Non-Equilibrium Statistical Theory For Evolution Of River Networks

With the great development of complex systems physics, which is a hot field of complexity science concerning open systems far from equilibrium, theoretical researches are sharply increasing in the past decade. Investigation on the evolution of river networks is a typical topic in this filed. In this thesis,distributions of river network parameters, transport of sediment and the relative changes of river channel patterns are discussed to reveal the get understandings to the evolution of river networks.1) Statistical dynamical of early river networks: Based on the random prolongation of river channels, a generalized Langevin equation is proposed by abstracting the determinate and stochastic factors. This equation can be transformed into the Fokker-Planck equation to describe the changes of the distributions of river network parameters, channel lengths or river basin areas. The Fokker-Planck equation can be solved by using the Fourier integral transform to obtain a generalized exponential power distribution, which may be common to the systems bearing both random and determinate properties. As tâ†'0, the generalized exponential powers degenerates the Gaussian distribution, while tâ†'t0it turned into powers distribution. This distribution can be easily extrapolated to infinity, which might correspond to the field observations conducted nowadays. This change demonstrates the time evolution of river networks, and displays the historical picture of a river network from the initial phase to the mature phase:the variation of the distribution from the Gaussian to the power law displays a gradual developing progress of the river network. The distribution of basin areas is obtained by means of Hack’s law. These provide us with new understandings towards river networks.2) Transformation of river patterns:The essence of river morphology transformation is the self organization of the channel structure so as to adapt to the change of the input or discharge. This variation is performed by feedback between the fluid and the channel. The feedback relates not only to the flow regime and the sediment concentration, but also to the geometry of channel and the bank material. It is very difficult to express such a complex coupling relation(by Described by complex coupling structure). However, if the idea of the so-called generalized flow in which the coupling is included can be put to use and with which the theory of Synergetics is combined, one may investigate the transformation of river channel pattern via revealing the changes of the generalized flow.and examine "flow" of the changes to reveal its channel structure phase transitions."Flow" from low-order phase transition to the high-order phase process. In the two points, mutation from minimal to maximal, means that the two input parameters of the river between water and sand suddenly enhanced by coupling, resulted to increase capacity that the water carries sand, and the river channel is possible to alter deposited state into scour state.3) The formation process of the turbulence structure, by view of the Lie algebra, which analysis Naveir-Stokes equation, in order to understand that the sand transport and trajectories in the river and the sandstorm. By calculating, we obtained the solution of formal. Then, we investigate the form of solution by numerical. We obtain amount of streamline structure that include that laminar flow, eddy, and similar to the center point and saddle point in statistical. We examine these structures, which related with the vertical pressure gradient, fluid density and Reynolds et al. To obtain the structure of the formation conditions, and to reveal the sandstorm sand process or between the flow regime and the sand-carrying capacity relationship as is very important in the future.