In this paper we introduce adaptive time step control for simulation of evolution of ice sheets. The discretization error in the approximations is estimated using "Milne's device" by comparing the result from two different methods in a predictor-corrector pair. Using a predictor-corrector pair the expensive part of the procedure, the solution of the velocity and pressure equations, is performed only once per time step and an estimate of the local error is easily obtained. The stability of the numerical solution is maintained and the accuracy is controlled by keeping the local error below a given threshold using PI-control. Depending on the threshold, the time step $\Delta t$ is bound by stability requirements or accuracy requirements. Our method takes a shorter $\Delta t$ than an implicit method but with less work in each time step and the solver is simpler. The method is analyzed theoretically with respect to stability and applied to the simulation of a 2D ice slab and a 3D circular ice sheet. %The automatically chosen $\Delta t$ is either restricted by accuracy or stability depedning on an error tolerance. The stability bounds in the experiments are explained by and agree well with the theoretical results
Accurate description of snow and firn processes is necessary for estimating the fraction of glacier surface melt that contributes to runoff. Most processes in snow and firn are to a great extent controlled by the temperature therein and in the absence of liquid water, the temperature evolution is dominated by the conductive heat exchange. The latter is controlled by the effective thermal conductivity k. Here we reconstruct the effective thermal conductivity of firn at Lomonosovfonna, Svalbard, using an optimization routine minimizing the misfit between simulated and measured subsurface temperatures and densities. The optimized k * values in the range from 0.2 to 1.6 W (m K) −1 increase downwards and over time. The results are supported by uncertainty quantification experiments, according to which k * is most sensitive to systematic errors in empirical temperature values and their estimated depths, particularly in the lower part of the vertical profile. Compared to commonly used density-based parameterizations, our k values are consistently larger, suggesting a faster conductive heat exchange in firn.
Abstract. The friction coefficient and the base topography of a stationary and a dynamic ice sheet are perturbed in two models for the ice: the full Stokes equations and the shallow shelf approximation. The sensitivity to the perturbations of the velocity and the height at the surface is quantified by solving the adjoint equations of the stress and the height equations providing weights for the perturbed data. The adjoint equations are solved numerically and the sensitivity is computed in several examples in two dimensions. A transfer matrix couples the perturbations at the base with the perturbations at the top. Comparisons are made with analytical solutions to simplified problems. The sensitivity to perturbations depends on their wavelengths and the distance to the grounding line. A perturbation in the topography has a direct effect at the ice surface above it, while a change in the friction coefficient is less visible there.
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