Icarus 207 (2010) 834–844

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Icarus

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Coupled convection and tidal dissipation in Europa’s ice shell

Lijie Han a,*, Adam P. Showman b

a Planetary Science Institute, 1700 E. Fort Lowell, Suite 106, Tucson, AZ 85719, USAb Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 May 2009Revised 29 October 2009Accepted 22 December 2009Available online 6 January 2010

Keywords:EuropaIcesJupiterSatellites

0019-1035/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.icarus.2009.12.028

* Corresponding author.E-mail addresses: han@psi.edu (L. Han), showman

man).

We performed 2D numerical simulations of oscillatory tidal flexing to study the interrelationshipbetween tidal dissipation (calculated using the Maxwell model) and a heterogeneous temperature struc-ture in Europa’s ice shell. Our 2D simulations show that, if the temperature is spatially uniform, the tidaldissipation rate peaks when the Maxwell time is close to the tidal period, consistent with previous stud-ies. The tidal dissipation rate in a convective plume encased in a different background temperaturedepends on both the plume and background temperature. At a fixed background temperature, the dissi-pation increases strongly with plume temperature at low temperatures, peaks, and then decreases withtemperature near the melting point when a melting-temperature viscosity of 1013 Pa s is used; however,the peak occurs at significantly higher temperature in this heterogeneous case than in a homogeneousmedium for equivalent rheology. For constant plume temperature, the dissipation rate in a plumedecreases as the surrounding temperature increases; plumes that are warmer than their surroundingscan exhibit enhanced heating not only relative to their surroundings but relative to the Maxwell-modelprediction for a homogeneous medium at the plume temperature. These results have important implica-tions for thermal feedbacks in Europa’s ice shell.

To self-consistently determine how convection interacts with tidal heating that is correctly calculatedfrom the time-evolving heterogeneous temperature field, we coupled viscoelastic simulations of oscilla-tory tidal flexing (using Tekton) to long-term simulations of the convective evolution (using ConMan).Our simulations show that the tidal dissipation rate resulting from heterogeneous temperature can havea strong impact on thermal convection in Europa’s ice shell. Temperatures within upwelling plumes aregreatly enhanced and can reach the melting temperature under plausible tidal-flexing amplitude forEuropa. A pre-existing fracture zone (at least 6 km deep) promotes the concentration of tidal dissipation(up to �20 times more than that in the surroundings), leading to lithospheric thinning. This supports theidea that spatially variable tidal dissipation could lead locally to high temperatures, partial melting, andplay an important role in the formation of ridges, chaos, or other features.

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1. Introduction

Galileo data show that Europa’s surface consists primarily ofsubdued plains, ridges, bands, chaos, pits, domes, and craters(Greeley et al., 1998, 2000, 2004; Senske et al., 1998; Head et al.,1999; Prockter et al., 1999; Figueredo and Greeley, 2000, 2004).Ridges on Europa’s surface can extend several hundreds of kilome-ters with widths of a few kilometers (Smith et al., 1979; Lucchittaand Soderblom, 1982) and heights up to a few hundred meters(Greeley et al., 2004; Malin and Pieri, 1986). Chaos regions varyfrom a few kilometers to hundreds of kilometers across (Greenberget al., 1999; Riley et al., 2000) and up to 250 m in height (Schenk

ll rights reserved.

@lpl.arizona.edu (A.P. Show-

and Pappalardo, 2004). Galileo images indicate that Europa’s sur-face has numerous small (3–30 km-diameter) landforms, includinguplifts, pits, and irregular lobate features (Pappalardo et al., 1998;Greeley et al., 1998; Spaun et al., 1999; Spaun, 2002; Riley et al.,2000; Greenberg et al., 2003). The uplifts (domes) and pits canreach 100–300 m or more topographically (Schenk and McKinnon,2001; Schenk and Pappalardo, 2004; Greenberg et al., 2003),although debate exists about the size distribution of the features.

Two end-member models have been suggested for the thicknessof the ice shell and the formation mechanism of the surface fea-tures. Greenberg et al. (1998, 1999, 2002, 2003) and O’Brien et al.(2002) preferred a thin-shell model (�few kilometers thickness).In this scenario, non-synchronous and tidal stress dominate thefracturing on Europa’s surface, and diurnal stress makes the exist-ing fractures open and close daily to form ridges and, eventually,bands. Chaos results from local melting of the ice shell due to the

Fig. 1. Model set up. A circular plume of temperature Tplume is encased in aconvective background with different temperature ðTbackÞ. The domain is20 � 20 km, and the plume has a radius of 1 km. The side walls are displacedsinusoidally with a strain amplitude of 1:25� 10�5 and orbital period of 3.5 days.

L. Han, A.P. Showman / Icarus 207 (2010) 834–844 835

local concentration of (tidal) heating at the bottom of a thin-iceshell, while incomplete or aborted melt-through events wouldproduce pits (Greenberg et al., 2003). An alternate formation sce-nario proposes that Europa has a thick ice shell (>10 km). Pits, do-mes, and chaos resulted from solid-state convection in the ice shell,perhaps aided by partial melting from tidal dissipation (Pappalardoet al., 1998; Rathbun et al., 1998; Head and Pappalardo, 1999;Collins et al., 2000; Spaun et al., 1998; Spaun, 2002). In thisscenario, domes would occur when ascending diapirs of buoyantice impinge against the lithosphere, while pits would form whencold, dense plumes detach from the lithosphere and descend intothe interior of the ice shell. Surface disruption (leading to chaos)would naturally result if the convective stresses exceeded thelithospheric yield strength.

In either scenario, heterogeneous tidal heating could play animportant role in generating surface features. Heterogeneous, tem-perature-dependent heating has been suggested to promote theformation of Europan ridges (Nimmo and Gaidos, 2002), pits anduplifts (Sotin et al., 2002), and chaos (Collins et al., 2000). In thesemodels, modest temperature increases that occur locally can softenthe ice, enhance the tidal dissipation, and potentially lead to a run-away heating that enhances horizontal temperature differences, al-lows localized partial melting, and promotes surface disruption.

Several authors have made attempts to test the hypothesis thatshear heating could cause Europa’s ridges under plausible assump-tions about the tidal-flexing amplitude and other parameters.Nimmo and Gaidos (2002) demonstrated that frictional heatingcan indeed localize along tidally sheared faults, and using a 2D dif-fusion model, they estimated that temperature increases along thefault could reach �60 K. Simple buoyancy arguments suggest thatthese temperature increases can lead to flexural uplift of up to�100 m, which is encouragingly close to typical ridge heights onEuropa. However, their shear-heating calculation did not consideradvection of the ice and so could not rigorously determine howthe ice deforms in response to the shear heating. Moreover, theycalculated the shear heating under the assumption that the shearvelocity along the fault zone is constant rather than oscillatory,as would occur with tidal flexing. This forces the full strain ratein the brittle layer to occur along the fault rather than partitioningbetween fault slip and distributed elastic deformation (Han andShowman, 2008) and changes the temperature dependence ofthe heating in the viscous layer. Taking a different strategy, Hanand Showman (2008) modeled the convective response to an im-posed shear heating, and they showed that the shear heating canlocally thin the stagnant lid, potentially trigger linear diapirism,and generate ridge-like topographic features �100 m tall. How-ever, their shear heating was imposed by hand rather than self-consistently determined from the heterogeneous temperaturefield. Finally, based on finite-element calculations of tidal‘‘walking,” Preblich et al. (2007) performed order-of-magnitudeestimates suggesting that only 3–5 K of heating would result fromshear heating. However, they did not calculate the dissipationrigorously, nor did they model the long-term thermal evolution,so their estimate does not account for the positive feedbackwhere modest initial temperature increases induce softening thatstrengthens the localization of strain below the fault, leading togreater heating rates as the runaway progresses.

Many authors have performed convection simulations to studythe thermal evolution of Europa’s ice shell (Sotin et al., 2002; Tobieet al., 2003; Barr et al., 2004; Barr and Pappalardo, 2005; Showmanand Han, 2004, 2005; Han and Showman, 2005; see Barr and Show-man (2009) for a review). So far, all these studies have included atidal heating that is either constant or depends only on local tem-perature, following the predictions of an isothermal Maxwell mod-el with zero spatial dimensions (Sotin et al., 2002; Tobie et al.,2003; Showman and Han, 2004, 2005; Han and Showman, 2005).

However, this is not self-consistent: in reality, tidal dissipation de-pends not only on local temperature, but also on surrounding tem-perature (Mitri and Showman, 2008). Mitri and Showman (2008)constructed an idealized 2D analytical model to show that a con-vective plume encased in a background of different temperaturecan experience strongly temperature-dependent tidal dissipation.However, because of their analytic approach, their calculationwas limited to Newtonian rheology; moreover, they did not coupletheir heating calculation to a convection calculation. To fullyunderstand the impact of tidal dissipation on convection and tec-tonics on Europa, it is important to solve the fully coupled problemof thermal evolution and oscillatory tidal flexing.

Here, we perform several related calculations to study tidalheating and its impact on convection in Europa’s ice shell. InSection 2, we present 2D viscoelastic numerical simulations ofthe tidal oscillation process in isolation, to determine how a heter-ogeneous temperature structure—such as an isolated warm or coldconvective plume—affects the spatially variable tidal heating. Thisis similar in approach to Mitri and Showman (2008) but allows usto explore a range of creep-flow mechanisms, both Newtonian andnon-Newtonian. In Section 3, we self-consistently couple these vis-coelastic tidal-heating calculations to simulations of the convectiveevolution, which reveal how the feedback between convection andtidal dissipation affects the state of the ice shell. In Section 4, wefocus on the shear-heating problem by including a localized weakzone to represent a pre-existing fracture, and determine how thecoupled convection and tidal dissipation responds to such a weakzone. Section 5 concludes and discusses geological implications.The emphasis at this stage is exploratory, with the primary goalof identifying the effects that coupling has in idealized models.The model set ups are thus intentionally kept simple.

2. Tidal dissipation with heterogeneous temperature structure

We use the 2D finite-element code Tekton (Melosh and Raefsky,1980) to solve for the oscillatory tidal stress, strain, and tidal heat-ing rate in the presence of temperature heterogeneities in Europa’sice shell. Following Mitri and Showman (2008), we adopt a 2Dmodel with a circular region of temperature Tplume surrounded bybackground ice of a different temperature Tback, representing a hor-izontal cross section through an isolated, vertically oriented con-vective plume. See Fig. 1 for the model set up. We adoptviscoelastic rheology, which is important because the tidal periodis close to the Maxwell time. The Young’s modulus is 1010 Pa andPoisson ratio is 0.25 (Gammon et al., 1983).

836 L. Han, A.P. Showman / Icarus 207 (2010) 834–844

The viscous rheology of ice can be described using a power-lawrelationship (Durham et al., 1997; Goldsby and Kohlstedt, 2001)

_� ¼ Arn

dp exp � QRT

� �ð1Þ

where _� is strain rate, A is a material constant, r is stress, n is stressexponent, d is grain size, p is grain-size exponent, Q is activation en-ergy, R is the gas constant, and T is temperature. We explore threedifferent creep-flow mechanisms: diffusion, grain boundary sliding(GBS), and dislocation creep. See Table 1 for the rheological param-eters. While the grain sizes are unknown, estimates have suggested0.1–1 mm (Kirk and Stevenson, 1987; Barr and McKinnon, 2007). Tobracket this range, we explore values spanning 0.1–1 mm. For thepresent study, we neglect pre-melting, which could lead to a viscos-ity drop and greater strain rates near the melting temperature thanare captured in Eq. (1) (e.g., Dash et al., 1995; Duval, 1977; De LaChapelle et al., 1999; Tobie et al., 2003).

In the numerical models, the calculation domain is a squarewith dimensions L of 20 km � 20 km and a resolution of100 � 100 finite elements (200 m grid spacing). To simulate the ti-dal-flexing process, we specify a displacement boundary conditionwherein the position of the sidewalls at x ¼ L=2 and �L=2 are dis-placed sinusoidally in time as n0 sin xt and �n0 sin xt, respectively,where n0 is 0.125 m and x ¼ 2:07� 10�5 s�1, corresponding toEuropa’s orbital period of 3.5 Earth days. The amplitude is chosenso that the domain-averaged peak-to-peak tidal-strain amplitudeis 2n0=L ¼ 1:25� 10�5, which is within the range of global valuesof �0 � 1–2� 10�5 calculated for Europa’s expected internal struc-ture (i.e., an ice shell, internal ocean, and silicate core) and currentorbital eccentricity (Ojakangas and Stevenson, 1989; Ross andSchubert, 1987; Moore and Schubert, 2000). Note, however, thatdepending on the temperature structure, the local value of tidal-flexing strain within the domain can deviate from the spatial-meanvalue 2n0=L; this is an important aspect, since strain localization isa major mechanism for generating spatially concentrated tidalheating. The horizontal boundaries at y ¼ �L=2 are free (deform-able) surfaces, along which the stress-tensor components ryy andrxy are taken to be zero. The calculations adopt plane strain.

Globally tidal potential varies with latitude and longitude (Kau-la, 1964). However, our models only consider a regional geometry,less than 10�4 of the entire spherical shell in area. The variations inthe mean tidal-flexing strain that occur across distances compara-ble to Europa’s radius (e.g., between the equator and pole) cantherefore be neglected on the scale of our domain (20 km-width).Likewise, Europa’s tidal strain may vary in radius (Kaula, 1964; To-bie et al., 2005). But our models only consider the top 20 km of theice layer, representing about 1% of Europa in radius. Thus, we ne-glect the radial tidal strain variation within this layer.

We start the simulations with zero stress, and run the simula-tion for 5–10 tidal cycles until the initial transients die out and atemporally periodic solution is achieved. Each tidal cycle is re-solved with 85 timesteps. We then calculate the tidal dissipationrate at each cell of the 2D domain by integrating stress times strainrate over the last tidal cycle, yielding a heating rate per volume:

Table 1Rheological parameters.

Creepmechanism

log(A) n p Q (kJ/mol)

Reference

Diffusion �3.46 1 2 60 Goldsby and Kohlstedt(2001)

GBS �2.4 1.8 1.4 49 Goldsby and Kohlstedt(2001)

Dislocation 5.1 4.0 0 61 Kirby et al. (1987)

qðx; yÞ ¼ 1Ds

Irijðx; y; tÞ _�ijðx; y; tÞdt ð2Þ

where q is the tidal heating rate, rij is stress tensor, _�ij is strain-ratetensor, t is time, and Ds is the tidal cycle period. Subscripts corre-spond to the x and y coordinate axes; repeated indices implysummation.

Fig. 2 displays the tidal dissipation rates versus temperaturefrom models with a homogeneous (spatially independent) temper-ature distribution. Each point corresponds to a distinct Tekton sim-ulation; the curves connect families of simulations with the samerheology but different temperatures. Under diffusion-creep rheol-ogy with an ice grain size of 0.5 mm (corresponding to melting-temperature viscosity at 1014 Pa s), the tidal dissipation rate peaksat a temperature �270 K. If the ice grain size decreases to 0.15 mm(corresponding to melting-temperature viscosity of 1013 Pa s), thetidal dissipation rate peaks at temperature of �250 K. This is con-sistent with previous estimates of tidal dissipation of uniform tem-perature structure with diffusion-creep rheology (Mitri andShowman, 2008). Under grain boundary sliding (GBS) with an icegrain size of 0.1 mm, our results show that the tidal dissipationrate is comparable to that obtained from models using diffusionrheology with a melting-temperature viscosity of 1013 Pa s. Whenthe ice grain size increases to 1 mm, GBS models show that tidaldissipation is very close to diffusion models with a melting-tem-perature viscosity of 1014 Pa s. The tidal dissipation rate is abouttwo orders of magnitude smaller if dislocation creep is imple-mented. For the temperatures, stresses, and grain sizes that arelikely in a convecting Europa’s ice shell, deformation by diffusionand grain boundary sliding (GBS) are most relevant (Goldsby andKohlstedt, 2001; Durham and Stern, 2001; McKinnon, 1998, 1999).

Note that, although previous studies have explored the tidaldissipation rate under the assumption of Newtonian rheology(which in the case of sinusoidal forcing leads to a sinusoidal re-sponse that is straightforward to treat analytically), this is the first

Fig. 2. Volumetric tidal dissipation rate as a function of temperature. Differentsymbols represent results from models with different rheological parameters. Star –diffusion-creep rheology with ice grain size of 0.5 mm; circle – diffusion-creeprheology with grain size of 0.15 mm; square – GBS creep rheology with grain size of0.1 mm; triangle – GBS creep rheology with grain size of 1 mm; plus – dislocationcreep rheology.

Fig. 3. Volumetric tidal dissipation rate in a plume versus the plume temperature.Diffusion-creep rheology (with a grain size of 0.15 mm) is used in the models.Different symbols represent the results from different background temperature.Stars, circles, and triangles represent cases with background temperatures of 200,250, and 260 K, respectively; plus signs represent cases with homogeneoustemperature.

Fig. 4. Volumetric tidal dissipation rate in a plume versus the plume temperatureusing grain boundary sliding (GBS) creep rheology with a grain size of 0.1 mm.Different symbols represent the results from different background temperature.Stars, circles, and triangles represent cases with background temperatures of 200,250, and 260 K, respectively; plus signs represent cases with homogeneoustemperature.

L. Han, A.P. Showman / Icarus 207 (2010) 834–844 837

study to rigorously quantify the tidal dissipation rate for non-New-tonian rheology.

Fig. 3 shows that the tidal dissipation rate in a convective plumedepends strongly on plume temperature when the background icetemperature remains fixed and diffusion creep is implemented. Ti-dal dissipation rates depend on both local and background temper-ature. Diffusion-creep rheology with a melting-temperatureviscosity of 1013 Pa s is implemented in the simulations. Fig. 3 con-siders a heterogeneous ice shell containing a plume embedded inan environment of different temperature as in Mitri and Showman(2008). The plume is 1 km in radius within a domain of 20 km inlength and width (see Fig. 1). Each point corresponds to a simula-tion with a convective plume at one temperature and backgroundice at a different temperature, and the curves connect simulationsthat hold the background temperature constant (at 200, 250, or260 K) yet consider different plume temperatures. For comparison,Fig. 3 also includes simulations with homogeneous temperature inthe calculation domain (plus signs). The tidal dissipation rate in aplume with lower temperature than the background is smallerthan the tidal dissipation rate calculated from a homogeneoustemperature. Tidal dissipation rate in a plume with higher temper-ature (than the background) is greatly enhanced compared withthe results from a homogeneous temperature. Furthermore, tidaldissipation rate in a convective plume at one particular tempera-ture decreases as the surrounding temperature increases. Forexample, the tidal dissipation rate in a plume of 255 K decreasesfrom 2� 10�5 to 4� 10�6 W m�3 if the surrounding temperatureincreases from 200 K to 260 K. When the surrounding ice is warm,the viscosity in the surrounding ice is smaller compared with thatin the plume. The strain is mostly taken up in the surroundingwhen they are warm and soft (limiting the available strain andhence tidal dissipation that can occur in the plume).

Fig. 4 shows that the tidal dissipation rate in a convective plumedepends strongly on the plume temperature when the backgroundice temperature remains fixed and grain boundary sliding (GBS)creep rheology is implemented. An ice grain size of 0.1 mm is usedin these simulations. Analogously to Fig. 3, each point correspondsto a simulation with a convective plume at one temperature and

background ice at a different temperature, and the curves connectsimulations that hold the background temperature constant (at200, 250, or 260 K) yet consider different plume temperatures.The plus signs represent simulations with homogeneous tempera-ture ice when grain boundary sliding creep rheology is used. Aswith our simulations using diffusion-creep rheology, the tidal dis-sipation rate in a hot convective plume surrounded by cooler icecan become greatly enhanced relative to the dissipation rate forhomogeneous temperature. The tidal dissipation rate in a plumeat one particular temperature decreases as the surrounding icetemperature increases.

In our simulations, the strong dependence of the tidal dissipa-tion rate on the temperature of the plume and its surroundingsis qualitatively consistent with Mitri and Showman’s (2008) resultsand supports the idea that runaway heating can potentially occurin convective plumes for appropriate parameters, as originally sug-gested by Sotin et al. (2002). Nevertheless, our results exhibit somemodest differences with Mitri and Showman in terms of dissipa-tion magnitude and the temperature at which the dissipationpeaks. These differences may be caused by several factors: First,our models implemented displacement boundary conditions,while Mitri and Showman (2008) used stress boundary conditions.Second, an infinite domain was used in Mitri and Showman’s(2008) model, while our numerical simulations implement a finitedomain. Third, several physical parameters used in the models areslightly different (for example, the Poisson ratio is 0.314 in Mitriand Showman (2008), and it is 0.25 in our models).

3. Coupled thermal convection with tidal dissipation

Next, we perform fully coupled numerical simulations of ther-mal convection and tidal heating that we self-consistently calculatefrom the time-evolving temperature structure. We again adopt 2Dcartesian (rectangular) geometry, with the dimensions in this caserepresenting horizontal position x and height z. Cartesian geometryis appropriate for regional studies of Europa’s ice shell becauseEuropa’s ice-shell thickness is much smaller than its radius. We ne-glect inertia and adopt the Boussinesq approximation. We use the

838 L. Han, A.P. Showman / Icarus 207 (2010) 834–844

finite-element code ConMan (King et al., 1990) to solve the dimen-sionless equations of momentum, continuity, and energy, respec-tively given by

@rij

@xjþ Rahki ¼ 0 ð3Þ

@ui

@xi¼ 0 ð4Þ

@h@tþ ui

@h@xi¼ @

2h

@x2i

þ q0 ð5Þ

where rij is the dimensionless stress tensor, ui is dimensionlessvelocity, h is dimensionless temperature, q0 is dimensionless heatingrate, ki is the vertical unit vector, t is dimensionless time, xi and xj

are the dimensionless spatial coordinates, and i and j are the coor-dinate indices. Repeated spatial indices imply summation. The Ray-leigh number Ra is given by

Ra ¼ gqaDTd3

jg0ð6Þ

where g is gravity, q is density, a is thermal expansivity, DT is thetemperature drop between the bottom and top boundaries, d isthe depth of the system, j is the thermal diffusivity, and g0 is thedynamic viscosity at the melting temperature. For reference, themodel parameters are presented in Table 2.

The purpose of the numerical simulations here is to study theimpact of tidal heating resulting from heterogeneous temperatureon thermal convection in Europa’s ice shell. We run two sets ofsimulations. For one set, we adopt a tidal dissipation rate depend-ing on local temperature only, based on the zero-dimensional iso-thermal Maxwell model prediction as follows (Sotin et al., 2002;Tobie et al., 2003; Showman and Han, 2004, 2005).

q ¼ �20x2g

2 1þ x2g2

l2

h i ð7Þ

where x is Europa’s tidal-flexing frequency, l ¼ 4� 109 Pa is therigidity of ice, and �0 is the amplitude of tidal-flexing strain (as-sumed constant throughout the domain). While this is not fullyself-consistent, this approach has been adopted by most previousauthors and thus this set of simulations serves as a reference.

For the other set of simulations, we study the tidal dissipationrate and its impact on thermal convection in fully coupled modelsof thermal evolution and tidal heating. This set of simulations in-cludes two components: convection (which is solved using Con-Man and involves timescale of millions of years) and oscillatorytidal flexing (which is solved using the finite-element code Tektonand involves timescale of days to weeks).

We adopt a 2D approach (with dimensions corresponding tohorizontal distance, x, and height, z). This is equivalent to assumingno variation in properties along the coordinate (y) perpendicular to

Table 2Model parameters.

Name Symbol Value

Acceleration of gravity g 1:3 m s�1

Density q 917 kg m�3

Thermal expansivity a 1:65� 10�4 K�1

Thermal diffusivity j 1� 10�6 m2 s�1

Specific heat cp 2000 J kg�1 K�1

Surface temperature Tt 95 KBottom temperature Tb 270 KMelting-temperature viscosity g0 1013 or 1014 Pa sThickness of ice shell d 15 or 20 km

the domain. Because Europan ridges are linear structures withminimal variation along strike, this approach is ideally suited toinvestigate shear heating as a ridge-formation mechanism. Per-forming coupled 3D simulations is a long-term goal but is compu-tationally extremely expensive; the 2D simulations described hererepresent an important first step.

The coupled calculation proceeds as follows. (1) For a giventemperature structure Tðx; zÞ throughout the ice shell, we performviscoelastic calculations assuming oscillatory tidal flexing to deter-mine the stress/strain response throughout a tidal cycle. We thendetermine the tidal dissipation qðx; zÞ for that thermal structureby integrating stress and strain rate over a cycle (Eq. (2)). (2)Including this dissipation as a heat source in Eq. (5), we then inte-grate forward the thermal-evolution equations, accounting forboth thermal diffusion and advection/convection in the ice shell,until the temperature has changed by a modest amount. (3) Thisnew thermal structure Tðx; zÞ is then used to recalculate the tidaldissipation, hence repeating step (1), and the cycle repeats. Theviscoelastic tidal-heating calculation can thus be viewed as aself-consistent means for specifying the time and spatially varyingtidal-dissipation source term in the thermal evolution calculation.We carry the calculation forward until the system reaches a statis-tical steady state.

Because the tidal-oscillation model needs to run�5–10 tidal cy-cles (each cycle being solved with 85 timesteps) it is very compu-tationally expensive to calculate the dissipation rate at everythermal-convection timestep in the coupled model. Moreover,the change in thermal structure (and the resulting change in tidaldissipation rate) per convective timestep is small. We ran testcases to calculate the tidal dissipation rate every 20, 50, 100, or200 timesteps in the thermal convection model, and the thermalstructure is identical in each case. Therefore, in practice, we gener-ally run the tidal-oscillation model to calculate the dissipation rateevery 100–200 timesteps in the thermal convection model.

We perform (both Tekton and ConMan) simulations with ice-shell thicknesses of 15 or 20 km, consistent with ice-shell thick-ness estimated from evolution, crater morphology, and cratersize-distribution studies (Hussmann et al., 2002; Ojakangas andStevenson, 1989; Turtle and Ivanov, 2002; Schenk, 2002). Wemaintain constant ice-shell thickness in any given simulation.The simulations are performed with aspect ratio (ratio of widthto depth) of 1 or 3. The aspect ratio has little effect on the qualita-tive results. The resolution is typically 100 elements in the verticaldirection and 100 or 300 elements in the horizontal directiondepending on the geometric ratio.

As we have described in Section 2, diffusion creep and grainboundary sliding (GBS) creep (the two creep mechanisms most rel-evant for Europa) have qualitatively the same effects on tidal dissi-pation rates under the same heterogeneous temperature structure.Because the current simulations are exploratory attempts tounderstand the coupling between convection and tidal dissipation,for simplicity, here we only consider diffusion creep (i.e. Newto-nian, temperature-dependent viscosity) appropriate for ice (Sotinet al., 2002; Tobie et al., 2003; Showman and Han, 2004, 2005;Han and Showman, 2005, 2008):

gðTÞ ¼ g0 exp ATm

T� 1

� �� �ð8Þ

where T is temperature, Tm is melting temperature (here assumedconstant at 270 K) and g0 is the viscosity at the melting tempera-ture. We fix g0 at 1013 Pa s or 1014 Pa s, appropriate to ice with grainsizes of � 0:1 mm or � 0:5 mm, respectively (Goldsby and Kohl-stedt, 2001). The value of A is assumed constant at 26, correspond-ing to activation energy of 60 kJ mol�1, which implies a viscositycontrast of 1020. We neglect the influence of partial melting on

Fig. 5. Temperature distribution from a fully coupled ConMan/Tekton model ofthermal convection and oscillatory tidal flexing. The model implements a domain-averaged tidal-flexing amplitude of 1:25� 10�5 and tidal period of 3.5 days. Thethickness of the ice shell is 15 km, and the Rayleigh number is 1:81� 107. Top:temperature range of 90–270 K. Bottom: temperature range of 260–270 K.

Fig. 6. Temperature distribution from a model of thermal convection driven bytidal-heating calculated using Eq. (7). The model implements a tidal-flexingamplitude of 1:25� 10�5 and tidal period of 3.5 days. The thickness of the ice shellis 15 km, and the Rayleigh number is 1:81� 107. Top: temperature range of 90–270 K. Bottom: temperature range of 260–270 K.

L. Han, A.P. Showman / Icarus 207 (2010) 834–844 839

viscosity in Eq. (8). Modest increase in melt fraction greatly increasethe percolation rate (Stevenson and Scott, 1991), which tends tostabilize the melt fraction at small values of 0.001–0.01 for condi-tions relevant to Europa (Showman et al., 2004; Gaidos and Nimmo,2000). Therefore we assume the effect of partial melting on viscos-ity can be neglected. Nevertheless, pre-melting may cause addi-tional decrease in viscosity near the melting temperature (Dashet al., 1995; Duval, 1977; De La Chapelle et al., 1999) beyond thatrepresented by Eq. (8) and should be considered in future work.

In the thermal convection models, we fix the bottom boundaryto a temperature of 270 K, the melting temperature. The top surfaceis maintained at 95 K, 35% of the basal temperature. The tempera-ture boundary condition is periodic on the sides. The temperaturewas initialized as a linear function of depth (i.e., a conductive pro-file), with a weak perturbation near the bottom to break the lateralsymmetry. The velocity boundary conditions are periodic on thesides and free-slip rigid walls on the top and bottom.

In the Tekton calculation, we implement a displacement bound-ary condition where the sidewall positions at x ¼ L=2 and �L=2vary in time as n0 sin xt and �n0 sin xt, respectively, with a dis-placement amplitude n0 specified to give a spatial-mean peak-to-peak strain amplitude 2n0=L (where L is the width of the domain)of 1:25� 10�5. The oscillation period is 3.5 Earth days. This bound-ary condition forces the domain to experience a temporally peri-odic cycle of extension/contraction. On Europa, the stress fieldgenerally rotates in time such that a tidal cycle involves bothextension/contraction and shear (e.g., Preblich et al., 2007). Thiscould be taken into account by including in the displacementboundary condition a component of oscillation perpendicular tothe domain, but we would not expect this phenomenon to qualita-tively alter the dissipation patterns, and we leave an exploration ofit for future work. The top/bottom boundaries are open (deform-able) surfaces where rzz and rxz are taken to be zero. In compari-son, in our reference simulations where we determine tidalheating using Eq. (7), we adopt �0 ¼ 1:25� 10�5 or 2:1� 10�5.The strain magnitude of 2:1� 10�5 is chosen so that the peak val-ues of tidal dissipation rates from the Tekton solution and zero-dimensional estimate are identical.

Fig. 5 shows the temperature distribution from a fully coupledsimulation with a mean strain amplitude of 1:25� 10�5 and amelting-temperature viscosity of 1013 Pa s. The domain has a thick-ness of 15 km, with an aspect ratio (width versus depth) of 3. Athick thermal conductive layer is formed at the top of the domain,and the temperature reaches 260–270 K throughout the bottomtwo-thirds of the domain (top panel in Fig. 5). To examine the tem-perature field in more detail, we replot the temperature distribu-tion using a modified scale (bottom panel in Fig. 5) thathighlights the structure in the convection region. Two strongupwelling plumes and several downwellings dominate the lowerpart of the domain, with temperature reaching melting tempera-ture (270 K) in these upwelling plumes.

Fig. 6 shows the temperature distribution from a convectionsimulation using the exact same parameter set up as the modelshown in Fig. 5, but determining the tidal dissipation rate usingEq. (7) rather than Tekton. The model implements a strain magni-tude of 1:25� 10�5. Results show that temperature within the topone-third of the domain is similar to that in Fig. 5, forming a ther-mal conductive layer. However, the temperature in the bottom halfof the domain is about 5 K lower, and the melting temperature isnot reached anywhere in the simulation. On the other hand,Fig. 7 shows the temperature distribution from a simulation simi-lar to the model in Fig. 6 (implementing tidal dissipation rate usingEq. (7)), but with strain magnitude of 2:1� 10�5. The temperatureis 2–3 K lower than the model using the fully coupled convectionand tidal dissipation with a strain magnitude of 1:25� 10�5

(Fig. 5). The melting temperature is not reached in this simulation.

To understand what causes the temperature differences in dif-ferent models, we plot out the tidal dissipation rates for a givenheterogeneous temperature structure using different dissipationcalculation methods (2D tidal oscillation simulation using Tektonversus simple estimation by Eq. (7)) (Fig. 8). Fig. 8a displays the

Fig. 7. Temperature distribution from a model of thermal convection with tidalheating rate calculated using Eq. (7). Tidal flexing amplitude of 2:1� 10�5 and atidal period of 3.5 days are used. The thickness of the ice shell is 15 km, and theRayleigh number is 1:81� 107. Top: temperature range of 90–270 K. Bottom:temperature range of 260–270 K.

Fig. 8. Tidal dissipation rates for a given heterogeneous temperature structure(shown in Fig. 5) using different dissipation calculation methods. (a) Tidaldissipation rates using Tekton simulation based on a tidal-flexing amplitude of1:25� 10�5. (b) Tidal dissipation rates calculated using Eq. (7) with a strainmagnitude of 2:1� 10�5. (c) Tidal dissipation rates calculated using Eq. (7) with astrain magnitude of 1:25� 10�5.

Fig. 9. Temperature distribution from a fully coupled ConMan/Tekton model ofthermal convection and oscillatory tidal flexing. Tidal flexing amplitude of1:25� 10�5 and tidal period of 3.5 days are used. The thickness of the ice shell is15 km, and the Rayleigh number is 1:81� 106. Top: temperature range of 90–270 K.Bottom: temperature range of 260–270 K.

840 L. Han, A.P. Showman / Icarus 207 (2010) 834–844

tidal dissipation rate associated with the temperature structureshown in Fig. 5 using a 2D Tekton simulation based on a meanstrain amplitude of 1:25� 10�5. Tidal dissipation rates under thesame temperature structure using Eq. (7) with different strainmagnitudes of 2:1� 10�5 and 1:25� 10�5 are shown in Fig. 8band c, respectively. The tidal dissipation rate in Fig. 8a considersthe effects from both local and surrounding temperature. On theother hand, the tidal dissipation rates in Fig. 8b and c depend onlyon local temperature. We see that tidal dissipation rate peaks atthe location where the temperature is about 250 K, because theMaxwell times in those locations are close to the tidal oscillationperiod. However, the peak tidal dissipation rate and average tidaldissipation rate in the upwellings are larger (by a factor of 2–5)in Fig. 8a compared with those rates in Fig. 8b and c. This explainswhy the mean temperature is higher in the fully coupled tidal dis-sipation and convection simulation (Fig. 5).

The patterns of thermal convection and tidal dissipation changedramatically if the melting-temperature viscosity is altered. Fig. 9shows the temperature distribution from a fully coupled simula-tion, with the same set up as that in Fig. 5, but with a melting-tem-perature viscosity of 1014 Pa s. A thick thermal conductive layer isformed at the top of the domain, and several strong upwellings anddownwellings dominate the lower part of the domain, with tem-perature reaching melting (270 K) in these upwellings. Fig. 10 dis-plays the tidal dissipation rate for a given heterogeneoustemperature (shown in Fig. 9) using different dissipation calcula-tion methods (2D tidal oscillation simulation using Tekton inFig. 10a versus simple estimation by Eq. (7) in Fig. 10b and c).The tidal dissipation rate within the upwellings (265–270 K) isgreatly enhanced due to the fact the Maxwell time in these areasare very close to the tidal oscillation period.

What overall implication do these results have? An examinationof Figs. 8 and 10 reveals that the horizontal variation of tidal dissi-pation in the fully coupled case (top panels) exceeds that calcu-lated from Eq. (7) with identical temperature structure (bottomtwo panels). In Fig. 8, the peak dissipation occurs at the base of

Fig. 10. Tidal dissipation rates for a given heterogeneous temperature structure(shown in Fig. 9) using different dissipation calculation methods. (a) Tidaldissipation rates using Tekton simulation based on a strain amplitude of 1:25�10�5. (b) Tidal dissipation rates using Eq. (7) with a strain magnitude of 2:1� 10�5.(c) Tidal dissipation rates using Eq. (7) with a strain magnitude of 1:25� 10�5.

L. Han, A.P. Showman / Icarus 207 (2010) 834–844 841

the stagnant lid, but the fully coupled case (Fig. 8a) shows that thedissipation in this layer is significantly greater over ascendingplumes than in other regions; in contrast, the dissipation at thebase of the stagnant lid as estimated by Eq. (7) is more horizontallyuniform (Fig. 8b and c). Likewise, in Fig. 10, the ratio of dissipationin convective upwellings and downwellings is greater in the fullycoupled case than that estimated by Eq. (7). These results wouldsuggest that the fully coupled case might be able to develop greaterspatial heterogeneity than the non-coupled counterparts, perhapspromoting surface disruption. Examination of the temperaturefields (Figs. 5–7, 9) suggests that the fully coupled cases more read-ily reach the melting temperature within local regions such asplumes. This is consistent with our horizontal 2D models fromFigs. 3 to 4, which show that, for a given plume temperature andspatially-averaged tidal-flexing amplitude, a warm plume experi-ences greater dissipation when the dissipation is calculatedself-consistently than when calculated via Eq. (7). Thus, our resultssupport the idea that tidal heating could be sufficiently heteroge-neous to allow (for example) localized partial melting, increasingthe likelihood of chaos formation or other surface disruption. Nev-ertheless, additional work will be needed to fully quantify theimplications of self-consistent coupling for surface tectonics aswell as its effect on metrics such as Nusselt number–Rayleighnumber relationships.

4. Coupled convection and tidal dissipation in the presence of aweak zone

Next, we explore the effect of a weak zone—representing a pre-existing fracture—on the coupled convection and tidal dissipation.

The motivation here is to better understand the extent to whichlocalized heating, modulated by a fracture, could play a role inthe formation of Europan ridges. A weak zone can lead to litho-spheric thinning and regulate thermal convection patterns withinthe ice shell (Tobie et al., 2005; Han and Showman, 2008). Fracturezones in Europa’s ice shell may weaken the material and causeconcentration of tidal stress and/or strain, which may result instrongly localized tidal dissipation (Nimmo and Gaidos, 2002).

To study the impacts of fracture zones on tidal heating and con-vection, we impose a narrow (few kilometer wide) weak zoneextending vertically downward from the surface to the base ofthe lithosphere. The weak zone is implemented by simply usingweaker material (one tenth of the viscosity calculated from Eq.(8)) compared with the surroundings. This approach has beenwidely used to study weak zones (i.e. ridges or subduction zones)in the Earth mantle convection community (e.g., Zhong and Gurnis,1994; Zhong et al., 1998). For simplicity, we did not consider fric-tional heating of the fault movement in our simulations.

We perform simulations with ice shell thicknesses 20 km, withan aspect ratio (ratio of width to depth) of 1. The weak zone is lo-cated at the center of the domain horizontally, with a width of2 km wide, extending from the surface to the depth of 6–10 km.The resolution is 100 elements in the vertical direction and 100elements in the horizontal direction in the domain.

Fig. 11 shows the results from a fully coupled convection and ti-dal dissipation model with a weak zone 2 km wide, extending fromthe surface to a depth of 10 km. The melting-temperature viscosityis 1013 Pa s in this case. The pre-existing fracture zone promotesthe concentration of tidal dissipation. The tidal dissipation rate isgreatly enhanced at temperature of about 250 K within the weakzone (see Fig. 11a). The upwelling plume reaches much shallowerdepth in the weak zone location. This supports the idea that en-hanced heating along pre-existing fractures may be important forthe formation of tectonic features (e.g., ridges). Simulations varyingthe depth of weak zone show that the weak zone has to be at least6 km deep (i.e. fracture zone extending from the surface to a depthof 6 km) for it to have a visible influence on the temperature field.The viscosity near the surface is extremely large and tidal dissipa-tion rate is very small due to the low local temperature.

For comparison, we ran a model implementing the same param-eters as those in Fig. 11, but with tidal dissipation calculated by Eq.(7). The tidal dissipation rate and temperature structure are shownin Fig. 12. The tidal dissipation rate peaks over the top of theupwelling plume (the location where temperature is 250 K), dueto the fact tidal dissipation depends on local temperature only.The high tidal dissipation rate (top panel in Fig. 12) is slightly ele-vated to a shallower depth within the weak zone because of theviscosity decrease there. The upwelling plume penetrates to a shal-lower depth compared with the model without weak zone. How-ever, the tidal dissipation rate is not greatly enhanced in theweak zone as that in Fig. 11. The melting temperature is notreached within the plume in this simulation.

5. Conclusions and discussions

We presented three calculations of the influence of tidal heatingon Europa’s ice shell, emphasizing the possible role that spatiallyheterogeneous temperatures play in affecting the tidal dissipationand its feedback with convection. First, we used the 2D viscoelasticcode Tekton to explore the tidal dissipation that occurs in the pres-ence of an isolated cold or warm plume surrounded by ice of differ-ing temperature. We explored three different creep-flowmechanisms: diffusion, grain boundary sliding (GBS), and disloca-tion creep. Our results show that, if the temperature is spatially uni-form ðTplume ¼ TbackÞ, the tidal dissipation peaks when the Maxwell

Fig. 11. Temperature distribution from a fully coupled model of thermal convectionand tidal oscillation process with a 2 km-width weak zone extending from thesurface to the depth of 10 km. Tidal oscillation process was solved using 2D finite-element code Tekton. Tidal flexing amplitude of 1:25� 10�5 and tidal period of3.5 days are used. The thickness of the ice shell is 20 km, and the Rayleigh number is4:3� 107. Top: temperature range of 90–270 K. Bottom: temperature range of 260–270 K.

Fig. 12. Temperature distribution from a fully coupled model of thermal convectionand tidal oscillation process with a 2 km-width weak zone extending from thesurface to the depth of 10 km. Tidal oscillation process is calculated using Eq. (7).Tidal flexing amplitude of 1:25� 10�5 and tidal period of 3.5 days are. The thicknessof the ice shell is 20 km, and the Rayleigh number is 4:3� 107. Top: temperaturerange of 90–270 K. Bottom: temperature range of 260–270 K.

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time is close to tidal period, consistent with previous estimates oftidal dissipation (Kirk and Stevenson, 1987; Tobie et al., 2003; Mitriand Showman, 2008). Under grain boundary sliding (GBS) rheologywith an ice grain size of 0.1 mm or 1 mm, our results show that thetidal dissipation rate is comparable to that obtained from modelsusing diffusion rheology with a melting-temperature viscosity of1013 Pa s, or 1014 Pa s, respectively. The tidal dissipation rate inEuropa’s ice shell is about two orders-of-magnitude smaller if dis-location creep rheology is implemented in our model.

A heterogeneous temperature structure has a strong impact onthe tidal dissipation rate. The tidal dissipation rate depends notonly on local temperature, but also on surrounding temperature.Our simulations show that tidal dissipation rate in a convectiveplume encased in a different background temperature dependson both plume and background temperature, consistent with theresults of Mitri and Showman (2008). For a given plume tempera-ture, the dissipation rate in the plume decreases as the backgroundtemperature is increased. The dissipation rate is smaller than thehomogeneous prediction (Eq. (7)) when the plume is cold relative

its surroundings. On the other hand, tidal dissipation rate in aplume of higher temperature is significantly enhanced comparedwith the results from the homogeneous prediction.

Second, we performed numerical simulations of coupled con-vection and oscillatory tidal flexing to study the interrelationshipbetween tidal dissipation and the thermal evolution of Europa’sice shell. The tidal dissipation rate based on a simple zero-dimen-sional isothermal Maxwell model (see Eq. (7)) exhibits qualitativedifferences from the self-consistently calculated dissipation rate;the former underestimates the horizontal variation of tidal dissipa-tion, and such spatial variations could play an important role in thegeophysics of the ice shell.

Nevertheless, the temperature dependences indicate that forsmall grain size (corresponding to g0 less than 1013 Pa s), the peakdissipation in a convecting ice shell will occur in cold plumes andat the base of the stagnant lid (where temperature is about 250 K).In contrast, for large grain size (corresponding g0 larger than1014 Pa s), the peak dissipation will occur in warm plumes and nearthe base of the ice shell. These differences have implications forgeophysical feedbacks depending on the grain size (Barr and Show-man, 2009).

L. Han, A.P. Showman / Icarus 207 (2010) 834–844 843

Third, our simulations demonstrate that pre-existing fracturezones or weak zones (at least �6 km deep) promote the concentra-tion of tidal dissipation. The tidal dissipation rate is increased fivetimes or more at the weak zone (Fig. 10a). Upwelling plumes reachshallower depth at the location of the weak zone, leading to en-hanced lithospheric thinning. Melting temperature is reachedwithin certain locations of the upwelling plume. This supportsthe idea that enhanced heating along pre-existing fractures maybe important for the formation of tectonic features (e.g., ridges).

As we noted earlier, the purpose of this study is to test the influ-ence of heterogeneous temperature structures on the tidal-heatingpatterns, and explore the effects that such dissipation has on the con-vective structure in fully coupled models. Our study here is not in-tended to describe the detailed convection patterns in Europa’s iceshell, for which more detailed models will be required in the future.

In our coupled thermal convection and tidal oscillation processmodels, we implemented 2D cartesian geometry by assuming novariation in temperature and tidal oscillation perpendicular tothe plane of the simulation. This is sufficient for idealized tests,especially of the efficacy of the localization of tidal heating in gen-erating Europan ridges. However, 3D geometry is needed to distin-guish cylindrical plume structures from linear structures such asridges. Furthermore, 3D models will allow a better representationof the cyclical tidal stresses, which involve tension, compression,and shear that may have any orientation to pre-existing fractures.In the future, fully coupled convection and tidal oscillation processshould be studied in models with 3D geometry.

Our coupled models neglect the impact of melting on viscosityand effects of latent heating of melting. Melting causes viscosity todrop, and allowing larger strain rate (Dash et al., 1995; Duval,1977; De La Chapelle et al., 1999), which in turn influences the ti-dal dissipation rate. Latent heating of melting buffers the temper-ature increase within the Europa’s ice shell. Future studies shouldinclude these effects.

Although our coupled convection/tidal-oscillation models as-sumed a diffusion-creep rheology, the non-Newtonian grainboundary sliding (GBS) rheology may play an important role andshould be explored in future work. Other areas for future investiga-tion include allowing the grain sizes to evolve, with growth ratesdepending on temperature and strain-rate conditions (cf. Barr andMcKinnon, 2007), and incorporating plastic or damage rheologyinto the simulations to allow for the effects of brittle deformation.

Acknowledgments

This work was supported by Grant NNX06AC41G from theNASA Planetary Geology and Geophysics Research Program to LH.APS was supported by PG&G Grant NNX07AR27G.

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