On the equations of the large-scale ocean
TL;DR: In this article, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e., primitive equations (the PEs), the PEV2s, and the Boussinesq equations.
Abstract: As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV2s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV2s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the Earth's climate.
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TL;DR: In this article, a computational scheme suitable for numerical weather prediction and climate modelling over a wide range of length scales is described, which is non-hydrostatic and fully compressible, and shallow atmosphere approximations are not made.
Abstract: A computational scheme suitable for numerical weather prediction and climate modelling over a wide range of length scales is described. Its formulation is non-hydrostatic and fully compressible, and shallow atmosphere approximations are not made. Semi-implicit, semi-Lagrangian time-integration methods are used. The scheme forms the dynamical core of the unified model used at the Met Office for all its operational numerical weather prediction and in its climate studies. © Crown copyright, 2005. Royal Meteorological Society
1,000 citations
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TL;DR: In this paper, the authors prove the global existence and uniqueness of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics, and show that strong solutions can be found in the real world as well.
Abstract: In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.
269 citations
TL;DR: In this paper, a review of the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean is presented.
Abstract: This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions 2 and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 1990s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction ( Section 1.4 ).
245 citations
Book•
01 Jan 2002
TL;DR: The Navier-Stokes System in Domians with Cylindrical Outlets to infinity (Konstantin Pileckas) and periodic homogenization problems in Incompressible Fluid Equations (Carlos Conca and M.R. Vanninathan).
Abstract: Preface On the Contact Topology and Geometry of Ideal Fluids (Robert Christ) Shock Reflection in Gas Dynamics (Denis Serre) The Mathematical Theory of the Incompressible Limit in Fluid Dynamics (Steven Schochet) Local Regularity Theory of Navier-Stokes Equations (Gregory Seregin) On the Influence of the Earth's Rotation on Geophysical Flows (Isabelle Gallagher and Laure Saint-Raymond) The Foundations of Oceanic Dynamics and Climate Modelling (George R. Sell) Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities (Josef Malek and K.R. Rajagopal) Navier-Stokes System in Domians with Cylindrical Outlets to Infinity (Konstantin Pileckas) Periodic Homogenization Problems in Incompressible Fluid Equations (Carlos Conca and M. Vanninathan) Author Index Subject Index
206 citations
TL;DR: In this article, the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography was proved.
Abstract: We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain (Cao and Titi 2007 Ann. Math. 166 245–67).
192 citations
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
2,597 citations
Book•
01 Jan 1965TL;DR: In this article, the authors present an introduction to the theory of higher-order elliptic boundary value problems, and a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher order elliptic edge value problems.
Abstract: This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality.
1,568 citations
Book•
01 Jan 1987
TL;DR: The second edition of the Navier-Stokes Equations as mentioned in this paper provides an overview of its application in a variety of problems, including the existence, uniqueness, and regularity of solutions.
Abstract: Preface to the second edition Introduction Part I. Questions Related to the Existence, Uniqueness and Regularity of Solutions: 1. Representation of a Flow: the Navier-Stokes Equations 2. Functional Setting of the Equations 3. Existence and Uniqueness Theorems (Mostly Classical Results) 4. New a priori Estimates and Applications 5. Regularity and Fractional Dimension 6. Successive Regularity and Compatibility Conditions at t=0 (Bounded Case) 7. Analyticity in Time 8. Lagrangian Representation of the Flow Part II. Questions Related to Stationary Solutions and Functional Invariant Sets (Attractors): 9. The Couette-Taylor Experiment 10. Stationary Solutions of the Navier-Stokes Equations 11. The Squeezing Property 12. Hausdorff Dimension of an Attractor Part III. Questions Related to the Numerical Approximation: 13. Finite Time Approximation 14. Long Time Approximation of the Navier-Stokes Equations Appendix. Inertial Manifolds and Navier-Stokes Equations Comments and Bibliography Comments and Bibliography Update for the Second Edition References.
1,342 citations
TL;DR: In this paper, a model for studying ocean circulation problems taking into account the complicated outline and bottom topography of the World Ocean is presented, and the model is designed to be as consistent as possible with the continuous equations with respect to energy.
1,048 citations