A mathematical model for the dispersion of air pollutants in low wind conditions

Pergamon Atmospheric Environment Vol. 30, No. 8, pp. 1209-1220, 1996 Copyright © 1996 Elsevier Science Ltd

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1352-2310(95)00442-4

A M A T H E M A T I C A L M O D E L FOR THE DISPERSION OF AIR P O L L U T A N T S IN LOW W I N D C O N D I T I O N S

MAITHILI SHARAN, ANIL KUMAR YADAV, M. P. SINGH, P. AGARWAL and S. NIGAM

Centre for Atmospheric Sciences, Indian Institute of Technology, New Delhi-110016, India

(First received 12 June 1995 and in final form 5 October 1995)

Abstract--For the atmospheric dispersion of pollutants in low wind conditions, a steady-state mathemat- ical model has been formulated which takes into account the diffusion in all the coordinate directions and the advection due to mean wind. The analytical solution has been obtained by assuming constant eddy diffusivity coefficients in the advection-diffusion equation. The solution reduces to a familiar form which yields the standard Gaussian plume solution when slender or thin plume approximation is invoked. The analytical solution obtained matches with the Gaussian plume solution for ground-level concentration at the plume centreline.

The model has been evaluated with data collected from the experiments conducted at the sports ground of the Indian Institute of Technology, Delhi (India). The results are reasonably good keeping in view the limitations of the data and the model. The model performs well as indicated by the agreement between the observed concentrations and those calculated using 30 min and 3 min averaging.

Key word index: Mathematical model, low wind dispersion, tracer experiment.

I N T R O D U C T I O N

Air pollutants released from various sources affect directly or indirectly man and his environment. The resulting ground level concentration patterns have to be estimated for a wide variety of air quality analyses for social planning and industrial growth. Air pollut- ants emitted from different sources are transported, dispersed or deposited by meteorological and topo- graphical conditions. Dispersion of pollutants in the atmosphere is governed by the following dominant mechanisms (Wark and Warner, 1981):

(1) mean air flow that transports the pollutants downwind, and

(2) turbulent velocity fluctuations that disperse the pollutants in all directions.

Wind speeds less than 2 m s-1 are generally con- sidered to be low, as most of the conventional models for dispersion are to some extent suspect because of their assumptions when the wind speed falls below about 2 m s-1 (Smith, 1983). These are of frequent occurrence at many sites, particularly in tropical re- gions and have a great potential for pollution epi- sodes. Pasquill (1961) asserted that diffusion under low wind conditions is very irregular and indefinite. Later, several diffusion experiments under light-wind, stable conditions supported Pasquill's assertion and indicated that effective estimates of dispersion co-

efficients (a's) can correspond to any atmospheric stability. Under low wind conditions during daytime, instantaneous plume usually does not spread much horizontally whereas during night it can meander over a wide angle (Etling, 1990). The resulting ob- served concentration distribution is non-Gau~sian and multi-peaked. As is evident from field e~peri- ments and theoretical considerations (Kristens¢n et al., 1981; Hanna, 1983), averaged concentration of pollutants under meander conditions can be a factor 2-6 lower than for the situation with straight plumes downwind of the point sources. In most of the Gaus- sian plume models, when mean wind speed becomes very low ( < 2 ms -1) the pollutant concentration tends to go exceptionally high because U appea~rs in the denominator. Thus, it has become necessary to study dispersion patterns during low wind conditions.

Generally, under moderate to strong winds, diffu- sion in the direction of wind is neglected in compari- son to advection. But this may not be the case in low winds and, consequently, diffusion could be compara- ble with advection in the direction of wind. Arya (1995) has shown that the conditions under Which downwind diffusion might be important are essential- ly strong convection and weak winds (U < 1.5 m S- 1).

In the past, different modelling approaches have been adopted to deal with low winds. Various investi- gators have attempted modifications at the applica- tion level (i.e. in terms of the dispersion parameters).

1209

1210 M. SHARAN et al.

Sagendorf and Dickson (1976) used schemes like split-sigma and segmented plume to improve upon the standard method to deal with nighttime low wind cases. For the estimation of a's, the split-sigma scheme involves the splitting of stability into horizontal and vertical whereas the segmented plume method con- sists in splitting the test period into sub-intervals. Zannetti (1981) has suggested a modification for com- putation of dispersion parameters (tr) in low winds. He introduced the concept of Umi, in order to artificially dilute the plume. Recently, Cirillo and Poli (1992) have intercompared four semi-empirical schemes with the diffusion data in low wind speed, inversion condi- tions. In a more recent study, Sharan e t al. (1995a) have suggested "short-term averaging" approach which is quite useful in the absence of detailed wind data.

Sirakov and Djolov (1979) made an attempt to model the dispersion of pollutants from a continuous source in the absence of the wind. However,the para- meterization of turbulent exchange coefficients to compute concentrations for practical purposes is left undone. Demuth et al. (1978) have proposed an ana- lytical model for calm wind situations when there is a finite mixing height. For unstable cases, they have taken the model by Berlyand and Kurenbin (1970) defined by diffusivity coefficients varying in space, and introduced a reflection at the top of the mixing layer. For the stable cases, they have taken a model with constant diffusivity coefficients. Yokoyama et al.

(1979) have derived a puff formula for computing the concentration of smoke emitted from a point source in calm wind conditions by expressing the dispersion parameters as linear functions of time. However, it requires a prior knowledge of the time for which calm conditions are persistent. In some cases, an arbitrary value of the persistent time was assumed. Yamamoto et al. (1986) have simulated the lateral profiles of concentration in weak and variable wind conditions by calculating the trajectories of the virtual particles emitted from a ground level source. These trajectories are computed using wind speed and direction ob- tained from tower observations. However, this method provides the proportional (relative) concen- tration at a point rather than the actual concentra- tion. They also developed a non-Gaussian model for low and variable wind conditions by assuming that the lateral concentration profiles are proportional to the angular distribution of the wind direction and inversely proportional to the product of wind speed and distance from the source. Okamoto and Shiozawa (1987) have proposed a trajectory plume model to handle calm and weak wind conditions. They have integrated three-dimensional puff model with respect to time by assuming the growth rate of spreading plume to be linear with diffusion time. Vari- ous aspects of atmospheric diffusion in low winds have recently been reviewed by Yadav et al. (1995).

The currently available dispersion parameters are based on field experiments conducted mostly in mid-

latitudes. These may not be applicable to tropical conditions. Therefore, a series of SF6 (tracer) field experiments were conducted at liT, Delhi, in order to formulate the horizontal dispersion parameter for different stabilities in tropical conditions. At this loca- tion, low winds occur frequently during day as well as nighttime. Another objective is to develop a suitable mathematical model to study dispersion patterns in low wind conditions. In the present study, an attempt has been made in this direction. A mathematical model for low wind dispersion is formulated and is evaluated with the field data.

MATHEMATICAL FORMULATION

Dispersion of a pollutant in the atmosphere is gov- erned by the basic atmospheric diffusion equation. Under non-isotropic conditions, the atmospheric dif- fusion equation satisfying the equation of continuity can be written as

[KYl at

a c l (1)

where C is the pollutant concentration, S represents the source term, R is the removal term, and u, v and w are the wind components and K~, Ky and Kz are the eddy diffusivity coefficients along the x, y and z direc- tions, respectively.

Equation (1) forms the basis of most air pollu- tion models. On the left-hand side of equation (1), the first term is time-dependent which accounts for non- stationary conditions whereas the remaining terms represent the transport due to wind. The first three terms on the right-hand side of equation (1) represent eddy diffusion in the x, y and z directions, respectively. The following assumptions are made:

(1) Steady-state conditions are considered. (2) There is no removal or washout of the pollu-

tants, i.e. R = 0. (3) As the magnitude of the vertical component of

velocity is smaller than that of the horizontal, advec- tion in the vertical direction is neglected in compari- son to that in the horizontal direction. This assump- tion is valid except for particulate pollutants (with appreciable settling velocities) under low wind speed conditions.

(4) The emission source is assumed to be located at the origin and accordingly

S = q f ( x ) f i ( y ) f i ( z ) (2)

where q is the emission rate, and 6 is the Dirac's delta function.

(5) Eddy diffusivities are considered to be constant.

A mathematical model for the dispersion of air pollutants 1211

Based on the above assumptions, equation (1) be- comes

~C ~C ~2C ~2C + = r T;-cx + K, U~X dy2

~2C + K,7"5- + qb(x)f(y)~(z). (3)

0 z "

Boundary conditions

Equation (3) is subject to the following boundary conditions:

(1) Pollutant concentration tends to zero at large distances from the source

C--,O aslxl , lYl, z ~ oo. (4)

(2) The pollutant is not absorbed at the ground and therefore there is no diffusion flux at the surface, i.e.

\ O z j = O at z =0 . (5)

Equation (3) is a three-dimensional linear elliptic partial differential equation. We solve it analytically, using the boundary conditions (4) and (5), in the next section.

SOLUTION

Using the transformation:

X = xK~ 1/2, Y = yK~ 1/2,

equation (3) becomes

U (~C (~C ~2C ~2C V - - = + - - + aY ~X 2 ~y2

Z = zKF 1/2

c~2C + ~-~ + q'~(X)~(Y)6(Z) (6)

where U = uK~ 1/2, V = vKf 1/2, q, =q(KxKrK~)- 1/2.

Using the cosine transform

C(X, Y, 23) = C(X, Y, Z)cos(),3Z)dZ

equation (6) reduces to

u~C_ _ ~C 02C ~2C o x + v-#-f = -g-k~ + -g-~ - ,~ ~ c

?; + q'tS(X)3(Y). (7)

Now, using double Fourier transform

(~(2t, 22, 2a) = C(X, Y, 23) -oo -oo

x e 2ni(21 x + 22 Y) dX d Y

equation (7) becomes

~ = ~ _ 2 q' 4n2(2~ + 22 ) + 22 - 27ri(21U + 22V)" (8)

On inverting (8) with respect to 21 and 22, using 632,207 and 917 (Campbell and Foster, 1957, pp. 74,

39, 125) we obtain

× Ko[(X2 + y2)1/2(2 2 1 2

where Ko is the modified Bessel function of the second kind of zero order.

Inverting equation (9) with respect to 23

fo [ x Ko (X 2 + y2)1/2

Using ~ 6.677.5 (Gradshteyn and Ryzhik, 1981, p. 736), we get

.UX + VY- C(X, Y, Z) = 2~(X 2 + y2 + z 2 ) e x p 2 _

E 1 2 V2)1/2 xexp - ~ ( U +

x (X 2 + y2 +Z2)1 /2 ] . (10)

In original coordinates (x, y, z)

C(x, y, z) = 2~(KxK, Kz)l/2reXp[ ~ ~ -~ +

2 \ K + ~ r r (ll)

where

I X y2 Z211/2 E+E+Ej •

Equation (11) allows us to compute the concentra- tion at any point (x, y, z) with reference to fixed coor- dinate axes. Assuming x-axis to be in the direction of wind, i.e. v = 0, and taking, u =/.7, we have

q C(x, y, z) = 2n(K~KrKz)l/2r

x exp 2-Kx 2(K~)1/2 r (12)

where r is same as in equation (11).

1212 M. SHARAN et al.

The solution (12) can also be obtained by integrat- ing the solution (i.e. puff solution) of the correspond- ing time-dependent equation with respect to time from 0 to ~ . For a similar problem, Roberts (1923) and Seinfeld (1986) have used the boundary condition for z from - ~ to oo, probably in analogy to puff approach, which does not seem to be realistic phys- ically in the case of a plume.

Slender plume approximation (a particular case) For the case of emission from a point source, we

can use the slender or thin plume approximation when the crosswind spread of the plume is small compared to the downwind distance it has travelled. In other words, slender plume approximation indi- cates that only the concentrations close to the plume centerline are of importance. Equation (12) can be rewritten as

C(x, y, z) =

where

2n(KrK,)l/2x(1 + p)1/2

x exp [2UK x Ux(1 "9"gxx -[- P)1/2 1 _]

K /'y z

For thin plume approximation, we should have

p , ~ l

which is equivalent to having

y2 + z 2 (1) ~ ,~ 1

(13)

(14)

gx gx (2) ~ ~< O(1), ~ ~< O(1) (15)

where "O" represents the order. The condition (15) seems plausible under moderate

to strong winds. Equation (13) can be simplified using binomial expansion and condition (15) to give

C = 2 n ( K r K z ) l / 2 x e X p [ - £ ( Y 2 z 2 4 X \ ~ r + ~ ) 3 (16,

which is the classical solution in the Gaussian form, The same result can also be obtained as a limiting case by taking Kx --* 0 which is the same as neglecting downwind diffusion.

COMPARISON WITH THE TRADITIONAL APPROACH

It is clear now that for low wind conditions the formula yielding the Gaussian plume solution is ap- proximate whereas the solution (12) seems more accu- rate as it accounts for the downwind diffusion also.

Thus, it is worthwhile to examine theoretically the situations of overprediction/underprediction by for- mula (16) in treating weak wind cases. This becomes important as formula (16) is simple, conceptually ap- pealing, and computationally efficient. Referring to equation (13) which is just a rearrangement of equa- tion (12), we can write the ratio, R, of equations (16) and (12) as

R=x/-~+p)expI~((l +p)l/2-(l +P)) 1 (17)

where p and fl are dimensionless parameters defined by

,18, + = x '

Notice that fl resembles the well known Peclet number, Pe, and it essentially represents the ratio of advective (convective) transport to diffusive transport. Here, both p and fl can take only positive values. Regarding the physical interpretation of p, one can think of the magnitude of p as giving the region of interest relative to the plume axis. The values of p close to zero represent the region in the proximity of the plume centreline. For application purposes, one can safely restrict oneself to the range (0,1) for p. On the other hand, fl can be physically interpreted as the parameter whose magnitude indicates the atmo- spheric conditions in terms of the strength of winds. Small values of fl can be related to the weak winds when the downwind diffusion becomes important and the region of interest remains close to the source, whereas large values of fl imply moderate to strong winds when the downwind diffusion is neglected in comparison to the advection and the region of interest extends to a larger distance from the source. For p = 0, we have R = 1 for all finite values of ft. How- ever, p equals zero in two cases: (1) y = 0 and z = 0, which implies that the ground-level concentration along the plume centreline is not affected by the inclu- sion of the downwind diffusion term in the formula- tion; (2) Kx ~ 0 (neglecting diffusion along the axis of the wind).

For other cases, we have studied the sensitivity of R with respect to the parameters p and ft. Figure 1 shows the variations of R with the parameter fl for various values of p. The figure reveals that for values of fl less than two, the ratio R increases with the increase in p. That is, for the condition of weak winds (when the downwind diffusion is expected to contribute) the application of formula (16) leads to overestimation, for example, to the extent of about 25% when p = 0.6. Even for the higher values of fl, there is an overpredicting trend for a small range of p (upto 0.4 in this figure). Finally, the situation with higher values of fl and p (close to 1) are physically uncommon and the variation of R is found to be in reverse order.

A mathematical model for the dispersion of air pollutants 1213 1.50~ p = o . o

~= :._- ~. p=0.2 ~ ,A~p=04 ill v w ~ v v •

• ~ - - - - - - - . . . . . . . . . _ ~ - ~ ~ : : : : :¢: : p = 0.6 • , .. ~ ~ . ~ ****, p = o . s

1"25 F

L 0

1.00

0.75 0.1

I . 1 I I I I 1 I I 1 I i I

!.0 10.0

Fig. 1. Variation of the ratio R with the parameter fl for various values of the parameter p.

PARAMETERIZATION OF EDDY DIFFUSIVITIES

For comput ing the concentration using formula (12), we require suitable parameterizat ion of the eddy diffusivity coefficients Kx, Ky and Kz. We express K r and Kz in terms of % and o-z, the standard deviations of the crosswind and vertical Gaussian concentration distribution respectively, as follows (Rao, 1983):

0 day 2 /.7 d0. 2 (19) K, = ~ - d x ' K~ = ~ - d--x-

In analogy to Ky and Kz, we express Kx in terms of 0-x (Llewelyn, 1983).

In order to make use of the large amount of empiri- cal data on the dispersion parameters, try and 0.,, available in the literature for various meteorological and terrain conditions, Ky and Kz can be expressed as (Rao, 1983)

~x t7 2 Ky = 0 "2, Kz = "~x 0.=. (20)

The studies related to dispersion parameters have been reviewed extensively by Gifford (1976), Hanna et al. (1977), Irwin (1983) and Yadav et al. (1995). Thus, in terms of 0.x, 0.y and o-z, solution (12) can be written as"

C(x, y, z) = ~U0.~0.y0.: L0.~ 0.~

where

i X x ~ y 2 2.211/2 s = +_-25+_-25 .

O-y o- z

It may be noted that solution (12) does not resemble the normal distribution. The parameterizat ion of eddy diffusivity in terms of dispersion parameters through equat ion (20) is usually invoked, in a math- ematical sense, when the solution is of Gaussian type. However, the Fickian diffusion approximat ion (i.e. equat ion (20)) which is also the solution to Taylor 's

statistical diffusion equation at large t is used in this study.

EXPERIMENTAL SETUP FOR VALIDATION

As mentioned in the Introduction, the field experiment was conducted at the sports ground, l iT Delhi to formulate the horizontal dispersion parameter in the tropics (Singh et al., 1991; Agarwal et al., 1995). We have utilized this diffusion data set to validate our model. For the sake of completeness, we describe here briefly some of the relevant experimental details.

The experimental site lies in the city, has fiat terrain but surrounded by buildings and other structures on almost all sides, and hence has been chosen to study dispersion in urban terrain. The sampling grid involved receptors on 50, 100 and 150 m and in some cases on 200 m circular arcs with 45 ° angular spacing between them. The layout plan is shown in Fig. 2. Samples were collected at various times of the day to cover almost all atmospheric stabilities. S F 6 tracer was released at a height of 1 m above the ground and the Sam- plers were also placed approximately at the same height. Thus, for computational purposes the release can be con- sidered as a ground source. On the first day, tracer: was released continuously at the rate of 50 ml rain--~. On sub- sequent days, release periods were of 60 rain duration with sampling during the later half, and the release rate varied between 30 to 50 ml min-1. The relevant details regarding the experiment are given in Table 1. Meteorological !data (wind speed, wind direction, etc.) have been obtained from a 30 m multi-level micrometeorological tower installed dose to the experimental site. It was instrumented mainly with cup anemometers, wind vanes and temperature sensors at 1, 2, 4, 8, 15 and 30 m levels. A sonic anemometer was mounted at 4 m but unfortunately it had some technical problems re- garding its operations during the tracer experiments. Wind instruments supplied mean variables every 3 min.

MODEL PARAMETERS

For the concentrat ion computations, we require the knowledge of wind speed, wind direction, soUrce strength and the dispersion parameters. Traces of wind speed and direction of a particular day on which the experiment was conducted are given in Fig. 3. I t is

1214 M. SHARAN et al.

Metetrw° leOrgical

• : Receptor locations Wel, Pump house

~ ) : Release point I t I i I 0 100 200

m

Fig. 2. Layout of the site for tracer diffusion experiments.

Table 1. Details of the tracer experiments conducted at the sports ground, l iT Delhi in February 1991

Run no.

Meteoro. conditions

Sampling Release time Date Release rate W/S W/D (h) (m-d-y) period (mlmin- 1) Layout (m s- 1) (deg)

P-G stab. class

1 2 3 4" 5 •

6 7 8 9"

I(Y'

11 12 13" 14"

1200-1230 02-13-91 continuous 50 1.56 343 1530-1600 02-13-91 release from 0.74 291 1850-1830 02-13-91 02-13-91 0.39 302 2330-2400 02-13-91 (1100 h) to 0.41 10 0400-0430 02-14-91 02-14-91 0.29 313

(0430 h)

1000-1030 02-19-91 02-19-91 30 1.40 286 1245-1315 02-19-91 02-19-91 30 1.54 284 1645-1715 02-19-91 02-19-91 50 b 0.89 301 0000-4)030 02-20-91 02-20-91 30 0.35 289 0400-0430 02-20-91 02-20-91 30 0.61 265

1000-1030 02-21-91 02-21-91 30 1.07 320 1215-1245 02-21-91 02-21-91 50 b 1.55 334 1530-1600 02-21-91 02-21-91 50 b 1.08 331 1830-1900 02-21-91 02-21-91 30 0.73 13

A-B B D F F

B B B F F

A-B B B D

a 15 m level meteorological data are used in the computations. bRelease point has been shifted 100 m towards northwest.

evident f rom the figure tha t mos t of the t ime and, in part icular , du r ing the test runs, the wind speed was less t h a n 2 m s - 1 at 4 m height. Windspeeds are less t h a n 3 m s -1 mos t of the t ime even at 15 m level. Fur ther , the var ia t ion in the wind direct ion with t ime is also visible.

The existing formula t ions for the dispersion para- meters can be broadly classified into the following three groups:

(a) me thods based on power law funct ions (Briggs, 1973);

A mathematical model for the dispersion of air pollutants 1215

.--.

400

I I I I I 1 I

i i I i J I I I I I I I ~,

- --I - - I-- - -q - - I- - -I - - I- - - I I I I I I A

I : l ; , . . / ' A , / [

"J" i I,t i l l

i

i

I

I

- 7 ~ ' I

I

I I l l l l l l l

8 ~ 1 0 0 0

_ _ _ 1 ~

I I I I

7--I---I--

I I

1

iV l i 14

I

I l l l l l l l

1200 1400

I I

I I

m I I

I I

I

I

!h ,

--~. qq.j' - - I - - L._ . . i n , " . , t t I i J i l l ,

1600 1800 2000

Local time (hours) 600 2200 2400 260O 2-800 30OO

350

3 0 0 ~'~ o

25o ..~

200

4 m l e v e l . . . . . . 15m level

Fig. 3. Time traces of wind speed and wind direction for a day on which diffusion experiments were conducted.

(b) methods based on statistical parameters such as horizontal and vertical wind direction variances (tr0, a~) (Draxler, 1976);

(c) methods based on similarity theory (Hanna et al., 1982).

Methods based on similarity theory are difficult to use as they require the knowledge of parameters such as friction velocity (u.), convective scaling velocity (w.), mixed layer height (Zi). The formulations based on statistical methods involve a0 and a , which need large amount of meteorological data for their compu- tations. Thus, in the present study, we have adopted dispersion parameters for urban terrain by Briggs which are based on power law functions. These are analytical expressions depending upon downwind dis- tance and atmospheric stability. The atmospheric stability has been calculated from Pasquilrs turbu-

lence typing scheme (Gifford, 1976) based on wind speed, solar insolation, and cloud cover.

RESULTS AND DISCUSSION

The concentration is computed using data collected at 4 m height (wherever 4 m data are not available, 15 m data are used) of a 30m multi-level micro- meteorological tower. In all, 16 test runs were conduc- ted for the purpose of computation, out of which 2 l~ave been dropped due to loss of data. Some of the receptors falling close to the well, stadium and the wall (Fig. 2) in the field have been discarded because of the possibility of unusual accumulation, stagnation or re-circulation, etc. Minimum threshold value of the instrument used for wind speed is 0.3 m s - 1 and the observations below this value have been replaced by 0.3 m s - 1.

AE 30:8-C

1216 M. SHARAN et al.

Table 2. Observed and predicted concentrations for Run 7 obtained with 3 and 30 min averaging

Predicted conc. (ppt) Sample Observed no. conc. (ppt) 30 min 3 min

1 31 - - 23 2 248 411 362 3 190 122 146 9 6 - - 6

10 288 103 91 11 38 30 36 17 20 - - 2 18 143 46 40 19 72 13 16 20 0 - - 1

ppt--parts per trillion.

The micrometeorological tower provided the aver- age wind data for every 3 min period. However, the sampling period for each test run was 30 min. Thus, the concentration at a receptor can be computed in the following two ways:

(i) applying formula (21) with z = 0 for every 3 min and then averaging out for the test period. This is referred to as 3 min averaging.

(ii) using formula (21) with z = 0 for half hourly aver- age of the wind data. This is 30 min averaging.

As an illustration, results computed from these ap- proaches are shown in Table 2, for a typical test Run

7. This shows that short-term averaging (i.e. 3 min) covers a larger number of receptors in comparison to 30 min averaging. This is due to the fact that wind direction variation is taken into account by dividing the sampling period into smaller intervals. We point out that there is no sanctity about the 3 min averaging period. Depending upon the nature of the available wind data, the averaging period can be so chosen as to satisfy the condition t~ > Xmax/U, where t~ is the aver- aging time, [7 is the wind speed averaged over the period t~, and Xm~x is the distance of the receptor farthest from the release point in the direction of the wind.

The maximum concentrations computed using 3 and 30 min averaging are shown in Fig. 4 along with the corresponding observed maximum values. The comparison shows that in most of the cases predic- tions are reasonably close to the observations for both the approaches. Also, there is no significant difference between the predictions by the two methods (except in one or two cases). Because of the wide scatter in the field data, we could not capture properly the role of short-term averaging. However, it has been observed that this approach is quite often helpful in dealing with weak and variable wind conditions (Sagendorf and Dickson, 1976; Sharan et al., 1995a). Therefore, in the rest of the analysis, we restrict ourselves to 30 min averaging. Here, we would like to mention that in one or two cases, the predicted maximum does not co- incide with the observed one. This may be due to the fact that wind measuring instruments such as wind

1o 5

o

10 4

10 3

~ a e - ~ Predicted ( 3 0 - m i n ) :. :. -~ :. :- O b s e r v e d *-**'** Predicted (3-rain)

.'*" Z/

I0 2 I I I I I I I I I I I I I I 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Run no.

Fig. 4. Maximum computed concentrations compared with the observed maximum value for each test run.

A mathematical model for the dispersion of air pollutants 1217

10 4

lo 3

~ 1° 2

101

* ° @ o * * 4 *

. '"

i i l l l J J I I I I l l l l J I l I I I I I l i 1 I l 1 t l l l J I I I I

1 0 ] 1 0 2 1 0 3 1 0 4

Observed cone. (ppt)

Fig. 5a. Scatter diagram showing the observed concentrations vs those predicted using 30 min averaging.

1oo

@ IO

U

0 . 1

., ~ roe qr

. • . . - . • , , , , , , , , , , , , "_ . j , . *

• *ee* e*,,,,* * s , 8

--'8** 8 0 0 *

U

0 . 0 1 t * l J l l t J a J J l J t l 1 I * * * * i l l 1 t I I i I I l i I I

I 0 1 1 0 2 1 0 3 1 0 4

Observed cone. (ppt)

Fig. 5b. Scatter diagram showing the ratio ( = predicted/observed) vs the observed values of concentration; solid lines indicate a factor of two, and dashed lines a factor of four.

vane and cup anemometer do not respond reliably for very low wind speeds. This can be improved upon by making correction in the wind direction. Further- more, in Runs 2, 7 and 14, the maximum concentra- tion is observed at the outer arc (100 m). Stagnation, re-circulation, puddling and meandering are signifi- cant processes observed in low wind con- ditions which might have been responsible for con- centration at the outer arcs being greater than at the inner arcs.

Not only the maximum concentration, but also the overall situation is reasonably good as evident from

the scatter diagram of predicted vs observed concen- tration (Fig. 5(a)). Majority of the points lie around the line of perfect correlation within a factor of 2. This is also evident from the plot of the ratio (predicted/ observed) against the observed values (Fig. 5(b)). Fig- ure 6 shows that more than 65 % of the predicted cases are within a factor of 2.

As pointed out earlier, the concentration distribu- tion given by equation (12) is non-Gaussian. After comparing the concentrations computed froml our model with those obtained from the Gaussian model, we found that the difference between the two iS not

1218 M. SHARAN e t al.

1011

80

e ~

20

N \ N x x . N x x x x x x " x N \

x x x x x \ \ x \ \ \ \ x x ~ x x \

x x x

2

x \ \ ' ~

X \ \ ' ~ x x x , ~ . N . \ ' ~ , x x ,

,\\'~ , x x ~

" \ \ " ,x,\"R , x x ,

. . . . ,N\"q , x x ~ , \ \ ,

• x x ~ x N . \ ' ~

• x x • ~ . \ N ' ~

4 6 Factor

>6

spheric conditions. Thus, the highest concentrations are expected near the ground for a ground-level non- buoyant source. The concentration trend is not uni- form with the atmospheric stabilities for higher values of z. This is justified since the area under the concen- tration distribution curve is constant due to mass conservation for all stabilities.

Similarly, the crosswind and the downwind behav- iour of the concentration distribution can be plotted and examined for the general characteristics.

LIMITATION AND IMPROVEMENTS

Fig. 6. Bar graph showing the percentage of predicted cases There are some limitations of this study from ex- (using solution (21)) within a factor of 2, 4, 6 and more than 6. perimental and modelling points of view. These are:

too significant. This may be due to the fact that the difference in the two solutions is negligible close to plume centreline for ground level concentration from a ground level non-buoyant source. Further, the esti- mates of dispersion parameters utilized here are valid for moderate to strong winds.

Now, we try to examine certain important features of the concentration distribution given by equation (21) which depends on the wind speed, atmospheric stability, source strength and the spatial coordinates (x, y, z). For this purpose, we define the relative con- centration, C* as l?C/q.

Figure 7 reveals that the relative concentration decreases with height for all stabilities. Further, the concentration near the ground increases with the stability as one moves from unstable to stable atmo-

(a) less number of samplers, (b) angular spacing between the samplers being too

large, (c) 30 m tower not on-site, (d) poor response of the wind measuring instruments

during light winds, and (e) using the classical approach of converting eddy

diffusivities (K's) to Gaussian dispersion para- meters, although the concentration distribution is non-Gaussian.

Smaller number of samplers and large angular spacing between the adjacent samplers would be less useful in situations when there is almost a fixed plume direction with little lateral spread. So, to capture the concentration pattern better, we should have more samplers with lesser spacing between the consecutive ones. Although not on-site, 30 m micrometeorological

35 [- x = 100m

I ~ " = = = = Unstable y = 0 :- =- =- .~ ~- Neutral 30 . . . . . Stable

~ 25 N

~ 20

1o

15

o 0.000 0.001 0.002 0.003

Relative cone. (m "2)

0.004

Fig. 7. Vertical behaviour of the relative concentration at y = 0, x = 100 m for stable, unstable and neutral atmospheric stabilities.

A mathematical model for the dispersion of air pollutants 1219

tower could be taken as representative since it is located only 300 m (approx.) away from the release point. Data from fast responding and more sensitive instruments such as sonic anemometer could have improved the reliability of the meteorological information in low wind conditions. With the availability of faster data, we could have explained phenomena like meandering, puddling and re-circula- tion, etc., which are generally responsible for observation of higher concentration at the outer arcs than at the inner arcs and non-zero concentration in the upwind side. Moreover, with the calculation of tr0 and a~ we could have used more realistic estimation of dispersion parameters based on direct observations.

The approach of taking K's to be constant initially in the formulation and relaxing it later, although mathematically inconsistent, it has been widely accep- ted from the application point of view. As a simple case, Sharan et al. (1995b) have solved the problem with eddy diffusivities as linear functions of the down- wind distance.

CONCLUSIONS

A mathematical model has been developed for low wind conditions by taking into account the diffusion in the downwind direction. The analytical solution so obtained turns out to be non-Gaussian. However, for practical application of the model we have used the conventional conversion from eddy diffusivities to Gaussian dispersion parameters. The standard Gaus- sian plume solution has been shown as a limiting case of the model when the concentration close to the plume centreline is desired. Equivalent is the situation when the downwind diffusion is neglected (K~ ~ 0). The ratio R of the two solutions has been expressed as a function of the dimensionless parameters p and fl and it is suggested that theoretically the Gaussian plume equation leads to considerable overestimation during low wind conditions. The model has been evaluated with tracer data from IIT-SF6 experiment. Although the data set collected from this experiment has some limitations, we could get reasonable agree- ment between the observed concentrations and those calculated from the model.

Short-term averaging (3 min) was not very useful in the present study; however, it provides good explana- tion for the situations where the observed concentra- tion pattern is wide and has multiple peaks (Sharan et al., 1995a) due to fluctuations in the wind. In the existing framework, the model performed well with the given data set. However, this model needs testing with more data sets under different conditions for better understanding of its performance and domain of applicability.

Acknowledgements--The authors would like to thank the reviewers for valuable suggestions.

REFERENCES

Agarwal P., Yadav A. K., Gulati A., Raman S., Rao S., Singh M. P., Nigam S. and Reddy N. (1995) Surface layer turbu- lence processes in low wind speeds over land. Atmospheric Environment 29, 2089-2098.

Arya S. P. (1995) Modeling and parameterization of near- source diffusion in weak winds. J. appl. met. 34, 1112-1122.

Berlyand M. Y. and Kurenbin O. I. (1970) In atmospheric diffusion of impurities during a calm. American Irlstitute of Crop Ecology, Survey of USSR. Air Pollution Literature (edited by Nuttonson), Vol. IV.

Briggs G. A. (1973) Diffusion estimation for small emissions. U.S. National Oceanic and Atmospheric Administration, ERL Report, ATDL-106.

Campbell G. A. and Foster R. M. (1957) Fourier Integrals for Practical Applications. Van Nostrand, New York.

Cirillo M. C. and Poli A. A. (1992) An intercomparison of semi-empirical diffusion models under low windspeed, stable conditions. Atmospheric Environment 26A, 765-774.

Demuth CI., Berger A., Jacquart Y. and Legros Ch. (1978) Analytical modeling of pollutant dispersion during calm wind situations. In Proc. 13th Int. Colloquim, Paris, France.

Draxler R. R. (1976) Determination of atmospheric diffusion parameters. Atmospheric Environment 10, 99-105.

Etling D. (1990) On plume meandering under stable stratifi- cation. Atmospheric Environment 24, 1979-1985.

Gifford F. A. (1976) Turbulent diffusion-typing schemes: A review. Nuclear Safety 17, 68-86.

Gradshteyn I. S. and Ryzhik I. M. (1980) Table of Integrals, Series and Products. Academic Press, New York.

Hanna S. R. (1983) Lateral turbulence intensity and plume meandering during stable conditions. J. Clim. appL Met. 22, 1424-1431.

Hanna S. R., Briggs G. A. and Hosker Jr R. P. (1982) Handbook on Atmospheric Diffusion. U.S. Dept. of En- ergy report DOE/TIC-11223, Washington, D.C.

Hanna S. R., Briggs G. A., Deardorff J., Egan B. A., Gifford F. A. and Pasquill F. (1977) AMS workshop on stability classification schemes and sigma curves--Summary of rec- ommendations. Bull. Am. Met. Soc. 58, 1305--1309.

Irwin J. S. (1983) Estimating plume dispersion--A compari- son of several sigma schemes. J. Clim. appl. Met. 22, 92-114.

Kristensen L., Jensen N. O. and Peterson E. L. (1981 ) Lateral dispersion of pollutants in a very stable atmosphere. At- mospheric Environment 15, 837-844.

Llewelyn R. P. (1983) An analytical model for the transport, dispersion and elimination of air pollutants emitted from a point source. Atmospheric Environment 17, 249-256.

Okamoto S. and Shiozawa K. (1987) A trajectory plume model for simulating air pollution transients. Atmospheric Environment 21, 2145-2152.

Pasquill F. (1961) The estimation of the dispersion of wind- borne material. Meteor. Mag. 90, 33-49.

Rao K. S. (1983) Plume concentration algorithms with de- position, sedimentation, and chemical transformation. NOAA Technical Memorandum ERL ARL-124.

Roberts O. F. T. (1923) The theoretical scattering of smoke in a turbulent atmosphere. Proc. Royal Soc. (London), Ser, A 1114, 640q554.

Sagendorf J. F. and Dickson C. R. (1976) Diffusion under low windspeed, inversion conditions. In Preprints of Third Syrup. on Atmospheric Turbulence, Diffusion, and Air Qual- ity, Raleigh, North Carolina, pp. 277-282.

Seinfeld J. H. (1986) Atmospheric Chemistry and PhySics of Air Pollution. Wiley, New York.

Sharan M., Yadav A. K. and Singh M. P. (1995a) Compari- son of sigma schemes for air pollutant dispersion in low winds. Atmospheric Environment 29, 2051-2059.

Sharan M., Singh M. P. and Yadav A. K. (1996) A math- ematical model for the atmospheric dispersion in low

1220 M. SHARAN et al.

winds with eddy diffusivities as linear functions of down- wind distance. Atmospheric Environment 30, 1137-1145.

Singh M. P., Agarwal P., Nigam S. and Gulati A. (1991) Tracer experiments--a report. Tech. Report, CAS, IIT Delhi.

Sirakov D.E. and Djolov G. D. (1979) Atmospheric diffusion of admixtures in calm conditions. Geophysique 32, 891-892.

Smith F. B. (1983) Low windspeed meteorology. Personal communication.

Wark K. and Warner C. F. (1981) Air Pollution. l ts Origin and Control. Harper and Row, New York.

Yadav A. K., Sharan M. and Singh M. P. (1995) Atmospheric dispersion in low wind conditions. In Proc. World Con- gress on Nonlinear Analysis, (in press).

Yamamoto S., Mizuno T., Hayashi M. and Yokoyama O. (1986) An atmospheric dispersion model for weak and variable wind conditions. In Proc. 7th World Clean Air Congress, Sydney, Australia.

Yokoyama O. et al. (1979) Study on atmospheric dispersion of pollutants. Rep. National Res. Inst. for Pollution and Resources, MITI, 15, 408.

Zannetti P. (1981) An improved puff algorithm for plume dispersion simulation. J. appl. Met. 20, 1203-1211.