Theoretical and numerical studies of the primitive equations of the ocean without viscosity

Abstract : This work is dedicated to the Primitive Equations (PEs) of the ocean, both from the theoretical and numerical viewpoints. The PEs are fundamental equations of geophysical fluid dynamics, based on the hydrostatic and Boussinesq approximations. Here we consider them without viscosity, in a bounded domain, which actually makes the problem ill-posed with any set of local boundary conditions, as shown in the introduction. In the first part (chapters 1 to 4), we study a slight modification of the hydrostatic equation, adding a friction term of size delta, which is a small parameter. We establish existence, uniqueness and regularity results, and study the asymptotic behaviour of the solutions as delta goes to 0. The numerical simulations evidence some boundary layers and reflexion phenomena at the boundary of the domain. We then confirm the numerical observations with a rigorous proof, obtained with the help of the corrector theory. In the second part (chapters 5 and 6), we go back to the original hydrostatic formulation of the PEs, and propose a set of transparent boundary conditions for the linearized equations. We prove the well-posedness of the corresponding boundary value problem, and perform numerical simulations in which we implement the boundary conditions that we introduced, both in the linear and nonlinear cases. As expected, the boundary layers and reflexion phenomena encountered before disappear.
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Contributor : Antoine Rousseau <>
Submitted on : Wednesday, June 15, 2005 - 9:47:30 PM
Last modification on : Tuesday, May 7, 2019 - 6:30:09 PM
Long-term archiving on : Friday, April 2, 2010 - 10:09:17 PM


  • HAL Id : tel-00009504, version 1


Antoine Rousseau. Theoretical and numerical studies of the primitive equations of the ocean without viscosity. Mathematics [math]. Université Paris Sud - Paris XI, 2005. English. ⟨tel-00009504⟩



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