The fractional advection–dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. While the fractional ADE is analyzed as a stochastic process in the Fourier and Laplace space so far, in this study a fractional ADE for describing solute transport in rivers is derived in detail with a finite difference scheme in the real space. In contrast to the classical ADE, the fractional ADE is expected to be able to provide solutions that resemble the highly skewed and heavy-tailed time–concentration distribution curves of water pollutants observed in rivers.

In order to account for subgrid-scale spatial variability (heterogeneity) of land surface characteristics in regional-scale hydrological-atmospheric models, a land surface parameterization of areally-averaged sensible heat and evapotranspiration fluxes which is based upon areally-averaged hydrological soil water flow and soil heat flow equations, was developed. This land surface parameterization is fully coupled in a two-way interaction with the atmospheric boundary layer and the regional atmospheric model's first layer. The Monin-Obukhov similarity theory which is utilized in modelling the atmospheric boundary layer, and the areallyaveraged hydrological conservation equations are strictly valid only over stationary-heterogeneous areas where the fluctuations of the hydrological and boundary layer state variable values, parameter values, and of boundary conditions have spatially invariant means, spatially invariant higher moments and spatially invariant probability distributions. Therefore, the coupled two-way interactive model was run first for the computation of land surface fluxes over stationary-heterogeneous land patches, each corresponding to a single soil texture-vegetation class. Then, utilizing a mosaic scheme, the land surface fluxes over a regional model grid cell were obtained by numerical probabilistic averaging. The application of the regional model with this land surface parameterization to California during April 1989 has produced promising results.