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Agricultural and Forest Meteorology 194 (2014) 118–131

Contents lists available at ScienceDirect

Agricultural and Forest Meteorology

j o ur na l ho me pag e: www.elsev ier .com/ locate /agr formet

nterpreting three-dimensional spore concentration measurementsnd escape fraction in a crop canopy using a coupledulerian–Lagrangian stochastic model

imone C. Gleichera, Marcelo Chameckia,∗, Scott A. Isarda,b, Ying Pana, Gabriel G. Katulc,d

Department of Meteorology, Pennsylvania State University, University Park, PA 16802, USADepartment of Plant Pathology and Environmental Microbiology, Pennsylvania State University, University Park, PA 16802, USANicholas School of the Environment, Duke University, Durham, NC 27708, USADepartment of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA

r t i c l e i n f o

rticle history:eceived 3 February 2014eceived in revised form 25 February 2014ccepted 25 March 2014

eywords:anopy turbulenceulerian closure modelsscape fractionagrangian stochastic modelspore dispersion

a b s t r a c t

Plant disease epidemics caused by pathogenic spores are common threat to agricultural crops. Pathogenicspores are often produced and released inside plant canopies but are transported out of the canopy regionby turbulent motions and advected longitudinally over long distances so as to infect other fields. Hence,the fraction of spores that “escape” the canopy sets the effective source strength that determines howfast and far plant diseases can spread. To explore the governing variables of this escape and spatialspread of spores, extensive spore release and recapture experiments were conducted in a maize fieldand interpreted using a coupled Eulerian–Lagrangian stochastic model (LSM). Spores were released frompoint sources located at three prototypical depths inside the canopy. Concentration measurements wereobtained inside and above the canopy with a 3-dimensional grid of spore collectors. The experimentalmeasurements of mean spore concentration were then used to evaluate an LSM for spore dispersion. Thedrift and dispersion terms of the LSM were predicted employing a conventional second-order closure

model of turbulence within plant canopies thereby allowing this combined Eulerian–Lagrangian formu-lation to be applied to a broad set of agricultural crops. The dispersion model includes spore depositionon and rebound from canopy elements. The combination of experimental and numerical simulations wasthen used to quantify the fraction of spores that escape the canopy. Effects of release height and frictionvelocity on the escape fraction of spores were explored with the LSM.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Many fungal diseases of important food crops, such as cerealnd legume rusts, spread across landscapes through atmosphericransport of spores (Stakeman and Harrar, 1957). Generally, thesepores are released into the air stream within the canopy region andheir fate is dependent on their trajectory. Spores that travel withinhe canopy region typically deposit very close to their release sitehile spores that escape from the canopy region into the free flow

bove are subjected to higher wind speeds and travel long distancesAylor, 1986; Nathan et al., 2002). The latter are responsible for the

pread of plant diseases at scales that range from tens of meters toundreds of kilometers (Aylor, 1999). It is of no surprise that the

raction of spores that escape from the canopy region (hereafter

∗ Corresponding author. Tel.: +1 814 863 3920; fax: +1 814 865 3663.E-mail address: chamecki@psu.edu (M. Chamecki).

ttp://dx.doi.org/10.1016/j.agrformet.2014.03.020168-1923/© 2014 Elsevier B.V. All rights reserved.

referred to as the escape fraction Ef) sets the the effective sourcestrength that determines how fast and far plant diseases can spread.

Within a given canopy, escape fraction depends on the heightat which spores are released (source height, zsrc) and the rela-tive importance of gravitational settling and turbulence diffusion(ws/u∗, where ws is the spore settling velocity and u* is the frictionvelocity above the canopy) (Aylor et al., 2001; Nathan et al., 2002).However, due to the lack of robust formulation of escape fraction,regional scale models used to predict spread of plant diseases suchas the Integrated Aerobiology Modeling System (IAMS, Isard et al.,2005) employ ad-hoc parameterizations of escape fraction as afunction of mean wind speed (Isard et al., 2007). Incorporation of anescape fraction that includes the effects of canopy structure, sourceheight, and turbulence into these models would lead to improved

regional forecast of disease spread.

Estimating the escape fraction from field experiments is nottrivial since the number of spores released (source strength)from an infected area cannot be readily determined. Even in field

Forest Meteorology 194 (2014) 118–131 119

ertaew1tLm(AtNzwrf(i2

tcwom2dt1tfltsLiwessa2LpsoibAarptaf

spw(t(htc

Table 1Characterization of experimental runs, including canopy height (h), leaf area index(LAI), wind direction with respect to the centerline of the experimental setup (seeFig. 1), friction velocity (u*), and source strengths at the three vertically alignedrelease heights z/h = 1/3, 2/3, and 3/3.

Run 1 2

Date 9 July 10 JulyTime (EDT) 1455–1525 1143–1213h (m) 1.93 2.05LAI 3.3 3.3Wind dir. (◦) 5.8 −2.4u* (m s−1) 0.41 0.51

−1 −4 −4

S.C. Gleicher et al. / Agricultural and

xperiments where spores are released at controlled rates, itemains difficult to estimate the fraction of spores that escapehe canopy. Most attempts to estimate escape fraction (e.g., Aylornd Taylor, 1983; Aylor and Ferrandino, 1985) have relied on theddy-diffusivity approach (i.e., K-theory), which is known to failithin plant canopies (see Raupach, 1989a; Kaimal and Finnigan,

994). Hence, there is a need for estimates of escape fractionhat do not employ K-theory when modeling turbulent fluxes.agrangian stochastic models (LSMs) have been the preferredethod to explore dispersion within plant canopies for this reason

Flesch and Wilson, 1992; Reynolds, 1998; Siqueira et al., 2000;ylor and Flesch, 2001; Nathan et al., 2002; Poggi et al., 2006) and

o estimate spore escape fraction (Aylor, 1999; Aylor et al., 2001;athan et al., 2002). Aylor (1999) studied the effects of ws/u∗ and

src/h (where h is canopy height) on escape fraction for a line sourceithin a wheat canopy using a two-dimensional LSM. These LSM

esults were then used to develop an empirical model for escaperaction of spores that captures the effects of ws/u∗ for zsrc/h ≈ 0.5the model does not contain explicit dependence on zsrc/h, whichs known to be significant (Nathan et al., 2002; Nathan and Katul,005; Katul et al., 2005)).

Most LSMs do not explicitly account for some characteris-ics of canopy turbulence (e.g., the non-Gaussian statistics, theycles of sweeps and ejections) and therefore should be evaluatedith field experiments. The inclusion of skewness and higher-

rder statistics into LSMs appear not to improve predictions forean concentration profiles (Flesch and Wilson, 1992; Reynolds,

012). For dispersion of passive scalars, LSMs also appear to repro-uce well experimental observations of mean concentration (oremperature) profiles (e.g., Flesch and Wilson, 1992; Reynolds,998; Poggi et al., 2006). The recent links between (Eulerian) flux-ransport terms in scalar flux budgets (e.g., the vertical turbulentux-transport ∂w′w′C′/∂z, appearing in the balance equation forhe mean vertical flux w′C′ for a concentration field C) and meancalar source distribution within canopies (explicitly resolved viaSMs) suggest that LSMs’ accounting of near-field effects makesmprovements to some (but not all) of the limitations associated

ith K-theory (Raupach, 1989a; Siqueira and Katul, 2002; Cavat al., 2006; Francone et al., 2012). Likewise, canopy turbulencetudies have shown that a finite flux-transport term does encodeignificant information about the relative importance of ejectionsnd sweeps in the flux budget (Cava et al., 2006; Francone et al.,012), thereby lending further theoretical support for the use ofSMs inside canopies. However, the uncertainty in modeling dis-ersion of heavy particles (such as seeds or spores) is even greater,ince very little is known about the processes of particle depositionn and resuspension from ground and canopy elements. Compar-sons between spore dispersion in the field and LSMs results haveeen performed by Aylor and Flesch (2001) and Aylor et al. (2001).ylor and Flesch (2001) show good agreement between modelednd measured spore concentrations for a line source of sporeseleased from two heights in a wheat canopy. However, the com-arison is limited to a single vertical profile located relatively closeo the source (x/h < 2.35). Aylor et al. (2001) also present fairly goodgreement between modeled and measured spore concentrationsor a vertical profile located at the center of a small area source.

New three-dimensional mean concentration measurements ofpores within and above a maize canopy are presented in thisaper. Cohorts of yellow, red and blue colored Lycopodium sporesere released simultaneously from 3 heights within the canopy

a single spore color cohort from each height). Mean concentra-ions were sampled at 45 locations inside and above the canopy

1/3 ≤ z/h ≤ 5/3) near the source (1 ≤ x/h ≤ 4). The data are usedere to evaluate results of a three-dimensional LSM parame-erized to estimate Lycopodium spore dispersal. A second-orderlosure model is used to compute the necessary flow statistics

Q1/3 (spores s ) 4.18 × 10 9.40 × 10Q2/3 (spores s−1) 8.36 × 10−4 4.18 × 10−4

Q3/3 (spores s−1) 6.27 × 10−4 6.79 × 10−4

for the LSM, thereby expanding the utility of such combinedEulerian–Lagrangian models to a broad range of crop canopies.These computed flow statistics are then compared to velocitymeasurements within and above maize canopies. The combinedEulerian–Lagrangian model is then used to interpret spore escapefractions from the measured three-dimensional mean concentra-tion field. Experimental methods, the LSM, and the second orderclosure model are described in Section 2. Comparisons betweenexperimental data and model results are presented in Section 3. TheLSM is then used to study escape fraction of rust spores in Section 4,and final discussions are presented in Section 5.

2. Materials and methods

2.1. Experimental Methods

A spore release and recapture experiment was conducted in alarge flat field planted with maize (Zea mays L.) near Mahomet, ILduring late June through early July 2011. Wind direction was fairlyconsistent and well aligned with the rows of fully grown maizeplants on 9 and 10 July and only data from experimental runs onthese days are used here. The experimental site was 120 m from thesouth edge and 500 m from the west border of the field (distancesto the other two sides were much larger). The rows of maize wereoriented in the north-south direction. Plant density was approxi-mately 10.5 plants m−2 with row spacing of 0.762 m and an averageinter-plant spacing of 0.125 m. The maize crop was managed usingstandard production practices. Five maize plants were marked atthe beginning of the experimental period and their heights weremeasured on days when spores were released and recaptured. Asixth plant, with a height approximately equal to the average of the5 marked plants, was pulled up from the field on days when runswere conducted and the area of each leaf was determined usinga LICOR Area Meter (LI3000). Total leaf area was used to calculateone-sided leaf area index (LAI). Canopy height and LAI for 9 and 10July are given in Table 1.

Five three-dimensional Campbell CSAT3 sonic anemometerswere aligned vertically at heights corresponding approximatelyto z/h = 1/3, 2/3, 3/3, 4/3, and 5/3 that were 0.7, 1.4, 2.1, 2.8, and3.5 m above the ground on 9 and 10 July. The three components ofwind velocity and air temperature were simultaneously sampledat 20 Hz.

2.1.1. Particle release and collectionDuring each 30-min experimental run, red, blue and yellow

Lycopodium spores (Carolina Biological Supply Company) werereleased simultaneously from three different heights, arrayed ver-

tically, within the maize canopy (Fig. 1). Approximately 500 gof Lycopodium spores were stained red with fuchsin basic thenwashed with a solution of 95% ethyl alcohol and distilled water(Aylor and Ferrandino, 1989). A similar quantity of spores was

120 S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131

Fig. 1. A schematic of the experimental setup inside the maize field. There were nine poles, each with five rotorrod collectors. The poles were spaced 2, 4, and 8 m, respectively,from the release mechanism in the x direction (down row direction). They were spaced 0.762 m in the y direction, in the center of the spaces between adjacent maize rows.T nd the

smtrp

nffittocfl(tssmktosCd1raiwp

t

he ground deposition network is also shown. The angle between the mean wind a

tained blue with methylene blue chloride and washed in the sameanner as the red spores. A third cohort of spores was washed with

he mixture of ethyl alcohol and distilled water but not stained thusetaining their natural yellow color. The three sets of spores werelaced in a drying oven (50 ◦C) until fully desiccated.

The spore release mechanisms consisted of three 150 ML Buch-er funnels connected to a large tank of compressed dry air. The

unnels had fine porous ceramic filters near their base and weretted with solid rubber stoppers as lids. A piece of stainless steelubing (0.032 m OD, 0.0018 m ID) extended 0.02 m through the cen-er of the stopper into the cavity below and 0.1 m above the topf the stopper. The narrow drain at the bottom of the funnel wasonnected to the compressed air source by PVC tubing. The air-ow through the device was controlled by an Acrylic Flowmeter0.058 m scale, range 1–10 LPM). Air flowed from the pressurizedank through the flow meter into the funnel where it suspendedome of the Lycopodium spores and carried them out the thinteel tube. A small vibrating motor (3 V) attached to the woodenounts used to position the funnels at specified heights on a pole

ept the layer of spores uniformly distributed across the filterhroughout the 30-min spore release run. Eighteen grams of col-red spores were placed on the ceramic filter in each funnel at thetart of an experimental run and the stopper securely attached.alibration of the three devices in the laboratory revealed slightifferences between release mechanisms and flow rates of 1.5,.3, and 1.7 LPM were selected for the red, yellow and blue sporesespectively to obtain a constant release rate over a 30 min periodnd a total release of 1–2 g of spores from each mechanism. Testsn the laboratory confirmed that the vast majority of the spores

ere released as single grains, and only very few clumps wereresent.

The spore release devices were attached to a single pole cen-ered in the space between two rows of maize and on 9 and 10 July

sampling grid � was approximately zero for the runs used here.

the upper orifices of the steel tubes were positioned 0.7, 1.4 and2.1 m above the ground (corresponding approximately to z/h = 1/3,2/3, and 3/3). At the end of each experimental run, the weight ingrams of the spores remaining in each funnel was determined. Thenumber of spores released NQ was estimated by multiplying theweight of the released spores m by a spore number density

N0 = 1

�p�d3p/6

= 4.71 × 107 spores g−1, (1)

where the spore diameter dp = 32.8 �m (Ferrandino and Aylor,1985) and density �p = 1.15 g cm−3 (Petroff, 2005) were used.Note that this value is in agreement with the estimateN0 = (4.89 ± 0.29) × 107 spores g−1 proposed by Wanner and Pusch(2000). The resulting source strengths are displayed in Table 1.

Rotorod spore collectors with retracting rods mounted on poles(6.096 m tall, 0.019 m OD) arranged in a 3 × 3 grid were deployedin the maize field as illustrated in Fig. 1. The first, second, andthird rows of poles were placed 2, 4, and 8 m north of the releasedevice, respectively. The middle column of poles was positionedin the center of the space between the maize rows in which thespores were released. The other two poles were in the center ofthe spaces one maize row to either side of the release position.The rotorod spore collectors on each pole were mounted at thesame heights as the sonic anemometers. Thus, the grid of 9 polesconsisted of 45 rotorod spore collectors. The rotorod spore col-lectors were calibrated in the field with an Extech High PrecisionContact Tachometer (Watham, Massachusetts). Glass slides for cap-turing spores deposited on the ground were placed on wood blocks

(0.0762 m ×0.0762 m ×0.019 m) positioned at 1 m intervals down-wind from the release pole along the centerline of the collectiongrid. An additional ground deposition slide was placed 1 m upwindfrom the release pole.

Fores

rdetLmrrcsttTcsclr

wrla

2

sancorovtl2daWf

2

sTswmcwir

�

wcmio

S.C. Gleicher et al. / Agricultural and

Two retractable greased (Vaseline) rods were mounted on eachotorod arm. The rotorod spore collectors, operated with an on/offuty cycle (fR, calculated as the fraction of time the rods werexposed) to prevent overloading of spores on the rods, were con-rolled by CR1000 Microloggers [Campbell Scientific, Inc. (CS),ogan, Utah]. The rotorod spore collectors closest to the releaseechanism were turned on for 15 s (rods exposed) and off (rods

etracted) for 45 s during each minute of a T = 30-min experimentalun. The duty cycle was increased to 30 s on and 30 s off for sporeollectors mounted on poles at the sides of the grid. The rotorodpore collectors attached to the back row of poles were exposed forhe duration of the experimental run. At the end of each experimen-al run, the 90 sampling rods were stored for spore enumeration.he number of Lycopodium spores of each color on the rods wereounted using a microscope at 40× after the field experiment. Meanpore concentration normalized by source strength C/Q for eacholor of spore at each sampling location on the grid was calcu-ated as the mean of the counts from the two rods attached to eachotorod collector NR using:

C

Q= NR/VR

NQ, (2)

here VR = �dRARfRT × RPM, dR = 0.086 m is the diameter of theotorod arm, AR = 2.5 × 10−5 m2 is the area of the rod used for samp-ing, and RPM is the number of rotations per minute of the rotorodrm.

.2. Lagrangian stochastic model

A three-dimensional LSM was employed to model the disper-ion of spores within and above the maize canopy. The flow wasssumed to be statistically stationary and horizontally homoge-eous. LSMs produce an ensemble of stochastic spore trajectoriesonstrained by imposed turbulence statistics. Vertical profilesf the Lagrangian timescale (TL) and six Eulerian statistics areequired: mean horizontal velocity (u), the vertical turbulent fluxf momentum (u′w′), the variances of the three components ofelocity fluctuations (�2

u , �2v , and �2

w), and the viscous dissipation ofurbulent kinetic energy (�). Note that v′w′ was assumed to be neg-igible. These quantities depend on canopy structure (Poggi et al.,004b). To keep the modeling approach flexible and adaptive toifferent canopies, the flow statistics were computed from the leafrea density profile a(z) using the second-order closure model ofilson and Shaw (1977). The main components of the modeling

ramework are described below.

.2.1. Turbulence parameterizationFor a stationary and planar homogeneous flow in the absence of

ubsidence, only vertical variations of the flow statistics are needed.he second-order closure model of Wilson and Shaw (1977) yields aystem of five equations for u, u′w′, �2

u , �2v , and �2

w . The canopy dragas parameterized by a constant coefficient (Cd) and a(z). While Cday vary within the canopy due to several (interconnected) pro-

esses such as sheltering, vegetation deflection, viscous drag, etc., itas assumed here that the vertical variation of the product Cda(z)

s primarily controlled by the vertical variations in a(z), not Cd. Theate of dissipation of turbulent kinetic energy was modeled as

= 2TKE3/2

a3lm(3)

here TKE is the turbulent kinetic energy, a3 ≈ 71.8 is a closure

onstant (Katul et al., 2011) that can be derived by matching theodel to its near-neutral atmospheric surface layer state, and lm(z)

s an effective mixing length. In the second-order closure model, notnly the dissipation but also the flux transport and pressure terms

t Meteorology 194 (2014) 118–131 121

were parametrized in terms of the mixing length. Consequently thesolution depends on the imposed shape of lm(z).

The baseline model for the mixing length adopted here was theone described by Katul et al. (2004) and used in Katul et al. (2011)for aerosol sized particle deposition onto vegetation. This modeluses a constant mixing length within most of the canopy regionseparating two regions of linear growth with z:

lb =

⎧⎨⎩

�z, z < (˛′h/�)

˛′h, (˛′h/�) ≤ z < h,

�(z − do), z ≥ h

(4)

where � = 0.4 is the von Karman constant, ˛′ = �(1 − do/h) is theparameter ensuring continuity for the mixing length, and do is thezero-plane displacement defined as the centroid of the momentumsink and is determined as part of the solution in the closure model(Katul et al., 2011).

Inside the canopy region, the mixing length is further con-strained by the drag force exerted by the canopy elements. Finniganand Belcher (2004) argue that the mixing length inside a uniformcanopy is given by

ld = 2ˇ3

Cda(z)(5)

where ̌ = u∗/u(h) ≈ 0.25 is a constant whose value varies with theleaf area density (Finnigan and Belcher, 2004; Poggi et al., 2006)saturating at about ̌ ≈ 0.3 for dense canopies (Massman and Weil,1999; Poggi et al., 2004b,a). Therefore, within the canopy region(z ≤ h) the most restrictive mixing length between Eqs. (4) and (5)was used

lm = min [lb; ld] . (6)

For the maize canopy studied here, Eq. (5) is the most restric-tive condition and defined the mixing length within the range0.2 ≤ z/h ≤ 0.9.

No analytical solution exists to the second-order closure modelof Wilson and Shaw (1977) except for some limited cases (Massmanand Weil, 1999). Here, the equations were discretized in z and aniterative solution was obtained. The mixing length was calculatedat each discrete point and a 5-point running average was usedto smooth the resulting profile. The profiles of a(z) and the cor-responding lm(z) used here are shown in Fig. 2(a)–(b). It is to benoted that closure schemes beyond second order have been devel-oped and used within canopies (Meyers and Paw, 1986). A numberof data-model comparisons suggest the added complexity in third-order closure schemes does not translate to improved predictionsabove second-order, at least for the flow statistics most pertinentto the LSM (Katul and Albertson, 1998; Juang et al., 2008).

2.2.2. Trajectory modelFor fungal spores (typical spore diameter d < 50 �m), particle

inertia can be neglected and spore trajectories can be computed bysuperimposing the settling velocity on the fluid velocity (Wilson,2000)

dxp

dt= up = u − wsk (7)

where u is the fluid velocity at the particle location xp, up is theparticle velocity, ws is the particle settling velocity in still fluid, andk is the unit vector in the vertical direction. Following Thomson(1987), the fluid accelerations are modeled by a stochastic differ-ential equation

dui = ai(xp, u, t)dt + bij(xp, u, t)dj (8)

where the coefficients ai and bij contain all the characterization ofthe turbulence and dj are increments of a Wiener process. The

122 S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131

Fig. 2. Comparison between the turbulence statistics from the field experiments and those computed from the second-order closure model. (a) Leaf area density a(z), (b)m spanw( ized b1 lid lin

ccetcecbbicd2

C

wTa

ixing length lm , (c) mean velocity u, (d) rms of streamwise velocity �u , (e) rms of

h) TKE dissipation rate �, and (i) Lagrangian timescale TL . All quantities are normal0 July (solid circles), data from Wilson (1988) (crosses), and model predictions (so

oefficients ai are the drift terms that were modeled using theanopy Eulerian statistics (and their vertical gradients) and thequations derived in Rodean (1995) so that the model satisfieshe well-mixed condition (Thomson, 1987). Most LSMs for plantanopies use this formulation for ai (Aylor and Flesch, 2001; Poggit al., 2006), even though the formulation is not unique and thehoice of ai impacts the results (Reynolds, 1998). The coefficientsij are the random acceleration (diffusion) terms modeled as

ij = (Co�)1/2ıij so as to ensure that u satisfies Kolmogorov’s similar-ty theory (Thomson, 1987). Here, Co= 3.125 is the inertial subrangeonstant that was adjusted to account for finite Reynolds numberependence following Lien and D’Asaro (2002) (see also Poggi et al.,008)

∗o ={

0.068Re1/2

Co, Re ≤ 100

1 − (0.1Re)−1/2Co, Re > 100(9)

here Re = �u/� is the Reynolds number based on theaylor microscale = �u

√15�/� and � is the kinematic viscosity of

ir.

ise velocity �v , (f) rms of vertical velocity �w , (g) turbulence momentum flux u′w′,y the friction velocity u* and canopy height h. Data from 9 July (open squares) ande).

The Lagrangian timescale for air parcels was estimated basedon the TKE and its dissipation rate as given by the second-orderclosure model

TL = 2TKEC∗

o�. (10)

Note that this choice is neither unique nor optimal (seeAppendix for further discussion on this choice), and that the spec-ification of TL is difficult in non-homogeneous turbulence becauseit can no longer be interpreted as an integral timescale (Wilson andSawford, 1996). To test the sensitivity of the model to the functionalform of TL, LSM runs were also performed using the parameteriza-tion proposed by Raupach (1989b):

TL = max[

0.3h

;� (z − do)

]. (11)

u∗ 1.25u∗

For spores, a reduction of the timescale TL is needed to accountfor the crossing trajectory effects (Csanady, 1973; Sawford and

Fores

Ga

T

wa(W

tr(

2

bIsqeBtabu

sp(d

S

w

tv

E

iSrtbfl

u(p

S

wpaltFc

2

s

S.C. Gleicher et al. / Agricultural and

uest, 1991). The spore decorrelation timescale Tp can be modeleds

p = 1√1 + (ˇLEws/�w)2

TL, (12)

here ˇLE = 1.5, which corresponds to a reduction of about 8%t the bottom of the canopy for the experimental conditionsu* = 0.41 m s−1), but can be larger for smaller values of u* (see

ilson (2000) for a discussion of the limitations of this approach).For more details about the trajectory model including the rela-

ions between ai and the Eulerian turbulence statistics, the reader iseferred to Rodean (1995), Aylor and Flesch (2001) and Poggi et al.2006).

.2.3. Particle deposition and resuspensionA key component of the LSM is the removal of ‘non-sticky’ air-

orne spores by deposition onto canopy elements and the ground.n addition, particles deposited onto leaves may be resuspended bytrong gusts. These complex physical processes are not completelyuantified and are parameterized from dimensional analysis andxperiments. Aylor and Flesch (2001), Dupont et al. (2006), andouvet et al. (2007) present models to represent canopy deposi-ion and resuspension based on the approach developed by Leggnd Powell (1979) and Aylor (1982). Here the approach describedy Aylor and Flesch (2001) and modified by Pan et al. (2014) wassed.

The probability of particle deposition on the plant canopy wasplit into deposition by impaction (SI) and sedimentation (Sd). Therobability of deposition by impaction in the horizontal directionimpaction in the vertical direction is neglected) during a timestept is given by

I = EI(Px + Py)a(z)up,hdt, (13)

here Px and Py are the projections of leaf area in the planes normal

o the x and y directions, up,h =√

u2p + v2

p is the particle’s horizontalelocity, and

I = 0.86

1 + 0.442St−1.967(14)

s the impaction efficiency adopted from Aylor (1982). Here, thetokes number St = �p/�f represents the ratio between the sporeesponse time scale (�p = ws/g) and the timescale associated withhe flow curvature around vegetation elements (�f = Lv/uf,h, Lveing a characteristic size of a vegetation elements and uf,h theuid’s horizontal velocity).

For particle sedimentation, it was assumed that the efficiency isnity if particles are moving downward and zero if moving upwardi.e., sedimentation only occurs on surfaces that are facing up). Therobability of particle deposition by sedimentation is given by

d ={

Pza(z)wsdt if wp < 0,

0 if wp ≥ 0,(15)

here Pz is the projection of leaf area in the horizontal plane. Resus-ension was modeled using an approach based on a critical velocitys discussed in Aylor (1999). If the fluid horizontal velocity wasarger than a critical value (uf,h > Vcrit), particles were not allowedo deposit (i.e., SI + Sd = 0). Following Aylor (1999), Vcrit = 0.45 m s−1.inally, spores were assumed to deposit onto the ground if theyrossed a threshold height of zdep = h/20 regardless of their velocity.

.2.4. Estimating concentrations and escape fractionsTo compare model results to field measurements, modeled

pore concentration was estimated for each sensor location (xs).

t Meteorology 194 (2014) 118–131 123

Following Flesch (1996), the mean concentration at the sensor loca-tion C(xs) is given by

C(xs)Q

= 1N x y z

NB∑i=1

TResi (16)

where x, y, and z define a box centered at the sensor location,N is the total number of spore particles released in the model, NB

is the total number of particles that enter the box, Q is the sourcestrength, and TRes

iis the estimated residence time of the ith particle

inside the box and it was calculated as

TResi = min

[ x

up,i;

y

vp,i;

z

wp,i

](17)

where up,i is the velocity of the ith particle when it is inside the box.Due to the mean flow in the x-direction, Eq. (17) will yield TRes

i=

x/up,i for most particles, which is equivalent to the equation usedby Aylor and Flesch (2001).

For a point source located at x = (0, 0, zsrc), escape fraction canbe defined as

Ef (x) ≡ 1Q

∫ x

−∞

∫ ∞

−∞Fz(x, y, z = h)dydx (18)

where Q is the spore release strength (spores s−1), Fz(x) is the ver-tical flux of spores (spores m−2 s−1), and h is the canopy height.Eq. (18) is similar to the definition of escape fraction from a linesource presented by Aylor (1999). The escape fraction for sporesreleased at a point in the canopy as defined above will increaserapidly with x near the source, and then decrease at a much slowerrate once most particles have escaped and the gravitational sett-ling becomes the dominant contribution to the flux. In this work,the maximum value of Ef(x) was used as an estimate of the escapefraction.

3. Comparison between observations and model results

3.1. Model parameters

For the model runs, a drag coefficient Cd = 0.3 was used (Wilson,1988). The vertical distribution measured by Wilson et al. (1982)was also used after adjusting for the small LAI differences betweenhis study and the present experiment. The resulting leaf areadensity profile is shown in Fig. 2(a). The projections of LAI ontothe three directions was achieved by setting Px = Py = 0.28 andPz = 0.44 according to measurements by Bouvet et al. (2007). A sett-ling velocity of ws = 0.0194 m s−1 was used for the Lycopodiumspores (Ferrandino and Aylor, 1984) and Lv = 0.02 m was usedas the characteristic size of vegetation elements (Bouvet et al.,2007). A timestep dt = 0.05TL was used in the LSM runs. A total ofN = 3 ×106 particles were released in each run, a number deter-mined to achieve statistical convergence of concentration on thefar poles. To calculate values of spore concentration, a box withsize x = y = 0.2 m and z = 0.1 m was used in Eqs. (16) and(17).

3.2. Turbulence statistics

The capabilities of the second-order closure model describe inSection 2.2.1 (solid lines in Fig. 2) were evaluated against the mea-sured profiles of the Eulerian statistics obtained from the fieldexperiment. On both 9 and 10 July, the wind was steadily from the

south. On 10 July, a 7.5 h period of statistically stationary conditionswas recorded, extending from 0930 to 1700 EDT (see Chamecki,2013). Flow statistics were calculated for the release experimentfor 9 July (1455–1525 EDT) and are shown as open squares in

124 S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131

Fig. 3. Comparison of normalized modeled (C/Q ) and observed (C/Q ) normalized concentrations for source at (a) z/h = 3/3, (b) z/h = 2/3, and (c) z/h = 1/3. Run 1 (squares)a es ind

Fiwwcfittsfl

po(pwsp(ctaatabnfaflo

(spf

meaidfrtts

Results presented in Fig. 3 display significantly less scattered thanobserved in most air quality models (Chang and Hanna, 2004) andeddy-diffusivity based approaches to modeling spore dispersioninside plant canopies (Skelsey et al., 2008). Concentrations from

Table 2Statistical performance measures for comparison of model spore concentrationswith observations.

Run zsrc/h MG VG FAC2

1 1/3 1.76 2.01 49%1 2/3 1.43 1.44 70%

LSM obs

nd run 2 (circles). The solid line indicates a one-to-one match while the dashed lin

ig. 2. For 10 July, the long stationary period was divided in 30-minntervals and the mean and standard deviation for each statistic

ere calculated from the sample of 15 runs. The mean is shownith solid circles and the standard deviation with errorbars. The

rosses indicate the data obtained using three-dimensional split-lm anemometers by Wilson (1988). The good agreement betweenhe present data set and the split-film data provides some supporto the use of sonic anemometers within maize canopies (the onlytatistic in which sonic resolution may be an issue is the momentumux at z/h=1/3 in Fig. 2(g)).

Overall there is good agreement between experiments and com-uted profiles obtained from the second-order closure model. Thenly exception is the rms of the spanwise velocity component �vFig. 2(e)), which is under predicted by the model. This is a commonroblem in such models likely originating from variations in meanind direction not accounted for in the closure model (see discus-

ion in Chamecki et al., 2009). To avoid an underestimation of laterallume spread in the LSM, �v was replaced by �adj

v = (�u + �w)/2shown with dashed line in Fig. 2(e)). It is to be emphasized that theoncentration downwind from point sources depends on u. Evenhough the general agreement appears reasonable between modelnd measurements of u/u∗ (Fig. 2(c)), the overestimation of u bothbove and below z/h ≈ 1 is expected to affect modeled concentra-ions (note that predicted mean velocities are about twice as larges measurements below z/h ≤ 1/2). To investigate the effects of thisias in mean velocity inside the canopy on spore dispersal, an alter-ative profile obtained by fitting to observed data an exponential

unction inside the canopy and a logarithmic function above waslso employed (shown with dashed line in Fig. 2(c)). The remainingow statistics are maintained as those predicted by the second-rder closure model.

The Lagrangian timescale estimated using Eq. (10) and Co = 3.125the correction C∗

o is not used in the figure, since it depends on thepecific value of u*) is shown in Fig. 2(i), where the parameterizationroposed by Raupach (1989b) and given by Eq. (11) is also shownor comparison.

The predicted triple moments from the second-order closureodel were complemented by an incomplete third order cumulant

xpansion and used to assess the relative importance of ejectionsnd sweeps on momentum fluxes ( So) within the CSL. It is shownn the Appendix that the second-order closure model correctlyescribes the So profile when compared to those determinedrom quadrant analysis and conditional sampling. The model cor-

ectly predicts that sweeps dominate momentum transfer withinhe canopy sublayer, with a near maximum So coinciding withhe peak in a(z). However, the predicted peak So appears to bemaller than determined by quadrant analysis.

icate a factor of 2 above and below.

3.3. Spore concentrations and deposition

In this section, the performance of the LSM is assessed with fieldobservations by comparing normalized values of means spore con-centrations C/Q and mean normalized ground deposition fluxes�/Q. Statistical performance measures traditionally employed inthe evaluation of air quality and dispersion models (Chang andHanna, 2004) are used to assess the accuracy of the model output.Here, the geometric mean bias (MG)

MG = exp

[1N

N∑i=1

ln

(C/Q

)obs(

C/Q)

LSM

], (19)

the geometric variance (VG)

VG = exp

⎡⎣ 1

N

N∑i=1

(ln

(C/Q

)obs(

C/Q)

LSM

)2⎤⎦ , (20)

and the fraction of predictions within a factor of two of observations(FAC2)

FAC2 = fraction of data that satisfy 0.5 ≤ (C/Q )obs

(C/Q )LSM

≤ 2.0, (21)

are used where N = 45 is the number of observations. For a perfectcorrespondence between model and observation, MG = 1, VG = 1,while FAC2 is equal to 100%. According to Chang and Hanna (2004),a ‘good’ model is expected to have about 50% of the predictionswithin a factor of 2 of the observations, a relative mean bias within30% (0.7 < MG < 1.3), and a relative scatter of about a factor of2 of the mean (VG < 1.6). Scatter plots comparing modeled andobserved normalized concentrations are shown in Fig. 3 and thestatistical measures of model performance are listed in Table 2.

1 3/3 1.93 1.88 49%2 1/3 1.02 1.45 71%2 2/3 1.13 1.80 67%2 3/3 1.87 2.05 55%

Fores

ttasuhoF

shzwuMiaptpp

prmm

Fli

S.C. Gleicher et al. / Agricultural and

he source at z/h = 1/3 present the most scatter, while for z/h = 3/3here is a clear low bias in model predictions. These results arelso reflected in the statistical measures of model performance pre-ented in Table 2. A relevant feature from Fig. 3 is the systematicnder prediction of small concentration values for all three sourceeights. This is the reason for the use of geometric statistics aspposed to regular bias and mean-squared error and log scales inigs. 3–7.

Vertical profiles of modeled spore concentration at the nineampling poles were compared to observations for the three sourceeights in Figs. 4–6. Inspection of the results for the source at/h = 3/3 shown in Fig. 4 suggests that, while concentrations wereell predicted at the canopy top (source height), the systematicnder prediction noted in Fig. 3 and reflected in the large values ofG occurs both inside and above the canopy, being more severe

nside the canopy. Three possible explanations for this behaviorre: (i) the model is under dispersive in the vertical, (ii) the overrediction of mean velocity both within and above canopy leadso under prediction in concentrations, and (iii) the deposition onlant canopies is over estimated, depleting too quickly the airbornelume.

A sensitivity study was conducted to identify the source of under

redictions displayed in Fig. 4. Three additional simulations forun 2 were performed. In the first one, an empirically adjustedean velocity profile was used, to test the effects of disagree-ent in predicted and observed mean velocity on concentrations.

ig. 4. Comparison between observed and modeled mean concentration profiles at eacines) and run 2 (circles and dashed lines). Mean wind direction is approximately from pos indicated by the circled S.

t Meteorology 194 (2014) 118–131 125

As earlier noted, the empirical profile was obtained by fitting anexponential function to the mean velocity inside the canopy anda logarithmic function above the canopy (the resulting curve isshown with a dashed line in Fig. 2(c)). The second simulation usedthe Lagrangian timescale parameterization proposed by Raupach(1989b), and the third one is a simulation in which deposition onthe canopy was not permitted. The resulting concentration pro-files for the centerline (poles 2, 5, and 8) are shown in the upperpanels of Fig. 7 for the three simulations (gray dashed, solid, anddotted lines, respectively). The changes in the mean velocity pro-file and the removal of canopy deposition had minor effects onthe concentrations for the source at canopy height. These resultsappear to indicate that the LSMs inability to reproduce verticaldispersion correctly is the main cause of under prediction of thespore concentrations. Indeed, the shape of the mean concentra-tion profiles was sensitive to changing the parameterization ofthe Lagrangian timescale. However, using the parameterizationproposed by Raupach (1989b) did not produce systematic improve-ment in model predictions. Finally, model results were insensitiveto reducing the height at which particles deposit onto the groundfrom z/h = 1/20 to z/h = 1/100.

Comparisons for spores released at z/h = 2/3 and z/h = 1/3 shown

in Figs. 5 and 6 are quite similar to each other, showing fairlygood agreement in the near poles (x/h ≤ 2) over the entire grid.Concentrations in the far poles (x/h ≈ 4) were consistently underestimated by the model, which caused the poor agreement in the

h sampling pole for source at the canopy top (zsrc/h = 1). Run 1 (squares and solidle 2 towards pole 8, with pole 2 being the closest to the source. The source height

126 S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131

but fo

laTbmtwowoipWsboBooftwa

mths

Fig. 5. Same as Fig. 4,

ow concentration end of Fig. 3(b)–(c). However, the well-mixedspect of the mean profile appears to be captured by the model.he underestimation in the far poles for all source heights coulde an indication that the model for deposition is removing tooany spores (the far poles are much more sensitive to parame-

erization of deposition). However, tests comparing runs with andithout deposition (see Fig. 7) showed that the effects of deposition

n the modeled concentrations within the grid of measurementsere much smaller than the differences between predictions and

bservations in the far poles (this result suggests that depositions not very important for predicting mean concentration of smallarticles near the source – the opposite behavior was reported byilson (2000) for large particles). Fig. 7 also shows that using the

maller value of �v shown in Fig. 2(e) improved the agreementetween model and observations in the far poles at the expensef significantly reducing model performance in the near poles.ased on this results, our best explanation for the underestimationf the mean concentration in the far poles is the overestimationf u. Comparison between the mean horizontal flux uC in thear poles shows that, despite the under prediction of concentra-ion, the flux is actually over predicted. This is indeed consistentith the under prediction of the vertical dispersion suggested

bove.As a final test of model performance, comparisons between

odeled and observed ground deposition fluxes along the cen-erline of the grid are shown in Fig. 8 for releases at all sourceeights. The best agreement between modeled and observed depo-ition was found for the source at z/h = 2/3 (Fig. 8(b)) while

r source at zsrc/h = 2/3.

deposition close to the source was over predicted for the low-est level (Fig. 8(c)). Comparison with the middle row of panels inFigs. 4–6 reveals a strong correspondence between the overpre-dictions of concentration and deposition with distance from thesource.

It is interesting to note the large values of deposition observedupwind from the source in the experimental data, in particular forthe source at z/h = 1/3. Legg et al. (1986) suggested that organizedrecirculating motions within the canopy (produced by canopyheterogeneity) explain some characteristics of heat dispersionobserved in a wind tunnel experiment. Flesch and Wilson (1992)suggested that these recirculations could cause transport of heatupwind from the source, and this possibility has been used tojustify under-prediction of mean concentration values by LSMsinside canopies (Flesch and Wilson, 1992; Reynolds, 1998). Thelarge values of spore deposition upwind from the source pre-sented in Fig. 8 provide possible experimental evidence for theimportance of these recirculations when the source is near theground.

The rigorous comparison of model results with field observa-tions presented in this section reveals that the model is ‘good’ (bythe metrics used in air quality) at reproducing spore concentra-tions in the near poles (i.e., x/h ≤ 2) from sources within the canopy(z/h < 1). This is the region where the largest concentrations were

observed. It is also the region that is most relevant for predict-ing escape fraction because the largest portion of spore escapetakes place near the source (typically x/h ≤ 3, not shown). Therefore,despite the less than optimal model predictions in the far poles and

S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131 127

but fo

fgc

4

oswwecpscossoifsTttsv

Fig. 6. Same as Fig. 4,

or sources at the top of the canopy, the model is likely to provideood estimates of escape fractions for sources of spores within theanopy.

. A study of escape fraction

In this section, the LSM model is used to characterize the escapef spores from the maize canopy with a focus on the effects ofource height and friction velocity. Model runs for ten values ofs/u∗ = 0.02, 0.03, 0.04, 0.06, 0.10, 0.19, 0.30, 0.39, 0.55, and 0.78ere performed. For each value of ws/u∗, eight source heights

qually spaced between zsrc/h = 1/8 and 8/9 were investigated. Goodonvergence in the escape fractions was obtained using N = 3 ×105

articles in the simulations. Escape fractions are displayed for eachource height as a function of ws/u∗ in Fig. 9(a), which can beompared to the results presented in Aylor (1999). The decreasef escape fraction with increasing ws/u∗ and increasing zsrc/h isimilar to the one presented by Aylor (1999), even though someignificant differences are also clear. In particular, the dependencef Ef on ws/u∗ found here is not as strong. However, this compar-son must be interpreted with caution because Aylor’s results areor a wheat canopy obtained using a 2-dimensional LSM. Results forelected values of ws/u∗ are shown in Fig. 9(b) as a function of zsrc/h.here is a region deep inside the canopy for which the escape frac-

ion increases linearly with increasing zsrc/h. In the upper canopy,his behavior transitions into a region of weaker increase in Ef formall values of ws/u∗ and only minor variation in Ef for the largeralues.

r source at zsrc/h = 1/3.

Overall, there are large variations of Ef as a function of ws/u∗and zsrc/h. This finding underscores the challenge of parameteriz-ing Ef in models of spore dispersal and disease spread on regionalscales. Katul et al. (2005) developed a model for the probability ofseeds escaping from forest canopies based on analytical solutionsto the Fokker-Planck equation. It is important to emphasize the dif-ference between the escape fraction as defined in Eq. (18) and theprocess described by Katul et al. (2005), which considers any seedthat crosses z/h = 1 as having escaped (here we represent that frac-tion as PE, to emphasize that this is the probability of escaping). Theexpression derived by Katul et al. (2005) can be recasted in termsof the parameter zsrc/h

PE =exp{

(zs/h)ws/(k�w,a)}

− 1

exp{

ws/(k�w,a)}

− 1, (22)

where k = 0.3 for dense canopies and �w,a is the average value of �w

over the canopy depth. Eq. (22) can be written in terms of ws/u∗ byreplacing �w,a = ˛u∗. For the present case, ̨ = 0.55 as determinedby integrating the profile of �w/u∗ produced by the closure model(see Fig. 2(f)). By virtue of its definition, PE → 1 when zsrc/h → 1underscoring differences from the definition of Ef that need notapproach unity as zsrc/h → 1.

Predictions from Eq. (22) are compared to the results from

the LSM in Fig. 9(b). Both models capture the increase in Ef withsource height and its decrease with settling velocity. However, thevalues of the escape fraction are about 20% less for the analyti-cal model than those predicted by the LSM. Note that based on

128 S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131

Fig. 7. Comparison between observed and modeled mean concentration profiles at centerline sampling poles for sources at zsrc/h = 1 (upper row), zsrc/h = 2/3 (middle row),and zsrc/h = 1/3 (lower row). The source height is indicated by the circled S. Observational data shown for run 1 (squares) and run 2 (circles). Model runs correspond to run2 and are shown using the base model configuration (black line), no deposition on plant canopies (gray solid line), the modified model for the Lagrangian integral timescale(gray dotted line), modified mean velocity profile (gray dashed line), and the original profile of �v (gray dot-dashed line).

tEEesd

Fd

he differences in the definitions of Ef and PE, one would expectq. 22 to produce larger estimates than the LSM. Derivation of

q. (22) does not take into account particle deposition on plantlements. However, results of additional LSM simulations (nothown) suggest that the effect on the escape fraction of sporeeposition on plant canopies is small. Note that the derivation of

ig. 8. Comparison of modeled and observed ground deposition densities for spores reeposition and the horizontal axis represents the downwind distance. Run 1 (squares and

Eq. (22) requires neglecting the vertical gradients in flow statis-tics. These gradients tend to increase the drift term, resulting in

more coherent trajectories. Therefore, the absence of the gradientsin the derivation of Eq. (22) leads to reduced integral timescalesas compared to the LSM, causing it to underpredict the escapefraction.

leased at in the lower canopy (h/3). The vertical axis represents the normalized solid lines) and run 2 (circles and dashed lines).

S.C. Gleicher et al. / Agricultural and Forest Meteorology 194 (2014) 118–131 129

F ack dol ray sor

5

tmtLstassscL

tousOmaswˇltmrtw

vaataahitthas

ig. 9. (a) Modeled escape fraction as a function of ws/u∗ for sources at zsrc/h =1/9 (bline), 5/9 (black dashed line), 6/9 (gray dashed line), 7/9 (black solid line), and 8/9 (gesults (black lines) and probability of escape predicted by Eq. (22) (gray lines).

. Discussion and conclusions

Lagrangian stochastic models are widely used in studies of par-icle and scalar dispersion inside plant canopies. However, most

odels rely on empirical parameterizations of vertical profiles ofurbulence statistics (first- and second-order moments and theagrangian timescale). This presents a challenge since turbulencetatistics are expected to change from one canopy to another dueo changes in canopy morphology and leaf area distribution. Here,

complete model with minimal information requirements is pre-ented: canopy height, vertical distribution of LAI, friction velocity,ource height and strength, and particle settling velocity. An exten-ive comparison between the velocity statistics produced by thelosure model and the mean concentration plume produced by theSM is provided.

The second-order closure model reproduces well the observa-ions of �u(z), �w(z), and u′w′(z). The mean velocity u(z) is slightlyver-predicted both inside and above the canopy, and �v(z) isnder-predicted. The former does not seem to impact the disper-ion model and the latter can be adjusted by using �v = (�u + �w)/2.verall, the second-order closure model provides adequate infor-ation to drive the LSM. Two main issues that still need to be

ddressed are the specification of the Lagrangian timescale and theensitivity of the closure to the value of ̌ in Eq. (5). Here, ̌ = 0.25as used, even though the field measurements yield ̌ = 0.29 (using

= 0.29 has a significant impact on the predicted profiles, creatingarger differences between measure model output and observa-ions). The need for a reduced ̌ is suggestive that the mixing length

ay be overestimated inside the canopy with ̌ = 0.29, and henceefined models for mixing length may improve results (note thathe model used here neglects any effects of thermal stratificationithin the canopy, which is known to impact the mixing length).

The overall agreement between LSM predictions and obser-ations of mean concentration is ‘good’ by air quality modelssessment metrics, but not as good as the result presented by Aylornd Flesch (2001). However, there are many differences betweenhe comparison presented here and the one performed by Aylornd Flesch (2001). In their work, only one profile is comparedgainst experiments for each run (in contrast to the 9 profiles usedere). Furthermore, their source strength was scaled by minimiz-

ng the mean-squared error between model and observations, sohat their model results cannot be considered a true prediction of

he concentration values (if a similar procedure were employedere, it would result in very large decreases in the values of MGnd VG and possibly FAC2 would increase). In addition, the casetudies in Aylor and Flesch (2001) are for line-source dispersion

tted line), 2/9 (gray dotted line), 3/9 (black dash-dotted line), 4/9 (gray dash-dottedlid line). (b) Escape fraction as a function of zsrc/h for selected values of ws/u∗: LSM

(a 2-dimensional problem), which excludes the difficult task ofmodeling crosswind spread associated with the lateral dispersion.The modeling of crosswind spread is quite a challenging aspectof dispersion modeling in the surface layer due to difficulties inreproducing wind meandering and slow changes in mean winddirection. In the model evaluation here, clear problems can be iden-tified for sources near the top of the canopy and for predictions inthe far poles. A large number of simulations were performed withadjustments to several coefficients present in the model to improveresults that include the Lagrangian time scale, particle rebound, andcanopy and ground deposition parameters. Modifying coefficientstend to improve results in part of the domain and increase disagree-ment in other regions. As an example, decreasing the lateral spreadby reducing �v does improve the predictions for the far poles, butit compromises model predictions near the source. This highlightsthe importance of using a large array of measurements for modelcalibration and validation.

Finally, the large differences between spore escape fractionsestimated here for a maize canopy and those by Aylor (1999) for awheat field may be evidence of the importance of canopy structure.Given the role of escape fractions in spore dispersal and diseasepropagation within models such as IAMS, this significant differencejustifies a comprehensive study of spore escape fractions and thedevelopment of individual parameterizations for major crops. Thisis a timely issue now with the rapid progress in measuring canopystructure from airborne remote sensing platforms such as canopyLiDARs (Lefsky et al., 2002) that now allow tracking a(z) in time andspace. IAMS would also benefit from models that present a unifiedtreatment of the effects of a(z) across all the transport, release, anddispersion processes. The a(z) variations should impact simulta-neously the flow field responsible for particle trajectories, criticalturbulent stresses responsible for releasing spores into the canopyvolume, and the depositional surface intercepting these trajecto-ries. The coupled Eulerian–Lagrangian framework proposed andemployed here shows how all these processes are interconnected.

Acknowledgments

This research is supported by National Science Foundation (NSF)grant AGS1005363. Katul also acknowledges support from NationalScience Foundation (AGS-110227), the U.S. Department of Agricul-

ture (2011-67003-30222), and the Binational Agricultural Researchand Development (BARD) Fund (IS-4374-11C) that resulted in thedevelopments of the Eulerian and Lagrangian components of theLSM employed here.

130 S.C. Gleicher et al. / Agricultural and Fores

F u

As

etie sr1pwtum

t1cfithioawn

A

tmte

D

wfOt

Bound.-Layer Meteorol. 128, 1–32.

ig. 10. Comparison between stress fractions carried by sweeps and ejectionsS0 = S2 − S4 from experimental data (symbols) and from the Eulerian closure model

sing Eq. (A.1) (gray line).

ppendix A. Characterization of sweeps and ejections fromecond-order closure modeling

One of the key characteristics of canopy turbulence is the exist-nce of cycles of sweeps and ejections responsible for most ofhe momentum transport within the canopy region. The relativemportance of ejections and sweeps can be quantified by the differ-nce between the stress fractions carried by sweeps and ejectionsS0 = S2 − S4. Here, S2 and S4 are the stress fractions carried by

weeps and ejections, respectively, and usually defined via quad-ant analysis (Lu and Willmarth, 1973; Willmarth, 1975; Katul et al.,997). Because of the availability of sonic anemometers at multi-le depths, quadrant analysis was used to determine the So profileithin the CSL. Interestingly, So can also be estimated based on

he modeled third-order moments of the Eulerian velocity statisticssing a truncated or incomplete third-order cumulant expansionethod (Raupach, 1981; Katul et al., 1997):

S0 = �u�w

2(2�)1/2u′w′

(u′u′w′�2

u �w− u′w′w′

�u�2w

). (A.1)

The third-order moments u′u′w′ and u′w′w′ were closed usinghe vertical gradients of second-order moments (Wilson and Shaw,977) and can be obtained from the statistics shown in Fig. 2. Theomparison between observed values of S0 and the one retrievedrom the closure model using the gradients in the second momentss shown in Fig. 10. The closure model appears to capture the correctrends in S0 but underestimated its magnitude. It can be surmisedere that the imbalance between sweeps and ejections is encoded

n gradients of the second moments that are resolved by the second-rder closure model and explicitly used in the calculations of driftnd dispersion coefficients of the LSM. This result may also explainhy the explicit addition of triple moments in LSM may not haveecessarily improved its performance as reported by other studies.

ppendix B. Model for TL

The most usual form of the relation between the Lagrangianimescale TL and the TKE dissipation rate � is obtained from Kol-

ogorov’s inertial subrange theory for homogeneous isotropicurbulence. In this case, the Lagrangian structure functions arexpressed as

uu(�) = Dvv(�) = Dww(�) = C0��, (B.1)

here Duu(�), Dvv(�), and Dww(�) are the second-order structureunctions of the three components of velocity and � is a lag in time.ne can assume that this scaling holds for timescales up to TL and

hen the structure functions flattens to a constant value equal to

t Meteorology 194 (2014) 118–131

twice the velocity variance �2u = �2

v = �2w . Therefore, continuity at

TL requires C0�TL = 2�2u = 2�2

v = 2�2w , which yields the usual model

TL = 2�2w

C0�. (B.2)

Turbulence within plant canopies is highly anisotropic and for athree-dimensional dispersion model, it may be more appropriate touse an average over the energy contained in fluctuations along allthree directions C0�TL = (2�2

u + 2�2v + 2�2

w)/3 = (4/3)TKE. There-fore, one can write

TL = 2TKEC′0�

, (B.3)

where C′0 is proportional to C0. This estimate of TL is also more

compatible with the Eulerian relaxation time scale often used inhigher order closure models.

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