Computation of a Combined Spherical-elastic and Viscous-half-space Earth Model for Ice Sheet Simulation

This report starts by describing the continuum model used by Lingle & Clark (1985) to approximate the deformation of the earth under changing ice sheet and ocean loads. That source considers a single ice stream, but we apply their underlying model to continent-scale ice sheet simulation. Their model combines Farrell’s (1972) elastic spherical earth with a viscous half-space overlain by an elastic plate lithosphere. The latter half-space model is derivable from calculations by Cathles (1975). For the elastic spherical earth we use Farrell’s tabulated Green’s function, as do Lingle & Clark. For the half-space model, however, we propose and implement a significantly faster numerical strategy, a spectral collocation method (Trefethen 2000) based directly on the Fast Fourier Transform. To verify this method we compare to an integral formula for a disc load. To compare earth models we build an accumulation history from a growing similarity solution from (Bueler, et al. 2005) and and simulate the coupled (ice flow)-(earth deformation) system. In the case of simple isostasy the exact solution to this system is known. We demonstrate that the magnitudes of numerical errors made in approximating the ice-earth system are significantly smaller than pairwise differences between several earth models, namely, simple isostasy, the current standard model used in ice sheet simulation (Greve 2001, Hagdorn 2003, Zweck & Huybrechts 2005), and the Lingle & Clark model. Therefore further efforts to validate different earth models used in ice sheet simulations are, not surprisingly, worthwhile. 1. Two linear earth models and their Green’s functions Lingle & Clark (1985) use as their fundamental tools the Green’s functions of two different linear earth models. The Green’s functions for these models are convolved with the load to compute (vertical) displacements of the earth’s surface. One finds an elastic displacement uE and a viscous displacement uV given a current load and a load history, respectively, as we will explain. The total displacement is then the sum u = uE + uV at any time. That is, the two linear models are superposed. The partial differential equations (PDEs) behind these Green’s functions are linear. In this report we state these PDEs, which is, in the case of the second model, a nontrivial accomplishment (see section 3). We then approximately solve these PDEs in a demonstrably efficient manner. First, however, we describe the two models and their sources in the literature. Draft February 2, 2008. Dept. of Mathematics and Statistics, Univ. of Alaska, Fairbanks AK 99775-6660. Email [email protected]. Geophysical Institute, Univ. of Alaska, Fairbanks AK 99775-6660.