Introduction

Images of Jupiter’s moon Europa from spacecraft show a bright young surface crisscrossed with linear structures, indicating an active environment. It is believed that Europa is covered by a water ice shell above an approximately 100km deep ocean of liquid water (Andersen and others, 1998; Reference PappalardoPappalardo and others, 1999).

The energy needed to keep the ocean from freezing under the cold surface conditions on Europa and for driving tectonic activity is believed to be dissipation energy from tidal forces acting on the ice shell. Europa is in synchronous rotation around Jupiter and should not experience time-varying deformation if the orbit is circular. However, an eccentric orbit is maintained by the Laplace resonance between the three inner Galilean satellites: Io, Europa and Ganymede. This causes the ice shell to flex 35 m with a period of 3.55 days (Reference GreenbergGreenberg and others, 1998).

The amount of tidal energy dissipated depends on the deformation properties of the ice shell, which are governed by ice thickness, temperature profile, ice crystal sizes, fabric and structure, and impurity content of the ice. These parameters have not been measured directly, so modelling, based on the data measured by spacecraft, is the only currently available way of gaining insight into possible processes and conditions in Europa’s ice and ocean.

The idea of this study is to estimate the sizes of ice crystals at Europa’s surface. It was inspired by looking at the Galileo images from the Conamara Chaos region, a region of disrupted surface, where polygonal blocks of ice up to 20 km in diameter have floated in a low-viscous layer below. Ice on Earth has a hexagonal crystal structure and is classified as ice Ih. This gives rise to the question: are the polygonal shapes of the ice blocks caused by the hexagonal structure of large ice crystals? It is plausible that large crystals could be found, because, unless disturbed, any crystal will continue to grow in time.

An Ice Crystal Model

Mass is lost from the surface of Europa due to sputtering by energetic ions (Reference Ip, Williams, Mcentire and MaukIp and others, 1998). If the ice thickness at a given position is assumed to be constant in time, there should be freezing of sea water to the lower boundary of the ice, to compensate for the mass lost from the surface. An element of ice would therefore move upwards, relative to the boundaries of the ice, with a velocity corresponding to the rate of mass loss. As new ice crystals formed at the base of the ice shell are transported upwards, the ice temperature changes, which influences crystal growth. The crystals grow fast near the bottom, where the ice is at the melting point.

Ice crystals grow by grain boundary migration, a process mainly driven by the curvature of grain boundaries and by differences in stored energy between the grains. In this study, it is assumed that crystals grow following the normal grain-growth law(Reference W. S. B.Paterson, 1994)

where D² is the cross-sectional area of crystals of age t and D ² ₀ is the initial crystal size. The growth rate k is temperature-dependent and given by

where E is the activation energy for grain boundary self-diffusion, T is the temperature and R is the gas constant.

For varying temperature through the ice, Equation (1) can be rewritten as

where d(D2) is the cross-sectional area by which the crystals grow.

For the ice crystal model for Europa, with v = dz/dt, the analytical expression for the crystal size at the surface is

The temperature profile T(z) of the ice is determined by the volumetric tidal heat dissipation q, the temperature at the surface Ts and at the base Tb. The temperature profile for the ice shell used in this study is given by (Reference Chyba, Ostro and EdwardsChyba and others, 1998)

where Ts is the temperature at the surface and h = bln(Tb/Ts).

Equations (4) and (5) are solved using a finite-difference scheme, and the final crystal size at the surface D ² _s is found. The sizes of the crystals are described by a length parameter ds corresponding to the diameter of a circle of area D2_s. In order to calculate ds the values of the following parameters are needed: the ice thickness b, the mass loss rate due to sputtering v, the surface temperature Ts, and the base temperature T^b.

For each parameter a reference value is chosen based on spacecraft measurements and models. Since there is an uncertainty in these values, tests are made for a range of values around the reference value. In these tests, the surface crystal size ds is found for the range of values listed in Table 1 for the examined parameter, and the rest of the parameters held constant at their reference values.

Table 1.

Values of the parameters used

The age of the crystals at the surface is the total time it takes for them to be moved from base to surface. The initial crystal size is assumed to be d0 = 10^–8 m. The model is not sensitive to this value, since the crystal typically grows several cm or tens of cm in the first iteration.

For the reference values, an ice shell with a net mass-loss rate of v = 0.2 m Myr–¹, an ice thickness b = 2 km, a surface temperature Ts= 100 K, and abase temperature T^b = 273 K, a crystal size in the upper part of the ice of ds = 7 m is found (Fig. 1).

Fig. 1.

(a) Plot of crystal size d(z) vs depth zfor an ice shell with thickness b = 2000 m and a mass-loss rate v = 0.2 m Myr_–1; inset is a close-up of the lowermost 20 m. (b) Plot of temperature T(z) vs depth z. The temperatures at the boundaries are T_s = 100K and T_b = 273K.

Discussion and Results for the Values of Parameters Used

Discussion

The broad range of the parameters used results in surface crystal sizes from d^min = 4cm to d^max = 80 m. It has been estimated from surface spectra that the crystal sizes are 50– 100 μm at the surface (Reference Shirley and CarlsonShirley and Carlson, 2000). However, sputtering redistributes material at a rate of 0.8 m Myr–₁ (Reference Ip, Williams, Mcentire and MaukIp and others, 1998), and a thin upper layer of small-grained redistributed material may exist above possible large-grained old ice.

A change in crystal diameter of about 1m occurs for a variation in the surface temperature of a few tens of °C. The same change in crystal diameter occurs for a variation in the base temperature of only a few °C. The reason for this is that the change of shape of the temperature profile by changing the surface temperature occurs in the upper region of the ice shell, where crystal growth is not as effective as it is near the base.

Lake Vostok on the Antarctic ice sheet has often been used as an analogy to the physical settings on Europa, since it is a large lake under 4 km of ice (Reference JouzelJouzel and others, 1999). Above Lake Vostok, a deep ice core has been drilled to 120 m above the ice–lake intersection. The glacier ice above is estimated to be 420 000 years old, and the lower 90 m of ice consists of frozen lake water and is estimated to be roughly 1 Myr old (Reference PetitPetit and others, 1999). In this part of the ice core, ice crystals of sizes>0.2–1.0m have been found (Reference JouzelJouzel and others, 1999). The large ice crystals from Vostok show that large crystals are indeed a possibility when the crystals have a long time to grow.

The impurity content of the ice on Europa is a very uncertain parameter, which influences the crystal growth rate k. At the surface, the relative concentration of non-ice and ice-rich components has been measured (McCord and others, 1998). This provides information on what chemical species may be expected, but their quantities are hard to deduce. Sputtering processes are likely to alter the chemical composition at the surface, so the composition of the ice below the surface may differ from this.

For the process of crystal growth, the content of soluble impurities in the ice is important, since it may impede crystal growth (Reference PatersonPaterson, 1991). However, the salt content in the ice would be less than that of the sea water, since part of the salt is expelled during freezing. The amount of salt expelled is controlled by the freezing rate of the ice.

The stresses in the ice shell are not included in the calculations, where crystal sizes are calculated on the basis of normal grain growth. The effect of the stresses may be that a fabric develops which makes deformation easier in some directions than others. A fabric may also be developed during formation of the ice crystals, as seen in sea ice on Earth. Bending stresses in the ice shell may lead to polygonization. However, the presence of cracks in the ice shell must absorb part of the deformation, so strains are decreased compared to an unfractured shell.

In order to determine how the different mechanisms may influence crystal growth, a stress-dependent growth law is needed. The fabric would influence the deformation properties of the ice, so an anisotropic flow law would have to be used to calculate the tidal heating. In case the crystal sizes influence the flow properties of the ice as suggested by Reference Goldsby and KohlstedtGoldsby and Kohlstedt (1997), this would also have to be included in the calculations.

In the crystal growth model, the travel time of the crystals through the ice shell is equivalent to the age of the crystals when they reach the surface, and can be calculated to be tice = b/v, where b is the total ice thickness and v is mass-loss rate. For an ice thickness of 2000 m and a mass-loss rate of 0.2 m Myr–₁, this yields t_ice = 2000 m/0.2 m Myr–₁ = 10 Gyr. This age is more than the age of the solar system and Europa. However, the initial freezing of the ice where the ice shell was built up must have taken place at a faster rate than the present rate of freezing. This would cause the age–depth relation to be non-linear, and the upper crystals to be younger than the values calculated as in the previous example. The freezing of the ice from the water must always have taken place at the base of the ice, where it is warm. So all the crystals will have experienced a period of high temperature and fast growth rates.

On Earth it is known that the size and orientation of ice crystals on the surface of the ice sheets and at the base of sea ice influence the scattering of radar (Reference ZwallyZwally, 1977; Reference Tucker, Perovich, Gow, Weeks and CarseyTucker and others, 1992). This is caused by the dielectric contrast between ice grains and their surroundings (e.g. air or impurities at grain boundaries). Thus it cannot be ruled out that the ice crystal sizes affect radar scattering on Europa.

Conclusion

Model calculations predicts that there are large ice crystals in Europa’s ice shell, 4 cm–80 m across, with a value of 7 m for the preferred parameter values.

The ice crystal sizes are important for the deformation and fracture properties of the ice shell, and will therefore affect the tidal heat dissipation.

The calculated large sizes of ice crystals make it more plausible that the polygonal shapes of the ice blocks are caused by the lattice structure of large ice crystals. Theblocks of ice may have separated along weak zones such as crystal boundaries.

As radar backscatter may be influenced by the crystal sizes, it will be important to know these in order to design an orbiting radar sounder, which is a possible method for detecting an ocean under the ice shell on Europa.