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

Transcription

1 Boundary-Layer Meteorol 2010) 135:67 88 DOI /s ARTICLE The Influence of Hilly Terrain on Aerosol-Sized Particle Deposition 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 Abstract Virtually all reviews dealing with aerosol-sized particle deposition onto forested ecosystems stress the significance of topographic variations, yet only a handful of studies considered the effects of these variations on the deposition velocity V d ). Here, the interplay between the foliage collection mechanisms within a dense canopy for different particle sizes and the flow dynamics for a neutrally stratified boundary layer on a gentle and repeating cosine hill are considered. In particular, how topography alters the spatial structure of V d and its two constitutive components, particle fluxes and particle mean concentration within and 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 principles that explicitly accounts for i) the flow dynamics, including the two advective terms, ii) the spatial variation in turbulent viscosity, and iii) the three foliage collection mechanisms that include Brownian diffusion, turbo-phoresis, and inertial impaction is developed and used. The model calculations suggest that, individually, the advective terms can be large just above the canopy and comparable to the canopy collection mechanisms in magnitude but tend to be opposite to each other in sign. Moreover, these two advective terms are not precisely out of phase with each other, and hence, do not readily cancel each other upon averaging across the hill wavelength. For the larger aerosol-sized particles, differences between flat-terrain and hill-averaged V d can be significant, especially in the layers just above the canopy. We also 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 , USA gaby@duke.edu G. G. Katul Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University, Durham, NC 27708, USA G. G. Katul Department of Physics, University of Helsinki, Helsinki, Finland D. Poggi Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, Torino, Italy

2 68 G. G. Katul, D. Poggi with the topography in the canopy sublayer, play a significant role in explaining variations in V d across the hill near the canopy top. Just after the hill summit, the model results suggest that V d fell to 30% of its flat terrain value for particle sizes in the range of 1 10 µm. This reduction appears consistent with maximum reductions reported in wind-tunnel experiments for 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 V d ) of aerosolsized particles on forested ecosystems primarily rely on stationary and planar homogeneous flow assumptions. As such, these micrometeorological measurements and V d formulations cannot account for the effects of local variations in topography. Hence, it is not surprising that virtually all recent reviews describing the various V d formulations acknowledge that the effects of topographic variability on the measuring and modelling of V d remain a major knowledge 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 of the order of magnitude estimate of how heterogeneity in terrain might affect the spatiallyaveraged V d, and highlighted recent advances in aircraft measurements that may be used to generate the requisite spatially-averaged datasets to address this problem. Topography perturbs the mean and turbulent flow fields that transport particles onto foliage sites; it also alters the immediate micro-climate including mean air temperature and humidity fields Raupach et al. 1992) in a manner that may modify particle coagulation or condensation processes. At longer time scales, the impact of topography on processes pertinent to dry deposition is quite extensive and includes canopy structural and morphological changes due 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 and its concomitant moisture, to name a few). Because of these numerous interactions and nonlinear feedbacks amongst all these processes, the way that topography alters V d for forested ecosystems 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 complex 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) proposed a one-dimensional approach that explicitly accounts for the balance between the total flux gradient and the vegetation collection mechanisms inside canopies as was done earlier by Slinn 1982), and discussed how various representations of these collection mechanisms can be formulated as closure models to terms arising from a multi-phase volume averaging operator applied to the continuity equation. Building on these separate advances, a natural starting point is to consider the interplay between the topographically induced advective terms, canopy turbulent transport processes and aspects of the foliage collection mechanisms commontomanyv d models. Condensation and coagulation processes, as well as any spatial variation in the collection mechanisms not introduced by spatial variations endogenous to the flow field, are ignored. To progress even within this restricted scope, many of the governing processes must be simplified and parameterized.

3 Complex Terrain and Aerosol-Sized Particle Deposition 69 The work here takes advantage of recent theoretical, numerical and experimental efforts on quantifying flows inside canopies situated over gentle hilly terrain Finnigan and Belcher 2004; Ross and Vosper 2005; Katul et al. 2006; Poggi and Katul 2007a,b,c, 2008a; Poggi et al. 2007, 2008). The primary goal is to examine biases that occur in determining V d values when assumptions appropriate to flat terrain are applied to measurements or models over forests on gentle hills. In particular, we seek to quantify where on the hill surface the advective terms tend to significantly disrupt the balance between the total flux gradient and the particle collection mechanism by the vegetation. This balance underpins much of the forest-atmosphere V d micrometeorological measurements and the multilayer models in use e.g. Slinn 1982; Petroff et al. 2008b; Pryor et al. 2008). This comparison is of particular significance to estimating aerosol-sized particle deposition rates onto forested ecosystems using micrometeorological methods. Given that these forested canopies are often situated on complex terrain, micrometeorological measurements unavoidably must be conducted in the canopy sublayer CSL) for such tall canopies e.g. Gallagher et al. 1997; Grönholm et al. 2007, 2009; Rannik et al. 2009). 2Theory 2.1 Overview of the Problem Set-Up As a case study for presenting the model calculations, the canopy is assumed to be uniform of height h c 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 h c,which is perhaps the most dynamically interesting case. When H h c, then these topographic variations are likely to be too small and can be ignored provided the topographic variations are assumed to be gentle. On the other hand, if H h c, then the problem is entirely pinned to the large mean pressure gradients because the differences between the turbulent stress gradient and the drag force inside the canopy become much smaller than the externally imposed horizontal mean pressure gradient. Hence, the internal flow dynamics within the canopy volume become less relevant for such conditions. For these reasons, we selected H = h c as the basis of our work here. For keeping the results as generic as possible and not tied to a particular foliage configuration or experimental site properties, it is also assumed that the canopy leaf area density is sufficiently dense and constant with depth. Above the hill-canopy system 3h c ), the atmosphere is assumed to be an infinite supply of aerosol sized particles of all sizes) that are transported by the flow and then collected by the foliage elements. All particles are assumed to have reached their equilibrium diameter d p ), and this equilibrium d p is no longer evolving on time scales pertinent to turbulent processes or their Reynolds average. To keep the parameters that may introduce variability in V d to a bare-minimum, the ground deposition is entirely neglected when compared to the integrated canopy collection mechanisms, though we are well aware that forest floor deposition rates can account for some 20% of the total deposition on the soil-canopy system Donat and Ruck 1999; Grönholm et al. 2009). Theflow field 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 and turbulent stress measurements collected in a flume in which the canopy was composed of densely arrayed rods and situated on repeated cosine hill modules Poggi et al. 2008). The agreement between model calculations and measurements was reasonable even though the measurements were carried out in a dynamical regime that does not conform entirely to

4 70 G. G. Katul, D. Poggi Fig. 1 Top left: The hill-canopy set-up and the origin of the coordinate system used. The hill height is H,the total length of the domain is 4L, wherel is the hill half-length, the canopy height h c = H, and the canopy top is shown by the dashed green line. The origin of the coordinate system x = 0, z = 0) is situated at the hill summit and the canopy top. The remaining panels show the two-dimensional variations of the mean longitudinal velocity labelled as u, ms 1 ), mean vertical velocity labelled as w, ms 1 ), turbulent shear stress labelled as uw, m 2 s 2 ), vertical velocity standard deviation labelled as σ w, ms 1 ), and eddy viscosity labelled as K t, m 2 s 1 ) obtained from FB04 see Appendix A for formulation) the FB04 assumptions see Fig. 1 in Poggi et al for a discussion). The FB04 model is a first-order two-dimensional closure for turbulent flow over and within a tall canopy on a low 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 assumed to be vertically uniform due to the hydrostatic assumption see Poggi and Katul 2007c for a discussion and experimental support for this assumption). The mean flow responds first to this vertically uniform pressure perturbation by generating horizontal and vertical gradients in 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 turbulent diffusivity for the size-resolved mean scalar continuity equation for the particles. The variable mean wind field also introduces both horizontal and vertical variations in the turbulent shear stress and the vertical velocity standard deviation thereby affecting the particle collection mechanisms due to the vegetation. To assess the relative importance of these hill-induced processes on particle concentration, canopy sources and sinks, flux distributions, and V d across the hill, all model results will be compared to the case of uniform flat terrain covered with a canopy having the same aerodynamic, geometric, and particle collection attributes.

5 Complex Terrain and Aerosol-Sized Particle Deposition 71 It is not our intent to model the absolute values of the fluxes and collection mechanisms but rather to quantify the relative magnitudes and the signs of the perturbations in these variables as introduced by gentle topographic variations. Hence, all calculations are referenced to this flat-terrain case scenario. Because the particle diameter can appreciably affect the vegetation collection mechanisms, we consider five particle-size classes ranging from 1 nm to 10 µm in factors of 10 increments. For these particle-size classes, the collection mechanisms that are significant 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 terrain have proliferated exponentially over the past 35 years following the pioneering work of Jackson and Hunt 1975). A partial list of diverse laboratory and field experiments and the wealth 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 Castro 2001; 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 not reviewed or repeated here. As mentioned by Katul et al. 2006), earlier models of airflow over hills of low slope = H/L, wherel is the half-length of the hill, Fig. 1) focused on momentum and scalar transfer 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 canopy layer, the two-dimensional mean continuity and momentum balance for stationary, high Reynolds number and neutrally stratified turbulent flows are given by FB04, u = 0, 1) u u x + w u z = 1 p ρ x x + w z u u x + u w z ) F d 1 H f z, h c ) ), 2) where x and z are defined in a hypothetical steamline or displaced) coordinate system set by the topographic shape and defined in FB04. This system reduces to terrain-following near the ground and rectangular Cartesian well above the hill surface e.g. in the middle layer). If the rectangular Cartesian system well above the hill is defined with X being the horizontal and Z being the vertical coordinates, then x = X + H/2) sin kx) exp kz + h c )) and z = Z H/2) cos kx) exp kz + h c )) with k = π/2l,andwhereu and w are the timeaveraged 1 velocities in the x and z directions, respectively, u and w are the corresponding turbulent velocity fluctuations, respectively, such that u = w = 0, p x, z) is the mean static pressure, and F d is the canopy drag exerted by the vegetation on the air flow, which is parameterized as, 1 Within the canopy, volume averaging over thin slabs containing many foliage elements is necessary to remove the large point-to-point variations Brunet et al. 1994). The extra dispersive flux terms arising from the volume averaging are lumped in the Reynolds stresses. Two experimental studies suggest that these dispersive fluxes are quite small when compared to the Reynolds stresses inside dense canopies Poggi et al. 2004b; Poggi and Katul 2008b).

6 72 G. G. Katul, D. Poggi F d = C d au 2, 3) in which C d is the dimensionless drag coefficient and az) is the foliage frontal area per unit volume. The origin is taken as the cross-stream coordinate z at the top of the canopy with x = 0 being the top of the hill see Fig. 1). The term H f is the Heaviside step function defined by { 1; z > 0 H f = 0; h c < z 0. 4) Using first-order closure principles, it is possible to derive analytical solutions for u, w and u w caused by the hill for the case of constant C d a and sufficiently small H/L so that the equations can be linearized and u u / x can be neglected. Appendix A presents the analytical solution for the mean flow field and the eddy viscosity used to drive the particle mean scalar conservation. 2.3 Scalar Mass Conservation For steady state conditions, the conservation of the mean concentration c of particles of size d p is given by, u c x + w c z = Fc z + F ) cx + S c 1 H f z, h c ) ) 5) x where F c and F cx are, respectively, the cross-stream and streamwise total fluxes of aerosol sized particles, and S c 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, whichis presented in Appendix A. Furthermore, model calculations in Katul et al. 2006) suggestthat for scalars in which the canopy is a major source or sink, F c z F cx x, and hereafter, this streamwise flux gradient term is ignored relative to its vertical counterpart. To close this problem mathematically, it is necessary that S c and F c be made dependent on c. Adopting first-order closure principles for the particle turbulent flux see reviews in Sehmel 1980; Slinn 1982; Pryor et al. 2008; Petroff et al. 2008b) yields F c x, z) ) cx, z) D p,m + D p,t x, z) V s cx, z), 6) z where D p,t and D p,m are the particle turbulent and Brownian diffusivities, respectively, and V s is the settling velocity Petroff et al. 2008b; Pryor et al. 2008). Limitations of gradient-diffusion approximations inside canopies are well-known Shaw 1977; Denmead and Bradley 1985; Finnigan 1985, 2000); however, these limitations may be less critical when assessing the spatial structure of the hill-induced perturbations from a mean background state for reasons discussed in Finnigan and Belcher 2004). For completeness, the two diffusivities and V s formulations are presented in Appendix B. The spatial variation in the deposition velocity can be readily computed from V d x, z) = F c x, z)/cx, z). The vegetation collection mechanisms are assumed to occur within a quasi-laminar boundary layer adjacent to the leaf surface, and are given as ) 2a cx, z) S c x, z) = π r b 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 r b is the local quasi-laminar boundary-

7 Complex Terrain and Aerosol-Sized Particle Deposition 73 layer resistance for particles of diameter d p given as Seinfeld and Pandis 1998; Wesely 1989): 1 r b x, z) = u w x, z) Sc 2/ /Stx,z)) + V t x, z)), 8) where Sc = ν/d m,p is the Schmidt number, St = V s u w x, z))/gν) is a turbulent Stokes number, and V t x, z) is the turbo-phoretic velocity. The inertial impaction term in r b is parameterized as 10 3/St, which is based on Slinn and Slinn 1980) formulation for water or smooth surfaces. This formulation was shown to be reasonably accurate for smooth boundary layers Aluko and Noll 2006). When such a formulation is applied to the entire canopy-soil system or a rough surface), then this inertial-impaction formulation underestimates depositional velocities e.g. see Discussion in Feng 2008). However, in a vertically resolved model of the foliage, it is not clear to what degree and precisely how the microroughness of the foliage alters this formulation given that the inertial impaction must be applicable to isolated leaf surfaces. Hence, for simplicity and consistency with formulations for the turbo-phoretic term see Appendix C), the quasi-laminar boundary layer over isolated leaf surfaces is assumed to be dynamically similar to a smooth boundary layer. Stomatal uptake, sedimentation, interception, rebound, and other phoretic processes are all ignored. We retained V t x, z) mainly because of its strong dependence on the flow dynamics; its estimation is presented in Appendix C. 3 Results and Discussion Before discussing how gentle topography affects particle deposition processes onto the canopy, 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 made to five regions: upwind region above the canopy x/l < 0, 0 < z/h c < 1); downwind region above the canopy x/l > 0, 0 < z/h c < 1); upwind region inside the canopy x/l < 0, 1 < z/h c < 0); downwind region inside the canopy x/l > 0, 1 < z/h c < 0); and the region in the vicinity of the hill summit x/l 0, z/h c 0). In some cases, the within-canopy region is further decomposed into two parts: the upper 0.3 < z/h c < 0) and lower canopy 1 < z/h c < 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, which are 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 mechanisms, iii) σ w, which is needed for computing the turbo-phoretic velocity see Appendix B), and iv) the eddy viscosity or diffusivity K t ), which is needed for computing the scalar turbulent 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 thereafter with increasing x. On the lee-side x/l 1.1), the flow experiences a negative u inside 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 gradient, though advection can modify its spatial extent Poggi et al. 2008). Note that, in the

8 74 G. G. Katul, D. Poggi absence of a canopy, this recirculation zone does not exist because the topography here is sufficiently gentle not to induce flow separation see Poggi et al. 2007; Poggi and Katul 2007a,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 negative velocity excursions in the lower layers of the canopy Poggi and Katul 2007b). It is worth pointing out here that the mixing length, when experimentally inferred from measured u w and u even in this recirculation zone, appeared to be nearly constant with depth and consistent with the FB04 assumption Poggiand Katul 2007b). Finally, because the vertical turbulent diffusivity is finite in this zone, passive scalar mass can diffuse into this recirculation zone, accumulate, and then eject out. This accumulation-ejection phase has been shown experimentally to be quasi-periodic in nature and contribute significantly to the local flux Poggi and Katul 2007b). For the case of aerosol-sized particles, the vegetation collection mechanism may entirely alter this accumulation phase, as aerosolsized particles diffusing into this zone may become partially trapped, thereby allowing the vegetation collection mechanism to remove them from the mean flow. The maximum u above the canopy occurs just in front of the hillcrest x 0) and is almost out-of-phase with the maxima in w. The fact that u and w are nearly but not precisely out of phase with each other due to nonlinearities in the mean momentum equation) has important implications 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 magnitude throughout the hill thereby preventing the deeper foliage 50% of the total leaf area) from being efficient deposition sites due to the large r b ). In the upper layers of the canopy, the minimum shear stress magnitude occurs near the hilltop x 0). Thisstressisusually out-of-phase with the topography. Because of a tight coupling between this stress and r b, the vegetation collection mechanism may be again less efficient at removing particles in this location. The implications of the reduction in collection efficiency on V d near the origin 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 d p = 1 nm. For this small d p,the vegetation collection mechanism in S c ) is primarily governed by Brownian diffusion. From Fig. 2, the large variations and the richness in the spatial patterns of normalized deposition velocities are rather striking. Gentle hills with a mean slope of 10% = H/L) can introduce spatial variations in V d that range from nearly 10 to 100%. Both mean scalar concentration and particle fluxes are altered by the canopy-hill system relative to the flat terrain); however, the emerging spatial patterns in V d appear to be more controlled by the fluxes a result already foreshadowed by Hicks 2008 for complex terrain). Moreover, the spatial variability in V d 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 concentration are 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/h c < 0), the deposition velocity is generally enhanced relative to its flatworld counterpart. However, progressing on the upwind side further 1 < x/l < 0), there is a decline in V d relative to its flat-terrain value, thereafter reaching a minimum just after the hill summit x/l 0.3).

9 Complex Terrain and Aerosol-Sized Particle Deposition 75 Fig. 2 Left: The hill-induced two-dimensional variations in normalized deposition velocity V d ) for d p = 1 nm, particle concentration labelled as C), and total vertical flux labelled as F c ). These flow quantities are normalized by their flat-terrain counterparts indicated by the subscript). Right: Comparison between the averaged quantities across the hill wavelength shown in the left panels denoted by and presented as solid lines) and the flat-terrain case dashed lines). The canopy top is also shown as a thin horizontal dashed line for reference 2. Within the middle layers of the canopy z/h c 0.5), V d is enhanced on the upwind side 2 < x/l < 0.5), but is then reduced to levels below its flat-terrain counterpart just after the hill summit x/l 0.1). The recirculation zone on the lee-side inside the canopy appears to leave a complex fingerprint inside the canopy, with enhanced deposition after the hilltop x/l ), followed by a reduced deposition on the remaining part of the lee-side 0.8 < x/l < 1.8), and then a recovery to flat-terrain values. We note that this pattern is strictly endogenous to the canopy-flow system because the ground deposition flux was set to zero throughout. 3. Above the canopy and on the upwind side, the deposition velocity is reduced. This pattern 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 due to the nature of the spatially constant mean concentration = C o ) imposed as an upper boundary condition at z/h c = 2.

10 76 G. G. Katul, D. Poggi While the analysis in Fig. 2 demonstrates by how much and where on the hill surface V d is reduced or enhanced, it remains to be seen whether these changes actually cancel each other out when a spatial average along the hill wavelength is computed. Again, recent advances in airborne aerosol-sized particle fluxes and concentration measurements can now sample such spatially-averaged quantities Hicks 2008). This averaging question is further explored by determining V d, c,and F c for the hilly case and comparing them to the flat-world model calculations. Hereafter, angular brackets denote a hill-averaged quantity i.e. ξx, z) = 1 4L 2L 2L ξx, z)dx). Figure 2 presents the outcome of this comparison for V d and its two constitutive terms. Inside the canopy, the hill-induced perturbations appear to nearly cancel each other for fluxes, concentrations, and deposition velocities at least for the smallest d p selected here). Near the canopy top, the hill-induced perturbations tend to reduce the overall deposition velocity by only 3%. Above the canopy, the advective terms do not entirely average 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 strong asymmetry in the flow dynamics, the advective terms did not entirely average out above the canopy. This last finding prompted further exploration of the individual components of the advective terms and the spatial variations of the remaining components of the mean continuity equation. We use again the d p = 1 nm as a case study to illustrate the two-dimensional variations in these terms. Figure 3 displays the two-dimensional spatial variations of the two advective terms, the flux gradients, and the canopy collection mechanism mainly due to Brownian diffusion). For reference, the computed V d 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-circulation zone 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 magnitude to the flux gradient and canopy collection mechanism. However, because they are not exactly out-of-phase with each other, they contribute to a non-zero flux gradient upon spatial averaging 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 upwind side, the canopy collection mechanisms are first enhanced due to a large enhancement in the magnitude of u w see Fig. 1), which in turn increases S c and c/ z near the canopy top. In this zone, c/ x has not been fully established P/ x is beginning to increase from a near-zero value here). When this enhancement in c/ z is subjected to w<0seefig.1), the vertical 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 orders of magnitude faster than the particle settling velocity). Near the canopy top and at the hill summit x 0, z 0), the canopy collection mechanism is reduced due to a reduction in 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 also increased 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 partially balancing the horizontal advection) still conspire to reduce the overall deposition velocity near the coordinate origin. Near the hill summit x/l = 0) and progressing down on the lee-side, the magnitude of u w increases again with increasing x see Fig. 1) and the canopy collection mechanisms regain their efficiency in the upper canopy layers, the vertical velocity is also reduced and then reverses from positive to negative in sign, and the deposition process re-establishes itself to near-background values by x/l = 1Fig.2). For larger particles, the

11 Complex Terrain and Aerosol-Sized Particle Deposition 77 Fig. 3 The two-dimensional variations of the components of the particle mean continuity equation for d p = 1 nm. The left panels show the individual advective terms and their sum zero for the flat-terrain case). The right panels show the flux gradient middle) and the particle collection bottom) mechanism their sum is balanced by the sum of the advection terms in the left panel). For reference, modelled V d in m s 1 ) is also shown top-right) deposition is enhanced beyond its background state by 50%, as we show later, especially near the canopy top. 3. Flux gradient: Above the canopy, the particle flux gradient is controlled by the relative importance of the horizontal and vertical advection terms recall that S c = 0 above the canopy). 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 terms do not entirely cancel out above the canopy see Fig. 2). The same analysis was repeated for d p = 10 nm, 100 nm, 1 µm and 10 µm, and the results appear to be qualitatively the same in terms of the spatial variability of the advective terms and canopy sinks. However, quantitatively, significant differences emerge and we summarize these differences in Fig. 4, which compares V d at each height z with its flatterrain counterpart for all five d p classes. For the larger particles, turbo-phoresis and inertial impaction significantly contributes to S c vis-à-vis Brownian motion). The V t value is sensitive to the spatial variability in σ w see Fig. 1) and the inertial-impaction term exhibits a

12 78 G. G. Katul, D. Poggi Fig. 4 Comparison between the flat-terrain and hill-averaged V d labelled as V d ) cases for all five particle sizes d p ) and for all z values. The 1:1 line is also shown for reference. Note that the major departures from the 1:1 line are for the larger d p 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 mechanisms, does not entirely average out and can lead to V d diverging from its flat-terrain counterpart see Fig. 4) depending on z. As noted in the Introduction, single-tower micrometeorological measurements of V d above tall 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 connection between these biases and the hill shape for a reference region in the CSL, Fig. 5 presents variations in V d x, 0) i.e. at the canopy top) normalized by its flat-terrain counterpart as a function of x for each of the five d p classes. For all d p values, the spatial patterns in deposition velocities 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 the hill summit x/l 0.3), followed by a recovery and then enhancement towards the mid section on the lee-side of the hill. It is also evident that the hill has a much more appreciable effect on deposition velocity when the inertial impaction term and turbo-phoresis dominate the collection mechanisms consistent with the analysis in Fig. 4). It is interesting to compare these model findings with a comprehensive wind-tunnel deposition study conducted by Parker and Kinnersley 2004) on an isolated pyramidal ridge with length-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 and 10% were in the size range µm in diameter. The V d measurements were conducted by collecting particles on filters placed on the ground surface. For this wind-tunnel experiment, there was a slight decrease in the deposition velocity upwind of the raised topography;

13 Complex Terrain and Aerosol-Sized Particle Deposition 79 Fig. 5 The modelled effects of particle diameter d p ) on the variability of the normalized V d x, 0) across the hill at the canopy top. The normalization uses V d, flat x, 0) as inferred for a flat surface for the same canopy and d p. For reference, the wind-tunnel WT) measurements of normalized V d for an isolated gentle pyramidal ridge are shown as symbols in the bottom panel for d p = µm. The wind-tunnel data here are not intended for a one-to-one comparison as the wind-tunnel experiments do not include a canopy effect. They are presented only for reference a maximum in deposition on the upwind face close to the peak; a region of decreased deposition on the leeward face, extending into the immediate wake of the obstacle, and gradually recovering with distance to upwind levels at the end of the pyramidal structure. Perhaps more intriguing in Fig. 5 is that, for the cosine hill-canopy system, variations in d p have more of an effect on the relative variation in deposition velocity across the hill when compared to the precise shape of the topography and the presence or absence of a canopy. Also, whether for the pyramidal structure with no canopy or the cosine hill structure with a tall canopy, the maximum decline in deposition velocity relative to the flat-terrain case appears to occur just in the vicinity of the maximum topographic excursion and this decline appears to be highly sensitive to d p. 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 V d for heavier particles, given that inertial impaction becomes the dominant particle collection mechanism inside the canopy, and this term decreases as a power law with decreasing u w. Even if other formulations

14 80 G. G. Katul, D. Poggi Fig. 6 Variations in Rsd p ) = 4L 1 2L 0 hc 2L S c x, z)dzdx, normalized by its flat-terrain counterpart as a function of particle diameter d p ). For reference, the domain-averaged V d, normalized by its flat-terrain counterpart is also shown, where Rvd p ) = 4L 1 2L 2hc 2L h V c d x, z)dzdx. 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 in a non-linear manner with the turbulent Stokes number and u w ), and hence the findings here on the reduction-enhancement patterns in V d x, 0) across the hill and its amplification with increasing d p is likely to hold for those formulations as well. To further assess the effects of the hill on the domain-integrated particle collection processes by the canopy, we consider the normalized variable given by 0 hc S cx, z)dz Rsd p ) = 0 h c S c,flat z)dz, 9) where is, as before, averaging across the hill wavelength L. Figure 6 presents the variations of Rs for the five diameter classes considered here. Not surprisingly, when d p is small and the collection mechanism is controlled by Brownian diffusion, the effects of the gentle hill on the integrated canopy particle removal rate is small <3%). For larger d p, the effects appear to be on the order of H/L. Hence, this finding suggests that, when model calculations are employed in assessing the overall rate of particle removal by the vegetation, gentle topographic variations on the vegetation sink term are not large <15%) and can be ignored for d p 10 µm. However, this finding cannot be extrapolated to the problem of inferring S c from micrometeorological flux measurements at a single tower. The analysis here e.g. Fig. 2) unambiguously shows that when inferring 0 h o S c z)dz at a single tower location, gentle topographic effects can be large even for the finest diameter. Moreover, for larger d p, the longitudinal variations in F c x, 0) or V d x, 0)) across the hill can far exceed the

15 Complex Terrain and Aerosol-Sized Particle Deposition 81 magnitude of the random flux error reported in Rannik et al. 2009) at a single location above the canopy, and hence, ought to be statistically discernable. 4 Conclusions The primary goal here was to examine biases that occur in determining V d values when assumptions appropriate to flat terrain are applied to measurements or models over forests situated on gentle hills. We focused on how the two advective terms disrupt the balance between the turbulent-flux gradient and the particle collection mechanism by the vegetation for a gentle cosine hill in which the hill height is comparable to the canopy height. We found that in the case of models, inferring 0 h c S c z)dz using flat-world assumptions is reasonable, and differences between the hill-averaged 0 h c S c x, z)dz and flat-world 0 h c S c z)dz are small, at least not exceeding H/L. However, estimating 0 h c S c z)dz from single tower-based micrometeorological flux measurements using flat-world assumptions is far more problematic. The model calculations here suggest that, individually, the advective terms are large but tend to be opposite in sign. 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 and vertical particle advective terms that, i) mirror those of u andw in their phase relationships, and ii) appear comparable in magnitude to the canopy collection or flux gradient terms for the layers above the canopy. Moreover, the two-dimensional spatial variations in u leads to spatial variations in u w that are in phase with u/ z, which is a minimum near the hill summit and nearly out of phase with the topography. These u w variations, along with particle sizes, dictate the spatial variation of the particle collection efficiency of the vegetation, especially in the upper canopy layers. A reduced u w and a positive vertical velocity near the canopy top conspire to appreciably diminish V d just after the hill summit. These reductions in V d can be a factor of 3 relative to their flat-terrain case depending on d p. Much of the work here focused on a cosine hill, and the logical follow-up question is how to account for real-world topographic variations when estimating 0 h c S c z)dz using micrometeorological flux measurements from a single tower. Given the recent advances in canopy Lidar systems e.g. see the review in Lefsky etal. 2002) from which the topographic variations and canopy leaf area distribution can now be simultaneously resolved, theoretical approaches that take advantage of such data must be developed to begin relaxing the flatworld assumptions. To be clear, the approach proposed here does not offer finality to this problem, but may provide a blueprint on how to proceed. First, as was the case with field studies and model calculations for CO 2 e.g. Feigenwinter et al. 2004; Katul et al. 2006), the model results here suggest that accounting for one of the two advective terms but ignoring the other may prove to be more problematic than ignoring both. Hence, some accounting of both advective terms is needed at least in a uniform canopy in the absence of density stratification). Second, for gentle topographic variations, the linearized pressure perturbations in FB04 can be superimposed. It is conceivable that some of the real-world topographic perturbations under consideration, if they are not too large relative to the canopy height, are amenable to being spectrally decomposed, and pressure perturbations be solved for each Fourier amplitude and wavelength. The linearity of the modified FB04 permits us to superimpose all these pressure perturbations together and analytically generate a realistic u and w that are in equilibrium with the complex topography. Such knowledge might allow some evaluation

16 82 G. G. Katul, D. Poggi of the first-order effects of particle advection on micrometeorological measurements of V d 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 all the helpful discussions that motivated this work. G. Katul acknowledges the support from the Department of Physics at University of Helsinki during his three month visit from Duke University, from the National Science Foundation Grants NSF-EAR , NSF-EAR , and NSF-ATM ), and the Binational Agricultural Research and Development BARD) Grant IS ) for the development of the canopy turbulence aspects of this work. D. Poggi acknowledges travel support from the European Cooperation in the field 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 be decomposed into an unperturbed or background equilibrium) state and a perturbation induced by topographic variations. Such decomposition allows tracking how the hill system modifies the flat-terrain solution. Mathematically, this decomposition results in ux, z) = U b z) + ux, z), wx, z) = wx, z) and u w x, z) = τ b z) + τx, z). The subscript b and the symbol indicate background and the hill-induced perturbations, respectively. The background mean velocity and turbulent stress, U b z) and τ b z), are modelled using a combination of logarithmic and exponential profiles, given by: U b z) u τ b z) u 2 = H f = H f [e [ 1 k v ln ] zβ l ef f z + hc d z o )] + 1 H f ) [ 1 β eβz/l ef f ], 10) + 1 H f ), 11) where u is the background friction velocity at the canopy top, d and z o are the zero-plane displacement and aerodynamic roughness length of the canopy, respectively, k v = 0.4isthe von Karman constant, U h is the mean velocity at the canopy top, β = u /U h is the dimensionless momentum flux through the canopy, and l ef f is a characteristic turbulent mixing length, equal to k v z + d) above the canopy and a constant 2β 3 L c inside the canopy, where L c = C d a) 1 is the adjustment length scale Belcher et al. 2003; Katul et al. 2004; Poggi et al. 2004a). Moreover, imposing the continuity constraints on U b z) and τ b z) at the canopy top results in z o = 2L cβ 3 e kv β, 12a) k v d = 2L cβ 3. 12b) k v The analytical solution to the mean momentum equations within the inner and canopy layers results in see Poggi et al. 2008): )] zβ l u u = {H f [U c A c e ef f + i U c β 1 Ai B 0 + ln +1 H f ) [U [ Li 2/d + z)z o) ]) ]} I 0 e isx, 13) { τ = u H f [2U c A c βe ] zβ l ef f ln [L i /z o ] + 1 H f ) [ ]} k v 2 + A i A r B 1 ) U I 0 e isx, 14) ln [L i /z o ]

17 Complex Terrain and Aerosol-Sized Particle Deposition 83 Table 1 Parameters used in the 2-D model calculations Parameters Values Hill attributes Z = f X) and px) s = 2L π H m), L m) 15, 150 Canopy attributes LAI m 2 m 2 ) 3.5 h c m), C d, a m 1 ) 15, 0.2, 0.23 Z = H 2 cos sx) 1) h c & px) = U o 2 H 2s cossx) The canopy attributes resemble those of a Boreal Scots pine forest Launiainen et al. 2007). The pressure variation formulation is also shown with U o being the outer-layer velocity determined as in FB04. The flat-world friction velocity u ) = 0.5ms 1 and the particle density ρ p = 1,500 kg m 3 where B 0 = K 0 A r ) and B 1 = K 1 A r ) are the modified Bessel function of the zeroth and first order with argument A r = 2 islz + d)/l i, A c and A i are the constants that permit the continuity of u and τ at the canopy top. Moreover, U c = s 2 L c H/2U 2 0 /U h) and U I 0 = sh/2u 2 0 /U i ) are the inside-canopy and above-canopy scaling velocities, where U i and U 0 are the characteristic velocities in the inner L i ) and outer layer L m ) depths see Finnigan and Belcher 2004; Poggi et al. 2008), and s = π/2l) is the hill wavelength see Table 1). The mean fluid continuity equation and u can now be used to derive w by imposing the boundary-condition wx, h c ) = 0. The resulting formulation is given by: H f su c h c + z) u U c β 2iA ce hc 2Lcβ e w = ]U I 0 A i A r 2 L i B 1 +isl 1 Li L ln[ zo ) hc+z 2Lcβ 2 d ln[d+z]+z Ln L c β )+ 2 1 H f ) [ ] ))) ) eisx. 2 + C z o d+z) L 2 i 15) Moreover, the eddy viscosity can be derived analytically and is given by [ ] [ zβ l K t = {H f 4A c e ff k L c U c β 4 2 ]} d + z) A i A r B1 2) + 1 H f ) e isx. 16) ln [L i /z o ] The unknown coefficients in the analytical solutions can be derived by matching the solutions for the velocity and stress above and inside the canopy to yield: A c = U k I 0 v Uc B 0 + A r 2 B 1 [ ] 1 ln dli β B 0 + k v A r 2 B 1 + ) 1 i U c U I 0 u U c β ) ln [ Li z o ])) ln [ Li z o ], 17)

18 84 G. G. Katul, D. Poggi A i = C = U I 0 L [ ] k v + β) β ln dli + β B 0 + k v A r Ai A r B 1 L i ln [L i /z o ] + sh cl Uc U I 0 ) [ ] 1 i U c u U I 0 U c β β ln Li z o ), 18) 2 B 1 u U c β isu I 0 d ln[d] ln [L i /z o ] + 2A cl c U c U I 0 β 2 ) 1 e Hcβ l ef 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) D p,m = k BT 3πµd p C c, 20) where k B = JK 1 is the Boltzmann constant, T is the absolute temperature, and µ = ρν is the dynamic viscosity of the air, where ρ and ν are the air density and kinematic viscosity, respectively. C c is the Cunningham coefficient given as: C c = 1 + λ d p exp d p λ )), 21) where λ is the mean free length of air molecules = µm at standard temperature and pressure). The Knudsen number Kn = 2λ/d p defines the nature of the suspending fluid air here) to the particle size. When d p λ or Kn 0, C c 1, and the standard formulation for molecular diffusion is recovered. The particle diffusivity is primarily dominated by the turbulent eddy viscosity and is given as Csanady 1963; Wilson 2000) D p,t = 1 + τ ) p 1, 22) K t τ where K t is the eddy viscosity of the flow, and for clarity, is repeated here and is given as K t = lm 2 u z, 23) and τ p is the particle time scale given by τ p = ρ pd 2 p 18µ C c, 24) where the Lagrangian turbulent time scale τ) is given as τ = K t σw 2, 25) and where σ w is the turbulent vertical velocity standard deviation. For small aerosol sized particles in the nanometer to micrometer diameter range,τ p /τ 1, and D p,t K t resulting in a turbulent Schmidt number of unity.