CONTENTS — A through E Numerical Implementation of Ice Rheology for Europa’s Shell

We present a discussion of approximations to the temperature dependent part of the rheology of ice. We have constructed deformation maps using the superplastic rheology of Goldsby & Kohlstedt, 2001, and find that the rheologies that control convective flow in the Europa's are likely grain boundary sliding (Q*=49 kJ/mol, n=1.8, p=1.4) and basal slip (Q*=60 kJ/mol, n=2.4) for a range of grain sizes 0.1 mm < d < 1 cm. We compare the relative merits of two different approximations to the temperature dependence of viscosity and argue that for temperature ranges appropriate to Europa, implementing the non-Newtonian, lab-derived flow law directly is required to accurately judge the onset of convection in the ice shell and temperature gradient in the near-surface ice. Deformation Maps: Deformation maps for ice I were constructed using the rheology of Goldsby & Kohlstedt, 2001, which expresses a composite flow law for ice I as the sum of four individual flow laws with different dependence upon temperature, strain-rate and grain size. The deformation map shows the locus of points in σ − T space where the strain rate from each pair of flow laws are equal. The flow law that contributes the largest strain rate is judged to be dominant in that region. Maps for d = 0.1 mm and d = 1 cm, which bracket common estimates of the grain size within Europa's ice shell [McKinnon, 1999], are shown in Figure 1 a,b. For the low (∼ 0.01 MPa) convective stresses in an ice shell 10's of km thick, the controlling rheology of the ice shell is grain boundary sliding, but basal slip can become important if the grain size is small. Implementing Temperature Dependence: There is a common method of approximating temperature-dependent rhe-ology used by the Earth mantle convection community [e.g. which approximates the lab-derived flow law (η ∼ exp(Q * /RT)) as η ∼ exp(−γT) where γ is the Frank-Kamenetskii (FK) parameter: γ = ∂(ln η) ∂T T i. (1) Here Ti is the interior temperature of the ice shell, which is not known a priori, but is typically close to the melting point (Tm) so it is assumed that γ = Q * /nRT 2 m. The viscosity contrast across the convecting layer ∆η = exp(E∆T). Use of this approximation results in lower surface viscosities than predicted by the lab-derived flow law. This does not affect the outcome of stagnant-lid (large ∆η) convection …