Europa's Double Ridges Produced by Ice Wedging

Double ridges are sprawling features observed globally across the icy surface of Europa. They consist of two topographic highs flanking a trough. The topographic relief of the ridges is approximately 100 m, and the ridges extend up to hundreds of kilometers in length. The interior structure and dynamics of Europa's ice shell are currently poorly constrained. Therefore, accurate models for the formation of these prominent surface features can be useful for determining how the ice shell operates. We hypothesize that double ridges form as a result of incremental ice wedging. We use both analytical and numerical finite element models to quantify the deformation that occurs as an ice wedge grows incrementally within the ice shell. We show that incremental growth of the ice wedge results in surface deformation that matches the size and shape of typical Europan double ridges, including their topographic relief and surrounding troughs. We find that as the depth of the ice wedge increases, double ridges become broader and shorter. We explore the possibility of local and non‐local sources for the liquid water that freezes to produce the wedge and ultimately argue in favor of local sources of liquid water within the ice shell.


Introduction
Europa is an ocean world with an outer shell of ice encapsulating its global ocean ∼100 km in depth (Kivelson et al., 2000;Schubert et al., 2009).The thickness of the ice shell is poorly constrained but is expected to be roughly on the order of 10-50 km (Howell, 2021;Nimmo & Manga, 2009;Schenk & Turtle, 2009;Vilella et al., 2020).The icy surface is notably lacking in impact craters and is littered with an abundance of diverse geologic features.Depending on the region, surface age estimates range from ∼100 Myr down to only a few tens of Myr, implying a strong likelihood of ongoing geologic activity (e.g., Kattenhorn & Prockter, 2014;Lesage et al., 2021;Nimmo & Manga, 2009;Roberts et al., 2023;Schmidt et al., 2011;Zahnle et al., 2003).As the most pervasive surface feature on Europa, double ridges appear so commonly over the body that they frequently crosscut one another (Prockter & Patterson, 2009).Double ridges consist of a pair of topographic highs, usually around 100 m tall, with a narrow trough running between them and broad shallow troughs bordering the outside.They are generally less than 5 km in total width from the outside of one ridge to the other, but can extend linearly for hundreds of km.Double ridges are observed globally and their formation may be genetically related to other surface features observed on Europa, such as lenticulae (Collins & Nimmo, 2009;Greenberg et al., 2003) and dilational bands (Manga & Sinton, 2004;Prockter & Patterson, 2009).Therefore, understanding the process by which double ridges form can help illuminate their relationship to these other features and provide constraints on the structure and thermodynamics of the ice shell conducive to their formation.Moreover, the surface of Neptune's moon Abstract Double ridges are sprawling features observed globally across the icy surface of Europa.They consist of two topographic highs flanking a trough.The topographic relief of the ridges is approximately 100 m, and the ridges extend up to hundreds of kilometers in length.The interior structure and dynamics of Europa's ice shell are currently poorly constrained.Therefore, accurate models for the formation of these prominent surface features can be useful for determining how the ice shell operates.We hypothesize that double ridges form as a result of incremental ice wedging.We use both analytical and numerical finite element models to quantify the deformation that occurs as an ice wedge grows incrementally within the ice shell.We show that incremental growth of the ice wedge results in surface deformation that matches the size and shape of typical Europan double ridges, including their topographic relief and surrounding troughs.We find that as the depth of the ice wedge increases, double ridges become broader and shorter.We explore the possibility of local and non-local sources for the liquid water that freezes to produce the wedge and ultimately argue in favor of local sources of liquid water within the ice shell.
Plain Language Summary Europa, the second Galilean satellite, is hypothesized to have a global salt-water ocean underneath its outer icy shell.Double ridges are a common feature on Europa's icy surface.They consist of a long trough bordered on either side by uplifted hills of ice.The height of the ridges above the surface is approximately 100 m, and the ridges may extend for hundreds of kilometers over the surface of the moon.Models that show how surface features, like double ridges, may form can tell us about characteristics of the ice shell and underlying ocean that are otherwise hard to measure.We use two different techniques to model a process for how double ridges might form.In this process, water, possibly from the subsurface ocean, enters a long vertical crack in the ice shell and freezes along the sides of the crack.Over time as water continues to enter and freeze in the same place, a new wedge of ice grows inside the ice shell and pushes on material around it.The ice wedge forces ice at the surface of the shell to deform into the same size and shape of double ridges that have been observed on Europa.

CASHION ET AL.
Triton is marked by double ridges with such morphologic similarity to Europan ridges that models of Europan double ridge formation have been applied to their formation (Prockter et al., 2005).Thus, advancements in modeling of Europan double ridges may be applicable to understanding Triton's ridges and interior as well.
Many models exist that attempt to explain the ridge formation, such as cryovolcanism, tidal squeezing, shear heating, and intrusive cryomagmatism (e.g., Greenberg et al., 1998;Han & Showman, 2008;Johnston & Montési, 2014;Kadel et al., 1998).The tidal squeezing model proposes that water is injected into an open crack and partially freezes, then is extruded to the surface during the compressional phase of diurnal tides (Greenberg et al., 1998).In the cryovolcanism model, water travels vertically upward to the surface through a crack in the ice shell.Once at the surface, the water freezes and builds up over time into double ridges flanking the fracture (Kadel et al., 1998).However, it may be difficult for pressurized water to rise along a crack to the surface, especially over the entire length of ridges which can extend for several hundred kilometers (Prockter & Patterson, 2009), and both cryovolcanism and tidal squeezing would be unlikely to produce the observed topographic profiles of the ridges (Dombard, 2013).For example, it is difficult to match the height of ridges with the cryovolcanism model because so much pressure is required to expel the water up to the surface.In the shear heating model (Nimmo & Gaidos, 2002), melt water from friction heating generated by strike-slip motion along fractures percolates downward and creates a void near the surface, which may be closed by the inward motion of ice.The compression may cause uplift of double ridges, although shear heating models that account for advection in the ice indicate that ridges would quickly relax away on timescales of ∼10 7 years without considering the effects of compositional heterogeneity and elasticity (Han & Showman, 2008).The intrusive cryomagmatism model of Johnston and Montési (2014) explores the scenario in which a dike of water is emplaced within the ice shell and then freezes.As the water freezes, it expands and causes stress on the surrounding ice that results in surface displacement consistent with double-ridge morphology but does not predict the troughs observed on the outsides of the ridges (Dombard, 2013;Johnston & Montési, 2014).
In this work, we build on initial research that proposed the incremental ice wedging model for ridge formation (Han & Melosh, 2010;Melosh & Turtle, 2004).The ice wedging model is similar to the model presented in Johnston and Montési (2014) but accounts for the gradual accumulation and freezing of water into an ice wedge within the shell.Europa's proximity to Jupiter and orbital resonance with Io and Ganymede result in vigorous and constantly changing tidal stresses.The cyclical tidal flexing results in tensile stress that can open cracks in Europa's ice shell.During the extensional phase of the tidal cycle when a crack is open, briny water from the pressurized ocean below or some other liquid water reservoir within the ice shell may be injected into the crack and start to freeze along the walls.As this process continues over many tidal cycles, a wedge of ice is formed inside the ice shell.The stresses induced by the emplacement and growth of the subsurface ice wedge cause uplift of the surface into double-ridge-shaped topography.The process of incremental ice wedging is shown schematically in Figure 1.This work is a culmination of findings since the ice wedging model was introduced in a series of conference abstracts (Han & Melosh, 2010;Melosh & Turtle, 2004;Melosh et al., 2017).Here, we build on previous work simulating double ridge formation and explore the parameter space using both analytical and numerical models.In Section 2, we present the results of an analytical model for ridge formation by incremental ice wedging.In Section 3, we present Finite Element Method (FEM) simulations of ridge formation with different sources of water.In Section 4.1, we discuss the implications and issues of wedge material being sourced from the ocean along an entire crack.In Section 4.2, we discuss the strengths associated with a more local source of wedge material.Finally, in Section 5 we discuss the implications of the incremental ice wedging model more broadly.

Analytical Edge Dislocation Models
Dislocation theory is the study of irregularities in the crystalline order of solids.A dislocation in a solid is conventionally defined as a disruption in the ordering of atoms in the lattice structure, but dislocation theory has been extensively applied to model large-scale defects on Earth (Steketee, 1958;Weertman & Weertman, 1992).Dislocation theory provides simplified but useful mathematical solutions of deformation in various geophysical problems related to faults and dikes (Battaglia et al., 2013;Bonaccorso & Davis, 1993;Du et al., 1994;Savage, 1983Savage, , 1987;;Savage & Burford, 1973).An edge dislocation is a type of dislocation characterized by the insertion of a half-plane of material between two lattice planes in a solid and can therefore represent an emplaced dike or sill.Savage (1998) finds solutions for the surface deformation field caused by edge dislocations within homogeneous, elastic half-space overlying another half-space of material.The elastic displacement due to an edge dislocation u is expressed from theory in the form, (1) In which i is a unit vector in the x direction, μ is the rigidity, κ = 3 − 4ν where ν is the Poisson ratio, and ϕ(x, y) and λ(x, y) are the Papkovich-Neuber potentials.The Fourier transforms of the Papkovich-Neuber potentials are expressed in terms of arbitrary functions that must be determined from the transformed boundary conditions of stress and displacement in the layers.Upon obtaining the completely transformed expressions of the potentials, the displacement field can be found from the original equation and inverted back to the spatial domain to find the displacement field.The complete description of this solution is described in Appendix I of Savage (1998).The rigidity variable factors out in our calculation and we assume the Poisson ratio of water ice is 0.25.
Incremental ice wedging in Europa's ice shell is analogous to the formation of a dike on Earth, so the concept of ice wedging can also be approximated by an analytical model of elastic deformation caused by edge dislocations in an elastic half-space.In this context, the ice shell is represented by a symmetrical three-dimensional lattice structure, and an edge dislocation is an extra half-plane of material inserted between rows of the crystalline ice structure.Ice wedges are represented by two opposing edge dislocations to represent their finite height.
Figure 2 shows schematics relating the emplacement of a physical ice wedge intrusion via a feeder dike (left) and the opposing edge dislocation positions (right) in an elastic half space representing the ice shell of Europa.
The ridge profiles are modeled using the solutions of surface deformation resulting from edge dislocations in an elastic half-space, presented in Savage (1998).The opposing edge dislocations are equivalent to an interjected wedge of finite length in the ice (Figure 2).The surface deformations due to opposing pairs of edge dislocations at 100 and 500 m, 500 and 900 m, and 900 and 1,300 m below the surface are shown in the top panel of Figure 3.The magnitude and direction that the original lattice is distorted due to the presence of the dislocation is measured by the Burgers vector.Each dislocation modeled is assigned a Burgers vector of 500 m in magnitude, which denotes the distortion in the ice shell caused by the presence of the dislocation (i.e., the ice wedge in this case) compared with a perfectly aligned lattice.In other words, here the Burgers vector represents the horizontal width of the wedge.Following Savage (1998), the top dislocations that face into Europa have a positive Burgers vector and the bottom dislocations facing the surface have a negative Burgers vector.The resulting ice wedge is rectangular.We calculate the surface displacement field generated by the emplacement of the dislocations in the half space.Our models give a purely elastic solution to the surface deformation.Therefore, the edge dislocations must remain stationary because any movement would contribute to plastic deformation.Edge dislocations are inherently restricted to move only in the direction of the Burgers vector (Hull & Bacon, 2001;Weertman & Weertman, 1992), so the dislocations remain fixed in place because lateral movement is restricted in the rigid half space representing Europa's ice shell.The superposition of the edge dislocations produces a surface feature that is topographically similar to a Europan ridge, as shown in the bottom panel of Figure 3.There is no positive vertical displacement directly above the edge dislocations because the dislocations are fixed in space and the surrounding mass must remain conserved.Thus, elastic uplift of ridges results in a central trough at the surface above the dislocations.10.1029/2023JE008007 5 of 17 The analytic model of the two shallowest opposing dislocations, with the top at 100 m and bottom at 500 m depths, produces two topographic highs of nearly 70 m surrounding a central trough.The top panel of Figure 3 shows the positive deformation caused by the upper, downward facing dislocations compared to the negative deformation caused by the lower, upward facing dislocations.The upward facing dislocation has the same magnitude of deformation as a downward facing dislocation at the same depth and with the same magnitude of Burger's vector, but the sign of displacement will be flipped.The surface deformation caused by increasingly deeper pairs of opposing dislocations results in shorter and broader corresponding ridges (Figure 3).There is no strict physical significance motivating the choice for a 400 m vertical extent of the wedges shown in the bottom panel of Figure 3.The surface deformation due to dislocations further than 400 m apart can also be extrapolated from the individual dislocation-driven deformations given in the top panel of Figure 3. Wedges taller than 400 m will result in ridges that are increasingly tall and wide.For example, opposing dislocations at 100 and 900 m result in ridges about 80 m tall.The width (of one ridge) at half maximum height due to the 800 m tall wedge is 2 km, compared to 1.4 km in the 400 m tall wedge case.Similarly, wedges shorter than 400 m in vertical extent will produce shorter, narrower ridges.However, it is unlikely that the vertical extent of wedges varies by more than an order of magnitude from our chosen parameters.In addition to the fact that extremely tall or short wedges will fail to produce appropriate double ridge topography, as we discuss in the following sections, either the thermal structure of the ice shell or depth to which the tensile crack propagates places physical constraints on the length of a wedge.The 500 m magnitude of the Burgers vector is also not strictly constrained and is subject to reasonable variability within an order of magnitude.Taking a Burgers vector wider than 500 m will enhance the resulting surface deformation and vice versa, but the rectangular wedge geometry imposed using the edge dislocation approach is already a rough approximation of the volume and shape expected for realistic wedges.
These results are based on purely elastic deformation from the emplacement of the edge dislocations and indicate that ridges may form exclusively from stress experienced within the ice shell without any flexural influence.However, the solutions presented in Savage (1998) assume infinitesimal strains and it is not clear if this assumption applies to wedges of this scale.Thus, the analytical solution suggests the formation of double ridges from ice wedging may be plausible, but FEM models are necessary because they allow us to simulate more realistic wedge geometries and accurately handle finite strains that determine the deformation resulting from large-scale wedges.

Finite Element Method Models
We created FEM models using the COMSOL Multiphysics software to study how the growth of an ice wedge generates stress and elastically deforms the surface of the ice shell.In our 2D COMSOL models, we used a rectangular ice shell mesh that is 8 km tall and 14 km wide.These dimensions were chosen to limit computational expense while still providing robust results such that boundary effects do not interfere with the formation of the ridges.We use a density of 1,000 kg/m 3 for ice, which assumes that the ice shell contains some impurities such as salts (e.g., Johnson et al., 2017).The Poisson ratio is set at 0.25 and the Young's modulus is 10 9 Pa (Nimmo & Gaidos, 2002).The top of the block is a free surface to allow for ridge formation, the sides have free-slip boundary conditions, and the base is held in place.We tested the effects of including gravitational acceleration and associated lithostatic stress in the model and found that it changed the resulting height of the double ridges by less than 1% compared to models without it.Therefore, we excluded compression from lithostatic stresses in our nominal simulations.
For simplicity, the intruding ice wedge is assumed to be completely rigid.We varied the depth of the top of the wedge from 100 m to 1 km, in 100 m increments, to test how this parameter affects the resulting surface features.
Hereafter, we will refer to the vertical extent of the wedge as its length.The total length of the wedge is 900 m in all these cases to best match observed ridges, because the length of the wedge controls the width of the resulting double ridges.If the wedge were much larger than 900 m, then the resulting double ridges would be much wider than observed double ridges, and similarly, a wedge length much smaller than 900 m would result in ridges that are not wide enough to match observations.The location of the bottom of the wedge depends on where water enters the crack (which is further discussed in Section 4).If ocean water is intruded along the entire crack, the bottom of the wedge is the depth where freezing becomes inefficient.If water comes from a local source within the ice shell instead, the bottom of the wedge is the bottom of the crack.
The growth rate for ice wedges in the shell may vary significantly based on the source of wedge material (e.g., in the case that wedge material is sourced from the ocean, the growth rate may be very slow due to the amount of latent heat associated with the volume of water injected into the shell during each cycle, as discussed in Section 4.1).However, the growth rate of the ice wedge is inconsequential to the ridge topography produced in these models because they measure the elastic deformation of the surface due to the stress of a wedge displacing subsurface material in the ice shell.Thus, the size, shape, and depth of the ice wedge are the relevant parameters for the resulting double ridges in this work.The ice wedge in these models is lens shaped and expands via a parabolic function with zero expansion at the top and bottom of the wedge where there is no growth.The wedge is widest at its center and reaches a final width of 1,366 m in every case.This function reflects the temperature decrease with height above the base of the shell that will initially cause the horizontal growth rate of the wedge to increase, but this rate must decrease to zero once again at the top of the wedge as material approaches the tip of the crack near the surface.For simplicity we use a symmetric lens shape where the maximum expansion occurs precisely at the vertical center of the wedge.More detailed thermal modeling in future work may reveal that a 10.1029/2023JE008007 7 of 17 peak width is reached within the top half of the wedge.Some of the differences between ridge profiles produced in the analytical and finite-element models may be caused by the assumed geometry of the wedge; in the analytical model the wedge is 500 m wide throughout (i.e., the wedge is rectangular), whereas in the finite element models the width varies and is thickest at the middle.
The horizontal growth of the ice wedge causes the ice surrounding the wedge to be compressed.This process results in a vertical expansion at the surface due to the Poisson effect.The deformation is completely elastic in these models, so they do not account for any plastic deformation that might occur due to the growth of the ridges.Ridges in the 100 m depth case grow to the scale of double ridges observed on Europa, about 100 m above the original position of the ice shell surface.
For all the depths of the wedge that we model (top of the wedge at 100 m to 1 km), the growing ice wedge causes two topographic highs with a topographic low in between, centered above the wedge, as shown in Figure 4. Examples of surface deformation resulting from ice wedge depths of 100 and 500 m are compared in Figure 5, where the shallower wedge creates taller and narrower ridges for the same wedge size.We define the width of the double ridge as the distance at which the outside slope ridge returns to the unperturbed elevation as shown in Figure 4.The width of the double ridge increases with increasing depth of the top of the ice wedge (Figure 6a): the case of the 100-m deep ice wedge results in a 2.4 km wide double ridge, while the 500-m deep wedge creates a double ridge that is 5.4 km wide (Figures 5 and 6).Assuming the same wedge growth rate, ridge formation occurs at a lower rate for deeper wedges.For the same wedge size, the final topographic relief generated by ice wedges decreases as the depth of the top of the ice wedge increases (Figure 6b).The 100-m deep ice wedge forms topographic highs over 100 m, while the 500-m deep ice wedge forms a ridge around 70 m in height (Figures 5 and 6b).A larger 500-m deep ice wedge could produce the same ridge height as the 100-m deep case, but it would produce an even wider ridge.Observations of ridges indicate that they tend to be <5 km in total width, with heights ranging from 10 to 100s of meters, up to ∼350 m (Prockter & Patterson, 2009).Our models are consistent with these constraints when wedges are relatively shallow at depths <500 m.In all of our FEM simulations, the double ridge is surrounded by a trough (e.g., Figure 5) consistent with observed ridges.We will discuss this aspect of our simulations in more detail in Section 5.
A direct comparison of a modeled double ridge profile from this work with examples of double ridge profiles derived from images of Europa by Dameron and Burr ( 2018) is shown in Figure 7.The modeled profile corresponds to the case of a 900-m-tall ice wedge 100 m below the surface (also shown in Figure 5).The model is compared to one system that is taller and asymmetric (∼150 m tall left ridge and ∼220 m tall right ridge) and one that is shorter (maximum height ∼80 m).The profiles are arranged such that the central troughs are aligned.
The interior angle of the modeled ridges aligns well with both observed profiles despite the height variation.The taller observed ridge profile has steeper exterior angles than the modeled profile, suggesting that a wedge closer to the surface and <900 m in length could produce a closer fit.The exterior slopes of the shorter observed system are still slightly steeper than the modeled profile, but exhibit more similarity to the model than the previously discussed profile.A shorter wedge that is deeper below the surface might match this observation more closely.

Source of Wedge Material
The source of the water that freezes into the ice wedge can significantly affect the process of ridge formation.
Water may be sourced from the ocean if the crack where the water intrudes reaches the bottom of the ice shell and extends along the entire length of the ridge, or it may be locally sourced if the vertical crack intersects a source of water within the ice shell or reaches the ocean at an isolated point or points.The cases of non-local and local sourcing of wedge material are discussed further in Sections 4.1 and 4.2, respectively.Both the analytical and FEM models presented here are purely elastic and therefore are not sensitive to where the wedge material comes from.We expect that our model results accurately represent surface deformation by material in the ice shell if the deformation happens on geologically rapid timescales for non-local sources (e.g., on the order of 10 3 years) or on potentially longer time scales if the ice wedge is locally sourced.

Water From Ocean Intruded Along the Entire Crack
For the case of ocean water intruded along the entire crack length, the bottom of the accumulation zone for the ice wedge occurs where the high latent heat of water makes it so that ice is unable to freeze.The precise interface between the base of the ice shell and the ocean will be at the melting point, 273 K.However, water injected into the crack from the ocean below the shell must be slightly above freezing temperature in order to avoid automatic supercooling upon injection into the shell.We assume injected water is experiencing a superheating of 1 K (Melosh et al., 2004) which ensures it will freeze above the bottom of the ice shell.Assuming that all sensible heat must be lost before accumulation begins, the minimum crack width w is related to the depth z of the accumulation zone (Melosh & Turtle, 2004) by where κ is the thermal diffusivity of water, H is the thickness of the ice shell, ΔT shell is the temperature difference between the bottom and top of the ice shell, ΔT water is the superheat of the water injected into the crack and p is the orbital period of Europa.The temperature at the top of the ice shell is 100 K and the temperature at the bottom of the ice shell is 273 K, assuming a subsurface ocean (Melosh & Turtle, 2004), so ΔT shell = 173 K. Assuming that ocean water experiences superheating of ΔT water = 1 K, an ice shell thickness H = 10 km, and z = 1 km, the necessary crack width is a few cm.This width allows water to easily rise in the crack over the 3.5-day tidal cycle.
The width of the crack becomes smaller for ice shells that are thicker than 10 km and/or when the accumulation zone is located at depths further than 1 km from the surface, implying that it is increasingly difficult for ocean water to rise and form shallow ice wedges on reasonable timescales in thicker ice shells.
For completeness, we also consider the case in which water injected into the crack from the ocean is not experiencing superheating.Following the solution for dike solidification by Turcotte and Schubert (2002), the width, w, of the horizontal crack is related to the timescale for solidification of water in the crack t s by where κ is the thermal diffusivity of water.If the water in the crack is experiencing 1 MPa of pressure, then κ = 1.32 × 10 −7 m 2 s −1 .The constant λ represents a complex expression describing the position of the time dependent dike solidification boundary and comes from solving The complete derivation of this expression is available in chapter 4 of Turcotte and Schubert (2002).In this study, L = 334 k J kg −1 is the latent heat of fusion of water, c = 4000 J kg −1 K −1 is the specific heat of water, T water = 273 K is the temperature of water entering the crack, and T shell is the temperature of the ice around the crack.We use a surface temperature of 100 K and a vertical thermal gradient that increases with depth at 25 K km −1 to describe the conductive portion of the shell (Silber & Johnson, 2017).With this linear gradient, the maximum shell temperature of 265 K is reached at 6.6 km depth, where the thermal regime transitions from conductive to convective (Silber & Johnson, 2017).The solidification timescales of the water emplaced in cracks ranging from 5 cm to 1 m wide are shown in Figure 8.Where the ice shell is colder than 265 K (depths shallower than ∼6.5 km), the solidification timescale exponentially decreases.Under the assumed thermal conditions, water within cracks greater than 40 cm wide will take longer than 3.5 days to fully solidify, which will limit the growth of the ice wedge during each cycle.When the solidification timescale is less than the diurnal period, the growth of the ice wedge will instead be limited by the volume of water injected into the crack.With each passing tidal cycle, latent heat will be added to the shell from the water passing through the crack, which will further lower the freezing rate of water in the crack.
The small number of observed impact craters on the surface of Europa indicates that the surface is very young, possibly on the order of tens of millions of years (Zahnle et al., 2003).This young surface age coupled with the global prevalence of double-ridge features indicates that the timescale for their formation must be short.
The incremental ice wedging model allows us to estimate the growth timescale for ridges to reach their full height.Advection from the ocean water injected into the crack warms the ice along the walls of the crack relative to the rest of the ice shell, and the gradual heating lowers the viscosity of the ice around the crack.If the material surrounding the ice wedge is too warm, further growth of the wedge could be accommodated by pushing the surrounding warm ice downward rather than pushing up a ridge.Therefore, the ridges must grow fast enough such that the warm channel of ice around the crack is too narrow for viscous flow to accommodate the volume expansion of the ice wedge.If they grow too slowly then the warm ice will expand allowing the ice to flow down instead of raising ridges.The minimum time required for the ridges to reach their full height and avoid this viscous relaxation is given by the analytical solution below, where V ridge is the cross-sectional area of the ridge,  is the thermal diffusivity of ice (which sets the width of the warm ice channel),  is the viscosity of warm ice, H the thickness of the ice shell and Y the yield stress of cold ice at Europa's surface, which resists the growth of the ridge.Assuming Newtonian rheology with viscosity assumed to be about 10 14 Pa-sec for warm ice (Nimmo, 2004), we derive an upper limit on the formation time of about 5,000 years for ridges to rise to their final height.If the complete ridge formation takes longer than this timescale, our purely elastic models are no longer applicable and full visco-elastic simulations are needed instead.
For ridges to rise instead of slumping back down under the surface due to viscous relaxation, they must grow relatively rapidly to their final height.We determine that the timescale for growth is within 5,000 years of initial ice wedge formation.We argue that it is plausible that the ridge formation is this geologically rapid because of the high frequency of ridges across the surface of Europa.Also, since the orbital period of Europa around Jupiter is only 3.5 days, around 500,000 tidal cycles occur over the course of 5,000 years and can contribute to the flow of ocean water into cracks in the ice shell that slowly grow an ice wedge.On this timescale, an ice wedge that is a few km below the surface can grow a double ridge a few 100 m in width if each tidal cycle deposits only 0.2 mm of ice (Melosh & Turtle, 2004).This rate allows for several generations of ridges to form over short geologic timescales, which may be consistent with surface observations that show crosscutting or overlapping ridges.
If ocean water is sourced along the entire crack length, cracks must propagate from the surface all the way down to the base of the ice shell or originate at the base of the ice shell and propagate to the near surface.However, Figure 8.The freezing timescale of the ice wedge when water is injected from the ocean into cracks ranging from 5 cm to 1 m wide and extending to a maximum of 10 km below the surface of Europa.The ice shell is assumed to be conductive with a thermal gradient of 25 K/km from the surface at 100 K to a maximum temperature of 265 K at depth.The orbital period of Europa, 3.5 days, is marked by the vertical dashed line.An average growth rate can be determined by dividing the width of the crack by the solidification timescale.
previous work suggests that it is quite difficult for cracks to penetrate to the base of the ice shell, where ice has very low viscosity (Nimmo & Manga, 2009).It is unclear exactly how thick the ice shell is, and consequently whether its entire depth is conductive, or only conductive near the top while the bottom is convective.If the shell is very thick with a convective layer, then it will be unlikely that cracks are able to propagate to the base of the ice shell.However, pure conduction in the shell is allowable within the current uncertainties (e.g., Hussmann et al., 2002).In a purely conductive case where cracks can form at the base of the ice shell, the crack would fill with water aiding propagation toward the surface (Crawford & Stevenson, 1988).
Double ridges are also observed in cycloidal shapes as well as more linear segments.Hoppa et al. (1999b) show that stresses from diurnal tides vary over time and space such that a crack formed during the extensional phase may propagate cycloidally.Hoppa et al. (1999b) demonstrate these cracks are observable at the surface, and likely only propagate to a depth of a km or so.Since cracks propagating from the base of the ice shell are not expected to reach the surface (Crawford & Stevenson, 1988), ocean material sourced along the entire length of the crack is difficult to reconcile with cycloidal double ridges.

Water Intruded Locally
Ice wedging may be more likely to present and produce double ridges if water intrudes locally instead of along the entire crack.Local intrusion of water could occur if a vertical crack reaches the ocean at one point or multiple discrete points, if a cryovolcanic dike intersects an existing crack, or if the crack intersects with a source of water within the ice shell.The water injected at the given point may then fill the rest of the crack.This water may form in relatively shallow pockets in the ice shell, such as the sills that are hypothesized to form pits and domes (Manga & Michaut, 2017).Water from a sill would be injected into the larger crack due to the overpressure caused by partial freezing.
For water intruded along the entire length of the ridge, as described in previous sections, the thermal structure of the ice shell surrounding the crack may be significantly warmed because the water entering it affects the ice shell along the crack for hundreds of km.However, if water is injected only at a point along the crack then the thermal effect would be limited to the ice directly adjacent to that point.Even if there are several points, the total thermal effect is confined enough such that the ice wedge that is formed by the water freezing will be unlikely to relax.Thus, the arguments in Section 4.1 need not apply and there would no longer be thermal constraints on the short timescale for forming the ridges.However, water that enters the crack via local sourcing will be unable to drain away to the ocean and will freeze quickly since the material surrounding the crack is colder than the injected material.Therefore, in the case of local sourcing, the growth of the ice wedge will be more efficient and could require fewer cycles to form the wedge.If the water intrudes locally from a sill-like source, then the issue of crack depth is also alleviated.Sills are expected to exist around 1-5 km below the surface (Manga & Michaut, 2017).The depth to which the tensile crack must propagate, and the maximum depth of the ice wedge formed, is set by the depth of the source sill.
Water will be forcefully injected into a thin crack in the shell due to the overpressure within the ice shell, which may be up to 1 MPa (Manga & Michaut, 2017).Again following the solution for dike solidification by Turcotte and Schubert (2002), the width, w, of the horizontal crack is related to the timescale for the solidification of water in the crack t s by Equation 3. If the water in the crack is experiencing 1 MPa of pressure and superheating of 1 K, then T water = 274 K and κ = 1.35 × 10 −7 m 2 s −1 .The temperature of the ice around the crack is T shell .For a vertical thermal gradient in the shell of 25 K km −1 , with a maximum of 265 K, and a surface temperature of 100 K (Silber & Johnson, 2017), the temperature at 1 km depth is T shell = 125 K.If the water injected into the crack freezes over the rest of the tidal cycle, t s = 3.5 days, then the width of the crack is 40 cm.Cracks at the surface of Europa may be a few meters in width (Poinelli et al., 2019), so 40 cm is a reasonable crack width at depth.The freezing timescale for water in the crack may be extended if the crack is wider (or shortened in thinner cracks).If the water flows smoothly through a horizontal crack with cylindrical symmetry, then the flow rate R of the water is related to the total length l that the water flows before freezing by We assume the crack to be a circular cylinder to produce a conservative estimate of the velocity of water through the crack.With an overpressure P = 1 MPa (Manga & Michaut, 2017), radius r = 20 cm, and the viscosity of water η = 1.792 mPa s.Double ridges may extend linearly across the surface of Europa for up to hundreds of kilometers.Taking the horizontal length of the crack to be l = 100 km gives a flow rate of 4 m 3 s −1 or a velocity of 30 m s −1 .At this rate, water would flow 100 km through the crack in only 56 min.In the case of a rectangular crack that is 40 cm wide in which water experiences the same overpressure, the larger cross-sectional area will result in an increased flow rate.Thus, the flow rate is unlikely to limit the lateral extent of a double ridge with localized water sources.The topography of double ridges observed on Europa is often quite uniform along their length (Prockter & Patterson, 2009).If water is injected into the crack and travels at the rapid flow rate expected from the overpressure of the ice shell, then the resulting ice wedge formed should be as uniform as the width of the crack where the wedge forms and it should not be difficult to achieve uniform ridges.In this scenario, the size of the wedge is more likely to be limited by the quantity of water within the source reservoir(s) rather than how much flow can occur in a short amount of time.

Discussion and Conclusions
Previous discussion of double-ridge formation models has discounted the viability of ice wedging based on the model's supposed inability to accurately replicate the observed topographic troughs that flank the outsides of the ridges (Dombard, 2013).However, in the FEM models (Figure 4) presented in this work, we find that these troughs are present.The troughs surrounding double ridges have previously been interpreted to form as a result of flexure from the topographic load of double ridges (Nimmo & Manga, 2009).The heat flow budget from within Europa necessary to facilitate the formation of these troughs, however, is higher than Europa's estimated heat budget (Dombard, 2013).Both our analytical model and FEM simulations assume an elastic half-space and ignore the effect of gravity, but the FEM simulations predict outer troughs.Thus, if double ridges and troughs are made by ice wedging, the troughs surrounding double ridges may not be the result of flexural effects, and appropriate caution should be exercised when attempting to use these features to constrain elastic thickness or the corresponding thermal structure of the ice shell.Other models of double ridge formation are consistent with the overall observed topographic relief of ridges but have not predicted the formation of these troughs (e.g., Han & Showman, 2008;Johnston & Montési, 2014).
The volume of water associated with building the ice wedges that we model in this work can be significant.The surface area of Europa is 3.09 × 10 7 km 2 .If the surface area modified by a double ridge system is approximately that of a triangular prism 100 m tall and 5 km wide, the length of the system would be 3 million km to be equivalent to the entire surface of Europa.If the subsurface ice wedge corresponding to 100-m of surface uplift is 500 m wide, 1 km tall, and extends under the entire length of the ridges, then the volume of new ice intruded into the ice shell by wedging would be 1.5 million km 3 , or 0.5% of the volume of the ice shell if it is only 10 km thick.Although this is a large volume of intruded ice, such a small fraction of the ice shell is not likely to have major implications on the ice shell recycling on the timescale of Europa's surface age ∼60 Myr (Prockter & Pappalardo, 2014;Zahnle et al., 2003).Locally, however, this recycling could be important in regions that have many generations of double ridges.
In addition to double ridges, the surface of Europa is covered with numerous lenticulae, such as pits, domes, and chaos regions (Culha & Manga, 2016;Fagents, 2003;Greenberg et al., 2003;Schmidt et al., 2011).The morphology of lenticulae is distinct from double ridges, but these quasi-elliptical features are also likely to be the result of emplaced subsurface water and may represent different evolutionary stages or outcomes of a common process (Collins & Nimmo, 2009;Greenberg et al., 1999).For example, pits and domes have been proposed to be formed when thin dikes within the ice shell provide water for forming saucer-shaped sills beneath the surface (Manga & Michaut, 2017).If double ridges are sourced from localized intrusions as discussed in Section 4.2, then double ridges and lenticulae may be related to their formation mechanism, wherein the resulting surface feature is dependent on the tensile stress induced in the ice.That is, it may be that double ridges form over sills in regions where tensile stresses from tides result in cracks, while pits and domes are more prevalent in regions where stresses are inadequate to open cracks.It is beyond the scope of this work to investigate whether there are patterns present which indicate potential relationships between these surface features and areas where water may have been emplaced or removed locally (e.g., double ridges where water may have been emplaced, and chaos, pits, or depressions where water may have been removed), but we suggest this possibility be considered in future work.
The elastic models presented in this work are unable to account for the growth of the crack due to the stress associated with the growing ice wedge, but stress at crack tips may have interesting consequences that should be considered in future work.For example, in the nonlocal case, stress at the top of the crack may propagate the fracture further toward the surface.Hydrated salt minerals are inferred to be present in low albedo regions of Europa's surface (Kargel et al., 2000;McCord et al., 1999) sometimes on double ridges (Prockter & Patterson, 2009) and may be sourced from the subsurface ocean (Zolotov & Shock, 2001).In the case of local sourcing from briny sills in the shell (Chivers et al., 2023), the crack is assumed to have originated at the surface and propagated downward.Salts may be more likely to be erupted onto the surface in this case because the conditions necessary for the eruption are more realistically achieved (Lesage et al., 2020(Lesage et al., , 2021;;Quick & Hedman, 2020).As previously discussed for the non-local sources, the pressures needed for water to travel vertically upward through the entire shell, perhaps greater than 10 km in thickness, could be difficult to achieve (Manga & Wang, 2007;Rudolph & Manga, 2009).
If the wedge reaches the surface and continues growing past what is modeled here, we can consider the possibility that dilational bands (Stempel et al., 2005;Tufts et al., 2000), another kind of prominent surface feature observed on Europa, are also produced by ice wedging.Many dilational bands exhibit similar morphology to double ridges, including bilateral symmetry of lineations that are elevated by ∼100 m (Nimmo et al., 2003) around a central trough (Prockter & Patterson, 2009).Lineated dilational bands appear to form episodically where the youngest features are nearest to the spreading center (Manga & Sinton, 2004), which also aligns with the cyclic, tidally driven process of incremental ice wedging.Thorough investigation of the continued growth of ice wedges including viscoelastic-plastic rheology is clearly warranted to determine the feasibility of this relationship and distinguish the differences between the formation of ridges versus bands.
Strike-slip faults often occur on Europa (Schenk & McKinnon, 1989) and frequently have been observed to be associated with double ridges (Hoppa et al., 2000).This motion has been attributed to diurnal stress, such that the faults arise along preexisting cracks that were opened by tidal stress (Greenberg et al., 1998;Hoppa et al., 1999a;Tufts et al., 1997).Hoppa et al. (2000) hypothesized that strike-slip faults with double ridges imply the crack must intersect with a low-viscosity decoupling layer, such as the subsurface ocean.Localized sources of water in the shell, as we discuss in Section 4.2, present an attractive alternative low-viscosity layer that allow for strike-slip motion to occur as well as act as a source for ice wedges that build double ridges.One distinction between the formation of double ridges and dilational bands by ice wedging may lie within this detail, where dilational bands could form in cases where the wedges reach very shallow depths and possibly even the surface, causing visibly spread iterations of lineations, while deeper wedges form one significant pair in the form of a double ridge that is susceptible to strike-slip motion with repeated cycles.
The recent discovery of an apparent Earth-analog for Europan double ridges on the Greenland ice sheet has been presented by Culberg et al. (2022).Radar analysis of this feature reveals its formation due to a subsurface wedge of ice growing over time as seasonal melt water percolates to the subsurface.The formation of the terrestrial ridges by elastic deformation of an expanding subsurface ice wedge is similar to the model presented in this work.The double-ridge analog in Greenland is of smaller scale than those observed in Europa, and the freezing water that creates the ridges runs along the length of the ridges.The radar observation of Culberg et al. (2022) suggests that the Radar for Europa Assessment and Sounding: Ocean to Near-surface (REASON) instrument aboard NASA's upcoming Europa Clipper mission (Blankenship et al., 2018;Pappalardo et al., 2022) will greatly improve our understanding of double ridge formation by providing radar sounding observations of emplaced ice wedges.In addition, actively forming wedges might also have a detectable thermal signature due to the sensible and latent heat associated with their emplacement.In tandem with REASON, the Europa Thermal Emission Imaging System (E-THEMIS) and stereo observations by the Europa Imaging System (EIS) aboard Europa Clipper (Rathbun et al., 2016;Turtle et al., 2023) could reveal extensive details of double ridges and the ice wedging process, including wedge properties, such as depth, geometry, and perhaps even source reservoir, that would allow for empirical characterizations of their associated surface features.In our models, the angle of the slope of the ridge interior and exterior is particularly sensitive to the depth of the wedge (Figures 3 and 5).Thus, the angle of observed ridges (Dameron & Burr, 2018) may correspond to the approximate depth of the ice wedge related to their formation and could be compared with the upcoming subsurface radar, surface topography, and thermal observations.
With the use of both analytical and numerical models, we have shown that the incremental ice wedging hypothesis is a viable mechanism for producing double ridges on the surface of Europa.Both types of our models consistently replicate the overall shape and size of the observed ridges on Europa, including the outer troughs flanking the ridges in the FEM models.The source of water for the ice wedge may be more likely to come from localized reservoirs of water within the ice shell rather than the subsurface ocean.Our results indicate that ice wedge properties directly affect the final topography of ridges.In turn, the wedge properties are governed by the width, length, and point(s) of the intersection of the crack with the wedge source reservoir, as well as the volume, composition, temperature, and pressurization of material in the source reservoir.The origins of local water sources in the shell are themselves debated (Fagents, 2003;Manga & Michaut, 2017;Manga & Wang, 2007;Schmidt et al., 2011), so the radar sounding and thermal observations from upcoming missions such as Europa Clipper should provide more insight into the global distribution-localized reservoirs, ice wedges, and the surface features that they generate.Whether Europa's ice shell is rigid and conductive throughout its entire depth or experiences varying regimes of thermal transport between different depths, incremental ice wedging is a natural process that is likely to occur as tidally driven cracks are forced to open within the shell.

Figure 1 .
Figure 1.The process of incremental ice wedging.In panel (a) a crack opens during the extensional phase of diurnal tides and pressurized water enters from the ocean or water reservoir below.In panel (b) compressional tides close the crack leaving only ice that has been deposited on the walls of the crack by freezing of the water.The presence of the frozen material creates stress that causes surface deformation.Panel (c) again shows the crack open due to extensional tides, allowing more water to rush in.In panel (d) compressional tides close the crack and the additional frozen material deposited contributes to wedge growth and more surface uplift.This process continues to repeat over each tidal cycle.Panel (e) shows a 3-dimensional view of the double ridge fully formed after the gradual surface deformation due to incremental growth of the ice wedge.

Figure 2 .
Figure2.Schematic depicting the model setup for opposing edge dislocations.On the left, the dashed line represents the dike that supplies ocean water to the ice wedge intrusion, represented by a thick black line.The schematic on the right shows the position of the opposing edge dislocations that represent the ice wedge in an elastic half space.

Figure 3 .
Figure3.The surface deformation results of an analytical model of opposing edge dislocations that approximate an ice wedge emplaced in the ice shell.In these models, the top dislocations are 100, 500, and 900 m below the surface, and the bottom dislocations are 500, 900, and 1,300 m below the surface.The Burgers vector of each edge dislocation is 500 m in magnitude.The top panel shows the surface deformation caused by the individual dislocations, and each curve in the bottom panel shows the total surface deformations caused by the superposition of two opposing dislocations as denoted by the legend.

Figure 4 .
Figure 4.A time series of the total displacement caused by a 100-m deep wedge, simulated in COMSOL.The ice wedge is allowed to grow for 1.5 × 10 12 s in model time.The color indicates the total displacement within the mesh.The total width of the double ridge is 3,200 m and is labeled in the figure.The height of each ridge is 97.3 m.

Figure 5 .
Figure5.Surface deformation profiles caused by ice wedge growth simulated using COMSOL.The topography is formed by a 900 m tall ice wedge at varying depths under the surface.The topography is symmetrical around the vertical axis.

Figure 6 .
Figure 6.Panel (a)  shows the relationship between the final width of deformation caused by an ice wedge at different depths beneath the surface of the ice shell.Panel (b) shows the relationship between the final topographic relief of the ridges and the depth to the top of the ice wedge.In both panels, each point comes directly from results of a COMSOL model.

Figure 7 .
Figure7.A comparison of a modeled double ridge profile (black line) with two double ridge profiles observed on Europa (blue dash-dotted line and red dotted line).The modeled profile, generated in COMSOL, is the surface deformation caused by a 900-m-tall ice wedge 100 m below the surface of Europa.