EPA-R4-73-030a
July 1973
ENVIRONMENTAL MONITORING SERIES
$i$l$j£^
'^^i^^^^^i^^^^^^^
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EPA-R4-73-030a
URBAN AIR SHED PHOTOCHEMICAL
SIMULATION MODEL STUDY
VOLUME I - DEVELOPMENT AND EVALUATION
by
S.D. Reynolds, Mei-Kao Lui,
T.A. Hecht, P.M. Roth, and J.H. Seinfield
Systems Applications, Inc.
9418 Wilshire Boulevard
Beverly Hills, California 90212
Contract No. 68-02-0339
Program Element No. 1A1009
EPA Project Officer: Herbert Viebrock
Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
July 1973
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This report has been review jd by the Environmental Protection Agency and
approved for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the Agency, nor does
mention of trade names or commercial products constitute endorsement
or recommendation for use.
11
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ABSTRACT
During the period July 1970 to June 1971, Systems Applications,
Inc., under Contract CPA 70-148, undertook "Development of a Simulation
Model For Estimating Ground Level Concentrations of Photochemical
Pollutants". The results of this work have been presented in Roth et
al. (1971), a seven volume project report which describes in detail
all aspects of the year's efforts. The abstract of that report states
the following:
"In this report we describe the progress that has been
achieved to date in the development and validation of
a simulation model for estimating ground level concen-
trations of photochemical pollutants. This model is
based on the finite difference solution of the equations
of conservation of mass, using the method of fractional
steps. The bulk of the effort reported here is develop-
mental, involving the compilation of a comprehensive
source emissions inventory, the development and valida-
tion of a kinetic mechanism for photochemical reactions,
the adaptation of the method of fractional steps for
use in the solution of the governing equations, and the
preparation of maps displaying spatial and temporal vari-
ations in wind speed and direction and in the height of
the inversion base. The details of these various efforts
are described in a series of appendices to this report.
Although a validated kinetic mechanism has been developed
and incorporated in the simulation model, validation
efforts have thus far been restricted to carbon monoxide.
Provisional validation results for the Los Angeles Basin
are presented."
The work that has been carried out under the present contract is
a continuation of the earlier developmental study. However, while
the emphasis in the initial project was on model development, we have
been concerned in the present study with
Limited model development and improvement
Extensive evaluation of the photochemical kinetics
mechanism, involving fourteen experimental studies
and four hydrocarbon systems
Extensive evaluation of the urban airshed model for the
Los Angeles Basin for six days on which pollution levels
were high. Five pollutants were considered — carbon monoxide,
nitric oxide, nitrogen dioxide, hydrocarbon, and ozone.
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It is the purpose of this report to fully document the procedures
and results of this effort.
This report consists of two parts — Volume I, which discusses
all technical aspects of the work (this document) and Volume II, which
presents a description of the computer programs that embody the model
and contains a user's guide.
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CONTENTS
Page
INTRODUCTION 1
I. AN OVERVIEW 3
II. THE AIRSHED MODEL 5
A. Theoretical Formulation of the Model . 5
B. The Model Developed in this Study 14
C. Modifications and Extensions to Original
Formulation of Model 23
1. Emissions 23
2. Photochemistry 24
3. Meteorology 24
4. Numerical Analysis 30
5. The Model 34
6. Modeling of Subgrid Scale Phenomena .... 37
7. Computer Programs 38
8. Air Quality Data 39
III. EVALUATION OF THE MODEL 42
A. The Evaluation Procedure 43
B. The Results 53
C. Discussion of Results 69
IV. RECOMMENDATIONS FOR FUTURE WORK 95
V. REFERENCES 98
THE EVALUATION RESULTS,
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CONTENTS (Continued)
This final report includes the following Appendices:
Appendix A Extensions and Modifications of a Contaminant
Emissions Model and Inventory for Los Angeles
Appendix B Further Validation of a Generalized Mechanism
Suitable for Describing Atmospheric Photochemical
Reactions
Appendix C A Microscale Model for Describing the Contribution
of Local Vehicular Sources to Pollutant Concentrations
Measured at Monitoring Stations
Appendix D Numerical Integration of the Continuity Equations
and
Volume II User's Guide and Description of the Computer
Programs
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INTRODUCTION
During the period July 1970 to June 1971, Systems Applications,
Inc., under Contract CPA 70-148, undertook development of a computer-
based mathematical model capable of estimating ground level concentra-
tions of photochemical pollutants. The results of this work have been
presented in Roth et al.(1971), a seven volume project report which
describes in detail all aspects of the year's efforts. The work that
has been carried out under the present contract is a continuation of
the earlier developmental study. However, while the emphasis in the
initial project was on model development, we have been concerned in
the present study with
• Limited model development and improvement
Evaluation of the photochemical kinetics
mechanism, involving fourteen experimental studies and
four hydrocarbon systems
Evaluation of the urban airshed model* for
carbon monoxide, nitric oxide, nitrogen dioxide, hydro-
carbon, and ozone for six validation days.
It is the purpose of this report to fully document the procedures and
results of this effort, which we will refer to hereafter as the
Phase II study.
The report describing Phase II consists of six parts. It is
broadly divided into two portions—Volume I, which discusses all
technical aspects of the work (this document) and Volume II, which
presents a description of the computer programs that embody the
model and contains a user's guide. Volume I, however, has four tech-
nical appendices, each of which is an independent report describing a
particular aspect of the work. The four appendices are:
Appendix A. Extensions and Modifications of a Contaminant Emissions
Model and Inventory for Los Angeles
*We will use the term "urban airshed model" both to refer to the model
developed in Phase I and validated in Phase II and to describe any
model capable of simulating transport, diffusion, and reaction in the
atmosphere on a regional scale.
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Appendix B. Further Validation of a Generalized Mechanism Suitable
for Describing Atmospheric Photochemical Reactions
Appendix C. A Microscale Model for Describing the Contribution of
Local Vehicular Sources to Pollutant Concentrations Measured at
Monitoring Stations
Appendix D. Numerical Integration of the Continuity Equations.
The main volume (I) itself is divided into two main parts.
The first (Section II) is concerned with the urban airshed model, its
theoretical foundation and its formulation, as well as with the modi-
fications and improvements made during Phase II. The second (Section
III) focuses on the validation of the model—procedures, results, and
discussion and evaluation of results. Discussion of the validation
of the photochemical kinetics mechanism is confined in its entirety to
Appendix B; it is not considered in the main text. Moreover, with the
exception of the commentary in this volume dealing with the theoretical
foundations and formulation of the urban airshed model, there has been
no attempt made to repeat any portion of the seven volume Phase I
report by Roth et al. (1971). The interested reader is referred to
this report if he wishes to obtain a full understanding of the model
and its initial development.
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I. AN OVERVIEW
Urban airshed models are mathematical representations of atmos-
pheric transport and chemical reaction processes which, when combined
with a source emissions inventory and pertinent meteorological data,
may be used to predict pollutant concentrations as a function of time
and location in the airshed. Models capable of accurate prediction over
a range of meteorological and source emission conditions will serve as
an important aid in urban and regional planning, including use in:
1. Simulating the effects of alternative emission control
strategies on pollutant concentrations in the airshed
2. Real-time prediction in an alert warning system
3. Examining the air pollution impact of new sources, such as
freeways and power plants.
A dynamic airshed simulation model that is to be generally useful
in urban planning studies must meet several requirements. First, it
should be capable of predicting accurately the ground level concentra-
tions of inert pollutants, as well as those formed in the atmosphere
by chemical reactions. Second, the model should have a spatial and
temporal resolution appropriate for the analysis of concentration vari-
ations which occur in a city throughout the course of a day. For a
typical large urban area, the horizontal spatial resolution may be of
the order of a mile, and the temporal resolution, of the order of an
hour. The resolution of the model will, of course, be influenced by
the availability of data of similar resolution. Third, the complexity
of the model, and thus the computing time and computer storage require-
ments, should be such that the model can be operated at a reasonable
cost using computers of general availability. The objective of the
combined Phase I and Phase II programs is to develop and validate an
airshed simulation model for photochemical air pollution that satisfies
the stated requirements.
As we have noted earlier, Phase I was primarily developmental.
We first selected a suitable modeling approach, one based on the numeri-
cal integration of the nonlinear, coupled equations of conservation of
mass. We then compiled a comprehensive source emissions inventory for
the region to be modeled. At the same time, we undertook the development
and validation of a kinetic mechanism for photochemical reactions and
selected and adapted a method for numerical integration of the continu-
ity equations. We concluded initial development with (1) preparation of
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hourly maps displaying spatial and temporal variations in wind speed
and direction and in the height of the inversion base for two "vali-
dation days" and (2) preparation and testing of the computer codes em-
bodying the model. In addition, validation runs were carried out for
carbon monoxide for the two days selected. We refer the reader to
Roth et al. (1971) for a full discussion of the development program.
Some comments are in order at this point as to the selection of
an urban area for the purpose of validation. The Los Angeles area was
chosen as the region for initial application of the model for three
reasons. First, the meteorological and pollutant data base in Los
Angeles is one of the richest available for any major urban center. A
network of nearly three dozen wind speed and direction sensors and
twelve air quality monitoring stations dot the Basin. In addition,
during the summer of 1969, the Scott Research Laboratories carried
out an extensive data gathering program in Los Angeles (1970). Particu-
larly valuable were the vertical temperature profile data they gathered
over three sites in the Basin, thereby permitting much more accurate
specification of the depth of the mixing layer than is normally possible.
Second, Los Angeles smog represents the most serious and persistent in-
cidence of photochemical air pollution in the United States. Third,
because of its lack of proximity to other large urban areas, Los Angeles
has an air pollution problem which is entirely locally generated. Thus,
we avoid the need to account for the influx of significant amounts of
pollutants from upwind areas.
In contrast to Phase I, the present, or Phase II, study has focused
primarily on model evaluation To be sure, a number of modifications
and improvements to the model have been made during this effort. For
example, the source emissions inventory was revised and extended (par-
ticularly for automotive emissions), the numerical integration procedure
was modified, a microscale model was developed for describing the con-
tribution of local vehicular sources to pollutant concentrations measured
at monitoring stations, treatment of both the chemistry and meteorology
were modified, and an additional computer program was written for the
purpose of data plotting and presentation. But, taken together, these
efforts constitute only a relatively minor portion of the overall study.
Primary attention was given to (1) evaluation of the photochemical kinetic
mechanism using smog chamber data and (2) evaluation of the airshed model,
first for an inert (carbon monoxide) to test the computer program and
to examine the treatment of meteorological variables, then for nitric
oxide (NO), nitrogen dioxide (NO ), hydrocarbon (HC), and ozone (0 ) for
six pre-selected "validation days". As we have indicated, evaluation
of the kinetic mechanism is discussed fully in Appendix B. In this
report, we devote our attention to evaluation of the airshed model. But
first we discuss and describe the model itself.
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II. THE AIRSHED MODEL
In this section we present a complete description of the model
we have developed. In the first part of the section we discuss the
theoretical foundation for the model, derive the governing equations,
and enumerate the assumptions upon which the model is based. In the
second part, we present the model as it has been developed for appli-
cation purposes. In particular, we transform the governing Equation (5)
into a form suitable for computation, describe the region that has been
modeled, and specify the spacing and extent of the grid of nodes that
has been overlaid on the region. We conclude the section by discus-
sing the various modifications and extensions of the original formula-
tion, as presented in Roth et al. (1971), that have been made during
the current effort.
A. Theoretical Formulation of the Model
The simulation of photochemical air pollution entails giving
description to the behavior of a number of chemically reactive species
in the turbulent atmospheric boundary layer. Consider N chemically
reactive species in a fluid. The concentration Ci(x,y,z,t) of each
constituent must satisfy the continuity equation,
3t 9x
\dx dy dZ /
(1)
+ R. (c.,-••,c ,T) + S. (x,y,z,t) i = 1,2,...,N
where u,v and w are the components of the wind velocity, D^ is the
molecular diffusivity of species i in air, R^ is the rate of forma-
tion of species i by chemical reaction, T is the temperature, and
S^ is the rate of emission of species i from sources. In most fluid
dynamics problems involving chemical reaction, it is necessary to carry
out the simultaneous solution of the coupled equations of mass, momen-
tum and energy to account properly for the changes in c^, u, v, w and T
and the effects of the changes in each of these variables on each other.
In considering air pollution, however, it is often quite reasonable to
assume that the presence of pollutants in the atmosphere does not
affect the meteorology to any detectable extent. An important excep-
tion is the attenuation of incoming radiation by photochemically-
generated haze, a common occurrence, for example, in the Los Angeles
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area during the summer and autumn months. While the variation in
pollutant concentrations may, in this way, alter the energy input to
the system, and thus affect both the energy and momentum equations, it
is possible to incorporate this effect in the equations of continuity
alone (by treating measured intensities as data) if it is not of in-
terest to predict temperature and velocity. Thus, the equations of
continuity (1) may be solved independently of the coupled Navier-Stokes
and energy equations.
Since atmospheric flows are turbulent, it is customary to represent
the wind velocity components as the sum of a deterministic and stochas-
tic component; e.g., u = u + u'. Substituting u = u + u', etc., into
Equation (1), taking the expected value of the equation and assuming
that molecular diffusion is negligible when compared with turbulent
dispersion yields the following equation governing the mean concentra-
tion ,*
a -
<> +
+ 4- + v- (2)
i 3y i 3z i
Si(x,y,z,t)
where c. = + c!, = 0, and turbulent fluctuations in temperature
have been neglected.
Equation (2) is the basic equation for the mean concentration of a
reactive pollutant species in the atmosphere. Its direct solution is
precluded by the appearance of the new dependent variables, ,
, and , as well as any variables of the form which arise
from . Considerable attention has been given to means for approximating
* We employ different notation for the mean wind components, u, v, and w,
and the mean concentrations . The bars are used for time-averaged
quantities, whereas the brackets are used for ensemble-averaged quantities.
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the variables , , and (see for example, Kraichnan (1962),
Saffman (1969), and Monin and Yaglom (1971)). The simplest and most pop-
ular means is through the so-called K-theory (Calder, 1949; Pasquill, 1962;
Monin and Yaglom, 1971) in which one sets
It is well known that in the proper general form of the K-theory, the
set of quantities Kvv, K..,,, etc. constitute the components of a second
A J\. ^*j
order tensor 1C. In most reported applications of the K-theory, off-
diagonal terms" of the form K , etc. are set equal to zero. For the
diagonal form of K to be valid at all points of the region it is neces-
sary for the tensCr to have the coordinate axes as principal axes at all
points of space. Such a situation can occur in the surface layers of
the atmosphere, where the mean wind vector can be regarded as everywhere
parallel to a given vertical plane (Calder, 1965). In a large airshed
this is clearly not the case. If it is possible to assume merely that
the mean velocity is parallel to the ground, with components u ^ 0,
v ^ 0, w = 0, then the only valid form of the K-theory, as given by
Equation (3), can be that in which K = K . Henceforth, we denote
K and K as K , and K as K . ^
xx yy H zz V
While there has been considerable study of means for relating the
variables , , and to the mean concentrations, there
has been comparatively little examination of approximations for terms
of the form which arise when chemical reactions are taking place
in turbulence. ^This lack of study is primarily a result of both the
enormous theoretical difficulties associated with the description of
turbulent chemical reactions and the lack of experimental data against
which to compare the predictions of the turbulence theories which have
been developed.* As a consequence of this situation, we make one final
*We refer the reader to Corrsin (1958), Lee (1966), and O'Brien (1966,
1968ab, 1969, 1971) for further information concerning initial studies
of means for approximating joint moments of the . In spite of
these studies, it is still not possible to assess the importance of
the contribution of fluctuating terms of the form to the mean
rate of reaction in atmospheric chemical reactions, although an
initial effort along these lines has been made by Donaldson and Hilst
(1972).
(3)
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simplifying assumption relative to Equation (2), namely that the mean
rate of reaction can be approximated by the rate based on the mean
concentrations; i.e.,
^,..., + c^, T)> = Ri(,..., ,T) (4)
Substituting Equations (3) and (4) into (2) yields
3
.) + —
3y i 9z
/ + ¥ \KH ""ay/ + ** \v ~^/
(5)
R (,..., , T) + S (x,y,z,t)
11 N 1
Contrary to the impression conveyed in a number of earlier air
pollution modeling studies, Equation (5) is not the fundamental
equation governing the dynamic behavior of air pollutants in the atmos-
phere; rather, by virtue of Equations (3) and (4), it is an approximate
equation, valid only under certain circumstances. We shall employ
Equation (5) as the basic model in this study. However, before doing
so, it is necessary to consider the limitations inherent in Equation
(5) that restrict its applicability in describing the transport and
reactions of air pollutants in the atmosphere.
Assessing the validity of Equation (5) for modeling air pollutant
dynamics has been the subject of two recent studies (Lamb, 1973; Lamb
and Seinfeld, 1973). It has been shown in these studies that Equation
(5) is a valid representation of atmospheric transport and chemical
reaction provided that the:
1. Time resolution of the model, At , is large compared with the
Lagrangian time scale of the turbulence.
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2. Characteristic temporal and spatial scales for gradients
in the mean velocity field are large compared with the
time resolution At and the average distance that a fluid
particle travels in At.
3. Characteristic temporal and spatial scales for gradients
in the turbulent velocity correlations are large compared
with the time resolution At and the average distance that
a fluid particle travels in At.
4. Characteristic temporal and spatial scales for gradients
in the source emission functions S^ are large compared with
At and the distance a particle travels in At.
5. Characteristic temporal scale for changes in the rate of
generation or depletion of species by chemical reaction,
R. , is large compared with At.
These conditions may be expressed more precisely in quantitative
forms. In particular, conditions 2 and 3 imply that each of the fluid
velocity components should satisfy the requirements*:
« -3- i » 1,2,3
=i3t
3ui
lj = 1>2>3
*For convenience in stating these conditions we employ the notation
u.,, u , u and KI , x_, x in place of u,v,w and x,y,z, respectively.
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where R. (3£,t) is the Lagrangian time correlation function,
oo
= C
J n
di k = 1,2,3
and v (sc,t) is the Lagrangian velocity at position x_ and time t.
Conditions 4 and 5 can be written as:
,
At] k=1'2'3
3R.
R. at At
These conditions, of course, impose restrictions on t and on the
temporal and spatial resolution of the velocity field and the source
emission functions that are to be used in Equation (5). Therefore, we
must determine the extent to which these restrictions apply for condi-
tions typically observed in the Los Angeles Basin. Only then can we
specify the appropriate spatial and temporal scales for the model.
In the Los Angeles airshed there are roughly three dozen wind
monitoring stations, with an average separation between them of roughly
seven miles. At most stations the wind speed and direction data are
averaged over a one-hour period. From these hourly-average values the
mean surface wind field, u(x,y,t) and v(x,y,t), can be constructed. Sub-
tracting the mean values from the instantaneous readings at each
10
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station and averaging over time yields the Eulerian correlations
, i,j = 1,2,3. An Eulerian time scale for the turbulence,
can be estimated from these correlations. Although the precise re-
lationship between the Eulerian and Lagrangian time scales, T and T ,
where E L
is unknown, a convenient estimate is that T =4r (Hay and Pasquill,
1959). Having estimated T from the wind station data, for the condi-
tion that At » T , we can place a lower limit on At, such as ST .
Using a At » T insures that the four conditions pertaining to the
fluid velocity components will be satisfied.
Once At has been selected, we must specify the spatial and temporal
resolutions for S^ and the temporal resolution for R^ that satisfy the
final three conditions. These conditions will determine the degree of
detail required for the source emissions inventory and the chemical
reaction mechanism, one which is commensurate with that of the meteoro-
logical data.
Unfortunately, data of the type needed to estimate the Eulerian
time scale of the turbulence in the Los Angeles Basin are not generally
available. In an effort to gather data of this type, Lamb and Neiburger
(1970) measured the turbulent structure of the atmosphere at a height of
20 meters in West Los Angeles. The wind velocities were averaged over
a period of T = 0.3 hours. From these data, TE was estimated to be 50
seconds. Assuming that TJ^ = 4TE, TL equals 200 seconds. We therefore
estimate that At >_ 103 seconds on the basis that At >_ STL- The
Lagrangian correlation functions were estimated to be: R^ - 100 m2sec~ ,
11
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2-1 2-1 3
R22 ~ 10° m sec ' and R33 ~ 10 m sec . With At = 10 seconds and
these values of the Rj^, the conditions limiting the validity of
Equation (5) assume the quantitative values shown in Table 1. Since
the conditions in Table 1 are based on data for which the averaging
time was 0.3 hours, they are less stringent than those based on data
for which the averaging time is 1 hour.
We have found that the minimum temporal and horizontal spatial
resolution in the source emission function must be 10 seconds and
2000 meters, respectively. Thus, the averaging time and distance
for source emissions should be, say, at least twice these values.
We note that, in spite of the fact that many major pollutant sources
are point and line sources, emissions must be averaged over relatively .
large distances to conform with the resolution of Equation (5). In the
source emissions inventory described in Appendices A of Roth et al. (1971)
and this report, we have spatially and temporally averaged source emis-
sions over 2 miles (approximately 3000 meters) and 1 hour, respectively
Thus, the spatial averaging we have employed is of somewhat finer reso-
lution than that suggested by the conditions in Table 1. The condition
on R£ states that the characteristic time scale for changes in the con-
centrations as a result of chemical reaction should be greater than 103
seconds—perhaps of the order of one hour. Finally, Lamb and Neiburger
(1970) estimated that the minimum vertical resolution of concentration
changes is of the order of 20 meters or greater. The minimum vertical
mesh spacing employed is roughly 20 meters.
In summary, we find that Equation (5) is applicable in resolving
those perturbations in the concentration field which have horizontal
scales greater than 2 kilometers, vertical scales greater than 20 meters,
and temporal scales greater than 10 seconds. These conditions serve
as a guide to the choice of grid size and averaging time to be used in
the solution of Equation (5).
*In the early morning, in order to account for the nonuniform distribu-
tion of trip starts, motor vehicle emissions are temporally averaged
over 15-minute periods for the first hour (6-7 a.m.).
12
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Table 1. Conditions Limiting Equation (5) Based on the Measurements
of Wind Turbulence of Lamb and Neiburger (1970).
1. Source emission function
as.
1 meter"1, k = 1,2
2ooo
2. Chemical reaction rate
«
3. Mean velocity components
i!^i
- 3t
u.
-1
u.
', k = 1,2
-rrr meter , k = 3
4. Turbulent velocity
components
8 ..-3 -1
i ^ 10 sec
9t
2000
meter , k = 1,2
-rr meter , k = 3
13
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B. The Model Developed in this Study
The model developed is based on the solution of the N coupled
partial differential equations (5) defined on the region, x^ _<_ x _<_ x
VS — y - VN'
h(x,y) ^ z ^ H(x,y,t) for t ^ t0, where
XE,
are
the west, east, south and north boundaries of the airshed, h(x,y) is
ground elevation above sea level at (x,y), and H(x,y,t) is the elevation
above sea level of the base of a temperature inversion or an assumed
upper limit for vertical mixing or transport. The initial condition on
Equations (5) is that the mean concentration be specified at all loca-
tions ,
(6)
(At this point and henceforth, for convenience, we omit brackets on
the concentrations and overbars on the velocities. However, all concen-
trations and velocities continue to be mean and time-averaged quantities,
respectively.)
The vertical boundary conditions are:
1. z = h(x,y) - K X^-n^ = Qi(x,y,t)
(7)
where K is the eddy diffusivity tensor,
K =
0 K 0
_° ° V
n. is the unit vector normal to the terrain directed into the
atmosphere, and QJ is the mass flux of species i at the surface.
2.
z = H(x,y,t)
= 0
if
if
(8)
> 0
14
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where V is the advective velocity of pollutants relative to the
moving inversion base, given by
V = ui + vi +
n^j is the outwardly directed unit vector normal to the surface
defined by the inversion base, and g. is the mean concentration
of species i aloft (just above the inversion base).
The condition V-njj ^ 0 in equation (8) applies when material is
transported into the modeling region from above the inversion base.
This boundary condition simply states that the normal component of the
mass flux is continuous across the upper boundary. The condition
V*ri > 0 applies when pollutants are transported up through the inversion
base. Because of the abrupt stability change associated with an in-
version layer, it is reasonable to assume that the turbulent diffusive
flux across the boundary is zero, thereby attributing any pollutant
transport into the inversion layer to advection alone. The second
boundary condition in Equation (8) expresses the negligibility of the
turbulent diffusive flux at the inversion base.
The horizontal boundary conditions are:
(Uc. - KVc.) 'n = £g. (x,y,z,t) •£ if £-n_ <_ 0
(9)
- KVc. -ri = 0 if IJ-ri > 0
where £ = ui^ + vj_, n_ is the outwardly directed unit vector normal to
the horizontal boundary, and g^ is the mean concentration of species
i just outside the airshed boundary. The first condition is, as
before, a statement of the continuity of mass flux across the boundary
when the flow is directed into the airshed. The second condition
specifies that the diffusive component of the total mass flux be set
equal to zero when the wind is directed out of the airshed. This con-
dition is equivalent to that conventionally employed at the exit of
tubular chemical reactors (Wehner and Wilhelm, 1956), although the
15
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conditions prevailing at the boundary of the region are not precisely
the same as those at the exit of such a reactor. Since the horizontal
advective component of the mass flux generally dominates the horizontal
diffusive component, the error incurred due to this approximation is
generally small.
In this study, we applied Equations (5) - (9) to the prediction of
pollutant concentrations over much of a fifty-mile square area that
includes virtually all centers of population in the Los Angeles Basin.
The region was divided into a grid of 625 2-mile x 2-mile squares,
198 of which lie over oceans or mountainous terrain having no pollutant
sources. These "source-free" grid squares were not included in the
region actually modeled, which is shown in Figure 1. Source emissions
and meteorological variables are distributed in conformance with this
grid; i.e., two miles is the resolution of the model, or the spatial
dimension over which all quantities are averaged. Furthermore, for
reasons to be discussed, the grid actually used in the solution of
Equations (5) is a three-dimensional array of five layers of cells
occupying the space between the ground and the base of the inversion
and lying directly over the area shown in Figure 1. Thus, each cell
has a base two miles square and a height of (H - h)/5. The center of
each cell, or node, is the point to which values of all variables are
assigned or referenced. Unfortunately, due to variations in both H and
h with x and y and, in the case of H, with t, the three-dimensional
modeling region has an irregular "roof" and "floor". To eliminate these
irregularities, which hamper the solution of the equations, we performed
the following change of variables:
z - h(x,y)
= x n = y r *
^ H(x,y,t) - h(x,y)
With these changes, Equations (5) becomes
•J- (AHc.) + -^r (uAHc.) + -j- (vAHc.) + -~- (We.)
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