On a Doubly Nonlinear Parabolic Obstacle Problem Modelling Ice Sheet Dynamics

Abstract

Abstract. This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet ow w e present the mathematical analysis and the numerical resolution of the second order, nonlinear, degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-newtoniandynamics. An obstacle problem is then deduced and analysed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to cope with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative a n d n umerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension. They can be applied succesfully to more realistic and sophisticated models related to other geophysical settings.

A moving-point approach to model shallow ice sheets: a study case with radially symmetrical ice sheets

“…As pointed out by Fowler (1992) and Calvo et al (2002), singularities can appear with the SIA at the margin of grounded ice sheets. The singularity arises because of the vanishing of h at the margin and the steepening of the slope ∂h/∂r.…”

“…Under hypotheses regarding the regularity of the ice thickness near the margin (see Calvo et al, 2002), we can differentiate Eq. (7) with respect to time.…”

Similarity, Mass Conservation, and the Numerical Simulation of a Simplified Glacier Equation

“…Introduction. In order to simulate the flow of large ice masses such as those in the Antarctic and Greenland [15,25]itis necessaryto makecrediblepredictionsabouttheevolution of glaciers, whichis essential foraccurately estimating future contribution to sea-levels.…”

“…The SIA is a zeroorder model based upon three assumptions; the ice flow is primarily horizontal, the ice width is large in comparison with the ice depth, and the change in stress (and consequently strain) is negligible in the horizontal directions in comparison to the vertical [18,23]. In this paper, a solution of a nonlinear diffusion equation is considered which is the isothermal, non-sliding SIA in the case of a flat bed, focusing on the concept of similarity [14,15,16]. It is called here the glacier equation for simplicity, although more complex dynamical models for glaciers are widespread in the glaciological literature [24,28].…”

“…Building on the scale invariance and conservation properties of this problem in [9,14,15,16]thispapergeneratesnodalvelocitiesbasedonalocalscale-invariantconservation principle and implements a similarity-based time-stepping scheme. It is shown that a moving-mesh constructed in this way can offer a relative error of less than 0.1% with as few as ten nodes.…”