Stream channel morphology from grain-scale roughness to large meanders drives hyporheic exchange flow. In practice, it is difficult to model hyporheic flow over the wide spectrum of topographic features typically found in rivers. As a result, many studies only characterize isolated exchange processes at a single spatial scale. In this work, we simulated hyporheic flows induced by a range of geomorphic features including meanders, bars and dunes in sand bed streams. Twenty cases were examined with 5 degrees of river meandering. Each meandering river model was run initially without any small topographic features. Models were run again after superimposing only bars and then only dunes, and then run a final time after including all scales of topographic features. This allowed us to investigate the relative importance and interactions between flows induced by different scales of topography. We found that dunes typically contributed more to hyporheic exchange than bars and meanders. Furthermore, our simulations show that the volume of water exchanged and the distributions of hyporheic residence times resulting from various scales of topographic features are close to, but not linearly additive. These findings can potentially be used to develop scaling laws for hyporheic flow that can be widely applied in streams and rivers.
 Improving our ability to predict pore water flow under streams is important, because it has a significant effect on solute transport and thus water quality and the fate of contaminants [Benner et al., 1995; Fuller and Harvey, 2000; McKnight et al., 2001; Medina et al., 2002]. The region of the subsurface that receives stream water is referred to as the hyporheic zone and the water flowing in and out of this zone is termed hyporheic exchange [Williams and Hynes, 1974; Winter et al., 1998; Harvey and Wagner, 2000; Packman and Bencala, 2000]. The hyporheic zone has high biological activity, which makes it an important region for uptake of nutrients and transformation of contaminants [Triska et al., 1993; Valett et al., 1996; Mulholland et al., 1997; Jones and Mulholland, 2000]. In order to properly design field measurements and estimate exchange rates, it is important to understand the effects of stream topography on both interfacial fluxes and hyporheic residence times of exchanged water.
 The multiple scales vof hyporheic flow that occur in rivers have only recently been described [Wörman et al., 2007; Poole et al., 2006; Cardenas, 2008]. It is now understood that meandering stream channels induce hyporheic exchange, as head gradients over stream channel boundaries and associated interfacial flux increase with sinuosity [Boano et al., 2006; Peterson and Sickbert, 2006; Cardenas, 2009; Revelli et al., 2008; Cardenas, 2009]. Bars and dunes have also been shown to induce hyporheic exchange through pressure variations induced by surface water flowing over these features [Stonedahl et al., 2012; Boano et al., 2007; Elliott and Brooks, 1997; Marzadri et al., 2006; Tonina and Buffington, 2007], but few studies have modeled hyporheic flow beneath both bars and dunes in sinuous channels. Thus, while ripples, dunes, bars, and meanders are all known to induce hyporheic exchange, little is known about the relative contributions of each of these scales of topography to overall residence time distributions and total exchange.
 The primary goal of this study is to determine the role and relative importance of various types of topographic features that drive hyporheic flow, and also to evaluate the degree of nonlinearity associated with interactions across scales of topography that determine flow paths, residence times, and total interfacial flux into the bed. We use the multiscale model of Stonedahl et al.  to assess the relative importance of meander-, bar-, and dune-driven hyporheic flow. Hypothetical rivers with five different single-thread planforms were simulated, and runs were then repeated after progressively adding realistic bar and dune features. The metrics examined were interfacial flux into the bed and hyporheic water residence time distributions.
 This study explores the relationship between topography and hyporheic exchange, which required some simplifying assumptions. It is important to note that this model assumed homogeneous sediment properties throughout the domain and an unchanging topography. In natural systems, hydraulic conductivity varies spatially by orders of magnitude [Landon et al., 2001]. Heterogeneity affects the subsurface flow particularly when the subsurface structure covaries with topographic features [Salehin et al., 2004; Cardenas et al., 2004]. We have not included heterogeneity because surface morphology represents first-order control on exchange and there is a lack of sufficient general knowledge base of near-stream permeability fields [Sawyer and Cardenas, 2009]. It is important to note that we chose to hold the channel slope constant as opposed to the valley slope because the primary goal of this paper was to investigate the effects of interactions across scales of topography, so we wanted the local slope to be the same for each case to facilitate comparisons. Natural streams contain many types of bars. Here we consider bars that are completely submerged, frozen in place, and represent features larger than dunes and smaller than meanders, which scale with stream width and depth [Nikora and Hicks, 1997].
2.1. Multiscale Model of Groundwater-Surface Water Interactions
 We employed a multiscale model to analyze topography-induced exchange between surface and subsurface water in streams. We only briefly review the model here, as the core elements have been published previously [Stonedahl et al., 2010, 2012]. Next we explain in detail how we generated the streambed planform and topography for modeling and follow with a discussion of three-dimensional flow modeling. Briefly, after generating topography we used the generated topography and river discharge to calculate the head at the stream channel boundaries. There are several steps involved in modeling the linked surface and subsurface flows. A conformal mapping is used to transform the stream channel into an orthogonal domain. The spectral components of the scaled topography are then found using Fourier analysis and transformed into a head distribution based on an extension of the two-dimensional bed form-induced exchange model of Elliott and Brooks . A finite difference approach is then applied to calculate subsurface flow using the head distribution along the stream channel as a boundary condition. This approach is described in detail in Stonedahl et al. , where we applied the multiscale model to laboratory flume data sets. We also used an extended version of the model to predict hyporheic exchange and solute transport in a small agricultural stream [Stonedahl et al., 2012].
2.2. Topography Generation and Flow Modeling
 Stream flow, bathymetry, and the hydrogeologic properties of the underlying sediments control the interfacial flux and hyporheic flow. Many different types of stream morphology and sediment structure can occur in nature. We chose to model small shallow low-gradient sand-bed streams because these streams are extremely common and are primary pathways for transmission of nutrient rich agricultural runoff [Peterson et al., 2001]. Considerable nutrient removal can occur in the hyporheic zone of low-order streams [Böhlke et al., 2009; Alexander et al., 2000, 2007]. Sinuosity is of particular interest with regards to these streams as many have been ditched and straightened to increase drainage efficiency, and this may have the unintended consequence of reducing interfacial flux and reaction time of polluted water spent within the hyporheic zone. We selected flow and sediment parameters that closely mirror conditions found in small streams.
2.2.1. Meandering Planform
 We generated a meandering planform morphology by connecting arcs of circles, as illustrated in Figure 1. For the sake of simplicity we created an equiwidth channel. The channel planform was created by placing points a distance of half the width from the centerline in both directions. Angles (ϕ) of 0°, 45°, 90°, 180°, and 270° were used to create channels with five different planforms with sinuosities varying between 1 and 3.33 (shown in Table 1 and Figure 2). These cases are henceforth referred to by the angles used to create the shape.
Table 1. Channel Geometries Used in Simulations
Arc angle: ϕ (degrees)
Wavelength: λ (widths)
Amplitude: A (widths)
 The average radius of curvature of small streams, which is also the radius of our circles, has been reported to be between 1.1 and 4.7 stream widths [Leopold and Langbein, 1966; Rinaldi and Johnson, 1997; Williams, 1986; Knighton, 1998]. We considered small streams to be streams with an average width less than 10 m. As a representative value, we chose to make our radius twice the width of the stream. This value allowed the wavelengths to vary from 0° (straight case) to 8 widths, and the amplitudes from 0 to 3.4 widths. Geomorphologically mature streams are reported to have wavelengths of 6.28–10 times their widths [Hey, 1976; Leopold and Wolman, 1960]. Our wavelengths ranged from 1 to 8, encompassing the range of straight to highly meandering streams.
 The thalweg of a natural stream oscillates from side to side as the stream bends, approaching the bank on the outside edge of each meander. In order to include this asymmetry in our simulated stream, we defined cross sections at each meander peak using piecewise parabolas given in equation (1) and illustrated in Figure 1 (right) at points a and c. The deepest point where the two parabolas meet was taken as a distance of 0.2 from the outer bank following stream geometries described by Leopold , Knighton , and Petts . The stream channel was constructed from the meander peak cross sections using the method described in Stonedahl et al. . This involves transforming the cross sections into their corresponding positions in a straightened domain with a Schwarz-Christoffel conformal mapping applying a piecewise cubic Hermite interpolation, and inverting the transformation to recover the original planform morphology. This procedure produced symmetric cross sections at the inflection points, as shown in the cross section (b) in Figure 1 (right). For the completely straight case, a symmetrical parabola was used with the same maximum depth as the asymmetrical cases, which results in each case having the same cross-sectional area. The mean depth-to-width ratio was selected to be 0.04 to match the geometries reported by Leopold and Maddock  for 18 small streams.
 The symbols used in equation (1) are shown in Figure 1. W refers to the stream width, s to the location of maximum depth, dmax to that depth, and η is the transverse coordinate.
2.2.2. Bars and Dunes
 In natural systems it has been found that bar and dune topography follow power law distributions of the form:
where S is the wave number spectrum, Kx is the wave number, and α is a constant that varies with the system [Hino, 1968; Nikora and Hicks, 1997]. We represented the topography with a two-dimensional Fourier series whose amplitude-to-wavelength ratio followed this relationship. In order to create the 2-D Fourier series, we used wavelengths that varied harmonically between π/70 widths and π/2 widths (a common factor of all of the meander lengths). Wavelengths between π/70 widths and π/16 widths were used to characterize dune-scale topography. For a simple sine curve the ratio between the amplitude and the standard deviation is (shown in Stonedahl et al. ). We used this relationship to create a topography with a ratio of approximately 0.17 between dune amplitude and maximum streamflow depth as suggested by Rubin and McCulloch  and used by Elliott and Brooks . This was done for the dunes by scaling the amplitudes until the standard deviation, σT, was 0.072 widths. After the amplitudes were set, a random phase shift was selected for each term of the Fourier series. This allowed for more natural topography shapes, while still maintaining the −3 power law distribution for dune topography (Figure 3a). Additional steps were required to create a reasonable three-dimensional bathymetry. For the dune topography, the 2-D function was applied in the conformal domain and the amplitudes of each point were scaled relative to stream depth. This caused the dune heights to decrease as they reached the edge of the stream and forced them to be fully submerged.
 We also generated a 2-D Fourier series for bar forms, following the procedure above. For the bars the wavelengths varied between π/15 widths and π/3 widths with a standard deviation of 0.11 widths, which translates to a ratio of approximately 0.25 between the bar amplitude and the maximum stream depth. We chose this value in order to qualitatively match the power spectrum for sand-bed rivers presented in Nikora et al. . The average slope of the dune topography's power spectrum was 0.320, which is 3.81 times larger than the average slope of the bars' power spectrum, 0.084. The final two-dimensional topography and the associated power spectra are shown in Figure 3.
 The 2-D bar function, like that of the dunes, needed to be adapted for the 3-D stream morphology. This was done in the conformal domain with the topographic function being scaled with the stream depth. Bars, however, do not span the river bottom as dunes do, but are instead frequently localized near the edges [Marzadri et al., 2006; Parker, 1976]. Therefore, we confined the bars to the outer 30% of the stream, with the following transverse variation in elevation,
where B(η) is the lateral bar elevation function before superposition with channel topography, and ξ and η are the downstream and transverse coordinates in the transformed domain. B(η) was scaled by the ratio of the depth of the meander topography at point (ξ,η) to the maximum depth. Then the product of the scaled B(η) and the 2-D longitudinal bar function at ξ were superimposed onto the meander topography. The resultant 3-D topography for all of the cases is shown in Figure 4 for the 90° case. Bars vary greatly in scale. We chose to include fully submerged small bars rather than larger morphological features associated with channel forming flows.
2.2.3. Flow and Sediment Parameters
 The mean stream velocity was taken as 0.003 widths/s based on the observations of small streams presented in Leopold and Maddock . Hydraulic conductivity values for clean sand range from 10−2 to 10−5 m/s [Freeze and Cherry, 1979]. Based on this we chose a nondimensionalized hydraulic conductivity value of 5 × 10−5 widths/s, which fits the above range for streams from 0.2 to 200 m.
 The channel slope was fixed at 0.001 because this falls at the intersection of the slopes predicted for pool-riffle and dune-ripple streams [Schumm, 2005; Montgomery and Buffington, 1997]. The valley slope was then calculated from the channel slope for each planform. We chose a porosity of 0.35, which is a common value for sandy sediments. For each of the five planforms shown in Figure 2 (ϕ = 0°, 45°, 90°, 180°, and 270°) we ran simulations with four types of topography: Meanders only, Dunes and Meanders, Bars and Meanders, and Dunes, Bars, and Meanders, as are shown in Figure 4.
2.2.4. Calculation of the Stream Channel Boundary Head Distribution
 The head distributions due to large and small topography were calculated independently. We classified small topography as features for which flow-boundary interactions are expected to be important (bars and dunes) and large topography as features for which only gravitational head gradients are important (meanders). Gravitational head gradients were determined by imposing a constant channel bed slope, SS, in a meandering stream. This corresponds to a constant downstream slope along the stream. The boundary head sloped with the valley slope, SV, such that total drop in head determined by SS and SV independently were the same. Head due to flow over small-scale features like bars and dunes was calculated using the method presented in Stonedahl et al. .
 A Schwarz-Christoffel conformal mapping was used to map topography points in the stream onto a rectangular grid of topography, [Driscoll, 1996]. Specifically, the elevation at each point in the orthogonal grid of small topography, , was multiplied by the scaling factor for the head variation over the boundary, . is a modified representation of the head distribution predicted by the basic pumping model for a single bed form [Elliott and Brooks, 1997],
 where g is the gravitational constant, dmax is the maximum stream depth, σT is the standard deviation of the 2-D small topography function, and is the velocity, which varies proportionally to the depth of the large topography such that the maximum velocity follows the thalweg and becomes zero at the banks. We imposed a constant mean stream velocity in all simulations. Head distributions over bed forms were measured by Fehlman , and the head was observed to be shifted relative to the topography by one-quarter wavelength upstream. Following this finding, the orthogonal grid defined by was fit with a Fourier function and each wavelength in the ξ direction was shifted upstream by a quarter of a wavelength in order to calculate the head distribution within the conformal domain.
 Subsurface flow was simulated using MODFLOW 2000 [Harbaugh et al, 2000] with steady flow. A three-dimensional grid of the subsurface was created, boundary conditions were defined, and then the subsurface head distribution, h1(x,y,z), was calculated using this finite difference model. The lateral span of the stream was twice the meander amplitude plus the stream width. The width of the model domain was seven times this lateral span, and the stream was centered laterally within the domain as illustrated in Figure 5. The distance from the stream to the lateral system boundaries was large enough to have a negligible effect on hyporheic flow. An impermeable bottom boundary was imposed parallel to the stream channel and 0.8 widths beneath the deepest part of the stream. This impermeable bottom boundary did not significantly affect interfacial flux. The upstream and downstream boundaries were set with constant heads to impose the desired head gradient over the system, and the head distribution along the lateral boundaries was set with a constant valley slope, SV, between the upstream and downstream boundary conditions. Note that the longitudinal domain shown in Figure 5 is 2 wavelengths long to avoid up-stream and down-stream edge effects in the center wavelength. The top boundary follows the stream channel bathymetry with the head distribution along the stream bank assigned according to the model calculations in section 2.2.4. The phreatic surface outside of the stream channel was obtained by an intermediate 2-D finite difference calculation to solve the Laplace equation based on the heads imposed on the domain boundary and stream channel boundary.
 The ability to resolve the system was limited by numerical constraints on the grid size of the compiled version of MODFLOW that we used. The 2 GB size limit was not large enough for us to characterize the desired range of topography. We circumvented this constraint by using two grids. The first was a coarse grid of the entire system with 401 × 2753 × 17 nodes and a horizontal spacing of no more than 0.04 stream widths. For the coarse grid the longitudinal boundaries were set to be constants and the lateral boundaries sloped with the valley slope at approximately the same level as the water. Then we reran a finer grid over a subdomain containing the stream using head values from the coarse grid as boundary conditions. The fine grid had dimensions of 401 × 1201 × 17 with a horizontal spacing of no more than 0.02 widths. The vertical spacing on both grids was the same. We used a higher vertical resolution near the surface to better resolve interfacial fluxes and shallow hyporheic exchange flow paths. The thickness of the top layer was 0.0002 widths and the bottom was 0.002 widths, with each intermediate layer 1.5 times thicker than the layer above it. The top layer was characterized as an unconfined layer and the rest were characterized as confined. No external stresses, such as recharge, were imposed in the model.
 Velocity fields were calculated directly from the head distribution. We calculated the specific discharge, qs, using Darcy's law, , where K is the hydraulic conductivity and is the head gradient. We treated both the hydraulic conductivity, K, and the porosity as homogeneous properties. The interfacial flux of water across the stream channel boundary was calculated as , where the unit-normal to the surface, , was obtained from the topography function. Interfacial flux into the subsurface is positive and flux out is negative. Water propagation through the subsurface was calculated using the seepage or pore water velocity, , where θ is the porosity. We determined subsurface flow paths and residence time distributions from these pore water velocity fields by particle tracking using a constant distance step of 0.001 widths, as described in Stonedahl et al. . Each flow path was characterized by a path length, total distance traveled, and a gradient, the difference between the head at the start and finish divided by the path length. It should be noted that although regional groundwater flow effects were not considered, such effects on hyporheic flow are generally very small [e.g., Wörmanet al., 2007].
3.1. Effect of Topographic Features on Hyporheic Exchange
 The average interfacial flux into the subsurface was calculated for each of the 20 cases shown in Table 2. The interfacial flux into the subsurface is defined as the total volume of flow into the subsurface per area of streambed per unit time. There is a trend toward more flux as more scales of topography are included in the model. However, bars added less flux than dunes. Interfacial flux also generally increases with sinuosity, indicating the effect of hyporheic flow through meander bends. The greater fluxes for Dune and Meander models compared to Bar and Meander models for the 45° and 90° cases is the only exception to this trend.
Table 2. Average Directly Modeled Interfacial Flux Values Into the Subsurface (Widths/Second)
Planform (Sinuosity) Topography
3.06 × 10−9
6.83 × 10−9
1.73 × 10−8
8.76 × 10−8
Dunes and Meanders
4.67 × 10−8
4.93 × 10−8
4.74 × 10−8
5.30 × 10−8
1.10 × 10−7
Bars and Meanders
3.04 × 10−9
1.03 × 10−8
1.26 × 10−8
2.10 × 10−8
8.90 × 10−8
Dunes, Bars, and Meanders
4.73 × 10−8
5.25 × 10−8
4.97 × 10−8
5.44 × 10−8
1.11 × 10−7
 Patterns of interfacial flux for all five planforms with four different topographies are shown in Figure 6. Meanders produce smooth periodic patterns of influx. The regions of efflux (dark) and influx (colored) are almost symmetrical. Increased sinuosity results in larger infiltration flux on the downstream side of meanders than at the apex. This is true in all of the topographies, but this effect is most pronounced in the case with meanders only. Dunes typically dominate flux when they are present. The pattern and rate of interfacial flux are most affected by the dunes in channels with low sinuosity. Dunes, without bars, increased the interfacial flux between 25.6% (270° case) and 1268% (45° case). Bars substantially contribute to local patterns of flux, but have significantly less effect on systems with larger sinuosity – note that the flux with bars and meanders closely resemble the meanders-only case for large sinuosity. Without dunes, bars increased the interfacial flux by 1.6% (270° case) to 184.5% (45° case). Bars add little exchange when dunes are also present: as can be seen in the Dunes, Bars, and Meanders case, the interfacial flux only increased by between 1.3% and 6.4% with the inclusion of bars relative to Dunes and Meanders.
 We calculated power spectra for the patterns of exchange shown in Figure 7 as the average of the 1-D power spectra associated with 25 longitudinal transects. These power spectra of interfacial flux demonstrate similar structure to the original topographic spectra, as expected [Nikora and Hicks, 1997]. In Figure 7, the light gray vertical lines delineate the divisions between wave numbers associated with each scale of topography. All four topographies shown (90° case) have a maxima at Kx = 1 (λ = 2π), the periodic channel length of this planform. The Bars and Meanders case (black) and the Dunes, Bars, and Meanders case (green) have higher spectral power in intermediate portion of the spectrum due to the inclusion of bar features. The power spectrum associated with Dunes and Meanders show that dunes also contribute to variation in bar-scale fluxes. This is due to interactions between multiple dunes or between dunes and meanders. Dune fluxes clearly dominate exchange at high frequencies, and cases without dunes decay to white noise at these higher frequencies. The relatively sharp changes at each topographic scale cutoff, denoted by gray vertical lines in Figure 7, demonstrate the close relationship between topography and the flux distribution, as well as the magnitude of the connection. They also emphasize the relative importance of each scale of feature in the resulting spatial patterns. For instance in spite of the relatively low flux induced by bars, and negligible impact of bars on the visual patterns of exchange in the Dunes, Bars, and Meanders case (Figure 6), the spectra (Figure 7) show quantitatively that the bars significantly impact interfacial flux patterns.
 Figure 8 (top) shows that path lengths do not change monotonically with sinuosity, as the 270° distribution falls between the 45° and the 90° case. This is because the distance between the meanders in the 270° case has decreased relative to 45° and 90° case. Figure 8 (bottom) shows the gradient and path length associated with each of 1000 particle paths for each meandering planform. Path start locations were selected on a flux-weighted basis. The largest gradients are associated with the 270° case, and the gradients decrease as the path lengths increase. Residence times are roughly proportional to path lengths and inversely proportional to the gradient. Thus, the large gradients at short path lengths of the 270° case should result in shorter residence times than the other cases. This was in fact observed for the meanders case (Figure 9).
 Cumulative residence time distributions for each case are shown in Figure 9. These results are normalized by the average flux found in the 270° case. It is clear that each scale of topography affects residence times in similar ways for all planform morphologies. The strong interfacial flux associated with dunes produces shorter residence times than larger bar and meander features, leading to a strong return flux between 104 and 106 s. This time scale of dune-induced subsurface flow occurs both with and without bars. The inclusion of bars is less significant, as the residence time distributions for all Dunes and Meanders cases (blue) are very similar to those for all Dunes, Bars, and Meanders cases (green). The shape of the residence time distribution for Bars and Meanders (black) is also similar to the one for more prominent Dunes and Meanders (blue) although the dip associated with bars is very small in the more sinuous cases. The residence time distributions for Bars and Meanders drop off at the time scale associated with flow paths induced by the bar features, which is in between that of the dunes and the meanders.
 The common time scale of exchange associated with meander-induced flow paths is reflected in all four of the sinuous planforms even when smaller features are present. The differences between the meander time scales are reflective of the varying flow path lengths and head gradients associated with each planform. Comparing differences between the five planforms, dunes contribute less to the 270° case distribution than they do in the other four cases, because there is significantly more meander-induced flux into the subsurface in the 270° case as compared to the other four cases (Figure 9). This flux results from the steep gradient between adjacent meanders occurring in this case, as shown in Figures 6 and 8. This behavior is also reflected in the residence time distributions, as the 180° and 270° case morphologies show steeper drops (more return flux) at time scales of 4 × 105 to 107 and 1.1 × 107 to 3.6 × 107, respectively, associated with narrowly focused hyporheic exchange at the sharp meander necks.
3.2. Direct Multiscale Predictions Versus Summations From Isolated Scales of Topography
 In Figure 10, we compare interfacial fluxes and cumulative residence time distributions generated by direct multiscale simulations and summations based on simulations of isolated component topographies. Interfacial flux for the summations and direct multiscale simulations differed by 1.7–35.2%. This shows that the interfacial flux and hyporheic exchange flows associated with each scale of topography are not linearly superimposable, but that reasonable predictions could be made in some cases from isolated effects of different topographies. The summation of fluxes from individual component topographies most closely matched the direct multiscale simulations when one scale of topography dominates the interfacial flux. Progressively larger discrepancies occurred when multiple topographic components produced similar fluxes. For the Dunes and Meanders as well as the Dunes, Bars, and Meanders the best predictions occur at low ϕ because the dunes dominate interfacial fluxes. For the Bars and Meanders, the higher ϕ values yield better predictions because the meanders dominate.
 The discrepancy between the multiscale model simulations and the predictions made from each isolated scale of topography demonstrates some nonlinearity of the flux induced by interactions between multiple scales of topography (Figure 10). This nonlinearity may stem from a variety of factors, since the interactions between scales are complex. The first explanation for nonlinearity stems from the varying impact of flux due to small topographic features (bars and dunes), depending on their placement relative to the large topographic features (meanders). Specifically, the impact of adding small topography is greater in regions of low flux because it creates interfacial flux (both influx and efflux) where there was little before, whereas in areas of high flux it locally increases and decreases the flux with little impact on the average interfacial flux in that region. In mathematical terms, this source of nonlinearity is because the reported value for the system is the absolute value, but for any local flux values Xsmall and Xlarge, |Xsmall|+|Xlarge| > |Xsmall+Xlarge| whenever Xsmall and Xlarge are nonzero numbers with opposite signs. This could account for why most of the combined scales of topography have lower average fluxes than predicted from the separate scales. However, a second source of nonlinearity may result in either increases or decreases in average flux. Imagine a system that consists only of large topography, for any location the local flux value is the component of the velocity vector in the direction of the surface normal. Now, picture the addition of small topographic features onto some portion of the large topography. The change in the surface topography affects the velocity field, the head function, and the normal vector to the surface, all of which influence the flux calculation. Thus, at a local level, the flux values do not correspond to a linear summation of the flux due to each scale of feature. Because of these (and other) issues of nonlinearity, averaging flux values at separate scales discards the spatially explicit information that is can be important for correctly predicting flux for systems composed of multiple scales.
 We compared cumulative residence time distributions generated by the two methods: directly modeled and summation from individual models. For the summation method the residence time distributions were evaluated for isolated topographies and then combined. For the directly modeled method the residence time distribution we obtained directly from multiscale modeling. We then generated flux-weighted residence time distributions for component topographies that could be compared between directly modeled and summed results using:
where τ is time, corresponds to the average flux into the subsurface, is is the normalized residence time distributions for each case with subscripts D for dunes, B for bars, and M for meanders followed by the planform number, ϕ.
 In Figure 11, we compare the directly modeled multiscale Dunes, Bars, and Meanders cumulative residence time results (dashed) to those predicted by summation of isolated scales of topography with equation (5) (solid). There is only a little discrepancy between the directly modeled and summation predictions 45° (black) and 90° (green) cases (Figure 11). This is because hyporheic exchange is dominated by dunes in each of these cases, resulting in similar predictions. In general, the residence times increase with sinuosity because both the flux into the longer meander-induced flow paths and the lengths of these paths increase with sinuosity. This second condition is not true for the 270° case because the high sinuosity reduces the distance between meanders. This reduced the residence times through this portion of the system by both increasing the subsurface velocity and reducing the length of the meander-induced hyporheic flow paths.
4. Discussions and Conclusions
 Here, we simulated hyporheic exchange in five equiwidth meandering planforms having sinuosity ranging from 1 to 3.33, with and without superimposed bars and dunes. The simulations were run using the multiscale model presented in Stonedahl et al. , which predicts subsurface flow from the channel topography, stream and sediment parameters. This model provided unique capability to simulate hyporheic flow in systems with varying sinuosities and local topographies. We used the model to evaluate the effects of sinuosity on hyporheic flow, consider how this varied when smaller topographic forms were present in the system, and predict hyporheic flow associated with isolated classes of topography.
 We found that interfacial flux increased with sinuosity regardless of dune morphology. This trend is consistent with the findings of Boano et al.  and Revelli et al. . We held the channel slope constant in these simulations. The flux increase with sinuosity would be less pronounced when comparing systems with constant valley slopes, in which case the channel slope would decrease with increasing sinuosity. The average interfacial flux due to meandering was significantly smaller than that induced by dunes until the sinuosity reached 3.33. This effect would have been larger had we held the valley slope constant instead of the channel slope constant. This emphasizes the importance of dunes and other bed forms in inducing hyporheic exchange relative to planform channel morphology in lowland rivers.
 While Gomez et al.  investigated sinusoidal stream planforms and found that increasing the sinuosity increased residence times, we did not find a direct relationship between sinuosity and residence times using our planforms. Generally our residence times increased with sinuosity because of the increased head gradients and thus increased flux through long meander-induced flow paths. However, the residence times associated with the 270° case (sinuosity 3.33) were shorter than those associated with the 180° case (sinuosity 1.57) because the distance between the meander peaks decreased. In the high-sinuosity case, shorter inter-meander distances were accompanied by larger head gradients, which further increased pore water velocities and decreased the residence times in these regions as shown in Figure 8.
 We found that dunes more significantly influence both interfacial flux and residence times in streams with small sinuosity. In fact dunes dominated both the interfacial flux and the cumulative residence time distributions for all but the most sinuous case (sinuosity 3.33), and even for this case the dune-induced exchange was significant. The interfacial flux resulting from each scale of topography is clearly delineated on the interfacial flux spectrum, with dunes contributing the largest values. Bar-scale topography did not significantly affect hyporheic flow. This finding is somewhat unexpected considering the bars modeled here were larger than the dunes. We believe the reason the bars were not as significant as dunes can be explained by the differences in their morphology. First the bars did not span the channel while dunes did, so bars occupied less area. Further, bars were restricted to the outer 30% of the stream, where the velocity is substantially lower than the average cross-sectional stream velocity. We also did not include larger-scale channel morphology such as pool-riffle sequences or variations of bed form morphologies with position relative to planform channel morphology. Finally, while the bars were taller than dunes, their wavelengths were proportionally even bigger, which resulted in significantly lower slopes in bars. The average dune slope was 3.8 times larger than the average bar slope, as can be seen clearly in Figure 3a. The lower bar slope produced a significantly lower boundary head gradient, which produced smaller interfacial flux distributions relative to dunes. A much wider range of bar forms can be found in natural systems including larger bars with significantly more flux [Tonina and Buffington, 2007, 2011].
 Most importantly, our results show that the exchanges induced by different scales of topography were close to, but not linearly additive. We compared interfacial flux and residence time distributions simulated for a particular degree of topographic complexity with those obtained by summing results from individual component topographies. The best flux predictions occurred where one scale of topography significantly dominated interfacial flux, as in the low sinuosity cases with dunes. This finding shows that neglecting a scale of topography or modeling each scale of topography as an isolated system can be justified only in systems where single scales of features dominate hyporheic exchange. In preliminary calculations, a series of single-scale models could be used to predict the relative importance of exchange induced by each scale of topography, and thereby to evaluate the need for more complicated multiscale models.
 Multiscale interactions were also apparent in the wide residence time distributions for multiscale topography relative to those associated with individual features. Distinct time scales of exchange were associated with dunes, bars, and meanders, but the residence times that occurred for the multiscale cases were not precisely predictable from the specific residence times and interfacial fluxes associated with each isolated scale. The degree of nonlinearity found in residence times across scales is shown in the discrepancies between the direct and the summation methodologies (Figure 11). Multiscale interactions are also apparent in the interfacial flux spectrum, which contribute to the residence time discrepancies. While the wavelengths of each topographic feature appeared in the flux spectrum, additional flux occurred at wavelengths produced by the combination of smaller scales of topography.
 Many field studies of solute transport in rivers only assess reach-scale behavior and neglect to characterize dune-scale topography, which we have found to dominate exchange in many systems. In fact our results suggest that planform features can often be neglected in low-sinuosity streams, but that small topographic features are essential to characterizing hyporheic exchange. Careful consideration of the system is essential to identify the topographical scales most likely to induce hyporheic exchange. Simulations using isolated scales of topography can be compared to determine the dominant scales of topography and under some circumstances can be summed to make good multiscale predictions. However, multiscale models are essential when multiple scales of topography contribute significantly to the total interfacial flux.
 The authors gratefully acknowledge U.S. National Science Foundation grants NSF EAR-0408733 and NSF EAR-0810270 and the U.S. Geological Survey's HR&D and NAWQA programs for financial support, and thank Northwestern's Quest HPCC for providing computational resources. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.