On Exact Solutions and Numerics for Cold, Shallow, and Thermocoupled Ice Sheets

This three section report can be regarded as an extended appendix to (Bueler, Brown & Lingle 2006). First we give the detailed construction of an exact solution to a standard continuum model of a cold, shallow, and thermocoupled ice sheet. The construction is by calculation of compensatory accumulation and heat source functions which make a chosen pair of functions for thickness and temperature into exact solutions of the coupled system. The solution we construct here is “Test G” in (Bueler et al. 2006) and the steady state solution “Test F” is a special case. In the second section we give a reference C implementation of these exact solutions. In the last section we give an error analysis of a finite difference scheme for the temperature equation in the thermocoupled model. The error analysis gives three results, first the correct form of the CourantFriedrichs-Lewy (CFL) condition for stability of the advection scheme, second an equation for error growth which contributes to understanding the famous “spokes” of (Payne et al. 2000), and third a convergence theorem under stringent fixed geometry and smoothness assumptions. 1. Derivation of the exact solution 1.1. Review: Equations of the continuum model. The flat bed, frozen base case of the cold shallow ice approximation is, for the purposes of this paper, taken to be the equations in the “Continuum Model” part of (Bueler et al. 2006); that paper is hereafter referred to as “BBL.” The notation used in, the physics of, and the boundary conditions for the continuum model are all laid out in BBL. The equations are repeated here for convenience: mass-balance: ∂H ∂t = M −∇ ·Q, (1) temperature: ∂T ∂t = k ρcp ∂2T ∂z2 −U · ∇T − w ∂T ∂z +Σ. (2) effective shear stress: 〈σxz, σyz〉 = −ρg(H − z)∇H, (3) constitutive function: F (T, σ) = A exp (