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, and 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 ``TestG'' in (Bueler and others, 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 Courant-Friedrichs-Lewy (CFL) condition for stability of the advection scheme, second an equation for error growth which contributes to understanding the famous ``spokes'' of (Payne and others, 2000), and third a convergence theorem under stringent fixed geometry and smoothness assumptions.Comment: 16 pages, two C codes; extended appendix to Bueler, Brown, and Lingle, "Exact solutions to the thermocoupled shallow ice approximation: effective tools for verification," submitted to J. Glacio

Conservation laws for free-boundary fluid layers

Time-dependent models of fluid motion in thin layers, subject to signed source terms, represent important sub-problems within climate dynamics. Examples include ice sheets, sea ice, and even shallow oceans and lakes. We address these problems as discrete-time sequences of continuous-space weak formulations, namely (monotone) variational inequalities or complementarity problems, in which the conserved quantity is the layer thickness. Free boundaries wherein the thickness and mass flux both go to zero at the margin of the fluid layer generically arise in such models. After showing these problems are well-posed in several cases, we consider the limitations to discrete conservation or balance in numerical schemes. A free boundary in a region of negative source -- an ablation-caused margin -- turns out to be a barrier to exact balance for a numerical scheme (in either a continuous- or discrete-space sense). We propose computable \emph{a posteriori} quantities which allow conservation-error accounting in finite volume and element schemes.Comment: 26 pages, 4 figure

A full approximation scheme multilevel method for nonlinear variational inequalities

We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonlinear partial differential equations, due to A.~Brandt, and the constraint decomposition (CD) method introduced by X.-C.~Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain function space subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and full multigrid cycles are optimal solvers. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure

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.Comment: 36 pages, 16 figures, 3 Matlab program

A Community Ice Sheet Model for Sea Level Prediction

Summary of a workshop that was held at Los Alamos National Laboratory, New Mexico, 18-20 August 2008, whose primary goal was to create a detailed plan for developing, testing, and implementing a Community Ice Sheet Model (CISM) to aid in predicting sea level rise

The shallow shelf approximation as a "sliding law" in a thermomechanically coupled ice sheet model

The shallow shelf approximation is a better ``sliding law'' for ice sheet modeling than those sliding laws in which basal velocity is a function of driving stress. The shallow shelf approximation as formulated by \emph{Schoof} [2006a] is well-suited to this use. Our new thermomechanically coupled sliding scheme is based on a plasticity assumption about the strength of the saturated till underlying the ice sheet in which the till yield stress is given by a Mohr-Coulomb formula using a modeled pore water pressure. Using this scheme, our prognostic whole ice sheet model has convincing ice streams. Driving stress is balanced in part by membrane stresses, the model is computable at high spatial resolution in parallel, it is stable with respect to parameter changes, and it produces surface velocities seen in actual ice streams.Comment: 12 pages of text; 4 tables; 27 figures; submitted to JGR Earth Surfac

Insights into Spatial Sensitivities of Ice Mass Response to Environmental Change from the SeaRISE Ice Sheet Modeling Project I: Antarctica

Atmospheric, oceanic, and subglacial forcing scenarios from the Sea-level Response to Ice Sheet Evolution (SeaRISE) project are applied to six three-dimensional thermomechanical ice-sheet models to assess Antarctic ice sheet sensitivity over a 500 year timescale and to inform future modeling and field studies. Results indicate (i) growth with warming, except within low-latitude basins (where inland thickening is outpaced by marginal thinning); (ii) mass loss with enhanced sliding (with basins dominated by high driving stresses affected more than basins with low-surface-slope streaming ice); and (iii) mass loss with enhanced ice shelf melting (with changes in West Antarctica dominating the signal due to its marine setting and extensive ice shelves; cf. minimal impact in the Terre Adelie, George V, Oates, and Victoria Land region of East Antarctica). Ice loss due to dynamic changes associated with enhanced sliding and/or sub-shelf melting exceeds the gain due to increased precipitation. Furthermore, differences in results between and within basins as well as the controlling impact of sub-shelf melting on ice dynamics highlight the need for improved understanding of basal conditions, grounding-zone processes, ocean-ice interactions, and the numerical representation of all three