Table Of ContentLunar and Planetary Science XXXVI (2005) 2146.pdf
CONVECTION IN ICE I WITH COMPOSITE NEWTONIAN/NON-NEWTONIAN RHEOLOGY:
APPLICATION TO THE ICY GALILEAN SATELLITES. Amy C. Barr1,2 and Robert T. Pappalardo,
Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, CO 80309
([email protected]); 2 now at Department of Earth and Planetary Sciences, Washington University,
Saint Louis, MO 63130 ([email protected])
Introduction: Ice I exhibits a complex rheology is used for each term in equation (2). To non-
at temperature and pressure conditions appropriate dimensionalize the rheology, we divide each term in
for the interiors of the outer ice I shells of Europa, equation (2) by the viscosity due to volume diffusion
Ganymede, and Callisto. We use numerical methods at the melting temperature of ice,
to determine the conditions required to trigger d2 # Q* &
" = exp% v ( . (4)
convection in an ice I shell with a stress-, o A $% RTm’(
temperature-, and grain-size-dependent rheology
The transition stresses between the deformation
measured in laboratory experiments by Goldsby and
mechanisms are mathematically represented by a
Kohlstedt [2001] (henceforth GK2001). Triggering
series of weighting factors (β) between the four
convection from an initially conductive ice shell with !
component rheologies, which govern the relative
a non-Newtonian rheology for ice I requires that a
importance of each mechanism as a function of
finite-amplitude temperature perturbation be issued to temperature and grain size [3]. Each viscosity
the ice shell [2]. Here, we characterize the amplitude function is expressed in non-dimensional coordinates
and wavelength of temperature perturbation required (primed quantities) as
to initiate convection in the outer ice I shells of 1 ’ E E *
Europa, Ganymede, and Callisto using the GK2001 # " = % "( 1&n)exp) & v , , (5)
rheology for a range of ice grain sizes. $ ( T " +T o" 1+To " +
where # " =#/($%˙ ) is the non-dimensional stress,
Numerical Implementation of Ice Rheology: o o
The strain rate for each mechanism in the composite E=Q*/nRΔT is the non-dimensional activation
rheology is described by !en ergy, Ev=Qv*/nRΔT, and T’o is the reference
temperature.
!
#n % $Q*( We use a reference Rayleigh number defined by
"˙ = A exp’ * , (1) "g#$TD3
dp & RT ) Ra= (6)
%&
where "˙ is the strain rate; A, n, p, and Q* are o
where ρ=930 kg m-3 is the density of ice, g is the
experimentally determined rheological parameters; d
acceleration of gravity, α=10-4 K-1 is the coefficient
is the ice grain size, R is the gas constant, and T is
tempe!ra ture (see Table 1). of t!he rmal expansion, ΔT is the temperature
!
GK2001 provide an alternate set of creep difference between the surface and the base of the
parameters for grain boundary sliding and dislocation shell, D is the thickness of the ice shell, and κ=10-6
creep in ice near its melting point, but we do not m2 s-1 is the thermal diffusivity. In a non-Newtonian
include this effect in our present models. If the fluid, viscosities in the layer may evolve to values
viscosities due to GBS and dislocation creep in the larger or smaller than η depending on the vigor of
ο
warm ice near the base of the shell are much smaller convection. The viscosity at the melting point near
than described here, convection might be possible in the base of the ice shell is 1013 Pa s when volume
ice shells thinner than described by our models. diffusion and dislocation creep are the dominant
To implement a viscosity due to all four rheologies. When GSS creep accommodates
deformation mechanisms simultaneously, we convective strain, the viscosity in the convecting
rephrase the composite flow law of GK2001 in terms interior of the ice shell is 1014 to 1015 Pa s.
of viscosities using "=#/$˙ , which allows an Initial Conditions: The approach we use to
approximate solution for the total viscosity [3], numerically determine the critical Rayleigh number
$ ’# 1 is similar to linear stability analysis [4,5]. The
" =& 1 + 1 +(" +" )#1) , (2) convection simulations are started from an initial
tot %&" !di ff "disl GBS bs () condition of a conductive ice shell plus a temperature
due to volume diffusion (diff), dislocation creep perturbation expressed as a single Fourier mode,
(disl), grain boundary sliding (GBS), and basal slip z#T ’ 2%D * ’ "z%*
T(x,z)=T " +$T cos) x, sin) , , (7)
(bs). An explicit stress-dependent rheology of form: s D ( & + ( D +
!
dp % Q*(
"= #(1$n)exp’ * (3)
A & RT)
!
!
Lunar and Planetary Science XXXVI (2005) 2146.pdf
where δT and λ are the amplitude and wavelength of maximum permitted shell thickness for each of the
the temperature perturbation, and z=-D at the warm Galilean satellites.
base of the ice shell. Future Work: GK2001 provide an alternate set
The simulation is run for a short time to of creep parameters for GBS and dislocation creep in
determine whether the initial perturbation grows and ice near its melting point, but we have not yet
convection begins, or decays with time due to included this effect in our models. If the viscosities
thermal diffusion and viscous relaxation, causing the due to GBS and dislocation creep in the warm ice
ice layer to return to a conductive equilibrium [Barr
near the base of the shell are much smaller than
et al., 2004]. For a given pair of δT, λ values, we run described here, convection might be possible in ice
a series of convection simulations with decreasing
shells thinner than found here.
values of Ra . The critical Rayleigh number is
o Acknowledgements: This work is supported by NASA
defined as the minimum value of Rao where the GSRP NGT5-50337 and NASA Exobiology Grant NCC2-
system convects for a given initial condition, and 1340. ACB wishes to thank Allen McNamara for valuable
here is determined to three significant figures. discussions.
Following the procedure described in [2], we References: [1] Goldsby, D. L., and D. L. Kohlstedt,
determine the wavelength of perturbation for which JGR v. 106, p. 11,017-11,030, 2001; [2] Barr, A. C., R. T.
Pappalardo, and S. Zhong, JGR, 109, E12008,
Ra is minimized, and investigate how Ra varies
cr cr doi:10.1029/2004JE0022962004; [3] Barr, A. C. and R. T.
with the amplitude of temperature perturbation. Over
Pappalardo, JGR submitted, 2004; [4] Turcotte, D. L. and
the range of grain sizes used in this study (d=0.1 mm
G. Schubert, Geodynamics: Applications of Continuum
– 10 cm), the critcal wavelength is constant at 1.75D.
Physics to Geological Problems, John Wiley & Sons, 1982;
Results: We find that the critical Rayleigh [5] Chandrasekhar, S., Hydrodynamic and Hydromagnetic
number for convection varies as a power (-0.24) of Stability, Oxford, Clarendon, 1961.
the amplitude of initial temperature perturbation,
$ "T ’* + Table 1. Rheological Parameters a
Ra =Ra & ) , (8) Rheology A (mp Pa-n s-1) n p Q* (kJ mol-1)
cr cr,0% #T ( Volume Diffusion 3.5 x 10-10 b 1 2 59.4
for perturbation amplitudes between 3 K and 30 K. Basal Slip 2.2 x 10-7 2.4 0 60
Based on the results of [2], we expect the critical GBS 6.2 x 10-14 1.8 1.4 49
Rayleigh number to reach a constant asymptotic Dislocation Creep 4.0 x 10-19 4.0 0 60
value for! δ T>0.25 ΔT. a After Goldsby and Kohlstedt [2001]. bFor Tm=260 K.
Values of the fitting coefficient Ra depend on
cr,0
the grain size of ice, and can be fit to a polynomial in
log-log space,
ln(Ra )="0.129(lnd)3"
cr,0 , (9)
2.98(lnd)2"22.6(lnd)"42.7
which can be combined with the definition of the
Rayleigh number to obtain a scaling between the
critical ice shell thickness for convection, the grain
!
size of ice, and the amplitude of temperature
perturbation.
Implications for the Icy Galilean Satellites:
Figure 1 illustrates the variation in critical shell
thickness (D ) for convection in Europa as a function
cr
of grain size. If the ice grain size is <1mm, D < 30
cr
km because relatively low thermal stresses are not
sufficient to activate weakly non-Newtonian GSS
creep, so plume growth is controlled by Newtonian Figure 1. Critical shell thickness for convection in Europa
volume diffusion. For grain size >1cm, Dcr< 30 km as a function of grain size, from equation (9). (a) Volume
because thermal stresses can activate strongly non- diffusion accommodates creep during initial plume growth,
Newtonian dislocation creep, and the ice softens as it and the critical shell thickness for convection increases as a
strong function of grain size. (b) GSS creep controls plume
flows. For intermediate grain sizes (1-10 mm),
growth, and the critical shell thickness achieves a
weakly non-Newtonian GSS creep controls plume
maximum value close to the maximum permitted shell
growth, yielding critical shell thicknesses close to the
thickness. (c) Dislocation creep accommodates flow,
yielding smaller values of critical shell thickness.
