s

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Icarus 177 (2005) 327–340www.elsevier.com/locate/icaru

Thermal and topographic tests of Europa chaos formation modelfrom Galileo E15 observations

F. Nimmoa,∗, B. Gieseb

a Department of Earth and Space Sciences, UCLA, 595 Charles Young Drive E, Los Angeles, CA 90095-1567, USAb DLR, Rutherfordstrasse 2, 12489 Berlin, Germany

Received 9 July 2004; revised 12 October 2004

Available online 31 May 2005

Abstract

Stereo topography of an area near Tyre impact crater, Europa, reveals chaos regions characterised by marginal cliffs and domicphy, rising to 100–200 m above the background plains. The regions contain blocks which have both rotated and tilted. We tested tof chaos formation: a hybrid diapir model, in which chaos topography is caused by thermal or compositional buoyancy, and blococcurs due to the presence of near-surface (1–3 km) melt; and a melt-through model, in which chaos regions are caused by mrefreezing of the ice shell. None of the hybrid diapir models tested generate any melt within 1–3 km of the surface, owing to the lowtemperature. A model of ocean refreezing following melt-through gives effective elastic thicknesses and ice shell thicknesses of 00.5–2 km, respectively. However, for such low shell thicknesses the refreezing model requires implausibly large lateral density(50–100 kg m−3) to explain the elevation of the centres of the chaos regions. Although a global equilibrium ice shell thickness of≈2 km ispossible if Europa’s mantle resembles that of Io, it is unclear whether local melt-through events are energetically possible. Thus, nemodels tested here gives a completely satisfactory explanation for the formation of chaos regions. We suggest that surface extrusiice may be an important component of chaos terrain formation, and demonstrate that such extrusion is possible for likely ice para 2005 Elsevier Inc. All rights reserved.

Keywords: Europa; Satellites of Jupiter; Ices; Geophysics; Tectonics

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1. Introduction

An important discovery of the Galileo spacecraft misswas the existence of areas of chaotic terrain on theface of Europa(Carr et al., 1998). These areas consistfractured blocks of pre-existing surface material, surrounby matrix material and often having sharp, cliff-like magins (seeFig. 1a). The blocks are commonly high-standirelative to the surrounding matrix material, by∼100 m inthe case of Conamara Chaos(Williams and Greeley, 1998).The mixture of matrix and pre-existing blocks varies fro

* Corresponding author. Present address: Department of Earth ScieUniversity of California Santa Cruz, 1156 High St., Santa Cruz, CA 950USA. Fax: +1 831 459 3074.

E-mail addresses: fnimmo@es.ucsc.edu(F. Nimmo),bernd.giese@dlr.de(B. Giese).

0019-1035/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2004.10.034

,

one chaos region to another, as does the size of thegions. Conamara Chaos is an area roughly 100 km acwhile patches of chaotic terrain have also been identifiethe interiors of 10 km scale lenticulae(Spaun et al., 1998Greenberg et al., 2003). The pre-existing blocks in Conamara appear to have both moved laterally, without leing any obvious sign of their motion in the surroundimatrix, and to have tilted sideways(Spaun et al., 1998Carr et al., 1998).

The importance of chaos terrain is that it potentially pvides clues to the nature of Europa’s ice shell, in partilar its thickness(Carr et al., 1998; Greenberg et al., 199Schenk and Pappalardo, 2004), subsurface composition antemperature structure. Unfortunately, there is as yet nosensus as to what the existence of chaotic terrain impliesthough the block topography and apparent motion in Comara Chaos are very suggestive of a “melt-through” ev

http://www.elsevier.com/locate/icarusmailto:fnimmo@es.ucsc.edumailto:bernd.giese@dlr.dehttp://dx.doi.org/10.1016/j.icarus.2004.10.034

328 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

esis

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sti--oth

adtate

itythe

fo-tiv-hto

dif-s ofnt

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(Carr et al., 1998; Greenberg et al., 1999), in which liq-uid water reached the surface of Europa, this hypothhas been challenged (e.g.,Collins et al., 2000), mainly be-cause of the enormous power required to cause sucevent.

Various aspects of chaos formation have been invegated recently.Pappalardo et al. (1998)suggested a solidstate diapirism mechanism for chaos formation, and bWang and Stevenson (2000)andSotin et al. (2002)arguedthat enhanced tidal dissipation within the diapir might leto increased thermal buoyancy and melting. Such solid-sdiapir models suffer from the high rigidity and viscosof the near-surface ice, which makes deformation ofnear-surface difficult(Showman and Han, 2004). Thomsonand Delaney (2001)suggested melt-through caused bycused upwelling plumes driven by seafloor magmatic acity. However,Goodman et al. (2004)have argued that sucplumes are likely to be both too broad and too diffuseaccount for observed chaos characteristics.Melosh et al.(2004)also argued for melt-through events, adopting aferent model for ocean stratification. The consequenceflow in the lower shell during a putative melt-through evehave been investigated byBuck et al. (2002)andO’Brien etal. (2002), although the conclusions of the latter may notcorrect (see Appendix inGoodman et al., 2004).

An important contribution byCollins et al. (2000)ap-plied simple quantitative tests to some of the hypothesechaos formation. These authors concluded that the enrequirements for a melt-through model were difficult to sisfy. However, they recognised that solid-state diapirisminconsistent with the observed size range of chaos blothe model predicts an absence of large (>1 km), elevatedblocks, in contradiction to the observations.Collins et al.(2000) proposed an intermediate or hybrid model, simto that ofHead and Pappalardo (1999), in which diapirismgave rise to shallow melting of salt-rich ices.

Here we adopt a similar approach toCollins et al. (2000),in that we develop simple quantitative models of chaosmation. We focus on two aspects: first, the thermal cotions required to satisfy the hybrid diapir model; and sond, the topographic consequences of both the melt-throand the diapir models. A recent survey of chaos topograhas been undertaken bySchenk and Pappalardo (2004). Ourtopographic data are obtained from one area only, but thsults should be applicable to other chaos areas which sthe same basic features (discussed below). We caution tis not yet clear whether all chaos terrains are formed insame fashion.

In Section2 we discuss the observations of our stuarea, and in particular the topography. We then proceeexamine two models of chaos terrain formation and topophy: the hybrid diapir model (Section3); and melt-through(Section4). In Section5 we discuss the relative meritsthe different hypotheses, and make suggestions for fuwork.

et

2. Observations

Fig. 1a shows a high-resolution mosaic of Galileo iages of an area SE of Tyre impact crater obtained duringE15 encounter [seeKadel et al. (2000)for a geological de-scription]. The area is dominated by cross-cutting ridgesalso contains several areas of chaos. The largest (CH2)irregularly-bordered area at the bottom (north) of the imacontaining several large blocks. There is also a 10 km wlobate area (CH1) at the top (south) of the image, coning almost entirely of matrix material, referred to bySchenkand Pappalardo (2004)as a lenticula and mapped byKadelet al. (2000)as knobbly chaos. A final area of chaos (CHtruncates the prominent diagonally-trending double ridgewards the top of the image. Small pit-like features inimage are secondary impact craters from Tyre.

Fig. 1b shows stereo-derived topography for the ashown in 1a, and profiles across the region are showFig. 2. Only the first two chaos regions referred to aboare covered by the stereo data. The stereo technique iscribed inGiese et al. (1998); the horizontal resolution othe topographic model is 200–400 m and the vertical acracy is generally better than 10 m. An independent stemodel of the same region has recently been publisheSchenk and Pappalardo (2004). The two models are in remarkably good agreement, although theSchenk and Pappalardo (2004)model has slightly fewer gaps which allowidentification of subtle features, such as a possible trougthe western margin of CH1, not identifiable in our model

Region CH2 has a complex short-wavelength topograsignature, visible in bothFig. 1b and profiles 12–14 inFig. 2.The east (left-hand) side of the region has a distinctvisible in both the image and the topography, with the baground material∼100 m higher than the matrix. Severchaos blocks are clearly elevated with respect to theroundings by similar amounts; one large block (at the Nof profile 13) appears to have rotated by about 30◦ and alsoto have tilted slightly. The overall topography of this regiis domical, and rises by 100–200 m above the base ocliff.

The second chaos region (CH1) is simpler. Again, this a pronounced cliff at the margins of the matrix mate(profiles 2–5), and again the centre of the region is domand elevated by 100–200 m with respect to the backgroplains. The margins of the region contain lobate matewith lineations suggestive of some kind of extrusion or floThe sinuous nature of the profiles outboard of the regionot noise, but is a result of crossing numerous ridges.western edge of the CH1 region is not obvious in the topraphy (profiles 7–9). The western topography appears tmain elevated beyond the edge of the mapped chaos reas also noted bySchenk and Pappalardo (2004); a similareffect for region CH2 is also seen in profile 13.

To isolate the shape of the trough at the eastern eof the CH1 region, profiles 1–6 were aligned on this mginal trough; the result of stacking these profiles is a

Tests of Europa chaos formation models 329

elytereo

Fig. 1. (a) Galileo high resolution image of area SE of Tyre impact crater. Image resolution 30 m/pix, taken on 31 May 1998, centred at approximat31◦ N, 142◦ W. Profiles shown are used in generatingFig. 2. Note that north is to the bottom of the image; illumination from the top of the image. (b) Stopography of same region, horizontal resolution 200–400 m, vertical accuracy 10 m.

e-eriamayds o

fre-chmi-

re-ack-s beevi-

haos

ted,ntl

rac-els

of.g.,

Theis ofune;

ently

shown in Fig. 2. Both the marginal trough and the elvated centre are apparent. Although the outboard matslopes downwards towards the chaos region, this resultbe an artefact caused by the elevated terrain at the enprofiles 2–4.

In summary, the chaos regions described here arequently characterised by a cliff-like marginal trough, whioften sharply truncates pre-existing features, and a docally uplifted centre. In particular, the centres of thegions are elevated by 100–200 m with respect to the bground plains. In some cases, the elevated area extendyond the edge of the chaos terrain. Similar features aredent at the much larger chaos feature termed Murias C(seeFigueredo et al., 2002). The short-wavelength (∼1 km)topography of the Conamara Chaos region is complicabut cliff-like margins and domical uplift are again evide(Schenk and Pappalardo, 2004). While it is not clear that al

l

f

-

chaos regions share similar characteristics, it is the chateristics outlined above that we will focus on in the modbelow.

3. Diapirism

The observed chaos morphology and the likelihoodice shell convection prompted a class of models (ePappalardo et al., 1998; Spaun et al., 1999) in which chaosregions are formed by the ascent of solid-state diapirs.existence of such diapirs is widely accepted on the bastheoretical arguments (e.g.,Pappalardo et al., 1998; Rathbet al., 1998; McKinnon, 1999) and the existence of surfacfeatures termed pits, spots or domes(Pappalardo et al., 1998Spaun et al., 1999; Greenberg et al., 2003). The relationshipbetween these latter features and chaos regions is presunclear(Greenberg et al., 2003).

330 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

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Thent toub-usesidity

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ir,ichwillthehaos

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Fig. 2. Topographic profiles across E15 area. Location of profiles shin Fig. 1. Successive profiles are offset vertically by 0.2 km and the melevation was removed. Crosses are the data, interpolated to 80 m splines are filtered using a 7 point moving average. Arrows denote cliff-margin of chaos regions (see text). Profiles p1–p6 are aligned on the stepoint of the filtered data within 6 km of the origin. The bottom profile isresult of stacking profiles p1–p6, with the thin lines showing± one standarddeviation (s.d.).

Fig. 3. Schematic of model for chaos formation by diapirism, afterCollinset al. (2000). Warm diapir ascends into salty ice and causes melting.melt zone may allow detachment faulting and surface block movemeoccur. If the melt drains laterally and/or downwards it will produce ssidence at the surface. The compositional buoyancy of the diapir catopographic uplift over a wider area than the subsidence, due to the rigof the overlying ice.

A disadvantage of the rising diapir model is that ithard to account for the observed lateral and rotationaltion of ice blocks in areas like Conamara Chaos(Collins etal., 2000). These authors therefore proposed a modificato the original hypothesis, similar to that ofHead and Pap

;

t

palardo (1999), in which the rising diapir melts overlying icwhich is contaminated with salts. This near-surface, intetial partial melt allows the surface ice blocks to detachmove as required.

Fig. 3 is a cartoon of the fundamental processes likto occur. The rising diapir is assumed to be thermabuoyant, though compositional buoyancy may also plarole (Pappalardo and Barr, 2004). The buoyant diapir willrise until either the overlying ice is sufficiently cold anrigid to prevent further movement, or the level of neutbuoyancy is reached. Once the diapir stalls it will beto propagate laterally. As it does so it will cool condutively, warming the surrounding material. If this materis contaminated with salts (e.g.,Kargel et al., 2000; Zolotovand Shock, 2001; Fanale et al., 2001), it will contain low-melting-temperature components, which will melt to formbrine. Since the centre of the diapir is likely to be closesthe surface, there will be a tendency for this melt to drlaterally; being denser than the surrounding ice, it will amove downwards. Although the diapir is likely to be cracfree due to its relatively high temperature, the surroundmaterial is probably fractured, permitting fluid flow. The rmoval of the melt will create subsidence at the surfaceoccurs with caldera collapses on Earth. As with caldethe margin of the collapse feature can be quite sharp. Mover, the production of melt may allow detachment faultand sliding of blocks to occur(Head and Pappalardo, 1999.Although the topography due to the thermal buoyancythe diapir will decay with time, any compositional varitions will induce permanent uplift(Nimmo et al., 2003aPappalardo and Barr, 2004). This uplift will be modified bythe rigidity of the overlying ice and may therefore extefor a greater lateral distance than the collapse feature.hybrid mechanism can thus in principle explain the mfeatures of the chaos terrains described.

The original analysis ofCollins et al. (2000)estimated amelt volume of the order of 1% of the volume of the diapbut did not include any predictions of the depth at whmelting was required to match the observations. Asbe seen below, it is this melting depth which providesstrongest constraint on plausible diapir models. Some cblocks are no more than 1 km in width (Collins et al., 2000,Fig. 1). The ratio of width:depth for the rotating ice blocksunlikely to greatly exceed 1, implying that partial melt muextend to within a few (≈1–3) km of the surface if detachment faulting along zones of melting is to occur. Two othweaker constraints may also be applied. First, if the margsubsidence of the matrix relative to the background mateis a result of melting, as suggested byFig. 3, then the in-tegrated thickness of melt generated must be of the sorder as the observed subsidence, that is∼100 m. Secondthe partially molten zone must persist for long enough tolow detachment and block rotation to take place.

Below, we will apply a quantitative analysis to the twmain processes in this model. First, we will examinedepths at which melt can be generated, and the rate at w

Tests of Europa chaos formation models 331

g-elyuffi-s is

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it is removed. Second, we will investigate the likely toporaphy created. We will conclude that even in extremfavourable circumstances, the generation of melt at sciently shallow depths (1–3 km) to match the constrainthighly unlikely.

Before embarking on a detailed model description, csideration of some simple physics will illustrate the badifficulty of generating diapir-induced melting at shallodepths. If a hot body (temperatureT1) is placed in contacwith colder background material (temperatureT0), then inthe absence of internal heating the initial temperature ainterface is(T1 + T0)/2 (e.g.,Turcotte and Schubert, 2002),and this interface temperature will subsequently decrewith time. As explained below, a lower bound on the meing temperature of salty ice is≈210 K; assuming a diapitemperature of 250 K, the minimum temperature of theroundings required to generate even infinitesimal meltin170 K, or about 40% of the depth to the base of the ice sGenerating melt at shallow depths is exceedingly difficsimply because the surrounding material is so cold.

3.1. Melting model

We adopt a simplified 2D Cartesian representation ofsituation shown inFig. 3. The diapir is modelled as a sttionary rectangular block of thicknesss with its top at deptht below the surface, and having a maximum initial tempature, tapered to the margins, ofTi . The near-surface (saltyice above a particular depth is assumed to have a metemperature ofTm, while ice in the diapir or below thisdepth has a melting temperature ofTb. The ice is otherwiseassumed to have uniform thermal properties, with the extion of thermal conductivity (see below). The shell thickneis tc and the bottom temperature is maintained at a consvalueTb. Since the conductive heat flux through an ice shas thin as 1 km is still≈102 times smaller than the incidensolar radiation, likely variations in subsurface heat flux wnot have any appreciable affect on the surface temperaTs, which we therefore assume constant. The initial thmal state is conductive everywhere except within the diaSince the diapir is assumed to be stationary (an assumdiscussed in Section3.4), it is equivalent to an intrusive siland the two terms will henceforth be used interchangeab

Near-surface ice on Europa is cold, heavily fracturand may contain substantial porosity(Nimmo et al., 2003a).Such porosity will significantly reduce the thermal condtivity, and will increase temperatures at depth, making ming more likely. Similar low thermal conductivity layers habeen proposed in the past (e.g.,Squyres et al., 1983; Fanaet al., 1990). Alternatively, the ice conductivity may be lobecause of the presence of clathrate hydrates (e.g.,Prieto-Ballesteros et al., 2004) or salts (e.g.,Prieto-Ballesteros anKargel, 2004). For instance, pure methane clathrate hathermal conductivity a factor of≈7 lower than water ice(Ross and Kargel, 1998). We will assume an exponential

varying near-surface conductivityk′(z) of the form

(1)k′(z) = k − �k exp(−z/δ),where z is positive downwards,k is the conductivity atdepth,�k = k − ks, whereks is the surface conductivity anδ is a decay length.

For a given heat fluxF the steady-state temperature strtureT (z) is given by

(2)T = Fδk

ln

[k − �k exp(−z/δ)

k − �k]

+ Fzk

+ Ts.At depths �δ, the effect of the reduced near-surfa

conductivity is to increase the temperature relative toconstant-conductivity case by

(3)�T = Fδk

ln

[k

ks

]= (Tb − Ts)δ

tcln

[k

ks

],

whereTb, Ts, andtc are defined above. This equation illutrates the fact that increasingδ or decreasingks increasesthe effect of the near-surface layer, as expected. Ascussed below, likely values ofk/ks andδ/tc are 20 and 0.1respectively. Under these circumstances, at depths�δ thetemperature increase is 30% of the total temperaturetrast (Tb−Ts), a significant effect. Note that for a conductishell, the effect of adding such a low-conductivity layer isreduce the equilibrium shell thickness.

The background initial temperature in our model is givby Eq.(2). The subsequent thermal evolution of the systemsolved using the finite-difference approach ofMcKenzie andNimmo (1999). This approach is described in more detaiAppendix A. The clean ice is assumed never to melt; to avnumerical instabilities, melting of the salty ice occurs ovetemperature range�Tm (usually set to 10 K), with the mefraction increasing from 0 to 100% in a linear fashion othis range. Note that this method is likely to overestimateamount of melt generated: in reality, a larger rise in tempature is likely required to generate additional melt as themelting temperature components are removed. Furthermmelt drainage is likely to occur on timescales short copared to the rate at which melting proceeds (see Section3.2).In general, grid points did not suffer complete melting;melt thicknesses given below were calculated by verticintegrating the melt fraction with depth.

The following parameters were adopted for the ice: hcapacity 2100 J kg−1 K−1, asymptotic thermal conductivity k = 3 Wm−1 K−1, density 900 kg m−3 and latent hea0.33 MJ kg−1. The brine produced had an assumed denof 1000 kg m−3 and a heat capacity of 4200 J kg−1 K−1.We adopted a 25 km thick ice shell, since this is rougconsistent with the existence of convection, at least for Ntonian behaviour (e.g.,McKinnon, 1999) and examined theeffect of varying this parameter below. Based on Galiphotopolarimeter–radiometer results(Spencer et al., 1999,the surface conductivityks was fixed at 0.15 W m−1 K−1.The thickness of the low-conductivity layer is poorly costrained. Theoretical models suggest that it is unlikely

332 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

ick-d

om-inre

l.,unt

ions

enthisthendThepletingrate

ingis

epervelsring

theon-the

t, asck-

ck-

edck-topusehetedin-

tionre

elt-andtion

l.ally

topp ofursters

s,

, weem-elt

s-ausi-

exceed one-quarter to one-third of the conductive shell thness(Nimmo et al., 2003a), while observations of isolatechaos blocks (e.g.,Collins et al., 2000, Fig. 2d) suggest rel-atively cohesive near-surface material. We adopted a ninal value for δ of 3 km and investigated variationsthis quantity below. The lowest likely melting temperatufor salty ice is 210 K(Carlson et al., 1999; Kargel et a2000)and this value was adopted to maximise the amoof melt generated. The initial temperature of the diapirTiwas set at 250 K, in agreement with convection calculat(McKinnon, 1999; Nimmo and Manga, 2002), and the sur-face and bottom temperaturesTs and Tb fixed at 110 and260 K, respectively.

Fig. 4a plots a typical model run, at the point whthe maximum amount of melt has been produced. Inmodel, the salty ice extends half-way down the side ofdiapir. However, melting only occurs above the diapir, adecreases with lateral distance from the diapir centre.total amount of melting is small, as expected from the simphysical argument given above. Despite the low ice meltemperature assumed, the mean thickness of brine geneabove the diapir is only 30 m, and the maximum meltfraction is 30%. More importantly, the zone of meltingconfined to an area immediately above the diapir, dethan 7 km. The same diapir emplaced at shallower lein the shell generates no melting, because of the buffeeffect of the (fixed) surface temperature.Fig. 4b plots thetime variation of a vertical temperature profile throughcentre of the diapir. The effect of the near-surface low cductivity layer is evident. The temperatures adjacent todiapir never exceed half the initial temperature contrasexpected. The timescale for the diapir to cool to the baground temperature is determined mainly by its initial thiness.

d

Fig. 5 plots the sensitivity of the maximum integratthickness of brine produced to variations in sill depth, thiness andδ. Fig. 5a shows that increasing the depth to theof the sill increases the amount of melt produced, becathe overlying ice is closer to its melting point. However, tmost important result is that in no case is melt generawithin 5 km of the surface. Increasing the sill thicknesscreases the total amount of melt produced (Fig. 5b), becausemore energy is available, and also prolongs the producof melt. Decreasingδ or increasing the melting temperatuTm reduces the amount of brine generated.

An additional source of energy available to generate mis tidal dissipation.Tobie et al. (2003)suggest that tidal dissipation may be concentrated in warm upwelling areas,that for equatorial areas the maximum tidal heat generais 2× 10−6 Wm−3. The dashed lines inFig. 5a show the ef-fect of adding a constant heating term of 2× 10−6 Wm−3within the diapir, and demonstrate that the effect is smal

For all the models examined, the thickness of the partimolten layer never exceeded 1 km, and the depth to theof this layer always exceeded 90% of the depth to the tothe sill. Thus, asFig. 5a demonstrates, melting never occwithin about 5 km of the surface for the range of parameexplored. Forδ � 5 km, melt thicknesses>100 m are onlyobtained for sills at depths>8 km, and even in these caseno melt is generated at depths 35 km or a melt-ing temperatureTm � 195 K. The available constraints, dicussed above, make such parameter values highly implble.

ature(m

e meltwaterime.

Fig. 4. (a) Typical outcome of melting models described in Section3.1, plotted at the instant of maximum melt production (40 kyr). Solid lines are tempercontours (in K). Initial diapir location denoted by dashed lines; initial diapir temperature 250 K. Light shaded area denotes location of salty iceeltingtemperature 210 K). Thin dark shaded area above the diapir depicts points at which partial melt generation occurs. The mean thickness of thlens overlying the diapir is 30 m. The conductivity length-scaleδ was 3 km. (b) Evolution of vertical temperature profile through centre of diapir with tCurvature of near-surface profile is due to variable thermal conductivity. Dotted line denotes melting curve of ice. Profiles are plotted at intervalsof 8 kyr.

Tests of Europa chaos formation models 333

dis

Fig. 5. (a) Variation of maximum melt thickness as a function of depth to top of diapir and conductivity length-scaleδ for a shell of thickness 25 km. Solilines are for no tidal dissipation within diapir (H = 0); dashed lines are forH = 2× 10−6 W m−3. Tm is ice melting temperature. The top of melting zonenever more than 1 km above the top of the sill. (b) Variation in melt thickness as a function of sill thickness andδ (depth to top of sill 7.5 km).

eles

r-rate

15-onirthe

bys

-

n is

-

ar).raincom

sude

m

ared, thean

willdsis

ng

ondingAsal

set

theice.er-thesi-

nd

thatiapir

iswn.ce

rustal

We also varied the shell thicknesstc from its nominalvalue of 25 km. Increasingtc results in colder material at thsame depth, and thus makes near-surface melting evenlikely; decreasingtc allows warmer material nearer the suface. Shells of 20 and 15 km thickness, respectively, genemelt with diapirs emplaced at depths�5 km and 3.5 km,respectively, for the nominal parameters. However, akm thick shell is unlikely to undergo thermal convecti(McKinnon, 1999). We therefore conclude that the diapmechanism is unable to generate melt within 3 km ofsurface for any likely combination of parameters.

3.2. Melt drainage timescales

The compaction timescale for melt has been derivedMcKenzie (1989)and was previously applied to problemon Europa byGaidos and Nimmo (2000). The timescaleτ isgiven by

(4)τ = Chmηa2φn−1�ρg

,

wherehm is the thickness of partially molten material,ais the grain size,φ is the melt fraction,�ρ is the densitycontrast between melt and matrix andg is the accelerationdue to gravity. For melt fractions>3%,C = 100 andn = 3(McKenzie, 1989). At low melt fractions, separation only occurs if the dihedral angle between melt and matrix is

334 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

irty.

kmex-nothe

ypi-this

fanaspo-

ionsthe

adctive

et-theen-e

0.3,

ick-r, ade-

Esti-iesr

-

in therates0.3,

aof

sis-vedrore,aosick-

heec-

kmthse

tionod-s

ince

un-ture.ur-

heuc-ices.n ise tooth

r, atre-that

)dis-res,

itionface

w-heigidticirnant4)akecaleur

density does not exceed that of water,≈1000 kg m−3. Thus,the maximum likely density contrast between clean and dice is≈100 kg m−3, and in practise�ρ is probably smallerFor the sake of argument, if�ρ is 50 kg m−3 then to pro-duce 100 m of uplift requires a diapir to penetrate 1.8into dirty ice. Since chaos regions on Europa commonlyceed 10 km in diameter, such a penetration distance isunreasonable. This distance also places a lower limit onthickness of contaminated ice, but since convection is tcally associated with ice thicknesses in excess of 20 km,limit is not very restrictive.

The amount of uplift will be affected by the rigidity othe overlying ice. Assuming an axisymmetric load, we ccalculate the resulting uplift as a function of ice rigidityfollows. The subsurface load (due to thermal and/or comsitional buoyancy) is expressed as a sum of Bessel functThis approach is the axisymmetric equivalent to takingFourier transform, and is described in more detail inBarnettet al. (2002). The deflection of the ice depends on the lodimensions and magnitude, the ice density and the effeelastic thickness of the iceTe.

Fig. 6shows the topography generated by an axisymmric subsurface load of width 10 km, cosine tapered overouter 20%. The magnitude of the load is equivalent to a dsity contrast of 45 kg m−3 extending over a vertical distancof 1.8 km. Here we assume an ice density of 900 kg m−3,a Young’s modulus and Poisson’s ratio of 1 GPa andrespectively, an acceleration due to gravity of 1.3 m s−2,and use the first 50 Bessel functions. For low elastic thnesses, the topography is essentially isostatic. HoweveTe increases, the maximum amplitude of the topographycreases, and the lateral extent of the uplift increases.mates ofTe on Europa vary over a rather wide range: studof Murias Chaos(Figueredo et al., 2002)and a plateau neaCilix impact crater(Nimmo et al., 2003b)yield values of4–6 km, while studies of double ridges (e.g.,Billings andKattenhorn, 2002; Hurford et al., 2004) and dome-like features (Williams and Greeley, 1998)yield values of order

Fig. 6. Variation in surface topography with elastic thicknessTe for a fixedaxisymmetric subsurface load, calculated using the method describedtext. Load width is 10 km, cosine tapered over the outer 20%, and gene90 m uplift in isostatic case. Young’s modulus is 1 GPa, Poisson’s ratioacceleration due to gravity 1.3 m s−2, ice density 900 kg m−3.

t

.

s

0.1 km.Fig. 6 shows that for relatively narrow features,value ofTe in excess of 1 km is likely to generate an areauplift significantly wider than the underlying load.

3.4. Comparison with observations

Several aspects of the hybrid diapir model are content with our calculations. The positive topography obserin the chaos region inFigs. 1 and 2can be accounted foby a diapir with a reasonable density contrast. Furthermtopographic uplift extending beyond the edge of the chregion can be explained for reasonable shell elastic thnesses (Section3.3).

The principal problem with the hybrid diapir model is textreme difficulty in generating melt at shallow depths (Stion3.1). While some melt can be generated at depth (Fig. 5),none of the models are capable of generating melt at 1–3depth. This inability to reproduce the inferred melting depis a significant shortcoming of the hybrid diapir model. Wnote that the convective models ofTobie et al. (2003)alsofail to generate any melt within a few km of the surface.

The degradation process responsible for the formaof matrix material is not understood. However, the melling of Section3.1 shows that diapiric activity increasethe near-surface heat flux by a factor of at most 2–3. Sthe background subsurface heat flux is at least 102 timessmaller than the incident solar flux, such variations arelikely to have any appreciable effect on surface temperaThus, diapirism will not generate any kind of enhanced sface sublimation. The brine remobilisation model ofHeadand Pappalardo (1999)suffers from the same problem as thybrid diapir model described above: shallow melt prodtion is simply not possible for reasonable parameter cho

One possible explanation for the surface degradatiothat heating could potentially cause a clathrate hydratdecompose, liberating vapour which would generated bthe inferred volume loss and the degradation. Howeveleast for CO2 and methane clathrates, the temperaturesquired to cause such decomposition are comparable torequired to generate liquid water(Ross and Kargel, 1998.While these hydrates therefore suffer from the problemscussed above of generating sufficiently high temperatuit is possible that similar species undergoing decomposat lower temperatures could be responsible for the surdegradation.

The model shown inFig. 4is a simplification of reality inthat it does not include vertical motion of the diapir. Hoever, such vertical motion is actually very unlikely in tshallow subsurface: the ice is probably too cold and rto permit upwards motion. One possibility is that plasyielding and/or brittle deformation might allow the diapto ascend closer to the surface than in the usual staglid models(Tobie et al., 2004; Showman and Han, 200.Even if such vertical motion does occur, it would have to tplace in a time short compared to the conductive timesof the overlying layer in order for melt generation to occ

Tests of Europa chaos formation models 335

is

tiveths

cesan-

theyiq-

rva-odi-h an

ellssf Iolmg-

seat

x,

ea-ate:

ts inate.ce

ons

not

nyinins,

tive.en-

e-ness

ind

ob-d by

otdelhewith

If a

-has0.5–

liffs

thez-

teds canll bell;

de-at

at shallow levels. It is not yet clear whether this conditionlikely to be satisfied.

Thus, although the diapir model contains some attracfeatures, the production of melt at sufficiently shallow depto match the hybrid diapir model ofCollins et al. (2000)does not appear possible. Either some fundamental prois missing from the model outlined above, or diapirism cnot be the sole explanation for chaos formation.

4. Melt-through model

Early observations of chaos terrain suggested thatmight be the outcome of “melt-through” events in which luid water reached the surface and then re-froze(Carr et al.,1998; Greenberg et al., 1999). As discussed in Section1,while this hypothesis apparently explained several obsetions of chaos regions, its great disadvantage is that prgious amounts of energy are required to generate sucevent(Collins et al., 2000).

Melt-through events are more plausible if the ice shis initially thin (1–2 km). Such a global ice shell thickneis in fact possible if Europa’s mantle resembles that o(Greenberg et al., 2002). The canonical value for Io’s globaheat flux based on ground-based observations is 2.5 W−2(Veeder et al., 1994), although more recent observations sugest a value closer to 0.2 W m−2 (De Pater et al., 2004).Assuming that the Love numbers and dissipation factorQof the two bodies were identical, then Europa’s surface hflux would be approximately 9% of Io’s (e.g.,Murray andDermott, 1999). Using the higher estimate of Io’s heat fluEuropa’s mantle heat flux would be 220 mW m−2, resultingin a conductive ice shell thickness of 2 km. It is not unrsonable that Europa’s mantle should be in an Io-like stas noted byPeale et al. (1979), tidal heating within a silicatemantle leads to a reduction in viscosity, a decrease inQ, andan increase in tidal heating; this positive feedback resulan extensively molten mantle and an Io-like dissipation rUnfortunately, inferring the state of the mantle from surfaobservations is very difficult; nor do models or observatiof orbital evolution yet place constraints on theQ of Eu-

s

ropa. Thus, the existence of a globally thin ice shell canbe ruled out on energetic grounds alone.

However, irrespective of the initial shell thickness, asource of heat internal to the shell is unlikely to resultcomplete melt-through, simply because as the shell thconductive heat removal becomes increasingly effecThis is essentially the same effect which makes melt geration so difficult at shallow depths in Section3.1. O’Brienet al. (2002)did obtain a melt-through event, but only bcause their model failed to adequately resolve the thickof the conductive shell (seeGoodman et al., 2004). The mostlikely mechanism for generating melt-through is some kof local heat pulse supplied by the underlying ocean(Meloshet al., 2004; Thomson and Delaney, 2001); however, it isunclear whether either the energetic requirements or theserved lengthscales of chaos regions can be reproducesuch models(Goodman et al., 2004).

In view of the large uncertainties involved, we will ndiscuss the energetic difficulties of the melt-through mofurther. Instead, we will focus on the extent to which tobservations of the E15 chaos region are compatiblelikely consequences of the melt-through model.

Fig. 7 shows a schematic of the sequence of events.melt-through event occurs (Fig. 7a), the initial level of wateris at a distanced below the surface of the ice, where

(6)d = tc(ρw − ρ)ρw

,

where tc is the ice thickness andρ and ρw are the densities of ice and water, respectively. This vertical offsetbeen used in the past to infer an ice shell thickness of2 km in chaos regions (e.g.,Williams and Greeley, 1998),and by this hypothesis is responsible for the marginal cobserved (Section2).

The exposed water will freeze rapidly, beginning atedges (Fig. 7b). A major assumption here is that this freeing ice is effectively welded to the neighbouring unmelice. We also assume that lateral shell thickness contrastpersist over the refreezing timescale. Such contrasts wiremoved by lateral flow in the lower portion of the ice shehowever, for the shell thicknesses of a few km which werive below, the rate of lateral flow is sufficiently slow th

blockonsequent

Fig. 7. Schematic showing development of chaos topography following “melt-through” event. (a) Melt-through leads to water (and remanent ice-s) atlevel d beneath the unmelted ice. (b) Refreezing ice is assumed to remain welded at initial level. (c) Thickening of ice results in upthrust and cup-doming of ice away from the edge.

336 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

;

p-es’p-in. Th

ex-

.

s

ter.enon

ionsingthe

ssedtolt.ss

ll beme-nce

he,ain

erf

sur-ofilethenalc-

ons

th-

e

ardob-

ionst

aut

3pec-

inas

cted

mtrast

in-esent

ess

oust

tothe

r-in

r or0–

hellr lat-shellposeen-

therdsalso,

it can be neglected(Buck et al., 2002; O’Brien et al., 2002Nimmo, 2004).

As the refreezing ice thickens, it will experience an uwards force proportional to its thickness due to Archimedprinciple. Normally this upwards force would result in uwards motion (isostatic uplift) of the thickening ice, butthis case the ice is assumed to be pinned at the edgesresult (cf.Allison and Clifford, 1987) is an inverted bowl-shaped profile, similar to that observed (Fig. 7c).

To quantify this effect, we assume that the ice has a flural rigidity D. The resulting effective elastic thicknessTeof the ice is given by

(7)D = ET3e

12(1− σ 2) ,

whereE is the Young’s modulus andσ is the Poisson’s ratioFor a Cartesian case, the upwards deflection of the icew isgiven by

(8)Dd4w

dx4+ ρwgw = �ρgh,

whereg is the acceleration due to gravity,x is horizontaldistance,�ρ = ρw − ρ and h is the local shell thicknes(which differs from the background ice shell thickness).

We take the boundary conditions to bew = d2w/dx2 = 0at the edge of the refreezing ice (wherex = ±L). It may beverified that a suitable solution to Eq.(8) is given by

(9)w = h�ρρw

[1− C coskx coshkx + S sinkx sinhkx

C2 + S2],

whereC = coskLcoshkL, S = sinkLsinhkL, and

(10)k =(

ρwg

4D

)1/4.

The quantityk is the inverse of the usual flexural parameWhenD = 0, Eq.(9) gives the isostatic response, and whD → ∞, w → 0, as required. The symmetry of the solutimeans thatdw/dx = 0 atx = 0.

Note that this expression does not make any predictabout the elevation of the ice relative to the surround(unmelted) material: this elevation contrast depends ondensities of the unmelted and refreezing ice, and is discubelow. Although the expression is Cartesian, it is unlikelydiffer except in detail to the equivalent axisymmetric resu

Equation(9) shows that for flexural parameters much lethan the domain width, the response of the ice sheet wisimilar to an isostatic case, while at large flexural paraters the vertical uplift is reduced. The characteristic distafrom the margin over which uplift occurs (Fig. 7c) is deter-mined by the flexural parameter. IfTe increases along withh, which is the likely situation in a refreezing ice layer, tuplift initially increases (becauseh is increasing). Howeverif the flexural parameter becomes comparable to the domwidth, the rigidity of the ice begins to dominate and furthincreases inTe andh result in reduced uplift. Irrespective o

e

Te, once the refreezing shell reaches the thickness of therounding, unmelted ice, the shape of the topography prwill not change subsequently. In particular, as long asmargin of the refreezing ice remains welded at its origilevel (seeFig. 7b), a marginal trough will remain irrespetive of the shell thickness. Thus,h provides only a lowerbound on the present-day shell thickness.

To compare our theoretical model with the observatiof an area in which the marginal trough is prominent (Fig. 2),we fix L at the observed trough distance and varyh andTesimultaneously until the minimum misfit of the model withe observations is obtained.Fig. 8a shows the stacked profile from Fig. 2 together with the minimum misfit profilthus obtained. The best-fit values ofh andTe are 1750 and250 m, respectively, and the minimum misfit is 0.32 standdeviations. The present model provides a good fit to theservations; note, however, that inclusion in the observatof profiles 7–9 fromFig. 1 (which do not show a prominenmarginal trough) would decrease the goodness of fit.Fig. 8bplots the variation in misfit withTe and shows that there isvery clearly defined minimum misfit position. Carrying othe same procedure for profile 12 (Fig. 1b) yields values forh andTe of 850 and 100 m, respectively, while for profile 1the (poorly-constrained) values are 400 and 350 m, restively.

In Fig. 8a the elevation which the surrounding terrawould occupy if it had the same density and thicknessthe refreezing ice is shown by the dotted line. This expeelevation exceeds that observed by≈100 m. This differ-ence could be explained if the refreezing ice were 50 kg3

less dense than the unmelted ice. Whether such a conis likely depends on the ability of the refreezing ice tocorporate contaminants, such as salts, which may be prin unmelted ice(Kargel et al., 2000). The resulting densitycontrasts are unlikely to exceed 100 kg m−3, so that compo-sitional contrasts alone cannot explain updoming in excof ≈200 m for shell thicknesses of�2 km. Updoming in ex-cess of this amount is observed bySchenk and Pappalard(2004), implying additional or alternative mechanisms mbe operating. An additional contribution could be duethe elevated temperature of the refreezing ice relative tosurroundings, but (as with Section3), such elevated tempeatures would diffuse away with time. Lateral variationsporosity can also generate uplift(Nimmo et al., 2003a), butsuch variations are typically confined to the top quartethird of the ice shell, and would have to be very large (340%) to explain the observations. Finally, a thicker ice swould generate the same elevation contrast with smalleeral density contrasts. However, the data suggest a thinat the time of refreezing, and there is no reason to supthat later shell thickening would preferentially generate dsity contrasts at the site of existing refrozen regions.

One aspect of the observations which is not fit bymelt-through model is the apparent downwarping towathe margins of the chaos region. Such downwarping hasbeen observed byFigueredo et al. (2002)at Murias Chaos

Tests of Europa chaos formation models 337

delsame

Fig. 8. Bold line is stacked topographic profile of E15 area, replotted fromFig. 2. Thin lines are±1 standard deviation (s.d.). Dashed line is best-fit moassuming melt-through [Eq.(9)]. Best-fit ice thicknessh is 1750 m andTe = 250 m. Dotted line indicates elevation of background terrain assumingdensity and thickness as the re-freezing ice (see text). (b) Misfit, normalised to the minimum value of 0.32 standard deviations, as a function ofTe. Misfit iscalculated over the domain indicated by the dashed line in (a). Bold line plots misfit whenh is allowed to vary for each value ofTe; thin line plots misfit whenh is fixed at 1750 m.

em-n at

any

no

frittlee-

to

eastof

ez-hell;-dayicew-ewed on

g’sver,f-wato

ell.sar-

tsidelyaterthe

ouldpre-s thehaos

bovere.

s ofthemim-nreas,p-

gion.’selt-

chas-

ikeck-con-d onrivednedsesd to

zing

ap-aos

and was there attributed to flexure due to surface loadplacement. Although there are hints of surface extrusiothe margins of CH1, we argued in Section2 that the down-warping observed there may be an artefact. Nor is theresimilar downwarping around CH2 (Fig. 1b). Thus, this dis-agreement between theory and observation is probablysignificant.

For Fig. 8a the value ofTe obtained is less than that othe ice thickness. This result is reasonable, since both bfailure and ductile yielding are likely to occur and will rduceTe relative toh (e.g.,Nimmo et al., 2003b). The localshell thicknessh of a few km obtained here is very similarthe estimates ofWilliams and Greeley (1998)andCarr et al.(1998), and suggests that the melt-through model is at linternally self-consistent. As explained above, the valueh obtained is probably set by the point at which the refreing material reaches the thickness of the surrounding sthis local shell thickness is not necessarily the presentvalue. Subsequent uniform cooling and thickening of theshell would not affect the pre-existing topography. If, hoever, the shell subsequently began to convect, then ntopography caused by diapirism might be superimposethe pre-existing, non-diapiric features.

The elastic bending stresses are proportional tod2w/dx2

in Eq. (9). The surface stresses are 4–6 MPa for a Younmodulus of 1 GPa, sufficient to cause fracture. Howethese bending stresses decrease to zero at a point halthrough the elastic portion of the ice shell, and are likelybe relieved by ductile creep in the lower portion of the shThus, Eq.(9) predicts surface failure but does not necesily imply that the shell will suffer throughgoing fracture.

The model presented here assumes that the ice outhe chaos region is stationary. Whether this is really likto be the case depends on its rigidity, which may be grethan that of the refreezing ice. The upthrust exerted on

t

r

y

e

chaos ice will also act on the unmelted ice beyond, and cpotentially dome these unmelted regions upwards. Thisdiction agrees with the observations that in some caseupdomed topography extends beyond the edge of the cregion (Section2). On the other hand, the profile inFig. 8adoes not support this prediction, although as discussed athe apparent downwarping is probably not a robust featu

5. Discussion and conclusions

This paper has produced simple quantitative modeltwo suggested chaos formation mechanisms, and testedagainst observations, primarily the E15 high resolutionages and topography. The principal observations (Sectio2)are that chaos regions can consist of elevated, domical aoften surrounded by an inwards-facing cliff. The domical ulift sometimes extends beyond the edges of the chaos re

While a ∼2-km thick ice shell is possible if Europamantle resembles that of Io, it is unclear whether local mthrough events are energetically possible. In Section4 weshow that a simple model of ice refreezing following sua melt-through event can plausibly reproduce severalpects of the observed topography, in particular the cliff-lmargin and the domical uplift. The predicted ice shell thinesses at the time of chaos formation of 0.5–2 km aresistent with previous estimates for chaos regions basea melt-through assumption. The elastic thicknesses de(0.1–0.3 km) are also similar to those previously obtaiin chaos regions. However, for the low shell thicknesderived, implausibly large density contrasts are requireexplain the observed elevation contrasts of>200 m betweenbackground plains and chaos centres. Thus, the refreemodel has difficulty reproducing all the observations.

The hybrid diapir model can reproduce the domicalpearance including uplift beyond the edges of the ch

338 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

lat-eltace.-pa

elyin-

woear

elysubtionwtheirage

atesandtion

inssedre-tiv-to

ob-eet-

byythity.

p,

ty

dthususi-t ofhis

nckre-er.

our

d

t

int

r-

the

of

r a

over

t

tothe

ingmi-est

per-

ries,

-inord to

a

regions. However, for the model to be able to explaineral chaos block motion and cliff formation, significant mneeds to be generated within about 1–3 km of the surfThermal modelling results (Section3.1) show that generating such melt appears to be impossible for reasonablerameter choices. This conclusion holds even if a relativthick insulating surface layer and tidal dissipation arecluded.

For the hybrid diapir model, there appear to be only tways of generating the required quantities of melt in the nsub-surface. One is to use a very thick (δ > 35 km), low-conductivity surface layer. However, such a layer is likto be only a transient phenomenon, since the increasedsurface temperatures will lead to rapid pore space reduc(Fanale et al., 1990); furthermore, the layer would have a lodensity and thus provide a barrier to further ascent ofdiapir. A slightly more likely explanation is that the diapmodel presented here does not fully account for the damand tidal dissipation generated by a rising diapir(Tobie et al.,2004). We also note that the presence of clathrate hydrmight significantly influence the temperature structure,provide a possible explanation for the surface degradaand volume loss inferred.

Any future model of chaos formation will have to explathe commonly elevated centres; occurrences of deprecliff-like edges; and variably-disrupted interiors of thesegions. It seems possible that some kind of diapiric acity which includes surface extrusion of material, leadingmarginal downwarping, could potentially explain theseservations (cf.Figueredo et al., 2002). To demonstrate thplausibility of such a model, we employ the axisymmric, spreading isoviscous fluid approach ofMcKenzie et al.(1992). In this model, the total extrusion timescale is setthe cooling time of the fluid drop, which is controlled bits thickness. The rate of lateral flow is controlled by bothe fluid thickness and its (constant, Newtonian) viscosEquation (21) ofMcKenzie et al. (1992)may be rewrit-ten to obtain the effective viscosityη of the spreading fluiddrop:

(11)η = cρgh50

κr20

.

Herer0 andh0 are the radius and thickness of the fluid droκ is the thermal diffusivity andc is a constant (= 0.18).FromFigs. 1 and 2we estimater0 = 5 km andh0 = 300 m,which givesη = 2 × 1013 Pas. Since the effective viscosiof ice close to its melting point is typically 1013–1015 Pas(Pappalardo et al., 1998), it is apparent that the observeshape could be generated by extrusion of warm ice andthat the proposed mechanism is at least potentially plable. Further work, in particular a more detailed treatmenthe ice rheology and cooling, will be required to place thypothesis on a firmer footing.

-

-

,

Acknowledgments

We thank Bob Pappalardo, Patricio Figueredo, FraMarchis, and Vicki Hansen for helpful discussions, and caful reviews by Geoff Collins and an anonymous reviewThis work funded by NASA-PGG NNG04GE89G.

Appendix A

Here we describe the numerical approach involved incalculations of diapir-induced melting (seeFig. 4). Furtherdetails may be found inMcKenzie and Nimmo (1999).

At each grid-point (i, j ), the upwards, downwards anlateral heat fluxesFN, FS, FE, andFW are calculated using

(A.1)FN�z = kj− 12[T (i, j) − T (i, j − 1)]

and similar expressions, wherekj is the (vertically-varying)thermal conductivity,T (i, j) is the temperature at grid poin(i, j ), j increases with depth and the vertical grid spacing�zequals the horizontal grid spacing�x. Note that a positivevalue denotes an outwards heat flow.

The total change in heat per unit length of a grid po�H is given by

(A.2)

�H(i, j) = −�t�x(FN + FE + FS + FW − Hint�x),

where�t is the time step andHint is the internal heat geneation rate in W m−3.

For ice which does not reach its melting temperature,total temperature change�T (i, j) is given by

(A.3)�T (i, j) = �H(i, j)/(ρCp�x2),whereρ andCp are the density and specific heat capacitythe material, respectively.

Melting is assumed to occur in a linear fashion overangeTm − �Tm/2, Tm + �Tm/2. If �T (i, j) is such thatthe solidus is exceeded, then the change in temperaturethe melting interval is calculated using Eq.(A.3) except thatCp is replaced byCp + L/�Tm, whereL is the latent heaof melting.

The initial temperature of the diapir is cosine-taperedthe background value over the top and bottom 15% ofblock, and laterally over the outer 30%, to mimic the coolof the diapir as it rises through the ice shell. For the nonal case (Fig. 4), removing the tapers changes the shallowdepth at which melt is produced by less than 2%. Tematures are updated each time step by using Eq. (A.3). Theboundary conditions are reflecting at the side boundafixed at the top and bottom.

A grid spacing of 0.125 km was used in both thex andz directions, and a fixed timestep of 108 s was adopted. Increasing the spatial or temporal resolution further had m(

Tests of Europa chaos formation models 339

onical3))d no

on

ianphys.

ck-Sci.

rsis-. 29,

od-

Eu-

n Eu-

d ed

ationhys.

ra-out

t ofdens

K.,ceanhys.

. Ge-o-

405,

lardogio

. Hy-ma-

999.

tonicrust.

lifts

ericula

ofV.

l his-pan2669.

row-Eu-life.

rys-

rac-

like

elt-

ell.

em-.idge

pa.

nse-. 29,

and

’s iceRes.

l for

chu-98.Na-

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sipa-

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ices,rgh,cht,

s on31,

tionhys.

llinghys.

.T.,obile

few percent during model runs. As an additional checkthe accuracy of the results, it was verified that the analytsolution ofCarslaw and Jaeger (1959, Section 2.2, Eq. (,was reproduced when the thermal gradient was zero anmelting occurred.

References

Allison, M.L., Clifford, S.M., 1987. Ice-covered water volcanismGanymede. J. Geophys. Res. 92, 7865–7876.

Barnett, D.N., Nimmo, F., McKenzie, D., 2002. Flexure of Venuslithosphere measured from residual topography and gravity. J. GeoRes. 107,10.1029/2000JE001398, 5007.

Billings, S.E., Kattenhorn, S.A., 2002. Determination of ice crust thiness from flanking cracks along ridges on Europa, Lunar Planet.XXXIII. 1813.

Buck, L., Chyba, C.F., Goulet, M., Smith, A., Thomas, P., 2002. Petence of thin ice regions in Europa’s ice crust. Geophys. Res. Lett10.1029/2002GL016171, 2055.

Budd, W.F., Jacka, T.H., 1989. A review of ice rheology for ice-sheet meling. Cold Reg. Sci. Technol. 16, 107–144.

Carlson, R.W., Johnson, R.E., Anderson, M.S., 1999. Sulfuric acid onropa and the radiolytic sulfur cycle. Science 286, 97–99.

Carr, M.H., 21 colleagues, 1998. Evidence for a subsurface ocean oropa. Nature 391, 363–365.

Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, seconOxford Univ. Press, Oxford, UK.

Collins, G.C., Head, J.W., Pappalardo, R.T., Spaun, N.A., 2000. Evaluof models for the formation of chaotic terrain on Europa. J. GeopRes. 105, 1709–1716.

De Pater, I., Marchis, F., Macintosh, B.A., Roe, H.G., Le Mignant, D., Gham, J.R., Davies, A.G., 2004. Keck AO observations of Io in andof eclipse. Icarus 169, 250–263.

Fanale, F.P., Salvail, J.R., Matson, D.L., Brown, R.H., 1990. The effecvolume phase-changes, mass-transport, sunlight penetration andfication on the thermal regime of icy regoliths. Icarus 88, 193–204.

Fanale, F.P., Li, Y.-H., De Carlo, E., Farley, C., Sharma, S.K., Horton,Granahan, J.C., 2001. An experimental investigation of Europa’s ocomposition independent of Galileo orbital remote sensing. J. GeopRes. 106, 14595–14600.

Figueredo, P., Chuang, F.C., Rathbun, J., Kirk, R.L., Greeley, R., 2002ology and origin of Europa’s “Mitten” feature (Murias Chaos). J. Gephys. Res. 107,10.1029/2001JE001591, 5026.

Gaidos, E., Nimmo, F., 2000. Tectonics and water on Europa. Nature637.

Giese, B., Oberst, J., Roatsch, T., Neukum, G., Head, J.W., PappaR.T., 1998. The local topography of Uruk Sulcus and Galileo Reobtained from stereo images. Icarus 135, 303–316.

Goodman, J.C., Collins, G.C., Marshall, J., Pierrehumbert, R.T., 2004drothermal plume dynamics on Europa: Implications for chaos fortion. J. Geophys. Res. 109,10.1029/2003JE002073, E03008.

Greenberg, R., Hoppa, G., Tufts, R., Geissler, P., Riley, J., Kadel, S., 1Chaos on Europa. Icarus 141, 263–286.

Greenberg, R., Geissler, P., Hoppa, G.V., Tufts, B.R., 2002. Tidal-tecprocesses and their implications for the character of Europa’s ice cRev. Geophys. 40, 1–33.

Greenberg, R., Leake, M.A., Hoppa, G.V., Tufts, B.R., 2003. Pits and upon Europa. Icarus 161, 102–126.

Head, J.W., Pappalardo, R.T., 1999. Brine mobilization during lithosphheating on Europa: Implications for formation of chaos terrain, lentictexture, and color variations. J. Geophys. Res. 104, 27143–27155.

Hurford, T.A., Preblich, B., Beyer, R.A., Greenberg, R., 2004. FlexureEuropa’s lithosphere due to ridge-loading. Lunar Planet. Sci. XXX1831.

.

i-

,

Kadel, S.D., Chuang, F.C., Greeley, R., Moore, J.M., 2000. Geologicatory of the Tyre region of Europa: A regional perspective on eurosurface features and ice thickness. J. Geophys. Res. 105, 22657–2

Kargel, J.S., Kaye, J.Z., Head, J.W., Marion, G.M., Sassen, R., Cley, J.K., Ballesteros, O.P., Grant, S.A., Hogenboom, D.L., 2000.ropa’s crust and ocean: Origin, composition and the prospects forIcarus 148, 226–265.

Mader, H.M., 1992. Observations of the water-vein system in polyctalline ice. J. Glaciol. 38, 333–347.

McKenzie, D., 1989. Some remarks on the movement of small melt ftions in the mantle. Earth Planet. Sci. Lett. 95, 53–72.

McKenzie, D., Ford, P.G., Liu, F., Pettengill, G.H., 1992. Pancake-domes on Venus. J. Geophys. Res. 97, 15967–15975.

McKenzie, D., Nimmo, F., 1999. The generation of martian floods by ming permafrost above dykes. Nature 397, 231–233.

McKinnon, W.B., 1999. Convective instability in Europa’s floating ice shGeophys. Res. Lett. 26, 951–954.

Melosh, H.J., Ekholm, A.G., Showman, A.P., Lorenz, R.D., 2004. The tperature of Europa’s subsurface water ocean. Icarus 168, 498–502

Murray, C.D., Dermott, S.F., 1999. Solar System Dynamics. CambrUniv. Press, Cambridge, UK.

Nimmo, F., 2004. Non-Newtonian topographic relaxation on EuroIcarus 168, 205–208.

Nimmo, F., Manga, M., 2002. Causes, characteristics and coquences of convective diapirism on Europa. Geophys. Res. Lett10.1029/2002GL015754, 2109.

Nimmo, F., Pappalardo, R.T., Giese, B., 2003a. On the origins of btopography, Europa. Icarus 166, 21–32.

Nimmo, F., Giese, B., Pappalardo, R.T., 2003b. Estimates of Europashell thickness from elastically-supported topography. Geophys.Lett. 30,10.1029/2002GL016660, 1233.

O’Brien, D.P., Geissler, P., Greenberg, R., 2002. A melt-through modechaos formation on Europa. Icarus 156, 152–161.

Pappalardo, R.T., Head, J.W., Greeley, R., Sullivan, R.J., Pilcher, C., Sbert, G., Carr, M.H., Moore, J.M., Belton, M.J.S., Goldsby, D.L., 19Geological evidence for solid-state convection in Europa’s ice shell.ture 391, 365–368.

Pappalardo, R.T., Barr, A.C., 2004. The origin of domes on EuroThe role of thermally induced compositional buoyancy. Geophys.Lett. 31,10.1029/2003GL019202, L01701.

Peale, S.J., Cassen, P., Reynolds, R.T., 1979. Melting of Io by tidal distion. Science 203, 892–894.

Prieto-Ballesteros, O., Kargel, J.S., Fernandex-Sampedro, M., HogenbD.L., 2004. Evaluation of the possible presence of CO2 clathrates inEuropa’s icy shell or seafloor. Lunar Planet. Sci. XXXV. 1748.

Prieto-Ballesteros, O., Kargel, J.S., 2004. Thermal state and complexogy of a heterogeneous salty crust of Jupiter’s satellite, Europa. IcIn press.

Rathbun, J.A., Musser, G.S., Squyres, S.W., 1998. Ice diapirs on EuImplications for liquid water. Geophys. Res. Lett. 25, 4157–4160.

Ross, R.G., Kargel, J.S., 1998. Thermal conductivity of Solar Systemwith special reference to martian polar caps. In: Schmitt, B., de BeC., Feston, M. (Eds.), Solar System Ices. Kluwer Academic, Dordrepp. 33–62.

Schenk, P.M., Pappalardo, R.T., 2004. Topographic variations in chaoEuropa: Implications for diapiric formation. Geophys. Res. Lett.10.1029/2004GL019978, L16703.

Showman, A.P., Han, L.J., 2004. Numerical simulations of convecin Europa’s ice shell: Implications for surface features. J. GeopRes. 109,10.1029/2003JE002103, E01010.

Sotin, C., Head, J.W., Tobie, G., 2002. Europa: Tidal heating of upwethermal plumes and the origin of lenticulae and chaos melting. GeopRes. Lett. 29,10.1029/2001GL013844, 1233.

Spaun, N.A., Head, J.W., Collins, G.C., Prockter, L.M., Pappalardo, R1998. Conamara Chaos Region, Europa: Reconstruction of m

http://dx.doi.org/10.1029/2000JE001398http://dx.doi.org/10.1029/2002GL016171http://dx.doi.org/10.1029/2001JE001591http://dx.doi.org/10.1029/2003JE002073http://dx.doi.org/10.1029/2002GL015754http://dx.doi.org/10.1029/2002GL016660http://dx.doi.org/10.1029/2003GL019202http://dx.doi.org/10.1029/2004GL019978http://dx.doi.org/10.1029/2003JE002103http://dx.doi.org/10.1029/2001GL013844

340 F. Nimmo, B. Giese / Icarus 177 (2005) 327–340

ae onanet.

era-ime

wate

Eu-. 106

ion:108,

thehell.

ress,

.D.,hys.

Eu-

na-.on106,

polygonal ice blocks. Geophys. Res. Lett. 25, 4277–4280.Spaun, N.A., Head, J.W., Pappalardo, R.T., 1999. Chaos and lenticul

Europa: Structure, morphology and comparative analysis. Lunar PlSci. XXX. 1276.

Spencer, J.R., Tamppari, L.K., Martin, T.Z., Travis, L.D., 1999. Temptures on Europa from Galileo photopolarimeter–radiometer: Nighttthermal anomalies. Science 284, 1514–1516.

Squyres, S.W., Reynolds, R.T., Cassen, P.M., Peale, S.J., 1983. Liquidand active resurfacing on Europa. Nature 301, 225–226.

Thomson, R.E., Delaney, J.R., 2001. Evidence for a weakly stratifiedropan ocean sustained by seafloor heat flux. J. Geophys. Res12355–12365.

Tobie, G., Choblet, G., Sotin, C., 2003. Tidally heated convectConstraints on Europa’s ice shell thickness. J. Geophys. Res.10.1029/2003JE002099, 5124.

r

,

Tobie, G., Choblet, G., Lunine, J., Sotin, C., 2004. Interaction betweenconvective sublayer and the cold fractured surface of Europa’s ice sWorkshop on Europa’s icy shell, LPI Contribution 1195, 89–90.

Turcotte, D.L., Schubert, G., 2002. Geodynamics. Cambridge Univ. PCambridge, UK.

Veeder, G.J., Matson, D.L., Johnson, T.V., Blaney, D.L., Goguen, J1994. Io’s heat flow from infrared radiometry: 1983–1993. J. GeopRes. 99, 17095–17162.

Wang, H., Stevenson, D.J., 2000. Convection and internal meting ofropa’s ice shell. Lunar Planet. Sci. XXXI. 1293.

Williams, K.K., Greeley, R., 1998. Estimates of ice thickness in the Comara Chaos region of Europa. Geophys. Res. Lett. 25, 4273–4276

Zolotov, M.Y., Shock, E.L., 2001. Composition and stability of saltsthe surface of Europa and their oceanic origin. J. Geophys. Res.32815–32827.

http://dx.doi.org/10.1029/2003JE002099

Thermal and topographic tests of Europa chaos formation models from Galileo E15 observationsIntroductionObservationsDiapirismMelting modelMelt drainage timescalesUpliftComparison with observations

Melt-through modelDiscussion and conclusionsAcknowledgmentsReferences