Semianalytical analysis of hyporheic flow induced by alternate bars

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1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 46,, doi: /2009wr008285, 2010 Semianalytical analysis of hyporheic flow induced by alternate bars A. Marzadri, 1 D. Tonina, 2 A. Bellin, 1 G. Vignoli, 3 and M. Tubino 1 Received 10 June 2009; revised 22 December 2009; accepted 24 March 2010; published 24 July [1] We investigate the effects of alternate bar morphology on the hyporheic flow in gravel bed rivers. Our goal is to investigate the relations between residence time distribution of a conservative tracer and the parameters controlling bed form morphology. We assume homogeneous, isotropic or anisotropic hydraulic properties of the streambed sediment and constant flow regime in equilibrium with the bed form, which is considered fixed because its formation timescale is much longer than that of the subsurface flow. Under these assumptions, we solve the in stream and hyporheic flow fields analytically in a three dimensional domain. We approximate the former with the shallow water equations and model the latter as a Darcian flow. The two systems are linked through the hydraulic head distribution, which is predicted at the streambed by the surface model and applied as a boundary condition to the hyporheic flow model. We solve the solute transport equation in the hyporheic zone for a conservative tracer by means of particle tracking. Our model predicts that the mean value and variance of the hyporheic residence time depend on the alternate bar amplitude at equilibrium. This result is found to be applicable also to discharges that are lower (70% in our simulations) than the formative and submerge the bars entirely. Moreover, our analysis shows that 95% of the hyporheic flow is confined in a near bed layer, whose depth is about the width of the channel and shallows from low to steep gradient streams. This causes the hyporheic mean residence time to reach a threshold when the alluvial depth is deeper than the channel width. Our results also show that as the stream slope increases, the streamlines compact near to the streambed, thereby reducing the mean residence time and its variance. Finally, we observe that the hyporheic residence time of pulse injections of passive solutes is lognormally distributed, with the mean value depending in a simple manner on the amplitude of the alternate bars. Citation: Marzadri, A., D. Tonina, A. Bellin, G. Vignoli, and M. Tubino (2010), Semianalytical analysis of hyporheic flow induced by alternate bars, Water Resour. Res., 46,, doi: /2009wr Introduction [2] The hyporheic zone is the saturated volume of sediment surrounding a stream that provides the linkage between rivers and their underlying aquifers [Edwards, 1998; Tonina and Buffington, 2009]. This connection originates from the uneven distribution of near bed pressure caused primarily by the interaction between stream flow and morphology [Elliott and Brooks, 1997a]. This process, known as the pumping mechanism, forces stream waters to enter the sediments in high pressure (downwelling) areas and to exit the sediments in low pressure (upwelling) areas [Savant et al., 1987; Elliott and Brooks, 1997a; Marion et al., 2002; Boano et al., 2007; Tonina and Buffington, 2007, 2009]. This mixing also 1 Department of Civil and Environmental Engineering, University of Trento, Trento, Italy. 2 Center for Ecohydraulics Research, University of Idaho, Boise, Idaho, USA. 3 Centro di Ingegneria e Sviluppo di Modelli per l Ambiente, Bolzano, Italy. Copyright 2010 by the American Geophysical Union /10/2009WR generates physical gradients that form the habitat for a rich ecotone exposed to nutrients and contaminant laden stream and pore waters within the streambed sediment [Sabater and Vila, 1992; Stanford and Ward, 1993]. Furthermore, recent studies show its important role in nutrient export and contaminant attenuation in streams [Bencala and Walters, 1983; Alexander et al., 2000; Mulholland et al., 2008]. Consequently, accurate predictions of nutrient cycling and solute transport along rivers require that the effect of the hyporheic zone be included. [3] In the last decades, several process based models have been proposed to investigate the interaction between surface and subsurface waters (for a review, see Packman and Bencala [2000]). Inanearlywork, Bencala and Walters [1983] proposed a two equation one dimensional model to account for the effects of in stream storage (i.e., dead zones) and the hyporheic zones on solute transport. In their Transient Storage Model (TSM/OTIS, One Dimensional Transport with Inflow and Storage), mass exchange is idealized in terms of a mean exchange rate with a well mixed hyporheic zone of constant volume [Runkel, 1998]. This model requires calibration of the transport parameters with tracer experiments and does not separate surface dead zones from the hyporheic zone 1of14

2 Figure 1. Sketch of the streambed topography with the reference coordinate system located along the centerline of the channel. The coordinates x, y, andz are positive downstream, leftward, and upward, respectively: (a) planar view, (b) cross section view of the bar front, and (c) details of the exchange area and of the flat bed approximation. [Briggs et al., 2009]. Subsequent studies focused on simplified two dimensional hyporheic flows induced by bed forms such as ripples and dunes [Elliott and Brooks, 1997a, 1997b; Marion et al., 2002; Boano et al., 2007; Packman et al., 2004], whose near bed pressure variations can be approximated by a sinusoidal distribution in a two dimensional domain. These studies investigated hyporheic flow in sand bedded streams through analytical solutions and laboratory experiments. On the other hand, studies on hyporheic flow induced by large scale three dimensional bed forms, such as those considered in the present work, were examined through field measurements and numerical models [Gooseff et al., 2003; Kasahara and Wondzell, 2003; Storey et al., 2003; Gooseffetal., 2005; Saenger et al., 2005; Kasahara and Hill, 2006; Wondzell, 2006], one set of laboratory experiments [Tonina and Buffington, 2007] and an analytical solution applied at a large watershed scale [Wörman et al., 2002]. These studies investigated the effects of stream morphology and flow regime on the hyporheic exchange. However, none of them provided an analytical relationship between bed form morphology and hyporheic flow nor linked the hyporheic flow with the parameters controlling the formation of the equilibrium configuration of bed forms. [4] Streambeds of non cohesive material develop different bed patterns characterized by various spatial scales that depend on the streamflow and sediment characteristics [see, e.g., Parker, 1976;Seminara and Tubino, 1989;Montgomery and Buffington, 1997; Seminara, 1998]. Consequently, gravel bed rivers are subject to a continuous morphologic evolution through the interaction between discharge and sediment transport. Common features emerging from this interaction are alternate bars, which develop spontaneously in almost straight reaches of both natural and rectified channels in urbanized areas. Alternate bars lead to a bed morphology composed of a sequence of riffles and pools separated by two consecutive diagonal fronts (see Figure 1) with a pool at the downstream face of each front along the channel banks [Buffington and Tonina, 2009]. Several studies investigated the formation of alternate bars in gravel bed rivers both under steady [e.g., Fredsøe, 1978; Ikeda, 1982; Blondeaux and Seminara, 1985; Colombini et al., 1987; Tubino et al., 1989; Lanzoni and Tubino, 1999] and unsteady [e.g., Tubino, 1991; Welford, 1994] flow conditions. The weakly non linear model of Colombini et al. [1987] shows that with stationary flow conditions alternate bars migrate downstream and grow until they reach an equilibrium amplitude. The resulting equilibrium topography is stable at the applied constant discharge and its associate pressure distribution (hydraulic head) at the streambed can be analytically predicted knowing the sediment representative diameter and density, bed slope, discharge, and channel width. [5] In the present work, we use the resulting hydraulic head distribution as boundary condition of a subsurface flow model [e.g., Gooseff et al., 2005; Wondzell, 2006]. In particular, we propose a process based model that relates the morphodynamic parameters controlling the formation and development of alternate bars directly to those characterizing the hyporheic flow. Our goal is to analyze the hyporheic exchange analytically in gravel bed rivers with complex three dimensional alternate bar topography. Specifically, we study the relation between the hyporheic residence time distribution and the parameters controlling the bed form mor- 2of14

3 phology. We also seek for a morphodynamic index that could characterize the first two hyporheic residence time moments entirely. Furthermore, we investigate the effect of discharge and the consequent change in bed form, streambed slope, and alluvial depth on the hyporheic flow when bed forms are submerged [Tonina and Buffington, 2007]. Although our model strictly applies to formative discharge, we test its applicability at lower stages that are able to keep the bars entirely submerged. 2. Methods 2.1. Morphological Model [6] Let us consider a straight alluvial channel with no erodible banks and assume that surface and subsurface flows can be decoupled with the near bed pressure distribution dictated by streamflow and bed characteristics. This is possible because only a small fraction, typically between 10 to 0.1%, of the water flowing along the stream enters the hyporheic zone [Thibodeaux and Boyle, 1987]. Under these assumptions we model the formation of alternate bars through the weakly nonlinear theory developed by Colombini et al. [1987], which predicts the bed form characteristics and stream water elevation and depth as a function of suitable morphological parameters. The model assumes hydrostatic pressure distribution, which is typical in gradually varying topography under the shallow water hypothesis. This condition has been applied in previous studies investigating hyporheic exchange [Kasahara and Wondzell, 2003; Storey et al., 2003; Zarnetske et al., 2008] and shown to hold for fully submerged alternate bars [Tonina and Buffington, 2007]. [7] According to the Colombini et al. [1987] model, alternate bars form when the aspect ratio b = B/Y 0, where Y 0 is the mean flow depth and 2B is the channel width (Figure 1), exceeds a threshold value b c. This threshold depends on two dimensionless quantities: the Shields parameter, = t 0 /(r S r)gd S, and the relative submergence, d S = D S /Y 0, where D S is the mean grain size of the streambed material, t 0 is the mean shear stress at the bed, g is the gravitational acceleration, while r and r s are the water and sediment densities, respectively. An estimate of b c can be found in Figure 6 of Colombini et al. [1987], where Meyer Peter Müller transport formula has been adopted. Colombini et al. [1987] derive the following expression for the bar amplitude H BM, which is defined as the difference between the maximum and the minimum bed elevations at equilibrium: " 1 2 # C C H BM ¼ Y 0 b 1 þ b2 ; ð1þ C C where b 1 = b 1 (, d S ) and b 2 = b 2 (, d S ) are suitable O(1) functions, which depend on the flow field and sediment characteristics [see Colombini et al., 1987, Figure 5]. [8] Notice that H BM depends on the morphologic and hydraulic parameters that define a bed form at equilibrium with the flow regime. Therefore, we expect that this parameter should be an important metric to characterize the hyporheic exchange, in line with the results of recent studies [e.g., Marion et al., 2002; Tonina and Buffington, 2007] that highlighted the importance of flow regime and bed form geometry on hyporheic exchange Near Bed Pressure Distribution [9] The theory of Colombini et al. [1987] allows us to represent the hydraulic head distribution analytically as superimposition of Fourier harmonics as follows: hx; ð yþ ¼ XNy X Nx n¼0;n6¼1 m¼0;m6¼1 a nm cosðmx nm Þcos ny 2B þ a 11 cos x 11 y sin ; ð2þ 2B where N x and N y are the number of harmonics considered in the expansion along the two horizontal coordinates (x and y), l is the wave number of the bar, a nm,and nm are the amplitudes, and phase differences of the transverse (n) and longitudinal (m) modes, respectively. In equation (2), we only retain the terms of order up to the second in both directions (Nx = Ny = 2), because according to the work of Colombini et al. [1987] higher order terms produce a negligible contribution to h Groundwater Flow Model [10] We consider the following governing equation for the three dimensional stationary flow in a homogeneous porous media with an anisotropic hydraulic conductivity tensor K = (K x, K y, K z 2 h K 2 þ 2 h 2 þ 2 h 2 ¼ 0; where h is the hydraulic head. The stationarity hypothesis implies that stream water discharge is constant, or slowly varying, over a timescale of the order of the residence time in the hyporheic zone. Notice that Elliott and Brooks [1997a] performed a similar analysis, which predicts a two dimensional flow field in the vertical plane for sandbedded rivers with dunes under the hypothesis of homogeneous and isotropic medium. Although, some investigators show that hydraulic conductivity heterogeneity of the streambed material may affect hyporheic flow [cf. Cardenas et al., 2004; Salehin et al., 2004] and others show it could drive the exchange [Vaux, 1962, 1968; Tonina and Buffington, 2009], we introduce the hypothesis of homogeneity to explore the impact of geomorphic features on residence time distribution analytically. Equation (3) is solved within a rectangular parallelepiped of length L, width 2 B and depth z d, where L is the bar wavelength, B is half channel width, and z d is the thickness of the alluvium. The bottom surface z = z d is impervious such as the two external surfaces parallel to the streamflow direction (see Figures 1a and 1b). The resulting boundary ¼ ¼ 0: ð4þ z¼ zd In addition, periodicity in the hydraulic head is imposed at the two external transverse surfaces: ð3þ hð L=2; y; zþ ¼ hl=2; ð y; zþ; ð5þ and the near bed hydraulic head, given by equation (2) is imposed at the horizontal surface z = 0. Strictly speaking, the boundary condition at the interface between the stream 3of14

4 Table 1. Channel and Alternate Bar Parameters Adopted for Testing the Validity of the Model at Lower Than the Formative Discharge Q form Parameter Description Q form 0.90Q form 0.80Q form 0.70Q form Aspect ratio b [ ] Shields number [ ] Relative submergence d S [ ] Mean sediment size Ds [m] Channel width 2B [m] Alternate bar amplitude H BM [m] Bar wavelength L [m] Stream slope s 0 [%] Mean water depth Y 0 [m] Mean flow discharge Qf [m 3 s 1 ] and the alluvium (equation (2)) should be imposed at z = h(x, y) (Figure 1c). However, this would preclude the possibility to obtain a close form solution of equation (3). In order to overcome this difficulty, we approximate the bed topography as a flat plane at z = 0, which coincides with the average elevation of the streambed, such that the boundary condition reads as in equation (2). Note that Elliott and Brooks [1997a] introduced the same assumption for the twodimensional case. In section 2.5 we show that although the computational domain differs slightly from the physical domain, the impact of assuming a flat bed instead of the real topography on the flow field, and consequently on the residence time, is negligible. [11] Subsequently, we find analytical solutions of the head field for both finite and infinite alluvium depths. The solution for the latter case follows from the former case letting z d. Once the pressure distribution is known within the hyporheic zone, we compute the velocity field through Darcy s equation, which relates velocity with the gradient of the hydraulic head rh as follows: u ¼ K rh; where is the alluvium porosity.the dimensionless solution of the flow equation (3), with the boundary conditions discussed above, and for the general case of finite alluvium depth is given by h* ðx*; y*; z* Þ ¼ a* 11 sin 2 y* cos *x* 11 ½coshðC* 11 z* ÞþtanhðC* 11 z d * ÞsinhðC* 11 z* ÞŠ þ XNy¼2 XN x¼2 n¼0;n6¼1 m¼0;m6¼1 a nm * cos n y* 2 ð6þ cosðm*x* nm Þ½coshðC nm * z* Þ þ tanhðc nm * z d * ÞsinhðC nm * z* ÞŠ s 0 x*; ð7þ where the superscript * denotes dimensionless quantities defined as follows: (x*, y*, z*, z d *) = (x/b, y/b, z/b, z d /B), h*= h/y 0, a* nm = a nm /Y 0,andl* =lb. Note that we superimpose the effect of alluvial valley slope (s 0 ) to account for the longitudinal groundwater flow. The average bed slope is an important controlling factor of both surface and subsurface flows because it affects streamflow characteristics (e.g., velocity and depth), sediment transport, alternate bar topography, and thereby hyporheic flow. Additionally it influences the strength of the groundwater flow and the magnitude of hyporheic velocity along the streambed gradient [Tonina, 2005]. Our model accounts for these effects by coupling the surface and subsurface models. [12] The differentiation of equation (7) with respect to the dimensionless spatial coordinates provides the dimensionless velocity field: u* ðu*; v*; w* @h* ; ; @z* where the velocity is made dimensionless with respect to u m = K x Y 0 /H BM, i.e. u*=u/u m. The derivation of equation (8) is outlined in Appendix A, which provides also the expressions of the coefficients C* nm, for m, n = 0, 1, 2. Notice that the hydraulic head (equation (7)) and consequently the three components of the velocity field (equations (A1), (A2), and (A3) in Appendix A) are obtained by superimposition of four components [n =1,m = 1], [n =2,m = 2], [n =0,m = 2], and [n =2,m = 0] with periodicity in both horizontal directions. [13] It is worth to note that the analytical solution (A1 A5) for the hyporheic flow fields is obtained considering the head distribution at the streambed interface (equation (2)) at the formative condition (i. e. the discharge for which bed forms form and reach their equilibrium amplitude H BM )[Colombini et al., 1987], which are not frequent in natural flows. In fact, alternate bars typically form during intermediate flow regimes, when the sediment is movable and the width to depth ratio is large enough to allow for bar formation. Thus we evaluate the flow field and near bed head distribution at low flows when bars are almost emerging using a threedimensional numerical shallow water model [Vignoli and Tubino, 2002; Toffolon and Vignoli, 2007]. We use the bed topography at equilibrium obtained through [Colombini et al., 1987] model as the bathymetry, which we keep constant for these simulations, and then we lower the discharge until the bars are barely submerged. The lowest discharge satisfying this condition for the analyzed topography (Table 1) is 70% of the formative discharge. [14] In case of an unbounded alluvium the solution can be obtained by computing the limit of equation (7) for z d. The resulting expression of the hydraulic head is given in the Appendix (equation (A5)). The analytical expressions of the flow field are subsequently used to model transport of a conservative tracer with the particle tracking method Transport of Conservative Tracers [15] The residence time concept of a conservative tracer is the building block of a class of Lagrangian models of solute transport widely applied in subsurface [e.g., Jury et al., 1986; Dagan et al., 1992] and surface hydrology [e.g., Rodríguez Iturbe and Valdés, 1979; Rinaldo et al., 1989; Gupta and Cvetkovic, 2002; Botter et al., 2005]. These models are founded on the hypothesis that the mass of a conservative tracer entering a control volume can be subdivided into a large number of non interacting particles, which move under the influence of the velocity field. Owing to the linearity of the transport equation, the solution can be described through a kernel, which is called transfer function in signal theory, representing the response of the system to 4of14

5 the downwelling area and large enough to stabilize the residence time CDF (equation (10)). We identify the downwelling areas as composed by all the grid nodes with negative vertical velocity and track each particle within the hyporheic zone until it exits from the upwelling area (positive vertical velocity). The particle originating from x 0 in the downwelling area is tracked with the following equation: Xðt; x 0 Þ ¼ Xðt Dt; x 0 ÞþuX ½ ðt Dt; x 0 Þ; Yðt Dt; x 0 Þ; Zt ð Dt; x 0 ÞŠ Dt; Yðt; x 0 Þ ¼ Yðt Dt; x 0 ÞþvX ½ ðt Dt; x 0 Þ; Yðt Dt; x 0 Þ; Zt ð Dt; x 0 ÞŠ Dt; Zt; ð x 0 Þ ¼ Zt ð Dt; x 0 ÞþwX ½ ðt Dt; x 0 Þ; Yðt Dt; x 0 Þ; Zt ð Dt; x 0 ÞŠ Dt; ð11þ Figure 2. Sketch of the parameters used to evaluate the curvature of the streamline. an instantaneous injection of a unitary mass of solute through the entry surface. This kernel can be interpreted as the probability density function (pdf) of the travel time, or hyporheic residence time as it is referred in the hyporheic literature [e.g., Elliott and Brooks, 1997a], of a conservative tracer (particle) within the hyporheic zone. In our case, the control volume is the hyporheic zone and the exit and entrance surfaces are the upwelling and downwelling areas, respectively. Additionally, the pdf of the hyporheic residence time reflects the variability of flow pathways leading from downwelling to upwelling areas due exclusively to streambed morphology, because of the homogenous hydraulic properties of the alluvium. Therefore according to this theory, the concentration breakthrough curve at the upwelling area, resulting from a continuous injection of solute at constant concentration through the downwelling area, as provided by a stream with constant water discharge and solute concentration, can be written as follows: C F ðþ¼ t Z t 0 C 0 ðþ fðt Þ d; ð9þ where f is the pdf of the residence time t and C 0 is the concentration flux, i.e., the mass flux divided by the water flux, entering through the downwelling area [Kreft and Zuber, 1978; Demmy et al., 1999]. Under the hypothesis that solute is well mixed in the stream and C 0 = const, equation (9) simplifies to C F ðþ t ¼ C 0 Z t 0 f ðþ d ¼ Ft ðþ; ð10þ where F is the cumulative distribution function (CDF) of the particle residence time t. Thus the residence time pdf embeds the dynamics controlling transport of conservative tracers in the hyporheic zone. Notice that F represents the fraction of the particles released with a pulse injection at time t = 0 that at time t are within the control volume [Dagan, 1989]. [16] We compute the pdf of the hyporheic residence time numerically through particle tracking. Following this approach, we release a number of particles, NP, uniformly at where X t (t; x 0 )=(X(t; x 0 ), Y(t; x 0 ), Z(t; x 0 )) is the trajectory of the particle released at time t = 0 at the location x 0 within the downwelling area. Notice that the set of equations (11) is the first order numerical approximation of the following integro differential equation, X t (t; x 0 ) = R t 0 X(t; x 0) dt, which is known as the Volterra equation. We compute the particle trajectories by applying the set of equations (11) recursively with a variable time step, Dt. As particles upwell in the stream, we record their residence time and approximate the pdf f with their frequency time distribution. In computing the time residence moments, we consider only the downwelling particles that exit the numerical domain from upwelling areas, whereas those particles that do not upwell at the surface are counted as entrained in the shallow groundwater flow [Tonina and Buffington, 2007]. [17] We perform the computations with dimensionless variables and with the number of particles, NP, ranging between and depending on channel size. Due to the variable curvature of the streamlines we adopt a variable time step. After a number of preliminary simulations, we define the following rule of thumb for selecting the time step dynamically:! 0:1 Dt* ¼ 0:7 jðþ s j 3 ; ð12þ þ 10 6 where Dt*=DtK h /H BM is the dimensionless time step, K h is the horizontal hydraulic conductivity (we set K x = K y = K h assuming isotropy in the horizontal plane) and k(s) is the streamline curvature, which is calculated as follows: jðþ s j ¼ d bt ds ¼ T b 2 bt 1 ; ð13þ ds 12 where bt 1 and bt 2 are the unit vectors tangent to the trajectory and ds 12 is the arc length between point 1 and point 2 along the streamline, as represented in Figure 2. After these preparatory steps, we approximate the residence time CDF as follows: R A Ft ðþ q* ðx 0 *; y 0 *; z 0 * ÞRx 0 *; y 0 *; z 0 R ð *; tþ da A q* x ; ð14þ ð 0*; y 0 *; z 0 * Þ da where A is the area of the downwelling surface. Additionally, R is a binary variable, which assumes the value of 1 if the particle released at the location x* 0 =(x* 0, y* 0, z* 0 )is within the hyporheic zone (control volume) at the time t since injection, and 0, if the particle exited the control 5of14

6 Table 2. Channel and Alternate Bar Parameters Adopted in the Tests Parameter Description Test 1 Test 2 Test 3 Aspect ratio b [ ] Shields number [ ] Relative submergence d S [ ] Mean sediment size Ds [m] Channel width 2B [m] Alternate bar amplitude H BM [m] Bar wavelength L [m] Alluvium depth z d [m] volume through the upwelling surface at an earlier time [Elliott and Brooks, 1997a]. The expression (14), which is the Cumulative Frequency Distribution (CFD) of the sample of residence times obtained with the particle tracking, converges to the residence time CDF as the number of particles grows large and the mass carried by a single particle tends to zero. Notice that due to the flat bed hypothesis the initial location of the particle coincides with the projection of (x* 0, y* 0, z* 0 ) over the horizontal plane (x* 0, y* 0, 0) such that equation (14) transforms to Table 3. Parameters of the Numerical Grid Used in MODFLOW Parameter Description Test 1 Test 2 Test 3 Number of cells along x Nx [ ] Number of cells along y Ny [ ] Number of cells along z (flat bed) [ ] Number of cells along z (bed 3D) [ ] Dimension of cells along x D x [m] Dimension of cells along y D y [m] not exceed 0.22% for very shallow alluvial and reduces rapidly to zero as z* d increases. [20] This exercise verifies that the numerical model based on MODFLOW predicts the head distributions well within the hyporheic zone and can be used as a benchmark for testing the flat bed hypothesis. Consequently we perform a second set of simulations, where we solve the flow field numerically with the real bed topography and the same boundary conditions used for the analytical solutions. The bed topography is represented by a layer with cells of variable height in the computational domain. Additionally, we R A Ft ðþ q* ð x 0*; y 0 *; 0ÞRx ð 0 *; y 0 *; tþ dx* dy* R A q* ðx 0 *; y 0 *; 0Þ dx* dy* ; ð15þ where the integral is performed over the horizontal projection of the downwelling area Plan Bed Model Verification [18] The analytical solutions of the hydraulic head (equation (2)) and the associated flow field (equation (8)) within the hyporheic zone are obtained under the hypothesis that the streambed is flat with constant elevation z =0, which corresponds to the mean bed surface (see Figures 1b and 1c). We verify the goodness of this hypothesis by comparing our analytical solutions with those obtained numerically with the finite difference groundwater flow model MODFLOW [McDonald and Harbaugh, 1996]. To make the comparison meaningful, we design the computational grid such as to obtain negligible differences between numerical, h num, and analytical, h an, solutions of equation (3) when a flat bed is assumed in both cases. We quantify the relative difference of heads at each grid node as follows: j h ¼ h num h an j ; h ¼ 1; ::; ng; ð16þ h an where ng is the number of grid nodes. We use the maximum h,max and mean, h, values as indexes of the maximum expected and global divergence between solutions, respectively. Tables 2 and 3 resume the main parameters adopted in the simulations. [19] Figures 3a and 3b show h,max and h as a function of alluvium depth z* d. We observe that both maximum and mean errors decrease as bar amplitudes increase (from Test 1 to Test 3), but in all cases they are small values. This suggests that the accuracy of the numerical model (MODFLOW) increases with stream dimensions. Additionally, our results show that the maximum percentage error is less than 1% and the mean percentage error is much smaller, since it does Figure 3. (a) Maximum and (b) mean percentage difference between numerical and analytical solutions of the hydraulic head for a flat bed topography as a function of the dimensionless alluvium depth. The analytical solution is exact and the relative difference represents the error associated to the numerical scheme used in the numerical solution of equation (3) with the boundary conditions defined by equations (5) and (2). 6of14

7 evaluate the relative error, h, for the bed form with the largest H BM considered in this work because of its largest protruding volume with respect to the plane at z =0. [21] Figures 4a and 4b shows the maximum percentage error and the mean error, respectively, which are computed from the set of e h values obtained by the equation (16), where the numerical solution is obtained by applying MODFLOW with the same grid used in Figure 3 but with the real topography, while the analytical solution is obtained under the flat bed hypothesis. As expected, the relative difference between the two solutions increases with respect to the previous case, but it remains smaller than 5% (the mean value does not exceeds 1.5%). Additionally, both the maximum and the mean relative difference reduce rapidly with alluvium depth and at a dimensionless depth of z* d = 0.3 they halve their values. Furthermore, they decrease with large bar amplitudes, suggesting that the flat bed approximation improves with stream dimensions. Therefore, our results support the flat bed hypothesis allowing the analytical solutions of both head and flow fields within the hyporheic zone. Figure 4. (a) Maximum and (b) mean percentage difference between analytical and numerical solutions of the hydraulic head for a three dimensional bed topography as a function of the dimensionless alluvium depth. The flat bed analytical solution is compared with the reference numerical solution obtained with the same numerical scheme reported in Figure 3. Given the small error introduced by the numerical scheme the relative difference can be attributed to the approximation introduced by adopting the flat bed hypothesis. 3. Results and Discussion 3.1. Steep Versus Low Gradient Streams [22] We analyze the effect of bed slope by keeping the same flow regime (Q = m 3 s 1 ), sediment size (D s = 0.01 m), and channel width (2B = 3 m) but varying the streambed slope s 0, from 0.53% (low), to 1.15% (medium), and 3.3% (steep). Figure 5 shows the main characteristics of the hyporheic flow. The first row shows a three dimensional view of streambed topography, while the three dimensional views of the hyporheic flow are depicted in the second row. The three columns from left to right refer to the above three slopes, respectively and for each slope we use different color scale. Streambed topography changes presenting longer wavelengths (l increases from 0.18 m 1 to 0.22 m 1, and 0.25 m 1 moving from the left to the right panels) and shallower bar amplitudes (H BM = 0.90 m, 0.25 m, and 0.13 m, respectively) as the stream becomes steeper, which influence downwelling and upwelling fluxes. This is because shear stress, and thus sediment transport, increases with slope, all other variables constant, and the streambed tends to the plane bed morphology, which is the threshold geometry between sediment and transport limited bed forms [Montgomery and Buffington, 1997]. We observe a complex three dimensional structure of the hyporheic flow field generated by the interaction of the streamflow with the alternate bar topography and the underlying groundwater flow, similar to that reported in flume experiments [Tonina and Buffington, 2007]. The velocity is periodic in both longitudinal and transverse directions due to the periodicity of the alternate bar morphology, and decays exponentially with depth (equation (7)). The decaying rate depends on the alluvium depth z d, the hydraulic head, and the ratios (K x /K z ) and (K y /K z ) (see equations (A1), (A2), and (A3)). [23] Moreover, the streamlines compact near the sediment surface as the stream slope steepens (compare the first with the third panel in the second row of Figure 5). High slopes cause very shallow hyporheic zones with a large number of streamlines that cross the downward section of the bar without upwelling. Particles following these trajectories are not accounted for in computing the residence time distribution because they depend on variations at scales larger than a single bar unit. These particles follow hyporheic streamlines mostly parallel to the streambed and emerge from the alluvium only at knickpoints, rock outcrops, changes in sediment and channels characteristics (e.g., hydraulic conductivity, and bed form dimensions) [Elliott and Brooks, 1997a, 1997b; Cardenas et al., 2004; Gooseff et al., 2005; Boano et al., 2007; Buffington and Tonina, 2009]. Additionally, these flows may develop further complex paths due to changes in discharge and sediment transport, especially of fine sediment. Fine sediment, which is mobile at low flows, may infiltrate and clog the pores among the coarse gravel particles causing variations in hydraulic properties [Hohen and Cirpka, 2006] Alluvium Depth [24] We analyze the effect of the alluvium depth z d on the residence time moments for isotropic and anisotropic hydraulic conductivity. In the latter case we assume K x = K y = 7of14

8 Figure 5. Streambed elevation h(x, y) and streamline distribution within the hyporheic zone for a bounded isotropic alluvium. The following three streambed slopes s 0 and water depths Y 0 are considered: (a) s 0 = 0.53%, Y 0 = m, (b) s 0 = 1.15%, Y 0 = m, and (c) s 0 = 3.3%, Y 0 = 0.10 m. In all cases Q = m 3 s 1 and 2B =3m. 8of14

9 Figure 6. (a) Mean value and (b) variance of the hyporheic residence time versus the dimensionless alluvium depth for isotropic and anisotropic (K z /K h = 0.35) alluvium. The short horizontal lines show the limit for unbounded alluvium depths. K h and K z = 0.35 K h, which is a typical value reported in literature for soils composed of gravel with coarse sand and pebble [Freeze and Cherry, 1979]. Furthermore, we select an alternate bar topography with parameters b = 15, = 0.08, d S = 0.10 and D S = 0.01 m that correspond to a bed form of length L = m and amplitude H BM = m in a channel of width 2B = 3 m and bank full water discharge equal to Q = m 3 s 1. [25] Both first and second residence time moments are larger in the anisotropic than isotropic case in unbounded domains (z d ), as shown in Figures 6a and 6b for the mean (*) and variance (S 2 t* ), respectively. This can be explained as follows. In an anisotropic alluvium, the smaller vertical than the other hydraulic conductivity components introduces two counteracting effects. The first effect is that the particle s velocity reduces with respect to the isotropic case because the equivalent hydraulic conductivity, K eq,is smaller than K h. The second effect is a compaction of the streamlines near the alluvium surface causing particles to travel shorter distances with respect to the isotropic case. This is due to the reduction of hydraulic conductivity, which bends the streamlines horizontally. The former effect dominates for deep streamlines, whereas the latter for shallow streamlines. Therefore, in an anisotropic alluvium short near surface streamlines have smaller residence times than in the companion isotropic alluvium with the same depth z d, while long deep streamlines have longer residence times. The resulting effect is a residence time pdf that is broad in the anisotropic case, which leads to large variances as shown in Figure 6b. [26] In an isotropic alluvium, both moments increase monotonically with z d approaching the asymptotic values, set by the unbounded case (z d ), for z d =2B. For an anisotropic alluvium both the moments first increase with z d, peaking at an intermediate depth, and then decrease to the corresponding asymptotic limits for z d >2B. This is due to a difference in the way streamlines change in response to a modification of z d. In an isotropic alluvium an increase of z d results in longer streamlines with the equivalent hydraulic conductivity that remains the same, thus leading to larger residence times. The same variation results in a more complex behavior in an anisotropic than isotropic alluvium; the tendency to increase the length of the streamlines is contrasted by the reduction of K eq, which bends the streamlines horizontally. This contrasting effect is weak at small z d values, when the streamlines are nearly horizontal. An increase of z d leads to a smaller variation of K eq with respect to the isotropic case, but becomes progressively stronger as z d increases because the streamlines tend to penetrate more deeply into the alluvium, such that a small variation in z d leads to a proportionally larger variation in K eq. Therefore, for a given z* d the residence time moments are larger for the anisotropic than isotropic alluvium, the difference reaching the maximum value at z* d = 3/4 for both * and S 2 t*. For alluviums deeper than z* d = 3/4 both moments converge rapidly to their asymptotic values, which are reached for z* d = 2. For a deep alluvium, the bottom impervious layer does not affect the development of the hyporheic zone. We calculate that 95% of the hyporheic flow develops within a sediment depth roughly equal to one channel width for low gradient streams and shallows with increasing stream bed slope due to the compression process caused by the groundwater flow. These results support the hypothesis that hyporheic flow is a near surface process Residence Time Moments [27] The bar amplitude H BM given by equation (1) is a function of the morphodynamic parameters, which affect the near bed pressure distribution. Thus, it is natural to hypothesize that the first two moments of the residence time distribution can be parameterized by H BM. In order to verify this hypothesis we perform a set of simulations with d S = 0.01, 0.05, and 0.10, and = 0.06, 0.07, 0.08, 0.09, and 0.10, for an isotropic alluvium. We report the results of this analysis in Figures 7a and 7b, which show the mean and variance of the dimensionless residence time, respectively, as a function of H* BM for different values of the relative submergence. [28] We define the dimensionless residence time as follows: * ¼ K hs 0 L ; ð17þ where the characteristic time of transport is given by the ratio between the hydraulic conductivity (K h ) and the mean length of the streamlines (L/s 0 ). Our results shows that the mean 9of14

10 of a power law relationship with the data provides the following expressions: and * ¼ 1:39ðH BM * Þ 0:60 ; R 2 ¼ 0:96 ð20þ S 2* ¼ 2:07ðH BM * Þ 0:89 ; R 2 ¼ 0:91 ð21þ Figure 7. (a) Mean value and (b) variance of the dimensionless hyporheic residence time t* versus the dimensionless amplitude of alternate bars H* BM = H BM /Y 0, where H BM is given by equation (1). The best fit of the power law equations, with the data are also shown as a function of the relative submergence d S. value and the variance of the dimensionless residence time distribution decrease with the dimensionless bar amplitude H* BM and increase with the relative submergence d S, all other parameters constant. We note that the d S has an important effect on the near bed pressure distribution and thus on the hyporheic exchange. The relative submergence is a surrogate of channel resistance, and can be related to the dimensionless Chezy coefficient through the following expression: 1 C z ¼ 6 þ 2:5ln : ð18þ 2:5d S [29] If we define the following dimensionless time: * ¼ K hs 0 C z ; ð19þ L data obtained for different d S collapse to two single power law relationships as shown in Figures 8a and 8b. The fitting for the mean and variance of the dimensionless residence time, respectively. Equations (20) and (21) are simple and powerful tools because they provide a good approximation of the moments with the only information given by H* BM, which is a quantity that can be measured in the field. Notice that although we obtained the expressions (20) and (21) under the assumption of equilibrium between streambed morphology and hydraulics, they can also provide a useful estimation of the first two residence time moments within the hyporheic zone in transient non equilibrium conditions as described in section 2.3. [30] Furthermore, Figures 8a and 8b show that both * and S t 2 * decrease as H* BM increases, such that wide channels with small H* BM values result in residence times, which are longer and have wider distributions than in narrow channels with large H* BM. This reflects the fact that wide channels are in general characterized by low gradient bed slopes and deep hyporheic zones, which provide long and slow flow paths and a large diversity among path lengths from the short near bed to the long and deep streamlines. On the other hand, small steep streams typically exhibit shallow hyporheic zones with short and meanly equivalent residence times because of streamline compaction near the streambed surface Residence Time Distribution [31] The residence time concept can quantify the transformation that a plume of a conservative tracer experiences from the entrance through the downwelling area to the exit through the upwelling area. As mentioned before, this is a well established methodology in subsurface hydrology, which allows to compute the Breakthrough Curve (BTC), i.e. the evolution in time of the concentration flux at the exit surface, as the time convolution of the release history at the entry surface and a transfer function, which coincides with the residence time pdf [Dagan et al., 1992]. Residence time distributions for the hyporheic zone have been modeled with Exponential [Bencala and Walters, 1983; Runkel, 1998], lognormal [Wörman et al., 2002], and Power law [Haggerty et al., 2002] probability distributions. For example, Wörman et al. [2002] showed that the hyporheic residence time of a dune like bed form follows a lognormal distribution. Given these previous results, we test whether the residence time distribution is also lognormal for hyporheic flows induced by alternate bars. The lognormal distribution assumes the following form: 2! f ð* Þ ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffi 2 2 * exp * ln * * 2* 2 : ð22þ 10 of 14

11 Table 4. Range of Variation of Hydraulic Parameters Adopted in the Tests Parameter Description Minimum Value Maximum Value Channel width 2B [m] Bar wavelength L [m] Mean water depth Y 0 [m] Alluvium depth z d [m] Water discharge Q [m 3 s 1 ] Stream slope s 0 [ ] that are close to the critical values condition for bar formation. This ensures that the alternate bars do not have complex characteristics such as multiple high or low points besides bar top and pool bottom. [33] We fit the lognormal model (22) to the residence time data obtained by releasing NP = particles uniformly over the downwelling area with the maximum likelihood method. Figure 9 shows the Q Q plot of the lognormal CDF Fð* Þ ¼ Z * 0 f ð 1 * Þ d 1 *; ð25þ Figure 8. (a) Mean value and (b) variance of the dimensionless hyporheic residence time t* versus the dimensionless amplitude of alternate bars H* BM = H BM /Y 0, where H BM is given by equation (1). The best fit of the power laws, equations (20) and (21), with the data are also shown. where f is the lognormal pdf given by the equation (22), versus the CFD of the sample composed by NP particle residence times. In the cases analyzed in the Q Q plots of Figure 9 b and d S are fixed (b = 13, d S = 0.10), while assumes the following values: = 0.06 (Test1), = 0.07 (Test2), and = 0.10 (Test3). [34] The Q Q plots are close to the 45 line (Bisect), which represents perfect match between CDF and CFD, therefore showing a good fit of the lognormal distribution with the data. The fit is better at small values, deteriorates as grows large from test 1 to test 3. This may be associated to the larger Shields number of test case 3, which leads to a more complex alternate bar topography. Although, the least squares method has a better match with the data than the According to the maximum likelihood method of estimation the parameters m* z and s 2 z * are given by 2 3 * h i * 6 7 ¼ ln4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; 2 * ¼ ln 1 þ CVð* Þ 2 ; ð23þ 1 þ CV ð* Þ 2 where * and CV(t*) are the mean residence time and its coefficient of variation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * ¼ 1 X NP j *; CV ð* Þ ¼ 1 u 1 X NP t j * 2 * ; ð24þ NP * NP j¼1 j¼1 and t* j is the dimensionless residence time of the j th particle in the collection of NP particles used to approximate the residence time pdf. [32] In the following, we compute the residence time pdf of bed forms with Shields number varying between 0.06 and 0.10, two values of relative submergence, d S = 0.05 and d S = 0.10, and b varying between 13 and 15, which correspond to the hydraulic parameters reported in Table 4. Notice that in all cases we investigate bed configurations Figure 9. Q Q plot of the CFD and lognormal CDF of the hyporheic residence time of three bar morphologies with = 0.06 (Test 1), = 0.07 (Test 2), and = 0.10 (Test 3). In all cases b = 13 and d S = According to the maximum likelihood method, the CDF is obtained by replacing the first two residence time moments obtained numerically by particle tracking into equation (22). 11 of 14

12 Figure 10. Q Q plot of hyporheic residence time CFD and lognormal CDF with parameters estimated with the maximum likelihood and minimum square techniques for Test 3(b = 13, d S = 0.10 and = 0.10). The theoretical lognormal CDF obtained by approximating the first two residence time moments with the expressions (20) and (21) is also shown (Power Law). maximum likelihood method, as shown in Figure 10 test 3, this method lacks of predictability. On the other hand, predictability can be gained by substituting into equation (9) the moments obtained with the expressions (20) and (21). This method provides a slightly better match than the maximum likelihood Extension of the Analytical Solution [35] Figure 11 shows the Q Q plot of the hyporheic residence time distribution for different percentage of the formative discharge Q form, keeping the bar topography obtained under formative conditions as explained in section 2.3. The data are close to the 1:1 line, which indicates a good match between CDFs predicted at lower and formative discharges. The correct prediction of the hyporheic residence time also at Figure 11. Q Q plot of the hyporheic residence time for different percentages of the stream formative discharge Q form. Figure 12. (a) Mean and (b) variance of residence time distribution for different values of water discharge. low discharges is due to the fact that the water surface elevation mainly depends on the local topography and Froude number under quasi stationary condition; hence, keeping the equilibrium bed topography and decreasing the discharge, the water profile translates almost vertically retaining the same shape. Numerical results show that the water profile evaluated under formative conditions is representative for water stages ranging from the formative conditions until the bed forms are close to exposure. [36] Accordingly, Figure 12 shows the mean and the variance of the residence time distribution as a function of the formative discharge Q form. We observe that both parameters slightly decrease with discharge, as pressure gradients caused by the topography decrease supporting the applicability of our model at low flows. 4. Conclusion [37] In this work, we analyze the residence time moments of in stream solutes within the hyporheic zone. Our first result shows the effectiveness of the assumption of planar streambed in analyzing the hyporheic flow induced by a complex three dimensional topography, provided that the streambed hydraulic head distribution is reproduced accurately. We also show that solute residence time within the hyporheic zone induced by the alternate bar topography is lognormally distributed with the mean and variance depending on the parameters that control the morphology of 12 of 14

13 the bed form through two power law functions. This holds as long as the alternate bar formation parameter b is close to the critical value. As b increases the accuracy of the maximum likelihood method deteriorates and the first two moments of the hyporheic residence time cannot explain its distribution entirely. On the other hand, the Least squares method provides the parameters for the lognormal distribution that fits the residence time CFD adequately, but it lacks predictability because it needs a large sample of residence times. Additionally, we consider the effects of the stream slope on the streamline distributions and on the residence time moments. We observe that when the stream slope increases streamlines compact near the bed surface and residence times decrease. In the case of steep slopes, the hyporheic flow is confined within a shallow volume with a significant number of streamlines that exit from the downward section of the bar without crossing the upwelling area. Results of our model also apply to stages lower than the formative conditions as long as bars do not emerge from the stream. We conclude that hyporheic residence time distributions of gravel bed rivers with alternate bars in equilibrium with their flow regime are close to be lognormal with the parameters depending through simple relationships such as equations (20) and (21) between the residence time moments and the bar amplitude H BM. Appendix A [38] The dimensionless velocity field u* =(u*, v*, w*) is obtained in an anisotropic alluvium of dimensionless depth z* d by substituting the hydraulic head, given by equation (7) into equation (8): u* ðx*; y*; z* Þ ¼ *a* 11 H BM * sin 2 y* ½cosh C 11 * z* þ H BM * XNy¼2 XN x¼2 n¼0;n6¼1 m¼0;m6¼1 sin *x* 11 ð ÞþtanhðC 11 * z d * ÞsinhðC 11 * z* ÞŠ m*a* nm cos n y* 2 sinðm*x* nm Þ½coshðC nm * z* Þ þ tanhðc nm * z d * ÞsinhðC nm * z* ÞŠ; ða1þ v* ðx*; y*; z* Þ ¼ 2 a 11 * H BM * cos 2 y* ½cosh C 11 * z* þ H BM * n¼0;n6¼1 m¼0;m6¼1 cos *x* 11 ð ÞþtanhðC 11 * z d * ÞsinhðC 11 * z* ÞŠ XN y¼2 XN x¼2 n 2 a* nm sin n y* 2 cosðm*x* nm Þ½coshðC nm * z* Þ þ tanhðc nm * z d * ÞsinhðC nm * z* ÞŠ; ða2þ and w* ðx*; y*; z* Þ ¼ K z H BM * C 11 * sin K x 2 y* cos *x* 11 ½sinhðC 11 * z* ÞþtanhðC 11 * z d * ÞcoshðC 11 * z* ÞŠ K XN y¼2 XN x¼2 z H BM * C nm * a* nm cos n y* K x 2 n¼0;n6¼1 m¼0;m6¼1 cosðm*x* nm Þ½sinhðC nm * z* Þ þ tanhðc nm * z d * ÞcoshðC nm * z* ÞŠ; ða3þ where the dimensionless coefficients assume the following expressions: 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 11 * ¼ ð* Þ 2 K x þ 3 2 K y K z 2 K rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z C 22 * ¼ ð2* Þ 2 K x þ 2 K y C nm * ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K z K z : ða4þ C 02 * ¼ ð2* Þ 2 K x 6 rffiffiffiffiffiffiffiffiffiffi K z 4 C 20 * ¼ 2 K 7 5 y K z In the particular case of an alluvium of infinite depth the solution of the flow equation simplifies to h* ðx*; y*; z* Þ ¼ a 11 * sin 2 y* cos *x* 11 exp ð C11 * z* Þ þ XNy¼2 XN x¼2 a nm * cos n y* 2 n¼0;n6¼1 m¼0;m6¼1 cosðm*x* nm ÞexpðC nm * z* Þ: ða5þ [39] Acknowledgments. 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