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Keywords:

  • cryovolcanism;
  • flood;
  • freezing

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[1] It is difficult for liquid water to erupt onto the surface of icy satellites, such as Europa and Enceladus, because liquid water is more dense than ice. If the ice shell thickens, the volume expansion of ice upon freezing increases pressure in the subsurface ocean. The excess pressure is determined by a balance between compression of ocean water and elastic expansion of the ice shell. We show that on Europa the freezing of ∼ 1–10 km of ice generates tangential stresses that exceed the tensile strength of ice. Excess pressure, however, is insufficient for liquid water to erupt to the surface. Within smaller icy satellites, such as Enceladus, ocean pressure can become large enough to cause an eruption of large amounts of liquid water.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[2] The past eruption of liquid water onto the surface of Europa is suggested by several observations. The most compelling features are smooth plains that fill topographic lows [e.g., Greeley et al., 2000; Prockter and Schenk, 2005] and some chaos features [e.g., Miyamoto et al., 2005]. The formation of all types of chaos may require large amounts of liquid water at or near the surface [e.g., Collins et al., 2000], though it remains controversial whether the liquid water originated in the ocean and chaos is caused by melt-through of the ice [e.g., Greenberg et al., 1999; O'Brien et al., 2002], or is generated within the ice shell by thermal and/or compositional diapirs [e.g., Sotin et al., 2002; Pappalardo and Barr, 2004; Han and Showman, 2005]. Some models for ridge building also require that liquid water ascends to at least near the surface [e.g., Greenberg et al., 1998].

[3] On Enceladus, liquid water may be erupting currently: one interpretation of the jets of icy particles and water vapor imaged by Cassini is that liquid water is being vented from the subsurface [e.g., Porco et al., 2006].

[4] Erupting liquid water is challenging because it is more dense than ice. This density difference is further enhanced if the ice is porous, as it might be near the surface of Europa [Nimmo et al., 2003a; Lee et al., 2005], and the water is salty. To overcome negative buoyancy, several mechanisms have been proposed, including the exsolution of volatiles [e.g., Crawford and Stevenson, 1988], overpressurization of water trapped in discrete reservoirs within the ice shell [e.g., Fagents, 2003] or generated by magma intrusion into the base of an ice crust [e.g., Wilson et al., 1997], and topographically-driven flow into low-lying regions [Showman et al., 2004].

[5] Here we consider the possibility that freezing of the global subsurface ocean increases pressure in the ocean. Such changes may arise from secular cooling, or the coupled orbital evolution and thermal evolution of the interior [e.g., Hussmann and Spohn, 2004]. We first experimentally show that the volume expansion of water as it freezes from the top increases pressure in the water confined below the ice. We then model the evolution of ocean pressure for satellites with properties and sizes representative of Europa and Enceladus and discuss the conditions under which this model applies.

2. Freezing of Ice and Pressurization of Water: A Demonstration

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[6] In order to show that freezing from the surface-down will increase the pressure in water trapped below the ice, we present in Figure 1 the results of a straightforward demonstration. A glass cylinder is filled with water. Glass is used because its coefficient of thermal expansion is very small. In order to freeze water at the surface, liquid nitrogen is poured into a plastic cup placed on the water surface. The evolution of pressure is monitored visually in this demonstration by the water level in a small capillary tube that enters the base of the cylinder. As freezing proceeds, the ice remains attached to the glass walls, so that the volume expansion upon freezing increases water pressure and causes the water level in the capillary tube to rise.

image

Figure 1. Experiment showing the evolution of pressure in water trapped below a freezing front; water is contained in a cylinder (7.5 cm diameter), open at the top and sealed at the bottom. The small capillary is connected to the cylinder and monitors its pressure. (top) Initial condition before freezing. (middle) Water level in the capillary rises 45 cm (well above the image) after a few mm of ice forms. (bottom) After a crack forms, inferred from acoustic emissions, water pressure returns to close to its original value. Horizontal white line indicate elevation of the water level in the capillary tube.

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3. Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[7] Consider a solid ice shell that thickens over time, with outer radius R, inner radius ri, and no radial stress at its outer surface. The ocean lies above a rocky interior with radius rc. We consider only freezing to ice I for which density decreases. The pressure increase Pex as the freezing depth z increases is given by [e.g., Wang et al., 2006]

  • equation image

assuming the upper surface of the ice does not move and rc does not change. Here, β is the compressibility of water, ρ is density, and the subscripts i and w refer to ice and water, respectively.

[8] The excess pressure in the ocean, however, will cause the ice shell to expand (in contrast to Figure 1) and Pex will not reach the limit given by (1). Over the time scales that the ice shell changes thickness (>(Rri)2/κ where κ is thermal diffusivity) the warmer bottom of the ice shell will deform in a viscous manner, and only the outer part of the ice shell, for radii ξ < r < R, will behave as an (assumed linear) elastic solid. Rather than apply a full viscoelastic model, we consider two models for the ice shell that should bound most reasonable scenarios. First, for a lower bound, the elastic layer maintains a constant thickness of R ξ = 1 km, a value consistent with flexure studies from regions with presumably high heat flow and hence thin elastic lithospheres [e.g., Billings and Kattenhorn, 2005]. Second, for an upper bound, the outer third of the ice shell is elastic, which follows from assuming ice behaves elastically on long times for T < 180 K and heat transfer occurs by conduction. In both models, the radial stress is Pex at the base of this elastic layer, at r = ξ. Lamé in 1852 [Sokolnikoff, 1956] obtained a solution to the Navier equations for the radial σr and tangential σt stresses:

  • equation image
  • equation image

and we neglect any compression in the rocky interior below the ocean. The corresponding radial displacement is

  • equation image

where E is Young's modulus and ν is the Poisson ratio. In contrast to (2), the model of ice shell freezing by Nimmo [2004] assumes that the radial stress at the base of the ice shell is zero and freezing can be represented by a displacement boundary condition at the upper surface of the ice shell.

[9] Expansion of the ice shell reduces the excess pressure by an amount δP = 3urri2/β(ri3rc3) so that

  • equation image

We solve (2)(5) numerically to obtain Pex for a given amount of freezing z. Unlike the analysis of freezing drops by King and Fletcher [1973], the equations of Lamé should remain a good approximation because the bottom of the ice shell deforms viscously on the freezing time scale and remains unstressed; nevertheless, the solution by King and Fletcher [1973], when applied to our problem, produces the same results for the volume of water that would be discharged following pressurization of the ocean.

[10] For the solid ice shell, we use E = 5 × 109 Pa [Nimmo, 2004], ν = 0.33 [Schulson, 2001], ρi = 910 kg/m3 and ρw = 1000 kg/m3. Using the model of Wager and Pruss [2002] the compressibility of water averaged over the depth of the ocean is β = 4 × 10−10 Pa−1.

4. Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[11] Figure 2a shows the evolution of excess pressure in Europa's ocean and the maximum tangential stress (at the base of the elastic part of the ice shell) as a function of the increase in ice shell thickness z. Here we use R rc = 120 km [Cammarano et al., 2006] and to be consistent with the analysis by Nimmo [2004] we start with Pex = 0 and an ice shell 2.4 km thick (results in Figure 2 are not highly sensitive to the assumed initial thickness). Pex is much smaller than would be predicted from (1), owing to radial expansion of the ice shell, and σt is smaller than the tangential stresses calculated in Nimmo's [2004] model of the volume change effect. Nevertheless, Figure 2a shows that Pex is still large enough that freezing of 0.1–5 km of ice is sufficient to generate tensional stresses that exceed the tensile strength σT of ice (value to be discussed later). It is important to note that there are other sources of stress, in particular, thermal contraction stresses if the elastic layer changes thickness [Nimmo, 2004], that are potentially larger than those induced by Pex.

image

Figure 2. Change of pressure and tangential stress (at the base of the elastic layer) with increasing freezing depth z on (a) Europa and (b) an Enceladus-sized satellite. Solid curves assume the elastic layer is 1/3 the ice shell thickness; dashed curves assume the elastic layer is always 1 km thick. The initial thickness of the ice shell is 2.4 km in Figure 2a and 50 km in Figure 2b. The pressure needed to erupt water to the surface is shown by the bold curve. The dotted region indicates the tensile strength of ice is for low porosity ice (porosity can weaken ice significantly, see Lee et al. [2005]), and its value is independent of strain-rate and depends only slightly on temperature [Schulson, 2001]; the dotted line is the tensile strength model of Bažant [1992] fit to field and lab data by Dempsey et al. [1999]. Also shown are the maximum diurnal tidal stresses and maximum stresses from nonsynchronous rotation [Leith and McKinnon, 1996]. The maximum volume of water that could erupt divided by the surface area of Enceladus, vmax, is shown in Figure 2b.

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[12] We now need to determine whether it is possible for tensile cracks, once formed at the base of the elastic layer, to propagate both up and down through the ice shell and hence to allow ocean water to rise to, or close to, the surface. The time for the relaxation of horizontal tension in the ice shell is tr ≈ 2πR/V, where V the compressional wave velocity. The time for the dynamic propagation of the crack through the ice shell (if it does) is tpb(R ri)/V, where b ranges between 0.4 and 0.9. Because tstp, we assume constant tension in the shell during crack propagation.

[13] The tensile stress at the crack tip is σtip = K/equation image, where a is the dimension of the region in which yielding occurs. The stress intensity factor K depends on the loading configuration and the ratio between crack length and thickness of the ice shell. For the present problem, a reasonable approximation is Mode I loading under uniform tensile stress σL because the magnitude of σL may not vary significantly over the elastic thickness of the shell. Under such conditions, K = mequation imageσL, where c is the crack length; m ranges from 1 to 3 for a wide range of ratios between crack length and ice shell thickness [Emery et al., 1969]. σL is the magnitude of the tangential stress with a maximum set by the tensile strength σT of the ice shell. Dempsey et al. [1999] report a ∼ 0.1 m based on lab and field experiments with sea ice. Tensile cracks formed in the elastic shell can thus propagate to a depth c = dmax where σtip becomes less than the overburden stress ρigdmax, where g = 1.3 m/s2 is gravity, i.e.,

  • equation image

The tensile strength is highly uncertain with intact lab-scale samples having strengths of ∼MPa (Figure 2), whereas porous ice and natural ice sheets on the ocean may have strengths as low as ∼104 Pa [e.g., Dempsey et al., 1999; Lee et al., 2005]. For tensile strengths and hence σL greater than a few 10s of kPa, we find that the maximum propagation depth is greater than the thickness of the ice layer. Propagation should occur on timescales shorter than the Maxwell time of even the base of the ice shell.

[14] If the ice shell ruptures, water will rise to a height of neutral buoyancy, referred to hereafter as the water-line, where the ocean pressure is balanced by the weight of water in fractures. The depth H of this water-line below the surface is

  • equation image

If H ≤ 0, the ocean overpressure exceeds the critical pressure image = (ρw ρi) (R ri)g needed to drive water to the surface. Figure 2a shows that Pex is always image and will not become large enough on Europa for ocean water to reach the surface.

[15] If water is intruded into the ice shell below the water line to make water-filled sills [Collins et al., 2000], overpressure in the water will create uplift. For sills with horizontal extents much greater than the elastic thickness of the lithosphere, topography will be ∼Pex/ρig. For Pex between 103 and 104 Pa, generated by freezing about 1 to 10 km of ice (Figure 2a), topography will be 1–10 m. If the water in such sills freezes after being disconnected from the ocean, the surface will rise still further ∼9% owing to the expansion upon freezing. The topography of Chaos regions, while irregular, is typically elevated above the surrounding plains by as much as 250 m [Schenk and Pappalardo, 2004] and thus and order of magnitude larger than that generated above sills filled with overpressured ocean water. Figure 2a suggests that Pex is unlikely to become large enough to create the nearly 1 km elevation of some domes [Prockter and Schenk, 2005], an amplitude that would seem to require compositional diapirsm within the ice shell [e.g., Pappalardo and Barr, 2004; Han and Showman, 2005].

[16] While ocean water is unlikely to be driven to the surface of Europa, higher water-lines and even eruption of water onto the surface will be favored in smaller satellites for two reasons. First, the smaller radius results in less expansion of the ice shell and hence larger Pex for a given amount of freezing. Tangential stresses will also be larger. Second, the smaller gravity allows water to rise further for a given Pex.

[17] Figure 2b shows Pex in the ocean and σt at the base of the elastic shell as a function of z for a geometry that might be representative of Enceladus [e.g., Kargel, 2006]. We use R = 252 km, rc = 161 km [Nimmo and Pappalardo, 2006], g = 0.113 m/s2 and assume Pex = 0 when the ice shell is 50 km thick. Unless the elastic lithosphere is very thin (<1 km), Pex becomes large enough for the water-line to reach the surface.

[18] If pressurized water flows to the surface, Pex > image discharge from the ocean will decrease Pex. There are two contributions to the discharge: (1) the volume expansion of the ocean during the pressure release, and (2) the change in volume of the ocean. The latter is governed by the contraction of the ice shell, with radial displacement urelax caused by the pressure decrease. We can calculate the maximum volume of water that can be expelled by calculating the volume of water that is removed from the ocean as Pex [RIGHTWARDS ARROW] image

  • equation image

where we divide the volume by the surface area of the satellite so that vmax has units of length. We obtain urelax from (4) assuming ξ = ri because on the short time scales of water discharge the entire ice shell should behave elastically. Figure 2b shows vmax as a function of the freezing depth z and indicates that substantial amounts of water may reach the surface. The 100 m of water in Figure 2b could sustain the eruption on Enceladus of 150 kg/s [e.g., Hansen et al., 2006] for > 104 years if ocean water actually is the source of the plume. We emphasize, however, that (8) provides a maximum estimate for two reasons. First, channels through which water flows to the surface may freeze before Pex returns to image Second, our estimate of Pex, as we discuss next, assumes the ice shell remains intact throughout its evolution.

[19] The most critical assumption in our model is that the outer part of the ice layer behaves as an intact, elastic shell, an assumption implicit in models of stresses from diurnal tides and nonsynchronous rotation [e.g., Leith and McKinnon, 1996] and other models of thickening ice shells [e.g., Nimmo, 2004]. On time-scales of diurnal tides, this may be a reasonable approximation. On much longer time-scales that might characterize changes in the ice shell thickness (>105 years for Europa, >108 years for Enceladus) the validity of this assumption is less obvious. Pervasive tectonic features such as ridges, cycloids and bands, likely reflect brittle or plastic failure and hence at least local stress release. The inferred strength of faults of ∼1 MPa [e.g., Nimmo and Schenk, 2006] is comparable to stresses generated by either freezing of the ice shell [Nimmo, 2004] and maximum stresses from non-synchronous rotation [e.g., Leith and McKinnon, 1996] making stress release possible at least on long time scales. Significant overpressure may still develop on these timescales. If the upper 2–3 km is too porous and hence weak to support stresses on long time scales [e.g., Nimmo et al., 2003a], and an elastic thickness of 6 km [e.g., Nimmo et al., 2003b] is more representative of a global value (rather than the ∼1 km inferred from regions that probably had high heat flow), the relevant elastic thickness is still a few km and within the range we considered.

[20] In summary, the challenge of delivering liquid water from a subsurface ocean to the surface of icy satellites is reduced if the ice shell thickens over time. On Europa, ocean pressure is unlikely to ever become large enough to erupt water at the surface; if injected as sills, however, the water could provide the low viscosity substrate needed to form chaos, as suggested by Collins et al. [2000]. On Enceladus-sized satellites, freezing of any ocean can more easily generate the overpressures needed for water to reach the surface.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References

[21] We thank A.C. Barr, F.N. Nimmo, R.T. Pappalardo, and A.P. Showman for constructive comments and suggestions, and NASA PGG NNG04GE89G and NAI NNA04CC02A for financial support.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Freezing of Ice and Pressurization of Water: A Demonstration
  5. 3. Model
  6. 4. Results and Discussion
  7. Acknowledgments
  8. References