On the development and analysis of coupled surface–subsurface models of catchments. Part 3. Analytical solutions and scaling laws

The objective of this three-part work is to formulate and rigorously analyse a number of reduced mathematical models that are nevertheless capable of describing the hydrology at the scale of a river basin (i.e. catchment). Coupled surface and subsurface flows are considered. In this third part, we focus on the development of analytical solutions and scaling laws for a benchmark catchment model that models the river flow (runoff) generated during a single rainfall. We demonstrate that for catchments characterised by a shallow impenetrable bedrock, the shallow-water approximation allows a reduction of the governing formulation to a coupled system of one-dimensional time-dependent equations for the surface and subsurface flows. Asymptotic analysis is used to derive semi-analytical solutions for the model. We provide simple asymptotic scaling laws describing the peak flow formation, and demonstrate its accuracy through a comparison with the two-dimensional model developed in Part 2. These scaling laws can be used as an analytical benchmark for assessing the validity of other physical, conceptual or statistical models of catchments.