THE MODIFICATION
OF PLUME MODELS TO ACCOUNT FOR DRY DEPOSITION
Thomas W. Horst Pacific Northwest Laboratory Richland, WA 99352, U.S.A.
and predictions of four diffusion-deposition ABSTRACT. The assumptions models are compared, and two simple plume depletion models are recomOne model applies an analytical, constant eddy-diffusivity mended. solution of the advection-diffusion equation as a deposition correction to the general Gaussian plume model. Predictions of this model compare moderately well with those of the surface depletion model, an exact treatment of plume depletion, and it is particularly useful for estimating the transport and deposition of settling particles. The second model is a correction to the simple source depletion model that also accounts for the change in the vertical concentration profile caused by deposition. The computational requirements of this model are similar to those of the unmodified source depletion model, while its predictions near the surface are very close to those of the surface depletion model. 1.
INTRODUCTION
Models for the estimation of airborne pollutant concentrations commonly focus on atmospheric transport and diffusion processes, and ideally these models are evaluated with data from experiments using conservative, nondepositing, nonreacting tracers. However, real atmospheric pollutants (and, often, convenient atmospheric tracers) usually deposit at some rate on the earth's surface. As a consequence, atmospheric diffusion models must be modified to account for the loss of airborne pollutants by dry deposition, as well as to estimate the resulting surface contamination. Methods for modifying atmospheric transport and diffusion models to account for deposition depend strongly on the type of model being considered. For this reason, it is useful to divide diffusion models into two broad classes. The first type estimates airborne pollutant concentrations by directly solving the advection-diffusion equation for all points in the spatial domain of interest, e.g., by using a gradienttransfer assumption. Since deposition is naturally expressed as a flux at the lower boundary of the atmosphere, it is readily incorporated
Boundary-Layer Meteorology 30 (1984) 413-430. o 1984 by D. Reidel Publishing Company.
0006-8314/84/0304-0413$02.70
414
THOMASW.HORST
into a solution of the advection-diffusion equation as one of the required boundary conditions. The other type of diffusion model, e.g., Gaussian plume, similarity, and statistical models, infers the shape and dimensions of the pollutant concentration distribution from the mean and turbulent structure of the atmosphere and/or from the behavior of atmospheric tracers. Since this type does not explicitly treat the vertical transport process, it is difficult to modify for deposition in a way that is physically realistic. Modification of this second type of diffusion model, which I will call plume models, is the main subject of this review. Depositing pollutants can also be divided into two classes, depending on the relative contributions of gravitational settling and turbulent diffusion to vertical atmospheric transport. For gases and for particles smaller than approximately 10 urn radius, settling may be neglected compared to turbulent diffusion. These nonsettling pollutants will be the chief concern of this review since they include respirable particles and since they are transported to greater distances from the source than settling particles. For nonsettling pollutants, modification of the diffusion model is principally confined to a layer near the surface that is depleted by deposition. For settling particles, the additional vertical transport caused by gravitational settling must be included throughout the plume. We begin with a short discussion of the inclusion of dry deposition in direct solutions of the advection-diffusion equation, followed by and then by the main topic of a general discussion of plume models, this review, a comparison of four plume depletion models for nonThese four models include an exact, surface settling pollutants. depletion model, two source depletion models, and a constant eddydiffusivity solution of the advection-diffusion equation. 2.
DIRECT SOLUTION OF THE DIFFUSION
EQUATION
Direct solution of the differential, advection-diffusion equation is a very fundamental approach to estimating concentrations of airborne With pollutants, and the inclusion of deposition is straightforward. gravitational settling and deposition and, e.g., the gradient-transfer assumption, the required boundary condition at the surface is (Calder, 1961), -K(z)%
- v,C
= F = -vdC(zd)
,
where K(z) is the vertical eddy diffusivity, C ismean airborne pollutant concentration, and vs is the pollutant settling velocity. The vertical pollutant flux F at the surface is a downward deposition flux, parameterized as a deposition velocity vd multiplied by the airborne concentration at some near-surface height zd. Solution of the advection-diffusion transfer assumption requires specification
equation with the gradientof the spatial distribution
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
FOR DRY DEPOSITION
415
of wind and eddy diffusivity, and realistic distributions of these variables generally require solution by a numerical (e.g., finiteSince computer costs increase rapidly with the difference) method. number of grid points, it is desirable to make the spacing of grid points as large as possible. This requires some caution in the application of Eq. (1). The first point to be noted is that vd is a function of the reference height zd. Assuming that the deposition flux near the surface is independent of height, integration of Eq. (1) shows that the concentration C increases with height and vd decreases C(z) v,(z) where
the
= C(zd)C1 = v&zd)/C1 atmospheric
R(Z,zd)= I’
+ vd(zd)R(z,zd)l + v&d)R(z,zd)] resistance
$+
,
,
R is (4)
and for clarity we have neglected gravitational settling. Thus for a finite vertical grid spacing, the deposition velocity must be reduced to account for the atmospheric resistance of the lowest grid interval, and the inverse of the atmospheric resistance provides an upper limit to the deposition velocity that can be used in the model (Doran and Horst, 1977). The second point is that the lowest grid interval should be small enough to neglect divergence of the vertical flux below the height where the deposition boundary condition is applied. This divergence is a necessary consequence of horizontal divergence or changes with time of the near-surface concentration (Slinn, 1984). Considering the horizontal divergence resulting from diffusion downwind of a point source, the requirement is roughly s/x<
416
3.
THOMASW.HORST
PLUME MODELS OF ATMOSPHERIC DIFFUSION
Because solution of the advection-diffusion equation is, in general, computationally complex and the assumpt ions necessary to obtain a solution are in many cases theoretically unsat isfactory, sev era1 methods have been developed to infer the shape and dimensions of a plume from the mean and turbulent structure of the atmosphere and/or from the behavior of atmospheric tracers. Often these models are analytical, and thus they are easier to apply and their predictions are easier to understand than is the case for a numerical solution of the diffusion equation. However, their analytical form and their empirical basis also makes them more difficult to modify to account for dry deposition. The most apparent example of a plume model is the Gaussian plume model. The binormal spatial distribution of the Gaussian model follows from the constant eddy-diffusivity solution of the advection-diffusion equation, but it is principally based on statistical arguments and direct observations of plumes from point sources of atmospheric tracers. The Gaussian model is easy to understand because the binormal distribution is an analytical function that is amendable to simple mathematical The input to the model is also quite simple: wind speed, analysis. wind direction and the dimensions of the plume as a function of distance from the source. Since estimates of the plume dimensions are based on observations, the model predictions can be fairly realistic in some situations. Despite the utility of the Gaussian plume model, it has been found in many cases to be only an approximate description of the vertical distribution, even for a nondepositing contaminant. Some examples are the vertical diffusion from a ground-level source (Neiuwstadt and van Ulden, 1978) and vertical diffusion during daytime convective conditions (Lamb, 1982). Because these non-Gaussian distributions are also empirical, it is quite important to note that many of the plume depletion techniques are adaptable to any plume model. Since the atmospheric motions contributing to the horizontal, crosswind diffusion of a plume are usually of a different nature from virtually all plume models treat those effecting the vertical diffusion, For example, the binormal Gaussian these two processes separately. distribution is easily divided into its horizontal and vertical components, and its horizontal and vertical dimensions are cotimionly estimated from different empirical curves or from quite different theoWith the assumption that retical analyses of the diffusion process. the vertical and horizontal distributions are independent of one another (neglecting, for one thing, the effects of vertical wind shear), it follows that deposition does not alter the horizontal distribution, and our discussion can deal solely with the crosswind-integrated distribution, C(x,z)
=
I m C(X,Y,Z) -a)
dy = Q(x)D(x,z)
.
(5)
THE MODIFICATION
The preceeding tant flux Q(x),
Q(x)
equation
OF PLUME MODELS TO ACCOUNT
also
Q, - /I
dz =
=fdU(L)C(X,Z)
and the vertical diffusion integrated concentration D(x,z)
introduces
horizontal,
vdC(x',zd)
downwind
pollu-
.
(6)
dx'
function D, defined as the crosswindC(x,z) normalized by the horizontal flux,
= c(x,z)ljz;u(z)C(x,z)
For a nondepositing
the
417
FOR DRY DEPOSITION
dz
pollutant,
we define
(7)
, D=D, and Q(x)=Qo.
Q(x) and D contain the two effects of dry deposition on the plume. describes the total loss of material The suspension ratio, Q(x)/Qo, from the plume by deposition, and D/D, (defined later as a profile correction P) describes the effect of deposition on the vertical distribution. Q(x) is related to D(x,zd) by the mass balance, Eq. (6), and hence Q(x)
= QoexpC-IX
vdD(x',zd,h)
dx'l
.
(8)
0
Thus the depleted 4.
principal diffusion
task of a plume function D.
depletion
model
is
to describe
the
SURFACE DEPLETION MODEL
The surface depletion model (Horst, 1977) is an exact modification It is a of nondepositing plume models to account for dry de osition. 1978 P , which estimates special case of the surface flux model (Horst, the airborne contamination resulting from a distributed, surface source If the source per unit area is S(x,y), then of depositing material. the net flux at the surface is F(x,Y)
= s(x,Y)
- vdc(x,y+,)
and the
crosswind-integrated,
A formal (1984), diffusion
discussion but it is equation
The surface ventional point
(9) airborne
contamination
is
of this equation can be found in Horst and Slinn based simply on the linearity of the advectionand the principle of superposition.
depletion source at
model height
for a plume follows by adding the conh and neglecting the surface source S,
418
THOMASW.HORST X C(x,z)
=
QODOh~z,h) - 1
vdC(x',zd)
Do(x-x',z,o)
dx'
.
(11)
0
The loss of airborne material by deposition is effected by a sink at the surface, whose influence is propagated vertically by the diffusion function for a ground-level source. Note that the depleted plume C(x,z) is determined directly from the diffusion function Do for a nondepositing pollutant. In general, the depletion integral in Eq. (11) must be evaluated numerically, but in contrast to Eq. (8) this task is exacerbated by the fact that the integrand of the depletion integral is a function of the receptor coordinate x. Yamartino (1981) has derived a formal solution of Eq. (11) for particular forms of oz(x). The Lagrangian similarity diffusion model (van Ulden, 1978; Horst, 1979) currently provides one of the best descriptions of diffusion from a ground-level source, and thus is a leading candidate for s ecifying the diffusion function DO(x,z,o) in the integrand of Eq. (11 P . However, PASQUlLL CATEGORY B 1 .o
E 4
E 5
-----
0.5’ PASQUILL CATEGORY D
PASQI JILL CATEGORY F 1.0
ii 2
E5 2 0.5 - -E? 3
-
-
0.2 - -
CONSTANT EDDY DIFFUSIVITY SURFACE DEPLETION
0
0.1
SOURCE DEPLETION
CORRECTED SOURCE DEPLETION 1
DOWNWIND DISTANCE (km) Figure
1.
The ratio of near-surface (zd=3cm) airborne concentration with deposition to that without deposition, as a function and h=lOm. downwind distance and stability, for Vd/=o.o02
of
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
FOR DRY DEPOSITION
419
to compare the plume depletion models discussed below, the Gaussian plume diffusion model will be used with the surface depletion model. The Gaussian diffusion function for a point source of nondepositing pollutant at height h is DO(x,z,h)
1
= m
UaZ
expC-(h-z)2/20z1
I
+ expC-(h+z)2/20zl
.
(12)
where aZ is the vertical dimension of the concentration distribution. The Gaussian model is used here because of its simplicity, generality The identical diffusion model is used with and widespread application. each plume depletion model in order to examine the differences in the depletion models, isolated from differences in deposition caused by differences in the diffusion models. Figures l-4 compare the predictions of three simpler plume depletion models to those of the surface depletion model. The discussions of these simpler models relate the differences in the model predictions PASQUILL
CATEGORY
B
PASQUILL 1 .o v-us
CATEGORY
D
1::
a” 2 a !2
h=lOm 0.2 -
5
z UJ
I
6
5 !z E
= 2X1 0-3
Vd/U
0.5 -
PASQUILL
1
II1111
CATEGORY
1
I1
11111
1
I
I
B”L
F
1 .o
0.5 - ---
SOURCE
--
DEPLETION
CONSTANT
0.2 - -
SURFACE
EDDY DIFFUSIVITY DEPLETION
CORRECTED
0
I
1
I
SOURCE ,101
0.1
1
2.
I
The suspension stability, for
0 \
1 I,,,,
1 DOWNWIND
Figure
\
DEPLETION 1
I
I
10 DISTANCE
1‘11 100
(km)
ratio, as a function of downwind vd/u=O.O02 and h=lOm.
distance
and
THOMASW.HORST
420
to the assumptions on which the models are based. The comparison in Fig. 1 and 2 is done for a source height of 10 m, Vd/U=o.o%!, z =3 cm and Pasquill diffusion categories B, D and F. With a wind spee 9 of 5 m/s, the ratio vd/u=O.O02 corresponds to a deposition velocity of 1 cm/s. Values for az were obtained from Briggs' formulas for rural dispersion (Gifford, 1976). Figure 1 shows the ratio of the nearsurface pollutant concentration with deposition to that without deposition, C(zd)/Co(zd), as a function of downwind distance, and Fig. 2 shows the suspension ratio. The dependence of plume depletion on deposition velocity and source height can be inferred from Fig. 3 and 4, which show similar model comparisons for Pasquill category D, vd/u=O.Ol, and h=lO m and 100 m. 5.
SOURCE DEPLETION MODEL
One of the earliest, and probably the most used, of the plume depletion models makes the simple assumption that the vertical distribution is unaffected by deposition; i.e., D=D, (Chamberlain, 1953). is called a source depletion model because it is described completely PASQUILL
CATEGORY
D
PASQUILL
CATEGORY
D
It
1 .o
0
5
0.2
a
a
h=lOm
- --
SOURCE DEPLETION -
CONSTANT
-SURFACE 0
EDDY DIFFUSIVITY DEPLETION
CORRECTED SOURCE DEPLETION 6 1 I I1111 1 DOWNWIND
Figure
3.
The ratio of near-surface deposition to that without wind distance, for Pasquill h=lOm and h=lOOm.
10
n, 1 I106
DISTANCE (km)
(zd=3cm) airborne concentration with deposition, as a function of downstability category D, vd/u=O.Ol,
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
FOR DRY DEPOSITION
by Eq. (8), which accounts for the deposition losses reducing the source strength with downwind distance. with the surface depletion model, the source depletion written in the form,
C(x,z) = QODOb,z,h) -
Do(x,z,h)~x
vdC(x',zd)
421
by appropriately For comparison model can be
dx'
.
(13)
0
Comparison of the diffusion functions in the depletion terms and Eq. (13) emphasizes the assumption of the source depletion that deposition is a loss at the source rather than locally, surface.
of
Eq. (11) model at the
The source depletion model instantaneously distributes the effect of deposition throughout the vertical extent of the plume, whereas in fact the plume is preferentially depleted at ground level. As a consequence, it overestimates the ground-level pollutant concentration, overestimates the deposition, and underestimates the suspension ratio. This is evident in the comparison with surface depletion, Fig. l-4. However plume depletion is a self-limitinq process that reduces errors PASQUILL 1.0 _
CATEGORY
D
0.5 = lo-’
V&U
\
h=lDDm Q
\
\
0.2 -
- - -SOURCE --CONSTANT
EDDY DIFFUSIVITY DEPLETION
CORRECTED I
0.1
I
\
SOURCE DEPLETION ,,I,
I
1
I
,111
1 DOWNWIND
Figure
4.
\
DEPLETION
0.2 - -SURFACE o
\
DISTANCE
10 (km)
\ \
1 ,I, 100
The suspension ratio, as a function of downwind distance, for Pasquill stability category D, vd/u=O.Ol, h=lOm and h=lOOm.
422
THOMASW.HORST
in the model: underestimation of the suspension ratio, caused by overestimation of the deposition, reduces overestimation of the ground-level concentration. It may be seen in Fig. 1 and 3 that, as a consequence, the ground-level concentration is overestimated near the source and is underestimated far from the source. From Fig. l-4 it is also evident that the errors of the source depletion model are small except for situations with limited atmospheric mixing (Pasquill F) or high deposition (vd/u=O.Ol). Source depletion is an acceptable approximation when the rate of deposition is small compared to the vertical growth rate of the plume, vd<
6. 6.1
A CORRECTION TO THE SOURCE DEPLETION MODEL The Profile
Correction
Horst (1980, 1983) has developed a modified source depletion model that gives results in close agreement with the surface depletion model, while the computational requirements are comparable to those of the This model calculates the suspension ratio source depletion model. using Eq. (8), but modifies the nondepositing diffusion function Do by a profile correction P(x,z) to account for the change in the vertical distribution caused by deposition, C(x,z)
= Q(x)D(x,z,h)
= Qb)DOb,z,h)P(x,z)
.
(14)
To estimate the profile correction P, we note that it will play the change in the vertical its most important role near the surface: distribution caused by deposition is greatest near the surface, and a correct value for the near-surface concentration is necessary for determining the deposition flux and the suspension ratio. When there is no deposition, there is a layer near the surface in which the pollutant concentration is independent of height. For a Gaussian plume at a downwind distance such that oZ>h, the depth of this layer is roughly With deposition, the vertical concentration profile in this 0.2az. layer will be determined solely by the deposition flux to the surface. Assuming that within this layer the deposition flux is independent of height, we can substitute Eq. (14) into Eq. (2) and use the near-surface approximation Do(x,z)=Do(x,zd) to obtain P(X,Z)
=
P(X,Zd)cl
+
VdR(Z,Zd)l
.
(15a)
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
Conserving mass by substituting of Eq. (6), we get
Eq.
(14)
423
FOR DRY DEPOSITION
and (15a)
into
the
first
half
(15b) Here the integrand has been simplified with the approximation This is justified by the observation that almost ru Do(x,z,O). D (x,z,h) occurs beyond the downwind distance where oZ=h. aP 1 deposition The diffusion function Do has its maximum value at that distance and goes rapidly to zero at smaller downwind distances, e.g., Do(oZ=h/4) = Do(az=h)/400. Thus Eq. (15b) need only be accurate for oZ>h, and h-0 is a reasonable approximation in that region. 6.2
Atmospheric
Resistance
The atmospheric resistance to vertical transport, Eq. (4), is familiar concept in deposition modeling (Wesley and Hicks, 1977). a passive contaminant, the eddy diffusivit.y near the surface is K(z)
a For
(16)
= u,kz/$
where u, is the eddy friction velocity, k is von Karman's constant, and 4 is a function of atmospheric stability often obtained from observations of the turbulent transport of sensible heat. For neutral stability $H iS a Constant a and thus
R(zszd) = 5 ln(z/zd) *
(17)
and .
Corresponding results the known dependence
for non-neutral of ?H on stability
atmospheres (Paulson,
(18) can be derived 1970).
from
For the corrected source depletion model to be physically realistic, it is important that the atmospheric structure used to calculate the resistance be consistent with that used to calculate the diffusion. This is not straightforward for the usual Gaussian diffusion model where the atmospheric structure is implied by the rate of growth of the plume. However, the integrand of Eq. (4) can be heuristically transformed from a function of K to a function of uz by considering the solution of the advection-diffusion equation for a constant eddy diffusivity. The solution for a point source is a Gaussian plume with o,(da,/dx) Further, the level source
= K/u
.
first and second are related by
(19) moments
of a Gaussian
plume
from
a ground-
THOMASW.HORST
424
i=Q7uz
.
Substitution
of
R(z,zd)
Eq.
=$;
(20) (19)
and (20)
I2
dz' z'(doz/dx)
into
Eq.
(4)
gives
(21)
'
'd As an example,
R(z,zd)
Examples
for
=
for
l
J
$&
other
,J~ = ax:
(22a)
ln(z/zd)
analytical
Il~llll
forms
II
of uz are
Il~llll
II
given
by Horst
Il~llll
(1983).
II
ll~llll
1.0 PASQUILL
3
CATEGORY
B
0.9 -
x‘ z 5i
0
0.8 -
0
E 8
0.7 -
Y x g
0.6SURFACE DEPLETION CORRECTED
MODEL
-
SOURCE DEPLETION
o .
A
0.5 v&u
= 2~10~~. h = 10m v
0.4 0.01
I
I
I
IllI
I
I
I
I
Illll
0.1
5.
I
I
I
illll
1 DOWNWIND
Figure
I
I
1
I
1 lllll
10
DISTANCE x(km)
The near-surface (zd-3cm) profile of downwind distance and Pasquill
correction stability
as a function category.
100
THE MODIFICATION
6.3
Comparison
OF PLUME MODELS TO ACCOUNT
to Surface
FOR DRY DEPOSITION
425
Depletion
The assumptions and approximations used to derive Eq. (15) for the profile correction and Eq. (21) for the atmospheric resistance can be tested by comparing the predictions of the corrected source depletion model with those of the surface depletion model. Figure 5 shows the near-surface profile correction P(x,zd) as a function of downwind distance and atmospheric stability, for the conditions of Fig. 1 and 2. The unmodified source depletion model assumes P is equal to one, corresponding to instantaneous vertical mixing of the losses due to Since atmospheric resistance is low for Pasquill category deposition. B and high for category F, the profile correction is close to unity for category B and furthest from unity for category F. The profile function predicted by the corrected source depletion model is in good agreement with that calculated from the surface depletion model, particularly near and beyond the distance where az=h (80 m for Pasquill B, 200 m for D and 800 m for F). This is consist1OL
I _ -
-
I
SURFACE
I DEPLETION
I
I
I
MODEL
0 CORRECTEDSOURCE DEPLETION
PASQUILL V&U =
CATEGORY
2X10-‘,
F
-
h = l&n
x=2km,uz=20m 0.001 I 0.6
I
I
I
I
I
I
1
0.7
0.8
0.9
1.0
1.1
1.2
1.3
PROFILE CORRECTION
Figure
6.
p(x,z)
The profile correction as a function of height for a case with a,/h=2.
426
THOMASW.HORST
ent with the assumptions made in deriving Eq. (15b). The errors in P(zd) upwind of that point are of little consequence because the elevated plume has not yet reached the surface and hence both the nearsurface concentration and the deposition are negligible. The differences beyond 10 km in P(zd) for Pasquill F are principally due to approximations used in deriving analytical expressions for the integrals in Eq. (21) and (15b) (Horst, 1983). Figure 6 shows the profile correction P(x,z) as a function of Z/O, for Pasquill category F and a distance of 2 km, which is downwind of the distance where oZ=h. The corrected source depletion model agrees quite well with the surface depletion model, and the agreement extends well above the height z=O.2oz to which Eq. (15a) is expected to be Note that the slope aP/az is predicted especially well. Since valid. aP/az=v aR/az, the good agreement of the two models below z=d supports Eq. (217, the expression for the atmospheric resistance as a f unction of oz. The near-surface concentration and suspension ratios predicted by the corrected source depletion model Eq. (8), (14) and (15) are compared to the predictions of the surface depletion model in Fig. l-4. Since these estimates are for a Gaussian plume, Eq. (21) has also been used The corrected source depletion to calculate the atmospheric resistance. model compares very well with the surface depletion model, particularly for uz/h>l, the region downwind of the source where most of the deposiNote in particular that the error in P(zd) beyond x=10 km tion occurs. does not produce 'comparable errors in c(Zd). This is again a consequence of the self-limiting nature of plume depletion. 7.
THE CONSTANT EDDY-DIFFUSIVITY
SOLUTION
There has been a continual interest in constant eddy-diffusivity solutions to the advection-diffusion equation with the deposition flux boundary condition (e.g., Smith, 1962; Ermak, 1977; Corbett, 1981; This interest is sustained by the existence of Llewelyn, 1983). solutions in an analytical form and by the similarity of the nondepositing solution to the Gaussian plume model. Ermak (1977) and Rao (1981) have suggested that these solutions provide a framework for modifying the Gaussian plume model for deposition that is preferable Despite its theoretical shortcomings, to the source depletion model. this approach has some appeal because of the simplicity of the analytical solution and because the solution also includes a plausible treatment of gravitational settling. Rao (1981) transforms the solution of the advection-diffusion equation from a dependence on the eddy diffusivity K to a dependence Since the solution assumes a on crz with the substitution az2=2Kx/u. constant eddy diffusivity, this procedure is correct only for the case The sole justification in the case of a general as x+. where u varies dz is t fl e recovery of the Gaussian plume when there is no deposition. Nevertheless, the result is,
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
c(x,z)= Qo exp A?7 uoz
exp[-
[
"s - u
(z-h;x aZ
[ 1 - 2&?$
*]
421
FOR DRY DEPOSITION
g]
{exp
$1
+ Z
Z
z expZerfc5
[
1)
(23a)
Z
v=v
d - “,I2
(23b) (23~)
Rae makes the interesting observation that for nonsettling particles, Eq. (23) accounts for deposition by reducing only the conventional image-source term as suggested by Csanady (1955) and Overcamp For settling particles, the leading exponential function that (1976). depends on vs tilts the plume downward at an angle whose tangent is "s/U. The constant eddy-diffusivity solution, Eq. (23), does not conserve mass with a general functional form for dZ(x). This can be corrected by dividing Eq. (23) by its vertical integral, Eq. (7), to get the normalized concentration D and then multiplying D by the actual horizontal flux Q(x) determined from the second half of Eq. (6). Rao (1981) provides analytical expressions for the vertical integral of Eq. (23). The constant eddy-diffusivity model , corrected for mass conservation, is compared to the preceeding models in Fig. l-4. In all cases it is at least as good as the source depletion model, and in most cases the difference between the constant eddy-diffusivity model and the surface depletion model is roughly one-half that of the source depletion model. The remaining differences between the constant eddy-diffusivity model and the surface depletion model are caused by the assumption of the constant eddy-diffusivity model that aZ varies as ~4. The constant eddy-diffusivity model specifies the correct value of aZ at the receptor location where the model is applied, but since uz is generally a linear function of x near the source, it overestimates uz upwind of the receptor. Thus, just as in the source depletion model, the deposition that occurred upwind of the receptor are mixed through a vertical losses depth that is too large, the near-surface airborne concentrations are overestimated, and the suspension ratio is underestimated. 8.
CONCLUSIONS
Four plume depletion models have been compared: a surface depletion model, two source depletion models, and an adaptation of the constant eddy-diffusivity solution of the advection-diffusion equation. The surface depletion model is an exact modification of nondepositing
428
plume models is applicable complex.
THOMASW.HORST
to account for the only to nonsettling
effect of dry contaminants
deposition. However and is computationally
it
The source depletion models are applicable to both settling and nonsettling contaminants, and they are easier to use than the surface depletion model. Chamberlain's (1953) source depletion model is biased in its estimates of deposition, but these errors are small for all situations except those with limited atmospheric mixing and high deposition. With the addition of a profile function to correct for alteration of the vertical distribution caused by deposition, the predictions of the source depletion model are in close agreement with those of the surface depletion model. The corrected source depletion model predic-tions are best for the near-surface airborne concentrations, particularly for o,/h>l. As a consequence it also gives a good prediction of the deposition and of the suspension ratio. The constant eddy-diffusivity plume-depletion model is also easier to use than the surface depletion model. It directly includes gravitational settling, but this aspect of the model has not been With a correction to assure that mass is conserved, this tested here. model is at least as good as the uncorrected source depletion model, and in most cases its predictions are closer to those of the surface depletion model by a factor of roughly l/2. In contrast to the other plume depletion models, however, the constant eddy-diffusivity model is applicable only to diffusion that can be adequately described by the Gaussian plume model. ACKNOWLEDGEMENTS The author is grateful for helpful discussions of these models with K. S. Rao and J. C. Doran. This research was supported by the U. S. Department of Energy, Office of Health and Environmental Research, and by the U. S. Environmental Protection Agency, under contract DE-AC06-76RL0 1830. REFERENCES Berkowitz, R. and Prahm, L. P.: 1978, 'Pseudospectral Simulation of Dry Deposition from a Point Source', &mos. Fnviron 12, 379-387. Calder, K. L.: 1961, 'Atmospheric Diffusion of Particulate Material Considered as a Boundary Value Problem', J.. 18, 413-316. Chamberlain, A. C.: 1953, Aspects of Travel and Deposition of Aerosol and Vapour Clouds, UKAEA Report No. AERE-HP/R-1261, Harwell, Berkshire, England. of Source-Depletion and Alternative Corbett, J. 0.: 1981, 'The Validity Approximation Methods for a Gaussian Plume Subject to Dry Deposition', Fnviron 15, 1207-1214. Travel of Doran, J. C. and'Horst, T. W.: 1977, Comments on 'Long-Range Airborne Material Subjected to Dry Deposition', Atmos. Environ. 11, 1246-1247.
THE MODIFICATION
OF PLUME MODELS TO ACCOUNT
FOR DRY DEPOSITION
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Csanady, G. T.: 1955, 'Dispersal of Dust Particles from Elevated Aust. J. Phvs. 8, 545-550. Sources', Ermak, D. L.: 1977, 'An Analytical Model for Air Pollutant Transport and Deposition from a Point Source', Atmos. Environ. 11, 231-237. Gifford, F. A.: 1976, 'Turbulent Diffusion Typing Schemes: A Review', Nucl. Saf. 17, 68-86. Horst, T. W.: 1977, 'A Surface Depletion Model for Deposition from a Gaussian Plume', Atmos. Environ. 11, 41-46. Horst, T. W.: 1978, 'Estimation of Air Concentrations Due to the Suspension of Surface Contamination', Atmos. Environ. 12, 797-802. Horst, T. W.: 1979, 'Lagrangian Similarity Modeling of Vertical Diffusion from a Ground-Level Source', J. Ao~l. Meteor. 18, 733-740. Horst, T. W.: 1980, 'A Review of Gaussian Diffusion-Deposition Models', Atmospheric Sulfur Deposition, D. S. Shriner, C. R. Richmond, and S. E. Lindberg (ed.), Ann Arbor Science, Ann Arbor, MI, 275-283. Horst, T. W.: 1983, 'A Correction to the Gaussian Source Depletion Model', PreCiDitatiOn Scavenclincl. Dry DeDosition and Resuspension, H. R. Pruppacher, R. G. Semonin and W.G.N. Slinn, Elsevier North Holland, Amsterdam, The Netherlands, 1205-1218. Horst, T. W. and Slinn, W.G.N.: 1984, 'Estimates for Pollution Profiles Above Finite Area-Sources', to appear in Atmos. Environ. Lamb, R. G.: 1982, 'Diffusion in the Convective Boundary Layer', pheric Turbulence and Diffusion Modeling, F.T.M. Nieuwstadt and H. van Dop (ed.), D. Reidel, Dordrecht, Holland, 159-230. Llewelyn, R. P.: 1983, 'An Analytical Model for the Transport, Dispersion and Elimination of Air Pollutants Emitted from a Point Source', Atmos. Environ. 17, 249-256. Nieuwstadt, F.T.M. and van Ulden, A. P.: 1978, 'A Numerical Study on the Vertical Dispersion of Passive Contaminants from a Continuous Source in the Atmospheric Surface Layer', Atmos. Environ. 12, 2119-2124. Gaussian Diffusion-Deposition Model Overcamp, T. J.: 1976, 'A General for Elevated Point Sources', J. ADDS. Meteor. 15, 1167-1171. Atmospheric Diffusion, 2nd Ed., John Wiley & Pasquill, F.: 1974, Sons, New York. Representation of Wind Speed Paulson, C. A.: 1970, 'The Mathematical and Temperature Profiles in the Unstable Atmospheric Surface Layer', J. Appl. Meteor. 9, 857-861. Analytical Solutions of a Gradient-Transfer Model Rao, K. S.: 1981, NOAA Tech. Mem. ERL ARL-109, for Plume Deposition and Sedimentation, Air Resources Laboratories, Silver Spring, MD. Slinn, W.G.N.: 1984, 'A Potpourri of Deposition and Resuspension Questions', Precipitation Scavenaina. Drv DeDosition and v H. R. Pruppacher, R. G. Semonin and W.G.N. Slinn, Elsevier Norih Holland, Amsterdam, The Netherlands, 1361-1416. Smith, F. B.: 1962, 'The Problem of Deposition in Atmospheric Diffusion of Particulate Matter', J. Atmos.-Sci 19, 429-434. van Ulden, A. P.: 1978, 'Simple Estimates for-tical Diffusion from Sources Near the Ground', Atmos. Environ. 12, 2125-2129.
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Wesely, M. L. and Hicks, B. B.: 1977, 'Some Factors that Affect the Deposition Rates of Sulfur Dioxide and Similar Gases on Vegetation J. Air Poll. Control Assoc 27, 1110-1116. to the Eauation for Surface Yamartino, R. J.: 1981. 'Solutions . Depletion of a Gaussian Plume', 12th NATOiCCMS International Technical Meeting on Air Pollution Modeling and Its Application, Palo Alto, CA. --___-
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