The irregular planform morphology of rivers leads to formation of hyporheic zones along its banks. This study investigates how hyporheic exchange in stream banks, whose planform is idealized as sinusoidal, is affected by net gains from and net losses of water to the adjacent aquifer. These effects are studied via numerical modeling of groundwater flow adjacent to sinuous channels across a broad range of sinuosity and gain/loss magnitude. Hyporheic zone areas and fluxes both decrease exponentially with increasing magnitude of net gain or loss relative to the case where the stream has no net flux of water (neutral). Residence time through the hyporheic zone also decreases with gain/loss magnitude. The hyporheic zones become constrained near the apex of bends, indicating that these areas could be hot spots for mixing and biogeochemical processing. Hyporheic zones in channels with smaller sinuosity are more prone to hyporheic flux and area reduction while very sinuous channels are able to maintain a hyporheic zone even under largely losing or gaining conditions. Equations fitted to the suite of simulation results allow for prediction of hyporheic flux, area, and residence time on the basis of aquifer hydraulic conductivity, channel sinuosity, and the ratio of along-valley and across-valley mean head gradients.
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 Although the hyporheic zone could be considered a “hot spot” [McClain et al., 2003] because of the steep ecological and biogeochemical gradients that are present, gradients would be steeper if deep groundwater interacts more with shallow hyporheic water. This occurs in rivers gaining groundwater while actively undergoing hyporheic exchange. Hyporheic exchange has been observed in base flow–influenced rivers such as low-order mountain streams [Harvey and Bencala, 1993] but also occurs in streams losing net water, such as in semiarid areas [Dent et al., 2007; Harvey et al., 2003]. In some cases, rivers switch between gaining and losing across the seasons [Wroblicky et al., 1998] or during floods. Ambient flow of deeper groundwater away from or toward the river influences and further complicates the physics and biogeochemistry of both the hyporheic zone water and surface water. Wroblicky et al.  showed how lateral hyporheic zone areas and vertical fluxes through the streambed of the Rio Calaveras in New Mexico change with the seasons; the river is gaining water during some parts of the year and loses water in other times. The water table and the irregular planform morphology of the Rio Calaveras both control where and when hyporheic zones occur. Kasahara and Hill [2007a] also showed how a remeandered channel induced lateral hyporheic exchange under both base flow and losing conditions. Counterintuitively, not just gains but also losses of net water lead to reduction of hyporheic zones induced by current–bed form interactions [Cardenas and Wilson, 2006, 2007a]. Boano et al.  theoretically showed the reduction of vertical hyporheic zone fluxes and depths across a transverse section of a channel. In this case, they assumed that there is no horizontal hyporheic exchange along the banks. Cardenas and Wilson [2006, 2007d] also did not investigate the dynamics of horizontal hyporheic exchange under gaining and losing conditions. However, horizontal channel sinuosity–driven hyporheic exchange [Boano et al., 2006; Cardenas, 2008b; Peterson and Sickbert, 2006] is likely to be just as prevalent as bed form–driven exchange, particularly in sandy higher-order streams.
 In this paper, results of numerical simulations of sinuosity-driven lateral hyporheic exchange under both gaining and losing river conditions are presented with the goal of deepening our understanding of the dynamics of these processes which ultimately leads to prediction. Thus, the paper concludes with equations for predicting hyporheic zone flux, area and residence time under the aforementioned scenarios.
2. Methods and Metrics
 Steady state, two-dimensional (2-D), horizontal groundwater flow through an aquifer bounded by a sinusoidal river on one side is simulated via finite element modeling of the Laplace equation (Figure 1) with COMSOL Multiphysics:
where h is the hydraulic head. The groundwater flow velocity is defined by Darcy's law:
where qx and qy are Darcy fluxes in the x and y directions, and K is hydraulic conductivity. The river is assumed to be sinusoidal with wavelength λ and amplitude α. The side bounded by the sinuous river is a constant-head boundary where head varies linearly along the channel:
where σ is the arc length along the boundary, S is sinuosity (length along channel divided by valley length), and Jx is the mean head gradient along the valley in the downstream direction. The left and right sides are periodic boundaries with a prescribed head change between them corresponding to the mean valley gradient:
Implied above is that the mean valley slope controls the mean water table configuration. In a previous related study [Cardenas, 2009], the top boundary (Figure 1) was considered a no-flow boundary effectively preventing any net losses or gains of groundwater. In this study, gaining and losing conditions are imposed by considering a head gradient along the y direction such that head at the top boundary located a distance equal to λ from the river is
where Jy is the mean head gradient across the valley. When the horizontal top boundary is a wavelength away from the stream, it no longer affects the near-stream solution for all cases of α considered (the largest α/λ is 0.375) making the solution equivalent to one for a semi-infinite domain. When Jy = 0, the river is effectively a neutral river and is neither losing nor gaining net water. This is considered as the base condition. When the right-hand side of equation (5) has a plus sign, the river is gaining; the river is losing when the equation has a minus sign. The degree to which the river is losing or gaining is indicated by the ratio of the mean head gradients in the principal directions perpendicular to each other:
The effects of gaining and losing conditions are investigated by running simulations at multiple S and Jy/x. Jy/x ranges from 0.0 to 4.0 in increments of 0.25.
S is varied from 1.02 to 1.87 by increasing α while keeping λ fixed. The stream is idealized as following a sinusoidal planform morphology. While this is reasonable for streams of small S (the largest we considered is 1.87), many rivers diverge from this simplification especially the more sinuous ones [Ferguson, 1975; Hooke, 2007]. Stream restoration efforts that focus on remeandering do tend to generate channels with sinusoidal planform [Kasahara and Hill, 2007b] thus making our study directly applicable for such scenarios.
 The dynamics of the stream-hyporheic zone-aquifer system is represented by the variation of area, flux, and characteristic residence time of the hyporheic zone for different S and Jy/x. The area of the lateral hyporheic zone is defined by the streamline which confines all streamlines beginning and ending at the channel, similar to the definition used in previous modeling studies [Cardenas and Wilson, 2007b, 2007d]. This area was delineated graphically using plots of streamlines plotted at high density and then by picking the streamline that best met the criterion. The hyporheic zone area is then normalized as follows:
where Ad is the normalized area, A is the area described above, and Ab is the area for the base case where Jy/x = 0.
 The dimensionless hyporheic flux per unit length for each run is defined as follows:
where Q is the dimensionless lateral hyporheic flux per unit length of stream bank, and qn is the Darcy flux normal to the channel. The division by 2 takes into consideration that half of the flux is going in while the other half is back to the river. Since hyporheic exchange occurs on both banks, the final value for hyporheic flux would have to be doubled as equation (8) only considers one side of the channel. For gaining and losing cases, the specific flux along the top boundary is subtracted from Q so that the flux only corresponds to hyporheic flux. This flux is further normalized as follows:
where F is the normalized dimensionless flux and Qb is the dimensionless flux for the base case.
 The bulk residence time of water flowing through the hyporheic zone is
where σλ is the arc length along the channel over one wavelength. The residence time for gaining and losing cases is normalized by the base case by
where Tb is the base case residence time. Ad, F, and Td are therefore constrained between 0.0 and 1.0, with 1.0 indicating base/neutral conditions.
3. Results and Predictive Models
3.1. Hyporheic Zone Area
 A good estimate for Ab is simply the integral of the sine curve, shifted by the amplitude, describing the river planform over one wavelength:
However, the graphically delineated Ab is somewhat larger than AT since the streamline delineating the hyporheic zone is not a straight line connecting two successive apices. The streamline curves away from the river a little bit (Figures 2a and 3a); equation (12) suggests that hyporheic zone area increases linearly with increasing wavelength or amplitude.
 Hyporheic zone area decreases as rivers become more losing with increasing Jy/x (Figure 2). At some point, a large Jy results in a stream that is losing water everywhere preventing any return flow or hyporheic exchange. However, more sinuous rivers are less prone to hyporheic zone area reduction (Figure 3), and it requires more pronounced losing conditions in order to completely diminish the hyporheic zone for a high-sinuosity river. In the cases where S = 1.74, there are still hyporheic flow paths at Jy/x = 4.0 (Figure 3) whereas there is hardly any at Jy/x = 0.6 for S = 1.14 (Figure 2). Hyporheic exchange is confined to near the bend apex and downstream of it under increasingly losing conditions.
 Hyporheic zone area also decreases as rivers become more gaining (Figure 4). A threshold also exists such that a large Jy results in a stream that is gaining water everywhere and no potential hyporheic water is able to leave the river. More sinuous rivers are similarly less prone to hyporheic zone reduction (Figure 5), and hyporheic exchange also becomes focused near the apex under increasingly gaining conditions. However, the bulk of the hyporheic zone is upstream of the apex unlike in the losing conditions where it forms downstream of this.
 The hyporheic zone area decreases exponentially with increasing gain/loss magnitude (Figures 6a and 6b), i.e., Jy/x. However, the exponential reduction is more rapid with small sinuosity channels. At higher sinuosity, it becomes difficult to identify a threshold Jy/x where A = 0 as the reduction becomes asymptotic; the hyporheic flow paths become parallel to the banks at large S and Jy/x (Figures 3 and 5). In order to compare across various S and Jy/x, the same exponential function is fitted to the normalized simulated hyporheic zone areas corresponding to different S cases:
For all values of S, the regression results in R > 0.994 even reaching as high as 0.9998. The scaling term a is considered as a fitting parameter, although it could have been fixed as 1.0, to get better fits. In any case, the regressed values were very close to 1.0.
 A predictive equation necessarily begins with calculating Ab since the areas are normalized versus this (equation (7)). The discrepancy between the areas predicted by equation (12) (AT) and the areas graphically delineated from simulated streamlines for both losing and gaining conditions generally increases with increasing area, and therefore increasing S or λ (Figure 7a). Instead of using equation (12), a model is alternatively fitted to the discrete simulation results that follow the following form:
where c = 185.064, d = 7.634, e = −679143.0 and Ap is now an empirically predicted or estimated hyporheic area. The regression has a R = 0.9995 (Figure 8). The unit for AP is the same as the length unit of λ2. Equation (14) allows prediction of the dimensional area for neutral conditions based on S and λ and it will be used as a basis for predictive equations for dimensionless areas under gaining and losing conditions. Prior to this, the model is tested first by comparing the predicted areas (AP) to the actual simulation results. The equation works well for estimating Ab (Figure 7b).
 Since Ab can now be predicted on the basis of channel planform morphology, any area normalized to Ab can be easily calculated. Fitting equation (13) to discrete simulation results leads to a suite of fitted coefficients a and b; one for each S. Hyporheic zone areas decrease in a similar fashion in both losing and gaining scenarios (Figures 6a and 6b). The scaling coefficient a slightly decreases with S in both scenarios but remains close to 1.0. A single linear function is therefore fitted to the combined losing and gaining simulation results:
where f = −0.07350 and g = 1.09658. The regression has a R = 0.9126 (Figure 9a). The exponent b in equation (13) becomes less negative with increasing S since more sinuous channels exhibit a gentler reduction of hyporheic zone area with increasing Jy/x. A function is also fitted to the resulting exponent b for each S case:
where h = −0.06986 and i = −1.5258. The regression has a R = 0.9999 (Figure 9b).
Equations (7) and (13)–(16) allow for prediction of hyporheic zone areas under neutral, gaining and losing conditions. The procedure is as follows: (1) Calculate the coefficients a and b on the basis of equations (15) and (16); (2) calculate the normalized dimensionless area Ad using equation (13); (3) calculate the dimensional base case area Ab using equation (14); and then, finally, (4) calculate the dimensional hyporheic zone area A using equation (7). The model only requires readily measurable or estimated quantities S, λ and Jy/x.
3.2. Hyporheic Exchange Flux
 Hyporheic flux decreases exponentially compared to neutral conditions as the river loses or gains more net groundwater (Figures 6c and 6d). The reduction in flux is more pronounced in straighter channels while very sinuous channels can still have a fairly significant amount of hyporheic circulation even at large Jy/x.
 A predictive equation for the base case hyporheic flux Qb has been presented previously and is as follows [Cardenas, 2009]:
where the coefficients are j = 1.4496, k = −1.3830, and l = −6.1844e-7. In a similar fashion to how the predictive model for A was developed, prediction of Q for the gaining and losing cases is founded on a priori knowledge of Qb.
 The model also requires representing the reduction in flux with increasing Jy/x for different S scenarios with a functional form. Since the normalized flux F also reduces exponentially with Jy/x (Figures 6c and 6d) for both gaining and losing cases, the following function of similar exponential form is fitted to the discrete simulated F(Jy/x) results:
For all S cases, the regression results in R > 0.990 even reaching as high as 0.9999. Functions are then fitted to the resulting suite of scaling coefficients n and exponents o. The reduction in flux is again similar for both gaining and losing conditions so only a single function is fitted to both the gaining and losing simulation results. The scaling coefficient m decreases linearly with S as follows:
where o = −0.09393 and p = 1.1382. The regression has a R = 0.952 (Figure 9c). The exponent n in equation (18) becomes less negative with increasing S; more sinuous channels exhibit a gentler reduction of hyporheic zone flux with increasing Jy/x. A function is also fitted to the resulting exponent n for each S case:
where q = −0.10862 and r = −1.8957. The regression has a R = 0.9968 (Figure 9d).
Equations (10), (11), and (17)–(20) are used for predicting hyporheic zone fluxes per channel length under neutral, gaining and losing conditions. The procedure is as follows: (1) Calculate the coefficients m and n on the basis of equations (19) and (20); (2) calculate the normalized dimensionless flux F using equation (18); (3) calculate the dimensionless base case flux Qb using equation (17); (4) calculate the dimensionless flux Q using equation (9); and finally, (5) calculate the dimensional flux by multiplying the Q by hydraulic conductivity (see equation (8)) and doubling the resulting number (note that Q is only for one side of the channel, hence it has to be doubled). In addition to parameters need for estimating area, Jx and K, are also needed in order to compute flux.
3.3. Residence Time Through the Hyporheic Zone
 Hyporheic flow paths result in broad residence time distributions which may exhibit power law tails [Cardenas, 2008b; Revelli et al., 2008]. Any process that is strongly time-dependent should therefore include a broader inspection of residence time distributions. However, for some applications, a characteristic residence time would suffice.
 The (mean) residence times also decrease exponentially with increasing Jy/x (Figures 6e and 6f). Using the same approach for flux and area, the following exponential functions are fitted to the each suite of calculated values of Td for each S:
For all S cases, the regression results in R > 0.990 even reaching as high as 0.9995. Functions are then fitted to the resulting suite of scaling coefficients s and exponents t. A single exponential function is fitted to both the gaining and losing simulation results owing to their similarity. The scaling coefficient s decreases linearly with S as follows:
where u = −0.09158 and v = 1.11849. The regression has a R = 0.877 (Figure 9e). A function is also fitted to the resulting exponent t for each S case:
where w = −0.07268 and x = −2.4334. The regression has a R = 0.9994 (Figure 9f).
Equations (10), (11), and (21)–(23) are used for predicting of residence times through the hyporheic zones under neutral, gaining and losing conditions. The procedure is as follows: (1) calculate the coefficients s and t on the basis of equations (22) and (23); (2) calculate the normalized residence time Td using equation (21); (3) calculate the base case residence Tb using equation (10); and, last, (4) calculate T using equation (11).
4. Discussion and Summary
 The mechanics of hyporheic exchange under both gaining and losing conditions has been mostly unexplored. Several studies have suggested that there can be substantial hyporheic exchange under such conditions. The dynamics of bed form–driven vertical fluid and heat exchange has been recently analyzed for such conditions [Cardenas and Wilson, 2007c, 2007d]. A comparison of the dynamics of bed form–driven and sinuosity-driven hyporheic zones follows. In previous studies at the bed form scale, losing and gaining conditions result in the competition between the driving force for hyporheic exchange, indicated by the Reynolds number of the flow over the bed forms, and ambient groundwater flow due to ambient vertical head gradients away from the sediment-water interface. Losing or gaining conditions leads to collapse of the hyporheic zone, with hyporheic zones becoming more extensive when the Reynolds number is large and the zone eventually occupies the entire space it would under neutral conditions. We show here that the hyporheic zones that are driven by channel planform sinuosity exhibit similar behavior. Small net losses or gains of water from the river leads to reduction of hyporheic zones areas and fluxes. Although the locations of the hyporheic zones are different for gaining and losing conditions, their geometry is similar and the magnitudes of decrease in area, flux and residence time are practically the same. This is similar to the case of bed form–driven hyporheic zones where the above metrics are practically the same for gaining and losing conditions with similar magnitudes. Hyporheic areas and fluxes decrease rapidly with initial increase in gain/loss magnitude and then the decrease becomes more gradual and somewhat asymptotic as the gain/loss magnitude increases further.
 There is a key difference between the bed form–driven and sinuosity-driven hyporheic zones under gaining/losing conditions. Hyporheic fluxes through bed forms are less sensitive to relative gain/loss magnitudes; they are reduced minimally. This is because the diminishing hyporheic zones become focused in areas with largest flux and where the pressure gradients are largest (either the trough or crest of the bed form). The rapid reduction of hyporheic zone area coupled to a less pronounced reduction of flux leads to a drastic reduction in residence time [Cardenas and Wilson, 2007d, Figure 6]. This is not the case for the sinuosity-driven cases where the hyporheic zone becomes constrained to the apices furthest away from the valley boundaries. Flux across the river-aquifer boundary is smallest in this area (Figure 10) under both gaining and losing conditions. The hyporheic zone is centered where the flux is 0 (around x = 30 and 70 m). This hinge point may move downstream under losing conditions and upstream when the river is gaining. However, its translation is dependent on the sinuosity of the channel. The center of the hyporheic zone tends to move less near sinuous channels. In fact, at S = 1.74 (Figures 3, 5, 10a, and 10b), it hardly moves at all. When the hyporheic zone area decreases, zones along the river that have larger fluxes across the boundary are excluded (compare the charts in Figure 10 to the corresponding flow fields); these are zones farthest away from the hinge point/apex. The hyporheic zone area reduces at a faster rate than flux and leads to the reduction of residence time. This reduction is more gradual than in the bed form–driven hyporheic zones.
 Although some studies have looked at how hyporheic zones are modified by large floods that also induce geomorphic adjustments in the channel [Wondzell and Swanson, 1999], to my knowledge, there has been not a lot of field characterization of dynamic hyporheic zones at the reach scale. Wroblicky et al.'s  study was the rare exception where they showed that hyporheic zones are persistent along tight bends under both gaining and losing conditions [see Wroblicky et al., 1998, Figures 3 and 4]. Recently, Kasahara and Hill [2007b] mapped lateral hyporheic zones along a restored channel in an urbanizing area. Coincidentally, their study site had a channel that was close to sinusoidal (it was engineered) with an amplitude of about 15 m and a wavelength of about 35 m [Kasahara and Hill, 2007b, Figure 6]; this is very similar to one of the scenarios we considered (Figures 3 and 5). Our simulated flow fields are similar to what they observed during gaining and losing conditions (compare Figures 3 and 5 in this manuscript to Figure 4 of Kasahara and Hill [2007b]). They found that during gaining/base flow conditions, regional groundwater flow is diagonal in relation to the regional slope (down valley and toward the river) and that most of the regional groundwater enters the river near the meander apex. The hyporheic zone becomes diminished and confined to just upstream of bend apices [Kasahara and Hill, 2007b, Figure 4a]; this is exactly what the models predict (Figure 5). During mildly losing conditions, some flow paths track the banks of the river but eventually veer away from the bank, while flow paths closer to the bank appear to define hyporheic zones [Kasahara and Hill, 2007b, Figure 4b]. This is also reflected in our models (Figures 3e and 3f). During strongly losing conditions, they observed that water infiltrated from the river converges toward the center of the point bar, and then apparently flows further away from the channel in a slight down valley direction. Confined between these converging flow paths appear to be hyporheic zones [Kasahara and Hill, 2007b, Figure 4c]. This is very similar to simulation results for strongly losing conditions (Figures 3g and 3h).
 The interaction of regional flow toward or away from sinuous rivers and their hyporheic zones has implications on biogeochemical and ecological processes along the fluvial corridor from the river to the riparian zones. Water with potentially different chemistry can still penetrate into the subsurface even under gaining conditions allowing for natural processes to occur, such as nitrate removal in riparian zones [Hill, 1996]. Moreover, aerobic zones may exist in these hyporheic zones whereas the groundwater along banks of strongly gaining rivers would typically be anaerobic since they are at the end of long flow paths in the subsurface. Duval and Hill  highlighted the role of seepage, i.e., hyporheic exchange, along banks of a meandering channel, especially during low base flow conditions, as a strong driver of biogeochemical activity in riparian zones. These areas receiving seepage are hotpots for aerobic microbial respiration and formation of ammonium. Under dominantly losing or gaining conditions, the meander apices closest to the valley boundary experience the largest flux (Figure 10, see flux at x ∼ 10 and 50 m). These areas may also serve key biogeochemical and ecological functions since reaction rates will be different here because of faster flow and possibly larger supply of reactants (in the losing case). Kasahara and Hill [2007b] also analyzed water samples along the lateral hyporheic zones that they mapped. They found that dynamic hyporheic flow paths that respond to changes in head gradients along the river and across the valley and hillslope lead to changes in hyporheic water chemistry; the size and location of hot spots for dissolved oxygen and nitrate track the flow paths.
 It is clear that more sinuous channels are favorable for formation of hyporheic zones and their hydrologic and biogeochemical importance is increased under gaining/losing conditions. River restoration typically involves remeandering of channels [Wohl et al., 2005]; this study's results show that another benefit of this is that hyporheic zones will form even under base flow and losing conditions. This modeling study shows that hyporheic zones that are driven by channel sinuosity form even under gaining and losing conditions. Hyporheic zones interact with ambient groundwater in a complex but predictable way. Equations for predicting hyporheic zone flux, area, and residence time were developed. These metrics may be readily correlated to parameters in in-stream solute transport models such as the transient storage model [Bencala and Walters, 1983] and its more recent varieties. Future work should be directed toward field or experimental verification of the cases modeled here and toward characterizing and quantifying the biogeochemical function of these hyporheic zones in gaining and losing rivers.
 Computing resources were provided by the University of Texas at Austin Jackson School of Geosciences. This manuscript benefited from comments from two anonymous reviewers and careful review by Fulvio Boano.