The 3D Primitive Equations in the absence of viscosity: boundary conditions and well-posedness in the linearized case

Antoine Rousseau 1 Roger Temam 2 Joe Tribbia 3
1 MOISE - Modelling, Observations, Identification for Environmental Sciences
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
3 CGD - Climate and Global Dynamics Division [Boulder]
NCAR - National Center for Atmospheric Research [Boulder]
Abstract : In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be \textit{nonlocal}. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.
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Antoine Rousseau, Roger Temam, Joe Tribbia. The 3D Primitive Equations in the absence of viscosity: boundary conditions and well-posedness in the linearized case. Journal de Mathématiques Pures et Appliquées, Elsevier, 2008, 89 (3), pp.297-319. ⟨10.1016/j.matpur.2007.12.001⟩. ⟨inria-00179961⟩

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