Modélisation mathématique et simulation numérique des phénomènes dynamiques et thermiques apparaissant dans un glacier

In this work, two problems linked to glacier modeling are investigated. We propose an optimisation method for studying the flow of the ice and we present a numerical study about glacier thermal phenomena. In the first chapter of this thesis, we expose the models of these two problems. On one hand, we note that the boundary conditions on the bedrock are misunderstood, which explains it is difficult to obtain an accurate simulation of the motion of the ice. Also we establish a mathematical model where the bedrock boundary conditions depend on a control parameter. The aim of this study is to minimize a cost functional describing the difference between the computed velocity at the surface and the measure done. We study the cost functional with respect to the control parameter and we detail an optimisation method to solve the optimal control problem. On the other hand, we introduce two thermodynamical model governing the temperature and the water content field. The models correspond to a Stefan problem for the temperature and a convection-diffusion equation for the water content. The second chapter deals with the numerical resolution of the optimisation problem. First, a Finite Element Method (FEM) is described to solve the partial differential equations. Then, the algorithms used for the optimal control problem are detailed. Finally, this techniques are applied on two glaciers : Griesgletcher for 2D and Storglaciaren for 3D. The third chapter deals with the numerical resolution of the temperature and the water content models. A FEM is used for each problem. Concerning the temperature problem, the Stefan problem is numerically solved and the results allow to detect a free surface between the temperated ice and the cold ice. The water content field is also simulated. Numerical results are discussed on the Storglaciaren.