In mountain rivers, the prevailing but coarse conceptual model for hyporheic exchange across a river step predicts downwelling flow paths upstream of the step and upwelling flow paths downstream of the step [see Kasahara and Wondzell, 2003, Figure 2; Gooseff et al., 2006, Figures 2 and 3; Harvey and Bencala, 1993, Figure 10; Tonina and Buffington, 2009, Figure 6a; Hester and Doyle, 2008, Figure 2]. This prevailing model, referred to as the coarse conceptual model, has focused on the control exerted by the large elevation and associated pressure gradient across the step. The coarse conceptual model has not considered how hydraulic features associated with a step such as a plunging cascade and hydraulic jump influence the local flow paths and potentially disrupt the flow path directions. In a companion paper [Endreny et al., 2011], we used flume and hydrodynamic model experiments to show how the plunging cascade and hydraulic jump downstream of the step significantly redirect hyporheic flow paths so that they deviate from the coarse conceptual model. We provide a refined conceptual model for hyporheic exchange across a river step in conditions of rapidly varied flow to resolve near-bed flow paths. Both conceptual models agree upstream of the step, but the refined model predicts downwelling downstream of the step, where the river plunges into the pool and beneath waves, and predicts upstream-directed upwelling along the longitudinal extent of the hydraulic jump. This paper examines how the hydrostatic model used in several of the above studies is capable of representing the refinements we have added to the conceptual model for hyporheic exchange at river steps.
1.1. Goal of Study
 The goal of this study is to examine how a standard hydrostatic groundwater model represents the hyporheic exchange flow paths across a step with rapidly varied flow including a hydraulic jump. Our first question is how well zones of downwelling and upstream-directed upwelling identified with the flume and hydrodynamic model are simulated by the hydrostatic model. Our second question is how sensitive these flow path predictions are to hydrostatic model parameterization of the water surface profile, step geology (e.g., shape and permeability), and riverbed topography. This research has the potential to identify and evaluate the significance of predictive errors in using a hydrostatic model when simulating hyporheic exchange at river steps with rapidly varied flow.
1.2. Models of Hyporheic Flow Paths and Hydraulics at a River Step
 The Toth  conceptual model of flow patterns within an undulating hillslope section illustrates how hydrostatic pressure can drive local upslope-directed flow paths nested within a deeper set of uniform downslope-directed intermediate and regional flow paths. Unfortunately, the Toth conceptual model cannot simply be rotated to characterize flow patterns beneath the longitudinal section of a river step because the river boundary delivers hydrodynamic as well as hydrostatic pressures. Hyporheic flow paths most influenced by the step are relatively short and shallow local flow paths nearest the riverbed. The deeper intermediate flow paths are generally considered to move in a downriver direction beneath the riverbed. The downriver pressure gradient is the physical driver governing the advective component of hyporheic exchange across a river step [Harvey and Bencala, 1993; Tonina and Buffington, 2009]. Other components of hyporheic exchange that are potentially active across the step include diffusion, momentum, and sediment turnover [Elliot and Brooks, 1997; Packman and Bencala, 2000]. We isolate and focus on the advective component of exchange in this paper but recognize that these other exchange components (e.g., turbulent momentum transfer) are potentially sensitive to river hydraulics and could respond to cascade and hydraulic jump dynamics. In the advective component of exchange, the pressure gradient is the driving force. The pressure terms are represented in length units by velocity head, pressure head, and elevation head, which combine to equal hydraulic head.
 The refined conceptual model for hyporheic flow paths around a river step considers the river with rapidly varied flow. The river's rapidly varied flow may contain a nappe and jet passing over the step and farther downstream may contain a hydraulic jump with rollers and a wave (Figure 1a). A single sketch of rapidly varied flow creates a steady state representation of the dynamic river flow, which can contain aerated rollers and undulating waves [Khatsuria, 2004]. In the refined conceptual model (Figure 1a), the jet region contains supercritical flow (Froude number, Fr1 > 1) with a flow depth y1 and relatively high velocity. Immediately downstream of the jump is subcritical flow (Fr2 < 1) with a flow depth y2 and relatively low velocity. The refined conceptual model focuses on hyporheic flow paths downstream of the step. There is a small zone of upwelling along the face of the step as a result of surface water vortices and lower pressure head. There is downwelling beneath the nappe and wave (Figure 1a, white star) as well as upstream-directed upwelling beneath the longitudinal length of the hydraulic jump (Figure 1a, white rectangle). The upstream-directed subsurface flow paths and downstream-directed river flow connect to form a vertical eddy (Figure 1a, dashed oval). A simplified representation of this conceptual model removes the nappe and curvature from the water surface profile (Figure 1b). This simplified sketch represents output from a commonly used water surface profile model (Hydrologic Engineering Center's River Analysis System (HEC-RAS)) used later in this research. The uniform intermediate flow paths oriented as downstream-directed upwelling are beneath the heterogeneous local hyporheic flow paths.
 The coarse conceptual model (Figure 1c) illustrates a rapid drop across the step but no hydraulic jump. We leave the nappe out of this illustration because the coarse conceptual model did not consistently include the nappe and never explicitly modeled or discussed its effect on flow paths. The water surface curvature over the step is considered convex, and the curvature into the pool is considered concave [Harvey and Wagner, 2000]. Earlier studies of exchange across steps focused on the change in river water elevation across the step [Gooseff et al., 2006; Harvey and Bencala, 1993; Hester and Doyle, 2008; Kasahara and Wondzell, 2003]. We identified two potential alternatives in the literature to this coarse conceptual model, but the ideas were not well developed. In a review paper, Buffington and Tonina  hypothesized a local flow path of upstream-directed flux within a step-pool sequence (see their Figure 3b) with a sketch but did not discuss drivers of this vector or explain why it was absent in their sketch of exchange in cascade morphology. In a companion paper, Tonina and Buffington  showed model output for a synthetic riffle-pool system with a hydraulic jump. In the porous matrix beneath this jump, they included vectors of upstream-directed upwelling but did not analyze or explain the flow paths. The research community has been developing an alternative to the coarse conceptual model, which is limited by its simplistic depiction of local hyporheic flow paths as uniform downstream-directed upwelling (Figure 1c).
 Studies generating flow paths conforming with the coarse conceptual model typically track only the deeper uniform flow paths and/or represent the river step with gradually varied flow. In gradually varied flow the flow depth generally follows the bed slope, changing with downstream distance but remaining either larger or smaller than the normal depth and not crossing the critical depth. While plunging cascades and hydraulic jumps are common to steps [Wilcox and Wohl, 2007], these earlier hyporheic modeling studies may not have identified rapidly varied flow features in their study sites or may have considered these features insignificant to their characterization of hyporheic exchange. Further, by employing hydrostatic models, these earlier studies accepted the tenet of the model as a simplification of the observed phenomenon [Beven, 1993; Hassan, 2004]. The hydrostatic model, by definition, neglects rapid changes in velocity head and stagnation pressure within a plunging cascade.