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imulationsdynamicagnitude

Icarus 177 (2005) 425–437www.elsevier.com/locate/icaru

Effects of plasticity on convection in an ice shell:Implications for Europa

Adam P. Showman∗, Lijie Han

Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA

Received 30 August 2004; revised 19 February 2005

Available online 26 May 2005

Abstract

Europa’s surface exhibits numerous pits, uplifts, and disrupted chaos terrains that have been suggested to result from convectioshell. To test this hypothesis, we present numerical simulations of convection in an ice shell including the effects of plasticity, whicha simple continuum representation for brittle or semibrittle deformation along discrete fractures. Plastic deformation occurs whereach a specified yield stress; at lower stresses, the fluid flow follows a Newtonian, temperature-dependent viscosity. Four distincbehavior can occur. For yield stresses exceeding∼1 bar, plastic effects are negligible and stagnant-lid convection, with no surface motiominimal topography, results. At intermediate yield stresses, a stagnant lid forms but deforms plastically, leading to surface velocseveral millimeters per year. Slightly smaller yield stresses allow episodic, catastrophic overturns of the upper conductive lid, with ()stagnant lids forming in between overturn events. The smallest yield stresses allow continual recycling of the upper lid, with simugradual ascent of warm ice to the surface and descent of cold, near-surface ice into the interior. The exact yield stresses overregime occurs depend on the ice-shell thickness, melting-temperature viscosity, and activation energy for viscous creep. To form hmatrix and translate chaos plates by several kilometers, substantial surface strain must accompany chaos formation, and the thredominated convection modes described here can provide such deformation. Our simulations suggest that, if yield stresses of∼0.2–1 barare relevant to Europa, then convection in Europa’s ice shell can produce chaos-like structures at the surface. However, our shave difficulty explaining Europa’s numerous pits and uplifts. When plasticity forces the upper lid to participate in the convection,topography of∼50–100-m amplitude results, but the topographic structures generally have diameters of 30–100 km, an order of mwider than typical pits and uplifts. None of our simulations produced isolated pits or uplifts of any diameter. 2005 Elsevier Inc. All rights reserved.

Keywords: Satellites of Jupiter; Europa; Ices; Tectonics; Surfaces; Satellite

ntoain.tledisednu-

ms,

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lt-mal

-;05)in

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1. Introduction

At low resolution, Europa’s surface can be classified itwo terrain types, the ridged plains and the mottled terrHigh-resolution Galileo images demonstrate that the motterrain consists predominantly of chaos regions, comprof hummocky material and disrupted crustal blocks, and

* Corresponding author. Fax: +1 520 621 4933.E-mail address: showman@lpl.arizona.edu(A.P. Showman).

0019-1035/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2005.02.020

merous small (3–30-km-diameter) pits, domes, platforirregular uplifts, and disrupted “microchaos” regions (Fig. 1)(Carr et al., 1998; Greeley et al., 1998, 2000; Pappalaet al., 1998; Greenberg et al., 1999). Debate exists regarding the formation mechanisms of these features.Greenberget al. (1999, 2003)proposed that chaos results from methrough of the ice shell, perhaps driven by oceanic therenergy(Melosh et al., 2004)or concentrations of heat fromthe silicate layer(O’Brien et al., 2002). Some of the features may result from icy volcanism(Greeley et al., 1998Fagents et al., 2000; Fagents, 2003; Miyamoto et al., 20.The most popular model, however, is that convectionthe underlying ice shell deforms the lithosphere, prod

426 A.P. Showman, L. Han / Icarus 177 (2005) 425–437

maraots ttion

,00;Papree-

hecon)

o-de-9)y

ma-elsalmelt;

and-icestsin

ead,ithnthe0–ver,er-eey

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helf,

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y

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rast

ara-0

ksre-aossur-ionslate

ur-s ofons

s.an

delsu-

eastheyersa-

;-

for-er,

efor-somevior

vi-

Fig. 1. Regional Galileo image of Europa’s surface showing the ConaChaos region at upper right and numerous uplifts, depressions, and spthe west and south of the chaos region. North is to the top and illuminais from the right.

ing chaos regions, pits, and domes (e.g.,Pappalardo et al.1998; Head and Pappalardo, 1999; Collins et al., 20Pappalardo and Head, 2001; Spaun, 2002; Schenk andpalardo, 2004; Figueredo et al., 2002; Figueredo and Gley, 2004).

Despite its popularity, quantitative investigations of thypothesis that chaos, pits, and domes resulted fromvection are, as yet, in their infancy.Pappalardo et al. (1998and McKinnon (1999)showed that convection could ptentially occur in an ice shell as thin as 10–20 km,pending on the ice-grain size.Head and Pappalardo (199andCollins et al. (2000)suggested that partial melting mapromote disaggregation of crustal blocks, aiding the fortion of chaos by ice-shell convection. Numerical modof convection in Europa’s ice shell indicate that, if tidheating is temperature dependent, then near-surfaceing may occur above ascending plumes(Sotin et al., 2002Tobie et al., 2003), promoting the formation of chaos.

To determine whether convection can produce pitsuplifts, Showman and Han (2004)performed two-dimensional numerical simulations of convection in Europa’sshell. The simulations showed that for viscosity contra(defined as the ratio of maximum to minimum viscositythe simulations) exceeding 106, a stagnant lid forms at thsurface and isolated pits or uplifts do not form. Instethe convection produces smooth topographic swells wheights<20 m and wavelengths of typically 60 km. Whethe viscosity contrast is 103–105, however, the cold ice a1–2-km depth deforms sufficiently well to participate in tconvection, leading to formation of 10–20-km-wide, 10300-m-deep depressions over the downwellings. Howenone of the simulations produced localized uplifts. Furthmore,Showman and Han (2004)showed that the convectivstresses can reach∼1 bar near the depressions, and thspeculated that disruption of the ice surface could result

o

-

-

-

The disrupted nature of chaos and many pits andlifts (e.g., Pappalardo et al., 1998) indicates that Europa’surface both fractured and deformed during chaos fortion; to explain chaos morphology, the deformation mbe sufficient to translate chaos plates several kilometersbreak up the intervening lithosphere into hummocky ma(Fig. 1). Although laboratory samples of ice generally hatensile strengths of∼10–30 bar, models for the formatioof europan cycloidal ridges require a fracture yield stresEuropa’s surface of just 0.4 bar(Hoppa et al., 1999). Fieldstudies on Earth’s closest Europa analog, the Ross Ice Salso indicate failure strengths of order a bar(Kehle, 1964).These studies suggest that Europa’s surface layer is wand raise the possibility that convection can induce brdeformation at the surface. On Earth, significant lithosphdeformation occurs along macroscopic faults and distribumicrofractures from the surface to∼40 km depth(Kohlstedtet al., 1995; Tackley, 2000a), and such deformation mabe important in allowing plate tectonics(Tackley, 2000b;Gurnis et al., 2000). A number of Earth mantle convetion models now include faults and simple representatof brittle deformation, which can push the convection awfrom stagnant-lid behavior even when the viscosity contis large.

Most of the 3–10-km-diameter plates within ConamChaos (Fig. 1) were translated horizontally from their original positions by 1–5 km and rotated by angles up to 6◦,with an average rotation of 11◦ (Spaun et al., 1998). The in-tervening matrix consists primarily of disorganized blocwith diameters of 0.1–1 km, which indicates that the pexisting lithosphere completely disintegrated during chformation. These characteristics imply that substantialface strain accompanies chaos formation. If chaos regneed 20 Myr to form, then a 2-km translation of a chaos prequires a mean velocity at the surface of 10−4 m year−1

over the interval of chaos formation. Furthermore, the sface velocity must vary substantially across lengthscale30–50 km to explain the distinct translations and rotatiof nearby chaos plates(Spaun et al., 1998). Any model forchaos formation must explain these surface deformation

Despite the importance of brittle deformation for europgeology, however, no published Europa convection mohave yet included brittle rheology. Unfractured ice at Eropa’s 100-K surface temperature has a viscosity at l1010 times that of the warm underlying ice, which, in tabsence of brittle effects, would preclude the surface lafrom participating in the convection and lead to the formtion of a stagnant lid at the surface (e.g.,Solomatov, 1995Moresi and Solomatov, 1995). The use of viscosity contrasts�106 in Showman and Han’s (2004)simulations wasintended as a simple means of allowing near-surface demation under controlled, well-defined conditions. Howevthis scheme cannot capture the expectation that brittle dmation generally occurs only when the stresses exceedthreshold. A simple scheme that incorporates this behais plastic rheology, which allows deformation only for de

Convection with plasticity on Europa 427

retes ish isthand anford

t-lid

onstheof

og-a’s

ua-wly

indiesex-

en-

,

ply

,top

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ep

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99, dy-nes

atoric stresses reaching a specified yield stressσY . Plasticityprovides a continuum representation of flow along discfractures; it is valid when the spacing between fracturemuch less than the lengthscales being modeled, whicprobably appropriate here. For deviatoric stresses lessσY , the temperature-dependent viscosity dominates animmobile stagnant lid would form at the surface, butstresses exceedingσY , plastic deformation would occur, anthe convection would be forced away from the stagnanregime.

Here we present two-dimensional numerical simulatiof solid-state convection in Europa’s ice shell includingeffects of plasticity. Our goal is to determine the effectplasticity on the convection, the resulting dynamic topraphy, and the implications for the formation of Europchaos, pits, and domes.

2. Model and methods

We used the ConMan finite-element code(King et al.,1990)to solve the incompressible (Boussinesq) fluid eqtions neglecting inertia, as appropriate to a viscous, sloconvecting system. The simulations were performedCartesian geometry, which is appropriate for regional stuof Europa’s ice shell because Europa’s radius greatlyceeds the ice-shell thickness.

The code solves the dimensionless equations of momtum, continuity, and energy, respectively given by

(1)∂σij

∂xj

+ Ra θki = 0,

(2)∂ui

∂xi

= 0,

(3)∂θ

∂t+ ui

∂θ

∂xi

= ∂2θ

∂x2i

+ q ′,

whereσij is the stress tensor,ui is velocity,θ is temperatureq ′ is the internal-heating rate,ki is the vertical unit vector,tis time, xi andxj are the spatial coordinates, andi and j

are the coordinate indices. Repeated spatial indices imsummation.

The Rayleigh numberRa is given by

(4)Ra = gρα�T d3

κη0,

whereg is gravity, ρ is density,α is thermal expansivity�T is the temperature drop between the bottom andboundaries,d is the depth of the system,κ is the thermaldiffusivity, and η0 is the dynamic viscosity at the meltintemperature. For reference, the model parameters aresented inTable 1.

The constitutive relationship between stresses and veities, in dimensional form, is given by

(5)σij = −pδij + ηeff

(∂ui

∂x+ ∂uj

∂x

),

j i

-

Table 1Model parameters

Name Symbol Value

Acceleration of gravity g 1.3 m s−2

Density ρ 917 kg m−3

Thermal expansivity α 1.65× 10−4 K−1

Thermal diffusivity κ 1× 10−6 m2 s−1

Specific heat cp 2000 J kg−1 K−1

Temperature contrast �T 175 KMelting-temperature viscosity η0 1013 Pa sThickness of ice shell d 15, 30, or 50 kmRayleigh number Ra 1.16× 107, 9.3× 107,

or 4.3× 108

wherep is pressure,δij is the Kronecker delta, andηeff is theeffective viscosity resulting from thermally activated creand plastic deformation.

The velocity boundary conditions are periodic onsides and free-slip rigid walls on the top and bottom. Ttemperature boundary condition is periodic on the sides.cause Europa’s ice shell is underlain by a liquid-water ocethe temperature at the base must remain at the meltingperature, and so we fix the bottom boundary to a temperaof 270 K. The top surface is maintained at 95 K, 35% ofbasal temperature.

The thickness of Europa’s ice shell is a major unknowwe perform simulations primarily with thicknesses of 130, or 50 km, consistent with estimates of the ice-sthickness from evolution, crater morphology, and crater sdistribution studies(Hussmann et al., 2002; Ojakangas aStevenson, 1989; Turtle and Ivanov, 2002; Schenk, 20.The ice-shell thickness in any given simulation remains cstant throughout the simulation. The simulations wereformed with aspect ratios (ratio of width to depth) rangfrom 3 to 6.

For the viscous creep, we adopt a Newtonian temperadependent viscosity relevant for ice:

(6)η(T ) = η0 exp

[A

(Tm

T− 1

)],

whereT is temperature,Tm is melting temperature (herassumed constant at 270 K) andη0 is the viscosity at themelting temperature. We fixη0 at 1013 Pa s, appropriate tice with grain sizes of∼0.1 mm (Goldsby and Kohlstedt2001). The value ofA is assumed constant at 9.83 or 26,which brackets the expected range for the relevant demation mechanisms at low stresses(Goldsby and Kohlstedt2001). These values imply a viscosity contrast betweentop and bottom boundaries of 7× 107 or 6× 1020, respec-tively.

In Earth mantle convection models, brittle/plastic behior has been incorporated in several ways. One clasmodels explicitly imposes faults within the system andvestigates how the faults interact with the fluid flow (Zhonget al., 1998; Zhong and Gurnis, 1994; Han and Gurnis, 19,and others). These models are useful for exploring geoidnamic topography, and flow geometry at subduction zo

428 A.P. Showman, L. Han / Icarus 177 (2005) 425–437

howlassgiesmi-

chly

sto bwithrhe-dardr-

eter

s-by

withallyt, or

reltstionngith

lau-our

ases

faceuntsere

nanalified

-ula-esro aIn

of

nt of a

be-zero

eg afeighti-ches

pli-less

tiontingses

yieldavior,cludeead-ress

y is

)-ndthe

eua-

te ofruc-

and other plate boundaries, but avoid the question ofthe faults were generated to begin with. A second cof models adopts strain-rate or strain softening rheoloand attempts to self-consistently generate brittle or sebrittle behavior from the simulations (e.g.,Bercovici, 1993;Moresi and Solomatov, 1998; Tackley, 2000a, 2000b; seeBercovici, 2003, for a review). Perhaps the simplest sumodel is plastic rheology, which allows deformation onfor deviatoric stresses exceeding a specified yield stresσY

(and because increases in strain rate are envisionedeasily accommodated by increased slip on fracturesminimal stress increase, the deviatoric stress in plasticology cannot exceed the yield stress). Plasticity is a stanmethod of dealing with brittle/semibrittle flow in numeous studies of crustal rifting on Earth (e.g.,Bassi, 1991;Govers and Wortel, 1995; Bott, 1997, to name a few).

We follow an approach similar to that ofMoresi and Solo-matov (1998)andTackley (2000a, 2000b)for incorporatingplastic rheology into the simulations. The relevant paramis the “effective viscosity,”ηeff, which we define as

(7)ηeff = min

[η(T ),

σY (z)

2ε̇

],

whereη(T ) is the thermally activated creep viscosity (asociated with grain-boundary sliding or diffusion) givenEq. (6), σY is the depth-dependent yield stress, andε̇ is thesecond invariant of the strain rate tensor (ε̇ = √

ε̇ij ε̇ij /2).In the terrestrial lithosphere, the yield stress increasesdepth near the surface following Byerlee’s law but genergives way, in a poorly understood manner, to constanat least more slowly increasing, yield stress at∼20–40 kmdepth (Kohlstedt et al., 1995). The increase of pressu(hence friction) with depth limits slip along discrete fauto the uppermost few kilometers. At depth, the deformaprobably results from volume-conserving deformation alodistributed microfractures. How the yield stress varies wdepth on Europa is unknown; we adopt a simple, yet psible, scheme consistent with the terrestrial behavior. Insimulations, we hold the yield stress constant atσ

deepY within

the bottom 90% of the domain and constant atσdeepY /2 in the

top 5% of the domain. In between, the yield stress increlinearly with depth fromσ

deepY /2 to σ

deepY . This behavior

crudely models a Byerlee-law type behavior near the sur(where yield stress increases with depth) but also accofor the flattening of the strength envelope at depth, whsemibrittle/semiductile flow takes over.

Although stresses within the convective interior in staglid convection in Europa’s ice shell are typically onlyfew hundredths of a bar, the stresses can become ampwithin a thin stress-boundary layer at the surface(Fowler,1985, 1993; Solomatov, 2004a, 2004b), which can promotethe possibility of surface fracture.Fig. 2illustrates the development of a stress-boundary layer for a 15-km-deep simtion performed usingA = 26. Such a boundary layer arisfrom the requirement that the shear stress must be zethe surface but is generally nonzero within the interior.

e

t-

t

Fig. 2. Horizontal normal stressσxx (solid), vertical normal stressσzz

(dashed), and shear stressσxz (dot-dashed) through a vertical transecta simulation in stagnant-lid convection withA = 26 (see Eq.(6)). Negativenormal stresses are compressional. The figure shows the developmestress boundary layer in the top 0.5 km of the domain.

stress equilibrium, a vertical difference in shear stresstweenσb at the base of the stress-boundary layer andat the surface leads to a horizontal normal stress∼Lσb/δ

(Melosh, 1977), where L is the horizontal length of thlithospheric region being stressed (e.g., that underlyinconvection cell, perhaps 10–30 km) andδ is the thickness othe stress-boundary layer (essentially a viscosity scale hin the lithosphere, which is∼1 km under europan condtions). The normal stress at the surface therefore reavalues of up to∼1 bar under europan conditions,∼20-foldgreater than typical stresses within the interior. This amfication can promote surface fracture for yield stressesthan∼1 bar, although we emphasize that finite deforma(e.g., strains approaching unity) is still needed for translachaos plates and forming hummocky matrix. The stresexpected within the stress-boundary layer suggest thatstresses less than 1–2 bar should lead to interesting behwhereas yield stresses exceeding a few bars should prefracture even within the stress-boundary layer (hence ling to stagnant-lid convection). We vary the deep yield stσ

deepY from 0.2–5 bar in our simulations.

The dynamic (i.e., convectively generated) topographcalculated from the relation(Zhong and Gurnis, 1994)

(8)h = σzz

�ρg,

whereh is the dynamic topography,σzz is the (dimensionalvertical normal stress at the surface,�ρ is the density contrast across the interface (here equal to density itself), agis gravity. The surface normal stress is calculated withconsistent boundary flux (CBF) method (seeZhong et al.,1993). Equation(8) is valid whenh is much less than thwavelength of the topography, which is true here. The eqtion neglects elasticity, which may cause an overestimathe topography for short-wavelength, large-amplitude st

Convection with plasticity on Europa 429

osex-so

or aa

ym-eso-al150the

e in-

m-wellap-lder,-

ordenhatthe

nandviorndss

gi-s oribits-lid.s

edsoseds

ndi-on-ursag-muc-itiesingnearrgoe

phy,lation

bar,ula-

h

the

for-

m 0redtan-valsts-rily-

emi-rgo

tures capped by an unfractured lithosphere; however, mof our simulations that produce substantial topographyperience brittle yielding in the region of the topography,Eq.(8) should provide an adequate approximation.

The temperature was initialized as either a constantlinear function of height (i.e., a conductive profile), withweak perturbation near the bottom to break the lateral smetry. The 30- and 50-km-thick cases generally used rlutions of 300× 100 elements in the horizontal and verticdirections, respectively, although a few simulations hador 200 elements vertically. For the 15-km-thick cases,resolution was usually either 300× 50 or 300× 100. Theseresolutions are sufficient to characterize the broad-scalteractions of plasticity with the convection.

3. Results

In a fluid with a viscosity that depends strongly on teperature but is independent of stress or strain-rate, it isknown that stagnant-lid convection results, wherein anproximately isothermal sublayer convects beneath a coimmobile “lithosphere”(Solomatov, 1995; Moresi and Solomatov, 1995). However, the addition of plastic behavicompetes with the tendency of the temperature-depenviscosity to form a stagnant lid. Our simulations indicate tfour distinct modes of behavior can occur depending onice-shell thickness and yield stress.

At the greatest yield stresses (>0.6–1 bar depending othe ice-shell thickness), the plastic effects are minimalstagnant-lid convection results. This first mode of behais illustrated inFig. 3, which shows the temperatures adynamic topography for a simulation with a layer thickneof 30 km, melting-temperature viscosity of 1013 Pa s, anddeep yield stress of 1 bar. Consistent withShowman andHan (2004), we find that the convective plumes have neglible influence on the surface topography; no localized pitdomes form at the surface. Instead, the topography exh∼40-km wavelength swells,∼20 m in amplitude, that correlate with horizontal thickness variations in the stagnantFor yield stresses exceeding∼2 bar, the surface velocitieare independent of yield stress and are<10−13 myear−1 forA = 26. Based on our discussion in the Introduction, speexceeding∼10−4 m year−1 are needed to produce chaover ∼20 Myr timescales; in contrast, the surface speshown inFig. 3are negligible.

At yield stresses of 0.3–0.8 bar, depending on the cotions, the behavior qualitatively resembles stagnant-lid cvection except that significant plastic deformation occat the surface; the lid is pliable rather than “dead” stnant.Fig. 4, which depicts a simulation in a layer 15 kthick, illustrates this behavior. In this simulation, a condtive lid forms at the surface and, as expected, the velocare small in the conductive lid and large in the convectsublayer (first and second panels). Nevertheless, thesurface stresses reach the yield stress and the lid unde

t

t

-s

Fig. 3. Temperature divided by melting temperature, dynamic topograsecond invariant of strain rate, and horizontal surface speed for a simuin a domain 180 km wide and 30 km deep. The deep yield stress is 1which is large enough that stagnant-lid convection results. For the simtion, melting-temperature viscosity is 1013 Pa s,A is 26, and basal Rayleignumber is 9.28× 107.

plastic deformation. At the surface, about two-thirds oflid behaves stiffly, with strain rates less than 10−15 s−1, butthe remaining one-third of the lid undergoes plastic demation with strain rates between 10−15 and 10−13 s−1 (thirdpanel). The horizontal speeds at the surface range froto 4 mm year−1 (fourth panel). Although these speeds a∼500 times slower than the 2 m year−1 peak speeds reachein the convective sublayer, they nevertheless imply substial surface strain over geological timescales. Over interof 106–107 years, the 4 mm year−1 speed differentials athe surface inFig. 4 imply a linear extension or compresion of 4–40 km. The speed differentials occur primaacross regions just∼5 km wide (Fig. 4), and in these regions the linear strain can easily exceed unity over 107-yearintervals. Because plastic rheology represents brittle or sbrittle deformation, these deforming regions would unde

430 A.P. Showman, L. Han / Icarus 177 (2005) 425–437

l ve-for as ofe

f the

theris-

ldnesss toinssbe-

plasely

in apshotsingsity

idndsenssessur-ns.ingentcedur-

anionlas-nis

Fig. 4. Temperature divided by melting temperature, two-dimensionalocity, second invariant of strain rate, and horizontal surface speedsimulation in a domain 45 km wide and 15 km deep with a yield stres0.29 bar and tidal-heating rate of 10−6 W m−3. The longest arrows in thsecond panel correspond to speeds of about 2 m year−1. The convection re-mains in stagnant-lid mode, but plasticity allows surface deformation ostagnant lid. The melting-temperature viscosity is 1013 Pa s,A is 9.83, andbasal Rayleigh number is 1.16× 107.

surface fracture and disruption, potentially producinghummocky material and kilometer-sized plates charactetic of many chaos regions.

In the third mode of behavior, which occurs at yiestresses of 0.1–0.6 bar depending on the ice-shell thicka stagnant lid forms but episodically overturns and sinkthe bottom of the ice shell. This behavior is illustratedFig. 5, which shows a simulation with an ice-shell thickneof 15 km and deep yield stress of 0.29 bar. The overturngins by necking of the upper lid (Fig. 5, first panel), whichexposes warm ice at the surface (second panel). At first,tic deformation is confined to only a few regions, effectiv

,

-

Fig. 5. Temperature divided by melting temperature for a simulationdomain 45 km wide and 15 km deep. The panels show successive snaof a single simulation, increasing in time from top to bottom, illustratnecking and overturning of the upper lid. The melting-temperature viscois 1013 Pa s,A is 9.83, and basal Rayleigh number is 1.16× 107.

dividing the remaining portions of the lid into quasi-rig“plates,” but eventually the entire lid deforms and desceto the bottom of the ice shell (third panel). The fluid thtemporarily becomes gravitationally stable and the stredecrease, allowing a new stagnant lid to form at theface (fourth panel) until a new overturning cycle begiThe process typically repeats at irregular intervals rangfrom 0.4 to 1.5 Myr. The overturns tend to be more frequwhen the yield stress is lower. Any region that experienan overturn of the stagnant lid would be completely resfaced.

In the fourth type of behavior, continual, rather thepisodic, recycling of the upper lid occurs. This deviatfrom stagnant-lid behavior results completely from the ptic rheology.Fig. 6illustrates the behavior with a simulatioidentical to that inFig. 3except that the deep yield stress

Convection with plasticity on Europa 431

phy,lation2 bar

erialra-

-mescon

that

cednessityvec-upofs—sub

lto

r in

heowsntalri-

heree

llicalle-

het lid

tion

m

chex-erally

de-. Forde-eralingfirst

n intionavendaryur-im-Foron-—

ro-pesara

aregesaos.

face

Fig. 6. Temperature divided by melting temperature, dynamic topograsecond invariant of strain rate, and horizontal surface speed for a simuin a domain 180 km wide and 30 km deep. The deep yield stress is 0.and other parameters values are identical to those inFig. 3.

0.2 bar. The persistent descent of cold near-surface matallowed by the plasticity, leads to an equilibrated tempeture in the convecting region only∼75–80% of that at thebase of the system (as compared to∼97% for the same parameters in the stagnant-lid case). As a result, the pluascending from the base of the system have temperaturetrasts, relative to the surrounding ice, about four timesin the stagnant lid case (i.e., ascending plumes inFig. 6 are∼20 K warmer than their surroundings while those inFig. 3are ∼5 K warmer than their surroundings). The enhanbuoyancy of ascending plumes and the extreme thickvariations of the upper lid both contribute to lateral denscontrasts that greatly exceed those in stagnant-lid contion. Dynamic topography with peak-to-peak amplitudesto ∼100 m results (Fig. 6). In these simulations, the shapethe upper lid and the locations of the convective plumehence the shape and amplitude of the topography—vary

,

-

-

stantially in time over 105–106-year intervals. The continuarecycling of the upper lid and ascent of fresh, warm icethe surface implies that complete resurfacing would occuany areas undergoing this type of convection.

Fig. 7 illustrates the magnitude of surface motion for tconvective modes discussed previously. The figure shthe time history of the root-mean-square (rms) horizosurface speed,urms, which provides a measure of the hozontally averaged surface motion and is defined as

(9)urms(t) =(

1

L

L∫0

u(x, t)2 dx

)1/2

,

where u(x, t) is the horizontal speed at the surface,x isthe horizontal coordinate, andL is the width of the do-main. When the convection enters stagnant-lid mode, tis a ∼5 × 105-year-long initial transient during which thstagnant lid forms; afterward,urms ranges from 10−12–10−14 m year−1 for A = 26 (Fig. 7a). These values are smaenough that negligible surface motion occurs over geologtimescales.Fig. 7b shows the surface speed for the pliabstagnant-lid case ofFig. 4, illustrating the∼mmyear−1 mo-tions that occur.Fig. 7c shows the surface speed for tepisodic-overturning convective mode. When a stagnanexists, the rms speed lies between 10−6 and 10−5 m year−1,but the speed gradually increases as plastic deformasets in, culminating in rms surface speeds of∼10 m year−1

during overturn events.Fig. 7d shows the rms speed frothe continual-recycling convective mode ofFig. 6. Substan-tial time variability exists, but the range of speeds is musmaller than that of the episodic-overturning mode. Aspected, the surface never stagnates; rms speeds genrange between 0.1–1 m year−1.

The lithospheric recycling and deformation processesscribed here can help to explain Europa’s chaos terrainschaos to form, the surface must not only fracture butform sufficiently to advect any existing chaos plates sevkilometers and produce hummocky matrix from pre-existlithosphere. These requirements cannot be met by themode of behavior (dead-stagnant-lid convection) showFigs. 3 and 7a, because in that case the surface deformais geologically negligible. (Some of these simulations hstresses that reach the yield stress within the stress-boulayer in upper portion of the lithosphere, implying that sface fracture would occur, but the lack of surface strainplies that chaos cannot result from these simulations.)the latter three regimes, however—pliable stagnant-lid cvection, episodic overturning, and continual overturningthe surface would fracture and deform sufficiently to pduce chaos. The mobile plates that exist within many tyof chaos, including the chaos-terrain archetype, ConamChaos (Fig. 1), provide a constraint. These plates, whichtypically ∼3–10 km across, often contain remnants of ridand other structures that predate the formation of the chTo explain these types of so-called “platy” chaos, the sur

432 A.P. Showman, L. Han / Icarus 177 (2005) 425–437

se-pli-

odes

be-vioror-but

rast

hist isogylas-

andy ofityction

eeda-

scaled

that,then-like

nc-u-

byedde-

ged

seseri-eedsr-

shelllidsls

man-on

Fig. 7. Root-mean-square horizontal surface speed vs time for aries of simulations. (a), (b), (c), and (d) show the dead-stagnant-lid,able-stagnant-lid, episodic-overturning, and continual-overturning millustrated inFigs. 3–6.

must break up but not fully descend into the interior. Thishavior is best explained by the pliable stagnant-lid behashown inFig. 4: the strains associated with plastic defmation are great enough for surface disruption to occur,the disrupted fragments remain at the surface. In contcomplete foundering of the upper lid (as occurs inFigs. 5and 6) would destroy all pre-existing surface landforms; tmight help to explain plate-free chaos terrains. A caveathat our simulations cannot predict the precise morphol(e.g., texture) of the terrains that form as a result of p

,

Fig. 8. (a) Summary of convective behavior vs ice-shell thicknessdeep yield stress for simulations with a melting-temperature viscosit1013 Pa s andA = 9.83. Triangles denote simulations where plasticcauses the upper lid to overturn, circles denote stagnant-lid convewith substantial surface motion (rms speed>10−4 m year−1) and crossesdenote stagnant-lid convection with minimal surface motion (rms sp<10−4 m year−1). Triangles and circles have sufficient surface deformtion to produce chaos. (b) Same, except that simulations have been reto a melting-temperature viscosity of 1014 Pa s.

tic deformation. Nevertheless, our simulations suggestif yield stresses of 0.2–0.8 bar are relevant to Europa,solid-state convection in the ice shell can produce chaosstructures at the surface.

Fig. 8a summarizes the convective behavior as a fution of ice-shell thickness and deep yield stress for simlations with a melting-temperature viscosity of 1013 Pa s.Triangles show simulations that recycled their upper lidsplastic deformation. Circles denote simulations that forma stagnant lid and experienced enough plastic surfaceformation to form chaos (defined here as time-averarms surface speeds>10−4 m year−1, which implies 2-kmsurface translation of chaos plates over 20 Myr). Crosshow simulations that formed a stagnant lid but expenced negligible plastic deformation (defined as rms sp<10−4 myear−1). The diagram indicates that plastic defomation can occur at larger yield stresses when the ice-thickness is greater. 15-km-thick shells recycle theironly at yield stresses<0.3 bar, whereas 50-km-thick shelcan recycle their lids at yield stresses up to∼0.6 bar. Theexact values of these cutoff yield stresses depend on thener in which plasticity is parameterized (i.e., they depend

Convection with plasticity on Europa 433

bly

r to

by

ionsla-mee-,

ly-

ionsty ofkmldg-up

,ressng-estng-i-for

n-s is

-eateld

ls beedMyrex-

peedins

edmo-

ionhinaylifts04)face

ll,a’s

ature

s withudesex-

hoseThe

ached

tom-ersrkmam-ithvelyroust)thenphy

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the type of failure criterion that is used) and are probauncertain by a factor of two. Nevertheless,Fig. 8 illustratesthat plastic deformation can occur at yield stresses similathose needed to form cycloidal ridges(Hoppa et al., 1999),indicating the plausibility of plastic surface deformationconvection in Europa’s ice shell.

The fact that ConMan solves the dimensionless equat(King et al., 1990)allows us to rescale the existing simutions to infer the convective behavior for cases with the saRayleigh number but differing melting-point viscosities, icshell thicknesses, and yield stresses (seeShowman and Han2004, for a description of the method). For example,Fig. 6also applies to a case withη0 = 1014 Pas,d = 64.6 km,width of 387.8 km, and yield stress of 0.435 bar. Apping this rescaling to our simulations, we plot inFig. 8b theexpected convective behavior predicted by our simulatwhen they are rescaled to a melting-temperature viscosi1014 Pa s. The panel illustrates that, for ice shells 100thick, recycling of the conductive lid can occur for yiestresses up to∼1.3 bar. In the stagnant-lid regime, geoloically significant deformation can occur at yield stressesto 1.7 bar. A comparison ofFigs. 8a and 8bsuggests thatat constant ice-shell thickness, the maximum yield stthat allows plastic deformation is larger when the meltitemperature viscosity is greater. This is a relatively modeffect, however; order-of-magnitude increases in meltitemperature viscosity cause∼30% increases in the maxmum yield stress that allows plastic-surface deformationa constant ice-shell thickness.

Fig. 9 illustrates how pliable-stagnant-lid behavior trasitions to dead-stagnant-lid behavior as the yield stresincreased. The figure shows the time-averagedurms (Eq.(9))for a series of simulations at different yield stresses.Figs. 9aand 9b show simulations forA = 9.83 and 26, respectively, and the solid, dashed, and dash-dot curves delinresults for 15-, 30-, and 50-km-thick ice shells. For yiestresses exceeding 1 bar, the rms surface speeds fallow 10−4 m year−1, which is the minimum speed needto translate chaos plates by several kilometers over 20-intervals. Chaos therefore cannot form at yield stressesceeding 1 bar forη0 = 1013 Pas (2 bar forη0 = 1014 Pas).At yield stresses below 1 bar, however, the rms surface srapidly rises to values that allow order-unity surface straover intervals of 107 years or less—as potentially requirto form chaos. Interestingly, the yield stresses needed tobilize the surface are a factor of∼2–3 smaller forA = 26than forA = 9.83. Our models assume thermal convectin pure ice, which leads to relatively small stresses witthe convective interior. The inclusion of salinity, which mbe necessary to explain the topography of pits and up(Pappalardo and Barr, 2004; Showman and Han, 20,would increase the internal stresses and might allow surmobilization at larger yield stresses than shown here.

Our simulations with plasticity do not produce the taisolated uplifts or deep pits that are common on Europsurface. When stagnant-lid convection occurs (Figs. 2–4),

-

Fig. 9. Time-averaged root-mean-square surface speedurms (see Eq.(9))vs deep yield stress for a range of simulations using a melting-temperviscosity of 1013 Pa s. (a) and (b) denote results forA = 9.83 and 26, re-spectively. Solid, dashed, and dot-dashed curves represent simulationice-shell thicknesses of 15, 30, and 50 km, respectively. The plot exclsimulations that exhibited episodic-overturning behavior, which havetremely variable speeds ranging up to 10 m year−1 during overturns. Sim-ulations that fall above the horizontal dotted line at 10−4 m year−1 havesufficient surface strain to produce chaos over 20-Myr intervals, while tbelow the dotted line have insufficient surface strain to explain chaos.calculations of surface speed were performed after the simulations requasi-equilibrium (i.e., after any start-up transients died away).

the surface topography<20 m. Even in simulations tharecycle their upper lids, the topography tends to be dinated by ∼50–100-m-tall structures that have diametof ∼30–100 km (Fig. 6). Although a wide range of othewavelengths occur, including some structures only 5–10across, these features are generally of low topographicplitude (<15 m tall or deep). These results contrast wconceptual and analytical models suggesting that positiand negatively buoyant diapirs can produce the nume5–10-km-diameter pits and domes on Europa(Pappalardo eal., 1998; Rathbun et al., 1998; Nimmo and Manga, 2002. IfEuropa’s ice shell contains substantial amounts of salt,horizontal salinity contrasts may help to produce topograup to∼300 m tall(Pappalardo and Barr, 2004), although itremains unclear even in this case whether the topograwould manifest as localized pits and domes or as lonwavelengths structures such as those inFig. 6. Future simu-lations are needed to address this issue.

434 A.P. Showman, L. Han / Icarus 177 (2005) 425–437

ces

m-on-ethe

uplythato-ter-

t forngs.

p-enles.

duf-

opan.

ingsesup-t-lidonsith

c-n.and

as

inestaosa-ressiorend

malon,thethe

oci-

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ities-

eric

iceanyur--onsu-

medceible,for-al.,ur-sur-yles

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Interestingly, the plastic rheology used here produtopography distinct from that described byShowman andHan (2004). They found that 100–300-m-deep, 10–20-kwide pits formed in a pure ice shell when the viscosity ctrast was low enough (103–105) for the cold, near-surfacice to participate in the convection. In their simulations,deep pits dominated the topography because the coldper (“sluggish”) lid exhibited thickness variations of on10–30% except in dense, 10–20-km-wide downwellingsextended from the sluggish lid to the bottom of the dmain. Because thickness variations in the upper lid demine the topography, the topography was minor excepthe deep pits that occurred over the dense downwelliIn contrast, the plastic rheology used here leads, inFigs. 5and 6, to order-unity variations in the thickness of the uper lid with length scales of 30–100 km; this situation thproduces topography primarily at 30–100 km length scaConsistent with the results described here, none ofShow-man and Han’s (2004)simulations produced tall, isolateuplifts. A key result is that thermal effects alone are insficient to produce topography exceeding∼300 m tall; com-positional effects are therefore necessary to explain eurtopographic structures with amplitudes exceeding 300 m

We find that tidal heating produces a modest inhibiteffect on the plastic deformation, which slightly decreathe maximum yield stress that allows recycling of theper lid to occur. Near the boundary between the stagnanand plastic-convection regimes, we found that simulatiwith no tidal heating exhibited overturning while those wtidal heating rates between 1× 10−7 and 1× 10−6 Wm−3

(for η0 = 1013 Pas) exhibited pliable-stagnant-lid convetion. Figs. 4 and 5provide an example of this phenomenoThese simulations have equal yield stresses (0.29 bar)ice-shell thicknesses (15 km); however,Fig. 5, which exhib-ited overturning behavior, had no tidal heating, whileFig. 4,which was tidally heated at a constant rate of 10−6 W m−3,remained in pliable stagnant-lid mode. Examples suchthose inFigs. 4 and 5appear inFig. 8 as overlapping cir-cles and triangles.

4. Discussion

We performed numerical simulations of convectionEuropa’s ice shell, including the effects of plasticity, to tthe hypothesis that convection can produce Europa’s chpits, and uplifts. The simulations allow plastic deformtion only when the stresses exceed a specified yield stOur simulations show that four distinct modes of behavcan occur. For yield stresses exceeding 0.6–1 bar, deping on the ice-shell thickness, plastic effects are miniand stagnant-lid convection, with negligible surface motiresults. At intermediate yield stresses of 0.3–0.8 bar,convection forms a quasi-stagnant lid that remains atsurface but deforms plastically, leading to surface velties of 0.1–3 mm year−1. Smaller yield stresses of∼0.2–

-

,

.

-

0.6 bar allow episodic, catastrophic overturns of the upconductive lid to occur, with (transient) stagnant lids foring in between overturn events. Still smaller yield stresof ∼0.1–0.2 bar allow continual recycling of the upper lwith simultaneous, gradual ascent of warm ice to the surand descent of cold, near-surface ice into the interior. Indead-stagnant-lid regime, the horizontal surface velocare typically<0.01 mmyear−1 (depending on the viscosity contrast), whereas the surface velocity in the lithosphoverturn/recycling modes can exceed 1 m year−1.

Our simulations suggest that, if yield stresses of∼0.2–1bar are relevant to Europa, then convection in Europa’sshell can produce chaos-like structures at the surface. Mof our simulations experience large-amplitude plastic sface deformation (e.g.,Figs. 4–6). Because plasticity represents brittle or semibrittle deformation, these simulatiimply that extensive surface disruption would occur on Eropa. Although the geological appearance of the deforterrains is difficult to predict, disaggregation of the surfainto chaos plates and hummocky terrain seems plausespecially if partial melting accompanies the plastic demation at depth(Head and Pappalardo, 1999; Collins et2000). The fact that the plastic strain in the deforming sface regions often exceeds unity supports the idea thatface disaggregation (rather than gentler deformation stsuch as normal faulting) could occur.

However, our simulations cannot explain the∼3–10-km-diameter pits and uplifts that pepper much of Europa’sface. When plasticity forces the upper lid to actively paripate in the convection, dynamic topography of∼50–100-mamplitude results, but the topographic structures genehave diameters of 30–100 km, an order of magnitude wthan typical pits and domes. No isolated, tall uplifts or depits ever formed. If convection indeed generated Europits and uplifts, then some physics is clearly missing frour model. The most obvious candidate is that we neglesalinity. Horizontal salinity contrasts could produce dynamtopography up to∼300 m(Pappalardo and Barr, 2004)and,by increasing the background density, may allow domeform more easily. Double-diffusive convection, as mightcur in salty ice, can also produce structures with horizolength scales much smaller than those in pure-ice contion at the same thermal Rayleigh number; therefore, salcould help to set the small widths of pits and domes (asas their heights). Furthermore, the results presentedand inShowman and Han (2004)demonstrate that dynamtopography and convective planform depend sensitivelythe rheology; an exploration of alternative, more exotic rologies may find parameter regimes that produce pit-dome-like morphology.

A difficulty in explaining chaos with lithospheric oveturning/recycling is that the pre-existing surface woulddestroyed, which would prevent the formation of the 1–km-wide plates, containing remnants of older terrain, comon in Conamara and other platy-chaos terrains. Theretwo possible resolutions to this problem. First, platy ch

Convection with plasticity on Europa 435

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may result from plastic deformation in a pliable stagnant(Fig. 4) rather than from complete foundering of the uper conductive lid. Surface fracture would occur, butfractured materials would remain near the surface, alling pre-existing landforms to be retained on the disrupfragments. Second, Europa could have a crust that is cpositionally distinct, and less dense, than the underlyingAlthough the crust would break up during any ice-shell ovturn events, the crustal plates would be buoyant and hunable to descend into the interior of the ice shell duringoverturns. Disrupted surface terrains, including chaos plamight then form during ice-shell overturns. In this scenathe most plausible ice-shell structure consists of a relatisalt-free crust overlying a denser, salty-ice substrate (Pappalardo and Barr, 2004). Exposure of the salty substraduring crustal breakup would also explain the high sulfconcentrations observed in many chaos regions(McCord etal., 1999; Carlson et al., 1999).

Although studies of Europa’s cycloidal ridges(Hoppa etal., 1999)and Earth’s Ross Ice Shelf(Kehle, 1964)suggestthe plausibility of∼1-bar yield stresses, two caveats exFirst, the yield stresses inferred byHoppa et al. (1999)as-sume the cycloidal ridges formed at Europa’s current orbeccentricity, but if they formed during a period of higheccentricity in the past, the inferred yield stress wouldlarger. The history of Europa’s eccentricity is unknown,this caveat is difficult to quantify, although some couporbital-thermal models allow temporal variations in thecentricity by factors of several (e.g.,Hussmann and Spohn2004). Second, several ridge-formation models suggestthe fractures that initiate ridge formation are fluid-filled (e.Greenberg et al., 1998; Melosh and Turtle, 2004), whichcould decrease the yield stress for ridges relative to thamelt-free europan ice. If so, the yield stresses inferredHoppa et al. (1999)for cycloidal ridges would then applonly to local regions and not to broad, melt-free expanof europan ice. However, regions undergoing partial melcould experience a yield-stress reduction analogous tofor fluid-filled fractures, in which case the low yield stressinferred byHoppa et al. (1999)might also apply to any partially molten regions. If so, convective stresses would tallow chaos formation in regions experiencing partial ming (where the yield stress is low) but not elsewhere (whthe yield stress is high). These ideas are consistent withHeadand Pappalardo (1999)andCollins et al. (2000), who arguedthat chaos terrains are best explained by surface disaggtion following near-surface partial melting.

In our simulations exhibiting episodic overturning (Fig. 5),the overturning always involves the entire upper lid. Thefore, if chaos terrains resulted from overturns, our sulations imply that chaos regions should be large, ifglobal; our overturns have difficulty explaining the smsize of many chaos terrains. However, this problem dispears if chaos resulted from plastic deformation of a (noverturning) stagnant or sluggish lid, as shown inFig. 4.When convection takes this form, the surface deforma

,

t

-

(hence chaos regions) can easily be confined to locagions; inFig. 4 the surface deformation primarily occurstwo local regions 5–10 km wide. These widths match thof many “microchaos” regions on Europa(Pappalardo et al.1998; Riley et al., 2000). Alternatively, if the yield stressvaries horizontally, overturns could potentially be confinto regions where the yield stress is lowest; the higher ystress in the surrounding terrains would inhibit overturnthose regions. Finally, we emphasize that our simulatare performed in regional domains only 45–180 km acrwhich is much less than Europa’s radius, so the fact thatoverturns span the entire domain does not necessarily imthat overturns on Europa would be global. Future simutions with spatially varying yield stresses and larger domaare needed to address these issues.

Laboratory experiments show that ice behaves as aNewtonian fluid. Under conditions relevant to Europa’sshell, grain-boundary sliding, for which the strain rateε̇ ∝σ 1.8, where σ is stress, may be the dominant deformtion mechanism(Goldsby and Kohlstedt, 2001; Durham aStern, 2001). This behavior implies that the effective vicosity decreases with increasing stress, although the eis milder than for terrestrial planets. Nevertheless, by aing the stress field, such rheology may influence the main which yielding occurs. Future studies will include noNewtonian rheology and investigate its interaction with btle deformation.

Lastly, the simulations presented here have astrobiolcal implications. If Europa recycles its upper conductivethen any disequilibrium compounds produced at the surfsuch as O2 and H2O2, would be transported into the ice-shinterior and might reach the liquid-water ocean, where tcould influence any biosphere that exists. For yield stresuch as inferred byHoppa et al. (1999), our simulationssuggest that surface materials undergo exchange with thterior on timescales of∼106 years.

Acknowledgments

We thank H.J. Melosh for useful discussions. V. Solomtov and J. Freeman provided helpful reviews. This work wsupported by Grant NNG04GI46G from the NASA PG&program.

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