## 1. Introduction

[2] To model and predict the flow of water through catchments we must choose a scale at which flows are resolved explicitly, and below which their variability is parameterized in some way [*Reggiani et al.*, 1998, 1999; *Beven*, 2006]. These parameterizations of the flow must capture in some parsimonious way how the landscape and climate properties interact to control the flow of water at scales below the resolution of the model. The shallow subsurface lateral flow that occurs in hillslopes when infiltrating water meets a low-permeability layer, such as the bedrock, is one such flow requiring parameterization. It is an important hydrologic flux both as a component of stormflow runoff generation in steep hillslopes with transmissive soils, as well as in lower gradient landscapes where saturated flow through an unconfined aquifer is the primary cause of variable source areas and saturation overland flow [*Dunne and Black*, 1970], and may provide baseflow to a stream, integrating the inputs over many storms.

[3] Parameterizations of this flux have been developed in the past both from physical arguments by upscaling Richards equation to the hillslope scale and approximating the solution in terms of the “characteristic response function” (CRF) [e.g. *Brutsaert*, 1994; *Berne et al.*, 2005; *Akylas and Koussis*, 2007; *Verhoest and Troch*, 2000; *Troch et al.*, 2003, 2004]. The CRF in this context is the discharge response normalized by the total discharge corresponding to the free drainage of a hillslope aquifer (with given initial and boundary conditions) [*Berne et al.*, 2005]. Other parameterizations of this flux have used simpler, empirical approaches that assume a power law relationship between storage and discharge in the hillslope [e.g., *Chapman*, 1999].

[4] While effective in many circumstances, both these methods have their drawbacks. In particular, neither provides an account of the way the climatic variability, boundary conditions and hillslope properties interact to control the subsurface runoff behavior. Climatic variability affects the initial storage in the hillslope, which is known to control the form of the CRF [*Brutsaert*, 1994]. The boundary condition at the base of a hillslope represents the connection of the hillslope to the remainder of the catchment. The role of the boundary condition is generally not explored in the CRF approaches, which usually adopt the mathematically convenient assumption that the saturated thickness at the foot of the slope is fixed at zero depth [*Brutsaert*, 1994; *Ram and Chauhan*, 1987; *Koussis and Lien*, 1982; *Troch et al.*, 2004] (an exception is *Akylas and Koussis* [2007]). In fact the thickness of a free seepage face is known to vary with the configuration of the hillslope [*Chapman*, 2005]. The empirical approach avoids all such considerations by subsuming the controls on hillslope behavior into the parameters and calibrating their values from data [e.g., *Wittenberg*, 1999]. A variety of exponents for these relationships has been observed from less than 1 to 9.1, with typical values around 2.5 [*Chapman*, 1999; *Wittenberg*, 1994; *Wittenberg and Sivapalan*, 1999].

[5] A full accounting of the physical mechanisms that control the variety of coefficients and exponents has not been made. Nor has the link between this and the CRF approach been fully articulated. These relationships are important for developing the closure relations necessary for distributed catchment modeling [*Reggiani et al.*, 1998, 1999; *Beven*, 2006]. In this work we will examine the dynamics underlying subsurface hillslope flow using an idealized model based on the Boussinesq equation and use the concept of dimensionless similarity and regimes [*Robinson and Sivapalan*, 1997] to resolve some of these issues.