To solve the momentum balance, ice sheet stresses need to be expressed as velocity fields, via a constitutive relation for ice. The rheology of polycrystalline glacier ice is well studied in laboratory and field environments, which reveal that ice deforms as a nonlinear viscous fluid. The original form of the flow law, proposed by John Glen and John Nye in the 1950s, is broadly supported by field studies of tunnel and borehole deformation, as well as observations and modeling of large-scale ice motion. This constitutive relation is known as Glen's flow law:
where etJ is the strain rate, T is ice temperature, and oi is the deviatoric stress tensor in the ice. The flow law is an empirical relation rather than a physical law, a la Newton, but it is rooted in the theoretical assumption that strain rates in ice should be a function of the stress tensor and its invariants. The deviatoric stress is used in (6.8) due to the observation that ice deformation is independent of confining pressure (normal stresses). For a linear (Newtonian) fluid, f(T) l/u (T), (6.9)
for the viscosity m. In glacier ice, the effective viscosity, meff, is represented as a function of the second invariant of the deviatoric stress tensor, = (olop 1/2/2, f(T, op = llHf = B (T) (6.10)
B(T) is an "ice softness" term that follows an Arrhenius temperature dependence,
B0 is called the Glen flow-law parameter, R is a constant, and Q is the creep activation energy. Ice deformation is typically modeled as an n = 3 process, giving
This formulation is an isotropic flow law that allows the first-order effects of ice temperature and deviatoric stress regime to be incorporated in estimates of ice deformation. Where shear stress and shear deformation are dominant, as is often the case, this is well approximated by
Glen's flow law is for pure, isotropic ice. There are numerous other complicating factors for ice deformation, such as anisotropic ice fabric, the potential impact of grain size, and the presence of impurities and intergranular liquid water content. These effects are not explicitly resolved in ice sheet models, so the flow rate parameter, B0, is typically tuned to approximate the bulk effects of crystal fabric, grain size, and impurity content.
Even without this level of detail or complexity in modeling ice rheology, there is tremendous variability in the effective viscosity of ice associated with the range of ice temperatures and stress regimes found in Earth's glaciers and ice sheets. Figure 6.4b plots the effective viscosity variation with depth at sample location in the Greenland ice sheet, as calculated from (6.10).
The strain rates in (6.12) or (6.13) can be expressed as velocity gradients and then vertically integrated or inverted and substituted into the momentum balance (6.6) to give a set of equations for the horizontal ice velocity. Various numerical solutions to these equations have been adopted in glacier and ice sheet modeling.
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