Normal and anomalous diffusion are ubiquitous in many physical complex systems. Here we define a system of diffusion equations generalized in time and space, using the conservation principles of mass and momentum at channel scale by a master equation. A numerical model for describing the steady one-dimensional advection-dispersion equation for solute transport in streams and channels imposed with point-loading is presented. We find the numerical model parameter as the solution of this system by estimating the transition probability that characterizes the physical phenomenon in the diffusion regime. The results presented (Part I) refer to the channel scale and represent the first part of a research project that has been extended to the basin scale.

At a channel (reach) scale, braided channels are fluvial, geomorphological, complex systems that are characterized by a shift of bars during flood events. In such events water flows are channeled in multiple and mobile channels across a gravel floodplain that remain in unmodified conditions. From a geometrical point of view, braided patterns of the active hydraulic channels are characterized by multicursal nature with structures that are spatially developed by either simple- and multi-scaling behavior. Since current studies do not take into account a general procedure concerning scale measurements, the latter behavior is still not well understood. The aim of our investigation is to analyze directly, through a general procedure, the scaling behavior of hydraulically active channels per transect and per reach analyzed. Our generalized stochastic approach is based on Taylor’s law, and the theory of exponential dispersion distributions. In particular, we make use of a power law, based on the variance and mean of the active channel fluctuations. In this way we demonstrate that the number of such fluctuations with respect to the unicursal behavior of the braided rivers, follows a jump-process of Poisson and compound Poisson–Gamma distributions. Furthermore, a correlation is also provided between the scaling fractal exponents obtained by Taylor’s law and the Hurst exponents.