Convective-conductive transitions and sensitivity of a convecting ice shell to perturbations in heat flux and tidal-heating rate: Implications for Europa

We investigate the response of conductive and convective ice shells on Europa to variations of heat flux and interior tidal-heating rate. We present numerical simulations of convection in Europa's ice shell with Newtonian, temperature-dependent viscosity and tidal heating. Modest variations in the heat flux supplied to the base of a convective ice shell, ΔF, can cause large variations of the ice-shell thickness Δδ. In contrast, for a conductive ice shell, large ΔF involves relatively small Δδ. We demonstrate that, for a fluid with temperature-dependent viscosity, the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number Racr. This jump implies that, for a range of heat fluxes relevant to Europa, two equilibrium states—corresponding to a thin, conductive shell and a thick, convective shell—exist for a given heat flux. We show that, as a result, modest variations in heat flux near the critical Rayleigh number can force the ice shell to switch between the thin, conductive and thick, convective configurations over a ∼10⁷-year interval, with thickness changes of up to ∼10-30 km. Depending on the orbital and thermal history, such switches might occur repeatedly. However, existing evolution models based on parameterized-convection schemes have to date not allowed these transitions to occur. Rapid thickening of the ice shell would cause radial expansion of Europa, which could produce extensional tectonic features such as fractures or bands. Furthermore, based on interpretations for how features such as chaos and ridges are formed, several authors have suggested that Europa's ice shell has recently undergone changes in thickness. Our model provides a mechanism for such changes to occur.     

The impact of methane thermodynamics on seasonal convection and circulation in a model Titan atmosphere

We identify mechanisms controlling the distribution of methane convection and large-scale circulation in a simplified, axisymmetric model atmosphere of Titan forced by gray radiation and moist (methane) convection. The large-scale overturning circulation, or Hadley cell, is global in latitudinal extent and provides fundamental control of precipitation and tropospheric winds. The precipitating, large-scale updraft regularly oscillates in latitude with seasons. The distance of greatest poleward excursion of the Hadley cell updraft is set by the mass of the convective layer of the atmosphere; convection efficiently communicates seasonal warming of the surface through the cold and dense lower atmosphere, increasing the heat capacity of the system. The presence of deep, precipitating convection introduces three effects relative to the case with no methane latent heating: (1) convection is narrowed and enhanced in the large-scale updraft of the Hadley cell; (2) the latitudinal amplitude of Hadley cell updraft oscillations is decreased; and (3) a time lag is introduced. These effects are observable in the location and timing of convective methane clouds in Titan’s atmosphere as a function of season. A comparison of simulations over a range of convective regimes with available observations suggest methane thermodynamic-dynamic feedback is important in the Titan climate.     

Thermal history of the Jovian satellite Io

The discovery of large volcanic eruptions on Io suggests that Io is one of the most geologically active planetary bodies. The energy source of this geologic activity is believed to be tidal heating induced by Jupiter. A number of thermal history calculations were done to investigate the effect of tidal heating on the thermal history of Io taking into account solid state convection and advective heat transfer. These simulations show that the total tidal heating energy in Io is almost equal to the advectively transferred heat, indicating that the observed heat flow from Io is nearly equal to the total tidal heating energy. Since total tidal heating energy is dependent on the radius of the liquid mantle and the internal dissipation factor (Q), the radius of the liquid mantle can be estimated for a given value of Q. Some reasonable thermal history models of Io were obtained using a model with Q ≈ 25–50 in which the magma source of Ionian volcanism is at a depth of 100–300 km. The models satisfy the heat flow data and the existence of a thick lithosphere. Using a model with Q = 25 and L = 300 km (thickness of the advective region) as the standard model (model II), we then studied the effect of convective heat transfer and the initial temperature distribution on the Ionian thermal history. In these calculations, the other parameters are the same as in the standard model (model II). These calculations show that although the temperature distribution in the central region reflects the difference in the efficiency of convective heat transfer and initial temperature distribution, the temperature distribution in the outer region does not changes appreciably.     

Self-excitation of motions in near-convection zones and the generation of solar magnetic field

In view of the recently discovered time variations in rotation velocity within the solar differentially rotating tachocline (Howe et al. 2000), we study conditions for the equilibrium and excitation of motions in nonrigidly rotating magnetized layers of the radiative zones located near the boundaries of the convection zone. The emphasis is on the possible relationship between quasi-periodic tachocline pulsations and the generation of a nonaxisymmetric magnetic field in the convection zone. This field generation is studied under the assumption that it results from a reduction in the expenditure of energy on convective heat transport. The (antisymmetric about the equator) field is shown to increase in strength if there are both a radial gradient in angular velocity and steady-state axisymmetric meridional circulation of matter. The sense of circulation is assumed to change (causing the sign of the generated field to change) after the maximum permissible field strength is reached. This is apparently attributable to the excitation of the corresponding turbulent viscosity of the medium. It is also important that the cyclic field variations under discussion are accompanied by variations in solar-type dipole magnetic field.     

Heat flow, lenticulae spacing, and possibility of convection in the ice shell of europa

Two opposing models to explain the geological features observed on Europa’s surface have been proposed. The thin-shell model states that the ice shell is only a few kilometers thick, transfers heat by conduction only, and can become locally thinner until it exposes an underlying ocean on the satellite’s surface. According to the thick-shell model, the ice shell may be several tens of kilometers thick and have a lower convective layer, above which there is a cold stagnant lid that dissipates heat by conduction. Whichever the case, from magnetic data there is strong support for the presence of a layer of salty liquid water under the ice. The present study was performed to examine whether the possibility of convection is theoretically consistent with surface heat flows of ∼100-200 mW m⁻², deduced from a thin brittle lithosphere, and with the typical spacing of 15-23 km proposed for the features usually known as lenticulae. It was obtained that under Europa’s ice shell conditions convection could occur and also account for high heat flows due to tidal heating of the convective (nearly isothermal) interior, but only if the dominant water ice rheology is superplastic flow (with activation energy of 49 kJ mol⁻¹; this is the rheology thought dominant in the warm interior of the ice shell). In this case the ice shell would be ∼15-50 km thick. Furthermore, in this scenario explaining the origin of the lenticulae related to convective processes requires ice grain size close to 1 mm and ice thickness around 15-20 km.     

On convection in ice I shells of outer Solar System bodies, with detailed application to Callisto

It has been argued that the dominant non-Newtonian creep mechanisms of water ice make the ice shell above Callisto's ocean, and by inference all radiogenically heated ice I shells in the outer Solar System, stable against solid-state convective overturn. Conductive heat transport and internal melting (oceans) are therefore predicted to be, or have been, widespread among midsize and larger icy satellites and Kuiper Belt objects. Alternatively, at low stresses (where non-Newtonian viscosities can be arbitrarily large), convective instabilities may arise in the diffusional creep regime for arbitrarily small temperature perturbations. For Callisto, ice viscosities are low enough that convection is expected over most of geologic time above the internal liquid layer for plausible ice grain sizes (?a few mm); the alternative for early Callisto, a conducting shell over a very deep ocean (>450 km), is not compatible with Callisto's present partially differentiated state. Moreover, if convection is occurring today, the stagnant lid would be quite thick (∼100 km) and compatible with the lack of active geology. Nevertheless, Callisto's steady-state heat flow may have fallen below the convective minimum for its ice I shell late in Solar System history. In this case convection ends, the ice shell melts back at its base, and the internal ocean widens considerably. The presence of such an ocean, of order 200 km thick, is compatible with Callisto's moment-of-inertia, but its formation would have caused an ∼0.25% radial expansion. The tectonic effects of such a late, slow expansion are not observed, so convection likely persists in Callisto, possibly subcritically. Ganymede, due to its greater size, rock fraction and full differentiation, has a substantially higher heat flow than Callisto and has not reached this tectonic end state. Titan, if differentiated, and Triton should be more similar to Ganymede in this regard. Pluto, like Callisto, may be near the tipping point for convective shutdown, but uncertainties in its size and rock fraction prevent a more definitive assessment.     

Saturn's icy satellites: Thermal and structural models

Thermal histories of the small icy Saturnian satellites Mimas, Tethys, Dione, Rhea, and Iapetus are constructed by assuming that they formed as homogeneous ice-silicate mixtures. The models include effects of radiogenic and accretional heating, conductive and subsolidus convective heat transfer, and lithospheric growth. Accretional heating is unlikely to have melted the water ice in the interiors of these bodies and solid state creep of the predominately ice material precludes melting by radiogenic heating. Mimas is so small that its thermal evolution is essentially purely conductive; at present it is a cold, nearly isothermal body. Any subsolidus convection or thermal activity in Mimas would have been confined to a brief period in its early history and would have been due to a warm formation. The four largest satellites are big enough and contain sufficient heat-producing silicates that solid state convection beneath a rigid lithosphere is inevitable independent of initial conditions. Dione and Rhea have convective interiors for most of their thermal histories, while Tethys and Iapetus have mainly conductive thermal histories with early periods of convective 0activity. The thermal histories of the five satellites for the last 4 by are independent of initial conditions; at present they have cold, conductive interiors. The model thermal histories are qualitatively consistent with the appearances of these satellites: Mimas has an ancient heavily cratered surface, Tethys and probably Iapetus have both heavily cratered and more lightly cratered areas, and Dione and Rhea have extensively modified surfaces. Because of their similar sizes and densities, Mimas and Enceladus are expected to have similar surfaces and thermal histories, but instead Enceladus has the most modified surface of all the small icy Saturnian satellites. Our results suggest a heat source for Enceladus, in addition to radiogenic and accretional heating; tidal dissipation is a possibility. Because the water ice in these bodies does not melt, resurfacing must be accomplished by the melting of a low-melting-temperature minor component such as ammonia hydrate.     

Scaling laws for convection and jet speeds in the giant planets

Three dimensional studies of convection in deep spherical shells have been used to test the hypothesis that the strong jet streams on Jupiter, Saturn, Uranus, and Neptune result from convection throughout the molecular envelopes. Due to computational limitations, these simulations must be performed at parameter settings far from jovian values and generally adopt heat fluxes 5-10 orders of magnitude larger than the planetary values. Several numerical investigations have identified trends for how the mean jet speed varies with heat flux and viscosity in these models, but no previous theories have been advanced to explain these trends. Here, we show using simple arguments that if convective release of potential energy pumps the jets and viscosity damps them, the mean jet speeds split into two regimes. When the convection is weakly nonlinear, the equilibrated jet speeds should scale approximately with F/ν, where F is the convective heat flux and ν is the viscosity. When the convection is strongly nonlinear, the jet speeds are faster and should scale approximately as (F/ν)^1/2. We demonstrate how this regime shift can naturally result from a shift in the behavior of the jet-pumping efficiency with heat flux and viscosity. Moreover, both Boussinesq and anelastic simulations hint at the existence of a third regime where, at sufficiently high heat fluxes or sufficiently small viscosities, the jet speed becomes independent of the viscosity. We show based on mixing-length estimates that if such a regime exists, mean jet speeds should scale as heat flux to the 1/4 power. Our scalings provide a good match to the mean jet speeds obtained in previous Boussinesq and anelastic, three-dimensional simulations of convection within giant planets over a broad range of parameters. When extrapolated to the real heat fluxes, these scalings suggest that the mass-weighted jet speeds in the molecular envelopes of the giant planets are much weaker—by an order of magnitude or more—than the speeds measured at cloud level.     

Solar Activity and Deep Convection Modeling

We suggest from synoptic charts of radial magnetic field and intensities of spectral lines (Fe?i, He?ii, and Fe?ix/x) over Carrington rotations 1942??C?2050 that deep convective layers control the pattern of large-scale solar activity. A new result is a Kolmogorov-type energy spectrum of the longitudinal variations of solar activity. This spectrum for nonphotospheric scales of convection (harmonic number m<100) is a new ??fingerprint?? of turbulence in the deep layers of the solar convection zone (CZ). The manifestation of one source of convective turbulence in the deep CZ is revealed as the excess in the power spectrum over the Kolmogorov spectrum. This source may be identified with giant convection cells at the CZ bottom. The convective cascade of the turbulence starts at the vortex size corresponding to the trans-CZ convective cells with the turnover time which the mixing length theory (MLT) predicts. This connection between the MLT formalism and real features in the Sun could account for the success of the MLT in stellar modeling.     

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The nature of the sunspot phenomenon

This paper points out the basic relation between the conversion of thermal energy into convective fluid motion (Alfvén waves when a strong vertical magnetic field is present) and the convective transport of thermal energy. It is shown that heat transport necessarily accompanies convective driving of fluid motion. Convective motions restricted to a layer whose thickness is a small fraction of the local scale height can divert no more than the same fraction of the energy into Alfvén waves. But if the convecting layer extends over many scale heights, then the convective forces may convert more energy into fluid motion than they transport. Hence the creation of a cool sunspot requires convection extending coherently over several scale heights, at least 500 km. This requirement is basically just the familiar thermodynamic efficiency of an ideal heat engine. The calculations establish that convection need not be much less efficient than the ideal.     

The dependence of the climate sensitivity on convective parametrization: statistical evaluation

Two sensitivity experiments, in which CO₂ is instantaneously doubled, have been performed with a general circulation model to determine the influence of the convective parametrization on simulated climate change. We have examined the spatial structure of changes in the annual mean and annual cycle for surface temperature and precipitation for both experiments; similarly we have examined changes in the variance for these two fields. We have also computed a range of test statistics in order to obtain reliable measures of the signal-to-noise ratio in the climate change signal from each experiment. We have computed test statistics for the entire globe and for five different region and we contrast the global response with the response in the Australian region taken as a representative sample.We find that the highest signal-to-noise ratios in the change from 1 * CO₂ to 2 * CO₂ are for the change in surface temperature for both experiments with little difference in the global averages between the experiments. Globally averaged precipitation shows a greater noise level but perhaps the greatest contrast between experiments. There are generally significant increases in the temporal and spatial variability of precipitation in the change from the 1 * CO₂ to 2 * CO₂ and with some differences apparent between the two experiments. The temporal variability of surface temperature does not change significantly in any of the 2 * CO₂ cases, and there is little difference between the experiments. There is a significant decrease in the spatial variability of surface temperature in all 2 * CO₂ experiments in all cases and with significant differences in the seasonal variations between different experiments. The spatial variability of precipitation increases in all 2 * CO₂ cases and also with substantial differences in the seasonal variations between the experiments. There are accompanying significantly different spatial pattern correlations for both surface temperature and precipitation. In general we find that the global changes are fairly robust with the differences associated with convective parametrization schemes being very small. However, at the regional level, there are marked differences between experiments with changes both in the means and in the spatial and temporal variances but often with low levels of significance.     

Shell thickness variations and the long-wavelength topography of Titan

The long-wavelength topography of Titan has an amplitude larger than that expected from tidal and rotational distortions at its current distance from Saturn. This topography is associated with small gravity anomalies, indicating a high degree of compensation. Both observations can be explained if Titan has a floating, isostatically-compensated ice shell with a spatially-varying thickness. The spatial variations arise because of laterally-variable tidal heating within the ice shell. Models incorporating shell thickness variations result in an improved fit to the observations and a degree-two tidal Love number h_2t consistent with expectations, without requiring Titan to have moved away from Saturn. Our preferred models have a mean shell thickness of ≈100 km in agreement with the observed gravity anomalies, and a heat flux appropriate to a chondritic Titan. Shell thickness variations are eliminated by convection; we therefore conclude that Titan’s ice shell is not convecting at the present day.     

Convection driven by tidal and radiogenic heating in medium size icy satellites

Convection is one of the most important processes responsible for the formation of the surface features on many planetary bodies. Observations of some icy satellites indicate that the satellites’ surfaces are modified due to the internally driven tectonic activity. The tidal heating could be an important source of energy responsible for such internal activity. This suggestion is supported by the correlation of the tidal parameter ψ and tectonic features. Consequently, the tidal and the radiogenic heat sources seem to be of primary importance for the medium size icy satellites. Our research deals with convection in a non-differentiated body. The convection is a results of both uniform radiogenic heating and non-uniform and non-spherically symmetric tidal heating. To investigate the problem a 3D model of convection is developed based on the Navier-Stokes equation, the equation of thermal conductivity, the equation of continuity, and the equation of state. The 3D formulae for the tidal heat generation based on the results of Peale and Cassen [1978. Icarus 36, 245-269] and others are used in the model. To measure the relative importance of radiogenic heating versus tidal heating a dimensionless number C_t is introduced. The systematic investigation of a steady-state convection is performed for different values of the Rayleigh number and for the full range of C_t. The results indicate that for low and moderate value of the Rayleigh number, convection pattern driven by the tidal heating and by the radioactivity in the medium size icy satellites consists of one cell or of two cells. For C_t>0 the critical value of Rayleigh number Ra_cr=0. The one-cell pattern is specific for low Rayleigh numbers but it could be observed for the full range of number C_t. It means that the pattern of convection does not fully follow the pattern of heating. This rather unexpected result could be of great importance for the final stage of convection. All patterns of tidally driven convection are oriented with respect to the direction to the planet. For two-cell patterns the regions of downward motion are situated in the centers of the near and far sides of the satellite, respectively.     

Convective cores in galactic cooling flows

We use hydrodynamic simulations with adaptive grid refinement to study the dependence of hot gas flows in X-ray luminous giant elliptical galaxies on the efficiency of heat supply to the gas. We consider a number of potential heating mechanisms including Type Ia supernovae and sporadic nuclear activity of a central supermassive black hole. As a starting point for this research we use an equilibrium hydrostatic recycling model. We show that a compact cooling inflow develops, if the heating is slightly insufficient to counterbalance radiative cooling of the hot gas in the central few kiloparsecs. An excessive heating in the centre, instead, drives a convectively unstable outflow. We model the onset of the instability and a quasi-steady convective regime in the core of the galaxy in two dimensions assuming axial symmetry.
Provided the power of net energy supply in the core is not too high, the convection remains subsonic. The convective pattern is dominated by buoyancy driven large-scale mushroom-like structures. Unlike in the case of a cooling inflow, the X-ray surface brightness of an (on average) isentropic convective core does not display a sharp maximum at the centre. A hybrid model, which combines a subsonic peripheral cooling inflow with an inner convective core, appears to be stable. We also discuss observational implications of these results.     

Oceans in the icy Galilean satellites of Jupiter?

Equilibrium models of heat transfer by heat conduction and thermal convection show that the three satellites of Jupiter—Europa, Ganymede, and Callisto—may have internal oceans underneath ice shells tens of kilometers to more than a hundred kilometers thick. A wide range of rheology and heat transfer parameter values and present-day heat production rates have been considered. The rheology was cast in terms of a reference viscosity ν₀ calculated at the melting temperature and the rate of change A of viscosity with inverse homologous temperature. The temperature dependence of the thermal conductivity k of ice I has been taken into account by calculating the average conductivity along the temperature profile. Heating rates are based on a chondritic radiogenic heating rate of 4.5 pW kg⁻¹ but have been varied around this value over a wide range. The phase diagrams of H₂O (ice I) and H₂O + 5 wt% NH₃ ice have been considered. The ice I models are worst-case scenarios for the existence of a subsurface liquid water ocean because ice I has the highest possible melting temperature and the highest thermal conductivity of candidate ices and the assumption of equilibrium ignores the contribution to ice shell heating from deep interior cooling. In the context of ice I models, we find that Europa is the satellite most likely to have a subsurface liquid ocean. Even with radiogenic heating alone the ocean is tens of kilometers thick in the nominal model. If tidal heating is invoked, the ocean will be much thicker and the ice shell will be a few tens of kilometers thick. Ganymede and Callisto have frozen their oceans in the nominal ice I models, but since these models represent the worst-case scenario, it is conceivable that these satellites also have oceans at the present time. The most important factor working against the existence of subsurface oceans is contamination of the outer ice shell by rock. Rock increases the density and the pressure gradient and shifts the triple point of ice I to shallower depths where the temperature is likely to be lower then the triple point temperature. According to present knowledge of ice phase diagrams, ammonia produces one of the largest reductions of the melting temperature. If we assume a bulk concentration of 5 wt% ammonia we find that all the satellites have substantial oceans. For a model of Europa heated only by radiogenic decay, the ice shell will be a few tens of kilometers thinner than in the ice I case. The underlying rock mantle will limit the depth of the ocean to 80-100 km. For Ganymede and Callisto, the ice I shell on top of the H₂O-NH₃ ocean will be around 60- to 80-km thick and the oceans may be 200- to 350-km deep. Previous models have suggested that efficient convection in the ice will freeze any existing ocean. The present conclusions are different mainly because they are based on a parameterization of convective heat transport in fluids with strongly temperature dependent viscosity rather than a parameterization derived from constant-viscosity convection models. The present parameterization introduces a conductive stagnant lid at the expense of the thickness of the convecting sublayer, if the latter exists at all. The stagnant lid causes the temperature in the sublayer to be warmer than in a comparable constant-viscosity convecting layer. We have further modified the parameterization to account for the strong increase in homologous temperature, and therefore decrease in viscosity, with depth along an adiabat. This modification causes even thicker stagnant lids and further elevated temperatures in the well-mixed sublayer. It is the stagnant lid and the comparatively large temperature in the sublayer that frustrates ocean freezing.     

Tidally-induced melting events as the origin of south-pole activity on Enceladus

The intense activity at the south pole of Enceladus hints at an internal water reservoir. However, there is no direct evidence of liquid water at present and its long-term stability in the interior remains problematic. By modeling heat production and transfer in the ice shell in a spherical geometry, we show that tidal heating naturally leads to a concentration of convective hot upwellings in the south polar region, favoring the preservation of liquid water at depth. We show that large volumes of water are produced within the ice shell at the south pole during periods of elevated orbital eccentricity (3–5 times the present-day value). Strong lateral variations in the melt production and crystallization rates result in stress concentration in the south polar region, thus providing an explanation for the tectonic activity observed today. We predict that an internal ocean may be sustained over the long term as the consequence of repeated periods with elevated orbital eccentricity, leading to episodic melting and resurfacing events.     

Possibility of Convection for Diffusion (Newtonian) Viscosity in the Ice Shell of Europa?

Recent investigations into convection in Europa’s ice shell have been based on non-Newtonian (stress-dependent) or Newtonian (stress-independent) viscosity for water ice. However, despite the wide use of Newtonian convection, experimentally observed water ice flow is non-Newtonian, and analysis of stability against convection of the ice shell using updated flow laws has been only performed for non-Newtonian rheologies. Here we use the flow law proposed for diffusion creep to analyze the possibility of the onset of convection for Newtonian viscosity in relation to the thermal state of Europa. Our findings indicate that for diffusion creep convection might have started, but that significantly lower heat flows (and equivalently higher shell thicknesses) and/or grain sizes are required than for superplastic flow, which is the most probable flow mechanism if the ice shell is convective.     

A model for the temperature-dependence of tidal dissipation in convective plumes on icy satellites: Implications for Europa and Enceladus

To explain the formation of surface features on Europa, Enceladus, and other satellites, many authors have postulated the spatial localization of tidal heating within convective plumes. However, the concept that enhanced tidal heating can occur within a convective plume has not been rigorously tested. Most models of this phenomenon adopt a tidal heating with a temperature-dependence derived for an incompressible, homogeneous (zero-dimensional) Maxwell material, but it is unclear whether this formulation is relevant to the heterogeneous situation of a warm plume surrounded by cold ice. To determine whether concentrated dissipation can occur in convective plumes, we develop a two-dimensional model to compute the volumetric dissipation rate for an idealized, vertically oriented, isolated convective plume obeying a Maxwellian viscoelastic compressible rheology. We apply the model to the Europa and Enceladus ice shells, and we investigate the consequences for partial melting and resurfacing processes on these bodies. We find that the tidal heating is strongly temperature dependent in a convective ice plume and could produce elevated temperatures and local partial melting in the ice shells of Europa and Enceladus. Our calculation provides the first quantitative verification of the hypothesis by Sotin et al. [Sotin, C., Head, J.W., Tobie, G., 2002. Geophys. Res. Lett. 29. 74-1] and others that the tidal dissipation rate is a strong function of temperature inside a convective plume. On Europa, such localized heating could help allow the formation of domes and chaos terrains by convection. On Enceladus, localized tidal heating in a thermal plume could explain the concentrated activity at the south pole and its associated heat transport of 2-7 GW.     

Thermal-orbital evolution of Io and Europa

Coupled thermal-orbital evolution models of Europa and Io are presented. It is assumed that Io, Europa, and Ganymede evolve in the Laplace resonance and that tidal dissipation of orbital energy is an internal heat source for both Io and Europa. While dissipation in Io occurs in the mantle as in the mantle dissipation model of Segatz et al. (1988, Icarus 75, 187), two models for Europa are considered. In the first model dissipation occurs in the silicate mantle while in the second model dissipation occurs in the ice shell. In the latter model, ice shell melting and variations of the shell thickness above an ocean are explicitly included. The rheology of both the ice and the rock is cast in terms of a viscoelastic Maxwell rheology with viscosity and shear modulus depending on the average temperature of the dissipating layer. Heat transfer by convection is calculated using a parameterization for strongly temperature-dependent viscosity convection. Both models are consistent with the present orbital elements of Io, Europa, and Ganymede. It is shown that there may be phases of quasi-steady evolution with large or small dissipation rates (in comparison with radiogenic heating), phases with runaway heating or cooling and oscillatory phases during which the eccentricity and the tidal heating rate will oscillate. Europa's ice thickness varies between roughly 3 and 70 km (dissipation in the silicate layer) or 10 and 60 km (dissipation in the ice layer), suggesting that Europa's ocean existed for geological timescales. The variation in ice thickness, including both convective and purely conductive phases, may be reflected in the formation of different geological surface features on Europa. Both models suggest that at present Europa's ice thickness is several tens of km thick and is increasing, while the eccentricity decreases, implying that the satellites evolve out of resonance. Including lithospheric growth in the models makes it impossible to match the high heat flux constraint for Io. Other heat transfer processes than conduction through the lithosphere must be important for the present Io.