EPA-600/4-78-003
January 1978
RESIDENCE TIME OF ATMOSPHERIC POLLUTANTS
AND LONG-RANGE TRANSPORT
by
Teizi Henmi, Elmar R. Reiter and Roger Edson
Department of Atmospheric Science
Colorado State University
Fort Collins, Colorado 80523
Grant 803685
Project Officer
George C. Holzworth
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, NC 27711
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NC 27711
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DISCLAIMER
This report has been reviewed by the Environmental Science Research Labora-
tory, U.S. Environmental Protection Agency, and approved for publication.
Approval does not signify that the contents necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does mention of
trade names or commercial products constitute endorsement or recommendation
for use.
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ABSTRACT
The Lagrangian trajectory model which is suitable for the study of long-
range transport of pollutants is developed. The model is aimed for use with
input parameters obtainable routinely from the national meteorological ser-
vices.
The horizontal dispersion of pollutants along the trajectory of a layer-
averaged wind is determined by the quantity T,., which is calculated from the
standard deviation of wind velocities. The vertical distribution of pollutants
is assumed to be uniform throughout the mixing layer.
The computer program is capable of calculating trajectories over the
region of the U.S. using routine sounding data. The output consists of tables
of locations of trajectory end points at each time-step, and dispersion widths
along a trajectory, and the plotting of trajectories.
Calculations of regional residence times, T, of SCL in the mixing layer
are based on the following assumptions: a) Dry deposition, precipitation
scavenging, and chemical transformation are the removal mechanisms of the
pollutant from the atmosphere, b) The deposition velocity is 1 cm/sec and the
chemical transformation rate is 1 x 10" sec" regardless of the season and the
location, c) The ratio of concentration of the pollutant in precipitation
water to that in air is 5 x 10 on a volume basis, d) There is no leakage of
the pollutant from the top of the mixing layer, and e) Import and export
fluxes due to turbulent diffusion through the boundaries of the region under
consideration are balanced.
Climatological data of the mixing layer depth (Holzworth, 1972) and
hourly precipitation data are used for the calculation of the mean regional
residence time, T. T is calculated for the region of the United States east
of 105°W longitude, for the cold season (November to April), and for the warm
season (May to October). The results are shown as isopleths of T over the
studied area.
In order to incorporate the scavenging due to cumulus cloud precipitation
into our trajectory model, a cumulus cloud model with detailed microphysical
processes was developed. The microphysical processes taken into consideration
are: Brownian diffusion, turbulence capture, impaction, autoconversion and
accretion. Using typical size distributions for maritime and continental
aerosols, scavenging characteristics in maritime and continental cumulus
clouds were studied. The results can be summarized as follows: The scavenging
coefficient of aerosols by cloud water droplets is one order of magnitude
larger in the continental cloud than in the maritime cloud. On the other
hand, the scavenging coefficient of aerosols by rainwater droplets is slightly
i i i
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larger in maritime clouds than in continental clouds. Similarly, the scaveng-
ing coefficient of cloudwater droplets by rainwater droplets is larger by
several factors in the maritime cloud than in the continental cloud- As a
whole, aerosols are more efficiently scavenged in the continental cloud than
in the maritime cloud.
IV
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CONTENTS
ABSTRACT iii
FIGURES vi
TABLES viii
SYMBOLS ix
ACKNOWLEDGEMENTS xiv
1. INTRODUCTION 1
2. CONCLUSIONS 3
3. RECOMMENDATIONS 5
4. LONG-RANGE TRANSPORT MODEL 6
5. REGIONAL RESIDENCE TIMES OF S02 OVER THE EASTERN U.S 45
6. SCAVENGING OF AEROSOL POLLUTANTS IN CUMULUS CLOUDS 61
REFERENCES 84
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FIGURES
Number Page
1. Schematic diagram of the model 8
2. Scheme of the calculation of standard deviations of wind components . 11
3. Surface synoptic charts for the AVE II, (a) for 12 GMT, 11 May 1974
(b) for 12 GMT, 12 May 1974 14
4. Surface synoptic charts for the AVE III, (a) for 00 GMT, 6 February
1975 (b) for 12 GMT, 7 February 1975 15
5. Trajectories of the layer-averaged wind for the AVE II. 12Z May 11 -
12Z May 12 16
6. Trajectories of the layer-averaged wind for the AVE III. 002 Feb. 6 -
12Z Feb. 7 17
7. Horizontal dispersion along the trajectory of the layer averaged
wind for the AVE II. May 11 - May 12.
(a) Time interval = 3 hours 21
(b) Time interval = 6 hours 22
(c) Time interval = 12 hours 23
8. Horizontal dispersion along the trajectory of the layer-averaged
wind for the AVE III. Feb. 6 - Feb. 7.
(a) Time interval = 3 hours 24
(b) Time interval = 6 hours 25
(c) Time interval = 12 hours 26
9. Configuration for determining a trajectory segment from observed
winds 29
10. An example of the plotting of trajectories. The trajectories or-
iginated at North Platte, Nebraska, on 11 May 1974 35
11. Horizontal dispersion as a function of travel^time (after Bauer,
1973). The calculated mean cloud widths for u = 1, lOm/sec
using the Ekman theory are shown by solid lines 37
12. The relationship between crv./u" and u calculated for the Atmospheric
Variability Experiment II 39
VI
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Number Page
13. The relationship between avh/u and u" calculated for the Atmospheric
Variability Experiment III 40
14. (a) Persistence Index, P.I. of crv. versus time step for trajectories
started at 12Z, 11 May 1974 42
(b) Same as Figure 14(a), except at 18Z, 11 May 1974 43
(c) Same as Figure 14(a), except at OOZ, 12 May 1975 44
15. Mean dry period, Td, (a) for the cold season and (b) for the warm
season.
53
16. Mean wet period, T , (a) for the cold season and (b) for the warm
season 54
17. The residence time due to dry deposition, t^, (a) for the cold
season and (b) for the warm season 55
18. Climatological scavenging coefficient, l(x 10" sec) for S02, (a)
for the cold season and (b) for the warm season 56
19. Turnover time, Te, versus residence time, T 57
20. The regional residence time, T, for SO,,, (a) for the cold season
and (b) for the warm season 58
21. Model size distributions of continental maritime air. (After
Junge and McLaren, 1971.) 72
22. Model sounding used as an input 74
23. Distribution of cloudwater, Q , and rainwater, Q, , with respect to
height 76
24. (a) Distributions of scavenging rates, A-,, A~, and X, in the con-
tinental cloud 77
(b) Same as Figure 24(a), except in the maritime cloud 78
25. (a) Distributions of mass fractions of aerosols in cloud, cloudwater
and rainwater in the continental cloud 80
(b) Same as Figure 25(a), except in the maritime cloud 81
VII
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TABLES
Number Page
1. The relative accuracy of trajectory analysis for the AVE II. (12Z
May 11 - 12Z May 12, 1974.) 18
2. The relative accuracy of trajectory analysis for the AVE III. (OOZ
Feb. 6 - 12Z Feb. 7, 1975.) 19
3. Horizontal dispersion and relative error 27
4. The number of reporting stations within a specified radius for each
time interval 31
5. The latitude and longitude of trajectory segment endpoints at each
time interval (tenths of degrees) 32
6. The height of the mixing layer (meters above ground) and dispersion
parameter 0v, (m/sec) at each time interval 33
7. The accumulated dispersion width, E^ (degrees latitude), at each
time interval 34
vm
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LIST OF SYMBOLS
A Cunningham correction factor
a aerosol radius
a1 coefficient in the autoconversion
b constant in the Ekman layer equation
C(x) concentration of a pollutant at the distance x
C-, source strength
C concentration of the pollutant
C1 entrainment constant
C specific heat at constant pressure
D diameter of raindrop
E(D/C) average collection efficiency between rain and cloud droplets
f(a) size distribution of aerosols
f(r) size distribution of cloudwater droplet
F(R) size distribution of rainwater droplet
f Coriolis parameter
f fraction of the distance x over which rain is falling
F constant defined by Eq. (42)
g acceleration of gravity
h height of the mixing layer
h1 coefficient defined in Eq. (48)
i aerosol size index
j cloudwater size index
ix
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k rainwater size index
k1 coefficient in the autoconversion Eq. (48)
k, Boltzmann's constant
k, total decay rate
k decay rate due to precipitation scavenging
K momentum eddy diffusivity
K constant defined by Eq. (43)
L latent heat of condensation
C£
L, source width
L(3), L(16) travel distance calculated with the 3-hours, 6- .ours, and 12-hours
L(12) respectively
ALg, AL-jp vectorial differences of trajectories obtained with the 3-hour
and 6-hour intervals, and with the 3-hour and 12-hour intervals
respectively
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Q total mixing ratio of condensed water substance
QT total moisture mixing ratio
q environmental vapor mixing ratio
q saturation mixing ratio
q cloud vapor mixing ratio
R rainwater droplet radius
R gas constant of air
a
r radius of cloudwater droplet
S coefficient defined by Eq. (62)
T residence time
T temperature in the cloud
Vx
Te turnover time
Te environmental temperature
t residence time due to precipitation scavenging
t residence time due to chemical transfer
L/
t, residence time due to dry deposition
u mean wind speed
V deposition velocity
V precipitation scavenging velocity
Vt terminal velocity of rainwater droplet
V y-component of the mean wind
w vertical velocity
a coefficient of Gamma distribution
(3 coefficient of Gamma distribution
r Gamma function
Y coefficient in the Marshall-Palmer distribution
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c ratio of the density of water to that of air
e' rate of energy dissipation
TI dynamic viscosity of air
nrf frequency distributions of dry period
n frequency distributions of wet period
6(z) wind direction at altitude z
K concentration of pollutant in rainwater
A-, scavenging rate of aerosols by cloudwater droplets
Ag scavenging rate of aerosols by rainwater droplets
AD Brownian diffusion scavenging rate of aerosols by rainwater
droplets
AT interception collection rate of aerosol by rainwater droplet
AJJ inertia! impaction rate of aerosols by rainwater droplets
Aj turbulent diffusion scavenging rate
A conversion rate of cloudwater into rainwater
X , decay rate due to dry deposition
A chemical transformation rate
A1 removal coefficient during wet period
X", removal coefficient during dry period
T mean precipitation scavenging coefficient
u entrainment parameter
v kinetic viscosity of air
v radius dispersion
p density of air
p, density of cloud air
a
p density of the environmental air
XII
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p density of water
p density of aerosol
Zih mean width of pollutant plume
AZ. relative error of horizontal dispersion
a coefficient of Gamma distribution
a dispersion factor defined by Eq. (12)
h
a standard deviation of x-component of wind
x
a standard deviation of y-component of wind
y
T , mean duration of dry period
T mean duration of wet period
T time
X concentration of pollutant in air
ABBREVIATIONS
DWi distance weighting factor
TSQ trajectory segment
TSi contribution to the trajectory segment from an observed wind to the
midpoint, of TSi
AWi alignment weighting factor
xm
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ACKNOWLEDGEMENTS
We are grateful to George C. Marshall Space Flight Center of NASA for
kindly supplying the Atmospheric Variability Experiment Data; and to The Air
Resources Laboratories of NOAA at Silver Spring, Maryland, for making it
possible to use the computer program of trajectory analysis.
xiv
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SECTION 1
INTRODUCTION
The transport of air pollutants across metropolitan, regional and nation-
al boundaries has received increasing attention by scientists and control
agencies. It is now recognized that long-range transport of polluted air
masses takes place well beyond 100 km, and that secondary air pollutants, such
as ozone and sulfate, are produced during the transport. Acid rain in the
northeastern United States is associated with air parcels that have traveled
through major industrial areas in the midwestern and east-central states one
or two days earlier (Dittenhoefer and Dethier, 1976). Thus, transport model
calculations play an important role in understanding the relation between
emissions and concentrations of pollutants in the atmosphere, and the forma-
tion of secondary air pollutants.
One of our principal efforts in the last two contract periods has been to
develop a long-range transport model suitable for keeping track of pollutants
downstream of large industrial complexes. The basic concepts of our model are
as follows: The mean motion of pollutants in the mixing layer is determined
by the mean wind field in the layer. In practice, the pollutant movement can
be followed by a step-by-step trajectory analysis, in which mean wind speeds
and directions in the mixing layer are estimated from available sounding data
and are applied for a period of several hours. From the time-dependent mean
wind fields, the trajectory of a large polluted air volume can be obtained.
The horizontal dispersion of pollutants is caused by horizontal turbulence and
by mean wind shear. However, for long-range transport, the mean wind shear is
by far the dominant cause of the dispersion of pollutants. In our transport
model, the horizontal dispersion of pollutants is determined by the quantity,
I, , which is calculated by the standard deviation of wind velocities in the
mixing layer. In our model, the physical removal processes due to dry de-
position and precipitation scavenging are included. In Section 4, the details
of our model are described.
A parameter which adequately characterizes the fate of pollutants over
longtime and space scales is the residence time in the atmosphere. We have
considered the determination of the residence time of S02 in the mixing layer.
Assuming that precipitation, deposition, and the transformation into sulfate
are the mechanisms of removal, the regional residence times of pollutants have
been calculated for the region of the United States east of 105°W longitude.
The details of this study are found in Section 5.
In pollution meteorology, cumulus clouds have two important roles. The
first role is to vertically transport heat, moisture, and momentum as well as
pollutants. The second role is to cleanse the polluted atmosphere by rainout
and washout processes. The best knowledge about these roles can be obtained
1
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from a cloud-modelling study. Taking into consideration the microphysical
processes such as Brownian diffusion, turbulent capture, impaction, and auto-
conversion, we have studied the in-cloud scavenging (rainout) characteristics
of clouds. Particularly, scavenging in the continental and maritime cumulus
clouds has been considered. This study is described in Section 6.
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SECTION 2
CONCLUSION
The Lagrangian trajectory model which is suitable for the study of long-
range transport of pollutants is developed. The model is aimed for use with
input parameters obtainable routinely from the national meteorological ser-
vices. The model equation contains the terms of physical removal processes,
so that the sink areas of pollutants which emanate from an industrial complex
can be determined. The horizontal dispersion of pollutants along the trajec-
tory of a layer-averaged wind is determined by the quantity £., which is cal-
culated from the standard deviations of wind velocities. A computer program
for trajectory analysis uses winds averaged through any desired layer above
average terrain. Computer output includes a listing of individual trajectory
endpoints after selected durations, dispersion factor £. , and plotted tra-
jectories.
The e-folding residence time T and turnover time Te of sulfur dioxide are
calculated over the region of the United States east of 105° W longitude. The
year was divided into the cold season (January - April, and November and
December), and the warm ':eason (May - October), and the T and Te for each
season are calculated. It is shown that, for practical purposes, either T or
Te can be used to represent the residence time for the studied area. The
following conclusions are obtained:
1. The residence time is, in general, longer in the warm season than in
the cold season over the studied area.
2. Short residence times characterize the region surrounding the Great
Lakes and the southern part of the United States.
3. Long regional residence times are found in the western parts of the
the studied area.
4. In the studied area, the regional residence time lies in the range
between 20 and 40 hours for the cold season and in the range between
30 and 60 hours for the warm season.
5. The dry deposition is the most dominant removal machanism in the
studied area.
In order to incorporate the scavenging due to cumulus cloud precipitation
into our trajectory model, the scavenging characteristics of continental and
maritime cumulus clouds are investigated, based on a numerical model of in-
cloud scavenging, combined with a cumulus model. It is found that the scaveng-
ing rate of aerosols by cloud water droplets is one order of magnitude larger
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in the continental cloud than in the maritime cloud. On the other hand,
scavenging rates of aerosols by rainwater droplets, and scavenging rates of
cloudwater droplets by rainwater droplets are larger in the maritime cloud
than in the continental cloud.
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SECTION 3
RECOMMENDATIONS
The Lagrangian trajectory model developed by this project is primarily
aimed for the study of transport of chemically inactive pollutants. It is
recommended that further efforts shall be made to incorporate into the model
the transformation of S02 to sulfate.
The residence time calculation of S02 was primarily based on the physical
removal processes. Sulfate removal is an additional important aspect of the
overall sulfur-removal process. Accordingly, an extension of the present cal-
culation to provide for sulfate residence time is recommended.
The study of scavenging characteristics by cumulus clouds was concerned
only with particulate pollutants. It is desirable to study the scavenging of
gaseous pollutants, such as S0?, by cumulus clouds, so that the precipitation
removal term in the transport model can be refined. The applications of a
method of calculating SO- washout from plumes, developed by Dana et al.,
(1976) should be considered. Furthermore, the study of scavenging character-
istics by other types of clouds, such as clouds associated with front forma-
tion, must be studied in the future.
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SECTION 4
LONG-RANGE TRANSPORT MuDEL
4.1 General Features of Long-Range Transport
In the long-range transport process of airborne pollutants, the follow-
ing physical features can be envisioned:
The general track of airborne pollutants is determined by the horizontal
wind field over the whole depth of the atmospheric layer through which the
material has been spread vertically. It is usually found that wind speed and
direction change substantially with respect to height. Ho-izontal spreading
of pollutants will be caused by horizontal turbulence, by tne systematic
change of wind with height, and by synoptic-scale variations in the wind
field. The horizontal spread of pollutants by systematic changes of wind with
respect to height will be particularly effective when the plume of pollutants
is subject to an alternation of nocturnal stable conditions and daytime con-
vective conditions. Mixing will also progressively extend the vertical spread
of pollution, ultimately to the top of the boundary layer and toward a state
of uniformity throughout the depth of this layer. This process will occur
rapidly in the course of transport during daytime convective conditions, but
will be slowed down during stable conditions at night. There is a diurnal
variation in the depth over which effective vertical mixing exists. The
pollution contained in the layer between the daytime and nighttime mixing
height will take a different track from that of pollutants in the mixing layer
during the nighttime hours. During the next day, far downstream from its
original source region, the pollution carried above the nighttime inversion
may again be mixed into the mixing layer as solar radiation spawns renewed
mixing processes during daytime. The transport of the pollution to even
greater heights is accomplished by penetrative convection (such as in Cb
clouds), by large scale ascent from horizontal convergence and by uplifting of
warm air over cold air.
In general, therefore, the transport of pollutants must be considered
separately in three main layers:
(1) Nighttime and daytime mixing layer containing a near uniform vertical
distribution of pollutants;
(2) The layer between daytime mixing height and nighttime mixing height,
in which pollutants are trapped during the nighttime;
(3) The free atmosphere containing pollution which has been carried up,
beyond the range of boundary layer mixing, by large-scale ascent or
by penetrative convection.
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In order to estimate the fate of pollutants in the atmosphere, the trans-
port in each of these three layers must be studied.
4.2 Trajectory Model for Long-Range Transport
One of the main objectives of this investigation is to determine the sink
areas of pollutants emanating from large industrial complexes. The pollutants
are removed from the atmosphere by mechanisms such as dry deposition and pre-
cipitation scavenging. Chemical transformation of pollutants during their
transport stage will have to be considered as well. In the following model,
the terms of chemical transformation are not included. Thus, the model in its
present stage is only applicable to the transport of particulate and chemically
inactive pollutants.
_ Let us assume that pollutants are transported with the mean wind speed
u, as shown in Figure 1, from an initial vertical plane at 0 in which the con-
centration c is uniformly distributed. The width of the source area perpen-
dicular to the mean wind is taken as L , its height is h, the height of the
mixing layer. After time t = x/u, the pollutants cross a vertical plane, P,
at a distance x from the source area. The width of the pollution plume at ?
is defined as
L(x) = LQ + 2[avh] 4- , (1)
where [av, ] is the quantity defined in the next section.
If C(x) is the concentration__at distance x, the rate at whj_ch pollutants
cross the vertical plane of P is u-h-L(x)'C(x). Assuming that u and h are
constant during the time interval required to travel the distance
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TOP OF MIXING LAYER
Lo
ovh-t
Figure 1. Schematic diagram of the model
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Therefore,
C(x) =
L + 2[ov,] -r- L h u -I (3)
u
A similar equation was derived by Scriven and Fisher (1974).
An application of equation (3) along the trajectory of the mean wind u,
which is defined in the next section, can be made as follows: After the first
interval, t, of trajectory analysis, the concentration of the pollutant under
consideration at the end point of trajectory, x-j, is given by
(4)
where subscript 1 refers to the first interval of trajectory analysis. For the
second interval, C-, is regarded as the concentration of the new hypothetical
xl
line source with the width of U = L + 2av, 1 . The concentration of
lo hi -
pollutant at the distance x from the new'hypothetical line source is
C, L, ( v + v , + f'v )
C(x) = exp \ ^ P~ -3- ( (5)
L, + 2[av, ] ( h u )
1 h u
where C, is given by equation (4). By repeating the above process, the con-
centration along the trajectory at any distance from the source area can be
calculated.
4.3. Vertical Dispersion of Pollutants in the Mixing Layer
If one attempts a comprehensive description of the transport process of
pollutants, one finds that little is known about the rate of vertical mixing
after tens of minutes and/or kilometers following the release of a pollutant
(Pasquill, 1974). This is due in part to the great difficulties in sampling
over a large volume of space within the mixing layer, and in part due to the
difficulties of measuring low concentrations existing at several tens of
minutes after release from a source. However, recent observations of pollutant
concentrations above and downwind of industrial complexes (Davis and Newstein,
1968; Georgii, 1969; Kocmond and Mack, 1972; Auer, 1975) have shown that, with-
in a mixing layer which is well defined by a capped inversion, the concen-
tration of pollutants is uniformly distributed in the vertical throughout the
mixing layer. Furthermore, computer and laboratory studies of the vertical
diffusion of nonbuoyant particles within the mixing layer by Deardorff and
Willis (1974) indicate that the distribution of particles released near the
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ground become vertically uniform after the dimensionless time, t*, has assumed
a value of about 3. t* is defined as t* = (w^/z^)t, where t is the time after
release, and w* = [(g/ej wT1" z.]1/3. g is the gravitational acceleration,
6m is the mean potential temperature in the mixing layer, and vTe7" is the
kinematic heat flux close to the surface.
In agreement with these findings, we assume that the concentrations of a
pollutant for mesoscale and synoptic scale trajectories are uniformly distri-
buted in the vertical throughout the mixing layer.
4.4 Horizontal Dispersion of Pollutants in the Mixing Layer
Physically, the horizontal dispersion of pollutants is caused by horizon-
tal turbulence and by mean vertical wind shear. However, for the mesoscale
and synoptic scale transport of pollutants it can be assumed that the mean
wind shear is the dominant cause for the dispersion (Tyld^ley and Wellington,
1965; Csandy, 1968). This dominance of vertical wind shear in the horizontal
dispersion, with the assumption of uniform distribution of pollutants in the
vertical throughout the mixing layer, brings us to the following arguments:
Let us take the X-axis along the east-west direction and the Y-axis
along the north-south direction. Then, within the mixing layer, as illustrated
in Figure 2, the following equations hold:
The mean mass-averaged velocity components are given by
h
{ pv(z) -cos 9 (z) dz
0
/pdz
0
h.
pv(z)-sin 0 (z)dz
v = - ^
Y hr
/pdz
0
where 7 and v are the x - and y - components of the mean wind, v(z) is the
wind speed at altitude z, 0(z) is the wind direction at altitude z, p is the
air density, and h is the height of the mixed layer.
The mean wind speed is given by
u =
* y
10
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Figure 2. Scheme of the calculation of standard
deviations of wind components.
11
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The mean veering angle is
0 = tan"
(9)
The deviation of wind from the mean wind is
av.
/ (vjz]-vv)2dz/h
L 0
x1- J x'
1/2
av
y
/ (v,,[z]-vJ2dz/h
y y
1/2
do)
(ID
The horizontal deviation of that wind component fron, '-he mean wind, which
is perpendicular to the mean wind, v, is
av
+ av
av
h
y
this is called a "dispersion factor".
The mean "width" of the pollutant plume along the mean wind is given by
av
v + av
ah = avh ' t =
X X
(13)
where t is the time step of the trajectory analysis, and x is the distance from
the source.
When the trajectory analysis is repeated for n time steps of equal length,
the accumulated width of the pollutant plume along the mean wind is given by
= 2
,1
ovh,2 ' t
2(5 a .) t+L
n
= 2 2 (av. .
'h,i -
1 vi
(14)
12
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where L is the width of the original source area perpendicular to the mean
wind.
4.5. Preliminary Studies of Trajectory and Dispersion of the Layer-Averaged
Wind in the Mixed Layer
In order to apply the long-range transport model to actual data, the
development of an accurate trajectory construction technique using the layer-
averaged wind is essential. For this purpose, the upper air sounding data
obtained during the NASA-sponsored Atmospheric Variability Experiments (AVE)
are used. The three-hourly sounding interval and the more detailed vertical
resolution of the data permit more accurate trajectory computation and mixing-
layer depth determination than is feasible with the customary twice daily
soundings.
In this study, three different trajectory analyses are made with 3-, 6-,
and 12-hourly time intervals using the AVE data.
4.5.1. Trajectory Studies
In this study, the height of the mixing layer is defined as the first
level (using 25 mb interval data) between 500 meters and 3000 meters from the
surface where a significant stabilization of temperature lapse rate occurs
3T
(-5- > -4°C/km). If no such level is found, the height of the mixing layer is
o Z
assumed to be at 3000 meters above the surface. After the layer-averaged
wind speeds and directions are calculated for all stations, the streamlines
of the layer-averaged wind are drawn, and trajectory analyses are performed.
The trajectory analysis technique employed is described in standard meteorolog-
ical analysis text books.
Figures 3 and 4 show the surface synoptic charts during the periods of
AVE II and III at the beginning (a) and at the end (b) of these two periods.
For the AVE II data, we chose North Platte, Nebraska; Monette, Missouri;
and Greensboro, North Carolina, as the starting points of air parcel trajec-
tories. For AVE III data, the trajectories of air parcels leaving from North
Platte, Nebraska; Green Bay, Wisconsin; and Buffalo, New York, were studied.
The reason for choosing these stations as initial points is that they are
situated under different synoptic weather conditions. In order to study the
effects on accuracy of the different time resolutions, three different trajec-
tory analyses are made with 3-, 6-, and 12-hour time intervals.
The results of the trajectory analysis for the AVE II and III periods
are shown in Figures 5 and 6. Assuming that the trajectories obtained with
the 3-hour time intervals are "exact", trajectories obtained with the 6-hour
and 12-hour intervals are compared with those using 3-hour intervals. Depar-
tures are obtained in terms of vectorial differences between the two trajector-
ies. The results are shown in Tables 1 and 2. In these tables, L(3) is the
travel distance calculated with the 3-hour time interval and A, and A, are
L6 L12
13
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12 08 04
04 00 00
04 08 12
00 04 08 12 16
(b)
Figure 3. Surface synoptic charts for the AVE II
(a) for 12 GMT, 11 May 1974
(b) for 12 GMT, 12 May 1974
14
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12
16
Figure 4.
Surface synoptic
(a) for 00 GMT,
(b) for 12 GMT,
(b)
charts for the AVE
6 February 1975
7 February 1975
III
15
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3hr
Figure 5.
Trajectories of the layer-averaged wind for the AVE III.
12Z May 11 - 12Z May 12.
16
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Figure 6. Trajectories of the layer-averaged wind for the AVE III.
OOZ Feb. 6 - 12Z Feb. 7.
17
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TABLE 1. The Relative Accuracy of Trajectory Analysis for the AVE II.
(12Z May 11 12Z May 12, 1974)
a. Trajectory of air parcel leaving North Platte, Nebraska
Travel Time (Hours)
L(3)
AL6
AL6/L(3)
AL12/L(3)
b. Trajectory of air
L(3)
AL6
AL12
AL6/L(3)
AL12/L(3)
c. Trajectory of air
L(3)
AL6
AL6/L(3)
AL12/L(3)
0 6
0 (km) 230
0 <15
0
0 <.07
0
(Average wind
12
420
25
110
.06
.26
speed = 10.5
18
670
55
.08
m/sec)
24
910
35
270
.04
.30
parcel leaving Monette, Missouri
Travel Time
0 6
0 (km) 110
0 0
0
0 0
0
(Average wind
(Hours)
12
210
80
95
.36
.43
speed = 8.1
parcel leaving Greensboro, North
Travel
0 6
0(km) 100
0 40
0
0 .40
0
Time (Hours)
12
210
70
70
.33
.33
18
420
95
.23
m/sec)
Carolina
18
430
70
.16
24
700
150
205
.21
.29
24
750
95
55
.13
.07
(Average wind speed =8.7 m/sec)
18
-------
TABLE 2. The Relative Accuracy of Trajectory Analysis for the AVE III.
(OOZ Feb. 6 12Z Feb. 7, 1975)
a. Trajectory of air parcel leaving North Platte, Nebraska
Travel Time (Hours)
L(3)
AU
0
AL]2
AL6/L(3)
AL12/L(12)
b. Trajectory
L(3)
AL6
AL12
AL6/L(3)
AL12/L(3)
0
0 (km)
0
0
0
0
of air
0
0 (km)
0
0
0
0
6
310
0
0
parcel
6
170
0
0
12
660
0
110
0
.12
(Average wind
leaving Green
Travel
12
370
0
70
0
.19
(Average wind speed
c. Trajectory
L(3)
AL6
AL12
AL6/L(3)
AL12/L(3)
of air
0
0 (km)
0
0
0
0
parcel
6
50
0
0
18
970
40
.04
speed
24
1180
50
180
.04
.15
= 12m/sec)
30 36
1370 1580
15 25*
250
.01 .02
.16
Bay, Wisconsin
Time (Hours)
18
610
70
.11
= 10.1
leaving Buffalo, New
Travel Time
12
100
0
45
0
.45
(Hours)
18
230
30
.13
24
850
100
145
.12
.17
m/sec)
York
24
430
35
25
.08
.06
30 36
1070 1320
80 55*
170
.07 .04
.13
30 36
670 910
<15 40*
60
<.02 .04
.07
(Average wind speed =6.9 m/sec)
In the AVE III, the data for the first and last 12-hour periods were taken
at 6-hour intervals while for the middle 12-hour period, the data were
taken at 3-hour intervals.
19
-------
the vectorial differences of trajectories obtained with the 3-hour and 6-hour
intervals, and with the 3-hour and 12-hour intervals, respectively. According
to these tables, trajectories obtained with the 12-hour intervals are, in
general, less accurate than those obtained with the 6-hour intervals. Least
accurate trajectories are obtained in the regions where the speed of the
layer-averaged wind is strongest. It is also r\otor* that, if the wind speeds
and directions vary from one time interval to the next, the trajectories
become less accurate. As can be expected, the relative errors defined by
ALg/L(3) and AL,2/L(3) are larger when a discontinuity, such as a cold front,
passes through the region.
This preliminary study indicates that trajectories obtained with 6-hour
intervals between observation times may be accurate enough so as not to warrant
the need of 3-hour intervals.
From the figures, it can be seen that the directions and distances of
trajectories obtained with 12-hour intervals agree, in general, with those ob-
tained with shorter intervals expept when a discontinuity passes through the
region. Therefore, trajectories calculated with 12-hour intervals may be re-
liable especially when the statistics of many trajectories is used.
4.5.2. Dispersion Study
Preliminary studies of horizontal dispersion, 2, , were also made along
the trajectories of layer-averaged winds in the mixing layer. The results are
shown in Figures 7 and 8 where (a), (b), and (c) are, respectively, the
trajectories analyzed with the 3-, 6-, and 12-hour intervals. The horizontal
dispersion is calculated as
2h = 2(avh,l ' *1 +avh,2 t2+ .... +avh>n -tn) + L0,
where L is the initial width of the source area perpendicular to the mean
wind and t is the time step of the trajectory analysis. In this study, L is
assumed to be zero. An average value of av, is determined for each trajectory
time step. In Table 3, the values of 2, calculated with different time
intervals for each trajectory analysis are shown. Also shown are the relative
errors defined by A2h/2h(3), where A2h is given by 2h(3) - 2h(6 or 12) |. It
can be seen that the agreement of the values of 5, calculated along trajector-
ies is good between different time intervals.
4.6. Computer Program for Trajectory Analysis
The computer program for trajectory analysis which was provided by NOAA's
Air Resource Laboratories has been modified for our purpose.
The details of this program are described by Heffer and Taylor (1975),
and can be summarized as follows.
20
-------
Figure 7 (a).
Horizontal dispersion along the trajectory of the layer
averaged wind for the AVE II. Time interval = 3 hours.
May 11 - May 12
21
-------
Figure 7 (b).
Horizontal dispersion along the trajectory of the layer-
averaged wind for the AVE II. Time interval = 6 hours.
May 11 - May 12
22
-------
Figure 7 (c).
Horizontal dispersion along the trajectory of the layer-
averaged wind for the AVE II. Time interval = 12 hours.
May 11 - May 12
23
-------
Figure 8 (a).
Horizontal dispersion along the trajectory of the layer-
averaged wind for the AVE III. Time interval = 3 hours.
Feb. 6 - Feb. 7
24
-------
Figure 8 (b).
Horizontal dispersion along the trajectory of the layer
averaged wind for the AVE III. Time interval = 6 hours.
Feb. 6 - Feb. 7
25
-------
Figure 8 (c).
Horizontal dispersion along the trajectory of the layer -
averaged wind for the AVE III. Time interval = 12 hours.
Feb. 6 - Feb. 7
26
-------
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A trajectory is composed of a series of three-hour segments. Each
segment is computed assuming persistence of the winds reported closest to the
segment time. For example, for a three-hour segment from OOZ to 03Z, three-
hour persistence of the OOZ winds is assumed. The three-hour segment from 03Z
to 06Z is computed assuming three-hour "backward" persistence of the 06Z
winds.
For each station, the average wind in the mixing layer is computed from
the reported winds linearly weighted according to height. Using the average
winds calculated, each trajectory segment is computed as (see Fig. 9).
R
Z DW. AW. TS.
Here TS is the trajectory segment, £ indicates the summation over all ob-
served winds within a radius R of the segment origin, DW. is the distance
weighting factor, TS.. = V^-At, is the contribution to the trajectory segment
from an observed wind to the mid-point of TS.; and AW. = f(9.), is the align
ment weighting factor, a function of 9^ the angle formed between TS. and a
line drawn from the segment origin to a wind observation point.
For the calculations of a and a , defined by Eqs. (10) and (11),
* y
similar equations as Eq. (15) are used:
R
L DW. AW. aV .
Z DW. AW..
R
Z DW, AW. aV
The dispersion factor a is calculated from Eq. (12). a along a trajectory,
h vh
say from OOZ to 06Z, is given as an arithmetic mean of two values at OOZ and
06Z.
The following are parameter values used in the program:
At = 3 hours
R = 300 nautical miles
28
-------
segment
origin
mid. .
point
TS:
Figure 9. Configuration for determining a trajectory segment from observed
winds.
29
-------
2
DW. = 1/dw. (the closest observation receives the greatest weight)
AW. = 1 - 0.5 sin 8. (observations upwind and downwind receive
the greatest weight)
The trajectory segments are linked together to produce a complete tra-
jectory. The first segment starts from the endpoint of the segment before it.
Trajectories terminate after the desired duration or when the specified
criteria are not met.
The input data necessary are as follows:
1. The origin of the trajectories.
2. The data trajectory computations begin and the number of days for
which trajectory computations are desired.
3. The number of days of wind input data.
4. The mixing layer height.
5. The geographical boundaries within which observed winds are con-
sidered for trajectory calculations.
6. The geographical boundaries for maps in the plotting subroutines.
In the following, we illustrate the outputs of the trajectory program
which is modified for the Atmospheric Variability Experiment II and III
(Scoggins and Turner, 1974; Fuelberg and Turner, 1975). The trajectories are
originated at North Platte, Nebraska on May 11, 1974, using AVE II data.
Table 4 shows the number of reporting stations within a specified radius
for each time interval. The reliability of a trajectory can be evaluated from
the number of reporting stations.
In Table 5 the latitude and longitude of trajectory segment endpoints at
each interval are shown. A trajectory that was terminated for not satisfying
operational criteria is identified by a latitude >^ 996 and longitude _> 9996.
Table 6 shows the height of the mixing layer and dispersion parameter
a at each time interval. In this particular example, the height of the
h
mixing layer is designated at 200 mb above the surface.
The accumulated dispersion width, Z^, defined by Eq. (14) is given in
Table 7. Figure 10 is an example of the plot of the trajectories. Trajec-
tories are coded A, B and C for starting time at 12Z, 18Z, and OOZ, respec-
tively.
30
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SYMBOLS
TRAJECTORY STARTING 12Z AT *
TRAJECTORY STARTING 18Z AT *
TRAJECTORY STARTING OOZ AT *
3 HOURS DURATIONS
6 HOURS DURATIONS
9 HOURS DURATIONS
12 HOURS DURATIONS
15 HOURS DURATIONS
18 HOURS DURATIONS
21 HOURS DURATIONS
24 HOURS DURATIONS
STATION LBF
INDIVIDUAL TRAJECTORIES FOR II MAY 74
110.0
106.6
103.2
99.9
96.5
93.1
43.0
42.7
42.4
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33.5
33.1*
ACBl
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Figure 10. An example of the plotting of trajectories.
The trajectories originated at North Platte,
Nebraska, on 11 May 1974.
35
-------
4.7. Characteristic of Dispersion Parameter a
h
4.7.1. Application to the Wind Profile Given by the Ekman Layer Theory
and the Wind Profiles of the AVE Data
The classical Ekman layer theory defines the wind field as
vv = Un(l - e"bz cos bz)
x 9
v = U e~ z sin bz
*/ j
where, b = /F72k, f is the Coriolis parameter and K is the momentum eddy
diffusivity. v is the wind component along the geostrophic wind and v is
x y
the wind component perpendicular to the geostrophic wind. The height of the
Ekman layer, h, is given by
h = Trt>.
Assuming that the height of the Ekman layer is identical with the height of
the mixing layer, the following quantities are obtained:
u" = 0.8638 U
av = 0.3026 U
x g
av = 0.1102 U
j y
av
av.
= 0.166 U
.193 u
For a steady state atmosphere which is also assumed by the Ekman theory, the
width of the pollutant plume along the mean wind after the dispersion time
T = nt is simply given by
ah = [avh] T = .193 u T
assuming that L = 0. This quantity is compared with observations of pollu-
tant plume widths in Figure 11 (Bauer, 1973). There is good agreement between
our estimates and experimental observations despite the assumption in the
Ekman theory of an idealized neutral environment within the mixing layer. The
agreement is especially good at travel times between one hour and one day.
av, calculated from wind fields obtained from actual radiosonde data are
in reasonable agreement with those from the classical Ekman layer theory.
36
-------
DISTANCE FROM POLE TO EQUATOR(103cm)
MEAN CURVE & BOUNDS
FROM HAGE (1964) -\P
icfsec 10 sec
10cm
Isec
lOsec I0"sec I03sec I0~sec
I hour
Idny lOdays Imonth 6rno lyecr lOyears
TRAVEL TIME
Figure 11. Horizontal dispersion as a function of travel-time (after
Bauer, 1973). The calculated mean cloud widths for v" = 1,
10, m/sec using the Ekman theory are shown by solid lines.
37
-------
Figures 12 and 13 show calculation results using the data of AVE II and III
which were obtained at weather stations in the eastern United States. In
these calculations the mixing layer height was assumed to be at 200 mb above
the ground. In Figures 12 and 13, each dot represents one observation. In-
cluded are observations from all locations in the eastern United States. The
scattering is, of course, caused by the different wind profiles at different
locations and tiroes. However, notice the magnitudes of av./u which are
mostly within the range of 0.1 through 0.4, except in the cases of low mean
wind speeds. The classical Ekman layer theory gives the value of .193.
Therefore, it seems that the horizontal dispersion calculated from
radiosonde data would give values compatible to those observed with the
spreading of pollutant plumes.
4.7.2 Persistence of Dispersion Parameter a Along a Trajectory
h
Our method of calculating the horizontal dispersion of pollutants along
the mean wind trajectory is based on the assumption that the wind-shear
structure throughout the mixing layer along the trajectory is persistent at
least for the period of the time step of the trajectory segment.
In order to examine the validity of this assumption, the persistence
index of a along a trajectory, defined as the ratio of the average value of
vh
the absolute difference of a between the time step to the average value of
h
a along a trajectory, is calculated. The persistence index P.I. is ex-
h
pressed as
P.I. =
N aVL
N
(18)
aV
hi
The calculations of this persistence index are conducted for different time
steps along trajectories originating from 36 different locations in the
eastern U.S.
The data used are AVE II. Trajectories are started at 12Z and 18Z, May
11, and OOZ, May 12, 1974. For each time step along each trajectory,
P.I. is calculated by Eq. (18). Then, the mean values and standard deviations
of the P.I. calculated for different time steps along different trajectories
are obtained.
The results are shown in Fig. 14(a),(b) and (c). In the figure dots are
the mean values of P.I., and bars are the standard deviations of P.I. It can
be seen that the values of the persistence index increase gradually with
38
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h
figures show that the values of a are about 0.3 for the time step of three
h
hours, and about 0.4 for the time step of 12 hours. This means that the
relative accuracy of o calculated with a time step of 12 hours is not too
different from the one with a time step of three hours.
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our method of calculating the horizontal dispersion of pollutants can be
applied.
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44
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SECTION 5
REGIONAL RESIDENCE TIMES OF S02 OVER THE EASTERN U.S.
5.1. Introduction
The present energy shortage will make it mandatory to develop fossil
fuels with a greater potential for polluting the environment than was here-
tofore the case. For an assessment of the environmental impact of this
energy development it will be essential to arrive at a better understanding
of the fate of pollutants, especially of sulfur dioxide, in the atmosphere.
A parameter which adequately characterizes the fate of pollutants over
long time- and space-scales is the "residence time or "turnover time" in the
atmosphere. The residence time of a pollutant can be defined as the time
required to decrease its burden in the atmosphere over a certain region by a
factor of 1/e, assuming that no further input into the atmosphere and no
import across the boundaries of the region occur. The turnover time is the
time equal to the total mass of a pollutant in the atmosphere over a certain
region divided by the removal flux of the pollutant.
There have been several reports on the residence time of sulfur dioxide.
Meetham (1950) studied the budget of industrial S02 over England and obtained
a residence time of sulfur dioxide of 11 hours. Junge (I960) reexamined
Meetham1s results and obtained a residence time for sulfur dioxide of four
days. In the same paper, from the data of emissions and depositions of
sulfur dioxide over the United States and the average horizontal transport
velocity, Junge obtained a residence time for anthropogenic S0? in the United
States of about five days. Rodhe (1970) found from a study in Sweden that
only a small part of the anthropogenic sulfur from a city in Sweden was
deposited within the nearest 10 km and the residence time for this sulfur
must have been at least five hours. From the atmospheric sulfur budget over
northern Europe, Rodhe (1972) estimated a turnover time for anthropogenic
sulfur of two to four days. Based on the emission data of S0~ over Europe
and calculations by a transport model, Eliassen and Saltbones (1974) arrived
at a residence time for SO- of about half a day.
These studies show that the residence time for SOp is different at
different locations and for different seasons. Most of the above studies
have been based on data obtained during several days when no precipitation
has occurred.
In this section, we report the residence time for sulfur dioxide in the
eastern United States, based on climatological data of the mixing layer
45
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height and of precipitation in the region. The reason for studying the resi-
dence time in this area is that most of the industrial activities are located
in this area.
Most of the pollutants have their sources near the surface of the earth
and are emitted into the mixing layer, where they eventually spread toward a
state of uniformity throughout the depth of the layer. The mixing layer
depth is an important parameter in air-pollution meteorology. For example,
seasonally averaged values of pollutant concentrations depend on seasonal
averages of the depth of the mixing layer over the region considered. Large
portions of the total burden of pollutants stay in this layer. Some of the
pollutant escapes by vertical transport into the free atmosphere above the
mixing layer through convection, through turbulent mixing and through large-
scale vertical motions. In this paper we are concerned only with the fate of
pollutants which remain in the mixing layer.
Calculations were made of the regional residence time of sulfur dioxide
over the region of the United States east of 105°W longitude where major
industrial activities are located. The year was divided into the cold season
(January - April, and November and December) and the warm season (May -
October), and the regional residence time for each of these two seasons was
calculated.
5.2. Data Used
5.2.1. Mean Mixing-layer Height, F.
The mixing-layer height is defined as the level above the surface which
limits the relatively vigorous vertical mixing near the ground. Holzworth
(1967 and 1972) defined the maximum mixing depth as the height at which the
adiabat through the surface temperature maximum observed between 1200 and
1600 1ST intersects the actual temperature sounding curve obtained from the
1200 GMT sounding. The morning mixing-layer height is calculated as the
height above ground which the dry adiabat through the minimum surface tem-
perature observed between 0200 at 0600 1ST, increased by 5°C, intersects the
observed 1200 GMT temperature sounding. Both mixing heights, thus defined,
have to be considered only as approximations to the actual depth of the
mixing layer. They are, however, thought to be reasonable estimates suited
for practical applications, especially when such applications are intended
for large regions and for climatological studies.
The report by Holzworth (1972) contains the isopleths and the tables of
these mixing-layer heights_for four seasons. For the purpose of this paper,
the mixing-layer heights, H, calculated for the cold and warm seasons separ-
ately, are defined as the averages of the afternoon mixing heights of winter
and spring, and summer and fall, respectively, as reported by Holzworth.
5.2.2. Precipitation data.
In order to calculate the mean durations of dry periods i^ and wet
periods T , and the mean scavenging coefficient A which will be described in
46
-------
the following section, the hourly precipitation data of the year 1974 (U.S.
Department of Commerce, 1974) were analyzed. The data for 61 stations located
in the study area were used for the computation of these parameters for each
season.
5.3. Approach
5.3.1. Regional residence time T and turnover time Te.
In order to calculate the residence time of a pollutant we assume that
it is distributed uniformly throughout the mixing layer, that there is no
leakage through the top of this layer, that the imported and exported amounts
of a pollutant due to turbulent diffusion across the boundaries of the region
are balanced, that it is removed from the layer by dry deposition and pre-
cipitation scavenging, and that it is transformed into other species by first
order reactions. Under these assumptions, we can write
dT = - <*d + kP + xc> c
C = CQ exp [-(Xd + kp + Xc)t]
The e-folding residence time T can be defined as follows:
T =
kt
C is the concentration of the pollutant under consideration, k. the total
decay rate, X the decay rate due to dry deposition, k the decay rate due to
precipitation scavenging, and AC the chemical transformation rate, t ,, t
and t would be defined as the residence times, if only one of the mechanisms,
\*
dry deposition, precipitation scavenging, or chemical transformation were
responsible for the removal.
Assuming a Markov process for the sequence of weather events, Rodhe and
Grande!! (1972) derived an expression for the expected "turnover time" Te of
a pollutant in the presence of precipitation. This "turnover time" is defined
as the total mass of the pollutant in the atmosphere divided by the removal
flux and can be written as:
T , + T + T ,T (P .X' + P X')
TP - d P d P d P P d
16 " T,A' + T A' + T,T X'X1
d d pp dpdp
Td -
0
47
-------
oo
tnp(n)dnp (22)
<23>
<24)
Here, T, and T are the mean durations of dry and wet periods, respectively,
nd and n are the frequency distributions of dry and wet periods, T is the
time, p . is the probability of dry periods and P is the probability of wet
periods. A' and A, are, respectively, the removal coefficients during wet
and dry periods.
The coefficients A' and A", can be written as:
xp = Xp+Xd+Xc (25)
M = xd + xc (26)
where, A is the rate of precipitation scavenging, A . is the rate of dry
deposition, and A is the rate of chemical transformation. In this paper, we
assume that Ad and A are constant, independent of weather.
The e-folding residence time T given by (19) and the turnover time, Te,
given by (20) are calculated and compared.
5.3.2. Dry deposition.
Dry deposition of pollutants subject to airborne transport can occur by
sedimentation and by retention at the ground through impaction or adsorption.
The mechanism of dry deposition is most conveniently expressed by the
concept of a deposition velocity, v . This velocity is defined as
v
deposition rate
g volumetric concentration
v is dependent on many factors, such as the surface roughness of the terrain,
the stability of the atmosphere, the chemical properties of the pollutants
and the biological properties of the plant canopy. There are ample data from
48
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field experiments of deposition velocities available for sulfur dioxide gas
(Garland et al., 1974; Owers and Powell, 1974; Shepherd, 1974; Whelpdale and
Shaw, 1974). The difficulty of analyzing such field data is that the factors
on which the deposition velocity depends enter into the data partially in a
controlled, and partially in an uncontrolled manner. Among these field
experiments the data by Whelpdale and Shaw (1974) show exceptionally systemat-
ic values. According to these authors, the deposition velocity of S02 is
larger under unstable conditions of the atmosphere than under stable condi-
tions. This is due to the fact that the stable atmosphere causes a suppres-
sion of pollution transfer between different atmospheric layers, whereas in
an unstable atmosphere there is a great deal of turbulent exchange. On the
other hand, the data obtained by Garland et al. (1974), Shepherd (1974), and
Owers and Powell (1974) do not show a dependence of deposition velocity of
sulfur dioxide on atmospheric stability. However, those results show that
the deposition velocity lies in the fairly broad range between 0.1 cm/sec and
3.0 cm/sec which could easily encompass the effects of different stabilities.
The deposition velocity of sulfur dioxide was also estimated from a mass
budget study. Meetham's (1950, 1954) analyses showed that the deposition
velocity is about 1.8 cm/sec. From aircraft sampling of sulfur dioxide and
sulfate off the east coast of England, Smith and Jeffrey (1974) concluded
that the loss of sulfur dioxide due to deposition was commensurate to a
velocity between 0.8 and 1 cm/sec over land, and between 0.6 and 0.8 cm/sec
over the sea.
Prahm et al. (1976) obtained a deposition velocity for sulfur dioxide of
2 cm/sec +; 50% over the Atlantic from the study of atmospheric transport of
sulfur oxides over the Atlantic.
In accordance with the above brief reviews, we assume that the dry
deposition velocity, v^, of sulfur dioxide is 1 cm/sec regardless of the
region and the season.
Given v., the decay rate A . due to dry deposition can be written as
A = ^ = f (27)
d H td
where IT is the mean mixing layer height.
5.3.3. Precipitation scavenging.
Climatological data of pollutants in precipitation show that their
concentrations in rain are large where the concentrations of pollutants in
the air at ground level are large, suggesting that the air mass in the
vicinity of the clouds from which the precipitation falls is responsible for
the concentration of pollutants in rainwater (Lodge et al., 1968; Stevenson,
1968; Andersson, 1969). Thus, the following assumptions can be made:
49
-------
1. The air in the mixing layer directly below the cloud is transported
into the cloud by large-scale vertical motion as well as by con-
vective motions.
2. The pollutants in the air are scavenged by both rainout and washout
processes.
3. The precipitation containing pollutants falls on the same general
region from which the polluted air was entrained into the cloud
system.
Under these assumptions it is practical to express both rainout and washout
processes in the single term of a "scavenging velocity", v . This parameter
is defined as p
Vp . (f ^ . p (28)
where K is the concentration of pollutants in rainwater, x the concentration
of pollutants in air, and p the mean precipitation rate. The subscript v
means that the ratio \\ is formed on a volume basis. This ratio is a func-
.X.
tion of many physical parameters such as the size distributions of the pollu-
tant and of precipitation particles, the chemical composition of the pollutant,
/K\
the precipitation rate, etc. The ratio varies with time even during the
\x /
same event of precipitation. Unfortunately, there are very few observations
available in which this ratio has been measured simultaneously with other
physical parameters.
Engelmann (1971) estimated from data by Georgii and Bielke (1966) that
(KC \
) for SOp is 19 for rains of 11 to 20 mm per day and 190 for
xAn
rain of 0.3 mm per day. Here the subscript m means that the ratio is formed
on a mass basis. These values on a mass basis can be converted into approxi-
3
mately those on a volume basis multiplying by a factor of 10 . Summers' data
(1970), obtained by flight observations of convective storms, show that the
ratio (-] for S02 is within the range of 2.7 x 104 to 8.6 x 104. For
\ X / v
typical values of S0? concentration in the air, the saturation value of
S07~ in distilled water is at least two orders of magnitude below the values
found in rainwater. The saturation concentration of SCL~ is determined,
however, by the pH, and oxidation will continue as long as the pH is kept
above the critical value. Adding NH, will accomplish this. Junge and Ryan
(1958) propose that there is enough NH3 in the atmosphere for this oxidation
process to account for the observed values of S0"~ in rainwater.
50
-------
However, recent investigation by Dana et al . (1976) showed that the oxi-
dation process due to NH~ is not solely responsible for the observed values
of SOT" in rainwater.
4 / K
In this paper, we chose the values of 5 x 10 as a typical value of
/v
for sulfur dioxide. The mean precipitation rate p was determined from the
hourly precipitation data, dividing the total precipitation amount during the
season of interest by the total hours of precipitation.
Given the values of (-\ and p", or the value of v , and the mean mixing-
layer height, F, the mean scavenging coefficient, T, can be calculated as
I = (£) p/iT = V /H (29)
\ A /y K
For the present purpose, the mean scavenging coefficient given by Eq.
(29) is used for X . The decay rate due to precipitation, k , in Eq. (19)
is given by
kp = Ppl (30)
5.3.4. Chemical transformation.
There are numerous research reports available on the reaction rates and
reaction kinetics of SCL with other pollutants (e.g., Urone and Schroeder,
1969; Bufalini, 1971). Some results are contradictory to others. However,
the following can be inferred: In the daytime and at low relative humidity
(below 70%), the photochemical reactions of SOp with (L and hydrocarbons will
be of primary importance. In this case SOp will be converted into H^SO^
aerosols. In the daytime when the relative humidity is between 70 and 100%,
and at night, the aerosols will absorb substantial quantities of water and
the aerosol size distribution will shift to larger sizes. The rate of S0?
conversion should be increased through solution oxidation mechanisms, aided
by the catalytic effects of metal salts present as nuclei. It is realized
that a full description of the transformation rate of S02 is difficult at
present.
From field observation data of the concentrations of S02 and particulate
SO, in Europe, Eliassen and Saltbones (1975) arrived at empirical values of
Xc of the order of 10"6 sec .
From the study of transport of sulfur oxides over the Atlantic Prahm et
al. (1976) arrived at XQ = 3 x 10'6 sec"1. In this paper, we assume that \c
in Eqs. (19) and (20) is 1 x 10"6 sec"1.
51
-------
5.4. Results
5.4.1. Mean dry period T, and mean wet period T .
In Fig. 15(a) and (b), the mean dry period, T,, is shown for the cold
season and for the warm season, respectively. In general, T, is larger in
the western part of the studied region than in the eastern part regardless of
the season. The region surrounding the Great Lakes and the coastal region
show small values of T,.
d
The mean wet period T is shown in Fig. 16(a) and (b) for the cold
season and for the warm season, respectively. T is within the range of two
w
to five hours. These values of T, and T are used for the following calcula-
tions.
5.4.2. Residence time due to dry deposition, t,.
In Fig. 17(a) and (b), the residence times due to dry deposition, t.,
are shown for the cold season and for the warm season, respectively.
From these diagrams, it can be seen that t, is larger in the warm season
than in the cold season over the whole region. This is simply due to the
fact that the mixing layer is deeper in the warm season than in the cold
season. The region surrounding the Great Lakes is shown to have the smallest
value of t,. On the other hand, the mean mixing-layer height in the western
part of the region studied is relatively high, resulting in the larger values
of td.
5.4.3. Mean scavenging coeffiecient T.
From Eq. (30), the mean scavenging coefficient T over the studied area
is calculated for each station, assuming that (K/X) = 5 x 10 . In Fig.
18(a) and (b), the distributions of A" are shown for the cold and the warm
seasons, respectively. The values of X are small in the western and the
northern parts of the studied area, and are large in the southern part of the
region. This is due to the_ moister climate in the southern area. Further-
more, it can be seen that T is larger in the warm season when the convective
activity is more frequent, than in the cold season so that, on the average,
the precipitation amount per precipitation event is larger in the warm season
than in the cold season.
5.4.4. Regional residence time T and turnover time Te.
Using Eqs. (19) and (20) and assuming k = 1 x 10" sec~ , the regional
residence time T and turnover time Te for the cold season and the warm season
52
-------
(a)
(b)
Figure 15. Mean dry period, Td (in hours), (a) for the cold season, and (b)
for the warm season.
53
-------
(a)
(b)
Figure 16. Mean wet period, TW (in hours), (a) for the cold season, and
(b) for the warm season.
54
-------
(a)
(b)
Figure 17. The residence time (in hours) due to dry deposition,
(a) for the cold season, and (b) for the warm season.
55
-------
(a)
(b)
Figure 18. Climatological scavenging coefficient, X, (x 10" sec) for
SO, (a) for the cold season and (b) for the warm season.
56
-------
SO}-
20
10
10 20 30 40 50 60 70
T ( hours) r
Figure 19. Turnover time, Te, versus residence time, T.
57
-------
(a)
30
(b)
\
30
Figure 20. The regional residence time, T (in hours), for S0?, (a)
for the cold season, and (b) for the warm season.
58
-------
are calculated for each station. In Fig. 19, Te versus T is plotted using
the values for both the cold season and the warm season at each station. It
can be easily seen that Te is slightly larger than T, and that, for practical
use, either Te or T can be calculated to represent the residence time for the
studied area.
Because of the close coincidence between Te and T, only the regional
residence times T for the cold season and the warm season are shown in Fig.
20(a) and (b).
The following can be seen from these diagrams;
1. The residence time is, in general, longer in the warm season than in the
cold season over the whole region studied. This is due to shorter dry periods
in the cold season than in the warm season, and due to the shallower depth of
the mixing layer in the cold season than in the warm season.
2. Short residence times characterize the region surrounding the Great Lakes
and the southern part of the United States.
3. Long regional residence times are found in the western parts of the
studied area, where the mixing layer height is large, the precipitation
frequency is small.
4. In the studied area, the regional residence time lies in the range be-
tween 20 and 40 hours for the cold season and in the range between 30 and 60
hours for the warm season.
5. A comparison of Fig. 20 with Fig. 17 shows a similar pattern of the
isolines, implying that in the studied area the dry deposition is the most
dominant removal mechanism.
As mentioned previously, Junge (1960) arrived at a residence time for
anthropogenic S02 of about five days over the United States. This value is
much larger than the values we obtained. The reason for this discrepancy is
that Junge did not take into account the removal due to deposition. Compar-
isons with the results obtained in Europe are difficult because of the differ-
ent methods of estimation and because of the different climate. However, it
should be noted that the residence times for SO- in the present study are in
agreement with those over Europe within a factor of two to three.
Our study has been based on the observed data available for v ,, ( )
and A . These parameters have been assumed to be constant over the studied
region, regardless of the season and the location. In reality, these para-
meters are, as has been mentioned previously, dependent on temperature,
humidity, wind speed, location and other factors. Therefore, the residence
times presented in the present paper should be regarded as approximate.
59
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5.5. Summary
Assuming that sulfur dioxide in the mixing layer is removed from the
atmosphere by dry deposition, by precipitation scavenging, and by transforma-
tion into S0.~, the regional residence time is defined and calculated for the
region of the United States east of 105°W longitude. The results have been
shown as the isopleths of the residence time. However, because of the assump-
/ K \ 4 6
tions that v, = 1 cm/sec, -) = 5 x 10 and k = 1 x 10 regardless of
VX /y C
season and location, the results should be regarded as approximate. Further
improvements must be delayed until these values are specified in more detail
for each season and location.
60
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SECTION 6
SCAVENGING OF AEROSOL POLLUTANTS IN CUMULUS CLOUDS
6.1. Introduction
Cumulus clouds are an important mechanism not only for transporting air
pollutants from the boundary layer into the free atmosphere, but also for
cleansing the atmosphere by precipitation processes.
Pollutants are removed by precipitation through rainout and washout
processes. "Rainout" comprises all processes within the clouds, and "washout"
constitutes the removal by precipitation below the clouds. It is important
to study the effects of the physical characteristics of clouds on rainout and
washout of pollutants. By understanding these effects, the vertical transport
processes of pollutants from the planetary boundary layer into the free
atmosphere and the cleansing mechanisms of the atmosphere will be estimated
quantitatively.
Recent studies by Dingle and Lee (1973) on rainout processes and by Dana
and Hales (1976) on washout processes of polydispersed aerosols make it
possible to investigate ":n detail the removal mechanisms of aerosol pollutants
by precipitation.
The study in this section is concerned with the rainout process in
cumulus clouds. A cumulus cloud model developed by Cotton (1972a,b) is
combined with the multi-rate model of in-cloud scavenging (Dingle and Lee,
1973), in order to investigate the effects of the physical characteristics of
clouds on in-cloud scavenging, particularly the characteristics of continental
and maritime cumulus clouds. The rain-out process in other types of clouds,
such as clouds associated with front formation must be studied in the future.
Studies carried out by Howell (1949), Mordy (1960) and Neiburger and
Chien (1960) have shown that the influence of cloud nuclei in determining the
number and size of cloudwater droplets is restricted to the lowest few meters
above cloud base. It is only in the base region that cloud nuclei are acti-
vated. Hence it is there that the concentration of cloudwater droplets is
determined. Subsequent condensation merely serves to increase the size of
droplets which are already present. In this study, we define a continental
cumulus as having a concentration of 300 droplets per cm and a radius dis-
persion of 0.25, and a maritime cumulus as having a concentration of 100
3
droplets per cm and a radius dispersion of 0.25. Here "radius dispersion"
is defined as the ratio of the standard deviation of droplet size to the mean
radius.
61
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The multi-rate model incorporates explicitly the rate of progress of each
process which contributes to the ultimate removal of contaminants from clouds.
The attachment of aerosols to cloudwater droplets and to rainwater droplets,
and the conversion of cloudwater droplets to rainwater droplets by autoconver-
sion and accretion are incorporated in the model. Water droplets in the cloud
are classified into two classes: cloudwater droplets refer to water droplets
having a diameter of smaller than 100 ym, and rainwater droplets refer to
water droplets having a diameter of larger than 100 ym. In the present study,
it is assumed that cloudwater droplets and rainwater droplets are kept un-
frozen even under temperatures below freezing.
In the case of a precipitating cumulus cloud, the discrete particle sizes
_2
of importance range from aerosol particles of the order of 10 ym in diameter
to precipitation drops as large as 5 x 10 ym in diameter. Therefore, in
order to study the interaction of pollutant particles and cloudwater droplets,
a model must be chosen in which the microphysical processes are incorporated
in as much detail as possible. One such model is the one-dimensional cumulus
model by Cotton (1972a,b). The model involves upward integration with height
following the rise of the convective bubble or plume. Although this cumulus
model deals only with the actively growing phase of a cloud, it is useful to
study the gross aspects of in-cloud scavenging in different clouds with differ-
ent physical characteristics.
6.2. The Model
6.2.1. Cumulus cloud.
In the following, the equations used in the model are cited. The details
will be found in Cotton (1972b).
1. Moisture continuity
The continuity equation of the total moisture mixing ratio QT
QT = % + Qc + QH
Q-j. = total moisture mixing ratio
q = cloud vapor mixing ratio
Q = cloudwater mixing ratio
QLI = rainwater mixing ratio
n
dQ, dq dQ dQ
H
3T 3 H - y - c
q = environmental vapor mixing ratio
y = entrainment parameter
62
- fallout (32)
-------
1 dM
^ - Mdl
M = cloud mass
dq eL a /dT \
(34)
q = saturation mixing ratio with respect to water
e = ratio of the density of water to that of air
L = latent heat of condensation
R = gas constant of air
a
g = acceleration of gravity
Here dT /dz is calculated from Eq. (39) below.
The continuity equation for Q is
dQ- dq
dz dz
Conversion =
Accretion
- conversion - accretion (35)
1_ oM
Padt
auto
w
1_ dM
p dt
accr
w
(36)
(37)
p = density of cloud air
a
The continuity equation of CL is
dQH
-TL = - yQu + conversion + accretion (38)
2. Cloud thermodynamics
The equation for the vertical lapse in temperature of a water-saturated
cloud is
63
-------
dz ,2
i * cqs (39)
C = specific heat at constant pressure
T = environmental temperature
3. Dynamics
The vertical motion equation used in the model is based on the derivation
by Squires and Turner (1962). The vertical change in momentum flux due to
buoyancy forces is
(40)
p, = density of cloud air
a
p = density of the environmental air
Q = total mixing ratio of condensed water substance
(Qs = QC + QH)
This equation can be rearranged as
If we let
and
Ka = 2y, (43)
equation (41) can be integrated analytically from z-j to z~, assuming F and
K are constant in the layer.
a
64
-------
w.
Az =
2F
a
2F
a
a
exp (-KaAz)
(44)
-
r
4. Autoconversion and accretion
The numerical calculations of Howell (1949), Mordy (1959) and Neiburger
and Chien (1960) show that droplet growth by condensation produces only a
relatively narrow distribution of small droplets during the lifetime of one
to two hours of a typical cumulus cloud. In warm clouds, a broadening of
this distribution to include some larger droplets can occur by collision and
coalescence. The occurrence of collision and coalescence among cloud parti-
cles is called "autoconversion".
Kessler (1967) hypothesized that the water converted to rainwater droplets
is size-distributed in the inverse exponential distribution formulated by
Marshall and Palmer (1948), given by
F(R) =
(45)
where y is a coefficient. Once these rainwater droplets have formed, they
can grow very rapidly by accretion of cloudwater droplets.
Using the stochastic collection model by Berry (1965), and assuming a
gamma distribution for cloudwater droplets, Cotton (1972a) derived the follow
ing equation for the autoconversion rate of cloudwater droplets to rainwater:
dM
dt
auto
= exp k' -
- h')2]
(46)
where t represents the age of a parcel of droplets and a1, k1 and h' are
coefficients which are functions of the cloudwater content for a given initial
concentration and dispersion.
When nQ = 100 cm"3 and vr = 0.25
h1
k1
a1
and when n
2.001 -0.478
- e m
9.63 -2.59
e m
300 cm'3 and v =0.25
r
(47a)
65
-------
h- = e
k. = _e2.46m-0.779
a1 = 15.6 x TO4 - 4.8 x 104 m.
Here n is the concentration of cloudwater droplets, v the radius dispersion
0 *3
and m is the liquid water content in gm-nf . The former set of parameters is
used for the maritime cloud and the latter is used for the continental cloud.
Kessler derived the following equation describing the accretion rate of
cloudwater droplets by rainwater droplets:
dH . ISOirpl^J ^-125^0707 r{3.5)mM0.875
(48)
where p is the density of water, E^D/C) represents an average collection
efficiency between the rainwater droplets and cloudwater droplets, r is the
_3
gamma function, and M is the rainwater content in gnvm .
5. Fallout of rainwater
Cotton (1972b) used the scheme suggested by Howell and Lopez (1968).
The scheme is to drop out the portion of water droplets that has a terminal
velocity larger than the updraft velocity, w. If D represents the raindrop
diameter falling at a terminal velocity equivalent to the updraft velocity,
then the water content that falls out is
^
7TQ D
MH(>DWJ = / -5 V dD (49)
6.2.2. Rainout model.
In the present model, the aerosols in the planetary boundary layer are
transported up into the cumulus cloud by convection. The cloud is envisaged
as an assembly of cloudwater droplets and rainwater droplets intermingled
with aerosol pollutants, some of which are free-floating in the cloud air and
some of which are collected by the droplets. Aerosol pollutants in the cloud
air and in the droplets are injected into the environmental air by detrainment.
Precipitating rainwater droplets also remove aerosol pollutants from the
cloud.
66
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After modifying the equations by Dingle and Lee (1973), the spatial
variation of the aerosol concentration in each category, while the aerosols
are carried up through the cloud, can be described by the following equations:
dN
a
dz
dN
ci
dz
N
E
M
E A,
- XN
N.
JH
w
1
w
for each i
(50)
(51)
dN
ri
dz
AN + Z A
. ci k=l
- yN
w
(52)
where
3
N = number concentration of aerosol pollutants in the cloud air (cm )
a
N = number concentration of aerosol pollutants attached to cloudwater
_o
droplets (cm~ )
N = number concentration of aerosol pollutants attached to the rainwater
droplets (cnf )
-1
A, = scavenging rate of aerosols by cloudwater droplets (sec" )
A2 = scavenging rate of aerosols by rainwater droplets (sec )
X = conversion rate of cloudwater droplets into rainwater droplets by
autoconversion and by accretion (sec~ )
i = aerosol size index
j = cloudwater droplet size index
k = rainwater droplet size index.
If A-,, A2 and X (the indices are omitted for brevity) are considered constants
in a single height interval, Eqs. (50), (51), and (52) can be integrated with
height, yielding the following solutions:
exp
w
(53)
67
-------
A)
N exp
A1 + A2
w
+ y Az ,
(54)
and
Al\
co A - ^
(IV + Nr + N ))exp (- yAz)
ro co ao
- N.
exp
- ~+V Az
A - A
exp
A1 +A2
w
u Az
(55)
- N.
X -
A2)
exp
2^
w
AZ
Here Az = z - zn ,, N , N , and N are the concentrations of aerosol in
n n~' ao co ro
cloud air, cloudwater, and rainwater, respectively. Equations (53), (54) and
(55) are evaluated for each size range of aerosol. In the present study, the
conversion rate X is parameterized as described in the next section, so that
the individual interactions between rainwater droplets and cloudwater droplets
cannot be distinguished. Therefore, it is assumed that aerosol pollutants
contained in rainwater are uniformly distributed in rainwater regardless of
the size of aerosols.
Based on this assumption, the effect of fallout is treated as follows:
In the evaluation of Nr of Eq. (55) at the height zn, N is replaced by
o
Nr (1 - P ), where P is the ratio of rainwater fallout to the total rainwater
o
at the height z ,.
6.2.3. The rate constants A-,, A-, and X.
1. The scavenging rate of aerosols by cloudwater droplets, A,
The scavenging of aerosols by cloudwater droplets occurs by Brownian and
turbulent diffusion (Greenfield, 1957, and Dingle and Lee, 1973), and by
phoretic diffusion (Slinn and Hales, 1971, and Young, 1974). Howell (1949),
Squires (1952), Mordy (1959) and Neiburger and Chien (I960) showed, from
theoretical calculations, that peak supersaturation (generally less than 1%)
is reached at a few tens of meters above cloud base and thereafter the cloud
droplets consume the moisture at such a rate that the supersaturation remains
below the 0.2% level. Warner (1968) estimated that the median supersaturation
68
-------
is O.l/o in small cumuli. The phoretic diffusion (diffusiophoresis plus
themophoresis) rate, compared with Brownian and turbulent diffusion rates, is
very small in the portion of cloud where the air is slightly supersaturated
with respect to water. Therefore, the phoretic diffusion effect is neglected.
The Brownian diffusion scavenging rate, Ag, is expressed by
AB(a,rs) = jf oM^-±^+ L±^-} (a + r) f(r)dr (56)
where a and r are aerosol and cloudwater droplet radii, respectively; A the
Cunningham correction factor (= 0.9); £ the mean free path of air molecules;
k, Boltzmann's constant; T the absolute temperature; and n. the dynamic vis-
cosity of air. f(r) is the size distribution of the cloudwater droplet. In
Eq. (56) and in the succeeding equations the subscript s refers to "the spec-
trum of", that is, A(a,r ) means the scavenging rate for aerosol of radius "a"
by cloudwater droplets having a spectrum defined by f(r)dr.
The turbulent diffusion scavenging rate, Ay, is expressed by
00 / I \ ''
AT(a,rs) = f 14,l(a + r)3 (J j f(r)dr (57)
o
where e1 is the rate of energy dissipation and v the kinetic viscosity of air
(Levich, 1962, Dingle and Lee, 1973). For a convective cloud such as cumulus,
? o
Ackerman (1968) obtained an average value for e of 46.2 cm sec" .
Considering that Brownian and turbulent diffusions are additive, the
scavenging rate of aerosols by cloudwater droplets A-, is given by
Al(a'rs) = Va'rs) + AT(a'rs) (58)
2. The scavenging rate of aerosols by rainwater droplets, A?
We take into consideration Brownian diffusion, interception collection,
and inertial impaction collection. Dana and Hales (1976) cited the following
approximate expressions for these collections.
The Brownian diffusion scavenging rate, AA, is expressed as
-i r\ o I 1 r\~~ i
(0.65 x
/\ v u w* A i u / iii\ ] 99 ' TT/T
LaV a^Rj
o
F(R)dR (59)
where R is the radius of rainwater droplets, a the radius of aerosol pollu-
tants, and F(R) the size distribution of rainwater droplets (Slinn, 1971).
69
-------
The interception collection rate, Aj, is
00
Aj(a,Rs) = | 3 | F(R)dR,
o
according to Fuchs (1964).
The inertial impaction rate, A,,, is given by
An(a,Rs) =
S - -
*12
3/2
F(R)dR
(60)
(61)
where
(62)
S =
p^ and p are the mass density of the aerosols and air, respectively, V. the
pa T,
terminal velocity of rainwater droplets, and v the kinematic viscosity of air.
Again considering these effects to be additive, the scavenging rate of
aerosols by rainwater droplets A2 is given by
A2(a,Rs) = Ag(a,Rs) + Aj(a,Rs) + Ajj(a,Rs) (63)
3. The conversion rate of cloudwater droplets into rainwater droplets, A
In cumulus clouds the rainwater droplets are generated by the processes
of autoconversion and accretion, as described previously. The growth due to
diffusion is small compared with these processes, so that it can be neglected.
Thus, the growth rate of rainwater can be expressed as
dM
dt
Therefore,
dM
dt
auto
dM
dt
= Am.
(64)
accr
A -
A " m dt
auto
dM
dt
(65)
accr.
Here, M and m are the contents of rainwater and cloudwater, respectively, in
dM
the cloud layer under consideration, and -^
auto
have been given
accr
70
-------
by Eqs. (47) and (48).
6.2.4. The size distribution of aerosols and cloudwater droplets.
1. The size distribution of cloudwater droplets
As mentioned previously, the size distribution of cloudwater droplets is
assumed to be given by a gamma distribution as
n r
f(r) = - 0 < r < - (66)
p(a)ea
_ p
The mean r and variance a of the distribution are given by
7 = ag (67)
and c2 = af (68)
The radius dispersion v specifies the parameter a by
vr = ? = of3* (69)
r
The mean radius is given by
and the parameter 3 is given by
B = I , _ 3m _11/3 (71)
P Urn* (a + 1) (a + 2) n^J (n)
Therefore, if the radius dispersion, v , the droplet concentration, n , and
the cloudwater content, m, are known, Eq. (67) completely specifies the distri-
bution.
2. The size distribution of aerosols
Model aerosol spectra for continental aerosols and for maritime aerosols
published by Junge and McLaren (1971) are used as input data for the present
study. In Fig. 21, the model aerosol distributions are shown.
71
-------
10
10
10"
10'
TJ
D
10'
10"
10"'
id2
10
continental
10
10
rp ( cm )
10
Figure 21. Model size distributions of continental and maritime air.
(After Junge and McLaren, 1971.)
72
-------
6.3. Numerical Procedures
1. In order to generate a cloud, the model sounding data shown in Fig. 22 are
used. This sounding is applied to both continental and maritime clouds.
In addition, the entrainment constant, c'(n = C'/RC» where R is the
cloud radius); the cloud radius, R ; the initial temperature perturba-
tion, AT; the initial updraft velocity, w-, ; the concentration of cloud-
water droplets, n ; the radius dispersion, vr; and the concentration of
rainwater droplets, N , are specified.
2. The vertical numerical integration scheme of the cloud parcel by height
steps, Az, is the same as suggested by Cotton (1972b). The integrations
of moisture continuity equations and vertical lapse rates in cloud tem-
perature are performed numerically with a first-order integration. The
vertical integration is repeated until the updraft velocity vanishes.
3. After each height step, Q (or m) and Qn (or M) are used to calculate the
size distributions of cloudwater droplets and of rainwater droplets, and
subsequently the scavenging coefficients, A,, A^, and X are derived. The
calculations of aerosol concentrations in cloud air, cloudwater, and
rainwater for each size class then follow.
4. In the next step the mass concentrations of aerosols in cloud air, cloud-
water, and rainwater are calculated as
Mx =
where the subscript x stands for cloud air, cloudwater or rainwater, i is
the size class of the aerosol and p is the density of the aerosols.
5. The next step is to calculate, at each height interval, the mass-integral
scavenging coefficients, A, (a ,r ) and A?(a ,R ), defined as
loo £.00
A-,(a,rs) a3f(a)da
j
(73)
I ajf(a)da
and
o
A2(a,Rs) a3f(a)da
A2(as,Rs) = ^,^ (74)
a3f(a)da
73
-------
300
400
500
600
Q.
700
800
temperature
- relative humidity
-20 -10
o
TCC)
10
20 40 60 80 100
RH(%) r
Figure 22. Model sounding used as an input.
74
-------
Equations (73) and (74) are for the scavenging of aerosols by cloudwater
and by rainwater, respectively. A,(a,r$) and A2(a,Rs) have been given by
Eqs. (58) and (63).
6.4. Results
6.4.1. Liquid water distribution with height in the clouds.
With the initial parameters of k1 = 0.2, Rr = 1 km, AT = 0.5°C, and
L»
w-, = 1 m/sec, and with the sounding of environmental air shown in Fig. 22,
the cloud tops were reached at 8300 m for both continental and maritime
cumulus clouds. The cloud bases are at 1500 m. As mentioned previously, for
the continental cumulus cloud, the concentration of cloudwater droplets,
nQ = 300 cm, the radius dispersion, vf = 0.25; and the parameter set given
by (47b) for the autoconversion equation (46) are used. For the maritime
o
cumulus, n = 100 cm, v = 0.25, and (47a) are used.
In Fig. 23, the distributions of cloudwater, Q , and rainwater, Qh, with
respect to height, are shown in terms of mixing ratio. The solid lines are
for a maritime cloud, and the broken lines are for a continental cloud. From
the figure, it can be seen that cloudwater conversion into rainwater is faster
in the maritime cloud than in the continental cloud.
6.4.2. The distribution of scavenging rates with height.
The scavenging rates, A-j and A2, defined by Eqs. (73) and (74), and A,
defined by Eq. (65), are shown in Fig. 24(a) and (b). Figure 24(a) is for the
continental cloud, and Fig. 24(b) is for the maritime cloud.
From these figures it can be seen that the scavenging rate of aerosols by
cloudwater droplets has a peak at a few hundred meters above the cloud base.
At this level, the conversion of cloudwater into rainwater has not started, so
that the scavenging of aerosols is only by cloudwater droplets. After the
rainwater starts to form, the scavenging rate, A-j, by cloudwater decreases.
The renewed increase in A-, above about 5000 m, for the case of the continental
cloud, can be explained by the decrease in the value of the denominator of Eq.
(69). In the case of the maritime cloud, a similar increase in A-, can be
seen, but it is not as obvious as in the continental cloud.
From a comparison of Fig. 24(a) with 24(b), it becomes clear that the
scavenging rate of aerosols by cloudwater droplets, A,, is about one order of
magnitude larger in the continental cloud than in the maritime cloud. On the
other hand, the scavenging rate by rainwater droplets, A2, is larger in the
maritime cloud than in the continental cloud. Similarly, the scavenging rate
75
-------
maritime
continental
aooo
£
s
2
eooo
4000
2000
cloud base
Qc andQh(x
Figure 23. Distribution of cloudwater, Q , and rainwater, Q. with respect to
height.
76
-------
8
^ 6
£
0
-C
"cloud base
345
A,, AZ,X (xio~B sec1)
10
Figure 24 (a). Distributions of scavenging rates, A-|, A2 and A, in the
continental cloud.
77
-------
8
I
cloud base
I 2 3 4 5 6 7 8 9 rO
M> ^z.x(x io"9 sec"1)
Figure 24(b). Same as Figure 24(a), except in the maritime cloud.
78
-------
of cloudwater by rainwater, X, is larger in the maritime cloud than in the
continental cloud. In these particular clouds, the average values of A,, A~
and X over the whole depth of the cloud are as follows:
Continental Maritime
A1 5.11 x 10"3 sec"1 4.88 x 10"4 sec"1
A2 1.76 x 10"3 sec"1 2.66 x 10"3 sec"1
X 3.47 x 10"3 sec"1 5.52 x 10"3 sec"1
Makhon'ko (1967) and Davis (1972) have obtained gross scavenging rates of
4 SI
10 ^ 10 sec . The values obtained in the present study can be favorably
compared with these observed values.
6.4.3. Mass fraction distributions of aerosols with height.
In Fig. 25(a) and (b), the mass fractions of aerosols in cloud air,
cloudwater and rainwater divided by the total mass of aerosols in the air
below the cloud base are shown. Figures 25(a) and (b) are for the continental
cloud and for the maritime cloud, respectively.
The theoretical calculations by Howell (1949), Mordy (1960), and Neiburger
and Chien (1960) show that the concentration of cloudwater droplets is deter-
mined by the number of effective condensation nuclei in the cloud base region.
Furthermore, the particles effective as condensation nuclei are, in general,
hygroscopic and larger than 0.1 microns, (Fletcher, 1966). Therefore, the
two cases are studied. In the first case, the simple assumption is made that
3
the 300 largest aerosol particles per cm are consumed as condensation nuclei
in the continental cloud (and the 100 largest in the maritime cloud) within
the first layer of integration above the cloud base. In the second case, con-
densation nuclei are assumed to be supplied by other sources. In Figs. 25(a)
and 25(b), the solid lines are for the case of condensation process assumed,
and the broken lines are for the case of no-condensation assumed for the
aerosols.
From a comparison of these two figures, the following can be inferred:
1. Aerosols in cloud air in the continental cloud are removed more
efficiently than those in the maritime cloud. This is due to more
efficient scavenging by cloudwater droplets in the continental cloud
than in the maritime cloud.
2. As a consequence, the mass fraction of aerosols in cloudwater is
larger in the continental cloud than in the maritime cloud.
3. Because of the more efficient conversion of cloudwater to rainwater
in the maritime cloud than in the continental cloud, the mass
79
-------
~ 6
I
condensation assumed
no condensation assumed
cloud base
MASS FRACTION
10'
Figure 25(a).
Distributions of mass fractions of aerosols in cloud,
cloudwater and rainwater in the continental cloud.
80
-------
8
I
I
condensation assumed
no condensation assumed
h dotld oir
L.
10-
cloud base
I0"
MASS FRACTION
Figure 25(b). Same as Figure 25(a), except in the maritime cloud.
81
-------
fraction of aerosols in rainwater is larger in the maritime
cloud than in the continental cloud.
4. In the continental cloud it does not make a significant difference
in the mass fractions of aerosols in cloudwater and in rainwater
whether condensation processes are assi.' d or not. On the other
hand, in the maritime cloud, the mass .factions of aerosols in the
cloudwater and in rainwater become significantly larger than those
of no assumption of condensation. This is again a consequence of
the fact that the scavenging rate, A-j, is larger in the continental
cloud than in the maritime cloud.
6.4.4. The ratio of aerosol mass in rainwater to that in the air at cloud
base, (K/x0)m.
Engelmann (1971) defines a "washout ratio" in terms of K/X , where K is
the concentration of a specific pollutant in rainwater, ant' x is the concen-
tration of the pollutant in surface air. According to the table of (K/X )
for various pollutants summarized by Engelmann, the values of (K/X ) vary, but
23 °
are within the orders of 10 to 10 . Here, the subscript rn means that the
ratio is taken on the basis of masses of rainwater and air.
In the present study, the following values of (K/X0)m are obtained:
Continental
Maritime
Total
Precipitation
Per Cloud
3.87 mm
5.82 mm
(K/x»)ra
- Condensation
Assumed
203
198
(K/X0)m
- No Condensation
Assumed
198
161
These values are in good agreement with values observed by various authors (see
Table 1, Engelmannn, 1971). It can be seen that (K/X0)m for the continental
cloud is larger than that for the maritime cloud.
6.5. Summary
The purpose of the present study was to investigate the scavenging char-
acteristics of continental and maritime cumulus clouds, based on a numerical
model of in-cloud scavenging, combined with a cumulus model. Using identical
sounding and initial parameters, w], AT, and R, two different clouds with
_ q
different numbers of cloudwater droplets (300 cm" for the continental cloud,
82
-------
and 100 cm for the maritime cloud) and with different autoconversion rates
of cloudwater to rainwater have been studied.
The scavenging rate of aerosols by cloudwater droplets, A,, is one order
of magnitude larger in the continental cloud than in the maritime cloud. On
the other hand, scavenging rates of aerosols by rainwater droplets, A^, and
scavenging rates of cloudwater droplets by rainwater droplets, X, are larger
in the maritime cloud than in the continental cloud.
Because of the larger value of A, in the continental cloud than in the
maritime cloud, the mass fraction of aerosols in cloudwater is larger in the
continental cloud than in the maritime cloud, and consequently the ratio of
(K/X ) is larger in the continental cloud than in the maritime cloud.
The application of results from the model calculation to the transport
model described in Section 4 can be made through the relationship
Vp = (K/XO)V P
where V is the scavenging velocity in equation 4, and P is the precipitation
rate. Therefore, if the precipitation rate in the area where the pollutant
plume passes through is known, the removal term of the pollutant due to pre-
cipitation can be estimated.
83
-------
SECTION 7
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88
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
NO
EPA-600/4-78-003
3. RECIPIENT'S ACCESSION-NO.
4 TITLE AND SUBTITLE
RESIDENCE TIME OF ATMOSPHERIC POLLUTANTS AND LONG-
RANGE TRANSPORT
5. REPORT DATE
January 1978
6. PERFORMING ORGANIZATION CODE
7 AUTHOR(S)
Teizi Henmi, Elmar R. Renter and Roger Edson
8. PERFORMING ORGANIZATION REPORT NO.
9 PERFORMING ORGANIZATION NAME AND ADDRESS
Colorado State University
Fort Collins, CO 80523
10. PROGRAM ELEMENT NO.
1AA603 AG-03 (FY-77)
11. CONTRACT/GRANT NO.
803685
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Environmental Sciences Research Laboratory - RTP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
Final 5/75-4/77
14 SPONSORING AGENCY CODE
EPA/600/09
15 SUPPLEMENTARY NOTES
16. ABSTRACT
The Lagrangian trajectory model which is suitable for the study of long-range
transport of pollutants is dev°loped. The computer program is capable of calculating
trajectories over the region of the U.S. using routine sounding data. The output
consists of tables of locations of trajectory end points at each time-step, dis-
persion widths along a trajectory, and the plotting of trajectories. The regional
residence times, T, of SO,, in the mixing layer are calculated for the region of
the United States east of 105CW longitude, based on climatological data of the
mixing layer depth and hourly precipitation data. The results are shown as isopleths
of T over the studied area for the cold season (November to April) and for the
warm season (May to October). Taking detailed microphysical processes into considera
tion, the scavenging due to cumulus cloud precipitation is studied. The results
can be summarized as follows: The scavenging coefficient of aerosols by cloudwater
droplets is one order of magnitude larger in the continental cloud than in the
maritime cloud. On the other hand, the scavenging coefficient of aerosols by
rainwater droplets is slightly larger in maritime clouds than in continental clouds.
As a whole, aerosols are more efficiently scavenged in the continental cloud than
in the maritime cloud.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
* Air pollution
* Transport properties
Meteorological data
* Atmospheric models
* Aerosols
* Sulfur dioxide
b.IDENTIFIERS/OPEN ENDED TERMS
c COSATI field/Group
i3B
04B
14A
07D
07B
13 DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report!
UNCLASSIFIED
21. NO. OF PAGES
103
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
89
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