s

warm,e buoyancysurfacing

kmgraben—the negativepressure

aries: waterthe surfaceter depthsescale ofte alternater through a

Icarus 172 (2004) 625–640www.elsevier.com/locate/icaru

On the resurfacing of Ganymede by liquid–water volcanism

Adam P. Showmana,∗, Ignacio Mosqueirab, James W. Head IIIc

a Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USAb NASA Ames Research Center 245-3, Moffett Field, CA 94035, USA

c Department of Geological Sciences, Brown University, Providence, RI 02912, USA

Received 6 December 2003; revised 19 July 2004

Available online 15 September 2004

Abstract

A long-popular model for producing Ganymede’s bright terrain involves flooding of low-lying graben with liquid water, slush, orsoft ice. The model suffers from major problems, however, including the absence of obvious near-surface heat sources, the negativof liquid water, and the lack of a mechanism for confining the flows to graben floors. We present new models for cryovolcanic reto overcome these difficulties. Tidal heating within an ancient Laplace-like orbital resonance (Showman and Malhotra 1997,Icarus 127, 93;Showman et al., 1997, Icarus 129, 367) provides a plausible heat source and could allow partial melting to occur as shallow as 5–10depth. Our favored mechanism for delivering this water to the surface invokes the fact that topography—such as a global set ofcauses subsurface pressure gradients that can pump water or slush upward onto the floors of topographic lows (graben) despitebuoyancy of the liquid. These eruptions can occur only within the topographic lows; furthermore, as the low areas become full, thegradients disappear and the resurfacing ceases. This provides an explanation for the observed straight dark-bright terrain boundcannot overflow the graben, so resurfacing rarely embays craters or other rough topography. Pure liquid water can be pumped tofrom only 5–10 km depth, but macroscopic bodies of slush ascending within fractures can reach the surface from much greadue to the smaller negative buoyancy of slush. A challenge for these models is the short predicted gravitational relaxation timtopographic features at high heat flows; the resurfacing must occur before the graben topography disappears. We also evaluaresurfacing mechanisms, such as pumping of liquid water to the surface by thermal expansion stresses and buoyant rise of watesilicate-contaminated crust that is denser than liquid water, and conclude that they are unlikely to explain Ganymede’s bright terrain. 2004 Elsevier Inc. All rights reserved.

Keywords: Satellites of Jupiter; Ganymede; Volcanism; Tides, solid body; Surfaces, satellite

ede,t ge-ithre-re-

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tier,

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ter-ands in

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1. Introduction

Voyager and Galileo images have shown that Ganymthe Solar System’s largest satellite, experienced a violenological history. About 35% of Ganymede is covered wancient, heavily cratered dark terrain that superficiallysembles the geologically dead surface of Callisto. Themaining 65%, however, consists of bright terrain with ratively low crater densities that indicates the occurreof a major geological upheaval(Pappalardo et al., 2004Showman and Malhotra, 1999; McKinnon and Parmen

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

0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2004.07.011

1986; Collins et al., 2000), perhaps as recently as 1 Ga a(Zahnle et al., 1998). In most places this bright terrainheavily grooved and tectonized (Fig. 1). Individual resur-faced units within bright terrain are typically∼ 100 kmwide, whereas the grooves within these units are∼ 1–10 kmwide (Shoemaker et al., 1982; Patel et al., 1999). The mech-anisms that replaced the ancient dark terrain with brightrain and deformed it with grooves are poorly understoodrepresent one of the most important unsolved problemicy satellite geology.

The prevailing Voyager-era view was that bright terrformed by flooding of a global set of graben—i.e., low-lyinfault-bounded blocks produced by lithospheric extensiowith liquid water, slush, or warm, soft ice(Golombek and

626 A.P. Showman et al. / Icarus 172 (2004) 625–640

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Allison, 1981; Parmentier et al., 1982; Murchie et al., 198.This model naturally explains the high albedo and low crdensities of bright terrain. Because the cryovolcanic flowould terminate against the fault scarps, it also explainsfact that bright–dark terrain boundaries are sharp andnot embay preexisting topography within the dark terr(Fig. 1). Such cryovolcanism also provides the best explation for observations that some lightly tectonized resurfalanes of width∼ 30–100 km are flat, horizontal, and lie 12 km lower than the surrounding terrain(Schenk et al., 2001Shoemaker et al., 1982)and that young bright-terrain lanesometimes embay ridges and troughs on adjacent obright terrains(Schenk et al., 2001).

High-resolution Galileo images have reopened thebate about bright-terrain formation mechanisms. Althoa few isolated cryovolcanic flow features have been discered(Schenk and Moore, 1995; Head et al., 1998; Schet al., 2001), there is little evidence for ubiquitous vocanic landforms on Ganymede. Furthermore, severagions where intense tectonism caused destruction of unlying craters or bright terrain have been found(Pappalardoand Collins, 1999; Pappalardo et al., 1998). The tectoni-cally resurfaced areas generally exhibit numerous triangridges and troughs of∼ 1-km wavelength, which probabresulted from tilt-block normal faulting that caused thestruction of the pre-existing landforms(Pappalardo et al1998; Pappalardo and Collins, 1999). These observationhave led to the suggestion that tectonic resurfacing maybeen widespread, hence mitigating or removing the ne

r

-

-

sity of cryovolcanism(Head et al., 1997, 2002; Prockte2001; Pappalardo et al., 2004). But many resurfaced terains are relatively smooth, suggesting that they have bonly lightly tectonized (Figs. 1, 2), and tectonic resurfacinhas difficulty explaining these regions unless cryovolcism occurred concurrently with the tectonism (e.g., HarpaSulcus; seeHead et al., 2002). Therefore, a mechanism focryovolcanism is still needed.

However, there are major difficulties in understandhow widespread cryovolcanic resurfacing can occur. Liqwater is denser than ice, so any liquid produced at evenlow depths will percolate downward, away from the surfaand be unavailable for volcanic resurfacing. This contrwith the situation on terrestrialplanets, where silicate meltless dense than the solid and volcanism can occur relateasily. Furthermore, the primary melt-production mecnism on Earth—pressure-release melting associatedadiabatic ascent of mantle material toward the surfaceimpossible on icy satellites. The decrease in ice-melting tperature with increasing pressure instead causesfreezing ofpartially molten ice advected toward the surface. Althothis freezing is incompletefor large melt fractions (∼ 1%freezing occurs for every 10 km of adiabatic ascent), itsures that melt can never be produced by adiabatic aalone, as occurs on terrestrial planets. A strong sourcinternal heating is instead required to produce liquid wnear the surface.

These problems persist even when alternate composiare considered. Ganymede’s ice may contain salts, bu

i

Fig. 1. Regional Galileo view of Ganymede bright and dark terrain. Many patches of the bright terrain are smooth, suggesting volcanism. The image, whch iscentered at 43◦ latitude and 194◦ longitude, covers an area approximately 664 by 518 km at 940 m pixel−1.

Resurfacing of Ganymede 627

npro

Fig. 2. High-resolution Galileo image of grooved terrain in Uruk Sulcus (11◦ N, 170◦ W) spanning 120 by 110 km at 74 m pixel−1. Some regions (A, B)contain numerous ridges and troughs of triangular cross section, which are apparently associated withstrains up to 50%. The tectonic formation of these terrainsapparently destroyed portionsother terrains (C, D), hence tectonically resurfacing them(Pappalardo et al., 1998). In contrast, (D) exhibits horst-and-grabefaulting and (E) exhibits faint fracturing with no obvious macroscopic strain. Removal of pre-existing craters and other landforms in (D) and (E)bably

required volcanic resurfacing acting in concert with the tectonism.

an

ally

wn oft the

iveaceingad-

odsirednd-

cific

ub-ver-er-ro-on

t ter-.

enl-

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plausible compositions, salty liquid water is still denser thsalty ice of the same composition(Kargel, 1991). And al-though a eutectic ammonia–water melt is almost neutrbuoyant relative to ice(Croft et al., 1988), Ganymede isunlikely to have formed with sufficient ammonia to allomassive ammonia–water flooding of the surface. Additiosalts or ammonia can lower the freezing temperature, buphenomenon of pressure-release freezing remains(Hogen-boom et al., 1995, 1997).

A mechanism for producing and transporting massamounts of liquid water, slush, or soft ice onto the surfis a necessary, but not sufficient, condition for explainGanymede’s bright terrain by cryovolcanic processes. Indition, the mechanism must explain how the volcanic flowere confined so completely to graben floors—as requto explain the generally straight dark–bright terrain bou

aries that are nearly universal on Ganymede. Two speconditions must be met:

(1) Although the eruptions must be effusive enough to smerge craters on the graben floors, they must not oflow the graben—otherwise, chaotic bright–dark train boundaries that embay topography would be pduced. Extremely few such features are observedGanymede, however(Allison and Clifford, 1987). Thissuggests the existence of a shut-off mechanism thaminated the eruptions before the graben overflowed

(2) Not only must graben be flooded, but theeruption lo-cations must be predominantly confined to the grabfloors. Any mechanism that allows eruption of cryvocanic materials within topographic highs is inconsistwith observations, because the erupted materials woul

628 A.P. Showman et al. / Icarus 172 (2004) 625–640

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flow downhill toward topographic lows, leaving rilleor other obvious flow markings (which are commonthe terrestrial planets but absent on Ganymede’sterrain). What is needed is a mechanism that allvolcanic eruptions in topographic lows but discourathem within topographic highs.

A buoyant melt could presumably rise to the surface awhere, so one might argue that the observations areexplained with a melt that is denser than the solid(Allison,1982). This argument is incomplete, however, becauseplausible mechanism for transporting large quantitiesdense melt onto Ganymede’s surface has been proposeis it clear why the eruptions would be confined to grabfloors.

Although Ganymede currently experiences negligtidal heating (due to its low orbital eccentricity of 0.0015)Ganymede’s eccentricity may have been high enougthe past for tidal heating and flexing to drive internaltivity. This heating might aid formation of bright terraand explain why Ganymede’s surface differs so dracally from Callisto’s.Showman and Malhotra (1997)andMalhotra (1991)describe the most plausible scenario,which Io, Europa, and Ganymede pass through a Lapllike resonance before evolving into the presently obseLaplace resonance. These Laplace-like resonancespump Ganymede’s eccentricity to∼ 0.01–0.04 dependinon the tidalQ of Ganymede and Jupiter, producing meheating up to∼ 1012 W and tidal flexing stresses upa bar. Showman et al. (1997)coupled an interior modeof Ganymede to the orbital evolution and showed thatancient tidal heating could cause formation of a massive internal ocean, satellite expansion up to∼ 1%, and consequenlithospheric deformation (e.g., production of graben).

For cryovolcanic resurfacing, the key effect of tidal heing is the likely production of near-surface liquid watslush, or warm, soft ice.Figure 3shows the steady-state ecentricities and tidal dissipation rates (expressed as flthrough Ganymede’s surface) for three different resonafrom Showman and Malhotra (1997). If Jupiter’s tidal dis-sipation factor,QJ , is near its time-averaged lower limof 3 × 104 (Goldreich and Soter, 1966), then tidal-heatingfluxes could have reached 80 mW m−2. In a conductiveice shell with a surface temperature of 120 K and a thmal conductivity of 567/T W m−1, appropriate to ice, thesfluxes would imply partial melting at depths as shallow6 km. Even if the time-averaged value ofQJ exceeds 105,orbital-geophysical feedbacks may have allowed tempo(107 year-long), order-of-magnitude increases in the tidheating rate(Showman et al., 1997). QJ may also vary intime (e.g.,Stevenson, 1983; Ioannou and Lindzen, 19Houben et al., 2001), and near-surface partial melting couhave occurred during intervals whenQJ was low. Further-more, if the lithosphere comprises rigid plates separby weak zones, then enhanced tidal heating, and shapartial melting, would occur at these weak zones (e

t

r

-

d

Fig. 3. Steady-state eccentricity and tidal-heating rate, expressed as aper area through the surface, experienced by Ganymede during pathrough a Laplace-like resonance. Horizontal axis is the ratio ofQ′ forGanymede to that for Jupiter, whereQ′ ≡ Q/k, Q is the tidal dissipationfactor, andk is the second-degree Love number. Solid, dashed, and dcurves refer toω1/ω2 ≈ 2, 3/2, and 1/2 Laplace-like resonances, respetively, whereω1 ≡ 2n2 − n1, ω2 ≡ 2n3 − n2, andn1, n2, andn3 are themean motions of Io, Europa, and Ganymede, respectively. AfterShowmanand Malhotra (1997).

Nimmo and Gaidos, 2002). Estimates of the elastic thickness at the edges of grooved terrain, based on flexmodels, leads to heat flux estimates at the time of brightterrain formation of∼ 100 mW m−2, implying thermal gra-dients of ∼ 30 K km−1 (Nimmo et al., 2002). Similarly,models for formation of the pervasive 5–10 km-wavelen“Voyager-scale” grooves from a necking instability can osucceed when the lithospheric thermal gradient exc∼ 20 K km−1 (Collins et al., 1998; Dombard and McKinnon, 2001). The bottom line is that partial melting at deptof 5–10 km is a plausible outcome of an ancient orbitalonance and seems consistent with inferences from sugeology. The question then becomes: how might this maial reach the surface?

In this paper, we evaluate possible mechanisms forovolcanic resurfacing. Our favored resurfacing mechaninvokes the fact that topography, such as a global segraben, induces subsurface pressure gradients that cannegatively-buoyant liquid water or slush upward into

Resurfacing of Ganymede 629

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graben floors. The model naturally confines cryovolcaeruptions to low-lying regions; furthermore, the pressgradients that cause resurfacing cease as the graben fill,ting off the resurfacing before the graben overflow. We aevaluate alternate resurfacing mechanisms, such as pumof liquid water to the surface by thermal expansion stresand buoyant rise of water through a silicate-contaminatecrust that is denser than liquid water, although these manisms are less favorable. We describe the “topograpumping” resurfacing scenario in Section2 and the alternateresurfacing mechanisms in Section3.

2. Topographic pumping of liquid water or slush

2.1. Topographic resurfacing by liquid water

When partial melting occurs, liquid water forms an intconnected network of pores, which allows the liquid toflushed from the solid matrix. Because the liquid water inegatively buoyant, the liquid would percolate downwardthe absence of external effects. However, topography,as a global set of downdropped graben, causes subsustress fields that, under appropriate conditions, can pnegatively buoyant liquid water upward. Here we illustrhow this process can occur.

Melting takes place first along grain boundaries, awe envision that the liquid water produced during tidheating initially exists within an interconnected network omelt-filled pores, at grainboundaries, that constitute ona small fraction of the total volume. Experiments with iindicate that when melting occurs the pores form chnels of triangular cross section at three-grain intersectiThe angle formed by the solid–liquid interfaces at the tipof the triangular channels (the dihedral angle) is ab25◦–40◦ (Ketcham and Hobbs, 1969; Walford et al., 198Walford and Nye, 1991; Mader, 1992), which implies thatthe channels form an interconnected network even for atrarily small melt fractions (e.g.,Stevenson and Scott, 1991).In salty ice, experiments show that the liquid may exisisolated pores (analogous to the bubbles in Swiss cheat melt fractions less than about 5%; melt-filled pores foan interconnected network only at melt fractions exceed5% (Golden et al., 1998). In either case, once an interconected network forms, thepermeability depends stronglyon the melt fraction, increasing as the melt fraction tosecond or third power(Stevenson and Scott, 1991). Oncethe melt fraction—hence permeability—is great enough,melt can be flushed from the matrix in response to mmatrix buoyancy or external forces. On Earth, such meltcolation, which is upward because the melt is buoyant, splies the magma that drives volcanism at mid-ocean ridand elsewhere. Beneath mid-ocean ridges, the balancmelt production and percolation leads to melt fractionsorder∼ 10−2 within the partially molten zone(Turcotte andSchubert, 2002, pp. 402–405).

t-

g

e

)

f

The equations governing the coupled flow of a melt amatrix have been derived by several groups over thetwenty years (e.g.,McKenzie, 1984; Scott, 1988; Stevensand Scott, 1991). The momentum equation governing trelative flow of melt and matrix, which is essentially Dacy’s law, can be written in general form as

(1)f(v

liqi − vsol

i

) = − k

ηliq

[∂pliq

∂xi

+ gρ liqδiz

],

wheref is the melt fraction (equivalent to porosity),k is thepermeability,g is gravity, pliq , ηliq, andρ liq are the pressure, viscosity, and density of the melt,δij is the Kroneckerdelta,vliq

i andvsoli are the true velocities of the melt and m

trix, respectively,xi are the spatial coordinates,z is height,and i = 1, 2, and 3 are the coordinate indices. If the mtrix is rigid, then large melt-matrix pressure differences cbe sustained. At the opposite extreme, if the matrix hasviscosity, then melt-matrix pressure differences will drdeformation that tends to lessen the pressure differenA useful limit to think about is one where this process iseffective that the pressure in the liquid is equal to the psure in the matrix. In this case, the vertical pressure gradin the melt is simply−ρg, whereρ ≡ fρ liq + (1 − f )ρsol

is the mean density of the matrix-melt mixture andρsol isthe density of the matrix. When summed with the weighthe melt, this gives a pressure gradient available for drivmigration equal to the buoyancy of the melt relative tomixture, which for small melt fraction is just−g�ρ, where�ρ ≡ ρ liq − ρsol. In the case of partially molten ice,�ρ ispositive, and the liquid would percolate downward.

However, in reality, the pressure in the liquid and soare not necessarily equal; furthermore, matrix deformacan occur, and this deformation alters the pressure fiethe melt.Scott (1988)shows that Eq.(1) can be written

f(v

liqi − vsol

i

) = k

ηliq

[−(1− f )g�ρδiz

(2)− ∂

∂xj

{(1− f )

(σ sol

ij + pliqδij

)}],

whereσ solij is the stress field within the solid. The expre

sion in square brackets is the pressure gradient availabdrive relative flow between the melt and the matrix. The firsterm is buoyancy and the second term is caused bytrix deformation and melt-fraction gradients(Scott, 1988;Stevenson, 1989).

The stress and velocity fields in the matrix are relatedthe constitutive law, which we take to be that of a viscosolid (e.g.,Scott, 1988):

(1− f )(σ sol

ij + pliqδij

)

(3)=(

ζ − 2

3η

)∂vsol

k

∂xk

δij + η

{∂vsol

i

∂xj

+ ∂vsolj

∂xi

},

where ζ and η are the bulk and shear viscosities of tmatrix. Repeated indices within any term imply summation over the three coordinates. The equations assume

630 A.P. Showman et al. / Icarus 172 (2004) 625–640

ver,s in

n ofoweme,ible.whe

ndcty-atrixd thefor

ond,

lastraduid

ra-warandich

postef-rcehictheselt

ted

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nalandin-ld,andfor

hisim-Ourha-.

the

for7)-eonn-ne.lu-uid

e

-om-s afor

odsast

-ceitepro-d by

ndi-riatere-

re-opo-

in

ohy—

-

n

the solid and liquid are separately incompressible; howewhen the melt fraction is nonzero, compressive stressethe matrix cause compaction of the matrix and expulsiomelt (hence altering the melt fraction). As a result, the fldivergence∂vsol

k /∂xk ≡ ∇ · vsol is generally nonzero. Thbulk viscosity is associated with these changes in voluand it is nonzero because expelling melt requires irreverswork. In general bothζ andη are functions of melt fraction

Using the constitutive law, we can write Darcy’s lasolely in terms of the velocity fields. In vector notation, tequation becomes

f(vliq − vsol) = k

ηliq

[−(1− f )g�ρz

(4)

− ∇{(

ζ + 1

3η

)∇ · vsol

}− ∇(

η∇vsol)],

wherevliq andvsol are the vector velocities of the melt amatrix andz is the upward unit vector. Three factors affethe migration of melt relative to solid: the melt-matrix buoancy, the pressure gradient caused by gradients of mvolume changes due to expansion and compaction, anpressure gradient caused by volume-conserving shear dmation of the matrix. These correspond to the first, secand third terms on the right side of Eq.(4). Matrix deforma-tion affects the pressure gradient in the melt through thetwo terms, and under appropriate circumstances these gents may overcome the negative buoyancy and allow liqwater to rise upward through the ice.

Consider the effects of surface topography. Topogphy causes subsurface stress fields that induce upmotion of the solid ice underneath topographic lowsdownward motion underneath topographic highs, whlessens the topography over time (crater relaxation andglacial rebound being standard examples). The flow isfectively driven by an ahydrostatic pressure-gradient fo−∇(η∇vsol) that points downward beneath topographighs and upward beneath topographic lows. Becausetopographically induced pressure gradients will affect mdrainage (Eq.(4)), the pressure-gradient force associawith matrix flow beneath topographic lowscounteracts thenegative buoyancy of the liquid water. If the ahydrostapressure gradients are strong enough, liquid water cancolate upward in topographic lows.

We illustrate this process with a simple two-dimensioanalytical solution that assumes constant melt-fractionviscosity. In reality, the behavior is complex and nonlear; variations in melt fraction are coupled to the flow fiemelting and freezing can occur, and the permeabilityviscosity depend strongly on the variables being solved(melt fraction and temperature). A realistic solution of tproblem is a major undertaking that requires numerical sulations, which is beyond the scope of the present study.current goal is simply to demonstrate the physical mecnism, and an analytical solution suffices for this purpose

e-

i-

d

-

e

-

When melt fraction is constant and melting is ignored,continuity equation for the matrix becomes

(5)∇ · vsol = 0.

This equation implies that we can define a streamfunctionthe matrix flow,ψ , andSpiegelman and McKenzie (198show that the problem of determiningψ reduces to solution of the biharmonic equation,∇4ψ = 0, under appropriatboundary conditions. In the limit where the melt fractif goes toward zero, the melt has negligible influence othe matrix force balance andvsol is governed by pressuregradient, viscous, and gravity forces in the matrix aloThe velocityvsol is then just determined by a standard sotion for gravitational relaxation of a one-phase, viscous fl(e.g.,Turcotte and Schubert, 2002, Chapter 6). Let the satel-lite interior and surface correspond to an infinite half-spacwith a sinusoidal surface topographyh = h0 cos(mx), wherem is the wavenumber equal to 2π over the wavelength,x ishorizontal distance, andh0m � 1 (the height of the topography is much less than its wavelength). Half-space geetry is approximately valid assuming the topography hawavelength smaller than the ice-shell thickness, which,ice shells∼ 100 km thick, implies wavelengths< 100 km.(Thermal models generally imply that, even during periof tidal heating, Ganymede’s ice shell was probably at le50 km thick (Showman et al., 1997); nevertheless, in future studies it would be worthwhile to relax the half-spaassumption and consider the flow in an ice shell of finthickness.) The sinusoidal topography considered herevides a crude representation of horst and graben formelithospheric extension. We apply a no-slip boundary cotion (horizontal speed equals zero) at the surface, appropbecause a stiff lithosphere overlies the partially moltengion. With this condition, the matrix velocityvsol is givenby (seeTurcotte and Schubert, 2002, pp. 238–240)

vsol = −ρgh0

2ηzemz sin(mx)x

(6)− ρgh0

2ηmemz[1− mz]cos(mx)z,

wherex andz are the horizontal and upward unit vectors,spectively. The pressure gradients associated with the tgraphically induced flow are

−∇psol = −η∇2vsol

(7)= ρgh0memz[sin(mx)x − cos(mx)z

],

wherez is height;z is zero at the surface and negative withthe interior.

Figures 4 and 5illustrate the pressure gradients for twspecific cases. The top panels illustrate the topograpa two-km peak-to-peak-amplitude sinusoid (h0 = −1 km)with 30-km wavelength inFig. 4a and a four-km peak-topeak-amplitude sinusoid (h0 = −2 km) with 60-km wave-length inFig. 5a. The middle panels,Figs. 4b and 5b, show−∇p as predicted by Eq.(7) based on the topography show

Resurfacing of Ganymede 631

hyog-

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rgest,

sure

-indi-owseci-

al tost

e-ped

avee

th aingofthehehon isce

ng

for

Fig. 4. −∇p, wherep is ahydrostatic pressure, for sinusoidal topograpwith 30-km wavelength and 2-km peak-to-peak amplitude. (a) The topraphy. (b) Pressure gradients caused by topography alone. (c) Topographpressure gradients plus the downward-pointing negative buoyancy (�ρg) ofliquid water assuming�ρ = 80 kg m−3. This is the pressure gradient felt bpore–space liquid water. The plot shows that, within topographic lows,uid water can be pumped to the surface from up to 5 km depth. The laarrows have amplitudes of 301 and 416 Pa m−1 in (b) and (c), respectivelyunder the assumption thatρ = 1000 kg m−3 andg = 1.44 m sec−2.

in the top panels. Because material flows down presgradients (i.e., from high to low pressure),−∇p gives ameasure of the direction in which fluid will flow. As expected for viscous relaxation, the pressure gradientscate that the ice flows upward underneath topographic land downward underneath topographic highs. For a spfied wavelength, the pressure gradients are proportiontopographic amplitude. The pressure gradients are greate

Fig. 5. Same asFig. 4, but for sinusoidal topography with 60-km wavlength and 4-km peak-to-peak amplitude. Here, liquid water can be puminto topographic lows from up to 10 km depth. The largest arrows hamplitudes of 301 and 416 Pa m−1 in (b) and (c), respectively, under thassumption thatρ = 1000 kg m−3 andg = 1.44 m sec−2.

near the surface and decay exponentially in depth wiscale height m−1. (This leads to the common understandthat the viscous relaxation of a landform involves motionmaterial primarily at depths comparable to or less thanlandform’s width.) For a given topographic amplitude, tsurface pressure gradients aregreatest for small wavelengtand weakest for large wavelength, but the reverse situatitrue at great depths because ofthe wavelength-dependenof the scale height.

Equation(4) implies that the net pressure gradient drivimotion in pore–space liquid water is that shown inFigs. 4band 5bplus the negative buoyancy of the melt, which

632 A.P. Showman et al. / Icarus 172 (2004) 625–640

nt

ofverr-netliq-relsobe-

graativegra-

po-

acepth

id is

ingped, thehicnic

ntord,tlyncema

bene

ex-

lyt-ich

e

ak-dpec

de-hicith

aveiateor a

f

ardot,raphic

gradi-d

g:y arerme-oughndl as-a-ucedthe-

e-for

de-ugher-um

km

small melt fraction is−�ρgz. This net pressure gradie(equal to the expression in square brackets from Eq.(4)) isshown inFigs. 4c and 5cassuming that�ρ = 80 kg m−3,relevant for pure liquid water and ice. In the absencetopography, the pore–space liquid water experiences atical pressure gradient−�ρg that causes downward pecolation of liquid water. When topography exists, thepressure gradient still causes downward migration ofuid water at great depths, because the topographic-pressugradients are weak at large depths. Liquid water will amigrate downward from any depth—even immediatelylow the surface—underneath topographic highs.

However, underneath topographic lows, the pressuredients caused by the topography can overwhelm the negbuoyancy of the liquid water, resulting in a net pressuredient that drives liquid waterupward into the topographiclows despite the liquid’s negative buoyancy. For the tographic amplitudes illustrated inFigs. 4 and 5, the maximumdepth from which liquid water can be pumped to the surfis 5–10 km. The fact that melting can occur at 5–10 km deduring plausible tidal heating events suggests that liquavailable to be pumped upward to the surface.

The key advantages of this mechanism for explainbright terrain are threefold. First, liquid water can be pumto the surface despite its negative buoyancy. Seconderuption locations are confined solely to the topograplows (graben), so the observed paucity of cryovolcaflow features extending from high-altitude dark terrain ilower-altitude bright terrain is naturally explained. Thias the graben fill with liquid water (which subsequenfreezes into fresh bright terrain), the topography—hepressure gradients—disappear and the resurfacing autoically ceases. Water therefore cannot overflow the graThis provides a natural shut-off mechanism that allows thstraight boundaries between bright and dark terrain to beplained.

For small, and constant, melt fraction, our simple anaical solution predicts that the maximum depth from whliquid water can reach the surface is

(8)dmax= 1

mln

(�ρ

ρ|h0|m)

,

which results from equating�ρg with the vertical pressurgradient atx = 0 from Eq. (7). Figure 6 illustrates thesemaximum depths—as a function of wavelength—for peto-peak sinusoidal topographic amplitudes of 4, 2, 1, an0.5 km (dash–dot, solid, dotted, and dashed lines, restively). It is clear from Eq.(8) that the maximum depth fromwhich liquid water can be pumped to the surface is inpendent of viscosity and gravity. For constant topograpamplitude, the maximum depth of pumping increases wwavelength at small wavelengths and decreases with wlength at large wavelengths, maximizing at an intermedwavelength of 50–100 km. The reason is as follows. Fgiven topographic amplitude, small (e.g.,< 10 km) wave-lengths do not favor pumping:although the amplitudes o

-

-

t-.

-

-

Fig. 6. Maximum depth from which liquid water can be pumped upwversus wavelength of the topography, assumed sinusoidal. Dashed-dsolid, dotted, and dashed curves correspond to peak-to-peak topogamplitudes of 4, 2, 1, and 0.5 km, respectively. Top:�ρ = 0.08 gm cm−3.Bottom:�ρ = 0.04 gm cm−3.

the surface pressure gradients are large, the pressureents decay rapidly with depth,so liquid can only be pumpeif it exists very close to the surface (< 1–2 km depth).Large (> 100 km) wavelengths also do not favor pumpinthe pressure gradients extend to great depth, but theweak because the topographic slopes are small. At intediate wavelengths, the pressure gradients are great ento overcome the negative buoyancy of liquid water, athey also extend to reasonable depths. The top panesumes�ρ = 80 kg m−3 as appropriate for pure ice and wter. Some salt systems of eutectic composition have redliquid–solid density differences, however; for example,ternary MgSO4–Na2SO4–H2O eutectic liquid, which contains ∼ 17% MgSO4 and ∼ 4% Na2SO4 by weight, hasa density only 60 kg m−3 greater than that of the corrsponding solid, and a similar density contrast holdsthe binary eutectic MgSO4–H2O (Kargel, 1991). Further-more, it is possible that silicate contamination helpscrease the average liquid–solid density difference (althothis configuration may be gravitationally unstable if undlain by cleaner ice). The bottom panel shows the maximdepth of pumping assuming that�ρ = 40 kg m−3. In thiscase, liquid water can be pumped upward from 10–20depth.

Resurfacing of Ganymede 633

p-f

30–

ves

og-

ace

ex-f dif-

lateeientut ace

a-the

r adtheidaltng-amtrucths

hor

ingam-

in th

ich

kmba-phy

ter.mus

ablefillhiching–20id–

s onrac-

enbutes

up-is

romrn will

The fact that the horizontal wavelength that favors puming is 50–100 km (Fig. 6) is consistent with the sizes oGanymede’s resurfaced domains, which are typically∼ 100 km wide(Shoemaker et al., 1982). However, it isunlikely that the topography associated with the groothemselves can allow resurfacing. Individual∼ 5–10 km-wide grooves within bright terrain have peak-to-peak topraphy typically ranging from 300–700 m(Squyres, 1981).This topography is insufficient to pump water to the surfunless melting occurs within the uppermost∼ 1 km, whichis unlikely.

Realistic topography for a flat-floored graben can bepressed as a Fourier superposition of many sinusoids ofering wavelengths, and, becauseour analytic solution is lin-ear, the corresponding pressure gradients can be calcufrom Eq.(7). Figure 7shows an example for a 20-km-widgraben 2 km deep. Near the surface, the pressure gradhave greatest magnitude at the edges of the graben, bdepth the greatest pressure gradients are at the grabenter (Fig. 7b). Interestingly, for�ρ = 80 kg m−3, liquid canbe pumped from up to 8 km depth (Fig. 7c), approximately50% deeper than the maximum depth from which liquid wter can reach the surface for sinusoidal topography ofsame amplitude (i.e., 2-km peak-to-peak). Similarly, fo4-km-deep graben that is 50km across, liquid can ascenfrom up to 15 km depth, again about 50% greater thanmaximum pumping depths for the corresponding sinusocase. This enhancement apparently results from the fact thathe Fourier integral representing a graben contains lowavelength sinusoidal components whose peak-to-peakplitudes (when summed) exceed the graben depth; constive and destructive interference with the short-wavelengcomponents then yields the final graben shape (Fig. 7a).Because the pressure gradients associated with the swavelength components decayrapidly with depth (Eq.(7)),the main factor determining the maximum depth of pumpis the long-wavelength components. Therefore the largeplitude of these components causes an enhancementmaximum depth from which liquid can ascend.

What are plausible depths for the graben within whthe bright terrain formed? Using stereo images,Schenk etal. (2001)identified smooth resurfaced areas that are 1–2lower than the surroundings. Viscous relaxation has probly lessened this topography over time, so the topograimmediately after resurfacing would have been even greaTo submerge pre-existing craters, the resurfaced areashave been flooded to depths of∼ 1 km, implying an origi-nal graben depth up to 2–4 km. The mechanism must beto supply liquid to the surface until the final stages of inas the resurfacing ceases. 4- and 2-km topography, ware plausible upper limits for the pre- and post-resurfacgraben depths, allow water to reach the surface from 10and 5–10 km depth, respectively, depending on the liqusolid density contrast.

Consider, to order-of-magnitude, the basic constraintmelt-production rate, permeability, and subsurface melt f

d

st

n-

--

t-

e

t

Fig. 7. Same asFig. 4, but for a 20-km-wide graben 2 km deep. The grabis composed of a superposition of Fourier modes; each mode contrito the total pressure gradient according to Eq.(7). Here, liquid water canbe pumped into the graben from up to 8 km depth if�ρ = 80 kg m−3.The largest arrows have amplitudes of 1090 and 1118 Pa m−1 in (b)and (c), respectively, under the assumption thatρ = 1000 kg m−3 andg = 1.44 m sec−2.

tion imposed by the requirement to form bright terrain. Spose the volumetric tidal heating rate that causes meltingq

and that liquid water pumped to the surface originates fa range of depthsd . If all of the liquid water produced ovethis range of depths reaches the surface, then the grabefill in a time

(9)τfill ∼ hρL

qd,

whereh is the depth to which the graben are flooded,L =3× 105 J kg−1 is the latent heat of melting andρ is ice den-

634 A.P. Showman et al. / Icarus 172 (2004) 625–640

al

port-se

chtion

ing-

r

ualck-he

ve-

up-e ofas0ds

t,etionhe

be

s,ro-they

ef-n iss.

acts

n

tenost

ra-cro-

ofthatr-uallycold

oc-tn isbe-and

owth,uallyt on

-t 5–imece

lat-

etheate

ticaltionde

e.thate, is-ntur-

it

thdctterionse the

sity. Usingd ∼ 10 km, q ∼ 10−6 W m−3, andh ∼ 1 km,then τfill ∼ 106 years. In other words, for reasonable tidheating rates, bright terrain units needed at least 106 yearsto form based purely on the requirement to melt, transto the surface, and then refreeze a∼ 1 km-deep layer of water. In reality, this lower limit may be even larger, becauonly a fraction of the liquid water that is produced will reathe surface and only a fraction of the total heat producwill lead to melting. There is also large uncertainty inq .The above value is appropriate for soft ice if the tidal-flexstrain amplitude is 0.5–1.5× 10−5, relevant to an orbital eccentricity of 0.01–0.02.

If there is a steady-state between the rate of liquid–wateproduction and transport to the surface, then from Eq.(4) therequired permeability is

(10)k ∼ hηliq

τfill

∂z

∂pa

,

where ∂pa/∂z is the ahydrostatic pressure gradient eqto the vertical component of the quantity in square braets in Eq.(4). If the deformation term dominates over tnegative buoyancy, then from Eq.(7), ∂pa/∂z ∼ hρgm. Theviscosity of liquid water isηliq = 0.002 Pa sec. Form ∼6 × 10−5 m−1, corresponding to a horizontal graben walength of 100 km, Eq.(10) implies that

(11)k ∼ 10−15(

106 years

τfill

)m2.

This is the permeability required to transport liquid waterward at the rate it is produced by tidal heating over a rangdepthsd . We can relate this permeability to melt fractionfollows. In real materials with dihedral angles less than 6◦,at small melt fractions, the permeability typically depenon melt fraction as

(12)k ∼ k0a2f 2,

wherea is grain size andk0 ∼ 10−3 (Stevenson and Scot1991). Equating Eqs.(10) and (12), we can therefore solvfor the average steady-state melt fraction if melt producwithin layerd is balanced by percolation of melt toward tsurface:

(13)f ∼ 1

a

(hηliq

k0τfill

∂z

∂pa

)1/2

.

If we again take∂pa/∂z ∼ hρgm, thenf ∼ 10−3(1 mm/a)×(106 years/τfill ). Grain sizes are unknown but may0.1–1 mm(Kirk and Stevenson, 1987)which, forτfill ∼ 106

years, implies thatf ∼ 0.001–0.01. For these melt fractionthe upward percolation rate of liquid water balances its pduction rate under plausible tidal heating. Physically,idea is that the melt fraction self-adjusts to reach a steadstate: if the melt fraction is too low, then percolation is inficient and the melt fraction increases. If the melt fractiotoo high, percolation is rapidand the melt fraction decreaseThe strong dependence of permeability on melt fraction

as a negative feedback that stabilizes the mean melt fractioto a value of 0.001–0.01.

Because the minimum depth of the partially molregions on Ganymede is 5–10 km, and the uppermlithosphere is extremely cold, the microscopic melt migtion we have described must eventually give way to mascopic flow along fractures within the uppermost 5–10 kmGanymede’s lithosphere. This situation is analogous tofor volcanism on Earth: melt is produced in rock at tempeatures exceeding 1000 K, percolates upward, and evententers lenses or fractures, which enables it to reach theupper lithosphere without freezing en route. Exactly howthe transition from microscopic to macroscopic transportcurs is poorly understood.Stevenson (1989)pointed out thaa partially molten region undergoing shear deformatiosubject to an instability that causes melt concentration,cause regions of high melt fraction have low pressurevice versa, which drives melt migration from regions of lmelt fraction to regions of high melt fraction. On Earthis process could produce magma lenses that eventfeed into fractures, and a similar process may be relevanGanymede.

Consider the conditions needed for liquid to reach thesurface without freezing. Suppose that a vertical fracture extends from the surface to a source region of liquid water a10 km depth and that liquid erupts onto the surface for a tτerupt. Over this time, heat will laterally diffuse a distan(κτerupt)

1/2 from the fracture, whereκ ≈ 10−6 m2 sec−1 isthe thermal diffusivity. The average heat flux conductederally out of the fracture walls isF ∼ ρcp�T (κ/τerupt)

1/2,where cp ≈ 2000 J kg−1 K−1 is the specific heat of icand�T ≈ 120 K is the temperature difference betweenwarm fracture and the surrounding cold ice. (This estimincludes only lateral heat conduction and neglects verheat loss from the fracture to the surface. This assumpis valid as long as(κτerupt)

1/2 is less than a few km anτerupt is less than∼ 3 × 105 years.) The heat lost from thfracture will cause the liquid water in the fracture to freezOver the course of the eruption, the volume of waterfreezes, per unit length of the fracture along the surfaccp�T d(κτerupt)

1/2/L, whereL is the latent heat of melting andd is the vertical length of the fracture. If we waa substantial fraction of the liquid water to reach the sface without freezing, then the total eruption volume per unlength of the fracture along the surface,vτerupt, must exceedthe volume of water that freezes. This implies

(14)v � cp�T d

L

(κ

τerupt

)1/2

≈ 0.7

(years

τerupt

)1/2

m2 sec−1,

wherev (m2 sec−1) is the volumetric eruption rate per lengof fracture along the surface andd = 5 km has been usefor the numerical estimate. (v can be viewed as the produof the fracture width and the vertical speed of liquid waascending through the fracture.) Longer-duration eruptrequire smaller effusion rates to prevent freezing becaus

Resurfacing of Ganymede 635

up-dinge,0.7

sith

pertimeendthes

om-h a

ssave

use

ini-ce

e

llssurepo-m

a-

hei-sed

c-fillving-

tes-l

a-meltan,

a

hicfluidthcessmaer-rceom

rategma

ug-,—in

im-ileben

at-

ionnis-the

zeso’sob-

ve;as

als

ainve

x inm8)ding

lax-er-

c

heat flux away from the fracture decreases with time. Ertions lasting 1 and 100 years require effusion rates excee0.7 and 0.07 m2 sec−1, respectively. For a fracture 1 m widthese rates correspond to vertical eruption velocities ofand 0.07 m sec−1.

Notice that the fracture width does not enter Eq.(14) di-rectly; liquid can reach the surface equally well in eruptionwith rapid flow through narrow fractures or eruptions wslow flow through wide fractures, as long asv is sufficientlylarge (all that is required is that the total mass of liquidtime that flows through the fracture exceed the mass perof liquid that freezes; the mass per time that freezes depon eruption duration but not on fracture width). However,fracture width plays an indirectrole because wider fracturemay allow larger effusion ratesv.

The above constraint on individual fractures can be cbined with the global need to resurface bright terrain witthicknessh ∼ 1 km of cryovolcanic flows. SupposeD is theaverage spacing between fractures that erupt, not neceily simultaneously, within bright terrain. Then we must havτerupt/D ∼ h. Given a mean fracture spacing, we canthis constraint to eliminateτerupt from Eq.(14):

(15)

v �(

cp�T d

L

)2κ

hD≈ 2× 10−6

(10 km

D

)m2 sec−1,

where the numerical estimate usesd = 5 km andh = 1 km.For a given fracture spacing, this equation gives the mmum effusion rates that delivers 1 km of liquid to the surfawithout freezing en route. For10-km fracture spacing, thcorresponding eruption times are� 2× 105 years.

We expect that, once negatively buoyant liquid fimacroscopic fractures, the topographically induced presgradients can drive this liquid onto the surface in tographic lows. Envision a fluid-filled fracture extending frothe partially molten region, at∼ 5–10 km depth, to thefree surface. Equation(7) implies that, at the center oftopographic low (i.e.,x = 0), the difference in ahydrostatic pressure between depthd and depth zero, caused by ttopography, isρgh0(1 − e−md). In comparison, the addtional pressure at the base of the fluid-filled fracture, cauby the additional weight per area of the liquid, is�ρgd .Therefore, the pressure withinthe ice at the base of the frature will exceed that of the liquid water in the fracture iρg|h0|(1 − e−md) > �ρgd . Such a pressure difference wcause the ice to deform, compacting the fracture and driliquid water onto the surface. For long wavelengths, this condition can be approximately written as|h0|m > �ρ/ρ. For100 km-wavelengths and�ρ/ρ = 0.08, ascent of the liquidwater in fluid-filled fractures occurs when|h0| � 1.3 km.

Can fluid realistically be delivered to the surface at raexceeding the lower limit of Eq.(15)? Many unresolved issues exist regarding melt segregation on Earth, but analyticaand numerical calculations demonstrate that melt segregtion into lenses and channels is a plausible end result ofmigration (e.g.,Stevenson, 1989; Wiggins and Spiegelm

s

r-

1995). Consider a cylindrically shaped fluid-filled “magmchamber” of diameterdc underlying a fluid-filled fractureextending to the surface within a graben. If the topograppressure gradients overwhelm the negative buoyancy ofinside the fracture, and the magma chamber lies depd

below the surface, then there will be a pressure ex∼ hρgmd between the ice and the fluid inside the magchamber (hereh is the graben depth). This pressure diffence will drive compaction of the magma chamber and fofluid onto the surface. The strain rate in the ice resulting frthis pressure difference ishρgmd/η, whereη is the viscosityof the ice surrounding the magma chamber. This straincan be equated to the fractional rate of change of machamber volume,v/d2

c . This implies that

(16)v ∼ hρgmdd2c

η.

For d ∼ 10 km,h ∼ 1 km, m ∼ 6 × 10−5 m−1, dc ∼ 1 km,andη ∼ 1014 Pa sec,v ∼ 0.01 m2 sec−1. This effusion rateexceeds the lower limit by four orders of magnitude and sgests that, if melt can segregate into∼ 1-km-wide lensesthen the melt can be easily delivered onto the surfacetopographic lows—before freezing.

The fact that warm, soft ice underlies the surfaceplies that the graben will be gravitationally relaxing whthe resurfacing occurs. To produce bright terrain, the gramust fill before they gravitationally relax, implying thτrelax > τfill . Therefore, for liquid–water cryovolcanic production of bright terrain to occur, the gravitational relaxattimescale must exceed 106 years. Gravitational relaxatiohas a contentious history for icy satellites. Early, purely vcous calculations of the relaxation process implied thatrelaxation time for craters is short (< 108 years) even fora cold, isothermal interior (an assumption that maximithe relaxation time), implying that Ganymede and Callistcraters should have all completely relaxed, contrary toservations(Thomas and Schubert, 1988). Studies attemptingto quantify the effects of elasticity on the relaxation haobtained contradictory results(Hillgren and Melosh, 1989Dombard, 2000). To date, the most thorough analysis hbeen performed byDombard and McKinnon (2000)andDombard (2000); they included the latest ice rheologicdata(Goldsby and Kohlstedt, 2001)and plastic effects awell as elasticity.Dombard and McKinnon (2000)showedthat a cold, isothermal icy satellite can indeed mainttopography over 1010 year timescales, helping to resolthe puzzle raised byThomas and Schubert (1988). How-ever, Ganymede thermal models predict that the heat fluabsence of tidal heating should be perhaps a few mW−2

(Kirk and Stevenson, 1987; Mueller and McKinnon, 198.When steep temperature gradients are included extendownward from a surface temperature of 120 K, the reation times are 103–108 years depending on the exact thmal profile and landform diameter(Dombard, 2000). Takenat face value, such short relaxation times are problemati

636 A.P. Showman et al. / Icarus 172 (2004) 625–640

un-owst foree ofasesinsur-

re-nhinn atsitynnel.g.,lon-e re,

fth,en-

aphyntlees

l-al-

on-inity) oran

iledto

vecyuidpth,ucere-in-

ze a——s.uresres-le touits

res-se ithe

rce atth ofpths

l-canpli-

udeativeely-

foratwardast

nceeateronlymelumeumben,

ghiallyentlid

sstrox-r

r

ac-toane,ime,us

atedp-forrps,

for our cryovolcanism model, as well as for a generalderstanding of crater retention under plausible heat flon the icy satellites. However, several mechanisms exislengthening the relaxationtimes, which may help resolvthe discrepancy. First, Dombard assumed a grain siz1 mm, but, as he pointed out, order-of-magnitude increin the grain size can cause order-of-magnitude increasesthe viscosity and hence relaxation time. Second, lowface temperatures, as may exist at high latitudes if thegolith is not strongly insulating, promote longer relaxatiotimes. Third, Dombard’s calculations assumed flow wita deep, constant-density layer; if the ice shell was thithe time of resurfacing, or if Ganymede has a low-dencrust, the near-surface flow would be confined to a chathat would increase the relaxation time substantially (eNimmo and Stevenson, 2001). Fourth, if the high thermagradients occurred only episodically or were spatially cfined to narrow regions underneath the graben (e.g., thsult of runaway heating along fractures;Nimmo and Gaidos2002), then the average thermal gradient affecting relaxationwould be smaller, promoting longer relaxation times. Fidynamically induced topography associated with lateral dsity contrasts can be retained much longer than topogrin a constant-density medium; for example, Earth’s maconvection can produce dynamic topography with lifetimof 108 years or more despite the fact that the postglaciarebound timescale (which is effectively what Dombard cculated) is 104 years. On Ganymede, lateral density ctrasts might be associated with lateral differences in sal(possibly coupled to thermal convection in the ice layersilicate loading of the ice layer (perhaps resulting fromuneven spatial distribution of impact-crater ejecta). Detanumerical studies of the relaxation process will be neededquantify these effects.

2.2. Topographic resurfacing by slush

Although liquid–water resurfacing is attractive, we hapointed out two difficulties: (i) the large negative buoyanof pure liquid water relative to pure ice ensures that liqcan only be pumped to the surface from 5–10 km dewhich in turn requires large tidal heating rates to prodliquid at such shallow depths, and (ii) the gravitationallaxation times are potentially shorter than the inferred mimum time to melt, transport to the surface, and refree∼ 1 km-deep layer of liquid water. Resurfacing by slusha matrix of ice with a few percent or less liquid waterprovides an alternative that helps ameliorate these problemIf crustal extension can produce open extension fract> 100 m wide on graben floors, then the topographic psure gradients associated with the graben may be abdrive macroscopic bodies of slush through these condonto the graben floors.

Slush can be close to neutrally buoyant, so the net psure gradients the slush experiences are close to thoFigs. 4b and 5b. Underneath topographic lows, therefore,

-

n

slush can experience a net upward pressure-gradient fodepths greater than 20 km, depending on the wavelengthe topography. Slush can thus be driven upward from de> 20 km, far exceeding the depth from which liquid watercan easily be pumped. Futhermore, because the slush is amost neutrally buoyant, the depth from which the slushbe pumped is almost independent of the topographic amtude (unlike the liquid–water case, where large-amplittopography was needed to counteract the large negbuoyancy of the liquid). Therefore, graben with a relativmodest depth (e.g.,∼ 1 km) can suffice for driving slush upward.

The gravitational relaxation problem is less severeslush resurfacing than liquid–water resurfacing. The fact ththe topographic pressure gradients can drive slush upfrom greater depth than liquid water implies that, at leinitially, the vertical thermal gradientdT/dz can be lowerfor slush resurfacing than for liquid–water resurfacing (simelting temperature is reached at∼ 20 km in the former casbut at 5–10 km-depth in the latter case). This allows gregravitational relaxation time scales. Furthermore, sincea few percent of the mass of slush is liquid, a given voluof slush can be produced more easily than the same voof liquid. Under plausible tidal heating rates, the minimtimescale needed to produce enough slush to fill the graτfill , may then be∼ 104–105 years rather than 106 years asfor the liquid case.

The key problem with slush is how to transport it throuthe uppermost cold lithosphere to the surface. Partmolten rocks and ice with melt fractions of a few perchave viscosities typically ten times lower than that of sorock or ice at the melting temperature(Kohlstedt and Zim-merman, 1996; De La Chapelle et al., 1999). Therefore theslush viscosity is still large,∼ 1012–1013 Pa sec, and thuthe conduits through which slush reaches the surface mube wide. The pressure gradient driving the slush is appimately ρghm (Eq. (7)), which implies a driving force pefracture-wall areaρghml, wherel is the width of the fracturethrough which slush ascends tothe surface. This force pearea is resisted by the viscous shear stress∼ ηw/l, wherew

is the vertical ascent velocity of the slush through the frture. Balancing these forcesand equating the ascent timed/w, whered is the vertical length of the fracture, givesascent time ofηd/(ρghml2). For slush to reach the surfacthe ascent time must be less than the thermal diffusion tl2/κ ; otherwise, the slush will cool, stiffen, and stall. Thslush can reach the surface only if

(17)l �(

ηdκ

ρghm

)1/4

,

which is∼ 102 m for slush viscosities of 1012–1013 Pa sec. Itremains unclear whether open conduits> 102 m wide wouldactually form; the extension might instead be accomodby slip along normal faults, effectively precluding the uward transport of slush. Additional issues also remain;example, the volcanism might produce km-tall lobate sca

Resurfacing of Ganymede 637

oww

theande ar—onceiredr the

ng, thetd b

t.er--86)srel-on-

teyed

di-and

sizeibu-dal

ese

tingpri-antac-ulduble

r di-rom

het

-sssthller

nedThe

ies of

The

f

ctorjectaeadity of

allther)

eded. Inma-

essed by

der.

ted,cta, al-oflu-

ionich

imitn,uffi-to

ally

finedov-e

which are not observed; furthermore, it is not obvious hlarge masses of slush will transition from diapiric-type floin the deep subsurface to flow along fractures withinlithosphere. But the mechanism warrants further study,we emphasize that despite the above challenges the samvantages exist as for topographic pumping of liquid wateeruptions are confined to topographic lows and ceasethe graben fill. Detailed numerical simulations are requto evaluate the mechanism further, and we leave these fofuture.

3. Alternate resurfacing mechanisms

3.1. Buoyant migration of liquid water to the surface

If the lithosphere above the minimum depth of meltihas an average density greater than that of liquid waterliquid can migrate to the surface, helping to produce brighterrain. Such a dense upper lithosphere may be causesilicate contamination from asteroidal or cometary impacGalileo gravity data imply that Ganymede is strongly diffentiated at present(Anderson et al., 1996), and thermal models(Schubert et al., 1981; McKinnon and Parmentier, 19suggest at least partial differentiation during accretion. Thiprocess would flush the surface layers of rock, leaving aatively pure ice lithosphere. Subsequent impacts would ctaminate the upper lithosphere with silicates (e.g.,Prockteret al., 1998). Our goal is to calculate the lithospheric silicamass loading by considering the impact histories displain the cratering record.

We start with crater statistics fromStrom et al. (1981),who presented crater counts as a function of craterameter for diameters exceeding 4 km on Ganymede11 km on Callisto. As yet, no updates to these globaldistributions have been published, although size distrtions for small craters(Prockter et al., 1998; Ivanov anBasilevsky, 2002)and photogeological studies of individucraters(Greeley et al., 2001; Schenk and Ridolfi, 2002)haverecently been published using Galileo data. Interestingly, thStrom et al. (1981)crater densities on Callisto exceed thoin Ganymede’s dark terrain by a factor of 2–3, suggesthat even Ganymede’s most ancient surfaces are notmordial. Callisto may therefore preserve the more relevrecord of total bombardment in the jovian system. Tocount for the fact that Jupiter’s gravitational focusing sholead to more impacts on Ganymede than Callisto, we doCallisto’s crater density for use with Ganymede.

For each crater size, we calculate the transient crateameter from the observed diameter using the relation fMcKinnon and Schenk (1995):

(18)Dtr = 0.864

(Df

km

)0.9025

km,

whereDtr andDf are the transient and final diameters. Ttransient crater volumeV is calculated from the transien

d-

y

diameter assuming the transientcrater is a paraboloid of revolution with a diameter 2

√2 times its depth, which implie

thatV = πD3tr/(16

√2). We then compute the impactor ma

m using Schmidt–Holsapple gravity scaling. Although bostrength and gravity effects are important for craters smathan 20-km diameter, the majority of the mass is contaiin the largest impactors, where gravity scaling suffices.impactor mass is then given by

(19)m = ρ

(V

K

)3/(3−α)[ g

u2

(3

4π

)]3α/(3−α)

,

whereV is the transient crater volume,ρ is target density,uis impact speed,g is surface gravity, andK andα are dimen-sionless parameters that depend on the material propertthe target. We use an impactor speed of 15 km s−1, the es-cape velocity of Jupiter at the distance of Ganymede.adopted material parameters,K = 0.2 andα = 0.65, are ap-propriate to a soft rock target (seeHolsapple, 1993) and areprobably relevant for ice, although we also test values oK

varying by a factor of∼ 2 andα varying by∼ 30%, sincethe uncertainties are of this order. We then sum the impamasses over all observed craters. By assuming that the eis evenly mixed to a depth of several km and evenly spraround the globe, we can calculate the increase in densthe uppermost lithosphere.

The calculations show that for crater diameters∼ 5–103 km, mass flux is dominated by the largest impacts. Incases, craters less than 256 km in diameter (taken togeproduce a density increase only a few percent of that neto allow liquid to reach the surface from several km depthall cases, the largest impactors also provide insufficientterial to allow liquid to reach the surface (by a factor∼ 10for nominalK andα). UsingK = 0.1 andα = 0.5 allowslarger impactors for a given crater size, but still providinsufficient silicate loading.If much of the impactor masescapes back to space during the impacts, as suggesthydrodynamic modeling(Pierazzo and Chyba, 2002), thenthe gap between the actual contamination and that needeto allow buoyant ascent of liquid is exacerbated still furthHowever, we assumed the material is globally distribuwhich is unlikely: material is probably dropped into an ejeblanket near the crater, rather than globally. Furthermorethough Callisto is geometrically unsaturated (its density> 30 km-diameter craters is much less than that in thenar highlands), an equilibrium could exist wherein formatof new craters causes obliteration of older craters, in whcase the adopted crater density provides only a lower lon the mass that has impactedthe surface. Possibly, thesome local ejecta blankets or other regions could have scient silicate loading to allow buoyant migration of waterthe surface from a few km depth.

However, bright–dark terrain boundaries are genersmooth and curvilinear over∼ 102–103 km distances. Thisobservation strongly suggests that resurfacing was conto a global system of graben. Lithospheric regions cered by thick, dense ejecta blankets would tend to becom

638 A.P. Showman et al. / Icarus 172 (2004) 625–640

benm-nd

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graben rather than horst, facilitating resurfacing on grafloors. However, very few regions where resurfacing ebayed craters and other topography have been found, ais doubtful that resurfacing by this mechanism would hoccurredonly at the graben floors, and would never haspilled over the top of graben.

3.2. Pumping of water to surface by compressive stress

When tidal heating occurs, spatial variations in the hing rate may cause spatial temperature variations, anddifferential thermal expansion associated with these temature variations can produce stresses available for pumliquid water to the surface. In the most extreme case, amal runaway occurs in a confined region that is fully encain cold, elastic ice. The thermal expansion that occurs duwarming will produce a compressive stress, which maylow cracks to penetrate to the surface and liquid water to bpumped to the surface. To order of magnitude, the presincrease within the region undergoing runaway is

(20)�p ∼ Kα�T,

whereK is the bulk modulus,α is thermal expansivity, an�T is the temperature increase occurring during theaway. For probable values (K ∼ 1010 Pa, α ∼ 10−4 K−1,and�T ∼ 50 K), this implies�p ∼ 500 bars. In practicaterms, the cold ice surrounding the runaway will fractonce the overpressurization reaches the 10–100-barstrength of the ice, so the maximum overpressurizatioactually 10–100 bars (or even less if the ice is already ftured). The maximum amount of water that can be pumto the surface is that which,because of the compaction asociated with mass loss, reduces the pressure increase�p tozero. If water pools on the surface in a linear region of wiL and depthh, and if the runaway occurs within a linear rgion a region of width and vertical extentl, then the depth oliquid water that reaches the surface is roughly

(21)h ∼ l2�p

KL.

If we assume the water pools only over the runaway (l ∼ L), let l ∼ 10 km, and use�p ∼ 10–100 bars, thenh ∼1–10 m. If instead we require the water to pool over a reg102 km wide (roughly the size of individual bright-terraunits), we geth ∼ 0.1–1 m. These are upper limit, for twreasons. First, much of the liquid may be pushed down rathan up. Second, the runaway can only lead to overpreization if it is fully encased within cold, elastic ice, but thcondition is severe and may not be achieved on GanymThe cold ice must be elastic on the timescale of the runawhich implies that the viscosityη > Kτrunaway. For a run-away time scale of 106 years, this requiresη > 1023 Pa sec,implying extremely low temperatures in the background(perhaps< 150 K). This condition is unlikely to be saisfied at depths greater than a few km even for mothermal gradients. The estimate in Eq.(21) is therefore an

it

-

.

upper limit. These amounts of liquid are significantly lethan the∼ 1 km inferred thickness of the resurfacedgions. Thus, although this process may have occurreGanymede, it is unlikely to explain bright terrain formtion.

4. Conclusions

Although cryovolcanic flooding of low-lying graben hbeen a long-favored model for producing Ganymede’s brterrain(Parmentier et al., 1982), there have been no plausble mechanisms to explain how vast quantities of matewere transported to the surface, why the eruptions werefined to graben floors, and why such floods did not overflthe graben. Here we presented models to overcomeproblems. In a global and time-averaged sense, tidal hing within an ancient Laplace-like resonance can causheat flux between 10–100 mW m−2, and even larger fluxemay occur episodically via geophysical feedbacks andvariability in the orbital history. Episodic melting at deptof 5–10 km is a plausible outcome of such a resonaThe tidal heating can produce a global subsurface oseveral hundred km thick, which allows the satellite topand by∼ 1% and provides a plausible mechanismproducing graben(Showman et al., 1997). The topographyassociated with these graben generates subsurface prgradients that can drive upward Darcy flow of subsurfpore–space liquid water beneath topographic lows dethe negative buoyancy of the liquid. This provides a reatic way of resurfacing Ganymede with liquid water. Onon the surface, the tops of the liquid–water flows wofreeze, providing an insulating crust and lengthening coolintimes enough for the flows to fill the graben floors(Allisonand Clifford, 1987). For plausible topographic amplitudesand ice–liquid density contrasts, the maximum depth fwhich liquid can be pumped to the surface is 10–20Because the Darcy flow is only upward underneath tographic lows, the model can explain why eruptions wconfined to graben floors (rather than occurring on adjacehigh-standing dark terrain) and ceased once the grabenfilled. The model therefore provides an attractive explation for the generally straight bright–dark terrain boundaand the absence of volcanic flow features across darkrain. A possible problem is that at least 106 years is re-quired to generate and transport to the surface sufficliquid water to submerge craters and other landforms,recent studies(Dombard, 2000)suggest that the graben mgravitationally relax before they can flood sufficiently. Trelaxation time depends on several poorly known paraters, however, and mechanisms exist that will allow lonrelaxation times; more detailed calculations are neederesolve this issue for our model. Resurfacing by slushvides an alternative that helps overcome some of theficulties associated with liquid–water volcanism, althougit is unclear whether wide enough lithospheric cond

Resurfacing of Ganymede 639

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existed for subsurface slush to actually get onto theface.

We also considered alternate mechanisms for transportinliquid water to the surface from∼ 10-km depths, including pumping by differential thermal expansion stressesbuoyant rise of liquid water to the surface through a silicaladen lithosphere that is denser than liquid water. Eacthese mechanisms has problems, however, leading usvor the “topographic pumping” mechanism.

The topography associated with craters also produsubsurface pressure gradients, and although some cfloors contain cryvolcanic flow-like features(Schenk andMoore, 1995), the general lack of bright-terrain patchesthe floors of dark-terrain craters needs explaining. A naural explanation is that newly formed graben were wzones, causing enhanced tidal flexing and heating and aing shallow melting underneath graben but not undernmost craters. This would allow topographic pumping of luid water onto graben floors while, underneath most crano liquid water existed at the appropriate depths (5–20for upward pumping of liquid water to occur.

The bright terrain formed over an extended∼ 108 yearperiod perhaps 1 to 3 byr ago(Shoemaker et al., 1982Zahnle et al., 1998), and relatively little, if any, cryovol-canic activity seems to have occurred since that time. Because the Laplace resonance does not pump Ganymeccentricity, Ganymede would naturally become dormafter the Laplace resonance was established, so the tnation of activity was probably a direct result of the obital evolution. What controlled the length of the resurfacepisode is less clear, but it may reflect the duration ofLaplace-like resonance that pumped Ganymede’s eccenity or, if the resonance lasted longer than the resurfacepoch, an episode where the tidalQ values of Jupiter, Ioand Ganymede conspired to produce unusually high ectricities and tidal heating rates in Ganymede. For examcoupled geophysical-orbital feedbacks can produce 107–108

year-long spikes in the tidal-heating rate(Showman et al.1997).

Acknowledgments

We thank D.J. Stevenson, W.B. McKinnon, F. NimmoSchenk, A. Dombard, and R. Strom for useful discussioBill Moore and an anonymous reviewer provided many heful comments. This work was supported by NSF grant A0206269 and by a grant from the NASA PG&G programA.P.S.

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