Dry deposition to a uniform canopy: Evaluation of a first-order-closure mathematical model

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ooo4-6!m1/%713.00+0.00 Pnpmon Journals Ltd.

Vol. 21. No. 7. pp. 15734585. 1987.

DRY DEPOSITION TO A UNIFORM CANOPY: EVALUATION OF A FIRST-ORDER-CLOSURE MATHEMATICAL MODEL B. Y. UNDERWOOD Safety and Reliability Directorate, UKAEA. Wigshaw Lane, Culcheth. Warrington. WA3 4NE, U.K. (First received 12 August 1986 and receioedfor publication 19 December 1986) Abstract-A mathematical model is developed for the transport of momentum and matter within a canopy consisting of identical ekmcnts protruding vertically from a smooth substrate. Turbulent flux is modellcd using a mixing-length approach. The loss of momentum (or matter) lo individual ekments is related 10 the mean wind speed. and the ekmcn+ckmcnt interaction via the turbulent wind field is represented by a sbeltcring factor. Careful consideration is given to the formulation of lower boundary conditions. The model assumptions arc compared with those of other models. ‘The model predictions arc compared with measurements on a vertically- and horizontally-uniform art&% canopy in a wind-tunnel. The model reproduces well the observed relationship between the parameters of the logarithmic wind speed profile above the canopy and the observed deposition velocities of th~riurn-B Cl’hB)atomsand particlesin thedieter range 0.08-32 m, using a sheltering factor which is little dependent on wind speed and has the same magnitude for momentum, gas and particles. The predicted dcpcndcnccs of &position velocity on friction velocity and, for particles, on diameter shed light on the performance of semLempirica1correlations proposed in the literature. For ThB atoms, the calculated deposition velocitiesare compared with those of other mathematical canopy models: a comparable degree of agreement is obtained here with fewer free parameters. The fraction of deposit on the substrate is underpredictedby an order of magnitude in some cases, pointing to the limitations of the modclling of conditions near the substrate in terms of quasi-shear flow. Key word index: Particle dry deposition, dry deposition lo vegetative canopies, gaseous transfer, mixingkngth model, sheltering factor.

I. INTRODUCT’ION

The dry deposition velocity, u,,, of an airborne contaminant to a surface, defined as the ratio of downward flux to concentration at a specified height, is known to depend on contaminant physicochemical properties, on meteorological variables (such as wind speed) and on aspects of the surface cover. Canopy models attempt to relate the bulk properties of surface cover consisting of separate elements protruding from a substrate to the properties of the individual elements and their spatial distribution on the substrate. SufBciently far from the edge of a horizontallyuniform canopy, for a source upwind of the canopy, the balance of momentum and contaminant within the canopy can be represented by ofiedimensional equations in which the change in vertical flux across a horizontal slice of canopy equals losses to elements within the slice. The flux and loss terms are Reynolds averages which have also been averaged over a horizontal length scale larger than the spacing between elements (Raupach and Thorn, 1981). The eddydifksivity approach, which relates the turbulent flux to the gradient of the mean field, has been widely used although there are theoretical reasons for questioning the applicability to canopies of this first-o& closure of the turbulent-transport equations (Lcgg and Monteith, 1975; Raupach and Thorn, 1981). For momentum, the appearance of

secondary wind speed maxima, particularly in tall canopies with dense upper crowns (Landsberg and James, 1971), has led to the introduction of secondorderclosure models (e.g. Wilson and Shaw, 1977) to predict the mean wind speed profile. However, similar profiles have been obtained (Kondo and Akashi, 1976) from a first-order-closure model which includes the horizontal pressure gradient and Coriolis force (although the secondary maxima do not occur for canopies with vertically-uniform leaf-area per unit volume), leaving open the question of the applicability of the eddydiffusivity approach. The turbulent fluxes of matter and momentum in a canopy are related in that transfer (outside the immediate vicinity of the boundary layers over individual elements) is effected by the same eddies in both cases. Canopy models for obtaining deposition velocity have usually applied semi-empirical exponential forms for the wind speed and eddydiffusivity profiles in the canopy (Belot et al, 1976; ShretRer, 1976; Brutsaert, 1979; Wiman and Agren, 1985). whereas potentially more information can be obtained from an attempt to model simultaneously both momentum and matter transport (Slinn, 1982). In parallel with the eddydiffusivity approach, a common method of parameterizing the losses at a given height in the canopy is in terms of the mean wind speed at that height (Inoue. 1963; Cowan, 1968; Belot e( al., 1974; Rache, 1986). Thus elements are treated as

1573

194

B. Y. UNDEIVOOD

if exposed to a steady wind equal to the mean wind speed; the loss of momentum or matter to an individual ekmmt is either measured in a separate experiment (e.g. Belot et af, 1976). or atimrta! theoretically (e.g. Davidson and FriadkDder 197%) for dcrncnts of sirnpie geometry. TiK dc;rikd effect of canopygenerated turbuknce in the tlow approaching an elewnt is not taken into account explicitly, but is incorporated into a ‘sbeltcring factor’ (Thorn. 1971X q. which modifies the calculated loua due to a canopy element compared to a’hn’ekment in an equivalent steady flow. It is convenient if q an be taken as a constmt deccrmincd by the structure of the given anopy @be rpdng sizq rhrpc of ekmnts). independeot of wind speal and of whether the quantity of interest is mownturn, contaminant gas or particle (Belot et d, 1976). Formulation of appropriate km boundary conditions on tbe equations of momentum or matter transport has usuaIly been treated in an od hoc manner on tbe grounds that. for typical canopy densities, the behaviour in tbe lower reachesof the canopy has little inlluence on tbe total drag or bulk deposition velocity (Cowan, 1968; Brutsaert_ 1979). However, the fraction of t&e deposit which is on the substrate rather than on the eknlents (termed the partition, p, below) is also of interest in some applications (Chamberlain, 19701 and for its prediction closer attention must be given to modclling the lower canopy. In the present work a canopy model is developed having the foliowing key features: Iirstorderclosure of the equations of turbulent transprt, with related lluxes of momentum and rnattrr, losses to ckments give0 in terms of man wind speed together with a sheltering factor; a consistent formulation of lower boundary conditions. The predictions of the model are compared with measurements on a vertically- and horizontally-uniform canopy of artificial grass in a wind tunnel (Chamberlain, 1966). The comparisons are facilitated by the simple geometrical structure of the canopy. and enhanced by the detailed kvel of expcrimeatal data available. The aim of the work is to investigate the extent to which the model can give a sia~Itaneous fit to the data on momentum and matter tfanspor& IO M whether a constant value of sheltering factor can be applial and to eucnit~ tbe performance of tbe rped6c extension to the addydffusivity appreach introducsd here to model transfer to the substrate. 2.

WIND SPEED PROFtLES

2.1. MolWuum qufzion

where K,

1 dP d +- P dx

(in uniform wind flow of rpeed u),

a is the ekmcnt area per unit volume (projected normal to Bow), 4 is the sheltering factor constant for given =DOPY. dP/dx is the mean pressure gradient in direction of flow. x, rssumed constant throughout the canopy, and P is air density. The Coriolis force an be neglectad in the wind tunnel situation here. The influence of the pressure gradient is govrmed by the dimensionkas quantity a( - h(dP/dx)/pv:),wherrhir~aaopyhei~tand Y, is the frktioo velocity for the wbok surf= layer. For short grass canopies in the bdd R is of order lo-‘-lo-‘, and the pressure-gradient term in (1) has negligible influena. In Chamberiain’s wind tunnel experiments, however, n is about - 0.2 (seemien 2.4) rnd annot be neglected 2.2. Analyhcal

solurions

For Kt - 0. analytical solutions to (1) have been given for a. K, and (qc#) constant (i.e. independent of z within canopy), Lendsberg and James (1971); b. KJu and (qc,a) constant, Cowan (1968); c. mixing lmgth (see section 2.3) and (qc& constant, Inouc (1963); d. K, constant (qc,o)u I/u, Siinn (1982). Under certain restrictions the last two give an exponential wind speed profile u = u,exp[-y(l

-r/h)]

(2)

where u, is tbc wind speed at I = hand y is a constant; also the first two typically givt near~xponential behaviour in the upper reachesof the anopy; this generic shape has empirical support (Cionco, 1978).

2.3. Mixing-length model and boundary conditions The Prandti mixing-kngtb representation of turbulent flux (Inoue, 1963) was chosen for the prcsen~ investigation. i.e. du du

F‘. - p/l dz I dr I

(3)

where F, is the downward turbulent fIux of momentum and I(r) is the mixing length. This is equivaknt to

K,

Momentum balance can be represcntai by - qc&)o

C&) is drag coeIEcicnt of individual ektncnt

(1)

I b vertical coordinate, is vertical diffusivity of momentum, Y is man wind speed.

(the modulus sign will be dropped siacc du/dx is always positive in this work). In the anopy mvironment a number of Lengthsala are potentially importlnr theassumption ofa singk domiuant length talc at each height is the essential feature of the first-order dosure approach. The well-known exponential sol-

ution (rsaioo 2.2) for 0 - 9 I qnstan(, which has 7-h

gcda t/s ( 2P 1

(5)

imly8p8rthhrintqnlof(l)whicbdoernotatiSfy 8

rmlisbc

lower

r0lUtiOo even for

bound8ry

condition;

the

@

I amtant is not cxponmtid in form

(AW 1) The prescription for the mixing-kngth profik is intriaricrlly linked to the bourxkry conditions applied which were as followr (8) The wind speed profile is mrtched to 8 logarithmic profik 8t t = h (since oeutral st8bility conditions rpphed in the aperiments~ i.e.

where to is the l oddprmic roughness kngth of the a~opy. d is xero-pkne dirpluxment md K is von Kumur’s constrot. In re8lity it is m ide8liz8tion to rpply (6) 8l1 the cmy down to h (R8up8ch cr al, 1980), but the masurements of Ch8mberkin (lp66) indicate th8t it is 8 rasorubk 8pproximrtion. Equation (6) also ignores the effect of the strumwise pressure gmdient on the pro& above the anopy, but this will generate only rbout 5% distortion in du/dr vs t over the intmrl L to 2h. The mixing kogth is then assumed to be const8ot in the upper raches of the unopy, rnd l?qti8)t0 K(h-d). (b) For the lower boundary, two possibilities are considered: first, the substrate pets as a fully rough surfr~e in qtursi-rhcar flow with its own oerodyn8mic roughoas leogth r& in which case II - 0 at z - &

(rough substrote)

(7)

This condition has been applied in both first-orderclosure (Seginer. 1974) and second-orderclosure modelling (Wilson aod Shaw, 1977) Then I(z) is taken equal to KZ io the region where turbulent flux is influenced by the presence of the substnrtc. This leads to the following mixing-length profile, ensuring continuity of du/dz md momentum flux I(Z) -

KZ

r,‘
-*(h-d)

h-d
-K(L-d)

r>h

rough substnte. (8) I

There is lomc experimenul support for such a profile (Cionco. I%$ Seginer cr a/, 1976). Wimrn 8nd Agrcn (1985) use a different form for I(r) within the anopy. which lads to 8 discontinuity at z - /I. However, for Y, or 16 mull enough or the canopy dense enough (I* du/dz) may become kss than Y (the mokcular diffusivity of momentum) for I > zb, in which case it is inconsistent to continue the above mixing-length prescription down to zb. For this situ8tion. an 8dditiotul anopy regime is introduad. extending from z,dowo to the substnte. in which the vertk~l diffusivity of momentum is KI equnl to v. Ihe

nlue of I, is found by 8ssutning tlut the substnte behpves with respect to momentum (rnd mrtter) tnnsport 8s 8n 8erodyn8mially-8mooth rurf8ce in shc8r flow, but with 8 ld v8hbe of u,, termed u;, correspoodiog to turbuknt wnditions in the vicinity of the substnte. (NB: A simikr u’, is introduced io the theory of momentum tmtlsfer to regukr 8rr8ys of more widely-sp8ced roughnessekments, e.g. Wooding d al, 1973.) The usual description of momentum tnnsfer to a ‘bare’ (i.e. with oo canopy) rerodynami&ly-smooth surface in shear flow (Schlichting, 1960) iovoka three kyers (turbulent core, tnosition layer 8nd ‘kmin8r’ Layer),but Brutsaert (1982) hss introduced 8 simplifiad prescription involviog two kyers (i.e. omittiog the transition layer), with 8 bounLry 81 z_- II v/u,, allowing a discontinuity in verticrl diffusivity-which f8llS from 1IKV to v 8s z drops across the bound8ry8nd a corresponding increclse in du/dz to preserve continuity in momentum flux. This prrrription is 8pplied 8s follows in the vicinity of the canopy substrate (since u; is uokoown a p&m), Equation (1) is solved subject to the lower boundary condition at z,

I’(&uz =1lKV.

(9)

II_

At z,. du/dz incm8sesdiscontiouously by 8 factor 11K falls, so the qu8tioo for the regioo below z_which hrr the LHS of (1) replaced by v dzu/dr’-is solved subject to the conditions that u - 0 at z - 0 rnd

a.s z

du z

1-t .

= l)K-

du

l

dz 1-I.

(10)

where + 8nd - refer to the rppro8ch to z,from rbove and below, respectively.z-is fouod from the cooditioo that u(z) is continuous 8cross z - z, Thus 2, is 8n unknown implicit parameter, but is readily obtained by choosing a numerical integration procedure which uses Newton iteration to converge on unknown generalixed parameters. For ‘a’ constant and cd 8 wcllbehaved function of u. the differentLl equation can be iotegmted once an8lytically both for z < z,8nd io the region of constant I in the upper anopy, thereby providing a useful check on the numtrial integration (Appendix 1). An example of the profik of K,. resultiog from the 8ppliation of the model to Cbuaberkio’s canopy 8t u. - 0.6 m s- ‘, is &town io Fig. 1 which also indicates the different regimes of behaviour of K_(z) and I(z) The rbove models of the lower boundary. rlthough selfconsistent and having the required behaviour as a+O.do oot noccrarily represent well the compkx turbulent environment in the vicinity of the substrate; they will be tested by comparisons with data. 2.4. Comportin with cxpMmmtd

results

The 8rtidcirl gr8ssaoopy (Chrmberkin, 1966) h8.s raztangular pkstic spills of width. w, 5mm and thickness 0.25 mm. protruding 75 mm from a smooth

B. Y.

1576

0.4 a3 t

UMKRWDDD

of Y, covered by Chamberfain: io all uses I, is kas tban (IA) giviog three dittinct regions of K, bctuviour in the canopy (wt Fig. 1). The mode) provides an inter-rdhonship between z. m.4 d for given Y. and 6xal q. CbaxnberkSs ruults show to and d constant within experimental error over the Y. range O.lS-Zms-‘; strict indepmdence of Y, for both z. and d an ooly be obtained if cc is indqmdmt of u and the ‘rough substrate’ boundary amditioo is applied (as io Seginer. 1974X ao tbe 6rst critical test of tbc model arises from the variation of values with u,. Tabk 1 rbows praMed d/h as a functioo of u, for 9 - 1, holding ro/h &al at tbe exp&nentat value 0.133 (aaauming the experimental error in to is likely to be ksa than that in 4 The variation is only 8 % across the whok range of II,, aod the mean value, 0.6, is equal to the measured value in Chamberlain (1967) (which differs by IO % from tbe value, 0.66. cited io Cbamberkin. 1946, for the same canopy). For q = 0.5, the man value drops to 0.48. Also given in Table 1 is tbc value of zJh sod the drag partition, pd (fraction of toti drag oo the substrate). Chamberlain measured the protik within tbe caoopy at U* = 0.6ms-’ only, aod Table 2 gives the comparison for q - 1. The agreemeat is good in the upper canopy, but there is a systematic divergence lower dowo; oo experimental uaaruiotiu are given. The measured lower profile appan anomalous compared to profiles measured within other artificial canopies (e.g. Thorn, 1971; Plate and Qunishi, l%S)_

----~-i~iz--f----j

z*d

0305

I

23

5

IO

203050

I002003ca5w

Fig. 1. Tbc mkutatal ntio of venial dillldvity. K,, to mokcuhr difTusivity.v. as a funuion of fmnional height rilhiatb~,a1~~-0.6ms-‘.

wax substrate with long axis vertical. Equal numbers of spills have their sbort axesat O”,90”, 45” and 135” to the bw direction;packingdensityisoncspill per 16Ommr of plan surface. Assuming that the streamwise pressure gradient is dominated by drag on the canopy covering the floor of the wind tunnel, n=

-h/H

(11)

where H is the height of the working section of tbe tunnel (380 mm), giving n - - 0.2. The combination (cp) has to be evaluated for the three distinct ategorier of spills (0”, 90”. 45” + IW), i.e.

(c,j&#J - ,i, (C&A.

(12)

For the 0” &meats, the drag is taken to arise purely from Bksius-type shin friclion, i.e. -I/2

(7

C&Y) = 1.33 -

V

(13)

(Batcbelor. 1967); tbe luodrted ‘o’is then basedon the frc area of tbe spills [i.e. l/4 x (S/180) 1. At tbe other an&q form drag dominata so that at high eoougb Reynolds number, Rr, (m uw/v), c, is approximately constant, and the incrax of cc with duxeasiog Rc, is expacted to be slow down to Rr, of order unity. Sensitivity studies cstablisbed that the &t&d nrirtioo of c,at bw RI, has little iotluence oo the results of ioterut, wberas its magnitude at bigh RI, is imporuot.T)lecumofc,vrRr,wu~eo tobavethe nme shape as that for a cyliodcr (Batchelor. 1%7), but renormali& at Re, - 1000 to the measurements of Tbom (1968) for a sic* hf. The anootb-substrace focmuktioo of the lower bowbdary condition is usumod to apply for llt values

Tabk 1. Predictedcbamcseristiaof the nod spad ptofik and drq putition for various f&on nlociticr un?dby Cbamhcrtain(1966)

,ln:- 1,

4h

0.128 0.185 0.28 0.40 0.60 0.83 1.30 I.30 2.00

0.629 0.62I 0.613 0.607 0.600 0.393 0.58s 0.383 0.382

‘(-2)

PI

0 0274 0228 0.186 0.1% 0.132 0.111 0.089 0.078 0.072

122(-2)’ l&(-2) 8.88(- 3) 7.63(- 3) 6.U( - 3) 5.60(- 3) 4.57(- 3) 4.W( - 3) 3.7S(- 3)

- lo-z.

TabkZA romp&on of wind qxat proftksa~ Y. =0.6ms-’

(m’m) m 60 SO 40 2 IO

u/u. CXpt. 2.x t .74 IS5 I.45

1.38 135 1.3s

2 2.36 1.75 1.22 1.01 0.80 0.6s 0.50

1577

Dry deposition to a uniform canopy

although the ckmcnts used were not the same shape as in Chambtrlain’s canopy. The influence of the strcamwise pressure gradient is illustrated by comparing the v&cal profile of u/u, at u, = 0.28 ms- ’ for R= -02withthatforR=O(Fig.2). A value of q close to unity gives the best fit to the momentum transfer data (although no formal optimization procedure was used to find the best value more precisely). Thorn (1971) found from detailed drag measurements in his artificial canopy that q = 1 applied, although the spacing/width ratio for his elements ( - 10) was much larger than that here. The results cited by tiginer et al. (1976), albeit for cylinders, would indite a value of q around 0.3 for Chamberlain’s canopy. but the same data would suggest a value of q w 0.75 for Thorn’s canopy, smaller than actually found. 3.

nB GAS DEPOSIttON

3.1. Equation and assumptions The differential equation for balance of thorium-B (I’ItB) atoms is 1

(

Kv$

>

=qE(u)auX

where x is contaminant concentration. E is the collcction efficiency for an individual element (defined as amount collected/amount incident) and K, is the vertical diffusivity of the gas. The values of E(u) were obtained from the measurements of Chamberlain (1966) of the collection efficiency of a single spill in a uniform flow as a function of wind speed. The data conformed to a Polhausen-type u - ‘I2 dependence for E(u) in agreement with the comparable experimental measurements of Thorn (1968). which also showed the cfiiency to be independent of the angle between the plane of the element and the wind flow. In this work K, is taken to be the same as the turbulent diffusivity for momentum, Km, for z > z,. Some authors have adopted a further idealization that both u and K” profiles have the exponential form of Equation (2). Equation (14) is then amenable to

analytical solution in terms of Bessel functions for a power-law dependence of E on u (Brutsaert, 1979although essentially the same solution was given earlier by Cowan, 1968, in terms of Hankel functions). Clearly the exponential form cannot be valid in the lower reaches of the canopy, and the accuracy of the analytical solution in predicting deposition velocity has been found to be governed by the degree to which the upper canopy region dominates collection of contaminant. A systematic assessment of the applicability of the analytical solution is not given here since it cannot readily represent realistic bchaviour in the lower region of the canopy nor does it apply when gravitational settling and non-power-law collection eiXciencics contribute (as is the case for particles-see below). Slinn (1982) has used an even simpler approach in which K v and the product (qaEu) are considered constant throughout the canopy, leading to a solution for x in terms of exponentials. Although this approach is able to illuminate certain broad features of deposition to canopies, it is not flexible enough for detailed comparisons with Chamberlain’s results. Here, Equation (14) is solved numerically using the wind speed profiles calculated in section 2. 3.2. Boundary conditions Since only the ratio of flux to concentration required, i.e. y&h)=KV$

=0.28mse’

forn=OandR=

-0.2.

(15)

the concentration can be normalized to unity at t p: /I. A smooth-substrate lower boundary condition was devised by analogy to the situation of gas deposition to a smooth surface in ordinary shear flow (Davies. 1966) for values of the Schmidt number u (E v/D, where D is the molecular diffusivity of the gas) of order unity (o = 2.778 for ThB atoms in airAppendix 2. An adaptation of the Davies prescription is applied to the canopy case by assuming that the total vertical diffusivity of ThB atoms falls linearly from 11 KV at z = z,to Dat z = az,(aconstant c l),and nmainsat D for z < az, a was chosen by requiring that this diffusivity prescription should give the best fit to the standard data when applied to a ‘bare’ smooth surface (i.e. a = O), yielding a = 0.55 (Appendix 2). The lower boundary condition then becomes X=Oatz=O.

Fig. 2. A comparison of wind speed profiles at u,

atz=h

is

(16)

Shrefiler (1976) used a lower boundary condition based on a smooth-substrate prescription, but chose zI,= 11 v/u,(z,J where u_(zd is an effective value of the friction velocity at z, determined by arbitrarily assuming u,(z) falls exponentially [i.e. as u does in Equation (2) 3 from its value of u* at z = h. He further assumes that the vertical diffusivity follows an cxponcntial profile from z = h down to z = z,, then drops to D. Brutsacrt (1979) applied the artificial condition that x -, 0 as z + - ag on the grounds that

lS78

B. Y.

thDE&WDDD

the magnitude of o1 is littk influenced by the lower boundary (though this cannot be true for p) Slim (1982)cxpr~s the lower boundary condition (in present notation) as

117) shifting the difIicuIties onto spec&ation arbitrarily assumes 0; =

E

0

of v,,‘. He

u2

-

A

cd

‘dh)

(18)

where (E/c&-a constant in Slinn’s model-takes value applicable to individual elements!

the

3.3. Comparison wi!h experimentu/ results and other model predictions Figure 3 compares the predicted values of od(defined as od(h)/uJ vs Y, for q = 1 with measured values. The agreement in overall magnitude is good; the slower fall-off of the experimental results with increasing u* requires comment. In terms of the three regimes in the relationship of v&,) to E (u* )-see Appendix 3-the values applicable here are around the point where n-r proportionality to E(u,) is lost. Approximating the results by a power-law dependence on u+, the model g&S Dd+U u;OJJ

(19)

rather than uda u;‘.‘. Cowan’s (1968) analytical calculations (on a different canopy) gave a corresponding u, exponent of -l/3 for values of E(u,) extending somewhat higher than in the present case. The deviation from an exponent of - 0.5 for gases can be clearly seen in the generalized fits to data of Schack cl al.

Table 3. A comparison of deposit profiles PI U. = 0.6ms-’

f EXPT -q*t

AZ (mm)

--po.5 I

0

(1985X indicating a limitation to their ‘universal’ uda utn correlation. Chamberlain’s resulU, however, lie much closer to a curve with exponent - 0.2 1. In terms of the parameter q, a better fit to data could be obtained by assuming q for ThB atoms iwith u, taking a power law exponent of around 0.15 (cf. the exponent 0.12 found by Laodsberg and Powell, 1973, for the leaves of young apple trees). A likely explanation is that turbulence in the flow approaching an element enhances the collection efficiency at higher u+ over that given by the simple model. Indeed, Shreffler (1976) has introduced a model in which the collection process is dominuted by microscale eddies; this approach is questionable in view of the likely eddy spectrum in the canopy and more especially in the manner in which the Kolmogorov scales were deduced, although the model has free parameters which can be adjusted to give a teasonable fit to Chamberlain’s results. Brutsacrt (1979) fitted the predictions based on his analytical solution (see section 3.1) to the ThB measurements in order to deduce qE(u) (in present notation), finding a u-o.35 dependence. Taking E a ueo,‘, this implies q a u”.I’ which is a somewhat faster dependence than obtained in the present work. This discrepancy can be explained, at least partly, by Brutsaert’s use of too large a value of y [in Equation (2)3 to represent the measured wind speed profile: his own analysis indicated that a smaller value of y would have led to the udependence of qE(u) having an exponent with larger absolute value, bringing the results more in line with the present work. The profile of deposit was measured by Chamberlain for lO-mm increments of canopy height at u5 = 0.6 m s-l. The predicted profile for q = 1 is compared with measured values in Table 3, showing good agrament. In this example, the concentration of ThB atoms falls by a factor of 2.5 from the centre of the top interval to the centre of the bottom interval. The measured partition, p, is stated to vary from 3 to 10 “/* with no further details. For q = 1 the predicted value of p varies from 0.4 % at 0.13 m s- ’ to 0.9 y0 at 2 m s- ‘, i.e. about an order of magnitude underprediction. Using q = 0.5 in both Equation (1) and Equation (14) the predictions range from 2.8 to 6.2”/, much closer to the observed range, although this leads to an underprediction in u; (see Fig. 3).

I

I

I

I

0.4

0.e

1.2

I.6

I 2.0

Relative amount CalC. Expt.

J

u l Ims”)

Fig. 3. The tYltio of depQsitioa velocity at canopy top, o,,(h), to friction velocity. II.. vs u,: a comparison of calculated values with mauumncnts for ThB atoms.

65-75 55-65 45-55 35-45 25-35 Is-25 O-15

0.24 0.21 0.15 0.13 0.11 0.09 0.06

0.251 0211 0.161 0.128 0.106 0.080 0.063

1579

~C@OlithtO8NifOlUlCMOpy . . to unity, the undaprad#r 00 io p proMAy indAta the limhtions of the aimplitkd tm8tmcnt of the flux io tbt vicinity of the substrate. Takioganothcrlincofrppro&cBklllationswere perforwd using a rough-substrate lomr bouodary amditioo on Equations (1) and (14). For convenience n was taken as zero in the fonacr &LX_ for small 2:. this lads to a logarithmic wind speed profile near the substrate, with corrcspoodiog II’. given by For q ckne

m

in m-up would 001 have had a significant impuzt, WI would atliog onto spill potcotiaI iIt&xuluks

toprr The principal ma%a&m of pltide oollaztion by spills is inertial impctioo. Experimental data arc availabk for impaction onto cktnents of regular shape, includiog data for Lycopodium-spore collection by a rectangular glass slide at various orientations to the Bow (Gregory rad Staiman, 1953). Rcpracntiog the dcpcndcocc of collaztioo ef6ciency on wind speed in terms of UK Stoker oumbcr

suggestinga lower boundary condition for (II) of the form

where r,is a suitably small value oft, and d,is obtained from correlations for rough surfin shear flow (e.g. Brutsacrt, 1982). & was increased until p was of the order 3-10% but it was found that this required tb to be of the same order u to, inconsistent with the assumption of simpk shear flow in the vicinity of z = iO. The magnitude of u;was virtually u~fkcted by this shift to a ‘rough substrate’ lower boundary condition. Omission of the pressure gradient in dcriving the wind speed profile lads to a 10y0 increase in the predicted value of u1 at Y, - 028 m s- ‘. ShreRkr (1976) was lbk to obtain values of p in the range 3-8 % using the parameters which optimixcd his fit to the depotition velocity masurcmen~ he had three free pammeters available compared to the single one here (4). BNLWX~ (1979) did not attempt to predict the partition from a physically realistic lower boundary condition, although he check& that his art&Al condition (see saztion 3.2) gave values of p of the correct magnitude.

4. DEPOSlllON OF LYMMDIUM

SMPEf

WZHAMIEILAIN. lU7) 4.1. Equation

ad

a.ssmpdons

Gravitational settliog has to be iociudcd in the balance equation for thae relatively large particles (diameter 32 m) which times

where V, is settling velocity. u, is horixontal surface area per unit volume and K&s the vertical diffusivity of the particles. The method of combining the efTcctsof horizontal and vertical motion impliad by the RHS of (223 used by w and Powell (1979), is not the only one advocated in the literature (e.g. Buhc, 19791,but it does have theoretical support under certain conditions (e.g, Robinson, 1956). For the artifkial unopy here. u, is taken as zero. (In reality all the spills may not have been vertical, but sensitivity studies have shown that

where g is the urtkratioo due to gravity and b is the width of the object normal to the flow. experimental data for cyliodus (axis normal to ilow) and glassslides (plane normal to flow) approximately collapse to a single curve (Legg and Powell, 1979). In the present work a semi-empirical paramcterimtion of the curve for cylinders (Davidson and Friedlander, 1978) was used: E(9) =

9’ St’ + 0.753 Sr’ + 2.796 St - 0.202

SI > 0.1. (24)

‘Ihc formula is not r&b& for St 4 1 (cvcntrrpllyyAding negative emcieocics) and the imprtion efficiency was taken as ttro for St less than 0.1. For the 0“ ekmcnts the simple axsumptioo is made that (24) holds with Sr based on the thickness of the spill (0.25 mm), but these ekmcnts do not have a major impact on the Lycopodium spore results bazause of their relatively small prrscotatioo area normal to the flow. (Setting this area to zero dacrtued the predicted vd by 10 % at 028ms-’ and 2% at 1.46ms-I). For the 45” (and 13s’) elements the 8pproximation is made that (24)can be used with Sr based on the projaztion of the short axis normal to the flow. The diEculty of particle bounce was circumvented in the experiments by using sticky surfaces: perfect sticking was assumed in the akzulations. Adding diffusion and interception contributions to the colkction efficknq (sa x&on 5) had negligible influence on predicted values of D( or p. Above L, tbc vertiul diffusivity for particles is assumed to be the saw as that for momentum and gas. However, below I, a somewhat different diffusivity prescription from that used for ThB atoms is required. since the point where eddy diffusivity becomes comparable IO mokcular diflusivity lies in the ‘laminar’ layer (Davies, 1966). The prescription&d in the latter is adapted to the canopy situation by allowing K p to fall linearly from its value at 2, to a value (0.04v + D)at I - SrJl1. and at lower x rc&cing the 0.04~ by a contribution proportio~l to 2’. with cocffkknt of proportionality chosen to prm continuity in Kpat the matching value of z - StJll. lo this form the prescription is used for the smalkst prtickr considered in taction 5. for which diffusive transfer to the substrate dominates (thus leading to the lower bound-

a Y.urcwrwooo

1m

ary condition that x - 0 at I - 0). For Lycopodium 1p0rq bowevcr, pivitatiod acttliag is far more importrnt than diffusion in transferring material to the xubatratc, and an efTa3ivc boundary condition of the fOrlIl

(25) is used.wkrc t,ir a distance from the substnte chosen so that the downward flux can be conddercd virtually conxtant for 2 < zb. Wiman and Agrcn (1985) choose the Ltter boundary coadition for all particle sizes. Inertial impaction is potentially the krgcst contributing mechanism of transfer of rporcx to the xubstnte apart from gravitational xettling, but akulations based on the picture of quaxi-shar flow near the subxtntc indicate that this would not make a cignifiant impact compared to titling even at the highest Y, values considered here. 4.2. Comparison wirh rxperimenrd data Measured and predicted values of DAYSu, for q - 1 are shown in Fig. 4. The experimental values correspond to the ratio of flux to the average of the spore concentrations at 50 mm and 75 mm above z = d. so the predictions have ken adjusted to give the xame prr;mcter. termed u,(50,75), using the relationship ktwcen o, and height above the canopy in Chamberlain (1967). Overall agreement is good, although there appears to be a systematic underprcdiction by about 30% at the lower values of u,. A test in which the modelled &St) was extended to yield values in the St < 0.1 regime in line with the experimental ruults of &lot and Gauthier (1975w saztion 5. I confirmed that this regime has no impact on the predictions of D, or p for Lycopodium xporcx. (NB: Even at the lowest Y,, St for the g(P elements does not drop below 0.1 until t/h < 0.2.)

Again, tk data is but fitted with q given tk xame value for mownturn a.bd autter. but in thix case o( is littk affaed by a change in tk magnitude of q because of near ampemadon due to iacrared wind xpteds in tk canopy when to is held Axed. Uxing q - 0.5 in both Equations (I) and (22X u&50,75) incraxai by only 4 % at u, -0.28ms-’ and dazruxai by 4% at u. - 1.46ms-’ The expcrimcntal results show a near linear dcpcndcncz of u&50,75) on u,. although a somewhat fasterthan-linear trend can k detected. Chamberlain interpreted his results to imply the validity of the simple axsumption of an infinite sink for particla at t - zd i.e. Reynoldsanalogy holds throughout. This would imply a constant value for vi of 0.44. and a more direct and xcnxitive test is obtained by reducing the experimental data IO give vi values (Table 4); these tend to increase with u.. although not quite u fut as do the prcdictions. &hack et 01. (1985) interpret tk fact that the measured values of (va- v,)are approximately proportional to u:‘* in terms of the domination of the interception mechanism (and the abscna of impaction) However, here the interception contribution is insignificant compared to the impaction, and impaction itselfcan kad to a near u:” khaviour as shown by the predictions of v&50,75). The dependence of E on u in Equation (24bwhich has E ac u approximately for St 2 0.1. rising to E a u”’ before xaturating+uggcstr that there would k a range of u, for which v&h) a u: or even a u: ’ if D; a E(u,) applied (Appendix 3). However. E is large enough that the depcndcna of vg(h) on u, is attenuated by the decrease of x in the canopy. ~~50.75) shows an even slower dependence on u. baaux of the rising contribution of aerodynamic resistance. Table 4 also givex akulatcd values of p. The experimental information provided by Chamkrlain is that p varied from 37% at u, = 0.28ms-’ to 4:; at 1.46 ms-‘.Theagrament isgoodattklowervalueof u, but there ix underprediction by a factor of 15 at the higher value. The latter is only reduced to a factor of 10 at q = 0.5.

03

i

o2

S. PARTICLES

of

OF 0THF.R SIZES

Chamberlain (1967) also measured tk deposition particlcx with diameters 0.08. I, 5 and 19 pm at

9

lab&

01

docity

4. Chnpriron and cuhtratc

and marurcd partition for

depositton

Lrcopodlum

rpo= (m’:- ‘)

m; apt

35 U 73

0.23 f 0.02 028 f 0.02 0.30 2 0.02 0.27 f 0.01

84

0.40 + o.(w

I20 I46

0.34 f 0.04 0.49 + 0.02

28

Fit 4. Dcpcdioa vrlocdy bd oo rvtrrp coaalwalioo .I 5OaUn aBd 75 mm above the LLIOplw dbphmall. 0, (50.75). n Y,: a annphsoo of akuhcd dues with m~lrcatcnts for Lycqdium qofes.

of pdictcd dcpositioo

0; ak

0.151

0.170

0202 0.313 0.350 0.440 0.482

P expl

P ak

0 37

0.371 0342 0.143 3.41q-2)

4.q -

2)

2168(-2) 6.149t- 31 2.643t- 3)

1%

DrJ&.pOSitiOOlO8UOifOlIUabopr

Y, - 0.35,0.7Oaod IAms-‘.&ertbacrangesofsixe and friction velocity 8 number of maAan&s make strongly varying relative c00tributions to the total deposition.

5.1. Sin&&nwnt collecrion nuchanisms boundary condirion

and lowrr

For particles of 5 crmdiameter and smalkr, there is a major uncertainty in estimating the contribution to singk&ment colkction eoiciency from inertial impaction, partly due to the influence of the Sr < 0.1 regime and partly due to the domination of the contribution from the 0” elements. The procedure adopted here is to assume an ctincy contribution in Ibe Sf < 0.1 regime having the form E(SI) a: sr*

(26)

in line with the experimental results of Belot and Gauthier (1975). The applicability of tbe latter matsurements can be questioned on the grounds that the experiments were performed on assembliescontaining 8 number of ekments in close proximity and at a range of &es to the flow, with ekmcnt shapes quite dilTerent from those in Chamberlain’s canopy; however, the data base for the Sr c 0.1 regime is much sparser than that at higher Sr. The coefficient of proportionality in (26) was adjusted to give a smooth fit to Equation (24) at Sr - 0.1. i.e. to have E(0.1) - 1.16~ lo-‘, a value not far from that found by &lot and Gauthicr for 2-mm wide pine needks. A dinirsion contribution to & was calculated by extending the singk~kmenl ThB measurements of Chmberkin to particles by using a u - I” dependence of E (I-born, 1968). This is consistent with the theoretical alcuktions of Pamas and Friedlander (1984) for cyliodar. 8put from a reduction in the numerical multiplier (from 1.88) to give -111

(1

E-1.5

w”

o-li3

(27)

Y

assumedto be the same for all spill orientations The ktta theory also provided an estimate of the ioteraptioo contribution due to finite particle size.The various contributions to E were simply added. However, for

TaMcS.

the 0.~m prtidcr (0 = 2.2 x lo*) test calculations shorad that the interception and imp&on contributions had negligibk inlluence on predictions. For the 19-lrm particks- the imprtioo contribution from the 45” (+ 135’) 8nd 90” elements 8bove Sr - 0.1 dominates Thus the grater uncertainties attach to the results for the l-pm and S-pm particles, which arc therefore shown in Tabk 5 both with rod without irnpection. The problem of how to combine mechanisms of transfer to the substrate does not arise sinceat each size a siogk process dominates. For the 0.08-m diameter part&s this is diffusive transfer. For the other sixes gravitational settling dominates if a smooth-substrate formulation is used to calculate other contributions. 5.2. Comparison with

deposition pfametm nrious Lizcs

-

0.35ms-’

t . 5 tG l

cxpt

for prticJ~ of

Y, - 1.4Oms-’

c,(ms-I)

Diamcta (Inn)

results

The predicted values of udat 75 mm above the xero plane art compared with experimental values in Tabk 5 at 0.35 ms-’ and 14Oms-’ for 9 - I. Also given are the calculated values of p. although no corresponding experimental data are available. For the O.O&pm particks. predictions of u1 are within a factor of two of measurements at both values of Y., and the predicted value increases from low to high U, by narly the same factor as that observed. No experimental uncertainties are given for the results so the dgnifiancc of a factor-of-two discrepancy is dif?kult to judge (especially since these particles had a polydispersesize distribution). For the I-pm and S-pm particks at U. - 0.35 m s- ‘, the values OTC, predicted without impaction are about a factor of two less than measured values whereas with impaction the predictions are too high, but by las than a factor of two. The obemed similarity in ud values for particles differing by over an order of magnitude in diameter (0.08~pm, 1-m) is reproduced by the model, and can be linked to the behaviour of the different contributions to E. At u, - 1.4 ms-‘. the inclusion of impaction has a more dramatic efkct: without it the predicted values of o( for the 1-m and S-pm particks art about an order of magnitude lower than measured values, but with impaction they are too high-by a factor of two for the S-_crm particks and nearly a factor of- 3O for the I-pm

com&orpsdicIat8odmewal Y,

0.08

lxperimenral

od(msS’) cxlc

CL

expt

ok

P cak

6.W (-3) 4.9 (-4) 2.86 (-4) 8.45 (-3) l-29 (-4) 1.85(-J) 1.87(-l) X4(-4) 9.67(-3) 3.06(-3) 6.8S(-5) 5.10(-l) 499(-J) 6.%(-l) t .4 ( - 3) 1.92 ( - 3) 2_63(- 1) 1.9 ( - 2) 3.88 ( - 2) 1.49 ( - 2)

I.9 (-4) t.O(-4)

71 (-2)

WIthout impaioa.

2.94(-2) 7.82 (-4)

9.92 2.42(-l) (- 1) 3.4(-l)

(-I) 2.77 (-9 8.08

9.69 5.72 (-2)

lg82

g. Y.

blXWtXJD

exalknt agramcnt with measurement. Since the put&s. lbe principal contribution to deposition in the kttcr ace arises from nluer of St just kss thrn 0.1, colJ&on ef?kiency depends on Y- ‘I’, sensitivity to the wind speed profik is attenuated but not lort indiatiog tht tl~ prrraiption dopted for E(Sf) compktely, and the good agreement g&s 8onu sup ovcrutimatcs in Ibis region. For the 19-pm ~wMu, the undcrprediction in D, is port to the akukted wind speed profile which falls off with decrasing L much more noidly then does the a factor of 25 at Y, -0.3Sms-‘. However, the masured value lppars anomalously high (exceeding observed profile. The deposit protik has a second the corresponding value for Lycopodium spores). dependence on the vertical diffusivity-besides its Fernandez de la Mora and Friedlander (1982) suggest dependence through the wind speedGnce the ThB that the warts on the surface of theseparticks (ragweed concentration falls off appreciably with depth in the canopy, so the comparisons give some support to the polkn) may incrase their dTective wrodyrumic diameter. At Y. - 1.4 ms- ‘, the underprediction is diffusivities employed. For particks, the principal fatures of the depenattenuated by the relatively large contribution of &zrodytmmic resistance. dence of deposition velocity on u, 8nd on particle diameter are well reproduced. In the main, these retkct In spite of the uncertaintia. at Y, = 0.35 m I- ’ the the behaviour of E(u), although at larger values of E observed increase in od by over two orders of magnitude as diameter increases from 1 pm to 19 pm is deposition velocity is also influenced by the attenuation in concentration through the canopy and by the retkcted in the prediction. and an be traced IO the rapid increaseof the impaction contribution. Wrack Ed increasing relative contribution of rerodynamic 01.(1985) proposed an increasein (cd- 0,) proportional tiSWMX. The comparisons 8iven here suggest that the model to d2 dw to interception without impaction, but the calculrta! interception contribution is much too small may be apable of providing reliable indications of the on its own to explain the observations. deposition velocity of gases and particles to natural canopies. However, a number of caveats are wry if the natural canopy differs signifiatntly in structure and size from the model canopy considered here. The C DISCUSSION AND CONCLUSIONS model canopy is vertically uniform; for canopies with dense upper crowns, a quite different aerodynamic The principal conclusion to be drawn from the comparisons between predictionsand measurements is regime may exist below the crown (Slinn, 1982). The that the model provides 8 consistent representation of model canopy has an element packing density low enough that 9- 1 applies; for uniform anopics of the data on total transfer of momentum and matter to the anopy. using a sheltering factor (9) which is little much greater density 9 is likely to be signifkantly less dependent on wind speedand has the same magnitude than unity, and in that cue may be more strongly for momentum, contaminant gas and particles. The dependent on wiad speed. etc. than found here. The measurements inch&d in this are the relationship anopy considered has a height much greater than the latenl clement dimensions; the model is conceptually btwecn z. and d and the values of deposition velocity inappropriate lo arrays of bluff elements (height for the ThB atoms and for particks in the diameter - width). Also, the absolute size of the individual range 0.08-32 pm. Landsberg and Thorn (197 1) found that for compkx pknt parts in a wind tunnel shelter ekments here is small enough that panmctcrintion of factors defined for water vapour and momentum the collection mechanisms in terms of the mean wind transfer were the same. Thorn (1972) and &lot lI 01. speed. with a correction factor 9. an be expected to have some validity; for much larger ekmentx, turbulent (1976) assumed that this equality holds for canopies. The predictions rekting to total transfer arc not deposition to quasi-plane surfaces might be a better sensitive to the detaikd shape of the vertical diffusivity starting point. In addition, natuml elements may have structural profik: in the momentum case most of the drag occurs properties which differ from those in the model anopy in the upper anopy and in the contaminant cue the rumciently to introduce new aerodynamic effects: colkction by ckmeats pclerally does not provide a strong enough sink to ause large conantntion they may have wter flexibility and, for eumpk. gradients in the upper canopy. A more detaikd test of be subject to streamlining and fluttering. Furtherthe momentum-transfer model is provided by a commore, the singk+kment colkction efliciency can be parison of inanopy wind speed profiles. Only one strongly influenced by surface micro-rough= for measured profile is anilabk. 81 Y, = 0.6 m s- r; there intermediate-sized particks (- I pm) and by bounceis ~CXAagreement in the top quarter of the canopy but off for larger particles ( 2 IO pmk for gasesin general signihcant diverrna thereafter. No experimental unthe rurfaa resistance Of nrtural ekments is often rn certainties are given; it is possibk that the masureimportant factor. For ThB atoms and Lycopodium spores, the prement technique used (a pitot tube--Garland. private communication) became unreliable in the lower dicted magnitude of o,is fairly insensitive to details of anopy due to grater turbukna intensity. the modelling ofconditions our the substrate, but this On the other hand. the predicted vertical profik of is no1 true of the prtition. p. For the assumptions the deposit of ThB atoms. also at Y. - 0.6 m s” ‘. is in made in this work, implying a quasi-shear structure IO

Dry deposition to a uniform canopy

the flow in the vicinity of the substrate, p is undcrpredicted in some cases by an order of magnitude for q = I. For the ThB atoms, predicted p values can be brought close to the measured range using q = 0.5, a value within the rangeconsistent with element packing density (although this value leads to underprediction of u,valucs). However, this improvement probably has no real significance since for Lycopodium spores a decrease in q does not significantly affect the large underprediction in p at the highest u+ value. Although there is only a limited amount of data on p. the comparisons indicate that a better representation of the transfer to the substrate is required, perhaps one which recognizes that the turbukncc in that vicinity may be determined more by the downward pcnttration of eddies generated by flow over the elements than by shear at the substrate. Ac&nowIe&menr-This work was partly funded by the Radiation Protection Programme, DirectorateGeneral for sdcacc, Research and Development, the Commission of EuropeanCommunities,undercontract B16.0131. UK(H).

REFERENCES Bathe D. H.

opia-I.

(1979) Particle trsnsport within plant canA framework for analysis. Atmospheric

Enuironmenr 13, 1257-1262. Bathe D. H. (1986) Momentum transfer to plant canopies: influcna of structure and variable drag. Atmospheric En~+onmenf 20, 1369-1378. Batchelor G. K. (1967) An Introduction IO Nuid Dynamics, Cambridge University Press, Cambridge. Belot Y, Baille A. and D&as J. L. (1976) Models numtriquc de dispersion dcs polluants atmosphtriques en pr&cna de couvcrts v#taux. Atmospheric Environment 10, 89-98. Belot Y. and Gauthicr D. (1975) Transport of micronic particles from atmosphere to foliar surfaas. In Hcor ond Mass Transjer in the Biosphere (edited by de VI& D. A. and Afgan N. H.). pp. 583591. Scripta Books, Washington, DC. Brutsaert W. (1979) Hat and ma.w transfer lo and from surfnccswith dense vegetation or similar permeable roughness. Boundary-hyer Met. 16.365-388. Brutsaert W. (1982) Euaporation into the Atmosphere. D.

Rcidcl. Dordrozht,Hollsnd. Chsmberlain A. C. (1966) Transporl of 8s~~ lo and from grass and grass-like surfam. Proc. R. Sot. Land. A290, 236265. Chamberlain A. C. (1967) Transport of Lycopodium spores aod other rmall part&s lo rough surfaces. Prcc R. Sot. Land. AX%, 45-70. Chamberlain A. C. (1970) lntcraptioa and retention of radiactive aerosols by vegetation. Afmospheric Enaironmwnr4.57-78. Cionco R. M. (1965) A mathematical model for air flow in a vegetative canopy. 1. appl. Met. 4, 517-522. Cionco R. M. (1978) Analysis of canopy index values for various canopy densities. Boundary-Layer Met. 15.81-93. Cowan 1. R. (1968) Mass, heat and momentum exchange between stands of plants and their atmospheric environment. Q, JI R. met. Sot. 94.523-544. Davidson C. 1.and Friedlander S. K. (1978) A filtration model for aerosol dry deposition: application lo traa metal deposition from the atmosphere. J. gcophys. Res. 83, 2343-2352. Davies C. N. (1966) Aerosol Scirncc. Academic Press, . . Lundon.

1583

Fcmandcz de la Mora J. and Friailander S. K. (1982) Aerosol and gas deposition 10 fully rough surfaces:filtration model for bldt-shapad ekmmts. Inf. J. Hear Mass nansjw 25, 1725-1735. Gregory P. H. and Stalman 0. J. (1953) &position of airborne Lycopodium spores on plane surfaces. Ann. appl. Bioi. 40,651-674.

Inouc E (lW3) On the turbulent structure of airflow within crop canopies. J. Met. Sot. Japan, Ser. I1 41.317-326. Kondo J. and Akashi S. (1976) Numerical studies on the twodimensional flow in horizontally homogeneous canopy __ layers. Boundary-Luyer Mer. 10.255-272 Landsbern J. J. and James G. B. (1971) Wind orof11a in slant canopia: studies on an analy&l model. j. appl. E&l. 8, 729-741.

Landsberg J. J. and Powell D. B. B. (1973) Surface exchange characteristics of leaves subject lo mutual interference. Agric. Met. 12, 169-184. Landsberg J. J. and Thorn A. S. (1971) Aerodynamic properties of a plant ofcomplex structure. Q. JI R. met. Sot. 97, 565-570. Lcgg B. and Monleith J. (1975) Heat and mass transfer within plant canopies. In Heat and Mass Transjer in de Biosphere (edited by de Vria D. A. and Afaan N. H.). DD. 167-186. Scripta &oks. Washington, DC.‘. ’ . Lcgg B. and Powell F. A. (1979) Spore dispersal in a barley crop: a mathematical model. Agric. Met. 20, 47-67. Parnas R. and Friedlander S. K. (1984) Particle deposition by diffusion and interception from boundary layer flows. Aerosol Sci. Techno!. 3, 38. Plate E. J. and Quraishi A. A. (1965) Modeltin of velocity distributions inside and above tall crops. J. appl. Met. 4, 4W408.

Raupach M. R. and Thorn A. S. (1981) Turbulence in and above plant canopies. Ann. Reu. Fluid Mech. 13.97-129. Raupach M. R., Thorn A. S. and Edwards 1. (1980) A windtunnel study of turbulent flow close IO regularly arrayed rough surfaces. Boundary-Lover Mel. 18. 373397. Robinson A. (1956) On th; m&ion of sm&l particles in a potential field of flow. Communs. pure appl. Marh. 9, 69-84. Schack C. J., Jr, Pratsinis S. E. and Friedlander S. K. (1985) A

g.cneral correlation for deposition of suspended particles from lurbulent gases to completely rough surfaces. Atmospheric Environment 19.953-960. Schlichting H. (1960) Boundary Layer Theory, 4th edn. McGraw-Hill, N.Y. Scginer I. (1974) Aerodynamic roughness of vegetated surfaces. Boundary-Layer Met. 5, 383393. Scgincr I, Mulhcarn P. J.. Bradley E. F. and Finnigan 1. J. (1976) Turbulent flow in a model olant canotw. Boundaw &ye; Met. 10.423-453. . ._ . Shreflkr J. H. (1976) A model for the transfer of gaseous pollutants lo a vegetational surface. J. appl. Met. 15, 744-746.

Slinn W. G. N. (1982) Predictions for particle deposition lo vegetative canopies. Atmospheric Environmenr 16. 1785-1794.

Thorn A. S. (1968) The exchange of momentum, mass, and heat between an artificial kaf and airflow in a wind-tunnel. Q. JI R. met. Sot. 94,44-55. Thorn A.S. (1971) Momentumabsorption by vegetation.Q. II R. met. Sot. 97.414-428. Thorn A. S. (1972) Momentum, mass and heat exchange of vegetation. Q. I/ R. met. Sot. 98, 124-134. Wilson N. R. and Shaw R. H. (1977) A higher order closure model for canopy flow. J. uppl. Mer. 16, 1197- 1205. Wiman B. L B. and Aaron G. I. (19851 Aerosol dca&ion and deposition in for&s-a model .analysis. &ospheric Environment

19. 335-347.

Wooding R. A, Bradley E. F. and Marshall J. K. (1973) Drag due to regular arrays of roughness ckments of varying gCOmetrY. Boundary-fuyer Mer. 5. 285-308.

1584

B. Y. UNDERWOOD APPENDIX

empirka1 fit:

I

Substituting (4) into (1) and introducing variables U = u/u1 and 9 = z)h gives

$[%Jl=

w

C(u,, U)U2+ n

u; (30) = 13.fia-2’3

and 2. = l/h.

where C(u,, U) = 917bc&,U)

-t

tt -4

For q > K( 1 -d/h), L is constant and Equation (Al)can be integrated once 10 give

(A91

demonstrating the adequacy of the Davies approach for (I in the range i-10. Applying the mod&d prescriptionin ation 3.2 to the a=ocascgivcs

(1lK - l/u)

h(iiK8)

whereas a plausible extension to Brutsacrt’s prescription using a z+ = 11 refercna level is v; (11) = llu-s’3.

where lJ W,,

U. U,) =

C(u,, U’)(U’)2dU

(A3)

s 0,

U0 is the wind speed at the lower boundary of where L is a constant, and (dU/dq), is the derivative of U at the pkcc where U = U,.

(AL01

(All)

Yalucs of a; based on (AlO) and (All) am compared in Tabk Al for the value of o1= 0.545 used in the present work, demonstrating that it gives an adequate rcprcacntation for u in the vicinity of 2.778. In the cnmopy case the z+ = 11 reference level is replaced by z L: I,.

APPENDIX

3

For Q = 0 and cd constant, (A3) gives

where 9c*oh ‘I3

Y =s

( >

WI

‘

This gives an exponential profile if (dU/d& = yti,. which does not apply in general (and cannot apply if L = const. is extended down to the substrate). However, for values of9cdah typical of some natural canopies, near exponential brhaviour obtains in the upper canopy. &low zID, the LHS of (I) is replaced by vd2u/dz2, which, with new variables, gives

&- =C(u,, u)U2+Q where Re = hu,/v, with first integral

=

2Re(l(u,.

U.

APPENDIX

In general, Equation (14) usually exhibits three regimes of bchaviour as the magnitude of E is varied for fixed substrate properties. a. As E becomes very small, rd approaches a value determined by deposition to the substrate alone, albeit with the aerodynamic contribution to the total rc&ana mod&d by the presence of the canopy: i.e. as E-r 0, Equation (14) becomes

(Al21 (i.e. flux is constant), subject to E = 1 at z = h. The following lower boundary condition can be applied at zb dose to the substrate (Al31

(A61

where u> is governed by substrate properties and conditions near the substrate. Thence it is easily shown that

+

01 QU).

i=LR(h,*,f a,(h) u;t

(A3

where

2

Davies’ (1966) prescription for calculating deposition to a smooth surface in shear flow assumesthat thcuidy diffusivity component of the total di~usivity falls linearly from its value atz+ = 3O(whcrcr’ = zu*/v)toa~ueO.~vatg’ =&and that the total diffusivity then drops to a constant, D, for z+
(AS)

In Tabk Al cakuktcd values of u; based on (A8) are camped with those based on 3rutsaert’s (1982) suni-

a

0; (30) Davies

t@I Brutsaert

t@l) Eqn (AlO)

a:(111 Eqn (Al 1)

1 3 10

13.8 28.3 68.0

13.8 28.3 63.1

8.2 21.2 64.0

29 51:i

(Al51

s ‘b Naturally, as E -+ 0, p + 1

b. For an intc~~iate range of E, where E is Largeenough for most of the flux to arise from transfer to ckments and not to the substrate and yet not so large that them is signiikant attenuation in conantration through the canopy, then a$ (u,) is closely proportional to E(u,) if E(u) is given by a power law: from Equation (14), the total flux at height h, FT, under these conditions is given by h

FtTable A 1. Comparison of values of u; for a smooth surface cakuktcd by various prescriptions

dz’/K,(t’).

Wt. r,,) =

i.e.

f0

qoE(ukydr

ud(r= FrMW)

_ 90x(h)

‘E(u)udz, f0

*“E(u)udu * 9a o (du/dz). I

SU-E(u)=Ku’-K~(~)I-Elu,t(~~.

qEtu,)oh

(A171

Wt8t

Transforming to U and n gives rulll‘.

v;tu.1-

(A%)

J,,

,I.+

v-

I

(dU/du,dU*

(Al9)

Dry deposition to a uniform anopy If the U(q) profile is independent of II, then

1585

For an exponential profile, i.e. dU/dq = gU _.#I’ 0; (U.) = 9Uu.W

The U(q) profile will be strictly independent of u* only if c&3 a constant 8nd the lower boundary condition is indqmknt of Y,. However. even in the ak~latioas of section 2 the profile is not a strong function of Y., erpsirlly in the upper anopy where much of the drag and deposition occurs.

“h

(8+ I)Y

(A211

where U, = U(I). The o; a E regime was noted by Slinn (1982). c. In the third regime, E is large enough that x falls apprcckbly through the canopy, and o;(uJ will show a different dependence on u, than does E(u, j