Infinite-Dimensional Dynamical Systems in Atmospheric and Oceanic Science

The following sections are included:

• Preface
• Contents

There are usually two methods for predicting long-term weather and climate. First, by statistical methods, we can use the current climate, the historical record and numerical analysis to predict the future climate and the possible global climatic changes. Second, because air is compressible, and seawater is incompressible, by dynamical methods, we consider that the future status of climate is a consequence determined by the current status and the physical principles dominating these changes, thus we study equations and models describing the atmospheric and oceanic motions. Regarding weather prediction as an initial-boundary value problem in mathematical physics, we can establish numerical weather prediction models based on mathematical physical equation…

As the basic equations and the primitive equations which describe the motions of the atmosphere and oceans are very complicated, researchers should find some models which are simplier but reflect some physical essences. In 1947, Charney established a quasi-geostrophic model in [32], which plays a crucial role in the first 24-hour numerical weather forecast on the ENIAV computer of Institute of Advanced Study in Princeton…

In 1979, Zeng discussed the well-posedness problem of the atmospheric primitive equations without viscosity with Galerkin method in reference [209], and obtained the existence of weak solutions. In the early 1990s, many mathematicians (such as Lions, Temam and Wang) started the mathematical research of the primitive equations of large-scale atmosphere, oceans and the coupled atmospheric and oceanic primitive equations (see [140,141,142,143,144,203] and the references therein). In [140], through the introduction of viscosity and some technical processes, Lions, Temam and Wang obtained the new formulation of the large-scale dry atmospheric primitive equations suitable for mathematical treatments. In phase space H, the initial-boundary value problem of the new formulation of the primitive equations of the large-scale dry atmosphere is abbreviated as

where U = (v, T), for more detail, see section 3.1. In the pressure coordinate frame, this new formulation of the primitive equations resembles the Navier-Stokes equation of incompressible uid (of course there are some differences, such as, the nonlinear term of Navier-Stokes equation is (u · ∇)u, but the nonlinear term of the new primitive equations expression consists of , where u is the three-dimensional velocity field of Navier-Stokes equation, and v is the horizontal velocity field of the atmosphere). By means of the methods used in studying Navier-Stokes equation in [137], they proved the global existence of weak solutions to the initial boundary value problem of the primitive equations (but they did not study the global well-posedness of the strong solutions). Under the assumption of global existence of strong solutions to the initial boundary value problem of the primitive equations of the atmosphere with vertical viscosity where the strong solutions satisfy the uniform boundedness of H¹ norm in time, they obtained Hausdorff and fractal dimension of the global attractors. With the same method, in references [141,142], they established the mathematical theories respectively about the primitive equations of the oceans and the coupled atmosphere-ocean model introduced in [142], where they mainly proved the global existence of weak solutions, and studied the estimates for the Hausdorff and fractal dimension estimations of the global attractors under the hypothesis of the global existence of strong solutions. In [130,131], in the hypothesis of global existence of strong solutions to the dry and moist atmospheric primitive equations, Li and Chou studied the asymptotic behavior of the corresponding solutions…

The effectiveness of the deterministic weather forecasting is bounded, which is mainly determined by the space scale and time scale of the forecast target. And it's impossible to accurately predict long-term weather and climate changes in a deterministic framework. To predict climate changes more objectively, after 1975, researchers proposed some stochastic climate models, set up Langevin equation describing weather changes and the corresponding Fokker-Planck equation. Efforts in this section can be seen in [74, 99, 129]. After 1980, researchers set up some simplified stochastic climate models, repealing the effect of the random force to the climate system changes, which can be seen in [86, 123, 134, 176, 187]…

In this chapter we recall some stability and instability theory of waves in the atmospheric and oceanic dynamics. We discuss neutral stability, linear stability, formal stability and nonlinear stability of waves in geophysics fluid dynamics. Here several methods are used in stability or instability of waves, such as normal mode approach, energy method, and variation method…

The following sections is included:

• References