Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox

We propose a mathematical framework for the general definition and computation of travel time distributions defined by the closure of a catchment control volume, where the input flux is an arbitrary rainfall pattern and the output fluxes are green and blue water flows (namely, evapotranspiration and the hydrologic response embedding runoff production through soil water dynamics). The relevance of the problem is both practical, owing to implications in hydrologic watershed modeling, and conceptual for the linkages and the explanations the theory provides, chiefly concerning the role of geomorphology, climate, soils, and vegetation through soil water dynamics and the treatment of the so called old water paradox. The work focuses in particular on the origins of the conditional and time-variant nature of travel time distributions and on the differences between unit hydrographs and travel time distributions. Both carrier flow and solute matter transport in the control volume are accounted for coherently. The key effect of mixing processes occurring within runoff production is also investigated, in particular by a model that assumes that mobilization of soil water involves randomly sampled particles from the available storage. Travel time distributions are analytically expressed in terms of the major water fluxes driving soil moisture dynamics, irrespectively of the specific model used to compute them. Relevant numerical examples and a set of generalized applications are provided and discussed.