The Inﬂuence of Hilly Terrain on Aerosol-Sized …Boundary-Layer Meteorol (2010) 135:67–88 DOI 10.1007/s10546-009-9459-2 ARTICLE The Inﬂuence of Hilly Terrain on Aerosol-Sized

Boundary-Layer Meteorol (2010) 135:67–88DOI 10.1007/s10546-009-9459-2

ARTICLE

The Influence of Hilly Terrain on Aerosol-Sized ParticleDeposition into Forested Canopies

G. G. Katul · D. Poggi

Received: 25 July 2009 / Accepted: 23 November 2009 / Published online: 10 December 2009© Springer Science+Business Media B.V. 2009

Abstract Virtually all reviews dealing with aerosol-sized particle deposition onto forestedecosystems stress the significance of topographic variations, yet only a handful of studiesconsidered the effects of these variations on the deposition velocity (Vd). Here, the interplaybetween the foliage collection mechanisms within a dense canopy for different particle sizesand the flow dynamics for a neutrally stratified boundary layer on a gentle and repeatingcosine hill are considered. In particular, how topography alters the spatial structure of Vd

and its two constitutive components, particle fluxes and particle mean concentration withinand immediately above the canopy, is examined in reference to a uniform flat-terrain case.A two-dimensional and particle-size resolving model based on first-order closure principlesthat explicitly accounts for (i) the flow dynamics, including the two advective terms, (ii) thespatial variation in turbulent viscosity, and (iii) the three foliage collection mechanisms thatinclude Brownian diffusion, turbo-phoresis, and inertial impaction is developed and used.The model calculations suggest that, individually, the advective terms can be large just abovethe canopy and comparable to the canopy collection mechanisms in magnitude but tend tobe opposite to each other in sign. Moreover, these two advective terms are not precisely outof phase with each other, and hence, do not readily cancel each other upon averaging acrossthe hill wavelength. For the larger aerosol-sized particles, differences between flat-terrainand hill-averaged Vd can be significant, especially in the layers just above the canopy. Wealso found that the hill-induced variations in turbulent shear stress, which are out-of-phase

G. G. Katul (B)Nicholas School of the Environment, Duke University, Box 90328, Durham, NC 27708-0328, USAe-mail: [email protected]

G. G. KatulDepartment of Civil and Environmental Engineering, Pratt School of Engineering, Duke University,Durham, NC 27708, USA

G. G. KatulDepartment of Physics, University of Helsinki, Helsinki, Finland

D. PoggiDipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, Torino, Italy

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68 G. G. Katul, D. Poggi

with the topography in the canopy sublayer, play a significant role in explaining variationsin Vd across the hill near the canopy top. Just after the hill summit, the model results suggestthat Vd fell to 30% of its flat terrain value for particle sizes in the range of 1–10 µm. Thisreduction appears consistent with maximum reductions reported in wind-tunnel experimentsfor similar sized particle deposition on ridges with no canopies.

Keywords Advection · Aerosol sized particle deposition · Canopy flow · Complex terrain ·Deposition velocity · Gentle hills

1 Introduction

Micrometeorological measurements and models of dry deposition velocity (Vd) of aerosol-sized particles on forested ecosystems primarily rely on stationary and planar homogeneousflow assumptions. As such, these micrometeorological measurements and Vd formulationscannot account for the effects of local variations in topography. Hence, it is not surprisingthat virtually all recent reviews describing the various Vd formulations acknowledge thatthe effects of topographic variability on the measuring and modelling of Vd remain a majorknowledge gap (Wesely and Hicks 2000; Holmes and Morawska 2004; Pryor et al. 2008;Petroff et al. 2008a). Another review by Hicks (2008) offered a speculative assessment ofthe order of magnitude estimate of how heterogeneity in terrain might affect the spatially-averaged Vd , and highlighted recent advances in aircraft measurements that may be used togenerate the requisite spatially-averaged datasets to address this problem.

Topography perturbs the mean and turbulent flow fields that transport particles onto foliagesites; it also alters the immediate micro-climate including mean air temperature and humidityfields (Raupach et al. 1992) in a manner that may modify particle coagulation or conden-sation processes. At longer time scales, the impact of topography on processes pertinent todry deposition is quite extensive and includes canopy structural and morphological changesdue to nutrient and soil moisture gradients that are known to affect the foliage distribution,canopy heights, and changes in forest floor properties (e.g. ice vs. ice free, litter thickness andits concomitant moisture, to name a few). Because of these numerous interactions and non-linear feedbacks amongst all these processes, the way that topography alters Vd for forestedecosystems at all spatial and temporal scales remains a vexing research problem. Naturally,exploring all these processes simultaneously is well beyond the scope of a single study.

A number of studies have already considered the problem of particle deposition on com-plex topography (Hill et al. 1987; Stout et al. 1993; Parker and Kinnersley 2004; Hicks 2008)but did not treat the flow field and the vertical variation in vegetation collection mechanisms,which can be significant inside tall canopies. On the other hand, Petroff et al. (2008b) pro-posed a one-dimensional approach that explicitly accounts for the balance between the totalflux gradient and the vegetation collection mechanisms inside canopies as was done earlierby Slinn (1982), and discussed how various representations of these collection mechanismscan be formulated as ‘closure models’ to terms arising from a multi-phase volume averagingoperator applied to the continuity equation. Building on these separate advances, a naturalstarting point is to consider the interplay between the topographically induced advectiveterms, canopy turbulent transport processes and aspects of the foliage collection mechanismscommon to many Vd models. Condensation and coagulation processes, as well as any spatialvariation in the collection mechanisms not introduced by spatial variations endogenous to theflow field, are ignored. To progress even within this restricted scope, many of the governingprocesses must be simplified and parameterized.

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Complex Terrain and Aerosol-Sized Particle Deposition 69

The work here takes advantage of recent theoretical, numerical and experimental effortson quantifying flows inside canopies situated over gentle hilly terrain (Finnigan and Belcher2004; Ross and Vosper 2005; Katul et al. 2006; Poggi and Katul 2007a,b,c, 2008a; Poggiet al. 2007, 2008). The primary goal is to examine biases that occur in determining Vd val-ues when assumptions appropriate to flat terrain are applied to measurements or modelsover forests on gentle hills. In particular, we seek to quantify where on the hill surface theadvective terms tend to significantly ‘disrupt’ the balance between the total flux gradientand the particle collection mechanism by the vegetation. This balance underpins much ofthe forest-atmosphere Vd micrometeorological measurements and the multilayer models inuse (e.g. Slinn 1982; Petroff et al. 2008b; Pryor et al. 2008). This comparison is of particularsignificance to estimating aerosol-sized particle deposition rates onto forested ecosystemsusing micrometeorological methods. Given that these forested canopies are often situated oncomplex terrain, micrometeorological measurements unavoidably must be conducted in thecanopy sublayer (CSL) for such tall canopies (e.g. Gallagher et al. 1997; Grönholm et al.2007, 2009; Rannik et al. 2009).

2 Theory

2.1 Overview of the Problem Set-Up

As a case study for presenting the model calculations, the canopy is assumed to be uniformof height hc and is situated on a cosine-shaped hill beneath a non-stratified atmospheric flow(see Fig. 1, top-left panel). The hill height (H) is also assumed to be comparable to hc, whichis perhaps the most dynamically interesting case. When H � hc, then these topographicvariations are likely to be too small and can be ignored provided the topographic variationsare assumed to be gentle. On the other hand, if H � hc, then the problem is entirely ‘pinned’to the large mean pressure gradients because the differences between the turbulent stress gra-dient and the drag force inside the canopy become much smaller than the externally imposedhorizontal mean pressure gradient. Hence, the internal flow dynamics within the canopy vol-ume become less relevant for such conditions. For these reasons, we selected H = hc as thebasis of our work here.

For keeping the results as ‘generic’ as possible and not tied to a particular foliage config-uration or experimental site properties, it is also assumed that the canopy leaf area density issufficiently dense and constant with depth. Above the hill-canopy system (≈3hc), the atmo-sphere is assumed to be an infinite supply of aerosol sized particles (of all sizes) that aretransported by the flow and then collected by the foliage elements. All particles are assumedto have reached their equilibrium diameter (dp), and this equilibrium dp is no longer evolv-ing on time scales pertinent to turbulent processes or their Reynolds average. To keep theparameters that may introduce variability in Vd to a ‘bare-minimum’, the ground depositionis entirely neglected when compared to the integrated canopy collection mechanisms, thoughwe are well aware that forest floor deposition rates can account for some 20% of the totaldeposition on the soil-canopy system (Donat and Ruck 1999; Grönholm et al. 2009). The flowfield computations are based on a variant of the wind-field model of Finnigan and Belcher(2004), (henceforth FB04). This model was recently compared against mean velocity andturbulent stress measurements collected in a flume in which the canopy was composed ofdensely arrayed rods and situated on repeated cosine hill modules (Poggi et al. 2008). Theagreement between model calculations and measurements was reasonable even though themeasurements were carried out in a dynamical regime that does not conform entirely to

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Fig. 1 Top left: The hill-canopy set-up and the origin of the coordinate system used. The hill height is H, thetotal length of the domain is 4L, where L is the hill half-length, the canopy height hc = H , and the canopytop is shown by the dashed green line. The origin of the coordinate system (x = 0, z = 0) is situated at thehill summit and the canopy top. The remaining panels show the two-dimensional variations of the mean lon-gitudinal velocity (labelled as u, m s−1), mean vertical velocity (labelled as w, m s−1), turbulent shear stress(labelled as uw, m2 s−2), vertical velocity standard deviation (labelled as σw, m s−1), and eddy viscosity(labelled as Kt , m2 s−1) obtained from FB04 (see Appendix A for formulation)

the FB04 assumptions (see Fig. 1 in Poggi et al. 2008 for a discussion). The FB04 model isa first-order two-dimensional closure for turbulent flow over and within a tall canopy on alow or gentle hill and serves as a logical starting point for the investigation here.

In the FB04 model, the presence of the hill creates a pressure perturbation that is assumedto be vertically uniform due to the hydrostatic assumption (see Poggi and Katul 2007c for adiscussion and experimental support for this assumption). The mean flow responds first tothis vertically uniform pressure perturbation by generating horizontal and vertical gradientsin the mean velocity across the hill via the advective terms in the mean momentum balance,and these gradients in turn introduce advective terms and spatial variability in the turbulentdiffusivity for the size-resolved mean scalar continuity equation for the particles. The variablemean wind field also introduces both horizontal and vertical variations in the turbulent shearstress and the vertical velocity standard deviation thereby affecting the particle collectionmechanisms due to the vegetation. To assess the relative importance of these hill-inducedprocesses on particle concentration, canopy sources and sinks, flux distributions, and Vd

across the hill, all model results will be compared to the case of uniform flat terrain coveredwith a canopy having the same aerodynamic, geometric, and particle collection attributes.

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It is not our intent to model the absolute values of the fluxes and collection mechanisms butrather to quantify the relative magnitudes and the signs of the perturbations in these variablesas introduced by gentle topographic variations. Hence, all calculations are referenced to thisflat-terrain case scenario. Because the particle diameter can appreciably affect the vegetationcollection mechanisms, we consider five particle-size classes ranging from 1 nm to 10 µm infactors of 10 increments. For these particle-size classes, the collection mechanisms that aresignificant range from Brownian diffusion (finest size), to inertial impaction (largest size),to turbo-phoresis (intermediate sizes).

2.2 Fluid Mass and Momentum Conservation

Numerical, experimental, and analytical approaches for turbulent flows on complex terrainhave proliferated exponentially over the past 35 years following the pioneering work ofJackson and Hunt (1975). A partial list of diverse laboratory and field experiments and thewealth of results from computational studies can be found elsewhere (Britter et al. 1981;Taylor et al. 1987; Zeman and Jensen 1987; Hunt et al. 1988; Carruthers and Hunt 1990;Raupach et al. 1992; Kaimal and Finnigan 1994; Ayotte 1997; Raupach and Finnigan 1997;Ying and Canuto 1997; Belcher and Hunt 1998; Wilson et al. 1998; Athanassiadou and Castro2001; Finnigan and Belcher 2004; Bitsuamlak et al. 2004; Ross and Vosper 2005; Katul et al.2006; Poggi et al. 2007, 2008; Poggi and Katul 2007a,b,c, 2008a) and their findings are notreviewed or repeated here.

As mentioned by Katul et al. (2006), earlier models of airflow over hills of low slope(= H/L , where L is the half-length of the hill, Fig. 1) focused on momentum and scalartransfer above the so-called roughness sub-layer of the atmospheric boundary layer (ABL).Hence, the novelty of FB04 is that the canopy sub-layer is treated as a “dynamically distinct”atmospheric layer interacting with the inner and outer layers of the ABL. Within the can-opy layer, the two-dimensional mean continuity and momentum balance for stationary, highReynolds number and neutrally stratified turbulent flows are given by FB04,

∂u

∂x+ ∂w

∂z= 0, (1)

u∂u

∂x+ w

∂u

∂z= − 1

ρ

∂ p

∂x−

(∂u′u′∂x

+ ∂u′w′∂z

)− Fd

(1 − H f (z, hc)

), (2)

where x and z are defined in a hypothetical steamline (or displaced) coordinate system setby the topographic shape and defined in FB04. This system reduces to terrain-following nearthe ground and rectangular Cartesian well above the hill surface (e.g. in the middle layer). Ifthe rectangular Cartesian system well above the hill is defined with X being the horizontaland Z being the vertical coordinates, then x = X + (H/2) sin (k X) exp(−k(Z + hc)) andz = Z − (H/2) cos (k X) exp(−k(Z + hc)) with k = π/2L , and where u and w are the time-averaged1 velocities in the x and z directions, respectively, u′ and w′ are the correspondingturbulent velocity fluctuations, respectively, such that u′ = w′ = 0, p (x, z) is the meanstatic pressure, and Fd is the canopy drag exerted by the vegetation on the air flow, which isparameterized as,

1 Within the canopy, volume averaging over thin slabs containing many foliage elements is necessary toremove the large point-to-point variations (Brunet et al. 1994). The extra ‘dispersive’ flux terms arising fromthe volume averaging are lumped in the Reynolds stresses. Two experimental studies suggest that these disper-sive fluxes are quite small when compared to the Reynolds stresses inside dense canopies (Poggi et al. 2004b;Poggi and Katul 2008b).

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Fd = Cdau2, (3)

in which Cd is the dimensionless drag coefficient and a(z) is the foliage frontal area per unitvolume. The origin is taken as the cross-stream coordinate z at the top of the canopy withx = 0 being the top of the hill (see Fig. 1). The term H f is the Heaviside step functiondefined by

H f ={

1; z > 00;−hc < z ≤ 0

. (4)

Using first-order closure principles, it is possible to derive analytical solutions for u, w andu′w′ caused by the hill for the case of constant Cda and sufficiently small H/L so that theequations can be linearized and ∂u′u′/∂x can be neglected. Appendix A presents the analyt-ical solution for the mean flow field and the eddy viscosity used to drive the particle meanscalar conservation.

2.3 Scalar Mass Conservation

For steady state conditions, the conservation of the mean concentration c of particles of sizedp is given by,

u∂c

∂x+ w

∂c

∂z= −

(∂ Fc

∂z+ ∂ Fcx

∂x

)+ Sc

(1 − H f (z, hc)

)(5)

where Fc and Fcx are, respectively, the cross-stream and streamwise total fluxes of aerosolsized particles, and Sc is the vegetation collection mechanism (Slinn 1982; Pryor et al. 2008;Petroff et al. 2008b). Here, u and w are given by the solution to Eqs. 1 and 2, which ispresented in Appendix A. Furthermore, model calculations in Katul et al. (2006) suggest that

for scalars in which the canopy is a major source or sink,∣∣∣ ∂ Fc

∂z

∣∣∣ �∣∣∣ ∂ Fcx

∂x

∣∣∣, and hereafter,

this streamwise flux gradient term is ignored relative to its vertical counterpart. To close thisproblem mathematically, it is necessary that Sc and Fc be made dependent on c. Adoptingfirst-order closure principles for the particle turbulent flux (see reviews in Sehmel 1980; Slinn1982; Pryor et al. 2008; Petroff et al. 2008b) yields

Fc(x, z) ≈ − (Dp,m + Dp,t (x, z)

) ∂c(x, z)

∂z− |Vs | c(x, z), (6)

where Dp,t and Dp,m are the particle turbulent and Brownian diffusivities, respectively, andVs is the settling velocity (Petroff et al. 2008b; Pryor et al. 2008). Limitations of gradient-dif-fusion approximations inside canopies are well-known (Shaw 1977; Denmead and Bradley1985; Finnigan 1985, 2000); however, these limitations may be less critical when assessingthe spatial structure of the hill-induced perturbations from a mean background state for rea-sons discussed in Finnigan and Belcher (2004). For completeness, the two diffusivities andVs formulations are presented in Appendix B. The spatial variation in the deposition velocitycan be readily computed from Vd(x, z) = −Fc(x, z)/c(x, z).The vegetation collection mechanisms are assumed to occur within a quasi-laminar boundarylayer adjacent to the leaf surface, and are given as

Sc(x, z) =(

2a

π

)c(x, z)

rb(x, z), (7)

where the factor π adjusts for the single-side projected leaf area to total surface area of leaves(assuming the foliage needles to be cylinders), and rb is the local quasi-laminar boundary-

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layer resistance for particles of diameter dp given as (Seinfeld and Pandis 1998; Wesely1989):

rb(x, z) =(√

−u′w′(x, z)(

Sc−2/3 + 10−3/St (x,z))

+ Vt (x, z)

)−1

, (8)

where Sc = ν/Dm,p is the Schmidt number, St = Vs(−u′w′(x, z))/(gν) is a turbulentStokes number, and Vt (x, z) is the turbo-phoretic velocity. The inertial impaction term inrb is parameterized as 10−3/St , which is based on Slinn and Slinn (1980) formulation forwater or smooth surfaces. This formulation was shown to be reasonably accurate for smoothboundary layers (Aluko and Noll 2006). When such a formulation is applied to the entirecanopy-soil system (or a rough surface), then this inertial-impaction formulation underesti-mates depositional velocities (e.g. see “Discussion” in Feng 2008). However, in a verticallyresolved model of the foliage, it is not clear to what degree and precisely how the ‘micro-roughness’ of the foliage alters this formulation given that the inertial impaction must beapplicable to isolated leaf surfaces. Hence, for simplicity and consistency with formulationsfor the turbo-phoretic term (see Appendix C), the quasi-laminar boundary layer over isolatedleaf surfaces is assumed to be dynamically similar to a smooth boundary layer. Stomataluptake, sedimentation, interception, rebound, and other phoretic processes are all ignored.We retained Vt (x, z) mainly because of its strong dependence on the flow dynamics; itsestimation is presented in Appendix C.

3 Results and Discussion

Before discussing how gentle topography affects particle deposition processes onto the can-opy, key properties of flows inside canopies situated on gentle hilly terrain are reviewed.When discussing the spatial variations of the flow statistics across the hill, reference is madeto five regions: upwind region above the canopy (x/L < 0, 0 < z/hc < 1); downwindregion above the canopy (x/L > 0, 0 < z/hc < 1); upwind region inside the canopy(x/L < 0, −1 < z/hc < 0); downwind region inside the canopy (x/L > 0, −1 < z/hc <

0); and the region in the vicinity of the hill summit (x/L ≈ 0, z/hc ≈ 0). In some cases, thewithin-canopy region is further decomposed into two parts: the upper (−0.3 < z/hc < 0)

and lower canopy (−1 < z/hc < −0.5) sub-regions.

3.1 Generation of the Flow Field

Figure 1 shows, (i) the spatial variations in u and w within and above the canopy, whichare needed for computing the two advective terms in the mean particle continuity equation,(ii) the turbulent stress (u′w′), which is needed for computing the particle collection mech-anisms, (iii) σw, which is needed for computing the turbo-phoretic velocity (see AppendixB), and (iv) the eddy viscosity or diffusivity (Kt ), which is needed for computing the scalarturbulent flux. We point out a number of features of these flow properties presented in Fig. 1:

1. Mean flow field: Inside the canopy and the first portion of the upwind region (x/L <

−1), u increases with increasing x until midway to the hilltop and then decreases there-after with increasing x . On the lee-side (x/L ≈ 1.1), the flow experiences a negative uinside the canopy, suggestive of a recirculation zone. This zone, first predicted by FB04,mainly occurs due to the interplay between the drag force and adverse pressure gradi-ent, though advection can modify its spatial extent (Poggi et al. 2008). Note that, in the

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absence of a canopy, this recirculation zone does not exist because the topography here issufficiently gentle not to induce flow separation (see Poggi et al. 2007; Poggi and Katul2007a,b for further discussion). In reality, this recirculation is not a classical ‘rotor’ type,but is characterized by a highly intermittent zone with alternating large positive and neg-ative velocity excursions in the lower layers of the canopy (Poggi and Katul 2007b). Itis worth pointing out here that the mixing length, when experimentally inferred frommeasured u′w′ and u even in this recirculation zone, appeared to be nearly constant withdepth and consistent with the FB04 assumption (Poggi and Katul 2007b). Finally, becausethe vertical turbulent diffusivity is finite in this zone, passive scalar mass can diffuse intothis recirculation zone, accumulate, and then eject out. This accumulation-ejection phasehas been shown experimentally to be quasi-periodic in nature and contribute significantlyto the local flux (Poggi and Katul 2007b). For the case of aerosol-sized particles, thevegetation collection mechanism may entirely alter this accumulation phase, as aerosol-sized particles diffusing into this zone may become partially trapped, thereby allowingthe vegetation collection mechanism to remove them from the mean flow. The maximumu above the canopy occurs just in front of the hillcrest (x → 0) and is almost out-of-phasewith the maxima in w. The fact that u and w are nearly but not precisely out of phasewith each other (due to nonlinearities in the mean momentum equation) has importantimplications to the two advection terms in the scalar continuity equation for particles.

2. Shear stress: In the deeper layers of the canopy, the shear stress is small in magnitudethroughout the hill thereby preventing the deeper foliage (≈50% of the total leaf area)from being efficient deposition sites (due to the large rb). In the upper layers of the canopy,the minimum shear stress magnitude occurs near the hilltop (x → 0). This stress is usu-ally out-of-phase with the topography. Because of a tight coupling between this stress andrb, the vegetation collection mechanism may be again less efficient at removing particlesin this location. The implications of the reduction in collection efficiency on Vd near theorigin (x = 0, z = 0) will be discussed later.

3.2 Particle Concentration, Fluxes, and Deposition

Figure 2 shows the two-dimensional variations of the total fluxes, mean particle concentration,and deposition velocity relative to the ‘flat-world’ case for dp = 1 nm. For this small dp , thevegetation collection mechanism (in Sc) is primarily governed by Brownian diffusion. FromFig. 2, the large variations and the richness in the spatial patterns of normalized depositionvelocities are rather striking. Gentle hills with a mean slope of 10% (= H/L) can introducespatial variations in Vd that range from nearly 10 to 100%. Both mean scalar concentrationand particle fluxes are altered by the canopy-hill system (relative to the flat terrain); how-ever, the emerging spatial patterns in Vd appear to be more controlled by the fluxes (a resultalready foreshadowed by Hicks 2008 for complex terrain). Moreover, the spatial variabilityin Vd above the canopy is larger than the spatial variability in fluxes or concentrations. This‘extra’ variability stems from the fact that zones enriched with particle mean concentrationare spatially co-located with zones of low particle fluxes (upwind), and conversely (lee-side).With respect to the various regions across the hill system, we note the following:

1. On the first portions of the upwind side (x/L < −1) and in the upper layers of the canopy(−0.3 < z/hc < 0), the deposition velocity is generally enhanced relative to its flat-world counterpart. However, progressing on the upwind side further (−1 < x/L < 0),there is a decline in Vd relative to its flat-terrain value, thereafter reaching a minimumjust after the hill summit (x/L ≈ 0.3).

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Fig. 2 Left: The hill-induced two-dimensional variations in normalized deposition velocity (Vd ) for dp =1 nm, particle concentration (labelled as C), and total vertical flux (labelled as Fc). These flow quantities arenormalized by their ‘flat-terrain’ counterparts (indicated by the subscript). Right: Comparison between theaveraged quantities across the hill wavelength shown in the left panels (denoted by 〈·〉 and presented as solidlines) and the flat-terrain case (dashed lines). The canopy top is also shown as a thin horizontal dashed linefor reference

2. Within the middle layers of the canopy (z/hc ≈ −0.5), Vd is enhanced on the upwindside (−2 < x/L < −0.5), but is then reduced to levels below its flat-terrain counterpartjust after the hill summit (x/L ≈ 0.1). The recirculation zone on the lee-side insidethe canopy appears to leave a complex fingerprint inside the canopy, with enhanceddeposition after the hilltop (x/L ≈ 0.2 − 0.7), followed by a reduced deposition onthe remaining part of the lee-side (0.8 < x/L < 1.8), and then a recovery to flat-ter-rain values. We note that this pattern is strictly ‘endogenous’ to the canopy-flow systembecause the ground deposition flux was set to zero throughout.

3. Above the canopy and on the upwind side, the deposition velocity is reduced. Thispattern reverses on the lee-side of the hill, as earlier noted, though not symmetrically.Some of the spatial variability in deposition velocity well above the canopy top is dueto the nature of the spatially constant mean concentration (= Co) imposed as an upperboundary condition at z/hc = 2.

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While the analysis in Fig. 2 demonstrates by how much and where on the hill surface Vd isreduced or enhanced, it remains to be seen whether these changes actually cancel each otherout when a spatial average along the hill wavelength is computed. Again, recent advances inairborne aerosol-sized particle fluxes and concentration measurements can now sample suchspatially-averaged quantities (Hicks 2008). This averaging question is further explored bydetermining 〈Vd〉, 〈c〉, and 〈Fc〉 for the hilly case and comparing them to the flat-world modelcalculations. Hereafter, angular brackets denote a hill-averaged quantity (i.e.〈ξ(x, z)〉 =

14L

∫ 2L−2L ξ(x, z)dx). Figure 2 presents the outcome of this comparison for Vd and its two

constitutive terms. Inside the canopy, the hill-induced perturbations appear to nearly canceleach other for fluxes, concentrations, and deposition velocities (at least for the smallest dp

selected here). Near the canopy top, the hill-induced perturbations tend to reduce the overalldeposition velocity by only 3%. Above the canopy, the advective terms do not entirely ‘aver-age out’, and an apparent ‘flux gradient’ persists for the hilly case. Recall that the hill and the‘flat-world’ cases were subjected to the same upper boundary condition, but due to the strongasymmetry in the flow dynamics, the advective terms did not entirely average out above thecanopy. This last finding prompted further exploration of the individual components of theadvective terms and the spatial variations of the remaining components of the mean continu-ity equation. We use again the dp = 1 nm as a case study to illustrate the two-dimensionalvariations in these terms.

Figure 3 displays the two-dimensional spatial variations of the two advective terms, theflux gradients, and the canopy collection mechanism (mainly due to Brownian diffusion).For reference, the computed Vd(x, z) (not normalized) is also presented.1. Advective terms: In the lower layers of the canopy, the advective terms are generally small,except near the ground on the lee-side of the hill. The upper ‘edge’ of the re-circulationzone (x/L ≈ 0.3) coincides with this positive longitudinal (and dominant) advective term.Above the canopy, the advective terms are large, opposite in sign and comparable in magni-tude to the flux gradient and canopy collection mechanism. However, because they are notexactly out-of-phase with each other, they contribute to a non-zero flux gradient upon spatialaveraging across the hill wavelength (as is evident in Fig. 2).2. Canopy collection mechanism: Near the canopy top and on the first-half of the upwindside, the canopy collection mechanisms are first enhanced due to a large enhancement in themagnitude of u′w′ (see Fig. 1), which in turn increases Sc and ∂c/∂z near the canopy top.In this zone, ∂c/∂x has not been fully established (∂ P/∂x is beginning to increase from anear-zero value here). When this enhancement in ∂c/∂z is subjected to w < 0 (see Fig. 1), thevertical advection term leads to an enhancement (i.e. more negative) in the overall deposition(i.e. w < 0 rapidly advects particles towards the canopy top, and in this case, many ordersof magnitude faster than the particle settling velocity). Near the canopy top and at the hillsummit (x → 0, z → 0), the canopy collection mechanism is reduced due to a reductionin the magnitude of u′w′ (see Fig. 1). Moreover, near this zone, w > 0 and becomes large,and the vertical advection reverses sign. The magnitude of the horizontal advection has alsoincreased in this region and partially cancels the overall impact of the vertical advection.The reduced canopy collection sink and the residual vertical advection (i.e. after partiallybalancing the horizontal advection) still conspire to reduce the overall deposition velocitynear the coordinate origin. Near the hill summit (x/L = 0) and progressing down on thelee-side, the magnitude of u′w′ increases again with increasing x (see Fig. 1) and the canopycollection mechanisms regain their efficiency in the upper canopy layers, the vertical velocityis also reduced and then reverses from positive to negative in sign, and the deposition processre-establishes itself to near-background values by x/L = 1 (Fig. 2). For larger particles, the

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Fig. 3 The two-dimensional variations of the components of the particle mean continuity equation for dp =1 nm. The left panels show the individual advective terms and their sum (zero for the flat-terrain case). Theright panels show the flux gradient (middle) and the particle collection (bottom) mechanism (their sum isbalanced by the sum of the advection terms in the left panel). For reference, modelled Vd (in m s−1) is alsoshown (top-right)

deposition is enhanced beyond its background state by 50%, as we show later, especiallynear the canopy top.3. Flux gradient: Above the canopy, the particle flux gradient is controlled by the relativeimportance of the horizontal and vertical advection terms (recall that Sc = 0 above the can-opy). Roughly, the phase relationships of these advective terms tend to track their u and w

counterparts (Fig. 1). The asymmetry in the flow is one reason why the large advective termsdo not entirely cancel out above the canopy (see Fig. 2).

The same analysis was repeated for dp = 10 nm, 100 nm, 1 µm and 10 µm, and theresults appear to be qualitatively the same in terms of the spatial variability of the advec-tive terms and canopy sinks. However, quantitatively, significant differences emerge and wesummarize these differences in Fig. 4, which compares 〈Vd〉 at each height z with its flat-terrain counterpart for all five dp classes. For the larger particles, turbo-phoresis and inertialimpaction significantly contributes to Sc (vis-à-vis Brownian motion). The Vt value is sen-sitive to the spatial variability in σw (see Fig. 1) and the inertial-impaction term exhibits a

123


Fig. 4 Comparison between the flat-terrain and hill-averaged Vd (labelled as 〈Vd 〉) cases for all five particlesizes (dp) and for all z values. The 1:1 line is also shown for reference. Note that the major departures fromthe 1:1 line are for the larger dp

power-law dependence on the Stokes number (and hence∣∣∣u′w′(x, z)

∣∣∣). Clearly, the asym-

metry in∣∣∣u′w′(x, z)

∣∣∣ shown in Fig. 1, when introduced into the non-linear collection mech-

anisms, does not entirely average out and can lead to 〈Vd〉 diverging from its flat-terraincounterpart (see Fig. 4) depending on z.

As noted in the Introduction, single-tower micrometeorological measurements of Vd abovetall forests are conducted in the CSL, and the presence of topography, even gentle topography,can lead to large biases at some locations, as evidenced by Fig. 2. To explore the connectionbetween these biases and the hill shape for a reference region in the CSL, Fig. 5 presentsvariations in Vd(x, 0) (i.e. at the canopy top) normalized by its flat-terrain counterpart as afunction of x for each of the five dp classes. For all dp values, the spatial patterns in depositionvelocities are similar: enhancement in the first region of the upwind slope (−2 < x/L < −1),followed by a reduction with increasing x/L with maximum reductions occurring after thehill summit (x/L ≈ 0.3), followed by a recovery and then enhancement towards the midsection on the lee-side of the hill. It is also evident that the hill has a much more appreciableeffect on deposition velocity when the inertial impaction term and turbo-phoresis dominatethe collection mechanisms (consistent with the analysis in Fig. 4).

It is interesting to compare these model findings with a comprehensive wind-tunnel depo-sition study conducted by Parker and Kinnersley (2004) on an isolated pyramidal ridge withlength-to-height ratio of 3:1 and with no canopy (also shown in Fig. 5). In these experiments,approximately 90% of the particle volume was in the size fraction from 0.75 to 1.0 µm and10% were in the size range 1.0–2.0 µm in diameter. The Vd measurements were conductedby collecting particles on filters placed on the ground surface. For this wind-tunnel experi-ment, there was a slight decrease in the deposition velocity upwind of the raised topography;

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Fig. 5 The modelled effects of particle diameter (dp) on the variability of the normalized Vd (x, 0) across thehill at the canopy top. The normalization uses Vd, f lat (x, 0) as inferred for a flat surface for the same canopyand dp . For reference, the wind-tunnel (WT) measurements of normalized Vd for an isolated gentle pyramidalridge are shown as symbols in the bottom panel for dp = 0.75 − 1.2 µm. The wind-tunnel data here are notintended for a one-to-one comparison as the wind-tunnel experiments do not include a canopy effect. Theyare presented only for reference

a maximum in deposition on the upwind face close to the peak; a region of decreased depo-sition on the leeward face, extending into the immediate wake of the obstacle, and graduallyrecovering with distance to upwind levels at the end of the pyramidal structure. Perhaps moreintriguing in Fig. 5 is that, for the cosine hill-canopy system, variations in dp have more ofan effect on the relative variation in deposition velocity across the hill when compared tothe precise shape of the topography and the presence or absence of a canopy. Also, whetherfor the pyramidal structure with no canopy or the cosine hill structure with a tall canopy, themaximum decline in deposition velocity relative to the flat-terrain case appears to occur justin the vicinity of the maximum topographic excursion and this decline appears to be highlysensitive to dp . We have shown earlier that this reduction is due to two factors—the large

positive w and the reduction in the particle collection mechanisms due to a reduced∣∣∣u′w′

∣∣∣.The effect of a reduction in |u′w′| ‘amplifies’ the reduction in Vd for heavier particles, giventhat inertial impaction becomes the dominant particle collection mechanism inside the can-opy, and this term decreases as a power law with decreasing |u′w′|. Even if other formulations

123


Fig. 6 Variations in Rs(dp) = 14L

∫ 2L−2L

∫ 0−hc

Sc(x, z)dz dx , normalized by its flat-terrain counterpart as afunction of particle diameter (dp). For reference, the domain-averaged Vd , normalized by its flat-terrain coun-

terpart is also shown, where Rv(dp) = 14L

∫ 2L−2L

∫ 2hc−hcVd (x, z)dz dx . Note that in both cases, the maximum

differences from unity are on the order of H/L

for the inertial impaction term are used (see Table 1 in Petroff et al. 2008a), they all vary ina non-linear manner with the turbulent Stokes number (and |u′w′|), and hence the findingshere on the reduction-enhancement patterns in Vd(x, 0) across the hill and its amplificationwith increasing dp is likely to hold for those formulations as well.

To further assess the effects of the hill on the domain-integrated particle collection pro-cesses by the canopy, we consider the normalized variable given by

Rs(dp) =⟨∫ 0

−hcSc(x, z)dz

⟩∫ 0−hc

Sc,Flat (z)dz, (9)

where 〈 · 〉 is, as before, averaging across the hill wavelength L. Figure 6 presents the varia-tions of Rs for the five diameter classes considered here. Not surprisingly, when dp is smalland the collection mechanism is controlled by Brownian diffusion, the effects of the gentlehill on the integrated canopy particle removal rate is small (<3%). For larger dp , the effectsappear to be on the order of H/L. Hence, this finding suggests that, when model calcula-tions are employed in assessing the overall rate of particle removal by the vegetation, gentletopographic variations on the vegetation sink term are not large (<15%) and can be ignoredfor dp ≤ 10 µm. However, this finding cannot be extrapolated to the problem of inferringSc from micrometeorological flux measurements at a single tower. The analysis here (e.g.Fig. 2) unambiguously shows that when inferring

∫ 0−ho

Sc(z)dz at a single tower location,gentle topographic effects can be large even for the finest diameter. Moreover, for largerdp , the longitudinal variations in Fc(x, 0) (or Vd(x, 0)) across the hill can far exceed the

123


magnitude of the random flux error reported in Rannik et al. (2009) at a single location abovethe canopy, and hence, ought to be statistically discernable.

4 Conclusions

The primary goal here was to examine biases that occur in determining Vd values whenassumptions appropriate to flat terrain are applied to measurements or models over forestssituated on gentle hills. We focused on how the two advective terms ‘disrupt’ the balancebetween the turbulent-flux gradient and the particle collection mechanism by the vegetationfor a gentle cosine hill in which the hill height is comparable to the canopy height. We foundthat in the case of models, inferring

∫ 0−hc

Sc(z)dz using ‘flat-world’ assumptions is reason-

able, and differences between the hill-averaged∫ 0−hc

Sc(x, z)dz and flat-world∫ 0−hc

Sc(z)dzare small, at least not exceeding H/L .

However, estimating∫ 0−hc

Sc(z)dz from single tower-based micrometeorological fluxmeasurements using ‘flat-world’ assumptions is far more problematic. The model calcu-lations here suggest that, individually, the advective terms are large but tend to be opposite insign. From the mean continuity equation, variations in u are out of phase with variations in w,but not exactly due to non-linearities in the mean momentum balance. This ‘misalignment’in the out-of-phase relationship between u and w is the ‘genesis’ of the longitudinal andvertical particle advective terms that, (i) mirror those of u and w in their phase relationships,and (ii) appear comparable in magnitude to the canopy collection or flux gradient terms forthe layers above the canopy. Moreover, the two-dimensional spatial variations in u leads tospatial variations in |u′w′| that are in phase with |∂u/∂z|, which is a minimum near the hillsummit and nearly out of phase with the topography. These |u′w′| variations, along withparticle sizes, dictate the spatial variation of the particle collection efficiency of the vegeta-tion, especially in the upper canopy layers. A reduced |u′w′| and a positive vertical velocitynear the canopy top conspire to appreciably diminish Vd just after the hill summit. Thesereductions in Vd can be a factor of 3 relative to their flat-terrain case depending on dp .

Much of the work here focused on a cosine hill, and the logical follow-up question ishow to account for ‘real-world’ topographic variations when estimating

∫ 0−hc

Sc(z)dz usingmicrometeorological flux measurements from a single tower. Given the recent advances incanopy Lidar systems (e.g. see the review in Lefsky et al. 2002) from which the topographicvariations and canopy leaf area distribution can now be simultaneously resolved, theoreticalapproaches that take advantage of such data must be developed to begin ‘relaxing’ the flat-world assumptions. To be clear, the approach proposed here does not offer finality to thisproblem, but may provide a blueprint on how to proceed. First, as was the case with fieldstudies and model calculations for CO2 (e.g. Feigenwinter et al. 2004; Katul et al. 2006), themodel results here suggest that accounting for one of the two advective terms but ignoringthe other may prove to be more problematic than ignoring both. Hence, some accounting ofboth advective terms is needed (at least in a uniform canopy in the absence of density strat-ification). Second, for gentle topographic variations, the linearized pressure perturbationsin FB04 can be superimposed. It is conceivable that some of the ‘real-world’ topographicperturbations under consideration, if they are not too large relative to the canopy height, areamenable to being spectrally decomposed, and pressure perturbations be solved for each Fou-rier amplitude and wavelength. The linearity of the modified FB04 permits us to superimposeall these pressure perturbations together and analytically generate a realistic u and w that arein equilibrium with the complex topography. Such knowledge might allow some evaluation

123


of the first-order effects of particle advection on micrometeorological measurements of Vd

collected at a single tower using the approach proposed here.

Acknowledgements The authors would like to thank T. Grönholm, S. Launiainen, and T. Vesala for allthe helpful discussions that motivated this work. G. Katul acknowledges the support from the Department ofPhysics at University of Helsinki during his three month visit from Duke University, from the National ScienceFoundation (Grants NSF-EAR 0628342, NSF-EAR 0635787, and NSF-ATM-0724088), and the BinationalAgricultural Research and Development (BARD) (Grant IS-3861-06) for the development of the canopy tur-bulence aspects of this work. D. Poggi acknowledges travel support from the European Cooperation in thefield of Scientific and Technical Research (COST) to visit the University of Helsinki.

Appendix A: The Mean Velocity Flow Field

According to Jackson and Hunt (1975), the mean longitudinal momentum balance can bedecomposed into an unperturbed (or background equilibrium) state and a perturbation inducedby topographic variations. Such decomposition allows tracking how the hill system modifiesthe flat-terrain solution. Mathematically, this decomposition results in u(x, z) = Ub(z) +�u(x, z), w(x, z) = �w(x, z) and u′w′(x, z) = τb(z) + �τ(x, z). The subscript ‘b’ andthe symbol � indicate background and the hill-induced perturbations, respectively.

The background mean velocity and turbulent stress, Ub(z) and τb(z), are modelled usinga combination of logarithmic and exponential profiles, given by:

Ub(z)

u∗= H f

[1

kv

ln

(z + hc − d

zo

)]+ (1 − H f )

[1

βeβz/ le f f

], (10)

τb(z)

u2∗= H f

[e

zβle f f

]+ (1 − H f ), (11)

where u∗ is the background friction velocity at the canopy top, d and zo are the zero-planedisplacement and aerodynamic roughness length of the canopy, respectively, kv = 0.4 is thevon Karman constant, Uh is the mean velocity at the canopy top, β = u∗/Uh is the dimen-sionless momentum flux through the canopy, and le f f is a characteristic turbulent mixinglength, equal to kv(z + d) above the canopy and a constant 2β3Lc inside the canopy, whereLc = (Cda)−1 is the adjustment length scale (Belcher et al. 2003; Katul et al. 2004; Poggiet al. 2004a). Moreover, imposing the continuity constraints on Ub(z) and τb(z) at the canopytop results in

zo = 2Lcβ3

kv

e− kvβ , (12a)

d = 2Lcβ3

kv

. (12b)

The analytical solution to the mean momentum equations within the inner and canopy layersresults in (see Poggi et al. 2008):

�u ={

H f

[Uc

(Ace

zβle f f + i

√u∗

Ucβ

)]

+(1 − H f )

[UI 0

(1 − Ai B0 + ln

[L2

i /((d + z)zo)])

ln [Li/zo]

]}eisx , (13)

�τ = u∗{

H f

[2Uc Acβe

zβle f f

]+ (1 − H f )

[UI 0

kv (−2 + Ai Ar B1)

ln [Li/zo]

]}eisx , (14)

123


Table 1 Parameters used in the 2-D model calculations

Parameters Values

Hill attributes

Z = f (X) and p(X) Z = H2 (cos (s X) − 1) − hc &

p(X) = − U2o H2s cos(s X)

s = π2L

H (m), L (m) 15, 150

Canopy attributes

LAI (m2 m−2) 3.5

hc (m), Cd , a (m−1) 15, 0.2, 0.23

The canopy attributes resemble those of a Boreal Scots pine forest (Launiainen et al. 2007). The pressure var-iation formulation is also shown with Uo being the outer-layer velocity determined as in FB04. The flat-worldfriction velocity (u∗) = 0.5 m s−1 and the particle density ρp = 1,500 kg m−3

where B0 = K0(Ar ) and B1 = K1(Ar ) are the modified Bessel function of the zeroth andfirst order with argument Ar = 2

√isL(z + d)/Li , Ac and Ai are the constants that permit

the continuity of �u and �τ at the canopy top. Moreover, Uc = s2 Lc H/2(U 20 /Uh) and

UI 0 = s H/2(U 20 /Ui ) are the inside-canopy and above-canopy scaling velocities, where Ui

and U0 are the characteristic velocities in the inner (Li ) and outer layer (Lm) depths (seeFinnigan and Belcher 2004; Poggi et al. 2008), and s = π/(2L) is the hill wavelength (seeTable 1). The mean fluid continuity equation and �u can now be used to derive �w byimposing the boundary-condition �w(x,−hc) = 0. The resulting formulation is given by:

�w =

⎡⎢⎢⎢⎣

H f sUc

((hc + z)

√u∗

Ucβ−2i Ace

− hc2Lcβ2

(−1 + e

hc+z2Lcβ2

)Lcβ

2)

+ (1−H f )(1

L ln[

Lizo

]UI 0

(− Ai Ar

2 Li B1+isL

(d ln[d+z]+ z

(Ln

[zo(d+z)

L2i

]−2

)))+ C

)⎤⎥⎥⎥⎦eisx .

(15)

Moreover, the eddy viscosity can be derived analytically and is given by

Kt ={

H f

[4Ace

zβl f f LcUcβ

4]

+ (1 − H f )

[k2(d + z) (Ai Ar B1 − 2)

ln [Li/zo]

]}eisx . (16)

The unknown coefficients in the analytical solutions can be derived by matching the solutionsfor the velocity and stress above and inside the canopy to yield:

Ac =−kv

UI 0Uc

(B0 + Ar

2 B1

(1 − ln

[dLi

]+

(1 − i Uc

UI 0

√u∗

Ucβ

)ln

[Lizo

]))(β B0 + kv

Ar2 B1

)ln

[Lizo

] , (17)

123


Ai =(kv + β) − β ln

[dLi

]+

(1 − i Uc

UI 0

√u∗

Ucβ

)β ln

[Lizo

](β B0 + kv

Ar2 B1

) , (18)

C = UI 0

L

(Ai Ar B1Li

ln [Li/zo]+ s Hc L

Uc

UI 0

√u∗

Ucβ

)

− isUI 0

(d ln[d]

ln [Li/zo]+ 2Ac Lc

Uc

UI 0β2

(1 − e

− Hcβle f f

)). (19)

Appendix B: Basic Formulations for the Diffusion Coefficients and Settling Velocity

B.1 The Diffusion Terms

The molecular diffusion term is standard and is given as (Seinfeld and Pandis 1998)

Dp,m = kB T

3πµdpCc, (20)

where kB = 1.38×10−23J K−1 is the Boltzmann constant, T is the absolute temperature, andµ = ρν is the dynamic viscosity of the air, where ρ and ν are the air density and kinematicviscosity, respectively. Cc is the Cunningham coefficient given as:

Cc = 1 + λ

dp

(2.514 + 0.8 exp

(−dp

λ

)), (21)

where λ is the mean free length of air molecules (= 0.066 µm at standard temperature andpressure). The Knudsen number K n = 2λ/dp defines the nature of the suspending fluid (airhere) to the particle size. When dp � λ or K n → 0, Cc → 1, and the standard formulationfor molecular diffusion is recovered.

The particle diffusivity is primarily dominated by the turbulent eddy viscosity and is givenas (Csanady 1963; Wilson 2000)

Dp,t

Kt=

(1 + τp

τ

)−1, (22)

where Kt is the eddy viscosity of the flow, and for clarity, is repeated here and is given as

Kt = l2m

∣∣∣∣∂u

∂z

∣∣∣∣ , (23)

and τp is the particle time scale given by

τp = ρpd2p

18µCc, (24)

where the Lagrangian turbulent time scale (τ ) is given as

τ = Kt

σ 2w

, (25)

and where σw is the turbulent vertical velocity standard deviation. For small aerosol sizedparticles in the nanometer to micrometer diameter range,τp/τ � 1, and Dp,t ≈ Kt resultingin a turbulent Schmidt number of unity.

123


B.2 The Settling Velocity

When the particle Reynolds number, Red = Vs dpν

≤ 1, the settling velocity for a sphericalparticle is given as

Vs = Cc(ρp − ρ)

ρ

gd2p

18ν, (26)

where ρp is the particle density (assumed here to be ≈1,500 kg m−3), and g = 9.81 m s−2

is the gravitational acceleration. It was assumed that the drag coefficient exerted by the fluidon the particle is given by Stoke’s equation (CD,p = 24/Red). The Vs formulation can berevised to include non-linearities in CD,p due to higher Reynolds numbers as

Vs =[

Cc4

3

(ρp − ρ)

ρ

gdp

CD,p

]1/2

, (27a)

CD,p ={

24/Red ; Red ≤ 1(24/Red)(1 + 0.15Re2/3

d ); 1 < Red < 1000.(27b)

Note here that Vs must be solved numerically when 1 < Red < 1000.

Appendix C: The Turbo-Phoretic Effect

Here, the incorporation of turbo-phoresis into the vegetation particle collection term isdescribed. The role of turbo-phoresis is gaining some attention in single-layer dry depo-sition models of the atmosphere-vegetation system as recently evidenced in Feng (2008).Moreover, turbo-phoresis has been used to refine models of particle deposition onto walls ofducts and was shown to improve the agreement between models and measurements (Zhaoand Wu 2006). The turbo-phoretic velocity can be approximated by (Caporaloni et al. 1975;Reeks 1983; Guha 1997; Young and Leeming 1997)

Vt = −τpdσ 2

w,p

dz, (28)

where,

σ 2w,p

σ 2w

=(

1 + τp

τ

)−1. (29)

The problem here is the choice of the appropriate length scale over which ∂σ 2w/∂z needs to

be determined in the particle collection mechanism. When treating the foliage as an equiva-lent rigid single-sided surface characterized by a thin quasi-laminar boundary pinned to thesefoliage elements, ∂σw

∂z ∼ σw

δ, where δ is a quasi-laminar boundary-layer thickness over which

the quantity σw → 0. That is, in a turbulent region above this thin layer, ∂σw/∂z ≈ 0 andonly within the viscous region does σw rapidly diminish.

In typical boundary-layer flows over smooth surfaces (i.e. the viscous sub-layer is muchthicker than the height of the roughness elements), the transition between this viscous bound-ary layer and the turbulent regime occurs at around z+ = u∗δ/ν ≈ b, where b = 5 − 50(depending on the flow statistic being analysed). Measurements of smooth channel-flow datasuggests b ≈ 25 (Poggi et al. 2002; see the data in their Fig. 11, but the vertical velocitykurtosis measurements in their Fig. 14 is more robust in offering a reasonable ‘threshold’

123


value for b). With this parameterization, Vt can be determined provided σ 2w(x, z) is known

assuming b ≈ 25 for smooth surfaces (i.e. neglecting the leaf surface micro-roughness).The problem then is determining σ 2

w(x, z) as this quantity is not modelled via first-orderclosure schemes and requires a more elaborate second-order closure scheme to compute.In lieu of this approach, we take advantage of the published flume experimental data for acanopy on a hill in Poggi and Katul (2007a) that suggest, when σw is normalized by the localfriction velocity at z = 0 and position x , these profiles collapse reasonably well (see theirFig. 5). Moreover, these normalized σw profiles appear to be quasi-linear inside the canopyand almost constant above the canopy. Hence, for the purposes of estimating σ 2

w for Vt (x, z),we assumed that

σw(x, z) = u∗(x)

[1.25 + (

1 − H f (z, hc)) z

hc

], (30)

where u∗(x) =√

−u′w′(x, 0) is computed from FB04.

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