Physically based modelling of water age at the hillslope scale: The Boussinesq age equations

Water age distribution is recognized as a key descriptor of hydrological systems, providing information on the natural phenomena taking place in such systems. It is widely used in the theoretical and experimental studies on hillslope and catchments, allowing a synthetic description of the complex dynamics by which the hillslopes store and release water and solutes at the outlet. In this study, we present a physically based framework for the description of the water or solute age within small catchments or hillslopes. The model matches the age equation with the three-dimensional flow equations under transient conditions, employing the Boussinesq approximation. The approach leads to a system of differential equations for the moments of water age, spatially integrated over cross sections normal to mean flow, denoted as Boussinesq-age-equations, that are both simple to use and to solve. The framework overcomes some of the limitations of previous studies, such as of steady state conditions or perfect mixing, merging the simplicity of 1D solutions with the complexity of the physically based models, being suitable for theoretical investigation and experimental data set analysis. Besides introducing the model, we provide some relevant examples for synthetic hillslopes, exploring the effects of the main topographic controls and of the main drivers of water flow such as the recharge.