GEOPHYSICALRESEARCH LETTERS, VOL. 13, NO. 5, PAGES448-451, MAY 1986

PHASE TRANSITIONS AND CONVECTION IN ICY SATELLITES

D. Bercovici, G. Schubert

Department of Earth and Space Sciences, University of California Los Angeles, California 90024

R. T. Reynolds

Theoretical and Planetary Studies Branch, Ames Research Center, NASA Moffett Field, California 94035

Abstract. The effects of solid-solid phase phase transitions (ice I - ice II, and ice VI - changes on subsolidus convection in the large ice VIII) and with two endothermic ones (ice I - icy moons of the outer solar system are ice III, and ice II - ice V). We carry out considered. Phase transitions affect convection linear stability analyses following the approach via processes that distort the phase change of Schubert et al. [1975] for the olivine-spinel boundary and/or influence buoyancy through (exothermic) and the spinel-postspinel thermal expansion. Linear stability analyses (endothermic) transitions in the earth's mantle. are carried out for ice layers with a phase By determining whether the specified ice phase change at the midplane. Two exothermic phase changes either enhance or impede convection we transitions (ice I - ice II, ice VI - ice VIII) can constrain evolutionary and structural models and two endothermic transitions (ice I - ice of the icy Jovian and Saturnian moons. In III, ice II - ice V) are considered. For the particular, we find that, as an icy moon cools, exothermic cases, the phase change can either the exothermic phase transitions change from impede or enhance whole-layer convection. For being stabilizing to destabilizing, whereas the the endothermic cases, the phase change always endothermic phase changes are always stabilizing inhibits whole-layer convective overturn and against convection. tends to enforce two-layer convection. These Phase Change Effects results place some constraints on possible models of icy satellite evolution and structure. Schubert et al. [1975] have discussed the Introduction physics of the interaction of solid-solid phase changes with convection. Two effects, advection The role of subsolidus convection in the icy of the ambient temperature and release or Galilean and Saturnian satellites has inspired a absorption of latent heat, distort the phase good deal of discussion, both before [Reynolds boundary. The distortion causes a hydrostatic and Cassen, 1979], and after [Parmentier and pressure head which enhances or impedes Head, 1979; Thurber et al., 1980; Schubert et convection across the boundary depending on the al., 1981, 1986; Cassen et al., 1981; Ellsworth sign of the pressure head. The third effect is and Schubert, 1983], the imaging experiments of also due to latent heat which influences the Voyagers 1 and 2 [Smith et al., 1979a, b, 1981, material's buoyancy through ordinary thermal 1982] revealed resurfacing and tectonic activity expansion. on several of these moons. However, the The advection of the ambient temperature influence of ice phase changes on convection has distorts the phase boundary via temperature received relatively little attention. Thurber et perturbations induced by upwelling and al. [1980] considered the ice II - ice V phase downwelling fluid. When hot upwelling material change and concluded that it impeded convection. rises through an exothermic (or endothermic) Reynolds et al. [1981] investigated the effects phase change the boundary is distorted on convection of several ice phase changes, both vertically since it must always lie along the exothermic and endothermic. They concluded that Clapeyron curve. The increase in temperature exothermic transitions are unstable to convection causes the boundary to distort downwards to for a wide range of temperature gradients and higher pressures if it is exothermic (with a that endothermic transitions are also unstable to Clapeyron curve whose slope is positive) and convection under conditions of high heat flow; upwards if endothermic (negative slope). the endothermic transitions are stable against Therefore, a hydrostatic head is developed which convective motion for lower heat flows that either enhances (for exothermic) or stabilizes correspond to later times in an icy moon's against (for endothermic) upward flow. evolution. In their thermal model of Ganymede, Similarly, downwelling, cold material causes the Kirk and Stevenson [1983] proposed that the ice boundary to distort upwards (exothermic) or I - ice III phase change is unstable to convec- downwards (endothermic) which affects the tion and would thereby initiate whole-layer stability of the flow similarly to the upwelling overturn within the outer strata of Ganymede case. early in its evolution. The latent heat also distorts the boundary In this paper, we examine the interaction of by increasing or decreasing the local subsolidus convection in ice with two exothermic temperature. Upwelling material will, upon passing through the equilibrium boundary, either Copyright 1986 by the American Geophysical Union. lose heat (exothermic) or gain heat (endothermic). By the nature of the Clapeyron Paper number 6L6077 curve, both cases will distort the boundary 0094-8276/86/006L-6077503.00 upwards, creating a hydrostatic head which

448 Bercovici et al.: Phase Changes and Convection in Icy Moons 449

TABLE 1. Properties of Ice at the Temperatures and Pressures of the Listed Phase Changes

I - II I -III II - V VI - VIII

m * 1.25x10' 4 1.4x10' 4 1.4x10' • 1.6x10' • (K-•)

g 1.4 1.4 1.4 1.4 (m s '• )

D 150 80 90 75 (km)

1.45x10 '6 9.2x10' ? 6.24x10' ? 4.2x10' ? (m•s -• )

8.1x10 s -2x10 s • -5x10 s -6.73x10 s 3.5x10 s (Pa K 'l )

240 204 61 ~ 200 (kg m'3 )

, P 1063 1042 1236 - 1700 (kg m'3)

Q/10a $$ 41.8 • 34.8 -0.987xT + 225.9 -66.9 • -64.4 65.5 (J kg 'l ) (T in K) Cp/10• $* 1.84 -> 1.51 0.0081xT - 0.094 1.92 -> 1.84 2.16 (J kg-lK -1 ) (T in K)

Q/cp 23 .... 34.9 30.3 (K) Temperatures and pressures lie along the Clapeyron curves of the respective phase changes. A range or function is indicated when a variation occurs along the Clapeyron curve. *Hobbs [1974]. $Hobbs [1974], Lupo and Lewis [1979]. $$Bridgeman [1912, 1935, 1937], Dorsey [1940], Hobbs [1974]. $*Lupo and Lewis [1979]. inhibits upward flow. The distortion of the change and shear stress-free boundaries was done boundary for downwelling material is in the by Schubert and Turcotte [1971]. The results of opposite sense and also impedes flow across the the stability analysis can be presented as a plot phase boundary. Therefore, the distortion of of the minimum super-adiabatic temperature the equilibrium boundary from latent heat is gradient (• - •a)crit at the onset of convection Wholly stabilizing against convection. versus kinematic viscosity •, (• is the ambient Finally, latent heat will contribute a temperature gradient and •a is the adiabatic stabilizing (exothermic) or destabilizing temperature gradient). This may be compared to (endothermic) influence on convection through similar plots for Rayleigh-B•nard(R-B) thermal expansion. Regardless of phase boundary convection with no phase change for either single distortions, upwelling material will heat up with an endothermic boundary, becoming less stable, or cool off for an exothermic boundary, 24j becoming more stable. Again, downwelling material will be influenced identically to upwelling material with respect to stability. To summarize the effects of phase changes, one can see that for exothermic boundaries the latent heat is stabilizing against convection in both manners of influence (boundary distortion and thermal expansion) while the advection of the ambient temperature is destabilizing. For an endothermic transition, the distortion of the boundary (for latent heat or temperature advection) is entirely stabilizing while the I0 I I I I I I I I I I influence on thermal expansion by latent heat is -6 -7 -8 -9 -D -II -12 -15 -14 -15 -16 -17 -18 destabilizing. It is apparent that the effect of phase changes on convective instability depends on several competing processes. Fig. 1. Viscosity •vs. strain rate • for ice I (dashed) and ice II (solid) at 200 K and Linear Stability Analysis ice I (dashed) and ice III (solid) at 250 K. The development of a linearized theory of Rheological laws from Durham et al. [1985] and stability for a fluid with a univariant phase Kirby et al. [1985]. 450 Barcovici et al.: Phase Changes and Convection in Icy Moons

iceSZ[-ice•TITiceI-ice]I entering into the stability criterion to the , , fourth power. In this model, D of each layer is characteristic of a differentiated, static, conducting moon the size of Ganymede with -2 8o' LIQUIDWATE• temperatures corresponding to conditions after a liquid water mantle has frozen out. Although the stability analysis is carried out only for ? -2 •-BDOUBLE / temperatures at which the considered phases exist, one must bear in mind that the CELL/• thicknesses of certain ice phases, namely ice iI III and ice V, will change markedly as the moon cools. P (kbar) Kinematic viscosity is calculated using the Arrhenius law [Schubert et al., 1981] '5-- v = O.139exp(8750/T) (1) where T is in Kelvins. Recent investigations [Kirby et al., 1984; Durham et al., 1985] have developed rheological laws for ices I, II, and o_56 -6 - III. These studies demonstrated that for -7C I I I I I lablaboratory strain rates greater than about 10 -4 13 14 15 16 17 18 19 s- i , ice III is weaker than ice I, which in turn is weaker than ice II. However, when these laws Iogo are used to calculate effective viscosities at Fig. 2. Minimum superadiabatic temperature geological strain rates, one finds that the gradient (• - •a)crit for the onset of viscosities for any two of these ice phases convection versus kinematic viscosity v for which co-exist at the same temperature are exothermic phase transitions ice I - ice II and similar (Figure 1). Thus, although there are ice VI - ice VIII. Viscosity and other presently only limited laboratory data on the parameters are calculated for the ices at rheological behavior of ices V, VI and VIII, we temperatures and pressures on the respective use a single viscosity law for the different Clapeyron curves (inset phase diagram). The phases. bold lines represent stability curves for single- The viscosities along each of the curves in cell (solid) and double-cell (dashed) Rayleigh- Figures 2 and 3 are calculated using (1) with Benard convection in ice without a phase change. temperatures that lie along the Clapeyron curve The thin solid curves represent the stability of each phase change. This assumes the curves for ice with a phase change. For a given occurrence of phase transitions at equilibrium viscosity, the curve with the smallest value of temperatures. However, below certain (• - •a)crit represents the most destabilizing temperatures, the reaction rates of most phase mechanism. A-B and C-D on the phase diagram changes are exceedingly slow [Bridgeman, 1912], delineate portions of the equilibrium phase and a phase may exist metastably in the boundaries along which the stability curves are stability field of an adjacent phase. computed. The (• - •a)crit versus v curves for the or double layer overturn (e.g., Figures 2 and 3). iceI-icellTiceTr-iceV For a given viscosity, the case with the smallest value of (• - •a)crit is the least stable. Thus, I if the curve of the fluid with a phase change lies abov& the double-cell R-B curve, it is more 1 80f LIQUID WATER• stable than two-layer convection and thereby inhibits whole- layer overturn. If the curve lies between the two R-B lines, the phase change allows single- cell convection but the fluid would convect more readily if there were no phase 'm r,,-',, / _ _ change. Whenthe curve is below the single cell \ / R-B line, the phase transition enhances whole- layer overturn. Thus, from this comparison, one may perceive the stability of a fluid with a • 0 • 46 8I0 I• 14 16 1820 2• •4 •• • phase change relative to the same fluid without a phase change. •- •/•OUBLE/

Results

Table 1 gives the values of mechanical and E thermal parameters used in the stability analyses. Most of the numerical constants do -7 CELL not vary greatly with temperature. However, the -8 I I I I relevant layer thickness D is strongly dependent 15 14 15 16 17 on the thermal evolution of a satellite (e.g., the ice III and ice V layers disappear completely Iogo at low temperatures). Therefore, D is the most uncertain of the parameters listed in Table 1, Chase •a•si•io•s ice • - ice • a•d ice • - but, it is also one of the most influential, ice V. Bercovici et al.: Phase Changes and Convection in Icy Moons 451

exothermic phase transitions ice I - ice II and Durham, W. B., S. H. Kirby, and H. C. Heard, ice VI - ice VIII are shown in Figure 2. For Rheology of the high pressure H•O ices II, the smaller viscosities (or warmer temperatures) III, and V (abstract), Lunar Planet. $ci. the phase changes are inhibiting to whole layer Conf., 16, 198-199, 1985. convective overturn. For viscosities greater Ellsworth, K., and G. Schubert, Saturn's icy than about 1016 m2s'l whole layer convection is satellites' thermal and structural models, preferred for both phase transitions. For Icarus, 54, 490-510, 1983. viscosities greater than about 3x10•? m2s'• (ice Hobbs, P. V., Ice Physics, 837 pp., Clarendon I - ice II) or 2x10 •? m•s '• (ice VI - ice VIII) Press, Oxford, 1974. the phase changes in fact enhance single layer Kirby, $. H., W. B. Durham, and H. C. Heard, convection. Rheologies of H20 ices Ih, II, and III at high The curves for the endothermic phase changes pressures' a progress report, in Ices in ice I - ice III and ice II - ice V (Figure 3) the Solar System, edited by J. Klinger et al., show that in the temperature ranges over which pp. 89-107, D. Reidel Publishing , Dordrecht, the transitions occur, the phase changes are Holland, 1984. inhibiting to single layer convective overturn Kirk, R. L., and D. J. Stevenson, Thermal evo- and that two layer convection is the preferred lution of a differentiated Ganymede and state. For the ice II - ice V transition, the implications for surface features (abstract), curve approaches single layer convective Lunar Planet. Sci. Conf., 14, 373-374, 1983. instability for viscosities around 10 l? m•s'•. Lupo, M. J., and J. S. Lewis, Mass radius relationships in icy satellites, Icarus, 40, Discussion 157-170, 1979. Parmentier, E. M., and J. W. Head, Some possible The effects of phase transitions on convective effects of solid state deformation on the instability may have a profound influence on the thermal evolution of ice-silicate planetary thermal histories of the larger icy moons. Early bodies, Proc. of Lunar Planet. Sci. in the evolution of a differentiated ice moon, after a water mantle between the ice I and ice Conf., 10, 2403-24!9, 1979. Reynolds, R. T., and P.M. Cassen, On the internal III layers has been frozen out, 2-layer structure of the major satellites of the outer convection will likely be preferred with the ice planets, Geophys. Res. Lett., •, 121-124, 1979. I - ice III boundary acting as a barrier to whole Reynolds, R. T., C. Alexander, A. Summers, and layer overturn. When the temperature of the moon P.M. Cassen, Solid state convection in icy decreases to the point where the ice I - ice II satellites' effects of phase transitions upon transition exists, 2-layer convection will stability, in NASA Tech. Memorandum84211' probably still•occur. However, as the Reports of Planetary Geology Program, temperature continues to decrease, the phase edited by H. E. Holt, 59-61, NASA Science change will •11ow single layer convection and as and Technology Information Branch, the temperature nears approximately 200K, the Washington D.C., 1981. phase change will enhance whole layer overturn. Schubert, G., and D. L. Turcotte, Phase changes Although these low temperatures are in the region and mantle convection, J. Geophys. Res., 76, of very slow reaction rates, the rate of overturn 1424-1432, 1971. is so small that the phase boundary will likely Schubert, G., D. A. Yuen, and D. L. Turcotte, be maintained at its equilibrium position. Role of phase transitions in a dynamic mantle, For the larger moons that may contain ice Geophys. J. R. Astron. Soc., 42, 705-735, 1975. polymorphs of density higher than that of ice II, the ice II - ice V phase change always Schubert, G., D. J. Stevenson, and K. Ellsworth, inhibits single layer convection. At depths Internal structures of the Galilean near possible silicate cores in these moons, the satellites, Icarus, 47, 46-59, 1981. ice VI - ice VIII phase transition will impede Schubert, G., T. Spohn, and R. T. Reynolds, single layer overturn while the moon is still Thermal histories, compositions and internal relatively warm. As the moon cools, however, structures of the moons of the solar system, single layer convection will be favored and in Natural Satellites, edited by J. Burns and eventually, at even colder temperatures,the D. Morrison, Universty of Arizona Press, phase change will enhance whole layer Tucson, in press, 1986. convection. Smith, B. A., eta!, The Galilean satellites and Jupiter' Voyager 2 imaging science results, References Science, 206 , 927-950, 1976b. Bridgeman, P. W., Water in the liquid and five Smith, B. A., et al., The Jupiter system through solid forms, under pressure, Proc. Am. Acad. the eyes of Voyager 1, Science, 204, 951-972, Arts Sci., 47, 441-558, 1912. 1979a. Bridgeman, P. W., The pressure-volume-temperature Smith, B. A., et al., Encounter with Saturn' relations of the liquid and the phase diagram Voyager 1 imaging science results, Science, of h•avy water, J. Chem. Phys., •, 597-605, 212, 163-191, 1981. 1935. Smith, B. A., et al., A new look at the Saturnian Bridgeman, P. W., The phase diagram of water to system' the Voyager 2 images, Science, 215, 45 000 kg/cm•, J. Chem. Phys., •, 964-966, 504-537, 1982. 1937. Thurber, C. H., A. T. Hsui, and M. N. Toksoz, Cassen, P.M., S. J. Peale, and R. T. Reynolds, Thermal evolution of Ganymede and Callisto' Structure and thermal evolution of the effects of solid-state convection and Galilean satellites, in The Satellites of constraints from Voyager imagery, Proc. Lunar Jupiter, edited by D. Morrison, pp. 93-128, Planet. Sci. Conf., 11, 1957-1977, 1980. University of Arizona Press, Tucson, 1981. Dorsey, N. E., Properties of ordinary water (Received February 19, 1986; substance, 673 pp., Reinhold, New York, 1940. accepted April 3, 1986.)