Click Here WATER RESOURCES RESEARCH, VOL. 46, W07535, doi:10.1029/2009WR008317, 2010 for Full Article A linear model for the coupled surface‐subsurface flow in a meandering stream Fulvio Boano,1 Carlo Camporeale,1 and Roberto Revelli1 Received 22 June 2009; revised 25 February 2010; accepted 5 March 2010; published 27 July 2010. [1] The interest about the exchange of water between streams and aquifers has been increasing among the hydrologic community because of the implications of the exchange of heat, solutes, and colloids for the water quality of aquatic environments. Unfortunately, our understanding of the relevance of the exchange processes is limited by the great number of coupled hydrological and geomorphological factors that interact to generate the complex spatial patterns of exchange. In this context, the present work presents a mathematical model for the surface‐subsurface exchange through the streambed of a meandering stream. The model is based on the linearization of the equations that govern the hydrodynamics and the morphodynamics of the system, and it provides a first‐order analytical solution of the coupled flow field of both the surface and the subsurface flows. The results show that stream curvature determines a characteristic spatial pattern of hyporheic exchange, with water upwelling and downwelling concentrated near the stream banks. The exchange can drive surface water deep into the sediments, thus keeping deep alluvium regions connected with the stream. The relationships between hyporheic exchange flux and the geometrical and hydrodynamical properties of the stream‐aquifer system are also investigated. Citation: Boano, F., C. Camporeale, and R. Revelli (2010), A linear model for the coupled surface‐subsurface flow in a meandering stream, Water Resour. Res., 46, W07535, doi:10.1029/2009WR008317. 1. Introduction [2] The exchange of water and solutes between rivers and aquifers is currently receiving a great attention by hydrologists, biologists, and ecologists, and its relevance for the riverine ecosystems is widely accepted by the hydrologic scientific community [e.g., Vaux, 1968; Stanford and Ward, 1993; Brunke and Gonser, 1997; Boulton et al., 1998; Jones and Mulholland, 2000]. The influence of the surface‐ subsurface exchange on the abundance of algae, plants, and invertebrates [Dent et al., 2000] and its contribution to the oxidation of organic matter within the biogeochemical cycle of carbon [Battin et al., 2008] are just two of the manifold examples of its role for the fluvial environment. [3] Many field studies have investigated the complex spatial and temporal patterns of surface‐subsurface exchange that exist in streams [Harvey and Bencala, 1993; Kasahara and Wondzell, 2003; Lautz and Siegel, 2006; Peterson and Sickbert, 2006; Poole et al., 2006; Kasahara and Hill, 2007]. These studies have shown that hyporheic exchange occurs on a wide range of spatial scales [Woessner, 2000; Wörman et al., 2007; Cardenas, 2008a], and the detailed features of these exchange patterns depend on the morphology of the stream‐aquifer system, the hydrodynamic characteristics of the surface and the subsurface flow, and on the degree of connectivity between the stream and the aquifer. The collection of a large quantity of data is required in order to 1 Department of Hydraulics, Transports, and Civil Infrastructures, Politecnico di Torino, Turin, Italy. Copyright 2010 by the American Geophysical Union. 0043‐1397/10/2009WR008317 characterize all these factors, and this procedure can thus be applied to evaluate exchange only for relatively short stream reaches. Caution is also required when trying to extrapolate the information about the exchange at one field site in order to evaluate the exchange at another site, because moderate differences in the hydraulic as well as the morphologic features between otherwise similar streams may cause different exchange processes to control surface‐subsurface exchange. Therefore, there is a need for modeling methods that can be applied in order to analyze the physical processes that control hyporheic exchange dynamics, and to predict how the ecosystem will respond to a particular anthropic modification (e.g., the alteration of the streamflow regime due to an upstream dam). [4] Because of the mentioned difficulties, a complementary approach to field studies is represented by the development of mathematical models based on the physical principles that govern the exchange. The efforts of researchers in the last decade have explained the fundamental mechanics of the exchange driven by different morphologic features including bed forms [Elliott and Brooks, 1997; Packman and Brooks, 2001; Marion et al., 2002; Cardenas and Wilson, 2007; Boano et al., 2007, 2008], channel bends [Cardenas et al., 2004; Boano et al., 2006; Revelli et al., 2008; Cardenas, 2008b, 2009a], in‐stream structures of logs, boulders, or wooden debris [Hester and Doyle, 2008] and large‐scale surface topography [Tóth, 1963; Sophocleous, 2002; Wörman et al., 2006, 2007; Cardenas, 2007]. These works have contributed to explain the physics of hyporheic exchange across many different spatial scales. Unfortunately, there are still some exchange processes for which predictive models are unavailable because of the complex nature of the morphological and hydrodynamical factors that control the W07535 1 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM Figure 1. Conceptual sketch of surface‐subsurface interactions in a meandering stream. The river planimetry (a) induces a complex pattern of water exchange that can be divided in (b) the quasi‐horizontal water exchange at scaleof the meander wavelength [Boano et al., 2006; Revelli et al., 2008; Cardenas, 2008b, 2009b], and (c) in the exchange flow at the scale of the channel width determined by the sediment point bars and the lateral slope of the stream surface (present work). (d) The overall pattern of exchange flow is given by the composition of the two flow fields. exchange. This lack of predictive tools represents an obstacle to our understanding of the ecological and biochemical processes in fluvial environments. [5] Among the exchange processes that have not been fully described it is possible to include the exchange induced by stream curvature of meandering streams (Figure 1a). Previous theoretical [Boano et al., 2006; Revelli et al., 2008; Cardenas, 2008b, 2009b] and experimental studies [Peterson and Sickbert, 2006] have stressed the existence of a horizontal flow at the scale of the meander wavelength, which is qualitatively depicted in Figure 1b (the shaded area denotes the investigated domain). The present work parallels and is complementary to the mentioned studies, as it investigates an exchange flow that occurs in sinuous streams at a different spatial scale than those already studied. In particular, we focus on the exchange that occurs at the smaller scale of the stream width because of the presence of point bars and of the W07535 lateral slope of the stream surface (Figure 1c). The resulting flow is markedly three‐dimensional and occurs in the sediment region beneath the streambed, which is not included in Figure 1b. An example of this exchange can be found in the work by Cardenas et al. [2004], where only a portion of the full meander wavelength has been examined. Both exchange flow fields pictured in Figures 1b and 1c are caused by the channel sinuosity, and they can be analyzed separately because of the difference between their spatial scales. The overall flow field in the complete domain (Figure 1d) can be thought as the composition of the two separate flow fields. [6] In order to explore the exchange flow shown in Figure 1d, the present work presents a mathematical solution for the 3D field of the hydraulic head in the porous medium and the resulting surface‐subsurface exchange flow in a sinuous stream flowing on gravel sediments. To this aim, we start from an existing perturbative solution for surface flow and bathymetry in a meandering stream and we extend it in order to obtain an analytical expression of the stream‐aquifer exchange. Although the approach necessarily relies on some simplifying assumptions, the derived solution presents a number of positive aspects that make it useful for the study of surface‐subsurface interactions. First, it predicts the spatial pattern of water flux through the streambed of a meandering river, providing indications for field investigators about where monitoring instruments should be placed in order to capture the essential features of the exchange pattern. Second, a fundamental advantage of the analytical method is that it identifies the geometrical and hydrodynamic quantities that control the exchange. Finally, the solution can be used as a benchmark case to test numerical approaches that are employed to model complex field settings. Therefore, the proposed solution provides significant insight on the considered exchange flow, and represents a further step toward a more comprehensive understanding of the interactions between surface and subsurface waters. 2. Method [7] In the present work, a perturbative approach is adopted to obtain analytical solutions for both the surface and the subsurface flow. This approach relies on the choice of a dimensionless perturbative parameter whose value must be smaller than unity. Here, the dimensionless stream curvature n is defined as the ratio between channel half‐width and twice of the minimum curvature radius, and it is adopted as perturbative parameter since its values are usually much smaller than unity, as discussed in more detail below. The small values of n imply that the perturbative approach represents a methodology that is particularly suited for the study of this problem, and that our findings will not be limited by the choice of n as perturbative parameter. [8] The stream‐aquifer system shown in Figure 2 is considered. The reference system {~s, ~n, ~z} is adopted, where ~s and ~n are the streamwise and spanwise curvilinear coordinates, respectively, and ~z is the vertical coordinate. The tilde symbol is hereinafter used to denote a dimensional quantity. [9] The stream is assumed to evolve in an aquifer composed by alluvial sediments with the hydraulic conductivity ~ In order to simplify the calculations the hydraulic conK. ductivity is treated as homogeneous, and its value is kept constant in both space and time. The aquifer sediments extend for a depth
~ on an impervious substratum that represents the 2 of 14 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 W07535 Figure 2. (a) Cross section of the surface‐subsurface system. (b) A 3D scheme of the subsurface domain only (without the free surface flow), with the streambed topography evidenced in shades of gray; the thicker lines delimit the cross‐section shown in Figure 2a. The scales of the axes are distorted in order to make the complex geometry of the domain more appreciable. lower limit of the aquifer. The main geometrical features of the stream are the average slope Sb, the constant half‐width ~b, ~ s), which is defined as the inverse of the and the curvature C(~ curvature radius of the river axis. In the present analysis, a simple sinusoidal function is adopted to describe the river curvature 1 ~ ð~sÞ ¼ 2 cosð~ ~ sÞ ¼ expði~ ~sÞ þ c:c:; C R~0 R~0 ð1Þ ~ = 2p/~ is where R~0 /2 is the minimum radius of curvature,  ~ the meander wave number,  is the meander wavelength, i is the imaginary unity, and c.c. denotes the complex conjugate. The complex exponential notation is here preferred to the trigonometric one because it simplifies the structure of the equations that are derived in our analysis. [10] In order to understand the relative importance of the various morphodynamic and transport processes, all the involved quantities are normalized using some characteristic scales. The original reference system {~s, ~ n, ~z} is thus converted to the new dimensionless reference system {s, n, z} that is defined as ~s s¼ ~b ~n n¼ ~b ¼ ( ~z ~ ~ D ~z ~ ~þ~
if ~z > ~ ; if ~z  ~ ð2Þ ~) is the local elevation of the streambed and where ~(~s, n ~ s, ~ D(~ n) is the local stream depth. The river half‐width is chosen as the typical horizontal length scale, while two different normalizations are used to define the dimensionless vertical coordinate z in the surface and subsurface domain, respectively. The other dimensionless variables are defined in a similar way to (2), namely ~  ¼ ~b
 ¼ ~b ~0 D D¼ ~ D D~0 ¼ ~ D~0
¼
~ ; ~0 D ð3Þ where D~0 is the average stream depth. Finally, the normalized curvature is ~ ¼  expði sÞ þ c:c: ¼ C1 ðsÞ þ c:c:; C ¼ ~b
C ð4Þ where n = ~b/R~0 is the maximum dimensionless curvature. It is important to notice that for natural rivers the curvature radius is always much larger than the river width (see section 2.4). [11] It is important to stress that the adoption of the dimensionless system {s, n, z} drastically simplifies the complex geometry of our surface‐subsurface domain. In fact, the domain becomes a rectangular parallelepiped with ~~ s 2 [0, l], n 2 [−1, 1], and z 2 [−1, 1], where l = / b = 2p/a. In particular, it follows from equation (2) that positive values of z denote the points within the water column, while negative values are associated with the subsurface domain. This geometry is definitely simpler than the one of the original domain, which is bounded by the wavy surfaces shown in Figure 2. This switch to a rectangular domain is fundamental to the derivation of the analytical solutions presented in the next sections. 2.1. Surface Flow and Morphology [12] In the last few decades several morphodynamic models have been developed for the solution of both the flow field and the bed topography of a meandering river under steady conditions [Ikeda and Parker, 1989; Seminara, 2006]. For the present analysis, we adopt a linear solution for the hydraulic head in the stream and for the bed topography. For this reason we only focus on the most complete known linear morphodynamic solution, as provided by the recent theory of Zolezzi and Seminara [2001]. Although the mathematical aspects of the theory have already been reviewed in previous papers [Camporeale and Ridolfi, 2006; Camporeale et al., 2007; Frascati and Lanzoni, 2009], we briefly recall its peculiar elements in order to make this paper self‐consistent. [13] Three fundamental hypotheses are assumed in the following. (i) The fluid is assumed to be incompressible, the flow to be fully turbulent, while the sediments of the river bed are considered cohesionless and with a uniformly distributed grain diameter, d~s. (ii) Since the typical vertical scale (i.e., the ~ is much smaller than the characteristic horiwater depth D) zontal scale (i.e., the river half‐width ~b), the vertical velocity component is neglected and a hydrostatic vertical pressure distribution can be adopted. (iii) It is assumed that both the 3 of 14 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 flow and bed topography instantaneously adjust to the planimetry, that is, the process is considered as quasi‐stationary [De Vries, 1965]. Point (i) supports Exner’s equation for bed sediment, hypothesis (ii) justifies the use of the shallow water equations which, thanks to point (iii), are made to be time‐independent. [14] Under the previous assumptions and introducing the velocity decomposition of the flow field [Kalkwijk and De Vriend, 1980], along with the no‐slip condition at the bottom and the no‐stress condition at the free surface, one obtains the depth‐averaged two‐dimensional equations for shallow water in curvilinear coordinates and in dimensionless form [Johannesson and Parker, 1989; Zolezzi and Seminara, 2001], as reported in Appendix. [15] The linearization of the morphodynamic problem is achieved through the perturbative expansion in the parameter n of the governing equations, which introduced in the shallow water equations (A1)–(A4), leads to a linear system with four first‐order PDEs. Zolezzi and Seminara [2001] solved the linear system using a Fourier expansion in the transversal direction, n, and obtained m systems of ODEs A dx1m ðsÞ þ Bx1m ðsÞ ¼ Am WkðsÞ ðm ¼ 0; . . . ; 1Þ; ds ð5Þ where Am = 2(−1)m/M 2, M = (2m + 1)p/2, and k¼  C; @C @ 2 C @ 3 C ; ; @s @s2 @s3 T ð6Þ ; whereas the vector x1m = {U1m, V1m, H1m, D1m}T contains the Fourier coefficients of the four unknowns of the problem (x1 = {U1, V1, H1, D1}T), namely the first‐order perturbations of the depth‐averaged velocity in both the longitudinal and spanwise directions, free surface P elevation and depth, respectively. It follows that x1(s, n) = 1 m¼0 x1m(s)sin(Mn). The entries of the 4 × 4 matrices A, B, W are reported by Camporeale et al. [2007]. [16] For the solution of the subsurface flow we need to impose a boundary condition on the bed topography (i.e., at z = 0) for the hydraulic head of the porous media, hp(s, n, z), namely the Dirichlet condition hp ðs; n; 0Þ ¼ H0 ðs; nÞ þ  1 X m¼0 H1m ðsÞ sin Mn; ð7Þ where the function H1m(s) is expressed, in analogy with equation (4), as H1m ðsÞ ¼ hm e is þ c:c: ð8Þ [17] In order to apply the normalization defined by equation (2) to the subsurface equations we also need the solution for the bed topography perturbations, h(s, n) = H(s, n) − D(s, n). For the herein considered case of sine‐ generated planimetry ‐ i.e., equation (4) ‐ the solution of hm and dm was provided by Seminara et al. [2001] in the following form 2 hm ¼ F Am j ðhÞ j¼0 ðiÞ jþ1 P5 j 1 j j¼1 ðiÞ P6 þ b2 þ ib3 2 !  b5 ; ð9Þ j ðd Þ jþ1 j 1 ð i Þ j j¼1 P6 j¼0 ðiÞ dm ¼ F 2 Am P5 ib6 F 2 2 þ b2 þ b4 F ! 2 W07535 þ ib3 ð10Þ  b5 ; where the coefficients bj, rj and sj depend on a, b, the Shield stress , and the dimensionless sediment diameter ds = d~s/d~0, and are reported by Camporeale et al. [2007, Appendix B]. [18] At this point, the equations (7)–(10) provide the morphodynamic forcing that must be imposed in order to solve the equations of subsurface flow, as shown in the following section. 2.2. Subsurface Flow [19] The water flow in a homogeneous and isotropic porous medium is governed by the Laplace equation, which in the intrinsic reference system {~s, ~n, ~z} is written as [Batchelor, 1967] N hp hp hp @2~ @2~ N 2 @2~ þ N3 þ 2 2 @~s @~ n ~ þ
~ @~z2 ~ n ~ ~ @~ @C hp ~ @ hp ¼ 0; þ N 2C @~s @~s @~ n ð11Þ where ~hp represents the hydraulic head in the subsurface porous domain and N (s, n) = 1 + n C(s) is a metric factor that arises from the change of coordinates. [20] The equation is solved in the subsurface domain shown in Figure 2. Tonina and Buffington [2007] have observed that pressure on sediment bars that are fully submerged is primarily hydrostatic with negligible contributions from dynamic pressure, in agreement with the hypothesis (ii) of Section 2.1. The reason for this behavior is that meander point bars have height‐to‐wavelength ratios that are much lower than dunes and that reduce the probability of flow separation. Therefore, we assume that the flow is forced by the hydrostatic head distribution at the streambed that derives from the level of the stream surface, and that is given by equation (7). This procedure reflects the assumption that the coupling between surface and subsurface flow occurs in one direction only, that is, surface flow drives water flow in the sediments but it is not significantly influenced by subsurface flow. This represents a common assumption in models of water flow over porous sediments [e.g., Cardenas and Wilson, 2007], and provides a good description of the main flow characteristics as long as exchange water fluxes are much smaller than stream discharge. [21] The bottom of the subsurface domain is treated as an impervious boundary, and periodic boundary conditions are imposed on s = 0 and s = l. Finally, no‐flow boundary conditions are imposed at the side boundaries n = ±1 in order to model a stream that does not gain or lose any net amount of water. Since equation (11) is linear with respect to ~hp, the case of a gaining or losing stream can be easily determined by the superposition of the present solution with a solution of the head and flow field in a stream in non‐neutral conditions [see Boano et al., 2008]. All the boundary conditions are summarized in Table 1. [22] The governing equation (11) is written according to the dimensionless form (2), and a solution is sought in the form hp(s, n, z) = hp0 + nhp1. In a similar way, the river cur- 4 of 14 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 W07535 Table 1. Boundary Conditions Coupled to Equation (11) for the Subsurface Flow Problema Boundary Top (z = 0) P hp = −bSbs + n eias m hm sin(Mn) + c.c. hp0 P = −bSb s hp1 = eias m hm sin(Mn) Complete O(n 0) O(n 1) a Bottom (z = −1) Sides (n = ±1) Upstream and Downstream (s = 0, s = l) hp,z = 0 hp0,z = 0 hp1,z = 0 hp,n = 0 hp0,n = 0 hp1,n = 0 hp|s=0 = hp|s=l + bSbl, hp,s|s=0 = hp,s|s=l hp0|s=0 = hp0|s=l + bSbl, hp0,s|s=0 = hp0,s|s=l hp1|s=0 = hp1|s=l, hp1,s|s=0 = hp1,s|s=l A comma followed by s, n, or z denotes the derivative with respect to the corresponding variable. vature and the streambed topography are expressed as C(s) = nC1(s) and h(s, n) = nh1(s, n), respectively. Substitution in equation (11) and the application of (2) yield, at the zeroth order O(n 0) 2
2 2 hp ðs; n;  Þ ¼ 2 @ hp0 @ hp0 @ hp0 þ
2 þ 2 ¼ 0: @s2 @n2 @ 2 ð12Þ The solution of equation (12) that satisfies the boundary conditions in Table 1 is hp0(s) = −bSb s. Thus, at the zeroth order (i.e., neglecting the effect of curvature) the hydraulic head in the subsurface simply decreases along the stream because of the average energy gradient Sb. [23] In order to be consistent with the hydraulic head at the streambed interface, the O(n) solution must also have a sinusoidal structure in s, namely hp1 = Hp1(n, z)exp(ias) + c.c. This analytical structure is consistent with the boundary conditions at s = 0 and s = l (see Table 1). This choice leads to a couple of equations of order n, the first of which is
2 2 @ 2 Hp1 2 @ Hp1 þ  @n2 @ 2 2
2 Hp1 þ i nSb 
2 ¼ 0 ð13Þ and the second one is its complex conjugate. The last term on the right hand side of (13) derives from the zeroth order solution hp0. [24] In order to achieve the solution, the unknown P function Hp1 is expressed as a Fourier series Hp1(n, z) = 1 m¼0 Hp1m(z) sin(Mn), which satisfies the no‐flow conditions on the side boundaries (see Table 1). We also make use of the expansion P n= 1 m¼0 Am sin(Mn), and from equation (13) we obtain 2 00  Hp1 m ð Þ 2 2 M
Hp1 m ð Þ 2 2 2 
Hp1 m ð Þ þ iAm Sb 
¼ 0: Hp1 m ¼ am þ bm cosh ½cm ð1 þ  ފ; ð15Þ where iAm Sb  ; M 2 þ 2 bm ¼ hm am ; coshðcm Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M 2 þ 2 cm ¼  ð16Þ are three algebraic coefficients whose values depend on m. Sb s þ expðisÞ
sinð MnÞ þ c:c: 1 X m¼0 ½am þ bm coshðcm ð1 þ  Þފ ð17Þ [26] It is interesting to notice that hp is related to the stream elevation hm through the coefficients defined in (16) but is apparently independent on the shape of the streambed. The hydraulic head hp1 actually depends on the streambed shape through the dimensionless coordinate z, but the adoption of the rectangular domain allows to express hp1 with the simple structure of equation (17). 2.3. Surface‐Subsurface Exchange Flux [27] Once the hydraulic head is known, it can be used to evaluate the magnitude of the exchange flux between the surface and the subsurface domain. The seepage velocities can be obtained from the Darcy law, which requires the knowledge of the hydraulic gradient. This gradient is given by [Batchelor, 1967] r~ hp ¼ ( N 1 ) @~ hp @ ~ hp @ ~ hp ; ; ; @~s @~ n @~z ð18Þ where the head is function of the intrinsic curvilinear coordinates {~s, ~n, ~z}. If we make use of the chain rule for derivation we can express the hydraulic gradient as a function of the dimensionless head in the rectangular domain hp(s, n, z) r~ hp ¼    1 @hp 1 þ  @hp @ 1 @hp ; N @s
þ  @ @s  @n  1 @hp :

þ  @  1 þ  @hp @ ;
þ  @ @n  ð14Þ This simple, second‐order ordinary differential equation in Hp1m(z) is coupled with the boundary conditions H′p1m(z = −1) = 0 and Hp1m(z = 0) = hm, the latter of which provides the link between the subsurface flow and the surface head variations induced by the stream curvature. Its solution is am ¼ [25] Putting together the previously described expression, the first‐order solution for the hydraulic head beneath the streambed can be expressed as ð19Þ [28] The Darcy velocity at the streambed interface is then given by ~ vj¼0 ¼  ~ r~ K hp ¼0 ¼ ~ v0 þ ~ v1 ; ð20Þ where the Darcy velocity is decomposed into its zeroth‐ and first‐order components ~v0 and ~v1 , respectively. These components can be evaluated after the introduction of the decomposition hp = hp0 + nhp1 in (19) and (20)  ~ @hp0 K ; 0; 0  @s    ~ @hp1 @hp0 K ~ v1 ¼ ; þ nC1  @s @s ~ v0 ¼ 5 of 14  ~ @hp1 K ;  @n  ~ @hp1 K ;
@ ð21Þ BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 Table 2. Range of Typical Values of Parameters n, a, b, ds, and for Gravel Bed Sinuous Streamsa n a b ds Range Source 0.07–0.10 0.22–0.31 2.6–171.9 1.5 · 10−3–2.0 · 10−1 0.05–0.89 1, 2, 3, 4 1, 2, 3 5 5 5 a Sources: 1, Leopold and Wolman [1960]; 2, Leopold et al. [1964]; 3, Braudrick et al. [2009]; 4, Camporeale et al. [2005]; 5, van den Berg [1995]. where we have made use of the first‐order Taylor expansion of the terms (1 + nnC1)−1 ≈ 1 − nnC1 and (g + nh1)−1 ≈ g −1(1 − nh1/g). [29] The analytical calculation of the exchange flux requires a mathematical description of the topographic surface of the streambed, which is provided by the function ~f (~s, ~ n, ~z) : ~(~s, ~ n) − ~z = 0. The normal vector to this surface that is directed toward the subsurface is given by N = r~f /~f . Following the same approach that has lead to (21) we can write N ¼ N0 þ N1 ¼ f0 ; 0 ; 1g þ    1;s 1;n ; ; 0 :   ð22Þ [30] The water flux through the bed surface is defined as ~qð~s; ~nÞ ¼ ~vj¼0
N ¼ ð~v0 þ  ~v1 Þ
ðN0 þ N1 Þ ð23Þ and from equations (21)–(22) it results ~ q0 ¼ ~v0
N0 ¼ 0 ; ~ q1 ¼ ~v1
N0 þ ~v0
N1 ¼ ~ @hp1 K
@ ~ @hp0 @1 K ;  2 @s @s ð24Þ where the two terms on the right hand side of the second equation represent the vertical flux due to the variations of the level of the stream surface and the horizontal flux due to the average stream slope, respectively. Finally, the introduction of equation (17) in (24) provides the desired analytical solution for the exchange flux between the surface and the subsurface domain ~ expðisÞ ~q ¼  K 1  X bm cm sinhðcm Þ m¼0
 iSb m þ sinð MnÞ þ c:c:  ð25Þ The exchange flux across the streambed surface can also be ~ expressed in dimensionless form as q = ~ q/K. 2.4. Controlling Parameters [31] The first‐order solution given by equation (25) shows that there are six dimensionless parameters whose values govern the dimensionless exchange flux q. For homogeneous sediments, these parameters are sufficient to completely describe the characteristics of the complex surface‐subsurface domain shown in Figure 2. In particular, there are only four parameters that summarize the geometrical features of the stream‐aquifer system, i.e., the stream sinuosity n, the meander wave number a, the stream aspect ratio b, and the depth of the impermeable bedrock g. Moreover, the influence W07535 of the hydrodynamical characteristics of the stream on the dimensionless exchange q is described by the relative roughness ds and the Shields stress . [32] Natural planimetries of unconstrained meandering streams commonly display an alternation of low‐sinuosity reaches and mature meander lobes. This alternating pattern results from the interplay between sedimentation and erosion processes, which tend to increase stream sinuosity and curvature, and the occasional occurrence of meander cutoffs, which suddenly reduce channel curvature. The effect of this morphodynamic activity is that the stream exhibits variations of local curvature among different reaches. [33] The study of the average values of curvature and other stream morphometric parameters is one the subjects of stream geomorphology. For instance, a number of researchers [Leopold and Wolman, 1960; Leopold et al., 1964; Braudrick et al., 2009] have proposed empirical scaling relationships that relate channel wavelength and width as ~  ð20 28Þ~ b: ð26Þ Additionally, Camporeale et al. [2005] have also found that ~ av j 1 ; ~  2:47jC ð27Þ ~ av| is the average absolute curvature. For a channel where |C with sinusoidal planimetry, integration of equation (1) shows ~ av| = 4/(pR ~ 0). ~ 0 is related to the mean curvature by |C that R Putting all these expressions together, values of n = 0.07– 0.10 are found, depending on the different coefficients in equations (26). These values represent average conditions from which curvatures values of single river reaches can diverge, but the order or magnitude is expected to hold for most meandering streams. This confirms that natural meandering streams usually have n  1, and that n is suited to be adopted as perturbation parameter. [34] Typical ranges of values for the controlling parameters n, a, b, ds, and are reported in Table 2. Values of b, ds, and have been calculated from a database of 60 rivers [van den Berg, 1995], considering only monocursal meandering streams flowing on sediments with median diameter larger than 2 mm (i.e., gravel bed rivers). Values of n and a have been obtained from the geomorphologic scaling equations (26)–(27). 3. Results [35] The properties of the exchange flow field predicted by the described modeling approach are now discussed. In the following examples we consider the reference case of ~ 0 = 1 m, a 30‐meters‐wide stream with an average depth D that flows on a gravel sediment bed (d~s = 10 mm) with slope ~ 0 = 500 m), Sb = 2 · 10−3. The stream is gently meandering (R with a wavelength ~ = 470 m. A large value of sediment layer thickness (~
= 1000 m) is chosen in order to reproduce the case of a hyporheic zone that is not constrained by an impervious bedrock. The dimensionless parameters that summarize the properties of this reference case are n = 0.03, b = 15, ds = 0.01, = 0.14, g = 1000, and a = 0.2. 3.1. Validity of the Linear Approach [36] Since our analysis relies on a perturbative approach, its validity is restricted to low values of the dimensionless 6 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 Figure 3. Comparison between the first‐order approximation (continuous lines) and the total reach‐averaged exchange flux (dashed lines) for the case of b = 15, g = 1000, ds = 0.01, = 0.14, and for two different values of a (0.2 and 0.8, respectively). scheme to solve the equation on a triangular mesh. Because of the domain shape an anisotropic mesh is adopted, with a higher density of nodes in the streamwise direction. The chosen number of tetrahedral elements ranges between 1000 and 1300 depending on the value of n. [38] The comparison between the average dimensionless flux q resulting from equation (25) and from the solution of the complete Laplace equation is shown in Figure 3 for two different sets of parameters. The first set corresponds to the chosen reference case, and the second is identical with the only exception of a higher dimensionless wave number a = 0.8, which corresponds to a stream with smaller meander wavelength. In both cases, the dimensionless curvature n is varied between 0 and 0.1, thus reproducing the effect of increasing channel sinuosity. [39] The comparison between the first‐order solution given by equation (25) and the numerical solution of the complete model shows that our solution is a good approximation of the exchange up to n = 0.10, with a 30% maximum relative error. It can be seen from Table 2 that typical values of dimensionless curvature are usually lower than this limit, which means that the higher‐order corrections that are neglected in our linear approach have a minor influence on the surface‐ subsurface exchange. maximum curvature n that is chosen as perturbative parameter. It is then necessary to assess the upper limit of n for which our first‐order solution can still be regarded as a good approximation of the complete problem, i.e., equation (11). [37] In order to achieve this aim, we first evaluate the ~ with the first‐order dimensionless exchange flux q = ~ q/ K solution provided by equation (25). The reach‐averaged dimensionlessR flux q that flows within the sediments is then evaluated as |q(x, y)|dxdy/(2A), where A is the streambed area and the factor 1/2 is introduced to avoid counting the upwelling and downwelling fluxes as separate contributions. This result is compared to the average dimensionless flux obtained from a numerical solution of the complete Laplace equation (11). This numerical solution is obtained with a finite element approach that adopts a LU‐factorization 3.2. Properties of the Exchange Flux Pattern [40] The solution given by (25) is now used to explore the main properties of the coupled surface‐subsurface exchange. The typical spatial pattern of the exchange flux for the chosen reference case is shown in Figure 4a. Positive and negative values of q represent water downwelling and upwelling, respectively. Figure 4a shows that the water exchange is mainly concentrated near the stream banks, and in particular in correspondence of the bends where the stream curvature is the highest. Conversely, in the straight parts of the stream the values of the exchange flux q are much lower than at the bends. [41] It is important to notice that this pattern is not directly related to the streambed morphology, which is presented in Figure 4b. Figure 4b shows that in the reference case sedi- Figure 4. Spatial patterns of (a) dimensionless exchange flux q and (b) streambed elevation ~ (m) for the reference case (n = 0.03, b = 15, g = 1000, ds = 0.01, = 0.14, and a = 0.2). The horizontal scales have been distorted in order to make the spatial variations more appreciable. 7 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 ~ (m) and (b) first‐order contribution Figure 5. Spatial patterns of (a) elevation of the stream surface H ~ 1 (m) for the reference case (n = 0.03, b = 15, g = 1000, ds = 0.01, = 0.14, and a = 0.2). The horizontal H scales have been distorted in order to make the spatial variations more appreciable. ment point bars develop at the inner banks just downstream of each bend, and they extend over the straight reaches up to the next bend. Simulations with different parameter values (not reported) have shown that the location of point bars is much more sensitive to channel curvature and geometry than the exchange flux pattern, which always retains the main features of Figure 4a. The comparison of the two panels of Figure 4 reveals that the exchange flux q is in phase with the curvature of the river even though the streambed elevation h is not. In other words, this example shows that the streambed morphology and the exchange flux are only partially correlated, although their features are both controlled by the surface flow. ~ which [42] The pattern of the total free surface elevation H, is the sum of the zeroth‐ and first‐order contributions, is displayed in Figure 5a. An arbitrary datum has been chosen in order to make stream elevation always positive over the stream reach. The free surface declines along the streamwise direction because of the channel slope. Moreover, the stream surface also presents slight spatial variations that are clearly evidenced in Figure 5b. Figure 5b shows the first‐order elevation of the free surface, which represents the relative difference between local and average stream depth. The presence of zones with higher (lower) surface elevation at the outer (inner) banks can be observed in Figure 5b. The magnitude of these local variations is of the order of a few centimeters, which makes them almost undetectable in Figure 5a. The pattern of first‐order stream surface elevation is clearly correlated to the pattern of exchange flux in Figure 4a. [43] It can be observed from equation (25) that the total exchange flux is given by the sum of two contributions, namely the vertical flux caused by the variations of the stream surface elevation (bmcm sinh(cm)/g) and the horizontal flux through the uneven bed surface caused by the average stream gradient (iaSbhm/b). These two contributions correspond to the first and second term in the last of equations (24), respectively. In order to thoroughly explore the connections between spatial patterns of river morphology and surface‐ subsurface exchange, we have randomly selected one hundred sets of values of the parameters {n, b, a, g, ds, } within the ranges shown in Table 2 and we have then evaluated the two terms of equation (25). The analysis of these results reveals that the second term is always much smaller than the first one. This means that the second term in equation (25) can be dropped without affecting the estimate of the exchange flux. Since the prevailing term in (25) is not explicitly related to the streambed elevation h, this finding implies that the magnitude of the exchange is not significantly influenced by the precise shape of the streambed. Even though the bed morphology is linked to the free surface by the shallow water equations (A1)–(A4), our results show that its influence on the exchange flux is only indirect (i.e., through the free surface elevation). This result suggests that the qualitative pattern of sinuosity‐induced vertical water exchange shown in Figure 4a is expected to hold in the majority of natural sinuous streams. [44] Figure 6a shows the pathlines of exchanged stream water particles for the chosen reference case. The absence of a shallow impervious layer allows water particles to penetrate deep into the sediments up to a depth of approximately 160 m. This deep exchange is a consequence of the large scales that characterize the meander geometry and the hydraulic gradient that drive the hyporheic flow. However, the depth of the hyporheic zone is only a fraction of the total thickness of the sediment layer
~ because of the confining action of the horizontal underflow. In the reference case, water particles that enter the hyporheic zone through the streambed mainly flow in the streamwise direction, as shown in Figure 6b. This behavior depends on the no‐flow boundary conditions imposed on the domain sides. If the intrameander flow field depicted in Figure 1c is summed to the present flow field, water particles that are close to the domain boundaries would actually flow outside the consider domain in the intrameander area as sketched in Figure 1b. Nonetheless, Figure 6 is representative of pathlines in the central part of the stream. 3.3. Influence of Controlling Parameters [45] A parametric analysis is performed to assess the influence of each dimensionless parameter on the exchange flux. Both terms of equation (25) are kept in the following computations in order to obtain the highest possible 8 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 Figure 6. Flow paths of sinuosity‐driven flow in the subsurface for the reference case (n = 0.03, b = 15, g = 1000, ds = 0.01, = 0.14, and a = 0.2). (a) Full subsurface domain and (b) detail of the zone interested by hyporheic flow. Flow direction is toward increasing x values. Scales have been distorted in order to better show the main features of the flow field. precision. The average dimensionless flux is plotted against the different dimensionless parameters in Figures 7a–7e. We have considered the reference case of n = 0.03, b = 15, a = 0.2, g = 1000, ds = 0.01, and = 0.14, and we have then varied the value of each parameter within the range reported in Table 2 in order to investigate its influence on the average dimensionless exchange q. [46] The geometry of the stream‐aquifer system is summarized by the curvature n, the aspect ratio b, the meander wave number a, and the sediment depth g. The stream curvature n is tightly related to the river sinuosity, which progressively increases because of the erosion of the outer banks and the sedimentation at the inner banks. This process leads to an increase in the amplitude of the sinusoidal planimetry of 9 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 Figure 7. Reach‐averaged dimensionless flux q as a function of (a) the stream half‐width‐to‐depth ratio b; (b) the dimensionless meander wave number a; (c) the dimensionless thickness of the sediment layer g; (d) the dimensionless Shields stress ; and (e) the relative roughness ds. (f) Relative error in the evaluation of q as a function of the number of modes used in equation (25). the river. Since the first‐order solution (25) for the dimensionless flux increases linearly with the curvature n, the evolution of the river morphology determines an increase in the average flux. [47] The relationship between the aspect ratio b and the average dimensionless flux q is pictured in Figure 7a, which shows that q is a decreasing function of b. If the other parameters are kept constant, this relationship implies that larger rivers tend to exchange less water with the sediments than smaller ones. This happens because the exchange is driven by the head gradient between the banks, and larger values of b correspond to an increase of the river width and thus to a lower hydraulic gradient. While the general rela- tionship between q and b can be approximated by a power law function, Figure 7a shows that this behavior is made somewhat irregular by the presence of a deviation from an otherwise straight line (in log‐log scales). This deviation is an example of the complex structure of equation (25), which predicts a nonlinear dependence of q on the aspect ratio b. [48] The response of the dimensionless flux to the river wave number a is presented in Figure 7b. It can be observed from Figure 7b that for meanders with longer wavelengths (i.e., decreasing values of a) the dimensionless flux q increases up to a maximum value. A further decrease in a leads to a slight decrease in q that eventually tends to a constant value. This happens because q represents the average 10 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM value of the flux over the streambed area, and both quantities increase with decreasing a. For large values of the wave number a the increase in the flux prevails over the increase in area, while the opposite occurs when a is low. [49] The role of the dimensionless thickness g of the subsurface domain is shown in Figure 7c. The flux first increases with the depth of the sediment layer and then reaches an asymptotic constant value. It is thus possible to make a distinction between shallow beds, for which q/K depends on the sediment depth, and deep beds, for which the bedrock is too deep to influence the exchange. The minimum value of the bed thickness of a deep bed must be proportional to the river half‐width ~ b, as it represents the characteristic length scale of the exchange. Thus, the dimensionless threshold value of g for a deep bed must be proportional to b, and Figure 7c b) the sediment suggests that for g > 2.5b (i.e.,
~ > 2.5~ thickness does no longer influence the exchange. [50] Beside the geometrical features, the exchange is also influenced by the hydrodynamical characteristics of the stream; these characteristics are summarized by the dimensionless Shields stress and the dimensionless roughness ds. Figure 7d shows that the average flux is a linear function of the Shields stress, with higher rates of exchange associated to higher values of the stress. Since the Shields stress is proportional to the stream slope, raising the value of while keeping constant the values of the other parameters results in an increase in the average stream velocity and thus in the dimensionless exchange flux. [51] Figure 7e shows that the dimensionless average flux increases with the dimensionless grain diameter ds, and it is approximately described by a power law relationship. The behavior of Figure 7e can be understood if we recall that the differences in the free surface elevation that drive the exchange are highly sensitive to the stream secondary currents, which in turn increase with the average stream velocity. The dimensionless grain diameter ds is a measure of the relative roughness of the river bed and is related to the Shields factor = Sb/(Dds), where D represents the specific weight of the submerged sediment grain relative to water. Thus, large values of the stream slope Sb must be considered in order to increase the value of ds while keeping constant the value of . Thus, the high values of ds in Figure 7e represent the behavior of steep streams, whose swift flows are characterized by intense secondary currents that enhance the water exchange through the sediment surface. 3.4. Convergence of Solution [52] The estimation of the average dimensionless flux q would require the evaluation of an infinite number of modes in the sum in equation (25), which is instead replaced by a sum over a finite number of modes. The influence of the truncation of the sum is shown in Figure 7f, which displays the relative error (q − q*)/q* between the dimensionless flux q and the exact value q* (calculated with 500 modes) as a function of the chosen number of modes. Even though it is better not to reduce the number of modes too much in order to limit the introduction of errors in the flux estimation, Figure 7f shows that the truncation only leads to a slight underestimation of q. It should be recalled that the errors in the estimates of the hydraulic conductivity and the other geometrical and hydraulic parameters are usually much larger W07535 than the errors introduced by the use of a finite number of modes, which are of the order of 10−2 or even less. 4. Discussion and Conclusions [53] In the present paper we have adopted a perturbative approach to solve the physically based equations that describe the surface and the subsurface flow of water in a sinuous stream‐aquifer system. In particular, we have derived an analytic first‐order solution for the water flux that is exchanged through the streambed. It is important to recall that the solution we have found is valid for a simplified stream‐aquifer system, without many of the complexities that are commonly found in natural meandering streams. The implications of these simplifications are now discussed in order to establish the relevance – as well as the limits – of our findings. [54] A first issue that characterizes the described method is that it only considers water exchange through the streambed that is induced by channel sinuosity. Thus, the predicted exchange pattern does not include the effect of other exchange processes that can be present. However, the overall exchange problem can always be seen as the sum of different exchange processes at different scales, and an estimate of the overall flow field can be obtained by composition of the single velocity fields. This is the case exemplified in Figure 1 for the presently examined exchange flow and lateral hyporheic exchange (as in the work by Revelli et al. [2008] or Cardenas [2009a]). Other exchange processes can be treated in the same way as long as predictive models for such processes are available. For instance, bed forms do not develop on the gravel bed streambeds considered in the present work, but turbulent eddy penetration can contribute to surface‐ subsurface exchange. In streams with finer sand sediments turbulent exchange would play a less important role, while bed forms would give a significant contribution to the overall hyporheic exchange. In this framework, the analysis presented in this work covers a type of surface‐subsurface interaction which has received very little attention so far. Although the feasibility of this approach has still to be rigorously tested, the increasing body of knowledge deriving from theoretical and field studies on surface‐subsurface interactions will hopefully make it a viable tool for prediction of hyporheic exchange in streams. [55] Among the model assumptions, the most critical one is probably the adoption of homogeneous hydraulic conductivity, that is implicit in the use of the Laplace equation (11). In contrast, most streambeds show some degree of heterogeneity [e.g., Ryan and Boufadel, 2007; Genereux et al., 2008], and the resulting spatial variations of hydraulic conductivity are expected to alter the subsurface head field as well as the exchange pattern. Deviations between modeled and actual values of the exchange flux will depend on the level of heterogeneity of the sediments, and will thus vary among different stream‐aquifer systems. Some comments on these deviations can be drawn from the comparison between ~ – which is equivalent to a our dimensionless flux q = ~q/K streambed head gradient ‐ and the patterns of head gradient observed by Kennedy et al. [2009] in West Bear Creek, a highly heterogeneous stream in an agricultural catchment. This stream gained water from the adjacent floodplain and was mostly straight, with only a small subreach (referred to as 11 of 14 W07535 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM “August small reach” in the work by Kennedy et al. [2009]) characterized by a mild sinuosity. In our framework, the effect of the gaining conditions would be to add an upwelling component to the pattern of q shown in Figure 4, leading to a higher dimensionless flux at the inner bank than at the outer bank. This behavior qualitatively agrees with the pattern of head gradient in the August small reach of West Bear Creek [Kennedy et al., 2009, Figure 5, Table 3], where the streambed head gradient exhibited a clear transversal profile with increasing values from the outer to the inner bank. It is also interesting to notice that this profile was not observed in the straight reaches of the stream. Instead, head gradient values in these reaches were always symmetrical with respect to the channel axis, coherently with the absence of sinuosity and with the modeling results described by Boano et al. [2009]. These considerations suggest that the streambed head gradients (i.e., the dimensionless flux q) predicted by equation (25) are qualitatively correct even for moderately heterogeneous streambeds. However, information about the ~ would be necessary in order to evaluate spatial pattern of K the actual fluxes ~ q. This implies that increasing our understanding on the factors that control streambed heterogeneity will be a crucial issue in future hyporheic zone research. [56] Despite the mentioned simplifications, the proposed solution retains many significant characteristics of the exchange. The comparison with a numerical solution of the complete problem has shown that for reasonable values of the stream curvature the first‐order solution represents a valid description of the exchange flow. The resulting exchange flux exhibits a peculiar spatial pattern, with the upwelling and downwelling of water concentrated near the outer and inner banks of the stream, respectively. While the exchange pattern always exhibits these qualitative features, the magnitude of the exchange flux is influenced by the geometry of the stream‐aquifer system and the hydrodynamic action of the surface flow. The present analysis has shown that these properties can be grouped together in order to form the dimensionless parameters that completely summarize the physics of the surface‐subsurface system (normalized stream curvature, stream half‐width‐to‐depth ratio, meander wave number, Shields stress, relative sediment roughness, normalized sediment thickness). These parameters control both the exchange flux and streambed morphology. [57] A careful inspection of the dependence of the exchange flux from the governing parameters has revealed that the exchange pattern is in phase with the stream curvature even when the streambed morphology is not. This behavior occurs because the major contribution to the flux is given by the spatial differences of the free surface elevation, while the role of streambed shape on the exchange is less relevant. This finding suggests that for sinuous rivers it is not necessary to obtain detailed information on the largest scales of streambed topography, i.e., meander point bars. However, it is clear that river beds can also present dunes and other morphological features that are known to determine phenomena of flow separation and are therefore more strongly coupled to the stream hydrodynamics. [58] It is interesting to notice that the differences in free surface elevation that drive the exchange flow can be so small to be hardly measurable in the field. Nonetheless, the discussed exchange process can drive water flow deep into the sediments, supporting the connectivity between surface water W07535 and the underlying alluvium. Thus, the description of the exchange at the scale of the meander wavelength presented in this work provides new insights on the complex spectrum of hydrologic processes that control the exchange with the hyporheic zone. Appendix A: Summary of the 2D Shallow Water Equations in Curvilinear Coordinates [59] In steady flow conditions, the depth‐averaged two‐ dimensional equations for shallow water in curvilinear coordinates and in dimensionless form are N UU ;s þVU ;n þ N CU ðV þ 2’Þ þ þ Cf þ  s D 1 ðUD’Þ;n ¼ 0 D N UV ;s þ VV ;n þ þ N H;s F2 ðA1Þ H;n N 2 n þ  þ ð DU ’Þ;s þ ðVD’Þ;n D F2 D D 1 ð ’1 DÞ;n þ N C’2 ¼ 0; D ðA2Þ where the comma followed by s or n denotes the derivative with respect to the corresponding direction. Equations (A1)– (A2) have to be coupled with the continuity equation for the water and bed sediment [Exner, 1925], respectively, N ð DU Þ;s þ ð DV Þ;n þ N CDV ¼ 0; ðA3Þ N qs;s þ qn;n þ N Cqn ¼ 0: ðA4Þ [60] In equations (A1)–(A4), U and V are the longitudinal and transversal depth‐averaged velocity, N (s, n) = 1 + n C(s) is the longitudinal metric factor, D is the depth, H is the free surface elevation, t ≡ {t s, t n} is the bed stress vector, q ≡ {qs, qn} is the volumetric vectorial bed load, Cf is the friction factor, and F is the Froude number. Moreover, ’ = hF v0i, ’1 = hv20i, and ’2 = 2V’ − U2 + ’1, where brackets refer to depth averaging, F (z) is the vertical profile of velocity, and v0(s, n, z) is the recirculating secondary current driven by curvature and with vanishing depth average. [61] The following boundary and integral conditions are also imposed to equations (A1)–(A4) V ¼ qn ¼ 0 Z 1 1 UD dn ¼ 2; Z 0  Z ðn ¼ 1Þ; ðA5Þ 1 1 ðH DÞ dn ds ¼ const; ðA6Þ where equation (A5) imposes the zero‐net‐flux condition between the center and the sidewall layers and no sediment transport across the sidewalls, whereas equations (A6) set the condition that the water discharge and the average reach slope are not influenced by perturbations in flow and topography (l = 2p/a is the dimensionless meander wavelength). [62] Finally, some closure relationships for the terms t, q, and v0 are required. In particular i) the dimensionless bed stress vector is considered aligned with the near‐bed velocity vector and it can therefore be expressed through a local friction coefficient; ii) the dynamic equilibrium of the bed 12 of 14 BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535 z Subscripts 0, 1 sediment written in an orthogonal reference system must be set; iii) the secondary currents v0 is resolved by an approximated iterative solution of the transversal momentum equation (A2) (see Zolezzi and Seminara [2001] for details). m Notation Latin symbols am ~ b bj bm cm dm d~s, ds hm ~ h p, h p i ~ n, n ~ q, q q ~s, s ~ b ~z A Am B ~ C C, ~ av| |C ~, D D ~0 D F ~ H H, H p1 ~ K M ~ N N Sb ~0 R U V Greek symbols ~ a , b D
~, g ~, h hm ~ l , n rj sj W coefficient of hp in equation (17). river half‐width. coefficient of hm in equation (9)–(10). coefficient of hp in equation (17). coefficient of hp in equation (17). m‐th mode of D solution, equation (10). sediment grain diameter (ds = d~s/D~0 ). m‐th mode of H solution, equation (9). hydraulic head in the porous medium hp/d~0 ). (hp = ~ imaginary unit. spanwise coordinate (n = ~ n/ ~ b). ~ hyporheic exchange flux (q = ~q/K). reach‐averaged dimensionless exchange flux. streamwise coordinate (s = ~s/~b). vector of Darcy seepage velocity. vertical coordinate. matrix of coefficients in equation (5). 2(−1)m/M 2. matrix of coefficients in equation (5). ~ ·~ b). stream curvature (C = C average absolute stream curvature. ~D ~ 0). stream depth (D = D/ average stream depth. Froude number of surface flow. ~ D ~ 0 ). stream surface elevation (H = H/ hp1(s, n, z) = Hp1(n, z)exp(ias). sediment hydraulic conductivity. (2m + 1)p/2. normal vector to streambed surface. metric factor for change of reference system. average streambed slope. twice of minimum radius of stream curvature. dimensionless streamwise velocity of surface flow. dimensionless spanwise velocity of surface flow. ~ ~ · b). meander wave number (a =  ~ 0). stream aspect ratio (~ b/D ratio between specific submersed weight of sediments and water specific weight. ~ 0). thickness of sediment layer (g =
~/D ~ 0). streambed elevation (h = ~/D m‐th mode of h solution (hm = hm − dm). vector of coefficients in equation (5). ~ ~b). meander wavelength (l = / ~ 0). dimensionless stream curvature (~b/R ~ 0Sb)/(Dd~s). Shields stress (D coefficient of hm in equation (9)–(10). coefficient of hm in equation (9)–(10). matrix of coefficients in equation (5). W07535 dimensionless vertical coordinate. Zeroth‐ and first‐order components (e.g., hp = hp0 + nhp1). m‐th Fourier mode (e.g., H p1 (n, z) = P 1 m¼0 Hp1m(z)sin(Mn)). [63] Acknowledgments. The suggestions and constructive comments provided by the Associate Editor Aaron Packman and by three anonymous reviewers are gratefully acknowledged. References Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge, U. K. Battin, T. J., L. A. Kaplan, S. Findlay, C. S. Hopkinson, E. Marti, A. I. Packman, J. D. Newbold, and F. Sabater (2008), Biophysical controls on organic carbon fluxes in fluvial networks, Nat. 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