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Barr and Showman: Heat Transfer in Europa’s Icy Shell 405

405

Heat Transfer in Europa’s Icy Shell

Amy C. BarrSouthwest Research Institute

Adam P. ShowmanUniversity of Arizona

Heat transport across Europa’s ice shell controls the thermal evolution of its interior andprovides a source of energy to drive resurfacing. Recent improvements in knowledge of icerheology, the behavior of convection, and the interaction between convection and lithosphericdeformation have led to more realistic and complex models of the geodynamics of Europa’sicy shell. The possibility of convection complicates efforts to determine the shell thicknessbecause a thin conductive shell can carry the same heat flux as a thick convective shell. Whetherconvection occurs depends on ice viscosity, which in turn depends on grain size. The grainsize may be controlled by internal deformation, or by impurities, depending on shell composi-tion. Creating the observed surface features with steady-state thermal convection is challeng-ing, even with tidal heating, because the near-surface ice is cold and stiff. Convection modelsthat include surface weakening and compositional buoyancy show promise in explaining somechaos terrains, pits, and uplifts, but new spacecraft and laboratory data and geophysical tech-niques are needed to match theory to observation.

1. INTRODUCTION

Images of the surface of Europa returned by the Galileospacecraft revealed that, in addition to its global networkof linear features, portions of its surface are covered by pits,spots, uplifts, and chaos regions suggestive of convection-driven resurfacing (Pappalardo et al., 1998; Greeley et al.,1998, 2004). The implication that Europa’s icy shell couldbe, or could have been convecting in the past, raised a num-ber of questions about its geodynamical behavior: Can theshell convect at present? How does tidal dissipation affectthe convection pattern? Can tidal dissipation and convectiondrive resurfacing? If the shell convects, can the ocean bethermodynamically stable? What role might compositionalheterogeneity play in driving motion in Europa’s icy shell?

The idea that solid-state convection may occur withinEuropa’s icy shell dates to the Voyager era when Consol-magno and Lewis (1978) suggested that the ice I layers oflarge icy satellites could convect. Reynolds and Cassen(1979) determined that an ice shell on Europa, Ganymede,or Callisto could convect if the shell were thicker than30 km, and that convection would rapidly freeze any liq-uid water ocean on a geologically short timescale. In thedecades since, advances in our knowledge about planetaryconvection, coupled with laboratory experiments and fieldstudies of ice deformation, have changed the way we thinkabout convection on Europa.

Recent laboratory and theoretical studies about the fluid-like behavior of solid water ice have clarified the micro-physical processes that accommodate deformation in ice(Goldsby and Kohlstedt, 2001; Durham and Stern, 2001)and the processes that control grain size (McKinnon, 1999;Schmidt and Dalh-Jensen, 2003; Barr and McKinnon,

2007), and thus viscosity and tidal dissipation in Europa’sice shell. Application of numerical terrestrial mantle con-vection models to Europa has shed light upon the conditionsrequired to trigger convection in Europa’s shell (McKinnon,1999; Ruiz and Tejero, 2003; Barr et al., 2004; Barr andPappalardo, 2005), the behavior of the icy shell close tothe onset of convection (Mitri and Showman, 2005, 2008a;Solomatov and Barr, 2007), the behavior of tidal dissipationand convection (Tobie et al., 2003, Mitri and Showman,2008b), the convective heat flux (Freeman et al., 2006), thepotential for convection-driven resurfacing (Showman andHan, 2004, 2005; Han and Showman, 2008), and the po-tential for driving resurfacing with thermochemical convec-tion (Pappalardo and Barr, 2004; Han and Showman,2005). Use of scaling relationships between the physicalproperties of the icy shell, the critical Rayleigh number, andthe convective heat flux has helped constrain the conditionsunder which the ocean can avoid freezing (Hussmann et al.,2002; Freeman et al., 2006; Moore, 2006).

Despite these important advances, the modes of heattransfer across Europa’s shell and the link between heattransfer and resurfacing remain unclear. Direct numericalsimulations of convection-driven resurfacing cannot easilymatch the observed morphologies of pits, spots, domes, andchaos — simulations find that thermal buoyancy stressescan create uplifts only approximately one-tenth the heightof those observed (Showman and Han, 2004). Composi-tional convection in a salty icy shell allows features withthe correct heights to be explained (Han and Showman,2005), but determining whether convection can cause chaosformation remains a challenge (Showman and Han, 2005).Using the spacing between quasicircular surface features onEuropa’s surface to constrain the physical properties, heat

406 Europa

flux, and thermal structure of the icy shell has proved prob-lematic (e.g., Nimmo and Manga, 2002). Compounding theuncertainty, Europa’s shell may oscillate between a conduc-tive and convective equilibrium, wreaking havoc on its sur-face (Mitri and Showman, 2005).

Here, we summarize recent breakthroughs in the under-standing of solid-state convection, numerical advances inmodeling the coupled processes of tidal dissipation andconvection, and laboratory experiments clarifying the duc-tile behavior of ice. We discuss implications of these ad-vances for the modes of heat transport and resurfacing inEuropa’s icy shell. In section 2, we provide an overview ofheat sources and transport mechanisms in a tidally heatedice shell. In section 3, we review recent experiments clari-fying how ice deforms in response to an applied stress,which is a long-standing source of uncertainty in europaninterior modeling. In section 4, we describe how the icyshell behaves close to the limit of convective stability. Insection 5, we discuss how convection and tidal dissipationmay contribute to the formation of Europa’s rich variety ofsurface features, including chaos, pits, spots, and uplifts.

2. HEAT GENERATION AND TRANSPORT

Europa’s ice shell is heated from beneath by radiogenic(and possibly tidal) heating from its rocky mantle, and fromwithin by tidal dissipation. Radiogenic heating within therocky mantle of Europa currently supplies roughly Fr ≈ (1/3)ρrHr(Rs – z)(1 – z/Rs)2 ~ 6 to 8 mW m–2 to its surface heatflux, where ρr = 3000 kg m–3 is the approximate density ofEuropa’s rocky mantle (comparable to its mean density),Hr = (4.5 ± 0.5) × 10–12 W kg–1 is the present chondriticheating rate (Spohn and Schubert, 2003), z ~ 120 km is thethickness of Europa’s H2O layer, and Rs is Europa’s radius.

The surface heat flux due to tidal dissipation in Europa(Ftidal) can be estimated as a function of its physical andorbital properties

Ftidal = =Etidal k2

Q

Rs3GMJ

2ne2

a6

⎛⎝⎜

⎞⎠⎟4πRs

2

21

8π(1)

where Etidal is the tidal dissipation rate (Cassen et al., 1979,1980); Rs is Europa’s radius; (k2/Q) is the ratio betweenthe degree-2 Love number (k2) and the tidal quality factor(Q), which describe how Europa’s interior deforms in re-sponse to the jovian tidal potential; MJ is the mass of Ju-piter; a is its semimajor axis; e = 0.01 is Europa’s orbitaleccentricity; n = 2π/P = 2.05 × 10–5 s–1 is Europa’s meanmotion; and P is its orbital period. For a nominal value ofk2 ~ 0.25 (Moore and Schubert, 2000) and Q ~ 100, Ftidal ~10–100 mW m–2, larger than radiogenic heating (Tobie etal., 2003) (see also the chapter by Sotin et al.).

How does Europa remove its tidal heat? The two pos-sible heat transport mechanisms within the outer ice I shellare conduction and solid-state convection. Figure 1a illus-trates the qualitative temperature structures that would ac-

company each of these states. In both cases, the top layerhas a steep, conductive temperature gradient; in the conduc-tive case (Fig. 1a, top), this layer extends to the top of theocean, whereas in the convective case (Fig. 1a, bottom), athick, nearly isothermal ice layer lies underneath the con-ductive lid. Despite greatly different thicknesses, these twosolutions can potentially have similar surface heat fluxes.This fundamental ambiguity makes it difficult to infer theice shell thickness from heat flow measurements. Here, wedescribe heat transport by conduction, and describe the ther-mal structure of an icy shell that generates heat with tidaldissipation and removes it by conduction. We then describethe governing equations of solid-state convection in Eu-ropa’s icy shell and define the fundamental quantities thatdescribe a convecting ice shell, including the Rayleigh num-ber and Nusselt number. More detail about the behavior of aconvecting icy shell will be provided in section 4 and sec-tion 5.

2.1. Conduction

The simple estimate of how much tidal dissipation occursin Europa from equation (1) provides us with an estimateof the heat flux carried across Europa’s outer ice I shell. Ifthe ice shell is in conductive equilibrium, Fourier’s law re-lates the heat flux to the shell thickness

Fcond =kΔT

D(2)

where ΔT = Tb – Ts is the temperature difference betweenthe temperature at its base (Tb) and its surface (Ts), and k ~3.3 W m–1 K–1 (here assumed constant for simplicity) is arepresentative thermal conductivity for cold ice. The prob-able detection of an ocean beneath Europa’s ice (Zimmer etal., 2000) suggests that the base of the icy shell should beat the melting temperature of water ice (for pure water ice,Tm = 253–273 K at pressures relevant to Europa’s interior).The global average surface temperature on Europa is Ts ~100 K (Ojakangas and Stevenson, 1989). Setting Fcond =Ftidal gives an ice shell thickness D ≈ 6 km (for Ftidal ~100 mW m–2) and D ≈ 60 km (for Ftidal ~ 10 mW m–2).

The thermal conductivity is temperature-dependent; coldice is a much better conductor of heat than warm ice (seePetrenko and Whitworth, 1999, for discussion), so the tem-perature gradient close to the surface of the icy shell issteeper than at its base. The heat flux across a conductiveicy shell with a temperature-dependent thermal conductiv-ity is (cf. Ojakangas and Stevenson, 1989)

F = lnTb

Ts

ac

D(3)

where k = ac/T and ac = 621 W m–1 (Petrenko and Whit-worth, 1999).


Fig. 1. (a) Schematic temperature structures for a conductive (top) and convective (bottom) ice shell on Europa from Mitri and Showman(2005). (b) Values of ice shell thickness where convection is possible in Europa as a function of grain size. For ice grain sizes d >2 mm, deformation during the onset of convection is accommodated largely by GSS creep (black), but for smaller grain sizes, defor-mation is accommodated by volume diffusion (gray). After Barr and Pappalardo (2005). (c) Behavior of an ice shell in Ra-Nu spaceduring the onset of convection in a basally heated fluid with θ = 18 (Δη = 108) (black arrows) and decay of convection (gray arrows).Diamonds illustrate location of simulations of the onset of convection by Mitri and Showman (2005), points, solid, dashed lines showlocations of simulations of the decay of convection by Solomatov and Barr (2007). When convection begins, Nu jumps from 1 to~1.6–1.7 (see section 4.2.3), depending on the form of temperature perturbation used. When convection stops, Nu can achieve verylow values for Ra < Racr,1, but ultimately stops when Ra < Ra*

cr, when Nu ~ 1.1–1.3 for rheological parameters for ice. (d) Heat fluxas a function of ice shell thickness for equilibrium configurations of Europa’s icy shell, illustrating the jump in heat flux at the con-vective/conductive transition D ~ 9 km. Tidal heating with a tidal-flexing strain amplitude 2 × 10–5 is assumed, and a Newtonian rhe-ology is used with a melting-temperature viscosity of 1013 Pa s and a viscosity contrast of 106. Triangles and diamonds show heat fluxinto the bottom and out the top of the ice shell, respectively. Solid curve shows relationship between flux and thickness for a conduc-tive solution with no tidal heating. From Mitri and Showman (2005).

408 Europa

2.2. A Tidally Heated Conductive Ice Shell

State-of-the-art models of tidal heating in Europa’s icyshell calculate the dissipation occurring in the shell due toits cyclical diurnal tidal flexure by modeling the icy shellas a Maxwell viscoelastic solid. For a general discussion ofthe Maxwell model, see Ojakangas and Stevenson (1989)and Turcotte and Schubert (2002). The energy dissipated ina Maxwell solid is maximized when the period of the forc-ing T ~ τM = η/μ, where τM is the Maxwell time, η is theviscosity, and μ is the shear modulus of the material. Bycoincidence (or orbital and geophysical “tuning” in the Ju-piter system), the orbital period of Europa (3.5 days) is veryclose to the Maxwell time for warm ice I (μ ~ 3.5 × 109 Paand η ~ 1015 Pa s, which gives τM ≈ 3 days). A warm, inter-nally heated ice shell on Europa ice shell may be close to amaximally dissipative state.

In a Maxwell viscoelastic solid, the volumetric dissipa-tion rate (q) is proportional to viscosity, q ~ η(T)e2 at hightemperatures, and inversely proportional to viscosity, q ~μ2e2/[ω2η(T)], at low temperatures. This implies a stronglytemperature-dependent dissipation rate with peak dissipa-tion occurring between ~220 and 270 K depending on theice grain size (see below). A hot ice shell can therefore bemore dissipative than a cold ice shell, and the dissipationshould depend on the shell thickness (Cassen et al., 1980).This will affect the existence of equilibria between tidaldissipation and heat transfer.

To date, the most physically realistic model of a tidallyheated conductive europan ice shell was proposed by Oja-kangas and Stevenson (1989), who related the thickness ofthe ice shell, D, to the heat flux from the deeper interiorFcore

D ≈

ac

2

ln(Tb/Ts)

+∫ q(T)dTT

Fcore

ac

Tb

0

2 1/2(4)

where ac = 621 W m–1 (Petrenko and Whitworth, 1999) (seealso equation (3)). Physically, this equation describes a con-ductive equilibrium in an internally heated ice shell withbasal heat flux Fcore and temperature-dependent thermal con-ductivity. If the ice is modeled as a Maxwell viscoelasticsolid, the tidal dissipation rate is (Ojakangas and Stevenson,1989)

1 + (ωτM)2q =

2μ⟨e2ij⟩ω

ωτM⎡⎣⎢

⎤⎦⎥

(5)

where ⟨e2ij⟩ is the time-average of the square of the sec-

ond invariant of the strain rate tensor, τM = η(T)/μ is thetemperature-dependent Maxwell time, and ω = n. The dis-sipation rate maximizes when ω ~ τ–1

M, corresponding totemperatures of ~220–270 K for ice viscosity of ~1013–1015 Pa s, is close to plausible viscosities for warm ice (see

discussion in section 3). Equation (5) suggests that tidal dis-sipation occurs in the warmest ice, and that tidal dissipa-tion in the cold, stiff portions of the icy shell is negligible.Assuming that the ice shell has horizontally uniform mate-rial properties, the quantity ⟨e2ij⟩ is a low-degree sphericalharmonic function that varies by a factor of 2 between theminimum at the subjovian point and the maximum at thepoles (see Fig. 1 of Ojakangas and Stevenson, 1989). Thesurface temperature on Europa also varies significantly,ranging from Ts ~ 52 K at the poles to Ts ~ 110 K at theequator. Integration of equation (4) with the tidal heatsource (equation (5)) and including the spatially varying sur-face temperature gives the equilibrium ice shell thicknessranging from ~15 to 30 km as a function of location onEuropa assuming radiogenic heating from the rocky mantleis 10 mW m–2 (Ojakangas and Stevenson, 1989).

Since its development, spacecraft data and advances inour understanding of the behavior of a floating ice shellhave questioned the applicability of the Ojakangas andStevenson (1989) model. Galileo images of pits, chaos, anduplifts on Europa’s surface suggesting a convecting icyshell, and the suggestion that a shell D ≥ 30 km thick couldconvect (see section 4), imply that the icy shell could bemuch thicker and still carry the tidal heat flux. The Oja-kangas and Stevenson (1989) model, however, is an ex-tremely valuable starting point for study of more compli-cated tidal/convective systems (e.g., Tobie et al., 2003)because it demonstrates that tidal dissipation in Europa’sshell is strongly rheology-dependent, and suggests that vari-ations in tidal dissipation and surface temperature may leadto variations in the activity within the shell and potentialfor resurfacing.

2.3. Convection

Europa’s ice shell may also transport heat by solid-stateconvection. The density of water ice, like most solids, de-creases as a function of increasing temperature. Therefore,an ice shell cooled from its surface and heated from withinand beneath is gravitationally unstable: Warm ice rises, coldice sinks, which transports thermal energy upward to thebase of a conductive “lid” at the surface of the icy shell.When this process is self-sustaining over geologically longtimescales, it is called solid-state convection.

Thermally driven convection in a highly viscous fluidwith no inertial forces is described by the conservationequations for mass, momentum, and energy (cf. Schubertet al., 2001)

∇ · v→ = 0 (6)

∇p – ρoα(T – To)gêz = ∇ · [η(∇v→ + ∇Tv→)] (7)

v→ · ∇T + ∂T∂t

= κ∇2T + γ (8)

where v→ is the velocity field, T is temperature, t is time, α


is the coefficient of thermal expansion, To and ρo are refer-ence values of density and temperature, κ is thermal dif-fusivity (here, assumed to be constant), p is the dynamicpressure (which excludes lithostatic pressure), êz is a unitvector in the vertical (z) direction, g is gravity, η is viscosity,and γ represents heat sources.

In a shell where convection occurs, the heat flux, Fconv,is enhanced relative to the conductive heat flux by a factorof Nu > 1

Fconv = NukΔT

D(9)

where the Nusselt number, Nu, is related to the vigor ofconvection, expressed by the Rayleigh number

ρogαΔTD3

κηRa = (10)

As we will describe in more detail in section 4, convec-tion can occur when the Rayleigh number of the icy shellexceeds a critical value, Racr. The value of Racr depends onhow the ice viscosity varies with temperature and stress(Solomatov, 1995; Solomatov and Barr, 2006, 2007). Therelative efficiency of convective heat transport over conduc-tion depends on the vigor of convection, an effect expressedin the relationship between Ra and Nu

Nu = cRaβn (11)

where the values of c and βn depend on the variation in iceviscosity as a function of temperature and stress (Solomatov,1995; Dumoulin et al., 1999; Solomatov and Moresi, 2000;Freeman et al., 2006) and (Ra/Racr), with extremely vigor-ous convection (Ra >> Racr) having different c and βn thansluggish convection (Ra > Racr) (Dumoulin et al., 1999;Mitri and Showman, 2005) (see also section 4.3). Thus, thepossibility of convection and the efficiency of convectiveheat transport depend critically on the viscosity of water iceand its variation as a function of temperature and appliedstress.

3. ICE RHEOLOGY AND GRAIN SIZE

A large volume of laboratory experiments and fieldmeasurements exist regarding the rheology of ice, mean-ing how solid ice flows in response to an applied stress.Durham and Stern (2001) provide an excellent review ofrecent developments in this area. Here, we focus on ad-vances most relevant to Europa. We discuss the rheologyof water ice and the role of impurities in modifying the iceflow law and in controlling ice grain size. We discuss con-trol of ice grain size by secondary phases, dynamic recrys-tallization, and tidal stresses. It should be noted that despitea number of important advances in the last decade, thedeformation mechanisms that accommodate large convec-tive strains in Europa’s ice shell and their descriptive pa-rameters are still uncertain. Further laboratory experiments

characterizing the behavior of ice at conditions relevant toEuropa’s ice shell are needed.

3.1. Rheology of Pure Water Ice

Over millions of years, solid water ice, like rock-form-ing minerals, can behave as a highly viscous fluid. Solidice is a polycrystalline material composed of individualgrains; within each grain, the orientation of the water crystallattice is constant. Like all polycrystalline solids, deforma-tion in ice is accommodated by the motion of defects in thepolycrystal, either within grains, or along grain boundaries.A voluminous literature dating back to the 1900s exists re-garding the behavior of water ice, much of it developed bythe glaciological community who sought to understand thefluid-like behavior of ice observed in large ice sheets andglaciers.

Ice rheology has traditionally been characterized in tworegimes: a high-stress regime (appropriate for glacial flowand the subject of considerable study in field and laboratorysettings), and a low-stress regime, wherein the relationshipbetween stress and strain rate has been estimated theoreti-cally. Laboratory and glacial studies typically character-ize ice behavior at stresses between 10–2 and 10 MPa andstrain rates between 10–7 s–1 to 10–4 s–1 (Durham and Stern,2001). Typical convective strain rates on Europa are ~10–13

s–1 (Tobie et al., 2003) and stresses ~10–3 MPa (Tobie et al.,2003) (see also section 3.1.1 here), so some extrapolationto lower stresses and strain rates is required to apply labo-ratory or field results to the study of satellite interiors. Therheology of ice at low stresses may be appropriate for mod-eling deformation in the warm interiors of convecting iceshells. The laboratory- and field-derived flow laws for ice Iat high stresses are most appropriate for modeling, for ex-ample, the onset of convection and lithospheric deforma-tion. The boundary between the “high” and “low” stress re-gimes depends on temperature and grain size (see Fig. 2).Both regimes of behavior may be relevant to calculatingtidal dissipation, but we note that the behavior of ice I un-dergoing cyclical deformation at europan frequencies is notwell-characterized: Further laboratory and field character-ization is urgently needed to advance our understanding oftidal heating.

3.1.1. Low stress regime: Diffusion creep. The behav-ior of ice in the low-stress regime is relevant to modelingflow within the warm regions of Europa’s icy shell (T >180 K), in locations where the ice grain size is small (d <1 mm), and/or where the driving stresses are relatively low(σ < 0.1 MPa). Owing to its low gravity, the thermal buoy-ancy stresses that drive motion within Europa’s icy shell aresmall. Within the warm, well-mixed interior of a convect-ing icy shell, thermal buoyancy stresses σi ~ ρgαΔTiδrh ~10–3 MPa (equation (25) of Solomatov and Moresi, 2000);see also Tobie et al., 2003), where ΔTi ~ 10 K is the magni-tude of temperature fluctuations driving convection and δrhis the rheological boundary layer thickness (Solomatov and

410 Europa

Moresi, 2000) ~O(1 km). Like most rock-forming minerals,ice is thought to deform by diffusion creep at low stresses,high temperatures, and in materials with small grain size.Diffusion creep occurs by two processes: volume diffusioncreep (Nabarro-Herring creep) and grain boundary diffusioncreep (Coble creep) (Goodman et al., 1981)

42Vmσ3RTd2

ediff = Dv + Dbπδd

(12)

where σ is differential stress, R = 8.314 J mol–1 K–1 is thegas constant, d is grain size, Dv is the rate of volume diffu-sion, δ = 2b is the grain boundary width, b is Burger’s vec-tor for ice, Vm is the molar volume, and Db is the rate ofgrain boundary diffusion (Goldsby and Kohlstedt, 2001).Each diffusion coefficient is strongly temperature-depen-dent, Dv = Do,vexp(–Qv/RT), and Db = Do,bexp(–Qb/RT).Note also that the strain rate from diffusion creep is grain-size dependent.

To date, diffusion creep in ice has not been directly ob-served in laboratory experiments, so values of its govern-ing parameters (summarized in Table 1) are calculated basedon microphysical models of the diffusion processes. Good-man et al. (1981) provide a comprehensive discussion ofdiffusion processes in ice, section 5.5 of Goldsby and Kohl-stedt (2001) gives recent updates for governing parameters,and Goldsby (2007) provides an update on efforts to observediffusion creep in the laboratory. At conditions appropriatefor a warm convecting ice shell with reasonable grain sizes~0.1 mm to 1 mm, the deformation rate from diffusioncreep is overwhelmingly dominated by the volume diffu-sion term. At T > 258 K, the rate of Coble creep in ice isexpected to increase by a factor of 1000, due to premeltingalong grain boundaries and triple junctions, which allowsfor more efficient grain boundary diffusion than a purelysolid grain boundary (Goldsby and Kohlstedt, 2001). Thisresults in a marked decrease in the viscosity of ice within10 K of the melting point (see deformation maps of Durhamand Stern, 2001), an effect that has been largely overlookedin current numerical studies (with the noted exception ofTobie et al., 2003).

An effective viscosity can be calculated from the stress-strain rate relationship (Durham and Stern, 2001; Ranalli,1987)

1

3(n + 1)/2η =

σe

(13)

where the factor of 3(n + 1)/2, where n is the rheological stressexponent, is included because the stresses that drive defor-mation have not been resolved into shear and normal com-ponents (Ranalli, 1987). For diffusion creep, n = 1. Thisgives an effective viscosity due to volume diffusion

RTd2

42VmDo,v

ηdiff = expQ*

v

RT(14)

The resulting behavior of ice is said to be “Newtonian,”meaning that the effective viscosity is independent of stress,

Fig. 2. Deformation maps for ice I using the rheology of Goldsbyand Kohlstedt (2001), for ice with grain sizes of 1.0 cm, 1.0 mm,and 0.1 mm. Lines on the deformation map represent the transi-tion stress between mechanisms as a function of temperature. FromBarr et al. (2004).


and that stress and strain rate are linearly related. The vol-ume diffusion flow law and variants of it have been widelyapplied to study of the interior of Europa’s icy shell sincethe 1980s. It is common to rewrite the flow law as

η = ηoexpTm

T– 1A (15)

which is equivalent to equation (14) if A = Q*/RTm ~ 26 forpure water ice and ηo is equal to ηdiff(do, Tm), where do isan assumed grain size. The value ηo is commonly assumedto be a free parameter ranging from ηo ~ 1013 to 1015 Pa s,corresponding to grain sizes of 0.1 mm and 1 mm, respec-tively (using equation (14)). The volume diffusion rheologyhas been used in all existing calculations of tidal dissipa-tion within Europa’s icy shell (Ojakangas and Stevenson,1989; Tobie et al., 2003; Showman and Han, 2004; Mitriand Showman, 2005, 2008b). We note that, at the stressesassociated with Europa’s daily tidal flexing, ~0.1 MPa, orduring the onset of convection, non-Newtonian deformationmechanisms could be relevant (Barr et al., 2004).

3.1.2. High stress regime: Grain-size-sensitive and dis-location creep. The behavior of ice I at relatively highstresses σ > 0.01 MPa is well-characterized by laboratory ex-periments and glacial measurements. Although the stressesin a convecting ice shell on Europa are relatively low, largerstresses (σ ~ 0.01 MPa) may build up in the icy shell dur-ing the onset of convection (see section 4.2.1) or by litho-spheric deformation. In ice with a grain size d > 1 mm, flowdriven by stresses of this magnitude is accommodated bydislocation creep and grain-size-sensitive creep.

In this regime, the stress-strain rate relationship for waterice is described as

Bdp

e = σnexp–Q*

RT(16)

where B and n are constants determined in laboratory ex-periments, or from measuring glacial flow, and p is the grainsize exponent. A summary of flow laws determined between

1952 and 1979 by Weertman (1983) reveals the level of un-certainty in ice rheology during the Voyager era. Values ofn ranging from 1.6 to 4 had been measured in differentcontexts: Creep in polycrystalline ice at low temperature(perhaps most appropriate for Europa) suggested n ~ 3 andactivation energies between 60 to 80 kJ mol–1 (Weertman(1983). Because n > 1, the effective ice viscosity in the high-stress regime depends on stress (i.e., ice is “non-Newtonian,”meaning that strain rate depends nonlinearly on stress).

Laboratory experiments reveal that for σ > 1 MPa, de-formation in ice occurs by dislocation creep, characterizedby equation (16) with n = 4 and Q* = 60 kJ mol–1 (Goldsbyand Kohlstedt, 2001). Dislocation creep has a high stressexponent n = 4, so the strain rate in cold ice with a largegrain size depends strongly on the applied stress (i.e., theice is highly non-Newtonian) and its strain rate is indepen-dent of grain size. Similar to the low-stress regime, strainrates from dislocation creep in ice with T > 258 K are alsoincreased due to premelting at grain boundaries and three-and four-grain junctions in ice (see section 5.4 of Goldsbyand Kohlstedt, 2001).

Recent laboratory experiments suggest the existence ofan intermediate regime for 0.01 MPa < σ < 1 MPa (seeFig. 5 of Goldsby and Kohlstedt, 2001), wherein deforma-tion occurs by weakly non-Newtonian deformation mech-anism(s) referred to collectively as “grain-size-sensitive”(GSS) creep. GSS creep is characterized by a relatively low-stress exponent n ~ 2, a relatively low grain-size exponent1 < p < 2, and a modest activation energy Q* ~ 49–60 kJmol–1 (Goldsby and Kohlstedt, 2001; Durham et al., 2001).GSS creep is of particular interest to the glacial community,because most large ice bodies on Earth, which have grainsizes ~1–10 mm and driving stresses ~0.1 MPa, are deform-ing in the intermediate stress regime.

Although the governing parameters of GSS creep aregenerally agreed upon, identification of the specific micro-physical process that accommodates strain in this regime isan open area of debate. The values of the governing param-eters strongly suggest that easy slip (equivalently basal slip),where ice grains deform along the basal planes of their hex-

TABLE 1. Rheological parameters for ice I after Goldsby and Kohlstedt (2001).

Parameter Basal Slip Grain Boundary Sliding Dislocation Creep

B (mp Pa–n s–1) 2.2 × 10–7 6.2 × 10–14 4.0 × 10–19

n 2.4 1.8 4.0p 0 1.4 0Q* (kJ mol–1) 60 49 60

Parameter Name Volume Diffusion

Vm (m3 mol–1) Molar volume 1.97 × 10–5

b (m) Burger’s vector 4.52 × 10–10

δ (m) = 2b Grain boundary width 9.04 × 10–10

Do,v (m2 s–1) Volume diffusion constant 9.10 × 10–4

Q*v (kJ mol–1) Volume diffusion activation energy 59.4

Do,b (m2 s–1) GB diffusion constant 7.0 × 10–4

Q*b (kJ mol–1) GB diffusion activation energy 49

412 Europa

agonal crystals, occurs in the intermediate regime (Goldsbyand Kohlstedt, 2001; Duval et al., 2000). A secondary proc-ess such as dislocation creep or grain boundary sliding mustoperate in tandem with basal slip to accommodate defor-mation in crystals whose planes are not oriented properlyfor basal slip to occur (Goldsby and Kohlstedt, 2002). Theidentification of this secondary process has been problem-atic. Scanning electron microscopy of deformed ice samplesallowed Goldsby and Kohlstedt (2001) to identify instancesof grain switching and occurrence of straight grain bound-aries and four-grain junctions, providing evidence that grainboundary sliding accommodates easy slip. This gives riseto a combined flow law in the intermediate regime (Goldsbyand Kohlstedt, 2001)

egss = +1egbs

1 –1

ebs(17)

where gbs stands for grain boundary sliding, and bs standsfor basal slip, with the strain rate from grain boundary slid-ing (egbs) dominating at conditions relevant to the interiorof Europa’s ice shell (Barr et al., 2004). Governing param-eters for GSS creep effectively controlled by GBS are sum-marized in Table 1.

However, it has been suggested that grain boundary slid-ing and basal slip acting together are not able produce thecrystal fabric (the coalignment of crystal lattices in adja-cent grains) observed in deformed sections of terrestrial icesheets (Duval et al., 2000; Duval and Montagnat, 2002).Montagnat and Duval (2000) suggest an alternate hypothe-sis: that grain boundary migration (essentially, grain growth)and associated recrystallization accommodates basal slip.However, Goldsby and Kohlstedt (2002) point out that grainboundary migration does not produce strain and thus is nota deformation mechanism. The identification of the micro-physical process accommodating deformation in ice at mod-erate stresses remains an active area of research.

3.1.3. A combined flow law. Goldsby and Kohlstedt(2001) propose that the behavior of ice I across the high-,intermediate-, and low-stress regimes can be described bya single governing equation

etotal = ediff + egss + edisl (18)

where, in application to Europa’s icy shell, the strain ratedue to GSS may be approximated by the governing param-eters of grain boundary sliding (Barr et al., 2004). Despiteuncertainties in the microphysical mechanisms at work inGSS creep, a combined flow law including both GSS anddislocation creep using governing parameters summarizedin Table 1 provides a good match to stress/strain rate/grainsize relationships deduced from previous laboratory experi-ments (Goldsby and Kohlstedt, 2001) and glacial measure-ments (Peltier et al., 2000). The composite flow law canbe used to determine the regimes of dominance in stress,temperature, and grain size space for each constituent de-

formation mechanism. By equating strain rates betweenpairs of mechanisms, one can construct a deformation mapfor ice that can be used to predict which rheology is appro-priate for a given application (see Fig. 2 and deformationmaps of Durham and Stern, 2001).

3.2. Effect of Impurities

The presence of substances other than water ice in Eu-ropa’s icy shell can have an important effect on its rheology.Here we summarize how the presence of various materials,including ammonia, sulfate salts, and dispersed particulates,may affect the rheology of Europa’s icy shell.

Ammonia dihydrate, NH3·2H2O, the stable phase in thewater-rich, low-pressure region of phase space in the am-monia-water system, melts at Tm = 176 K. Ammonia hasbeen suggested as a possible means to thermodynamicallystabilize liquid water oceans beneath convecting ice shells(e.g., Spohn and Schubert, 2003) and implicated as a pos-sible component of cryovolcanic magmas on icy satellites.The rheology of ammonia dihydrate has been measuredin laboratory experiments by Durham et al. (1993). Theflow law for ammonia dihydrate, with mole fraction xNH3

=0.3 (corresponding to a mole fraction of 90% dihydrate),can be expressed in similar form as equation (16), with B =10–15 Pa–5.8 s–1, n = 5.8, and Q* = 102 kJ mol–1. At its melt-ing point and a nominal stress of 0.01 MPa, ammonia dihy-drate is 2 orders of magnitude less viscous than water ice,but its large activation energy leads to a rapid increase inviscosity as the temperature is decreased.

Galileo NIMS data suggest that the surface of Europa’sicy shell is composed predominantly of water ice and non-ice materials that include one or more hydrated materials.Candidates for the latter include hydrated magnesium and/or sodium salts (McCord et al., 1999) or hydrated solidifiedsulfuric acid (Carson et al., 2005). Geochemical modelingof water-rock chemistry in Europa’s ocean also suggest theformation of magnesium and/or sodium salts, supporting theview that the icy shell may be salty throughout (Zolotov andShock, 2001; McKinnon and Zolensky, 2003). The sulfatesalts were found to have much higher viscosities than purewater ice at comparable temperatures (Durham et al., 2005).For example, the difference in hardness between mirabilitegrains and water ice is so high that the dispersed mirabil-ite particles can act as hard secondary phases, and have asimilar effect on the rheology of the bulk material as sili-cate grains.

Recent laboratory experiments by, e.g., McCarthy et al.(2007) have explored dissipation in mixtures of ice andmagnesium sulfate. Frozen eutectic mixtures of ice andmagnesium sulfate form a lamellar structure with layers ofice and magnesium sulfate sandwiched together. The result-ing mixture is stiff but highly dissipative due to the micro-scale boundaries between layers of magnesium sulfate andwater ice. Future laboratory experiments on pure water iceand water ice mixed with other materials may provide al-ternate models for dissipation than the Maxwell model, and


help bridge the gap between theory and data regarding tidaldissipation in Europa.

Europa’s icy shell may contain small amounts of silicatedust, but the dust content is limited by its bulk density —the icy shell must maintain a low enough density to remaingravitationally stable atop the ρ ~ 1000 kg m–3 ocean (ormore if the ocean is salty). Small rock particles (with sizesmuch less than the mean ice grain size) mixed with ice actas a barrier to flow within the ice and disturb the flow pat-tern — as a result, the effect of particles on ice viscos-ity depends critically upon the location of the particleswithin the polycrystalline structure of ice. Laboratory ex-periments characterizing the viscosity of mixtures of icewith a grain size of ~1 mm and silicate particles with grainsizes ~100 µm show that the increase in viscosity of waterice due to the presence of SiO2 and SiC/SiCaCO3 grains atlow volume fractions relevant to Europa’s icy shell, φ < 0.1,is negligible (see Fig. 14 of Durham et al., 1992). However,silicate particles may play a role in inhibiting grain growthin Europa’s icy shell, as will be discussed in section 3.3.1.

3.3. Ice Grain Size

Observations of grain size and the processes that con-trol grain size in terrestrial ice sheets experiencing stressand temperature conditions similar to the interior of Eu-ropa’s icy shell can provide estimates of likely grain sizes.Grain sizes in Europa’s shell are commonly assumed to beuniform and between 0.1 mm to 10 mm, by analogy withgrain sizes in terrestrial ice sheets (Budd and Jacka, 1989)[see also, e.g., Thorsteinsson et al. (1997) for a sample icecore grain size profile].

Two recent works cast doubt upon these estimates andhave led to a reevaluation of the plausibility of the 0.1–10-mm range for assumed ice grain sizes. Nimmo and Manga(2002) estimated the viscosity at the base of the icy shellby using the measured diameters of pits and uplifts (~4–10 km) to infer properties of the underlying convectionpattern. They obtained a basal viscosity for the icy shellbetween 1012 and 1013 Pa s, and suggested that ice grainsizes at the base of the shell should be between 0.02 and0.06 mm. Follow-on studies by Showman and Han (2004)suggest that near-surface ice grain sizes less than 0.04 mmare required to create small depressions such as those ob-served on the surface of Europa (see section 5.1). Schmidtand Dahl-Jensen (2003) applied a simple model of unim-peded grain growth in the low-stress, high-temperature, andliquid-rich environment at the base of Europa’s icy shell andsuggested that its grain size may be between 4 cm and 80 m.Because the rate of volume diffusion depends on grain sizesquared, the range of ice viscosity implied by all the aboveestimates is 15 orders of magnitude.

However, the estimates of Schmidt and Dahl-Jensen(2003) ignore processes known to modify grain sizes withinterrestrial ice cores. In addition, the geodynamical studiesdid not account for near-surface weakening by other proc-esses (e.g., microcracking), which could explain the low

effective near-surface viscosity required to create depres-sions, or the possibility of premelting at the base of Europa’sicy shell, which could yield low basal viscosities for a muchlarger grain size. Because stress and temperature conditionsexpected within Europa’s icy shell are similar to those ex-perienced by many ice bodies on Earth, processes control-ling grain sizes within terrestrial ice sheets may controlgrain size within Europa’s ice shell and act to self-regulateand/or limit its ice grain size to values closer to those ob-served in ice cores. Here, we discuss two possible methodsof controlling ice grain size in Europa’s icy shell: (1) Zenerpinning due to the presence of hard secondary phases (Kirkand Stevenson, 1987; Barr and McKinnon, 2007), and(2) dynamic recrystallization/tidal flexing (McKinnon, 1999;Barr and McKinnon, 2007).

3.3.1. Grain size control by secondary phases. Meas-ured grain sizes in many terrestrial ice cores indicate thatgrain size in impurity-laden ice are invariably smaller thanthose in clean ice (Alley et al., 1986a,b). In the absence ofimpurities, grains grow by grain boundary migration drivenby the free energy decrease associated with reduction ofgrain boundary curvature. Non-water-ice materials concen-trate on grain boundaries, and can decrease the grain growthrate and in some cases can even halt grain growth altogether(Poirier, 1985; Alley et al., 1986a,b). The role that any typeof impurity plays in inhibiting or preventing grain growthdepends on the location of the impurity within the struc-ture of the ice polycrystal: Impurities concentrated alonggrain boundaries and at grain junctions can be much moreeffective at inhibiting grain growth than impurities randomlydispersed in the ice (Durand et al., 2006). Recent advancesin SEM imaging of samples from ice cores shows the spa-tial correlation between silicate microparticles and kinks ingrain boundaries, providing compelling evidence that sili-cate particles can inhibit grain growth (Weiss et al., 2002).

On Europa, one could imagine that silicate or salt par-ticles might act as hard secondary phases (or pinning par-ticles) that could slow or halt grain growth. The effect ofpinning particles on grain size was modeled by Zener, whorelated the drag force exerted by hard secondary phases ongrain boundaries to the rate of grain growth dr/dt (Poirier,1985)

PZ

αGγgb

–EA

RT= Kg,0exp

drdt

–1r

(19)

where EA = 46 kJ mol–1 is the activation energy for grainboundary migration, γgb = 0.065 J m–2 is the grain bound-ary free energy (De La Chapelle et al., 1998), and αG = 0.25is a geometric factor. The pinning pressure PZ exerted onthe grain boundary is related to the number of particleson the boundary (fNx) (Poirier, 1985) (where f is the num-ber fraction of particles residing on the grain boundary),

13PZ = πγgbrxrfNx, where rx is the radius of the particles re-

siding on the grain boundary and Nx is the number of par-ticles per unit volume. Grain growth is completely stopped

414 Europa

when dr/dt = 0, which gives the Zener limiting grain size(cf. Durand et al., 2006)

3αG1/2

πrxfNxrz = (20)

where the numerical factor of 3 indicates that the particlesreside on grain boundaries, and f ~ 0.25 is estimated bycounting the number of particles residing on grain bound-aries in SEM images of the GRIP ice core (Barnes et al.,2002; Weiss et al., 2002). If we suppose that the ice shellof Europa is loaded with microscopic silicate particles ofdensity 3000 kg m–3 and radii of 10 µm, up to a total vol-ume percentage of 4%, the Zener limiting grain size is

rx1/2

10 µmrz = 0.1 mm

0.04φ

(21)

where φ is the volume fraction of silicate in Europa’s icyshell and 4

3Nx ~ φ/( πr3x). The upper limit on ice grain size

derived from the Zener pinning model is inversely propor-tional to impurity content — fewer impurities mean largergrain sizes. We note that Kirk and Stevenson (1987) con-structed a similar argument to estimate grain sizes in the icemantle of Ganymede: Their estimate has a similar depen-dence on radii of the silicate particles but assumes randomlydistributed particles and gives rz ~ φ–1. For Europa, we canput a plausible estimate on the upper limit of silicate con-tent based on the density of the icy shell — if the shell den-sity exceeds 1000 kg m–3 (or more for a salty ocean), it willbe gravitationally unstable atop the ocean.

Observations of grain sizes between 0.5 and 1 mm inimpurity-laden sections of terrestrial ice cores, coupled withour upper limit on ice grain size based on a Zener pinningmodel, suggest that grain sizes in Europa’s icy shell mayhover around the 0.1–1-mm range (McKinnon, 1999; Barrand McKinnon, 2007). At present, terrestrial observationsof grain growth and impurity distribution within polycrys-talline ice are limited to temperatures between 235 to 273 K.Therefore, knowledge of grain size and processes control-ling grain size gained through study of terrestrial cores maybe most applicable to the warm and convecting interior iceshell (Barr and McKinnon, 2007). Grain sizes closer to thesurface of Europa may be controlled by other, non-ther-mally activated processes such as cyclical tidal deformation(McKinnon, 1999).

3.3.2. Dynamic recrystallization. Observations of rel-atively impurity-free sections of terrestrial ice cores revealthat ice grain sizes are constant as a function of depth(equivalently, time) within sections of the core that haveexperienced significant strain (Thorsteinsson et al., 1997;De La Chapelle et al., 1998). If ice grains can grow unim-peded, one would expect grain size in the ice sheet to in-crease as a function of depth and time. This suggests thatdeformation acts to decrease grain size, thereby competingwith natural grain growth and allowing a roughly constantsteady-state grain size to be maintained over time. It hasbeen suggested that the accumulated strain due to verticallayer compaction results in grain size reduction due to a

process called dynamic recrystallization (Thorsteinsson etal., 1997; De La Chapelle et al., 1998). In dynamic recrys-tallization, the grain size in a deforming material is con-trolled by a balance between grain growth and the forma-tion of new grains (nucleation) by a process called subgrainrotation (Shimizu, 1998; DeBresser et al., 1998). Subgrainrotation can only occur if deformation is occurring in thematerial, leading to a threshold strain at which grain sizes ina deforming material achieve their steady-state recrystal-lized values. For temperature and strain rate conditions ap-propriate for the GRIP ice core, T ~ 240 K and e ~ 10–12 s–1,the threshold strain is about 25% (Thorsteinsson et al.,1997). A model of this process has been applied to estimategrain sizes within actively deforming regions of ice shellsby Barr and McKinnon (2007), who find that in the well-mixed, warm convective interior of an already convectingice shell, grain sizes will evolve to a steady-state value thatdepends on the applied stress (Derby, 1991; Shimizu, 1998;DeBresser et al., 1998; Barr and McKinnon, 2007)

σµ

drecrys = Kb–m

(22)

where K = 1–100 is a grouped material parameter, µ is theice shear modulus, and m = 1.25. Barr and McKinnon(2007) suggest that in the absence of impurities, recrystal-lized grain sizes in convecting ice shells will be large,drecrys ~ 30–80 mm, which may lead to highly viscous iceand a gradual shut-down of convection as the grains achievetheir recrystallized value. The implication is that withoutimpurities to limit grain growth, ice shells may convect slug-gishly, and may be limited to a small number of convec-tive overturns before transitioning to a conductive state.

On Europa, however, tidal flexing of the icy shell itselfmay control the ice grain size. McKinnon (1999) hypoth-esized that if cyclical straining in the presence of convec-tion of Europa’s icy shell has the same effect as the con-tinuous strain on terrestrial ice cores (i.e., driving dynamicrecrystallization), the grain size in the icy shell would de-crease as d ∝ e–1/2. The grain size controlled in this mannerwould have a maximum of 1 mm at the warm base of theicy shell. Thus, cyclical tidal flexing may prevent grains inEuropa’s icy shell from growing to the large values pre-dicted by continuous-deformation dynamic recrystallizationmodels (Barr and McKinnon, 2007), exempting Europa’sicy shell from being choked while convecting.

4. ICY SHELL CONVECTION

In section 2, we described two possible modes of remov-ing tidal heat generated in Europa’s icy shell: conductionand convection. But how do we decide whether convectioncan happen? Until recently, knowledge of how convectionstarts and stops in realistic planetary mantles was relativelylimited because terrestrial planet mantles are commonlyassumed to convect throughout most of their geological his-tory. Although many of the techniques developed for study-ing terrestrial planet mantle convection apply to Europa, theheat flow history in Europa’s icy shell sets it apart from


terrestrial planets. Unlike a terrestrial planetary mantle,Europa’s icy shell may receive periodic bursts of heat dueto tidal dissipation in addition to radiogenic heating from itsrocky mantle. As a result, the mode of heat transport acrossEuropa’s icy shell may change from conductive to convec-tive many times during its evolution (Mitri and Showman,2005). Here, we summarize several decades’ worth of studyof the issue of whether convection is possible in Europa’sice shell, how convection may start and stop, and the effi-ciency of convective heat transport.

4.1. Is Convection Possible?

Early efforts to judge whether convection could happenin Europa’s icy shell used two implicit assumptions. First,it was assumed that deformation during the onset of con-vection would be accommodated by Newtonian diffusioncreep, even though field and laboratory measurements ofice viscosity suggested that ice could be non-Newtonian ifthe grain size is sufficiently large. It was also assumed thatthe value of the critical Rayleigh number was independentof the type of temperature perturbations available to trig-ger convection in the icy shell. Here, we discuss the conse-quences of relaxing these assumptions. Recently developednumerical techniques allow for the study of how convectionmay be triggered from “realistic” temperature fluctuationsin the icy shells; we summarize the results of these studies.

The simplest representation of Europa’s icy shell in thelanguage of fluid dynamics is a layer of fluid cooled fromabove and heated from within and beneath. It is commonto assume that a conductive icy shell on Europa is heatedmostly at its base by radiogenic and tidal heating becausetidal dissipation is likely maximized there (however, thisassumption is not necessarily correct; see section 5.2.1). It isalso common to model the ice shell as using a two-dimen-sional Cartesian geometry. Although Europa’s icy shell is,truly, a spherical shell, plausible icy shell thicknesses aresmall compared to the radius of Europa, so for many pur-poses it is sufficient to think of it as a Cartesian box. Atpresent, most simulations of convection in Europa’s icy shellare performed in two dimensions because of limited com-puting resources.

The question of whether convection can occur in a fluidlayer has been studied in a variety of planetary and fluiddynamical contexts. It is a simple geophysical argument:Does the Rayleigh number of the fluid layer exceed a criti-cal value? Using the definition of the Rayleigh number(equation (10)), this can be phrased mathematically as

ρgαΔTD3

κη ≥ Racr (23)

where Racr is the critical Rayleigh number. The value ofRacr for convection in any fluid depends on the wavelengthof initial temperature perturbation within the fluid layer andthe geometry of the layer (see Turcotte and Schubert, 2002,for discussion). Because both the thickness of Europa’s iceshell and its viscosity are poorly constrained at present, wecannot definitively determine whether convection can occur:

The best we can do is determine a critical shell thicknesswhere convection is possible by rearranging equation (20)

Racrκη1/3

ρgαΔTDcr = (24)

and constrain values of Dcr as a function of physical prop-erties of the icy shell.

4.1.1. Newtonian viscosity. Early works such as that ofConsolmagno and Lewis (1978) and Reynolds and Cassen(1979) approximated ice as a constant-viscosity fluid. In aconstant viscosity fluid, Racr ≈ 1000 (Chandrasekhar, 1961),so estimates of the mean ice viscosity, thermal and physicalparameters, and surface temperature and ice melting tem-peratures on Europa could be used to determine Dcr fromequation (24) alone. Reynolds and Cassen (1979) deter-mined that convection could occur in a bottom-heated ice Ishell on a Europa-like satellite if D > 30 km.

A source of uncertainty in evaluating Dcr for a constant-viscosity ice shell is the appropriate choice of viscosityvalue. The viscosity of ice is strongly temperature-depen-dent, so do we evaluate Dcr using η(Ts), or η(Tm), or somewell-chosen mean? This is addressed using algebraic (“scal-ing”) relationships between the activation energy in the iceflow law (which controls ∂η/∂T) and the critical Rayleighnumber for a Newtonian fluid by Stengel et al. (1982) (forn = 1) and Solomatov (1995) (for general n).

The analysis of Solomatov (1995) focuses on the behav-ior of the bottom thermal boundary layer of a basally heatedfluid with a strongly temperature-dependent viscosity at theonset of convection. If the viscosity of the fluid dependsstrongly on temperature [if η(Ts)/η(Tm) > 104 (Solomatov,1995)], fluid motions in the upper part of the layer are mini-scule, and the upper part of the layer forms a lid of cold,high-viscosity fluid (referred to as a “stagnant” lid). In theso-called “stagnant lid regime,” convective fluid motions areconfined to a warm sublayer at the base of the fluid, wherethe temperature is approximately constant, and the tempera-ture dependence of ice viscosity can be neglected. With thisapproximation, the critical Rayleigh number for convectionin a fluid with a temperature-dependent viscosity can beestimated by determining when the warm sublayer beginsto convect, or determining when the local Rayleigh numberin the sublayer exceeds 1000. The result is a scaling rela-tionship between the critical Rayleigh number and rheologi-cal parameters of the fluid (Solomatov, 1995)

4(n + 1)Ra1,cr = Racr(n)

eθ 2(n + 1)/n

(25)

where the subscript 1 indicates that we will compare thiscritical Rayleigh number to the Rayleigh number at the baseof the ice shell (where T = Tb), n is the rheological stressexponent, θ = γΔT, where the Frank-Kamenetskii param-eter γ = –∂(lnη)/∂T|Ti

) = Q*/(RTi2), and Racr(n) ≈ Racr(1)1/n

Racr(∞)(n – 1)/n. In icy satellite convection studies, it is com-monly assumed that the warm, well-mixed convective in-terior of the ice shell has a temperature very close to the

416 Europa

ice melting point, so Ti ≈ Tb = Tm. This gives θ ≈ (Q*ΔT)/(RT2

m) (cf. McKinnon, 2006). The value Racr(n) representsthe critical Rayleigh number for convection in a fluid witha viscosity dependent solely on stress (i.e., η = Bσn), whichis estimated using the value for n = 1, Racr(1) = 1568, andthe limit of Racr(n) as n → ∞, or Racr(∞) ≈ 20 (see Fig. 5 ofSolomatov, 1995).

A large volume of work exists regarding the onset of con-vection in ice I shells of the satellites assuming a Newtonianrheology for ice: In these studies the viscosity of ice de-pends strongly on temperature, but is independent of stress.For a Newtonian ice rheology, n = 1, and equation (25) re-duces to (Stengel et al., 1982; Solomatov, 1995)

Ra1,cr = 20.9θ4 (26)

A surface temperature of Ts = 100 K and basal temperatureof Tm = 260 K implies θ ≈ 18, which gives Racr,1 = 2.2 ×106 (using equation (26)). Values of Racr,1 for a range of θvalues appropriate for volume diffusion and Europa’s icyshell are summarized in Table 2. Evaluating equation (24)using ρ = 920 kg m–3, κ = 2.6 × 10–6 m2 s–1 as a represen-tative value for warm ice, and evaluating the ice viscosityusing coefficients in the volume diffusion flow law (seeTable 1), gives an expression for the critical shell thicknessfor convection (cf. McKinnon, 1999)

d 2/3

0.4 mmDcr,diff = 31 km (27)

where d = 0.3 mm gives a basal ice viscosity of 1014 Pa s,midway between the values of 1013 to 1015 Pa s commonlyassumed in Europa studies.

4.1.2. Realistic ice rheology. For ice with small grainsizes d < 1 mm, diffusion creep likely accommodates strainduring the onset of convection, and the critical ice shellthickness for convection is given by equation (27). How-ever, the composite flow law for ice suggests that deforma-tion at stresses built up within ice shells during the onsetof convection may be accommodated by non-NewtonianGSS creep (Barr et al., 2004). If convection is triggered bytemperature fluctuations of δT ~ 5 K and height λ ~ D (seesection 4.2.1), the thermal stress due to a plume of this mag-nitude is approximately σth ~ ρgαδTλ ~ 0.02 MPa. For grainsizes >1 mm, strain due to stresses on the order of σth maybe accommodated by GSS creep (see Fig. 2). When GSS

creep accommodates deformation during the onset of con-vection

Raa,1(κdp)(1/n)exp

(3(n + 1)/2A)1/nρgαΔT

n/(n + 2)( )Dcr =

Q*

nRTm(28)

where Raa,1 = 3.1 × 104 is the absolute minimum Rayleighnumber where convection is possible (for an optimal per-turbation) in GBS with θ ≈ 15 appropriate for an ice shellwith Ts = 90 K and Tm = 260 K (Barr and Pappalardo,2005). Equation (28), which is for arbitrary n and can beused for non-Newtonian fluids, reduces to equation (24) forn = 1. Evaluating the rheological parameters for GSS us-ing the grain boundary sliding values (see Table 1), andusing nominal values for descriptive properties of the iceshell, the critical shell thickness becomes

d 1.4/3.8

1 mmDcr,GSS = 75 km (29)

The grain size at which GSS becomes the “controlling”rheology at the onset of convection can be determined bysetting equations (27) and (29) equal and solving for d. Thisgives d = 2 mm, indicating that in ice with a grain size≤2 mm, diffusion creep likely accommodates strain duringthe onset of convection. For d > 2 mm, GSS creep accom-modates strain during the onset of convection. Figure 1bsummarizes how the critical ice shell thickness where con-vection can occur depends on ice grain size in an ice shellwith a realistic Newtonian diffusion creep and non-Newto-nian GSS rheology (Barr and Pappalardo, 2005).

McKinnon (1999) argues that tidal stresses in Europa’sice shell may alter its rheology. In an ice shell deformingby GSS creep, if the tidal stresses are much greater thanthe thermal buoyancy stresses driving convection, the GSSviscosity law can become effectively “linearized.” The ef-fect is similar to the modification of mantle rheology dueto interaction between convection and postglacial reboundon Earth (Schmeling, 1987). Within Europa’s ice shell, thelow-stress convective flow field “sees” an effectively New-tonian rheology, but the high-stress tidal field “sees” a non-Newtonian viscosity. McKinnon (1999) estimates a basalviscosity of 8 × 1013 Pa s for tidally linearized GBS, whichdepends on grain size as η ∝ d1.4/1.8. Convection is possible

TABLE 2. Critical Rayleigh numbers for stopping and starting convectionin a Newtonian ice shell (after Solomatov and Barr, 2007).

log10(Δη) Equivalent θ Ra*cr Nu(Ra*

cr) Racr,1 Nu(Racr,1)

5 11.5 2.7 × 105 1.30 3.67 × 105 1.568 18.4 1.35 × 106 1.22 2.41 × 106 1.5310 23.0 2.94 × 106 1.19 5.88 × 106 1.4912 27.6 5.61 × 106 1.17 1.218 × 107 1.4716 36.8 1.58 × 107 1.14 3.59 × 107 1.4320 46 3.57 × 107 1.12 9.40 × 107 1.40


in a 25-km-thick ice shell with such a rheology if the grainsize is about 1 mm, which may be the case if grain size iscontrolled by tides (see section 3.3.2).

Here, we have calculated critical ice shell thicknesses forconvection assuming a constant thermal conductivity for theice shell. The critical ice shell thickness for convection ina variable-conductivity shell is larger than in a constant-con-ductivity shell. This effect can be estimated by equating theequivalent heat flow Fconv across a shell with variable con-ductivity to Fconv with a constant conductivity (McKinnon,1999, 2006; Tobie et al., 2003; Barr and Pappalardo, 2005)(see also equations (2) and (3)).

ac

kcΔT= ln

Dtrue

Dcr

Tm

Ts

(30)

where Dtrue is the actual critical shell thickness with variableconductivity taken into account, Dcr is the value obtainedassuming a constant conductivity of kc (here, 3.3 W m–1 K–1).For Ts = 100 and Tm = 260 K, Dtrue/Dcr ~ 1.17.

Finally, we note that although it is beyond the scope ofthis chapter, it is possible that the microphysical processesthat accommodate the first ~10% strain during the onset ofconvection are entirely different than those that govern well-developed convection (see, e.g., Birger, 1998, 2000). Meas-urements of ice behavior during transient or primary creep(Glen, 1955) may be relevant to the question of the onsetof convection in addition to flow laws for steady-state creep(Solomatov and Barr, 2007).

The results of recent efforts to refine the range of criti-cal ice shell thickness for convection, which predict criti-cal thicknesses from 10 km to a few tens of kilometers, gen-erally agree with the original estimates derived by Reynoldsand Cassen (1979): Dcr ~ 30 km. Although the value ofcritical ice shell thickness may not have changed much in30 years, the relationship between ice rheology, the criti-cal shell thickness for convection, and ice grain size hasbeen clarified.

4.2. Behavior of the Icy Shell Close to theCritical Rayleigh Number

4.2.1. Starting convection. In the previous section, wedescribed the results of recent studies attempting to narrowthe range of conditions where convection is possible in Eu-ropa’s ice shell. In a mathematically idealized scenario, anunperturbed and heated ice shell will sit quiescently unlesstemperature fluctuations drive flow and trigger convection.Since the earliest work on convective stability, it has beenknown that the critical Rayleigh number depends on theshape (e.g., Turcotte and Schubert, 2002) and amplitude oftemperature perturbation within the fluid layer (see, e.g.,Chandrasekhar, 1961; Stengel et al., 1982, for discussion).

A key open question is whether tidal dissipation can trig-ger convection in Europa’s icy shell. Because the Maxwelltime of warm ice near the base of Europa’s shell may beclose to Europa’s orbital period, a purely conductive ice

shell may be heated largely at its base. Recent numericalwork suggests that maximally effective perturbations forstarting convection are concentrated at the base of the fluidlayer and have wavelength λcr ~ 2(n + 3)θ–1D (Solomatovand Barr, 2006, 2007). In the absence of temperature fluc-tuations, zones of weakness, or other means of localizingtidal dissipation, tidal heating in a conductive ice shell isessentially constant over the horizontal scale of convectivecells because the r.m.s. strain rate varies by only a factor of~2 between the equator to pole. One could envision the tem-perature perturbation due to tidal dissipation as a smoothlyvarying harmonic function with a wavelength λtidal ~ REuropa/4 or so (see Fig. 1 of Ojakangas and Stevenson, 1989). Iftidal heating is the sole cause of the density differencesnecessary to trigger convection, tidal dissipation must gen-erate temperature perturbations on horizontal length scalesλ ~ λcr to trigger convection (Barr et al., 2004). If λtidal >>λcr, the critical Rayleigh number would increase substan-tially, perhaps by a factor of 100 or more, because perturba-tions with such long wavelengths are inefficient at triggeringconvection. This suggests that tidal dissipation, as envi-sioned by Ojakangas and Stevenson (1989), may not be ableto trigger convection in a purely conductive icy shell.

Other types of temperature fluctuations, e.g., bursts ofheat released close to the surface of the icy shell from largeimpacts, are essentially useless in triggering convection be-cause they diffuse away too quickly to warm the surround-ing ice enough to permit it to flow. Compositional varia-tions present in a realistic icy shell may be able to providethe necessary density contrasts to trigger convection (e.g.,Pappalardo and Barr, 2004). Understanding how convec-tion begins in Europa’s icy shell will require characteriza-tion of the types and locations of temperature fluctuationsnaturally present in the icy shell.

4.2.2. Stopping convection. If the Rayleigh number ofEuropa’s icy shell drops below the value where convectioncan be maintained, convection will cease. The Ra of the iceshell may change, for example, due to perturbations in thebasal heat flux (Mitri and Showman, 2005), or due to anincrease in ice grain size over time (Barr and McKinnon,2007). Gray arrows in Fig. 1c describe the path in Ra-Nuspace taken by an ice shell where convection is stopping.As the Rayleigh number is decreased, Nu decreases untilRa = Ra*

cr, the lowest value of Ra where convection is pos-sible. For Newtonian rheologies in the stagnant lid regime(θ > 8), the value of Ra*

cr ~ 12 Racr,1 (see Table 2). In the vi-

cinity of this point, (Nu – Nucr) ∝ (Ra – Racr)1/2, and at Ra =Ra*

cr, Nu ≈ 1.1 to 1.3, and convective motion is confined toa very thin layer at the base of the ice shell. Values of Ra*

crand Nu(Ra*

cr) for a range of parameters appropriate for avolume diffusion rheology and range of θ appropriate forEuropa’s icy shell are summarized in Table 2 (see also Ta-ble 1 of Solomatov and Barr, 2007). When convection stops,very low values of (Nu – 1) can be achieved, and the mini-mum value scales with θ–1 (Solomatov and Barr, 2007).

4.2.3. Conductive-convective switching. When convec-tion starts in an icy shell, it results in a reorganization of

418 Europa

heat and mass transfer in its interior. The spatially averagedtemperature in the shell increases significantly, as does theheat flux across the shell, resulting in thermal stresses thatmay be sufficient to drive lithospheric deformation (Mitriand Showman, 2005). Here, we summarize the results ofthese recent studies about convective turn-on and turn-off inan ice shell with a Newtonian rheology, and discuss implica-tions for resurfacing of Europa’s icy shell.

The black arrows in Fig. 1c describe the path in Ra-Nuspace traced out by a Newtonian ice shell during the onsetof convection. From a conductive equilibrium (Nu = 1), theice shell begins convecting when Ra = Racr,1 (see equa-tion (25)), and the heat flux jumps rapidly to Nu > 1 (Mitriand Showman, 2005; Solomatov and Barr, 2007). Values ofNu achieved for Ra = Racr,1 using a volume diffusion rheol-ogy for ice and numerical methods described by Solomatovand Barr, 2007) are summarized in Table 2.

The value of Nu achieved during the onset of convec-tion depends on the type of temperature perturbation thatexists or develops in the conductive ice shell (Solomatovand Barr, 2007). As described in section 4.2.1, the mostefficient perturbations at starting convection are confinedto the base of the ice shell: The optimal perturbation shape(for stagnant lid convection in a two-dimensional Cartesiangeometry) is a small convective roll obtained for Ra = Ra*

cr(see section 4.2.2). Using the optimal perturbation shapegives a lower bound on the jump in heat flux when con-vection begins; for θ = 18, Nu jumps from 1 to 1.56 whenconvection begins. Using a sinusoidal perturbation that addsδT = 0.175 to a background conductive equilibrium, Mitriand Showman (2005) find that Nu jumps to 1.7 in a tidallyheated ice shell. The difference between the two valuesprovides an estimate of the effect of perturbation geometryon the ΔNu associated with the onset of convection: ΔNuchanges by ~20% as the wavelength is varied by a factor of~2. This suggests that careful consideration of the types ofrealistic temperature perturbations available to trigger con-vection in icy satellites is required to obtain more accurateestimates of the heat flux jump when convection begins.

A Nu(Ra) plot, as shown in Fig. 1c, is the most straight-forward way of summarizing the heat flux across an iceshell during the onset and decay of convection. To under-stand the geological consequences of the onset of convec-tion, we need to trace variations in the heat flux as a func-tion of ice shell thickness. For Europa, this is most easilydone by assuming that the viscosity at the base of the iceshell (i.e., the melting-temperature viscosity) is indepen-dent of shell thickness; the Rayleigh number can then bedirectly translated into a measure of shell thickness. Like-wise, with equation (2) we can translate Nu into heat flux.Figure 1d shows an example of such a plot, from Mitri andShowman (2005), assuming the melting-temperature viscos-ity is 1013 Pa s. The existence of a heat flux jump impliesthat, for a range of heat fluxes relevant to Europa (basalheat fluxes between 35–60 mW m–2 in this case), two solu-tions exist for a given heat flux: a thin conductive shell and

a thick convective shell (Mitri and Showman, 2005). Mod-est variations in the heat flux can force the shell to switchbetween these states. This will have important geophysicalconsequences.

Imagine a thin, conductive ice shell, with its basal tem-perature held at the local melting temperature of waterice, which is in steady-state with an enormous basal heatflux. Imagine that this basal heat flux gradually declines intime. Such a system would begin in the upper left cornerof Fig. 1d and would gradually slide down the conductivebranch toward the right as the shell slowly thickened. Whenthe shell reaches 8 km thickness at a basal heat flux of35 mW m–2 (for the parameters in Fig. 1d), a crisis ensues:The critical Rayleigh number is reached and convectioninitiates, but the convection transports far more heat fluxthan is available from below. Thus, the shell cannot con-tinue to thicken while remaining in equilibrium. Instead,rapid thickening occurs until the shell reaches a new equilib-rium thickness of ~15 km at the heat flux of 35 mW m–2.Any continued reductions in the basal heat flux lead tocontinued shell thickening on the convective branch. Con-versely, suppose the shell lies on the convective branch andexperiences a gradually increasing basal heat flux. The shellcan gradually thin, maintaining equilibrium, until reachinga thickness of ~9 km at a basal flux of 60 mW m–2. Again,a crisis ensues: The shell cannot continue to thin whilemaintaining equilibrium with the heat flux available frombelow. Instead, the shell becomes conductive, and thentransports far less heat than is available from below. Melt-ing ensues, which rapidly thins the shell until a new con-ductive equilibrium is attained at a thickness of ~5 km atthe basal flux of 60 mW m–2.

Thus, the heat-flux jump implies that modest variationsin the basal heat flux can lead to large, and geologicallyrapid, changes in the ice shell thickness. These thicknesschanges occur over a timescale (Mitri and Showman, 2005)

LρΔDΔF

τ = ≈ 107 yr (31)

where L ≈ 3 × 105 J kg–1 is the latent heat, ΔD ≈ 10 km isthe thickness change resulting from the conductive-convec-tive transition, and ΔF ≈ 20 mW m–2 is the mismatch influxes between the heat flux transported by the ice shell andthat supplied from below. Importantly for tectonics, thistimescale is much shorter than typical orbital and thermalevolution timescales of 108–109 years.

These rapid changes in shell thickness cause rapidchanges in Europa’s volume, which can lead to stresses upto ~10 MPa and may induce surface fracture (Mitri andShowman, 2005; Nimmo, 2004). Thus, conductive-convec-tive switches may have important implications for europantectonics. The fact that Europa’s heat flux could vary epi-sodically (Hussmann and Spohn, 2004) introduces the pos-sibility that such conductive-convective switches may haveoccurred repeatedly in Europa’s history.


4.3. Convective Heat Flux

As introduced in section 2.3, the relationship betweenthe convective heat flux and the vigor of convection canbe expressed in a relationship of form, Nu = c Raβn. Forconvection in a fluid with a temperature-dependent viscos-ity, the value of c depends on θ (Solomatov, 1995), givingrise to a Ra-Nu relationship of form (cf. Solomatov andMoresi, 2000)

Nu = aθ–αnRaiβn (32)

where a, αn, and βn are constants that depend on n, and Raiis the value of Rayleigh number evaluated at Ti (and addi-tionally, a strain rate of κ/D2 for a non-Newtonian rheology)(Solomatov and Moresi, 2000). The grouped aθ–αn is analo-gous to the constant c used in equation (11) and in Voyager-era convection studies. Scalings between Nu and Rai can beused for both internally heated and basally heated ice shellsprovided Ti is properly estimated in the basally heated case(described in detail by McKinnon, 2006).

For steady convection in a Newtonian fluid at low Ra(for Ra/Racr < 103), Dumoulin et al. (1999) suggest a = 1.99,αn = 1, and βn = 1/5. For vigorous convection, where the ve-locity and temperature field are time-dependent (Ra/Racr >103), Solomatov and Moresi (2000) suggest a = (0.31 + 0.22 n),αn = 2(n + 1)/(n + 2), and βn = n/(n + 2), which for New-tonian ice, where n = 1, give a = 0.53, αn = 4/3, and βn =1/3. Numerical simulations of vigorous convection with amulticomponent rheology for water ice including termsfrom Newtonian diffusion creep and weakly non-NewtonianGBS give a = (0.82 ± 1.69), αn = 1.07 ± 0.19, and βn =0.25 ± 0.02, which are roughly similar to the values for dif-fusion creep alone (Freeman et al., 2006). The applicabil-ity of these relationships to tidally heated ice shells withviscosity-dependent tidal dissipation has not yet been dem-onstrated explicitly. However, it is likely that the coefficientsin the Ra-Nu relationship for tidally heated ice shells willbe similar to those derived for uniform internal heating (e.g.,Solomatov and Moresi, 2000). Because tidal dissipation inthe cold stagnant lid at the surface of the ice shell is negli-gible (Showman and Han, 2004), viscosity-dependent tidalheating does not fundamentally alter the value of θ, the be-havior of the stagnant lid, or the rheological boundary layerbetween the stagnant lid and the well-mixed convective inte-rior of the ice shell, which are the key controls on Nu (Solo-matov and Moresi, 2000; Barr, 2008).

5. CONVECTIVE-DRIVEN RESURFACING

Galileo observations showed that Europa’s mottled ter-rain consists predominantly of chaotic terrain, pits, domes,platforms, irregular uplifts, and lobate features (Carr et al.,1998; Pappalardo et al., 1998; Greeley et al., 1998; Green-berg et al., 1999, 2003). Although several formation mecha-nisms have been suggested, including melt-through of the

icy shell (Greenberg et al., 1999; O’Brien et al., 2002;Melosh et al., 2004) and cryovolcanism (Fagents, 2003),the most common suggestion is that these features formedfrom subsurface convection in the icy shell (Pappalardo etal., 1998; Collins et al., 2000; Head and Pappalardo, 1999;Spaun, 2002; Spaun et al., 2004; Figueredo et al., 2002;Figueredo and Greeley, 2004). More generally, tectonicfeatures on Enceladus, Miranda, Triton, and other moonshave also been suggested to result, directly or indirectly,from subsurface convection (e.g., Nimmo and Pappalardo,2006; Pappalardo et al., 1997; Schenk and Jackson, 1993).Here we review our current understanding of the extent towhich subsurface convection can induce surface tectonicson icy satellites.

Two basic routes exist whereby convection may modifya planetary surface. First, the convective stresses and strainsbelow the lithosphere (associated with the convective tem-perature and motion fields) could directly cause surfacefracture and deformation. Candidates for this type of directmodification include Europa’s chaos, pits, and uplifts;Triton’s cantaloupe terrain, and Miranda’s coronae. Second,the effects of convection on the internal density structureand long-term evolution could lead, indirectly, to noncon-vective stresses that induce surface tectonics. For example,time variation in convective density anomalies could leadto reorientation of the satellite figure relative to the rotationaxis; substantial surface stresses would occur as the rota-tional and tidal bulges shifted across the surface. This maybe relevant to Enceladus (Nimmo and Pappalardo, 2006).Alternately, convection could lead to changes in the thick-ness of the icy shell, and hence in satellite volume, leadingto global tensional or compressional stresses. This mecha-nism is potentially relevant to Europa, Ganymede, and otherbodies (Nimmo, 2004; Mitri and Showman, 2005; 2008a)(see also section 4.2).

Here, we focus on the first mechanism and discuss theextent to which convection can produce surface featuressuch as pits, uplifts, and chaos terrains. This problem hasbeen attacked with a variety of approaches, ranging fromsimplified analytical calculations (Rathbun et al., 1998;Nimmo and Manga, 2002) to full numerical simulations ofthe convection (Sotin et al., 2002; Tobie et al., 2003; Show-man and Han, 2004, 2005; Han and Showman, 2005,2008). We first address the production of topography (pitsand uplifts) and second discuss surface disruption (chaos).

5.1. Pits and Uplifts

A difficulty in explaining large-amplitude topographyand surface disruption by convection is the small expectedconvective stresses on icy satellites (Showman and Han,2004). Thermal-buoyancy stresses associated with convec-tive plumes are σ ~ ραghδT, where ρ, α, and g are density,thermal expansivity, and gravity; δT is the temperature dif-ference between a plume and its surroundings, and h is thevertical height of the plume. The temperature-dependent

420 Europa

viscosity leads to the development of a stagnant lid at thesurface; the convection then occurs in a nearly isothermalsublayer confined below the stagnant lid. Theoretical stud-ies imply that the total viscosity contrast across such a sub-layer is only a factor of ~10 (e.g., Solomatov and Moresi,2000), which for realistic activation energies (Q*) impliesthat convective plumes have temperature constrasts of only~10 K. As emphasized by Showman and Han (2004, 2005),this weak thermal buoyancy leads to small stresses of only~0.01 MPa (Tobie et al., 2003) (see also section 3.1.1). Thedynamic (i.e., convectively generated) topography inducedby such stresses is only σ/ρg ~ 10 m (Nimmo and Manga,2002; Showman and Han, 2004), which is far below the~100–300-m heights of typical europan uplifts. These stressesare also much less than the expected yield stress of ice, sug-gesting that pure thermal convection cannot easily fracturethe surface.

Several authors have performed analytical calculationsof the conditions required to explain the properties of pitsand uplifts by convection. Rathburn et al. (1998) adapteda simple model for the ascent of hot thermal diapirs througha cooler ice shell to study the formation of uplifts in Eu-

ropa’s shell. Based on the fact that diapirs spread laterallyas they impinge against the stagnant lid, Rathburn et al.(1998) suggested that the initial diapirs must have diam-eters of several kilometers or less to explain uplifts ~10 kmacross. They also suggest that, to remain coherent as theyascend, such diapirs must have originated from depths lessthan a few tens of kilometers. Using boundary-layer theory,Nimmo and Manga (2002) carried these arguments furtherby linking such diapiric behavior to the required convectiveproperties of the ice shell. Hot ascending diapirs presum-ably originate in the hot convective boundary layer at thebottom of the ice shell, and experimental results show thatthe initial diapir diameter is ~5 times the thickness of thebottom hot boundary layer (Manga and Weeraratne, 1999).Based on this result, Nimmo and Manga (2002) infer thatthe bottom boundary layer thickness needed to explain 10-km-diameter domes is ~1–2 km. This demands a smallmelting-temperature viscosity of 1012–1013 Pa s, imply-ing ice grain sizes of only 0.02–0.06 mm (see section 3.3).Nimmo and Manga (2002) also suggest that, for such adiapir to induce surface uplift, the stagnant lid thicknessmust be <2–4 km, implying heat fluxes of 100–200 mW m–2.

Fig. 3. See Plate 22. (a),(c) Temperature and (b),(d) dynamic topography from simulations of thermal convection in a 50-km-thickeuropan ice shell from Showman and Han (2004). Rheology is Newtonian with Q* = 60 kJ mol–1, ηo = 1013 Pa s (right), and ηo =1014 Pa s (left), and upper viscosity cutoff of 109 ηo. (a),(b) A high melting point viscosity leads to sluggish convection beneath athick stagnant lid, and topography on the order of tens of meters, much lower than observed on Europa. Domain is 150 km wide and50 km deep. (c),(d) Convection in an ice shell melting point viscosity of 1013 Pa s leads to vigorous convection characterized by nar-row upwellings and a thinner stagnant lid. Domain is 300 km wide by 50 km deep. Although the topography predicted by more vig-orous convection has a smaller wavelength, the plume buoyancy is unchanged between the two cases: the dynamic topography isapproximately tens of meters.


Consistent with the buoyancy arguments described above,their predicted dome heights are only ~5–30 m, far less thanthe observed heights of typical europan domes.

Although the above analytical studies are valuable, theyadopted simplified prescriptions of the dynamics that po-tentially exclude important effects. To determine whetherpit-and-dome-like surface topography can result from thefull convective dynamics, Showman and Han (2004) per-formed two-dimensional numerical simulations using a dif-fusion creep rheology. (We note that use of a non-Newto-nian rheology does not modify the fundamental buoyancyargument described above and is unlikely to, by itself, leadto convection-driven resurfacing.) Showman and Han (2004)found that, in stagnant-lid convection, ascending and de-scending plumes have essentially no surface expression —pits and uplifts do not form (see Fig. 3). This results directlyfrom the extremely small temperature contrasts (~10 K) ofthe ascending and descending plumes. The simulationsdeveloped modest surface topography of ~10–30 m, whichresulted from long-wavelength lateral variations in the thick-ness of the stagnant lid rather than from the plumes under-lying the lid. Showman and Han (2004) found, however,that if the viscosity contrast is Δη ~ 103–105, then the coldice at 1–2-km depth deforms enough to participate in theconvection (Δη is small enough that convection occurs inthe so-called “sluggish lid” regime), leading to formationof 100–300-m-deep pits over the downwellings. However,none of the simulations produced localized uplifts. The sim-ulated pits range in width from 5 to 100 km depending onthe melting-temperature viscosity, thickness of the ice shell,and other properties. Consistent with Nimmo and Manga(2002), Showman and Han (2004) found that explainingobserved pits with diameters of ~5–10 km requires melting-temperature viscosities of ~1012 Pa s or less, implying icegrain sizes of ≤0.04 mm. At these viscosities, the maximumheat flux transportable by convection is ~100–150 mW m–2.Explaining pits less than ~4 km in diameter is extremelydifficult unless the viscosities are unrealistically small.

Runaway tidal heating in hot convective plumes is some-times invoked as a mechanism for enhancing the internaltemperature contrasts, therefore increasing the amplitude ofsurface topography. However, Showman and Han (2004)pointed out that such runaways, if any (see section 5.2.1),cannot significantly enhance the thermal buoyancy in as-cending hot plumes. The mean ice temperature in the con-vective sublayer is only ~10 K less than the temperature atthe bottom of the ice shell, which for Europa is expected tobe the ~260-K melting temperature. Even accounting for thepressure-dependence of the melting temperature, this putsa fundamental limit of ~10–20 K on the maximum tempera-ture difference between ascending hot plumes and the back-ground ice through which they rise: Plumes simply cannotbe heated to temperatures exceeding the melting tempera-ture. Once a hot plume reaches the melting temperature, anyfurther heating will instead cause partial melting, whichwould increase the plume’s density and therefore decrease

its thermal buoyancy — lessening the topographic ampli-tude of any resulting uplifts.

Motivated by the insufficient buoyancy associated withthermal density contrasts, several authors have proposedthat compositional density contrasts are important in gen-erating the large (100–300 m) topography of typical pitsand uplifts (Nimmo et al., 2003; Showman and Han, 2004;Pappalardo and Barr, 2004; Han and Showman, 2005).The most plausible scenario for explaining uplifts is onewhere relatively salt-free (hence low-density) diapirs ascendthrough a saltier, denser environment. In this case the to-pography is ~hΔρ/ρ, where Δρ is the plume-environmentdensity contrast and h is the height of the plume. For aplume 10 km tall, explaining 300-m-tall uplifts would re-quire a plume/environment density difference of ~30 kg m–3,which could occur if the plume-environment salinity dif-ference were ~5–10% (Pappalardo and Barr, 2004), mar-ginally consistent with current estimates of the salinity ofEuropa’s ocean (cf. McKinnon and Zolensky, 2003; Handand Chyba, 2007; see also chapter by Zolotov and Kargel).However, as pointed out by Showman and Han (2004), itis difficult to understand how strong compositional contrastscan be maintained against mixing if the shell is convect-ing. Furthermore, any partial melting in the ice would tendto deplete the ice shell of salts (which percolate down intothe ocean with the melt), so maintaining such compositionaldensity contrasts over long timescales is difficult (Showmanand Han, 2004). Pappalardo and Barr (2004) proposed thatthe compositional convection is a transient process that be-gins with a recent onset of thermal convection and then diesoff as the ice shell becomes depleted in salts. If so, thenthe uplifts would be short-lived and would disappear as theshell became salt-free. However, they also suggested thatdiking from the base of the ice shell might replenish theshell with salts.

Han and Showman (2005) performed two-dimensionalnumerical simulations of thermo-compositional convec-tion to test the qualitative scenario of Pappalardo and Barr(2004). Because grid-based methods can cause an artifi-cial numerical diffusion of the salinity, Han and Showman(2005) treated the salinity using the particle-in-cell method,which allows advection of the salt with essentially no nu-merical diffusion. Following Pappalardo and Barr (2004),they initialized the simulations with a warm salt-poor icelayer underlying a colder, saltier, denser ice layer. In typi-cal simulations, a Rayleigh-Taylor instability developed be-tween the salt-poor and saltier layers, leading to composi-tionally driven diapirs that generated pits and uplifts withtopography of ~300 m or more (see Fig. 4). Because the in-stability involves the relatively cold near-surface ice, it oc-curs over a timescale η0χ/(gδΔρ), where η0 is the melting-temperature viscosity, χ is the assumed viscosity contrastacross the ice shell, δ is the thickness of the salty layer, andΔρ is the density difference between the salty and salt-poorlayers. This leads to pit-and-uplift formation timescales lessthan Europa’s known surface age of 40–90 m.y. (Zahnle et

422 Europa

al., 2003; see chapter by Bierhaus et al.) only for viscositycontrasts <107–108. For viscosity contrasts >108, pits anduplifts cannot form in timescales less than Europa’s surfaceage. The implication is that compositional convection canonly produce Europa’s pits and uplifts if Europa’s surfaceis weak. The simulated pits and uplifts were 10–30 km wide(see Fig. 4); explaining the many pits and uplifts with diam-eters <5 km is difficult.

As described above, matching the observed sizes of pitsand uplifts remains a challenge. Pits and uplifts range indiameter from ~3 to 50 km (Greenberg et al., 2003; Spaun,2002; Rathburn et al., 1998). Based on an early samplingof Galileo images, Pappalardo et al. (1998) suggested thata preferred diameter of ~10 km exists, which Spaun (2002)and Spaun et al. (2004) revised downward to ~4–8 km af-ter performing an exhaustive survey of images that became

available later in the mission. Greenberg et al. (2003) alsoperformed an exhaustive survey and suggested that, whena preferred diameter exists at all, it is ~3 km and reflectsthe limits of image resolution rather than a physical peak.However, these divergent results may reflect differences inanalysis methods rather than an actual discrepancy (seeGoodman et al., 2004, Appendix A; chapter by Collins andNimmo). For our purposes, the main point is that, regard-less of the preferred diameter, many pits and uplifts aresmall, with diameters of 3–10 km. Although convection(with salinity) can plausibly produce the largest of thesefeatures, explaining the smallest (3–5-km-diameter) featuresis difficult. It is plausible that multiple origins exist for pitsand uplifts, with some of the larger features resulting fromconvection and the smallest features resulting from someother process.

Fig. 4. (a) Topography, (b) composition, and (c) temperature for a numerical simulation ofconvection with salinity from Han and Showman (2005). Domain is 45 km wide and 15 kmthick. Black dots in (b) are tracers marking the locations of salt-poor, low-density ice, whichwas initially near the bottom of the ice shell but experiences diapirism. White regions aresalty, denser ice. The topography attains 200–300 m with widths of 15–20 km. This sug-gests that Europa’s widest pits and uplifts can form from convection with salinity.


5.2. Chaos

Based on observations of Europa’s chaotic terrain, sev-eral authors have proposed that chaos results from convec-tion within the icy shell, possibly aided by partial melting(Pappalardo et al., 1998; Head and Pappalardo, 1999;Collins et al., 2000; Figueredo et al., 2002; Spaun, 2002;Schenk and Pappalardo, 2004; Spaun et al., 2004). Herewe review theoretical work that investigates whether con-vection can lead to surface disruption and whether that dis-ruption has the properties of chaos (see also chapter by Col-lins and Nimmo).

5.2.1. Runaway heating in convective plumes. Moti-vated by the suggestion that the disaggregation and tiltingof chaos blocks would be aided by partial melting underthe lithosphere (Collins et al., 2000), several authors haveproposed that runaway tidal heating and localized partialmelting occurs within warm, ascending convective plumes,promoting the formation of chaos (Wang and Stevenson,2000; Sotin et al., 2002; Tobie et al., 2003) (see also chap-ters by Collins and Nimmo and by Sotin et al.). This at-tractive idea is based on the temperature-dependence of theMaxwell model for tidal heating (equation (5)), which pre-dicts that tidal heating increases strongly with temperatureat low temperature (scaling as η–1) and decreases stronglywith temperature at high temperature (scaling as η), peak-ing at intermediate values corresponding to ωτM ~ 1 (at tem-peratures of 220–270 K for melting-temperature viscositiesof 1012–1015 Pa s). The low-temperature behavior promotesrunaway: Increases in temperature enhance the tidal heat-ing rate, leading to further increases in temperature. For icegrain sizes exceeding ~0.5 mm (i.e., melting-temperatureviscosities ≥1014 Pa s), the peak heating occurs at tempera-tures exceeding the melting temperature, which suggeststhat under appropriate conditions this runaway can drivetemperatures all the way to the melting temperature withina localized, ascending warm plume.

While the idea merits further investigation, several pos-sible roadblocks exist. First, calculations by Tobie et al.(2003) and Nimmo and Giese (2005) suggest that meltingthe near-surface ice (at 1–3 km depth) is difficult becauseof the high power generation that is required. Tobie et al.’s(2003) tidally heated convection simulations, for example,produce partial melting only at depths exceeding 10 km,which may be too deep to allow disaggregation of the sur-face.

A potentially more serious issue is that for ice grain sizes<0.5 mm, the Maxwell model (equation (5)) predicts thatthe runaway changes sign near the melting temperature.Tobie et al. (2003) and Mitri and Showman (2005) pointout that if the melting-temperature viscosity is 1013 Pa s orless (implying ice grain sizes of ≤0.1 mm), then for plausi-ble ice-shell temperatures, the greatest tidal heating occursin cold plumes, not the warm plumes (see Fig. 3 of Mitriand Showman, 2005). In this case, warm plumes would beheated less than the background ice, implying a negative

feedback that reduces the thermal contrasts of plumes rela-tive to the background. This would effectively preclude run-away heating in warm plumes. Given the small viscositiesand grain sizes apparently required to explain pits and up-lifts via convection (Nimmo and Manga, 2002; Showmanand Han, 2004), this difficulty is relatively serious. However,the problem might be surmounted in the presence of low-eutectic-temperature contaminants (sulfuric acid or chloridesalts), which could allow melting at sufficiently low tem-peratures for the positive feedback to operate up to the (low-ered) melting temperature. Whether large pockets of meltwould result [sufficient to mobilize the overlying chaosblocks, as suggested by Collins et al. (2000)] would dependon the concentration of the impurities.

Finally, the Maxwell model used in convection studiesto date (Wang and Stevenson, 2002; Sotin et al., 2002; Tobieet al., 2003), which is essentially that of equation (5), isrigorously appropriate to an ice shell that exhibits no lat-eral variation in viscosity (i.e., it is a “zero-dimensional”Maxwell model). However, it is unclear whether this isappropriate for the heterogeneous conditions of Europa’sicy shell. A warm plume surrounded by colder, stiffer icemay exhibit cyclical tidal stress and strain patterns (hencedissipation as a function of temperature) different fromthose used to derive equation (5). Moore (2001) argued thatsmall-scale structures such as convective plumes would notcouple well to the large (hemispheric) scale of the tidal flex-ing, and that, as a result, minimal lateral variation in thetidal heating rate across convective plumes could occur. Ifso, the local runaways envisioned by Wang and Stevenson(2000), Sotin et al. (2002), and Tobie et al. (2003) wouldbe ruled out. More detailed analyses are needed to clarifythis issue.

Mitri and Showman (2008b) revisited this issue with atwo-dimensional model corresponding to a horizontal cross-section through a cylindrical, vertically oriented plume.A Maxwell viscoelastic rheology was adopted; the plumeand environment were allowed to take different viscositiesand elastic parameters. Given an imposed cyclical tidal-flex-ing stress and strain field at infinity, Mitri and Showman(2008b) solved for the stress, strain, and tidal dissipationwithin and surrounding the plume. These calculationsshowed that tidal dissipation does remain strongly tempera-ture-dependent inside a convective plume (even when thebackground temperature is held constant), broadly support-ing the idea that plumes can experience positive (or nega-tive) feedbacks between local temperature and local tidalheating rate. Nevertheless, it would be worthwhile to ex-tend these calculations to three dimensions and explore abroader range of geometries.

In summary, theoretical work to date supports the ideathat tidal dissipation depends strongly on temperature in aconvective plume, but whether the tidal heating in hotplumes is larger or smaller than the background heating ratedepends on the ice grain size. Near the melting tempera-ture, positive feedbacks (runaways) are possible for large

424 Europa

grain sizes, but negative feedbacks occur for small grainsizes.

5.2.2. The difficulty of fracturing the surface. Thermalconvection causes typical stresses of 10–3–10–2 MPa, whichis much smaller than the ~1–3-MPa failure strength ofunfractured ice. This discrepancy raises difficulties for un-derstanding how solid-state convection can induce surfacedisruption.

However, several factors could ameliorate this difficulty.First, Europa’s surface shows abundant evidence that tidalstresses, which reach ~0.1 MPa at Europa’s current eccen-tricity, have fractured the surface (Greenberg et al., 1998;Hoppa et al., 1999). For example, models for the forma-tion of Europa’s cycloidal ridges suggest a fracture yieldstress of just 0.04 MPa (Hoppa et al., 1999). Field studieson the Ross Ice Shelf, one of Earth’s closest analogs toEuropa’s ice shell, also exhibit failure strengths of order0.1 MPa (Kehle, 1964). While not definitive, these studiesare consistent with the idea that Europa’s near-surface iceis weak, but it is not clear whether these estimates are ap-propriate for the failure of Europa’s entire lithosphere. If,for example, cycloids are relatively shallow phenomena, therelatively low stress associated with cycloid propagationmay be relevant to Europa’s near-surface ice only. If Eu-ropa’s band topography forms in a manner similar to ter-restrial mid-ocean ridges (Prockter et al., 2002), the yieldstrength of Europa’s lithosphere at the time and location ofband formation is ~0.4 to 2 MPa (Stempel et al., 2005).Another approach may be to consider the effects of micro-cracking on the rheology of near-surface ice (Tobie et al.,2004). Between a depth of ~15–40 km on Earth, micro-cracks are expected to play a role in accommodating de-formation, and facilitate semibrittle-plastic behavior that isconductive to forming zones of weakness in the crust (Kohl-stedt et al., 1995; Tackley, 2000a; Bercovici, 2003). Fur-ther field characterization of relevant terrestrial analogs andstudies of flexure and failure on Europa constrained by newspacecraft data are needed to shed light upon this issue.

Second, Showman and Han (2005) pointed out thatstresses can become greatly enhanced within a thin “stressboundary layer” near the surface, promoting the likelihoodof surface fracture. This phenomenon results from the needto balance forces in a lithosphere whose width far exceedsits thickness (Melosh, 1977; Fowler, 1985, 1993; McKinnon,1998; Solomatov, 2004a,b). To illustrate, consider a two-dimensional lithosphere with horizontal dimension x andvertical dimension z. Horizontal force balance in the litho-sphere leads to the stress equilibrium condition

∂σxx

∂x= 0+

∂σxz

∂z(33)

where σxx and σxz are the horizontal normal and shearstresses, respectively. The shear stress is zero at the surfacebut due to convection (or other processes) is nonzero in theinterior. Suppose the shear stress at the base of the stressboundary layer is σb. For a stress boundary layer of thick-

ness h that experiences this shear stress over a horizonaldistance L, we have to order of magnitude

Lh

σxx ~ σb (34)

The appropriate value for h is the viscosity scale height inthe lithosphere (Solomatov, 2004a), which is ~1 km for eu-ropan conditions. The interior convective stresses shouldremain coherent over distances L comparable to the inter-plume spacing, which is similar to the ice-shell thickness.Adopting L ~ 20 km, we thus see that stresses can becomeenhanced by a factor of ~20 within the stress boundarylayer. In agreement with this estimate, numerical simula-tions by Showman and Han (2005) show that, althoughconvective stresses within the ice-shell interior are typically~10–3–10–2 MPa, the normal stresses due to thermal con-vection can exceed ~0.1 MPa near the surface.

Third, the stresses that occur during compositional con-vection in a heterogeneous salty ice shell would far exceedthose due to thermal convection alone. For the density con-trasts needed to explain ~300 m-tall uplifts (Δρ ~ 30 kg m–3

over a height range h ~ 10 km), typical convective stressesare ~Δρgh ~ 0.4 MPa. In the presence of a stress boundarylayer, near-surface stresses could be enhanced by an addi-tional order of magnitude or more. These values exceedthose needed to fracture ice. Thus, the idea that convectioncan fracture the surface seems reasonable.

5.2.3. Can convection produce a chaos-like morphol-ogy? To test the hypothesis that convection can cause for-mation of chaos-type terrains, Showman and Han (2005)performed two-dimensional numerical simulations of ther-mal convection in Europa’s ice shell including the effectsof plasticity, which is a continuum representation for de-formation by brittle failure. Plastic deformation occurs whenthe deviatoric stresses reach a specified yield stress σY; atlower stresses, the rheology corresponds to a Newtonian,temperature-dependent viscosity (cf. Trompert and Hansen,1998; Moresi and Solomatov, 1998; Tackley, 2000b). Par-tial melting and salinity were not considered. These simu-lations showed four regimes of behavior depending on theyield stress, thickness of the ice shell, and other parameters.At large yield stresses (≥0.1 MPa), the stresses never attainnecessary values for plastic deformation, and so stagnant-lid convection occurs. At modestly smaller yield stresses(~0.03–0.08 MPa), a thick, cold upper lid remains, but itdeforms via plastic deformation (see Fig. 5). Showman andHan (2005) dubbed this the “pliable lid” regime. Most ofthe plastic deformation is confined near the surface, as aresult of the stress boundary layer. At even smaller yieldstresses (<0.05 MPa), the convection moves away fromstagnant-lid regime, exhibiting either episodic founderingand regrowth of the lid (see Fig. 6) or continual recyclingof the lid.

What is the connection between these simulations andEuropa’s chaotic terrain? The formation of chaos requiresnot only surface fracture but sufficient strains to rotate andtranslate surviving chaos rafts and disaggregate the inter-


vening matrix. Yet the existence of chaos rafts suggests that,in many cases, surface materials remained near the surfaceeven as they were disrupted. Thus, modes of deformationinvolving complete foundering of the upper lid (e.g., Fig. 6)appear not to have occurred on Europa. On the other hand,the so-called “pliable lid” regime of Showman and Han(2005) (Fig. 5) seems to capture key aspects of the observedbehavior. In these simulations, the near-surface strain ratesexceed 10–14 s–1 in localized regions, implying that order-

unity strains would occur on timescales of several millionyears. This is sufficient to disaggregate the surface. Theabsence of foundering in these simulations suggests thatchaos rafts would remain at the surface, as observed. In-terestingly, these high-strain regions were localized, occur-ring in zones only ~5 km wide. This is an encouraging re-sult, although the simulations must be extended to threedimensions to determine whether the disrupted regionswould have quasicircular rather than linear (band-like) mor-

Fig. 5. Simulation of pure thermal convection including plasticity from Showman and Han (2005)with a yield stress of 0.03 MPa. Temperature divided by melting temperature (top), second invariantof strain rate (middle), and surface velocity (bottom). Domain is 45 km wide and 15 km deep. Plasticdeformation occurs in the upper lid, leading to significant surface deformation. This may be relevantto chaos formation on Europa.

426 Europa

Fig. 6. A time sequence (top to bottom) of a simulation of pure thermal convection includ-ing plasticity from Showman and Han (2005) with a yield stress of 0.03 MPa. This simula-tion illustrates necking and overturn of the upper lid, followed by reformation of a cold upperlid by conduction.

phology. However, a difficulty is that the pliable-lid regimeoccurs over only a narrow range of yield stresses; a yieldstress that is slightly too large or small pushes the behav-ior into stagnant-lid or lithospheric-foundering regimes,respectively. Potentially, partial-melting or porosity in thesubsurface could cause a density stratification that would

prevent lithospheric foundering (Collins et al., 2000) andallow the observed behavior to occur over a wider range ofyield stresses.

Although thermal and/or compositional convectionseems to be a viable mechanism for causing at least somechaotic terrain, explaining the specific observed aspects of


Europa’s chaos remains a challenge. A cautionary note isprovided by attempts to simulate Earth’s plate tectonicsfrom first principles, which show that the interaction ofconvection with brittle deformation can depend sensitivelyon the adopted formulation for the brittle rheology (Tackley,2000a,b,c; Bercovici, 2003). One may expect similar sen-sitivity in the interaction of convection with brittle defor-mation on Europa. A full understanding of whether, andhow, convection can cause chaotic terrain will require fu-ture numerical studies that investigate compositional effects,partial melting, and a wider range of brittle rheologies.

6. DISCUSSION

Despite many advances in the knowledge of ice rheol-ogy, the behavior of solid-state convection, and the inter-action between convection and lithospheric deformation,several aspects of the behavior of Europa’s icy shell remainunexplained. Here, we describe several key gaps in knowl-edge about Europa, which are required to address the fun-damental questions about Europa’s icy shell posed in sec-tion 1: Can Europa’s icy shell convect at present? How doestidal dissipation affect convection? Can convection driveresurfacing? What role does compositional heterogeneityplay in driving motion in Europa’s shell?

What is the thickness and thermal structure of Europa’sicy shell? Further spacecraft data are needed to constrainthe true thickness of Europa’s icy shell, to characterize thetopography inferred to form from convection (at both shortand long wavelengths), and to determine the thermal struc-ture of the icy shell (namely, the depth to warm ice). Glo-bal geophysical data obtained by an orbiter equipped with,for example, a laser altimeter, radar sounder, a near-infra-red mapping spectrometer, and high-resolution imagingsystem are needed to answer many of the basic questionsposed above. Such data would also provide constraints formore sophisticated modeling efforts suggested below.

Can convection cause resurfacing? Models published todate provide encouragement that at least some fraction ofEuropa’s pits, uplifts, and chaos could result from convec-tion in the icy shell, but the models are nevertheless far fromexplaining the actual observed properties of these features.Compositional convection allows pits and uplifts with theobserved topography to be produced, but the simulatedfeatures are wider than most of Europa’s pits and upliftsunless ice viscosities are extremely small. Convection mod-els including simple parameterizations of brittle failure canproduce some chaos-like behaviors, but they also producebehaviors that appear not to occur on Europa. A new gen-eration of three-dimensional models including salinity, par-tial melting, and more realistic parameterizations of brittlefailure can help determine whether pits, uplifts, and chaoscan actually result from convection.

How does tidal flexing on Europa affect the microphysi-cal structure and rheological behavior of ice? It is notknown how the cyclical tidal flexing of Europa’s ice shellaffects the ice in its interior. The Maxwell model is perhapsan overly simplistic description of the behavior of Europa’s

ice. Laboratory experiments are needed to clarify how tidaldissipation occurs on a microphysical scale in ice, and toclarify whether cyclical flexing affects ice microstructure.

How does tidal flexing interact with mechanical, thermal,and compositional heterogeneity in the ice shell? Implicitin our discussion of the effects of tidal flexing on heat trans-fer has been the assumption that tidal dissipation is hetero-geneous, and that tidal heating obeys the Maxwell model(equation (5)). The results of laboratory experiments mustbe combined with sophisticated geophysical techniques tostudy the localization of tidal strain and heating in modeleuropan ice shells with thermal, mechanical, and composi-tional heterogeneity to more accurately model tidal dissi-pation and its link to resurfacing.

Acknowledgments. A.C.B. acknowledges support from theSouthwest Research Institute and NASA OPR grant NNG05GI15Gto W. B. McKinnon. A.P.S. acknowledges support from NASAPG&G grant NNX07AR27G. We thank G. Tobie, W. B. McKin-non, and S. Solomatov for helpful comments.

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