A linear model for the coupled surface-subsurface flow in a meandering stream

ClickHere

for

FullArticle

A linear model for the coupled surface‐subsurfaceflow 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 beenincreasing among the hydrologic community because of the implications of the exchange ofheat, solutes, and colloids for the water quality of aquatic environments. Unfortunately, ourunderstanding of the relevance of the exchange processes is limited by the great numberof coupled hydrological and geomorphological factors that interact to generate the complexspatial patterns of exchange. In this context, the present work presents a mathematical modelfor the surface‐subsurface exchange through the streambed of a meandering stream. Themodel is based on the linearization of the equations that govern the hydrodynamics andthe morphodynamics of the system, and it provides a first‐order analytical solution of thecoupled flow field of both the surface and the subsurface flows. The results show that streamcurvature determines a characteristic spatial pattern of hyporheic exchange, with waterupwelling and downwelling concentrated near the stream banks. The exchange can drivesurface water deep into the sediments, thus keeping deep alluvium regions connected withthe stream. The relationships between hyporheic exchange flux and the geometrical andhydrodynamical 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 flowin a meandering stream, Water Resour. Res., 46, W07535, doi:10.1029/2009WR008317.

1. Introduction

[2] The exchange of water and solutes between rivers andaquifers is currently receiving a great attention by hydrolo-gists, biologists, and ecologists, and its relevance for theriverine ecosystems is widely accepted by the hydrologicscientific 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, andinvertebrates [Dent et al., 2000] and its contribution to theoxidation of organic matter within the biogeochemical cycleof carbon [Battin et al., 2008] are just two of the manifoldexamples of its role for the fluvial environment.[3] Many field studies have investigated the complex

spatial and temporal patterns of surface‐subsurface exchangethat exist in streams [Harvey and Bencala, 1993; Kasaharaand Wondzell, 2003; Lautz and Siegel, 2006; Peterson andSickbert, 2006; Poole et al., 2006; Kasahara and Hill,2007]. These studies have shown that hyporheic exchangeoccurs on a wide range of spatial scales [Woessner, 2000;Wörman et al., 2007; Cardenas, 2008a], and the detailedfeatures of these exchange patterns depend on the morphol-ogy of the stream‐aquifer system, the hydrodynamic char-acteristics of the surface and the subsurface flow, and on thedegree of connectivity between the stream and the aquifer.The collection of a large quantity of data is required in order to

characterize all these factors, and this procedure can thus beapplied to evaluate exchange only for relatively short streamreaches. Caution is also required when trying to extrapolatethe information about the exchange at one field site in orderto evaluate the exchange at another site, because moderatedifferences in the hydraulic as well as the morphologic fea-tures between otherwise similar streams may cause differentexchange processes to control surface‐subsurface exchange.Therefore, there is a need for modeling methods that can beapplied in order to analyze the physical processes that con-trol hyporheic exchange dynamics, and to predict how theecosystem will respond to a particular anthropic modifica-tion (e.g., the alteration of the streamflow regime due to anupstream dam).[4] Because of the mentioned difficulties, a comple-

mentary approach to field studies is represented by thedevelopment of mathematical models based on the phys-ical principles that govern the exchange. The efforts ofresearchers in the last decade have explained the fundamentalmechanics of the exchange driven by different morphologicfeatures including bed forms [Elliott and Brooks, 1997;Packman and Brooks, 2001; Marion et al., 2002; Cardenasand Wilson, 2007; Boano et al., 2007, 2008], channelbends [Cardenas et al., 2004; Boano et al., 2006; Revelliet al., 2008; Cardenas, 2008b, 2009a], in‐stream structuresof 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]. Theseworks have contributed to explain the physics of hyporheicexchange across many different spatial scales. Unfortunately,there are still some exchange processes for which predictivemodels are unavailable because of the complex nature of themorphological and hydrodynamical factors that control the

1Department of Hydraulics, Transports, and Civil Infrastructures,Politecnico di Torino, Turin, Italy.

Copyright 2010 by the American Geophysical Union.0043‐1397/10/2009WR008317

WATER RESOURCES RESEARCH, VOL. 46, W07535, doi:10.1029/2009WR008317, 2010

W07535 1 of 14

http://dx.doi.org/10.1029/2009WR008317

exchange. This lack of predictive tools represents an obstacleto our understanding of the ecological and biochemical pro-cesses in fluvial environments.[5] Among the exchange processes that have not been fully

described it is possible to include the exchange induced bystream curvature of meandering streams (Figure 1a). Previoustheoretical [Boano et al., 2006;Revelli et al., 2008;Cardenas,2008b, 2009b] and experimental studies [Peterson andSickbert, 2006] have stressed the existence of a horizontalflow at the scale of the meander wavelength, which is qual-itatively depicted in Figure 1b (the shaded area denotes theinvestigated domain). The present work parallels and iscomplementary to the mentioned studies, as it investigates anexchange flow that occurs in sinuous streams at a differentspatial scale than those already studied. In particular, wefocus on the exchange that occurs at the smaller scale of thestream width because of the presence of point bars and of the

lateral slope of the stream surface (Figure 1c). The resultingflow is markedly three‐dimensional and occurs in the sedi-ment region beneath the streambed, which is not included inFigure 1b. An example of this exchange can be found in thework by Cardenas et al. [2004], where only a portion of thefull meander wavelength has been examined. Both exchangeflow fields pictured in Figures 1b and 1c are caused by thechannel sinuosity, and they can be analyzed separatelybecause of the difference between their spatial scales. Theoverall flow field in the complete domain (Figure 1d) can bethought 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 solutionfor the 3D field of the hydraulic head in the porous mediumand the resulting surface‐subsurface exchange flow in asinuous stream flowing on gravel sediments. To this aim, westart from an existing perturbative solution for surface flowand bathymetry in a meandering stream and we extend it inorder to obtain an analytical expression of the stream‐aquiferexchange. Although the approach necessarily relies on somesimplifying assumptions, the derived solution presents anumber of positive aspects that make it useful for the study ofsurface‐subsurface interactions. First, it predicts the spatialpattern of water flux through the streambed of a meanderingriver, providing indications for field investigators aboutwhere monitoring instruments should be placed in order tocapture the essential features of the exchange pattern. Second,a fundamental advantage of the analytical method is that itidentifies the geometrical and hydrodynamic quantities thatcontrol the exchange. Finally, the solution can be used asa benchmark case to test numerical approaches that areemployed to model complex field settings. Therefore, theproposed solution provides significant insight on the con-sidered exchange flow, and represents a further step towarda more comprehensive understanding of the interactionsbetween surface and subsurface waters.

2. Method

[7] In the present work, a perturbative approach is adoptedto obtain analytical solutions for both the surface and thesubsurface flow. This approach relies on the choice of adimensionless perturbative parameter whose value must besmaller than unity. Here, the dimensionless stream curvaturen is defined as the ratio between channel half‐width and twiceof the minimum curvature radius, and it is adopted as per-turbative parameter since its values are usually much smallerthan unity, as discussed in more detail below. The smallvalues of n imply that the perturbative approach representsa methodology that is particularly suited for the study of thisproblem, and that our findings will not be limited by thechoice of n as perturbative parameter.[8] The stream‐aquifer system shown in Figure 2 is con-

sidered. 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 symbolis hereinafter used to denote a dimensional quantity.[9] The stream is assumed to evolve in an aquifer com-

posed by alluvial sediments with the hydraulic conductivity~K. In order to simplify the calculations the hydraulic con-ductivity is treated as homogeneous, and its value is keptconstant in both space and time. The aquifer sediments extendfor a depth ~� on an impervious substratum that represents the

Figure 1. Conceptual sketch of surface‐subsurface interac-tions in ameandering stream. The river planimetry (a) inducesa complex pattern of water exchange that can be divided in(b) the quasi‐horizontal water exchange at scaleof the mean-der wavelength [Boano et al., 2006; Revelli et al., 2008;Cardenas, 2008b, 2009b], and (c) in the exchange flow at thescale of the channel width determined by the sediment pointbars and the lateral slope of the stream surface (present work).(d) The overall pattern of exchange flow is given by the com-position of the two flow fields.

BOANO ET AL.: SURFACE‐SUBSURFACE FLOW IN A MEANDERING STREAM W07535W07535

2 of 14

lower limit of the aquifer. The main geometrical features ofthe stream are the average slope Sb, the constant half‐width ~b,and the curvature ~C(~s), which is defined as the inverse of thecurvature radius of the river axis. In the present analysis, asimple sinusoidal function is adopted to describe the rivercurvature

~C ~sð Þ ¼ 2~R0cos ~�~sð Þ ¼ 1

~R0exp i~�~sð Þ þ c:c:; ð1Þ

where ~R0/2 is the minimum radius of curvature, ~� = 2p/ ~� isthe meander wave number, ~� is the meander wavelength, i isthe imaginary unity, and c.c. denotes the complex conjugate.The complex exponential notation is here preferred to thetrigonometric one because it simplifies the structure of theequations that are derived in our analysis.[10] In order to understand the relative importance of the

various morphodynamic and transport processes, all theinvolved quantities are normalized using some characteristicscales. The original reference system {~s, ~n, ~z} is thus con-verted to the new dimensionless reference system {s, n, z}that is defined as

s ¼ ~s~b

n ¼ ~n~b

� ¼~z�~�~D

if ~z > ~�~z�~�~�þ~� if ~z � ~�

(; ð2Þ

where ~�(~s, ~n) is the local elevation of the streambed and~D(~s, ~n) is the local stream depth. The river half‐width ischosen as the typical horizontal length scale, while two dif-ferent normalizations are used to define the dimensionlessvertical coordinate z in the surface and subsurface domain,respectively. The other dimensionless variables are defined ina similar way to (2), namely

� ¼ ~b � ~� � ¼~b~D0

D ¼~D~D0

� ¼ ~�~D0

� ¼ ~�~D0

; ð3Þ

where ~D0 is the average stream depth. Finally, the normalizedcurvature is

C ¼ ~b � ~C ¼ � exp i�sð Þ þ c:c: ¼ �C1 sð Þ þ c:c:; ð4Þ

where n = ~b/ ~R0 is the maximum dimensionless curvature. It isimportant to notice that for natural rivers the curvature radiusis 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 thecomplex geometry of our surface‐subsurface domain. Infact, the domain becomes a rectangular parallelepiped withs 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 valuesof z denote the points within the water column, while nega-tive values are associated with the subsurface domain. Thisgeometry is definitely simpler than the one of the originaldomain, which is bounded by the wavy surfaces shown inFigure 2. This switch to a rectangular domain is fundamentalto the derivation of the analytical solutions presented in thenext sections.

2.1. Surface Flow and Morphology

[12] In the last few decades several morphodynamicmodels have been developed for the solution of both the flowfield and the bed topography of a meandering river understeady conditions [Ikeda and Parker, 1989; Seminara, 2006].For the present analysis, we adopt a linear solution for thehydraulic head in the stream and for the bed topography. Forthis reason we only focus on the most complete known linearmorphodynamic solution, as provided by the recent theory ofZolezzi and Seminara [2001]. Although the mathematicalaspects of the theory have already been reviewed in previouspapers [Camporeale and Ridolfi, 2006; Camporeale et al.,2007; Frascati and Lanzoni, 2009], we briefly recall itspeculiar 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, theflow to be fully turbulent, while the sediments of the river bedare considered cohesionless and with a uniformly distributedgrain diameter, ~ds. (ii) Since the typical vertical scale (i.e., thewater depth ~D) is much smaller than the characteristic hori-zontal scale (i.e., the river half‐width ~b), the vertical velocitycomponent is neglected and a hydrostatic vertical pressuredistribution can be adopted. (iii) It is assumed that both the

Figure 2. (a) Cross section of the surface‐subsurface system. (b) A 3D scheme of the subsurface domainonly (without the free surface flow), with the streambed topography evidenced in shades of gray; the thickerlines delimit the cross‐section shown in Figure 2a. The scales of the axes are distorted in order to make thecomplex geometry of the domain more appreciable.


3 of 14

flow and bed topography instantaneously adjust to the pla-nimetry, that is, the process is considered as quasi‐stationary[De Vries, 1965]. Point (i) supports Exner’s equation forbed sediment, hypothesis (ii) justifies the use of the shallowwater equations which, thanks to point (iii), are made to betime‐independent.[14] Under the previous assumptions and introducing

the velocity decomposition of the flow field [Kalkwijk andDe Vriend, 1980], along with the no‐slip condition at thebottom and the no‐stress condition at the free surface, oneobtains the depth‐averaged two‐dimensional equations forshallowwater in curvilinear coordinates and in dimensionlessform [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 parametern of the governing equations, which introduced in the shallowwater equations (A1)–(A4), leads to a linear system withfour first‐order PDEs. Zolezzi and Seminara [2001] solvedthe linear system using a Fourier expansion in the transversaldirection, n, and obtained m systems of ODEs

Adx1m sð Þ

dsþ Bx1m sð Þ ¼ AmWk sð Þ m ¼ 0; . . . ;1ð Þ; ð5Þ

where Am = 2(−1)m/M2, M = (2m + 1)p/2, and

k ¼ C;@C

@s;@2C

@s2;@3C

@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 perturba-tions of the depth‐averaged velocity in both the longitudinaland spanwise directions, free surface elevation and depth,respectively. It follows that x1(s, n) =

P1m¼0 x1m(s)sin(Mn).

The entries of the 4 × 4 matrices A, B, W are reported byCamporeale et al. [2007].[16] For the solution of the subsurface flow we need to

impose a boundary condition on the bed topography (i.e., atz = 0) for the hydraulic head of the porous media, hp(s, n, z),namely the Dirichlet condition

hp s; n; 0ð Þ ¼ H0 s; nð Þ þ �X1m¼0

H1m sð Þ sinMn; ð7Þ

where the function H1m(s) is expressed, in analogy withequation (4), as

H1m sð Þ ¼ hmei�s þ c:c: ð8Þ

[17] In order to apply the normalization defined byequation (2) to the subsurface equations we also need thesolution 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 hmand dm was provided by Seminara et al. [2001] in the fol-lowing form

hm ¼ F2Am

P6j¼0 i�ð Þ j hð Þ

jþ1P5j¼1 i�ð Þ j�1j

þ b2 þ i�b3 � �2b5

!; ð9Þ

dm ¼ F2Am

P6j¼0 i�ð Þ j dð Þ

jþ1P5j¼1 i�ð Þ j�1j

þ b2 þ b4F�2 þ i�b3

� i�b6F�2 � �2b5

!; ð10Þ

where the coefficients bj, rj and sj depend on a, b, the Shieldstress �, and the dimensionless sediment diameter ds = ~ds/~d0,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 tosolve the equations of subsurface flow, as shown in the fol-lowing section.

2.2. Subsurface Flow

[19] The water flow in a homogeneous and isotropicporous medium is governed by the Laplace equation, whichin the intrinsic reference system {~s, ~n, ~z} is written as[Batchelor, 1967]

N @2~hp@~s2

þN 3 @2~hp@~n2

þ N 2

~� þ ~�

@2~hp@~z2

� ~n@ ~C

@~s

@~hp@~s

þN 2 ~C@~hp@~n

¼ 0;

ð11Þ

where ~hp represents the hydraulic head in the subsurfaceporous domain andN (s, n) = 1 + n C(s) is a metric factor thatarises from the change of coordinates.[20] The equation is solved in the subsurface domain

shown in Figure 2. Tonina and Buffington [2007] haveobserved that pressure on sediment bars that are fully sub-merged is primarily hydrostatic with negligible contributionsfrom dynamic pressure, in agreement with the hypothesis (ii)of Section 2.1. The reason for this behavior is that meanderpoint bars have height‐to‐wavelength ratios that are muchlower than dunes and that reduce the probability of flowseparation. Therefore, we assume that the flow is forced bythe hydrostatic head distribution at the streambed that derivesfrom the level of the stream surface, and that is given byequation (7). This procedure reflects the assumption that thecoupling between surface and subsurface flow occurs in onedirection only, that is, surface flow drives water flow in thesediments but it is not significantly influenced by subsur-face flow. This represents a common assumption in modelsof water flow over porous sediments [e.g., Cardenas andWilson, 2007], and provides a good description of the mainflow characteristics as long as exchange water fluxes aremuch smaller than stream discharge.[21] The bottom of the subsurface domain is treated as an

impervious boundary, and periodic boundary conditions areimposed on s = 0 and s = l. Finally, no‐flow boundaryconditions are imposed at the side boundaries n = ±1 in orderto model a stream that does not gain or lose any net amount ofwater. Since equation (11) is linear with respect to ~hp, the caseof a gaining or losing stream can be easily determined by thesuperposition of the present solution with a solution of thehead and flow field in a stream in non‐neutral conditions [seeBoano et al., 2008]. All the boundary conditions are sum-marized in Table 1.[22] The governing equation (11) is written according to

the dimensionless form (2), and a solution is sought in theform hp(s, n, z) = hp0 + nhp1. In a similar way, the river cur-


4 of 14

vature and the streambed topography are expressed as C(s) =nC1(s) and h(s, n) = nh1(s, n), respectively. Substitution inequation (11) and the application of (2) yield, at the zerothorder O(n0)

�2@2hp0@s2

þ �2@2hp0@n2

þ �2 @2hp0@�2

¼ 0: ð12Þ

The solution of equation (12) that satisfies the boundaryconditions in Table 1 is hp0(s) = −bSb s. Thus, at the zerothorder (i.e., neglecting the effect of curvature) the hydraulichead in the subsurface simply decreases along the streambecause 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 asinusoidal structure in s, namely hp1 =Hp1(n, z)exp(ias) + c.c.This analytical structure is consistent with the boundaryconditions at s = 0 and s = l (see Table 1). This choice leads toa couple of equations of order n, the first of which is

�2@2Hp1

@n2þ �2 @

2Hp1

@�2� �2�2Hp1 þ i nSb��

2 ¼ 0 ð13Þ

and the second one is its complex conjugate. The last termon the right hand side of (13) derives from the zeroth ordersolution hp0.[24] In order to achieve the solution, the unknown function

Hp1 is expressed as a Fourier series Hp1(n, z) =P1

m¼0 Hp1m(z)sin(Mn), which satisfies the no‐flow conditions on the sideboundaries (see Table 1). We also make use of the expansionn =

P1m¼0 Am sin(Mn), and from equation (13) we obtain

�2H0 0p1m

�ð Þ �M2�2Hp1m �ð Þ � �2�2Hp1m �ð Þ þ iAmSb��2 ¼ 0:

ð14Þ

This simple, second‐order ordinary differential equation inHp1m(z) is coupledwith the boundary conditionsH′p1m(z =−1) =0 and Hp1m(z = 0) = hm, the latter of which provides the linkbetween the subsurface flow and the surface head variationsinduced by the stream curvature. Its solution is

Hp1m ¼ am þ bm cosh cm 1þ �ð Þ½ �; ð15Þ

where

am ¼ iAmSb��

M2 þ �2; bm ¼ hm � am

cosh cmð Þ ; cm ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 þ �2

p

�ð16Þ

are three algebraic coefficients whose values depend on m.

[25] Putting together the previously described expression,the first‐order solution for the hydraulic head beneath thestreambed can be expressed as

hp s; n; �ð Þ ¼ ��Sbsþ �exp i�sð ÞX1m¼0

am þ bm cosh cm 1þ �ð Þð Þ½ �

� sin Mnð Þ þ c:c: ð17Þ

[26] It is interesting to notice that hp is related to the streamelevation hm through the coefficients defined in (16) but isapparently independent on the shape of the streambed. Thehydraulic head hp1 actually depends on the streambed shapethrough the dimensionless coordinate z, but the adoption ofthe rectangular domain allows to express hp1 with the simplestructure of equation (17).

2.3. Surface‐Subsurface Exchange Flux

[27] Once the hydraulic head is known, it can be used toevaluate the magnitude of the exchange flux between thesurface and the subsurface domain. The seepage velocitiescan be obtained from the Darcy law, which requires theknowledge of the hydraulic gradient. This gradient is givenby [Batchelor, 1967]

r~hp ¼ N�1 @~hp@~s

;@~hp@~n

;@~hp@~z

( ); ð18Þ

where the head is function of the intrinsic curvilinearcoordinates {~s, ~n, ~z}. If we make use of the chain rule forderivation we can express the hydraulic gradient as a func-tion of the dimensionless head in the rectangular domainhp(s, n, z)

r~hp ¼ 1

�N@hp@s

� 1þ �

� þ �

@hp@�

@�

@s

� �;1

�

@hp@n

� 1þ �

� þ �

@hp@�

@�

@n

� �;

�

� 1

� þ �

@hp@�

�: ð19Þ

[28] The Darcy velocity at the streambed interface is thengiven by

~vj�¼0 ¼ �~K r~hp��¼0

¼ ~v0 þ �~v1; ð20Þ

where the Darcy velocity is decomposed into its zeroth‐ andfirst‐order components ~v0 and ~v1, respectively. These com-ponents can be evaluated after the introduction of the decom-position hp = hp0 + nhp1 in (19) and (20)

~v0 ¼ �~K

�

@hp0@s

; 0; 0

� �

~v1 ¼ �~K

�

@hp1@s

þ nC1@hp0@s

� �; �

~K

�

@hp1@n

; �~K

�

@hp1@�

� �; ð21Þ

Table 1. Boundary Conditions Coupled to Equation (11) for the Subsurface Flow Problema

Boundary

Top (z = 0) Bottom (z = −1) Sides (n = ±1) Upstream and Downstream (s = 0, s = l)

Complete hp = −bSbs + neiasP

m hm sin(Mn) + c.c. hp,z = 0 hp,n = 0 hp|s=0 = hp|s=l + bSbl, hp,s|s=0 = hp,s|s=lO(n0) hp0 = −bSbs hp0,z = 0 hp0,n = 0 hp0|s=0 = hp0|s=l + bSbl, hp0,s|s=0 = hp0,s|s=lO(n1) hp1 = eias

Pm hm sin(Mn) hp1,z = 0 hp1,n = 0 hp1|s=0 = hp1|s=l, hp1,s|s=0 = hp1,s|s=l

aA comma followed by s, n, or z denotes the derivative with respect to the corresponding variable.


5 of 14

where we have made use of the first‐order Taylor expan-sion 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 topographicsurface of the streambed, which is provided by the function~f (~s, ~n,~z) : ~�(~s, ~n) −~z = 0. The normal vector to this surface thatis directed toward the subsurface is given by N = r~f /~f .Following the same approach that has lead to (21) we canwrite

N ¼ N0 þ �N1 ¼ 0 ; 0 ; �1f g þ ��1;s�

;�1;n�

; 0

� �: ð22Þ

[30] The water flux through the bed surface is defined as

~q ~s; ~nð Þ ¼ ~vj�¼0 �N ¼ ~v0 þ �~v1ð Þ � N0 þ �N1ð Þ ð23Þ

and from equations (21)–(22) it results

~q0 ¼ ~v0 � N0 ¼ 0 ;

~q1 ¼ ~v1 � N0 þ ~v0 � N1 ¼~K

�

@hp1@�

�~K

�2

@hp0@s

@�1@s

; ð24Þ

where the two terms on the right hand side of the secondequation represent the vertical flux due to the variations ofthe level of the stream surface and the horizontal flux due tothe average stream slope, respectively. Finally, the introduc-tion of equation (17) in (24) provides the desired analyticalsolution for the exchange flux between the surface and thesubsurface domain

~q ¼ �~K exp i�sð ÞX1m¼0

bmcm sinh cmð Þ�

þ i�Sb�m�

� �sin Mnð Þ þ c:c:

ð25Þ

The exchange flux across the streambed surface can also beexpressed in dimensionless form as q = ~q/~K.

2.4. Controlling Parameters

[31] The first‐order solution given by equation (25) showsthat there are six dimensionless parameters whose valuesgovern the dimensionless exchange flux q. For homogeneoussediments, these parameters are sufficient to completelydescribe the characteristics of the complex surface‐subsurfacedomain shown in Figure 2. In particular, there are onlyfour parameters that summarize the geometrical featuresof the stream‐aquifer system, i.e., the stream sinuosity n, themeander wave number a, the stream aspect ratio b, and thedepth of the impermeable bedrock g. Moreover, the influence

of the hydrodynamical characteristics of the stream on thedimensionless exchange q is described by the relative rough-ness ds and the Shields stress �.[32] Natural planimetries of unconstrained meandering

streams commonly display an alternation of low‐sinuosityreaches and mature meander lobes. This alternating patternresults from the interplay between sedimentation and erosionprocesses, which tend to increase stream sinuosity and cur-vature, and the occasional occurrence of meander cutoffs,which suddenly reduce channel curvature. The effect of thismorphodynamic activity is that the stream exhibits variationsof local curvature among different reaches.[33] The study of the average values of curvature and

other stream morphometric parameters is one the subjects ofstream geomorphology. For instance, a number of researchers[Leopold andWolman, 1960; Leopold et al., 1964; Braudricket al., 2009] have proposed empirical scaling relationshipsthat relate channel wavelength and width as

~� � 20� 28ð Þ~b: ð26Þ

Additionally, Camporeale et al. [2005] have also found that

~� � 2:47j~Cavj�1; ð27Þ

where |~Cav| is the average absolute curvature. For a channelwith sinusoidal planimetry, integration of equation (1) showsthat ~R0 is related to the mean curvature by |~Cav| = 4/(p~R0).Putting all these expressions together, values of n = 0.07–0.10 are found, depending on the different coefficients inequations (26). These values represent average conditionsfrom which curvatures values of single river reaches candiverge, but the order or magnitude is expected to hold formost meandering streams. This confirms that natural mean-dering streams usually have n � 1, and that n is suited to beadopted as perturbation parameter.[34] Typical ranges of values for the controlling param-

eters n, a, b, ds, and � are reported in Table 2. Values ofb, ds, and � have been calculated from a database of 60 rivers[van den Berg, 1995], considering only monocursal mean-dering streams flowing on sediments with median diam-eter larger than 2 mm (i.e., gravel bed rivers). Values of n anda have been obtained from the geomorphologic scalingequations (26)–(27).

3. Results

[35] The properties of the exchange flow field predictedby the described modeling approach are now discussed. Inthe following examples we consider the reference case ofa 30‐meters‐wide stream with an average depth ~D0 = 1 m,that flows on a gravel sediment bed (~ds = 10 mm) with slopeSb = 2 · 10−3. The stream is gently meandering (~R0 = 500 m),with a wavelength ~� = 470 m. A large value of sediment layerthickness (~� = 1000 m) is chosen in order to reproduce thecase of a hyporheic zone that is not constrained by an im-pervious bedrock. The dimensionless parameters that sum-marize 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

Table 2. Range of Typical Values of Parameters n, a, b, ds, and �for Gravel Bed Sinuous Streamsa

Range Source

n 0.07–0.10 1, 2, 3, 4a 0.22–0.31 1, 2, 3b 2.6–171.9 5ds 1.5 · 10−3–2.0 · 10−1 5� 0.05–0.89 5

aSources: 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].


6 of 14

maximum curvature n that is chosen as perturbative param-eter. It is then necessary to assess the upper limit of n forwhich our first‐order solution can still be regarded as a goodapproximation of the complete problem, i.e., equation (11).[37] In order to achieve this aim, we first evaluate the

dimensionless exchange flux q = ~q/~K with the first‐ordersolution provided by equation (25). The reach‐averageddimensionless flux q that flows within the sediments is thenevaluated as

R|q(x, y)|dxdy/(2A), where A is the streambed

area and the factor 1/2 is introduced to avoid counting theupwelling and downwelling fluxes as separate contributions.This result is compared to the average dimensionless fluxobtained from a numerical solution of the complete Laplaceequation (11). This numerical solution is obtained with afinite element approach that adopts a LU‐factorization

scheme to solve the equation on a triangular mesh. Becauseof the domain shape an anisotropic mesh is adopted, with ahigher density of nodes in the streamwise direction. Thechosen number of tetrahedral elements ranges between 1000and 1300 depending on the value of n.[38] The comparison between the average dimensionless

flux q resulting from equation (25) and from the solution ofthe complete Laplace equation is shown in Figure 3 for twodifferent sets of parameters. The first set corresponds to thechosen reference case, and the second is identical with theonly exception of a higher dimensionless wave number a =0.8, which corresponds to a stream with smaller meanderwavelength. In both cases, the dimensionless curvature nis varied between 0 and 0.1, thus reproducing the effect ofincreasing channel sinuosity.[39] The comparison between the first‐order solution given

by equation (25) and the numerical solution of the completemodel shows that our solution is a good approximation of theexchange up to n = 0.10, with a 30%maximum relative error.It can be seen from Table 2 that typical values of dimen-sionless curvature are usually lower than this limit, whichmeans that the higher‐order corrections that are neglected inour linear approach have a minor influence on the surface‐subsurface exchange.

3.2. Properties of the Exchange Flux Pattern

[40] The solution given by (25) is now used to explore themain properties of the coupled surface‐subsurface exchange.The typical spatial pattern of the exchange flux for the chosenreference case is shown in Figure 4a. Positive and negativevalues of q represent water downwelling and upwelling,respectively. Figure 4a shows that the water exchange ismainly concentrated near the stream banks, and in particularin correspondence of the bends where the stream curvature isthe highest. Conversely, in the straight parts of the stream thevalues of the exchange flux q are much lower than at thebends.[41] It is important to notice that this pattern is not directly

related to the streambed morphology, which is presented inFigure 4b. Figure 4b shows that in the reference case sedi-

Figure 3. Comparison between the first‐order approxima-tion (continuous lines) and the total reach‐averaged exchangeflux (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).

Figure 4. Spatial patterns of (a) dimensionless exchange flux q and (b) streambed elevation ~� (m) for thereference case (n = 0.03, b = 15, g = 1000, ds = 0.01, � = 0.14, and a = 0.2). The horizontal scales have beendistorted in order to make the spatial variations more appreciable.


7 of 14

ment point bars develop at the inner banks just downstreamof each bend, and they extend over the straight reaches up tothe next bend. Simulations with different parameter values(not reported) have shown that the location of point bars ismuch more sensitive to channel curvature and geometrythan the exchange flux pattern, which always retains themain features of Figure 4a. The comparison of the two pa-nels of Figure 4 reveals that the exchange flux q is in phasewith the curvature of the river even though the streambedelevation h is not. In other words, this example shows thatthe streambed morphology and the exchange flux are onlypartially correlated, although their features are both con-trolled by the surface flow.[42] The pattern of the total free surface elevation ~H , which

is the sum of the zeroth‐ and first‐order contributions, isdisplayed in Figure 5a. An arbitrary datum has been chosenin order to make stream elevation always positive over thestream reach. The free surface declines along the streamwisedirection because of the channel slope. Moreover, the streamsurface also presents slight spatial variations that are clearlyevidenced in Figure 5b. Figure 5b shows the first‐orderelevation of the free surface, which represents the relativedifference between local and average stream depth. Thepresence of zones with higher (lower) surface elevation at theouter (inner) banks can be observed in Figure 5b. The mag-nitude of these local variations is of the order of a few cen-timeters, which makes them almost undetectable in Figure 5a.The pattern of first‐order stream surface elevation is clearlycorrelated 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 streamsurface elevation (bmcm sinh(cm)/g) and the horizontal fluxthrough the uneven bed surface caused by the average streamgradient (iaSbhm/b). These two contributions correspond tothe first and second term in the last of equations (24),respectively. In order to thoroughly explore the connectionsbetween spatial patterns of river morphology and surface‐subsurface exchange, we have randomly selected one hun-dred sets of values of the parameters {n, b, a, g, ds, �} withinthe ranges shown in Table 2 and we have then evaluated the

two terms of equation (25). The analysis of these resultsreveals that the second term is always much smaller than thefirst one. This means that the second term in equation (25) canbe dropped without affecting the estimate of the exchangeflux. Since the prevailing term in (25) is not explicitly relatedto the streambed elevation h, this finding implies that themagnitude of the exchange is not significantly influenced bythe precise shape of the streambed. Even though the bedmorphology is linked to the free surface by the shallow waterequations (A1)–(A4), our results show that its influence onthe exchange flux is only indirect (i.e., through the freesurface elevation). This result suggests that the qualitativepattern of sinuosity‐induced vertical water exchange shownin Figure 4a is expected to hold in the majority of naturalsinuous streams.[44] Figure 6a shows the pathlines of exchanged stream

water particles for the chosen reference case. The absence of ashallow impervious layer allows water particles to penetratedeep into the sediments up to a depth of approximately 160m.This deep exchange is a consequence of the large scales thatcharacterize the meander geometry and the hydraulic gradientthat drive the hyporheic flow. However, the depth of thehyporheic zone is only a fraction of the total thickness ofthe sediment layer ~� because of the confining action of thehorizontal underflow. In the reference case, water particlesthat enter the hyporheic zone through the streambed mainlyflow in the streamwise direction, as shown in Figure 6b.This behavior depends on the no‐flow boundary conditionsimposed on the domain sides. If the intrameander flow fielddepicted in Figure 1c is summed to the present flow field,water particles that are close to the domain boundaries wouldactually flow outside the consider domain in the intrameanderarea as sketched in Figure 1b. Nonetheless, Figure 6 is rep-resentative of pathlines in the central part of the stream.

3.3. Influence of Controlling Parameters

[45] A parametric analysis is performed to assess theinfluence of each dimensionless parameter on the exchangeflux. Both terms of equation (25) are kept in the follow-ing computations in order to obtain the highest possible

Figure 5. Spatial patterns of (a) elevation of the stream surface ~H (m) and (b) first‐order contribution~H1 (m) for the reference case (n = 0.03, b = 15, g = 1000, ds = 0.01, � = 0.14, and a = 0.2). The horizontalscales have been distorted in order to make the spatial variations more appreciable.


8 of 14

precision. The average dimensionless flux is plotted againstthe 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 thenvaried the value of each parameter within the range reportedin Table 2 in order to investigate its influence on the averagedimensionless exchange q.

[46] The geometry of the stream‐aquifer system is sum-marized by the curvature n, the aspect ratio b, the meanderwave number a, and the sediment depth g. The stream cur-vature n is tightly related to the river sinuosity, which pro-gressively increases because of the erosion of the outer banksand the sedimentation at the inner banks. This process leadsto an increase in the amplitude of the sinusoidal planimetry of

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, anda = 0.2). (a) Full subsurface domain and (b) detail of the zone interested byhyporheic flow. Flow direction is toward increasing x values. Scales have been distorted in order to bettershow the main features of the flow field.


9 of 14

the river. Since the first‐order solution (25) for the dimen-sionless flux increases linearly with the curvature n, theevolution of the river morphology determines an increase inthe average flux.[47] The relationship between the aspect ratio b and the

average dimensionless flux q is pictured in Figure 7a, whichshows that q is a decreasing function of b. If the otherparameters are kept constant, this relationship implies thatlarger rivers tend to exchange less water with the sedimentsthan smaller ones. This happens because the exchange isdriven by the head gradient between the banks, and largervalues of b correspond to an increase of the river width andthus to a lower hydraulic gradient. While the general rela-

tionship between q and b can be approximated by a powerlaw function, Figure 7a shows that this behavior is madesomewhat irregular by the presence of a deviation from anotherwise straight line (in log‐log scales). This deviation is anexample of the complex structure of equation (25), whichpredicts 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 observedfrom Figure 7b that for meanders with longer wavelengths(i.e., decreasing values of a) the dimensionless flux qincreases up to a maximum value. A further decrease in aleads to a slight decrease in q that eventually tends to aconstant value. This happens because q represents the average

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 evaluationof q as a function of the number of modes used in equation (25).


10 of 14

value of the flux over the streambed area, and both quantitiesincrease with decreasing a. For large values of the wavenumber a the increase in the flux prevails over the increase inarea, while the opposite occurs when a is low.[49] The role of the dimensionless thickness g of the sub-

surface domain is shown in Figure 7c. The flux first increaseswith the depth of the sediment layer and then reaches anasymptotic constant value. It is thus possible to make a dis-tinction between shallow beds, for which q/K depends on thesediment depth, and deep beds, for which the bedrock is toodeep to influence the exchange. The minimum value of thebed thickness of a deep bed must be proportional to the riverhalf‐width ~b, as it represents the characteristic length scale ofthe exchange. Thus, the dimensionless threshold value of gfor a deep bed must be proportional to b, and Figure 7csuggests that for g > 2.5b (i.e., ~� > 2.5~b) the sedimentthickness does no longer influence the exchange.[50] Beside the geometrical features, the exchange is also

influenced by the hydrodynamical characteristics of thestream; these characteristics are summarized by the dimen-sionless Shields stress � and the dimensionless roughness ds.Figure 7d shows that the average flux is a linear function ofthe Shields stress, with higher rates of exchange associated tohigher values of the stress. Since the Shields stress is pro-portional to the stream slope, raising the value of � whilekeeping constant the values of the other parameters results inan increase in the average stream velocity and thus in thedimensionless exchange flux.[51] Figure 7e shows that the dimensionless average flux

increases with the dimensionless grain diameter ds, and it isapproximately described by a power law relationship. Thebehavior of Figure 7e can be understood if we recall thatthe differences in the free surface elevation that drive theexchange are highly sensitive to the stream secondary cur-rents, which in turn increase with the average stream velocity.The dimensionless grain diameter ds is a measure of the rel-ative roughness of the river bed and is related to the Shieldsfactor � = Sb/(Dds), whereD represents the specific weight ofthe submerged sediment grain relative to water. Thus, largevalues of the stream slope Sb must be considered in order toincrease the value of dswhile keeping constant the value of �.Thus, the high values of ds in Figure 7e represent the behaviorof steep streams, whose swift flows are characterized byintense secondary currents that enhance the water exchangethrough the sediment surface.

3.4. Convergence of Solution

[52] The estimation of the average dimensionless flux qwould require the evaluation of an infinite number of modesin the sum in equation (25), which is instead replaced by asum over a finite number of modes. The influence of thetruncation of the sum is shown in Figure 7f, which displaysthe relative error (q − q*)/q* between the dimensionless flux qand the exact value q* (calculated with 500 modes) as afunction of the chosen number of modes. Even though it isbetter not to reduce the number of modes too much in orderto limit the introduction of errors in the flux estimation,Figure 7f shows that the truncation only leads to a slightunderestimation of q. It should be recalled that the errors inthe estimates of the hydraulic conductivity and the othergeometrical and hydraulic parameters are usually much larger

than the errors introduced by the use of a finite number ofmodes, which are of the order of 10−2 or even less.

4. Discussion and Conclusions

[53] In the present paper we have adopted a perturbativeapproach to solve the physically based equations that describethe surface and the subsurface flow of water in a sinuousstream‐aquifer system. In particular, we have derivedan analytic first‐order solution for the water flux that isexchanged through the streambed. It is important to recallthat the solution we have found is valid for a simplifiedstream‐aquifer system, without many of the complexitiesthat are commonly found in natural meandering streams. Theimplications of these simplifications are now discussed inorder to establish the relevance – as well as the limits – of ourfindings.[54] A first issue that characterizes the described method is

that it only considers water exchange through the streambedthat is induced by channel sinuosity. Thus, the predictedexchange pattern does not include the effect of otherexchange processes that can be present. However, the over-all exchange problem can always be seen as the sum of dif-ferent exchange processes at different scales, and an estimateof the overall flow field can be obtained by composition of thesingle velocity fields. This is the case exemplified in Figure 1for the presently examined exchange flow and lateralhyporheic exchange (as in the work by Revelli et al. [2008] orCardenas [2009a]). Other exchange processes can be treatedin the same way as long as predictive models for such pro-cesses are available. For instance, bed forms do not developon the gravel bed streambeds considered in the present work,but turbulent eddy penetration can contribute to surface‐subsurface exchange. In streams with finer sand sedimentsturbulent exchange would play a less important role, whilebed forms would give a significant contribution to the overallhyporheic exchange. In this framework, the analysis pre-sented in this work covers a type of surface‐subsurfaceinteraction which has received very little attention so far.Although the feasibility of this approach has still to berigorously tested, the increasing body of knowledge deriv-ing from theoretical and field studies on surface‐subsurfaceinteractions will hopefully make it a viable tool for predictionof hyporheic exchange in streams.[55] Among themodel assumptions, the most critical one is

probably the adoption of homogeneous hydraulic conduc-tivity, that is implicit in the use of the Laplace equation (11).In contrast, most streambeds show some degree of hetero-geneity [e.g., Ryan and Boufadel, 2007; Genereux et al.,2008], and the resulting spatial variations of hydraulic con-ductivity are expected to alter the subsurface head field aswell as the exchange pattern. Deviations between modeledand actual values of the exchange flux will depend on thelevel of heterogeneity of the sediments, and will thus varyamong different stream‐aquifer systems. Some comments onthese deviations can be drawn from the comparison betweenour dimensionless flux q = ~q/~K – which is equivalent to astreambed head gradient ‐ and the patterns of head gradientobserved by Kennedy et al. [2009] in West Bear Creek, ahighly heterogeneous stream in an agricultural catchment.This stream gained water from the adjacent floodplain andwas mostly straight, with only a small subreach (referred to as


11 of 14

“August small reach” in the work by Kennedy et al. [2009])characterized by a mild sinuosity. In our framework, theeffect of the gaining conditions would be to add an upwellingcomponent to the pattern of q shown in Figure 4, leading to ahigher dimensionless flux at the inner bank than at the outerbank. This behavior qualitatively agrees with the pattern ofhead gradient in the August small reach of West Bear Creek[Kennedy et al., 2009, Figure 5, Table 3], where the stream-bed head gradient exhibited a clear transversal profile withincreasing values from the outer to the inner bank. It is alsointeresting to notice that this profile was not observed inthe straight reaches of the stream. Instead, head gradientvalues in these reaches were always symmetrical with respectto the channel axis, coherently with the absence of sinuosityand with the modeling results described by Boano et al.[2009]. These considerations suggest that the streambedhead gradients (i.e., the dimensionless flux q) predicted byequation (25) are qualitatively correct even for moderatelyheterogeneous streambeds. However, information about thespatial pattern of ~K would be necessary in order to evaluatethe actual fluxes ~q. This implies that increasing our under-standing on the factors that control streambed heterogeneitywill be a crucial issue in future hyporheic zone research.[56] Despite the mentioned simplifications, the pro-

posed solution retains many significant characteristics of theexchange. The comparison with a numerical solution of thecomplete problem has shown that for reasonable values ofthe stream curvature the first‐order solution represents a validdescription of the exchange flow. The resulting exchange fluxexhibits a peculiar spatial pattern, with the upwelling anddownwelling of water concentrated near the outer and innerbanks of the stream, respectively. While the exchange patternalways exhibits these qualitative features, the magnitude ofthe exchange flux is influenced by the geometry of thestream‐aquifer system and the hydrodynamic action of thesurface flow. The present analysis has shown that theseproperties can be grouped together in order to form thedimensionless parameters that completely summarize thephysics of the surface‐subsurface system (normalized streamcurvature, stream half‐width‐to‐depth ratio, meander wavenumber, Shields stress, relative sediment roughness, nor-malized sediment thickness). These parameters control boththe exchange flux and streambed morphology.[57] A careful inspection of the dependence of the

exchange flux from the governing parameters has revealedthat the exchange pattern is in phase with the stream curvatureeven when the streambed morphology is not. This behavioroccurs because the major contribution to the flux is given bythe spatial differences of the free surface elevation, while therole of streambed shape on the exchange is less relevant. Thisfinding suggests that for sinuous rivers it is not necessary toobtain detailed information on the largest scales of streambedtopography, i.e., meander point bars. However, it is clear thatriver beds can also present dunes and other morphologicalfeatures that are known to determine phenomena of flowseparation and are therefore more strongly coupled to thestream hydrodynamics.[58] It is interesting to notice that the differences in free

surface elevation that drive the exchange flow can be so smallto be hardly measurable in the field. Nonetheless, the dis-cussed exchange process can drive water flow deep into thesediments, supporting the connectivity between surface water

and the underlying alluvium. Thus, the description of theexchange at the scale of the meander wavelength presentedin this work provides new insights on the complex spectrumof hydrologic processes that control the exchange with thehyporheic zone.

Appendix A: Summary of the 2D Shallow WaterEquations in Curvilinear Coordinates

[59] In steady flow conditions, the depth‐averaged two‐dimensional equations for shallow water in curvilinearcoordinates and in dimensionless form are

NUU ;s þVU ;n þNCU V þ 2’ð Þ þ NF2

H ;s � �Cf þ ��sD

þ 1

DUD’ð Þ;n ¼ 0 ðA1Þ

NUV ;s þVV ;n þH ;nF2

þ ��nDþN

DDU’ð Þ;s þ 2

DVD’ð Þ;n

þ 1

D’1Dð Þ;n þNC’2 ¼ 0; ðA2Þ

where the comma followed by s or n denotes the derivativewith respect to the corresponding direction. Equations (A1)–(A2) have to be coupled with the continuity equation for thewater and bed sediment [Exner, 1925], respectively,

N DUð Þ;s þ DVð Þ;n þNCDV ¼ 0; ðA3Þ

N qs;s þ qn;n þNCqn ¼ 0: ðA4Þ

[60] In equations (A1)–(A4), U and V are the longitudinaland transversal depth‐averaged velocity,N (s, n) = 1 + n C(s)is the longitudinal metric factor, D is the depth, H is the freesurface elevation, t ≡ {ts, tn} is the bed stress vector, q ≡{qs, qn} is the volumetric vectorial bed load, Cf is the frictionfactor, and F is the Froude number. Moreover, ’ = hFv0i,’1 = hv02i, and ’2 = 2V’ − U2 + ’1, where brackets refer todepth averaging, F (z) is the vertical profile of velocity, andv0(s, n, z) is the recirculating secondary current driven bycurvature and with vanishing depth average.[61] The following boundary and integral conditions are

also imposed to equations (A1)–(A4)

V ¼ qn ¼ 0 n ¼ �1ð Þ; ðA5Þ

Z 1

�1UD dn ¼ 2;

Z �

0

Z 1

�1H � Dð Þ dn ds ¼ const; ðA6Þ

where equation (A5) imposes the zero‐net‐flux conditionbetween the center and the sidewall layers and no sedimenttransport across the sidewalls, whereas equations (A6) set thecondition that the water discharge and the average reach slopeare 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 bedstress vector is considered aligned with the near‐bed velocityvector and it can therefore be expressed through a localfriction coefficient; ii) the dynamic equilibrium of the bed


12 of 14

sediment written in an orthogonal reference system mustbe set; iii) the secondary currents v0 is resolved by anapproximated iterative solution of the transversal momentumequation (A2) (see Zolezzi and Seminara [2001] for details).

Notation

Latin symbolsam coefficient of hp in equation (17).~b river half‐width.bj coefficient of hm in equation (9)–(10).bm coefficient of hp in equation (17).cm coefficient of hp in equation (17).dm m‐th mode of D solution, equation (10).

~ds, ds sediment grain diameter (ds = ~ds/ ~D0).hm m‐th mode of H solution, equation (9).

~hp, hp hydraulic head in the porous medium(hp = ~hp/~d0).

i imaginary unit.~n, n spanwise coordinate (n = ~n/~b).~q, q hyporheic exchange flux (q = ~q/~K).

q reach‐averaged dimensionless exchangeflux.

~s, s streamwise coordinate (s = ~s/~b).~b vector of Darcy seepage velocity.~z vertical coordinate.A matrix of coefficients in equation (5).Am 2(−1)m/M2.B matrix of coefficients in equation (5).

~C, C stream curvature (C = ~C · ~b).|~Cav| average absolute stream curvature.~D, D stream depth (D = ~D/~D0).~D0 average stream depth.F Froude number of surface flow.

~H , H stream surface elevation (H = ~H /~D0).Hp1 hp1(s, n, z) = Hp1(n, z)exp(ias).~K sediment hydraulic conductivity.M (2m + 1)p/2.~N normal vector to streambed surface.N metric factor for change of reference

system.Sb average streambed slope.~R0 twice of minimum radius of stream

curvature.U dimensionless streamwise velocity of

surface flow.V dimensionless spanwise velocity of

surface flow.Greek symbols

~�, a meander wave number (a = ~� · ~b).b stream aspect ratio (~b/~D0).D ratio between specific submersed weight

of sediments and water specific weight.~�, g thickness of sediment layer (g = ~�/~D0).~�, h streambed elevation (h = ~�/~D0).hm m‐th mode of h solution (hm = hm − dm). vector of coefficients in equation (5).

~�, l meander wavelength (l = ~�/~b).n dimensionless stream curvature (~b/~R0).� Shields stress (~D0Sb)/(D~ds).rj coefficient of hm in equation (9)–(10).sj coefficient of hm in equation (9)–(10).W matrix of coefficients in equation (5).

z dimensionless vertical coordinate.Subscripts

0, 1 Zeroth‐ and first‐order components (e.g.,hp = hp0 + nhp1).

m m‐th Fourier mode (e.g., Hp1(n, z) =P1m¼0 Hp1m(z)sin(Mn)).

[63] Acknowledgments. The suggestions and constructive commentsprovided by the Associate Editor Aaron Packman and by three anonymousreviewers are gratefully acknowledged.

ReferencesBatchelor, 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 controlson organic carbon fluxes in fluvial networks, Nat. Geosci., 1(2), 95–100.

Boano, F., C. Camporeale, R. Revelli, and L. Ridolfi (2006), Sinuosity‐driven hyporheic exchange in meandering rivers, Geophys. Res. Lett.,33, L18406, doi:10.1029/2006GL027630.

Boano, F., R. Revelli, and L. Ridolfi (2007), Bedform‐induced hyporheicexchange with unsteady flows, Adv. Water Resour., 30(1), 148–156.

Boano, F., R. Revelli, and L. Ridolfi (2008), Reduction of the hyporheiczone volume due to the stream‐aquifer interaction, Geophys. Res. Lett.,35, L09401, doi:10.1029/2008GL033554.

Boano, F., R. Revelli, and L. Ridolfi (2009), Quantifying the impactof groundwater discharge on the surface‐subsurface exchange, Hydrol.Processes, 23, 2108–2116, doi:10.1002/hyp.7278.

Boulton, A. J., S. Findlay, P. Marmonier, E. H. Stanley, and H. M. Valett(1998), The functional significance of the hyporheic zone in streams andrivers, Annu. Rev. Ecol. Syst., 29, 59–81.

Braudrick, C. A., W. E. Dietrich, G. T. Leverich, and L. S. Sklar (2009),Experimental evidence for the conditions necessary to sustain meander-ing in coarse‐bedded rivers, Proc. Natl. Acad. Sci. U. S. A., 106(40),16,936–16,941.

Brunke, M., and T. Gonser (1997), The ecological significance of exchangeprocesses between rivers and groundwater, Freshwater Biol., 37, 1–33,doi:10.1046/j.1365-2427.1997.00143.x.

Camporeale, C., and L. Ridolfi (2006), Convective nature of the planimet-ric instability in meandering river dynamics, Phys. Rev. E, 73(2),023611, doi:10.1103/PhysRevE.73.026311.

Camporeale, C., P. Perona, A. Porporato, and L. Ridolfi (2005), On thelong‐term behavior of meandering rivers, Water Resour. Res., 41,W12403, doi:10.1029/2005WR004109.

Camporeale, C., P. Perona, A. Porporato, and L. Ridolfi (2007), Hierarchyof models for meandering rivers and related morphodynamic processes,Rev. Geophys., 45, RG1001, doi:10.1029/2005RG000185.

Cardenas, M. B. (2007), Potential contribution of topography‐drivenregional groundwater flow to fractal stream chemistry: Residence timedistribution analysis of Tóth flow, Geophys. Res. Lett., 34, L05403,doi:10.1029/2006GL029126.

Cardenas, M. B. (2008a), Surface water‐groundwater interface geomor-phology leads to scaling of residence times, Geophys. Res. Lett., 35,L08402, doi:10.1029/2008GL033753.

Cardenas, M. B. (2008b), The effect of river bend morphology on flow andtimescales of surface water‐groundwater exchange across pointbars,J. Hydrol., 362(1–2), 134–141.

Cardenas, M. B. (2009a), A model for lateral hyporheic flow based onvalley slope and channel sinuosity, Water Resour. Res., 45, W01501,doi:10.1029/2008WR007442.

Cardenas, M. B. (2009b), Stream‐aquifer interactions and hyporheicexchange in gaining and losing sinuous streams, Water Resour. Res.,45, W06429, doi:10.1029/2008WR007651.

Cardenas, M. B., and J. L. Wilson (2007), Dunes, turbulent eddies, andinterfacial exchange with permeable sediments, Water Resour. Res.,43, W08412, doi:10.1029/2006WR005787.

Cardenas, M. B., J. L. Wilson, and V. A. Zlotnik (2004), Impact of heteroge-neity, bed forms, and stream curvature on subchannel hyporheic exchange,Water Resour. Res., 40, W08307, doi:10.1029/2004WR003008.

Dent, C., J. Schade, N. Grimm, and S. Fisher (2000), Subsurface influenceson surface biology, in Streams and Ground Waters, edited by J. B. Jonesand P. J. Mulholland, chap. 16, pp. 381–405, Academic, San Diego,Calif.


13 of 14

De Vries, M. (1965), Considerations about non‐steady bed‐load transportin open channels, paper presented at 11th Congress, Int. Assoc. ofHydraul. Res., Leningrad, Russia.

Elliott, A. H., and N. H. Brooks (1997), Transfer of nonsorbing solutes to astreambed with bed forms: Theory, Water Resour. Res., 33(1), 123–136.

Exner, F. M. (1925), Uber Die Wechselwirkung Zwischen Wasser undGeschiebe in Flussen, Sitzungsber. Akad. Wiss. Wien, Math.‐Naturwiss.Kl., Abt. 2A, 134, 165–180.

Frascati, A., and S. Lanzoni (2009), Morphodynamic regime and long‐termevolution of meandering rivers, J. Geophys. Res., 114, F02002,doi:10.1029/2008JF001101.

Genereux, D. P., S. Leahy, H. Mitasova, C. D. Kennedy, and D. R. Corbett(2008), Spatial and temporal variability of streambed hydraulic conduc-tivity in West Bear Creek, North Carolina, USA, J. Hydrol., 358(3–4),332–353.

Harvey, J. W., and K. E. Bencala (1993), The effect of streambed topogra-phy on surface‐subsurface water exchange in mountain catchments,Water Resour. Res., 29(1), 89–98.

Hester, E. T., and M. W. Doyle (2008), In‐stream geomorphic structures asdrivers of hyporheic exchange, Water Resour. Res., 44, W03417,doi:10.1029/2006WR005810.

Ikeda, S., and G. Parker (1989), River Meandering, Water Resour. Monogr.Ser., vol. 12, AGU, Washington, D. C.

Johannesson, H., and G. Parker (1989), Linear theory of river meanders,in River Meandering, Water Resour. Monogr. Ser., vol. 12, edited byS. Ikeda and G. Parker, pp. 181–214, AGU, Washington, D. C.

Jones, J. B., and P. J. Mulholland (Eds.) (2000), Streams and GroundWaters, Academic, San Diego, Calif.

Kalkwijk, J. P. T., and H. J. De Vriend (1980), Computation of the flow inshallow river bends, J. Hydraul. Res., 18, 327–342.

Kasahara, T., and A. R. Hill (2007), Lateral hyporheic zone chemistry in anartificially constructed gravel bar and a re‐meandered stream channel,southern Ontario, Canada, J. Am. Water Resour. Assoc., 43(5), 1257–1269, doi:10.1111/j.1752-1688.2007.00108.x.

Kasahara, T., and S. M. Wondzell (2003), Geomorphic controls on hypor-heic exchange flow in mountain streams, Water Resour. Res., 39(1),1005, doi:10.1029/2002WR001386.

Kennedy, C. D., D. P. Genereux, D. R. Corbett, and H. Mitasova (2009),Spatial and temporal dynamics of coupled groundwater and nitrogenfluxes through a streambed in an agricultural watershed, Water Resour.Res., 45, W09401, doi:10.1029/2008WR007397.

Lautz, L. K., and D. I. Siegel (2006), Modeling surface and ground watermixing in the hyporheic zone using MODFLOW and MT3D, Adv. WaterResour., 29(11), 1618–1633.

Leopold, L. B., and M. G. Wolman (1960), River meanders, Bull. Geol.Soc. Am., 71, 769–794.

Leopold, L. B., M. G. Wolman, and J. P. Miller (1964), Fluvial Processesin Geomorphology, Freeman, San Francisco, Calif.

Marion, A., M. Bellinello, I. Guymer, and A. Packman (2002), Effect of bedform geometry on the penetration of nonreactive solutes into a streambed,Water Resour. Res., 38(10), 1209, doi:10.1029/2001WR000264.

Packman, A. I., and N. H. Brooks (2001), Hyporheic exchange ofsolutes and colloids with moving bed forms, Water Resour. Res.,37(10), 2591–2605.

Peterson, E. W., and T. B. Sickbert (2006), Stream water bypass through ameander neck, laterally extending the hyporheic zone, Hydrogeol. J.,14(8), 1443–1451.

Poole, G. C., J. A. Stanford, S. W. Running, and C. A. Frissell (2006),Multiscale geomorphic drivers of groundwater flow paths: Subsurfacehydrologic dynamics and hyporheic habitat diversity, J. North Am.Benthol. Soc., 25(2), 288–303.

Revelli, R., F. Boano, C. Camporeale, and L. Ridolfi (2008), Intra‐meanderhyporheic flow in alluvial rivers, Water Resour. Res., 44, W12428,doi:10.1029/2008WR007081.

Ryan, R. J., and M. C. Boufadel (2007), Evaluation of streambed hydraulicconductivity heterogeneity in an urban watershed, Stochastic Environ.Res. Risk Assess., 21(4), 309–316.

Seminara, G. (2006), Meanders, J. Fluid Mech., 554, 271–297.Seminara, G., G. Zolezzi, M. Tubino, and D. Zardi (2001), Downstreamand upstream influence in river meandering. Part 2. Planimetric develop-ment, J. Fluid. Mech., 438, 213–230.

Sophocleous, M. (2002), Interactions between groundwater and surfacewater: The state of the science, Hydrogeol. J., 10(1), 52–67.

Stanford, J. A., and J. V. Ward (1993), An ecosystem perspective ofalluvial rivers: Connectivity and the hyporheic corridor, J. North Am.Benthol. Soc., 12(1), 48–60.

Tonina, D., and J. M. Buffington (2007), Hyporheic exchange in gravel bedrivers with pool‐riffle morphology: Laboratory experiments and three‐dimensional modeling, Water Resour. Res., 43, W01421, doi:10.1029/2005WR004328.

Tóth, J. (1963), A theoretical analysis of groundwater flow in smalldrainage basins, J. Geophys. Res., 68, 4795–4812.

van den Berg, J. H. (1995), Prediction of alluvial channel pattern ofperennial rivers, Geomorphology, 12(4), 259–279.

Vaux, W. (1968), Intragravel flow and interchange of water in a streambed,Fish. Bull., 66(3), 479–489.

Woessner, W. W. (2000), Stream and fluvial plain ground water interac-tions: Rescaling hydrogeologic thought, Ground Water, 38(3), 423–429.

Wörman, A., A. I. Packman, L. Marklund, J. W. Harvey, and S. H. Stone(2006), Exact three‐dimensional spectral solution to surface‐groundwaterinteractions with arbitrary surface topography, Geophys. Res. Lett., 33,L07402, doi:10.1029/2006GL025747.

Wörman, A., A. I. Packman, L. Marklund, J. W. Harvey, and S. H. Stone(2007), Fractal topography and subsurface water flows from fluvialbedforms to the continental shield, Geophys. Res. Lett., 34, L07402,doi:10.1029/2007GL029426.

Zolezzi, G., and G. Seminara (2001), Downstream and upstream influencein river meandering. Part 1. General theory and application to over-deepening, J. Fluid Mech., 438, 183–211.

F. Boano, C. Camporeale, and R. Revelli, Department of Hydraulics,Transports, and Civil Infrastructures, Politecnico di Torino, Corso Ducadegli Abruzzi 24, I‐10129, Turin, Italy. ([email protected])


14 of 14

Documents

A linear model for the coupled surface-subsurface flow in a meandering stream