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Mathematical theory for the coupled atmosphere-ocean models (CAO III). (English) Zbl 0866.76025

(Authors’ summary.) In order to understand the mechanism of atmospheric and oceanic turbulence and the climate, it is necessary to simulate properly the interaction mechanism of the atmosphere and the ocean. As a first step for us towards this long range project, our aim in this article and two companions articles [the authors, Comput. Mech. Adv. 1, No. 1, 5-54 (1993; Zbl 0805.76011); ibid. 55-119 (1993; Zbl 0805.76052)](also referred to as (CAO I) and (CAO) II)) is to derive some mathematical models for the coupling of the ocean and the atmosphere and to study them from the numerical viewpoint as well as from the mathematical viewpoint (existence and uniqueness of solutions, attractors \(\dots\)). The present article is devoted to the mathematical analysis of the Coupled Atmosphere and Ocean Models (CAO models). The companion articles (CAO I) and (CAO II) contain respectively the presentation of the models and their numerical study.
The models introduced in (CAO I) are based on the primitive equations (PEs) and on the primitive equations with vertical viscosity (\(PEV^2s\)) of the atmosphere-only and the ocean-only systems. The resulting models are called the primitive equations of the coupled atmosphere-ocean (PEs of CAO) and the primitive equations with vertical viscosity of the coupled atmosphere ocean (\(PEV^2s\) of CAO). Both models are studied here. In order to highlight the main mathematical aspects of the PEs and the \(PEV^2s\) of CAO, two simplified models, the simplified PEs and the simplified \(PEV^2s\), are also considered in this article.
First of all, we present in this article the variational formulations of the equations, and obtain the existence and time-analyticity of solutions to the equations. We then establish some physically relevant estimates for the dimension of the attractors of the problems.

MSC:

76D99 Incompressible viscous fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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