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

Hodge theory and cohomology with compact supports

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian \lap_\mu associated to a measure $d\mu$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to \lap_\mu which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures

Lessons from the short history of ice sheet model intercomparison

International audienceIntercomparison should include measurement of differences, between model results and observations, among the model results themselves, or between model results and exact solutions. The processes of measuring differences and critically analyzing those differences are vital. Without such measurement as a component of intercomparison, the only expected benefits of an intercomparison project are participation, possibly the discovery of communal confusion, and the establishment of public, non-proprietary data sets

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

Understanding the Difference between the Right to Subrogation and Assignment of an Inusrance Claim - Keisker v. Farmer

Trinity Universal Insurance Company wrote a policy that did not expressly create an assignment of its policyholder’s future claims and, as a result, recovered only a fraction of the amount it paid to the policyholder. Had Trinity carefully drafted its policy to create an assignment of the insured’s claims, it might have recovered the entire amount from those responsible for the damages. For this reason, insurance companies need to understand the difference between assignment and subrogation. Furthermore, insured individuals need to understand this distinction so that they are aware of their own rights and obligations

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 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