a flow resistance model for assessing the impact of ... · a flow resistance model for assessing...

A flow resistance model for assessing the impact of vegetation onflood routing mechanics

Gabriel G. Katul,1,2,3 Davide Poggi,3 and Luca Ridolfi3

Received 25 November 2010; revised 17 June 2011; accepted 28 June 2011; published 27 August 2011.

[1] The specification of a flow resistance factor to account for vegetative effects in theSaint-Venant equation (SVE) remains uncertain and is a subject of active research in floodrouting mechanics. Here, an analytical model for the flow resistance factor is proposed forsubmerged vegetation, where the water depth is commensurate with the canopy height andthe roughness Reynolds number is sufficiently large so as to ignore viscous effects. Theanalytical model predicts that the resistance factor varies with three canonical length scales:the adjustment length scale that depends on the foliage drag and leaf area density, thecanopy height, and the water level. These length scales can reasonably be inferred from arange of remote sensing products making the proposed flow resistance model eminentlysuitable for operational flood routing. Despite the numerous simplifications, agreementbetween measured and modeled resistance factors and bulk velocities is reasonable across arange of experimental and field studies. The proposed model asymptotically recovers theflow resistance formulation when the water depth greatly exceeds the canopy height. Thisanalytical treatment provides a unifying framework that links the resistance factor to anumber of concepts and length scales already in use to describe canopy turbulence. Theimplications of the coupling between the resistance factor and the water depth on solutionsto the SVE are explored via a case study, which shows a reasonable match betweenempirical design standard and theoretical predictions.

Citation: Katul, G. G., D. Poggi, and L. Ridolfi (2011), A flow resistance model for assessing the impact of vegetation on flood

routing mechanics, Water Resour. Res., 47, W08533, doi:10.1029/2010WR010278.

1. Introduction[2] Equations describing the bulk flow velocity (Ub),

such as Manning’s equation, are widely used in hydraulicengineering and surface hydrology, especially in the con-text of flood routing [Dooge, 1992; French, 1985; Hauser,1996; Hornberger et al., 1998]. A range of ecological andenvironmental applications of such routing problems hasrenewed interest in the theoretical prediction of bulk flowvelocity, especially in the field of ecohydraulics [Green,2005]. These applications include urban wetland construc-tion or restoration [Kadlec, 1994; Nepf, 1999], grass swaledesign to maximize pollution control in urban storm waterrunoff [Kirby et al., 2005], and linking tidal hydrodynamicforcing to flow and sediment transport over coastal wet-lands [Christiansen et al., 2000; Koch and Gust, 1999;Leonard and Luther, 1995; Shi et al., 1995; Wang et al.,1993]. The behavior of extreme overbank flows, wherefloodplain vegetation may be inundated [Wilson and Hor-ritt, 2002], is also of interest due to its socio-economic

implications and impact on large-scale biogeochemicalcycling.

[3] In flood routing, the flood mechanics are generallydescribed by the Saint-Venant equations, which are mathe-matically ‘‘closed’’ by various approximations to the fric-tion slope (Sf). In numerous applications, the Sf is inferredfrom Manning’s equation by assuming that the flow islocally steady and uniform [Ajayi et al., 2008; Moussa andBocquillon, 2000, 2009; Parlange et al., 1981; Richardsonand Julien, 1994; Singh and Woolhiser, 1976; Smith andWoolhiser, 1971; Tayfur et al., 1993; Wang et al., 2002;Woolhiser and Liggett, 1967; Woolhiser, 1975; Yen,2002]. The key parameter to be specified in this ‘‘closure’’is the absolute surface roughness, which may be encodedin the momentum roughness height (zo) or Manning’sroughness coefficient (n). These two roughness measuresdepend on the mean height of the roughness elements pro-truding into the flow (D) and their geometric arrangement.The convention is to assume that these two roughness meas-ures are constant when the roughness Reynolds numberzþo ¼ u�zo=� is sufficiently large, where u� is the friction ve-locity and � is the kinematic viscosity of water [Brutsaert,1982; Gioia and Bombardelli, 2002; Huthoff et al., 2007].For channels in which u� can be determined from measuredwater depth (Hw) and bed slope (So), the zo and n can beanalytically linked to each other provided that Hw/D is large[Chen, 1991]. When Hw/D is of order unity, however,which is generally the case for flood routing over sub-merged vegetation, a priori specification of n pose unique

1Nicholas School of the Environment, Duke University, Durham, NorthCarolina, USA.

2Department of Civil and Environmental Engineering, Pratt School ofEngineering, Duke University, Durham, North Carolina, USA.

3Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnicodi Turin, Turin, Italy.

Copyright 2011 by the American Geophysical Union.0043-1397/11/2010WR010278

W08533 1 of 15

WATER RESOURCES RESEARCH, VOL. 47, W08533, doi:10.1029/2010WR010278, 2011

http://dx.doi.org/10.1029/2010WR010278

challenges. This approximates the situation for open chan-nel flow over vegetated surfaces, where the canopy height(Hc) can be comparable to Hw. For such cases, the flow re-sistance formulation is problematic, and to date, has resistedcomplete theoretical treatment thereby inviting the use ofempirical formulations such as flow retardance curves[Kouwen and Unny, 1969].

[4] For predicting the resistance parameters for flowthrough vegetation, an analysis of the mean momentumbalance inside and just above dense and rigid canopies areproposed when Hw/Hc > 1. Because flood-routing overvegetated surfaces is often characterized by Hw/Hc > 1, thesubmerged vegetation case is considered here. The emer-gent vegetation case, while important in a number of sce-narios, is beyond the scope. Next, the resulting meanvelocity profile from the solution of the mean momentumbalance is used to analytically describe n and the Darcy-Weisbach friction factor f as a function of three lengthscales. These length scales are Hw, Hc, and Lc where Lc ¼(Cda)�1 is the so-called adjustment length scale [Belcheret al., 2003], which is widely used in canopy turbulence toparameterize the loss of turbulent kinetic energy fromadvecting eddies due to their dissipation by drag elements[Belcher et al., 2003; Katul et al., 2004]. The Lc is depend-ent on a dimensionless drag coefficient Cd, and the meanleaf area density a � LAI/Hc, where LAI is the one-sidedleaf area index.

[5] The predictions of the resistance factor from the pro-posed analytical model here are compared with a largenumber of flow experiments through vegetated canopiesreported elsewhere [Poggi et al., 2009]. The model resultsshow explicitly how n and f nonlinearly decrease withincreasing Hw/Hc for a given Lc/Hc, and with increasing Lc/Hc for a given Hw/Hc. The implications of these nonlineardependences for flood routing dynamics are then explored.Using the one-dimensional Saint-Venant equation, thehydrographs and water depths along a stream covered uni-formly with rigid vegetation with a constant Lc/Hc are com-puted using both a constant and a dynamically evolving nas a function of Hw/Hc. The constant and dynamic n casesare contrasted in terms of the timing and modulation of thepeak flow rate longitudinally.

2. Theory[6] A brief review of the basic definitions and relation-

ship between zo and n (or f) for large Hw/Hc and for rigidvegetation is first discussed. An analysis of how this emerg-ing picture is altered when Hw/Hc is of order unity is thenpresented. Even within this restrictive scope, n is shown tovary, at minimum, with Hw, Hc, LAI, and Cd. The conse-quences of these alterations to n(or f) on the flood-routingmechanics are then analyzed via a case study using scalingarguments and numerical runs.

2.1. Basic Definitions and the Deep Layer Formulation[7] The balance between frictional and gravitational

forces in a wide rectangular channel whose control volumeis shown in Figure 1a with Hw/Hc� 1 is discussed. In thisanalysis, it is assumed that (1) the bed slope angle (�) issmall enough so that sinð�Þ � tanð�Þ ¼ S0, (2) the channelwidth is sufficiently large so that the hydraulic radius Rh �

Hw, (3) the flow is fully turbulent, statistically stationaryand planar homogeneous, and (4) the surface can be treatedas hydrodynamically rough (zþo > 2). This last conditionimplies that the log-law with a constant zo is a reasonablerepresentation of the time-averaged velocity ( �U ) profileover Hw as shown in Figure 1a. Under these conditions, theforce balance between frictional forces at the canopy topand gravitational forces simplifies to (Figure 1a)

� ¼ �g ðHw � HcÞ sinð�Þ ¼ �g ðHw � HcÞSo; ð1Þ

where � � �t ¼ � u�2 is the total stress approximated by theturbulent stress (�t) shown in Figure 1a given that zþo > 2, gis the gravitational acceleration, and � is the water density.With these definitions, equation (1) provides an estimate ofu� given as

u� ¼ffiffiffi�

�

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig ðHw � HcÞ So

p: ð2Þ

[8] The logarithmic mean velocity profile or variants onit for flexible canopies [Stephan and Gutknecht, 2002] isexpressed as a function of zo and u� using

�UðzÞ ¼ u��

lnz� d

zo

� �; ð3Þ

where overbar denotes time-averaging, z is height from thechannel bottom, and �ð¼ 0:4Þ is Von Karman’s constant[e.g., Izakson, 1937], and d is the zero-plane displacement(Figure 1a). The depth-averaged mean velocity or bulk ve-locity (Ub), is given as

Ub ¼1

Hw

ZHw

zoþd

�UðzÞdz¼ u��Hw

ZHw

zoþd

lnz� d

zo

� �d½ðz� dÞ�

¼ u��

1Hwðd�Hwþ zoÞþ ðHw� dÞ ln Hw� d

zo

� ��

� u��

ðHw� dÞHw

� lnðeÞþ lnHw� d

zo

� �� u��

lnHw

e zo

� �;

ð4Þ

where ln(e) ¼ 1 and for a deep-layer formulation, it wasassumed Hw � zo þ d. Rearranging equation (4) into adimensionless form commonly used for flow resistanceequations gives

Ub

u�¼ 1�

lnHw

ezo

� �: ð5Þ

[9] Manning’s equation for a wide rectangular channel is

Ub ¼1n

Hw2=3So

1=2: ð6Þ

When Hw� Hc, and upon combining equations (2) and (6)gives

Ub

u�¼Hw

1=6

nffiffiffigp : ð7Þ

W08533 KATUL ET AL.: FLOW RESISTANCE OF VEGETATION W08533

2 of 15

[10] Another flow resistance measure generally employedin flood routing is the Darcy-Weisbach friction factor f,which can be related to n via

Ub

u�¼

ffiffiffi8f

s¼Hw

1=6

nffiffiffigp : ð8Þ

[11] To establish a simplified relationship between n andzo, it is noted that for large values of Hw/(ezo), the logarith-mic function in equation (5) can be approximated by theclassical 1/7 power law [Chen, 1991; Katul et al., 2002]

lnHw

ezo

� �� 5

2Hw

ezo

� �1=7

: ð9Þ

Figure 1. (a) Conceptual model for the force balance above the canopy and the assumed mean velocityprofile �UðzÞ in streams where Hw/Hc� 1 (deep layer formulation). The Hw is the water depth, Hc is thecanopy height, and So is the bed-slope, also shown. (b) Same as Figure 1a but 1 < Hw/Hc < 10 (shallowlayer formulation). The flow inside the canopy (canopy layer) is labeled as CL while the flow above thecanopy is denoted as surface layer and labeled as SL. The difference in the mean flow between the twolayers is due to (1) the presence of a drag force assumed to be much larger than the ground shear stressand (2) a constant mixing length leff. The determination of d and zo in SL are based on the continuity andsmoothness conditions applied to �UðHcÞ. The parameter � ¼ u�= �UðHcÞ, where u� is the friction velocityat the canopy top. The mixing length leff is continuous but not smooth in this formulation. (c) Examples ofmodeled �UðzÞ=u� using equation (18) for (left) atmospheric and (right) aquatic boundary layers. For theatmospheric boundary layer (ABL) experiments, the data sets include a Loblolly pine forest, a southernmixed hardwood forest, a spruce canopy, a Boreal pine forest, an alpine coniferous forest, a rice canopyand a corn canopy (data sources are described in the work of Katul et al. [2004]). The aquatic boundarylayer experiments include steel rods [from Poggi et al., 2004b] (densest canopy), flexible plastic vegeta-tion [from Nepf and Vivoni, 2000], and wooden dowels [from Lopez and Garcia, 2001]. The adjustmentlength scale Lc ¼ (Cda)�1 used in equation (14) is also presented. For the ABL experiments, the boundarylayer height is much larger than Hc. For the aquatic boundary layer, Hw/Hc varies between 1.5 and 5.


3 of 15

[12] Combining equations (9) and (5) results in

Ub

u�� 5

2�Hw

ezo

� �1=7

; ð10aÞ

n¼ H1=6w

H1=7w

!2�e1=7

5g1=2

� �zo

1=7 � 0:06zo1=7; ð10bÞ

ffiffiffif8

r¼ 2�e1=7

5

� �zo

Hw

� �1=7

� 0:18zo

Hw

� �1=7

: ð10cÞ

[13] Equations (5) and (7) can also be combined toyield n¼ �=

ffiffiffigp� �

H1=6w ln Hw=zoð Þ� 1½ ��1 or

ffiffiffiffiffiffiffiffi8=f

p¼ ð1=�Þ

ln Hw= ezoð Þ½ � without invoking any power law approxima-tion to the log-profile. The dependence of n on g�1/2 waspointed out by a number of authors [Chow, 1959; Gioiaand Bombardelli, 2002; Yen, 1992], and it was used toadjust tabulated n values to the gravitational field of Marsin the study of Martian channel formations [Carr, 1979].The relationship n � 0:06zo

1=7 is also consistent with anumber of other studies that employed power law approxi-mations for �UðzÞ to predict Ub for hydro-dynamicallyrough surfaces [Katul et al., 2002]. Recall that for large zþo ,zo, and n are independent of zþo . Hereafter, the results inequations (10) are referred to as the deep-layer formulationgiven that Hw � Hc [Chen, 1991; Katul et al., 2002; Yen,1992, 2002].

2.2. Canopy Effects and the Shallow LayerFormulation (1 < Hw/Hc < 10)

[14] For a stationary and planar-homogeneous flow inthe absence of a mean vertical velocity, the time and pla-nar-averaged mean momentum equation along the longitu-dinal direction is given by

0 ¼ �g So þ@�

@z� �

�U2

Lc�sf ; ð11Þ

where � ¼ �m þ �t, � is the total stress, �m is the viscousstress, and �t ¼ �u0w0 is the turbulent stress, where u0w0 isthe momentum turbulent flux at height z, z is the heightfrom the channel bottom, and �sf is the Heaviside step func-tion given by

�sf ¼1 z=Hc � 1

0 z=Hc > 1

:

[15] In the aquatic vegetation literature, it is common todefine the drag force as 0:5 �Uð Þ2=Lc rather than �Uð Þ2=Lc,which implies that the inferred dimensionless drag coeffi-cient Cd from such studies should be halved prior to usingequation (11). In the derivation of equation (11), a numberof approximations were invoked, most of which are dis-cussed elsewhere [Finnigan, 2000; Lopez and Garcia,2001; Nikora et al., 2001; Raupach and Shaw, 1982; Shi-mizu and Tsuiumoto, 1994; Wilson and Shaw, 1977].

[16] Two simplifications are worth noting: (1) dispersivefluxes, which are formed when spatial correlations exist inthe time averaged mean momentum equation that are

subsequently spatially averaged within the canopy volume;and (2) finite porosity effects. While the dispersive fluxesappear to be small in dense canopies, at least compared tou0w0, they are significant in sparse canopies [Cheng andCastro, 2002; Poggi et al., 2004a; Poggi and Katul, 2008a,2008b]. We are mindful that their parameterization remainsin its infancy and no attempts have been made here toinclude them. Equation (11) also neglects any finite poros-ity effects arising from spatial averaging, though such finiteporosity effects can be corrected for. In essence, the dragthat the canopy elements exert on the fluid leads to a decel-eration of the fluid only within the fraction of the volumeoccupied by the fluid, which is 1 � p, where P is the pro-portion of volume occupied by canopy elements. Upon vol-ume averaging the concentrated drag force induced by thefoliage within 1 � p, the bulk volume-averaged drag forceshould be divided by 1 � p and is equivalent to reducingLc by 1� p. If p � 1, then finite porosity effects can beignored. Alternatively they may be absorbed in Lc using areduction factor of 1 � p, leaving the form of equation(11) unchanged.

[17] The depth-averaged mean momentum balance isgiven as

0 ¼ZHW

0

g S dzþ �ðHW Þ�� ð0Þ

��ZHc

0

L�1c

�UðzÞ½ �2dz ð12Þ

where, after neglecting the molecular stresses relative toturbulent stresses, results in �ðzÞ � �tðzÞ ¼ �u0w0ðzÞ. In theabsence of any wind-induced or rain-induced shear stresseson the free surface, u0w0ðHwÞ � 0, and for a dense canopy,

�u0w0ð0ÞRHc

0L�1

c�UðzÞ½ �2dz

<< 1

so that the force balance reduces to the interplay betweenthe canopy drag force and the weight of the fluid given as

ZHc

0

L�1c

�UðzÞ½ �2dz � g SoðHw � HcÞ: ð13Þ

[18] Hypothetically, if the mean velocity profile is uni-form with �UðzÞ ¼ Ub and Lc is independent of z, then equa-tion (13) leads to Hc=Lcð Þ Ub½ �2 � u2

� and 8=f ¼ U2b=u2

� �Hc=Lc [Poggi et al., 2009]. This analysis demonstrates thatHc/Lc must be one of the dimensionless quantities to beretained in any flow resistance formulation.

[19] Since �UðzÞ is not uniform, a number of approachescan be used to model �UðzÞ within and immediately abovecanopies, including first-order and higher-order closuremodels [Baptist et al., 2007; Defina and Bixio, 2005;Neary, 2003; Poggi et al., 2009]. In the work of Poggi et al.[2009], it was shown that a first-order closure model withan imposed mixing length (leff) that remains constant insidethe canopy, hereafter referred as to as the canopy layer or


4 of 15

CL but varies linearly as �ðz� dÞ above the canopy, here-after referred to as surface layer or SL, reproduced meas-ured �UðzÞ and u0w0 profiles from a wide range ofexperiments reasonably well. Figure 1b shows the assumedshape of leff in CL and SL. When variations in Lc with z aresmall, semianalytical models can also be used to estimate�UðzÞ. One such model can be expressed as follows [Finni-gan and Belcher, 2004; Harman and Finnigan, 2007;Massman, 1997; Massman and Weil, 1999; Poggi et al.,2008]:

�UðzÞ ¼USLðzÞ ¼

u��

lnz� d

zo

� �; z=Hc > 1;

UCLðzÞ ¼u��

exp �ðz� HcÞ

leff

� �; z=Hc < 1

8>><>>: ; ð14Þ

where leff ¼ 2�3Lc is the constant mixing length inside thecanopy and � ¼ u�= �UðHcÞ is a momentum absorptioncoefficient. The continuity (i.e., USL(Hc) ¼ UCL(Hc)) and

smoothnessd �USL

dz

z=Hc¼1

¼ d �UCL

dz

z=Hc¼1

(i.e., ensuring a

continuous u0w0) of the mean velocity profile at z/Hc ¼ 1necessitates unique relationships between zo and d as afunction of � and Lc/Hc given by

dHc¼ 1� 2�3Lc

�Hc

zo

Hc¼ 1� d

Hc

� �exp ��=�ð Þ ¼ 2�3Lc

�Hcexp ��=�ð Þ:

ð15Þ

[20] A two-layer (i.e., z/Hc > 1 and z � Hc) representa-tion for the mean flow field has been used by a number ofauthors to arrive at approximate flow resistance formula-tions [Huthoff et al., 2007; Yang and Choi, 2010]. In deriv-ing the effective flow resistance, these studies primarilyfocused on determining the depth-averaged velocity ineach of the two layers without using any smoothness condi-tions imposed on �UðzÞ as done in equations (14) and (15).Also, it should be emphasized that the mixing length iscontinuous but not smooth thereby resulting in a continuousand nonsmooth turbulent viscosity at z/Hc ¼ 1.

[21] The near-exponential mean velocity profile shapeinside the canopy and the near-logarithmic mean velocityprofile shape above the canopy are supported by atmos-pheric boundary layer (ABL) and flume experiments for

dense canopies in which 1 < Hw/Hc < 5 (Figure 1c). More-over, they represent well zones often used to characterizethe mean flow through submerged vegetation [Baptist et al.,2007] except immediately close to the ground and the bot-tom layers of the canopy. In the case of ABL experiments,the leaf area density profiles, especially for the forested eco-systems in Figure 1c, are far from vertically uniform, andyet the canonical mean velocity profile shapes remain near-exponential inside the canopy and near-logarithmic above.For the aquatic vegetation, departures from the exponentialform in the CL can occur in the mid to bottom layers of thecanopy (z/Hc < 0.4), where a is generally low [Nepf andVivoni, 2000]. This is also the case for the hardwood can-opy for the ABL experiment of Figure 1c. These secondarypeaks are often connected with the presence of a meanpressure gradient or large vertical gradients in the flux-transport terms [Katul and Albertson, 1998; Katul andChang, 1999; Shaw, 1977; Wilson and Shaw, 1977]. How-ever, much of the bulk velocity within the aquatic vegeta-tion remains dominated by the higher �UðzÞ in the uppercanopy layers, where the exponential mean velocity profileappears to be reasonable (Figure 1c). Furthermore, themean velocity profile immediately above the canopyappears to follow a logarithmic behavior even though Hw/Hc is not too large. With this formulation for �UðzÞ, the nor-malized bulk velocity is given by

Ub

u�¼ Hc

Hw

Ub;CL

u�þ Hw � Hc

Hw

Ub;SL

u�; ð16Þ

where, as before, u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigSoðHw � HcÞ

pand

Ub;CL

u�¼ 1

u�Hc

ZHc

0

UCLðzÞdz¼ 2�Lc

Hc1� exp � 1

2�2

Hc

Lc

� �� ;

Ub;SL

u�¼ 1

u�ðHw�HcÞ

ZHw

Hc

USLðzÞdz¼ 1�

(�1þ ln

Hw�dzo

� � Hw�dð Þ= Hw�Hcð Þ Hc�dzo

� �� Hc�dð Þ= Hw�Hcð Þ" #)

;

ð17Þ

where zo and d are given by equation (15) as a function of� and Lc/Hc and f (or n) can be analytically expressed as

Ub

u�¼

ffiffiffi8f

s¼ Hw

1=6

nffiffiffigp ¼ 2�

Lc

Hw1� exp � 1

2�2

Hc

Lc

� �� zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Direct Canopy Layer Effects

þ 12

1�

1� Hc

Hw

� ��1þ ln

Hw � dzo

� � Hw�dð Þ= Hw�Hcð Þ Hc � dzo

� �� Hc�dð Þ= Hw�Hcð Þ" #( )

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Perturbed Log: ProfileDue to Canopy Layer Effects

:ð18Þ


5 of 15

[22] Hereafter, the formulation in equation (18) isreferred to as the shallow-layer formulation. It asymptoti-cally converges to the deep layer formulation when Hw �Hc and Hw � �Lc. Equation (18) makes explicit the directeffects of the canopy on Ub in the CL and their modulatingeffect on the log-profile in SL. The momentum absorptioncoefficient � in equation (18) is known to vary with thecanopy density for small a but saturates at about 0.33 fordense canopies as discussed elsewhere [Katul et al., 1998;Massman, 1997; Massman and Weil, 1999; Poggi et al.,2004b; Poggi and Katul, 2008a; Raupach, 1994]. Figure 1calso shows that �UðHcÞ=u� is about 3 for the ABL experi-ments, which implies that � ¼ u�= �UðHcÞ � 1=3, a constantfor these dense-canopy experiments. For sparser canopies, �may be derived from flume experiments on rods with variousdensities [Poggi et al., 2004b]. These experiments suggestthat for a fixed canopy height, � ¼ min 0:135

ffiffiffiap

; 0:33ð Þ,which is the relationship used here for the aquatic vegeta-tion. However, the formulation by Massman and Weil[1999] can also be used if the effect of vegetation shelteringis known, and is given by

� � c1 � c2 exp c3 CdLAI=Pmð Þ; Pm � 1þ c4LAI; ð19Þ

where c1 ¼ 0.33, c2 ¼ 0.264, c3 ¼ 15.1, and c4 2 ½0; 0:4�,and Pm is a sheltering coefficient (c4 ¼ 0 when no shelter-ing corrections are employed). The specification of � isfundamentally connected to the penetration depth (�w), orthe depth referenced from the canopy top at which 90% themomentum flux (¼ u0w0) is extracted by the canopy [Nepfand Vivoni, 2000]. The penetration depth is often used tocategorize aquatic vegetation into sparse (�w=Hc ¼ 1,where the eddy penetration occurs throughout Hc) or dense(�w=Hc < 1) [Nepf et al., 2007]. Using the mean velocityprofile in equation (14) and a first-order closure approxima-tion for u0w0 ¼ �l2

eff ðdUCL=dzÞ2, the momentum flux insidethe canopy is given by

u0w0

u2�¼ � exp

z� Hc

�2Lc

� �: ð20Þ

[23] Upon setting z ¼ �w, u0w0=u2� ¼ �0:1, and solving

for �w results in a linear dependence between �w=Hc and(Cd LAI)�1 given as

�w

Hc� � lnð0:1Þ�2 Lc

Hc� 2:3�2

CdLAIð21Þ

when � is constant. The linear relationship between �w=Hcand (Cd LAI)�1 has also been reported across a wide rangeof experiments for aquatic vegetation [Nepf and Ghisal-berti, 2008]. In airflow within terrestrial vegetation, LAImust exceed 2:3�2=Cd for the ‘‘dense canopy’’ criterion tobe met, i.e., �w=Hc � 1. If � � 0:3 (see ABL experimentsin Figure 1c) and for a typical Cd � 0.2, this result suggeststhat ‘‘dense’’ canopies must have LAI 1 m2 m�2, a rea-sonable choice given that the data sets for the ABL in Fig-ure 1c all have LAI > 3 m2 m�2. Moreover, for LAI > 3.5(typical of forested ecosystems), equation (21) suggeststhat almost 90% of the momentum is extracted in the

top 30% of the canopy, which is consistent with empiricalfindings for a number of atmospheric flow experimentsthrough dense canopies [Katul et al., 2004].

3. Data Sets[24] The data sets used to evaluate the model in equation

(18) spans a wide range of canopy types, Hc, Hw, and a assummarized in Table 1 of Poggi et al. [2009]. They includewooden dowels, stainless steel rods, plastic plants, and realplants. For rigid canopies, 53 data sets were used and aredescribed elsewhere [Ghisalberti and Nepf, 2004; Lopezand Garcia, 2001; Meijer and Van Velzen, 1999; Murphyet al., 2007; Poggi et al., 2004b].

[25] For the flexible vegetation datasets, a variety ofexperiments were also employed that include data sets onwheat stems, mixed grasses, Spartina anglica, plasticaquarium plants, and natural reeds [Baptist, 2003; Carolloet al., 2002; Ciraolo and Ferreri, 2007; Jarvela, 2005;Kouwen and Unny, 1969; Meijer and Van Velzen, 1999;Nepf and Vivoni, 2000; Shi et al., 1995], and one data setfrom a plastic plant prototype [Nepf and Vivoni, 2000]. Inthese cases, the reported deflected height rather than thegeometric height of the vegetation was used as Hc. Whencomparing model calculations with the experiments, meas-ured So, Hc, Hw, a and an estimate of Cd, are needed. Thesevariables are presented in Table 1 of Poggi et al. [2009]and are not repeated here. The � ¼ min 0:135

ffiffiffiap

; 0:33ð Þand Lc ¼ (Cda)�1 can be directly computed from suchmeasurements. The values of zo and d are then determinedfrom equation (15), and Ub/u�, n, and f are computed fromequation (18).

4. Results[26] As earlier noted, in studies that utilize �UðzÞ, the sur-

face roughness is generally characterized by zo while instudies that utilize Ub (as may be the case in flood routing),f or n are often preferred. To explore biases arising fromusing the deep layer formulation when linking these twosurface roughness measures, the following analysis wasconducted. The value of zo was determined from equation(15) using inferred Lc, Hc, and � for all the data sets. Then,equation (10c) was used to estimate f. The computed andmeasured f values are compared in Figure 2. Figure 2shows that the deep-layer formulation recovers the approxi-mate 1/7 power law evidenced by the overall patterns in thecombined data set, but perhaps most evident in the rod can-opy experiments [Poggi et al., 2004b]. This rod canopydata [Poggi et al., 2004b] was collected at Hw/Hc ¼ 5, andhence, the experiment is transitioning toward a deep layerbehavior. The mixed grasses experiment [Carollo et al.,2002], which includes the flexible canopy runs with mostbending (and possibly reduced drag) appears to have areduced friction factor when compared to the deep-layerformulation predictions. This data set is the only data setwith measured f smaller than predictions from equation(10c). When comparing the deep layer formulation predic-tions with the predictions from the shallow layer formula-tions, it is evident from Figure 2 that the shallow layerformulation always yields a larger f than its deep layercounterpart. In Figure 2, the deep-layer expressionffiffiffiffiffiffiffiffi

f =8p

¼ ð1=�Þ ln Hw= ezoð Þ½ �f g�1 is also presented for


6 of 15

reference. As earlier noted, the power law approximation inequation (9) is only valid for small zo/Hw. However, as zo/Hw > 0.01, the power law approximation underestimatesthe measured

ffiffiffiffiffiffiffiffif =8

p, as expected.

[27] A one-to-one comparison between measured andmodeled f, n, and Ub using the shallow layer formulationis presented in Figure 3. Despite numerous simplificationsmade in its derivation, the agreement between measuredand modeled variables, especially for Ub, is reasonablethough clear model biases do emerge. These biases werequantified using the relative error eR ¼ �Ub=Ub, where� is the difference between measured and modeled Ub.Figure 3 presents eR against Hc/Lc, the submergence depthHw/Hc, and the canopy Reynolds number Rec ¼ u�Hc=�.It is clear from the eR analysis that the model performsbetter for rigid canopies when compared to flexible cano-pies. The largest relative error was for Spartina anglica[Shi et al., 1995], which has a variable leaf area densityfollowed by the highly flexible mixed grasses [Carolloet al., 2002]. The trends in eR were not explained by var-iations in Hc/Lc, Hw/Hc, or Rec ¼ u�Hc=�. If viscouseffects did impact � or Cd, then a bias in eR at the lowerend of Rec would have emerged, which was not the casehere. The variations in eR were also explored against thebulk Reynolds number defined as Reb ¼ UbHw=� and theoutcome was similar to the analysis conducted againstRec. Despite these biases, agreement with the data in Fig-ure 3 are comparable to other first-order closure modelresults [Poggi et al., 2009] that did not a priori assumethe shape of �UðzÞ.

[28] While these analytical model-data comparisons areencouraging, a number of issues remain problematic andneed to be confronted. First, the analysis here focused onrigid canopies, and flexible canopies were treated as rigidwith a modified canopy height set to the specified bendingheight. The immediate consequences of this approximationare that the model suppresses any interactions between thebulk flow and vegetation bending. Correcting for such aninteraction is difficult within large-scale hydrologic model-ing as the stiffness and bending properties of the vegetationand its multiscale architecture become essential. Second,the closure model primarily dealt with submerged canopies,though the flow through emergent canopies can also play arole in flood routing, especially for overbank flow problemsand flow in wetlands. Third, first-order closure modelswere employed throughout and any nonlocal contributionsto momentum transfer originating from the flux-transportterm were neglected. Likewise, the dispersive stresses inthe volume-averaged momentum balance were ignored.Fourth, the adjustment length scale was assumed to be con-stant independent of z, which is clearly not realistic for com-plex canopy morphology. Fifth, the mean momentum balancetreatment was assumed to be entirely one-dimensional (verti-cal), though in typical flood routing cases, the advectiveand nonsteady terms inside the canopy can be significantand contribute to the force imbalance in equation (12).Sixth, u0w0ðHwÞ was neglected, which is not likely to holdduring rain or strong wind events. Despite all assumptionsand limitations discussed above, equation (18) does providea physically based relationship between n or f and Hw under

Figure 2. A comparison between predicted (dashed and dotted lines) and measuredffiffiffiffiffiffiffiffif =8

p(symbols)

as a function of the dimensionless momentum roughness length (zo/Hw), where zo is computed fromequation (15). Predictions (dashed and dotted lines) are based on the deep-layer formulations for a powerlaw and a log-law mean velocity profile. Note that the deep-layer formulations (dashed and dotted lines)underestimate the majority of the experiments. The prediction from the shallow-layer formulation (equa-tion 18) is also presented as blue open circles.


7 of 15

a set of restrictive conditions. The significance of this de-pendence and its consequence on flood routing mechanicsare discussed next.

5. Application to Flood Routing[29] The basic equations that describe the one-dimensional

flood routing mechanics along the longitudinal direction (x)are the continuity and Saint-Venant’s equations given by

@A@tþ @Q@x¼ 0

@Q@tþ @

@xQ2

A

� �þ gA

@Hw

@x� So

� �þ gASf ¼ 0

ð22aÞ

with

Sf �n Q

AR2=3h

!2

; ð22bÞ

where t is time, Q is the total flow rate, and A is the cross-sectional area. This representation of Sf via Manning’sequation assumes that the frictional forces along the wettedperimeter of A are responsible for the energy losses, whichis not the case for shallow water flow over vegetated surfa-ces due to the drag force imposed by the vegetation on theflow. However, the effects of the canopy can still beaccounted for in equation (22b) by an equivalent or effec-tive n that is only acting on the wetted perimeter yet repro-duces the same Ub/u� as the vegetated surface (whose Hc

and Lc are a priori specified) at a given Hw. Likewise, thederivation of equation (18) assumed that u� is driven by theturbulent shear stress at the canopy top, while equation(22b) places that same shear stress on the wetted perimeter.The two formulations make the same assumptions thatu� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðHw � HcÞSo

p. In essence, for operational flood

routing approaches employed for vegetated surfaces, theeffects of the canopy elements on n are replaced by ahydro-dynamically equivalent n generated from the wetted

Figure 3. Comparison between measured and modeled n, f, and Ub for all the experiments using theshallow layer formulations. Symbols are the same as in Figure 2. The one-to-one line (dashed) is alsoshown. The relative error eR ¼ �Ub=Ub as a function of the dimensionless adjustment length scale Hc/Lc, the relative submergence depth Hw/Hc and the canopy Reynolds number Rec ¼ u�Hc=� are also pre-sented. A 620%relative error bounds (horizontal lines) are shown for clarity.


8 of 15

perimeter and responds to variations in Hw as in equation(22b). This is precisely how the effective n was defined anddetermined in equation (18). As earlier noted, operationalflood routing models assume n to be a constant independentof Hw [HEC-RAS, 2002]. However, coupling the frictionfactor with Hw to generate what can be termed as stage-dependent friction factor has been successfully employed insurface water routing [Casas et al., 2010]. To explore howsuch nonlinear coupling between n and Hw affects the floodrouting mechanics, two separate analyses are conducted anddiscussed. The first is a scaling analysis on a similarity solu-tion to equation (22), and the second is centered on a numer-ical analysis of the full Saint-Venant equations with twosets of model calculations contrasted: one with n main-tained as a constant in x and t, while another evolves n in xand t based on the dynamics of Hw given by equation (18).Hereafter, scenarios with constant and evolving n arereferred to as ‘‘static’’ and ‘‘dynamic,’’ respectively.

5.1. Scaling Analysis[30] The scaling analysis here explores how perturba-

tions introduced by the presence of vegetation in the expo-nent describing the rating curve of Q � Hw impacts thespace-time evolution of Hw. For illustration, a rectangularchannel section whose width B is much larger than Hw isconsidered. Moreover, the mean momentum balance inequation (22) is considerably simplified by assuming that Sf

� So (i.e., locally steady and uniform). Under theseassumptions, Q and the flow rate per unit width q arerelated by Q ¼ B q ¼ B(VHw) ¼ B a1 (Hw)m, where for thestatic cases a1 ¼ ðS1=2

o =nÞ and m ¼ 5/3 so that the meancontinuity equation reduces to

@Hwðx; tÞ@t

þ @

@xa1Hwðx; tÞm½ � ¼ 0: ð23Þ

[31] In the static case, n � no and is assumed constant,while in the dynamic case, it is approximated byn � no Hc=Hwð Þ! so that when ! � 0 the static case isrecovered. This approximation is not exact but captures thecanonical shape of equation (18). Hence, a finite ! leads toan a1 ¼ ðS1=2

o =noÞ=H!c and an m ¼ 5=3þ !. By propagat-

ing the effects of a finite ! on a1 and m, and subsequentlyon the solution to equation (23), a description of how adynamic n modifies flood routing can be explored withoutexplicitly solving equation (23). It should be noted here thatfor ! ¼ 1=3 and m ¼ 2, equation (23) reduces to an inviscidBurger equation known to admit similarity solutions andshockwaves [Barenblatt, 2003]. When Hw(x, t) > 0 for (x,t) > 0 and m = 1, one plausible solution can be derivedusing the scaling arguments in Appendix A and is given as

Hwðx; tÞ a1m t

x

� 1=ð1�mÞ¼ no

S1=2o

H!c

5=3þ !xt

!1=ð2=3þ!Þ

: ð24Þ

[32] Hence, a finite ! modifies both the exponent and theamplitude of the speed of flood propagation that scales asx/t. A formal analysis that includes the effects of a dynamicinflow hydrograph is discussed using numerical analysis,where all the terms in the Saint-Venant equations areretained.

5.2. Case Study: Flood Routing Over Grass Swales[33] For illustrating the effects of a dynamic n on flood

routing, a grass swale configuration is employed as a case-study. The setup is not intended to replicate a particularsystem but resembles designs employed in grass swalesalong major highways [ISU, 2008]. In grass swale designs,the So should be steep enough to ensure adequate velocity,but usually not steeper than 4% to reduce the possibleoccurrence of a hydraulic jump, with 1% being a recom-mended design value. Moreover, for storms having a 10-yearreturn period, the expected design Ub should not exceed 1 to2 m s�1, and to enhance particle removal efficiency, a mini-mum of 10 min residence time for water is recommended.These two constraints imply that the swale length should beon the order of few hundred meters to about kilometer. Forthe model runs here, a length Lx ¼ 1000 m and a bed slopeof So ¼ 1.0% are selected as typical values. However,unlike the parabolic cross-sectional area used in grassswale design, a rectangular cross sectional area with a con-stant width B ¼ 2.5 m is employed throughout to simplifythe depth-area relationship and minimize parameter specifi-cations. This B is commensurate with a recommendedbottom width that varies from a minimum of 0.6 m to amaximum of 2.6 m.

[34] Optimal grass swale designs recommend that thegrass be regularly mowed to maintain Hc between 0.10 and0.15 m. Hence, the 1 km channel is assumed to be covereduniformly with rigid vegetation having an Hc ¼ 0.15 m, anLAI ¼ 0.5 m2 m�2, and a Cd ¼ 0.5. These canopy charac-teristics closely follow values reported for short mowedgrass [Novick et al., 2004; Thompson and Daniels, 2010],though the grass canopy is not typically rigid as assumedhere. Interestingly, the zo predicted by equation (15) usingthese assumed Hc, Cd, and LAI is 0.009 m, which maintainsa hydrodynamically rough surface because zþo >zo

ffiffiffiffiffiffiffiffiffiffiffiffigHcSop

=� � 2. Employing the deep layer formulationðn ¼ 0:06z1=7

o Þ to such a zo value yields an n � 0.031. Thisvalue of n is in excellent agreement with the minimum n ¼0.03 recommended in grass swale designs when the flowdepth is expected to exceed 0.3 m (i.e., Hw/Hc 2) as dis-cussed elsewhere [ISU, 2008]. For Hw/Hc! 1, the shallowlayer formulation in equation (18) predicts an n � 0.14,which is also in good agreement with the maximum recom-mended design value for n � 0.15 when the water level atthe design flow rate are comparable to or smaller than Hc[ISU, 2008]. Moreover, this n is in good agreement withthe measured value reported for short grass prairie [Eng-man, 1986]. Hence, as logical bounds to the static n modelruns, these two extreme n values are employed. For initialconditions, it is assumed that the flow is uniform with Hw/Hc ¼ 1.0 in all model runs.

[35] Boundary conditions are given by an idealizedinflow hydrograph approximated by

Qinð0; tÞ ¼ Qo exp ��1ðt ��2Þ2h i

þ Qo;i; ð25Þ

where Qo is the maximum amplitude, �1 and �2 are pa-rameters describing the spread and time to peak of theinflow hydrograph, respectively. The model parametersused in this setup are listed in Table 1.


9 of 15

[36] The modeled hydrographs Q(x,t) and water depthHw(x, t) are presented, respectively, in Figures 4 and 5 forspecified locations along the channel. The dynamic n val-ues at the same spatial locations are also shown in Figure 6.The model results are conducted for a Qo ¼ 4 m3 s�1.These results suggest that the flow rate and concomitantwater depth differences between the dynamic and the staticrun employing the minimum n are, to a first approximation,

minor across the various stream locations. The opposite istrue when comparing the dynamic n with the static runemploying the maximum n. For this large Qo, the dynamicn rapidly drops to values comparable to the minimum staticn as the flood wave advances (Figure 6), resulting in floodrouting mechanics resembling a static case with a frictionfactor given by the deep layer formulation. The flood prop-agation velocity and maximum Hw are comparable for

Table 1. Flood Routing Model Parameters Used in the Grass Swales Case Study

Variable Description Value

Channel AttributesB Channel width (m) 2.5Lx Channel length (m) 1000So Bed slope (m m�1) 0.01

Vegetation AttributesLAI Leaf area index (m2 m�2) 0.5Cd Drag coefficient 0.5Hc Canopy height (m) 0.15n Manning’s roughness Two static cases: 0.03 (deep layer formulation); 0.14

(shallow layer formulation). Dynamic case: Seeequation (18).

Initial and Boundary ConditionsHw(0, x) Initial uniform water level (m) Hc

Qin Inflow hydrograph (m3 s�1)Qðt; 0Þ ¼ Qo exp ��1ðt ��2Þ2

� þ Qo;i

Qo, Qo,i Amplitude of flood inflow hydrograph, andQi is the Q at the initial condition Hw(0,x).

Qo 2 ½2:5; 12:5� Unless otherwise stated, Qo ¼ 4 m3 s�1

�1, �2 Spread and time to peak of Qin. �0.0001 s�2, 250 s

Figure 4. The effects of dynamic n on the hydrographs along various positions across the channellength for the setup in Table 1 and for Qo ¼ 4 m3 s�1. Model calculations with dynamic n, computedusing equation (18), are shown in solid lines and the ones computed with a constant n are shown indashed lines for the deep layer formulation (minimum n) and in dots for the shallow layer formulation(maximum n). The inflow hydrograph (thick-solid) is also shown in the top left plot for reference.


10 of 15

Figure 5. Same as Figure 4 but showing the profile of Hw rather than Q.

Figure 6. Same as Figure 4 but showing the space-time evolution of the dynamic Manning’s roughnessn. The dashed and dotted horizontal lines are for the two ‘‘static’’ n cases derived from deep layer formu-lation (minimum n) and shallow layer formulation for Hw ¼ 1.01 Hc (maximum n).


11 of 15

these two end-member cases, at least when compared to theresults for the static n inferred from the shallow layer for-mulation. In essence, these model results lend some theo-retical support to the choice of a constant n set to theminimum recommended for the high flood flow rate [ISU,2008]. Naturally, with such a high Qo, Hw/Hc at the peak ofthe advancing wave becomes sufficiently large so that n (1)becomes nearly independent of Hw and (2) approaches anear constant value (¼ 0:06z1=7

o ) as evidenced by Figure 6.[37] A natural follow-up question then is to what degree

this emerging picture is altered for various Qo. To explorethis point further, the above model runs were repeated for Qo

that varied from 3 to 13 m3 s�1. For each Qo, the maximumflow rate at each x location along the channel was computedfor the dynamic and the two static runs. The similarity in

these maxima was then assessed using the dimensionlessratios �Q=Q, where � is the different between the dynamicsolution and one of the two static solutions, normalized bythe dynamic solution. A negative dimensionless ratio impliesthat the static runs over-predict their dynamic counterpart.

[38] For each Qo, the regions along the stream where thedynamic and static predictions of maximum flow ratesdiverged most are presented in Figure 7. It is clear from Fig-ure 7 that the static solution with n determined from the deeplayer formulation notably diverged from the dynamic solu-tion at very low Qo (<2 m3 s�1) and at large x/Lx (>0.7).Otherwise, the dynamic and static solutions for this mini-mum n differ by no more than 20%, with the static solutionover predicting these extreme flow rates. When Qo > 7m3 s�1, the over predictions diminish to less than 20%. For

Figure 7. Differences between static and dynamic n formulations of the peak flow rates at normalizeddownstream distance (x/Lx) for various inflow hydrograph amplitudes Qo. The relative differences (col-ors) are defined as departures from the dynamic case and referenced to the dynamic case. (left) Differen-ces in modeled maximum flow rate for the dynamic and static case with n set to its deep layer(or minimum) value. (right) Same as Figure 7 (left) but static case is with n set to its maximum value.


12 of 15

the static runs at maximum n, the agreement with thedynamic solution is primarily confined to the upstream por-tion (x/Lx < 0.2) but for all Qo. Beyond this range, the differ-ences between this static and dynamic case can exceed 90%.

6. Discussion and Conclusions[39] The use of dimensional analysis to describe the fric-

tion factor over rough surfaces as a function of the Reynoldsnumber is one of the main successes in hydraulic research.Extending the dimensional analysis to canopy flows remainsa major scientific challenge because the height and spacingof the vegetation elements can be comparable to the largesteddy and water depth, and the dissipation of turbulent kineticenergy by drag elements can occur over distances that canbe comparable to the canopy height rather than the viscousdissipation length scale. Such a setup does not readily admitto a clear scale separation between the eddy sizes transport-ing momentum to a rough surface (e.g., attached eddies tothe boundary) and their interaction with sizes or spacing ofthe protruding canopy elements [Huthoff et al., 2007].

[40] The case of submerged vegetation flow at sufficientlylarge Reynolds number was considered so that viscouseffects can be neglected. Even for this limiting case, an anal-ysis on the governing terms for the mean momentum balancesuggest that the canopy height, the water depth, and theadjustment length scale must be included in any dimensionalconsideration. Using first-order closure principles, an analyt-ical model for the flow resistance factor was proposed andtested across a wide range of data sets that includes thesethree length scales. Despite the numerous simplificationsmade, the agreement between measured and modeled resist-ance factors and bulk velocities were reasonable to within20% for several data sets that employed rigid canopies. Themodel recovers the flow resistance formulation when thewater depth is much larger than the canopy height, at leastfor hydrodynamically rough surfaces. The model also pro-vides a unifying framework that links a number of parame-ters and length scales often used in canopy turbulencestudies such as the penetration depth, the momentum absorp-tion coefficient, the adjustment length scale, the momentumroughness length, and the zero-plane displacement height.While simplifications were made for analytical tractability,the model performance remained comparable to first-orderclosure model results reported in the literature for the samedata sets. The implication on routing mechanics of a frictionfactor varying with water depth emerging from this theoreti-cal consideration was also explored. Again, commencingwith a scaling analysis, it was shown that a water depth de-pendent friction factor slows down the propagation velocityof an advancing flood wave relative to the deep layer case.Numerical simulations to the full Saint-Venant equationsdemonstrate that as the flood wave advances, the Hw/Hc

increases locally thereby diminishing the value of n, whichspeeds up the peak of the flood wave compared to the neigh-boring fluid velocities. The numerical results show a reason-able match between empirical design standards for grassswales and predictions from the model here.

[41] From a broader perspective, advancements in Inter-ferometric Radar measurements of water level fluctuationsfrom space [Alsdorf et al., 2000, 2001, 2005, 2007a,2007b; LeFavour and Alsdorf, 2005; Mason et al., 2003;

Smith, 1997] and the rapid progress in air-born canopyLidar measurements [Lefsky et al., 2002a, 2002b; Casaset al., 2010] permit (1) the key canopy attributes (LAI andHc) and (2) repeated measurements of water level maps intime to be performed at unprecedented spatial resolutionover large basins. Some preliminary attempts using suchdata for flow rate estimates over large basins are promising[Alsdorf et al., 2007a]. As earlier mentioned, the variablesneeded to predict the flow resistances here include leaf areaindex, canopy height, water depth, and estimates of the fo-liage drag coefficient. While the first three variables can bedetermined from remote sensing platforms, the approachproposed here is ‘‘tempered’’ by the fourth variable (Cd).Nonetheless, the proposed approach here is analyticalthereby permitting efficient implementation in flood routingacross large basins, and specifying Cd (a foliage attribute)remains constrained (in both space and time).

Appendix A: Validation of the Proposed Similar-ity Solution

[42] A possible solution to the partial differential equation(pde) given by

@Hwðx; tÞ@t

¼ � @

@xa1Hwðx; tÞm½ �

was proposed for m = 1 and was expressed as

Hwðx; tÞ �a1m t

x

� 1=ð1�mÞ:

[43] The genesis of this solution is a scaling analysis onthe pde that commences with the argument that

�Hw

�t a1mHw

m�1 �Hw

�x;

which can be expressed as

�t�x � 1

a1mHwm�1

tx;

resulting in

1

Hwm�1

� �1=ð1�mÞ¼ Hw a1m

tx

� 1=ð1�mÞ:

[44] To verify that this scaling result satisfies the abovePDE, it is replaced in the left-hand side (LHS) and theright-hand side (RHS) for an equality check. This replace-ment leads to the following:

let � ¼ a1mtx

> 0;

@Hw

@t¼ a1m �ð Þ�1þ 1=ð1�mÞ½ �

ð1� mÞx ;

� @

@xa1Hm

w

� �¼ a1mð Þ2t �ð Þ�1þ 1=ð1�mÞ½ � �ð Þðm�1Þ=ð1�mÞ

ð1� mÞx2:


13 of 15

[45] For LHS to be identical to the RHS, the condition

a1mð Þtx

�ð Þðm�1Þ=ð1�mÞ ¼ 1;

must be satisfied for all �, x, and t > 0. Provide m = 1,this condition is always satisfied since

� �ð Þðm�1Þ=ð1�mÞ ¼ 1:

This completes the proof that the proposed similarity solu-tion satisfies the PDE. It must be emphasized that this solu-tion was intended to show how a finite ! impacts thecanonical scaling laws emerging from the PDE rather thanany precise matching to specific initial and boundary condi-tions imposed on the flood routing problem.

[46] Acknowledgments. The authors thank Sally Thompson and Cos-tantino Manes as well as three anonymous referees for all the helpful com-ments. Support from the Fulbright-Italy distinguished scholars programand the National Science Foundation (NSF-EAR-10-13339) are acknowl-edged. Poggi also acknowledges support from the Commission of the Eu-ropean Communities’ WARECALC program (PIRSES-GA-2008-230845).

ReferencesAjayi, A. E., et al. (2008), A numerical model for simulating Hortonian

overland flow on tropical hillslopes with vegetation elements, Hydrol.Processes, 22(8), 1107–1118.

Alsdorf, D. E., et al. (2000), Interferometric radar measurements of waterlevel changes on the Amazon flood plain, Nature, 404(6774), 174–177.

Alsdorf, D., C. Birkett, T. Dunne, J. Melack, and L. Hess (2001), Waterlevel changes in a large Amazon lake measured with spaceborne radarinterferometry and altimetry, Geophys. Res. Lett., 28(14), 2671–2674,doi:10.1029/2001GL012962.

Alsdorf, D., et al. (2005), Diffusion modeling of recessional flow on centralAmazonian floodplains, Geophys. Res. Lett., 32, L21405, doi:10.1029/2005GL024412.

Alsdorf, D., et al. (2007a), Spatial and temporal complexity of the Amazonflood measured from space, Geophys. Res. Lett., 34, L08402,doi:10.1029/2007GL029447.

Alsdorf, D. E., E. Rodr�ıguez, and D. P. Lettenmaier (2007b), Measuringsurface water from space, Rev. Geophys., 45, RG2002, doi:10.1029/2006RG000197.

Baptist, M. J. (2003), A flume experiment on sediment transport with flexi-ble and submerged vegetation, in International Workshop on RiparianForest Vegetated Channels: Hydraulic, Morphological and EcologicalAspects, Int. Assoc. for Hydraul. Res., Trento, Italy.

Baptist, M. J., et al. (2007), On inducing equations for vegetation resistance,J. Hydraul. Res., 45(4), 435–450.

Barenblatt, G. I. (2003), Scaling, 171 pp., Cambridge Univ. Press, NewYork.

Belcher, S. E., et al. (2003), Adjustment of a turbulent boundary layer to acanopy of roughness elements, J. Fluid. Mech., 488, 369–398.

Brutsaert, W. (Ed.) (1982), Evaporation into the Atmosphere: Theory, His-tory, and Applications, 299 pp., Kluwer Acad. Publ., New York.

Carollo, F. G., et al. (2002), Flow velocity measurements in vegetated chan-nels, J. Hydraul. Eng., 128(7), 664–673.

Carr, M. H. (1979), Formation of Martian flood features by release of waterfrom confined aquifers, J. Geophys. Res., 84(B6), 2995–3007, doi:10.1029/JB084iB06p02995.

Casas, A., et al. (2010), A method for parameterising roughness and topo-graphic sub-grid scale effects in hydraulic modelling from LiDAR data,Hydrol. Earth Syst. Sci. Discuss., 7, 2261–2299.

Chen, C. L. (1991), Unified theory on power-laws for flow resistance,J. Hydraul. Eng., 117, 371–389.

Cheng, H., and I. P. Castro (2002), Near wall flow over urban-like rough-ness, Bound. Layer Meteorol., 104(2), 229–259.

Chow, V. T. (1959), Open-Channel Hydraulics, 680 pp., McGraw-Hill,New York.

Christiansen, T., et al. (2000), Flow and sediment transport on a tidal saltmarsh surface, Estuarine Coastal Shelf Sci., 50(3), 315–331.

Ciraolo, G., and G. B. Ferreri (2007), Log velocity profile and bottom dis-placement for a flow over a very flexible submerged canopy, paper pre-sented at Proceedings of 32nd Congress of IAHR—Harmonizing theDemands of Art and Nature in Hydraulics, Venice, Italy.

Defina, A., and A. C. Bixio (2005), Mean flow and turbulence in vegetatedopen channel flow, Water Resour. Res., 41, W07006, doi:10.1029/2004WR003475.

Dooge, J. C. (1992), The Manning formula in context, in Channel Flow Re-sistance: Centennial of Manning’s Formula, edited by G. T. Yen, p. 135,Water Resour. Publ., Littleton, Colo.

Engman, E. T. (1986), Roughness coefficients for routing surface runoff,J. Irrig. Drain. Eng., 112(1), 39–53.

Finnigan, J. (2000), Turbulence in plant canopies, Annu. Rev. Fluid Mech.,32, 519–571.

Finnigan, J. J., and S. E. Belcher (2004), Flow over a hill covered with aplant canopy, Q. J. R. Meteorol. Soc., 130(596), 1–29.

French, R. H. (1985), Open-Channel Hydraulics, 705 pp., McGraw Hill,New York.

Ghisalberti, M., and H. M. Nepf (2004), The limited growth of vegetatedshear layers, Water Resour. Res., 40, W07502, doi:10.1029/2003WR002776.

Gioia, G., and F. A. Bombardelli (2002), Scaling and similarity in roughchannel flows, Phys. Rev. Lett., 88(1), 14,501–14,504.

Green, J. C. (2005), Modelling flow resistance in vegetated streams:Review and development of new theory, Hydrol. Processes, 19(6),1245–1259.

Harman, I. N., and J. J. Finnigan (2007), A unified theory for flow in thecanopy and roughness sub-layer, Boundary Layer Meteorol., 129(3),323–351.

Hauser, B. A. (1996), Practical Hydraulics Handbook, 368 pp., LewisPubl., Los Angeles, Calif.

HEC-RAS (2002), HEC-RAS Reference Manual, U.S. Army Corps of Eng.,Washington, D.C.

Hornberger, G. M., et al. (1998), Elements of Physical Hydrology, 302 pp.,Johns Hopkins Univ. Press, Baltimore, Md.

Huthoff, F., D. C. M. Augustijn, and S. J. M. H. Hulscher (2007), Analyticalsolution of the depth-averaged flow velocity in case of submerged rigidcylindrical vegetation, Water Resour. Res., 43, W06413, doi:10.1029/2006WR005625.

ISU (2008), Grass Swales, in Iowa Stormwater Management Manual,Section 2I-2, 25 pp., Iowa State Univ. Inst. for Transp., Ames, Iowa.

Izakson, A. (1937), Formula for the velocity distribution near a wall, Zh.Eksp. Teor. Fiz., 7, 919–924.

Jarvela, J. (2005), Effect of submerged flexible vegetation on flow structureand resistance, J. Hydrol., 307(1–4), 233–241.

Kadlec, R. H. (1994), Detention and mixing in free-water weltands, Ecol.Eng., 3(4), 345–380.

Katul, G. G., and J. D. Albertson (1998), An investigation of higher-orderclosure models for a forested canopy, Boundary Layer Meteorol., 89(1),47–74.

Katul, G. G., and W. H. Chang (1999), Principal length scales in second-order closure models for canopy turbulence, J. Appl. Meteorol., 38(11),1631–1643.

Katul, G. G., et al. (1998), Active turbulence and scalar transport near theforest-atmosphere interface, J. Appl. Meteorol., 37(12), 1533–1546.

Katul, G., P. Wiberg, J. Albertson, and G. Hornberger (2002), A mixinglayer theory for flow resistance in shallow streams, Water Resour. Res.,38(11), 1250, doi:10.1029/2001WR000817.

Katul, G. G., et al. (2004), One- and two-equation models for canopy turbu-lence, Boundary Layer Meteorol., 113(1), 81–109.

Kirby, J. T., et al. (2005), Hydraulic resistance in grass swales designed forsmall flow conveyance, J. Hydraul. Eng., 131(1), 65–68.

Koch, E. W., and G. Gust (1999), Water flow in tide- and wave-dominatedbeds of the seagrass Thalassia testudinum, Marine Ecol. Prog. Ser., 184,63–72.

Kouwen, N., and T. E. Unny (1969), Flow retardance in vegetated channels,J. Irrig. Drain. Div., 95(IR2), 329–342.

LeFavour, G., and D. Alsdorf (2005), Water slope and discharge in theAmazon River estimated using the shuttle radar topography mission digi-tal elevation model, Geophys. Res. Lett., 32, L17404, doi:10.1029/2005GL023836.

Lefsky, M. A., et al. (2002a), Lidar remote sensing of above-ground bio-mass in three biomes, Global Ecol. Biogeogr., 11(5), 393–399.


14 of 15

Lefsky, M. A., et al. (2002b), Lidar remote sensing for ecosystem studies,Bioscience, 52(1), 19–30.

Leonard, L. A., and M. E. Luther (1995), Flow hydrodynamics in tidalmarsh canopies, Limnol. Oceanogr., 40(8), 1474–1484.

Lopez, F., and M. H. Garcia (2001), Mean flow and turbulence structure ofopen-channel flow through non-emergent vegetation, J. Hydraul. Eng.,127(5), 392–402.

Mason, D. C., et al. (2003), Floodplain friction parameterization in two-dimensional river flood models using vegetation heights derived fromairborne scanning laser altimetry, Hydrol. Processes, 17(9), 1711–1732.

Massman, W. J. (1997), An analytical one-dimensional model of momen-tum transfer by vegetation of arbitrary structure, Boundary Layer Mete-orol., 83(3), 407–421.

Massman, W. J., and J. C. Weil (1999), An analytical one-dimensional sec-ond-order closure model of turbulence statistics and the Lagrangian timescale within and above plant canopies of arbitrary structure, BoundaryLayer Meteorol., 91(1), 81–107.

Meijer, D. G., and E. H. Van Velzen (1999), Prototype scale flume experi-ments on hydraulic roughness of submerged vegetation, paper presentedat 28th International IAHR Conference, Graz, Austria.

Moussa, R., and C. Bocquillon (2000), Approximation zones of the Saint-Venant equations for flood routing with overbank flow, Hydrol. EarthSyst. Sci., 4(2), 251–261.

Moussa, R., and C. Bocquillon (2009), On the use of the diffusive wave formodelling extreme flood events with overbank flow in the floodplain,J. Hydrol., 374(1–2), 116–135.

Murphy, E., et al. (2007), Model and laboratory study of dispersion inflows with submerged vegetation, Water Resour. Res., 43, W05438,doi:10.1029/2006WR005229.

Neary, V. S. (2003), Numerical solution of fully developed flow with vege-tative resistance, J. Eng. Mech., 129(5), 558–563.

Nepf, H. M. (1999), Drag, turbulence, and diffusion in flow through emer-gent vegetation, Water Resour. Res., 35(2), 479–489, doi:10.1029/1998WR900069.

Nepf, H., and M. Ghisalberti (2008), Flow and transport in channels withsubmerged vegetation, Acta Geophys., 56(3), 753–777.

Nepf, H. M., and E. R. Vivoni (2000), Flow structure in depth-limited,vegetated flow, J. Geophys. Res., 105(C12), 28547–28557, doi:10.1029/2000JC900145.

Nepf, H. M., et al. (2007), Transport in aquatic canopies, in Flow andTransport with Complex Obstructions: Applications to Cities, VegetativeCanopies, and Industry, NATO Sci. Ser., edited by, Y. A. Gayev and J. C.R. Hunt, pp. 221–250, Springer, Berlin.

Nikora, V., et al. (2001), Spatially averaged open–channel flow over roughbed, J. Hydraul. Eng., 127(2), 123–133.

Novick, K. A., et al. (2004), Carbon dioxide and water vapor exchange in awarm temperate grassland, Oecologia, 138(2), 259–274.

Parlange, J. Y., et al. (1981), Kinematic flow approximation of runoff on aplane: An exact analytical solution, J. Hydrol., 52(1–2), 171–176.

Poggi, D., and G. G. Katul (2008a), The effect of canopy roughness densityon the constitutive components of the dispersive stresses, Exp. Fluids,45(1), 111–121.

Poggi, D., and G. G. Katul (2008b), Micro- and macro-dispersive fluxes incanopy flows, Acta Geophys., 56(3), 778–799.

Poggi, D., et al. (2004a), A note on the contribution of dispersive fluxes tomomentum transfer within canopies: Research note, Boundary LayerMeteorol., 111(3), 615–621.

Poggi, D., et al. (2004b), The effect of vegetation density on canopy sub-layer turbulence, Boundary Layer Meteorol., 111(3), 565–587.

Poggi, D., et al. (2008), Analytical models for the mean flow inside densecanopies on gentle hilly terrain, Q. J. R. Meteorol. Soc., 134(634), 1095–1112.

Poggi, D., C. Krug, and G. G. Katul (2009), Hydraulic resistance of sub-merged rigid vegetation derived from first-order closure models, WaterResour. Res., 45, W10442, doi:10.1029/2008WR007373.

Raupach, M. R. (1994), Simplified expressions for vegetation roughnesslength and zero-plane displacement as functions of canopy height andarea index, Boundary Layer Meteorol., 71(1–2), 211–216.

Raupach, M. R., and R. H. Shaw (1982), Averaging procedures for flowwithin vegetation canopies, Boundary Layer Meteorol., 22(1), 79–90.

Richardson, J. R., and P. Y. Julien (1994), Suitability of simplified overlandflow equations, Water Resour. Res., 30(3), 665–671, doi:10.1029/93WR03098.

Serrano, S. E. (2006), Development and verification of an analytical solu-tion for forecasting nonlinear kinematic flood waves, J. Hydrol. Eng.,11(4), 347–353.

Shaw, R. H. (1977), Secondary wind speed maxima inside plant canopies,J. Appl. Meteorol., 16(5), 514–521.

Shi, Z., et al. (1995), Flow Structure in and above the various heights of asalt-marsh canopy: A laboratory flume study, J. Coastal Res., 11(4),1204–1209.

Shimizu, Y., and T. Tsuiumoto (1994), Numerical analysis of turbulentopen-channel flow over a vegetation layer using a k-epsilon turbulencemodel, J. Hydrosci. Hydraul. Eng., 11, 57–67.

Singh, V. P., and D. A. Woolhiser (1976), A nonlinear kinematic wavemodel for watershed surface runoff, J. Hydrol., 31, 221–243.

Smith, L. C. (1997), Satellite remote sensing of river inundation area, stage,and discharge: A review, Hydrol. Processes, 11(10), 1427–1439.

Smith, R. E., and D. A. Woolhiser (1971), Overland flow on an infiltratingsurface, Water Resour. Res., 7, 899–913, doi:10.1029/WR007i004p00899.

Stephan, U., and D. Gutknecht (2002), Hydraulic resistance of submergedflexible vegetation, J. Hydrol., 269(1–2), 27–43.

Tayfur, G., et al. (1993), Applicability of the St-Venant equations for2-dimensional overland flows over rough infiltrating surfaces, J. Hydraul.Eng., 119(1), 51–63.

Thompson, S. E., and K. E. Daniels (2010), A porous convection model forgrass patterns, Am. Nat., 175(1), E10–E15.

Wang, F. C., et al. (1993), Intertidal Marsh suspended sediment transport proc-esses, Terrebonne Bay, Louisiana, USA, J. Coastal Res., 9(1), 209–220.

Wang, G. T., et al. (2002), Modelling overland flow based on Saint-Venantequations for a discretized hillslope system, Hydrol. Processes, 16(12),2409–2421.

Wilson, C., and M. S. Horritt (2002), Measuring the flow resistance of sub-merged grass, Hydrol. Processes, 16(13), 2589–2598.

Wilson, N. R., and R. H. Shaw (1977), Higher-order closure model for can-opy flow, J. Appl. Meteorol., 16(11), 1197–1205.

Woolhiser, D. A. (1975), Simulation of unsteady one-dimensional flowover plane: The rising hydrograph, Water Resour. Res., 3, 753–771.

Woolhiser, D. A., and J. A. Liggett (1967), Unsteady, one-dimensional flowover a plane: The rising hydrograph, Water Resour. Res., 3(3), 753–771,doi:10.1029/WR003i003p00753.

Yang, W., and S. Choi (2010), A two-layer approach for depth-limited open-channel flows with submerged vegetation, J. Hydraul. Res., 48, 466–475.

Yen, B. C. (1992), Hydraulic resistance in open channels, in Channel FlowResistance: Centennial of Manning’s Formula, edited by B. C. Yen, p.135, Water Resour. Publ., Littleton, Colo.

Yen, B. C. (2002), Open channel flow resistance, J. Hydraul. Eng., 128(1),20–39.

G. G. Katul, Nicholas School of the Environment, Duke University,Box 90,328, Durham, NC 27708-0328, USA. ([email protected])

D. Poggi and L. Ridolfi, Dipartimento di Idraulica, Trasporti ed Infra-strutture Civili, Politecnico di Turin, Corso Duca degli Abruzzi, 24,10137, Turin, Italy.


15 of 15

a flow resistance model for assessing the impact of ... · a flow resistance model for assessing...

Documents