not useful in a mechanically and gravitationally coupled
system (mantle+ocean+shell). The ultimate source of energy is the orbital and
rotational energy of the interacting bodies.

  4. 4.

It is misleading to say that the crust does not directly feel the tidal
potential when it is modelled as a massless membrane. In the membrane model,
the gravitational response of the crust is separated from its elastic
response: the former is modelled as a fluid layer added to the ocean, whereas
the latter is modelled as a zero-thickness elastic membrane without mass. If
one is not happy with this approach, the thick shell model of Matsuyama et al.
[2018] provides a treatment where the shell is massive and directly coupled to
the tidal potential.

  5. 5.

Besides the cases of thin or stratified oceans considered in previous papers,
Tyler proposes that dissipation of ocean tidal potential energy provides a
plausible high dissipation state for Enceladus. As discussed above, this kind
of dissipation actually occurs within the crust and has been the focus in the
last 40 years of many studies of icy satellites with subsurface oceans, e.g.
Cassen et al. [1980], Ross and Schubert [1987, 1989], Ojakangas and Stevenson
[1989], Hussmann et al. [2002],Tobie et al. [2003, 2005], and Moore [2006] (to
cite only significant papers before 2006). The convective model of the crust
used by Tyler was initially tailored [Beuthe, 2016] to illustrate strong
damping of dynamical effects by crustal dissipation. More realistic models of
the crust predict much less crustal dissipation [Souček et al., 2019; Beuthe,
2019]. As a caveat, I do not exclude the possibility that new dissipation
processes could operate at the crust-ocean boundary, such as the tidally
induced porous flow discussed by Vance et al. [2007].

I will now give the technical details supporting my arguments. In Section 2, I
rederive the energy balance equation on which Tyler’s interpretation is based.
This is necessary because Tyler formulated his results in a very abstract way
which is not easy to relate to the rest of the literature on ocean
dissipation. I also discuss the ambiguous meaning of the various terms in the
energy balance. These ambiguities can only be removed by computing each kind
of dissipation (mantle, ocean, crust, or total) with another method. In
Section 3, I show that dissipation of ocean potential energy tends in the
deep-ocean/static limit to the total work of tidal forces. In Section 4, I
prove the equality between dissipation of ocean potential energy and crustal
dissipation by computing the latter as the product of stress and strain rate
within the crust. A side result is the depth-dependence of the volumetric
dissipation rate within the crust. The notation for the Laplace Tidal
Equations is defined in Appendix A.

##  2 Energy balance

Tyler computes dissipation by building an averaged energy balance equation
from the equations of motion, as is common in oceanography (e.g. Hendershott
[1972]; Egbert and Ray [2001]). Doing the same, I take the scalar product of
the ocean momentum $\rho\mathbf{u}$ ($\rho$ being the ocean density) with the
equation of tangential motion (Eq. (24)), use the vector identity
($\mathbf{u}\cdot\nabla f=\nabla\cdot(f\mathbf{u})-f\nabla\cdot\mathbf{u}$),
and substitute the equation of radial motion (Eq. (25)) in order to get

$\partial_{t}{\cal E}_{k}+2\alpha\,{\cal
E}_{k}=-\mathbf{\nabla}\cdot\mathbf{P}-\frac{\rho}{D}\,{\rm
Re}\bigg{(}\Big{(}g\sum_{n}\beta_{n}\,\eta_{n}-\upsilon_{2}\,U_{2}^{T}\Big{)}e^{i\omega
t}\bigg{)}\partial_{t}\eta\,,$ (1)

where $\alpha$ is the coefficient of linear drag, $\eta$ is the radial tide
(or radial displacement of the top of the ocean relative to its bottom,
separated on average by the ocean depth $D$), while $(\beta_{n},\upsilon_{2})$
are complex parameters coupling the ocean to the crust and to the tidal
potential $U_{2}^{T}$ (see Appendix A). The rotation is synchronous so that
$\omega$ represents both the mean motion and the angular velocity of rotation.
In the LHS, ${\cal E}_{k}=\rho\mathbf{u}\cdot\mathbf{u}/2$ is the ocean
kinetic power density while $2\alpha{\cal E}_{k}$ represents the ocean
dissipation due to linear drag. In the RHS, the horizontal power flux vector
$\mathbf{P}$ is equal to

$\mathbf{P}=\rho\,{\rm
Re}\bigg{(}\Big{(}g\sum_{n}\beta_{n}\,\eta_{n}-\upsilon_{2}\,U_{2}^{T}\Big{)}e^{i\omega
t}\bigg{)}\mathbf{u}\,.$ (2)

Integrating Eq. (1) over the volume of the ocean $V_{o}$ and averaging over
one orbital period $T=2\pi/\omega$, we see that the terms $\partial_{t}{\cal
E}_{k}$ and $\nabla\cdot\mathbf{P}$ vanish. The averaged energy balance
equation can be written as

$\dot{E}_{O}+\dot{E}_{\eta}=\dot{E}_{W}\,,$ (3)

where

$\displaystyle\dot{E}_{O}$ $\displaystyle=$ $\displaystyle\frac{\rho\alpha
D}{T}\int_{T}dt\int_{S}\left(\mathbf{u}\cdot\mathbf{u}\right)dS\,,$ (4)
$\displaystyle\dot{E}_{\eta}$ $\displaystyle=$ $\displaystyle\frac{\rho
g\omega}{2}\sum_{n}{\rm Im}(\beta_{n})\int_{S}|\eta_{n}|^{2}\,dS\,,$ (5)
$\displaystyle\dot{E}_{W}$ $\displaystyle=$
$\displaystyle\frac{\rho\omega}{2}\int_{S}{\rm
Im}\Big{(}\upsilon_{2}\,U^{T}_{2}\eta_{2}^{*}\Big{)}\,dS\,.$ (6)

where $S$ is the surface of the satellite. Tyler interprets $\dot{E}_{O}$ as
dissipation of ocean kinetic energy due to linear drag, which does not pose
any problem. Besides, he interprets $\dot{E}_{W}$ as the work performed on the
ocean by the tidal forces. This is correct as long as the mantle is non-
deformable or both the mantle and the crust are elastic, as I will now show.

The mean rate of working by tidal forces throughout the body can be computed
as a surface integral of product of the primary tidal potential $U^{T}$ and
the time derivative of the secondary potential $U^{\prime}$ induced by the
deformations within the body [Zschau, 1978; Platzman, 1984]. When applied to
the LTE coupled to a thin shell and a viscoelastic mantle, this formula yields
the total dissipated power as a sum of three terms (Eq. (H.4) of Beuthe
[2016]). If the mantle is non-deformable, only the third term is non-zero so
that

$\displaystyle\dot{E}_{T}=\frac{\rho\omega}{2}\int_{S}{\rm
Im}\Big{(}\upsilon_{2}^{*}\,U^{T}_{2}\eta_{2}^{*}\Big{)}\,dS\,.$ (7)

Therefore, $\dot{E}_{W}$ coincides with the total tidal work as long as
$\upsilon_{2}$ is real, which is for example true if the mantle is non-
deformable and the shell is modelled as a membrane (Eq. (27)). But in the
general case (thin viscoelastic shell with deformable mantle, or thick
viscoelastic shell), $\dot{E}_{W}$ is not equal to the total tidal work
$\dot{E}_{T}$.

Finally, Tyler interprets $\dot{E}_{\eta}$ as dissipation of ocean potential
energy with the following justification: $\dot{E}_{\eta}$ is proportional to
the potential energy of the ocean if the coupling constants $\beta_{n}$ are
the same at all degrees (one should also assume that the mantle does not
deform and neglect self-gravity). He also interprets $\dot{E}_{\eta}/(SD)$ as
the volumetric dissipation rate of ocean potential energy, which is an
arbitrary choice because the term $|\eta_{n}|^{2}$ is not a depth-dependent
variable. Although Tyler considers dissipation of ocean potential energy as a
generic mechanism, i.e. without specifying at first the origin of ${\rm
Im}(\beta_{n})$, the only example proposed in his paper is the coupling to a
viscoelastic membrane (Eq. (27)). Since ${\rm Im}(\beta_{n})=0$ if and only if
the shell is elastic (Eq. (29)), Tyler’s interpretation leads to a
contradiction: ocean dissipation of potential energy is due to viscosity
inside the shell. Actually, the proportionality of $\dot{E}_{\eta}$ to the
squared radial tide $|\eta_{n}|^{2}$ can be understood in terms of the product
of the stress and strain rate within the shell. In Section 4, I will show that
$\dot{E}_{\eta}$ coincides with the crustal dissipation rate $\dot{E}_{C}$
computed from the microscopic dissipation rate in the solid shell, based on
the standard approach of dissipation within solid layers. As for the equality
$\dot{E}_{W}=\dot{E}_{T}$, the identification $\dot{E}_{\eta}=\dot{E}_{C}$
depends on the assumptions that the mantle is non-deformable and that the
shell can be approximated as a membrane.

##  3 Deep-ocean limit

If the ocean is much deeper than the range in which ocean resonances occur,
dynamical effects due to fluid motion become negligible and the degree-2
radial tide tends to the equilibrium (or static) tide:

$\eta_{2}\approx\frac{\upsilon_{2}}{\beta_{2}}\,\frac{U_{2}^{T}}{g}\,,$ (8)

while non-degree-2 components are negligible. Crustal dissipation should also
tend in that limit to the classical formula derived for static tides. As in
Tyler [2020], I assume that the mantle is not deformable. In that case, the
degree-2 tidal Love numbers of the whole body (including the ocean and crust)
can be computed in the static limit with the membrane approach (Eq. (43) of
Beuthe [2015]):

$\displaystyle k_{2}^{T}$ $\displaystyle=$
$\displaystyle\xi_{2}\,h_{2}^{T}\,,$ (9) $\displaystyle h_{2}^{T}$
$\displaystyle=$ $\displaystyle\frac{1}{1-\xi_{2}+\Lambda_{2}}\,,$ (10)

where $\xi_{2}$ is the degree-2 density ratio (defined after Eq. (27)) and
$\Lambda_{2}$ is the degree-2 membrane spring constant (Eq. (28)). Putting
these relations together, I obtain

${\rm Im}(k^{T}_{2})=-\xi_{2}\,|h^{T}_{2}|^{2}\,{\rm Im}(\Lambda_{2})\,,$ (11)

while Eq. (8) combined with Eq. (27) becomes

$\eta_{2}\approx h_{2}^{T}\,\frac{U_{2}^{T}}{g}\,.$ (12)

Substituting this equation into Eqs. (5)-(6), using Eq. (11) and
$g=(4\pi/3)GR\rho_{bulk}$ ($G$ is the gravitational constant, $R$ is the
surface radius and $\rho_{bulk}$ is the bulk density), I can write that, in
the deep-ocean limit,

$\dot{E}_{\eta}\approx\dot{E}_{W}\approx-\frac{5}{2}\,\frac{\omega R}{G}\,{\rm
Im}\big{(}k_{2}^{T}\big{)}\left(\frac{1}{4\pi
R^{2}}\int_{S}|U_{2}^{T}|^{2}\,dS\right).$ (13)

For eccentricity tides, the term within brackets is equal to $(21/5)(\omega
R)^{4}e^{2}$ (e.g. Table 1 of Beuthe [2013], $e$ being the eccentricity), so
that $\dot{E}_{\eta}$ tends to the classical formula for the total dissipation
rate due to equilibrium or ‘static’ eccentricity tides [Segatz et al., 1988]:

$\dot{E}_{\eta}\approx\dot{E}_{W}\approx-\frac{21}{2}\,{\rm
Im}\big{(}k_{2}^{T}\big{)}\,\frac{(\omega R)^{5}}{G}\,e^{2}\,.$ (14)

Since this formula does not include fluid motion effects (at least if
$k_{2}^{T}$ is computed in the static limit), it shows that $\dot{E}_{\eta}$
does not result from ocean dissipation.

Figure 1 illustrates these conclusions by showing ocean and crustal
dissipation (more precisely the corresponding surface fluxes) in Enceladus due
to eccentricity tides. In each panel associated with a different value of the
shell thickness, the deep-ocean limit corresponds to the range of ocean
thicknesses at the right-hand-side of the peaks. Crustal dissipation tends to
the constant value predicted from equilibrium tides, whereas ocean dissipation
tends to zero. The bottom right panel (very thin shell) is a case where the
ocean is driving tidal dissipation in the resonance range. The other panels
show that a thicker shell shifts the resonance peaks to the left and strongly
damps them.

Figure 1: Tidal dissipation in the crust and ocean of Enceladus as a function
of ocean thickness. Each panel corresponds to a different value of the crustal
thickness. The forcing is due to eccentricity tides. The crust is made of an
upper conductive layer (40% of crust thickness) and a lower convective layer
(60% of crust thickness). Crustal rheology is modelled as in the convective
model used in Figure 13 of Beuthe [2016] (the asymptotes are described in the
same paper).

##  4 Crustal dissipation

In Beuthe [2016], I obtained the crustal dissipation rate by computing the
average work done by the bottom load on the shell. This procedure can give the
wrong impression that this kind of dissipation occurs at the ocean-crust
interface and is thus a new kind of ocean dissipation. I will now show that
the same result is obtained by integrating the microscopic dissipation rate
over the volume of the crust. For simplicity, I will assume that the material
is incompressible, but the end result (Eqs. (22)-(23)) is the same if
compressibility is included (see Beuthe [2015, 2019] for similar computations
including compressibility).

At each point within a viscoelastic solid, the dissipation rate due to
friction is given by the rate at which stresses do work per unit volume,
averaged over a tidal period [Kaula, 1963, 1964; Peale and Cassen, 1978]:

$P(r,\theta,\varphi)=\frac{1}{T}\int_{0}^{T}\sigma_{ij}(t)\,\partial_{t}{\epsilon}_{ij}(t)\,dt\,,$
(15)

where the summation convention applies. Next, stress and strain are Fourier-
transformed in the frequency domain (e.g. $\sigma_{ij}(t)={\rm
Re}(\sigma_{ij}\,e^{i\omega t})$) where the correspondence principle can be
applied. In an incompressible viscoelastic solid, stress is related to strain
by $\sigma_{ij}=2\mu\epsilon_{ij}$, where $\mu$ is the complex shear modulus
of the material which generally depends on position. The volumetric
dissipation rate becomes

$P(r,\theta,\varphi)=\omega\,{\rm
Im}(\mu)\,\epsilon_{ij}\,\epsilon_{ij}^{\,*}\,.$ (16)

The application to a thin shell is best done in several steps. First, the 3D
strain in a thin shell can be computed in terms of two 2D functions: the
stress function $F(\theta,\phi)$ and the radial displacement
$w(\theta,\phi)=\eta^{top}$. The dissipation rate expressed in terms of these
variables is given by Eq. (19) of Beuthe [2019]. In the membrane limit, this
equation greatly simplifies because the bending and mixed terms that depend on
$w$ are negligible. In that case, the volumetric dissipation rate within the
crust reads

$P_{C}(r,\theta,\varphi)=\frac{\omega\,{\rm
Im}(\mu)}{4\left(1+\nu\right)\mu_{0}^{2}\,d^{2}}\left(\left|\Delta^{\prime}F\right|^{2}-(1+\nu)\,{\cal
A}(F\,;F^{*})\right),$ (17)

where $\nu=1/2$ is Poisson’s ratio for an incompressible material. The
volumetric dissipation rate depends only on the radial coordinate $r$ through
$\mu$ which can vary a lot from the surface to the bottom of a viscoelastic
shell (it also varies laterally to a lesser extent). By definition, the
effective shear modulus $\mu_{0}$ ($\mu$ integrated over the shell thickness
$d$) does not depend on depth. The differential operators $\Delta^{\prime}$
and ${\cal A}$ are defined in Appendices A and B of Beuthe [2018]. Expanding
the stress function in spherical harmonics ($F=\sum_{n}{}F_{n}$), I can write
the action of these operators as (see Eq. (B.11) of Beuthe [2018]):

$\displaystyle\Delta^{\prime}F_{n}$ $\displaystyle=$
$\displaystyle\delta_{n}^{\prime}\,F_{n}\,,$ (18) $\displaystyle{\cal
A}(F_{n}\,;F_{p}^{*})$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left(\Delta\Delta+\kappa_{np}^{\prime}\,\Delta+\lambda_{np}^{\prime}\right)(F_{n}F_{p}^{*})\,,$
(19)

where $\delta_{n}^{\prime}=-(n-1)(n+2)$,
$\kappa_{np}^{\prime}=6-2(\delta_{n}^{\prime}+\delta_{p}^{\prime})$, and
$\lambda_{np}^{\prime}=(\delta_{n}^{\prime}-\delta_{p}^{\prime})^{2}-2(\delta_{n}^{\prime}+\delta_{p}^{\prime})$.

Since we are not interested here by the spatial pattern of heating, I compute
the angular average of the volumetric dissipation rate over the sphere
$\Omega$ of radius $r$: $\langle
P_{C}\rangle=(4\pi)^{-1}\int_{\Omega}P_{C}\,\sin\theta\,d\theta\,d\phi$. The
orthogonality of spherical harmonics and the divergence theorem result in
great simplifications:

$\langle P_{C}(r,\theta,\varphi)\rangle=\frac{\omega\,{\rm
Im}(\mu)}{16\pi\mu_{0}^{2}\,d^{2}}\,\sum_{n}\frac{\delta_{n}^{\prime}\left(\delta_{n}^{\prime}-1-\nu\right)}{1+\nu}\int_{\Omega}|F_{n}|^{2}\,\sin\theta\,d\theta\,d\phi\,.$
(20)

If the shell is laterally uniform, the spherical harmonic component $F_{n}$ of
the stress function is related to the spherical harmonic component $w_{n}$ of
the radial displacement by Eq. (59) of Beuthe [2018] (in which $\chi=\psi=1$
in the membrane limit):

$F_{n}=\frac{2\left(1+\nu\right)\mu_{0}}{\delta_{n}^{\prime}-1-\nu}\,\frac{d}{R}\,\eta^{top}_{n}\,.$
(21)

Substituting Eq. (21) into Eq. (20) and using Eq. (28) for the membrane spring
constant $\Lambda_{n}$ (recall that $\nu=1/2$), I can write the angular
average of the volumetric dissipation rate as

$\langle P_{C}(r,\theta,\varphi)\rangle=\frac{\rho g\omega}{8\pi
d}\,\frac{{\rm Im}(\mu)}{{\rm Im}(\mu_{0})}\,\sum_{n}{\rm
Im}(\Lambda_{n})\int_{\Omega}|\eta^{top}_{n}|^{2}\,\sin\theta\,d\theta\,d\phi\,.$
(22)

This formula does not change if compressibility is included, except that one
should use the formula for $\Lambda_{n}$ depending on the effective Poisson’s
ratio (Eq. (B.6) of Beuthe [2016]). Figure 2 illustrates the depth-dependence
of the volumetric dissipation rate inside the crust. Dissipation is large in
the convective layer and drops to zero in the conductive layer.

Figure 2: Tidal dissipation rate per unit volume of Enceladus’s crust as a
function of depth from the surface. The forcing is due to eccentricity tides
and the dissipation rate is averaged over the angles. The ocean thickness is
much larger than the resonance range. The crust is made of an upper conductive
layer ($10\,$km thick) and a lower convective layer ($15\,$km thick). The
rheology is modelled as in the convective model used for Figure 13 of Beuthe
[2016].

Finally, the crustal dissipation rate is obtained by integrating the averaged
volumetric dissipation rate over the volume of the shell
($\dot{E}_{C}=\int_{V}\langle P_{C}\rangle\,dV$):

$\dot{E}_{C}=\frac{\rho g\omega}{2}\sum_{n}{\rm
Im}(\Lambda_{n})\int_{S}\big{|}\eta^{top}_{n}\big{|}^{2}\,dS\,,$ (23)

where the surface integration is done at $r=R$. This expression is identical
to the work performed on the crust by the bottom load $q_{n}=\rho
g\Lambda_{n}\eta_{n}^{top}$ (Eq. (76) of Beuthe [2016]). It is also identical
to $\dot{E}_{\eta}$ (Eq. (5)) if $\eta_{n}=\eta_{n}^{top}$ and ${\rm
Im}(\beta_{n})={\rm Im}(\Lambda_{n})$, which is true if the mantle is not
deformable (see Eq. (27)).

## Acknowledgments

I thank an anonymous reviewer for carefully reading the manuscript. This
research has been supported by the Belgian PRODEX program managed by the
European Space Agency in collaboration with the Belgian Federal Science Policy
Office.

##  Appendix A Laplace Tidal Equations for an ice-covered ocean

Consider an icy satellite with a global subsurface ocean and submitted to
tides synchronous with the rotation. The dissipative Laplace Tidal Equations
for the ocean sandwiched between a viscoelastic mantle and a viscoelastic
shell (membrane, thin shell, or thick shell) are given by

$\displaystyle\partial_{t}\mathbf{u}+2\mathbf{\Omega\times
u}+\alpha\,\mathbf{u}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho}\mathbf{\nabla}p\,,$ (24)
$\displaystyle\partial_{t}\eta+D\,\mathbf{\nabla\cdot u}$ $\displaystyle=$
$\displaystyle 0\,.$ (25)

The fluid motion variables are $\mathbf{u}(t,\theta,\phi)$, defined as the
depth-averaged horizontal velocity vector, and the radial tide
$\eta(t,\theta,\phi)=\eta^{top}-\eta^{bot}$, defined as the difference between
the radial displacements at the top and the bottom of the ocean. The term
$\alpha\mathbf{u}$ describes dissipation due to linear drag with coefficient
$\alpha$. Other parameters are the body rotation vector $\mathbf{\Omega}$ with
angular velocity $\omega$ (equal to the tidal frequency), the ocean depth $D$,
and the ocean density $\rho$. The operators $\nabla$ and $\nabla\cdot$ denote
the surface gradient (tangential vector operator) and divergence (dimension
$\sim m^{-1}$). In the RHS, the forcing term is the surface gradient of a
pressure-like potential $p$ given by

$p=\rho\,{\rm
Re}\bigg{(}\Big{(}g\sum_{n}\beta_{n}\,\eta_{n}-\upsilon_{2}\,U_{2}^{T}\Big{)}e^{i\omega
t}\bigg{)}\,,$ (26)

where $g$ is the surface gravity. The radial tide is Fourier-transformed in
the frequency domain and decomposed over the sphere in spherical harmonics of
degree $n$: $\eta=\sum_{n}{\rm Re}(\eta_{n}e^{i\omega t})$. The tidal forcing
is represented in the same way, but only the harmonic degree 2 contributes:
$U^{T}={\rm Re}(U_{2}^{T}e^{i\omega t})$. The complex parameters
$(\beta_{n},\upsilon_{2})$ are defined by Eq. (28) of Beuthe [2016] if the
shell is thin, and by Eq. (22) of Matsuyama et al. [2018] if the shell is
thick.

The LTE have often been studied for an ice-free ocean above a non-deformable
mantle, and without taking into account the self-gravity of the ocean, in
which case $\beta_{n}=\upsilon_{2}=1$. In general, both coupling constants
differ from 1 and are complex if either the shell or the mantle is
viscoelastic. If the shell is thin and the mantle does not deform (as in
Tyler’s paper), $\upsilon_{2}=1$ but $\beta_{n}\neq 1$ because of the
ocean+shell self-gravity and the (visco)elastic response of the shell:

$\displaystyle\upsilon_{2}=1\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\beta_{n}=1-\xi_{n}+\Lambda_{n}\,,$ (27)

where the ocean+shell self-gravity depends on the degree-$n$ density ratio
$\xi_{n}=(3/(2n+1))(\rho/\rho_{bulk})$. The shell density is assumed to be
equal to the ocean density, so that gravitation does not ‘see’ any difference
between shell and ocean, and the mass of the shell can thus be included in the
mass of the ocean. The degree-$n$ spring constant $\Lambda_{n}$ can be
computed given the rheology of ice and the shell thickness:

$\Lambda_{n}=6\,\frac{(n-1)(n+2)}{2n^{2}+2n-1}\,\frac{\mu_{0}}{\rho
gR}\,\frac{d}{R}\,,$ (28)

where $R$ is the surface radius and $\mu_{0}$ is the effective shear modulus
(the subscript stands for zeroth-moment), i.e. the complex shear modulus of
ice integrated over the shell thickness $d$. This formula is valid for an
incompressible membrane, otherwise the spring constant also depends on the
effective Poisson’s ratio and on an additional bending term proportional to
$d^{3}$ (Eqs. (52)-(54) of Beuthe [2018]). Eqs. (27)-(28) imply that

${\rm Im}(\beta_{n})\neq 0\hskip 14.22636pt\mbox{iff}\hskip 14.22636pt{\rm
Im}(\mu_{0})\neq 0\,.$ (29)

## References

  * Beuthe [2013] Beuthe, M. (2013), Spatial patterns of tidal heating, Icarus, 223, 308–329, doi:10.1016/j.icarus.2012.11.020.
  * Beuthe [2015] Beuthe, M. (2015), Tides on Europa: The membrane paradigm, Icarus, 248, 109–134, doi:10.1016/j.icarus.2014.10.027.
  * Beuthe [2016] Beuthe, M. (2016), Crustal control of dissipative ocean tides in Enceladus and other icy moons, Icarus, 280, 278–299, doi:10.1016/j.icarus.2016.08.009.
  * Beuthe [2018] Beuthe, M. (2018), Enceladus’s crust as a non-uniform thin shell: I tidal deformations, Icarus, 302, 145–174, doi:10.1016/j.icarus.2017.11.009.
  * Beuthe [2019] Beuthe, M. (2019), Enceladus’s crust as a non-uniform thin shell: II tidal dissipation, Icarus, 332, 66–91, doi:10.1016/j.icarus.2019.05.035.
  * Cassen et al. [1980] Cassen, P., S. J. Peale, and R. T. Reynolds (1980), Tidal dissipation in Europa - A correction, Geophys. Res. Lett., 7, 987, doi:10.1029/GL007i011p00987.
  * Egbert and Ray [2001] Egbert, G. D., and R. D. Ray (2001), Estimates of M2 tidal energy dissipation from TOPEX/Poseidon altimeter data, J. Geophys. Res., 106, 22,475, doi:10.1029/2000JC000699.
  * Hendershott [1972] Hendershott, M. C. (1972), The effects of solid Earth deformation on global ocean tides, Geophys. J. R. astr. Soc., 29, 389–402, doi:10.1111/j.1365-246X.1972.tb06167.x.
  * Hussmann et al. [2002] Hussmann, H., T. Spohn, and K. Wieczerkowski (2002), Thermal equilibrium states of Europa’s ice shell: Implications for internal ocean thickness and surface heat flow, Icarus, 156, 143–151, doi:10.1006/icar.2001.6776.
  * Kaula [1963] Kaula, W. M. (1963), Tidal dissipation in the Moon, J. Geophys. Res., 68, 4959–4965, doi:10.1029/JZ068i017p04959.
  * Kaula [1964] Kaula, W. M. (1964), Tidal dissipation by solid friction and the resulting orbital evolution, Rev. of Geophys., 2, 661–685, doi:10.1029/RG002i004p00661.
  * Matsuyama et al. [2018] Matsuyama, I., M. Beuthe, H. C. F. C. Hay, F. Nimmo, and S. Kamata (2018), Ocean tidal heating in icy satellites with solid shells, Icarus, 312, 208–230, doi:10.1016/j.icarus.2018.04.013.
  * Moore [2006] Moore, W. B. (2006), Thermal equilibrium in Europa’s ice shell, Icarus, 180, 141–146, doi:10.1016/j.icarus.2005.09.005.
  * Ojakangas and Stevenson [1989] Ojakangas, G. W., and D. J. Stevenson (1989), Thermal state of an ice shell on Europa, Icarus, 81, 220–241, doi:10.1016/0019-1035(89)90052-3.
  * Peale and Cassen [1978] Peale, S. J., and P. Cassen (1978), Contribution of tidal dissipation to lunar thermal history, Icarus, 36, 245–269, doi:10.1016/0019-1035(78)90109-4.
  * Platzman [1984] Platzman, G. W. (1984), Planetary energy balance for tidal dissipation, Rev. Geophys. Space Phys., 22, 73–84, doi:10.1029/RG022i001p00073.
  * Ross and Schubert [1987] Ross, M. N., and G. Schubert (1987), Tidal heating in an internal ocean model of Europa, Nature, 325(6100), 133–134, doi:10.1038/325133a0.
  * Ross and Schubert [1989] Ross, M. N., and G. Schubert (1989), Viscoelastic models of tidal heating in Enceladus, Icarus, 78, 90–101, doi:10.1016/0019-1035(89)90071-7.
  * Segatz et al. [1988] Segatz, M., T. Spohn, M. N. Ross, and G. Schubert (1988), Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io, Icarus, 75, 187–206, doi:10.1016/0019-1035(88)90001-2.
  * Souček et al. [2019] Souček, O., M. Běhounková, J. Čadek, O.and Hron, G. Tobie, and G. Choblet (2019), Tidal dissipation in Enceladus’ uneven, fractured ice shell, Icarus, doi:10.1016/j.icarus.2019.02.012.
  * Tobie et al. [2003] Tobie, G., G. Choblet, and C. Sotin (2003), Tidally heated convection: Constraints on Europa’s ice shell thickness, J. Geophys. Res., 108, 5124, doi:10.1029/2003JE002099.
  * Tobie et al. [2005] Tobie, G., A. Mocquet, and C. Sotin (2005), Tidal dissipation within large icy satellites: Applications to Europa and Titan, Icarus, 177, 534–549, doi:10.1016/j.icarus.2005.04.006.
  * Tyler [2020] Tyler, R. (2020), Heating of Enceladus due to the dissipation of ocean tides, Icarus, 348, 113,821, doi:10.1016/j.icarus.2020.113821.
  * Vance et al. [2007] Vance, S., J. Harnmeijer, J. Kimura, H. Hussmann, B. deMartin, and J. M. Brown (2007), Hydrothermal systems in small ocean planets, Astrobiology, 7(6), 987–1005, doi:10.1089/ast.2007.0075.
  * Zschau [1978] Zschau, J. (1978), Tidal friction in the solid earth: Loading tides versus body tides, in Tidal Friction and the Earth’s Rotation, edited by P. Brosche & J. Sündermann, Springer-Verlag, pp. 62–94, New York, doi:10.1007/978-3-642-67097-8_7.