Conservation equations governing hillslope responses: Exploring the physical basis of water balance

A unifying approach for the derivation of watershed-scale conservation equations governing hydrologic responses has recently been introduced. The approach is based on space-time averaging of the corresponding point-scale conservation equations over an averaging region called the representative elementary watershed (REW). The conservation equations are supplemented by constitutive relationships needed for the closure of various mass and momentum exchange terms. In this paper, we present a summary of the resulting equations for a somewhat simpler problem and show how these equations can be employed to model the long-term water balance of a single, hypothetical REW. The governing equations are formulated in terms of an average saturation and a flow velocity of the unsaturated zone soils and the average thickness of the saturated zone soils. The equations are coupled ordinary differential equations and are nonlinear. Their solution is carried out simultaneously for a variety of combinations of watershed geometry, soil type, and atmospheric forcing. We show how a similarity analysis of the governing equations can lead to a general classification of REWs in terms of meaningful dimensionless similarity variables. In addition, we investigate the long-term water balance and show that the governing equations are able to provide a realistic picture of the impact of changing climate, soil, and topographic controls on long-term water balance.