Convection in Ice I With Non-Newtonian Rheology: Application to the Icy Galilean Satellites
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Transcript of Convection in Ice I With Non-Newtonian Rheology: Application to the Icy Galilean Satellites
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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
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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.
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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
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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.
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Dedication
For Bernice Pedersen Courtright and Alberta Engvall Siegel
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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.
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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)
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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.
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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.
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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)
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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)
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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.
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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.
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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-
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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
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67
020406080
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)
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020406080
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)
0 5 10 15T (K)
020406080
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(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.
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68
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)
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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.
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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
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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,
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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
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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
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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
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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)
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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
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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.
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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