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Advances in Water Resources

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River banks and channel axis curvature: Effects on the longitudinaldispersion in alluvial rivers

Stefano Lanzoni⁎,a, Amena Ferdousib, Nicoletta Tambronic

a Department of Civil, Environmental and Architectural Engineering, University of Padua, Italyb Faculty of Engineering and Technology, Eastern University, Dhaka, Bangladeshc Department of Civil, Chemical and Environmental Engineering, University of Genova, Genova, Italy

A R T I C L E I N F O

Keywords:Alluvial riversDispersionMeandering rivers

A B S T R A C T

The fate and transport of soluble contaminants released in natural streams are strongly dependent on the spatialvariations of the flow field and of the bed topography. These variations are essentially related to the presence ofthe channel banks and to the planform configuration of the channel. Large velocity gradients arise near to thechannel banks, where the flow depth decreases to zero. Moreover, single thread alluvial rivers are seldomstraight, and usually exhibit meandering planforms and a bed topography that deviates from the plane con-figuration. Channel axis curvature and movable bed deformations drive secondary helical currents which en-hance both cross sectional velocity gradients and transverse mixing, thus crucially influencing longitudinaldispersion. The present contribution sets up a rational framework which, assuming mild sloping banks andtaking advantage of the weakly meandering character often exhibited by natural streams, leads to an analyticalestimate of the contribution to longitudinal dispersion associated with spatial non-uniformities of the flow field.The resulting relationship stems from a physics-based modeling of the flow in natural rivers, and expresses thebend averaged longitudinal dispersion coefficient as a function of the relevant hydraulic and morphologicparameters. The treatment of the problem is river specific, since it relies on an explicit spatial description,although linearized, of the flow field that establishes in the investigated river. Comparison with field dataavailable from tracer tests supports the robustness of the proposed framework, given also the complexity of theprocesses that affect dispersion dynamics in real streams.

1. Introduction

Estimating the ability of a stream to dilute soluble pollutants is afundamental issue for the efficient management of riverine environ-ments. Rapidly varying inputs of contaminants, such as those associatedwith accidental spills of toxic chemicals and intermittent dischargefrom combined sewer overflows, as well as temperature variationsproduced by thermal outflows, generate a cloud that spreads long-itudinally affecting the fate of the pollutant.

The classical treatment of longitudinal transport in turbulent flowsrelies on the study put forward by Taylor (1954) for pipe flows, andextended to natural channels by Fischer (1967). Taylor’s analysis in-dicates that, far enough from the source (in the so called equilibriumregion), the cross-sectionally averaged tracer concentration, C, satisfiesa one-dimensional advection-diffusion equation, embodying a balancebetween lateral mixing and nonuniform shear flow advection (Fickiandispersion model). Under the hypothesis that the velocity field is sta-tistically steady and the investigated channel reach is geometrically

homogeneous and extends far inside the equilibrium region, the ad-vection-diffusion equation prescribes that the variance of C in the alongstream direction s* increases linearly with time and any skewness, in-troduced by velocity shear close to the contaminant source (i.e., in theadvective zone) or by the initial distribution of contaminant, begins toslowly decay, eventually leading at any instant to a Gaussian distribu-tion of C(s*) (Chatwin and Allen, 1985). The coefficient of apparentdiffusivity K* governing this behavior, usually denoted as dispersioncoefficient, is much greater than the coefficient for longitudinal diffu-sion by turbulence alone.

Many engineering and environmental problems concerning the fateand transport of pollutants and nutrients are tackled resorting to theone-dimensional advection-diffusion approach (Botter and Rinaldo,2003; Revelli and Ridolfi, 2002; Rinaldo et al., 1991; Schnoor, 1996;Wallis, 1994) and, therefore, require a suitable specification of thelongitudinal dispersion coefficient K*. Also within the context of muchmore refined models developed to account for the formation of steepconcentration fronts and elongated tails caused by storage and delayed

https://doi.org/10.1016/j.advwatres.2017.10.033Received 26 July 2017; Received in revised form 27 October 2017; Accepted 28 October 2017

⁎ Corresponding author.E-mail address: stefano.lanzoni@unipd.it (S. Lanzoni).

Advances in Water Resources 113 (2018) 55–72

Available online 31 October 20170309-1708/ © 2017 Elsevier Ltd. All rights reserved.

T

release of pollutant in dead zones (see, among many others, Bencalaand Walters, 1983; Czernuszenko et al., 1998 and Bear andYoung, 1983), a reliable estimate of longitudinal dispersion in the mainstream (quantified by K*) is fundamental to properly account for che-mical and biological processes acting in different channel regions(Lees et al., 2000).

Several procedures have so far been proposed to estimate thelongitudinal dispersion coefficient from either tracer data(Rutherford, 1994) or velocity measurements at a number of crosssections (Fischer, 1967). These approaches are usually expensive andtime consuming. The lack of experimental data which characterizesmany applications, as well as the necessity of specifying K* when car-rying out preliminary calculations, has thus stimulated the derivation ofvarious semi-empirical and empirical relationships (Deng et al., 2002;2001; Disley et al., 2015; Etamad-Shahidi and Taghipour, 2012;Fischer, 1967; Iwasa and Aya, 1991; Kashefipour and Falconer, 2002; Liet al., 2013; Liu, 1977; Noori et al., 2017; Sahay and Dutta, 2009; Sattarand Gharabaghi, 2015; Seo and Cheong, 1998; Shucksmith et al., 2011;Wang and Huai, 2016; Zeng and Huai, 2014), which can be cast in thegeneral form:

=K κ βc

B U*( )

* *,κ

fuκ u0

1

2 (1)

where β is the ratio of half channel with, B*, to mean flow depth, D*,u cfuis the friction coefficient, U*u is the mean value of the cross-sectionally

averaged flow velocity within the reach of interest and κi =i( 0, 1, 2)are suitable constants, specified in Table 1. Note that in Table 1 we havejust reported the main formulas; for sake of completeness a wider list ofrelationships is given in the Supplementary Information.

The empirical parameters that are introduced in these relations toaddress the complexity embedded in the mixing process still make thequantifying of K* a challenging task. In many cases, the proposedpredictor provides only a rough estimate, and the discrepancy betweenthe predicted values of K* and those determined from tracer tests isquite high. Among the many reasons responsible for this high scatter,one may be the prismatic character assumed as the basis of the Fickiansolution (Wang and Huai, 2016). Nevertheless, natural channels are

Notations

A*[m2] local cross sectional area.A m* [ ]u

2 mean value of the cross sectional area in the reach.B*[m] free surface half width of the channel.B m* [ ]c half width of the central channel region.=C c [/] cross sectionally averaged concentration

c[/] depth averaged concentrationcfu[/] friction coefficient

−m[ ]1C channel curvatureD*[m] local flow depth.D m* [ ]c flow depth of the central region of the channel.D m* [ ]u cross-sectionally averaged flow depth.D m* [ ]z flow depth measured normally to the bed.d m* [ ]gr sediment grain size.dr[/] discrepancy ratio.e m s* [ / ]t

2 depth averaged transverse eddy diffusityg[m/s2] gravitational acceleration.H*[m] elevation of the water surface with respect to an hor-

izontal reference planeH m* [ ]u

1.5 cross sectional average of D*1.5

(hs, hσ)[/]metric coefficients.K*[m2/s] mean value of the longitudinal dispersion coefficient in

the reach.K m s* [ / ]u

2 dispersion coefficient scale.k m s* [ / ]nu

2 transversal mixing coefficient for a straight channel.k k m s( *, *)[ / ]s n

2 longitudinal and transversal mixing coefficient.k m s* [ / ]t

2 transverse dispersion coefficient.L*[m] average intrinsic meander length.L m* [ ]c contaminant cloud lengthn*[m] horizontal coordinate normal to s*.P m* [ ]0 wetted perimeter of each bank region of the channel.Q*[m3/s] flow discharge.R m* [ ]0 twice the minimum value of the radius of curvature.r*[m] local radius of curvature.S[/] longitudinal channel slope.s*[m] longitudinal curvilinear coordinate coinciding with the

channel axis.

T s* [ ]0 longitudinal dispersion time-scale.T s* [ ]1 differential advection time-scale.T s* [ ]2 transverse mixing time-scale.t*[s] time.U*[m/s] depth averaged longitudinal velocity.U m s* [ / ]u mean value of the cross sectionally averaged longitudinal

velocity in the reach.u*[m/s] longitudinal component of the velocity.u m s* [ / ]f local friction velocity.u m s* [ / ]fc scale for the friction velocity in the central region of the

channel.u m s* [ / ]fu scale for the friction velocity under uniform flow condi-

tions.V*[m/s] depth averaged transverse component of velocity.z*[m] upward directed axis.β[/] half free surface width to uniform depth ratio.βc[/] half central region width to uniform depth ratio.βf[/] cross section shape parameter measuring the steepness of

the bank.γ[/] relative importance of transverse mixing and nonuniform

transportΔ[/] immersed relative sediment densityδ[/] relative variation rate of the cross section in the transverse

direction.m sϵ*[ / ]n

2 transverse mixing coefficient contribution due to disper-sion.

λ[/] dimensionless meander wave numberν[/] curvature ratioν m s* [ / ]T

2 turbulent viscosity.ξ*[m] pseudo-lagrangian coordinateφ angle that the vertical forms with the normal to the bedρ[kg/m3] water density.ρs[kg/m3] sediment density.σ*[m] transverse curvilinear coordinate.ζ*[m] coordinate normal to the bed.τ*u[/] Shields parameter

Table 1Values attained by the constants of the generalized formula (1) and by the associatedmean value of the discrepancy ratio dr (defined in Section 4.2) for various predictorsavailable in literature, namely: (1) Fischer et al. (1979); (2) Seo and Cheong (1998); (3)Liu (1977); (4) Kashefipour and Falconer (2002); (5) Iwasa and Aya (1991); (6)Deng et al. (2001); (7) Wang and Huai (2016). The dimensionless transverse eddy dif-fusivity, et, and mixing coefficient, kt, are defined in Section 2.

(1) (2) (3) (4) (5) (6) (7)

κ0 0.044 9.1 0.72 10.612 5.66 0.06/( +e kt t) 22.7κ1 1.0 −0.38 1.0 −1.0 0.5 0.67 −0.64κ2 1.0 0.428 −0.5 1.0 −1.0 1.0 0.16< dr> 1.05 1.24 1.10 1.05 1.04 0.63 1.13

S. Lanzoni et al. Advances in Water Resources 113 (2018) 55–72

56

usually characterized by a complex bed topography, which stronglyaffects the flow field and, hence, the longitudinal dispersion(Guymer, 1998), but is only roughly accounted for in the various ap-proaches. In addition, in some cases a suitable tuning of the empiricalparameters is needed in order to achieve a good agreement with theexperimental data (Deng et al., 2001).

Many of the existing expressions for predicting the longitudinaldispersion in rivers have been developed by minimizing the error be-tween predicted and measured (through tracer tests) dispersion coef-ficients. These relations generally differ in terms of the relevant di-mensionless groups (determined through dimensional analysis) and ofthe optimization technique (e.g., nonlinear multi regression, geneticand population-based evolutionary algorithms) used to calibrate thecoefficients of the predictor (Disley et al., 2015; Kashefipour andFalconer, 2002; Noori et al., 2017; Sahay and Dutta, 2009; Seo andCheong, 1998). More recently, the dataset provided by the dispersioncoefficients measured in the field has been used for training and testingartificial neural networks or bayesian networks (Alizadeh et al., 2017).A less few attempts were devoted to derive analytical relationships bysubstituting in the triple integral ensuing from Fischer analysis of shearflow dispersion the flow field that, under the uniform-flow assumption,establishes in stable straight channels (Deng et al., 2001) and inmeandering rivers (Deng et al., 2002). In the present contribution wefollow this latter approach, which has the advantage of being riverspecific, i.e., to relate the dispersion coefficient to the shear flow dis-persion that actually takes place in the river under investigation. Theimprovement with respect to the contributions by Deng et al. (2001,2002) is essentially related to the morphodynamics-based modelling ofthe flow that establishes in alluvial rivers. In the case of straight rivers,rather than using the general hydraulic geometry relationship for stablecross sections, we propose a specific treatment of the shear flow effectsby dividing the cross section into a central flat-bed region and twogently sloping banks computing the flow field therein. In the case ofmeandering rivers, the flow field outside the boundary layers that formnear to the banks is solved explicitly, although in a linearised way,accounting parametrically for the secondary flow circulations inducedby streamline curvatures, and computing the bed topography by solvingthe two-dimensional sediment balance equation.

The aim of the present contribution is thus to develop physics-based, analytic predictions of the longitudinal dispersion coefficient,accounting for the cross-sectional morphology occurring in alluvialrivers. More specifically, we intend to relate the estimates of K* to therelevant hydraulic, geometric and sedimentologic parameters (flowdischarge, bed slope, representative sediment size, bank geometry)governing the steady flow in an alluvial river. First, we apply to theflow field which establishes in sinuous, movable bed channels theperturbative procedure developed by Smith (1983), that accounts forthe fast variations of concentration induced across the section by irre-gularities in channel geometry and the presence of bends. This meth-odology, introducing a reference system moving downstream with thecontaminant cloud and using a multiple scale perturbation technique,allows the derivation of a dispersion equation relating entirely to shearflow dispersion the along channel changes in the cross-sectionallyaveraged concentration. Next, we take advantage of the weaklymeandering character of many natural rivers to clearly separate thecontributions to longitudinal dispersion provided by the various sourcesof nonuniformities. At the leading order of approximation, corre-sponding to the case of a straight channel, we consider the differentialadvection related to the presence of channel banks, solving the flowfield by means of a rational perturbation scheme (Tubino andColombini, 1992). At the first order of approximation, we introduce thecorrection to K* due to the presence of bends by using the hydro-morphodynamic model of Frascati and Lanzoni (2013).

The proposed methodology is finally validated through the compar-ison with the tracer test data collected in almost straight and in mean-dering rivers. Among others, we consider the dataset provided by

Godfrey and Frederick (1970), which includes detailed measurements offlow depth, longitudinal velocity, and the temporal evolution of thetracer concentration at different cross sections, as well as estimates of K*based on the method of moments. These concentration data are herereanalyzed by considering the Chatwin’s method (Chatwin, 1980), whichindicates if and where a Fickian model likely applies, and the routingmethod, based on the Hayami solution (Rutherford, 1994). We anticipatethat the proposed framework provides estimates of K* that are in rea-sonable good agreement with the values computed from tracer tests.

The paper is organized as follows. The mathematical problem isformulated in Section 2, with particular emphasis on the typical tem-poral and spatial scales which allow to set up a rational perturbativeframework and eventually determine the overall structure of the dis-persion coefficient in alluvial channels. The analytical solutions of thedepth averaged flow field used to compute K* are described inSection 3. The comparison with available field data is reported inSection 4, while Section 5 is devoted to the discussion of the results.Finally, Section 6 summarizes the concluding remarks.

2. Formulation of the problem

We consider the behavior of a passive, non-reactive contaminantwhich (e.g., due to an accidental spill) is suddenly released in an allu-vial channel with a compact cross section and, in general, a meanderingplanform. The river cross section bed is assumed to vary slowly in thetransverse direction as the banks are approached. This assumption al-lows for solving the flow field by adopting a closure model of turbu-lence in which the turbulent viscosity ν *T is a function of the local flowcondition (see Section 3.1). The channel has fixed banks, a constant freesurface width 2B*, a longitudinal mean slope S, and conveys a constantdischarge Q* (hereafter a star superscript will be used to denote di-mensional variables). The reach averaged value of the flow depth is D*u .The corresponding cross-section area is =A B D* 2 * *,u u while the cross-sectionally averaged mean velocity is =U Q A* */ *u u . The erodiblechannel bed is assumed to be made up of a uniform cohesionless se-diment with grain size d* ,gr density ρs, and immersed relative density= −ρ ρ ρΔ ( )/ ,s with ρ the water density. Moreover, we denote by=β B D*/ *u the half width to depth ratio, =u gD S* ( * )fu u

1/2 the frictionvelocity (with g the gravitational constant), and =c u U( * / *)fu fu u

2 thefriction coefficient. These two latter quantities are influenced by thebed configuration, which can be either plane or covered by bedformssuch as ripples and dunes, depending on the dimensionless grain size

=d d D* / *,gr gr u and the Shields parameter, =τ u gd* * /(Δ * )u fu gr2 .

2.1. The 2-D dimensionless advection-diffusion equation

The problem can be conveniently studied introducing the curvi-linear orthogonal coordinate system (s*, n*, z*) shown in Fig. 1a, wheres* is the longitudinal curvilinear coordinate coinciding with the channelaxis, n* is the horizontal coordinate normal to s*, and z* is the upwarddirected axis. The two-dimensional advection-diffusion equation for thedepth-averaged concentration c(s*, n*, t*) reads (Yotsukura, 1977):

⎜ ⎟

∂∂

+ ∂∂

+ ∂∂

= ∂∂

⎛⎝

∂∂

⎞⎠

+ ∂∂

⎛⎝

∂∂

⎞⎠

h D ct

D U cs

h D V cn s

k Dh

cs

nk h D c

n

**

* **

* ** *

* **

** *

*

s s ss

n s(2)

where t* denotes time, D* is the local flow depth, U* and V* are thedepth-averaged longitudinal and transverse components of the velocity,and k*s and k *n are longitudinal and transverse mixing coefficients whichaccount for the combined effect of vertical variations of velocity andturbulent diffusion. Moreover, hs is a metric coefficient, arising from thecurvilinear character of the longitudinal coordinate, defined as:

= + = +h nr

ν n1 **

1 ,s C (3)

S. Lanzoni et al. Advances in Water Resources 113 (2018) 55–72

57

where r*(s*) is the local radius of curvature of the channel axis, as-sumed to be positive when the center of curvature lies along the ne-gative n*-axis, =ν B R*/ *0 is the curvature ratio, =n n B*/ * is the di-mensionless transverse coordinate, = R r*/ *0C is the dimensionlesschannel curvature, and R *0 is twice the minimum value of r* within themeandering reach.

In meandering channels the cross-sectionally averaged concentra-tion undergoes relatively small and rapidly changing gradients, asso-ciated with the spatial variations of the flow field along the bends, and aslower evolution due to longitudinal dispersion. In order to deal withthe fast concentration changes acting at the meander scale, it proves

convenient to introduce a pseudo-lagrangian, volume following co-ordinate, ξ*, which travels downstream with the contaminant cloud andaccounts for the fact that the cross-sectionally averaged velocity Q*/A*is not constant along the channel (Smith, 1983). This coordinate isdefined as:

∫= −ξA

ds U t* 1*

* * * *u

su0

*A

(4)

where the integral on the right side is the water volume from the originof the coordinate system to the generic coordinate s*, while

∫ ∫= =− −

A D dn h D dn* * *, * * *.B

B

B

Bs*

*

*

*A (5)

Clearly, *A and A* can vary along s* as a consequence of the var-iations of section geometry induced by the bed topography that

establishes in the meandering channel. The derivation chain rule im-plies that:

∂∂

= ∂∂

+ ∂∂

∂∂

= ∂∂

− ∂∂s s ξ t t

Uξ* * *

,* *

**uA

(6)

where = A*/ *uA A . Consequently, for an observer moving with velo-city U*u (i.e., with the advected pollutant cloud) the dilution of theconcentration associated with longitudinal dispersion is accounted forby the coordinate ξ* and occurs at a length scale comparable with thelength of the contaminant cloud, L*c . It then results that=c c s n ξ t( *, *, *, *), and Eq. (2) can be rewritten as:

In order to better appreciate how transverse mixing, differentialadvection, longitudinal dispersion and spatial changes in bed topo-graphy contribute to dilute the pollutant concentration, Eq. (2) is madedimensionless introducing the following scaling:

= = = =s L s ξ L ξ n B n D D D* * , * * , * * , * * ,c u (8)

= = ⎛⎝

⎞⎠

=t T t U V U U LB

V k k k k k* * , ( *, *) * , **

, ( *, *) * ( , )u s n nu s n0(9)

where L* is the average intrinsic meander length within the in-vestigated reach (see Fig. 1a), k *nu is the transverse mixing coefficient fora straight channel configuration, and T*0 is the typical timescale atwhich longitudinal dispersion operates within the contaminant cloud.

Besides the timescale =T L K* * / *c u02 (where K *u is a typical dispersion

coefficient), other two timescales, =T L U* */ *c u1 and =T B k* * / * ,nu22

Fig. 1. Sketch of a meandering channel and notations. a) Plan view. b) Typical cross-section in a neighborhood of the bend apex. Note the scour at the outer bank and the depositioncaused by the point bar at the inner bank. c) Average cross-section, typically occurring nearby the inflection point of the channel axis.

⎜ ⎟

⎜ ⎟

∂∂

+ ∂∂

− ∂∂

⎛⎝

∂∂

⎞⎠= − ∂

∂− ∂

∂+ ∂∂

⎛⎝

∂∂

+ ∂∂

⎞⎠

+ ∂∂

⎛⎝

∂∂

+ ∂∂

⎞⎠

D U cs

h D V cn n

k h D cn

D h U U cξ

h D ct s

k Dh

cs

k Dh

cξ

ξk D

hcs

k Dh

cξ

* **

* ** *

* **

* ( * *)*

** *

* **

* **

** *

** *

*

s n s s u s ss

ss

ss

ss

A A

A A(7)

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58

characterize the processes (longitudinal dispersion, differential advec-tion and transverse mixing) that govern the concentration dynamics ofthe pollutant cloud. In order to ensure that they are well separated(Fischer, 1967; Smith, 1983), we introduce the small parameter

=k

B Uϵ

** *

,nu

u (10)

and recall the relationships usually adopted to predict the transverseand longitudinal mixing coefficients k *nu and K *u .

The rate of transverse mixing is determined by turbulent diffusion,quantified by the depth averaged transverse eddy diffusivity e *,t andvertical variations in the transverse velocity, quantified by the trans-verse dispersion coefficient k *t (Rutherford, 1994). Both coefficientsscale as u D* *fu u and, consequently, the transverse mixing coefficient canbe expressed as:

= +k e k u D* ( ) * *nu t t fu u (11)

Recalling that =u D B U c β* * * * / ,fu u u fu it results that= +e k c βϵ ( ) /t t fu . Experimental observations in straight rectangular

flumes indicate that et usually falls in the range (0.10 - 0.26), with amean value equal to 0.15 (Rutherford, 1994). On the other hand, forlarge rivers the transverse dispersion coefficient kt has been related tothe mean flow velocity and the channel width through a relation of theform (Smeithlov, 1990):

⎜ ⎟ ⎜ ⎟= ⎡

⎣⎢

⎛

⎝

⎞

⎠⎛⎝

⎞⎠

⎤

⎦⎥k

Uu

BD

13520

**

2 **t

u

fu u

1.38

(12)

Observing that the ratio c β/fu attains values of order −O (10 )2 and−O (10 )3 in gravel and sandy rivers (Hey and Thorne, 1986; Parker,

2004), it results that the parameter ϵ is indeed small.According to the semi-empirical relationship developed by

Fischer et al. (1979), =K B U u D* 0.044 ( * *) /( * *)u u fu u2 . This functional de-

pendence is confirmed by the dispersion data reported byRutherford (1994), indicating that the dimensionless ratio K B U*/ * *u umostly falls in the range −0.14 36, with mean 4.4 and standard de-viation 5.0. We can then write:

= + ∼kK

k k cβ

** 0.044

ϵ ,nu

u

n t fu2

2

(13)

Consequently, =T T*/ * ϵ1 0 and =T T*/ * ϵ ,2 02 provided that =B L*/ * ϵ ,c

2

that is the contaminant cloud has reached a length of order of hundredof meters or kilometers, depending on the width of the channel section.This result implies that the three time scales are well separated, i.e.

< <T T T* * *0 1 2 . In other words, the longitudinal dispersion operates on atimescale much slower than the timescale characterizing transversemixing which, in turn, is much faster than nonuniform advection(Fischer, 1967; Smith, 1983).

The derivation of the longitudinal dispersion coefficient takes ad-vantage of the small character of the parameter ϵ, ensuring the se-paration of the three timescales. Substituting the dimensionless vari-ables (8) and (9) into Eq. (7), we obtain:

⎜ ⎟⎜ ⎟= − ∂∂

− ∂∂

+ ⎛⎝

∂∂

+ ∂∂⎞⎠⎛⎝

∂∂

+ ∂∂⎞⎠

c D h U cξ

h D ct

γs ξ

γk D

hcs

k Dh

cξ

ϵ ( ) ϵ ϵ ϵϵs s s

ss

s

2 2L A A A

(14)

where the differential operator L reads:

= ⎛⎝

∂∂

+ ∂∂

⎞⎠− ∂∂

⎛⎝

∂∂

⎞⎠

γ D Us

h Vn n

h D kns s nL

(15)

The additional parameter =γ L Lϵ */ *c arises because of the presenceof two spatial scales. The spatial variations of c associated with long-itudinal dispersion at the scale of the contaminant cloud are describedby the slow variable ξ, whereas the comparatively small and rapidlychanging variations in concentration across the flow associated withstream meandering are accounted for through the fast variables s, n.

The parameter γ describes the relative importance of transverse mixing,which tends to homogenize the contaminant concentration, and non-uniform transport at the bend scale, which, on the contrary, enhancesconcentration gradients. It is readily observed that = −γ λ πϵ /2 ,1 wherethe dimensionless meander wavenumber =λ πB L2 */ * typically rangesbetween 0.1 and 0.3 (Leopold et al., 1964). The product γϵ then turnsout of order −O (10 )2 and, hence, gives rise to higher order terms in theperturbation analysis described in the next Section.

2.2. The longitudinal dispersion coefficient

The presence of different spatial and temporal scales can be handledemploying a multiple scale technique (Nayfeh, 1973). To this purposewe expand the concentration =c c s n ξ t( , , , ) as:

= + + + ⋯c c c cϵ ϵ0 12

2 (16)

We substitute this expansion into (14), and consider the problemsarising at various orders of approximation:

=O c(ϵ ) 000L (17)

= − ∂∂

O c D h U cξ

(ϵ) ( )s10L A

(18)

= − ∂∂

− ∂∂

O c D h U cξ

h D ct

(ϵ ) ( ) ,s s2

21 0L A

(19)

coupled with the requirements that ∂ ∂ =c n/ 0i ( =i 0, 1, 2) at the channelbanks, where the normal component of the contaminant flux vanishes.

The partial differential Eqs. (17)–(19) provide a clear insight intothe structure of the contaminant concentration. Recalling that, for asteady open channel flow, the depth-averaged (i.e., two-dimensional)continuity equation, written in dimensionless form, reads:

∂∂

+ ∂∂

=UDs

h VDn

( ) ( ) 0,s(20)

we integrate (17) across the section and find that c0 does not depend ons, n and, hence, it is not affected by the fluctuations induced by flowmeandering. Eq. (18) suggests a solution of the form = ∂ ∂c g s n c ξ( , ) / ,1 1 0

with g1 a function describing the nonuniform distribution across thesection of the contaminant concentration. Similarly, Eq. (19) indicatesthat = ∂ ∂c g s n c ξ( , ) /2 2

20

2. The depth-averaged concentration then re-sults:

= + ∂∂

+ ∂∂

+c s n ξ t c ξ t g s n cξ

g s n cξ

O( , , , ) ( , ) ϵ ( , ) ϵ ( , ) (ϵ )0 10 2

2

202

3

(21)

and clearly discriminates the slower evolution due to longitudinal dis-persion, embodied by the terms c0, ∂c0/∂ξ, ∂2c0/∂ξ2, from the small andrapidly varying changes associated with the spatial variations of theflow field, described by the functions g1 and g2.

Integrating (21) across the section and along a meander, the cross-sectionally averaged concentration can be approximated as= +c c O (ϵ )0

3 provided that < > =g 0i ( =i 1, 2), where:

∫ ∫ ∫= = =− − −

+c h D c dn g h D g dn g g ds1

2, 1

2,s i s i i s

si1

1

1

1

1/2

1/2

A A

(22)

It is important to note that only averaging (21) along the entiremeander length ensures that the arbitrary constant embedded in gi doesnot actually depend on s.

We are now ready to derive the advection-diffusion equation, gov-erning the evolution of the cross-sectionally averaged concentration c ,and the related longitudinal dispersion coefficient. We sum togetherEqs. (18) and (19), integrated across the section and along a bend, andrequire that the flux of contaminant vanishes, i.e., ∫ =− DUg dn 0i1

1

( =i 1,2), a condition needed in order to eliminate secular terms whichwould lead c2 to grow systematically with s. We eventually obtain:

S. Lanzoni et al. Advances in Water Resources 113 (2018) 55–72

59

∂∂

= ∂∂

+ct

K cξ

O (ϵ)02

02 (23)

where:

∫= = ⎛⎝

− ⎞⎠−

K D h U g dn, 12

,s21

11A K K

A A (24)

while the function g1(s, n), describing the cross-sectional distribution ofthe concentration, results from the solution of the O(ϵ) equation:

= −g D h U( ),s1L A (25)

with the requirements that ∂ ∂ =g n/ 01 at the channel banks, where thenormal component of the contaminant flux vanishes, and =g 01 .

Before proceeding further, some observations on the expression (24)are worthwhile. In accordance with Fischer (1967), the contribution tolongitudinal dispersion provided by vertical variations of the velocityprofile (embodied by the terms of (14) containing ks) is of minor im-portance. Longitudinal dispersion is essentially governed by shear flowdispersion induced by the nonuniform distribution across the section ofboth the contaminant concentration, accounted for through the func-tion g1(s, n), and the flow field, quantified by −D h U( )s A . This latterterm, however, differs from the much simpler term − U(1 ) that wouldarise in the classical treatment pursued by Fischer (1967), as a con-sequence of the fact that here the mean flow velocity can in generalvary along the channel, as accounted for through the volume-followingcoordinate ξ. In addition, it is important to observe that the bendaveraged coefficient K is always positive while, in the presence of riverreaches characterized by rapid longitudinal variations of the flow field,the coefficient K can also attain negative values, thus favoring spur-ious instabilities (Smith, 1983).

Finally, it is useful to relate the local and the bend averaged dis-persion coefficients,K and K, to the local dispersion coefficientD thatarises when considering only the fast coordinate s. Decomposing theconcentration c and velocity U* as the sum of their cross-sectionallyaveraged values, c , U *, plus the corresponding fluctuations c′, U′*, theclassical one-dimensional advection-dispersion equation results:

∫∂∂

+ ∂∂

= − ∂∂

′ ′−

ct

Q cs s

D U c dn*

** *

1* *

* * *B

B

*

*

A A (26)

Setting ′ = ∂ ∂c B U k g c s( * */ * ) / *,u nu2

1 after some algebra it can be de-monstrated that:

∫

=

= −−

U Bk

D U U g dn

* ( * *)*

,

12

( ) ,

u

nu

2

1

11

D D

DA (27)

and, consequently, =K D and =K 2A D .

2.3. Structure of the longitudinal dispersion coefficient in meanderingchannels

Natural channels are seldom straight. In the general case of ameandering planform configuration, the problem can be faced bytaking advantage of the fact that the curvature ratio ν appearing in (3)is typically a small parameter, ranging in the interval −0.1 0.2(Leopold et al., 1964). This evidence is widely used to describe the flowfield in meandering channels (Seminara, 2006) and to model their longterm evolution (Frascati and Lanzoni, 2010; 2013). It implies that theflow field and the bed topography of a meandering channel can bedetermined by studying the relatively small perturbations associatedwith deviations from a straight channel configuration. We then in-troduce the expansions:

=

+

+ +

U s n D s n s U n D n

ν U s n D s n s

ν U s n D s n s O ν

[ ( , ), ( , ), ( )] [ ( ), ( ), 1]

[ ( , ), ( , ), ( )]

[ ( , ), ( , ), ( )] ( )

0 0

1 1 1

22 2 2

3

A

A

A

(28)

where the unperturbed O(ν0) state corresponds to a straight channel.Similarly, we expand in terms of ν the function g1 and the dimensionlesstransverse mixing coefficient kn:

= + +k s n g s n k n g n ν k s n g s n O ν[ ( , ), ( , )] [ ( ), ( )] [ ( , ), ( , )] ( )n n n1 0 10 1 112

(29)

The longitudinal dispersion coefficient in meandering channels isdetermined by substituting (28) and (29) into (24), and recalling (3).We obtain:

= + + +K K ν K ν K O ν( )0 12

23 (30)

where

∫= −−

K U D g dn12

(1 )0 1

10 0 10 (31)

∫∫

= − < + >

+ < − − >

−

−

K U D g D g dn

n U U D g dn

12

(1 )

12

1 1

10 0 11 1 10

1

11 0 1 0 10C A

(32)

∫∫∫

= − < + + >

+ < − − + >

− < + + >

−

−

−

K U D g D g D g dn

n U U D g D g dn

U U U D g dn

12

(1 )

12

( ) ( )

12

2 1

10 0 12 1 11 2 10

1

11 0 1 0 11 1 10

1

12 1 1 0 2 0 10

C A

A A(33)

It is immediately recognized that the leading order contribution K0

corresponds to the classical solution obtained by Fischer (1967). Itaccounts for dispersion effects which arise in a straight uniform flow asa consequence of the transverse gradients experienced by U0 and theconcentration distribution in the bank regions (Fig. 1). Note that ne-glecting these effects is equivalent to set = =U D 1,0 0 such that =K 00 .It is also easy to demonstrate that the O(ν) correction K1 is identicallyzero. Indeed, = 0,1A and the various integrals involve products of even( − U1 ,0 D0, g10) and odd (D1, U1, g11, n) functions that, integratedacross a symmetrical section, yield a zero contribution. Finally, the O(ν2) term K2 includes the effects of the near bank velocity and con-centration gradients, mainly represented by the first integral on theright hand side of Eq. (33), and those due to the complex structure ofthe flow field, the bed topography and the spatial distribution of theconcentration induced by the meandering stream. The former con-tribution to K2 is likely of minor importance when dealing with wideand shallow sections (i.e., with large β), as often occurs in alluvial riversand, in the following will be neglected in order to keep the model at thelower level of complexity. In fact, as it will be seen in the next Section,the solution of the flow field in a meandering channel is available inclosed form only by neglecting the boundary layers that form near tothe banks (Frascati and Lanzoni, 2013).

The functions g1i(s, n) ( =i 0, 1) that describe the cross sectionaldistribution of c are obtained by solving the partial differential equa-tions that arise by substituting from (28) and (29) into (25). They read:

⎜ ⎟

∂∂

− ∂∂

⎛⎝

∂∂

⎞⎠= =γD U

gs n

D kgn

f s n i( , ) 0, 1in

ii0 0

10 0

11

(34)

and are subject the constraints that ∂ ∂ =g n/ 0i1 at the walls and

∫

∫

< > =

= = − < + >

−

−

D g dn b

with

b b g D D n dn0, ( ) .

i i1

10 1 1

10 11 1

110 1 0 C (35)

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60

The forcing terms f1i are obtained recalling the expression of themetric coefficient hs, and read:

= −f D U(1 )10 0 0 (36)

= − − + − −∂

− ∂∂

⎡⎣⎢

+ + ⎤⎦⎥

f n U U D U γ D Vdg

n

nk n k D k D

dgdn

( ) (1 )

( ) )n n n

11 1 0 1 1 0 0 110

1 0 0 0 110

C A

C(37)

where =k D ,ni i having assumed that = +k e k u D* ( ) * *n t t fu (Deng et al.,2001).

The solution of the boundary value problem given by (34) and itsconstraints is in general given by the sum of a homogeneous solution,common to any order of approximation, and a particular solution re-lated to the forcing term f1i. The homogeneous solution can be writtenin term of Fourier series, and generally depends on the transverse dis-tribution of concentration at the injection section. However, in the caseof a sudden release of contaminant treated here, it tends to decreaseexponentially with the coordinate s and, hence, vanishes far down-stream of the input section (Smith, 1983). This condition is equivalentto impose that the O(ϵ) and O(ϵ2) pollutant fluxes vanish∫ =− DUg dn( 0i1

1 for =i 1, 2), as required in the derivation of Eq. (23).Finally, note that, for the uniform flow in a straight channel, the

along channel gradient of g10 is identically zero and (34) yields theclassical relation (Fischer, 1967; 1973; Rutherford, 1994):

∫ ∫= ⎡⎣⎢

− ⎤⎦⎥

+− −

g nD k

D U dn dn α( ) 1 ( 1)n

n

n10 1 0 0 1 0 0 2 1 0

1

(38)

where the constant α0 allows g10 to satisfy the integral condition (35),but does not give any contribution to K0.

3. Depth averaged flow field in alluvial channels

The characteristics of the steady flow that establishes in alluvialchannels are determined by the form of the cross section and theplanform configuration of the channel. The governing two-dimensionalequations of mass and momentum conservation are in general obtainedby depth-averaging the corresponding three-dimensional equations,and by accounting for the dynamic effects of secondary flows inducedby curvature and of the boundary layers that form near to the channelbanks.

The complexity of the problem prevents the derivation of generalsolutions in closed form. Also numerical solutions are non straightfor-ward, owing to the difficulty of modeling secondary circulations(Bolla Pittaluga and Seminara, 2011). However, the governing equa-tions can be linearized in the presence of gently sloping channel banksand meandering channels with wide and long bends, such that the flowfield can be solved by perturbing the uniform flow solution in terms oftwo small parameters δ and ν. We resort just to these solutions, whichhave the pratical advantage to explicitily account, although in a sim-plified form, for the effects exerted on the base flow field by the bankshape and the channel axis curvature. We then estimate analytically thelongitudinal dispersion coefficient through (31) and (33).

In the following, we first derive the cross-sectional distribution ofthe longitudinal velocity U0 in a straight channel with gently slopingbanks (small δ). Next, we briefly recall the structure of U1 in wide andlong meander bends (small ν) with either an arbitrary or a regulardistribution of the channel axis curvature.

3.1. Straight channels

The uniform turbulent flow field that establishes throughout a crosssection of a straight channel can be conveniently studied introducingthe local orthogonal coordinate system (s*, σ*, ζ*), where s* is thelongitudinal (in this case straight) coordinate (directed downstream),

σ* is the transverse curvilinear coordinate aligned along the cross-sec-tion profile (with origin at the channel axis), and ζ* is the coordinatenormal to the bed (pointing upward) (see Fig. 1c). The curvilinearnature of σ* is accounted for through the metric coefficient:

⎜ ⎟= +∂∂

= − ⎛⎝

∂∂

⎞⎠

hζ

φD

σφ

Dσ

1*

cos*

*, cos 1

**

,σ

202

02

(39)

with D σ* ( *)0 the local value of the flow depth, φ the angle that thevertical forms with ζ*, and =D D φ* */cosz 0 the flow depth measurednormally to the bed (Fig. 1).

The uniform character of the flow implies that, on average, the flowcharacteristics do not vary in time and along the direction s*. Hence,denoting by u*(σ*, ζ*) the corresponding component of the velocity, thelongitudinal momentum equation, averaged over the turbulence, reads(Appendix A):

⎜ ⎟⎜ ⎟+ ∂∂

⎛⎝

∂∂

⎞⎠+ ∂∂

⎛⎝

∂∂

⎞⎠=S h g

σνh

uσ ζ

ν h uζ*

* ** *

* **

0,σT

σT σ

(40)

where ν *T is the eddy-viscosity used to express the turbulent Reynoldsstresses through the Boussinesq eddy viscosity approximation.

In general, the channel cross section is assumed to consists of(Fig. 1c): i) a central region of width B2 *c and constant depth depth D*,cand ii) two bank regions, each one characterized by a width ( −B B* *c )and wetted perimeter P*0 . In natural channels the flow depth is usuallymuch smaller than the wetted perimeter and, consequently, the di-mensionless parameter

=+

δD

P B*

* *u

c0 (41)

is small. We will take advantage of this for solving Eq. (40). To this aim,we introduce the scaling:

= =+

= =ζζ

D σσ σ

P BD

DD

u uu

** ( *)

, ** *

,**

, **

,z c u fu0

00

(42)

= =νν

D uu

uu

** *

,**

,TT

u fuf

f

fu (43)

where =u gD S* ( * )f 01/2 is the local value of the friction velocity, related

to the local bed shear stress τ*b by the relation =u τ ρ* ( */ )f b1/2. Note that,

having normalized ζ* with D*,z it turns out that:

⎜ ⎟∂∂

= ∂∂

∂∂

=+

⎛⎝

∂∂

− ∂∂⎞⎠ζ D ζ σ P B σ

ζFζ*

1*

,*

1* *

,z c0

1(44)

with

= ∂∂

⎡⎣⎢

+ ∂ ∂− ∂ ∂

⎤⎦⎥

FD

Dσ

δ D D σδ D σ

1 1 ( / )1 ( / )

.10

02

02

02

20

2 (45)

Substituting (42) and (43) into (40), the dimensionless longitudinalmomentum equation results:

⎜ ⎟ ⎜ ⎟∂∂⎡⎣⎢

∂∂⎤⎦⎥+ ⎛

⎝

∂∂

− ∂∂⎞⎠⎡⎣⎢

⎛⎝

∂∂

− ∂∂⎞⎠⎤⎦⎥ + =

D ζh ν u

ζδ

σζF

ζνh σ

ζFζ

u h1 0z

σ TT

σσ2

21 1

(46)

Under the assumption that the transverse slope of the channel bankvaries slowly, such that the normals to the bed do not intersect eachother, it is possible to express the dimensionless eddy viscosity νT as:

=ν ζ u D ζ( ) ( ),T f z N (47)

The simplest model for the function ζ( )N is that introduced byEngelund (1974), whereby = 1/13N . In the following, we will adoptthis scheme which allows for an analytical solution of the problem, and,as shown by Tubino and Colombini (1992), leads to results that agreeboth qualitatively and quantitatively with those obtained with a more

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61

accurate model for the function ζ( )N . Under the assumption of aconstant ,N a slip condition has to be imposed at the bed, such that:

⎜ ⎟= ⎡

⎣⎢ + ⎛

⎝⎞⎠

⎤

⎦⎥=u u ln

Dd

2 2.5 .ζ fz

gr0

(48)

The other two boundary conditions to be associated to Eq. (46)require that the dimensionless shear stress vanishes at the water surfaceand equals uf at the bed:

⎜ ⎟⎡⎣⎢

⎛⎝

∂∂

− ∂∂

⎞⎠⎤⎦⎥ = ⎡

⎣⎢∂∂⎤⎦⎥

== =

νD

uζ

δh

uσ

νD

uz

u1 0, .Tσ ζ

T

z ζf

0

2

1 0

2

(49)

The presence of the small parameter δ allows the expansion of theflow variables as:

= + +u u u u δ u u δ( , ) ( , ) ( , ) ( ).f f f0 02

14

1 O (50)

The cross sectional distribution u(σ, ζ) of the longitudinal velocity isobtained by substituting this expansion into Eqs. (46), (48) and (49), bycollecting the terms with the same power of δ2, and by solving the re-sulting differential problems (see Appendix A). Integrating u along thenormal ζ to the bed, the local value U0(σ) of the depth-averaged long-itudinal velocity results:

= + +U U δ U O δ( )0 002

014 (51)

where U00 is a function of the local flow depth D0(σ) and the relativegrain roughness dgr, while U01 depends also on ∂D0/∂σ (i.e., the localslope) and ∂2D0/∂σ2 (Appendix A).

The cross sectional distribution of U0 needed to compute the long-itudinal dispersion coefficient is then determined by specifying the re-lative bed roughness dgr and, more importantly, the across sectiondistribution of the flow depth D0(σ). In the absence of experimentaldata, we need to describe the bank geometry. Here, we propose tohandle empirically the problem assuming a transverse distribution ofthe flow depth of the form:

= − ∈ −D σ D erf β σ σ* ( ) * [ ( 1 )], [ 1, 1],c f0 (52)

with βf a shape parameter measuring the steepness of the banks. Notethat, according to (52), erf(βf) should be equal to 1 in order to ensurethat =D D* (0) *c0 . The latter requirement is fulfilled only asymptotically,for βf tending to infinity. For this reason, in the following we will

consider only values of ≥ =−β erf (0.999) 2.32675,f1 corresponding to

< ≤D D D* * (0) 0.999 *c c0 . Note also that βf is related to the parameter δβcthrough the relation:

= ⎡

⎣⎢ − ⎤

⎦⎥

−δ β

erfβ

1(0.999)

cf

1 2

(53)

where =β B D*/ *c c u and having assumed ≃D B D* ( *) 0.999 *c c0 . As βf in-creases also δβc increases, resulting in progressively steeper cross sec-tions (Fig. 2a). Note that increasing values of δ imply higher bankslopes. In the limit of =β 2.32675,f it results δβc = 0, corresponding to

=B* 0c (no central region), while as βf→∞, the classical rectangularcross-sectional configuration ( =P* 00 ) is recovered (Fig. 2a).

Interestingly, the distributions of uf1(σ) shown in Fig. 2b indicate anincrease of the friction velocity uf, with respect to the uniform flow, inthe steeper portion of the bank, and a corresponding decrease in thepart of the bank adjacent to the central region. This trend, due to thelongitudinal momentum transfer from the center of the cross section(where flow velocities are higher) to the banks, implies that the channelcan transport sediments even though the bank toes are stable. Note alsothat uf1(σ) vanishes towards the center of the cross section, where thebottom is flat, and at the outer bank boundary, where D0 tends to zero.

3.2. Meandering channels

The flow field that takes place in a meandering channel with acompact cross section is strictly related to the secondary flow circula-tions driven by the curvature of streamlines and the deformation of thechannel bed, which generally exhibits larger scours in the correspon-dence of the outer bank of a bend (Seminara, 2006). Although nu-merical models have the advantage to overcome the restrictions af-fecting theoretical analyses (e.g., linearity or weak non-linearity,simplified geometry) they still require a large computational effort tocorrectly include the effects of secondary helical flow and to reproducethe bed topography of movable bed channels (Bolla Pittaluga andSeminara, 2011; Eke et al., 2014). That is why linearized models havebeen widely adopted to investigate the physics of river meandering(Seminara, 2006), the long-term evolution of alluvial rivers (Bogoniet al., 2017; Frascati and Lanzoni, 2009; Howard, 1992), and the pos-sible existence of a scale invariant behavior (Frascati andLanzoni, 2010). These models, owing to their analytical character, not

Fig. 2. (a) Cross-sectional bed profile for various βf and for =δ 0.1 and =δ 0.2. (b) Cross-section distributions along the σ coordinate of the δ( )2O corrections provided to the di-mensionless friction velocity uf1, for the values of βf considered in plot a) and for =d 0.02gr .

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62

only provide insight on the basic mechanisms operating in the processunder investigation, but also allow to develop relatively simple en-gineering tools which can be profitably used for practical purposes.

In the following we refer to the linearized hydro-morphodynamicmodel developed by Frascati and Lanzoni (2013) that, in the mostgeneral case, can manage also mild along channel variations of thecross section width. The model is based on the two-dimensional, depth-averaged shallow water equations, written in the curvilinear co-ordinates s, n and, owing to the large aspect ratio β usually observed innatural rivers, neglects the presence of the near bank boundary layers.The flow equations, ensuring the conservation of mass and momentumand embedding a suitable parametrization of the secondary flow cir-culations, are coupled with the two-dimensional sediment balanceequation, complemented with the relation describing the rate of sedi-ment transport. The solution of the resulting set of partial differentialequations takes advantage of the fact that, in natural channels, thecurvature ratio ν is small (ranging in the interval 0.1–0.2), and assumethat flow and topography perturbations originating from deviations ofthe channel planform from the straight one are small enough to allowfor linearization. In the case of a constant width rectangular section (forwhich =D D* *c u ), the dimensionless flow field yields:

= + +U D ν U D O ν( , ) (1, 1) ( , ) ( )1 12 (54)

where

∑

∑

=

= + ′ + ″ +

=

∞

=

∞

U s n u M n

D s n h h h n d M n

( , ) sin( )

( , ) ( ) sin( )

mc c

mc

10

1 1 2 30

m

mC C C(55)

Here, s( )C is the local curvature of the channel, ′ s( )C and ″ s( )C itsfirst and second derivatives, hi and di ( =i 1, 3) are constant coefficients,

= +M m π(2 1) /2, and u s( )cm , d s( )cm are functions of the longitudinalcoordinate s:

∫∑ ∑ ⎡⎣ ⎤⎦

∑ ∑

= + +

= +

= =

−

=

−

−=

−

−

u c e A g ξ e dξ g

d d d uds

A d dds

( )cj

cλ s

cj

cs λ s ξ

c

cmj

mjj

mj m

jmjc

j

j

1

4

1

4

0( )

1

4 1

11

5 1

1

m mjcmj

m jcmj

j0 1C C

C

(56)

We refer the interested reader to Frascati and Lanzoni (2013) forfurther details about the model, its derivation and implementation,while all the coefficients needed to compute U1 and D1 are reported inthe Supplementary Information. It is worthwhile to note that the re-levant dimensionless parameters (geometric, hydraulic and sedi-mentological) needed as input data to the model are the width to depthratio β, the dimensionless grain size dgr, and the Shields parameter forthe uniform flow conditions, τ*u.

The expressions (54) are used to compute the forcing term f11,needed to solve the boundary value problem (34) for g11. Note that bysubstituting (54) into (38) yields =g 010 (owing to the neglecting ofbank effects). The particular solution of (34) is obtained by writing theforcing term as =f p s q n( ) ( )11 (i.e., separating the variables throughFourier series), and by introducing the appropriate Green function(Morse and Feshbach, 1953). We obtain:

∫∑= − + −=

∞+

+− − +g s n

γμ n p s χ e dχ( , ) 1 ( 1) cos[ ( 1)] ( )

m

mm

s sm

μ χ γ11

0

12 1 0

/m0

2 12

(57)

where =μ mπ/2,m

= − −p sM

s u s( ) 2 ( 1) ( ) ( ),mc

mc2 mC

(58)

and s0 denotes the position of the injection section. By assumption, thelength scale over which the contaminant cloud has evolved, L*,c is well

in excess of the transverse mixing distance, ∼ U B k* * / *n02 . Consequently,

the position s0 of the injection section can be set arbitrarily far up-stream, taking = −∞s0 . Physically, this is equivalent to assume that thesolution depends only on values of −p s χ( ) upstream of s over a dif-fusion length scale. Indeed, the integral with respect to the dummyvariable χ decays as −μ χ γexp( / ),m

2 and hence depends on the values of pclosest to s.

The solution (57) is in general valid for an arbitrary, althoughslowly varying, spatial distribution of the channel axis curvature. Ittakes a particularly simple form in the schematic case of a regular se-quence of meanders with the axis curvature described by the sinegenerated curve = +s e c c( ) . .π s2 ıC (Leopold et al., 1964), where i is theimaginary unit, and c.c. denotes complex conjugate. In this case theflow field reads (Blondeaux and Seminara, 1985):

= + + += + + +

U n d n d sinh n d sinh n e c cD n d n d sinh n d sinh n e c c

( ) [ (Λ ) (Λ )] . .( ) [ (Λ ) (Λ )] . .

u u uπis

d d dπis

1 0 1 1 2 22

1 0 1 1 2 22 (59)

The constant coefficients =d d j, ( 0, 1, 2),uj dj Λ1, Λ2 (reported in theSupplementary Information) depend on β, dgr, τ*u, and λ. The aboverelationships indicate that both the flow depth and the velocity tend toincrease towards the outside channel bank. The deepening of the outerflow that takes place in a movable bed, in fact, pushes the thread ofhigh velocity towards the outside bank, unlike in the fixed bed case,where the predicted thread of high velocities is located along the insideof the bend.

The forcing term = −f n U11 1C can be written as = +f p s q n c c( ) ( ) . . ,11with =p s e( ) πis2 and =q n( ) − − −n d n d sinh n d sinh n(Λ ) (Λ )u u u0 1 1 2 2 . Itfollows that:

∑=+

+ +=

∞

g s nγ

bπ M γ

M n e c c( , ) 12 ı /

cos[ ( 1)] . .m

m π s11

02

2 ı

(60)

where bm are constant coefficients (see Supplementary Information).Substituting (60) into (33) and recalling (24) we finally obtain the re-lationship giving the bend averaged O(ν2) correction to the longitudinaldispersion coefficient associated with a regular sequence of meanders:

∑=+=

∞

K ν M b bM πγ(2 )

m

m m2

0

24 2 (61)

where a tilde denotes complex conjugate.

4. Comparison with tracer field data

4.1. The considered dataset

In order to test the validity of the proposed theory, we need in-formation not only on the dispersion coefficient and the average hy-drodynamic properties of the considered river reach, but also on theplanform shape of the channel, on the geometry of the cross sectionsand, possibly, on the cross sectional velocity distribution. Despite thenumerous tracer experiments carried out on river dispersion (McQuiveyand Keefer, 1974; Nordin and Sabol, 1974; Seo and Cheong, 1998;Yotsukura et al., 1970), only a few report also this type of information.In particular, the data collected by Godfrey and Frederick (1970) in-clude the time distribution of the local tracer concentration C at anumber of monitoring sections and the cross-section distributions of theflow depth, D*(n*), and of the vertical profiles of the longitudinal ve-locity u*(n*, z*). This dataset therefore provides all the informationneeded to assess the robustness of the present modeling framework. Ineach test a radiotracer (gold-198) was injected in a line source acrossthe stream. About 15ml of the tracer, a highly concentrated solution ofgold chloride in nitric and hydrochloric acid, was diluted to a volume of2 l. The injection was made at a uniform rate over a 1-min period. Theconcentration of radionuclide used in each test was proportional to thedischarge (about 2.6 GBq m−3 s−1). The concentrations near to the

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63

stream centerline were observed by a scintillation detector. The resol-ving time for the entire system was found to be 50 s. The error due tothe resolving time was about 5–10%.

Among the five river reaches considered by Godfrey andFrederick (1970), three exhibit almost straight planforms: the CopperCreek below gage (near Gate City, Va), the Clinch River above gage(hereafter Clinch River a.g., near Clinchport, Va), and the Clinch Riverbelow gage (hereafter Clinch River b.g., near Speers Ferry, Va). Theother two river reaches, the Powell River near Sedville (Tenn) and theCopper Creek above gage (near Gate City, Va) have meandering plan-forms. This data set has been integrated with the estimates of K* ob-tained from tracer tests carried out in other five straight streams andeight meandering rivers, namely the Queich, Sulzbach and Kaltenbachrivers (Noss and Lorke, 2016), the Ohio, Muskegon, St. Clair and RedCedar rivers (Shen et al., 2010), the Green-Duwamish River(Fischer, 1968), the Missouri River (Yotsukura et al., 1970), the LesserSlave River (Beltaos and Day, 1978), and the Miljacka River(Dobran, 1982). Fig. 3 shows the planform configurations of the in-vestigated reaches, extracted from topographic maps, while the geo-metrical, hydraulic and sedimentologic parameters of each stream arereported in Table 2. In particular, the curvature ratio ν and the wave-number λ have been determined from the spatial distribution ofchannel axis curvature through the automatic extraction proceduredescribed by Marani et al. (2002). The mean grain size estimates havebeen obtained on the basis of information available from literature(Beltaos and Day, 1978; Godfrey and Frederick, 1970; Yotsukura et al.,1970), from the USGS National Water Information System (http://www.waterdata.usgs.gov/nwis ), or from direct inspection (Dobran2007, personal communication). In addition to real meandering streamdata, Table 2 reports the laboratory data characterizing the longitudinaldispersion experiment carried out by Boxall and Guymer (2007) in aflume with a sine generated meandering planform and a sand bed thatwas artificially fixed by chemical hardening after the initially uniform

trapezoidal cross section was shaped by the flow.For a given test i, we used the data collected by Godfrey and

Frederick (1970) to compute at each monitored cross section j the areaA *ij and the total wetted perimeter P*ij . We then used the cross sectionalvelocity data u n z* ( *, *)ij to compute the depth averaged velocity U n* ( *)ijand the flow discharge Q*ij . All the relevant quantities deduced from theexperimental dataset are collected in Table 2 reported in the Supple-mentary Information. In particular, the values of the mean slope arethose provided directly by Godfrey and Frederick (1970), while thefriction velocities have been estimated under the hypothesis of a locallyuniform flow field.

The dispersion coefficients by Godfrey and Frederick (1970) havebeen obtained by applying the method of moments. However, thepresence of a relatively long tail in the temporal distribution of theconcentration (Fig. 4a) and the sensitivity of small concentrations tomeasurement errors limit the accuracy of this method(Rutherford, 1994). For this reason, we have recalculated the dispersioncoefficients by considering the Chatwin’s method (Chatwin, 1980),which has also the advantage to give an indication whether a mon-itoring section is located or not whitin the equilibrium region, where aFickian dispersion model can be applied. Fig. 4b shows an example ofthe application of the method to the tracer data collected in the ClinchCreeek (test T10). The Chatwin method introduces the transformedvariable C t( *), defined as:

=±C tC t

C t*ln

*

*max max

(62)

where =C C s t( *, *)j is the temporal distribution of the cross-sectionallyaveraged concentration measured the at jth cross section, Cmax is thecorresponding peak concentration, t *max is the peaking time, andthe + and - signs apply for ≤t t* *max and >t t* * ,max respectively. In thetransformed plane C t, *, a temporal distribution of tracer concentrationfollowing a Gaussian behavior should plot as a straight line. The slope

Fig. 3. a) Plan view of the river reaches investigated by Godfrey and Frederick (1970) and location of the monitored cross sections. b) Planforms of further meandering streams (Beltaosand Day, 1978; Dobran, 1982; Fischer, 1968; Noss and Lorke, 2016; Shen et al., 2010; Yotsukura et al., 1970) considered for testing the present theoretical approach.

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− Q A K0.5 ( */ *) / *j j and the intercept s K0.5 */ *j j of this line allow one to

estimate the dispersion coefficient K *j and the cross sectionally aver-aged velocity Q A*/ *j .

The data suggest that, for all the sections, only the rising limb andthe near peak region of the concentration time distribution are ap-proximately linear, and hence can be described by a Gaussian

distribution. Conversely, a departure from the linear trend is evident inthe correspondence of the tails, indicating a deviation from the Fickianbehavior.

The values of K *j estimated by considering the linear part of C t( *)for all the data collected by Godfrey and Frederick (1970) turn outinvariably smaller than those calculated according to the method ofmoments (see Table 3 of the Supplementary Information). In order to

Table 2Tracer tests considered to assess the present theoretical framework. Definitions are as follows: B*, half cross-section width; Q*, flow discharge; S, longitudinal channel slope; δ, relativevariation rate of the cross section in the transverse direction= +D P B*/( * *)u c0 ; ν, curvature ratio = B R*/ *,0 with R *0 twice the minimum radius of curvature of the channel axis within ameandering reach; λ, dimensionless meander wavenumber, = πB L2 */ *, with L* the intrinsic meander length. All the quantities are averaged along the investigated river reach.

River B* Q* S u*fu Planform δ ν λ(m) (m3/s) (%) (m/s)

Clinch River a.g.a 17.3 6.8 0.03 0.045 straight 0.032 0 –Clinch River b.g.b 30 9.1,85,51 0.04 0.05,0.085,0.076 straight 0.04,0.07,0.07 0 –Copper Creek a.g.c 8.5 1.5,8.5 0.13 0.08,0.104 straight 0.06,0.09 0 –Copper Creek b.g.d 8.5 0.9 0.30 0.104 meandering 0.044 0.11 0.04Powell Rivere 17.2 4 0.03 0.052 meandering 0.047 0.15 0.036Green-Duwamishf 20.0 12 0.02 0.049 meandering 0.07 0.13 0.090Lesser Slaveg 25.4 71 0.01 0.055 meandering 0.17 0.2 0.063Missourih 90 950 0.01 0.055 meandering 0.06 0.05 0.04Miljackai 5.7 1 0.11 0.055 meandering 0.08 0.09 0.05Exp. Flumej 0.5 0.025 0.12 0.031 meandering 0.19 0.08 0.157Queich 1k 1.52 0.21 0.12 0.048 meandering 0.25 0.13 0.32Queich 2k 0.95 0.25 0.19 0.068 straight 0.42 0. –Sulzbach 1k 1.315 0.16 0.32 0.079 meandering 0.25 0.11 0.21Sulzbach 2k 0.72 0.16 0.26 0.025 straight 0.6 0. –Kaltenbachk 1.01 0.15 0.52 0.103 straight 0.4 0. –Muskegonl 35 48.41 0.6 0.24 straight 0.06 0. –Ohiol 235 1405 0.007 0.061 meandering 0.04 0.05 0.1St Clairl 276.6 5000 0.088 0.083 straight 0.06 0 –Red Cedarl 6.33/12.34 2.7/19.8 0.2 0.11/0.14 meandering 0.19/0.15 0.01 0.02/0.04

a Godfrey and Frederick (1970), test T5;b Godfrey and Frederick (1970), tests T2, T7, T10;c Godfrey and Frederick (1970), tests T1, T6;d Godfrey and Frederick (1970), test T3;e Godfrey and Frederick (1970), test T4;f Fischer (1968);g Beltaos and Day (1978);h Yotsukura et al. (1970);i Dobran (1982);j Boxall and Guymer (2007);k Noss and Lorke (2016);l Shen et al. (2010);

0 2 4 6 8

10 12 14 16 18

1 2 3 4 5 6

K*

[m2 s-1

]mea

sure

d

section

c) Chatwincalculated

Hayami

-200-150-100-50

0 50

100 150 200

0 3000 6000 9000 12000

C[s

ec0.

5 ]

t*[sec]

b)

0 2 4 6 8

10 12

0 2000 4000 6000 8000 10000 12000 14000 16000

C

t*[sec]

a) measureHayami

Fig. 4. a) Temporal distributions of the concentration measured by Godfrey and Frederick (1970) in the tracer test T10 carried out along the Clinch River b.g.: black circles indicate themeasured concentration; continuous lines denote the concentration profiles predicted by applying the Hayami’s calibration method. b) The Chatwin’s transformation is applied to the datashown in a): black circles indicate the measured data; continuous straight lines are regression lines fitted to concentrations near the peak. c) Comparison between the dispersioncoefficients estimated by means of the present theoretical approach and Chatwin and Hayami methods.

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assess the sensitivity of these estimates to the method used to derivethem, we applied also the routing method based on the Hayami solution(Rutherford, 1994). This method, provided that the dynamics of thecross-sectionally averaged concentration is Gaussian, takes advantageof the superimposition of effects to determine the temporal distributionof C in a section, given the local values of U*, K* and the concentration-time curve in an upstream section. The resulting solution has the ad-vantage that it can be used to route downstream a given temporaldistribution of concentration without invoking the frozen cloud ap-proximation. In fact, for moderately large values of s* and t* it gives aconcentration profile similar to that provided by the classical Taylorsolution. For all the monitored sections, except the first one, it is thuspossible to estimate the values of U* and K* which ensure the bestagreement between the measured and predicted concentration profiles.

Fig. 5 shows the results of the application of the Chatwin andHayami methods. In some cases the Hayami method tends to yieldlarger values of K*. Possible reasons of this behaviour are the poorfitting of the routed solutions and the significance of the tail, due to theentrapment and retarded release of the tracer into dead zones, ab-sorption on sediment surfaces, hyporheic fluxes. Nevertheless, the mostsignificant differences between the two approaches generally occur insections where the estimate deviates significantly (longer error bars inFig. 5a) from the average value in the considered river reach, i.e. wherethe dynamic of the tracer cloud is likely influenced by some localisedeffects, such as irregularities along the channel sides, or the channelbed, determining the retention of a certain amount of tracer. On the

other hand, the average velocity estimated with the two methods arevery similar (Fig. 5b). In the following, we will consider the estimates ofthe dispersion coefficients provided by the Chatwin method when re-ferring to the measured values of K*.

4.2. Comparison with straight river dispersion data

Before pursuing a comparison between observed and predicteddispersion coefficients, it is worthwhile to test the reliability of the flowfield model described in Section 3.1. Fig. 6 shows the cross sectionaldistribution of the flow depth (left panels) and depth averaged velocity(right panels) measured by Godfrey and Frederick (1970) in six loca-tions along the Clinch River b.g. (test T10). The theoretically predictedvelocities, shown in Fig. 6, have been obtained either by introducinginto equation (51) the observed flow depth, or by considering thesimplified cross sectional geometry described by Eq. (52) and selectingthe value of βf which better interpolates the measured depth profile.The agreement between measured and computed velocity distributionsis in general reasonably good (correlation coefficient, =R 0.81U

2 ).The comparison between the estimates of K* obtained from the

tracer data of Godfrey and Frederick (1970) and those predicted byinserting in Eq. (31) the flow field described by Eq. (51) are shown inFig. 7a. This figure also shows in white squares a comparison betweenthe predicted dispersion coefficients and those estimated from mea-surements by Noss and Lorke (2016) and Shen et al. (2010) for theQueich, Sulzbach, Kaltenbach and Muskegon Rivers.

The theoretical estimates are reasonably good, with about 70% ofpredictions ensuring an error smaller than ± 30% (dotted lines inFig. 7a). Overall, the model tends to underestimate the dispersioncoefficient for the larger values of K*, which, usually occur in the mostdistant sections from the injection, where the measured concentrationprofiles are particularly flat.

In any case, the present estimates of K* are definitely more accuratethan those provided by other predictors available in literature(Fig. 7c), as documented by the values of the discrepancy ratio,=d log K K( * / * ),r pred meas plotted in Fig. 7b) and d). To give a visual per-

ception of the goodness of the different models, Fig. 7 shows the datarelative to the models displaying the best (smaller mean discrepancyratio) and worst (larger mean discrepancy ratio) performance accordingto Table 1 (a Figure reporting the predictions of all models is providedin the Supplementary Information). Note that the coefficient κ0 of theformula proposed by Deng et al. (2001) and reported in Table 1, hasbeen here reduced by 1/15. Indeed, this coefficient was originally de-termined by introducing a multiplicative empirical constant ψ (= 15,according to Deng et al. (2001)) in order to achieve a better agreementwith the observed dispersion coefficients. These coefficients, at least inthe specific case of the data provided by Godfrey and Frederick (1970),were calculated through the method of moments that, as discussedabove, tends to overestimate K* with respect to the Chatwin or therouting methods. Nevertheless, even by reducing the value of κ0, thepredictions of K* obtained from the formula by Deng et al. (2001) aresignificantly less accurate (mean discrepancy ratio < > =d 0.63r ) thanthose resulting from the present theoretical approach (< > =d 0.19r ).Even worse results are attained when considering other predictors (seeTable 1).

It is important to stress that the present methodology, being physicsbased, does not need the introduction of any fitting parameter. Theinput data are simply the flow discharge, the free surface channelwidth, the longitudinal slope, the friction velocity (strictly associatedwith the sediment grain size and the type of bed configuration, i.e.,plane or dune covered), and the cross-sectional distribution of the flowdepth or, alternatively, its simplified analytical description (Eq. (52)).Finally, we observe that including the higher order effects that thepresence of the channel banks exert on the transverse gradient of U0

(associated to the O(δ2) contribution in Eq. (51)) always leads to im-prove the estimate of K* (< > =d 0.190,r instead of 0.188).

Fig. 5. a) The dispersion coefficients predicted by the Hayami method are plotted versusthe values provided by the Chatwin method. The error bar measures the scatter of thelocal value of the dispersion coefficient provided by the Haymi method with respect to theaverage value of each river reach. b) The average velocity values of each cross sectionpredicted by the Hayami and Chatwin methods are plotted versus the measured value.The continuous line denotes the perfect agreement; the dashed lines correspond to a± 50% error.

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4.3. Comparison with meandering river dispersion data

In the case of meandering streams, besides Q*, B*, S, u* ,fu additionalinput information to the present model is the spatial distribution ofchannel axis curvature. These data are used to determine the di-mensionless parameters β, τ*u, dgr, ν, as well as the along channel dis-tribution of the channel axis curvature C (s) needed to compute theflow field through Eqs. (55). The expressions of U1(s, n) and D1(s, n) arethen employed to compute f11 and to solve the problem (34) for g11,and, ultimately, to obtain the O(ν2) correction (33) to the longitudinaldispersion coefficient.

In order to test the reliability of the flow field model described inSection 3.2, a comparison with the flow depths (left panels) and depthaveraged velocities (right panels) measured by Boxall andGuymer (2007) at the apex and the cross-over sections of an experi-mental meandering channel is reported in Fig. 8. The theoreticallypredicted velocities have also been compared with the velocity profilescalculated according to Smith (1983) for the theoretical flow depthdistributions (continuos lines on the left panels):

=UD U D

H*

* * **u u

u

0.5

(63)

where H*u is the cross-sectional average of D*1.5. The cross sectionalshapes predicted by the present model reproduce correctly (RD

2=0.90)

the topography variations induced by alternating bends (possible de-partures being related to the presence of bedforms not accounted for inthe model). The overall comparison appears reasonably good also interms of depth integrated longitudinal velocities (RU

2 = 0.93) and thetheoretically predicted profiles yield a better performance with respectto those calculated according to the approximate method proposed bySmith (1983) (RU

2=0.91).Fig. 9a shows the comparison between the bend averaged values of

the longitudinal dispersion coefficient estimated from the measurescarried out by Godfrey and Frederick (1970) in the Copper Creek and inthe Powell River and those predicted by either the leading term K0

(Eq. (31)) entailing a straight channel, or by considering also the cor-rection ν2 K2 (Eq. (33)), accounting for the presence of river bends. Thiscorrection turns out to pick up the right order of magnitude and, on thewhole, ensures a degree of accuracy greater than that attained whenneglecting curvature effects (< > =d 0.32r instead of < > =d 0.76r ). Asexpected, when treating the river as straight, the predicted values of K *0are systematically lower than those observed in the field. In is worth-while to note that the points corresponding to cross sections T3-S1, T3-S2 and T4-S1, for which the theory tends in any case to overestimateK*, are quite close to the injection section and, therefore, likely falloutside the zone where a Fickian dispersion model holds.

The ability of the present theoretical framework to give robust es-timates of K* is confirmed by Fig. 9b, reporting the predicted values ofK* against those resulting from the tracer test data for all the consideredmeandering streams. On the whole, the effect of the curvature is toslightly improve the degree of accuracy (< > =d 0.22,r instead of 0.29),although sometimes the theoretical coefficients turn out to be lowerthan those observed in the field. This can be partly explained with thefact that the predicted O(ν2) correction does not account for the nearbank velocity gradients associated with the presence of a boundarylayer and has been obtained on the basis of a linearized treatment of theflow field, which tends to underestimate the intensity of both secondarycirculations and transverse bed deformations forced by the meanderingstream. Clearly, a number of other processes act in the field to makedispersion not entirely Fickian, contributing to the data scatter. Wereturn later on this issue. Finally, note that for the considered set ofrivers, the results remain basically unaltered (< dr>=0.225, insteadof 0.22) when, instead of considering the observed spatially varyingcurvature signal, we consider a sequence of regular meanders withmaximum curvature equal to the inverse of the mean minimum radiusof curvature within the river reach.

5. Discussion

The rational perturbative framework developed in the previoussections, based on a suitable scaling of the two-dimensional advection-diffusion equation and on the introduction of a reference system, tra-veling downstream with the contaminant cloud and accounting for thealong channel variability of the cross-sectionally averaged velocity,provides a clear picture of the processes affecting the spreading of acontaminant in alluvial rivers.

The velocity gradients that characterize the near bank regions ofnatural streams, where the flow depth progressively vanishes, influencethe longitudinal dispersion at the leading order of approximation(equation (31)), corresponding to a straight channel planform. Sec-ondary circulations driven by centrifugal and topographical effects ty-pical of meandering channels provide a second order correction(equation (33)). The presence of a secondary helical flow enhancestransverse velocity gradients which, in turn, tend to increase thelongitudinal dispersion coefficient. On the contrary, the increasedtransverse mixing promoted by secondary currents (e.g., Boxall et al.,2003) would lead to a reduction of longitudinal dispersion. This be-havior is summarized in the analytical relation (61), obtained by con-sidering a regular sequence of sine generated bends.

Bend effects are explicitly accounted for through the dependence on

Fig. 6. Cross sectional distributions of the flow depth D* and of the depth averaged ve-locity U *0 across six sections of the Clinch River b.g.. Black circles correspond to the datameasured by Godfrey and Frederick (1970) in test T10. Continuous lines represent thesmoothed cross section described by (52) and the corresponding velocity profiles( =R 0.95D

2 ; =R 0.84U2 ). Dotted lines represent the velocity profile predicted by sub-

stituting into Eq. (51) the actual flow depth distributions ( =R 0.81U2 ).

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67

ν2 while the characteristics of the flow field and the bottom topographyaffect the coefficients bm. Moreover, as observed by Fischer (1969) andSmith (1983), the ratio γ of cross-sectional mixing timescale to long-itudinal advection timescale, accounts for the frequency of alternatingbends along the meandering reach. In the case of long enough bends, i.esuch that γ is much smaller than 1, the term (2πγ)2 at the denominatorof (61) can be neglected with respect to M4. On the contrary, if γ in-creases, K tends to decrease. This is the case of short bends, for whichthe changes in the flow field associated with alternating curves are toofast to allow cross-sectional mixing to eliminate concentration gra-dients.

Fig. 10 shows two typical examples of the variations of K as a

function of the aspect ratio β for either plane (Fig. 10a) or dune-coveredbed (Fig. 10b). In both cases, for given values of the dimensionlessparameters ν, dgr and γ, the bend averaged longitudinal dispersioncoefficient increases with the Shields parameter, τ*u. On the other hand,for a given τ*u, the values of K corresponding to quite different di-mensionless grain sizes dgr exhibit a relatively narrow range of varia-tions, as shown in Fig. 10c. Finally, Fig. 10d demonstrates that K tendsto increase significantly when approaching the resonant conditions(see, e.g., Lanzoni and Seminara, 2006). Nevertheless, it must be re-called that the meandering flow field in a neighborhood of the resonantstate cannot be described by the linear model adopted here, but itwould require a weakly nonlinear approach.

Fig. 8. Cross sectional distributions of the flow depth D* and of the depth averaged velocity U* across two sections of the experimental meandering channel of Boxall and Guymer (2007).Black circles correspond to the data measured by Boxall and Guymer (2007). Continuous lines represent the theoretical cross section ( +D νD0 1), described by (52) and (59), and thecorresponding velocity profiles. Dotted lines represent the velocity profile predicted according to Smith (1983) for the theoretical flow depth distributions (continuous lines on the leftpanels).

Fig. 7. On the left: Comparison between the dispersion coefficients predicted theoretically and those estimated from tracer test data (Godfrey and Frederick, 1970) in almost straightchannels by applying the Chatwin method. a) Present theoretical approach. c) Deng et al. (2001)’s (ψ=1) and Seo and Cheong (1998)’s predictors. White squares in a) are the dispersioncoefficients evaluated according to the present theoretical approach versus the values estimated from measurements by Noss and Lorke (2016) and Shen et al. (2010) for the Queich,Sulzbach, Kaltenbach and Muskegon Rivers. The continuous line denotes the perfect agreement; the dotted lines correspond to a ± 30 % error. On the right, b) and d): values of thediscrepancy ratio associated with the data plotted in figures a) and c). The continuous line denotes the perfect agreement; the dashed lines correspond to a ± 50 % error.

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In general, the linearized treatment of the flow field set as the basisof the present theoretical framework holds for relatively wide bends(small ν), long enough meanders to ensure slow longitudinal variationsof the flow field (small λ), small intensity of the centrifugally drivensecondary flow, a condition met for small values of ν β c/( fu ), and smallamplitude of bed perturbations with respect to the straight configura-tion (small ν τ cfu* /u and λβ τ*u ) (Bolla Pittaluga and Seminara, 2011;Frascati and Lanzoni, 2013). These intrinsic limitation of the theory canpartly explain the deviations of the predicted dispersion coefficientsfrom the values estimated from tracer test data. Other physical pro-cesses however concur to the scatter of data. The bed configurationpredicted by the considered hydro-morphodynamic model stems fromthe imposed flow discharge, corresponding to that actually observedduring the tracer tests. Nevertheless, this discharge can differ from theformative discharge producing the actual river bed configuration. Thepresence of regulation works and human activities (e.g., sedimentmining, dredging) can modify the bed topography and, consequently,the structure of the flow field controlling shear flow dispersion. Finally,the presence of bedforms, width variations, islands, and dead zonesconcurs to a non-perfectly Fickian behavior, enhancing the rate ofdispersion and causing the long tails usually observed in concentration-time curves. The Gaussian solution resulting from a Fickian approach todispersion can then be used to describe only the upper portion of theconcentration-time curves, as indicated by the tracer data plotted usingChatwin’s transformation. Other approaches are needed to fit entirelythese curves, such as the transient storage model that accounts for theeffects of temporary entrapment and subsequent re-entrainment ofpollutants (Cheong and Seo, 2003), the adoption of a fractional ad-vection-dispersion equation (Deng et al., 2004), or the asymptotic

treatment of one-dimensional solutions from an instantaneous pointsource (Hunt, 2005).

6. Conclusions

We set a physics-based theoretical framework to estimate thelongitudinal dispersion coefficient on the basis of the hydro-morpho-dynamic modeling of the flow field and the bed topography that es-tablish in alluvial rivers. The rational perturbative framework has beendeveloped on the basis of a suitable scaling of the two-dimensionaladvection-diffusion equation, and by the introduction of a referencesystem moving with the contaminant cloud with a velocity that variesaccording to the river cross sectional geometry. This framework pro-vides a clear picture of the processes affecting the spreading of a con-taminant in natural streams, that can be summarized as follows.

The longitudinal dispersion dynamics in alluvial rivers is controlledby velocity shear at the banks and secondary circulations driven bycentrifugal and topographical effects. In particular, the helical flowassociated to these circulations enhances relatively small and rapidlychanging velocity and concentration gradients, both in the transverseand in the longitudinal directions, which in general lead to an increaseof the longitudinal dispersion coefficient. Nevertheless, the planformshapes of meandering channels are usually characterized by relativelysmall values of the curvature ratio ν, implying that the increasedtransverse mixing, also promoted by secondary flows, affects the con-centration distribution only at higher orders of approximation.

Another consequence of the small values typically attained by ν is

β

0.09

0.07

0.110.130.15

τ∗

0.3

0.2

0.40.6

β

λ

Fig. 10. The theoretical values of bend averaged longitudinal dispersion coefficient Kpredicted by (61) are plotted versus the aspect ratio β for =ν 0.1 and =k 0.225n0 . a)=λ 0.1, =d 0.01,gr plane bed;. b) =λ 0.1, =d 0.001,gr dune covered bed; c) =*τ 0.09u ;

=λ 0.1, plane bed; d) =*τ 0.09,u =d 0.01s , plane bed.

Fig. 9. Comparison between the dispersion coefficients predicted by equations (31), (33)and those estimated from the tracer test carried out in meandering rivers: a) Copper Riverb.g. and Powell River; b) Copper b.g., Powell, Green-Duwamish, Lesser Slave, Missouri,Miljacka, Queich, Sulzbach, Ohio and Red Cedar rivers, and in the experimental flume ofBoxall and Guymer (2007). The sources of data are reported in Table 2. The continuousline denotes the perfect agreement; the dashed lines correspond to a ± 50 % error.

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the possibility to separate the contribution to shear flow dispersionprovided by near bank velocity gradients of the unperturbed straightconfiguration (equation (31)) from that induced by streamline curva-tures and by the alternating sequence of bars and pools which estab-lishes in the perturbed meandering configuration (equation (33)). Theformer contribution can be accounted for analytically for gently slopingchannel banks.

The longitudinal dispersion coefficient, averaged over the meanderlength in order to deal with longitudinal variations of the flow field,depends on the relevant bulk hydrodynamic and morphologic di-mensionless parameters, β, dgr, τ*u, λ, ν and γ. The latter parameter,accounting for the ability of cross-sectional mixing to adapt to along-channel flow changes, could lead to a reduction of the longitudinaldispersion coefficient in the presence of a sequence of relatively shortbends (Eq. (61)).

The comparison with field data obtained from tracer tests indicatesthat the proposed approach provides robust estimates of the reach

averaged longitudinal dispersion coefficient. The residual scatter can bepartly explained by the linearized character of the hydro-morphody-namic model used to compute K*. Flow nonlinearities, enhancing bothtransverse mixing and shear flow dispersion, induce opposite effects onlongitudinal dispersion. Other possible causes of the departures be-tween predicted and estimated coefficients are associated with the notentirely Fickian behavior of the dispersion process, whereby the con-centration-time curves decay more slowly than if they were Gaussian.

Acknowledgments

This work has been partially supported by Cariparo within theproject “GIS-based integrated platform for Debris Flow Monitoring,Modeling and Hazard Mitigation” and by the University of Paduawithin the project (Progetto di Ateneo 2010) “Morphodynamics ofmarsh systems subject to natural forcings and climatic changes”.

Appendix A. Cross-sectional distribution of a uniform turbulent flow in a straight channel

Let us consider the longitudinal momentum equation averaged over the turbulence, written terms of the local curvilinear orthogonal coordinatesystem (s*, σ*, ζ*) (Lanzoni and D’Alpaos, 2015):

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= − ∂∂

+ ⎡⎣⎢∂∂

+∂∂

+∂∂

⎤⎦⎥ −

+ ∂∂

ut

u us

vh

uσ

w uζ

g Hs ρ h

h Ts

Tσ

h Tζ

v T ρh

hs

**

* **

* **

* **

**

1 ( *)*

**

( *)*

/*σ σ

σ ss σs σ ζs σσ

σ

σ2

(A.1)

where hσ is the metric coefficient associated with the curvilinear transverse coordinate σ, u*, v*, w* are the components of the velocities along thethree coordinate axes, H* is the elevation of the water surface with respect to an horizontal reference plane, g is the gravitational constant, ρ is thewater density andT T T*, * , *,ss σs ζs T*σσ are components of the turbulent Reynolds stress tensor. In the case of uniform flow conditions, as those occurring ina straight channel with a compact cross section (Fig. 1c), the relevant variables do not vary in time and along the main flow direction s*.

Expressing the components Tσs and Tζs of the Reynolds stress tensor through the Boussinesq eddy-viscosity approximation:

= ∂∂

= ∂∂

T ρνh

uσ

T ρ ν uζ

* **

* **σs

T

σζs T (A.2)

equation (A.1) simplifies to:

⎜ ⎟⎜ ⎟+ ∂∂

⎛⎝

∂∂

⎞⎠+ ∂∂

⎛⎝

∂∂

⎞⎠=S h g

σνh

uσ ζ

ν h uζ*

* ** *

* **

0,σT

σT σ

(A.3)

where = −∂ ∂S H s*/ * is the longitudinal water surface slope that, for uniform flow conditions, coincides with the bed slope and the energy slope, andν *T is the turbulent eddy viscosity.

Under the assumption of a constant vertical distribution of ν *,T the longitudinal slip-velocity at the bottom must satisfy the condition:

⎜ ⎟= ⎡

⎣⎢ + ⎛

⎝

⎞

⎠

⎤

⎦⎥

=

u u lnDd

* * 2 2.5**

ζ

fz

gr* 0 (A.4)

where =u gD S* ( * )f 01/2 is the local value of the friction velocity. In addition, the shear stress must vanish at the water surface and take the value ρu*f 2 at

the bed, namely:

⎜ ⎟⎡

⎣⎢

⎛⎝

∂∂

−∂∂

∂∂

⎞⎠⎤

⎦⎥ = ⎡

⎣⎢∂∂

⎤⎦⎥

== =

ν uζ h

Dp

up

ν uζ

u* **

1 **

**

0, * **

* .Tσ

z

ζ DT

ζf2

* * * 0

2

z (A.5)

In terms of the dimensionless variables (42) and (43), the problem described by equations (A.3)–(A.5) leads to equations (46), (48) and (49).The solution of this boundary value problem is obtained by expanding u(ζ, σ) and uf(σ) in terms of the mall parameter δ:

= + +u u u u δ u u δ( , ) ( , ) ( , ) ( ).f f f0 02

14

1 O (A.6)

Substituting this expansion into equations (46), (48), (49), and collecting the terms with the same power of δ2, we obtain a sequence of ordinarydifferential problems that can be readily solved in closed form. After some algebra we find:

• O(δ0)

⎜ ⎟= ⎛⎝− + + + ⎞

⎠u ζ σ

ζζ D

dD( , )

132

13 2 52

lngr

0

20

0(A.7)

=u σ D( )f 00 (A.8)

S. Lanzoni et al. Advances in Water Resources 113 (2018) 55–72

70

• O(δ2)

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟

=⎧⎨⎩

⎡

⎣⎢⎢

⎛

⎝⎜

⎛⎝

⎞⎠+ + ⎛

⎝⎞⎠

⎞

⎠⎟ + ⎛

⎝⎜− − ⎛

⎝⎞⎠

⎞

⎠⎟ + ⎛

⎝⎜ + ⎛

⎝⎞⎠

⎞

⎠⎟

+ ⎛⎝

⎞⎠+ + ⎛

⎝− + − ⎞

⎠

⎤

⎦⎥⎥

∂∂

+⎡

⎣⎢⎢

⎛

⎝⎜

⎛⎝

⎞⎠+ + ⎛

⎝⎞⎠

⎞

⎠⎟

− + ⎛

⎝⎜ + ⎛

⎝⎞⎠

⎞

⎠⎟ + + ⎛

⎝⎞⎠+ ⎛

⎝− − + ⎞

⎠

⎤

⎦⎥⎛⎝

∂∂

⎞⎠⎫⎬⎭

u ζ σ D Dd

Dd

Dd

ζ Dd

ζ

Dd

ζ ζ ζ ζD D

σDd

Dd

ζ Dd

ζ Dd

ζ ζ ζ Dσ

( , ) 458

ln 72

2516

ln 113

78

516

ln 74

58

ln

58

ln 12

134 16 4

38

20516

ln 334

258

ln 113

516

338

54

ln 74

58

ln 138 16 4

gr gr gr gr

gr gr gr

gr gr

1 00 0

20 2 0

04 3 2

02

02

0 02

20 0

2 40

2

(A.9)

⎜ ⎟ ⎜ ⎟= ⎡

⎣⎢⎛⎝

+ ⎞⎠

∂∂

+ ⎛⎝

+ ⎞⎠⎛⎝∂∂

⎞⎠⎤

⎦⎥u σ

D Dd

D Dσ

Dd

Dσ

( )13

5 58

ln 598

54

lnfgr gr

0 00

20

20 0

2

1(A.10)

It is worthwhile to note that, at the leading order of approximation, the friction velocity is proportional to the square root of the local flow depth(equation (A.8)), as it occurs under uniform flow conditions, while the first order correction (equation (A.11)) quantifies the effects due to the crossslope and curvature of the cross-section bed profile.

The local value of the depth-averaged longitudinal velocity = +U σ U σ δ U σ( ) ( ) ( )0 002

01 is determined by integrating u0 and u1 along the normal ζ.It results:

⎜ ⎟= ⎛⎝

+ ⎞⎠

UuU

ln Dd

D**

193

2.5fu

u gr00

00

(A.11)

∫

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟

= ⎧⎨⎩

⎡

⎣⎢ + ⎛

⎝⎞⎠+ ⎛

⎝⎞⎠

⎤

⎦⎥ + ⎡

⎣⎢ + ⎛

⎝⎞⎠+ ⎛

⎝⎞⎠

⎤

⎦⎥

⎫⎬⎭

+ ⎧⎨⎩

⎡

⎣⎢ + ⎛

⎝⎞⎠

⎤

⎦⎥ + ⎡

⎣⎢− ⎤

⎦⎥⎫⎬⎭

UuU

D ln Dd

ln Dd

D D ln Dd

ln Dd

D

uU

D ln Dd

DD

D dσ D

**

781390

395312

25208

, 174373120

465208

25104

,

** 2

52

54

, 1 , ,

fu

u gr grσσ

gr grσ

fu

u grσ

σσ σ

01 00 2 0

0 00 2 0

02

0 00

0 0 02

0

Finally, we convert the coordinates σ to the corresponding Cartesian coordinates n by observing that:

∫ ∫= − ⎛⎝∂∂ ′

⎞⎠

′ = ⎡⎣⎢

− ⎛⎝∂∂ ′

⎞⎠

′ + ⎤⎦⎥

n σβδ

δ Dσ

dσβδ

σ δ Dσ

dσ δ( ) 1 1 12

( )σ σ

00

2 2

00

24O

(A.12)

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.advwatres.2017.10.033.

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