On the numerical computation of temperature in an ice sheet

Quadrature solutions for temperature fields in the central regions of ice sheets where flow is by internal deformation are presented and compared with numerical computations and the European Ice Sheet Modelling Initiative (EISMINT) Benchmark standards. The solution appears to be more accurate than the EISMINT Benchmark standards for thermomechanically uncoupled temperature distributions. The ability of finite-difference and pseudo-spectral methods to reproduce this solution is considered. Errors in the EISMINT Benchmark are larger than expected. The possibility that they could arise from inaccurate evaluation of the vertical velocity is considered, and found to be unlikely. Formulae for computing the vertical velocity in Jenssen's sigma coordinate are complicated and may lead to programming errors. A simple form of the heat-transport equation in the sigma coordinate for flow by sliding and internal deformation is derived and presented, and it is shown how this form is particularly suitable for finite-element computations. It is also shown that only two quadratures are necessary to compute the vertical and horizontal heat transport and the ice flux.