The appearance of alluvial channel networks is similar to that of trees and their roots what have self-similarity. In the process of a fluvial evolution, alluvial channels are inner-shaping, adjust to achieve a stable equilibrium state, and form some channel patterns that establish balance relationships between the parameters of the local turbulence field of the channel flow and the supplies of the water discharge, sediment transport rate and sediment composition in a river basin. Therefore, the characteristic dimension lΩ of the representing eddy of the turbulence field of the channel flow and the dynamic characteristic length ld of a catchment are supposed to respectively represent the local dynamic factor and the dynamic factor stream along. The geometric dimension of the cross section of a channel is deduced. And then, it is clarified that the self-similarity equation of cross section morphology has similarity relationship conjugated with the self-similarity equation of the widths of a channel network. One of the hydraulic geometry relationship equations reveals that the nominal dimension lΩ and the gravitational acceleration g have the same effects on shaping channel patterns in the fluvial processes. Finally, the governing equations for channels in regime are derived. It is considered that the regime situation can be regarded approximately to that of the bankfull water level of channels, and exponential relationship equations of channels in regime are given systematically. All the results of the study are consistent with the actual statistical data of the channels of Songhua River Basin.

In the process of a fluvial evolution, the water discharge, sediment charge and stream energy expenditure dominant the channel patterns of a river. Given water and sediment, an alluvial channel is self-organizing, adjust to achieve a stable equilibrium state, and form a characteristic channel geometry (channel width, depth and slope). In the equilibrium condition which is also said to be in regime or graded, the flux of the water and sediment from the watershed should be equal to the flux of the downstream channel(s). By studying bed load transportation and stream power conversion on a steady and uniform stream, we suggest two characteristic parameters that are energy conversion length and regime transportation length of sediment. The regime equation and equations of fractal features are set here. All cross-section variables (stream width, depth and velocity) of a regime stream, who theoretically derived under the equilibrium of sediment transportation and the conversion of stream power, are exclusively determined by the two lengths and the water discharge.