Tidal Heating in Icy Satellite Oceans ⇑ E.M.A

Icarus 229 (2014) 11–30

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Icarus

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Tidal heating in icy satellite oceans ⇑ E.M.A. Chen, F. Nimmo , G.A. Glatzmaier

Dept. Earth and Planetary Sciences, University of California, Santa Cruz, 1156 High St., Santa Cruz, CA 95064, USA article info abstract

Article history: Tidal heating plays a significant role in the evolution of many satellites in the outer Solar System; how- Received 14 May 2013 ever, it is unclear whether tidal dissipation in a global liquid ocean can represent a significant additional Revised 21 October 2013 heat source. Tyler (Tyler, R.H. [2008]. Nature 456, 770-772; Tyler, R.H. [2009]. Geophys. Res. Lett. 36, Accepted 22 October 2013 doi:10.1029/2009GL038300) suggested that obliquity tides could drive large-scale flow in the oceans Available online 30 October 2013 of Europa and Enceladus, leading to significant heating. A critical unknown in this previous work is what the tidal quality factor, Q, of such an ocean should be. The corresponding tidal dissipation spans orders of Keywords: magnitude depending on the value of Q assumed. Satellites, dynamics To address this issue we adopt an approach employed in terrestrial ocean modeling, where a significant Tides, solid body Jupiter, satellites portion of tidal dissipation arises due to bottom drag, with the drag coefficient O (0.001) being relatively Saturn, satellites well-established. From numerical solutions to the shallow-water equations including nonlinear bottom Triton drag, we obtain scalings for the equivalent value of Q as a function of this drag coefficient. In addition, we provide new scaling relations appropriate for the inclusion of ocean tidal heating in thermal–orbital evolution models. Our approach is appropriate for situations in which the ocean bottom topography is much smaller than the ocean thickness. Using these novel scalings, we calculate the ocean contribution to the overall thermal energy budgets for many of the outer Solar System satellites. Although uncertainties such as ocean thickness and satellite obliquity remain, we find that for most satellites it is unlikely that ocean tidal dissipation is important when compared to either radiogenic or solid-body tidal heating. Of known satellites, Triton is the most likely icy satellite to have ocean tidal heating play a role in its present day thermal budget and long-term thermal evolution. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction on or within solid bodies may also be a source of dissipation. On the Earth it is well-known that tidal dissipation occurs mainly in Tidal heating influences the present-day behavior of some plan- the oceans (Munk and MacDonald, 1960; Egbert and Ray, 2000; etary bodies, such as Io (Peale et al., 1979) and Enceladus (Spencer Ray et al., 2001). Global subsurface oceans are thought to occur et al., 2006; Howett et al., 2011). It probably also played a role at on at least Europa, Ganymede, Callisto, Titan and perhaps Encela- earlier times elsewhere, including Europa (Hussmann and Spohn, dus (Khurana et al., 1998; Kivelson et al., 2002; Bills and Nimmo, 2004), Ganymede (Showman et al., 1997), Triton (Jankowski et al., 2011; Iess et al., 2012; Postberg et al., 2011); our focus in this paper 1989), and the Moon (Garrick-Bethell et al., 2010), and may be is to examine tidal dissipation within such oceans. important in some super-Earth exoplanets (Henning et al., 2009). In a prescient paper, Ross and Schubert (1989) discussed the pos- For solid bodies, the effects of tides and their associated dissipa- sibility of tidal heating on Enceladus arising from turbulent dissipation are typically calculated assuming a viscoelastic rheology, such tion in a subsurface ocean. More recently, Tyler (2011) expanded an as Maxwell, Andrade or Burgers (e.g. Ross and Schubert (1989); To- analysis initially developed by Longuet-Higgins (1968) to investi- bie et al. (2005); Efroimsky and Williams (2009); Castillo-Rogez gate energy dissipation in tidally-driven satellite oceans. In Tyler et al. (2011); Nimmo et al. (2012)), though other processes (such (2011), a key free parameter is the linear drag constant a, which as frictional heating, e.g. Nimmo and Gaidos (2002)) may also play can be related to a tidal quality factor Q. It is worth noting that the a role. For primarily fluid bodies, such as giant planets, a significant value of a or Q is a priori very poorly known and the total energy dis- component of dissipation is likely to be due to the breaking of sipation scales linearly with the model’s prescribed value. internal gravity waves (Ogilvie and Lin, 2004). Lastly, fluid layers In this paper, we follow an analysis similar to that of Tyler (2011). However, we depart from his approach in two important

⇑ Corresponding author. respects. First, we provide an estimate for Q using an approach E-mail address: [email protected] (F. Nimmo). and parameter values developed in studies of the Earth’s oceans.

Second, we present approximate scaling relationships which allow These equations ignore ocean stratification, and thus do not include fluid dissipation to be calculated in a manner analogous to the the effects of internal tides. In addition, they do not include overly- well-known equations for solid body dissipation (e.g. Segatz et al. ing ice shell rigidity, though this effect should be small (Matsuyama, (1988); Ross and Schubert (1989); Wisdom (2008)). This will facil- 2012). itate investigation of long-term satellite evolution, in which the thermal and orbital histories are coupled (e.g. Ojakangas and Ste- 2.1. Tidal potentials venson (1989); Hussmann and Spohn (2004); Bland et al. (2009); Meyer et al. (2010); Zhang and Nimmo (2012)). For synchronously rotating satellites, such as the regular sat- The rest of this paper is organized as follows. Section 2 reviews ellites of Jupiter and Saturn, we are concerned with the ocean the shallow water equations appropriate for flow in global fluid lay- flow driven by the eccentricity of the orbit and the obliquity, ers on a rotating spherical shell. For clarity, we summarize a semi- the tilt of the rotational axis relative to the orbital axis. The forc- analytic solution to these equations, similar to that adopted by ing tidal potentials can be derived by assuming the planet is a Longuet-Higgins (1968) and Tyler (2011), and in addition, explicitly point mass and calculating the resulting gravitational potential present the method and equations used to calculate quantities such at every point on the satellite (cf. Kaula (1964); Murray and Der- as the average kinetic energy and energy dissipation. Section 3 sim- mott (1999)). plifies this system and carries out an analytical study of the response of a shallow global ocean to tidal forcing, building on the 2.1.1. Obliquity tides method presented in Section 2. This novel analysis derives approx- The obliquity tidal potential at a point of colatitude h and longi- imate scaling relationships for ocean tidal flow and the resulting tude / on a synchronously rotating satellite with small obliquity h0 dissipation under typical icy satellite parameters. The algebra in- (in radians) is a standing wave and can be written as the sum of an volved can be tedious; to aid clarity, many details have been rele- eastward and a westward propagating potential (cf. Tyler (2011) gated to Appendices E and F, while Table 4 summarizes the key Eq. (34)) results. The advantage of these relationships is that they retain 3 2 2 the fundamental physical effects while being somewhat simpler Uobl ¼ X R h0 sin h cos hðcosð/ XtÞþcosð/ þ XtÞÞ: ð3Þ to implement than the method presented in Section 2. The results 2 m of Section 3 are expressed in terms of an unknown effective (pre- We define Laplace spherical harmonics of degree l and order m, Yl , sumably turbulent) viscosity. In Section 4, we present an estimate as for this viscosity using a numerical technique based on analogy to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frictional ocean dissipation on Earth. We discuss the applications m ð2l þ 1Þ ðl mÞ! m Y ðh; /Þ P ðcos hÞeim/ ð4Þ and implications of these results in Section 5. In particular, ocean l 4p ðl þ mÞ! l dissipation is unlikely to be a significant heat source unless the m orbital eccentricity is very small; Triton is thus the most likely can- employing a Condon–Shortley phase factor of ð1Þ for m > 0. didate for a satellite in which ocean tidal dissipation is significant. These spherical harmonics are orthogonal under Z Z 2p p m m0 2. Ocean tidal dissipation Y l Yl0 sin hdhd/ ¼ dl;l0 dm;m0 ð5Þ 0 0

Here we briefly review the equations of motion for a shallow where denotes the complex conjugate and d is the Kronecker delta. global satellite ocean. These equations are equivalent to Eqs. (3) The obliquity tidal potential thus can be expressed in spherical and (4) presented in Tyler (2011); we present fully dimensional harmonics as a westward propagating potential Uobl;W , rffiffiffiffiffiffiffi equations and explicitly expand these equations for the solutions 3 2p 2 2 iXt 1 iXt 1 to unknown spherical harmonic coefficients. Uobl;W ¼ X R h0ðe Y2 e Y2 Þ 2 15rffiffiffiffiffiffiffi ! The forced, dissipative, shallow water equations on a rotating sphere are (cf. Longuet-Higgins (1968) Eqs. (13.1)–(13.3) or Tyler 3 2p 2 2 iXt 1 1 iXt 1 ¼ 2 X R h0 Rðe Y Þ2U R e Y ; ð6Þ (2011) Eqs. (3) and (4), noting the sign difference in the tidal po- 2 15 2 2;W 2 tential term) and a symmetric eastward propagating potential Uobl;E for which the 1 1 @~u 2 eiXt term is replaced by eiXt and U U . þ 2X cos h^r ~u ¼gr~g r~U a~u þ mr ~u ð1Þ 2;E ¼ 2;W @t and 2.1.2. Eccentricity tides The eccentricity tidal potential can be expressed as (Kaula, @g þ hr~ ~u ¼ 0; ð2Þ 1964) (cf. Tyler (2011) Eq. (35)) @t 3 where ~u is the radially-averaged, horizontal velocity vector, X is the U ¼ X2R2e½ð3 cos2 h 1Þ cos Xt þ sin2 hð3 ecc 4 constant rotation rate, ^r is the unit vector in the radial direction, g is / / : the surface gravity, g is the vertical displacement of the surface, and cos 2 cos Xt þ 4 sin 2 sin XtÞ ð7Þ h is the constant ocean depth. The dissipation can be represented as For the subsequent analysis, the eccentricity tidal potential can be either a linear process with a linear coefficient a or a Navier–Stokes split into three separate components. There is an axisymmetric type viscosity with a viscous diffusivity of m. We assume no radial component, Uecc;rad,(Tyler (2011) calls this the ‘‘radial’’ component) gradients in our shallow-water model such that the Laplacian oper- rffiffiffiffi 2 ator, r , has no radial contributions and thus, m is an effective hor- p 2 2 0 U ¼ 3 X R e cos Xt Y izontal diffusivity. U represents the forcing potential due to tides, ecc;rad 5 2 rffiffiffiffi either eccentricity- or obliquity-related. Eqs. (1) and (2) are valid 1 p 2 2 iXt iXt 0 0 iXt iXt 0 for incompressible flow under the assumptions that the thickness ¼ 3 X R e ðe þ e ÞY U ðe þ e ÞY ð8Þ 2 5 2 2 2 of the fluid layer is much smaller than the radius of the body

ðh RÞ, the vertical displacement is much smaller than the layer There is also an asymmetric librational component, Uecc;lib, that can thickness ðg hÞ and fluid properties are constant (e.g. a and m). be split into a westward propagating potential E.M.A. Chen et al. / Icarus 229 (2014) 11–30 13 rffiffiffiffiffiffiffi ! with equivalent expansions for U and g. Here the coefficients Wm 3 2p 2 2 i t 2 i t 2 l X X m Uecc;lib;W ¼ X R e e Y 2 þ e Y2 m 4 15 (and their equivalents Ul , and gl ) are complex, thereby permitting rffiffiffiffiffiffiffi ! a phase lag between the tidal forcing and the response. The response 3 2p frequency is assumed to be the same as the forcing frequency. ¼ 2 X2R2e R eiXtY2 2U2 R eiXtY2 ð9Þ 4 15 2 2;W 2 Using Eq. (18) and the wave solutions for W (19) and its equivalents, we have and an eastward propagating potential h rffiffiffiffiffiffiffi ! gm ¼i lðl þ 1ÞUm: ð20Þ l R2X l 21 2p 2 2 iXt 2 iXt 2 Uecc;lib;E ¼ X R e e Y þ e Y 4 15 2 2 Plugging these wave solutions into Eqs. (16) and (17) and pro- rffiffiffiffiffiffiffi ! jecting the equations onto each spherical harmonic, we obtain 21 2 m m p 2 2 iXt 2 equations describing solutions for the coefficients W and U that ¼ 2 X R e R e Y2 l l 4 15 are coupled in l but separable in m, 2 iXt 2 mlðl þ 1Þ 2Xim l 1 l þ 2 2U2;ER e Y2 : ð10Þ iX þ a þ Wm 2X CmUm þ Cm Um ¼ 0; ð21Þ R2 lðl þ 1Þ l l l l1 l þ 1 lþ1 lþ1 2.2. Semi-analytic method of solution mlðl þ 1Þ 2Xim ighlðl þ 1Þ iX þ a þ Um 2 lðl þ 1Þ 2 l R XR The horizontal velocity can be expressed using a Helmholtz l 1 m m l þ 2 m m decomposition as the sum of the gradient of a scalar potential U þ 2X Cl Wl 1 þ Cl 1Wl 1 l l þ 1 þ þ and the curl of a streamfunction W^r where (cf. Tyler (2011) Eq. (5)) no ¼U1 d d ; U2 d d ð22Þ ~ ~ ^ ^ 2;W l;2 m;1 2;W l;2 m;2 ~u ¼ rU þ r W^r ¼ uhh þ u//; ð11Þ with the term on the right hand side of (22) corresponding to the with the components of velocity given by forcing potential of interest, either westward obliquity or westward 1 @U 1 @W m librational eccentricity. The constants Cl are due to differential uh ¼ þ ð12Þ R @h R sin h @/ operators of the spherical harmonics, and these are given by and ðl þ mÞðl mÞ 1=2 Cm ¼ : ð23Þ 1 @U 1 @W l ð2l þ 1Þð2l 1Þ u/ ¼ : ð13Þ R sin h @/ R @h Eqs. (21) and (22) are effectively Eq. (3.19) of Longuet-Higgins (1968) The divergence and radial component of the curl of (11) are with correction of the sign inconsistencies described in Appendix A.1 of Tyler (2011). The differences in the constants between the equa- ~ 2 r ~u ¼ r U; ð14Þ tions presented here and those of Longuet-Higgins (1968) and Tyler ^r ðr~ ~uÞ¼r2W: ð15Þ (2011) are primarily due to the use of normalized spherical harmonics throughout this work and a difference in the sign of the tidal Taking the divergence of the momentum Eq. (1) and making the potentials. For helpful relations, similar to (3.17) and (3.18) of Long- substitution from (14) results in uet-Higgins (1968), we refer the reader to Appendix A. The linear system defined by Eqs. (21) and (22) can be easily solved numerically for @ 2 2 2X @ 2 sin h @ þ a mr r þ U þ 2X cos hr W the coupled coefficients Um and Wm, repeating for each potential of @t R2 @/ R2 @h l l interest. While the series is infinite, typically the coefficients drop 2 2 ¼gr g r U: ð16Þ off rapidly with increasing l (Fig. 1). We exploit this fact in Section 3 Similarly, the radial component of the curl of the momentum equa- to derive analytical scalings that reduce the complexity of imple- tion with the substitution (15) is menting ocean tidal effects in thermal evolution models. A similar projection can be obtained for the eastward forcing @ 2X @ sinh @ þamr2 r2 þ W2X coshr2 U¼0: ð17Þ potentials by changing the direction of the response in (19) and @t R2 @/ R2 @h its equivalents (see details in Appendix B). Additionally, in (19) we have neglected the contributions from the radial component Eqs. (16) and (17) are equivalent to Eq. (3.7) of Longuet-Higgins of the eccentricity tide ðm ¼ 0Þ. These coefficients must be real (1968), with the dissipative terms added. Eqs. (16) and (17) are and thus, different wave solutions are required. We present the also equivalent to Eqs. (12) and (13) of Tyler (2011) if one ignores solution method explicitly in Appendix C. We have presented the the residual terms (R1 and R2) and does not make the substitution westward response because for satellite oceans the dominant re- for g using the continuity equation (see (18) below). sponse is likely to be that to the westward propagating component Substituting (14) into (2), the surface displacement is depen- of the obliquity tide (Tyler, 2008); for further details, we direct the dent only on U, reader to Section 3.1. @g þ hr2U ¼ 0: ð18Þ Given the spherical harmonics coefficients for U and W, the aver- @t age kinetic energy and energy dissipation can then be calculated. It For the response to a westward propagating tidal potential, we can be shown (see details in Appendix D) that the kinetic energy seek wave solutions of the form averaged over the satellite surface and the orbital period is given by: " X1 X1 Xl 0 2 0 2 ÂÃ E ¼ qh lðl þ 1ÞjU j þjW j Wðh; /; tÞ¼ WmeiXtY m þ WmeiXtY m tot;avg l;W l;W l l l l l¼1 l¼1 m¼1 # ! X1 Xl X1 Xl ÂÃ þ lðl þ 1ÞjUm j2 þjUm j2 þjWm j2 þjWm j2 : ð24Þ 2 WmeiXtYm ; 19 l;W l;E l;W l;E ¼ R l l ð Þ l¼1 m¼1 l¼1 m¼1 14 E.M.A. Chen et al. / Icarus 229 (2014) 11–30

Fig. 1. Power spectrum with varying spherical harmonic degree l of the full semi-analytic solution described in Section 2.2 using parameters relevant to an ocean on Europa (Tables 1 and 2), here with a viscosity of m ¼ 1:0 102 m2=s for both the obliquity and eccentricity tides. The response of each tidal component is separated. Note the signiﬁcant separation between degrees 1 and 2 in the case of the westward obliquity response and degrees 3 and 4 for all other tidal components, justifying the scaling arguments presented in Section 3 and Appendices E and F.

The corresponding average ﬂow speed u is then given by et al., 1988). It will be shown in Section 3 and Appendices E and F ! that for a wide range of parameters, the primary role of the viscosity 1=2 E is to set the tidal phase lag without altering the amplitude of the u ¼ tot;avg : ð25Þ 2pR2hq response.

The energy dissipation rate similarly averaged (and taken to be positive) is 2.3. General features of the shallow-water tidally-forced system " X1 In Fig. 1, we show a solution for the spectral coefﬁcients from E_ ¼ 2aE ¼ 2aqh lðl þ 1ÞjU0 j2 þjW0 j2 lin;avg tot;avg l;W l;W the semi-analytic method for a model global ocean on Europa l¼1 # (see parameters in Tables 1 and 2) including all tidal components. X1 Xl m 2 m 2 m 2 m 2 Even though in this case the eccentricity is ten times larger than þ lðl þ 1ÞjUl;W j þjUl;Ej þjWl;W j þjWl;Ej ; ð26Þ l¼1 m¼1 the assumed obliquity, the dominant tidal component is the westward obliquity response ðl ¼ 1Þ, as pointed out in Tyler (2008). The for linear dissipation and spherical harmonic modes are coupled in degree because of the " X1 Coriolis force. This coupling is fairly weak, and the response has _ 2qhm 2 2 0 2 0 2 ENS;avg ¼ l ðl þ 1Þ jU j þjW j power primarily in the modes around the large scale tidal forcing. R2 l;W l;W l¼1 # For the westward propagating obliquity response, the responses at X1 Xl degrees 2 and 3 have much smaller amplitudes than that at degree 2 2 m 2 m 2 m 2 m 2 þ l ðl þ 1Þ jUl;W j þjUl;Ej þjWl;W j þjWl;Ej : 1. Degrees 2 and 3 dominate the eccentricity response. For degrees l¼1 m¼1 l > 3, all responses are negligible. This separation between degrees ð27Þ 3 and 4 allows us to carry out the scaling arguments presented in Section 3. for Navier–Stokes type dissipation. A key input parameter to the system is the value of the dissipa- An alternative method of thinking about tidal dissipation is that tion coefﬁcient, a or m. We present the average kinetic energy and in the time- and spatially-averaged sense it is equivalent to the work done by the tides. It can be shown that this work is related to distorting the surface (see details in Appendix D.2.2): "# X2 Table 1 _ 0 0 m m m m Constant parameters used in thermal energy calculations. Eavg ¼2ð2Þð3ÞqhU2R U2;W þ U2;W RðU2;W ÞþU2;ER U2;E : ð28Þ m¼1 Parameter Value First reference

Note that dissipation depends only on the real part of U and is inde- H 4.5 1012 Wkg1 Section 5.1 3 pendent of W. Two physical consequences follow immediately. First, qsil 3500 kg m Eq. (65) 3 situations in which there is no surface displacement (g ¼ U ¼ 0as qice 950 kg m Eq. (65) in Tyler (2008)) will result in no dissipation. Second, while the Q sol 100 Eq. (66) G 6.67 1011 m3 kg3 s2 Eq. (66) amount of kinetic energy depends on the magnitude of U (see l 4 109 Nm2 Eq. (67) (24)), the energy dissipation rate depends on only the real compo- c0 0.33 Eq. (68) nent of U, and thus is sensitive to the phase lag between the tidal q (ocean) 1000 kg m3 Eq. (24) forcing and the ﬂuid response, similar to viscoelastic tidal dissipa- h 30 km Eq. (2) c 0.002 Eq. (54) tion in solid bodies (Kaula, 1964; Peale and Cassen, 1978; Segatz D E.M.A. Chen et al. / Icarus 229 (2014) 11–30 15

Table 2 Satellite speciﬁc parameters used in thermal energy calculations. The physical parameters on the left half of the table are taken from Schubert et al. (2004) for the Galilean satellites, Thomas (2010), Zebker et al. (2009) and Jacobson et al. (2006) for the saturnian satellites, Thomas (1988) and Jacobson et al. (1992) for the uranian satellites, and Thomas (2000) and Jacobson (2009) for Triton. The orbital parameters are taken from JPL satellite ephemerides (http://ssd.jpl.nasa.gov/?sat_elem). Calculations for the quantities on the right side of the table are described in Section 5.1; measured (as opposed to calculated) quantities are indicated with ⁄. The gravity coefﬁcients for the Galiean satellites are from Galileo measurements (Schubert et al., 2004); others are calculated from a hydrostatic assumption. The obliquity for Titan is taken to be the observed value (Stiles et al., 2008) and is not a theoretical Cassini state value.

3 2 R (km) qsat (kg m ) g (m/s ) X (rad/s) pei(deg) k2 J2 C2;2 h0 (deg) Europa 1565.0 2989 1.31 2.05 105 3.10 103 0.0094 0.466 2.08 101 4.36 104⁄ 1.32 104⁄ 0.053 Ganymede 2631.2 1942 1.43 1.02 105 6.77 103 0.0013 0.177 2.42 101 1.28 104⁄ 3.83 105⁄ 0.033 Callisto 2410.3 1834 1.24 4.36 106 7.42 103 0.0074 0.192 1.89 101 3.27 105⁄ 1.02 105⁄ 0.24 Mimas 198.2 1150 0.064 7.72 105 3.82 102 0.0196 1.574 5.73 104 1.43 102 4.30 103 0.041 Enceladus 252.1 1610 0.11 5.31 105 6.29 102 0.0044 0.003 1.81 103 4.83 103 1.45 103 0.00014 Tethys 531.0 985 0.15 3.85 105 9.64 102 0.0001 1.091 3.01 103 4.17 103 1.25 103 0.039 Dione 561.4 1478 0.23 2.66 105 1.56 103 0.0022 0.028 7.56 103 1.32 103 3.97 104 0.0020 Rhea 763.5 1237 0.26 1.61 105 2.90 103 0.0002 0.333 9.78 103 5.73 104 1.74 104 0.030 Titan 2574.73 1882 1.35 4.56 106 1.61 104 0.0288 0.306 2.21 101 – – 0.32⁄ Miranda 235.8 1200 0.079 5.15 105 4.58 103 0.0013 4.338 8.84 104 6.10 103 1.83 103 0.021 Ariel 578.9 1665 0.27 2.89 105 8.30 103 0.0012 0.041 1.02 102 1.39 103 4.16 104 0.00050 Umbriel 584.7 1399 0.23 1.76 105 1.12 104 0.0039 0.128 7.35 103 6.13 104 1.84 104 0.0026 Titania 788.9 1714 0.38 8.35 106 8.20 103 0.0011 0.079 1.99 102 1.13 104 3.38 105 0.014 Oberon 761.4 1630 0.35 5.40 106 5.30 103 0.0014 0.068 1.68 102 4.13 105 1.48 105 0.075 Triton 1353.4 2060 0.78 1.24 105 4.27 104 0.000 156.87 8.11 102 2.07 104 6.20 105 0.35

energy dissipation for a range of hypothetical viscosities in Figs. 2 more importance for models of thermal evolution, the energy dis- and 3. The value of the viscosity is important in this model for two sipation rate depends on the viscosity. Thus, determination of a sa- main reasons. First, the value of the viscosity determines the tellite’s thermal and orbital history hinges on the prescribed value behavior of the response to the westward propagating obliquity for the viscosity, which is in reality very poorly constrained. We tide. For the tidal responses not due to the westward propagating address this issue in Section 4. obliquity tide, the behavior is fairly straightforward. The average kinetic energy is relatively independent of the value of the dissipation coefficient prescribed for a large range of values (m K 106 m2=s 3. Scaling laws for tidally-driven ocean flows in Fig. 2). For the westward obliquity response, the behavior is different; there is a parameter regime where the amount of kinetic The shallow-water system presented in Section 2 can be solved energy, and thus the flow velocities, are sensitive to the prescribed using a semi-analytic method with computational ease. The chal- value of m. Additionally, in this regime, the obliquity response may lenge in understanding the ocean response to the tidal forcing is be less important than the eccentricity response. Second, and of determining how the various input parameters affect the resulting

Fig. 2. Average kinetic energy associated with each of the tidal responses for an ocean on Europa with parameters from (Tables 1 and 2), where symbols indicate the values calculated from scaling arguments (Section 3 and Appendices E and F) and the solid lines indicate values calculated through the semi-analytic solution (Section 2.2). The shaded region indicates where the Reynolds number is less than 50 for this model ocean and the scaling argument predictions are no longer within 10% of the semi-analytic solution value (see (29)). The dashed line represents the effective viscosity for the response to the eccentricity tide and the dot-dashed line represents that for the obliquity tide, both of which are numerically-derived (see Section 4.2) using the same coefﬁcient of drag ðcD ¼ 0:002Þ. Note that these effective viscosities are not the same, making the obliquity tide the likely dominant contributor to the ocean energy dissipation (see Section 5.2 for further details). 16 E.M.A. Chen et al. / Icarus 229 (2014) 11–30

Fig. 3. Similar to Fig. 2 but for average energy dissipation.

flow and choosing an appropriate value for m or a. We address the of the linear rotational speed to the surface gravity wave speed, , first issue here and discuss an estimate for the effective ocean vis- called the Lamb parameter in Longuet-Higgins (1968), cosity in Section 4. 4X2R2 The shallow water system presented in Section 2.2 involves ser- : ies solutions that are coupled in spherical harmonic degree. How- gh ever, as can be seen in Fig. 1, the power decreases quite rapidly Below, we present scaling arguments which allow us to analyt- with increasing spherical harmonic degree l. The tidal responses, ically calculate the ocean response, the average kinetic energy and previously represented as infinite sums or for numeric purposes average energy dissipation rate in the case of Navier–Stokes type as a truncated series, can be reasonably estimated by using a very dissipation. The algebra in the following section can be tedious; limited number of modes (typically 2–3) without any significant we present scalings for the response to the westward propagating loss of accuracy. Instead of solving the system of equations given obliquity tidal potential here and direct the reader to Appendices E by (21) and (22) numerically, we now solve the system and F for details associated with the other components of the tide. algebraically. For those uninterested in the details, we summarize all of the scal- The benefits of these solutions are multifold. First, the simplifi- ing results in Table 4. These estimates are accurate to 10% of the cations we present make the underlying physics more transparent full solutions from Section 2.2 if the following limits are satisfied, than the fully coupled problem. Second, these solutions provide high Reynolds number, relations for the kinetic energy and energy dissipation in terms of the various input parameters (e.g. R; g; h; X), instead of the Re 50; ð29Þ spherical harmonic coefficients. These relations provide insight and small Lamb parameter, into the sensitivity of the solutions to the various input parameters 4 that themselves may have a degree of uncertainty. Additionally, 6 : ð30Þ these relations are simple to implement in orbital evolution mod- 15 els and are likely accurate in most scenarios, similar to the familiar For typical satellite oceans, where the rotation rate and satellite ra- solid-body tidal heating equations (Segatz et al., 1988; Wisdom, dius are well-known, these limits should be thought of as low effec- 2008). These scalings are limited in that they do not address the is- tive viscosity (m K 106 m2=s) and thick oceans (h J 10 km), sue that the ocean’s effective viscosity is essentially unconstrained, respectively. Because the molecular viscosity of water is very small, but we will use these scalings to inform the estimates of the viscos- the scaling constraint (29) should be appropriate for all satellite ity that we present in Section 4. oceans. Earth lies in a high Lamb parameter regime because the In addition to the tidal forcing, the ocean responds to three oceans on Earth are far shallower than those presumed to exist on forces (cf. momentum Eq. (1)): pressure gradients in the form of the icy satellites (Khurana et al., 1998). In the small Lamb parameter surface gravity waves, the Coriolis force, and the drag. For the sca- regime, the time lag between the forcing potential and the ocean’s lings we suggest below, we need the ratios of these forces. We adopt response is small because the gravity wave speed is fast in compar- a Navier–Stokes viscosity such that the role of viscosity can be com- ison to the surface rotational speed; as the tidal potential changes, pared to that of rotation through a rotational Reynolds number Re, the gravity waves can rapidly propagate this disturbance. Under these limits, we now present the analysis for the re- XR2 sponse to the westward propagating obliquity tide. This is likely Re ; m to be the dominant ocean response for many satellite parameters (as recognized in Tyler (2008)) but for completeness, we also direct where the characteristic velocity is taken to be the linear rotational the reader to Appendices E and F for details related to the other speed, not the typical ocean flow velocity, and the characteristic tidal components. As discussed below, the obliquity tide involves length scale is R. We are also concerned with the square of the ratio a resonant response which can drive large flows. Other resonant E.M.A. Chen et al. / Icarus 229 (2014) 11–30 17 modes exist, but typically require very thin oceans (Tyler, 2011). In similar fashion, the average energy dissipation (27) can be We therefore ignore these modes, because of the expected thick- approximated as nesses (tens of km) of most icy satellite oceans (see Section 5.5). 2 2qhm 2 2 1 2 1 1 2 E_ ð1Þ ð2Þ W ¼ 18 C qhXRe U : ð38Þ R2 1 2 2 3.1. Scalings for the response to the westward obliquity tidal potential Substituting (6) and (36) into (38), the energy dissipated by the response is given as Looking at the response to the westward obliquity tide (Fig. 1), 2 2 the dominant component of the power spectrum is the l ¼ 1 term 1 1 72 C2 qhU2;W (i.e. W1). This response is coupled through the Coriolis force to U1 E_ 1 2 4 ÀÁ and the forcing tide. We thus carry out this scaling analysis with 1 1 2 XRe 9 C2 þ 2304 Re the assumption that the response is accurately captured by only 1 2 the l 1 and l 2 terms. The vorticity Eq. (21) for W here written 1 ¼ ¼ 1 8 10 2 108 C2 p qhmX R h in terms of the Reynolds number and Lamb parameter is ¼ 0 : ð39Þ 4 5 1 6 8 2 2 2ð1Þ 3 9 C X R þ 144m2g2h i þ i W1 2 C1U1 ¼ 0: ð31Þ 2 Re ð1Þð2Þ 1 2 2 2 We may also use Eq. (37) to derive an average flow speed u: The first imaginary term in (31) represents the ocean response

1 1 while the second imaginary term represents a Rossby–Haurwitz 3 C2 U2;W wave. In this instance, these two exactly cancel, and we obtain u ¼ ffiffiffiffi 4 ÀÁ 1=2 p 1 1 2 1 pXR 9 C2 þ 2304 Re 1 3C2 1 W1 ¼ ReU2; ð32Þ rffiffiffiffiffiffi 2 4 5 27 X R h0 ¼ : ð40Þ where the amplitude of the response grows as the viscosity de- 50 4 1=2 1 6 8 2 2 2 creases, and Reynolds number increases; this is the resonance that 9 C2 X R þ 144m g h Tyler (2008) refers to. For other tidal responses (see Appendices E and F), the imaginary terms do not disappear and in fact, are much The scalings (37), (39) and (40) are the general form for high larger than the Reynolds number term. These terms limit the re- Reynolds number and small Lamb parameter as defined in (29) sponse amplitude and, therefore, the overall behavior of the other and (30). While the scalings are cumbersome, these values are sim- responses differ dramatically from that of the westward obliquity ple to calculate in a numerical code. As mentioned previously, it is response. clear from Figs. 2 and 3 that the response to the westward propa- 1 1 The amplitude of W1 is related to U2 by the horizontal diver- gating obliquity tide can be split into two behavioral regimes; gence Eq. (22) which regime applies depends on the product of the Reynolds 1 1 number and the Lamb parameter, Re. We present further simplifi- 2 6 24 3 2 U i þ i U1 þ ðC1Þ ReU1 2 ð33Þ cations for these two regimes in order to highlight the underlying 3 Re 2 2 2 2 X force balances and behavioral physics.ÀÁ 1 For high Reynolds numbers 1 K 1 , the Rossby–Haurwitz when the relation (32) is substituted and the contribution from W3 Re 80 is neglected. In the appropriate scaling limits (large Reynolds num- wave response dominates the bulk behavior. Ignoring the gravity 1 ber and small Lamb parameter), the dominant terms of (33) are wave contribution in (34), U2 is rffiffiffiffiffiffiffi 1 U1 2p 3 1 2 24 1 U2;W 1 2;W ðC Þ Re i U : ð34Þ U2 2 ¼5 mh0; ð41Þ 2 2 2 X 3 1 15 X 2 ðC2Þ Re The total kinetic energy (24) of the response is dominated by W1 1 which in turn yields (Fig. 1) and can be approximated as rffiffiffiffiffiffiffi 1 2 U 3 2p 1 1 2;W R2h : 42 1 2 9ðC2Þ 2 1 2 W1 1 ¼ X 0 ð Þ Etot 2qh W ¼ qhRe U ; ð35Þ ðC ÞX 2 3 1 2 2 2 when the relation (32) is included. Recasting (34), we have This solution asymptotes to the inviscid solution presented in Tyler (2008). In this regime, the kinetic energy is independent of 1 2 1 2 jU2;W j viscosity, jU2j 2 2 3 1 24 2 1 2 2 4 2 X j 2 ðC2Þ Re i j Etot 2qh W1 ¼ 3pqhX R h0; ð43Þ 4jU1 j2 ¼ 2;W : ð36Þ and we can derive an average flow speed u of 4 ÀÁ 2 2 1 1 2 X Re 9 C þ 2304 1=2 2 Re 3 u ¼ XRh : ð44Þ 2 0 Substituting (6) and (36) into (35), the kinetic energy of the response is given as This makes intuitive sense: if viscosity is not impeding flow signif-

2 icantly, the flow speed is controlled by the rate at which the tidal 18ðC1Þ qhjU1 j2 2 2;W bulge is moving around the body. From (27) or (28), the associated Etot 4 ÀÁ 2 1 1 2 energy dissipation scaling scales linearly with the viscosity X 9ðC2Þ þ 2304 Re 2 2 2 2 2 E_ 12 h X R h 8 h u : 45 1 pq m 0 ¼ q m ð Þ 8 12 2 27 C2 p qhX R h ÀÁ ¼ 0 : ð37Þ For lower Reynolds numbers 1 J 1 , while still remaining in 4 Re 80 5 1 6 8 2 2 2 9 C2 X R þ 144m g h the scaling limits defined by (29) and (30), the gravity waves dominate the response. Then, 18 E.M.A. Chen et al. / Icarus 229 (2014) 11–30 rffiffiffiffiffiffiffi 1 U i 2p X3R4h The effective Q of the Earth is known (Q ¼ 12; Murray and Der- 1 ÀÁ2;W 0 46 U2 i24 ¼ ð Þ mott (1999)) from lunar laser ranging, and is due mainly to ocean X 4 15 gh dissipation. Unfortunately, the Earth’s ocean is in many ways quite and therefore different to that on an icy satellite: it spins non-synchronously, it rffiffiffiffiffiffiffi has bottom topography that is comparable to the ocean thickness, C1U1 Re 4 6 1 3 2 ÀÁ2;W i3 2p X R h0 and it is stably stratified in some regions (see Section 5.5). As a result, W1 i24 ¼ : ð47Þ 2 X 40 3 mgh it is hard to use this particular observational constraint to derive an effective Q for icy satellites. Instead we adopt an approach based on In this regime, the viscosity has altered the natural frequency of the an empirically-derived friction coefficient, as described next. system, and thus the response amplitudes no longer behave reso- nantly. The kinetic energy now depends on the value of the Rey- nolds number, 4. Effective viscosity in a global ocean 2 1 8 12 2 The scalings presented in Section 3 provide a description of glo- 2 C2 p 3p qX R h 1 2 4 2 2 0 bal ocean behavior; however, to estimate the energy dissipation in Etot 2qh W1 qhX R h0ðReÞ ¼ 2 2 ð48Þ 320 400 m g h the ocean, these scalings require a prescribed value for the effec- with an associated average flow speed of tive viscosity. This viscosity is unlikely to be as small as the molecular viscosity, presumably due to turbulence (Pope, 2000), but it is 1=2 4 5 3 X R h0 unclear how large an effective viscosity one should adopt (Lumb u ¼ : ð49Þ 800 mgh and Aldridge, 1991). While there is potential to infer from measurements what the effective viscosity is for a global liquid layer, In this limit flow speeds decrease as viscosity increases. The associ- such as for the liquid core of Earth (Smylie, 1999; Mound and Buff- ated energy dissipation is now inversely proportional to the ett, 2007; Buffett and Christensen, 2007) or Mercury, it is unlikely viscosity, that we can do so for icy satellite oceans. Also, while these mea- 3p qX8R10h2 surements can give some insight into the effects of turbulence, E_ 0 : ð50Þ 100 mg2h the effective viscosity is likely to be time-variable, changing with flow conditions and/or degree of turbulence (Brito et al., 2004). The shape of the energy dissipation curve (Fig. 3) can be thought of Because of the high degree of variability amongst satellites of as the product of two effects: the bulk flow velocity and the value of the outer Solar System both in the past and at present, it is unlikely the viscosity. This peak in dissipation rate is similar to that seen for that a single value for the effective ocean viscosity can be broadly Maxwell viscoelastic dissipation in solid bodies (Ross and Schubert, applied. Thus, we approach the question of what the effective vis- 1989). For the high Reynolds number regime, the velocities are high cosity in a satellite ocean is by analogy with the Earth’s oceans. On but the viscosity is too low to generate significant dissipation. In the Earth, there is a long history of estimating dissipation of tides due lower Reynolds number regime, the viscosity is large and thus can to bottom boundary friction (Taylor, 1920; Jeffreys, 1921; Munk dissipate energy. But in analogy with forced-damped harmonic and MacDonald, 1960). While the drag coefficient cD (see Eq. (54) oscillators, the viscosity has moved the natural frequency of the below) was originally empirically-based (Taylor, 1920), a value cD system sufficiently away from the forcing frequency that the re- of O (0.001) is still commonly used, even in more modern numer- sponse no longer has large resonant amplitudes. ical ocean models (Jayne and Laurent, 2001; Egbert and Ray, 2001).

Sohl et al. (1995) discuss what value of cD to apply to Titan’s ocean.

3.2. Effective tidal quality factor Q in a satellite ocean Taking the weak dependence of cD on Reynolds number into ac- count, these authors conclude that values in the range 0.002– Throughout this section, we have considered viscous effects 0.01 are likely appropriate for liquid water. through a Navier–Stokes viscosity, primarily because this allows We incorporate a cD-based parameterization into the shallow for parameterization in the form of the Reynolds number. These water model for a global ocean. This approach is undeniably simplis- scalings can also be expressed in terms of an effective tidal quality tic. However, when given the alternative of choosing a value for the factor Q, which can in turn be related to the linear drag coefficient effective viscosity where the uncertainty is approximately twelve or- a mentioned in Section 2. Q is defined as (Murray and Dermott, ders of magnitude (Lumb and Aldridge, 1991), we favor our approach, 1999), although there are caveats. We address these in Section 5.5. While we 2pE XE adopt the value cD ¼ 0:002 in this work, the results and scalings that Q ÀÁtot ¼ tot ð51Þ _ 2p _ we present in Section 4.2 are general (and linear) for cD,andthus,per- E X E mit the effect of uncertainties in cD to be readily explored. and is related to the linear drag coefficient a by (Tyler, 2011), 4.1. Numerical method X a ¼ : ð52Þ 2Q The momentum equation for an ocean subject to tidal forcing For the westward propagating obliquity tide, by substituting (43), and bottom boundary friction is (cf. (Jayne and Laurent, 2001)) (45), (48) and (50), we find that the tidal quality factor is simply @~u c ! þ 2X~ ~u ¼gr~g r~U D j~uj~u: ð54Þ 1 XR2 1 @t h Q ¼ Re ð53Þ obl;W 4 m 4 Similar to the process in Section 2.2, we can employ a Helmholtz decomposition and write the divergence and the radial component for all regimes. We emphasize that this is an effective Q, in attempt- of the curl of (54) as ing to maintain consistency with the terminology used in Tyler @ 2X @ sin h @ (2011). This Q is different from that typically defined for solid-body r2 þ U þ 2X cos hr2 W @t 2 @/ 2 @h tidal heating (Peale et al., 1979; Murray and Dermott, 1999) in that R R c the dissipation is referenced to the kinetic energy of the ocean, in- ¼gr2g r2U D r~ ðj~uj~uÞ ; ð55Þ stead of the stored elastic energy due to the tide. h E.M.A. Chen et al. / Icarus 229 (2014) 11–30 19 @ 2X @ sin h @ dissipated for our model Europan ocean is approximately 109 W r2 þ W 2X cos hr2 U @t 2 @/ 2 @h total, and the amount dissipated from the enhanced dissipation R R c term is less than 0.1 W. ¼ D ^r ðr~ ðj~uj~uÞÞ : ð56Þ h We have run a series of numerical integrations varying the se- ven input parameters (i.e. R; g; h; X; h0; e, and cD) to understand While the solutions to (55) and (56) are likely oscillatory, the non- the sensitivity of the system to each. These numerical integrations linearity of the dissipative term requires that we integrate the sys- are initialized to either zero or the semi-analytic solution; the stea- tem numerically. We seek solutions of the form dy-state solution is independent of these values. We evolve the XNL Xl ÂÃ equations in time until they produce a constant solution for the ki- m m m m Wðh; /; tÞ¼ Wl Yl þ Wl Y l netic energy and energy dissipation when averaged over an orbital l¼1 m¼1 ! period, similar to the behavior of the linear solution (cf. (24) and XNL Xl ÂÃ (27)). m m ¼ 2R Wl Y l ; ð57Þ l¼1 m¼1 4.2. Scaling arguments m and similarly for U and g. Here the coefficients Wl (and their equiv- m m The shallow water system including nonlinear bottom drag set- alents Ul , and gl ) are still complex but are now functions of time, and NL is the series truncation level. Plugging in (57) and its equiv- tles on a wave solution. While the dissipation processes are not ex- alents into (55), (56) and (18), and projecting the resulting equa- actly alike, we can assume that the previous linear Navier–Stokes tions onto individual spherical harmonics, the system of equations formulation is equivalent, in a time-averaged sense, to the nonlin- to solve is ear bottom drag with c @ 2Xim l 1 l þ 2 D j~uj~u mr2~u: ð61Þ Wm 2X CmUm þ Cm Um h @t lðl þ 1Þ l l l l1 l þ 1 lþ1 lþ1 which implies that 2 m R cD ~ ¼ ^r ðr ðj~uj~uÞÞ ð58Þ 2 lðl þ 1Þ h l c R j~uj m D ; ð62Þ lðl þ 1Þh @ 2Xim m l 1 m m l þ 2 m m U þ 2X C W þ C W wherepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the lengthscale associated with the Laplacian is taken to be @t lðl þ 1Þ l l l l1 l þ 1 lþ1 lþ1 R= lðl þ 1Þ. Assuming that j~uj scales with the average velocity, we 2 m m R cD ~ 0 find the scalings ¼ggl þ r ðj~uj~uÞ 2U2 cosðXtÞdl;2dm;0 lðl þ 1Þ h l 4 7 cD X R h0 1 meff ;obl ¼ kobl ð63Þ 2U2;W cosðXtÞdl;2dm;1 h 4 1=2 hi ð9ðC1Þ X6R8 þ 144m2 g2h2Þ 2 2 2 2 2 eff ;obl U2;W þ U2;E cosðXtÞþ U2;W U2;E i sinðXtÞÞ dl;2dm;2; by using the relation (40) for the obliquity-related response and ð59Þ 3 5 cDX R e m meff ;ecc ¼ kecc 2 ð64Þ @g hlðl þ 1Þ m gh l U ¼ 0: ð60Þ @t R2 l by using (F.10), (F.20) and (F.29) for the eccentricity response. From

These equations are coupled in both spherical harmonic degree l our numerical runs, the least squares fit for kobl is 0.40 and kecc is and order m through the dissipative terms on the right hand side 0.13. of Eqs. (58) and (59). We include all components of the tide in The value of the scaling parameters in Eqs. (63) and (64) may be (59) because nonlinearity has the ability to couple different spher- approximated analytically by taking the flow speeds from Table 4 ical harmonic orders, while previously these were separable. Be- and substitutingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi into Eq. (62). For the eccentricity tide, this yields cause we have not assumed wave solutions, we now have to solve kecc 57=112=plðlffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ 1Þ0:12, taking l ¼ 2. For the obliquity tide, explicitly for the spherical coefficients of g, whereas we previously we obtain kobl 27=50=lðl þ 1Þ0:37, taking l ¼ 1. These analyt- assumed the relationship (20) between g and W. ical approximations agree very well with the numerical results. Eqs. (58)–(60) are nonlinear and thus cannot be solved with the The effective viscosity for obliquity (63) can be solved quadrat- semi-analytic method previously employed in Section 2.2. Instead ically (see Table 4), and displays simpler behavior in the low and of assuming wave solutions as in Section 2.2, we employ a numer- high Reynolds number limits discussed in Section 3.1. For a model ical integration method where the time derivatives are explicitly ocean on Europa, the numerically-derived effective viscosity is 3 2 treated using the Adams–Bashforth method. We calculate the non- meff ;obl ¼ 3:3 10 m =s for the obliquity-related flow and 2 2 linear dissipative terms using a spectral-transform method. The meff ;ecc ¼ 1:7 10 m =s for the eccentricity-related flow. horizontal velocities are calculated on a spherical grid which con- sists of 100 evenly spaced grid points in longitude and 50 Gauss– 4.3. Summary Lobatto nodes in colatitude. The frictional dissipation contributions are calculated on this grid, and subsequently, transformed back The numerically derived scalings (63) and (64) provide esti- into their spherical harmonic decomposition. To reduce the re- mates for the effective viscosity given a set of input parameters. quired number of spectral coefficients, we added an enhanced Na- These viscosities can be input into either the semi-analytic system vier–Stokes dissipative term of the form mlðl þ 1Þ=R2 with (Section 2.2) or the scaling laws (see Table 4) in order to determine m ¼ 1 108 m2=sto(58) and (59) for all coefficients with l =NL the flow response and energy dissipation rate of a satellite ocean. It (the series truncation level) and l =NL 1. We adopt a series trun- is worth noting that the effective viscosities for the obliquity and cation level of NL = 30. The enhanced dissipative terms reduce the eccentricity responses scale differently with the input parameters required truncation level by an order of magnitude without affect- because the velocities scale differently between the two responses. ing the results presented here; for example, the amount of energy These values for a model Europan ocean are displayed in Figs. 2 and 20 E.M.A. Chen et al. / Icarus 229 (2014) 11–30

3 for reference. In the context of thermal–orbital evolution models, Titan is currently the only icy satellite with a measured obliq- where the input parameters change over time, the scalings for the uity (Stiles et al., 2008). In order to estimate the contributions to viscosities are critical to better estimating the long-term impact of other satellites from the obliquity tide, we have assumed obliqui- ocean tidal heating. ties appropriate for satellites in damped Cassini states with a single orbit precession frequency; interactions with other satellites do 5. Implications for the outer Solar System satellites not generally change the results significantly (Bills, 2005; Baland et al., 2012). In a Cassini state, the invariable pole, orbit normal, With estimates of effective viscosity in hand, it is now possible and spin pole remain coplanar as they precess (Colombo, 1966; to complete a survey of the thermal heat budgets for various satel- Peale, 1969). For this coplanar precession to occur, the obliquity lites of the outer Solar System. Specifically, we seek to answer two must satisfy the relation (Peale, 1969; Ward, 1975; Bills and Nim- fairly broad questions: should ocean tidal heating be an important mo, 2008) heat source at present for any of the icy satellites; and could ocean 3 ½ðJ þ C Þ cos h þ C p sin h ¼ c0 sinði h Þð68Þ tidal heating have been a significant heat source in the past? These 2 2 2;2 0 2;2 0 0 questions are difficult to answer in that there are many uncertainties in the parameters that go into the following calculations. Be- where J2 and C2;2 are the degree-2 gravity coefficients, i is the orbital cause our goal was to understand broad trends and not to inclination, and p is the ratio of the orbital motion to the orbit plane explain specific observations or quantities, we have used simplified precession and is given by theoretical models in order to calculate the thermal budgets, with X the understanding that these results may diverge from what is ob- p ¼ ð69Þ dX =dt served, as for example in the long-recognized anomalous cases of orb 0 Mimas or Enceladus (Squyres et al., 1983; Meyer and Wisdom, where Xorb is the longitude of the ascending node. The quantity c is 2007; Howett et al., 2011). the normalized polar moment of inertia, with the prime denoting the fact that the relevant moment of inertia may be that of a decou- 5.1. Calculating the available thermal energy pled shell rather than the entire body. We introduce this distinction because of the case of Titan, where the moment of inertia c esti- The primary heat sources available to a satellite are radiogenic mated from gravity measurements (see below) does not equal that heat and tidal dissipation, either in the solid ice and rock layers or inferred from Titan’s obliquity (Bills and Nimmo, 2011). in a liquid ocean layer. For the radiogenic component of the ther- While the orbital parameters are well-established, observations mal budget, we assume a chondritic heating rate of of the gravity coefficients and moment of inertia for many satel- H ¼ 4:5 1012 Wkg1 (Ellsworth and Schubert, 1983). We calcu- lites are limited. We estimate the unmeasured gravity coefficients late an equivalent silicate mass, Msil, assuming a two-component assuming a hydrostatic satellite using the Darwin–Radau relation satellite (Darwin, 1899; Radau, 1885; Hubbard and Anderson, 1978) ! ! 2 4 qsat qice 3 1 X 5 Msil ¼ qsil pR ð65Þ ÀÁ ÀÁ 3 C2;2 ¼ 4 2 1 ð70Þ qsil qice 4 G pq 5 15 3 sat 1 þ 2 4 c 3 where the silicate density is taken as qsil ¼ 3500 kg m , the ice 3 where c denotes the normalized polar moment of inertia of the en- density is taken as qice ¼ 950 kg m , and the satellite mean density tire body. For a hydrostatic body J2 ¼ 10C2;2=3. qsat as presented for various satellites in Table 2. To estimate the amount of solid body tidal heating, we employ For the ocean contributions to tidal heating, we estimate the the standard viscoelastic model (Segatz et al., 1988; Ross and Schu- effective viscosities from (63) and (64) for the obliquity and eccen- bert, 1989; Wisdom, 2004). In this model, the rate of solid body tricity tides separately. We use values of h = 30 km and cD ¼ 0:002 dissipation is given by for all satellites. After calculating a satellite-specific effective viscosity, the associated tidal dissipation is calculated from the ana- 5 _ 3 k2 ðXRÞ 2 2 lytical scalings summarized in Table 4; the semi-analytic E ¼ ð7e þ sin h0Þ; ð66Þ 2 Q sol G approach from Section 2.2 produces nearly identical results. Table 1 summarizes all of the common parameters used in this where the satellite structure and dissipation characteristics are analysis, while Table 2 shows the parameters used and calculated parameterized by the tidal Love number, k , and the tidal quality 2 quantities that are satellite specific. In preparing this Table, we factor, Q , and G is the gravitational constant. For the unknown sa- sol took c ¼ c0 ¼ 0:33, representative of a differentiated body with no tellite parameters, k and Q , we adopt a value of Q of 100 2 sol sol shell decoupling. Increasing c0 to 0.67 (a decoupled shell) while (Goldreich and Soter, 1966), and we estimate the Love number keeping c ¼ 0:33 results in an increase in the predicted obliquity using Kelvin’s formula (Love, 1944), h0 by a factor of roughly 2, and the obliquity-related heating by a 3=2 factor of roughly 4. k ¼ ð67Þ 2 19l The various contributions to the thermal energy budget are pre- 1 þ 2q gR sat sented for each satellite in Table 3.InFigs. 4–6 we highlight key re- for a homogeneous body with rigidity, l, assumed to be sults that are tabulated in Table 3.InFig. 4, we present the relative l ¼ 4:0 109 Nm2. We also calculate the dissipation with a contributions to the ocean tidal dissipation from the eccentricity homogeneous fluid k2 of 3=2 to understand the energy budget in and obliquity tides. While the obliquity tidal response likely dom- the limit of a fully deformable body, such as one with an ocean. inates ocean dissipation, there are examples of satellites where the In previous works, the dissipation related to the obliquity tide eccentricity response could be more important. The full thermal has typically been neglected, primarily because it is assumed that budgets of many satellites are presented in Figs. 5 and 6 where the obliquity is small and also, until recently, there have been no the solid-body tidal dissipation is calculated using illustrative val- measurements of satellite obliquity. While for most satellites the ues of Q sol ¼ 100 and k2 corresponding to an elastic homogenous obliquity solid body tidal dissipation is negligible when compared body or a fully fluid body, respectively. The actual values for these to the eccentricity driven solid body dissipation, this relationship is are likely to depend on the details of internal structure and reversed in the case of ocean tidal heating. rheology (Ross and Schubert, 1989), and in particular whether a E.M.A. Chen et al. / Icarus 229 (2014) 11–30 21

Table 3 Energy contributions calculated from parameters presented in Tables 1 and 2 as described in Section 5.1. Entries marked [] in ﬁnal column have > 4=15 for the nominal parameters.

Homogeneous k2 k2 ¼ 3=2 ______EradðWÞ Esolid;eccðWÞ Esolid;oblðWÞ Esolid;eccðWÞ Esolid;oblðWÞ Eocean;eccðWÞ Eocean;oblðWÞ Europa 2.02 1011 9.77 1011 1.34 109 7.05 1012 9.70 109 1.47 107 3.1 109 Ganymede 4.67 1011 8.79 109 2.49 108 5.46 1010 1.55 109 1.66 104 1.3 109 Callisto 3.20 1011 2.08 109 9.30 107 1.65 1010 7.40 108 8.80 102 3.8 109 Mimas 4.02 107 2.90 108 5.46 104 7.59 1011 1.40 108 2.38 107 [2.5 106] Enceladus 2.73 108 2.37 107 1.04 100 1.96 1010 8.64 102 2.32 104 1.2 101 Tethys 1.34 108 1.70 105 1.11 106 8.45 107 5.51 108 2.57 101 [3.7 107] Dione 2.42 109 4.25 107 1.52 103 8.44 109 3.02 105 4.46 103 2.2 103 Rhea 3.31 109 1.73 105 1.79 105 2.65 107 2.75 107 7.35 101 7.5 106 Titan 4.11 1011 6.43 1010 3.46 108 4.37 1011 2.35 109 1.23 105 1.1 1010 Miranda 8.50 107 6.17 105 7.10 103 1.05 109 1.21 107 6.37 102 2.6 105 Ariel 3.59 109 3.00 107 2.31 102 4.42 109 3.40 104 1.36 103 5.1 101 Umbriel 2.32 109 2.00 107 4.07 103 4.08 109 8.29 104 9.69 102 1.8 103 Titania 9.71 109 4.71 105 3.27 103 3.55 107 2.46 105 1.63 101 1.3 105 Oberon 7.76 109 6.10 104 7.81 103 5.44 106 6.98 105 5.82 103 3.3 106 Triton 7.12 1010 0 8.70 108 0 1.60 1010 0 6.6 1010

Fig. 5. Thermal heat budget for various outer Solar System satellites; solid body tidal heating components are calculated assuming a homogeneous elastic structure (Eq. (67)). Details of the calculations are presented in Section 5.1, while the numerical quantities are presented in Table 3. Pie charts indicate relative Fig. 4. Energy dissipated in various satellite oceans due to the obliquity and contributions from radiogenic heating (blue), solid body eccentricity tidal heating eccentricity tides calculated from the scalings summarized in Table 4. The (green), solid body obliquity tidal heating (yellow), ocean obliquity tidal heating parameters used are h = 30 km, cD ¼ 0:002, and satellite parameters from Table 2. (red), and ocean eccentricity tidal heating (gray). The size of the pie charts are For satellites lying to the right of the dotted line, the obliquity response is the scaled by logðRÞ. (For interpretation of the references to colour in this ﬁgure legend, dominant contribution to dissipation. Here for plotting purposes, Triton has energy the reader is referred to the web version of this article.) dissipation associated with the eccentricity, but in reality, this value should be zero. decoupling ocean exists. In choosing the ﬂuid and elastic limits, we likely bracket the actual value of the tidal dissipation, while providing insight into the degree of uncertainty associated with the tidal model. It is clear that for most of the satellites the thermal budgets are dominated by two components: radiogenic heating and eccentricity solid-body tidal heating. Triton and Tethys are likely the only satellites with large present-day heat contributions from obliquity tides, because of their relatively small eccentricities. Triton might be primarily heated by ocean obliquity tides, while Callisto and Titan could experience minor ocean heating.

5.2. Relative contributions from eccentricity and obliquity

While Tyler (2008, 2011) recognized that a satellite ocean response is typically dominated by the westward obliquity response, Fig. 6. Similar to Fig. 5, here with the solid body tidal heating components this is not always the case. Fig. 4 shows the energy dissipated in an calculated assuming a completely fluid response ðk2 ¼ 3=2Þ. ocean calculated with the input parameters h = 30 km, cD ¼ 0:002, and the satellite-appropriate parameters from Table 2. For most sa- assuming the obliquity response is in the low viscosity (nearly tellite oceans, the obliquity tidal heating contribution is larger than inviscid) regime, the average velocity will be much higher for the that from the eccentricity. The reason for this is twofold. First, obliquity related flow due to the resonant response (Tyler, 2008), 22 E.M.A. Chen et al. / Icarus 229 (2014) 11–30 if the obliquity and eccentricity are roughly comparable. Second, 5.4. Triton because higher velocities equate to higher effective viscosities (see Section 4.2), dissipation is further enhanced in the obliquity The thermal history of Triton is a puzzle; if satellite ocean tidal response. heating is significant for any icy satellite, it is most likely to be Tri- Taking the scalings for the energy dissipation from Section 3.1 ton. Observations of the surface features suggest that Triton has and Appendix F (summarized in Table 4) and the numerical results been quite active in the past (Smith, 1989; Croft et al., 1995; from Section 4.2, it can be shown, here in the low viscosity regime, McKinnon and Kirk, 2007), and this activity could have been as re- that the obliquity likely dominates unless there is a significant dis- cent as 10 Ma (Schenk and Zahnle, 2007). Eccentricity tides have parity between the eccentricity and obliquity: been evoked for heating mechanisms like frictional shear heating (Prockter et al., 2005) or ‘‘thermal blanketing’’ (Gaeman et al., E_ 44; 800 1 3 k h 3 obl obl 0 : ð71Þ 2012), but because the eccentricity at present is zero, there is no _ Eecc 519 kecc e clear reason why geologic activity should be recent. A significant amount of tidal heating likely occurred during Triton’s capture Recall that for these scalings to be applicable, the Lamb parameter, and subsequent orbital circularization (Ross and Schubert, 1990); , must be low (30), and additionally, kobl is approximately 3 (cf. Sec- kecc however, the likelihood that Triton’s capture is recent is very low tion 4.2). Oceanic energy dissipation is clearly biased towards the (Agnor and Hamilton, 2006; Schenk and Zahnle, 2007) and addi- obliquity response. Because of this, oceans on satellites with low tionally, the circularization timescale, while dependent on the orbital inclination, and thus, low Cassini state obliquity, are unlikely interior structure, is likely to be less than 1 Gyr (Ross and Schubert, to experience significant dissipation (Chen and Nimmo, 2011). 1990). Prior to observations from Voyager 2, Jankowski et al. (1989) 5.3. Global tidal heating pattern suggested that solid-body obliquity tidal heating could play a significant role in the thermal history of Triton, specifically if Triton Thus far we have primarily discussed the time- and spatially- occupied Cassini state 2. From imaging it was shown that this state averaged energy dissipation rate. While this is an important quan- was unlikely (Smith, 1989); however, the role of the obliquity tide tity when thinking about the overall thermal and orbital evolution, on the evolution of Triton should not be discounted. Ultimately, the the spatial distribution of tidal heat can also have implications for obliquity is related to the inclination of the orbit (Peale, 1969), and spacecraft observables. As suggested for solid-body tidal heating, unlike eccentricity, the inclination is not damped rapidly (Murray spatial variations in the heat flux can generate lateral variations and Dermott, 1999). It was recognized that the solid-body tidal in conductive ice shell thickness (Ojakangas and Stevenson, heating contribution is small (Jankowski et al., 1989) if Triton is 1989). These variations can potentially generate long-wavelength in Cassini state 1; however, the ocean can effectively contribute topography that can be detected through topography or gravity heat at a rate of approximately 66 GW if this is the case – compa- (Schenk and McKinnon, 2009; Nimmo and Bills, 2010). In addition, rable to radiogenic heat production (Figs. 5 and 6). Because the the spatial pattern of the heating at the base can affect convection inclination has remained fairly constant, ocean tidal heating could in the ice shell (Tobie et al., 2005; Roberts and Nimmo, 2008). The have been involved throughout the thermal history of Triton, spe- effects of this convection may potentially manifest as geologic fea- cifically aiding in keeping an ocean from freezing and providing a tures on the surface (Tackley et al., 2001; Besserer et al., 2013). moderate heat source for recent geologic activity. Coupled ther- We present the time-averaged global heating patterns in the mal–orbital models that include ocean tidal heating could provide ocean for the obliquity response and eccentricity response in further insight into the long term evolution of Triton. Fig. 7. The primary power for both responses occurs in degrees l = 0, 2, and 4, as would be expected from tidally-driven flow. How- ever the relative power in each degree and thus the location of hot 5.5. Caveats and cold spots differ from the case for solid-body tidal heating (Ojakangas and Stevenson, 1989; Beuthe, 2013). In particular, In estimating the thermal budgets for the outer Solar System ocean obliquity heating is maximized at the pole (as is solid-body satellites (Figs. 5 and 6), we have used fairly simplistic models in eccentricity heating in a thin shell), while ocean eccentricity heat- order to calculate the amount of tidal dissipation both in the so- ing, although much smaller, is maximized at the equator. Obliquity lid-body and ocean. These models are sufficient to capture and tidal heating will thus tend to reinforce shell thickness variations highlight broad trends. Given the general lack of data and large arising because of solid-body eccentricity heating. uncertainty in physical parameters, we do not attempt to carry

Table 4 3 3 6 8 2 _ 2 Summary of scalings presented in Section 3 and Appendices E and F. Here we deﬁne: u0 ¼ðX R Þ=ðghÞ; E0 ¼ qX R =ðg hÞ and E0 ¼ E0m=R . These scalings are accurate to 10% of the XR2 4X2 R2 K 4 full semi-analytic solution of Section 2.2 when Re ¼ m > 50 and ¼ gh 15. The values presented for the coefﬁcients in m are numerically obtained (see Section 4.2).

Tidal component Average velocity Kinetic energy E_ m Q ÀÁ qffiffi 1 1 2 4 2 2 1 Obliquity-westward, 3 3 qhX R h2 12 qhX R h2 – Re Re 80 XRh0 p 0 p m 0 4 ÀÁ qffiffiffiffiffiffi2 1 1 2 2 Obliquity-westward, J 3 3p E Re h2 3p E_ Re h2 – Re 80 u0Reh0 400 0 0 100 0 0 q800ffiffiffiffiffiffiffiffiffiffi Obliquity-eastward 2397p 2 9237p _ 2 – 12;316 2397 E0h0 E0h0 Re 54880u0h0 27;440 6860 799 qffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 3pqhX2 R4 h2 12pqhX2 R2 mh2 hi1=2 1 Obliquity-general 3 XRh0 0 0 2 Re 1=2 2 2 3 4 4 2 2 2 2 ð1þ½400m2 =u2 R Þ ð1þ½400m2 =u2 R Þ X Rffiffi 200 0:40cD gh0 ðÞ1þ½400m =u R 0 0 p 1 þ 1 þ 0 20 2gh 3 X2 R qffiffiffiffiffiffi 57p 2 39p 2 19 Eccentricity-radial 57 E0e E_ e Re u0e 280 14 o 260 q560ffiffiffiffiffiffi 3p 2 69p 2 5 Eccentricity-westward, librational 3 E0e E_ e – Re u0e 112 140 o 92 q224ffiffiffiffiffiffi Eccentricity-eastward, librational 63p 2 231p _ 2 - 3 63 80 E0e E0e 44 Re 160u0e 20 57 2 3 5 95 Eccentricity-total – p 519p _ 2 cD X R e 56 E0e E0e 0:13 1384 Re 35 gh2 E.M.A. Chen et al. / Icarus 229 (2014) 11–30 23

Fig. 7. Time-averaged energy dissipation pattern associated with (a) obliquity tidal response and (b) eccentricity tidal response for a model ocean on Europa. These are calculated from the numerical model presented in Section 4 with cD ¼ 0:002 and h = 30 km.

out models of specific bodies in this work. With regard to the solid Eqs. (1) and (2) may also not be valid. Long-wavelength topogra- body dissipation calculated here, it is recognized that the Maxwell phy of either the ice shell above the ocean or the silicate mantle be- viscoelastic model is insufficient to explain the tidal dissipation ob- low the ocean can have amplitudes on the kilometer scale (Schenk served, for example for Enceladus (Meyer and Wisdom, 2007; and McKinnon, 2009; Nimmo and Bills, 2010). The dynamics of the Howett et al., 2011). ocean may be far more complex in the case of a very shallow ocean, In adopting a simple model for the ocean behavior by analogy to and the shallow-water scalings we have described do not capture Earth’s oceans, we have neglected a significant dissipation mecha- this complexity. nism, internal tides, which arise because of compositional stratification (Munk and MacDonald, 1960; Egbert and Ray, 2000). While 6. Conclusion Europa’s ocean, at least, is electrically conductive, presumably due to brine (Khurana et al., 1998; Kargel et al., 2000), it is unclear It was recognized by Tyler (2008, 2009, 2011) that tidal dissipa- whether satellite oceans are compositionally stratified (Goodman tion in a global ocean could provide a heat source that had not been et al., 2004). The major drivers of compositional variations in previously recognized; however, the amount of dissipation could Earth’s oceans, such as evaporation or addition of fresh water, are vary over orders of magnitude depending on the value of the effec- likely much less important on icy satellites. Because internal tide tive viscosity (parameterized as either ; a,orQ) implemented in generation is largely due to ocean bathymetry (Baines, 1973; Jayne m the model. We have expanded that work in two ways. First, we and Laurent, 2001), a property which is completely unconstrained provide scalings for the tidal responses, specifically the average ki- in the context of satellite oceans, we have ignored the effects of a netic energy and energy dissipation, in terms of the model input stratified ocean. Additionally, our shallow-water approach cannot parameters. These can be easily implemented into thermal–orbital capture the effects of convection. Convection will alter the dynam- evolution models for many satellites, similar to relations used for ics of the ocean flow to some extent (Chen et al., 2010; Soderlund solid-body tidal dissipation. Second and more importantly, we esti- et al., 2013). However, the magnitude of the tidally-driven veloci- mate the effective viscosity of a satellite global ocean based on ties are likely comparable to those generated through convection analogy to frictional dissipation in oceans on Earth. Both of these and also, convection has a tendency to destroy stratification. Thus, results are summarized in Table 4. the shallow-water model may be sufficient to predict the bulk sa- Based on these scaling relations and estimates for the effective tellite ocean behavior. Unfortunately, due to computational limita- viscosity, we find that for almost all satellites the thermal budget is tions, three-dimensional models that can include convection may dominated by the radiogenic and solid-body eccentricity tidal not necessarily be more accurate than our shallow-water model, heating, as long recognized. For completeness, evolution models because these models are unable to resolve the processes that pre- can include ocean tidal dissipation; however, this is unlikely to cre- sumably dissipate energy in the system (Glatzmaier, 2002). ate large deviations from previous results, unless high obliquities As discussed in Section 4, the appropriate value of the drag coef- existed in the past. In the special case of Triton, ocean tidal heating ficient cD is probably in the range 0.002–0.01. In using the lower may play a large role in keeping a primordial ocean warm, primar- bound, we are being conservative in our estimate of ocean heating. ily because eccentricity tidal heating, either in the ocean or solid- Similarly, our approach to calculating the obliquity h0 might be too body, is likely small after rapid circularization of the orbit post- low by a factor of 2 if decoupled shells are present (Section 5.1). capture. Because the inclination evolves on a much longer time- Taken together, our ocean heating estimates could be low by at scale than the eccentricity, obliquity tidal heating in the ocean most a factor of 20. In most cases, this uncertainty would not can be significant even up to the present and may provide a heat change our conclusions – Mimas and Europa would join the exist- source for the apparent recent geologic activity on the surface. ing candidates (Triton, Titan and Callisto) as possibly having expe- rienced significant ocean obliquity heating. Throughout this work, we assume a small Lamb parameter; the Acknowledgments uncertainty in the Lamb parameter arises from the uncertainty in the ocean depth. The scalings we present in Section 3 are likely We thank Mikael Beuthe and Isamu Matsuyama for their not applicable if the ocean depth is less than 10 km. However, thoughtful and thorough reviews. Chris Edwards and Doug Lin pro- if this is the case, the assumption of uniform ocean thickness in vided helpful insights and discussions. This work was supported by 24 E.M.A. Chen et al. / Icarus 229 (2014) 11–30

NASA’s Planetary Geology and Geophysics Program Appendix C. Response to the radial eccentricity tide (NNZ09AD89G) and a UCSC Dissertation-Year Fellowship. For the prior wave solutions (e.g. (19) and its equivalents or (B.1)), we have not included the m ¼ 0 terms. The solutions for Appendix A. Useful spherical harmonic relations W; U, and g must be real, and because the spherical harmonic 0 0 0 0 Yl is real, the corresponding coefficients Wl ; Ul , and gl must be m real. Because the linear system is separable in degree m, this only To obtain equations for the spherical harmonic coefficients Wl m applies specifically for the radial component of the eccentricity and Ul (e.g. Eqs. (21) and (22)), we used the following spherical harmonic relations: forcing. To solve for this response, we can break the tidal potential (8) into a westward and eastward potential in a manner sim- m m m ilar to that for the obliquity tide and librational component of the Yl ¼ð1Þ Yl ; ðA:1Þ eccentricity tide with a constraint that the total response must be lðl þ 1Þ r2Ym ¼ Ym; ðA:2Þ real, i.e. l R2 l m m m m m X1 hi cos hY ¼ C Y þ C Y ; ðA:3Þ 0 0 0 l lþ1 lþ1 l l1 Wðh; /; tÞ¼ W eiXt þ W eiXt Y 2 R; ðC:1Þ m l;W l;E l @Y l¼1 sin h l ¼ lCm Ym ðl þ 1ÞCmY m : ðA:4Þ @h lþ1 lþ1 l l1 and similarly for U and g. Here the coefficients are still complex and These are similar to Eq. (3.17) of Longuet-Higgins (1968) and (23)– the subscripts W and E denoting the westward and eastward re- (25) of Tyler (2011) for normalized Laplace spherical harmonics. We sponse, respectively. It can be shown that for (C.1) to be true, the m m m relation use the convention that W ¼ð1Þ W and likewise with gm l l l and Um. In addition, a useful relation when calculating spatially l W0 ¼ W0 ; ðC:2Þ averaged quantities, such as the kinetic energy, is l;W l;E Z Z "# must be true, and similarly for U and g. We can solve either for 2p p m m0 m m0 @Y @Y 0 1 @Y @Y 0 0 l l l l the westward or eastward response, by substituting U2dl;2dm;0 þ 2 sin hdhd/ 0 0 @h @h sin h @/ @/ for the right hand side of either (22) or (B.3) with the understanding that ¼ lðl þ 1Þdl;l0 dm;m0 : ðA:5Þ X1 hi 0 iXt 0 iXt 0 Wðh; /; tÞ¼ Wl;W e þ Wl;Ee Yl l¼1 Appendix B. Response to eastward propagating tidal potentials X1 X1 0 iXt 0 0 iXt 0 ¼ 2 R Wl;W e Yl ¼ 2 R Wl;Ee Y l ðC:3Þ In the linear system, we can separately solve for the response to l¼1 l¼1 the eastward propagating potentials. In this case, we seek wave and likewise for U and g. solutions of the form: Appendix D. Derived quantities Xl X1 ÂÃ m iXt m m iXt m Wðh; /; tÞ¼ Wl e Yl þ Wl e Y l For ease of solution, the fluid dynamic equations are expressed m¼1 l¼1 ! Xl X1 ÂÃ in terms of the streamfunction W and scalar potential U. However m iXt m ¼ 2R Wl e Y l ðB:1Þ the quantities of interest are those such as kinetic energy or energy m¼1 l¼1 dissipation rate. Here we explicitly present the process to move from the spherical harmonic coefficients associated with the solu- and similarly for U and g. tion of the fluid dynamic equations to physical quantities. The resulting coupled equations from substituting (B.1) and similar expressions for U and g into (16)–(18) are D.1. Kinetic energy mlðl þ 1Þ 2Xim iX a þ Wm Once the coefficients W and U have been determined for the R2 lðl þ 1Þ l ocean response, the kinetic energy of the ocean flow, Etot, can be l 1 m m l þ 2 m m calculated from þ 2X Cl Ul1 þ Clþ1Ulþ1 Z l l þ 1 q ~ ~ ¼ 0; ðB:2Þ Etot ¼ u udV: ðD:1Þ 2 V We ignore the potential energy contribution to the total energy be- mlðl þ 1Þ 2Xim ighlðl þ 1Þ iX þ a þ þ Um cause it is negligible when compared to this. The kinetic energy ex- 2 lðl þ 1Þ 2 l R XR pressed in terms of W and U is l 1 l þ 2 Z "# þ 2X CmWm þ Cm Wm q @U 1 @W 2 1 @U @W 2 l l l1 l þ 1 lþ1 lþ1 E ¼ þ þ dV: ðD:2Þ tot 2 @h sin h @/ sin h @/ @h 1 2 2R V ¼U2;Edl;2dm;1 U2;Edl;2dm;2 ðB:3Þ Because the energy is a nonlinear product, we must include both with the terms on the right hand side of (B.3) corresponding to the the westward and eastward responses, here denoted with the sub- eastward obliquity tidal potential and eastward librational eccen- scripts W and E, respectively. Expanding (D.2) in terms of the wave tricity tidal potential. solutions from (19), its equivalents and (B.1) results in E.M.A. Chen et al. / Icarus 229 (2014) 11–30 25 Z Z (" qh 2p p Xl X1 @Ym @Y m we can take the time-averaged volumetric integral of the scalar E ¼ Um eiXt l þUmeiXt l tot 2 l;W @h l;W @h product of the momentum equation and the horizontal velocity 0 0 m¼1 l¼1 vector. In this section, we present the calculation of both of these @Ym @Ym 1 @Y m þUm eiXt l þUmeiXt l þWm eiXt l quantities; the calculation for viscous dissipation is far more l;E @h l;E @h l;W sinh @/ straightforward, while the calculation of the tidal work provides 1 @Y m 1 @Y m 1 @Y m 2 þWmeiXt l þWm eiXt l þWmeiXt l more physical insight. We consider the two linear models for vis- l;W sinh @/ l;E sinh @/ l;E sinh @/ " cous dissipation presented in (1). Xl X1 1 @Y m 1 @Ym þ Um eiXt l þUmeiXt l l;W sinh @/ l;W sinh @/ m 1 ¼ l¼1 D.2.1. Viscous dissipation in the ocean 1 @Y m 1 @Y m @Y m þUm eiXt l þUmeiXt l Wm eiXt l The energy dissipated in the ocean is dependent on the viscous l;E sinh @/ l;E sinh @/ l;W @h ) term employed. Tyler (2011) employs a linear drag (Rayleigh dissi- m m m 2 m iXt @Y l m iXt @Y l m iXt @Yl pation) formulation, in which the global dissipation rate is given by W e W e W e sinhdhd/:ðD:3Þ l;W @h l;E @h l;E @h Z _ Elin ¼ q ða~u ~uÞdV: ðD:9Þ By orthogonality of the spherical harmonics (5), the non-zero terms V are Z Z ( where here we take the dissipation to be positive. This essentially 2 Xl X1 h p p scales with the kinetic energy given by (D.1). Averaged over an E ¼ qh Um Um þUm Um þWm Wm þWm Wm tot l;W l;W l;E l;E l;W l;W l;E l;E orbital period (cf. (24)), the viscous dissipation in terms of W and 0 0 m¼1 l¼1 U is m m m m m m m m þ U U þW W ei2Xt þ U U þW W ei2Xt " l;W l;E l;W l;E l;E l;W l;E l;W X1 2 2 @Ym @Y m 1 @Y m @Ym _ 0 0 l l l l Elin;avg ¼ 2aEtot;avg ¼ 2aqh lðl þ 1Þ Ul;W þ Wl;W þ 2 sinhdhd/:ðD:4Þ @h @h sin h @/ @/ l¼1 # X1 Xl 2 2 2 2 Substituting (A.5) into (D.4) and averaging over an orbital period, m m m m þ lðl þ 1Þ Ul;W þ Ul;E þ Wl;W þ Wl;E : we find that the average kinetic energy is l¼1 m¼1 l D:10 X1 X 2 2 2 2 ð Þ m m m m Etot;avg ¼ qh lðlþ 1Þ Ul;W þ Ul;E þ Wl;W þ Wl;E ; ðD:5Þ l¼1 m¼1 For a Navier–Stokes type viscosity, the solution to the integral Z where the time-dependent components of (D.4) integrate to zero. _ 2~ ~ This expression makes physical sense: the kinetic energy of the flow ENS ¼q ½ðmr uÞudV ðD:11Þ V depends on the amplitude of the tidal response, but not on the 2 phase. is also similar to (D.1) with an additional prefactor of lðl þ 1Þ=R While (D.1) is nonlinear, the solution is separable in m. For ease introduced to the summation by the Laplacian operator. The associ- in presentation, we calculate the m ¼ 0 component separately. The ated time-averaged energy dissipation is thus, " kinetic energy Eq. (D.2) can be expanded further in terms of the ra- X1 2 2 dial response from (C.3) resulting in _ 2qhm 2 2 0 0 ENS;avg ¼ l ðl þ 1Þ U þ W 8"# 2 l;W l;W Z Z 2 R l¼1 2p p < X1 0 # 0 qh 0 iXt 0 iXt @Yl X1 Xl E ¼ U e þ U e 2 2 2 2 tot 2 : l;W l;W @h þ l2ðl þ 1Þ2 Um þ Um þ Wm þ Wm : 0 0 l¼1 l;W l;E l;W l;E "#9 l¼1 m¼1 2= X1 @Y 0 ðD:12Þ þ W0 eiXt þ W0 eiXt l sin hdhd/: ðD:6Þ l;W l;W @h ; l¼1

From the relation (A.5), the non-trivial terms of (D.6) are D.2.2. Work done by the tide The work done by the tidal potential can be calculated from X1 qh Z Z E ¼ ðlÞðl þ 1Þ rad;tot 2 _ l¼1 E ¼q ðrU ~uÞdV ¼qh ðrU ~uÞdS V S 2 2 Z 0 iXt 0 iXt 0 iXt 0 iXt 2p Ul;W e þ Ul;W e þ Wl;W e þ Wl;W e ðD:7Þ ¼qhR2 Z 0 which include both time-dependent and time-independent terms. p 1 @U 1 @U When averaged over the orbital period, the kinetic energy associ- uh þ u/ sin hdhd/; ðD:13Þ R @h R sin h @/ ated with the radial response is 0 X1 2 2 The full tidal potential U is the sum of each of the components pre- 0 0 0 Erad;tot;avg ¼ qh lðl þ 1Þ Ul;W þ Wl;W : ðD:8Þ sented in Section 2.1, l¼1 U ¼ U0ðeiXt þ eiXtÞY 0 The sum of (D.5) and (D.8) is presented in (24). 2 2 1 iXt 1 iXt 1 iXt 1 iXt 1 þ U2;W e Y2 e Y2 þ e Y2 e Y 2 D.2. Energy dissipation 2 iXt 2 iXt 2 2 iXt 2 iXt 2 þ U2;W e Y2 þ e Y2 þ U2;E e Y2 þ e Y2 : ðD:14Þ Additionally, the amount of energy dissipation is related to the ﬂow response. Integrated over an orbital period, the amount of Similar to (D.1), the integral (D.13) is separable in m. Therefore, for work done by the tides should be equivalent to the amount of fric- clarity, we present the dissipation associated with radial compo- tional energy dissipated in the ocean. To calculate these quantities, nent ðm ¼ 0Þ separately. 26 E.M.A. Chen et al. / Icarus 229 (2014) 11–30

For the obliquity tide and the librational components of the regime, and the approximations made here break down at small eccentricity tide, the integral (D.13) in terms of the tidal potentials Reynolds numbers (see criteria given in (29)). and response coefficients is The vorticity Eq. (B.2) provides the relationships between Z Z ("#W1; W1 and U1, here assuming power existing only in the three 2p p X2 @Ym @Ym @Ym @Ym 1 3 2 E_ ¼qh Um eiXt 2 þUm eiXt 2 þUm eiXt 2 þUmeiXt 2 2;W @h 2;W @h 2;E @h 2;E @h largest modes and expressed in terms of the Reynolds number: 0 0 m¼1 " 1 X1 Xl m m m m @Y @Y 1 @Y 1 @Y 1 3 C2 1 1 Um eiXt l þUmeiXt l þWm eiXt l þWmeiXt l W ¼ ÀÁ þ i U ðE:1Þ l;W @h l;W @h l;W sinh @/ l;W sinh @/ 1 2 2 l¼1 m¼1 2 1 Re Re þ 1 @Ym @Ym 1 @Ym 1 @Ym þUm eiXt l þUmeiXt l þWm eiXt l þWmeiXt l l;E @h l;E @h l;E sinh @/ l;E sinh @/ " # and X2 1 @Ym 1 @Ym 1 @Ym 1 @Ym þ Um eiXt 2 þUm eiXt 2 þUm eiXt 2 þUmeiXt 2 2;W sinh @/ 2;W sinh @/ 2;E sinh @/ 2;E sinh @/ 4 C1 12 7 "m¼1 1 3 1 W3 ÀÁ þ i U2: ðE:2Þ X1 Xl 1 @Ym 1 @Ym @Ym @Y m 3 12 2 49 Re 6 Um eiXt l þUmeiXt l Wm eiXt l WmeiXt l þ l;W sinh @/ l;W sinh @/ l;W @h l;W @h Re 36 l¼1 m¼1 m m m m m iXt 1 @Yl m iXt 1 @Yl m iXt @Yl m iXt @Yl The response coefficients are related to the forcing potential þUl;Ee þUl;E e Wl;Ee Wl;E e sinhdhd/: sinh @/ sinh @/ @h @h through the horizontal divergence of the momentum Eq. (B.3), here ðD:15Þ written in terms of Reynolds number and Lamb parameter, Using the relation (5), the non-trivial components of (D.15) are U1 Z Z " 4 24 6 1 1 1 8 1 1 2;W 2p p X2 i þ i þ U2 þ C2W1 þ C3W3 : ðE:3Þ _ m m m m m m m m 3 Re 3 X E ¼qh U2;W U2;W þ U2;W U2;W þ U2;EU2;E þ U2;E U2;E 0 0 m¼1 For large Re, the Reynolds number terms in the denominator of (E.1) þ Um Um þ UmUm ei2Xt þ Um Um þ Um Um ei2Xt and (E.2) can be neglected. The dominant terms of (E.3) are 2;W 2;E 2;E 2;W 2;E 2;W 2;W 2;E 2 3 2 2 m m m m 1 1 @Y l @Y l 1 @Y l @Y l C C þ sin hdhd/: ðD:16Þ 6 6 3 2 1536 3 247 @h @h 2 @/ @/ 4 þ þ þ i 5U1 sin h Re 2 Re 49 Re 2 Making use of (A.5) and recalling that the coefficients describing the 1 1 tides (e.g. U2;W ) are real, (D.16) simplifies to 9237 1 24 1 U2;W ¼ þ i U2 ; ðE:4Þ X2 686 Re X E_ ¼2ð2Þð3Þqh Um R Um þ Um R Um 2;W 2;W 2;E 2;E after making the substitutions for W1 and W1 and neglecting their m¼1 1 3 imaginary components for small Lamb parameter. The resulting m m m m i2Xt m m m m i2Xt þ U2;W U2;E þ U2;EU2;W e þ U2;EU2;W þ U2;W U2;E e : flow is given by rffiffiffiffiffiffiffi ðD:17Þ 1 3 4 1 1 U2;W 1 2p X R h0 U I U ¼ ; ðE:5Þ Thus, the time-averaged tidal dissipation for the obliquity and libra- 2 2 24X 4 15 gh tional tides is 1 1 3 1 1 3 1 W I W C U ¼ pffiffiffi U ; ðE:6Þ 1 1 2 2 2 2 X2 2 r5ffiffiffiffiffiffi _ m m m m Eavg ¼2ð2Þð3Þqh U2;W R U2;W þ U2;ER U2;E : ðD:18Þ 1 1 8 1 1 8 8 1 m¼1 W I W C U ¼ U : ðE:7Þ 3 3 7 3 2 7 35 2 The dissipation associated with the radial component is given Combining (E.5) with the relations (E.6) and (E.7) and the ki- by netic energy Eq. (24), we find that the kinetic energy is indepen- Z Z ( ! 2p p 0 dent of viscosity, _ 0 iXt iXt @Y2 hi Erad ¼qh U ðe þ e Þ 2 1 2 1 2 1 2 0 0 @h E h 2 W 6 U 12 W "#) ! tot;obl;E q 1 þ 2 þ 3 X1 @Y0 0 iXt 0 iXt l 9 1 2 1 2 6144 1 2 U ; e þ U ; e sin hdhd/: ðD:19Þ qh U þ 6 U þ U l W l W @h 10 2 2 1715 2 l¼1 6 8 2 The solution of (D.19) can be used to derive a time-averaged dissi- 7191 1 2 2397p qX R h0 ¼ qh U2 : ðE:8Þ pation of 686 27; 440 g2h _ 0 0 0 The kinetic energy in turn may be used to determine an averaged Erad;avg ¼2ð3ÞqhU U þ U 2 2;W 2;W flow speed u: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 2 3 hU : D:20 3 3 ¼ ð Þð Þq 2R U2;W ð Þ 2397 X R h u 0 : ðE:9Þ The sum of (D.18) and (D.20) is presented in (28). 54; 880 gh The associated energy dissipation calculated from (28) and the solu- Appendix E. Scalings for the response to the eastward obliquity tion of (E.4) is given by tide _ 1 1 E ¼12qhU2;W R U2;E While the magnitudes of the forcing potentials for the eastward !! U1 2 and westward propagating obliquity tides are the same, the re- 1 3079 2;W 12qhU2;W sponse of the ocean is fundamentally different. For the westward 131; 712 XRe response, the kinetic energy and energy dissipation are dominated 9237p qX6R6mh2 by the lowest mode, W1 . For the eastward response, the kinetic ¼ 0 : ðE:10Þ 1 6860 g2h energy is significant in the lowest three modes with the largest 1 amplitude being U2 (see Fig. 1). There only exists a single scaling The tidal quality factor as defined by (51) is E.M.A. Chen et al. / Icarus 229 (2014) 11–30 27 qX6R6mh2 ! _ X 9237p 0 2 (21) and ignoring the contributions from the Reynolds number XEobl;E 6860 g2h 12; 316 XR Q obl;E ¼ ¼ terms in the denominator, similar to the process used in Appendix X6R8h2 Etot;obl;E 2397p q 0 799 m 2 E, the relations between the three modes are 27;440 g h 12; 316 2 ¼ Re: ðE:11Þ W0 3C0 i U0 ðF:1Þ 799 1 2 Re 2 and E.1. Relative contribution from eastward obliquity 0 4 0 12 0 W3 C3 i U2 ðF:2Þ Throughout this paper, we have ignored the response to the 3 Re eastward propagating obliquity tide. Under the assumptions of These modes are related to the tidal forcing by high Reynolds number and low Lamb parameter, it can be formally "# ! ! shown that the contribution from the eastward obliquity response 6 i24 C0 2 4C0 4 12 iþ þ2 2 3C0 i þ2 3 C0 i U0 is negligible when compared to that of the westward propagating Re 2 2 Re 3 3 3 Re 2 response. 0 In the high Reynolds number regime, the kinetic energy ratio is U 2 ; independent of the ocean viscosity. From Section 3.1 and Appendix X E, the relative contributions to the kinetic energy from the west- ðF:3Þ ward and eastward obliquity responses are where the dominant real and imaginary terms end up being E 3 hX2R4h2 439; 040 1 2 tot;W pq 0 J 0 ¼ 7700: ðE:12Þ 130 1 i24 0 U2 qX6R8h2 Etot;E 2397p 0 799 U2 : ðF:4Þ 27;440 g2h 7 Re X

0 In the low Reynolds number regime, this ratio is sensitive to the The forced response jU2j is thus, rffiffiffiffi Reynolds number with 0 3 4 0 0 U2 1 p X R e U I U ¼ : ðF:5Þ qX8R12h2 2 2 3p 0 24X 4 5 gh Etot;W 400 m2g2h 343 ¼ Re2 J 1700: ðE:13Þ X6R8h2 Etot;E 2397p q 0 495 For large Reynolds numbers, the relations (F.1) and (F.2) are 27;440 g2h dominated by the imaginary term, such that rffiffiffiffiffiffi Similarly, the ratio of energy dissipation is independent of viscosity 4 0 0 0 0 0 in the high Reynolds number regime, W1 I W1 3C2 U2 ¼ 3 U2 ; ðF:6Þ 15 E_ 12pqhX2R2mh2 439; 040 1 2 obl;W 0 ¼ J 2000 ðE:14Þ and _ qX6R6mh2 rffiffiffiffiffiffi Eobl;E 9237p 0 3079 6860 g2h 0 0 4 0 0 1 0 W3 IðW3Þ C3 U2 ¼ 4 U2 : ðF:7Þ and scales inversely with the Reynolds number in the low Reynolds 3 35 number regime, Using the lowest three modes, the average kinetic energy associated

8 10 2 with the radial tide can be approximated from (24) as 3p qX R h0 E_ 2 343 obl;W 100 mg h Re2 J 55: E:15 hi 6 6 2 ¼ ð Þ 0 2 0 2 0 2 _ 9237p qX R mh0 15; 395 Eobl;E Etot;rad qh 2 W1 þ 6 U2 þ 12 W3 : ðF:8Þ 6860 g2h Typical satellite parameters (e.g. Tables 1 and 2) usually lie far Combining the relations (F.1), (F.2) and (F.5) with (F.8), we find that from the scaling limits used to calculate these ratios, and the ratios the average kinetic energy is "# presented in Eqs. (E.12)–(E.14) and (E.15) are typically much lower 2 2 than those for actual satellite parameters. Therefore, we reason- 0 0 0 2 4 0 0 Etot;rad qh 23C2 U2 þ 6 U2 þ 12 C3 U2 ably ignore the contribution of the eastward obliquity response 3 in all obliquity-related scalings. 6 8 2 114 0 2 57p qX R e ¼ qh U ; ðF:9Þ 7 2 280 g2h Appendix F. Scalings for the response to the eccentricity tide and thus the associated average velocity is given by The responses to the eccentricity tide is similar to that of the rffiffiffiffiffiffiffiffiffi 57 X3R3e eastward propagating obliquity tide, in that there is a single scaling u : ðF:10Þ 560 gh regime characterized by a kinetic energy profile that is independent of viscosity and energy dissipation linearly varying with vis- 0 From (F.4), we can also calculate R U2 , cosity. Unlike the unequal responses of the eastward and rffiffiffiffi 0 4 4 westward propagating obliquity tide, the magnitudes of the re- 0 U2 130 1 65 p X R me RðU2Þ ÀÁ ¼ : ðF:11Þ sponses to each of the three components of the eccentricity tide 24 2 7 Re 84 5 2 2 X g h are comparable. We present scalings for each tidal component separately, as well as including scalings for the total response. The resulting dissipation from (D.20) is thus

6 6 2 0 39p qX R me F.1. Scalings for the response to the radial eccentricity tide E_ 12qhU R U0 : ðF:12Þ rad 2 2 14 g2h The response to the radial eccentricity tidal potential can be Combining (F.9) and (F.12), we find that the tidal quality factor 0 0 0 approximated with the lowest three modes, W1; U2, W3. From associated with the radial response is 28 E.M.A. Chen et al. / Icarus 229 (2014) 11–30 ! X6R8e2 57p q 2 same as that for the westward response. The relationship between X 2 280 g h 19 XR 19 2 2 Q rad 6 6 ¼ ¼ Re: ðF:13Þ U2 and W3 as given by (B.2) is 39p qX R me2 260 m 260 2 14 g h 3C2 12 4 W2 3 þ i U2 ðF:23Þ 3 4 Re 3 2 F.2. Scalings for the response to the westward propagating component with of the librational eccentricity tide rffiffiffi

2 2 2 2 1 2 The response of the ocean to the librational eccentricity tidal W3 IðW3Þ C3 U2 ¼ U2 : ðF:24Þ 2 2 7 potential is dominated by the lowest two modes, U2 and W3. To calculate the westward response, we return to the vorticity equation The dominant terms in the horizontal divergence Eq. (B.3) are "# 2 given by (21). It can be shown that for large Reynolds numbers 2 U2 6 24ðC3Þ 24 2 66 1 24 2 2;E þ þ i U2 ¼ þ i U2 : ðF:25Þ 2 2 12 2 2 Re Re 7 Re X W 3C i U ðF:14Þ 3 3 Re 3 2 The resulting flow is given by and 2 rffiffiffiffiffiffiffi rffiffiffi U 3 4 2 2 2;E 7 2p X R e 2 2 2 2 1 2 U I U F:26 W IðW Þ 2C U ¼ 2 U : ðF:15Þ 2 ð 2Þ ¼ ð Þ 3 3 3 2 7 2 24X 8 15 gh From (22) and including the dominant real and imaginary terms as with rffiffiffiffiffiffiffi ! 2 before, the tidal response is given by U 4 4 2 3 2 2;E 66 1 11 2p X R me R U2 ÀÁ ¼ : ðF:27Þ 2 24 2 2 2 2 2 X 7 Re 8 15 g h 6 6 96 C3 247 138 1 24 U 4 þ i 5U2 ¼ i U2 2;W : ðF:16Þ Re Re 2 7 Re 2 X Similar to (F.19), the average kinetic energy is 2 2 2 2 2 54 2 2 E qh 6 U þ 12 C U ¼ qh U Similar to the response to the radial component, the kinetic energy tot;lib;E 2 3 2 7 2 is related to 63p qX6R8e2 2 rffiffiffiffiffiffiffi ðF:28Þ U 3 4 80 g2h 2 2 2;W 1 2p X R e U I U ¼ ðF:17Þ 2 2 24X 8 15 gh and the flow speed is rffiffiffiffiffiffiffiffiffi while the dissipation is related to 63 X3R3e rffiffiffiffiffiffiffi ! u ðF:29Þ 2 4 4 160 gh 2 U2;W 138 1 23 2p X R me R U2 ÀÁ ¼ : ðF:18Þ 24 2 7 Re 56 15 2 2 Note that this value exceeds the equivalent westward flow speed X g h (Eq. (F.20)), as expected. The corresponding energy dissipation for The average kinetic energy associated with the response can be the eastward flow (28) is approximated from (24), (F.15) and (F.17) as 6 6 2 hi 2 2 231p qX R me 2 2 _ : : 2 2 Elib;E 12qhU2;ER U2;E 2 ðF 30Þ Etot;lib;W qh 6 U2 þ 12 W3 20 g h 2 2 2 2 2 90 2 2 The tidal quality factor is qh 6 U þ 12 2C U ¼ qh U 2 3 2 2 ! 7 qX6R8e2 X 63p 2 6 8 2 80 g2h 3 XR 3 3p qX R e Q lib;E ¼ ¼ Re: ðF:31Þ : X6R6 e2 2 ðF 19Þ 231p q m 44 m 44 112 g h 20 g2h with a corresponding average flow speed of rffiffiffiffiffiffiffiffiffi F.4. Scalings for the total response to the eccentricity tide 3 X3R3e u : ðF:20Þ 224 gh The three components of the eccentricity-driven flow behave From the relations (28) and (F.18), the associated energy dissipation quite similarly, and as seen in Appendices F.1, F.2 and F.3, the main is difference exists in the pre-factors defining each of the flows. From (F.9), (F.19) and (F.28), the ratios of the kinetic energies are 6 6 2 _ 2 2 69p qX R me Elib;W 12qhU2;W R U2;W : ðF:21Þ 147 147 140 g2h E E E ðF:32Þ tot;lib;E 38 tot;rad 5 tot;lib;W The associated Q is therefore with a total kinetic energy associated with the eccentricity tide of 6 8 ! 3p qX R e2 X 2 6 8 2 112 g2h 5 XR 5 57p qX R e Q lib;W 6 6 ¼ ¼ Re: ðF:22Þ Etot;ecc : ðF:33Þ 69p qX R me2 92 m 92 56 g2h 140 g2h The energy dissipation behaves similarly with the eastward librational response contributing a bulk of the dissipation with non-neg- F.3. Scalings for the response to the eastward propagating component ligible contributions from the other components. The relative of the librational eccentricity tide dissipation rates from (F.12), (F.21) and (F.30) are

The process to calculate the response to the eastward propagat- _ 539 _ 539 _ Elib;E Erad Elib;W : ðF:34Þ ing librational component of the eccentricity tide is essentially the 130 23 E.M.A. Chen et al. / Icarus 229 (2014) 11–30 29

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