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W.K. authors and The research; data; performed analyzed W.K. W.K. tools; research; reagents/analytic designed new G.F. contributed and W.K. contributions: Author density, equation, above the In dH https://github.com/wanyingkang/icy-moon-shell-evolution owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To dt atlvlo smer a ul poe iloso years. open. crack of could poles millions two over the signifi- of up one a build over history, ice can the Eventually, moon’s asymmetry of the level in cant early infinitesimal perturbation an from unexpected random that, this than found for mystery. 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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES contributes an additional −0.6 (see SI Appendix for details). Besides these, a more dissipative ice shell would have an even stronger rheology feedback (26), and ocean circulation may also contribute (Concluding Remarks and Discussion). Given these uncertainties, we leave pα as a tunable parameter. The schemat- ics depict a situation where the ice shell is thinner over the south pole and thus tidal heating is amplified there. The idealized Maxwell rheology used here has been shown to underestimate the total tidal-heat generation (28, 29). To com- pensate for this underestimation, a multiplicative factor, γ, is incorporated in the tidal-heating formula. Since the total tidal heating in the ice shell is poorly constrained, we leave γ as a tunable parameter. Symmetry Breaking of Enceladus’s Ice Shell

The equilibrium ice topography Heq can be obtained numeri- cally by evolving the ice-thickness model from an initial value H (θ) = H0 for a long enough period of time until the tendency terms exactly compensate each other. Heq and the corresponding melting/freezing rate induced by F, T , Q are shown in Fig. 2A. The tidal heating T peaks at the two poles. Compensated by a relatively faster heat loss to space F in the polar regions and a poleward ice flow, the system reaches equilibrium, with thin- ner ice at the two poles. Here, we set γ = 38, which yields a 24.8 mW/m2 tidal-heating rate on global average. Since this heating rate is lower than the average heat-loss rate at F = 33.8 mW/m2 given by the heat conduction model, a positive constant balanc- Fig. 1. Schematics to demonstrate the physics processes we consider in this model. See Overview of Our Ice Evolution Model for descriptions. ing heating B is required to keep the global ice-shell thickness unchanged. Physically, this term could correspond to dissipation in the ocean or the core (23, 30, 31). In the real world, we would not expect a perfectly uniform ice denotes the radius of Enceladus. F, T , Q, B, C are all func- shell to begin with. If the initial ice-shell thickness H is slightly tions of the ice thickness H . Their definitions are given in SI and randomly perturbed around the mean H0, as shown by the Appendix. thin dashed curve in Fig. 3, a break in symmetry between the two The heat conduction to space F is inversely proportional to hemispheres appears spontaneously. After 100 My, the ice-sheet H . The ice flow Q flows down-gradient, smoothing the ice-shell thickness reaches a final state with a significant tilting from one inhomogeneities, particularly those at small scales. Q weakens pole to the other, as shown by the thick solid curve in Fig. 3. This in regions with thin ice, allowing the formation of ice “holes.” final state captures the main characteristics of ice-shell geometry The crack-induced cooling C allows extra heat loss when the ice constrained by observation (27, 32, 33): the ice shell is thinner at thickness is below the crack threshold Hcrack = 8 km (to account the poles than the , with geysers concentrated at one of for the strong heat flux from the geysers on the south pole; the poles. refs. 7–10). The balancing term B is introduced to keep the This symmetry breaking arises from a normal-mode instabil- global-mean ice thickness H0 unchanged, as our focus here is to ity induced by the concentration of tidal heating in the regions understand topography formation on the ice shell rather than the where the ice shell is already thinner than elsewhere. By lin- maintenance of the ice shell at a certain thickness. earizing the ice-evolution model around the unperturbed equi- Tidal heating generated in the ice shell and the core domi- librium state Heq , we obtain the linear tangential system (for nantly balances heat loss by conduction F (22, 23). As Enceladus orbits Saturn, the change of geopotential induced by orbital eccentricity deforms the ice shell and the core and generates heat. Here, we focus on the tidal dissipation in the ice shell, which is sketched as two reddish patches over the poles in Fig. 1 to demonstrate the polar-amplified heating profile, as suggested by previous studies (24–27). In a laterally varying ice shell, tidal heating T would be con- centrated in regions where the ice shell is thinner and thus more mobile (geometric effect), just like a rubber band with a weak point would generate more heat there. The enhanced heating rate over the thin-ice regions can, in turn, warm up the ice shell, making it even more mobile (rheology feedback). To make the problem analytically approachable, we account for this effect by amplifying the membrane-mode heat generation by a fac- Fig. 2. The equilibrium ice thickness starting from a perfect uniform ice pα ∗ shell and the most unstable mode based on linear analysis. A shows Heq as a tor of (H /H0) . According to the calculation in ref. 26, the function of colatitude θ as a solid black curve with the y axis on the left. The geometric effect contributes −1 to pα, and rheology feedback equilibrium-state melting/freezing rate induced by tidal heating T (H)/Lf ρi 1 (red dashed), ice flow a sin θ ∂θ (sin θQ(H)) (blue dashed), and heat loss to space F(H)/Lf ρi (green dashed) are shown on the right axis. A minus sign is *Among the three tidal-dissipation modes, the membrane mode is the dominant one. multiplied to the tidal-heating curve and heat-loss curve to show the curves Physically, it corresponds to the dissipation induced by extension/compression and in the same axis. B shows the normalized structure of the most unstable tangential shearing of the ice membrane (26). mode against colatitude θ and the corresponding growth rate.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.2001648117 Kang and Flierl Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 erpa h aecluainfrvros(H various for calculation same parameters, the constrained poorly repeat of we choice the to mode) sensitive pletely degree-2 particular while (in grow to modes mode other decay. degree-1 to for the both for enough in enough small large form be would to needs km) range 8 narrow relatively of than a within occurs thinner breaking Symmetry poles. (ice force geysers in degree-1 would result, pattern over profile degree-2 tidal-heating dominate a become polar-amplified could would the mode mode because degree-2 degree-2 the only The also the not unstable. but range, that mode this strong degree-1 Above so the anywhere. becomes sym- geysers self-amplification hemispheric no topography a with at stabilize state, would metric shell ice the mW/ m of 25.1 asymmetry to 23.8 of rate signifi- 38. (p [36.9, concen- a setup being present with the pole temperature With state one geysers. with a with and trated into asymmetry evolve hemispheric cant naturally asymmetry, priori could a without Enceladus that, suggests calculation above The Discussion 3. of Fig. 20, set in ref. see in parallel condi- proposed we a initial mechanism as finally the the and through on develop thinner, may (depending and geysers poles thinner get the unstable would of the tion) one time, most with over grow shell The to continues ice structure Mode). 2B this If Fig. structure. Unstable in tilting shown the is structure and eigenmode Evolution ness see derivation, perturbations. initial distribution uniform a initial from drawn The between identically and comparison. independent is for point curve grid dashed thin a (Eq. model thickness 3. Fig. agadFlierl and Kang loss heat global-mean the global-mean exceed space.† the not to that would guarantees rate meanwhile, tidal-heating and and, 34) breaking a try 32, for (1, search we constraints different observational three for by guided nations, † hslae omfrohrha sources. heat other for room leaves This oso httesmer-raigmcaimi o com- not is mechanism symmetry-breaking the that show To γ o oto h aaeesw xlr eebecause here explore we parameters the of most for h nlietopography ice final The .2 n .2.Tesse ol qiirt atrwt larger with faster equilibrate would system The 0.025. and −0.025 hc orsod oagoa-enice-dissipation global-mean a to corresponds which 1], h obntosta lo st n uha such find to us allow that combinations The T s IAppendix, SI 80 = γ p o 0 y h nta odto of condition initial The My. 100 for 1) α p htcnla otehmshrcsymme- hemispheric the to lead can that sstto set is α ) ymtybekn cuswhen occurs breaking symmetry K), aus o each For values. H eq 2 vnwtotprubtos sa As perturbations. without even , −1.5. eo hsrne h hemispheric the range, this Below . H ierzdEuto o c Thick- Ice for Equation Linearized tiksldcre fe vligteice- the evolving after curve) solid (thick n a pole-to-pole a has and (H h 0 0 = , T (H α s = − ) 0 surface −1.5, , H combination, 0 H T )/H s ssonas shown is combi- ) 0 teach at γ γ are ∈ γ ed oa diinlncsaycniinfrieselsymmetry ice-shell for condition breaking, necessary additional an to leads Eq. of bound of side Eq. more low breaking is regime the symmetry symmetry-breaking on when actual the 1 therefore, than occurs; smaller significantly becomes for especially well, (T space the eter in Eq. regime 4, symmetry-breaking Fig. the in of curves location dashed red by (see shown As tions). different across mean the and planetary-scale the topography, change significantly ice that flow ice and heating, Here, symmetry- (no bound for upper found the is beyond regime thickness breaking ice an With T necessary two the happen. of to one breaking is symmetry which for topography, elements ice of growth greater the a of because unsurprising range is This the breaking. does as widens, n uineeg,temlcnutvt,addniyo the of density and conductivity, includ- thermal constants, physical energy, of fusion combination a ing in from given comes right is definition whose and (i.e., producer heat major of a definitions is same shell the ice and parameters whose Eq. orbital Jupiter Saturn their around moons if and icy For moons, known. are icy characteristics other physical to applied be rhe- body dissipation 29). Maxwell the (28, the underestimate rate significantly that could given here satisfied used possible be ology be a would could by condition enhanced which this is 14.8, is exam- results, rate the that ice-dissipation the For the loss in shell. if heat show ice we of the as in percentage heating ple, the tidal for the bound by compensated lower a sets This analy- criteria: following stability the the satisfy crack-free to a in According maintain shown lower perturbations. to sis weak the of too absence below efficient‡ is the thickness flow too ice ice becomes the With flow bound, topography. ice the any because maintain occur not does 4, Fig. 4. Fig. in p color white the by marked where of are inequality that of bound upper get we and lower the ‡ o eoe oe(es fcetwe c hl stik(thin). thick is shell ice when efficient (less) more becomes flow smnindin mentioned As α s o uhasmer-raigrgm oeit h upper the exist, to regime symmetry-breaking a such For rmFg ,oecnceryseta,fraygiven any for that, see clearly can one 4, Fig. From h c-hl ymtybekn rtra(Eqs. criteria symmetry-breaking ice-shell The As ymtybekn a nyocri pcfi ag of range specific a in occur only can breaking symmetry , = 2. 3 η −1. |p τ sstse) ymtybekn ol cu troughly at occur would breaking symmetry satisfied), is g F sawihe-vrg icst costeieshell, ice the across viscosity weighted-average a is α and 5, β , | H (τ τ nrae,tesmer-raigrgm gradually regime symmetry-breaking the increases, τ τ −2, 0 T Q F F h ymtybekn eieshould regime symmetry-breaking the Appendix, SI eadeso te aaees usiuigthe Substituting parameters. other of regardless , a 4 vriwo u c vlto Model Evolution Ice Our of Overview , /τ 2 ∝ τ r h ufc rvt n aiso Enceladus, of radius and gravity surface the are τ F T respectively. −2.5, Q a ob agrta h oe on,which bound, lower the than larger be to has (a ) τ τ and r h ieclsfrha odcin tidal conduction, heat for timescales the are F T < H 2 /g β 0 > τ τ oprdwt h rdcingvnby given prediction the with compared , Q F )( τ stepretg aito of variation percentage the is −p Q η < o (T log H α noEq. into −p 1) + 2 (β/ 0 p 5 H α γ The Appendix. SI α 0 = (τ m htcntigrtesymmetry the trigger can that ≤ 1 o greater For −1.5 F /T 36 /τ 2 s T )) − m,smer breaking symmetry km), n oiigta both that noticing and 1) + 2 )(β/ NSLts Articles Latest PNAS IAppendix SI ∼ 4 β/ n hw in shown and A–C γ O atrgetrthan greater factor . (10 |p hw eut for results shows −8 α s | 10 2 2 − ), , a enhance can IAppendix SI | rdcsthe predicts H and p −8 1. o defini- for α 0 T param- ) |, p nthe on τ | can 3) H α about O F eq f5 of 3 /τ and (1), H [2] [3] [4] ice , to in 0 T .

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 4. (A–C) Diagrams showing the combinations that permit symmetry breaking of the global-mean ice-shell thickness H0 and the surface temperature Ts. White denotes the parameter regimes that allow us to find a γ that can lead to symmetry breaking. Red curves show the symmetry-breaking criteria given by Eq. 2 with β = 1, τF /τT = 1.

ice. The fact that τF / τQ is proportional to the fifth power mechanism works and the rate at which hemispheric asymme- of H0 implies the symmetry-breaking regime is always cen- try grows. In particular, tidal heating from the core is found to tered around H0 ∼ 10 to 30 km. Even if the radius of the be necessary to counterbalance the heat conduction to space, moon is enlarged by 10 times (while keeping the bulk density as heating generated in the ice shell alone is not enough, given unchanged), a2/g would increase by a factor of 10, and the our current understanding of ice dynamics (26, 27). The heat- symmetry breaking permitting H0 would only change by a fac- ing generated in the core is carried upward by eddies, convection tor of (10)1/5 ∼ 1.6. The Galilean satellites are warmer than plumes, and large-scale circulation in the ocean. It is possible that Saturnian satellites by about 30 K, and the associated impact some of these processes could respond to changes in ice topogra- on H0 is no larger than a factor of 2 as well. This leads to phy and form a feedback loop, as suggested by ref. 44. We spec- an even simpler empirical rule for the mechanism proposed ulate that freezing (melting) at the ocean–ice interface may be here: ice-shell symmetry breaking would only occur on Galilean able to reduce (increase) the local salinity and drive an upwelling and Saturnian icy moons whose ice-shell thickness is around 10 (sinking) motion there, which in turn enhances (reduces) the to 30 km. upward heat transport from the core and leads to further freez- Our calculation assumes the ice shell is conductive. However, ing (melting). Feedback like this can be accounted for in our besides Enceladus, the only place inside our solar system that framework by a greater pα factor. Processes that are not directly satisfies this assumption is ; other icy moons (such as affected by the ice topography are represented by the balancing Titan, Ganymede, and Callisto) have an ice shell that is too term B. Our work assumes a globally uniform profile of B, but it is highly possible that B is concentrated in a few hotspots (30). If thick (35–38) for ice convection to be triggered (39). Repeating 0 the same calculation for Europa, we found symmetry breaking that is true, the equilibrium state heq would change, which would could only occur when the ice-shell thickness is 20 to 26 km, in turn affect the growing mode structure and its growth rate. assuming the mean surface temperature is around 100 K (40) Even a tiny hemispheric asymmetry in B would help determine and pα = −1.5. Based on magnetic conduction, Europa’s ice- the direction of symmetry breaking, and it could significantly shell thickness is constrained to be below 15 km (41), indicating affect the parameter regime where symmetry breaking could that Europa could have narrowly missed the symmetry-breaking happen. regime. However, the estimation of Europa’s ice-shell thickness Concluding Remarks is not yet conclusive (35, 42, 43), so more studies are needed to improve the thickness estimation and the symmetry-breaking Our work takes an idealized framework to demonstrate the criteria. feasibility of spontaneous hemispheric symmetry breaking. To make the problem analytically approachable, we made Over billions of years, Enceladus could have gone through many simplifications and assumptions, which could have an large eccentricity variations (45–47), leading to secular melt- impact on the symmetry-breaking regime. For example, when ing/freezing over time. As the ice-shell thickness varies, it considering the tidal heating generated in an inhomogeneous ice seems unavoidable that Enceladus would pass through the mem pα symmetry-breaking regime shown in Fig. 4. When Enceladus shell, we multiply T0 by a factor of (H /H0) , rather than solve the thin shell ordinary differential equation with variable gets into the symmetry-breaking regime, hemispheric asymme- coefficients. As a result, we ignore the fact that small-scale per- try would accumulate; when it gets out, hemispheric asymmetry turbations would be suppressed by the greater bending rigidity would gradually decay. Before the historical orbital record is associated with it. Taking this into consideration would lead to obtained, understanding the stochastic dynamics of this sys- a stronger scale selectivity. In addition, the slow circulation in tem would shed light on how much hemispheric asymme- the slush zone at the ocean–ice interface can also help damp try of Enceladus can be developed through the mechanism the small-scale topographies, widening the symmetry-breaking proposed here. regime. We also ignore the decrease of surface temperature ACKNOWLEDGMENTS. Most of this work was carried out during the 2019 toward the poles, as the role played by this factor is less clear Geophysical Fluid Dynamics (GFD) Summer School at Woods Hole. We and will be explored in future work. thank the GFD faculty for organizing the program and the Woods Hole Oceanographic Institution for hosting it. During the review process, the Other ignored mechanisms, such as ocean circulation, sur- two anonymous reviewers gave us extremely helpful comments, which we face snow cover, dynamics of ice convection, and non-Newtonian deeply appreciate. We also thank Prof. Geoff Vasil, Prof. Mikael Beuthe, and ice rheology, could also affect the parameter regime where this Prof. Ming Cai for helpful discussions.

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