Mass Transport in Vegetated Shear Flows

Mass transport in vegetated shear flows

Marco Ghisalberti and Heidi NepfRalph M. Parsons Laboratory, Department of Civil and Environmental

Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts,

USA

June 10, 2005

Abstract. Submerged aquatic vegetation has the potential to greatly improvewater quality through the removal of nutrients, particulates and trace metals. Theefficiency of this removal depends heavily upon the rate of vertical mixing, whichdictates the timescale over which these constituents remain in the canopy. Con-tinuous dye injection experiments were conducted in a flume with model vegeta-tion to characterize vertical mass transport in vegetated shear flows. Through theabsorbance-concentration relationship of the Beer-Lambert Law, digital imaging wasused to provide high-resolution concentration profiles of the dye plumes. Verticalmass transport is dominated by the coherent vortices of the vegetated shear lay-ers. This is highlighted by the strong periodicity of the transport and its simplecharacterization based on properties of the shear layer. For example, the verticalturbulent diffusivity is directly proportional to the shear and thickness of the layer.The turbulent diffusivity depends upon the size of the plume, such that the rate ofplume growth is lower near the source. In the far-field, mass is mixed more thantwice as rapidly as momentum. Finally, plume size is dictated predominantly by X, adimensionless distance that scales upon the number of vortex rotations experiencedby the plume.

Keywords: vegetated flow, shear layer, vortex, mass transport, turbulent Schmidtnumber, turbulence

1. Introduction

Submerged vegetation is a critical component of many aquatic ecosys-tems. Aquatic canopies provide habitats for macrofauna (Edgar [6])and can be responsible for significant nutrient, particulate and tracemetal removal in wetlands (Kadlec and Knight [15], Silvan et al. [26]).The efficacy of these functions depends heavily upon the rate of ex-change between the water within the canopy and the overlying water.Therefore, to fully describe the impact of submerged vegetation onwater quality, we must be able to quantify the rate of vertical mix-ing. Through modification of the flow, vegetation significantly affectsvertical transport relative to a bare bed, tending to increase verticaldiffusivity above the canopy and to decrease it within (see, e.g., Finni-gan [7], Ackerman [1]). The goal of this paper is to use dye injectionexperiments to characterize vertical mass transport in vegetated shearflows.

c© 2005 Kluwer Academic Publishers. Printed in the Netherlands.

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2 Ghisalberti and Nepf

The vertical discontinuity of drag in flows with submerged vege-tation creates a shear layer across the top of the canopy. As in amixing layer, this shear layer contains an inflection point, renderingthe flow inherently unstable to Kelvin-Helmholtz vortices (Raupachet al. [24], Ikeda and Kanazawa [14]). Through the “pinching” effectsof streamwise vorticity, these vortices evolve to form three-dimensionalstructures (Finnigan [7]). Within these structures, there remains highlyorganized lateral vorticity (Ikeda and Kanazawa [14]). Vertical trans-port in a vegetated shear layer is dominated by these coherent vortexstructures (Gao et al. [9], Ghisalberti and Nepf [10]). The vertical eddyviscosity (νtz) of the layer scales upon its thickness (tml, with referenceto Figure 5) and the shear across it (∆U), and peaks in the middle ofthe layer (Ghisalberti and Nepf [11]). As shown by Ghisalberti and Nepf[11], vortex growth ceases once the production of vortex-scale energyis exactly balanced by drag dissipation. Thus, in dense canopies, thevortex structures often do not extend fully to the bed, segregating thecanopy into a upper region of rapid exchange (the “exchange zone”)and a lower region with negligible shear and more limited water re-newal (the “wake zone”) (Nepf and Vivoni [21]). These two regionsdiffer significantly in their rates of turbulent diffusion, as illustratedby the following example. In the wake zone, we assume that the flowresembles that through an emergent array of cylinders, with verticaltransport dominated by wake turbulence. In an emergent array thevertical turbulent diffusivity, Dtz , scales upon the cylinder Reynoldsnumber (Red = Ud/v, where U is the longitudinal velocity and d thecylinder diameter) and the areal fraction occupied by wakes (Figure 7in Nepf et al. [20]). Consider a typical canopy with a plant fraction, P ,of O(1%) in a mean flow of O(10 cms−1). Based on the observationsof Nepf et al. [20], the vertical diffusivity in the wake zone is O(0.1cm2s−1). Assuming that the turbulent Schmidt number (Sct = νtz/Dtz)is of order unity, the mean vertical diffusivity in the exchange zone isO(1 cm2s−1) (Ghisalberti and Nepf [11]). The order of magnitude dif-ference between the diffusivities in these two regions of the same canopyhighlights the impact of the coherent vortices on vertical transport.

2. Methodology

Laboratory experiments were conducted in a 24-m-long, glass-walledrecirculating flume with a width (wf ) of 38 cm. The experimental setupis summarized in Figure 1. Flow scenarios of varying flowrate (Q) andplant density (a) replicate those used in Ghisalberti and Nepf [11], astudy conducted with the same flume and canopy, such that details of

EFMC154.tex; 10/06/2005; 9:43; p.2

Mass transport in vegetated shear flows 3

velocity structure can be drawn from that study. As in Ghisalberti andNepf [11], a constant flow depth (H) of 46.7 cm was employed, witha canopy height (h) of 13.9 cm. Model canopies consisted of circularwooden cylinders (d = 0.64 cm) arranged randomly in holes drilledinto Plexiglas boards. The range of dimensionless plant densities (ad =0.016 − 0.051) is representative of dense aquatic meadows (see, e.g.,Chandler et al. [3]). The total canopy length was 10 m. Table I detailsthe relevant flow parameters, with each run named as in Ghisalbertiand Nepf [11]. In this table, z1 is the location of the bottom of theshear layer (see Figure 5) and Red is evaluated using the velocity belowthis level (U1). The parameter De represents the vertical diffusivityin an emergent array of dimensionless density ad and flow speed U1,as prescribed by Figure 7 of Nepf et al. [20]. Note that Nepf et al.[20] focuses on conditions of Red > 200, for which the wakes are fullyturbulent. In this study, 50 < Red < 240, such that the values of De

given in Table I may be overpredictions. It is important to note thatthis analysis pertains only to canopies of sufficient density that theshear layer does not penetrate to the bed and canopy drag dominatesbed drag. This is shown to be true for CDah & 0.1 (as seen in the dataof Dunn et al. [5] and Poggi et al. [23]), where CD is the drag coefficientof the canopy.

2.1. Image analysis

Digital imaging provides an unintrusive method of generating high-resolution concentration data in the laboratory. Data can be gatheredsimultaneously at all points in a vertical profile, and at several longi-tudinal locations. To quantify mixing in vegetated flows using digitalimage analysis, a link between pixel intensity and dye concentrationis required. The Beer-Lambert Law (BLL) provides this link, and hasbeen used to determine concentration profiles in flows through porousmedia (see, e.g., Schincariol et al. [25], Gramling et al. [12]). The BLLstates that when light is incident upon a sample of an absorbing species,the relationship between the incident intensity (Io) and the transmittedintensity (I) is given by

log

(Io

I

)= ǫbC, (1)

where ǫ is the absorptivity of the absorbing species, b the path lengthover which the light is attenuated and C the concentration of theabsorbing species. If C is non-uniform along the path length, the con-centration extracted from (1) will be the mean along the path.

The BLL was used to measure concentration downstream of a contin-uous dye injection from a line of ports spanning the flume. The flume

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was backlit by 120-cm-long, 2.5-cm-wide, 32 W fluorescent lamps. Ablue gel food coloring (Country Kitchen, Inc.) was used as the dye.The lamps were wrapped in amber cellophane to maximise the con-trast between undyed and dyed fluid. Digital movies were captured bytwo digital cameras (a Sony DSC-S85 (DC1) and a Canon PowershotG3 (DC2)) and a digital video camcorder (Canon ZR45 (DC3)). Fromeach digital movie, 640×480 bitmap images were acquired using AppleQuicktime (DC1, DC2) and ULead Video Studio (DC3). MATLAB’sImage Processing Toolbox was used to convert images to grayscaleand evaluate pixel intensity (0 to 255) to determine spatial fields of I(during dye injection) and Io (before injection). The intensity of thelamps was uniform and constant, log Io varying by approximately 0.2%in the innermost 50 × 1 cm of the tubes and by 0.1% between images.

To determine the validity of (1) in these experiments, a simplecalibration was conducted using a right-angled isosceles acrylic tank(l = 36 cm, Figure 2). When placed in the flume, the path lengththrough the triangle between the horizontal lamp and the camera variedalong the length of the tank. Each camera was paired with a lampand used to capture four digital movies of the backlit triangle: onewhile filled with water, and three while filled with dye solutions (0.017,0.045 and 0.136 gL−1). The distance from the camera to the flume(dc = 210 cm) was much greater than the width of the triangle. Usingthe definitions in Figure 2, the length of dye (b) between lamp andcamera at pixel position ξ is given by

b ≈ξ

cos(α/n) + sin(α/n)≈

ξ(1 + (l/2)−ξ

ndc

) , (2)

where n is the relative refractive index of water. This approximationis valid in the limit of small α, a restriction imposed to minimize un-certainty in b. Equation (2) was used to determine b for each pixelin the range 7.5 cm < ξ < 28.5 cm (|α| ≤ 0.05). In each pixel, theintensities in the absence of dye (Io) and in the presence of the threelight-attenuating dye solutions (I) were evaluated. Figure 3 shows thecalibration plot of absorbance (A = log(I/Io)) versus bC for cameraDC2. The linear relationship described by the BLL is clearly evidentin the range 0 ≤ bC ≤ 1.9 cmgL−1 (0 ≤ A ≤ 1.4). Similarly, theBLL was shown to be valid for DC1 and DC3 in the ranges bC ≤ 1.4cmgL−1 and bC ≤ 1.3 cmgL−1 respectively. Note that ǫ does nothave to be maintained exactly between experiments; rather, (1) willbe used to determine relative concentrations, with absolute concentra-tions obtained through conservation of mass. This calibration merelyimposes the range over which there is a linear relationship betweenconcentration and absorbance for each camera.

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2.2. Experiments

In these canopies, shear layer growth is arrested, and fully-developedflow conditions (i.e. ∂/∂x = 0) established, within the first 5 m ofthe canopy (Ghisalberti and Nepf [11]). As our analysis is concernedwith mass transport in fully-developed vegetated shear layers, dye wasinjected into the flume 6 m into the canopy. We will define this pointas x = 0. Twelve 0.9-mm-diameter needles (separated by 3.5 cm) con-stituting a line injection were affixed to the top of dowels spanning theflume at x = 0. The needles were connected by 1/8” Polyurethane tub-ing to three 50 mL syringes. A syringe pump maintained the injectionvelocity at the local mean velocity, Uh. The volumetric injection rate,Vi, varied between 0.17 − 0.50 mLs−1. At six measurement locations(xl = [19 cm, 54 cm, 92 cm, 150 cm, 250 cm, 380 cm]), dowels within a4-cm-long slice across the channel were removed and redistributed alongthe canopy. This was done to allow a line of sight between each lamp(now vertically aligned) and its corresponding camera. The length ofthis slice is equivalent to 0.8−1.4 times the inter-cylinder spacing, ∆S.As shown in Ikeda and Kanazawa [14], the removal of canopy elementsover a short length (7∆S in their study) has little impact upon the flowconditions.

The dye solutions used in these experiments had concentrations(Ci) of 120 − 250 gL−1. Through the addition of isopropyl alcohol,the buoyant velocities of the dye solutions were brought to less thanone-fortieth of the vertical turbulent velocity at the point of injection(wrms(h)), ensuring that buoyant effects were negligible. For each flowscenario, the three cameras were used to capture movies of the plume atmeasurement locations 1, 2 and 3. The flume was then re-filled, the dyeinjection repeated, and movies captured at measurement locations 4, 5and 6. To keep all images within the linear ranges described in §2.1, lessconcentrated dyes were used when capturing digital footage at positions1, 2 and 3. Likewise, DC2, which had the most extensive linear range,was always used in the measurement location nearest to the source. Thecamera-lamp pairings and the camera settings were unchanged from thecalibration. The duration of dye injection (td) varied between 3 and 8minutes. This time was chosen carefully so as to be large enough thata steady state was reached at all locations (td & xl,max/U1) and smallenough that no dye recirculated to the most upstream location. Notethat Runs B, J and K from Ghisalberti and Nepf [11] have not beenincluded in these experiments. In such low flows, the buoyant velocityof the dye and the time taken to reach steady state made accurateanalysis difficult.

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In the determination of concentration at heights not equal to theheight of the camera (hc), the line between camera and lamp was nothorizontal. The distance between the cameras and the flume (dc =170− 200 cm) was kept much larger than the flume width to minimizethe resultant parallax error in the measurements. For the same reason,the cameras were positioned at roughly mid-depth (hc = 19 − 23 cm).At the outer edges of the flow, the path from lamp to camera traversedheights within the flume approximately 2 cm greater than and less thanthe (quoted) mean height.

Frames were captured from the digital movies at 1 Hz and convertedto grayscale. The pixel size in the experiments varied between 0.8−1.4mm; near the source, the size of the plume allowed us to zoom inon the region around z = h. Figure 4 shows a sample backgroundimage, an image with dye and the resultant C/Cmax profile from RunE. Sixteen images preceding dye injection (Figure 4(a)) were used todetermine Io(z); this quantity was evaluated by averaging the logarithmof light intensity horizontally (over the innermost 1 cm of the lamp) andtemporally (over the 16 images). During dye injection (Figure 4(b)),I(z, t) was also evaluated as a horizontal average over 1 cm. The BLL(1) was then used to evaluate ǫC(z, t), accounting for the weak heightdependence of the path length through the flume (wf ≤ b < 1.01wf ).Once a steady state was reached in all pixels (i.e. ∂ (ǫC) /∂t = 0), valuesof ǫC(z) were temporally averaged over a minimum of 100 images.Values were then normalized by the maximum value in the profile toyield steady-state values of C/Cmax(z) (Figure 4(c)). As the camerasettings were fixed during each experiment, ǫ was assumed invariant.Absolute concentrations (C(z)) were determined using the conservation

of mass requirement that∫ H0 UCdz = ViCi/wf (under the assumption

that Pe = Uxl/Dtx ≫ 1).

3. Results

Figure 5 displays steady-state concentration profiles (C(z)) at all mea-surement locations in Run I. All concentration data have been normal-ized by the maximum value at measurement location 1. Over distance,the height of maximum concentration falls from the injection height(h) to the bed, initially due to the velocity shear and then due to theno-flux boundary at the bed. The data do not extend to z = 0 andz = H because of the parallax error in the measurements; a linearextrapolation of concentration to the boundaries was applied whenrequired.

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Intrinsic to this method of image analysis is the lateral averaging ofconcentration. However, flows with submerged vegetation have signifi-cant lateral heterogeneity. In strong currents, flexible submerged vege-tation exhibits a pronounced waving (monami) at the vortex frequency,fv. This waving is clearly confined to lateral subchannels (Ghisalbertiand Nepf [10]); in each subchannel, the waving (and hence vortex street)is out of phase with that in neighboring subchannels. This lateral het-erogeneity is demonstrated in the photographs of Figure 6, taken duringRun H. This figure clearly shows the presence of two subchannels. Inthe left-hand image, dye streaks in the subchannel nearer to the cameraundergo a sweep into the canopy (u′ > 0, w′ < 0, using the standardReynolds decomposition). Simultaneously, the dye streaks in the sub-channel farther from the camera undergo an ejection (u′ < 0, w′ > 0).In the right-hand image, taken half a vortex-period ( 1

2fv≈ 7 s) later, the

situation is reversed. Since the plume in each subchannel experiences aperiodic sequence of sweeps and ejections, the laterally-averaged profileprovides a good estimate of the temporal average in each subchannel. Itis important to note that the subchannel width appears to be a functionof the vortex size, such that in shallower flows in the same flume, threesubchannels have been observed (Ghisalberti and Nepf [10]).

The influence of the coherent vortices in vegetated flows is elucidatedin Figure 7. Figure 7(a) captures a dye injection, distinct from theexperiments described in §2.2, during Run H. The incorporation of thedye streak into a growing vortex at the front of the canopy (roughly1.5 m in) is clearly evident. Such clarity of visualization of the vorticalstructure was only possible near the front of the densest canopy, wherethe rate of vortex rotation is highest. Figure 7(b) shows the smoothedpower spectrum of the depth-averaged in-canopy concentration at mea-surement location 1 in Run E. There are clear peaks at both fv (thevortex frequency, 0.068 Hz) and 2fv, as there are in spectra of momen-tum transport, u′w′ (Ghisalberti and Nepf [10]). The strong periodicityof in-canopy concentration is indicative of the dominance of the vorticesin vertical transport. The presence of two subchannels in these flows,combined with the lateral averaging of concentration, augments thepeak at 2fv. The average Strouhal number (St = fvθ/U , where θ isthe momentum thickness of the shear layer and U is the average of thevelocities at the top and bottom of the layer) is 0.037 ± 0.002. Thisis slightly higher than the value observed for bent, flexible canopies(0.032 ± 0.002) but in agreement with the value observed when thesame flexible canopy was erect under low flows (0.037, Ghisalberti andNepf [10]).

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3.1. Two-box model

To characterize vertical transport in vegetated shear flows, the validityof a two-box model was examined. As shown in Figure 8, we seek toquantify the mass flux (mz(h)) between the upper, unvegetated box(h < z < H) and the lower, vegetated box (0 < z < h). Velocity andconcentration are considered uniform within each box (Ua and Ca inthe upper box, Uc and Cc in the lower box). The volume of the canopyelements in the lower box (4% at most in the experimental flows) isnegligible. We assume a constant exchange coefficient (k, with unitsof LT−1) such that mz(h) is proportional to k and the difference inconcentration between the boxes. That is, over a distance dx,

mz(h) = k (wfdx) (Cc − Ca). (3)

While this exchange parameter doesn’t take into account vertical vari-ation in velocity and diffusivity, it is a useful tool when consideringchemical and biological processes that occur fully within the canopyand not at all above it (e.g., nutrient uptake).

Using this two-box formulation, the exchange coefficient can be eval-uated by tracking the longitudinal variation in the concentration of oneof the boxes. Over the distance dx,

dCa =mz(h)

(H − h)wfdx

dx

Ua=

kdx

(H − h)Ua(Cc − Ca) (4)

using (3) and assuming Pe ≫ 1. Therefore,

dCa

dx=

k

(H − h)Ua(Cc − Ca). (5)

Similarly,dCc

dx= −

k

hUc(Cc − Ca). (6)

By defining ∆C = Cc − Ca as the difference in concentration betweenthe two boxes, subtracting (5) from (6) gives

d∆C

dx= −k

(1

hUc+

1

(H − h)Ua

)∆C. (7)

Integration of (7) yields

∆C(x) = ∆C(0) exp

(−kx

(1

hUc+

1

(H − h)Ua

)). (8)

By substitution of (8) into (6),

dCc

dx= −

k∆C(0)

hUcexp

(−kx

(1

hUc+

1

(H − h)Ua

)). (9)

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Therefore,

Cc(x) =∆C(0)(

1 + hUc

(H−h)Ua

) exp

(−kx

(1

hUc+

1

(H − h)Ua

))+ Cc(∞),

(10)where Cc(∞) is the concentration achieved when the dye has becomeperfectly well-mixed over depth. That is,

Cc(∞) =ViCi

wf (hUc + (H − h)Ua). (11)

From (10) we define C∗, the normalized in-canopy concentration, asfollows:

C∗(x) =Cc(x) − Cc(∞)

Cc(0) − Cc(∞)= exp

(−kx

(1

hUc+

1

(H − h)Ua

)). (12)

That is, under the assumptions of uniform velocity and concentra-tion within each box, C∗ decays exponentially with distance along thecanopy (from unity to zero). As the injection is at the interface betweenthe two boxes,

Cc(0) =ViCi

2wfhUc. (13)

To account for concentration and velocity gradients in the experi-ments, quantities within boxes are represented by vertically-averagedexperimental values (i.e. Gc = 1

h

∫ h0 G dz; Ga = 1

(H−h)

∫ Hh Gdz, where

G = [U ,C]). As described in Ghisalberti and Nepf [11], velocity mea-surements could not be taken within approximately 7 cm of the freesurface. The experimental velocity profiles were extrapolated to thesurface (see Figure 8) to best match near-surface profiles taken abovevegetated shear layers [M. Ghisalberti, unpublished data, 1999]. Figure9 shows the observed exponential decay of C∗ over distance. Data withthe same plant density (but different flow speeds) are grouped. Thevertical bars represent the standard uncertainty in each point and thusreflect the variability within each group. This variability is small, witha mean of roughly 0.02 on the C∗ scale. The legend of Figure 9 displaysthe mean in-group values of k∗, the fitting coefficient in the curveC∗(x) = exp (−k∗x). Clearly, the well-mixed condition is approachedmore rapidly, and k∗ is higher, for denser canopies. Denser canopiesgenerate vortices with a greater rotational speed (which scales upon∆U), relative to the mean flow (≈ Uh) (Ghisalberti and Nepf [11]).The higher rates of rotation result in a more rapid flushing of thecanopy over distance. The exponential relationship in (12) is clearlyinvalid near the source, where the assumption of a uniform in-canopy

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concentration breaks down. While the theoretical concentration at thesource is given by (13), the true value is much lower (ViCi/2hwf Uh)since, in reality, the injection is at the point (z = h) of the highestin-canopy velocity. The concentration profiles show that the in-canopyconcentration becomes uniform (varying by less than 15% around themean) by x = 54 cm (for ad = 0.025, 0.051) and x = 92 cm (forad = 0.016, 0.022). The curve fit is only applied beyond this pointwhere, despite the fact that the upper, unvegetated box isn’t similarlywell-mixed, C∗ clearly decays exponentially. Although the curves ofbest fit were determined by prescribing the intercept (i.e. C∗(0) = 1),the exponential fits to the data alone had, on average, an intercept ofalmost exactly unity. This suggests that the deviation from a uniformin-canopy concentration in the near-field does not significantly affectthe bulk exchange between the canopy and the overlying fluid.

We would expect the exchange coefficient (k) to simply scale uponthe shear across the layer (∆U), as this dictates the vortical velocityof the structures. From (12),

k =k∗hUc(

1 + hUc

(H−h)Ua

) . (14)

As shown in Figure 10 (in which the data are no longer grouped), thisexpectation is very closely met (k ≈ ∆U/40). Therefore, for a givenmean flow, k will increase with canopy density. It is important to notethat in the experimental flows, the exchange zone encompasses mostof the canopy (70− 90%), such that a relatively uniform concentrationcan be achieved within the canopy and a two-box model is appropriate.With increasing canopy density, the exchange zone thickness decreases.Therefore, in dense arrays, the overall flushing rate of the canopy maybegin to decline with canopy density. In such dense arrays, a two-boxmodel is likely to be invalid as the greatly differing diffusivities withinthe canopy will preclude the attainment of an approximately uniformconcentration.

The observed exchange coefficient is lower than that predicted byBentham and Britter [2] for unbounded obstructed shear flows. Using aboundary layer framework to describe the flow, the authors derive thetheoretical relationship k ≈ 0.3u∗ for momentum exchange across thetop of sparse arrays, where u∗ = (−u′w′|z=h)1/2 is the friction velocity.Turbulent statistics from Ghisalberti and Nepf [11] show that in shallowvegetated shear flows, u∗ ≈ (0.14±0.01)∆U , such that k ≈ 0.18u∗. Thediscrepancy between these two estimates of the exchange coefficientmost likely arises from the assumption of a logarithmic (boundarylayer) velocity profile by Bentham and Britter [2]. Shallow vegetated

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flows more closely resemble mixing layers, rather than rough boundarylayers (Ghisalberti and Nepf [10]).

3.2. Flux-gradient model

While a two-box model is convenient for describing vertical transport,it fails to encapsulate any vertical variability, revealing little of thestructure of mass transport. Consequently, a profile of vertical turbulentdiffusivity (Dtz(z)) throughout the shear layer was sought. Similarly tothe eddy viscosity, the turbulent diffusivity is expected to scale upon thesize of the vortices (tml) and their rotational speed (which in turn scalesupon ∆U). Although a flux-gradient model is not strictly valid in theseflows (refer to Corrsin [4]), it provides a simple means of characterizingvertical transport.

The mass balance for the control volume shown in Figure 8 is

wf

(∫ z

0UCdz

)

A= wf

(∫ z

0UCdz

)

B− wf∆x

(Dtz

⟨∂C

∂z

⟩)

z. (15)

The terms represent, respectively, the inward and outward advectivefluxes and the vertical diffusive flux. The angular brackets denote a lon-gitudinal average between A and B. Experimental values of Dtz(z) wereevaluated by considering (15) between adjacent measurement locations,i.e.

Dtz(z) = −∆ (

∫ z0 UCdz)

〈∂C/∂z〉z ∆x. (16)

We assume that the longitudinally-averaged gradient can be approxi-mated by the mean of the gradients at the two measurement locations(i.e. 〈∂C/∂z〉 ≈ (∂C/∂z|A + ∂C/∂z|B)/2). This assumption is best ad-hered to in the slowly-varying far-field, so only measurement locationsin the range xl ≥ 92 cm were used in the evaluation. Furthermore, thedetermination of Dtz was restricted to heights at which ∂C/∂z changedby less than a factor of 3 between adjacent measurement locations.The vertical profiles of ∂C/∂z were smoothed with a 2 cm movingwindow before finding the mean of the concentration gradients at ad-jacent locations. To minimize the uncertainty in Dtz, only heights withsignificant concentration gradients (∂C/∂z > 0.05max(∂C/∂z)x=92cm)were considered. This meant that the diffusivity was not evaluatednear the (no-flux) boundaries, where the vertical concentration gradientapproaches zero.

Figure 11(a) shows the collapse of Dtz , when normalized by ∆Utml,across the range of plant densities. Data of equal plant density havebeen grouped, with horizontal bars representing the standard uncer-tainty in each point. The vertical axis of Figure 11(a) (z∗ = (z − z1) /tml)

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represents the dimensionless height above the shear layer bottom. Thecollapse of the diffusivity data throughout the entire shear layer is good,confirming that the coherent vortices dominate transport throughoutthe layer. There is significant vertical variability in Dtz , which peaksat the top of the canopy (Dtz(h) ≈ 0.032∆Utml). In contrast, the eddyviscosity peaks in the middle of the shear layer, which lies above thecanopy (Ghisalberti and Nepf [11]). Note that in the mass balance usedto evaluate Dtz (15), an absence of vertical advective mass transportwas assumed. The secondary circulation presumed to exist above theshear layer (Ghisalberti and Nepf [11]) may drive a small advective fluxin the uppermost 20% of the shear layer. In this region, the calculatedvalue of Dtz , which would incorporate this advective transport, doesnot decrease with height as the eddy viscosity does. Rather, it takes anapproximately uniform value (Dtz ≈ 0.013∆Utml).

The relationship between the eddy viscosity and turbulent diffusiv-ity is shown in Figure 11(b), which details the form of the turbulentSchmidt number (Sct = νtz/Dtz). There is significant uncertainty inSct in the uppermost 20% of the shear layer, due to both the spread ofthe Dtz data in this region and the possible incorporation of a verticaladvective flux into this data. This uncertainty notwithstanding, theaverage observed turbulent Schmidt number in the range 0.1 < z∗ < 0.9is 0.47, with the minimum value occurring just below top of the canopy.This indicates that the transport of mass is more than twice as ef-ficient as the transport of momentum in vegetated shear layers. Theobserved turbulent Schmidt number is lower than those in boundarylayers (0.8 ± 0.1, Launder [17]; Hassid [13]; Koeltzsch [16]) and planarmixing layers (≈ 0.54, from data in Raupach et al. [24]). The differencebetween vegetated shear layers and unobstructed mixing layers maybe explained by numerical results discussed in Fitzmaurice et al. [8].Ensemble averages of the velocity and pressure fields around majorsweep events in vegetated flows reveal that when the sweep approachesthe canopy, the encounter with the region of high drag generates regionsof high dynamic pressure. So, while the sweep event carries momentumand scalars downward via identical motions, the transfer of momentummay be offset by the pressure field induced by the motion at the canopyboundary.

3.3. Similarity of plume behavior

Within the shear layer, the coherent vortices dominate vertical trans-port. Thus, we anticipate that the timescales of plume advection (x/Uh)and vortex rotation (tml/∆U) govern the growth of a scalar plume. The

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ratio of these timescales defines a dimensionless distance,

X =x∆U

tmlUh, (17)

that scales upon the number of vortex rotations experienced by theplume. We expect that the concentration profiles will be similar acrossall runs at a given value of X . That is, a given number of vortex rota-tions are expected to have a similar effect on scalar plumes, irrespectiveof the canopy density. For example, consider the exponential decaycurves of in-canopy concentration as a function of x (Figure 9) and as

a function of X (Figure 12). Whilst the behavior of C∗(x) is strongly

dependent upon ad, the values of C∗(X) collapse to a single curve. Wetherefore expect plume spread and structure, within the shear layer atleast, to depend primarily on a single variable, X.

4. Particle tracking model

Due to the difficulty of determining vertical diffusivity in the near-field, where the concentration profile changes rapidly, the longitudinaldependence of Dtz was not directly calculated. As diffusivity in oceansand lakes is known to increase with the size of dye patches (see, e.g.,Okubo [22], Lawrence et al. [18]), it was thought that Dtz would in-crease with distance from the dye source. Additionally, we sought todetermine the accuracy of a simple model that assumes a diffusivity (D)that is constant in the vertical above z = z1. Below this point, verticaltransport is assumed to be dominated by wake turbulence, with a diffu-sivity (De) given by Figure 7 in Nepf et al. [20] (Table I). Accordingly,we employed a Lagrangian particle tracking model (LPTM, developedby Peter Israelsson, Massachusetts Institute of Technology) to evaluatethe longitudinal dependence of D.

The LPTM was employed as a two-dimensional (x and z) particletracking model in which particles are advected by the mean velocityfield (U(z)) and diffused vertically by a random walk process. Lon-gitudinal diffusion is neglected with respect to advection; assumingthat Dtx ∼ O(Dtz), the lowest Peclet number in this system is O(10).The model released 1000 particles/s (found to be sufficient to obtainconverged statistics) at x = 0, z = h and tracked their position for 1000s over the domain 0 < x < 380 cm. The cell size in the model was 2cm (in x) × 1 cm (z); the number of particles in each cell was dividedby the cell area to yield an areal concentration (ML−2). Once a steadystate was reached at each xl, vertical concentration profiles (sampledat 0.1 Hz) were temporally averaged to provide the steady-state profile.

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In flows with inhomogeneous diffusivity, particle tracking models oftensuffer from the over-prediction of concentration in regions of low dif-fusivity (Weitbrecht et al. [27]). In this study, the order of magnitudebetween D and De had the potential to result in an unrealistic build-up of concentration in the wake zone. However, this was not observedin the LPTM, which predicts a uniform steady-state concentration towithin 1% as x → ∞.

The LPTM was used to determine the values of D that best repli-cated the experimental concentration profiles of each run (i.e. the valuethat maximized r2 between the model result and experimental profile).For each measurement location (xl), D was assumed spatially invariant,both vertically and longitudinally in the range 0 < x < xl. That is, thebest-fit value of D best describes the rate of plume growth between thesource and the measurement location. Based on the collapse of data inFigure 11(a), it was prescribed that D = γ∆Utml where γ is a constant.

From the best-fit values of D at each measurement location, γ(X) wasdetermined for each run, as shown in Figure 13. This figure clearlyshows that the diffusivity near the source is much less than that inthe far-field. However, the plume’s ‘memory’ of the reduced near-fielddiffusivities is erased by X ≈ 7. The mean values of D in the far-field(i.e. X & 7) for each run are detailed in Table I. The rapid rate ofmixing associated with the vortices is highlighted by the ratio of D toDe, which ranges from 15 − 31 in this study.

In a flow with vertically uniform velocity and diffusivity, it can beeasily shown that the relationship between D(x) (the effective constantdiffusivity between the source and position x) and D(x) (the localdiffusivity at x) is given by

D(x) =∂

∂x

(Dx

). (18)

The velocity shear and step-profile of diffusivity in these flows wereshown by model runs not to impact this relationship. Application of(18) to the data in Figure 13 shows that the local diffusivity is constant

(0.019∆Utml, ±15%) beyond X ≈ 2.5. The experimental concentration

profiles reveal that X ≈ 2.5 is almost exactly the location at which theplume has grown to encompass the entire shear layer. For example,Position 2 in Run E (Figure 14) is located at X = 2.2; at this position,the plume extends almost completely to the top of the shear layer (z2 ≈38 cm). These observations lend weight to the concept of a diffusivity

that is dependent upon the plume size. In the near-field (X < 2.5), theplume is smaller than the mixing (vortex) structures and the diffusivityis low. Once the plume reaches the size of the vortices, the diffusivityis maintained at a constant, maximum value.

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In Figure 13, the darkness of the markers is proportional to thegoodness of fit (r2) of the modeled concentration profile to that ob-served. To focus upon the accuracy of the assumption of a uniformdiffusivity above z = z1, the goodness of fit was evaluated in the rangez1 < z < H (which comprised 42−45 data points). The mean value of r2

was more than 0.98, indicating that a constant diffusivity above z = z1

is a suitable approximation. This is further demonstrated in Figure 14,which compares the experimental and modeled concentration profiles ofRun E. The shape of the profile is predicted well at all locations. Inter-estingly, the model overpredicts the concentration gradients below z1 inthe far-field. This suggests that at the same flow speed, the diffusivityin the wake zone is greater than that in equally dense emergent arrays.The apparently enhanced diffusivity in this region is due possibly tosecondary vertical flow behind individual canopy elements in the smallboundary layer near the bed (Nepf and Koch [19]). The measurementsof Nepf et al. [20] were centered at mid-depth and therefore affectedlittle by this vertical flow.

5. Conclusion

Through the absorbance-concentration relationship of the Beer-LambertLaw, digital imaging has been used to provide high-resolution concen-tration profiles of scalar plumes in vegetated flows. It has been clearlyshown that the coherent vortices of a vegetated shear layer dominatevertical transport, such that transport is easily characterized by prop-erties of the shear layer. Firstly, using a two-box model, the exchangevelocity between the canopy and overlying water is shown to scale uponthe total shear (k ∼ ∆U). Secondly, using a flux-gradient model, thevertical turbulent diffusivity scales upon the shear and size of the layer(Dtz ∼ ∆Utml). Our results suggest that plume size is dictated predom-

inantly by X , a dimensionless distance that scales upon the number ofvortex rotations experienced by the plume. The turbulent diffusivitydepends upon the size of the plume (and thus X), such that the rate of

plume growth is lower near the source. In the far-field (X & 7), massis mixed more than twice as rapidly as momentum. Finally, images ofdye injection demonstrate the lateral variability of the vortex structure,which causes transport to be instantaneously nonuniform in the lateraldirection.

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Acknowledgements

This material is based upon work supported by the National ScienceFoundation under Grant No. 0125056. Any opinions, findings, and con-clusions or recommendations expressed in this material are those ofthe author(s) and do not necessarily reflect the views of the NationalScience Foundation. The authors would like to thank Peter Israelssonfor configuring the LPTM, Paula Deardon for her help in the laboratoryand Eric Adams and Ole Madsen for their insightful comments.

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dimensional scalar microfront systems in a large-eddy simulation of vegetationcanopy flow’. Bound.-Layer Meteorol. 112, 107–127.

9. Gao, W., R. Shaw, and K. Paw U: 1989, ‘Observation of organized structure inturbulent flow within and above a forest canopy’. Bound.-Layer Meteorol. 47,349–377.

10. Ghisalberti, M. and H. Nepf: 2002, ‘Mixing layers and coherent structures invegetated aquatic flows’. J. Geophys. Res. 107(C2), 3–1–3–11.

11. Ghisalberti, M. and H. Nepf: 2004, ‘The limited growth of vegetated shearlayers’. Water. Resour. Res. 40. W07502, doi:10.1029/2003WR002776.

12. Gramling, C. M., C. F. Harvey, and L. C. Meigs: 2002, ‘Reactive transport inporous media: A comparison of model prediction with laboratory visualization’.Environ. Sci. Technol 36, 2508–2514.

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14. Ikeda, S. and M. Kanazawa: 1996, ‘Three-dimensional organized vortices aboveflexible water plants’. J. Hydraul. Eng. 122(11), 634–640.

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15. Kadlec, R. H. and R. L. Knight: 1996, Treatment wetlands. Boca Raton, FL:Lewis Publishers.

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17. Launder, B.: 1976, Topics in Applied Physics, Vol. 12, Chapt. 6. Heat and masstransport, pp. 231–287. Springer-Verlag.

18. Lawrence, G. A., K. I. Ashley, N. Yonemitsu, and J. R. Ellis: 1995, ‘Naturaldispersion in a small lake’. Limnol. Oceanogr. 40(8), 1519–1526.

19. Nepf, H. and E. Koch: 1999, ‘Vertical secondary flows in submersed plant-likearrays’. Limnol. Oceanogr. 44(4), 1072–1080.

20. Nepf, H., J. Sullivan, and R. Zavistoski: 1997, ‘A model for diffusion withinemergent vegetation’. Limnol. Oceanogr. 42(8), 1735–1745.

21. Nepf, H. and E. Vivoni: 2000, ‘Flow structure in depth-limited, vegetated flow’.J. Geophys. Res. 105(C12), 28547–28557.

22. Okubo, A.: 1971, ‘Ocean diffusion diagrams’. Deep-Sea Res. 18, 789–802.23. Poggi, D., A. Porporato, L. Ridolfi, J. Albertson, and G. Katul: 2004, ‘The

effect of vegetation density on canopy sub-layer turbulence’. Bound.-Layer

Meteorol. 111, 565–587.24. Raupach, M., J. Finnigan, and Y. Brunet: 1996, ‘Coherent eddies and turbu-

lence in vegetation canopies: the mixing-layer analogy’. Bound.-Layer Meteorol.

78, 351–382.25. Schincariol, R. A., E. E. Herderick, and F. W. Schwartz: 1993, ‘On the appli-

cation of image analysis to determine concentration distributions in laboratoryexperiments’. Journal of Contaminant Hydrology 12, 197–215.

26. Silvan, N., H. Vasander, and J. Laine: 2004, ‘Vegetation is the main factor innutrient retention in a constructed wetland buffer’. Plant Soil 258, 179–187.

27. Weitbrecht, V., W. Uijttewaal, and G. H. Jirka: 2004, ‘2-D Particle tracking todetermine transport characteristics in rivers with dead zones’. In: G. H. Jirkaand W. S. J. Uijttewaal (eds.): Shallow flows. pp. 477–484, A. A. Balkema.

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Table I. Summary of experimental conditions and vegetated shear flow parameters.Runs B, J and K from Ghisalberti and Nepf [11] were not replicated, as the low velocitiescomplicated the experimental technique.

Run A C D E F G H I

Q (×10−2cm3s−1) 48 74 48 143 94 48 143 94

ad 0.016 0.022 0.022 0.025 0.025 0.025 0.051 0.051

tml (±1.0cm) 32.8 31.4 30.7 35.4 33.5 28.8 33.9 32.7

∆U (cms−1) 3.2 4.9 3.5 9.5 6.0 3.3 11 7.4

Uh (cms−1) 2.5 3.5 2.4 6.7 4.6 2.3 6.3 4.0

z1 (±0.5cm) 1.4 2.2 2.6 2.5 2.9 3.3 3.2 4.2

Reda 91 110 70 240 170 78 170 110

De (cm2s−1)b 0.11 0.13 0.11 0.20 0.16 0.12 0.23 0.18

D (cm2s−1)c 1.9 3.0 1.9 6.2 3.9 1.7 7.2 4.5

a In contrast to Ghisalberti and Nepf [11], Red is evaluated here using the velocitybelow the shear layer.

b De, the vertical diffusivity in an emergent stand with equal density and flowspeed, was estimated from Figure 7 in Nepf et al. [20].

c The value in the far-field (X & 7).

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Figure 1. Side view of dye injection in the flume. Twelve needles spanning the widthof the flume were affixed to the tops of dowels 6 m into the canopy; by this point,the flow was fully-developed. The injection velocity was kept at the local velocity(Uh) by a syringe pump. Vertical light sources were placed at 6 locations along thecanopy (x = 19, 54, 92, 150, 250 and 380 cm), where digital footage of the dye plumewas captured.

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Figure 2. Plan view of the calibration used to determine the validity of theBeer-Lambert Law (1) in these experiments. Digital movies were captured of thebacklit isosceles tank (l = 36 cm) filled with water and three dye solutions. Thepath length of light through the tank (b) varied with distance along the front ofthe tank (ξ, (2)). This allowed examination of the validity of (1) fully in the range0.12 < bC < 4.0 cmgL−1. The variation of attenuated light intensity along the tankis shown in the lower graphic. To minimize uncertainty in b, only central pixels(|α| < 0.05) were examined.

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Figure 3. The relationship observed by DC2 between absorbance (A = log(I0/I))and bC for three dye solutions in the triangular tank. The linear relationship pre-scribed by the Beer-Lambert Law is valid in the range 0 < bC < 1.9 cmgL−1. Thevertical error bars represent the uncertainty in the mean due to spatial (over theheight of the tank) and temporal (between images) variability.

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Figure 4. Sample evaluation of an instantaneous concentration profile in Run E(captured on DC2). The figures are: (a) A sample background image, used todetermine Io(z), (b) a sample image with dye, used to determine I(z), and (c)the resultant profile of log Io − log I , averaged horizontally over the innermost 1cm of the 2.5-cm-wide lamp. Ignoring the weak height-dependence of path lengththrough the flume, log Io − log I is directly proportional to concentration (C). Asrequired, max(log Io−log I) ≈ 0.9 < 1.4, the linear range end-point (Figure 2). Oncea steady state was reached in all pixels, profiles of concentration were averaged overa minimum of 100 images.

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Figure 5. Steady-state concentration profiles and velocity profile (taken from Ghisal-berti and Nepf [11]) of Run I. All concentration data have been normalized by themaximum value at measurement location 1. The velocity shear and, subsequently,the bed result in a concentration profile that is asymmetric about the injection point(z = h). The markers are used simply to identify each profile; the true vertical reso-lution of the concentration data is approximately 1 mm. The estimated uncertaintyin each data point is roughly 10%.

Figure 6. Evidence of the lateral heterogeneity in vegetated shear flows. The flow isdivided laterally into (in this case, two) subchannels of the vortex width (≈ tml/2).In the left-hand picture, the dye in the vortex street nearer the camera is sweptinto the canopy simultaneously to the vortex street in the neighboring subchannelejecting dye above the canopy. In the right-hand picture, taken half a vortex period( 1

2fv

) later, the reverse is true.

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Figure 7. The importance of Kelvin-Helmholtz vortices in vertical transport in veg-etated shear flows. (a) The negative image of a dye injection made at the front ofthe canopy in Run H. The dye rises (as fluid is redirected over the drag elements)before being incorporated into the growing vortex street approximately 1.5 m intothe canopy. (b) The smoothed power spectrum of the depth-averaged in-canopyconcentration at measurement location 1 in Run E. There are clear, pronouncedpeaks at the vortex frequency (fv , 0.068 Hz) and 2fv.

Figure 8. Definitive diagram of the two-box and flux-gradient models used to de-scribe vertical transport. In the two-box model, concentration and velocity areconsidered uniform in both the upper (h < z < H) and lower boxes (0 < z < h).Extrapolation of the experimental concentration and velocity profiles was requiredat the outer edges of the flow. These extrapolations are represented with dottedlines.

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Figure 9. The exponential decay of normalized in-canopy concentration (C∗) overdistance. Data with the same plant density (but different flow speeds) are grouped,with vertical bars representing the variability within each group. For clarity, thead = 0.022 data (which falls alongside the ad = 0.025 data) has been omitted.Exchange between the canopy and overlying water occurs more rapidly over distancefor dense canopies. This is highlighted by the mean in-group values of k∗, the fittingcoefficient in the curve C∗(x) = exp (−k∗x). This exponential relationship is invalidnear the source, where the assumption of a uniform in-canopy concentration breaksdown.

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Figure 10. The direct proportionality between the exchange velocity, k, and thetotal shear, ∆U . The vertical bars represent the 90% confidence intervals of eachpoint, based on the uncertainty of the exponential fit through each data set.

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Figure 11. (a) The collapse of Dtz, when normalized by ∆Utml, across the rangeof plant densities. Data of equal plant density have been grouped, with hori-zontal bars representing the inter-run variability in each point. The vertical axis(z∗ = (z − z1) /tml) represents the dimensionless height above the shear layer bot-tom. The thick grey bars show the average location of canopy height in z∗ space foreach plant density; the greater the density, the darker the bar. For each density, theturbulent diffusivity peaks near the top of the canopy. (b) The vertical profile of theturbulent Schmidt number (Sct = νtz/Dtz) in the shear layer. Sct has an averagevalue of approximately 0.47 in the shear layer and reaches a minimum just belowthe top of the canopy.

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Figure 12. The collapse of the decay curves of in-canopy concentration when plottedagainst a dimensionless distance, X. This suggests a strong similarity of plumestructure in X , which scales upon the number of vortex rotations experienced bythe plume.

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Figure 13. The longitudinal variation of normalized best-fit diffusivity,γ = D/∆Utml). Near the source, the diffusivity is low compared to that inthe far-field. The darkness of each point is proportional to the goodness of fit(r2) between the experimental and model concentration profiles. A model thatassumes a vertically uniform diffusivity above z1 generally predicts the shape of theconcentration profile well (r2 > 0.98).

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Figure 14. The good agreement between the experimental concentration profiles ofRun E and those predicted by the LPTM. Both profiles have been normalized by themaximum observed concentration at Position 1. The experimental profiles exhibitsmaller gradients below the shear layer (z1 < 2.5 cm) in the far-field, suggestingthat the diffusivity is actually greater than De (0.20 cm2s−1) in this zone.

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Mass Transport in Vegetated Shear Flows

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