Convection in Ice I With Non-Newtonian

Rheology: Application to the Icy Galilean

Satellites

by

Amy Courtright Barr

B.S., California Institute of Technology, 2000

M.S., University of Colorado, 2002

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Geophysics Graduate Program

Department of Astrophysical and Planetary Sciences

2004

This thesis entitled:Convection in Ice I With Non-Newtonian Rheology: Application to the Icy Galilean

Satelliteswritten by Amy Courtright Barr

has been approved for the Geophysics Graduate ProgramDepartment of Astrophysical and Planetary Sciences

Robert T. Pappalardo

Dr. Robert Grimm

Dr. Bruce Jakosky

Dr. John Wahr

Dr. Shijie Zhong

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly

work in the above mentioned discipline.

iii

Barr, Amy Courtright (Ph. D, Geophysics)

Convection in Ice I With Non-Newtonian Rheology: Application to the Icy Galilean

Satellites

Thesis directed by Prof. Robert T. Pappalardo

Observations from the Galileo spacecraft suggest that the Jovian icy satellites

Europa, Ganymede, and Callisto have liquid water oceans beneath their icy surfaces.

The outer ice I shells of the satellites represent a barrier between their surfaces and their

oceans and serve to decouple fluid motions in their deep interiors from their surfaces.

Understanding heat and mass transport by convection within the outer ice I shells of

the satellites is crucial to understanding their geophysical and astrobiological evolution.

Recent laboratory experiments suggest that deformation in ice I is accommodated

by several different creep mechanisms. Newtonian deformation creep accommodates

strain in warm ice with small grain sizes. However, deformation in ice with larger

grain sizes is controlled by grain-size-sensitive and dislocation creep, which are non-

Newtonian. Previous studies of convection have not considered this complex rheological

behavior.

This thesis revisits basic geophysical questions regarding heat and mass trans-

port in the ice I shells of the satellites using a composite Newtonian/non-Newtonian

rheology for ice I. The composite rheology is implemented in a numerical convection

model developed for Earths mantle to study the behavior of an ice I shell during the

onset of convection and in the stagnant lid convection regime. The conditions required

to trigger convection in a conductive ice I shell depend on the grain size of the ice, and

the amplitude and wavelength of temperature perturbation issued to the ice shell.

If convection occurs, the efficiency of heat and mass transport is dependent on

the ice grain size as well. If convection occurs, fluid motions in the ice shells enhance the

iv

nutrient flux delivered to their oceans, and coupled with resurfacing events, may provide

a sustainable biogeochemical cycle. The results of this thesis suggest that evolution of

ice grain size in the satellites and the details of how tidal dissipation perturbs the ice

shell to trigger convection are required to judge whether convection can begin in the

satellites, and controls the efficiency of convection.

Dedication

For Bernice Pedersen Courtright and Alberta Engvall Siegel

vi

Acknowledgements

I would like to thank Bob Pappalardo for sharing his excellence and creativity

with me for four years. The motivation for this thesis stems from a conversation with

Dave Stevenson that occurred when I was a freshman at Caltech. Shijie Zhong and his

post-docs Jeroen Van Hunen and Allen McNamara helped me turn my pile of ideas into

numerically tractable projects. Bill McKinnon, Don Blankenship, Francis Nimmo, and

Bill Moore have repeatedly raised the bar for success by asking tough questions and

listening patiently as I stammered out the answers.

I would not have made it through grad school without an incredible support

network of friends, family, and faculty members. The core of this network is Bernadine

Barr, who served both as mother and seasoned academic advisor. Thanks to the faculty

at CU, especially Fran Bagenal, Jim Green, Bruce Jakosky, Mike Shull, and John Wahr.

Special thanks to Louise Prockter, Geoff Collins, and Jeff Moore, for providing assurance

that there will be life after grad school. Thanks to Erika Barth, David Brain, Shawn

Brooks, G. Wesley Patterson, James Roberts, Andrew Ste, Dimitri Veras, and Arwen

Vidal. Thanks to my -friends Catherine Boone, Kjerstin Easton, Sarah (DEI)

Milkovich, Brian Platt, David (this is all his fault) Tytell, Travis Williams, and Adrianne

and Yifan Yang.

Support for this work was provided by NASA Graduate Student Researchers

Program grant NGT5-50337 and NASA Exobiology grant NCC2-1340.

vii

Contents

Chapter

1 Introduction 1

1.1 Questions Addressed in this Thesis . . . . . . . . . . . . . . . . . . . . . 3

1.2 Geological and Geophysical Setting . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Tidal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Astrobiological Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Rheology of Ice I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Convection in Ice I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.2 Non-Dimensional Coordinates . . . . . . . . . . . . . . . . . . . . 26

1.5.3 Viscosity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5.4 Composite Rheology for Ice I . . . . . . . . . . . . . . . . . . . . 29

1.6 The Onset of Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 33

1.6.2 Non-Newtonian Rheologies . . . . . . . . . . . . . . . . . . . . . 34

1.7 Previous Studies of Convection in the Icy Satellites . . . . . . . . . . . . 36

viii

2 Convective Instability in Ice I with Non-Newtonian Rheology 39

2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.1 Numerical Implementation of Ice I Rheology . . . . . . . . . . . 43

2.3.2 Numerical Convection Model . . . . . . . . . . . . . . . . . . . . 46

2.3.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.1 Critical Rayleigh Number . . . . . . . . . . . . . . . . . . . . . . 50

2.4.2 Critical Shell Thickness . . . . . . . . . . . . . . . . . . . . . . . 59

2.4.3 Variation of Melting Temperature . . . . . . . . . . . . . . . . . 59

2.5 Comparison to Existing Studies . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Implications for the Icy Galilean Satellites . . . . . . . . . . . . . . . . . 65

2.6.1 Conditions for Convection in Callisto and Ganymede . . . . . . . 66

2.6.2 Conditions for Convection in Europa . . . . . . . . . . . . . . . . 71

2.7 Discussion: The Role of Tidal Dissipation . . . . . . . . . . . . . . . . . 73

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Onset of Convection in Ice I with Composite Newtonian and Non-Newtonian

Rheology 78

3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.1 Numerical Implementation of Composite Rheology for Ice I . . . 80

3.3.2 Numerical Convection Model . . . . . . . . . . . . . . . . . . . . 84

3.3.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

ix

3.5 Implications for the Icy Galilean Satellites . . . . . . . . . . . . . . . . . 100

3.5.1 Conditions for Convection in Europa . . . . . . . . . . . . . . . . 101

3.5.2 Conditions for Convection in Ganymede and Callisto . . . . . . . 103

3.5.3 Role of Tidal Heating . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5.4 Evolution of Grain Size and Orientation . . . . . . . . . . . . . . 106

3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4 Implications for the Internal Structure of the Major Satellites of the Outer Plan-

ets 110

4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3.1 Numerical Implementation of Ice Rheology . . . . . . . . . . . . 111

4.3.2 Numerical Convection Model . . . . . . . . . . . . . . . . . . . . 113

4.3.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.4 Thermodynamic Stability of Oceans . . . . . . . . . . . . . . . . . . . . 114

4.4.1 Critical Rayleigh Number . . . . . . . . . . . . . . . . . . . . . . 115

4.4.2 Efficiency of Convection . . . . . . . . . . . . . . . . . . . . . . . 115

4.4.3 Ocean Stability Without Tidal Heating . . . . . . . . . . . . . . 119

4.4.4 Presence of Non-Water-Ice Materials . . . . . . . . . . . . . . . . 120

4.4.5 Tidal Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Implications for Astrobiology 128

5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3 Astrobiological Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4 Onset of Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

x5.5 Convective Recycling of the Ice Shell . . . . . . . . . . . . . . . . . . . . 138

5.5.1 Geophysical Descriptive Parameters . . . . . . . . . . . . . . . . 139

5.5.2 Astrobiologically Relevant Parameters . . . . . . . . . . . . . . . 140

5.6 Endogenic Resurfacing Events on Europa . . . . . . . . . . . . . . . . . 149

5.6.1 Domes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.6.2 Ridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.7 Ocean Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6 Conclusions and Future Work 155

6.1 Answers to the Key Questions . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.2.1 Grain Size Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.2.2 Tidal Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2.3 Premelting in Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Bibliography 179

Appendix

A Thermal, Physical, and Rheological Parameters 186

B Selected Input Parameters 189

xi

Tables

Table

2.1 Variation in critical Rayleigh number with perturbation amplitude . . . 55

2.2 Numerically determined fitting coefficients for Racr . . . . . . . . . . . . 59

2.3 Comparison to analysis of Solomatov (1995) . . . . . . . . . . . . . . . . 62

4.1 Convective heat flux and Nu for 20 km < D < 100 km . . . . . . . . . . 118

4.2 Orbital parameters for Ganymede and Europa . . . . . . . . . . . . . . . 125

6.1 Rheological parameters for T Tm from Goldsby and Kohlstedt (2001) . 169

A.1 Thermal and physical parameters of the satellites . . . . . . . . . . . . . 187

A.2 Rheological parameters, after Goldsby and Kohlstedt (2001) . . . . . . . 188

B.1 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with GBS rheology . . . . . . . . . . . 190

B.2 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with GBS rheology (continued) . . . . 191

B.3 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with basal slip rheology . . . . . . . . 192

B.4 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with basal slip rheology (continued) . 193

xii

B.5 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with composite rheology . . . . . . . . 194

B.6 Selected input parameters for simulations used to determine the critical

Rayleigh number and wavelength with composite rheology (continued) . 195

B.7 Weighting values for the composite rheology of ice I . . . . . . . . . . . 196

B.8 Input parameters used in Chapters 4 and 5 . . . . . . . . . . . . . . . . 197

B.9 Input parameters used in Chapters 4 and 5 (continued) . . . . . . . . . 198

xiii

Figures

Figure

1.1 The Galilean satellites of Jupiter . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Phase diagram of water . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Interiors of the icy Galliean satellites . . . . . . . . . . . . . . . . . . . . 6

1.4 High resolution image of a double ridge on the surface of Europa . . . . 9

1.5 Pits, spots, and domes on the surface of Europa . . . . . . . . . . . . . . 10

1.6 Chaos terrain on Europa . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Grooved terrain on Ganymede . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Conceptual diagrams of deformation mechanisms in ice I . . . . . . . . . 20

1.9 Initial temperature perturbation issued to the ice shell . . . . . . . . . . 25

2.1 Onset of convection in ice I with basal slip rheology . . . . . . . . . . . 51

2.2 Evolution of kinetic energy with time . . . . . . . . . . . . . . . . . . . . 52

2.3 Critical Rayleigh number as a function of wavelength . . . . . . . . . . . 54

2.4 Critical Rayleigh number as a function of perturbation amplitude . . . . 56

2.5 Asymptotic and power law regimes . . . . . . . . . . . . . . . . . . . . . 57

2.6 Comparison of Raa to values from Solomatov (1995) . . . . . . . . . . . 64

2.7 Critical ice shell thickness for convection in Callisto . . . . . . . . . . . . 67

2.8 Critical ice shell thickness for convection in Ganymede . . . . . . . . . . 68

2.9 Critical grain size for convection in Callisto . . . . . . . . . . . . . . . . 69

xiv

2.10 Critical grain size for convection in Ganymede . . . . . . . . . . . . . . . 70

2.11 Critical ice shell thickness for convection in Europa . . . . . . . . . . . . 72

2.12 Critical grain size for convection in Europa . . . . . . . . . . . . . . . . 74

3.1 Deformation maps for ice I . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2 Composite viscosity of ice I as a function of stress . . . . . . . . . . . . 85

3.3 Determination of Racr for convection in ice I with d = 3.0 cm . . . . . . 91

3.4 Temperature and viscosity fields for convection in ice I with composite

rheology and d = 3.0 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Example of determination of cr for ice with composite rheology . . . . 94

3.6 Variation of Racr with perturbation amplitude . . . . . . . . . . . . . . 95

3.7 Variation in Racr,0 as a function of grain size . . . . . . . . . . . . . . . 96

3.8 Activation of non-Newtonian creep mechanisms in ice with 0.1 mm < d 255 K, Goldsby and Kohlstedt (2001) present an alternate set of creep

parameters, which yield a faster creep rate for GBS in ice near the melting point,

consistent with terrestrial observations. The enhancement of creep rate is caused by

pre-melting of the ice at grain boundaries and grain edges which causes the ice to have

a low viscosity. We do not include the creep enhancement near the melting point of ice

for numerical simplicity. We briefly discuss the effects of including the high temperature

creep enhancement term in section 2.6.

The strain rate from diffusion creep is described by

=ADFVm

RTmd2

(Dv +

dDb

)(2.4)

where ADF is a dimensionless constant, Vm is the molar volume, Tm is the melting

temperature of ice, Dv is the rate of volume diffusion, is the grain boundary width,

and Db is the rate of grain boundary diffusion. For small strains (1%), ADF = 42, but

larger strains may yield larger values of ADF and enhanced creep rates due to diffusional

flow (Goodman et al., 1981); here, we use ADF = 42 (Goldsby and Kohlstedt , 2001).

For a range of grain sizes close to values estimated for the Galilean satellites ice

shells (0.1 to 100 mm), the grain size is much larger than the grain boundary width

(9.04 1010 m) (Goldsby and Kohlstedt , 2001), so volume diffusion dominates over

grain boundary diffusion, and we may ignore its contribution to the strain rate. The

45

strain rate for volume diffusion is:

=42Vm

RTmd2Do,v exp

(QvRT

)(2.5)

where Do,v is the volume diffusion rate coefficient and Qv is the activation energy. The

viscosity of ice for volume diffusion is Newtonian, but does depend on grain size. The

parameters for volume diffusion are listed in Table A.2, where we have grouped the

pre-exponential parameters to calculate an effective A = (42VmDov/RTm).

The deformation mechanism that yields the highest strain rate for a given temper-

ature and differential stress is judged to dominate flow at that temperature and stress

level. At low stresses, Newtonian diffusional flow is dominant, but at higher stresses,

the non-Newtonian creep mechanisms are activated. The transition stress between dif-

fusional flow and grain boundary sliding is

T =

(AGBSAdiff

dpdiff

dpGBSexp

((Qdiff Q

GBS)

RT

)) 1ndiffnGBS, (2.6)

and a similar expression can be obtained for the transition stress between diffusional

flow and basal slip. The transition stress between GBS and diffusional flow for ice near

the melting temperature with a grain size of 1.0 mm is 0.02 MPa. If the grain size of

ice is 0.1 mm, the transition stress increases to 0.1 MPa; with a grain size of 100 mm,

the transition stress is 6 104 MPa.

The non-Newtonian deformation mechanisms will control the growth of convective

plumes if the thermal stress due to a growing plume exceeds the transition stress between

diffusional flow and GSS creep. The thermal stress due to a growing plume of height ,

warmer than its surroundings by T , is approximately th gT. In an ice shell

50 km thick on Europa, Ganymede, or Callisto, a plume with = D and T = 5 K can

generate a thermal stress of 0.03 MPa. In an ice shell 25 km thick, a plume of height

approximately 25 km can generate 0.015 MPa. For reasonable plume sizes and grain

sizes of ice, the thermal stress associated with a growing plume exceeds the transition

46

stress between GBS and diffusional flow, indicating that GBS can control plume growth

in ice with a grain size of order 1.0 mm.

The thermal stress associated with the onset of convection in ice with a plausible

range of grain sizes is close to the transition stress between the Newtonian and non-

Newtonian deformation mechanisms. For this reason, the composite Newtonian and

non-Newtonian rheology for ice I is implemented in Chapter 3. In this initial study, we

focus on the growth of initial convective plumes large enough to activate GBS and basal

slip, rather than growth of perturbations by diffusional flow. In this way we begin to

characterize the behavior of a non-Newtonian ice shell during the onset of convection.

2.3.2 Numerical Convection Model

The dynamics of thermal convection are controlled by the Rayleigh number, a

single dimensionless parameter that expresses the balance between thermal buoyancy

forces and the viscous restoring force. Large values of Ra indicate vigorous convection;

convection cannot occur unless the Rayleigh number exceeds the critical Rayleigh num-

ber (Racr). We adopt a reference Rayleigh number for the ice shell from Solomatov

(1995)

Ra1 =gTD(n+2)/n

(dpA1)1/n exp( QnRTm

) (2.7)where Tm is the melting temperature of the ice shell, and values of the rheological

parameters are taken directly from the lab-derived flow laws from Goldsby and Kohlstedt

(2001). An explicit temperature- and strain-rate-dependent rheology of form

=

(dp

A

)1/n(1n)/nII exp

(Q

nRT

)(2.8)

is used, where II is the second invariant of the strain rate tensor. Thermal and physical

parameters used in our models are summarized in Table A.1. The reference Rayleigh

number is obtained from the nominal definition of Rayleigh number (2.1) by explicitly

evaluating the non-Newtonian viscosity of ice at a reference strain rate of o = /D2

47

and a reference temperature equal to the melting temperature of ice. The convective

strain rates in the ice shells are not well-constrained, so we choose this definition of

reference strain rate to reduce the number of free parameters in the Rayleigh number.

When a stress is applied to non-Newtonian ice, the strain rate increases as the

ice flows to relieve the stress, and as the ice flows, its viscosity decreases. This feedback

causes the strain rates in the warm convecting sublayer of the ice shell to naturally evolve

to values some 103 times higher than the reference strain rate, and the viscosity of the

ice shell to evolve to values substantially lower than the reference viscosity. Typical

values of viscosity at the melting point during the onset of convection are of order 1014

Pa s for basal slip, and 1015 Pa s for GBS (see Figure 2.1).

A more physically intuitive effective Rayleigh number for the ice shell can be

obtained after the convection simulation is completed, by re-evaluating the Rayleigh

number using the viscosity value during the onset of convection, rather than the ref-

erence viscosity (Malevsky and Yuen, 1992). In our simulations, the melting point

viscosities are smaller by a factor of 100 than the reference viscosity, yielding effective

Rayleigh numbers of order 106 to 107.

The above rheology has been incorporated into the finite-element convection

model Citcom (Moresi and Gurnis, 1996; Zhong et al., 1998, 2000), which solves the

governing equations of thermally-driven convection in an incompressible fluid. Our sim-

ulations are performed in a 2D Cartesian geometry, free-slip boundary conditions are

used on the surface (z = 0), base (z = D), and side walls of the domain (x = 0, xmax).

All simulations in this study were performed in a domain with 32 x 32 elements, chosen

to resolve the bottom thermal boundary layer while allowing sufficient coverage of our

large parameter space given limited computational resources.

The domain is basally heated so we do not include the effects of tidal dissipation,

but discuss its probable role in triggering convection in section 2.7. The surface of the

convecting region is held constant at a temperature appropriate for the temperate and

48

equatorial surface of a jovian icy satellite, which we vary in our study from 90 K to

120 K. The base of the domain is held at a constant temperature equal to the melt-

ing temperature of the ice shell, Tm. We use a value of Tm = 260 K for the majority

of simulations shown here, but discuss the effects of varying the melting temperature

by 10 K in section 2.4.3. We have not taken into account the thermal or rheologi-

cal effects of potential contaminant non-water-ice materials such as hydrated sulfuric

acid, or hydrated sulfate salts, which have been suggested to exist on Europas surface

based on near-infrared spectroscopy (Carlson et al., 1999; McCord et al., 1999), or high

temperature creep enhancement (see section 2.3.1).

With these modifications in place, our model was benchmarked using results for a

Newtonian, temperature-linearized flow law with large viscosity contrasts (Moresi and

Solomatov , 1995). Results using a non-Newtonian rheology were compared to results for

a temperature-linearized flow law with n = 3 and large viscosity contrasts (Christensen,

1985). In the vast majority of cases, our results for convective heat flux (Nu) and the

internal average temperature agree with published results to within 1%.

2.3.3 Initial Conditions

The approach we use to numerically determine the critical Rayleigh number is

similar to linear stability analysis (Turcotte and Schubert , 1982; Chandrasekhar , 1961).

The convection simulations are started from an initial condition of a conductive ice shell

plus a temperature perturbation expressed as a single Fourier mode:

T (x, z) = Ts zT

D+ T cos

(2D

x

)sin

(z

D

)(2.9)

where T and are the amplitude and wavelength of the perturbation, and z = D at

the warm base of the ice shell. Use of free-slip boundary conditions requires that the

width of the computational domain (xmax) be equal to one half the wavelength of initial

perturbation. The simulation is run for a short time to determine whether the initial

49

perturbation grows and convection begins, or decays with time due to thermal diffusion

and viscous relaxation, causing the ice layer to return to a conductive equilibrium. For a

given initial condition, we run a series of convection simulations with decreasing values

of Ra1. The critical Rayleigh number is defined as the minimum value of Ra1 where

the system convects for a given initial condition, and here is determined to within two

significant figures.

The kinetic energy of the fluid layer is used as a diagnostic for the vigor of

convection. The kinetic energy is

E

xmax0

D0 (v

2x + v

2z)dxdz xmax

0

D0 dxdz

(2.10)

where xmax is the width of the numerical domain and vx, vz are the horizontal and

vertical fluid velocities, respectively. If the kinetic energy of the fluid grows with time

during the opening stages of the simulations when initial plumes develop, the layer is

judged to convect; if the kinetic energy decays with time, the layer does not convect

and the system returns to conductive equilibrium.

For simple rheologies (isoviscous, only temperature- or stress-dependent), the

kinetic energy of the fluid layer grows exponentially or quasi-exponentially with time as

the initial perturbation grows and convection begins. This quasi-exponential behavior

forms the basis for existing numerical methods of determining Racr for fluids with

simpler rheologies (Zhong and Gurnis, 1993; Korenaga and Jordan, 2003). For a non-

Newtonian fluid, we find that the growth of kinetic energy with time is more complex,

and is not readily analyzed mathematically. Although the kinetic energy may increase

initially, indicating growth of the initial perturbation, after some time has elapsed, the

fluid velocities can decrease as the system returns to conductive equilibrium. As a result,

the outcome of the simulation cannot be judged by looking solely at the initial growth

or decay of the kinetic energy. Therefore, we run our simulations for roughly 20% of

the thermal diffusion time (diff D2/), to determine whether the layer ultimately

50

returns to a conductive equilibrium or convects. The key advantage of this procedure is

that the final outcomes of our simulations are clearly self-sustaining convective states,

and not transient, quasi-stable states that convect briefly and return to conductive

equilibrium at a later time. The temperature field, velocity vectors, and viscosity fields

for a sample simulation where Ra = Racr for the basal slip rheology is shown in Figure

2.1. A sample graph of the evolution of kinetic energy over time is shown in Figure 2.2.

2.4 Model Results

2.4.1 Critical Rayleigh Number

The viscous restoring force that counteracts the buoyancy of a growing plume is

wavelength-dependent, so the critical Rayleigh number for convection will depend on

the wavelength of the perturbation, regardless of the rheology of the fluid. The critical

values of Rayleigh number (Racr) reported here are critical values of Ra1. We first

determine the wavelength that minimizes the value of Racr, then investigate how Racr

for that specific Fourier mode with = cr varies with T .

We find two regimes of behavior of the non-Newtonian ice shell. For small tem-

perature perturbations less than the rheological temperature scale (Trh), the critical

Rayleigh number depends on the amplitude of perturbation to a power . This is desig-

nated the power-law regime. For temperature perturbations greater than the rheological

temperature scale, the critical Rayleigh number approaches a constant value and is in-

dependent of the perturbation amplitude. This is designated the asymptotic regime.

The transition between the two regimes of behavior occurs when T > Trh,

Trh =1.2(n + 1)RT 2i

Q, (2.11)

where Ti is the roughly constant temperature in the convective interior (Solomatov

and Moresi , 2000). Approximating Ti Tm, the rheological temperature scale is ap-

proximately 37 K for both rheologies, which corresponds to perturbation amplitudes of

51

-40

-30

-20

-10

0

Dep

th (k

m)

0 10 20 30

X (km)

-40

-30

-20

-10

0

Dep

th (k

m)

0 10 20 30

X (km)0 10 20 30

X (km)0 10 20 30

X (km)

-40

-30

-20

-10

0

Dep

th (k

m)

0 10 20 30

X (km)

120 140160 180200 220 240260

T (K)

0 20 0 10 20 30

X (km)0 10 20 30

X (km)

14 15 16 17 18 19 20 21 22 23

log10 (Pa s)

a) t=0 b) t=0

c) t=5.8 Myr d) t=5.8 Myr

Figure 2.1: Temperature field (panel a) with superimposed velocity vectors, and vis-cosity field (panel b) with superimposed contours of constant viscosity for a sampleinitial condition from our study. The simulation is started with an initial temperatureperturbation of 15 K, and Ra = Racr = 4.0 10

4. This corresponds to an ice shell 49km thick on Ganymede with a surface temperature of 110 K, a melting temperature ofice of 260 K, and a grain size of ice of 1.0 mm. The initial condition evolves over 5.8Myr to generate the temperature and viscosity fields in panels c and d.

52

10-2

10-110-1

100

101

102

0.00 0.05 0.10 0.15 0.20t

EEEEEEConvection

No Convection

Figure 2.2: Growth of kinetic energy (E) with non-dimensional time (t=t/diff ) for icewith GBS rheology with Ts = 110 K and Tm = 260 K, given an initial perturbation ofamplitude 0.75 K. Each line represents the evolution of kinetic energy for a simulationwith a different Rayleigh number from Ra1 = 1.8 10

5 (top line) to Ra1 = 1.3 105

(bottom line). After an initial phase of quasi-exponential growth of kinetic energy (fort < 0.05), the kinetic energy grows super-exponentially as convection begins. Wherekinetic energy does not grow, convection did not initiate. The highest Rayleigh numberthat resulted in convection, 1.6 105, is the critical Rayleigh number for this rheologyand set of boundary temperatures.

53

0.22T to 0.25T for the range of boundary temperatures considered here.

For a nominal set of boundary temperatures Ts = 110 K and Tm = 260 K, the

wavelength that minimizes Racr for both GBS and basal slip rheologies in the power law

regime is 1.5D, which does not change with the amplitude of perturbation. Figure

2.3 shows how Racr varies with wavelength for both rheologies, for the nominal set of

boundary temperatures. These values are substantially lower than cr for an isoviscous

fluid (Turcotte and Schubert , 1982). This is likely because in a fluid with strongly

temperature-dependent rheology, initial fluid motions are confined to the bottom 30%

of the shell, decreasing the effective aspect ratio of the convecting region. The critical

Rayleigh number for the GBS and basal slip rheologies varies by a factor of two between

the minimum value when = cr and the maximum value of wavelength used, = 3D.

In the asymptotic regime, 1.8D < cr < 2.2D, and the critical Rayleigh number is very

weakly dependent on wavelength, varying by only 20% as is increased from 1.2D to

2.2D.

As discussed in section 2.3.1, the non-Newtonian deformation mechanisms begin

to control the growth of a perturbation at the base of the ice shell when the thermal

stress associated with the plume (th gT) exceeds 0.02 MPa in ice with a

nominal grain size of 1.0 mm. For the average maximum permitted ice shell thickness

in Ganymede and Callisto of 170 km, a perturbation of 0.75 K above the ambient

conductive equilibrium spread across a horizontal distance D can generate 0.02

MPa, sufficient to activate grain boundary sliding and basal slip in ice with a grain size

of order 1 mm. In a relatively thin ice shell with D 20 km, a perturbation of 15

K is required to activate the non-Newtonian deformation mechanisms. These values

supply the minimum and maximum perturbation amplitude T that we use, 0.005T

and 0.1T .

In the power law regime, the critical Rayleigh number varies as a power of the

54

0

25000

50000

75000

100000

Ra1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Wavelength (/D)

0

25000

50000

75000

100000

Ra1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Wavelength (/D)

0

25000

50000

75000

100000

Ra1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Wavelength (/D)

0

25000

50000

75000

100000

Ra1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Wavelength (/D)

GBS

Basal Slip

Figure 2.3: Critical Rayleigh number as a function of dimensionless wavelength forbasal slip rheology (diamonds) and grain boundary sliding rheology (GBS, dots) withTs = 110 K and Tm = 260 K. A constant perturbation amplitude of T = 7.5 K is usedhere.

55

Table 2.1: Variation in Critical Rayleigh Number with Perturbation Amplitude

Rheology T/T RacrBasal Slip 0.005 1.2 105

0.010 8.0 104

0.025 4.6 104

0.050 3.1 104

0.075 2.4 104

0.100 2.0 104

Grain Boundary Sliding 0.005 1.6 105

0.010 1.2 105

0.025 7.7 104

0.050 5.5 104

0.075 4.6 104

0.100 4.0 104

amplitude of initial perturbation, obeying a relationship of form

Racr = Racr,0

(T

T

)(2.12)

where Racr,0 and are determined with a least-squares fit to values of Racr in log-log

space. Figure 2.4 shows a sample set of Racr data for Ts = 110 K and Tm = 260 K for

both the GBS and basal slip rheologies, with values used in the plot listed in Table 2.1.

Figure 2.5 shows values of Racr in the power law and asymptotic regimes for basal slip

rheology with Ts = 110 K and Tm = 260 K.

Regardless of the boundary temperatures, the critical value of Ra1 varies by

approximately an order of magnitude over the range of T explored. The onset of

convection is governed largely by the viscosity structure near the base of the ice shell,

which is controlled by the rheological temperature scale (Davaille and Jaupart , 1994):

T =(Tm)

T

Tm

. (2.13)

For the form of rheology used here, the rheological temperature scale is given by

T =nRT 2mQT

, (2.14)

56

104

105105

Ra 1

1 2 5 10 20T (K)

104

105105

Ra 1

1 2 5 10 20T (K)

104

105105

Ra 1

1 2 5 10 20T (K)

104

105105

Ra 1

1 2 5 10 20T (K)

GBS

Basal Slip

Figure 2.4: Critical Rayleigh number as a function of the amplitude of initial tem-perature perturbation (T ) for GBS (dots) and basal slip (diamonds) rheologies withTs = 110 K and Tm = 260 K. Lines are least squares fits to the data, where the sloperepresents the fitting coefficient in equation (2.12). For GBS, = 0.6, for basal slip, = 0.5.

57

104

105105

Ra 1

11 10 100T (K)

104

105105

Ra 1

11 10 100T (K)

Power Law

Asymptotic

Figure 2.5: Critical Rayleigh number as a function of the amplitude of initial tempera-ture perturbation (T ) for basal slip rheology with Ts = 110 K and Tm = 260 K. In thepower law regime, for perturbation amplitudes less than 37 K, the critical Rayleighnumber is a function of perturbation amplitude. For perturbation amplitudes largerthan 37 K, the critical Rayleigh number reaches a constant value of 1.2 104.

58

and can be used to scale Racr,0 using:

Racr,0 = Ra0,0 +MT (2.15)

where M and Ra0,0 are the derived fitting coefficients.

In the asymptotic regime, the critical Rayleigh number does not depend on the

amplitude of temperature perturbation, and approaches an asymptotic value Raa. Val-

ues of Raa using cr = 2.0D and T = 0.35T are listed in Table 2.3.

Given a set of boundary temperatures, and amplitude of temperature pertur-

bation, the critical Rayleigh number in the power law regime can be estimated by

combining equations (2.12), (2.14), and (2.15):

Racr =

[Ra0,0 +M

(nRT 2mQT

)](T

T

). (2.16)

Values of the fitting coefficients Ra0,0, andM for both grain boundary sliding and basal

slip rheologies are shown in Table 2.2. We report Ra0,0 and M values for Tm = 260 K

only, and briefly discuss the effects of varying the melting temperature in section 2.4.3.

The expression for Racr in the power law regime is likely only valid when the ice

shell is in the stagnant lid convection regime, where viscosity contrast across the layer

is large and convective instability is limited to the warm, low-viscosity sub-layer near

the base of the ice shell. For the ice shell to be in the stagnant lid regime, the viscosity

contrast due to temperature alone, T = ((Ts)/(Tm)) exceeds exp(4(n + 1)), or

7104 for GBS and 8105 for basal slip (Solomatov , 1995). For the range of boundary

temperatures used here, T ranges from 2 106 to 2 1010 for GBS and 7 105 to

3 109 for basal slip.

59

Table 2.2: Numerically Determined Fitting Coefficients for Racr

Rheology Ra0,0 M

Grain Boundary Sliding 5.1 104 0.6 2.7 105

Basal Slip 1.7 104 0.5 7.7 104

2.4.2 Critical Shell Thickness

The critical shell thickness for the onset of convection due to small temperature

perturbations T < Trh can be obtained using the definition of Ra1:

Dcr =

(Racr

(dpA1

)1/nexp

( QnRTm

)gT

)n/(n+2), (2.17)

where the value of Racr can be estimated using equation (2.16). The values of critical

Rayleigh number in the asymptotic regime can be used to determine an absolute lower

limit on the ice shell thickness required for convection. The lower limit on shell thickness

is obtained from Raa using:

Da =

(Raa

(dpA1

)1/nexp

( QnRTm

)gT

)n/(n+2). (2.18)

In the power law regime, the critical grain size required to initiate convection in an ice

layer with thickness D is

dcrit =

(gD(n+2)/n(

A1)1/n

exp( QnRTm

)Racr

)n/p. (2.19)

For d < dcr, convection can occur; for d > dcr the ice is too stiff to convect for the

given initial condition. The asymptotic value of Rayleigh number can also be used to

determine an upper limit on the grain size that can permit convection in a layer of

thickness D:

da =

(gD(n+2)/n(

A1)1/n

exp( QnRTm

)Raa

)n/p. (2.20)

2.4.3 Variation of Melting Temperature

Two sets of simulations were run to quantify how much the critical Rayleigh

number is influenced by changing the melting temperature. In the case of GBS, Ts = 110

60

K and Tm = 270 K were used to obtain a relationship between T and Racr. The

resulting values of Racr,0 and were compared to the values obtained when Tm = 260

K. For basal slip, procedure was repeated, using Tm = 250 K. In both cases, the fitting

coefficients obtained were different from their Tm = 260 K counterparts by only 1%. Use

of equation (2.16) for alternative melting temperatures between 250 K and 270 K is valid

for Racr to two significant figures, provided the high temperature creep enhancement

in ice near its melting point is not included in the rheology.

2.5 Comparison to Existing Studies

For simple rheologies, the critical Rayleigh number for convection in a fluid can be

obtained using linear stability anaylsis (Turcotte and Schubert , 1982; Chandrasekhar ,

1961). However, the critical Rayleigh number for the onset of convection in a non-

Newtonian fluid cannot be determined using linear stability analysis (Tien et al., 1969;

Solomatov , 1995). The viscosity of a non-Newtonian fluid depends on both temperature

and strain rate, so the viscosity in the perturbed layer of fluid depends on the amplitude

of the initial perturbation and becomes infinite as the amplitude of perturbation becomes

small (Solomatov , 1995). Convection in a non-Newtonian fluid with a temperature- and

strain-rate-depdendent rheology is always a finite-amplitude instability, and cannot be

readily analyzed analytically (Solomatov , 1995).

Analysis of the onset of convection in a fluid with stress-dependent (but not

temperature-dependent) rheology can provide constraints on how the non-Newtonian

behavior affects Racr. An alternative method of determining Racr for a non-Newtonian

fluid stems from a physical argument put forth by Chandrasekhar (1961), who pos-

tulated that the critical Rayleigh number occured at a critical temperature gradient

where the dissipation of energy by viscous forces in the system exactly balanced the

release of energy from the rising, thermally buoyant plume. Using an energy balance

argument, Tien et al. (1969) were able to calculate the critical Rayleigh number for

61

non-Newtonian fluids with a range of values of stress exponent, which compared fa-

vorably to their laboratory measurements of critical Rayleigh number for fluids with

stress-dependent rheologies.

The most widely-used results for the critical Rayleigh number for convection in a

non-Newtonian fluid arise from the pivotal study of Solomatov (1995), who built upon

the analysis of Tien et al. (1969) plus additional studies by Ozoe and Churchill (1972)

to consider a stress- and temperature-dependent rheology. With the knowledge that

the critical Rayleigh number for a non-Newtonian fluid depends on initial conditions,

Solomatov (1995) characterized the value of Rayleigh number where convection could

not occur, regardless of initial conditions.

The analysis of Solomatov (1995) focused on the behavior of the bottom thermal

boundary layer at the onset of convection. If the viscosity of the fluid depends strongly

on temperature, there are no fluid motions in the upper part of the convecting layer,

forming a stagnant lid. In the stagnant lid regime, convective motions are confined to

a warm sub-layer of the ice shell, where the temperature dependence of viscosity can

be neglected by evaluating the viscosity of the material at the mean temperature in the

sub-layer.

With this approximation, the critical Rayleigh number of the sub-layer can be

evaluated by assuming that the viscosity of ice depends only on stress, thus using the

results of Tien et al. (1969) and Ozoe and Churchill (1972). Convection in the entire

layer initiates when the local Rayleigh number of the bottom thermal boundary layer

exceeds a critical value. The critical Rayleigh number for entire fluid layer can therefore

be related to the critical Rayleigh number of the sub-layer.

To closely follow the analysis of Solomatov (1995), we non-dimensionalize our

rheology (equation 2.8) as

(T , ) = C1/n(1n)/n exp

(E

T + T o

E

1 + T o

)(2.21)

62

Table 2.3: Comparison to Analysis of Solomatov (1995)

Rheology Ts (K) Raa (our study) Racr (Solomatov (1995))

Grain Boundary Sliding 90 3.1 104 2.3 104

100 2.7 104 1.9 104

110 2.2 104 1.5 104

120 1.9 104 1.2 104

Basal Slip 90 1.4 104 8.4 103

100 1.2 104 7.1 103

110 9.8 103 5.9 103

120 8.6 103 4.8 103

where C represents the pre-exponential parameters in the laboratory-derived flow law,

E = Q/(nRT ) is the non-dimensional activation energy, and T o = Ts/T is the

non-dimensional reference temperature.

The Rayleigh number of the unstable sub-layer of thickness zsub at the base of

the fluid layer is given by Solomatov (1995) as:

Rasub =gTz

2(n+1)/nsub

D(C)1/n

exp

(E

1 + T o

E

(1 (zsub/2) + T o

)(2.22)

where the viscosity is evaluated at the mean temperature in the sub-layer, T = 1

(zsub/2), and the strain rate has been evaluated at /z2sub, the characteristic strain rate

in the sub-layer. The sub-layer reaches its maximum thickness and becomes convectively

unstable when the local Rayleigh number in the sub-layer is equal to the critical Rayleigh

number for a fluid with stress-dependent rheology:

Rasub(zmax) = Racr(n). (2.23)

The results of Tien et al. (1969) and Ozoe and Churchill (1972) are summarized and

extrapolated by Solomatov (1995) to obtain an approximation for the critical Rayleigh

number of a fluid with an arbitrary stress exponent:

Racr(n) Racr(1)1/nRacr()

(n1)/n (2.24)

63

with Racr(1) = 1568, and Racr() 20 represents the formal asymptotic limit of

Racr(n) for n.

The maximum sub-layer thickness (zmax) is obtained by solving for the value of

zsub that yields Rasub/zsub = 0. For the form of temperature dependence used here,

we obtain a quadratic equation for zmax as a function of the non-dimensional activation

energy, stress exponent, and reference temperature. The quadratic equation yields two

results, but only the negative root yields physically applicable solutions where zsub < D:

zmax =D

2(n+ 1)

(4(n+ 1)(T o + 1) + En (8En(n + 1)(T

o + 1) + E

2n2))1/2

. (2.25)

Substituting this value of zmax into equation (2.22) we obtain

Rasub =gTD(n+2)/n(

C)1/n

(zmaxD

)2(n+1)/nexp

(E

1 + T o

E

1 zmax2D + To

). (2.26)

When using the non-dimensional rheology of form eq. (2.21), the viscosity at the melting

point and reference strain rate is equal to C1/n. Therefore, the first term in the above

equation is simply the critical Rayleigh number of the entire fluid layer, with (Tm, o).

Setting the expression for Rasub = Rasub(zmax) and solving for Racr we obtain:

Racr = Racr(n)

(zmaxD

)2(n+1)/nexp

(E

1 zmax2D + To

E

1 + T o

). (2.27)

Values of Racr from this analysis are compared to our numerically determined values of

critical Rayleigh number in the limit of the maximum permitted temperature pertur-

bation, T Trh, Raa. The values of Raa from our study are summarized in Table

2.3. Agreement between our values of critical Rayleigh number and values obtained

using the method of Solomatov (1995) agree to within 35 to 60%. The variation in

Racr according to equation (2.27) is compared to numerically calculated values of Raa

in Figure 2.6.

64

0

10000

20000

30000

Ra c

r

80 90 100 110 120 130 140Ts (K)

0

10000

20000

30000

Ra c

r

80 90 100 110 120 130 140Ts (K)

0

10000

20000

30000

Ra c

r

80 90 100 110 120 130 140Ts (K)

GBS

Basal Slip

Figure 2.6: Comparison of our values of asymptotic critical Rayleigh number (Raa)calculated using = 2.0D and T = 0.35T (dots=GBS, diamonds=basal slip) tocritical Rayleigh numbers calculated using the analysis of Solomatov (1995) (curves,bold=GBS, thin=basal slip), for various surface temperatures. Agreement between ourvalues and the analysis of Solomatov (1995) ranges from 35% to 60% as a functionof surface temperature.

65

2.6 Implications for the Icy Galilean Satellites

Gravity data do not place tight constraints on the thickness of the ice shells of any

of the icy Galilean satellites. The maximum thickness of Europas H2O layer is 170

km, but the fraction of the layer that is liquid is poorly constrained (Anderson et al.,

1998). The upper bounds on ice I shell thickness for all the icy satellites are obtained

by estimating the depth to the minimum melting point of ice I. The minimum melting

point occurs at a depth of approximately 170 km in Europa, 160 km in Ganymede, and

180 km in Callisto, if the density of the ice shell is 930 kg/m3 (Kirk and Stevenson,

1987; Ruiz , 2001). The grain sizes in the icy satellites are poorly constrained as well,

with estimates of grain size spanning eight orders of magnitude, from microns (Nimmo

and Manga, 2002) to meters (Schmidt and Dahl-Jensen, 2004). Conclusions regarding

the convective stability of the ice shells made here may not be correct if the grain sizes

in the satellites are much larger than 1 cm or smaller than 0.1 mm. Additionally, it is

plausible that the Goldsby and Kohlstedt (2001) rheology does not adequately describe

the true behavior of the ice shells of the Galilean satellites, for example, if impurities

have a significant effect on rheology. Moreover, we have ignored internal heating by

tidal dissipation in these calculations, a topic addressed in section 2.7.

If the high-temperature creep enhancement described in section 2.3.1 were in-

cluded in our models, the viscosities of ice at the base of the ice shell would be much

smaller, potentially permitting convection in significantly thinner ice shells. As the be-

havior of the convecting layer transitioned from initial plume growth to well-developed

convecting cells, the entire convecting sublayer of the ice shell could have a very low

viscosity due to the high-temperature softening. Because we have not included this

term, the critical ice shell thicknesses calculated using our models yield upper limits

on the shell thicknesses required for convection. More detailed calculations should be

performed in the future including this term in the rheology to investigate how high-

66

temperature softening of the ice affects both the onset of convection and the pattern of

convection.

In the likely event that the lab-derived flow law does not perfectly match the

true behavior of ice in the Galilean satellites, and that tidal dissipation plays a role in

modifying the thermal structure of the ice shells during the onset of convection, future

modeling efforts can use methods similar to those discussed here, to investigate more

thoroughly the conditions required to trigger convection in ice I shells.

2.6.1 Conditions for Convection in Callisto and Ganymede

Figure 2.7 shows the critical layer thickness for the onset of convection in Callistos

ice I shell for both grain boundary sliding and basal slip rheologies, if the ice has a grain

size of 1.0 mm. Similarly, Figure 2.8 shows the critical shell thickness on Ganymede.

For GBS, if the ice has a grain size of 1.0 mm, the critical shell thickness for convection

in Callistos ice shell varies between 103 km and the maximum permitted shell thickness

of 180 km for grain boundary sliding, and 32 km and 80 km for basal slip. In Ganymede,

if the ice has a grain size of 1.0 mm, the critical shell thickness ranges from 96 km to

greater than the maximum allowed ice shell thickness of 160 km, depending on surface

temperature. If flow is controlled by basal slip (which seems unlikely because the rate-

limiting flow law in the GSS deformation mechanism is GBS), the critical shell thickness

in Ganymede ranges from 30 km to 74 km.

In the more likely case that GBS is the controlling rheology, the largest initial

perturbation in this study (0.1T ) cannot trigger convection in either Ganymede or

Callistos ice shells with the nominal boundary temperatures if the ice near the base

of the ice shell has a grain size d > 3 mm (Figures 2.9 and 2.10). If the ice in either

satellite has a smaller grain size, convection can occur provided the requirements on shell

thickness and temperature perturbation are met. For GBS in an ice shell with a d = 1.0

mm and Ts = 110 K, a 5 K temperature perturbation can trigger convection in an ice

67

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

DmaxDmax

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

90 K100 K

110 K120 K

Figure 2.7: Critical ice shell thicknesses (eq. 2.17) for the onset of convection in Callistosice shell, with grain boundary sliding (bold curves) or basal slip (thin curves) rheologies,for various surface temperature values. A constant grain size of 1.0 mm for the ice shellsis assumed for GBS, and a constant melting temperature of 260 K is assumed for bothrheologies. The maximum permitted ice shell thickness on Callisto, 180 km, is indicatedby the horizontal dashed line. The critical shell thickness predicted by the basal sliprheology ranges from 32 to 80 km over the range of T considered.

68

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

DmaxDmax

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200

Dcr

(km

)

0 5 10 15T (K)

90 K100 K

110 K120 K

Figure 2.8: Similar to Figure 2.7, but for Ganymede. Over the range of T considered,the critical shell thickness ranges from 96 km to the maximum permitted shell thicknessof 160 km for GBS, which is the rate-limiting creep mechanism.

69

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

Convection

No Convection

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150 180D (km)

90 K

100 K

110 K

120 K

Figure 2.9: Critical grain size for convection as a function of ice shell thickness (equation2.19) in Callistos ice shell with GBS rheology for various surface temperatures. Aconstant perturbation T = 5 K is used here.

70

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

Convection

No Convection

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

90 K

100 K

110 K

120 K

Figure 2.10: Similar to Figure 2.9, but for Ganymede.

71

shell on Callisto 150 km thick. Under identical circumstances in Ganymede, Dcr is

141 km. The lower limit on ice shell thickness (Da) in the limit of large temperature

perturbations (in the asymptotic regime) varies from 50 to 57 km in Ganymede and 53

to 60 km in Callisto, as a function of surface temperature, if the ice has a grain size of

1.0 mm.

The equilibrium thicknesses for a conductive ice shell in Callisto and Ganymede

(in the absence of tidal dissipation) given the expected present-day radiogenic heating

rate of 4.5 1012 W kg1 (Spohn and Schubert , 2003), are 148 km, and 128 km

respectively. Triggering convection at present would require a temperature perturbation

of only 5 to 7 K, issued in the mathematical pattern described by equation (2.9) if =

cr. If the perturbation is issued with a larger or shorter wavelength, the temperature

perturbation required to trigger convection will be larger.

Roughly 1.5 billion years ago when concentrations of 40K were higher, and radio-

genic heating rates were twice their present values, the equilibrium ice shell thicknesses

of Callisto and Ganymede would have been 74 km and 64 km, respectively. Triggering

convection in these ancient, thin ice shells of Callisto or Ganymede was only possible if

the grain size of ice was less than 2.5 mm, even if the amplitude of the temperature

perturbation was greater than Trh. Therefore, initiating convection in an ice shell

may be easier later in the satellites history when decreased radiogenic heating allows

for a thicker ice shell.

2.6.2 Conditions for Convection in Europa

Figure 2.11 shows the critical layer thickness for convection in Europas ice shell,

with the simplifying assumption that the rapid tidal flexing of the shell does not affect

its rheology and merely results in tidal dissipation that perturbs the temperature field.

If the ice has a grain size of 1.0 mm, the critical shell thickness for the GBS rheology

ranges from 100 km to greater than the maximum permitted shell thickness of 170

72

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

DmaxDmax

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

020406080

100120140160180200220

Dcr

(km

)

0 5 10 15T (K)

90 K100 K

110 K120 K

Figure 2.11: Similar to Figure 2.7, but for Europa. The critical ice shell thickness rangesfrom 100 km to the maximum permitted shell thickness of 170 km for the GBS rheology,and from 31 to 78 km for basal slip.

73

km; for the basal slip rheology, the critical shell thickness ranges from and 31 km to

78 km. Triggering convection in an ice shell with the nominally accepted thickness

of 20-25 km (Pappalardo et al., 1999; Nimmo et al., 2003) with GBS rheology in the

asymptotic regime with a large temperature perturbation requires the ice has a grain

size 0.07 0.1 mm, respectively. Larger grain sizes lead to stiffer ice, and convection

is not permitted, even if T Trh. Figure 2.12 demonstrates that for the GBS

rheology, triggering convection with a temperature perturbation of amplitude 5 K in

the thickest possible ice shell in Europa requires a grain size 2.0 mm. This conclusion

regarding the grain size is qualitatively similar to the conclusions made by McKinnon

(1999), but consideration of the non-Newtonian rheology adds an additional constraint:

a temperature perturbation must be issued to the ice shell to soften the ice in order to

trigger convection.

2.7 Discussion: The Role of Tidal Dissipation

Tidal dissipation is a likely mechanism to generate temperature anomalies of

order 1-10s K within the ice shells of tidally flexed satellites. Although estimates of

the total amount of dissipation within Ganymede and Europa exist, how this heat is

distributed within their ice shells is a poorly constrained problem. If tidal dissipation is

concentrated on spatial scales much longer than cr, triggering convection with may not

be possible even in the thickest ice shells in Ganymede and Europa if ice flows by GBS

only. Tidal heating may concentrate in zones of weakness in the ice shell, providing a

laterally heterogeneous heat source within the ice shell [e.g. Tobie et al., 2004]. Zones

of weakness could form beneath double ridges on Europa, whose upwarped morphology

may be due to thermal and/or compositional buoyancy driven by localized shear heating

generated by cyclical lateral motion along strike-slip faults (Nimmo and Gaidos, 2002).

If the tidal dissipation is concentrated within the ice shells on spatial scales similar to

cr, convection could be triggered by tidal heating in shells thinner than the maximum

74

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

Convection

No Convection

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

0.0010.001

0.01

0.1

1

d cr (m

m)

0 30 60 90 120 150D (km)

90 K

100 K

110 K

120 K

Figure 2.12: Similar to Figure 2.9, but for Europa.

75

allowed shell thickness of 160 km in Ganymede and 170 km in Europa.

Tidal dissipation may change the mode of heat transfer across the outer ice I

shells of tidally flexed icy satellites such as Ganymede or Europa during past epochs

of increased tidal activity (Showman and Malhotra, 1997; Hussmann and Spohn, 2004).

We envision two possible scenarios. If the ice shell is initially in conductive equilibrium

when tidal dissipation begins, dissipation would be concentrated where the viscosity of

the ice is such that the tidal forcing time scale is equal to the Maxwell time of the ice,

likely at the warm base of the shell (Ojakangas and Stevenson, 1989). This addition

of heat would raise the local temperature above the conductive equilibrium, potentially

causing the bottom layer of the ice shell to become convectively unstable. Conversely,

if the ice shell is initially convecting when tidal dissipation begins and the heat flux

due to tidal dissipation exceeds the convective heat flux, the ice shell would thin by

melting, and convection would cease McKinnon (1999), and convection would be only

a transient phenomenon occurring only in the beginning stages of passage through an

orbital resonance. The existence of an equilibrium between tidal dissipation and the

convective heat flux is controlled by the actual rheology of the ice shell and the details

of tidal dissipation, both of which are not well constrained.

Given the requirement of a finite-amplitude temperature perturbation to initiate

convection in a non-Newtonian ice shells, tidal dissipation could be required to initiate

convection in all icy satellites. A causal relationship between tidal dissipation and

endogenic resurfacing is supported by the observation that all endogenically-resurfaced

icy satellites in the solar system are presently in or have passed through, an orbital

resonance (Dermott et al., 1988; Showman and Malhotra, 1997; Goldreich et al., 1989).

If this is the case, the endogenic resurfacing on Europa and Ganymede could have

been formed during a brief transient period during which tidal dissipation occurred,

triggering convection. Because Callisto has apparently not undergone tidal dissipation,

its non-Newtonian outer ice I shell may have never convected, and therefore has never

76

experienced endogenic resurfacing.

2.8 Summary

The laboratory-derived composite flow law for ice I implies that the growth of

modest-amplitude ( 1-10s K) temperature perturbations in an ice shell is governed

by non-Newtonian creep mechanisms. Therefore, the initiation of convection depends

on the success of plume growth under the influence of these non-Newtonian deforma-

tion mechanisms, which place stringent requirements on the thickness and grain size of

an ice I shell. In the absence of tidal dissipation, the initiation of convection depends

on growth of temperature perturbations governed by the non-Newtonian rheology of

grain boundary sliding. For temperature perturbations larger than the rheological tem-

perature scale (> 37 K), the critical Rayleigh number is independent of perturbation

amplitude and yields an lower limit on the shell thickness required for convection if ice

deforms by GBS or basal slip only.

In Callisto, the critical shell thickness ranges between 103 km and the maximum

permitted shell thickness of 180 km. In Ganymede, the critical ice shell thickness for

convection controlled by GBS in ice with a nominal grain size of 1.0 mm is between

96 km and the maximum permitted ice I shell thickness of 160 km. In both satellites,

convection can only be triggered by modest temperature perturbations of 1-10s K if

the grain size is less than 1.0 mm. If larger temperature perturbations are issued to the

ice shell by, for example, tidal dissipation, convection may occur in ice shells with larger

grain sizes.

In Europa, the critical shell thickness for convection ranges from 100 to the max-

imum permitted shell thickness of 170 km, for GBS and a grain size of 1.0 mm. Con-

vection in a Europan ice shell thicker than 100 km can be initiated from modest 1-10s

K temperature perturbations if the grain size of ice is small, less than 2.0 mm.

Extrapolations of these results to other icy satellites, boundary temperatures,

77

grain sizes, and rheologies can be made using the derived relationships among the phys-

ical, thermal, and rheological parameters of the system and the critical Rayleigh num-

ber. Convection can be initiated from a conductive equilibrium in the non-Newtonian

ice shells of Europa, Ganymede, and Callisto if a temperature perturbation is issued to

the ice shell to soften the ice and permit fluid motion. The critical Rayleigh number and

conditions permitting convection depend on the amplitude and wavelength of tempera-

ture perturbation issued to the ice shell. For the Galilean satellites, large temperature

perturbations of order 10s K are required to initiate convection in ice shells thinner

than 100 km, regardless of grain size. For perturbation amplitudes greater than 37 K,

the critical Rayleigh number is constant, indicating that regardless of the amplitude of

perturbation, convection may not be possible in ice shells with large grain size. Re-

gardless of the critical ice shell thickness required for convection, the non-Newtonian

behavior of ice requires that a finite-amplitude temperature perturbation be issued to

the shell to trigger convection. Tidal dissipation may be required to generate initial

temperature perturbations, suggesting that convection may only occur in thin outer ice

I shells of satellites when the shell is tidally flexed.

Chapter 3

Onset of Convection in Ice I with Composite Newtonian and

Non-Newtonian Rheology

This chapter has been submitted to the Journal of Geophysical Research:

Barr, A. C. and R. T. Pappalardo (2004), Onset of Convection in Ice I with

Composite Newtonian and non-Newtonian Rheology: Application to the Icy Galilean

Satellites J. Gephys. Res., 2004JE002371, submitted.

3.1 Abstract

Ice I exhibits a complex rheology at temperature and pressure conditions appro-

priate for the interiors of the outer ice shells of Europa, Ganymede, and Callisto. We

use numerical methods to determine the conditions required to trigger convection in

an ice I shell with the stress-, temperature- and grain size-dependent composite rheol-

ogy measured in laboratory experiments by Goldsby and Kohlstedt (2001). The critical

Rayleigh number for convection varies as a power (0.2) of the amplitude of initial tem-

perature perturbation, for perturbation amplitudes between 3 K and 30 K. The critical

Rayleigh number depends strongly on the grain size of ice, which governs the transi-

tion stresses between the Newtonian and non-Newtonian deformation mechanisms. The

critical ice shell thickness for convection in all three satellites is < 30 km if the ice

grain size is

79

(GSS) creep, so plume growth is controlled by Newtonian volume diffusion. The critical

shell thickness is 1 cm, where thermal stresses can activate

strongly non-Newtonian dislocation creep, and the ice softens as it flows. For interme-

diate grain sizes (1-10 mm), weakly non-Newtonian grain-size-sensitive creep controls

plume growth, yielding critical shell thicknesses close to the maximum permitted shell

thickness for each of the Galilean satellites. Regardless of the rheology that controls

initial plume growth, a finite amplitude temperature perturbation is required to soften

the ice to permit convection, and this may require tidal dissipation.

3.2 Introduction

Chapter 2 examined the convective stability of an initially conductive ice I shell

under the influence of two of the weakly non-Newtonian deformation mechanisms in ice,

namely grain boundary sliding (GBS) and basal slip (bs). That work characterized the

critical Rayleigh number in two regimes of behavior. For modest amplitude temperature

perturbations, the critical Rayleigh number was found to be a function of the amplitude

of initial temperature perturbation. In the limit of large amplitude perturbations (> 37

K), the critical Rayleigh number was found to approach a constant, asymptotic value. In

the power law regime, the critical Rayleigh number for convection in ice with a rheology

of only GBS or only basal slip was found to vary by an order of magnitude as the

amplitude of initial temperature perturbation varies from 0.7 K to 17 K. In Chapter 2, we

concluded that convection could occur in the outer ice I layers of Europa, Ganymede, and

Callisto provided stringent requirements on shell thickness, perturbation amplitude, and

grain size are met simultaneously. If deformation in ice was accommodated by the GBS

deformation mechanism alone, then convection in Europa, Ganymede, or Callisto could

only occur in an ice shell with a grain size of 1 mm or less, triggered by a temperature

perturbation of order 1-10 K in shell greater than 100 km thick.

However, GBS and basal slip accommodate deformation in ice I only for a small

80

range of temperatures, grain sizes, and stresses, and the roles of the Newtonian deforma-

tion mechanism of diffusional flow and the highly non-Newtonian mechanism dislocation

creep were left unaddressed in Chapter 2. Using similar numerical methods, we extend

the results of this previous work to determine the conditions required to trigger con-

vection in an ice I shell using a composite Newtonian and non-Newtonian rheology for

ice.

3.3 Methods

3.3.1 Numerical Implementation of Composite Rheology for Ice I

Laboratory experiments indicate that deformation in ice I is accommodated by

four creep mechanisms, resulting in a composite flow law (Goldsby and Kohlstedt , 2001):

total = diff + disl +

(1

bs+

1

GBS

)1. (3.1)

The composite flow law includes contributions from diffusional flow (diff ), dislocation

creep (disl), and grain-size-sensitive creep (GSS), where deformation occurs by both

basal slip (bs) and grain boundary sliding (GBS ) (Goldsby and Kohlstedt , 2001). Basal

slip and GBS are dependent mechanisms and both must operate simultaneously to

permit deformation. When responsible for flow, the total strain rate for GSS is controlled

by the slower of the two constituent mechanisms (Durham and Stern, 2001).

The vertical viscosity structure near the base of the ice shell controls the viscous

restoring forces that retard growth of initial convective plumes, so estimates of the grain

size near the melting point of ice are useful for evaluating the conditions required to

permit convection. The grain sizes of ice in the satellites are not well constrained, with

estimates spanning eight orders of magnitude, from microns (Nimmo and Manga, 2002)

to meters (Schmidt and Dahl-Jensen, 2004). Terrestrial ice sheets under similar stress

and temperature conditions as the base of Europas ice shell exhibit grain sizes of order 1

mm (De La Chapelle et al., 1998). Grain growth in Europas ice shell would be limited

81

by rapid tidal flexing of the ice shell and by the presence of non-water-ice materials

(McKinnon, 1999). The presence of non-water-ice materials in the shells of Ganymede

and Callisto might similarly limit grain growth in these satellites as well.

To account for the uncertainty in the grain size of ice within the icy Galilean

satellites, we use grain sizes ranging from 0.1 mm to 10 cm. We characterize the

conditions required for convection as a function of grain size. We assume that the

ice shells have a uniform grain size, which is implausible in a real ice shell.

The strain rate for each creep mechanism in the composite rheology (equation

3.1) is described by

= An

dpexp

(Q

RT

), (3.2)

where is the strain rate, A is the pre-exponential parameter, is stress, n is the stress

exponent, d is the ice grain size, p is the grain size exponent, Q is the activation energy,

R is the gas constant, and T is temperature. Rheological parameters after Goldsby and

Kohlstedt (2001) are summarized in Table A.2.

Goldsby and Kohlstedt (2001) provide an alternate set of creep parameters for

GBS and dislocation creep for ice near its melting point. For T > 255 K, deformation

rates in ice due to GBS are increased by a factor of 1000, in response to melting at

grain boundaries and edges, resulting in very low viscosities near the melting point.

This behavior is consistent with observations of grain size, temperature, stress, and

strain rate for terrestrial ice cores (De La Chapelle et al., 1998). A similar effect occurs

for dislocation creep at T > 258 K. We have not included the high temperature creep

enhancement in our initial numerical models. Use of the creep enhancement for warm

ice alone will result in extremely low viscosities near the base of the ice shell, which

presents a difficulty for our numerical model.

The strain rate from diffusion creep is described by

=42Vm

RTmd2

(Dv +

dDb

)(3.3)

82

where Vm is the molar volume, Tm is the melting temperature of ice, Dv is the rate of

volume diffusion, is the grain boundary width, and Db is the rate of grain boundary

diffusion (Goodman et al., 1981; Goldsby and Kohlstedt , 2001).

The grain sizes we consider are much larger than the grain boundary width

(9.04 1010 m) (Goldsby and Kohlstedt , 2001), so volume diffusion dominates over

grain boundary diffusion, and we may ignore the contribution of grain boundary diffu-

sion to the strain rate. The strain rate for volume diffusion is:

diff =Adiffd2

Do,v exp

(QvRT

)(3.4)

where Do,v is the volume diffusion rate coefficient and Qv is the activation energy. The

viscosity of ice deforming by volume diffusion is Newtonian, but it does depend on grain

size. The parameters for volume diffusion are listed in Table A.2, where we have grouped

the pre-exponential parameters to calculate an effective Adiff = (42VmDo,v/RTm).

The deformation mechanism that yields the highest strain rate for a given tem-

perature and differential stress is inferred to accommodate deformation in ice at that

temperature and stress level. The transition stress between any pair of flow laws, for

example, GBS and dislocation creep, is described by

T =

[AGBSAdisl

dpdisl

dpGBSexp

((Qdisl Q

GBS)

RT

)] 1ndislnGBS. (3.5)

The expressions for the transition stresses between the various deformation mechanisms

can be used to construct deformation maps showing the boundaries of regimes of domi-

nance for each constitutent creep mechanism. Deformation maps for ice with grain sizes

0.1 mm, 1.0 mm, 1.0 cm, and 10 cm are illustrated in Figure 3.1.

The strain from a growing convective plume will be accommodated by the de-

formation mechanism that is dominant near the melting temperature of ice and the

thermal stress exerted by the growing plume. The thermal stress due to a plume of

height , warmer than its surroundings by T , is approximately th gT. In an

83

-4

-2

0

2lo

g 10

(M

Pa)

120 150 180 210 240

-4

-2

0

2lo

g 10

(M

Pa)

120 150 180 210 240

-4

-2

0

2lo

g 10

(M

Pa)

120 150 180 210 240

-4

-2

0

2lo

g 10

(M

Pa)

120 150 180 210 240

Disl

GBS

d=10 cm

120 150 180 210 240

120 150 180 210 240

120 150 180 210 240

120 150 180 210 240

Disl

DiffGBS

d=1.0 cm

-4

-2

0

2

log 1

0

(M

Pa)

120 150 180 210 240T (K)

-4

-2

0

2

log 1

0

(M

Pa)

120 150 180 210 240T (K)

-4

-2

0

2

log 1

0

(M

Pa)

120 150 180 210 240T (K)

-4

-2

0

2

log 1

0

(M

Pa)

120 150 180 210 240T (K)

Disl

GBS

BS Diff

d=1.0 mm

120 150 180 210 240T (K)

120 150 180 210 240T (K)

120 150 180 210 240T (K)

120 150 180 210 240T (K)

Disl

GBSBS

Diff

d=0.1 mm

Figure 3.1: Deformation maps for ice I using the rheology of Goldsby and Kohlstedt(2001), for ice with grain sizes of 10 cm, 1.0 cm, 1.0 mm, and 0.1 mm. Lines on thedeformation maps represent the transition stress between mechanisms as a function oftemperature. A melting temperature of 260 K is assumed. For large grain sizes, dislo-cation creep (n=4) dominates the rheological behavior for thermal stresses associatedwith initial plume growth in the icy satellites ( 104 102 MPa). The weakly non-Newtonian deformation mechanisms play important roles for intermediate grain sizes(1.0 cm and 1.0 mm). Diffusional flow, which is a Newtonian deformation mechanism,becomes important at small stresses, small grain sizes, and temperatures close to themelting point.

84

ice shell 50 km thick on Europa, Ganymede, or Callisto, a plume with height D and

T 5 K can generate a thermal stress of 0.03 MPa. In an ice shell 25 km thick, a

plume of height approximately 25 km can generate 0.15 MPa. For the range of grain

sizes considered, the thermal stresses associated with initial plume growth can activate

any of the four deformation mechanisms, with dislocation creep controlling initial plume

growth for grain sizes of 10 cm, and diffusional flow controlling plume growth in ice with

a grain size of 0.1 mm.

Each deformation mechanism in the composite rheology has a distinct stress ex-

ponent and activation energy, so inversion of equation (3.1) for an exact expression for

viscosity ( = /) is not possible. van den Berg et al. (1995) implement a compos-

ite rheology for mantle materials, including a term for Newtonian diffusional flow and

non-Newtonian dislocation creep. To derive an expression for the total viscosity due to

all four deformation mechanisms, we follow the procedure described by van den Berg

et al. (1995) by expressing the composite flow law (equation 3.1) in terms of viscosities

as = /. This procedure yields an approximate solution for the effective viscosity

due to all four deformation mechanisms as

eff =

[1

diff+

1

disl+

(bs + GBS

)1]1. (3.6)

This approximation provides a good estimate of the total v