Qualitative Theory Of The Dynamical Equations Of Atmospheric And Oceanic Motion And Its Applications

On the basis of the known results, the global qualitative characteristics of the complete dynamical equations of atmospheric (dry air and moist air) and oceanic motion and its applications are studied by new theory and new methods of infinite dimensional dynamical system. The major conclusions of this study may be summarized as:1. The existence of global attractor of the complete dynamical equations of atmospheric motion is proved in the infinite dimensional Hilbert space. That is to say, no matter how initial values are, any state of the system is bound to evolve to the global attractor(we call it the atmosphere attractor or the climate attrctor). Based on the above result, we reveal that the asymptotic behavior of solutions of nonlinear atmospheric system with forcing and dissipation shows itself on the structure of attractor, and that the system has nonlinear adjustment process to the external forcing and the characteristic of decay of effect of initial field. The existence of global attractor and its finite dimension show that the long range behavior of solutions of complete partial differential equations of atmospheric motion can be accurately described by a set of finite ordinary differential equations. On the basis of the above results, we may establish attractor viewpoint of process of the long-range atmosphere and climate.2. The results about the asymptotic behavior of solutions of the dynamical equations of atmospheric motion that are obtained under the stationary external forcing are extended to the case of non-stationary external forcing. Therefore, the long-range behavior of the dynamical equations of the atmosphere with non-stationary external forcing can be described by a finite ordinary differential equations with the same non-stationary external forcing.3. The asymptotic behavior of solutions of the dynamical equations of the moist atmospheric motion is studied in the infinite dimensional Hilbert space. A reasonable description of the latent heat is given by first introducing the discriminant functionδ. Based on it and after introducing suitable Hilbert space, the existence of global attractor of the moist atmosphere is obtained. So, the global qualitative characteristics of the dynamical equations of the atmosphere that are got in the case of dry air are extended to the case of moist air.4. The global qualitative characteristics of the complete dynamical equations of oceanic motion are studied. The difficulty that is caused to the theoretical analysis by the complete equation of liquid state without the explicit formula is surmounted by introducing the suitable system of independent variable. On the basis of the above work, the existence of global attractor of the complete dynamical equations of oceanic motion is proved. Thus, the conclusions on the global qualitative characteristics got in the dynamical equations of atmospheric motion are extended to the complete dynamical equations of oceanic motion.5. On the basis of known understanding about time boundary layer, three classes of time boundary layer in the forced-dissipative nonlinear system are first raised. The inside of first class time boundary layer is quick adjustment process to attractor, outside it, the process is the evolution on attractor. Under the circumstance, the system should be regarded as a forced-dissipative nonlinear system with stationary external forcing. The outside of second time boundary layer is slower evolution of macroscopical field as external parameter, and in the circumstance, the system should be regarded as a forced-dissipative nonlinear system with non-stationary external forcing. Besides, there exists the third class time boundary layer, i.e. inner time boundary layer, and inside it, the system can be treated as a adiabatic non-dissipative system.6. It is first raised that splitting algorithm must obeys the principle that the properties of operator cannot be changed. Based on the principle, the splitting equations can basically keep the overall properties of original equations so that both computation time-saving and good computational results would be got.7. On the basis of known idea that the difference scheme is designed by using the properties of operator, the concept of the computational quasi-stability is first raised , and is used for studying the nonlinear computational stability of nonlinear evolution equations.8. The mechanism of atmospheric multiple equilibria is studied. It is proved that the stationary solution for the equations of stationary atmospheric motion is either unique or non-existential and in any case there does not exist multi-solution if anyone of nonlinearity, dissipation and external forcing is omitted. From this we conclude that the joint action of nonlinearity, dissipation and external forcing is source of the atmospheric multiple equilibria. Besides, it is discussed that the asymptotic behavior of solutions of the forced dissipative nonlinear system is essentially different from that of the adiabatic non-dissipative system, the adiabatic dissipative system, the forced non-dissipative system and the forced dissipative linear system. Based on the above results, we point out that external forcing, dissipation and nonlinearity are the fundamental factors that must be considered in long-range process. Therefore, a simplified dynamical model that describes the long-range behavior must be a forced dissipative nonlinear system, neither a adiabatic non-dissipative system nor a linear system.