Construction of steady state solutions for isothermal shallow ice sheets

Exact solutions for ice sheet equations can and should play an important role in numerical model validation. In constructing a time-dependent exact solution to the thermocoupled shallow ice approximation, it was discovered that the existing analytical steady solutions are a poor basis [1]. This prompted a search for cleaner steady solutions to the isothermal shallow ice equation. In particular, we seek smooth solutions, at least between central peak and margin, which have the physically realistic property, for a grounded ice sheet, of margin–in–ablation–zone. The source of cleaner solutions is identified here as a flux function Q whose nth root is integrable, where n is the Glen exponent. Radially symmetric and one-horizontal dimension examples are identified. Introduction The steady, isothermal cold shallow ice equation [7] is (1) a = ∇ ·Q, Q = − Γ n+2 hn+2|∇h|n−1∇h, h ≥ 0 as an equation for surface elevation h = h(x, y) in a region in R. Equation (1) is the special case of an ice sheet on a flat, rigid bed, so that surface elevation and thickness coincide. Here a = a(x, y) is the ice–equivalent accumulation/ablation function and Q = Q(x, y) is the vertically-integrated horizontal ice flux. The constants Γ > 0 and n > 1 represent material properties. The former depends strongly on the constant value of temperature. Equation (1), for h given a, is second order in space and is elliptic as a PDE for h. More precisely, in regions where h > 0, (1) can be written as a quasilinear PDE for h by eliminating Q, that is, a = −∇· ( Γ n+2 hn+2|∇h|n−1∇h). (A PDE is quasilinear if it is linear in the highest derivative.) In regions where h|∇h| > 0 the PDE is second–order in space and strictly-speaking elliptic. Generally, we say that the PDE is “degenerate elliptic.” Most essentially, (1) is a free boundary value problem with a one–sided constraint and more-or-less a “variational inequality” [2, 3]. One horizontal dimension Suppose h = h(x) and therefore ∇ (h(x)) = h′(x) and ∇ · (Q(x)x̂) = Q′(x). Let us also suppose that x = 0 represents an ice–ridge and therefore Q(0) = 0. If there is