Keywords: shallow ice models, numerical simulation, nonlinear PDE system, free boundaries, finite elements, duality methods, fixed point iteration.

The formulation of appropriate global mathematical models for large ice masses and its numerical simulation has been the subject of many research works in last decades (see, for example, Hutter [1] and the references therein). Particularly, the idea of an ice sheet is a large highly viscous ice drop, where the steady state equilibrium configuration can be briefly explained as a consequence of the balance between the ice accumulation taking place at the top and the ablation processes that mainly occur at the margins. Typical examples of present ice sheets are Antarctic and Greenland ones.

In this paper we propose a global coupled model which governs the different thermal, hydrodynamic and elastic processes which take place in the evolution of large ice sheets. The model can be framed into the shallow ice approximation. The idea of this approach is to appropriately scale (shallow ice scaling) the coordinates and unknowns appearing in the conservation and constituive laws so that in the resulting scaled equations some terms could be neglected, as a consequence of the fact that in a typycal ice sheet geometry length and width are far larger than depth. For the Antarctic ice sheet, typical length and width values are meters while depth is about meters, so that the aspect ratio is small enough to neglect terms in the original conservation laws to state simpler asymptotic models. Moreover, as in ice sheets an almost bidimensional flux occurs, we can restrict ourselves to a longitudinal section of the initial 3-d ice geometry. For a justified statement of the shallow ice models here proposed for the profile, velocity and temperature we address the reader to the work [2] based on the seminal paper of Fowler [3].

Thus, the proposed highly nonlinear system of pde's governs three main problems: the upper profile evolution, the ice velocity field and the temperature distribution. Each problem requires the solution of the other two ones, so that a fixed point iteration between them seems a possible numerical technique. Indeed, the profile and temperature models are posed as free boundary problems whereby the ice sheet extent and the interface between cold and temperate ice are additional unknowns. Concerning to the numerical techniques, for the profile problem a Lagrange-Galerkin approximation is combined with a duality method for maximal monotone operators. In the thermal problem, besides the appropriate time and space discretizations, nonlinear viscous terms are treated by a Newton method, and two phase Stefan formulation and Signorini boundary condition by duality methods. The computation of velocity field inside the ice in the framework of shallow ice models is performed by numerical quadrature.

The different partial models have been previously solved separately by the authors in previous works: a simpler version of the thermal problem in [4], two approaches of the profile problem in [5,6] or the coupling between both for prescribed velocity field in [7], for example. In this paper we propose global model fully included in the shallow ice approximation setting. Particularly, the velocity field inside the fluid is obtained from the momentum equations and the profile and thermal models include the presence of realistic moving boundary features. A particular novelty is the fully implicit treatment of the nonlinear Signorini boundary condition when including additional nonlinear sliding terms.

The combination of these techniques allows for the numerical simulation of Antarctic ice sheet by considering a real data set. Finally, results concerning to basal magnitudes, profile, temperature and velocity fields are shown by means of an original and specific numerical simulation toolbox (GLANUSIT).

1

K. Hutter, "Theoretical Glaciology", Reidel, Dordrecht, 1981.

2

N. Calvo, J. Durany, C. Vázquez, Un modelo de Stefan-Signorini no-newtoniano con efectos basales para la termodinámica de grandes masas de hielo, "Actas del V Congreso de Métodos Numéricos en Ingeniería", Madrid (2002), CD-ROM.

3

A.C. Fowler, "Modelling ice sheet dynamics", Geophys. Astrophys. Fluid Dynamics, 63, 29-65, 1992. doi:10.1080/03091929208228277

4

N. Calvo, J. Durany, C. Vázquez, "Numerical approach of temperature distribution in a free boundary polythermal ice sheet", Numerische Mathematik, 83, 557-580, 1999.

5

N. Calvo, J. Durany, C. Vázquez, "Numerical computation of ice sheet profiles with free boundary problems", Applied Numerical Mathematics, 35, 111-128, 2000. doi:10.1016/S0168-9274(99)00052-5

6

N. Calvo, J.I. Díaz, J. Durany, E. Schiavi, C. Vázquez, "On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics", SIAM J. of App. Math., 63, 683-707, 2002.

7

N. Calvo, J. Durany, C. Vázquez, "Numerical approach of thermomechanical coupled problems with moving boundaries in theoretical glaciology", Math. Mod. and Meth. in App. Sci., 32(2) 229-248, 2002. doi:10.1142/S0218202502001623

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