Manifestations of ice microphysical processes at the scale of whole ice sheets
Department of Geography, University of California-Berkeley, 507 McCone Hall, Berkeley, CA 94720-4740, USA
Ice sheets live and die according to a grand contest between climate and ice flow. Specifically, the configuration and evolution of ice sheets is governed by the competition between net accumulation rate and the divergence of ice flux arising from gravitationally induced flow. Ice microphysical processes strongly regulate the character of this flow, especially the dependence of flow rate on ice-mass geometry. Thus the great ice sheets are not only an imprint of climatic forcing, they are also continental-scale manifestations of molecular-scale and grain-scale processes.
This chapter provides a conceptual review of the most important of these manifestations. Ice sheets have a profound impact on the global environment by modifying sea level, planetary albedo and atmospheric circulation. The magnitude of these impacts is related to ice-sheet geographical characteristics: volume, area and topographic form. The first and third of these are most cleanly determined by microphysical processes, if icesheet span and climatic forcing are given. The ice-sheet area (and hence span) are determined in a substantially more complex fashion by feed-backs between climatic forcing and ice-sheet topography (and hence microphysical processes), and geographical contingencies such as large-scale influences on climatic forcing, and the distribution of land mass and ocean. The present discussion will focus mostly on ice-sheet volume and profile.
At present, it is not possible to connect microphysics to whole-ice-sheet properties in a manner that is simultaneously direct and convincing. Instead, the connection is made using, as intermediary, a phenomenological description of the constitutive properties of polycrystalline ice derived from laboratory bench-scale experiments and in situ measurements in glacier boreholes and tunnels. In this article, I will explore the ice-sheet-scale implications of four primary factors that embody the consequences of microphysics at this phenomenological scale. These are the effective viscosity of ice h, the stress exponent n in the constitutive relation, the temperature-dependence of viscosity (related to the activation energies for ice deformation), and the viscosity variations due to other ice properties (collectively known as 'enhancement', meaning enhancement of ice fluidity). I will finish the article with succinct summaries of how these intermediate-scale properties are thought to arise from molecular and grain-scale processes.
The approach adopted here is to analyse and use one-dimensional, steady-state ice-sheet models to examine controls on ice-sheet character (following Vialov, 1958; Nye, 1960; Haefeli, 1961; Weertman, 1961; most of the relevant physics is summarized in Paterson, 1994). In terms of geographical realism, such a model is inferior to three-dimensional, whole-ice-sheet models but represents a distillation of the relevant physics and geography to their essentials. The aim is to be illustrative but quantitative, with illustration provided both by formulae and numerical results. None of the conceptual results presented here are altered by relaxing the simplifying assumptions used.
57.2 Model framework 57.2.1 Foundations
The formal statement of the governing competition for ice sheets is dH
wherein H is the ice thickness, b the net accumulation rate (ice-equivalent thickness per time added to the glacier surface) and (J the ice flux. The ice flux in a horizontal direction x arises from basal velocity (rate ub) and internal shear strain rate (exz)
where z is the vertical coordinate (z = b at the bed). A massive amount of empirical evidence shows that the deformation of ice is a form of power-law creep (Glen, 1955; Nye, 1957; Barnes et al., 1971; Weertman, 1983; Budd & Jacka, 1989; Goldsby & Kohlstedt, 2001) meaning that exz is a function of powers of the deviatoric stress, the deviation of stress from its mean normal value. Most commonly, ice deformation is treated using the phenomenologi-cal relation where A0 = 4.9 x 10 25 s 1 - Pa 3, and the activation energy is Ea = 60kJ mol-1 for T < 263K and Ea = 139 kJ mol-1 for T > 263 K.
In ways that are even less completely understood, microphysi-cal processes also help control ub, which depends in part on deformation properties of ice, thermal properties of ice and interactions of ice with melt water. High basal water pressure (Pw close to the ice overburden pressure Pi) lubricates basal motion (increases ub), as does the presence of unconsolidated substrate. Although not a predictive relationship, the statement ub = 1
is consistent with our understanding of basal motion and is useful for analyses (Bindschadler, 1983), bearing in mind that variations in the lubrication parameter 1b can largely determine the variations of ub in some cases.
Ice sheets flow because gravity induces pressure gradients in the ice, resulting from the ice-air surface slope. The flow gives rise to deformations (including exz) which are associated with resisting stresses (including txz) via the ice constitutive relation approximated by Equation (3). These resisting stresses balance the 'driving stress' td, the net horizontal force-effect of gravity, per unit horizontal area (Van der Veen & Whillans, 1989). The td is a simple function of ice thickness and surface slope, and is (given bed elevation b)
57.2.2 The strong-bed ice sheet
Throughout much of the ice sheets the ice-bed interface is mechanically strong and the flow is not confined in narrow channels. Consequently, the driving stress is balanced dominantly by the primary down-flow shear stress txz acting at the bed (value tb), and the txz varies approximately as a linear function of depth
Other strain-rate components are similarly proportional to the corresponding deviatoric stress component. Here t(2) is the second invariant of the deviatoric stress tensor, n is the stress exponent (usually estimated as 3 for rapidly deforming ice; data reviewed by Weertman, 1983), A(T) is the temperature-dependent ice softness (or fluidity), and E is the enhancement factor, a correction factor used to account for anisotropic effects and other variables not otherwise included. In ways that are not yet completely understood, microphysical processes determine this functional form, and values for E, A(T) and n. From the compilation due to Paterson (1994, p. 97), the function A(T) is well approximated by
Much can be learned by making this and further approximations (Nye, 1959, 1960; Haefeli, 1961). Substitutions into Equation (2) give
The multiplicative relation of variables in the integrand of Equation (8) has important implications for large-scale ice-sheet flow, in that values for E and A in ice close to the bed where the shear is concentrated are most significant. One can replace the term E A(T) with an effective value A*, a constant defined to yield the equivalent flux as Equation (8), so that
The case of a uniform accumulation rate b illustrates the properties of this curve most simply (Vialov, 1958), and in particular gives the ice thickness in the centre of the ice sheet (the divide thickness Hd) as
where ud is the depth average of the deformational velocity. The effective temperature T* is the temperature for which A(T) has the value A*/E. For a strong-bed system the ratio of ub to ud typically varies in a range between zero and order one (Paterson, 1994, p. 135), so the factor 1 + ub /ud varies between one and a few. For subsequent use, I define the combined softness as
If there is non-trivial basal motion (ub ^ 0) the basal temperature is at the bulk melting point, and the T* will always be greater than —10°C. In this range, a factor of two change in S (as when the ub changes from being 20% to 140% of ud, a large change) is equivalent to an effective temperature change of only approximately 3°C.
Further, defining the fraction of the total ice flux due to internal deformation as u
and the surface slope as a, the ice-sheet evolution Equation (1) for one-dimensional flow is dH
Because m and n are similar numbers and the qd is of order one, the presence of basal motion in a strong-bed ice sheet does not significantly alter its properties.
The steady-state cross-sectional profile of an ice sheet determines the volume of stored ice per length of ice divide. Integrating Equation (9) by specifying the flux as the integrated net accumulation up-flow of location x yields an estimate for this profile, which for an ice-sheet of span L, a flat bed, a spatially uniform S, and a zero thickness at the ice margin (x = 0) is
V npg J
The ice-sheet volume scales with b and S in similar fashion, and with i3'2. Values for Cj and c2 given here are for n = 3.
126.96.36.199 Perspectives from the strong-bed case
An ice sheet is a forced diffusive system (Equation 13), with the forcing being net accumulation determined by climate and curvature of underlying topography, modulated by gradients in effective softness S. Ice sheets respond to variable forcings via both diffusive and wave-type behaviours. The variables g1 and g2 are the non-linear diffusivity and wave speed, respectively (Equations 14 & 15). Both vary spatially. Diffusive behaviour dominates where surface slope is small, in the extensive thick interior regions of ice sheets. Ice-sheet thickness evolves in a manner analogous to temperature evolution in a plate with spatially varying energy source terms.
The non-linearity of the constitutive relation (n,m values) affects the response time of ice sheets to variable forcings; higher non-linearity (larger n,m) enables faster response. This is, in essence, a consequence of the fact that the stress change needed to accommodate a change in ice flux is smaller if the non-linearity is higher. The stress change, in turn, is a function of the redistribution of mass in the glacier body, and this redistribution is less extensive in the higher non-linearity case and thus achieved more rapidly. The non-linearity also directly increases the magnitude of the topographic forcing term for a given bed curvature.
Closely related is that the steady-state ice volume becomes more weakly dependent on the net accumulation rate and softness as the non-linearity increases (Equation 18). For n = 3, the value thought to be most appropriate for the ice sheets, a doubling of accumulation rate, or halving of softness, changes the steady-state volume by only 9%, if L is fixed. This is a tremendous insensitiv-ity and largely explains why the modern Greenland and Antarctic ice sheets are only modestly different from their Last Glacial Maximum versions. Also, a uniform increase of the basal motion from zero to ub = ud would decrease ice volume by only ca. 8%. This explains why ice-sheet models do a very good job of explaining topography of the Greenland and East Antarctic ice sheets even though we have no predictive understanding of basal motion and limited knowledge of its actual spatial distribution in the ice sheets. Similarly, the temperature of deep ice layers need not be known precisely, although it is essential to know it approximately.
57.2.3 The weak-bed, ice-stream-dominated ice sheet
The important case for which the assumptions tb ~ td and ub ~ ud are completely wrong is that for which the ice flux is accom
x modated almost entirely by ice streams that have minimal basal strength (Whillans & Van der Veen, 1997; Raymond et al., 2001). This situation, typified by the modern Siple Coast of West Antarctica and possibly applicable to sectors of the Pleistocene ice sheets, is made possible by high basal water pressures in combination with a weak, unconsolidated, deformable substrate (high 1b and Pw in Equation 5). In the limiting case, the driving stress is entirely balanced by wall stresses (average magnitude tw) in the shear margins bounding the ice streams.
To explore how this situation differs from the previous one, consider an approximation in which the shear stress t^ varies linearly across the ice stream from a value +T„ to a value -tw. The ice stream has a width W and half-width jw, and the cross-glacier coordinate is y (zero at the margin). Then ffltd = Htw
For comparison to the strong-bed case, consider a hypothetical situation for which the ice-stream flow spans the entire ice sheet from divide to margin, and the marginal thickness is zero. The thickness at the ice divide for a constant accumulation rate case would be
Values for c3 and c4 given are again those for n = 3. 188.8.131.52 Perspectives from the weak-bed case
The flux per unit width in the ice stream is (Raymond, 2000) 2H r® r • , „ , , 2Htw t® fy
«s = £ f £ e,dy " dy ' = ^ f j>T )(1 - W dy "dy '
Defining the fractional area (or cross-flow length) of the ice stream within the total ice area as fs and the number density of ice streams per cross-flow distance as Ns, and defining an effective uniform A* as before, gives the ice flux per unit width as
24 Qx=—2 fsA* tdnH i-nw n+i=—n s a* tih i-nw n+2 (23)
and the governing equation for evolution of the thickness as dH (s) d2b
dt 11 dx2
gi d2 H dH
Analogous calculations to those above yield the thickness profile, given a flat bed, uniform w and fs, and a marginal (grounding-line) thickness Hog n+i n+i
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