Elsevier

Icarus

Volume 180, Issue 1, January 2006, Pages 251-264
Icarus

Thermal convection with a water ice I rheology: Implications for icy satellite evolution

https://doi.org/10.1016/j.icarus.2005.07.014Get rights and content

Abstract

We model stagnant–lid convection for water ice I using a multicomponent rheology, combining grain boundary sliding, dislocation and diffusion creep mechanisms. For the superplastic flow–dislocation creep rheology, dislocation creep (n=4) dominates the deformation within the actively convecting sublayer whilst superplastic flow (n=1.8) is the dominant process within the stagnant–lid whilst for the superplastic flow–diffusion creep rheology, superplastic flow is the dominant deformation mechanism within the convecting sublayer while diffusion creep (n=1) is the dominant deformation process in the stagnant–lid. These results suggest deformation in the actively convecting sublayer is likely to be dominated by the mechanism with the largest stress exponent. We also provide heat flux scaling relationships for the superplastic flow, basal slip, dislocation creep–superplastic flow and superplastic flow–diffusion creep rheologies and provide a simple parameterized convection model of an icy satellite thermal evolution.

Introduction

Systematic laboratory investigations concerning the deformation behavior of water ice has resulted in the construction of a detailed deformation map. Three distinct mechanisms have been identified as functions of strain rate and temperature. These are dislocation creep, superplastic flow and a basal slip regime. The dislocation creep regime is characterized by a high stress exponent of 4.0 and is independent of grain size. In the grain boundary sliding-accommodated basal slip, or, superplastic flow regime, deformation is characterized by a stress exponent of 1.8 and an inverse relationship on grain size (to the power of 1.4). Basal slip-accommodated grain boundary sliding is characterized by a stress exponent of 2.4. A fourth regime, that of grain boundary diffusion with a stress exponent of 1, has not been observed experimentally even at low stress and very fine grain size (Goldsby and Kohlstedt, 2001).

Deformation of materials under planetary conditions is a complicated process with transitions in the deformation style occurring as functions of many parameters. The deformation is most realistically described as a sum of contributions from the many available deformation processes. Under various stress and temperature conditions one mechanism may be identified to contribute the majority of strain and, therefore, will dominate the deformation. This point has allowed the investigation and characterization of individual creep mechanisms. Application of this water ice deformation map has resulted in numerous interpretations concerning the geodynamic evolution of icy moons. Whilst the applicable deformation regime for an icy satellite interior is still subject to large uncertainties, the laboratory data places useful bounds on the rheology. In addition, the determination of flow parameters under varying strain rate and thermal conditions allows for the incorporation of these flow laws into thermal convection and geodynamic evolution models.

Thermal convection with in an icy satellite is characterized by low strain rates (ɛ˙10−14s−1 or slower for the convective strain rate in a large moon like Ganymede) and low stress (σ0.10.01MPa in a convective sublayer). A further complication is introduced by the grain size dependence of some deformation mechanisms. It is assumed that the grain size within an icy mantle will be relatively uniform, with current estimates suggesting a value of ∼0.1 mm. Pappalardo et al. (1998) considered the europan ice shell and demonstrated that a change in grain size from 0.1 mm at a strain rate of 3×10−10s−1 to a grain size of 1 cm at a strain rate of 1×10−10s−1 resulted in up to an order of magnitude change in the effective viscosity. In the Pappalardo et al. (1998) study a grain size of 1 mm was chosen based upon Galileo imaging of Europa's surface. Terrestrial ice sheets and glaciers have typical ice grain sizes of ∼1 mm to several centimeters over temperature ranges from 233 to 273 K and stresses typically ⩽0.1 MPa (Goldsby and Kohlstedt, 2001). McKinnon (1999) suggests that polar glacial ice are potentially more relevant when extrapolating terrestrial ice grain sizes to the icy satellites and uses the data of De La Chappell et al. (1998) to infer that the grain size in the europan ice shell should not exceed 1 mm at 260 K and will be smaller at lower temperatures.

In any case, the applicable deformation regime to icy satellite interiors has been in question. Nimmo and Manga (2002) considered both superplastic flow and diffusion creep as the deformation mechanisms in their investigation concerning convective diapirism within the europan ice shell. These researchers conclude that deformation within the ice shell is in the diffusion creep regime and suggest an ice grain size of 0.02–0.06 mm. This result is in disagreement with the study performed by Ruiz and Tejero (2003) who concluded that superplastic flow would be the dominant deformation mechanism in the europan ice shell with a grain size close to 1 mm.

Nimmo (2004) considered both basal slip and superplastic flow mechanisms in his study of rifting end extension on icy satellites, deciding upon the dominant mechanism as a function of strain rate and grain size. He concludes that for the likely range of grain sizes, 1–10 mm based on the range observed in terrestrial sea ice (Budd and Jacka, 1989) that the basal slip mechanism will dominate the rheology. Dombard and McKinnon (2001) employed the superplastic flow law when investigating the formation of Ganymede's grooved terrain, suggesting that strain rates of 10−1610−14s−1 and low stress quantify the observed surface topography. Grain sizes in the Dombard and McKinnon (2001) study were modeled at being in the range 0.1 mm–1 cm based upon glacial ice studies (De La Chappell et al., 1998). The Ganymede crater relaxation modeling by Dombard and McKinnon (2000) considered various rheological laws describing dislocation creep, basal slip and superplastic flow and demonstrated that with a realistic water ice rheology, crater relaxation times for Ganymede were very long. This result did not agree with earlier attempts to model relaxation times of icy satellite topography where the rheological properties of water ice were less well known.

Ruiz (2001) considered both dislocation and superplastic flow in modeling the stability against convection of an outer ice shell on Callisto. He showed that with a non-Newtonian rheology this outer shell would be stable against convection, implying that a subsurface ocean may have survived to the present. Early models of icy satellite thermal evolution approximated the ice rheology with a Newtonian temperature-dependent law only. These models resulted in rapid differentiation and freezing of the satellite interior (Reynolds and Cassen, 1979).

This subset of icy satellite modeling demonstrates the large uncertainty present when estimating the deformation regime within an icy satellite. Whilst consensus amongst the literature does not seem apparent, insights into the nature of icy satellite geodynamics have still been achieved, in some cases with extraordinary outcomes. In any case, a systematic numerical investigation of thermal convection with a water ice rheology has not been undertaken, except for the dislocation creep regime (Freeman et al., 2004). In this study we investigate thermal convection with either a superplastic flow law or a basal slip flow law. We also model thermal convection with a multicomponent rheology. These multicomponent rheological laws consist are (1) dislocation creep–superplastic flow and (2) superplastic flow–diffusion creep.

Section snippets

Rheology

A constitutive relationship for the steady state deformation of a crystalline material is (Durham et al., 1992)ɛ˙=Adpτnexp[(E+PV)RT], where P is a pressure (mean of the principal stresses), d is the grain size, T is the temperature, R is the gas constant, E and V are the activation energy and activation volume respectively, A, p, and n, are flow constants particular to the deformation mechanism, τ is the second invariant of the deviatoric stress tensor and ɛ˙ is the second invariant of

GBS deformation

We first determine the boundary between the convective regimes by studying non-Newtonian viscosity convection with a stress exponent of n=1.8 and n=2.4 over a range of basal Rayleigh numbers and viscosity contrasts. To characterize the deformation regime, we calculate the following quantities; the Nusselt number (Nu=Fd/kΔT), the convective internal temperature (Ti), the surface and basal r.m.s. velocities (u0 and u1), the upper and lower boundary layer thickness (δ0 and δ1) and the surface

Dislocation creep and superplastic flow

We model the multicomponent system of superplastic flow and dislocation creep over a range of viscosity contrasts and basal Rayleigh numbers. The dynamics of this system is shown in Fig. 5, Fig. 6 where we present thermal convection snapshots detailing the temperature field and stream function, second invariant of the deviatoric stress tensor, second invariant of the strain rate tensor, viscous dissipation (τ2/η) and a logarithmic viscosity ratio (ηr) of each deformation mechanism. This

Application to icy satellites

Many studies of the thermal evolution of the inner planets of the Solar System have employed a parameterized approach Schubert et al., 1979, Spohn and Schubert, 1982, Stevenson et al., 1983, Honda, 1995, Honda and Iwase, 1996, McNamara and van Keken, 2000. Direct application of the method to modeling the thermal evolution of an icy satellite have also been previously undertaken Ellsworth and Schubert, 1983, Deschamps and Sotin, 2001, Hussmann et al., 2002. An energy conservation equation is

Grain boundary sliding mechanisms

We have characterized the evolution into the stagnant lid regime with a non-Newtonian water ice rheology. The rheology is descriptive of grain boundary sliding creep mechanisms even though the grain size evolution of the material was not explicitly included in our models. The numerical models describing the non-Newtonian grain size sensitive deformation mechanisms demonstrated that the regime boundaries are similar to those determined for Newtonian viscosity convection.

For the superplastic

Acknowledgements

We thank D.R. Stegman for many helpful discussions. Computational resources were provided by Monash Cluster Computing and the Victorian Partnership for Advanced Computing.

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