United States
Environmental Protection
Agency
Environmental Sciences Research
Laboratory
Research Triangle Park NC 27711
EPA-600/4-78-049
August 1978
Research and Development
Select Research
Group in Air
Pollution
Meteorolgy
Third Progress Report
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields
The nine series are.
1 Environmental Health Effects Research
2 Environmental Protection Technology
3 Eco'ogical Research
4. Environmental Monitoring
5 Socioeconomic Environmental Studies
6 Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9 Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-78-049
August 1978
SELECT RESEARCH GROUP IN AIR POLLUTION METEOROLOGY
THIRD ANNUAL PROGRESS REPORT
by
R. A. Anthes
A. K. Blackadar
R. L. Kabel
J. L. Lumley
H. Tennekes
D. W. Thomson
The Pennsylvania State University
University Park, Pennsylvania 16802
Grant No. R800397
Project Officer
Francis Binkowski
Meteorology Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, 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.
11
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FOREWORD
The set of papers comprising this particular report by members of the
Select Research Group (SRG) in Air Pollution Meteorology at the Pennsylvania
State University was prepared at the request of K. Calder, the program's EPA
project monitor.
After 3 years of research, the scientific results of the EPA-supported
program at Penn State were clearly becoming evident in the air pollution
meteorology research community. For example, at the Third Symposium on Atmo-
spheric Turbulence, Diffusion and Air Quality held in Raleigh, North Carolina
on 19-22 October 1976, ten scientific papers were presented by members of
SRG. Numerous other publications are now in press or preparation for a
variety of the regular air-pollution-related scientific journals.
Rather than duplicate publication of SRG's diverse scientific results, it
was felt that the members would perform an important service by writing their
views, principally in a review sense, on topics related to their project
responsibilities. The following papers are the results of that exercise.
111
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ABSTRACT
In lieu of a scientific report of the detailed investigations of the
Select Research Group in Air Pollution Meteorology (SRG) (which are being
published elsewhere), a series of reviews on topics of concern to SRG has been
prepared at the request of the EPA scientific monitor. These reviews are
presented in this report.
The overall problem of constructing a general, predictive air-quality
model is considered in an article by R. A. Anthes. Emphasis is placed on the
meteorological aspects of the modeling problem. A scale analysis of the pol-
lutant conservation equation indicates that different physical processes are
important on the urban, regional, and global scales of pollutant transport and
diffusion. Therefore, different types of models are appropriate for these
different scales. The physical and numerical aspects of predictive meteor-
ological models on the regional scale are reviewed in detail. Next, some
general classes of air pollution transport and diffusion models are discussed.
Finally, an example of a simple combined meteorological air-quality model is
given.
Most past attempts to model the distribution of meteorological quantities
and mixing processes within the planetary boundary layer have featured a
gradient- or K-type closure. A review of these closure specifications, parti-
cularly as they relate to the Richardson number, is given by A. K. Blackadar.
The implications of second-order closure methods for K-type approximations are
discussed, and a nocturnal boundary-layer model based on this approximation is
presented.
H. Tennekes considers the question of how mixing-height variations affect
the pollutant concentrations predicted by air-quality simulation models. It
IV
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is found that the variations of the mixing height have significant effects on
the climatology of box models, and that the behavior of a box model during
episodes of poor ventilation is modified profoundly by the amplitude of the
diurnal cycle in the mixing height.
J. Lumley considers two types of urban air pollution models according to
the way in which turbulent transport is simulated. It is shown that gradient
transport models cannot predict the countergradient fluxes of heat and con-
taminants that are observed in convective situations. Second-order models are
examined in detail, and recently raised questions regarding the accuracy
requirements for these models are put to rest. The details and limitations of
second-order models are discussed.
The importance of accounting for natural sources and sinks of atmospheric
pollutants in air-quality simulation models is discussed by R. Kabel. Several
removal processes are described and illustrated with examples from the litera-
ture. Emphasis is placed on those mechanisms active at the earth's surface.
In particular, a brief evaluation of methods of predicting gas- and liquid-
phase mass transfer coefficients for an atmosphere-water interface is attempted.
Alternative i-H-situ and remote-probing measurement techniques suitable
for application to model validation studies are reviewed by D. W. Thomson.
Locally applied inversion-rise, box, and Gaussian plume models may be verified
using conventional micrometeorological measurements supplemented by acoustic
sounder (sodar) observations. Definitive evaluation of urban- and regional-
scale models will require a variety of special surface and aerological measure-
ments .
v
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CONTENTS
FOREWORD iii
ABSTRACT iv
FIGURES ix
TABLES xii
METEOROLOGICAL ASPECTS OF REGIONAL-SCALE AIR-QUALITY MONITORING .... 1
R. A. Anthes
Introduction 1
Potential Uses for Predictive Air-Quality Models 3
Meteorological Models on the Regional Scale 18
Transport and Diffusion Models 36
Box Models 36
A Combined Meteorological Air-Quality Model 45
Acknowledgments 56
References 56
HIGH-RESOLUTION MODELS OF THE PLANETARY BOUNDARY LAYER 63
A. K. Blackadar
Introduction 63
Empirical Methods. 64
Implicit K Models 67
Approximations Based on a Second-Order Closure 76
Predictive Modeling of the Nocturnal Boundary Layer 83
References 100
Appendix: Soil Slab Model 105
THE EFFECTS OF MIXING-HEIGHT VARIABILITY ON AIR-QUALITY
SIMULATION MODELS 109
H. Tennekes
Introduction 109
Inversion-Rise Parameterization Ill
The Dynamics of Box Models 113
The Climatology of Box Models 126
Mode-Switching Problems 134
Conclusions 137
Acknowledgments 138
References 138
VI1
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SIMULATING TURBULENT TRANSPORT IN URBAN AIR POLLUTION MODELS 141
J. L. Lumley
Introduction 141
Classification of Urban Pollution Models 142
Drawbacks of Gradient-Transport Models 146
The Second-Order Models: Stability Considerations 150
Second-Order Modeling Technique and Pitfalls 153
Verification of Second-Order Models 157
What Can We Expect from Second-Order Models? 166
Acknowledgments • 169
References 169
NATURAL REMOVAL OF GASEOUS POLLUTANTS 175
R. L. Kabel
Introduction 175
Natural Removal Processes 176
Principles 182
Gas Phase Mass Transfer Coefficient 186
Liquid Phase Mass Transfer Coefficient 189
Overview 191
Acknowledgments 192
Nomenclature 192
References 193
OBSERVATIONAL REQUIREMENTS FOR VALIDATION OF AIR POLLUTION
METEOROLOGY MODELS 197
D. W. Thomson
Introduction 197
General Considerations 198
One-Dimensional Mixing-Layer Models 199
Box Models 203
Gaussian Plume Models 203
Lagrangian Puff Models 205
Grid-Point Diffusion Models 207
Grid-Point Dynamical Models 209
Considerations for the Organization of Mesoscale Experiments . 212
Nomenclature 213
References 214
Vlll
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FIGURES
Number Page
METEOROLOGICAL ASPECTS OF REGIONAL-SCALE AIR-QUALITY MONITORING
R. A. Anthes
1 Schematic view of the components of a general air-quality
model 7
2 Error in the geostrophic wind as a function of horizontal
interval produced when a temperature difference error of
1° C is integrated hydrostatically over a 200-mb depth
centered at pressure levels of 300, 500, 700, and 900 mb . 24
3 Temperature changes produced by vertical circulations
associated with a propagating wind maximum (jet streak)
at a level above the jet 27
4 Structure of a mixed-layer model 48
5 West-east cross section showing wind direction and potential
temperature structure at 9 hours from the mixed-layer
model 48
6 Horizontal profiles of the west-east component of velocity, u;
vertical velocity, w, at the top of the mixed layer; and
the height of the mixed layer at 0 hours (initial condi-
tions) , the 5- to 6-hour average, and the 11- to 12-hour
average . 52
7 Forecast SO concentrations after 3 hours using combined
meteorological (mixed-layer) and particle-in-cell
diffusion model 53
8 Forecast SO concentrations after 6 hours in the experiment
described in the text and in the caption for Figure 7 . . 55
9 Forecast SO concentrations after 12 hours in the experiment
described in the text and in the captions for Figures 7
and 8 55
IX
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Number Page
HIGH-RESOLUTION MODELS OF THE PLANETARY BOUNDARY LAYER
A. K. Blackadar
2 2
1 K /£ s and 1C /£ s according to various formulations 71
2
2 K /£ s as function of Richardson number according to various
m
formulations 72
3 Level 2 approximation for K and 1C for positive Richardson
numbers. , 82
4 Level 2 approximation for positive Richardson numbers 82
5 Atmospheric layers and grid nomenclature for the nocturnal
boundary-layer model 85
6 Spiral wind hodograph for a steady-state neutral simulation . . 92
7 Vertical distribution of K for a steady-state neutral
simulation , 92
8 Predicted and observed wind speed and potential temperature
profiles at O'Neill, Nebraska 94
9 Predicted and observed surface air temperatures as a function of
time for the O'Neill composite 94
10 Evolution of wind speed and potential temperature profiles for
conditions listed in Table 1; roughness parameter 0.1 m and
geostrophic wind speed 8.0 m/s 95
11 Evolution of wind speed and potential temperature profiles for
conditions listed in Table 1; roughness parameter 1.0 m and
geostrophic wind speed 8.0 m/s 95
12 Evolution of wind speed and potential temperature profiles for
conditons listed in Table 1; roughness parameter 1.0 m and
geostrophic wind speed 9.4 m/s 97
13 Wind vector hodograph calculated for Run 2 in Table 1 97
14 6 and u# as function of time during the night for conditions
listed in Table 1; roughness parameter 0.1 m and geostrophic
wind speed 8.0 m/s 98
15 6 and u^ as function of time during the night for conditions
listed in Table 1; roughness parameter 1.0 m and geostrophic
wind speed 8.0 m/s 98
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Number Page
16 6 and u^ as function of time during the night for conditions
a listed in Table 1; roughness parameter 1.0 m and geostrophic
wind speed 9. 4 m/s 99
THE EFFECTS OF MIXING-HEIGHT VARIABILITY
ON AIR-QUALITY SIMULATION MODELS
H. Tennekes
1 Typical 1-day emission curve and associated concentration
curves 120
SIMULATING TURBULENT TRANSPORT IN URBAN AIR POLLUTION MODELS
J. L. Lumley
la Observed profiles of turbulence quantities in buoyancy-driven
mixed layers 147
2 2
Ib The fluxes q u and -3q u /3x as calculated by a scalar
transport model 147
NATURAL REMOVAL OF GASEOUS POLLUTANTS
R. L. Kabel
1 Mass transfer coefficient correlations 190
xa
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TABLES
Number Page
METEOROLOGICAL ASPECTS OF REGIONAL-SCALE AIR-QUALITY MONITORING
R. A. Anthes
1 Scales of Air-Quality Models 11
2 Meteorology Classes Used in Scale Analysis 11
3 Results of Scale Analysis 13
4 Typical Parameterization of Physical Processes in
Mesoscale Models 31
5 Classes of Diffusion and Transport Models 37
HIGH-RESOLUTION MODELS OF THE PLANETARY BOUNDARY LAYER
A. K. Blackadar
1 Parameters and Initial Conditions Used in the Third Experiment . 96
THE EFFECTS OF MIXING-HEIGHT VARIABILITY
ON AIR-QUALITY SIMULATION MODELS
H. Tennekes
1 Amplitude Attenuation and Phase Lag for a Diurnal Emission
Cycle (u) = 27T/T) 118
2 Amplitude Attenuation and Phase Lag for a Semidiurnal Emission
Cycle (u) = 4TT/T) 119
SIMULATING TURBULENT TRANSPORT IN URBAN AIR POLLUTION MODELS
J. L. Lumley
1 Experiments That Elucidate Phenomena of Turbulence 161
XII
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Number Page
NATURAL REMOVAL OF GASEOUS POLLUTANTS
R. L. Kabel
1 Removal Processes 176
2 Solubility in Water and Uptake Rate of Pollutants 180
OBSERVATIONAL REQUIREMENTS FOR VALIDATION OF
AIR POLLUTION METEOROLOGY MODELS
D. W. Thomson
1 Data Requirements for Mixing-Layer Models 202
2 Data Requirements for Box Models 204
3 Data Requirements for Plume Models 205
4 Data Requirements for Trajectory and Puff Models 206
5 Data Requirements for Grid-Point Diffusion Models 208
6 Data Requirements for Regional Grid-Point Dynamical Models. . . 211
Xlll
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METEOROLOGICAL ASPECTS OF REGZONAL-SCALE AIR-QUALITY MONITORING
Richard A. Anthes
Department of Meteorology
INTRODUCTION
As urban industrial areas expand, air pollution problems extend farther
from localized pollution sources and become regional, national, and even
global in extent. The scientific, political, economic, and pathological
aspects of large-scale pollution problems are enormously complex. Unlike
local problems (in which those who do the polluting suffer diminished air
quality), on the regional and larger scales individuals far downwind pay the
price.
Besides the obvious economic, biological, and aesthetic penalties of
increased regional pollution, the earth's climate may be vulnerable to increas-
ing emissions of gases and particulates. One example is the documented (SMIC,
1971) increase in CO over the last 20 years and the associated modifications
to the global radiation balance. Another example currently receiving much
attention is the emission of freon at the ground with subsequent diffusion to
the stratosphere, where chlorine atoms formed by its photodissociation threaten
the ozone layer (Basuk, 1975). On a smaller scale, there is growing concern
that the use of cooling towers in giant energy parks may significantly affect
the temperature, cloud cover, and precipitation for tens of kilometers around
a site (Hanna and Gifford, 1975) . Already, considerable evidence exists that
warm-season rainfall is increased by as much as 30% within 50 km of major
urban areas (Huff and Changnon, 1973; Semonin and Changnon, 1974). Harnack
and Landsberg (1975) note cases wherein the urban thermal effect was the
likely triggering mechanism for shower development over Washington, D.C.
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The significance of possible changes to weather and climate and the
association between decreasing air quality and increasing morbidity and mortal-
ity (e.g., Landsberg, 1969) demand quantitative studies of the fate of pollu-
tants that have left a source. Such studies require models, since we wish to
be able to predict future air quality from a given set of initial conditions,
or to predict the outcome of a given pollution control strategy. Simply
monitoring a given pollutant or atmospheric condition, while a necessary task
in pollution control, does not permit predictions hours, days, or years in
advance.
"Air-quality model" can describe many models of varying types and com-
plexities. Even a qualitative assessment of meteorological conditions and
their probable effect on pollutant transport and diffusion represents a simple
model — an imprecise conceptual model built in the mind of the forecaster by
prior experience. At the other end of the complexity scale, we can imagine an
enormous mathematical, physical, and chemical model which predicts all scales
of atmospheric motion affecting contaminant transport and diffusion (and the
chemical reactions modifying each species). This model is well beyond our
present scientific and computational skills. Still, there must be models that
can provide acceptably accurate quantitative answers at reasonable cost.
This paper considers the overall regional-scale air-quality prediction
problem and summarizes potential uses and limitations of various available
mathematical models. Because the problem is very large, only a brief look at
each modeling component is possible. (Many of the individual components are
so complex as to deserve separate, detailed reviews.) After briefly enumerat-
ing some potential uses for accurate models, we discuss the general problem of
forecasting the concentration of a contaminant at a given point. This fore-
cast involves two major components: a meteorological model and a pollutant
transport and diffusion model. The development of a realistic, predictive
meteorological model is considered first. Modeling of pollutant transport and
diffusion by the wind field is then discussed. A specific example (based on
one aspect of current mesoscale modeling work at The Pennsylvania State Univer-
sity) of the combination of a time-dependent meteorological model and a trans-
port and diffusion model to produce an air-quality model is presented.
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POTENTIAL USES FOR PREDICTIVE AIR-QUALITY MODELS*
Before discussing the various types of air-quality models, it is useful
to speculate on the potential uses of a truly predictive air-quality model —
one which starts with an observed state of the atmosphere and, together with
the proper boundary conditions (including known emission rates), accurately
predicts the behavior of the pollutant in space and time. The question is
not whether a perfect forecast of this type is possible; we know the answer
to that question is no. The question is: How close, and at what cost, can
we approach this ideal?
One of the most useful applications of a predictive air-quality model
would be the prediction of the concentration of a particular pollutant at a
receptor in order to provide a rational basis for an air pollution control
strategy. Models for this purpose could be used in real time during air
pollution episodes to regulate industrial and domestic uses of energy or on
a long-term basis to study the consequences of proposed changes in pollutant
emissions. Such a prediction on the regional scale demands an accurate ac-
counting of contributions from possibly many sources upwind of the receptor.
With the knowledge of a relative contribution from each source, the efficacy
of a proposed control measure can be evaluated. An example of the use of an
air-quality model to test particular emission control strategies for the New
York City region is given by Slater (1974).
A second application of air-quality models is in land-use planning. As
major new polluting sites are proposed, it may be possible to take advantage
of local meteorological conditions to minimize the effect of emissions on the
local population. Models may be used to determine which location of major
transportation systems, fossil-fuel and nuclear power parks, or industries
will least affect the environment. In some cases, models may indicate that
the proposed construction should not proceed at all in the region, because the
cost to the local air quality or climate would be too great. Obviously, the
*Fortak (1974) provides an excellent comprehensive discussion of the applications
of air-quality modeling.
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models whose results determine the fate of such million- or billion-dollar
projects must be accurate. It is also obvious that, when changes to the micro-
and mesoscale climate are considered, such as the possibility of dramatically
increasing the frequency of winter fogs and summer thunderstorms by the con-
struction of a major cooling tower complex (e.g., Hanna and Gifford, 1975),
the models must be sufficiently general so as to handle the complicated feed-
back between the pollution and the atmosphere. It is in such problems asso-
ciated with land-use planning and the preparation of environmental impact
statements that the most complicated air-quality models will probably be most
useful.
A third use of air-quality models is the real-time forecasting of air
quality. Real-time forecasting of air quality can be useful to people planning
outdoor activities and domestic chores even during relatively non-critical
situations, as well as during the critical air pollution episodes when health
warnings may become necessary. The benefits of accurate real-time forecast-
ing in determining the impact of controls (such as a switch to cleaner fuels
or a curtailment of industrial activities) have been discussed above.
Finally, a fourth potential use of air-quality models is the evaluation
of a particular control strategy. In this role, the model may be utilized to
determine whether the pollution problem has improved because of the controls
or because of changing meteorological conditions.
An air-quality model that also includes time-dependent meteorological
parameters can be very useful as a research tool to determine the minimum
meteorological conditions for a given location that will ensure that concentra-
tions do not exceed a prescribed standard. Thus, many experiments with a model
under varying large-scale meteorological conditions may determine the necessary
local conditions for a 3-hour average to exceed the standard for a given
emission rate. For example, on a coastal city it might be found that for a
standard to be exceeded it is necessary that all of the following occur:
(a) the large-scale geostrophic wind speed is less than 5 m/s;
(b) the lapse rate near the ground is less than 3 °C/km;
2
(c) the surface heat flux is less than 0.2 cal/cm min.
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Forecasting in real time would be greatly aided at a very low cost by statis-
tical statements of this kind.
In a related example of the utility of a combined meteorological air-
quality model, we consider the possibility of generating local climatological
statistics in data-poor regions as a function of known, large-scale climato-
logical parameters as measured by standard observing systems. In a related
use, we might imagine "filling in" observations between stations through the
use of the models. Such local meteorological statistics could then be used
in simple air-quality models, such as the Gaussian model. Statistics of the
type included in a stability wind rose (joint probability distribution of wind
directions, speeds, and stability class) or which describe the mean afternoon
mixing depth and mean depth of the nocturnal inversion could be generated by
the model and used to provide estimations of annual average concentrations.
The input would be derived from each possible large-scale condition; the out-
put would be a probability distribution of the expected air quality.
Generalized Air-Quality Models
The general goal of an air-quality model is to forecast the concentration
(in dimensions of mass per volume) of a contaminant, Q, over space and time
given the initial conditions on the atmospheric structure and on the distribu-
tion of Q, and given the boundary conditions. For limited domains, the bound-
ary conditions generally consist of the meteorological and concentration
values on the upwind side of the domain (lateral boundary conditions), the
conditions at the surface including surface heat, moisture and momentum fluxes,
the emission rates in space and time over the domain, and appropriate upper
boundary conditions. Mathematically, we wish to solve the equation for the
time rate of change of Q(x,y,z,t):
f? = -
+ Sources + Sinks
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given Q(x,y,z,t ), the time-dependent mean* horizontal and vertical velocities
(V and w), the horizontal and vertical eddy fluxes represented by V .V' Q'
~H H ^ H
9w'O'
and —-—, and the volume sources (emission rates) and sinks (e.g., deposition,
oZ
rainout, reactions).
It is obvious from Equation 1 that the mean wind and the turbulent fluxes
play a major role in determining the behavior of the concentrations. The
truly predictive model will consist of forecast equations for these meteoro-
logical variables as well as the prediction of Q itself. For example, Pandolfo
and Jacobs (1973) describe experimental forecasts of CO over the Los Angeles
area utilizing a combined meteorological-pollutant model. However, solutions
of Equation 1 alone for specified wind distributions may be of some use in
strategy planning (for example, in answering "what if" questions, such as "What
would the local concentration be if the wind behaved in a certain way?"). Real-
time forecasting of particular pollution episodes, however, requires a predic-
tive model for the meteorology as well as the pollutant.
Because it is possible and sometimes useful to utilize fairly sophisticated
models to predict air quality without simultaneously predicting the meteorology
(for example, by solving Equation 1 for specified or observed wind distribu-
tions) , it is convenient to break the general air-quality model into two major
components: the meteorological model and the pollutant model (Figure 1) . The
meteorological model, which may range in complexity from a simple empirical
model to a vast, sophisticated, computer-oriented model, provides the meteoro-
logical variables that affect the transport, diffusion, and reaction of Q.
These variables are then utilized in the pollutant model to forecast the
advection and diffusion of Q. If Q is passive (that is, if its behavior does
not appreciably affect the meteorology), the meteorological and pollutant
models may be run in series. The meteorological model is run first; the
appropriate data are stored and then used in the subsequent pollutant model.
*Here the mean refers to averages over appropriate space and time scales. In
a grid-point numerical model, for example, the spatial average is the average
over a single mesh volume; the time average is over one time step.
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In the more difficult situation in which the pollutant affects the weather.
both models must be run simultaneously. Fortunately, for many air pollution
problems, the pollutant does not significantly affect the dynamics and thermo-
dynamics on the regional scale. An important exception, however, occurs in
the stratosphere, where several pollutants threaten the ozone layer.
The meteorological and pollutant components of the general air-quality
model are discussed later in this paper. At this point we discuss the special
requirements for air-quality models that arise on different scales of motion
and for different meteorological conditions. This analysis demonstrates that
different types of air-quality models are required for different conditions.
Thus, in building air-quality models it is very useful for the modeler to
know the specific requirements for the model, including the horizontal scale
of interest, the length of the forecast, the pollutant to be modeled, the
available input data, and the computation resources available. That is, the
particular problem must be defined as precisely as possible.
Scale Analysis of the Pollutant Conservation Equation
Some of the fundamental differences between air-quality predictions on
the regional (400 x 400 km), urban (5x5 km), and single-plume scales may be
seen from a scale analysis of Equation 1. Gr^nskei (1974) has made a similar
calculation. This analysis for the regional and urban scales assumes that the
variation of the pollutant concentration is of the same scale throughout the
domain and that the characteristic scale of variation is comparable to the
size of the domain itself. In other words, the typical distance from a rela-
tive maximum in concentration to a relative minimum is of the same order of
magnitude as the domain size. Thus this analysis does not apply to the problem
of tracing an individual plume on the regional scale. This problem, which
should be approached by a trajectory-diffusion model, is discussed in "A Com-
bined Meteorological Air-Quality Model," below.
Without loss of generality, we may assume that the average v-component
of the horizontal wind is 0 and combine the sources and sinks of Q into a
single term, S. We then define the nondimensional variables:
-------
Q' = Q/Q
u' = u/u
w' = w/w
x' = x/L
(Eq. 2)
y y y
z' = z/L
t? = t/T
where u and w are the magnitudes of the typical mean horizontal and vertical
velocity components, Q is the amplitude of the variation in pollution concen-
tration over the domain, L and L are the typical horizontal scales associated
with the spatial variation of Q, L is the vertical scale of variation, and T
is the time scale of the variation of Q. We also assume, for order of magnitude
purposes, that the horizontal and vertical turbulent fluxes can be represented
by gradient-transport or K theory, and that the term including the three-
dimensional divergence is small compared to the advection terms. With these
definitions and assumptions, Equation 1 may be written as
T
C/X
1 a l)
dx
&'
3y'2
W / 1 l
- - (w'-j
Z
K
L
i) '
j Z
)2Qf
t £•
3z
f —
L
x
)+$
Q
1 ,3Q\ u
= - 2
. .
(Eq. 3)
where K and K are the horizontal and vertical diffusivities, respectively.
By the definition of the scaling parameters, all terms in parentheses are non-
dimensional and of order of magnitude one. Therefore, the relative importance
of the horizontal and vertical advection, horizontal and vertical diffusion,
and source/sink terms are given by the coefficients of the nondimensional
terms. Furthermore, the time scales of each process are given by the inverse
of the coefficients. Therefore, given certain space scales of variation and
particular meteorological conditions, the importance of each process can be
ascertained, and the overall time scale of the pollution fluctuation at a given
point can be estimated.
We analyze Equation 3 for three space scales and six meteorological
conditions. The three space scales are the regional, urban, and single-plume
scales, summarized in Table 1. For the regional and urban scales, a uniform
-------
square grid array of dimensions 40 x 40 is assumed to cover the domain. This
array size, together with the size of the domain (given by L and L ), deter-
mines the mesh size, As. For these horizontal scales, the vertical scale, L ,
z
is assumed to be equal to the mixing depth, H. The horizontal diffusion
coefficient, K , is the most arbitrary and difficult-to-determine parameter.
Here we relate K to a perturbation horizontal velocity, V (arbitrarily set
equal to 1 m/s), and a mixing length which is taken to be the grid size of
the mode1:
K = V As (Eq. 4)
H
For the single-plume scale, the horizontal dimension, L , along the
direction of the flow is greater than the scale across the flow, and the ver-
tical scale is usually smaller than the mixing depth (Gifford, 1973, p. 8).
Furthermore, on this small scale the order of magnitude of the horizontal
diffusivity is taken equal to the magnitude of the vertical diffusivity, K ,
Z
which will be related to the meteorological parameters.
The six meteorological categories utilized in this analysis are listed
in Table 2. Moderate wind conditions are represented by u equal to 4 m/s
and w equal to 1 cm/s. The height of the mixed layer is 1 km, except for the
stable categories, when it is reduced to 200 m.
The magnitude of the vertical diffusivity, K , is assumed to be equal to
the vertical eddy coefficient for momentum. In the surface layer, under neu-
tral conditions, K is given by
KZ = k U*Z (Eq. 5)
where k is the Karman constant (~ 0.4) and u# is the friction velocity,
ul = T/p (Eq. 6)
where T is the surface stress and p is the density. The surface stress is
usually related to the "anemometer-level" wind speed, V |, by the relation
10
-------
TABLE 1. SCALES OF AIR-QUALITY MODELS
Regional
Urban
Plume
L
X
400 km
5 km
1000 m
L
y
400 km
5 km
100 m
L
z
H
H
100 m
As K
H
10 km 104 m2/s
125 m 125 m2/s
K
— — Z
TABLE 2. METEOROLOGY CLASSES USED IN SCALE ANALYSIS
(a) typical: moderate
wind, neutral
stability
(b) moderate wind,
convective
(c) light wind,
convective
(d) light wind,
stable
(e) calm,
convective
(f) calm,
stable
Approximate
Pasquill (1961)
Stability Class u(m/s) w(cm/s)
D
0
0.25
0
H (m) K (m /s)
Z
1000
1000
0.25 1000
200
0 1000
200
25
250
250
250
11
-------
T = PS'Ys1 (Eq. 7)
where C is the dimensionless drag coefficient which ranges from 1 x 10
over a smooth sea to 16 x 10 over thick grass (Button, 1953).
Equation 5 for K cannot hold as z becomes comparable to the height of
the planetary boundary layer (PEL), since the size of the mixing eddies is
ultimately limited by this depth. Thus K reaches a maximum somewhere near
the middle of the PEL and decreases to a small value at the height h. For
estimation of the order of magnitude of K under neutral conditions, Equatioi
7 is combined with Equations 6 and 5 and then evaluated at 100 m:
Kz(100 m) = 100 k C^ |yj (Eq. 8)
With k = 0.4, C = 10 x 10 , and M = 4 m/s, u. equals 0.4 m/s and the
2
value of K is 16 m /s. The maximum value of K in a PEL of 1-km depth is
z z
probably greater than this value. For example, Deardorff (1972), in his
three-din
given by
three-dimensional boundary-layer model, found the maximum value of K to be
z
KmaX = .02 u
Z — (Eq. 9)
-4
where f equals the Coriolis parameter. For f equal to 10 /s and the above
value of u., the maximum K is 32 m /s.
* z
Based on the above estimates, a typical order of magnitude for K under
condition (a) in Table 2 is 25 m /s. Under "convective" conditions of
strong upward heat flux, K is increased by a factor of 10. This gives the
same order of K found by Yamada and Mellor (1975) in their model of the
Z
afternoon of Day 33 of the Wangara Experiment. Under stable conditions (d
2
and f), the vertical diffusivity is reduced to 1 m /s.
The results of the analysis are presented in Table 3. Because of the
extreme variability in the source/sink term, we neglect this term in the
12
-------
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-------
general analysis and consider only the relative importance of horizontal and
vertical transport and diffusion. For quasi-steady-state conditions, this
term may be assumed to balance the largest terms in the equations.
On the regional scale, the horizontal time scale is greater than the
vertical time scale for all meteorological situations. The next-to-the-last
column, labeled "u-At ," provides an estimate of the horizontal distance down-
wind from a source at which vertical mixing through the planetary boundary
layer is essentially complete. For example, for case (a), the vertical mixing
of a contaminant would be complete through a depth of 1 km at a distance of
about 160 km downwind of a ground source. This result agrees with the down-
wind growth of the vertical dispersion coefficient, a , in the Gaussian plume
z
model discussed by Turner (1970, pp. 5-9) for a Pasquill stability class of D.
The last column shows the ratio of this horizontal length to the horizontal
grid size. For all meteorological classes except class "a" (moderate winds
with typical vertical mixing), the vertical mixing may be considered complete
(vertical distribution of Q uniform) beyond 4 grid lengths downwind of the
source. Thus, for these conditions it is not necessary to include in the
model a high vertical resolution of the PEL for the prediction of Q.
On the regional scale, horizontal diffusion by subgrid-scale eddies may
be neglected (compared to horizontal advection) for all meteorological situa-
tions except perfectly calm, since the time scale for this process is about 28
weeks. However, the vertical transport by the mean vertical motion is of the
same order as the horizontal transport, which indicates that vertical motions
must be considered on this scale. These results are supported by the sensi-
tivity experiments of Liu et al. (1974), who found vertical motions and ver-
tical wind shear far more important than horizontal mixing in determining the
accuracy of trajectory and grid models. Models which did not include vertical
motions and vertical wind shear overpredicted ground-level concentrations by
over 50 percent.
The data used by Nordlund (1975) in his study of SO transport over
northern Europe may be used as an example of the order of magnitude of the
14
-------
source term, S/Q, compared to the other terms in Equation 3 for the regional
scale. Typical emissions of SO into a volume 150 km x 150 km x 1 km are
5 4
5 x 10 ton/yr, or 1.4 x 10 g/s. A representative concentration over
3 S
Europe in Nordlund's example is 30 yg/m . For these values, — equals
-5 2
2.1 x 10 /s, which is comparable in magnitude to the advective terms (Table
3).
The time scales for the variation of Q on the regional scale range from
about a day (for moderate winds) to several days (for very light winds).
Because these time scales are fairly long, there is a possibility of a deter-
ministic prediction of air quality at a point under specific conditions.
Also, the long time scales indicate that initial conditions must be specified
accurately, as they will affect the predictions for long periods of time.
Finally, steady-state conditions on this horizontal scale may be assumed for
several hours, since the time scale of the variation is about a day.
On the urban space scale, the time scales of air quality variation in
the horizontal become much shorter, on the order of an hour for all except
perfectly calm conditions. The time scale for vertical diffusion exceeds the
horizontal time scales, which indicates that a detailed vertical resolution of
the PEL is necessary over the city. Complete vertical mixing, under most
conditions, will not occur until tens of kilometers downwind of the urban area.
This result helps explain why urban concentrations are poorly correlated with
local mixing depths (Gifford, 1973, p. 7).
In contrast to the regional scale, mean vertical advection is negligible
compared to horizontal advection on the urban scale. However, under some
conditions vertical advection may produce an effect comparable to vertical
mixing. Horizontal diffusion by subgrid-scale eddies is small compared to the
advective effects, except under nearly calm conditions, a conclusion supported
by Shir and Shieh's (1974, p. 190) results. Thus, for the urban scale, an
approximate balance exists between horizontal advection, sources/sinks, and
vertical diffusion.
15
-------
A situation quite different from the regional and urban scales exists on
the single-plume scale. As is well known, for the single plume, horizontal
diffusion across the axis of the plume becomes important, and tends to balance
horizontal advection from the plume's source and vertical diffusion. The mean
vertical motion is negligible compared to these effects , as is the horizontal
diffusion along the axis of the plume. This balance is the basis of Gaussian
plume models. For example, in Roberts' (1923) model, the concentration is
given by
Q = - § - ^ exp { - f [ + |] }
47Tx (KR 4 "Si z (Eq. 10)
where S is the emission rate [mass time ]. Equation 10 is a solution to the
steady-state equation,
+K
2 z 8 2 (Eq. 11)
Y z
Other empirically determined Gaussian plume models, which do not imply con-
stant horizontal and vertical diffusivities, are available (e.g. Turner, 1970,
P. 5).
The very short time scales of variations in the concentrations associated
with single plumes indicate that deterministic forecasts — i.e., forecasts of
instantaneous concentrations at a point — are impractical, if not impossible.
Instead, therefore, time-averaged (typically 1-hour) estimates of Q are
obtained at a point. Instantaneous values may easily differ by an order of
magnitude or more from these average values.
In summary, the scale analysis has shown that different types of air-
quality models are required for regional, urban, and single-plume predictions.
For the regional scale, horizontal and vertical advection are important, and
16
-------
vertical diffusion throughout the PEL is usually rapid enough so that a de-
tailed vertical resolution of this process is often unnecessary. The time
scales are long, so deterministic forecasts are practical. An accurate
specification of the initial distribution of Q is necessary on this scale.
For the urban prediction problem, a detailed vertical resolution in a
grid-point model is necessary, since the time scale of vertical diffusion is •
long compared to the horizontal transport time scale. Furthermore, turbulence
models that are more accurate than K-theory models are needed to treat the
general urban problem, since the size and structure of the turbulent eddies
become important. The K-theory models become inadequate in the presence of
buoyancy effects. Models that produce the same turbulent structure as the
atmosphere (in a statistical sense) may be utilized to estimate the prob-
ability of a local (point-value) concentration exceeding a prescribed standard
over a given time interval.
Because the horizontal time scales associated with urban pollution pro-
blems are of the order of an hour, initial conditions are of less importance
than an accurate knowledge of the sources and the wind field; i.e., if the
sources are known correctly and the winds forecast accurately, a correct
forecast will evolve in roughly an hour, no matter what the initial conditions.
This result was also found by Shir and Shieh (1974) in their urban model.
For the prediction of concentrations associated with a single plume, the
time scales are so short that only time-averaged predictions are possible.
The scale analysis indicates an approximate balance between horizontal advec-
tion, crosswind diffusion, and vertical diffusion, in agreement with the
applicability of the Gaussian plume model.
The following section reviews the requirements of meteorological models
on the regional scale and discusses some of the models currently under develop-
ment.
17
-------
METEOROLOGICAL MODELS ON THE REGIONAL SCALE
In this section, we discuss the most important aspects of predictive
meteorological models on the regional scale. As shown in Figure 1, the com-
ponents of a general meteorological model consist of (1) measuring initial
values of the dependent variables, such as winds, surface pressure, temperature,
and moisture, (2) analysis of these data to produce consistent three-dimensional
initial fields, (3) treatment of boundary conditions on the edges of the do-
main, (4) modeling of important physical processes such as advection; turbulent
fluxes of heat, moisture and momentum; radiation; and condensation; and (5)
the numerical solution of finite difference equations that represent the pro-
cesses in (4). Each of these component problems is complicated enough when
considered by itself. When taken together in a complete model, as is necessary
for a realistic treatment of general atmospheric conditions, the problem is
truly enormous. It is therefore impossible in this brief review to give de-
tailed descriptions of each component. Instead, the essential aspects of each
component are introduced and references are given to other, more complete dis-
cussions.
The scale analysis of the preceding section indicated that pollutant
variations on small space scales were associated with short time scales, so
that deterministic predictions of instantaneous values of concentrations on
horizontal scales of less than about 5 km were impractical. The same general
rule is also true of the meteorological variables: small space scales are
associated with very short time scales. Therefore, we restrict the following
remarks to numerical models with horizontal resolutions of at least several
kilometers which are capable of resolving the flow in a deterministic sense.
For these models, the hydrostatic assumption is valid.
The basic set of equations utilized in time-dependent hydrostatic models
(the so-called primitive equations) include the horizontal equation of motion,
9" 3V
— = (V-V)V - w ^ - -Vp - fkxV + F (Eq. 12)
9t ~ ~ 9z p ~
18
-------
the thermodynamic equation,
3T .. „_ 9T . o)
the continuity equation,
_ _ y.ov _
3t ~ P~ 9z (Eq. 14)
the hydrostatic equation,
3z = ~ ^ (Eq. 15)
the equation of state,
p = RT p
v (Eq. 16)
and the continuity equation for water vapor,
11 = - V-Vq - w 11 - C + E
where V is the horizontal vector velocity, p is density, p is pressure, f is
the Coriolis parameter, T is temperature, u is •—, C is the specific heat at
constant pressure for day air, T is virtual temperature, g is the acceleration
of gravity, and q is specific humidity.
The term F in Equation 12 represents all frictional accelerations asso-
ciated with turbulent motions; Q in Equation 13 represents diabatic effects,
and C and E in Equation 17 represent condensation and evaporation, respectively.
The modeling of these "sources and sinks" consists of relating these compli-
cated processes to the dependent variables in the model. This modeling is
often termed parameterization.
With the parametric modeling of the "source and sink" terms, the basic
Equations 12-17 may be written in finite difference form and solved for future
states of the atmospheric variables, subject to initial conditions over the
19
-------
entire domain and time-dependent conditions on the vertical and horizontal,,
boundaries of the domain.
Initialization of Mesoscale Regional Models
The relative importance to the forecast of the initial analyses of wind,
temperature, and moisture compared to the other components of the meteorolog-
ical model varies considerably with horizontal scale and the meteorological
situation. A detailed representation of the initial conditions is most im-
portant on large scales and when the local forcing functions (represented by
F, Q, C and E in Equations 12, 13, and 17) are weak. Under these conditions,
potential vorticity tends to be conserved and future states of the atmosphere
are governed by a redistribution of the initial mass and momentum fields.
As the horizontal space scale decreases, a detailed representation of
the initial conditions becomes somewhat less important. The specified condi-
tions on the boundaries assume greater importance, since under all but very
light wind conditions the variations associated with the initial conditions
are advected away from the domain early in the forecast. Furthermore, the
local forcing by terrain, frictional, and diabatic effects becomes more impor-
tant on the smaller scales. On the urban scale, therefore, detailed initial
conditions are unnecessary; the solutions will be determined almost entirely
by the boundary conditions (possibly time-dependent) and the modeling of the
local forcing functions. On the large regional scale (400 x 400 km or greater),
however, the initial conditions must be resolved in some detail.
While initial three-dimensional analyses of wind, temperature, pressure,
and moisture are required by the models, it is not necessary to measure all
of these variables independently. For deep motions (vertical scales comparable
to the tropopause height), the most important variable to measure and analyze
accurately on the mesoscale is the horizontal wind. This result follows from
adjustment theory (Rossby, 1938; Cahn, 1945; 0kland, 1970), which states that
on this scale the mass field (temperature and surface pressure) will quickly
adjust through gravity waves to whatever initial wind field is provided. Thus
it makes no sense to analyze extremely detailed temperature variations unless
20
-------
a correspondingly detailed wind analysis is also provided. Warner (1976) has
recently considered the transfer of initial errors between the mass and momen-
tum variables in mesoscale models. He concluded that errors in the mass and
momentum initial data are not strongly coupled, in the sense that there is very
little transfer of error-related energy or uncertainty from one variable to
another. Wind errors remain in the wind field while small mass field errors
are removed through the adjustment process.
Although there is general agreement that the horizontal wind is the most
important variable to initialize mesoscale models, it is not known how much of
the mesoscale variability in the wind field must be included in the initial
analysis for useful 12- to 24-hour mesoscale forecasts. It is the hope that
the small-scale perturbations in the winds will evolve with time from the
measured larger-scale circulations as a result of the physical forcing func-
tions in the models.
Given the initial wind analysis, it is desirable to obtain a dynamically
consistent temperature and surface pressure analysis. If the real atmosphere
is balanced in a quasi-geostrophic sense, the temperature analysis may be ob-
tained from a solution of the balance equation for geopotential height as a
function of the winds (Fankhouser, 1969). Under some conditions, however,
the atmosphere may not be balanced and the divergent and ageostrophic compo-
nents of the wind may be significant. Solutions of a quasi-geostrophic 1-equa-
tion (e.g., O'Brien, 1970) to obtain the divergent component of the wind may
not be applicable on the mesoscale. Therefore, iterative initialization
methods which utilize the model's predictive equations themselves to produce
a consistent analysis of all the variables may be necessary (Nitta and Nover-
male, 1969; Miyakoda and Moyer, 1968; Anthes and Warner, 1974; Rao and Fishman,
1975). The advantage of these schemes is their generality — all the physical
processes in the model are incorporated into the initial analysis. The dis-
advantage is their cost — in some cases, the initialization process may con-
sume a significant fraction of the total computer time required to make the
forecast.
21
-------
Lateral Boundary Conditions
The numerical treatment of the lateral boundaries is a difficult but very
important aspect of a limited area forecast model. The problem on the interior
of the domain is well posed only when the proper set of boundary conditions is
prescribed. The solution on the interior may be completely different for ap-
parently minor variations in the boundary conditions. Much discussion has ap-
peared in the literature concerning the proper specification of variables on
the open boundaries (Moretti, 1969; Shapiro, 1970). As discussed by Moretti,
the only straightforward case occurs when the flow is supersonic, in which
case the value on an inflow boundary may be arbitrarily specified, and the
values on an outflow boundary obtained by a linear extrapolation outward from
the interior of the domain.
In the subsonic case, however, waves may propagate upstream, so that
values at inflow boundaries are partially dependent upon the flow on the in-
terior of the domain. Thus the arbitrary specification of either constant
or time-dependent boundary values may make the problem ill-posed. However, in
the absence of proven computational methods to compute boundary variables cor-
rectly under general conditions, modelers are forced to take the pragmatic view
that, if the lateral boundaries are located far enough away from the region of
interest, the errors introduced at the boundaries will remain within some ac-
ceptable tolerance in the interior of the domain during the forecast period.
We must therefore search for a set of conditions which minimizes the errors
generated by the boundaries and their feedback into the interior.
Although it is normally recognized that a set of boundary conditions that
minimizes the generation of high temporal and spatial frequency components to
the numerical solution is desirable (if not absolutely necessary), it is less
generally recognized that smoothly varying solutions near the boundary are not
sufficient to guarantee accurate solutions on the interior of the grid. It is
apparent that, as the size of the horizontal domain decreases, the specifica-
tion of the velocity components and temperature along the boundaries affects
the mean (wave number zero) values of these quantitites over the entire domain
to an ever-increasing degree. Thus, on a mesoscale domain of 600 x 600 km, a
22
-------
set of boundary conditions may be computationally "stable" and produce "smooth"
results, but even small errors in the treatment of temperature or velocity may
profoundly affect the mean kinetic and internal energy budgets over the domain.
The importance of specifying accurate values of temperature, surface
pressure, and velocity components on the lateral boundaries of mesoscale models
can be shown by considering the domain-averaged equation of motion (Anthes
et al., 1974). Here, the approximate time rate of change of the average u-
component of velocity (for frictionless flow) was shown to be
3uD
3t
L
2 _ 2 7
UE "w
L
x
L
(uv)N - (uv)s
L
y
(Eq. 18)
where the ( ) operator denotes an average over the domain, the E, W, N, and S
subscripts denote east, west, north and south boundaries respectively, and
L and L are the lengths of the domains in the east— west and north— south
x y
directions.
The importance of the specification of the mass (given by the surface
pressure and vertical temperature structure) appears in the term fv , which
g
is mainly determined by the difference in surface pressure and geopotential
across the domain. As the domain size is decreased, the error in the mean
geostrophic wind for a given temperature error increases rapidly. This point
is illustrated in Figure 2, which shows the errors in the calculated geo-
strophic wind as a function of horizontal distance when a temperature dif-
ference error of 1° C is integrated hydrostatically over a 200-mb depth
centered at the given pressure level. For synoptic scale domains (L >~ 3000 km),
the error is about 1 m/s at all levels. The error increases rapidly as the
domain size decreases below 1000 km. For a regional-scale model, with a do-
main size of 600 x 600 km, a 1° C temperature error on the lateral boundaries
will produce large (~ 5 m/s) errors in the mean geostrophic wind across the
grid, especially in the upper levels. These errors, if they persist in time,
23
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may produce large erroneous accelerations of the mean motion over the domain.
-4
For example, with f = 10 /s, a 5 m/s error in geostrophic wind which per-
sists for 6 hours would produce an erroneous change of 11 m/s in the mean
velocity over the domain.
The preceding analysis indicates that, although it might be tempting for
computational stability purposes to calculate the temperature on the lateral
boundaries by some type of outward extrapolation from the interior of the do-
main, or a one-sided differencing scheme, the extreme sensitivity of the mean
acceleration to small temperature errors leads us to reject these possibilities.
Instead we conclude that, for mesoscale domains, it is preferable to specify
the mass variables at all boundary points in a realistic and accurate way, in
order to insure that the mean geostrophic wind and associated linear accelera-
tion term remain accurate during the integration. This specification may come
from a large-scale model (one-way interaction), or, in the research mode, from
consistent analyses of observations.
The importance of accurately specifying the velocity components on the
boundary is also discussed by Anthes et al. (1974). As the domain size be-
comes smaller than 1000 x 1000 km, net horizontal fluxes of momentum across
the lateral boundaries may produce mean accelerations over the domain that are
comparable in magnitude to the linear accelerations associated with the pres-
sure gradient and Coriolis forces.
The preceding remarks indicate that the accurate calculation of lateral
boundary conditions is a major component of the predictive mesoscale modeling
problem. One attractive approach to the solution of this problem is to nest
the fine-scale prediction model within a larger-scale prediction model. In a
one-way interacting system, the large-scale model is integrated separately and
provides time-dependent boundary conditions to the fine-mesh model. The fore-
casts on the fine mesh do not feed back into the large-scale model forecasts.
Theoretical analyses (Phillips and Shukla, 1973), however, suggest that a
better approach is to allow the two meshes to interact dynamically, which re-
quires a simultaneous integration of the model equations on both meshes. An
25
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example of a mesoscale forecast produced by such a two-way interacting system*
is shown in Figure 3. Here a small-scale jet streak (local wind maximum),
which is superimposed on a larger-scale jet stream, is forecast by a fine-mesh
of 20-km horizontal resolution. The coarse mesh is 80 km. The isotachs (in
m/s) are shown by solid lines; the temperature changes associated with vertical
circulation in the vicinity of the jet are shown by dashed lines. This mete-
orological system consisting of strong winds normal to the lateral boundaries
was chosen to provide a severe test of the numerical technique of meshing the
two grid systems.
Finite Difference Schemes
Even if the initial and boundary conditions were known precisely and the
forcing functions in Equations 12-17 represented by perfect mathematical models,
the approximation of the partial derivatives in the governing equation would
lead to errors in the forecasts. However, in contrast to the complicated
feedbacks in the other components of the model that make a general analysis
of errors very difficult, the errors introduced by the finite difference
schemes are relatively well understood. There exist a vast number of temporal
and spatial finite difference schemes to tempt the modeler. Many have special
advantages in terms of accuracy, numerical stability, speed of computation, or
simplicity in programming. In general, the most accurate schemes are the most
complicated and time-consuming; the simple, fast schemes usually produce large
truncation errors. For example, one of the simplest time and space differenc-
ing methods is the forward-in-time, upstream-in-space scheme, which may be
illustrated for the simple advection equation
!?--"&
by the finite difference approximation
*This forecast was produced by a nested grid version of the Penn State
University mesoscale model (Anthes et al., 1974).
26
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Figure 3. Temperature changes (dashed lines in °C) produced by vertical
circulations associated with a propagating wind maximum (jet
streak) at a level above the jet. The basic flow is from west
to east (left to right in figure). Isotachs (solid lines) are
labeled in m/s. The forecast was made on a two-way interacting
nested grid.
27
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_ 2Q)
n (Qj H- 1 - QI) p
u. ^-^ - T --- ^ for u. < 0
Here the n superscript denotes the time level (t = nAt) and the j subscript
indicates the horizontal position on the x axis (x. = jAx) . This scheme is
popular because it requires only two time levels of information and is very
simple to use. However, as shown by Molenkamp (1968), the truncation errors
associated with this scheme produce a strong false dispersion of Q. The
computational pseudo-diffusion coefficient is
where -" — j- — <_ 1. The magnitude of the horizontal mixing associated with
ZAX
this computational K usually exceeds the magnitude of the physical mixing
process by an order of magnitude or more. For example, for a grid size of
5 km, a velocity of 5 m/s, and a ratio 7 — equal to 1/2, the computational
3 2
diffusion coefficient is 6.25 x 10 m/s. In the meteorological prediction
equations, the use of such a damping integration scheme may be beneficial in
maintaining numerically smooth and stable solutions. In modeling phenomena
with strong local sources of energy, such as hurricanes (Rosenthal, 1970),
cumulus clouds (Ogura, 1963), or sea breezes (Pielke, 1974), this scheme has
produced useful results at a small cost. However, for less energetically
active systems, and especially for predicting the transport of conservative
quantities, more accurate schemes are necessary.
The forward-in-time, upstream-in-space finite difference scheme has only
first-order accuracy in time and space (that is, the errors are proportional
to the first powers of the time step, At, and grid spacing, Ax). More accurate
finite difference schemes may be derived by including more terms in the Taylor's
series expansions (e.g., Haltiner, 1971, p. 91). These higher-order schemes
contain more points in the approximate expressions and require more computa-
tional time. Because the order of the finite difference equations is increased
28
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beyond the order of the original partial differential equation, purely
numerical modes may be introduced into the finite difference solutions. An
example is the "computational mode" associated with the leapfrog (or centered
in time and space) differencing scheme (e.g., Haltiner, 1971, p. 95). This
particular mode of the numerical solution has no physical counterpart, and
hence its amplitude must be controlled during the forecast.
The leapfrog scheme has second-order accuracy in space and time, which
means that the errors in the differences are proportional to the second power
of the step and grid size. Although these truncation errors are less than
those associated with the first-order scheme, they do produce significant
errors in the phase speed and rate of energy propagation of waves shorter than
about 8Ax. If advection or wave propagation is a dominant part of the meteor-
ologically interesting portion of the solution, these errors will be signifi-
cant, causing the short waves to propagate too slowly compared to their counter-
parts in the real atmosphere.
Other more accurate, finite difference schemes are available (e.g.,
Richtmyer and Morton, 1967; Crowley, 1968) at the expense of increased com-
plexity. An increasingly popular scheme for evaluating space derivatives
has fourth-order accuracy and was shown by Crowley (1968) to give superior
results in the advection of a conservative quantity.
Because of the difficulty that traditional finite difference schemes have
in reproducing faithfully the behavior of short wavelength features, alterna-
tive approaches are now receiving attention in research models. An example is
the Galerkin method utilizing Chapeau basis functions for the independent
variables (Long and Hicks, 1975). By expanding the variables in terms of the
Chapeau functions, a more efficient finite difference scheme is generated.
This scheme is able to resolve wavelengths as small as 4Ax with better ac-
curacy than the conventional schemes. Use of these newer methods may make
numerical models considerably more efficient in the future.
29
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Parameterization of Physical Processes
One of the most important components of the meteorological model is the
representation of physical processes in the atmosphere by mathematical rela-
tionships. These processes include:
(a) complex terrain effects,
(b) condensation heating and evaporative cooling,
(c) infrared and shortwave radiation,
(d) conduction of heat from the earth's surface to the air in the
contact layer, and
(e) turbulent transfers of heat, moisture, and momentum through the
boundary layer.
Complex terrain effects enter the mesoscale model as a lower boundary
condition; the other processes involve models in themselves. These sub-models
may become nearly as complex as the overall model and consume a major fraction
of the computing time. Ideally, however, simple models which represent the
essential physics associated with each process through parameterized relation-
ships can be used to keep the model economically feasible. Although the de-
tails of the parameterization of physical processes may vary considerably
among models, the concepts tend to be rather similar. This section reviews
typical methods for representing the above physical processes in mesoscale
models. The nature of these parameterizations and the model variables upon
which they depend are given in Table 4. Blackadar (1978) presents a more
complete review of the surface and boundary layer processes (d and e) which
are most important to air-quality modeling.
Complex Terrain Effects—
Inhomogeneities at the earth's surface generate mesoscale and microscale
circulations in a number of ways. Long, deep valleys tend to channel the
surface wind, giving a strong bias to certain wind directions over the year.
Differential heating of the sides of basins leads to oscillations in the
surface winds (Staley, 1959). When statically stable air flows over ridges
30
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TABLE 4. TYPICAL PARAMETERIZATION OF PHYSICAL PROCESSES IN MESOSCALE MODELS
Process
Important Parameters
Vertical flux of momentum,
heat and moisture in PEL
(a) Lower boundary condition
(surface stress, surface
heat flux, surface moisture
flux)
H height of PEL
9 mean potential temperature in PEL
mean wind at anemometer level
V
~a
D
R.
mean wind in PEL
drag coefficient
heat transfer coefficient
friction velocity
roughness parameter
average potential temperature at
ground
~~ _ 0 )
s bulk Richardson number
IT7I 2
(b) profiles in surface layer
(nominal depth ~ 100m)
(c) fluxes from top of surface
layer to top of PEL
L = - i
z , L
o
- u'w'
- w
- K
m
9u
3z
] Monin-Obukhov length
„ K
t t i\. ^
w q q oz
K , K , K may be formulated in a number
m h q
of ways by relating them to
(a) height of PEL
(b) local vertical shear of wind
averaged over area and time
(c) local lapse rate, averaged over
area and time
(d) surface heat flux
(continued)
31
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TABLE 4 (continued)
Process
Important Parameters
Cumulus convection (effect of
cumulus clouds on environment)
Radiation (shortwave)
Radiation (longwave)
Terrain effects
vertical velocity: w , w
temperature: T , T
specific humidity: q , q
horizontal velocity: VG, V
c subscript denotes cloud variable,*.
e subscript denotes environmental
value.
( ) represent average over lifetime of
cloud (10-60 min.)
surface albedo, percentage and type of
clouds.
T(z)
q"(z)
percentage and types of clouds
(including heights).
( ) represents average over horizontal
areas comparable to grid size
height of terrain with wavelengths less
than ~ 4Ax eliminated. Low-level
stability, mean wind speed in PEL, and
vertical profile of mean wind are
important atmospheric parameters to
define response of atmosphere to flow
over rough terrain.
32
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that are oriented more or less normally to the flow, a variety of gravity-wave
phenomena are generated. Notable among these phenomena are the lee-slope
windstorms that occur in the Rockies (Klemp and Lilly, 1975). However,
mountains as high as the Rockies are not necessary to produce significant
mesoscale perturbations; even hills of a few hundred meters in height can
drastically modify the local rainfall distribution (Huff et al., 1975).
Under light prevailing winds, variations in terrain elevation produce
elevated heat sources and sinks, which lead to vertical circulation on a
small scale. Cold air at night drains down sloping surfaces and accumulates
in valleys. This effect played a crucial role in the Donora, Pennsylvania
disaster of 1948 (Williamson, 1973, p. 180). Knowledge of the frequency and
intensity of these valley inversions, which can be monitored using indirect
sensing techniques, would be extremely useful in land-use planning. Urban-
scale models may be of some use here in real-time forecasting as well as in
the research mode.
Gifford (1975) reviews some of the effects of rugged terrain on diffusion.
In general, diffusion rates appear to be larger for a given meteorological
situation than would be expected over homogeneous terrain. Differential
heating and cooling, vortex shedding, and gravity waves are the probable
mechanisms for increasing the diffusion rates.
Condensational Heating and Evaporational Cooling—
The treatment of the water cycle in a regional-scale model is important,
both in the effect of condensation and evaporation on the dynamic and thermo-
dynamic variables, and in the important role precipitation has in removing
pollutants from the air. Even nonprecipitating clouds and fog greatly affect
the surface heat flux and low-level stability. (Again, the Donora episode of
1948 may be cited; the formation of a deep fog layer prevented the solar
radiation from burning off the nocturnal inversion.)
Cumulus and cumulonimbus convection, besides playing important roles in
the energetics and dynamics of mesoscale systems, also act as very efficient
33
-------
mixers of lower and upper tropospheric air. For example, it is common to
observe "dirty plumes" in the otherwise clean upper tropospheric air over
Miami where a cumulus cloud has dissipated. These links of the PEL with the
middle and upper tropospheric layers are quite important in convective situa-
tions in determining the fate of pollutants on the regional scale.
Because the hydrostatic models cannot explicitly consider moist convective
circulations/ the cumulative effect of these cloud-scale motions on the tempera-
ture, moisture, and momentum structure of the mesoscale atmosphere must be
related to the circulations that are resolvable by the model. Both theoretical
and observational results indicate that the mesoscale horizontal convergence
of water vapor is the most important parameter in determining when and where
organized moist convection will occur (Hudson, 1971). Values of moisture
-4
convergence as high as 10 g/kg s may occur on the mesoscale.
The mesoscale moisture convergence over an area provides an integral
constraint on the amount of precipitation and associated latent heating that
is possible. The vertical distribution of the heat, moisture, and momentum
fluxes by the convection is determined by the spectrum of clouds present.
Current cumulus parameterization schemes vary from simple ones in which only
a single cloud type is considered (e.g., Kuo, 1974) to more sophisticated ones
in which a number of cloud types are permitted (Arakawa and Schubert, 1974;
Ooyama, 1971).
Infrared and Shortwave Radiation—
Most mesoscale models do not yet contain the effects of radiation flux
divergence in the free atmosphere; only the radiation budget at the ground is
considered. This neglect is probably justified above the PEL for most meso-
scale predictions because of the short time scales involved and the dominance
of other physical processes in determining temperature changes in the free
atmosphere. Radiation is probably most important in the evolution of the
nocturnal PEL, although — in specific air pollution problems — attenuation of
the daytime radiation by particles or other pollutants may also be important.
Reasonably simple radiation models are available to treat this process (e.g.,
34
-------
Sasamori, 1968). The most important variables that determine the longwave
flux are cloud cover (depth and height) and the average vertical profiles of
temperature and moisture.
The Planetary Boundary Layer—
Because knowledge of the time variation of the structure and height of
the PEL is essential in the regional air pollution problem, the realistic
parameterization of the PEL is of primary importance in the models. While
the use of simple "mixed-layer" models (see, for example, "Transport and
Diffusion Models," below) may be considered, a number of levels within the
PEL may be necessary to resolve possible important vertical variations of
momentum, moisture, and pollutants. With surface fluxes of heat, moisture,
and momentum computed from a surface energy budget calculation and established
similarity theory, vertical transfers of these quantities are calculated in
most models by "K" theory, in spite of the nonrigorous nature of the flux-
gradient relationship. The vertical mixing coefficients depend on such para-
meters as the height of the PEL, surface heat flux, and (locally) on the vertical
wind shear and stability.
The predictive, as opposed to diagnostic, calculation of the surface
fluxes of heat and moisture requires a simple energy budget which utilizes
for input only external parameters such as time of day, year, latitude, and
characteristics of the soil (Carlson, 1976). This budget is necessary to
calculate the portion of the insolation that is utilized to evaporate surface
moisture and heat the air in the contact layer. The surface flux of heat is
necessary to calculate the Monin-obukhov length, L, which is important in
establishing the momentum and temperature structures in the surface layer.
The surface flux of moisture is important in augmenting the water supply in
the PEL, which in turn exerts a strong influence on the evolution of moist
convection. Blackadar (1978) discusses the modeling of the PEL in greater
detail.
35
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TRANSPORT AND DIFFUSION MODELS
The second major component of the predictive air-quality model (Figure 1)
is the transport and diffusion model, which utilizes the meteorological vari-
ables generated by the meteorological model to advect and diffuse a pollutant.
This model may also calculate chemical reactions with other species or removal
by such processes as washout or dry deposition. Some of these nonconserva-
tive aspects of the pollutant prediction problem are discussed by Kabel (1978).
Recently there have been numerous excellent reviews of the various trans-
port and diffusion techniques that are utilized in air-quality models. Nota-
ble contributions are articles by Eschenroeder (1975), who gives 129 refer-
ences, Hoffert (1972), Johnson (1972), Seinfeld (1970), and Lamb et al. (1973).
Because of the existence of these comprehensive reviews, it would be redundant
to give another detailed discussion of the various types of pollution trans-
port and diffusion models. Instead, I will summarize the types of models
available for this purpose and briefly discuss their relative strengths and
weaknesses.
The general classes of air pollution transport and diffusion models are
presented in Table 5, which gives a brief description of the important aspects
of each model type, the horizontal scale of the model's applicability, an
estimate of the computational expense in using the model (which is also a good
measure of the model's complexity), and a single example. Eschenroeder (1975)
gives many other examples and a more detailed description of each type of
model.
BOX MODELS
The "box model" is one of the oldest air-quality models and is based on
the assumption that the pollutant is uniformly mixed between the ground and a
mixed layer depth, and that the area source rate and the wind speed are con-
stant. These assumptions lead to very simple expressions for local concentra-
tions, such as the model of Gifford and Hanna (1973):
36
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38
-------
Q = C -p (Eq. 22)
where S is the area source rate (mass/area«time), and the value of C, a
dimensionless parameter, depends on the city size and on the stability class.
For typical city sizes and stabilities, C is about 250. In many tests of the
performance of this simple model compared to other more complex models in pre-
dicting average concentrations over an area, the simple model generally does
as well or better than the complex models in obtaining high correlation co-
efficients between observed and predicted averages. This fact should cause
urban modelers to ask why a very simple model can perform, on the average,
so well when compared to complex models, which are theoretically capable of
treating more complicated episodes. The answer, I believe, lies simply in the
relative importance of the terms in the general pollutant prediction equation.
On the urban space and time scales, the dominant terms are the area sources,
advection, and vertical mixing. The simple box model considers these effects,
and (when they are known) good average predictions of Q result. However, such
a simple model is incapable of treating situations in which large horizontal
gradients of Q exist, or of treating the many complicated problems on the
larger regional scale. Furthermore, the model breaks down as the wind speed
approaches 0. Finally, a high correlation of observed and predicted values
does not necessarily imply a good performance in individual cases, as shown by
Anscombe (1973) and summarized by Eschenroeder (1975). Nevertheless, the
practical utility of such simple models under the proper conditions is in-
disputable .
Gaussian Plume Models
The best known and most widely used air-quality models are variants of
the Gaussian plume model. These models consider the vertical and cross wind
dispersion of a single plume under steady-state conditions. Time-averaged
(about 1 hour) concentrations downwind of a single source are obtained as
functions of the mean wind speed and stability class. Multiple sources can be
treated with Gaussian plume models by adding the plumes associated with sepa-
rate sources. The concentrations from each plume are then combined in overlap
39
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regions. Gaussian plume models are ideally suited for use with climatological
n
data for producing annual estimates of concentrations near a particular source.
Turner (1970) presents a user's workbook which describes the use of the Gaus-
sian plume model. It is noteworthy that most Gaussian plume formulas are
steady-state, empirical models based on the assumption of Gaussian distribu-
tions in the vertical and the horizontal. For example, a commonly used Gaus-
sian plume model (Calder, 1970) is
2
S exp [- -^ (r~z
y
z2
z
irUa a
y z
Q = _ y z (Eq. 23)
where S is the emission rate (mass/time) from a continuous point source at ,,
an elevation Z , U is the mean wind speed in the x-direction, and a and a
o y z
are empirically-determined dispersion coefficients which depend on the dis-
tance, x, from the source and on the static stability. In many applications,
the stability dependence of a and a is estimated in terms of the local
meteorological conditions based on a scheme introduced by Pasquill (1961) .
Lagrangian Puff Models
When the behavior of the pollutants emitted from a single source is over
long periods of time (greater than 1 hour) or distances longer than those for
which the Gaussian plume models are valid, a Lagrangian puff model becomes
applicable. These models consider the behavior of a single "puff" of con-
taminated air after it leaves its source. The center of mass of the pollu-
tant concentration is advected along a trajectory which may be computed from
a three-dimensional wind field. The wind and stability along the trajectory
may vary, and so the model is potentially applicable under a wide variety of
meteorological conditions. The expansion and deformation of the puff and
the diffusion within the puff by turbulence and wind shear may be computed as
the puff moves along the trajectory. Because the downwind transport of the
puff is accomplished by a Lagrangian calculation, there is no artificial dif-
fusivity associated with the advection calculation. Although the puff model
approach is most suitable for tracing individual emissions, a number of puffs
40
-------
originating at different sources could be followed in order to obtain spatial
variations at a later time. Examples of puff models are given by Roberts et
al. (1970) and Sheih and Moroz (1972).
Eulerian Grid-Point Models
For forecasting temporal and spatial variations of pollutant concentra-
tions over urban, regional, and larger scales, Eulerian grid-point models be-
come useful, although three-dimensional models are quite expensive because
they require large amounts of computer time. In the grid-point models, a
finite-difference version of Equation 1 is solved as an initial-boundary value
problem at many points over the domain of interest. The method is quite
general, and may incorporate sources, sinks, and chemical reactions. The
primary disadvantage, besides cost, lies in the truncation errors associated
with the advection calculation and the resultant pseudo-diffusion (see "Meteo-
rological Models on the Regional Scale," above; also, Liu and Seinfeld [1975]).
However, this is an economic, not a theoretical difficulty. Finite difference
methods of higher accuracy (Egan and Mahoney, 1972) and/or a larger number of
grid points may be employed to reduce the truncation errors to any reasonably
small value. An excellent example of the use of an Eulerian grid-point model
is given by Shir and Shieh (1974), who solve the conservation equation for SO
over a three-dimensional array over the St. Louis area. The 30 x 40 horizontal
array utilizes a constant mesh size of 1524 m along a side; the vertical
structure of the model consists of 14 levels from the ground to the height of
the mixed layer. Their results were better in general than those from the
Gaussian plume model.
Particle-In-Cell (PIC) Models
An alternative to the Eulerian grid-point model which is also well
suited for the time-dependent calculation of a pollutant concentration in
three dimensions is the PIC model. In these models, particles represent a
mass or a concentration of a species. These particles are transported by a
specified wind field. In the absence of diffusion or other nonconservative
41
-------
processes, the concentration or mass of a contaminant represented by a par-
ticle does not change; thus, the numerical diffusion associated with the
advection calculation in a grid-point model is eliminated. However, for
accurate resolution of gradients, a larger number of particles is necessary,
which may make the PIC method more costly than grid-point models for a given
accuracy. Nonconservative processes, usually defined at grid points, may
modify the concentrations of nearby particles. Examples of the PIC technique
are given by Sklarew et al. (1972) and Teuscher and Hauser (1974). An ex-
ample of the PIC method used in conjunction with a mesoscale meteorological
model is presented in this section.
Trajectory Models
Trajectory models are useful in tracing individual pollutants forward
from their source to downwind receptors, or in identifying the sources of a
given pollutant by computing backward trajectories from a particular receptor.
When using real meteorological data, a wide variety of computational techniques
is available to compute trajectories. These range from simple horizontal kine-
matic techniques in which an analyzed wind field on a constant level or con-
stant pressure surface is used to compute successive positions of air parcels
in space and time, to more complicated methods in which three-dimensional mo-
tions are considered. Because of the importance of vertical motions on the
movement of air over periods of a day or more, trajectory models applied to
these time scales must consider the three-dimensional atmospheric motions
unless the pollutants are trapped under a well-defined mixed layer. The com-
bination of typical synoptic-scale vertical motions (5 cm/s) in the presence
of moderate vertical wind shear can lead to significant differences between
horizontal and three-dimensional trajectories over 24 hours, especially when
the wind changes direction with height. For example, let the wind change
linearly from west at 10 m/s at a height of 1 km to east at 10 m/s at a height
of 3 km. If the vertical velocity were constant at 2.3 cm/s during the 24-hour
period, the parcel would rise from 1 km to 3 km in 24 hours. Its average
east-west velocity during this period would be 0, and so its true horizontal
42
-------
displacement would be 0. A constant-level trajectory, however, would er-
roneously place the parcel 864 km downwind! Utilizing real data, Danielsen
(1961) has found similar errors.
The most accurate trajectory method in the free atmosphere for use with
real data is probably some variation of the isentropic trajectory method. Very
useful in tracing middle and upper tropospheric pollutants such as ozone, this
method requires the computation of trajectories along isentropic — or constant
potential temperature — surfaces (Danielsen, 1961; Bleck, 1968). Under adiabat-
ic conditions, pollutants are confined to isentropic surfaces and vertical dis-
placements are readily calculated from the displacements in the isentropes.
Although some dynamical constraints may be incorporated into the basic kine-
matic calculation on isentropic surfaces, a comparison of three-dimensional
trajectories computed from the more complicated methods showed no obvious
improvement over the simple kinematic method (Hoke, 1973).
Quasi-Lagrangian Models
When a well-defined mixed-layer height exists over a regional domain,
quasi-Lagrangian models are suitable for modeling the regional transport of
pollutants. For example, Nordlund (1975) utilized this method to calculate
the time-dependent behavior of SO over northwestern Europe. Liu and Seinfeld
(1975) compared the validity of Eulerian and Lagrangian models. In the quasi-
Lagrangian cell model, the atmosphere from the ground to the mixed-layer height
is divided into more or less rectangular cells, with horizontal sides of the
order of 50 km. These cells represent material volumes of the atmosphere that
are transported by the mean wind in the PEL. In the presence of horizontal
divergence or deformation, the cells may shrink, expand, or become distorted
in shape. However, because the transport is done in a Lagrangian rather than
Eulerian framework, numerical diffusion is absent. As the cells pass over
sources, they may be enriched by additional pollutants. Removal processes
and chemical reactions may be calculated within the individual cells. The
quasi-Lagrangian method becomes less applicable in situations in which a
clearly defined mixed-layer top does not exist, because the pollutants are not
43
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confined by an upper surface. Multi-level grid-point models or PIC model^--
would be more appropriate under these conditions.
A Brief Look at the Accuracies of Air-Quality Models
The ultimate test of any air-quality model is how well it performs its
assigned task. Because the assigned tasks may vary so much from one air-
quality modeling problem to another (e.g., compare the problem of estimating
annual average SO concentrations at a stack base to the problem of modeling
ozone transport and reactions), a general comparison of the models is impos-
sible. Before a comparison is meaningful, the problem and the constraints
must be precisely defined. Eschenroeder (1975) has summarized some compari-
sons between simple box models, PIC models, Eulerian grid-point models, ariS
trajectory models. He emphasizes that simple statistics (such as temporal
correlation coefficients) can be very misleading, and that prospective users
must critically examine the model's output in many different individual
situations, as well as the statistics, to judge fairly a model's utility.
Regarding the overall accuracy of air-quality models, it appears that the
limiting variables in all the models are (1) the accuracy of the emissions
inventory, (2) the accuracy of the meteorological variables, particularly the
mesoscale wind field, (3) the accuracy of the parameterization of meteorolog-
ical processes such as turbulent mixing, and (4) the accuracy of the representa-
tion of the chemical reaction processes. The computational difficulties asso-
ciated with the grid-point and Lagrangian models can be eliminated immediately,
if resources permit, by simply increasing the resolution or the efficiency of
the numerical scheme. It is relatively straightforward to calculate the cost
of reducing the numerical truncation errors by a given amount.
The problem of estimating and improving the accuracy of the emissions
inventory is very important, but the solution appears to be mainly technical.
The problem of calculating an accurate emissions inventory is mammoth; for
example, Shenfield and Boyer (1974) estimated that 10,000 man-hours were re-
quired to obtain an inventory of 5 pollutants for Toronto. In spite of the
44
-------
enormous amount of work required to establish accurate emissions inventory,
this problem, like the accuracy of the computational scheme, is limited by
lack of resources, not by a fundamental gap in basic knowledge.
In contrast to the problems of obtaining the emissions inventory and
improving the numerical accuracy of the computation scheme (which are at least
theoretically solvable), very basic questions remain concerning the limiting
accuracy of modeled mesoscale meteorological processes and the models of the
important chemical reactions. It is not known for certain that useful pre-
dictions are possible on the regional scale, much less the urban scale. In
my opinion, therefore, the long-range improvements in predictive air-quality
models will depend on progress in these two areas of research, including re-
search in understanding the basic physical processes discussed above ("Mete-
orological Models on the Regional Scale"). These efforts will undoubtedly
lead to more complicated models than some might wish, however. In the words of
Shir and Shieh (1974), "After all, any attempt at research modeling should
first aim toward understanding the overall phenomenon rather than emphasizing
the simplicity and convenience of the model."
A COMBINED METEOROLOGICAL AIR-QUALITY MODEL
The problems associated with meteorological and air pollution modeling
have been discussed very generally. In the balance of this paper, a specific
example of how meteorological models and transport models may be combined to
produce an air-quality model will be presented. The example uses one of the
mesoscale meteorological models under development at The Pennsylvania State
University.
While progress continues on the development of the general, multi-level
hydrodynamic model described by Anthes et al. (1974), we realize that similar
meteorological models of the planetary boundary layer would be useful in some
air-quality modeling problems. Therefore, we have developed and are currently
testing a relatively simple mixed-layer model which may have the potential of
modeling mesoscale perturbations to larger-scale flows in the boundary layer.
45
-------
This section describes the model, shows some of the results, and demonstrates
how its meteorological output can be utilized for an air-quality prediction.
This latter example illustrates the relative ease with which mesoscale meteor-
ological models may be combined with transport and diffusion models to produce
a completely time-dependent air-quality model.
Mesoscale Mixed-Layer Model
The disadvantage of any multi-level primitive equation model is its
requirement for large amounts of computer time. It is, therefore, important
to investigate the conditions under which simpler, more economic models can
duplicate the main results from the more complicated models. Since the major
mesoscale forcing occurs at the ground, it might be expected that under most
conditions the mesoscale perturbations will have their greatest amplitude in
the PEL. Thus it is tempting to consider mesoscale models of the PEL alone,
assuming that the middle and upper tropospheric circulations are adequately
predicted by large-scale models.
Lavoie (1972, 1974) developed a prototype model of the PEL which has been
applied quite successfully to mesoscale studies of Great Lakes snowstorms and
convective precipitation over Hawaii. Application of this mixed-layer model
to two such diverse situations suggests that this type of model may have other
applications to the modeling of low-level flows. Because of its simplicity,
we wish to ascertain the meteorological conditions under which the simple model
may be substituted for the complex model. With this in mind, we have modified
Lavoie's model, which was originally designed with steady-state problems in
mind, to more realistically accommodate time-dependent solutions. The major
modifications include:
(a) Changing the first-order time and space differencing schemes, which
heavily damp the solutions, to a conservative second-order scheme;
(b) Incorporation of a more realistic parameterization of entrainment
at the top of the inversion;
(c) Modification of the parameterization of the response of the stable
layer above the inversion.
46
-------
The vertical structure of the mixed-layer model is shown in Figure 4. A
thin (50 m) surface layer, which contains most of the vertical wind shear,
follows the variable terrain. The next layer, which extends from Z equal to
5
50 m above the terrain to the base, h, of a stable upper layer, is assumed to
be well mixed with nearly uniform horizontal winds and potential temperature,
9. The forecast equations for horizontal velocity, V, and 6 apply to this
layer. The layer above the PBL is approximated by a constant (in the vertical)
lapse rate. This layer initially may contain synoptic-scale variations in
temperature and wind. Later, the perturbations that develop in the PBL may
produce perturbations in the upper stable layer which feed back into the PBL.
The height, h, of the PBL is assumed to be a material surface with the
important exception that turbulent entrainment of mass, heat, and momentum may
occur across h. This entrainment is related to the surface fluxes of heat
and momentum.
The isentropic surfaces depicted in Figure 4 indicate possible variations
in the thermal structure of the model. The 0 isentrope intersects the height
of the PBL at a first-order discontinuity in potential temperature. The 0
isentrope intersects h at a zero-order discontinuity in 0. Mesoscale varia-
tions in potential temperature are possible within the PBL and between h and
H, the undisturbed level. Above H, only synoptic-scale variations are per-
mitted.
A Forecast of Boundary Layer Flow Over Complex Terrain
A preliminary experiment conducted with the mixed-layer model is discussed
in this section to illustrate its potential. The model domain consists of a
west-east cross section through the Appalachian mountains that extends over
the Atlantic Ocean (Figure 5). To test the stability of the model under strong
wind conditions, the initial conditions consist of a mean west wind of 30 m/s,
a constant potential temperature of 290 K, and a stable layer with no large-
scale temperature variations above h. The height of the inversion, h, is
initially 1425 m.
47
-------
Figure 4. Structure of a mixed-layer model. The dashed lines are isentropes,
the top of the mixed layer is denoted by h. Above the level H,
the atmosphere is assumed to be undisturbed.
2500
400 500
X(km)
Figure 5. West-east cross section showing wind direction and potential tem-
perature structure at 9 hours from the mixed-layer model.
48
-------
The wind and temperature structure of the model after 9 hours is shown in
Figure 5. The inversion has been lifted by air flowing over the mountains and
to a lesser extent by entrainment due to mechanical mixing (the surface heat
flux was 0 in this experiment). A perturbation exists near the coast (780 km)
where the ground surface becomes suddenly smoother. Strong upward motion
exists upwind of the mountains with downward motion extending to the lee.
The isentropic surfaces in the upper stable layer reflect the deformation
of the PEL height. The perturbed surfaces show how significant vertical mo-
tions (and presumably clouds and precipitation) can be created in a deep layer
above the PEL. Finally, entrainment has produced a downward heat flux at h,
resulting in a maximum of 2° C warming in the PBL (note the 290 K and the
291 K isentropes).
A Particle-In-Cell Transport and Diffusion Model
Once the time-dependent meteorological fields are available, a transport
and diffusion model is necessary to complete the prediction of a passive con-
taminant. As discussed in "Transport and Diffusion Models," above, the PIC
model is one way of calculating regional-scale transports without large numer-
ical diffusive effects. The transport model presented here treats the advec-
tion of a pollutant (SO in this example) in a Lagrangian sense by adopting
the PIC technique of Teuscher and Hauser (1974). Although the advection is a
Lagrangian calculation, the effects of vertical diffusion, divergence, and
deposition are calculated on an Eulerian grid. For this calculation, the con-
centrations represented by the particles are interpolated to the Eulerian grid
where the appropriate contribution to the time rate of change of the concen-
tration by each process is computed. These changes are then interpolated back
to the particles. In the absence of nonconservative physical processes, the
concentrations are preserved exactly as they move downwind. The computational
details will be presented in future publications.
To resolve possibly important vertical variations in the pollutant con-
centrations, the vertical grid length in this experiment is 200 m. The meteor-
ological data, however, are derived from the mixed-layer model described
49
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earlier. These data are interpolated in the vertical as needed by the trans-
port model. It is important to note that any model data (or observations)
could be utilized in the transport model.
It may be argued that meteorological data from a mixed- layer model, which
represent averages over the depth of the PEL, do not justify a high-resolution
(in the vertical) transport model. While this is undoubtedly true for many
meteorological situations in which the pollutant is uniformly mixed between
the ground and the top of the PBL, the scale analysis (in "Potential Uses for
Predictive Air-Quality Models," above) indicates that there are regional-scale
meteorological situations in which pollution will not be mixed uniformly for
a significant (100-200 km) distance downwind of the source. Another poten-
tial application of a high-resolution transport model in a meteorologically
well mixed boundary layer is the prediction of low clouds. Treating water
vapor as a "pollutant," a high-resolution transport model could be useful
even within a fairly well mixed PBL to predict fogs, stratus, and stratocumulus
clouds. In any case, the large number of layers here are included mainly for
illustration purposes; simplificaion to a single uniformly mixed layer is
quite simple.
The conservation equation for S0_ solved in this experiment is
. 24)
2
where F is the vertical flux of Q in yg/m s. The flux at the lower
boundary (50 m in this experiment) is calculated from
F50m = Vd ^ - Q50m) + S <**• 25)
where V, is the deposition velocity, Q is the "background" concentration,
d
Q,.^ is the value of Q at 50 m, and S is an arbitrary source rate. The
X50 m
vertical flux at levels above 50 m is calculated from the expression
50
-------
where K(z) is given by a parabolic expression
„, , 4 Kmax ,1 z .
K(z) = = Z(1 ~ h^} (Eq. 26)
2
where Zo is the terrain elevation and Kmax is 20 m /s.
The meteorological data for the SO prediction experiment were derived
from a 12-hour forecast utilizing the mixed-layer model. The initial condi-
tions for this forecast for horizontal velocity, u, and the height of the
mixed layer are shown in Figure 6. The forecast data were interpolated lin-
early between stored values at 1-hour intervals for use in the transport
model.
The hourly-averaged variations of u, w, and h between 5-6 hours and 11-
12 hours are depicted in Figure 6. The conditions are fairly steady during
the period. High wind velocities occur immediately downwind of the peak. The
mixing depth tends to follow the terrain. Downward motion occurs over most of
the domain.
The initial conditions for the SO transport model consist of a uniform
concentration of 50 yg/m . Particles which enter through the upstream boun-
dary are assigned this value. In order to generate a plume, an upward flux
Df SO of 3;
(Figure 7) .
2
of SO of 35 yg/m s occurs throughout the forecast between 40 and 60 km
Deposition velocities of SO are of the order of 1 cm/s (Smith and Jeffrey,
1975; Rasmussen et al. 1975). There is evidence that the deposition velocities
under some conditions are as much as an order of magnitude higher over water
than land (Kabel, 1978). For this experiment, the deposition velocity is
assumed to be 0 over land (x < 280 km) and 0.025 m/s over water (x > 280 km).
The background concentration, Q , is taken to be 0.
Starting from homogeneous conditions at t - 0, a well developed plume is
moving downwind at 3 h (Figure 7). The initially uniform mixed-layer height
has quickly adjusted to the variable terrain. The advancing pollutant "front"
51
-------
10
5-6 h
Oh
200
4OO
X(Km)
E
.o
*
Oh
. /
X
II-I2H
200
400
X(Km)
1600
1200
800
5-6 h'
Oh
200
X(Km)
400
Figure 6. Horizontal profiles of the west-east component of velocity, u;
vertical velocity, w, at the top of the mixed layer; and the height
of the mixed layer at 0 hours (initial conditions), the 5- to 6-hour
average, and the 11- to 12-hour average. These profiles were com-
puted from the mixed-layer model.
52
-------
E
JC
X
ID
CM
O
TJ CO
C ^-1
(d m 01
0 >
~ o
M
o o o
o
fi
•H
W
•H O
3 -P O
2°
•H C
o
O id u si
fa Oi 0 -P
0)
M
3
cn
•rH
53
-------
is nearly vertical, and deposition over the water has significantly reduced
the initial concentration.
At 6 hours (Figure 8), the plume is advancing down the mountainside
and starting to move out over the water. Very strong gradients of S09 are
resolved near the source. Deposition over the water has caused a positive
vertical gradient of SO near the surface. A steady-state concentration,
representing a balance between the emission rate, advection, and diffusion,
has been reached near the mountain peak.
At 12 hours (Figure 9), the leading edge of the plume has moved off the
domain. The deposition process has produced a weak minimum at the lower out-
flow corner of the domain. Particles in this area have had the largest
trajectory in contact with the water surface. Vertical mixing of SO by the
turbulence parameterization is essentially complete throughout the mixed layer
at a distance of 200 km (10 horizontal grid units) downwind of the source.
This hypothetical experiment illustrates the generality of the PIC model-
ing technique when utilized with a mesoscale meteorological model. Time-
dependent winds, mixed-layer heights, and sources and sinks are easily com-
bined to produce a completely predictive air-quality model if the initial and
boundary conditions are known. The meteorological model of the PEL is rela-
tively inexpensive (compared to multi-layer models) — the computation time for
this 12-hour forecast on an IBM 370/168 was 18 s. For the two-dimensional
version of this model on a 40 x 40 grid, the time for a 12-hour forecast would
be about 8 min.
Future research should be directed toward improving the physics in the
meteorological model and the parameterization of the reaction and sink proces-
ses in the pollutant model, and in the testing of both models against observa-
tions. For conditions in which the mixed-layer model is inapplicable, the
pollutant transport model could obtain the data from more general, multi-
level models.
54
-------
1500
1000
500
100
ZOO 300
X (km.)
400
500
Figure 8. Forecast SO concentrations after 6 hours in the experiment described
in the text and in the caption for Figure 7. Note that the plume
"front" is continuing its eastward advance (compare with Figure 7)
while a steady-state concentration, representing a balance between
source, advection, and diffusion, has been reached near the moun-
tain peak.
1500
1000
5OO
100
200 300
X (km.)
400
500
Figure 9. Forecast SO concentration after 12 hours in the experiment described
in the text and in the captions for Figures 7 and 8.
55
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ACKNOWLEDGMENTS
C-
Daniel Keyser contributed substantially to the mixed-layer model and to
the PIC model. Joseph Sobel contributed the illustration of the meshed model
results (Figure 3) from his work. Henk Tennekes and Dennis Thomson critically
read the manuscript and made useful comments. Robert Kabel contributed to
the modeling of the SO deposition parameterization. Betty Bunnell typed the
manuscript and Gary Fried drafted the figures. This work was supported by
the U.S. Environmental Protection Agency through Grant R800397.
REFERENCES
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HIGH-RESOLUTION MODELS OF THE PLANETARY
BOUNDARY LAYER
Alfred K. Blackadar
Department of Meteorology
INTRODUCTION
It may be taken as axiomatic that all transport models used in air-quality
simulation applications require a knowledge of the mean wind distribution with-
in some portion of the boundary layer. The great number of air-quality simula-
tion models that have evolved within the last few years differ rather funda-
mentally in the way this information is expected to become available. In the
simplest class of models, it is merely assumed that the required wind distribu-
tion is provided directly from observations or less directly through some inter-
polative analysis of observations made at a number of places. The models com-
prising the most complicated class provide for the prediction of the complete
mean wind distribution from an observed antecedent state and from boundary
conditions based on predictable external physical features of the environment
such as solar radiation, soil characteristics, and surface roughness. Between
these two extremes lies an intermediate class of models characterized by a
partially known wind and temperature field, consisting perhaps of only surface
values. In this intermediate class, the dynamical equations are used primarily
to enable the unknown portions of the three-dimensional field to be estimated.
Models are also required to give information about the diffusion of
contaminants or other properties into or out of the path of the mean wind.
The demands that must be met by models are of two different kinds: diffusion
transports that are required for the determination of contaminant distributions,
and those that are needed for the correct modeling and prediction of the mean
wind distribution. Both demands must be kept in mind when planning how the
boundary layer and the boundary conditions are to be treated. For example, a
63
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knowledge of the mean wind and lapse rate near the ground may be all that is
needed to estimate the concentration of contaminants downwind of an elevated
point source; however, it is probably necessary to model the diffusion of heat
and momentum in a fairly sophisticated way throughout the PEL in order to
provide information about the surface wind distribution, if this needs to be
known at some future time. The solar radiation that enters into the esti-
mation of stability class needed for many diffusion estimates poses special
problems for predictive models, for the state of cloudiness almost certainly
is quite sensitive to the accumulations caused by the convergence fields of
water vapor diffusion.
For many purposes, it is sufficient to predict the distribution of wind
and turbulence in the lowest 100 m of the atmosphere. If this is the case,
aerodynamic theory can provide the needed information if the surface fluxes of
heat and momentum are predicted by the model. There are basically two ways of
providing these fluxes. The first is to abandon a detailed description of the
boundary layer in favor of parametric relations between the surface fluxes
and a combination of free-atmosphere and surface variables. The second is to
provide a sufficient number of levels in the boundary layer to resolve the signi-
ficant features of the distributions of wind and temperature, and to deduce
from these the resulting fluxes of physical quantities and changes of mean
variables. The parametric treatment of the boundary layer is described by
Tennekes (1978). The present report deals with the second approach. A more
thorough discussion of recent work on second-order closure approximations in
high-resolution treatment of the boundary layer is given by Lumley (1978).
EMPIRICAL METHODS
Although the incorporation of the boundary layer into simulation models
of the atmosphere on various scales is a relatively new requirement, the theory
on which modern solutions are based goes back to the early twentieth century.
Not surprisingly, the fluxes of heat and momentum were patterned after the
kinetic theory of gases which, during the latter half of the nineteenth century,
had been triumphantly successful in explaining and predicting the molecular
fluxes of heat and momentum in gases. Following the early theory of Ekman for
64
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the current distribution in the surface layer of the ocean, and the adaptation
of this theory by Taylor (1915) to the atmospheric boundary layer, it came to
be realized that the effective viscosity of the atmosphere is not a property
of the medium but of the state of turbulence and the size distribution of the
turbulent eddies existing in the boundary layer.
Thus, at an early date the stage was set for the classical treatment of
the fluxes, in which they are assumed to be determined by the gradients of the
mean state variables, providing only that these variables are chosen to be
represented in a form that is independent or conservative with respect to ver-
tical motions. With considerable justification, the turbulent fluxes in hori-
zontal directions have generally been assumed to be negligible in practical
applications. Accordingly, the equations governing the variation of mean
variables have been written as follows:
M 4-nM j- v H + w ^2- = - - Q + fV + — K —
3t 3x 3y 3z - 3x 3z m 3z
1Y iY. v— +W— =- — -^E- - f U + — K —
~8~t 3z 3y 3z — 3y 3z m 3z (Eq. l)
36 A TT 39 4. W 36 4. TT 39 - _L- ^ 3_ jl
__ + U _- + v-^ + W -97 - - c p 3z +3z Kh3z
3U 3V , 3W _ n
3x 3y 3z
3C 3C 3C 3C 3C^
~ + U -z-2- + V -~ + W -—• = S + -vr-
3t 3x 3y 3z a
in which U, V, w, and 8, p, and C are the mean velocity components, potential
temperature, pressure, and pollutant concentrations, respectively; f is the
Coriolis parameter; S is the source strength for the a pollution component;
and K , K and K are the respective exchange coefficients for vertical turbu-
lent transfer.
65
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Although there is a great deal of interest and effort being placed in non-
K-type theories at the present time (see Lumley, 1978), the proven and tested
state of the art has not progressed much beyond K-type theories. The progress
that has been made over the last 50 years has been mainly centered around
methods of predicting the magnitude and distribution of the various K values
either explicitly as a function of z or implicitly as a function of the dis-
tribution of mean quantities under the full range of meteorological conditions
found in nature.
Completely explicit models of K were the first to be used and are still
m
being used in some air-quality simulation models. An extensive bibliography
of explicit treatments of K has been given by Wippermann (1973). Usually, the
form of the vertical distribution is specified with a magnitude scale left to
be determined in an appropriate way by meteorological conditions. Taylor's
model consisted of the Ekman (constant K) layer matched to a surface layer in
which the wind shear is parallel to the wind stress at the surface. From
observations of the surface wind direction and height of the gradient wind
2
level, Taylor determined that the value of K varies from 2.8 m /s during light
2
wind conditions to 6.2 m /s in strong winds. Later models of the PEL in which
a surface layer is matched to an upper layer with a prescribed K distribution
have been studied by Rossby and Montgomery (1935), Yudin and Shvetz (1940),
Ellison (1955), and Blackadar (1974a). Models such as these have been success-
ful in relating the surface wind stress to the geostrophic wind speed and
direction. They have thus contributed usefully to parametric (i.e., low-
resolution) models of the boundary layer. Explicit constant K air pollution
models have been formulated by Lee and Olfe (1974) and Yamada (1972).
O'Brien (1970) has proposed an explicit distribution for K that has been
widely used in air-quality simulation models. It may be viewed as a cubic
polynomial designed to satisfy the values of K and its derivative at key points:
K(z) = KA + [(z - h*)2 / (h* - zB)2]
(Eq. 2)
X (
-------
In this equation, h* is the height of the top of the boundary layer, B refers
to the surface layer, and the prime refers to the first derivative with respect
to height. K and K ' are assumed to be known from surface boundary-layer
theory. The O'Brien distribution has been used by Bornstein (1972 and 1975)
and by Pielke and Mahrer (1975). Of a similar nature but differing in detail
is the distribution used by Shir and Shieh (1974):
K = U.J. k z exp (-4z/h*) (Eq. 3)
where u^ is the friction velocity given by
u* = ^T0/P (Eq' 4)
and k is von Karman's constant. In a similar way, a trapezoidal function for
K(z) was used by Bankoff and Hanzevack (1975). Although the use of explicit
distributions of K has obvious advantages of simplicity, they must be scaled
at the surface by parameters (K, u^, etc.) that are known a priori- at every
time and place. This requirement tends to restrict their use to diagnostic
models.
IMPLICIT K MODELS
Implicit models are designed to remove the necessity of making ad hoe
assumptions about the distribution of K. Instead it is assumed that K should
be determined by the distribution of wind shear and lapse rate. By introducing
these relationships one can leave the model to generate the K distribution
step by step in accordance with the evolving wind and temperature distributions.
The method has the advantage that it is unnecessary to make ad hoc assumptions
or definitions about the location and extent of the boundary layer. In a sense,
the whole atmosphere becomes the boundary layer, and the model itself deter-
mines the domain where variables are influenced by the boundary conditions.
Implicit models are also useful for a different reason. It is almost
impossible to determine the actual K distributions from observed velocity and
temperature distributions, except in the lowest 100 m of the atmosphere. In
67
-------
the case of the eddy viscosity, K , this situation results from the fact that,
m 5)
where Ri is the Richardson number, s is the wind shear
-i2 + 2]"
and £ is a length that is presumed to characterize the energy-containing
turbulence. Within the surface layer, H is proportional to height:
£ = kz (Eq. 7)
It is known that H does not continue to increase linearly upward above the
surface layer, but little is actually known about its distribution in the
expanded atmospheric boundary layer. The most widely used distribution is
that suggested by Blackadar (1962) on the basis of Mildner's Leipzig wind
observations:
68
-------
I = kz /(I + kz/A) (Eq> 8)
where
A = .00027G/f (Eq. 8a)
or, as suggested more recently,
A = .0063u*/f (Eq. 8b)
and G is the geostrophic wind speed. This formulation has been used by Estoque
and Bhumralkar (1970), Delage and Taylor (1970), Wagner and Yu (1972), Delage
(1974), Lee and Olfe (1974), Gutman and Torrance (1975), Sheih and Moroz (1975),
Yu and Wagner (1975) , Torrance and Shum (1976) , Yu (1976) , and, with small
modifications, by Fiedler (1972) and Mellor and Yamada (1974). Other explicit
£ formulations which are, in effect, rather similar to the above have been
suggested by Lettau (1962) and Appleby and Ohmstede (1964).
Not all authors have chosen to make £ independent of lapse rate, and from
the point of view of Equation 7 it is immaterial whether the dependence of K
on lapse rate is placed in £ or in f(Ri), or divided between them. However,
for the present discussion, the work of the various contributors will be pre-
sented in such a way that the entire dependence on lapse rate is contained in
f(Ri). We have chosen to present the discussion in terms of Richardson numbers
rather than the similarity function z/L because the latter cannot as a rule
be computed from model variables such as wind shear and lapse rate.
One of the simplest empirical determinations of f (Ri) follows from the
work of Panofsky et al. (1960) . Good agreement with observations was found
over negative Richardson numbers for the equation
- 18 *3 ' l (Eq' 9)
69
-------
in which
(Eq>
and
, = ._6u*s
kg3g/3z (Eq. 11)
2
Since K is by definition u* /s and since in these cases £ = kz, one can
m
easily show that
£ (Ri) = Km = (1 - ISM)1*
m —;r
£ s
This function is shown in Figure 1 along with other formulations that have been
suggested. Although the study discussed above was completed more than 15 years
ago, it appears to be consistent with a great deal of data that have been col-
lected since then, and may therefore be considered the best empirical basis
for judging the validity of other functions that have been used. Also, as far
as it is known at present, the function in Equation 12 has the correct asymp-
totic behavior for large negative Richardson numbers (though many uncertainties
about this remain) and therefore represents the best basis for an empirical
extrapolation into the extended planetary layer.
Under stable conditions the observations scatter so much that it is al-
most impossible to justify any particular empirical law. K should fall to
0 when the Richardson number approaches the critical value, but there is
no agreement what this value is. The consensus favors a value in the range
from 0.20 to 0.25. However, it must be kept in mind that, when the Richardson
number is measured from observations that span an extended layer, the measured
value is usually larger than the largest values occurring within the layer and
larger even than the mean value of all the Richardson numbers occurring in the
layer. Thus the "practical value" of the critical number may be larger than
the value that would be appropriate if the fine structure of each layer could
be determined.
70
-------
O
(£
y
2
O
CM
rH
&
•H
a
»-* h*-l ' '
O1 ft
o . e^
rH -H
CN
o o
Ti -H
04
3
Q)
71
-------
The most thorough summary of all of the nighttime wind and temperature
€
profiles available in published sources is that of Dyer (1974). His conclu-
sion is that the universal function 4> for all quantities is the same, implying
the equality of the K values for all stable Richardson numbers. In this case,
the adopted relation
5z/L'
(Eq. 13)
may be rewritten as
f(Ri) = (1 - 5Ri)'
(Eq. 14)
implying a critical Richardson number of 0.20.
Figure 2.
I
\
This function is plotted in
Ri/Rc
1.0
2
Figure 2. K /£ s as function of Richardson number according to various formula-
m
tions: a. empirical function following Dyer (1974); b. approximation
adopted for the nocturnal boundary-layer model; o. energy equation
with a = 1; d. Estoque and Bhumralkar (1970) as corrected; e. Gut-
man et al. (1973).
72
-------
With these empirically determined functions in mind, we may now look at
some of the functions that have been used in boundary layer simulation models
by various authors. The first is that of Estoque and Bhumralkar (1970). (Note
that the equations that appear below are those that appear in corrected form
in Gutman et al. [1973], rather than those given in the original article.)
K = £2s (1-3 Ri)2 for Ri < 0
2 -2
= £ s (1+3 Ri) for Ri > 0
These equations are shown in Figures 1 and 2 as curves c and d, respectively.
It is possible that these are not the functions that the authors intended to
be used; in any event, they do not conform acceptably to the empirical func-
tions. Several later papers by Wagner and Yu (1972), Gutman et al. (1973),
Gutman and Torrance (1975) , and Torrance and Shum (1976) employ what appear to
have been the originally intended functions:
K = £2s (1-3 Ri) for Ri < 0
2 -1 (Eq. 16)
= JTs (1 + 3 Ri) for Ri > 0
These are shown as d in Figure 1 and e in Figure 2. They appear to give accept-
able values for small positive and negative Richardson numbers but may not have
the correct asymptotic behavior under very stable or very unstable conditions.
Pandolfo and his colleagues have published a number of model studies of
the boundary layer. Of particular interest for air-quality simulation model-
ing is a study by Pandolfo, Atwater, and Anderson (1971). The forms used in
this method are:
-dRir
4
— Ri)1
k
for
for
Ri >
Ri <
- 0.48
- 0.48
where a is the Monin-Obukhov constant (the value of which is in the range of
3 to 5) and h is Priestley's constant (with a value close to unity). The func
tion for the stable case corresponds exactly with the empirical function shown
73
-------
as a in Figure 2. Pandolfo, however, did not allow K to fall below the value
2
1 m /s. For negative Richardson numbers, the function falls far below the
empirical values. The systematic error is offset by the fact that Pandolfo's
£ increases linearly upward throughout the entire boundary layer instead of
approaching a constant (as it is usually treated in other models).
In the semi-empirical method of Zilitinkevich and Laikhtman (1965) , it
is assumed as a starting point that
Km = Cl lz q ' *h = "T Km (Eq. 18)
where £ is a length to be discussed later and
z
2 ,2 . ,2 . ,2
= u + v + w
(Eq. 19)
The quantity q is determined from the turbulent energy equation which, for
stationary homogeneous turbulence without turbulent transport of energy, may
be written
O rr ^ A
KmS ^T Km 9 Jz ~ e ~ (Eq. 20)
The dissipation rate e is assumed to be given by
e = C~q3/ fc (Eq. 21)
J Z
Substitution gives the result
Km = £z S (1 ~ "T Rl) (Eq. 22)
after absorbing the ratio c /c into definition of £ . If £ is defined in
J_ j> z z
the customary way (so as to be independent of the stratification), this result
falls very close to curve b in Figure 1 and o in Figure 2. In the latter case,
the implied critical Richardson number is unity — a value that is almost cer-
tainly too large. In the unstable case, a has a value that ranges from about
1.3 to a limiting value of a little over 2 for indefinitely large negative
74
-------
values of Ri. Comparison with the empirical equation (Equation 12) reveals
that the energy equation yields the correct form for f (Ri), but the constant
is too small. This fact was first pointed out by Panofsky (1961), who explained
that the large constant in the empirical equation apparently results from the
high sensitivity of K to thermally induced motions, both because of their large
size and tneir vertical orientation.
The method of Zilitinkevich and Laikhtman (1965) is interesting also
because of the implicit method of determining £ :
Z
£ (Eq. 23)
where
= s2(l - CLRi)
1 (Eq. 24)
The method may be viewed as an extension of von Karman's expression
„ _ , , ds
' dz (Eq. 25)
to nonneutral layers. £ is a function of Richardson number, and before com-
z
parison with other methods can be accomplished it is necessary to evaluate
£ in terms of the usually defined £ that is equivalent to kz in the surface
z
layer. This has been done, assuming (as did Zilitinkevich and Laikhtman) that
a = 1, and assuming that <|> satisfies the KEYPS equation
4 vz 3
^m ~ L *m = 1 (EC^ 25)
in which L is the Monin length defined by
L = - 6cpu
H is the surface turbulent heat flux, and y is a constant equal to about 12.
75
-------
With these assumptions, one can obtain the result
(1 - Rir (1 - ~ yRi)
f (Ri) = - - -- 2 (Eq. 28)
(1 - JgRir (1 - y
This function appears in Figure 1 as curve e. Although the result does not
compare favorably with the empirical curve a, the difference is offset in the
planetary boundary model by the fact that £ is kz and thus increases linearly
upward throughout the entire boundary layer. This method was broadened to
include the effects of the turbulent diffusion of turbulent energy and applied
to the entire planetary boundary layer by Bobileva et al. (1965). The pre-
dicted geostrophic drag coefficients and surface wind directions appear to
display the same systematic errors that led Blackadar (1962) to reject Equation
25 for £ in favor of Equation 8.
Another implicit formulation of £ has been suggested by Wippermann (1971)
on the basis of the vertical distribution of stress.
Rossby number similarity demands that — in a stationary, homogeneous,
neutral atmosphere — £ should be scaled by the length u*/f . This view is
compatible with both Equations 8 and 11. The view has recently come to be
widely accepted that, as the atmosphere becomes increasingly convective, the
appropriate scale length shifts rapidly from u*/f to the actual depth of the
mixed layer. A few authors, including Busch et al. (1976), have actually used
such a specification for £ in convective cases. There is no agreement on how
to treat the nighttime stable layer; logical possibilities include some adapta-
tion of Equation 8 and a shift toward the Monin length L as the length scale.
APPROXIMATIONS BASED ON A SECOND-ORDER CLOSURE
Recently, a number of authors, including Donaldson (1973) , Deardorff
(1973), Mellor (1973), Lewellen and Teske (1973), Lumley and Kajeh-Nouri (1973),
Blackadar (1974b) , Yamada and Mellor (1975), and Wyngaard and Cote (1974) have
applied second-order closure methods to simulation models of the atmospheric
boundary layer. A review of the development of these methods is presented by
76
-------
Lumley (1978). The methods are still experimental in nature, but they have
nevertheless led to considerable success in deriving relationships within the
surface boundary layer that had previously only been known from observational
experience. In this section, we shall make inferences about the fluxes of
heat and momentum by applying to the respective equations approximations of
horizontal homogeneity and stationarity as they are usually applied in boundary-
layer models.
The nature of the closure approximations that have been applied to the
equations for the turbulent fluxes has been described by Donaldson (1973) and
Deardorff (1973). Mellor and Yamada (1974) have considered the relative order
of magnitude of the terms in these equations and have shown that three levels
of approximation can be achieved in a systematic way. We shall base the en-
suing discussion on the Level 2 approximation, which is achieved by disregarding
advective and diffusive terms in the equations for the second moments of the
turbulent fluctuations. The equations as written below apply to the situation
in which all of the mean quantitites are functions of height only — conditions
which are generally assumed to be acceptable for the purpose of calculating
fluxes in boundary- layer models:
C r1 a ™ = - c^ a 2 |5- + - u^ , OQX
m w ^ w 8z g (Eq. 29)
C l~l a w - - a^o 2 ~ + ^ W
m w i" w dz Q
"
2/3 (™
- - f
32)
f - ^ H (Eq. 33)
C, £ l a w6' = OL -I 6'2 - a 2 -?1 (Eq. 34)
n w ho w dz
77
-------
V1 °w q2 • f W0f - (™ + ™ l } (E(J- 35)
Cal"1 a 6'2 = - "w0T |^ (Eq. 36)
6 w 3z
In obtaining these equations from the Mellor and Yamada Level 2 approxima-
tion, we have made certain changes that appear to improve their usefulness.
Using what is essentially the Level 3 approximation, which includes the dif-
fusive terms, Lewellen and Teske (1973) achieved Monin-Obukhov similarity when
the equations were applied to a surface-layer model. With the Level 2 approxi-
mation, however, Monin-Obukhov similarity cannot be achieved, because q/a
w
cannot be eliminated from the equations and, as is well known, q/a does not
w
obey Monin-Obukhov similarity in the surface layer. As has been shown by
Blackadar (1974b), Monin-Obukhov similarity is restored by writing the equations
in the above form, making them equivalent to the Mellor-Yamada equations if the
following substitutions are made for the original length scales £ , £ , A ,
-L ^ _L
and A :
q *•
= ~
"1 3a C -WE
w m
(Eq. 37)
£ £_ £_
°w h w 6
We have also put
2
"m E (1 " Cl ^2 } (Eq. 38)
w
The quantities a and a, originate from the effects of vortex stretching and
m n
buoyancy adjustment on the pressure correlation terms. Under isotropic condi-
tions they have the values 2/5 and 2/3, respectively. Under boundary-layer
conditions the turbulence is not (in general) isotropic; therefore, these
quantities must be determined empirically. Mellor and Yamada did not include
the effects of buoyancy adjustment; therefore, in their equations a is re-
placed by unity.
78
-------
The eight equations represent a closed set that may be solved algebraically
to obtain explicit relations between the fluxes and the gradients of mean
quantities. We note that Equations 6 and 8 together yield a gradient- type
heat flux such that
K, 2a y _!
_£ = k = (C + _n_ )
0 £ Kh ^ h C0 } (Eq. 39)
W D
where
;2s2
y E ±-2- Ri
a 2 (Eq. 40)
w
Similarly, Equations 1 and 4 or 2 and 5 together yield flux-gradient expressions
for the momentum flux components with an effective exchange coefficient given
by
K
- Kn = (Ch am - khy) / (CmCh + y) (Eq.
Finally, an independent expression relating p and Ri is obtained by eliminating
q between Equations 3 and 7 an<
equation is quadratic in form:
q between Equations 3 and 7 and then eliminating a using Equation 12. This
2
(a1Ri - a2) y + (t^Ri - b^ y + CRi = 0 (Eq. 42)
where ar , a0, b., , b , and C are positive functions of C , C, , C , C., a, , and
1^1^ mhE6h
a .
m
Certain properties are easily seen from these expressions. First, when
the Richardson number is 0, p is also 0; thus a (though indeterminate in
w
Equation 12) is not necessarily 0 in neutral conditions. The solution further
shows that y always has the same sign as the Richardson number. Second, as y
becomes infinitely positive, the Richardson number approaches a limiting
value. This is the critical Richardson number, and its value is given by
a (C - 2c ) (2a a C - c )
_ . _ 2. _ m E h m h o
c ~ IT ~ (Eq. 43)
1 6a, C C + CD (C + 4O
h E m 6m E
79
-------
The values of C , C, , Cn, and C are easily derived from neutral surface-
m n U E g,
layer observations if a is known; their values are independent of the value
m
of a . Using published results of the 1968 Kansas field experiment, Blackadar
(1974b) determined the values
C = 0.52
m
C = 1.10
(Eq. 44)
C0 = 0.19
CE = .10
assuming
a = .40
m
k = .40 (Eq.
a = kz
He also showed that these values give excellent predictions of the asymptotic
relations for free convection if a, in this case approaches the value 1.
The set of Equations 11 through 14 allows K, , K , and a to be completely
h m w
determined from the Richardson number, wind shear s, and an appropriate length
H if a and a, can be assumed to be fixed. Although the value of a. near
m h ^ h
the neutral state has not been determined, its effect in this part of the
range is minimal. The relationships for the unstable (negative) range of
Richardson numbers are given in Figure 1. The curves are well represented
by the functions
K /£2s = (1-21 Ri)'5
m
s = (1-87 Ri)^ ^ 46)
The predicted curve for K is very close to the empirical function, as may be
m
seen from Equation 12.
The fact that a appears to approach unity in free convection is consis-
tent with the hypothesis that buoyancy compensation becomes negligible as the
vortices become greatly elongated in the vertical. Buoyancy compensation arises
80
-------
when temperature anomalies set up a gradient of the pressure anomaly that op-
poses the buoyant acceleration. With chimney-type convection, it is reason-
able to believe that these opposing forces are at a minimum. By the same
reasoning, it should be expected that, with stable stratification and the
strong tendency for vertical motions to be suppressed, the buoyancy compensa-
tion would be enhanced and that a, should accordingly become quite small.
We can make an inference about the behavior of a, under stable stratifi-
h
cation in the following way. Experience has shown that under very stable con-
ditions K, and K approach equality. If we invoke this requirement on Equations
n m
11 and 13 as p tends to infinity, we obtain the condition
a a, = Cfl / C
m h 8 h
The quantity a does not appear to change over the entire range of negative
m
Richardson numbers; therefore, we might reasonably expect that it is also
invariant under stable conditions. Using the value 2/5 for a in Equation 46
m
results in the estimate of 0.437 for a, which, as expected, is smaller than the
theoretical value of 0.667 for isotropic conditions. Putting this value into
Equation 43 leads to a predicted value of the critical Richardson number equal
to 0.207, which is almost exactly the value that was determined from the Kansas
field program observations. Again, we do not know how a, varies with Richard-
son number over the whole stable range, but we are aided by the fact that the
distributions of K, , K , and a in the vicinity of the neutral state are com-
h m w
pletely independent of a,. Accordingly, we may hope to obtain a reasonable
prediction of the distributions of these quantitites by holding a, fixed over
the entire range. The resulting predictions of the distributions of these
quantities are given in Figures 3 and 4. It may be noted that, with negligible
error over the range of positive and slightly negative Richardson number, 1C
and K are given quite well by the simple formulas:
Ri - Ri ?
Km = ^ = 1.1 ( ~-fj~- ) £ s If Ri<_ Ric (Eq, 4y)
=0 if Ri > Ri
- c
81
-------
-.10
0 .10
RICHARDSON NUMBER
.20
Figure 3. Level 2 approximation for K and 1C for positive Richardson numbers.
1.5
1.0- •
0.5 •
-JO 0 .10 .20
RICHARDSON NUMBER
Figure 4. Level 2 approximation for positive Richardson numbers.
82
-------
Unfortunately, the approximations of the second-order equations that have
been used do not give any insight as to how the value of £ is to be found in
the higher portions of the boundary layer where the size distribution of the
eddies ceases to be determined by the distance to the ground. We are thus
forced to rely on empirical evidence for this length as in the more primitive
K theories. One hypothesis is that, under stable conditions, H might reason-
ably be identified with the Monin length, which typically has a magnitude of
the order of 10 to 50 m. Since this is of the same order as the typical free-
atmospheric value of SL under neutral conditions, one might expect that £ is
not strongly affected by the stability over most of the stable range.
Up until now we have considered only the solutions leading to the calcula-
tion of the fluxes of heat and momentum. It needs only to be pointed out that,
as soon as these fluxes have been determined, one can return to Equations 32,
33, 35, and 36 to determine the remaining statistical quantities. The applica-
tion in the surface boundary layer has given good results in the prediction of
0' , and — if the extended boundary-layer models are also at least moderately
successful in the application of this equation — will be of great interest
for the interpretation of accoustic sounding observations. Equation 35 has
not been very successful in the surface layer for predicting q, apparently
because of the strong influence of the neglected diffusion term. The importance
of the diffusion terms away from the immediate influence of the boundary and,
consequently, the ability of Equation 35 to produce acceptable results in the
extended layer are not known.
PREDICTIVE MODELING OF THE NOCTURNAL BOUNDARY LAYER
The model that has been developed for insertion into the regional-scale
three-dimensional model has had as its goal the achievement of maximum simpli-
city consistent with the requirements of predicting the distributions of wind,
temperature, and other mean quantities relevant to air-quality simulation. The
accuracy required for the predicted distributions does not need to be greater
than that which is generally available for verification. By the same token,
the accuracy need not eliminate the likely effects (on the solution) of un-
certainties in the initial state and boundary conditions. The effect of such
83
-------
uncertainties has not been quantitatively determined, and one of the important
applications of our predictive model will be to determine its sensitivities to
such uncertainties.
The requirements of a predictive model demand that the soil heat budget
be included, for only in this way can the interactions of the predictable long-
wave and shortwave radiation fluxes at the surface with the lower atmosphere
be properly taken into account. Normally, models designed to include the soil
heat budget have had to incorporate a considerable number of levels within the
soil. From the point of view of predicting the interactions with the atmo-
spheric layer, however, it is necessary only to know the soil surface tempera-
ture as a function of time. As a result, it is possible to reduce the number
of levels very greatly; in fact, we have shown that — for a pure sinusoidal
heat input at the surface — the surface temperature amplitude and phase can
be exactly calculated with a simple model consisting of a uniform slab resting
on a substrate of constant temperature. This analysis is given in the Appendix.
Let 0 be the surface temperature of the soil, which is also the uniform
g
temperature of the slab. We shall designate the heat capacity per unit area
of the slab by C . it is shown in the Appendix that the appropriate value of
g
C is given by
Cg = 0.95 Ac-ffi (Eq. 48)
where X is the thermal conductivity of the soil, C the heat capacity of the
s
soil per unit volume, and u> is the angular velocity of the earth's rotation.
If 9 is the temperature of the substrate, the rate of heat transfer from the
m
substrate to the slab is given by K C (6 - 6 ), where K is shown in the
m g m g m
Appendix to be appropriately given by
K = 1.18 to
m
(Eq. 49)
The heat budget of the slab then yields an equation for the soil surface
temperature:
84
-------
if* = f (ig + u - aeg4 - HO) - Km (eg - em) (Eq. so)
o
where I is the solar insolation absorbed in the surface, 14- is the longwave
s
back radiation from the atmosphere, a is the Sefan-Boltzmann constant, and HQ
is the heat flux carried away from the surface by turbulence. The value to
be used for 9 is the mean tempe
m
air) during the most recent past.
be used for 9 is the mean temperature of the slab (and thus of the surface
m
For computational purposes, we divide the domain of the atmosphere to be
modeled into layers of thickness Az. The average properties of each layer
are designated by an odd value of the index i, as shown in Figure 5:
6
5
4
3
2
1
I I I I I I I 1 I I I I I I I I I I I I I I I I
Figure 5. Atmospheric layers and grid nomenclature for the nocturnal boundary-
layer model.
Quantities such as the Richardson number
R1 „ ._a_ ( ei+i - ei-i
' 2
-------
si2 = [(vi+l ~ vi-l)2 + (vi+l - vi-l)2] /(AZ)2 (Eq- 52)
will normally be calculated only at even-numbered values of i. The equations
for the mean velocity and the mean potential temperature are applied at each
odd numbered value of i greater than or equal to 3. We assume that the mean
vertical velocity is 0.
9U. 9U. 9U. K .. nN K .. ..
*r + "i ar + \ w - f
-------
surface fluxes, which have yet to be determined. Since the layer thickness
Az is normally kept rather small (50 m has typically been used) it is reason-
able to assume that the wind direction does not change between the ground and
z = Az/2. The surface stress is then in the same direction as the wind at
z . Thus the equations for the lowest layer are:
u-TUn u*2 Ui
3U1
at
9vi
3t
9u au
, /> 1' V, ,. —
1 3x 1 9y
9V 9V
1 TT J- W
1 U. „ 1 V, r.
1 3x 1 oy
V -V
(Eq. 54)
39. 90 96_ 9-9. H
3t 1 0x 1 3y 2V 2 ' C pAz
^ Az p
where
W E
(Eq. 55)
In order to solve the atmospheric equations it is necessary to have, in
addition to the initial and boundary conditions, the components of the geo-
strophic wind as function of height and time at each place. After insertion
of the boundary- layer model into the three-dimensional regional-scale model,
this information will be available; when used as a one-dimensional model, the
boundary-layer model must rely on observations to estimate the geostrophic
wind. In a similar way, the back radiation from the atmosphere, which is
needed in Equation 50, will eventually be calculated from the temperature and
water vapor distributions provided by the model.
The final requirement of the model is to specify how the surface fluxes
of heat and momentum are to be calculated from the variables of the model. In
the past, modelers have generally solved this problem by assuming that the
lowest layer satisfies Monin-Obukhov similarity. The wind and temperature pro-
files are then specified according to empirical equations in terms of the
87
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boundary fluxes, which are determined by fitting the profiles at the levels
where the temperature and velocity components are known — usually at the
nominal surface z and at the first level z,. This is the procedure that was
o 1
followed by Pandolfo et al. (1971) and by Yu and Wagner (1975). The procedure
is open to some question, however, particularly at night, because it is known
that — as one approaches the surface — the processes dominating the temperature
distribution change quite rapidly from turbulent exchange to radiation. Thus,
extrapolations of the Monin-Obukhov temperature profiles to the surface (which
may have a different z from that applicable to the wind) may lead to a gross
error in the estimate of the ground surface temperature, an error that is dif-
ficult to verify observationally because of the notorious difficulty of measur-
ing the actual air temperature at the ground surface.
K
For the purposes of our model, we assume that the temperature profile is
dominated by turbulence and therefore obeys Monin-Obukhov similarity through-
out the first layer down to a level a , which has a nominal height of 1 m and
3.
which may be identified with the height of the thermometer in an ordinary
instrument shelter. Below z , we assume that temperature changes are deter-
3.
mined by the radiative flux divergence as well as the turbulent flux conver-
gence. For the calculation of the wind distribution, we assume that the wind
profile follows Monin-Obukhov similarity down to the height z , where the wind
speed is 0.
As far as we are aware, satisfactory empirical equations for the radia-
tive and turbulent flux divergences in the surface layer have never been deter-
mined. We have adopted a suggestion by Sellers (1965, p. 234) that the radia-
tive cooling rate is proportional to (6 - 9 ) and that, in the equilibrium
a g
case, this cooling rate is compensated by the effects of turbulent flux con-
vergence. The latter is assumed to be proportional to H . This view of the
operation of these flux divergences appears to be supported by the observation
of Rider and Robinson (1951) that the radiative flux divergence tends to in-
crease with the amount of turbulent mixing near the surface. Accordingly,
the "surface" air temperature is governed by the equation
90
= a (8 -0)-bH / c p z (Eq. 56)
3t ^ g a o ' p ^ a
88
-------
The behavior of this equation is such that the existence of turbulence at night
causes the surface air temperature to be elevated above the ground surface tem-
perature; when the wind becomes calm, the surface air temperature decays ex-
ponentially to the ground temperature. Sellers, using data of Rider and Robin-
-4
son, suggests a coefficient that would make a = 3.6 x 10 /s. We have experi-
-4
mented with a variety of values and have adopted a = 8.3 x 10 /s and b = 0.200.
Since we are assuming that Monin-Obukhov similarity prevails in the lowest
layer, well known laws would permit (8 - 9 ) and W to be determined from
JL 3. -L
T. = -H /c k pu. and from u.. In principle, the problem can be inverted; in
•X Q p ft ft
practice, this is infeasible due to the nature of some of the functions in-
volved in the relationship. Fortunately, the problem is simplified by the fact
that good initial estimates of u^ and T^ are available from the previous time
step and a double iteration routine is sufficient to produce accurate updated
values. The first step in this routine is to estimate the bulk Richardson
number
B=t 72 [^-e^+T^ln^] (Eq. 57)
y w o
using the most recently available values of 0 , 9 , W , and T . The form of
J- cl JL
the profile equations varies according to the value of B, and three possibi-
lities exist. The first case occurs when B is positive and exceeds the value
1/5. In this case the lowest layer becomes nonturbulent, and {even though
the wind at the level z is not 0) the momentum and heat fluxes at the surface
are set equal to 0. The second case is the stable case in which 0 < B ^ 1/5.
Here we proceed to calculate the Monin length L from the formula
1 1 -, Zl r B ,
L = I, ^ 7" [T^T] ^q. 58)
1 o
and then derive u and T from
z
(In — - 59)
o
89
-------
(Eq. 60)
in which
= H* = -c; <, /T
m h 3 z1 / L (Eq- 61)
In the unstable case, B < O is not so likely to occur at night. When it
occurs, the following routine is executed. Using the last estimate of the
Monin length L, one determines
x = (1 - 16 — ) * (Eq. 62)
and then proceeds, in sequence, to obtain
¥, - 2 In ^L
m
(Eq. 64)
u* = kWx / (In — - y (Eq. 65)
o
a = (9 - 9 ) x (In — - ¥ ) (EcJ- 66)
a
Zl 2
(in — - y r
1 B r Zo m -,
L = I [ z ] (Eq' 67)
_L /i _L it* \
The functions in Equations 63 and 64 were derived by Paulson (1970).
Having thus arrived at new values of urt and T^, one may return to recal-
culate B (and so forth) as many times as desired. A single reiteration has
proved to be entirely satisfactory in our one-dimensional model experiments.
Having obtained acceptable values of u. and T., one obtains H from
~ * o
Ho = ^ % PU* T* (Eq. 68)
90
-------
To ensure computational stability, it is necessary at all times to satisfy
the inequality
2 E6t
To choose a time step that is small enough to ensure that this inequality is
always satisfied would be unnecessarily expensive in computer time. Ordinarily,
a 2-min time step is sufficient for the level of accuracy demanded of the
model. To cope with occasional situations where shorter time steps are needed,
provision is made for a variable time step to be calculated from the equation
i- - m
6t - m — - (Eq. 70)
max
where m is less than 1/2 and K is the largest value of K found during the
max
previous time step. A value of m = 1/4 has performed very well. The variable
value thus calculated is used only if it is less than 2 min.
The remainder of this report will deal with several experiments that have
been performed with the model. In the first of these, the atmosphere was
initially set at a constant potential temperature of 25° C, and the same tem-
perature was assigned to 0 and 6 . I , 1 4-, and a were all set equal to 0.
cr m s
Thus, there was no heating or cooling throughout the experiment and the stabil-
ity was neutral. The geostrophic wind was set equal to 20 m/s parallel to
the x-axis, and the wind was set equal to the geostrophic value at every level.
The model, of course, constrains the wind at the ground to 0, and (since
there is a momentum flux at the surface) a boundary layer grows rapidly upward.
The model was allowed to run for a period equivalent to about 2 pendulum days,
at which time the solution had nearly approached a steady state. The resulting
wind hodograph and the vertical distribution of K are shown in Figures 6 and
7. It is possible that the distribution of K is affected by the upper boundary.
Although the wind is not constrained at the top level, the momentum flux has
been arbitrarily made equal to 0 there. The small value of K, however,
results mainly from the fact that the shear is approaching 0 as one ascends to
greater heights. The spiral and the K distribution agree at least qualita-
tively with what is normally observed in neutral barotropic boundary layers.
91
-------
10 r
500
5 -
1000
1500
Figure 6. Spiral wind hodograph for a steady-state neutral simulation.
1500
1000
5OO
10
Km (m*s')
15
Figure 7. Vertical distribution of K for a steady-state neutral simulation.
92
-------
The second experiment attempted to predict the evolution of wind and tem-
perature distributions observed at O'Neill, Nebraska during the Great Plains
Field Observation Program in 1953. The data used for the initial condition
and for the verification were from the 1st, 3rd, 6th, and 7th general observa-
tion periods. A composite formed from the average of the four observations was
used. Averages of U, V, and 9 were tabulated from the surface up to 2000 m
above the surface, at 2-hour intervals from 1835 CST to 0635 CST. The advec-
tive terms in the equations were all set equal to 0, and the ground slab
characteristics were determined from the published measurements (Lettau and
Davidson [1957]). The geostrophic wind was held fixed during the period of
integration. The components adopted at 1500 m were those estimated by Lettau
and Davidson, but the geostrophic magnitude at the surface had to be increased
from 10.8 to 14.5 m/s to obtain satisfactory agreement between predicted and
observed changes. A comparison of the observed and predicted wind speed and
temperature profiles is shown in Figure 8. Figure 9 shows a comparison of
the predicted ind observed values of "surface" air temperature 0 . The poor
performance of the measured geostrophic winds is a serious problem in the
prediction of the boundary layer evolution from observed conditions, just as
it has always rendered determination (in practice) of the vertical distri-
bution of stress and exchange coefficient by the geostrophic departure method
nearly impossible.
In the third experiment, the integration was begun at 1600 Local Apparent
Time with a neutral equilibrium wind distribution calculated from a simple
two-layer model. 6 and the potential temperature of all layers of the atmo-
g
sphere were initially set equal to 25° C. The values of other important para-
meters are given in Table 1. Solar heating was calculated at each time step
from the hour angle and declination of the sun using an atmospheric transmis-
sion coefficient of 0.8 and a surface albedo of 0.20. Although the solar zenith
angle is increasing, the surface heating is still sufficiently strong (initally)
to cause the entire layer to become unstable; there is an immediate and rapid
adjustment of all of the variables during the first hour or so. Thereafter,
all quantities are evolving slowly and smoothly. The results are shown for
three runs. Runs 1 and 2 differ only in the value of the roughness of the
surface; Runs 2 and 3 differ only in the geostrophic wind. Figures 10 and 11
93
-------
1900
1000
<9
X
900
19 20 29
POTENTIAL TEMPERATURE
19
20
WIND SPEED
-------
J
1-
S2
i
900
800
700
600
500
400
aoo
.
1
1
M
I
1
M
It
K
6;
'
1 I
, /
'/
[ m\
5 10 15
WIND SPEED (ms")
20
25
e (eo
30
Figure 10. Evolution of wind speed and potential temperature profiles for
conditions listed in Table 1; roughness parameter 0.1 m and
geostrophic wind speed 8.0 m/s.
90
5 10 15 20 25
WIND SPEED (ms"1) 6 (°C)
Figure 11. Evolution of wind speed and potential temperature profiles for
conditions listed in Table 1; roughness parameter 1.0 m and
geostrophic wind speed 8.0 m/s.
95
-------
TABLE 1. PARAMETERS AND INITIAL CONDITIONS USED IN THE THIRD EXPERIMENT
JL-
Downward longwave radiation I
(ly/min)
Roughness parameter z (m)
Geostrophic wind U (m/s)
V (m/s)
g
Substrate Temp. 6 (°C)
m
Latitude
Solar declination
Cg/Cp p (m)
Critical Richardson number
i (m)
1
0.42
0.10
8.0
0.0
20
45N
+10°
221
0.25
71
Run
2
0.42
1.0
8.0
0.0
20
45N
+10°
221
0.25
71
3
0.42
1.0
9.4
0.0
20
45N
+10°
221
0.25
71
show very clearly how an increase of the surface roughness from 0.1 to 1.0 m
increases the height of the layer of mixing during the night. There is also
a slight increase of wind at 0 hours at the upper levels due to the fact that,
with the larger roughness, the initial wind at these levels is more greatly
retarded from the geostrophic wind, and consequently the amplitude of the
inertia oscillation is increased. Comparison of Figures 12 and 13 shows a
still greater growth of the depth of the mixed layer caused, in this case,
by an increase of the geostrophic wind from 8.0 to 9.4 m/s. Figure 13 shows
an example of the vector variation of the wind during the night at a level
near the top of the mixed layer; the vector wind hodograph at 03 hr is interest-
ing in that (rather typically) it bears a resemblance to a wind spiral, even
though the wind is not in equilibrium.
Figures 14, 15, and 16 show the course of the "surface" air temperature
and u^ during the night for each of the three runs. The most rapid cooling
96
-------
900
800
TOO
~ 900
E
~ ,00
(9
U 400
900
200
100
0
5 10 15
WIND SPEED (nwT')
20
23
6CC)
30
Figure 12. Evolution of wind speed and potential temperature profiles for
conditions listed in Table 1; roughness parameter 1.0 m and
geostrophic wind speed 9.4 m/s.
-51-
Figure 13. Wind vector hodograph calculated for Run 2 in Table 1. Dashed
line shows the wind vector variation with height at 0300 Local
Apparent Time. Solid line shows the wind vector variation with
time at a height of 275 m.
97
-------
30
_ 25
of
20
15
JO
— .50
.20
.10
rv/1
16 18 20 22 00 02
LOCAL APPARENT TIME
(HOURS)
04
06
06
Figure 14. 8 and u as function of time during the night for conditions
a *
listed in Table 1; roughness parameter 0.1 m and geostrophic wind
speed 8.0 m/s.
30
25
of
20
15
.60
— .90
.20
.10
U.
.T.
• * •
16 18 20 22 00 02
LOCAL APPARENT TIME
(HOURS)
04
06
08
Figure 15. 6 and u as function of time during the night for conditions
cL
listed in Table 1; roughness parameter 1.0 m and geostrophic wind
speed 8.0 m/s.
98
-------
i r
CO
00
UJ
M 2
0
UJ O)
CC. QC.
CM <
CM O
O
o
CM
CO
ID
CM
O
CVJ
O O O O
«> « ^t. rQ
w
c
o
•H
-P
•H
•o
o
o
4J
CO
O
Q)
cn
c
(0
•H
C O
c
•ri
8
•H
-P
§
•H
-P
O
4-t
tn
•H
CD
-P
fl
ft
0)
C
<1)
H
•9 •
fo CT>
CO
rd
* C 0)
^ -H a)
_ ft
0)
4J 13
W C
m -H -H
99
-------
occurs immediately after sunset, in accordance with what is most often actually
observed. The rapidity of cooling at this time is partly associated with the
fact that the upper level winds have not had time to accelerate, and (accord-
ingly) the wind shear and turbulent mixing in the ground-based nocturnal in-
version are still weak. Later, the increasing wind shear causes a higher level
of turbulence to prevail, thus lifting the inversion and maintaining a higher
u^ at the surface and a somewhat reduced surface cooling rate. In the case
of the second run, the particular combination of parameters results in the
occurrence of a remarkable series of alternations between calm and turbulent
episodes. During the calm episodes, the "surface" air temperature decays to
the ground temperature. Initially, the result is to increase the Richardson
number aloft and thus to reduce the downward flow of heat and momentum. At
the same time, however, the dynamical terms are increasing the shear aloft,
thus tending to reduce the Richardson number. Because the latter process has
a time scale that is controlled by the Coriolis parameter, however, the onset
of the subcritical Richardson number is delayed. When it does happen, there
is a catastrophic breakdown of several levels almost simultaneously. The
event is followed by several minutes of warmer temperatures and gusty winds
at the surface. Such turbulent episodes were first observed and discussed by
Durst (1933). A well observed case has also been given by Gifford (1952). It
is a curious fact that, in spite of the increased roughness of the second run,
the minimum "surface" air temperature was less than in the first run; the aver-
age "surface" air temperature, however, was increased. In the case of the
third run, the higher geostrophic wind did not permit the wind to become com-
pletely calm at the surface. Once the higher wind shear aloft became estab-
lished, the surface winds remained fairly strong and gusty throughout the night,
and the minimum "surface" air temperature stayed quite high.
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104
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APPENDIX: SOIL SLAB MODEL
Statement of Problem
Given a soil slab of uniform properties exposed to a periodic input of
heat from above given by
4
H = (I + 14- - a9 - H ) = H cos cot ._ _n.
v S go (Eq. Al)
and in thermal contact with a heat reservoir at constant temperature 8 below,
m
we seek to determine the values of C and K in the equation
g m
36
C -s-2- = H - K C (9 - 0 )
g 3t o m g g m
that will enable the amplitude and phase of the slab temperature 0 to be
identical to the surface temperature of a real soil layer of uniform thermal
conductivity X and heat capacity per unit volume C . The outcome of this
5
analysis is (1) to show that the slab model is capable of providing a ground
surface temperature that imitates the real soil temperature, and (2) to
enable C and K to be calculated from u, A, and C .
g m s
Actual Temperature Distribution Within the Soil
Let T represent the temperature as function of depth -z and time t in
the soil layer. The Fourier heat equation governing the temperature distri-
bution is
9T _ 1_ 82T
3t " C ,2 (Eq. A3)
s oz
We require that at infinite depth the temperature approaches a constant value
T and that at z = O the heat flux is continuous. Thus we have
J\m -^
(X ^) = H cosut (Eq> M)
z=0
105
-------
The trial solution
/\ Q
T = T + TepZcos (ut - 3 + vz) (Eq> A5)
satisfies the boundary condition at infinite depth and also satisfies Equation
A3 provided
V = 3 = /Cgu> / 2A (Eq> A6)
Negative values of v and 3 are ruled out by the boundary conditions.
Substitution of the solution into Equation A4 places an additional con-
straint on the solution, which may be expressed in the form
A /A
H cosoot = ATB [(sin $ + cos 8) cosut + (sin & - cos 6) sin ut] (Eq. A7)
Since the coefficient of sin u>t must be identically 0, we have
6 = TT/ 4
Thus the surface temperature peaks at 1/8 period after the peak heat flux at
the surface. Equating coefficients of the cos u>t term gives
T = H / /AC W (Eq. A8)
s
Requirements of the Slab Model
The slab model is not intended to provide any information about the
temperature or heat flux distribution within the soil. Its only function is
to present the atmosphere with a surface temperature 0 that has the same
amplitude and phase as the actual soil temperature at z = O. Accordingly, it
is required that
/^
H
9g = To + /A Cs W C°S (Wt ~ f} (Eq. A9)
106
-------
be a solution of Equation A2. Substitution gives
/\
H ^
-C sin (ut - -r) = H cos tot + K C 6 - K C T
8 A^T 4 m gm m g o
s
(Eq. A10)
H ^
-C K /TT; - COS (U)t - 7- )
g m /AC 0) 4
which must be satisfied for all values of t. One requirement is
em = TO (Eq.
With this substitution and further expansion one obtains the equation
C u> „ KmC H
. (coswt - sinwt) = Hcoscot - — 8 • (cosut + sinwt) (Eq. A12)
/2A C oT /2,C w"
s As
which is satisfied identically only if
K = 0)
m (Eq. A13)
and if
= /AC/2co _ w. ..
g s (Eq. A14)
In reality, the forcing function H is not a simple cosine function (as
assumed here). However — since the equations are linear — we can represent the
functions H and T as a Fourier series, each mode of which satisfies the above
sets of equations individually, with an appropriate multiple of oj in place of
the fundamental. These considerations suggest that it is possible to choose
C and K in such a way as to give a perfect representation of any single
g m
desired mode, but no single values of C and K can give a perfect realization
g m
of all of the modes together. Presumably, the best performance of the slab
model results if one uses an average value of these parameters weighted in
proportion to the energy contained in the various modes comprising the forcing
107
-------
function. The solar insolation at the time of the equinoxes, normalized by
r
its midday value, is well represented by the series
R
^ = .3183 + .5000 cos cot + .2122 cos 2wt - .0424 cos 4cot
K
noon
+ .0182 cos 6 cot - .0101 cos 8 ut + .0064 cos 10 cot -...
When weighted by the squares of the coefficients of each harmonic, the values
of the parameters become
K =1.18 00
m
C = 0.95 AC / 2 u
g s
(Eq. A15)
Since these so closely approximate the values applicable to a pure cosine
forcing, and since the actual heating function is somewhat purer than that of
the insolation, it is to be expected that the slab model will imitate the
actual soil surface with acceptable accuracy.
108
-------
THE EFFECTS OF MIXING-HEIGHT VARIABILITY
ON AIR-QUALITY SIMULATION MODELS
Henk Tennekes
Department of Aerospace Engineering
INTRODUCTION
The structural properties of the atmospheric boundary layer and the
turbulent motion occurring within it determine the vertical transport of
momentum, heat, moisture, and pollutants to and from the earth's surface.
It is evident that the height to which turbulent mixing processes extend is
a key parameter in air pollution meteorology. For example, seasonally aver-
aged values of pollutant concentration depend on seasonal averages of the mix-
ing height over the geographical region concerned. Some of the parameters in-
volved in the computation of plume dispersion from "point" or "line" sources
are related to the mixing height; convective activity in a shallow boundary
layer, for example, is less vigorous than that in a deep one with the same
surface heat flux. Since the dispersion of pollutants is controlled not only
by diffusion but also by advection, air-quality simulation models depend on
accurate forecasts of the flow field. Mixing-height parameterization is an
integral part of the boundary-layer and cumulus-cloud parameterization schemes
needed in advanced numerical weather predictions on regional and synoptic
scales (Deardorff, 1972; Arakawa and Schubert, 1974).
The mixing height does not play a major role in the prediction of pollut-
ant dispersion over relatively short distances. As long as the pollution
from surface sources and sources in the lower part of the boundary layer does
not reach the inversion that normally caps the atmospheric boundary layer, the
influence of the "lid" on vertical dispersion is relatively small. A rough
estimate (Deardorff and Willis, 1975) for the horizontal distance covered be-
109
-------
before pollution from a ground source first reaches the top of a convective
boundary layer is:
x = hUaw (Eq. 1)
where h is the mixing height, U is the wind speed, and a is the vertically
w
averaged standard deviation of the vertical velocity fluctuations. If, for
example, a =0.2 m/s, U = 2 m/s, and h = 1000 m, then pollutants released at
vf
the surface travel 10 km downwind before they reach the top of the boundary
layer, and probably another 10 or 20 km are required before the vertical
concentration distribution becomes reasonably uniform. It is, therefore, not
surprising that the air-quality index in urban environments is rather poorly
correlated with the mixing height (Eschenroeder, 1975). However, that con-
clusion is valid only for average conditions over a medium-size urban area; it
cannot, and should not, be applied to the regional scale or to stagnation
episodes on the urban scale.
The mixing height limits vertical dispersion over distances that are long
compared to those given by Equation 1. The effects of an inversion lid can be
studied best if an air-quality simulation model that includes the mixing height
as an explicit variable is used. A box model for pollution in a large urban
area with distributed sources appears to be ideally suited for this purpose.
This paper explores some of the problems associated with the effects of mixing-
height variations on box models. It is not our intent to replace the detailed
numerical computations that are needed for accurate predictions of pollutant
concentrations. Instead, we wish to demonstrate the impact of mixing-height
variations on average concentration levels and on the concentration levels
encountered in severe pollution episodes. In this way, we illustrate the
sensitivity of air quality to the probability distribution of the mixing height
in any given geographical area, and call attention to some of the statistical
measurement problems caused by the interplay among diurnal, synoptic, and
seasonal variability of the parameters involved.
110
-------
INVERSION- RISE PARAMETERIZATION
We begin our analysis with a brief review of the state of the art in
inversion-rise parameterization. For many years, maximum-temperature forecasts
and forecasts of the mixing height at sunset have been based on the premise
that the total amount of heat added to the atmosphere between sunrise and sun-
set modifies the vertical temperature distribution in such a way that heat is
conserved while the lapse rate in the daytime mixed layer remains adiabatic.
Writing this conservation statement in the form of a differential equation, we
obtain
1 AV,2 H
JL dh _ _ s
2 Y dt " pc (Eq. 2)
where y is the lapse rate of potential temperature in stable air just above
the top of the mixed layer (z > h) , and H is the surface heat flux. The cor-
S
responding potential temperature in the mixed layer is taken to be equal to
the potential temperature of the sunrise (or midnight) sounding at the height
z = h.
The current generation of inversion-rise models takes the effect of en-
trainment at the top of the boundary layer into account (Betts, 1973; Carson,
1973; Deardorff, 1972; Tennekes, 1973). In convective conditions, the dif-
ferential equation governing inversion rise is surprisingly similar to Equa-
tion 2:
2 H
Y _ = Q , __
2 T dt U + e; c (Eq. 3)
P
Here, e depends on the efficiency of the conversion of turbulent kinetic energy
to potential energy. The most commonly used value of e is 0.4, corresponding
to conditions in which the downward heat flux at the inversion base is 20% of
the upward heat flux from the surface. Note that e = 0.4 corresponds to a 40%
increase in the rate of growth <
effect on h itself is only 20%.
2
increase in the rate of growth of h over that predicted by Equation 2; the
111
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Expressions such as Equation 3 must be modified if there is substantial
mechanical production of turbulent kinetic energy (as in high wind conditions);
estimates for the required correction term have appeared in the literature
(e.g., Tennekes, 1973). Corrections are also needed in case large-scale sub-
sidence and/or advection of horizontal changes in mixing height are present.
These effects require information from numerical prognostic models of the
three-dimensional low-level wind field (Anthes, 1978). The correction terms
in the mixing-height parameterization, however, are quite straightforward
(Deardorff, 1972); they need not concern us at this point.
In the present context, the accuracy and reliability of forecasts based
on Equation 3 and its generalizations are of great importance. The accumulated
experience of the last 3 years indicates that current inversion-rise models
perform exceedingly well. They predict the daytime temperature cycle to within
2° C, and the mixing height at noon to within 100 m or so (Tennekes and
Van Ulden, 1974); they are used for regional pollution-index forecasts for the
greater London area (Pasquill, 1974); and they are supported by field and
laboratory data from a wide variety of sources. Detailed and comprehensive
research by Zeman (1975) and Stull (1975) suggests that the current generation
of inversion-rise parameterization schemes successfully handles all of the
conditions that occur in practice. Thus, for most practical purposes the
inversion-rise problem may be regarded as solved. We can move on to the next
problem.
As a preliminary, we need to derive the steady-state advective counter-
part of Equation 3. If an urban heat island is surrounded by rural areas with
negligible heat flux, and if the wind speed over the region is U, Equation 3
becomes
2 Y U 3x pc (Eq. 4)
We have used e = 0.4. The mixing height thus increases as
h(x) = (2.8H /pc YU)1/2x1/2 (Eq. 5)
s p
112
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where x is the distance from the upwind edge of the city. If the diameter of
the urban area is D, the mixing height at the downwind edge of the heat island
is given by
£.on I/O 1/9
h(D) = fe^ °
P
This may be regarded as an updated version of Summers' (1965) heat-island
formula. As the analysis in the next section will show, steady-state
expressions (such as Equation 6) are virtually useless in stagnation episodes
because the dynamic response of the pollutant concentration under the inver-
sion lid is quite sluggish if the wind speed is small.
If the population density of urban areas is roughly independent of the
2
diameter D, then the total population N is proportional to D . The mixing
height over urban heat islands thus is proportional to the 1/4 power of the
population, clearly a rather weak dependence (Panofsky, 1976). Since the
mixing height tends to increase with city size, the parameter h/D occurring in
the steady-state solution of box-model equations will tend to depend only
weakly on size. This factor contributes to the poor correlation between pol-
lutant concentration and mixing height over most cities (Gifford and Hanna,
1973). However, as Gifford has pointed out repeatedly (e.g., Gifford, 1973),
the pvinsipal reason that urban ground-level concentration is poorly corre-
lated with mixing height is that urban receptors perceive mainly nearby sources.
The effective diffusion height of pollution from sources that are capable of
influencing a particular receptor is almost always small compared to the mixing
height.
THE DYNAMICS OF BOX MODELS
Box models are formulated in such a way that the effective height of ver-
tical pollutant dispersion appears as an explicit parameter. We have to make
a clear distinction between the two major types of urban box models. If the
area concerned is not too large, if it is not walled in by topographical ob-
structions, and if the wind speed is not too small, then vertical dispersion
113
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is limited by the finite diffusivity of the turbulence in the mixed layer. In
that case, pollutants are unlikely to disperse to the top of the mixed layer
before they are carried out of the box by the wind. The mixing height conse-
quently plays no explicit role in the formulation of the problem under these
conditions. It does play an implicit role, however, because the average tur-
bulent diffusivity in the boundary layer depends on the integral scale of the
turbulent motion (which, obviously, is larger if the boundary layer is thick)
and on the turbulence intensity (which, at least in convective conditions, is
also a function of the boundary-layer thickness). In these cases, the proper
procedure for formulating the problem is to define the upper boundary of the
box as a sloping surface corresponding to the effective height of turbulent
dispersion (Gifford and Hanna, 1973; Hanna, 1975). For a variety of reasons,
some of which have been mentioned above, the effective dispersion height at
the downwind edge of the area concerned tends to be roughly proportional to
its diameter, so that it is not useful to make a major issue of possible scale
effects (Gifford and Hanna, 1973; Gifford, 1973).
Diffusion-limited box models become rather cumbersome if time-dependent
phenomena have to be studied, because the upper boundary then has to be re-
garded as a function both of position and of time. Fortunately, as we shall
see shortly, the concentration in a diffusion-limited box model tends to be
very close to its moving equilibrium value (except in stagnation conditions),
so that a quasi-steady analysis often suffices.
The other major class of box models is that in which vertical dispersion
is limited by an inversion lid. If the wind speed is low, if the area con-
cerned is very large, or if the ventilation is limited by mountains, we may
assume that turbulent dispersion is so rapid that the vertical concentration
distribution becomes roughly uniform a short distance from the upwind side of
the city. Inversion-limited box models thus are appropriate for the study of
severe air pollution episodes over very large urban areas. This is the problem
we investigate below.
We consider a shallow box over a large urban area covered by uniformly
distributed surface sources of pollution. The source strength per unit area
114
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2
is denoted by q (kg/m s). The box is square; the length of each side is D.
The upper boundary of the box is at the current mixing height h; we take the
inversion lid to be independent of position. The wind speed is U; since the
ventilation cross section is hD, the pollutant flux out of the downwind side
of the box is cUhD (c is the average pollutant concentration [kg/m ] in the
box). The average pollutant concentration left above the current mixing
height by the mixed layer from previous days is denoted by c^. Clearly, c^ =
0 if there is adequate ventilation, or if the current mixing height happens to
be larger than the sunset values of the last few days. We assume that there
is sufficient mixing within the box to make the concentration inside essen-
tially uniform. Since D is large, we ignore turbulent diffusion out of the
lateral sides of the box. We also ignore the background concentration outside
the box, because it is advected through the area without change and thus adds
nothing to the dynamics of the problem.
The conservation equation for the total amount of pollution inside the
box now may be written as
—• (chD2) = qD2 - UDhc + c*
Dividing this by D , we obtain
dch Uhc ^ dh
~ZF = q " ~ + °* aF
or
dc _ Uhc , N dh
^dt q D U C*; dt (Eq. 8)
These equations are not valid if dh/dt < 0 for causes other than subsidence,
because the pollutants deposited at higher levels by a daytime mixed layer are
left there when the mixing height collapses around sunset.
Equation 8 is quite similar to the one proposed by Lettau (1970). The
principal difference is that the last term of Equation 8 is a parameterized
115
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version of the corresponding term in Lettau's equation. The term at issue^is
the turbulent pollutant flux w'c' caused by entrainment at the inversion lid.
The parameterization involved in the last term of Equation 8 is identical to
that employed for the turbulent fluxes of heat, moisture, and momentum in
inversion-rise models. A straightforward application of those ideas gives
<>z,h -
-------
ch - crhr = (^ ~ crhr) { 1 - exp (- ~) } (Eq. 10)
where t is counted from sunrise. At sunset (subscript s) that same day, Equa-
tion 10 gives
Cshs - crhr = (^T - CrV { l ~ ex? (~ §> } (Eq' U)
where T is 24 hours (we are taking days and nights of equal duration). These
equations demonstrate the effect of the ventilation time D/U: the exponential
decay in Equations 10 and 11 is measured in ventilation units. The equations
also suggest that there is little advantage to be gained by incorporating the
inversion-rise term into the definition of "flushing frequency," as proposed
by Lettau (1970). We note further that the integration cannot be continued
beyond sunset because the mixing height rapidly decreases as the surface heat
flux changes sign, so that the upper boundary of the box has to be relocated
at a height appropriate to nighttime conditions.
The behavior of Equation 1 also shows under what kind of conditions the
pollutant conservation equation may be approximated by its steady-state form.
If UT/2D » 1, so that it takes much less than 12 hours to ventilate the urban
area, the effects of transients in the variables involved are damped out very
quickly. The actual concentration at all times, then, is very close to its
moving equilibrium value:
c = qD/hU (Eq. 12)
The quasi-steady approximation is also valid if the effective turbulent dis-
persion height is used instead of the mixing height; the only important restric-
tion is that UT/2D must be large. Clearly, this condition is satisfied more
easily in a diffusion-limited box model. Formally, there is also a restric-
tion on high-frequency components or transients in the emission rate q; in
practice, however, that is of no real concern.
117
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Unfortunately, Equation 12 cannot be used in stagnation episodes with
severe pollution accumulation. In fact, the nature of Equations 8 and 10 and
the observation that yesterday's pollution carries over into today if UT/2D < 1
suggest that a stagnation episode may be defined as one in which UT/2D < 1
(that is, one in which the region concerned cannot be fully ventilated in half
a day). In a stagnation episode, it will be necessary to use differential
equations (such as Equation 8) because the ventilation is so slow that tem-
porary storage effects become of key importance.
A deeper appreciation for these effects can be obtained by examining the
frequency response of a box with fixed lid (h = H), constant wind speed, and
sinusoidal forcing. The appropriate equation is
H^df" + (D)c^ = qi + q2 Sln "* (Eq> 13)
The phase lag of the oscillations in c is then given by
tan = OD/U (Eq> 14)
and the amplitude of the concentration oscillations is attenuated relative to
that of the emission cycle by a factor A, given by
A = (1 + (WD/U)2} ^ (Eq. 15)
Examples are given in Tables 1 and 2.
TABLE 1. AMPLITUDE ATTENUATION AND PHASE LAG FOR A
DIURNAL EMISSION CYCLE (01 = 2ir/T)
D/U (hours) 24.0 12.0 6.0 3.0
(hours) 5.5 4.8 3.8 2.5
A(n/d) 0.16 0.30 0.54 0.79
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TABLE 2. AMPLITUDE ATTENUATION AND PHASE LAG FOR A
SEMIDIURNAL EMISSION CYCLE (OJ = 47T/T)
D/U (hours)
(hours)
A(n/d)
24.0
2.9
0.08
12.0
2.7
0.16
6.0
2.4
0.30
3.0
1.9
0.54
These data show clearly that the amplitude of the concentration fluctua-
tions is severely attenuated if the ventilation time D/U is large. The storage
of pollution inside the box in cases with poor ventilation almost completely
suppresses diurnal and semidiurnal concentration fluctuation. The amplitude
of the concentration cycle is cut roughly in half (compared to that of the
emission cycle) when the latter is diurnal and the ventilation time is 6 hours,
or when it is semidiurnal and the ventilation time is 3 hours. The correspond-
ing phase lags are about 4 hours and 2 hours, respectively.
A typical emission pattern would be one with a diurnal cycle having a
maximum at noon and a semidiurnal cycle that peaks at sunrise and sunset.
This gives an emission curve that peaks at the morning and afternoon rush
hours (Figure 1). The corresponding concentration curves, for a ventilation
time D/U = 6 hours, show that the effect of the morning rush hour is partially
canceled by the phase delay in the diurnal concentration cycle, causing a
shallow concentration maximum between 10 and 11 a.m. The highest concentration
is reached in the early evening, when the concentration increase due to the
afternoon rush traffic adds to the delayed maximum in a diurnal cycle. A
similar example is given by Lettau (1970). If a complicated emission cycle
is expanded in a Fourier series, Equations 14 and 15 can be used to calculate
the Fourier coefficients of the concentration cycle. Such an extension is
straightforward but of limited practical use, because the wind speed is un-
likely to remain constant over extended periods of time.
The response time of the box model equals the ventilation time, and the
latter is large in stagnation episodes. This means that emission control
measures (such as the reduction of S0_ emissions from fossil-fuel-burning
119
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0 2 4 6 8 10 12 14 16 18 20 22 24
EMISSION COMPONENT
"SEMIDIURNAL COMPONENT
q(t)
SEMIDIURNAL DIURNAL CONCENTRATK
CONCENTRATION COMPONENT ( COMPONENT
CONCENTRATION (AVERAGE REMOVED)
8
10 12 14
HOURS
16 18 20 22 24
Figure 1. Typical 1-day emission curve and associated concentration curves.
120
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power plants) must antiai-pate a stagnation episode by a time interval compar-
able to the critical ventilation time. If the ventilation time of a large
urbanized region gradually increases to several days as the center of a high-
pressure system approaches, emissions have to be decreased before the ventila-
tion time becomes too large. This conclusion is obvious enough; the point is
that the box model equation demonstrates it so clearly.
If, for example, U decreases as
U = UQ(1 - t/S) (Eq. 16)
then the solution of the homogeneous box model equation under a fixed lid is,
for t < S,
U t
C = CQ{ exp - ~_ (1 - ± |)} (Eq. 17)
Let us apply this to a case in which emissions are cut back to 0 when the
ventilation time becomes 12 hours. We take S = 48 hours. Twenty-four hours
after the control measures U t/D = 2, but the factor in the parentheses be-
o
comes 3/4, so that the concentration decay factor c/c is not 0.13 but 0.22.
o
Forty-eight hours later, the decay factor is not 0.02 but 0.13. If the emis-
sion control during the same episodes is delayed by 12 hours, the ventilation
time at the time of cutback has increased to 16 hours and S has decreased to
36 hours. The concentration decay factor at the time the wind speed becomes
0 is then 0.32. since c was allowed to increase by a factor of approximately
16/12, the final concentration is 0.43 of the concentration corresponding to
D/U = 12 hours. The 12-hour control delay thus increases the final concentra-
tion (at t = S) by a factor of 3.
Again, in an episode with S = 48 hours and D/U = 24 hours, the concentra-
o
tion decay factor is 0.46 after 24 hours and 0.37 after 48 hours (when U be-
comes 0). Obviously, control measures have to be taken well before the
ventilation time becomes too large. In this last case, the concentration at
t = S is comparable to that in a steady-state situation with D/U = 12 hours.
121
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We have used S = 48 hours as a typical example. If the wind speed de-
creases more rapidly, early emission control measures are even more urgent.
It goes without saying that a thorough study of box models with parametric
excitation in wind speed and mixing height would be of substantial interest.
The examples given above support our definition of stagnation episodes
as periods in which the ventilation time exceeds 12 hours. For a town 6 mi
in diameter, the corresponding wind speed is 0.5 mi/hour; for a metropolis
with a diameter of 24 mi, the critical wind speed is 2 mi/hour. If the prob-
ability distribution of the wind speed is known, it is not difficult to com-
pute the frequency of occurrence of stagnation episodes. The box model gives
simple but profound answers to urban air pollution problems. The model cannot
become confused by details of source distributions and turbulent dispersion
patterns, for it does not allow those complications to enter into the govern-
ing equation. Of course, the model can handle only one class of problems; it
is of no use, for example, when the impact of a strong isolated source (such
as a power plant or energy park) must be evaluated.
The dominant role of the advective time constant in the dynamic response
of the box model resembles that in hot-wire anemometry (Hinze, 1975) and in
the theory of the Eulerian time spectrum of turbulence (Tennekes, 1975).
We now turn to the qualitative study of the effects of inversion rise.
This is accomplished conveniently by writing Equation 8 in the form:
h dc _ q Uh ° " °* dh
c~dT~c~D~ c dt (Eq' 18)
One effective way to store pollution without rapidly increasing concentration
levels is to increase the height of the mixed layer. If c^ = 0, this makes
clean air available for dilution at a speed dh/dt; this may compare favorably
with the equivalent ventilation speed Uh/D, especially if the size D of the
area concerned is very large. Unfortunately, this mechanism, which is poten-
tially quite effective, is virtually useless in stagnation episodes, because
after one or two stagnation days the air above the mixed layer is just as
polluted as the air inside it.
122
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It is also instructive to nondimensionalize Equation 18 with the period
T/2 of the semidiurnal cycle. This yields
_!£*£.= qT _ EE L^--~ 1_ dJi
2c dt 2hc 2D " c 2h dt (Eq> 19)
The left-hand side of Equation 19 is the fractional increase in c in diurnal
time units. The factor (T/2h) dh/dt is of order 1 if the mixing height
increases by 1000 m from sunrise to sunset, with an average value of 500 m.
These numbers are typical of summer conditions; they demonstrate that the con-
tribution made by inversion rise can be comparable to that of a wind speed
barely capable of ventilating the box in half a day. For an urban area with
D = 40 km, the corresponding wind speed is 1 m/s. The effects of dh/dt on
episodes with relatively poor ventilation thus should not be neglected, but we
must remember that recapture of polluted air from above does tend to diminish
the beneficial effects of inversion rise in stagnation episodes lasting several
days.
Since the wind speed is a function of time, and since the response of the
box model depends on the wind speed, it seems natural to normalize the time t
with the current flushing frequency. The normalized clock then runs slow if
the flushing frequency is small, and fast when the latter is large. Lettau
(1970) approaches this issue by writing Equation 8 as
dc q
I? = h " V is defined by
U ^ ° ~ C* dh
Wf = D + — dE (Eq' 21)
Here we have used the parameterized form of Lettau's flux term given by Equa-
tion 9. Entrainment at the rising inversion lid increases the effective flush-
ing frequency. However, we prefer not to follow Lettau1s example on this point,
because the entrainment term in Equation 21 contains a combination of dependent
and independent variables. The rate at which normalized time advances then is
a function of the unknown concentration; that seems a major disadvantage.
123
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We propose instead to use the product = ch as the principal output-
variable (h, of course, is an input variable, but that should cause us no con
fusion) . The value of corresponding to the quasi-equilibrium solution of
the box-model equation will be denoted by :
= (ch) = qD/U (Eq. 22)
S 5
This quasi- steady solution is a simple algebraic function of the input vari-
ables. The time increment dz on the normalized clock is defined by
dT = (U/D)dt (Eq. 23)
Rewriting Equation 7 in terms of (the value of ch corresponding to an exact balance
S
between emission and ventilation at all times) serves as an equivalent source
strength. The normalized recapture term, c^ dh/dt, is best interpreted as an
additional source term; it is 0 most of the time (whenever TU/2D > 1) but
makes occasional brief contributions during stagnation episodes.
If c^ happens to be 0, Equation 24 can be integrated (Lettau, 1970) to
yield
= 4>Q exp(-T) + /exp(T' - T)(|)s(TI)dTI (Eq. 25)
Since the mixing height is discontinuous around sunset, the maximum permissible
integration period is from just after sunset today to sunset tomorrow (we take
h to have a small constant value during the night) . This restriction severely
limits the range of applications of a formal solution such as Equation 25.
Still, this equation very clearly demonstrates the memory effects inherent in
the box model. The effects of the initial condition are completely negli-
gible after T = 3, and the effective integration interval in the second term
124
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of Equation 25 is a few units of normalized time before the current time T.
The box has a "fading memory;" in normalized time units, the time constant of
the memory equals 1.
More detailed and comprehensive studies of the dynamics and statistics
of box models probably should include an analysis of the practical benefits
of working with normalized time. In this context, it seems worthwhile to
discuss some of the major properties of the ventilation clock. The time x on
that clock is obtained by integrating Equation 26, yielding
T = i/T U dt = ~ + 5/Vdt (Eq. 26)
o o
We have written the wind speed U as the sum of its long-term average value U
and the fluctuations u' . The latter represent diurnal and synoptic oscilla-
tions; turbulent wind fluctuations have been removed in the way the box model
was defined (the ventilation speed is the average wind speed over the ventila-
tion cross section) .
Equation 26 contains the integral of a stationary random function with
0 mean. The second term of Equation 26 thus becomes a Gaussian random vari-
able after sufficient time has passed (Lumley, 1970; Tennekes and Lumley,
1972). For time values which are large in comparison to 1 (in ventilation
units), Equation 26 behaves as
T = +- (2ttf2f(t) (Eq. 27)
Here, f (t) is a normalized Gaussian variable, a is the standard deviation of
u
the wind speed fluctuations, and T is the time integral scale associated with
2
the correlation function pu(t - t1) = u' (t)u' (t - t')/a (Tennekes and Lumley,
1972, Chapter 6). In the long run, the ventilation clock thus runs at a rate
proportional to the mean wind (seasonal or longer average) , but it has Gaussian
1/2
"jitter" whose amplitude increases as t as time proceeds. This property
may be useful in studies of the climatology of box models; we have not yet
explored that possibility.
125
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Another interesting property of T is that it increases quite slowly
during stagnation episodes. The time series of the concentration c, taken as
a function of normalized time, thus tends to have a brief spike of very large
amplitude whenever stagnation conditions occur. Such an "intermittent" time
series provides opportunities for certain simplifications; a related example
is given in the next section.
Expressions such as Equations 24 and 25 provide a basis for studies of the
interactions between the dynamical and statistical (climatological) features
of box models. For example, the memory integral in Equation 25 presumably
reduces the variance of relative to that of the equilibrium solution d> ,
s
and increases the integral scale (correlation period) of the former compared
to that of the latter. Further exploration of these issues, however, would
carry us far beyond the scope of this paper. This does not mean that the
climatology of box models is of no importance. Quite on the contrary, a close
look at some key statistical features of box models is of considerable interest.
That is the subject of the next section.
Let us recapitulate. Through a variety of examples we have illustrated
the potential of a box model that takes inversion rise and response times into
account. We have seen that the use of a differential equation is essential
during stagnation episodes because the model responds so slowly; control mea-
sures during stagnation episodes thus cannot be based on the quasi-steady
solution (Equation 12). It appears that a detailed comparison of the model
with actual air pollution episodes over urban areas would be a worthwhile
exercise. The box model, of course, would be used in conjunction with the
inversion-rise models discussed in "Inversion-Rise Parameterization" (above).
THE CLIMATOLOGY OF BOX MODELS
Over sufficiently long time periods, a box model has to maintain a balance
between emission and ventilation. This is true both for diffusion-limited box
models and for inversion-limited ones. The statistical analysis presented in
this section applies to both types, though details of the interpretation differ.
126
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We begin our analysis by considering an urban area in which UT/2D » 1
most of the time during the course of a year. To be specific, we assume that
stagnation episodes (in which the equilibrium solution of the box model equa-
tion is not valid) constitute a set of measure 0. This implies that stagna-
tion episodes do not have a significant impact on the statistics of the pro-
blem. We further assume that the effects of sunset discontinuities in the
effective ventilation height h are negligible. With these restrictions (which
we shall relax shortly), we can write
qD = Uhc (Eq. 28)
and assume it will be valid almost all of the time. As we have seen before,
the applicability of the equilibrium solution is not restricted to inversion-
limited box models. Indeed, the height h in Equation 28 may be taken as the
mixing height or as the effective dispersion height, whichever is smaller. A
box model for any given area is diffusion-limited most of the time, but inver-
sion-limited during stagnation episodes. If the urbanized area is large, or
happens to have a high incidence of low inversion lids, the fraction of time
spent in the inversion mode is relatively large. That issue is discussed in
"Mode-Switching Problems" (below).
We now take a long-term average of Equation 28. This could be a monthly
average, a seasonal or annual mean, or a formal climatological average. The
only restriction is that the averaging time should be long enough to produce
stable statistics. We obtain
qD = Uhc
... (Eq. 29)
= U h c + U h1 c' + h u'c1 + c u'h1 + c'h'u'
In order to simplify the discussion, we ignore the triple correlation in
Equation 29 and assume that mixing-height variations are uncorrelated with
wind speed variations. Both of these assumptions should be experimentally
tested. The parameters h, c, and U all are positive, and are likely to have
strongly skewed probability distributions. Triple correlations, therefore,
127
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may not be negligible. The wind speed spectrum tends to have a peak corres-
ponding to the synoptic cycle (3-5 days), while the mixing-height variations
occur mostly on a diurnal cycle. This would make the correlation h'u' small.
However, in areas with sea or lake breezes, the wind speed does have a signi-
ficant diurnal cycle which would be well correlated with the mixing-height
variations. Furthermore, the effective dispersion height during diffusion-
limited periods depends on the Richardson number (stability category); the
correlation between the dispersion height and u, therefore, is not likely to
be negligible.
With the simplifying assumptions outlined above, Equation 29 becomes
qD = U h c + U h'c' + h u'c' (Eq. 30)
We introduce the correlation coefficients r, and r , which are defined by
h u
u'c' = -r a a ,
u u c (Eq.
Minus signs are used in order to make r > 0 and r > 0 (clearly, the two
il C
correlations involved are negative, because c1 > 0 if u1 < 0 or h' < 0) . In
order to facilitate comparison with the Gif ford-Hanna (1973) model, we define
the reference height h by
R
& = "V (Eq. 33)
Substituting Equations 31, 32, and 33 into Equation 29, we obtain
qD = UtLC = U h "c - Ur a o- - hr a a
K h h c u u c
This yields
hn a, a a a
^ = 1 - r -h-C - r -^ (Eq. 35)
h hhc uu"c"
128
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Let us apply this relation to diffusion-limited box models. The reference
height h implied in the Gifford-Hanna model is seen to be considerably smaller
R
than the mean vertical dispersion height at the downwind edge of the urban area
(which itself is smaller than the average mixing height in a diffusion-limited
box model). Obviously, one of the strong points of the Gifford-Hanna scheme
is that it avoids the issue of fluctuations and their correlations altogether
by simply using an empirically determined value of h /D, without bothering
R
about the actual statistics of h.
We have been unable to find reliable data on the correlation coefficients
and standard deviations occurring in Equation 35. The discussion by Eschen-
roeder (1975) suggests that r - 0.2 and r - 0.6. We anticipate that a /c -I,
m u c
while a,/h and a /u are likely to be about 0.5. In that case, h/h is about
h u R
0.6; clearly, the effects are significant. It would be of considerable interest
to collect such statistical data, if only to determine if there are any syste-
matic trends depending on the size D of the urban area. For example, it is
conceivable that a /c decreases with city size; if that were the case, h
C -t\
would tend to increase with D, making the constant K = D/h in the Gifford-
Hanna expression less dependent on the scale of the urban area. Very large
urban areas are always somewhat polluted; this tends to increase c without a
corresponding increase in a .
c
The value of K = D/h proposed by Gifford and Hanna (1973) is 225 for
R
cities ranging from about 15 to 48 km in diameter. The corresponding values
of h are in the vicinity of 100 m. Clearly, this is much smaller than the
R
actual mean mixing height over the urban areas concerned. This confirms the
hypothesis that urban box models are, statistically speaking, almost always
(but not in stagnation episodes 1) diffusion-limited. Climatological applica-
tions of a ventilation index or "flushing factor" based on seasonal averages
of the mixing height thus make little sense for most cities.
The value of h corresponding to a large metropolis is about 200 m. The
X\.
numbers given above suggest that the corresponding average dispersion height
at the downwind side might be about 400 m. This value is fairly close to
typical mixing heights; it is, in fact, larger than the mixing height one
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might encounter on clear nights with little wind. The box model for a large
urbanized region thus operates fairly frequently in the inversion-limited
mode. This implies that climatological studies of urban box models will have
to deal with the problem of switching from one mode to the other. That pro-
blem is discussed in "Mode-Switching Problems" (below).
Since the mean concentration c in an urban area is adversely affected by
the correlations that occur in Equation 35, we must raise the issue of how
these correlations might be determined in an observational program. Clearly,
the measurement of a correlation requires spectral fidelity in the frequency
ranges that contribute most to the covariance. Now, h has a spectral peak at
the diurnal cycle, while much of the variance of U is associated with the
typical 3- or 4-day period of synoptic variability. This implies that the
variance of c has significant contributions at both these frequencies, and
suggests that a sampling interval of half a day might be sufficient to obtain
accurate values of the two correlation coefficients. It should be noted that
daily mean values are altogether inadequate, because they would correspond to
severe aliasing (Nyquist folding) of the spectra involved. The problem, how-
ever, is very much complicated by the fact that the ventilation height tends
to decrease rather abruptly around sunset each day. As the convective activity
in the afternoon's mixed layer dies out because the surface heat flux changes
sign, the ventilation height reestablishes itself at a much smaller value,
appropriate to nighttime conditions. The pollutants brought to higher levels
by the mixed layer in the afternoon are left behind and may contribute to
tomorrow's pollution if the ventilation time is large. Since all of this
typically happens within the 1-hour period preceding sunset, it adds high-
frequency components to the time series of the dispersion height. Often,
these changes in h are accompanied by an equally rapid change in U, because
the downward transport of geostrophic momentum also virtually ceases when
convective mixing stops as day changes into night.
Similar considerations apply to the sampling rate needed to determine the
statistics of q. The emission rate obviously exhibits diurnal and seasonal
cycles, but it also has significant variance at periods corresponding to the
length of the morning and afternoon traffic rush hours. Clearly, hourly mean
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values would be most suitable for an analysis of this sort. Since the 2 hours
around sunset are crucial, it might be sufficient to sample every hour around
sunset, and every 3 or 4 hours at all other times. The time series then can
be filled in by linear interpolation for all hours during which no data are
taken. Still, time series consisting of sets of hourly means would be prefer-
able. It is evident that this leads to a potentially valuable application of
acoustic sounders: remote sensing of h(t) by acoustic radar is relatively
cheap and convenient; in conjunction with hourly observations of c and U,
this would yield extremely valuable information for air pollution climatology.
We note, in passing, that adequate time series of all parameters involved
also would provide an opportunity to study the statistical relations between
the variances of those parameters. We have refrained from writing a formal
equation for a because of the large number of correlations involved, but it
c
is clear in principle that adequate time series would allow a thorough study
of the effects of the variability of h and U on the variability of c. The
issues involved are related to exceedance statistics: how likely is it that
c exceeds c by a factor of 10, say?
We now turn a statistical analysis of box models in which the pollutant
concentration c^ above the current mixed layer cannot be ignored. This is
not just a matter of poor ventilation, because the air above the thin mixed
layer in the early evening is always polluted, even if the wind speed is
relatively large. At sunset the daytime mixed layer collapses, leaving a
concentration c in a layer whose depth h is often appreciably larger than
that of the nighttime mixed layer. If we call the latter h , we have a layer
of depth h - h filled with a pollutant concentration c... The initial value
s n *
of c^ is c ; the concentration c^ decreases during the night as ventilation
5
carries it out of the layer above the nighttime box. We define
hs ~ h = h* (Eq. 36)
so that the rate at which the remnants of daytime pollution are carried off by
ventilation equals h^c^U^. We allow U^ to be different from U; the wind speed
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inside the mixed layer may differ from that above it, especially during stagna-
tion episodes. The term h^c^U^ makes a significant contribution to the over-
all ventilation rate in the early evening hours, even if there is adequate
ventilation. If UT/2D » 1, that contribution is different from 0 only
during a small fraction of each diurnal cycle. Therefore, it tends to be quite
intermittent, being equal to 0 most of the time. In episodes with poor
ventilation, however, the contribution made by h^c^U^ is appreciable during
the entire night and well into the morning hours of the next day. If the long-
term average concentration is constant, the balance between emission and mean
ventilation now is given by
qD hUc + h^U^c^ (Eq. 37)
The influence of the second term (which we propose to call the "sunset detrain-
ment effect") is most important during nights with poor ventilation, because
h^ then is large compared to h, while c^ remains comparable to c over an ex-
tended period of time (in particular, when the emission rate at night is small
compared to that in the daytime) .
Applying the same averaging methods that were used for Equation 28, and
making the same assumptions about the various triple correlations and the
correlation between wind speed and mixing height, we obtain
qD = U h c + U h'c' + h u'c' + U^V* + U^ h^ + h^ u^ (Eq. 38)
The variable c^ is equal to 0 most of the time if the urban area is well
ventilated on the average, and different from 0 mainly in the early evening
hours. If non-0 values are encountered over a small fraction, say e, of
the entire averaging period (e includes both the early evening hours of each
day and the stagnation episodes), then
c^ = c^e (Eq. 39)
where c# is the mean value of c^ for the periods in which it is not negligible
(presumably, comparable to c). The fluctuations c^ then are given by
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c! = c. - c = C. - c e (Eq. 40)
so that the variance becomes
, ,.2 2 ~2 „ ~ ^ ~ 2 2 2~22
-------
adequate ventilation and not on the difference between the mixing heights of
today and yesterday.
Contributions to u^c^ are made by the strong negative correlation between
u^ and c^ during stagnation episodes, but also by the we 11- ventilated periods,
because the mean value of u^ is larger than 0 if stagnation periods are
excluded.
How large is e? If an urban area is well ventilated on the average
(UT/D » 1) , the time D/U it takes to remove c^ by ventilation in the early
evening corresponds to a fraction D/UT of each diurnal cycle (T = 24 hours) .
The average value of this fraction is proportional to the average value of
the reciprocal wind speed. Therefore, the value of e for a well ventilated
urban area is given by
£ = DCU")/! = D/UT (Eq> 43)
The ventilation terms involving c^ thus are proportional to the average venti-
lation time of the area concerned. Their contributions to Equation 38 cannot
be neglected if a city is very large or is located in a region with a low
average wind speed. This is a rather surprising conclusion; intuitively, one
would not expect that the climatological features of a box model are dependent
on the response time. Clearly, the dynamical properties of box models are
related to their climatology in unforeseen ways, and the effects of mixing-
height variability require careful study. Further analysis of these issues
appears to be worthwhile; however, that is beyond the scope of these explora-
tory investigations.
MODE-SWITCHING PROBLEMS
The box model for a typical urban area functions in the diffusion-limited
mode most of the time. The corresponding values of UT/D are typically rather
large; the analysis given in the preceding section has shown that quasi-steady
solutions of the governing equations tend to be adequate in these circumstances.
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During stagnation episodes, however, the box model switches to an inversion-
limited mode; since the corresponding values of UT/D are less than 2 (recall
that we have defined a stagnation episode as one in which UT/2D < 1) , the be-
havior of the box then generally cannot be approximated with the aid of
equilibrium solutions. Roughly speaking, the box model is either in a quasi-
steady diffusion-limited mode or in a slowly responding inversion-limited
mode.
We need to investigate under what conditions the box model switches from
one kind of behavior to the other. On the average, the effective dispersion
height z at the downwind edge of a diffusion- limited box model is about twice
the reference height h of the Gifford-Hanna model (see "The Dynamics of Box
R
Models," above). Therefore, the mean value of z/D is about 0.01. We now
assume that z/D is proportional to the relative turbulence intensity a /U,
and that the average value of the latter is about 0.02 (corresponding roughly
to neutral conditions) . This implies that
z/h = (a /U)D/h (Eq. 44)
/ w
which is consistent with the estimate given in the Introduction to this report.
The ratio of interest is z/h, where h is the mixing height; it is given
by
z/D = 2" aw/U (Eq. 45)
Whenever z/h exceeds 1, the ventilation term in the box model equation
is proportional to the actual mixing height. This occurs when a /U is rela-
tively large (convective conditions) , when h/D is relatively small (as in an
episode with appreciable subsidence, or on clear nights with low wind speeds) ,
or when D/U is relatively large (stagnation periods) . If z < h, the box model
is diffusion-limited, and z has to replace h in the ventilation term of the
governing equation.
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For very large urbanized areas, the ventilation cross section is limited
by an inversion lid much of the time. This does not mean that no correction
is needed for the effective volume of the box. Near the leading edge of the
area, the effective height z of the box is a function of x; with the aid of
Equation 44 we may approximate this dependence as
•7 (v^ — •—< (r\ /TT^-v
200 ~ 2 1VU;X (Eq. 46)
The effective lid reaches the inversion lid at a distance x^, such that
x, = ZhCa/U)"1 (Eq. 47)
n w ^
The nominal volume of the box (hD ) is thus greater than the effective volume
by an amount equal to
4 x, hD = h2D U/a
2 h w (Eq. 48)
and the ratio of the effective volume V to the nominal one becomes
e
- 49)
hD w
Whenever necessary, this correction term can be included in the left-hand side
of the differential equation for the box model. However, Equation 49 is quite
simple; the correction term will not complicate matters very much. It should
be noted also that the volume of the box does not enter into the equilibrium
solution of the box model equation. The latter depends only on the ventilation
cross section hD.
In this context, it is of interest to realize, again, that the response
time of the box model equals the ventilation time D/U. Our discussion points
to applications over very large urbanized areas with large response times.
This means that transient effects are likely to be quite important, and that
the equilibrium solution of the box model equation is not going to give an
adequate description of the pollutant concentration in the box. Thus a thor-
ough study of the dynamic equation previously discussed ("The Dynamics of Box
136
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Models") cannot be avoided. By way of illustration, consider an area with
D = 60 km, over which the annual mean wind speed is 5 km/hour. Such an area
has a mean response time of 12 hours, so that it will have a relatively high
frequency of stagnation episodes (recall that we have defined a stagnation
episode as one in which the pollution in the thick mixed layer at sunset today
is not flushed away by tomorrow morning).
CONCLUSIONS
The box model is one of the simplest conceivable air-quality simulation
models. We agree with Gifford (1973) that "simple" is the antonym of "complex,"
not of "sophisticated." The antonym of "sophisticated" is "naive," and the
box model is ideally suited for the study of simple, sophisticated questions
about pollution control during stagnation episodes and about many aspects of
air pollution climatology and environmental impact assessment.
The analysis presented in this paper suggests that the time has come for
a marriage between box models and inversion-rise models. Both are simple,
reliable, and frugal with computer time; used together with appropriate observa-
tional programs, they appear to be ideal vehicles for exploratory studies of
key issues related to air pollution control strategies for large cities and
urbanized regions.
One of the principal conclusions that can be drawn from the results
presented here is that Lettau's "flushing frequency" is indeed a key parameter
in urban air pollution meteorology. It is no exaggeration to claim that the
ventilation time D/U is, in many ways, of much greater importance than the
ventilation index hu. The "flushing factor" is a steady-state concept, which
seems rather inappropriate in stagnation episodes (because transient effects
are quite large if the wind speed is small).
We have repeatedly drawn attention to the consequences of the sunset
collapse of the mixing height. This effect leaves pollution (which may be
entrained — recaptured — again during the next day in a stagnation episode)
behind at elevated levels, and leads to additional terms in the climatological
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balance equation for the box model. Thus, the behavior of the mixing height
around sunset must be answered in the design of observational programs, the
maximum permissible integration time for the differential equation involved
is limited, there are severe restrictions on studies of the statistical
effects of parametric excitation, and so on. At sunset, the mixed layer has
to be redefined; in effect, the problem has to be restarted every day.
Lettau's box model contains a term representing the turbulent flux of
pollution at the top of the box. In this paper, that term has been replaced
by a parameterized one, borrowed directly from the "jump conditions" employed
in the current generation of inversion-rise models. In this way, we have
forged a strong and direct link between inversion-limited box models and
inversion-rise models, and paved the way for further studies of box model
dynamics.
ACKNOWLEDGMENTS
An early draft of this paper was read by F. A. Gifford, D. W. Thomson,
and R. A. Anthes. Their comments helped to clarify several issues. (It is
rather ironic that I managed to become confused by a simple model; I hate to
think what I might have done with a complicated one!)
This research was supported by the U.S. Environmental Protection Agency
under Grant No. R800397 with The Pennsylvania State University, and by the
Atmospheric Sciences Section of the U.S. National Science Foundation through
Grant DES75-13357.
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Monitoring. In: Select Research Group in Air Pollution Meteorology,
Third Annual Progress Report. U.S. Environmental Protection Agency,
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Arakawa, A., and W. H. Schubert. 1974. Interaction of a Cumulus Cloud
Ensemble with the Large-Scale Environment, Part I. J. Atmos. Sci. 31:674-
701.
Betts, A. K. 1973. Non-precipitating Cumulus Convection and its Parameteri-
zation. Quarterly J. Royal Meteor. Soc. 99:178-196.
Carson, D. J. 1973. The Development of a Dry, Inversion-Capped, Convectively
Unstable Boundary Layer. Quarterly J. Royal Meteor. Soc. 99:450-467.
Deardorff, J. W. 1972. Parameterization of the Planetary Boundary Layer
for Use in General Circulation Models. Mon. Wea. Rev. 100:93-106.
Deardorff, J. W., and G. E. Willis. 1975. A Parameterization of Diffusion
into the Mixed Layer. J. Appl. Meteor. 14:1451-1458.
Eschenroeder, A. 1975. An Assessment of Models for Predicting Air Quality.
ERT Document ERTW-75-3. Environmental Research and Technology, Inc.,
Santa Barbara, California. 83 pp.
Gifford, P. A. 1973. The Simple ATDL Urban Air Pollution Model. ATDL
Contribution File No. 78 Revised, Air Resources Atmospheric Turbulence
and Diffusion Laboratory, Oak Ridge, Tennessee. 17 pp.
Gifford, F. A., and S. R. Hanna. 1973. Modeling Urban Air Pollution. Atmos.
Environ. 7:131-136.
Hanna, S. R. 1975. Urban Diffusion Models. ATDL Contribution File No. 75-8,
Air Resources Atmospheric Turbulence and Diffusion Laboratory, Oak Ridge,
Tennessee.
Hinze, J. O. 1975. Turbulence (2nded.). McGraw-Hill, New York.
Lettau, H. H. 1970. Physical and Meteorological Basis for Mathematical
Models of Urban Diffusion Processes. In: Proceedings of the Symposium
on Multiple-Source Urban Diffusion Models, A. C. Stern, ed. APCO
Publication AP-86, U.S. Environmental Protection Agency, Research Tri-
angle Park, North Carolina.
Lumley, J. L. 1970. Stochastic Tools in Turbulence. Academic Press, New
York.
Panofsky, H. A. 1976. Personal communication.
Pasquill, F. 1974. Atmospheric Diffusion (2nd ed.). John Wiley and Sons,
New York.
Stull, R. B. 1975. Temperature Inversions Capping Atmospheric Boundary
Layers. Ph.D. Dissertation. University of Washington, Seattle,
Washington.
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Summers, P. 1965. An Urban Heat Island Model. Paper presented at the Fi#st
Canadian Conference on Micrometeorology, Toronto, 12-14 April.
Tennekes, H. 1973. A Model for the Dynamics of the Inversion above a
Convective Boundary Layer. J. Atmos. Sci. 30:558-567.
Tennekes, H. 1975. Eulerian and Lagrangian Time Microscales in Isotropic
Turbulence. J. Fluid Mech. 67:561-567.
Tennekes, H., and J. L. Lumley. 1972. A First Course in Turbulence. MIT
Press, Cambridge, Massachusetts.
Tennekes, H., and A. P. van Ulden. 1974. Short-Term Forecasts of Temperature
and Mixing Height on Sunny Days. In: Preprints, Symposium on Atmospheric
Diffusion and Air Pollution (American Meteorological Society), Boston,
Massachusetts, Sept.
Zeman, O. 1975. The Dynamics of Entrainment in the Planetary Boundary Layer:
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The Pennsylvania State University, University Park, Pennsylvania.
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SIMULATING TURBULENT TRANSPORT IN
URBAN AIR POLLUTION MODELS
John L. Lumley
Department of Aerospace Engineering
INTRODUCTION
An urban air pollution model is made up of many things: specification of
emissions, with regard to type, strength, and distribution in time and space;
specification of surface absorptivity and washout for the various pollutants;
modeling of chemical reactions; and specification of the flow field, including
the turbulent transport of the various pollutants. We intend here to limit
our attention to the last aspect, turbulent transport. In view of the uncer-
tainties of the first three aspects in present models, it is legitimate to
ask whether there is any point in giving detailed consideration to turbulent
transport.
Although crude transport models may be satisfactory for many purposes at
the present state of development, we may hope that there will be a continual
upgrading in all aspects of urban air pollution modeling, and that before too
long we will reach a stage in which better models of the transport will be
desirable. There is, in addition, evidence that the crude models used at
present are inadequate in certain situations even by present standards, al-
though this tends to be obscured because model predictions are not compared
with well documented experimental data.
In what follows, we will examine the problem of modeling turbulent
transport as though the other aspects of the air pollution modeling problem
did not exist, and we will attempt to establish a general program for the
development of an ideal model. Needless to say, real models in use at any
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stage will be only crude approximations to such an ideal model, but the ex-
istence of the ideal may help us to avoid some of the more serious pitfalls
in cruder models.
CLASSIFICATION OF URBAN POLLUTION MODELS
Urban pollution models may be conveniently classified by the way in which
they treat the turbulent transport of contaminant and any other quantities
which appear in the equations used. Turbulence, of course, is characterized
by its enhanced transport (Tennekes and Lumley, 1972), and the proper treat-
ment of this phenomenon is vital to the success of a model.
The majority of pollution models in use at the present time employ some
form (more or less disguised) of gradient transport of contaminant. By this
we mean that the predicted concentration distributions could be obtained from
solution of a diffusion equation with variable diffusivities, whether in
practice they are or not. The existence of a diffusion equation is evidence
that a gradient-transport assumption has been made for the turbulent trans-
port. The straightforward K-theory models (Turner, 1964; Lamb and Neiburger,
1971), of course, are of this type. Typically, in such a model the mean
winds, diffusivities, and height of the inversion base are direct inputs, or
must be predicted from relatively simple equations in which complex dynamical
effects are parameterized. The specification by whatever means of the ver-
tical and horizontal diffusivities always presents a problem; many semi-
empirical ways of specifying the variation with height exist in the literature
(Jaffe, 1967; Agee et al., 1973). None of these is completely satisfactory;
in particular (as will be shown later), it is possible in convective situa-
tions to have countergradient flux of contaminant which cannot be described by
these simple theories. A more fundamental problem, though less easy to docu-
ment, is the fact that diffusivities pep se are only appropriate to a homo-
geneous situation (Tennekes and Lumley, 1972). When terrain is changing
rapidly in the wind direction, it is unlikely that any specification of the
diffusivities will be satisfactory. It is only fair to say that these models
have the advantage of conceptual and computational simplicity; their accuracy
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is probably not too bad after dispersion has continued long enough for the
contaminant to uniformly fill the mixed layer (although this does not occur on
the urban scale as a general rule — this phase is more appropriate to a re-
gional model such as that of Bowne, 1968). In the earlier phases, before the
contaminant has filled the mixed layer (particularly in the presence of mul-
tiple sources), it is probably not possible to get closer than a factor of 2
to observed contaminant levels with this type of model (Turner, 1964). In
addition, of course, the quantities which gradient-transport models can pre-
dict are limited. Essentially, they can predict only mean concentration,
whereas we would like very much to predict, for example, concentration fluc-
tuation variance, so as to be able to say something about the likelihood of
maxima exceeding predetermined levels. Although we will probably continue to
rely on this type of model for some time, particularly with better parameter-
izations of such quantities as inversion rise (Tennekes and Zeman, 1975), it is
clear that it is nevertheless desirable to replace these models (if possible)
with one which predicts more and which does not require such extensive para-
meterization and/or empirical input. We feel that the second-order models
generally fall into this category.
The Gaussian plume technique (Pasquill, 1961) is a variant of gradient-
transport modeling, in an indirect way. As it is usually applied, the user
never comes in contact with the basic equations. However, the prototypical
Gaussian plume is a solution of a diffusion equation with variable diffusi-
vities. Hence, the Gaussian plume technique will have the same basic limita-
tions as any other gradient-transport technique — primarily, poor prediction
of the diffusivities in nearly windless convection. In addition, as applied
in practice, the diffusivities are often allowed to vary in such a way that
the resulting plume is not the solution of any reasonable equation; hence,
it may not do as well as a more rational gradient-transport model. Overlooking
this aspect, the Gaussian plume may be regarded in general as a solution ap-
propriate to a homogeneous turbulent field; the Gaussian plume approach for
multiple sources patches together homogeneous solutions to construct a solu-
tion in an inhomogeneous situation. This technique is valid if the scales of
the inhomogeneity are large relative to the scales of the plumes (i.e., in
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quasi-homogeneous situations). The Gaussian plume approach, of course, has
numerous computational advantages in certain simple situations (particularly
those with isolated sources).
The Lagrangian approach derives from Lamb and his co-workers (e.g., Lamb
and Seinfeld, 1973). The techniques discussed above are Eulerian; that is,
they describe events at a fixed point in space, past which matter is swept
by the moving air. The entire turbulent transport problem can, of course, be
cast in Lagrangian terms (that is, following individual particles). The pre-
diction of the mean concentration then hinges on knowing the transition proba-
bilities (see Tennekes and Lumley, 1972) . Although this method is potentially
exact if one knows the transition probabilities, in practice these are far more
difficult to predict than diffusivities. Essentially, one always ends by
assuming Gaussian transition probabilities. Gaussian transition probabilities
correspond to a kinetic-theory type of situation (Lumley, 1975) and imply
gradient transport; that is, they imply homogeneity and equilibrium, implying
that time and length scales of the turbulence are small relative to those of
the mean motions. In fact, Lamb (1974) has shown that the Lagrangian technique
with Gaussian transition probabilities produces a Gaussian plume. Thus, the
method reduces to an elegant and complicated version of the gradient-transport
approach. Gifford (1973) comments on the desirability of sophisticated
(rather than naive) models, and points out that this variable is not neces-
sarily well correlated with the simple-complex variable. It is possible to be
sophisticated only about things with which one has a great deal of experience.
If a model is cast in terms which are unfamiliar, one runs the risk of having
naive assumptions hidden by the unfamiliarity. We feel that the Lagrangian
models as used are as naive as any other gradient-transport model, the appear-
ance of sophistication being due to complexity.
It can hardly be emphasized too strongly that how a particular model is
applied in practice (i.e., whether a diffusion equation actually appears and
is solved explicitly) is irrelevant; if the model satisfies a diffusion equa-
tion, then it suffers from the inadequacies detailed above.
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The only models which do not suffer from the defects touched on above,
and which show promise of being able to calculate a greater range of quantities
in more extreme (less homogeneous) situations, are the so-called second-order
models. In these, no attempt is made to close the equations for the mean
quantities by relating the second-order quantities to the mean quantities (the
process which results in gradient transport in its many variants). The second-
order quantities, of course, are the variances and the fluxes of momentum,
heat, and contaminant. Instead, the equations for the second-order quantities
are used, and closure assumptions are made regarding the third-order quanti-
ties which appear in the equations. These quantities are the fluxes of the
variances and fluxes, and the terms which interchange kinetic energy among
components, as well as the terms which describe the maintenance of the rate
of destruction of the variances. Thus, the closure assumptions are made at
a level that is one order higher. These second-order methods were first
developed by Donaldson; for the elaboration to atmospheric flows, see Donaldson
(1973). There are now many workers in the field (see Reynolds, 1976, for a
partial list). The differences among these workers lie in the closure assump-
tions. There is no reason a priori to expect a closure assumption made at
third order to be better than one made at second order, except for an unjusti-
fied hope that if the assumption is made at a sufficient remove from the quanti-
ties in which one is interested, it will do less harm. In fact, however, it
turns out that — although the third-order closure assumptions are also elabora-
tions of gradient transport, or kinetic theory approximations — there are var-
ious approximations which can legitimately be made at third order, but not
at second, so that the closure assumptions are much better at third order.
The second-order models result generally in a collection of several tens of
nonlinear partial differential equations, first order in time, which can be
solved with appropriate boundary conditions by time-stepping. In other words,
when the values of all the field variables at an instant are known, the equa-
tions can be used to predict the values at the next instant. These techniques
were not, of course, developed in the first instance for urban pollution
predictions; rather, they were developed for prediction of flows of tech-
nological importance. Thus, the meteorological community benefits from about
a decade of development of models for the turbulent momentum flux.
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In the next section, we examine more closely some of the ways in which
the gradient-transport models may fail (considering K theory, Gaussian plume,
and Lagrangian transport together with Gaussian transition probabilities).
DRAWBACKS OF GRADIENT-TRANSPORT MODELS
The ways in which gradient-transport models fail are well known (see
Tennekes and Lumley, 1972, Sections 2.3-2.5, 3.4). Generally speaking, gra-
dient-transport models assume that the length and time scales of the turbulence
are small relative to the length and time scales of the mean motion; that is,
that there is a spectral gap (Lumley and Panofsky, 1964). This, of course, is
rarely the case in a turbulent flow, where the length and time scales of the
turbulent motion are usually of the same order as the length and time scales
of the mean motion. Nevertheless, there are situations where gradient-trans-
port ideas do work; it turns out that these are situations in which there is
locally only one length and time scale in the flow, so that the flux must be
proportional to the gradient (and to anything else which has the same dimen-
sions) . Hence, we may expect gradient-transport ideas to break down in situa-
tions in which we have several length or time scales; typically, this means
that we have several physical effects going on simultaneously. We would
consequently expect violations of gradient transport not only in situations
which are rapidly changing in the streamwise direction (introducing another
scale, the rate of change of the surface conditions), but in convective situa-
tions, in which buoyancy as well as wind shear is important. The failures in
the former case are well documented (Kline et al., 1969). In the latter
situation, simple examples have been worked out (Tennekes and Lumley, 1972, p.
101). We will present two examples of the latter situation, the first taken
from Zeman (1975). Figure la shows the vertical distribution of vertical
variance and turbulent energy in a convectively driven atmospheric mixed
layer, together with the vertical fluxes of vertical variance and energy and the
divergence of the flux of turbulent energy. The vertical turbulent transport
of turbulent energy must remove turbulent energy from the region near the
surface, and transport it to the vicinity of the inversion base. It is this
process that is responsible for the growth in thickness of the surface mixed
146
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Top of mixed layer
flux
loss-gain
Figure la. Observed profiles of turbulence quantities in buoyancy-driven
mixed layers (Zeman, 1975).
2 2
Figure Ib. The fluxes q u and -3q u /3x as calculated by a scalar transport
model (Zeman, 1975).
147
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layer, and for the entrainment of stable overlying fluid. Figure Ib illus-r
trates the form of the divergence of turbulent energy flux which would be
produced by a gradient-transport model. Energy is now removed from the center
of the layer, and only a fraction is sent up to the inversion base, the re-
mainder being sent down to the surface. Since — in a surface mixed layer
driven entirely by convection — the vertical transport of turbulent energy
(and temperature variance, and other second-order quantities) is entirely
responsible for the dynamics, a layer powered by a gradient-transport model
cannot behave properly (in fact, the rise of the inversion base is very poorly
predicted, while the vertical distribution of turbulent energy is wildly in
error) .
We turn now to a somewhat more familiar situation that was discussed
first by Priestley and Swinbank (1947) and later by Deardorff (1966, 1972).
This is the matter of the countergradient vertical heat flux. Deardorff
(1972) finds that, with negligible vertical wind shear, the vertical heat flux
is related to the vertical potential temperature gradient by a form like
(Eq.
There is a certain amount of disagreement about the constant before the second
term, although most authors would accept a number between 2/3 and 4/5 (see
Lumley, 1975) . Here w is the fluctuating vertical velocity, 9 is the fluctua-
tion in potential temperature, 1C is an eddy diffusivity for heat, 0 is the
mean potential temperature, 96/9z is an average potential temperature through
the layer in question, and g is the acceleration of gravity. Hence, if the
vertical gradient of potential temperature is small and positive, the second
term in parentheses can become more important, resulting in a positive (counter-
gradient) heat flux. It has recently been pointed out (Warhaft, 1976) that
the same effect takes place for a passive contaminant. Proceeding in exactly
the same way as Deardorff (1972) , we can obtain
we = -K,(8C/8z - ag~6c/0 w (Eq. 2)
148
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where c and C are the fluctuating and mean values of a passive scalar contam-
inant. We may expect the 6c correlation to be positive if both 6w and we are
positive, so that the behavior of the term will be approximately the same as
that of Equation 1. Since 6c can reverse sign in other situations, the be-
havior in general is somewhat more complicated than that of Equation 1. It
is difficult to make general statements about the effects of such a form as
Equation 2, beyond saying that it will — in circumstances of intense fluctua-
tions of potential temperature and concentration and relatively weak concentra-
tion gradients — cause an •intens'if'iaa'bi.on of the gradient (due to the counter-
gradient flux) resulting in higher concentration levels in certain regions
than would be predicted by simple gradient-transport techniques.
From the point of view of urban air pollution modeling, it is legitimate
to ask whether the failure of gradient-transport models in convective situa-
tions matters. That is, are urban pollution episodes associated with convec-
tive conditions? Urban pollution episodes do not often occur in the bright,
sunny conditions ordinarily associated with convection. They are usually
associated with a relatively low inversion base, and hence weak mixing and
entrainment. They are associated, however, with relatively low wind, which
would otherwise flush the volume. Hence, the mixing that does occur is pri-
marily driven by convection, and not by wind shear. Other cases are easy to
find: in the morning, in summer, during the commuting hours, the relevant
turbulent transport will often be almost entirely convective, even under cir-
cumstances which may not constitute a pollution problem later in the day; at
night, in the inner city, what transport exists will be driven by convection
induced by the urban heat island. Thus, we must conclude that the ability to
predict turbulent transport under conditions dominated by convection is a very
important property of an urban air pollution model.
It might be suggested that a countergradient flux could be produced
within the framework of conventional gradient transport by consideration of
negative diffusivities. Unfortunately, negative diffusivities are violations
of the second law of thermodynamics, and are violently unstable both analyti-
cally and numerically. The slightest irregularities in an initial profile
are amplified rapidly, producing catastrophic breakdown of the calculations.
149
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Equations 1 and 2 were generated by using a rudimentary form of second-
order model. The type of effect represented (as well as other, less familiar
effects) is incorporated automatically in the second-order models. Let us turn
now to a more detailed examination of these models.
THE SECOND-ORDER MODELS: STABILITY CONSIDERATIONS
Smith and Pasquill (1974) have raised some questions about second-order
modeling pep se. They point out that one is frequently computing a situation
which is nearly stationary. Let us consider the equation for the vertical
heat flux:
8w6/3t = K - U (Eq. 3)
K stands for the various terms which are known on the right-hand side, while
U stands for the unknown terms that must be modeled. Smith and Pasquill
(1974) reason that, if the time derivative on the left-hand side is 10 of
the magnitude of K, then U must be modeled to an accuracy of 10 for the
equation to balance.
If this were true, of course, second-order modeling would be a lost cause;
there is no hope of modeling the various third-order terms to such accuracy.
Fortunately, the difficulty is a chimera. Let us consider a simplified situa-
tion:
df/dt = A - Bf (Eq. 4)
This equation has the steady-state solution f = A/B. Suppose that we are
interested in an almost-steady-state situation. We could simply neglect the
time derivative and use the steady-state solution as an approximation: if A
and B are known to within 5%, then the value of f will be within 10%. This
is essentially what was done in analyzing the countergradient heat flux
(Deardorff, 1972). One of the unknown terms on the right-hand side of Equa-
tion 3 is modeled as proportional to the heat flux; if the time derivative is
150
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neglected, one can solve for the heat flux. This sort of approach is not seen
by Pasquill (1975) to raise doubts to nearly the same extent.
Certainly, if it is only the steady state that interests one, this ap-
proach seems simplest at first glance. However, one is often interested in
situations in which the time derivative is small, but not negligible. In
addition, since one is in reality usually dealing with a large set of nonlinear
equations which are extensively cross-coupled (even if only the steady state
is of interest) , it may be simpler to solve the system of equations as dif-
ferential equations in time, allowing the system to approach its steady state
if it will, putting in current values of all the variables on the right-hand
side, and obtaining the new values from the time derivatives. If the system
approaches a steady state, the values obtained will stop changing.
Let us see how this would work for the model Equation 4. We can write
the solution in time as
f = fe"Bt + Cl - e~Bt)A/B (Eq. 5)
where f is the initial value chosen. Let us suppose that B > 0 (we will
o
consider the opposite case later) . Then f will approach A/B for long time,
no matter what the value of f , and will have covered 99% of the distance
o
between f and A/B by Bt = 4.61. Of course, the final value will again be
within 10%. What happens, of course, is that — the initial choice of f = f
being very much in error — the time derivative of f is initially not small.
During a dimensionless time of 4.61, the value of f adjusts itself to bring
the equation back in balance. The accuracy of the final value is of the order
of the accuracy of the individual terms; in terms of Equation 3, neither the
known nor the unknown terms have precisely the right values, but the values of
all the variables have been adjusted to bring the equation into balance, and
the amount they have had to be adjusted is of the order of the inaccuracy of
the modeled terms.
The time required in this simple case is about 5 characteristic times to
arrive within 1% of the answer. Although the real situation is much more
151
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complicated, the behavior is in fact quite similar. The second-order equa-.
2 2
tions are characterized by a time scale of roughly q /3e, where q is the
turbulent mean square fluctuating velocity, and e is the mean rate of dissipa-
tion of turbulent kinetic energy per unit mass. Any given problem, of course,
may have other time scales, although there is a tendency for all time scales
in a turbulent problem to become comparable as time goes on. Zeman (1975)
uses h /w^ as his time scale for modeling inversion rise, where h is the
initial height of the mixed layer, and w^ is the scaling velocity for verti-
cal fluctuating velocity determined from the surface heat flux. This is
approximately equal to the scale given above, and Zeman finds that the initial
transient is no longer perceptible after 3 characteristic times. The disparity
between 3 and 5 probably only indicates that the eye cannot discern departures
of much less than 5%. Pasquill (1975) estimates a "settling-down" time of
some tens of minutes in a practical situation. This is a little long; our
criterion above would suggest 10 min for convergence to within 5%, 15 min
for convergence to within 1%. This is probably just short enough to give some
assurance that the solution will have "settled down" during the transit time
over an area source of pollution of typical size. Pasquill (1975) suggests
that the "settling-down" time may be improved by better modeling; from our
discussions above, this is clearly not the case. The only thing that would
reduce the "settling-down" time would be a better initial guess at the solu-
tion, even if the terms were all modeled perfectly. Since we may presume that
a perfect initial guess at the solution is too much to hope for (obviating the
necessity for a prediction scheme), we must learn to live with the "settling-
down" time.
It is instructive to consider the case of B negative. Here, even though
Equation 4 has a steady-state solution A/B, the solution will not approach
this value for any positive time (in fact, f would go to this value only if
time could go backward). If the initial value of f were precisely equal to
A/B, f would remain at that value; practically, of course, the slightest dif-
ference from A/B would be amplified, and f would diverge to positive or nega-
tive infinity. The equation is now unstable. In modeling, it is of the utmost
importance to be sure that the form of the modeled terms produces a stable set
152
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of equations, in order to benefit from the behavior described earlier. If a
term which has a destabilizing effect is inadvertently introduced, the system
will immediately diverge and will not approach a steady state (even if one
exists). Common sense is often a good guide in this; for example, it is clear
in Equation 4 (even without the analysis) that B must be positive for relaxa-
tion; in systems of equations, however, the situation is not so clear if the
term to be added contains a variable from another equation. The instability
may become evident only by careful analysis of several of the equations taken
together.
This discussion would be precisely the same if one were discussing
mesoscale modeling or global-scale weather forecasting. The assumptions,
the parameterization, and the modeling of various terms would be different, but
the behavior of sets of partial differential equations of this type would be
the same. The discussion above is also independent of the differencing scheme
employed; there will be separate stability problems associated with the dif-
ferencing scheme, which must be dealt with in the usual manner (Roache, 1972).
SECOND-ORDER MODELING TECHNIQUE AND PITFALLS
The heart of the second-order modeling problem is the modeling of the
various unknown third-order terms in the equations. Two examples of these
are u.9p/3x. and 68p/9x. (these are designated as third-order terms because
the pressure is quadratic in the velocity; see Tennekes and Lumley, 1972).
Donaldson (1972) was the first to suggest that such terms be represented by
tensorially and dimensionally correct combinations of second-order variables,
which are (potentially) known. For this reason, second-order modeling is
often called (as Donaldson suggested) "invariant" modeling, suggesting that
the terms be modeled in a tensorially invariant form (wherein the model adopted
retains its correctness regardless of the coordinate system). This is a require-
ment that has been applied in many other fields (and in fact is not optional;
if an equation is correct, it must be independent of the coordinate system).
The field to make greatest use of the idea is probably continuum mechanics;
there one searches for a constitutive relation (relating stress and deformation
153
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history) for various materials. Requiring that the relation in question be
tensorially invariant under certain transformations has shed light on the
possible forms of relations.
In second-order modeling, we attempt to do the same thing. We are not
relating stress and deformation history, but we are relating physically mean-
ingful tensors of various kinds. In addition to being tensorially and dimen-
sionally correct, these relations must also satisfy various other requirements:
they must have the right limiting values for large and small Reynolds numbers,
for large and small anisotropy, and so forth. From a stability point of view,
they must satisfy certain requirements, sometimes called realizability condi-
tions (Schumann, 1976). An example of the latter is the requirement that the
various terms modeled in the equations for the variances of the turbulent
velocities do not permit negative variance values. That is, if the variance
is 0, the time derivative of the variance must be 0 also (so that the variance
cannot cross over to negative values). This is not always easy to arrange.
One is inclined to feel that it is physically unlikely that the variance of a
component will vanish, and that a model which is satisfactory for ordinarily
observed values of the variances will therefore suffice; experience indicates,
however, that (particularly during the convulsive recovery from a set of un-
realistic initial conditions) if the equations are capable of producing non-
physical values of the variables, there is a chance that they will.
The literature abounds with examples of requirements that have not been
met. Since the modeling was originally conceived as invariant modeling, all
of the models are properly invariant in some sense. Thus, all models pro-
posed for p8u./3x. are second-rank tensors with zero trace (since p9u./9x.
must vanish within the Boussinesq approximation [Lumley and Panofsky, 1964]).
Rotta (1951) showed that this term may be split into several parts, one of
them corresponding to the interaction of the turbulence with itself (referred
to as "nonlinear scrambling" or "return to isotropy," because this is the
effect of the term), and others proportional to the mean velocity gradient,
the buoyancy, and the Coriolis parameter. The latter are linear in the turbu-
lent fluctuating velocity. They have the property of being anisotropic even
154
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in an isotropic turbulence (Lumley, 1975). To illustrate, imagine a com-
pletely isotropic velocity and temperature field. At t = 0, apply a strong
velocity gradient (or gravitational field, or Coriolis acceleration). Aniso-
tropy of the velocity and temperature fields will immediately begin to develop.
It is surprising, however, that these linear parts of the pressure (and hence
of its correlations with other quantities) are -immediately anisotropic while
the velocity field is still isotropic (no development time is required).
Hence, a model for pdu./dx. which vanishes when the turbulence is isotropic
is not likely to be satisfactory. This was recognized by Rotta (1951), and
various models for these parts have been suggested (Hanjalic and Launder,
1972). However, there are authors who continue to suppress these parts
(Lewellen, 1975). The experience of most authors is that it is impossible
to reproduce the rate of increase of momentum or heat flux in rapidly changing
situations without these terms. The effect of these contributions to the pres-
sure is to partially cancel terms present in the equations (Lumley, 1975).
Thus, the buoyant contribution to the pressure has the effect of reducing the
buoyant contribution in the vertical energy and heat flux equations, while
introducing terms in vertical heat flux in the horizontal energy equations.
The terms thus redistribute some of the production, so that it is less aniso-
tropic. These buoyant terms (and their analogs in contaminant concentration)
are, however, neglected by some authors.
In the past, some authors did not use a complete set of equations; that
is, the number of variables was greater than the number of equations, so that
at least one variable had to be specified as input. The variable in question
was usually a length scale, or some other quantity equivalent to a length
scale. Lewellen and Teske (1973), Donaldson (1972), and Mellor (1973) all
input a distribution for the length scale. This has proved quite satisfactory
in well-understood situations, which are the kind usually employed in the test-
ing of models (i.e., in horizontally homogeneous boundary layers). In these
cases, one can easily predict a realistic distribution of length scale. There
are, in fact, more than enough data to predict a realistic distribution of
all the variables. Applied to the kind of situations that are of practical
interest, however (with strong horizontal inhomogeneity of skin friction and
155
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surface heat flux), it is not clear that one can so easily guess at a length
scale distribution. Considerable effort on the part of many workers has gone
into the development of a satisfactory equation for a length scale or equiva-
lent quantity (and this effort continues); for a discussion, see Reynolds
(1976). Use of a length scale equation (or equivalent) is now almost universal;
Lewellen (1975) now uses a variant of the length scale equation used by other
workers.
The dissipation term in the Reynolds stress tensor equations
) (9u./8x ) has been modeled by some authors as proportional to
K. J K.
u.u. (Daly and Harlow, 1970; Donaldson, 1972, 1973). However, from Kologorov's
ideas of local isotropy (see Tennekes and Lumley, 1972) , the off-diagonal terms
must vanish as the Reynolds number becomes large (see Corrsin, 1972). That
is, at very large Reynolds numbers, there must be a dissipation term in the
component energy equations but not in the equation for the Reynolds stress
uw. The inclusion of such a term creates serious difficulties requiring com-
pensation elsewhere in the equation. Most authors now realize this, and use
a form for this term which will become diagonal as the Reynolds number goes to
infinity (Lewellen, 1975).
There are probably almost as many forms for the turbulent transport
terms as there are authors. We are considering here the turbulent transport
of the variances and fluxes, typically u.u.u, . As we have seen earlier,
1 3 K.
simple gradient-transport ideas for these quantities are unlikely to work in
situations that are dominated by buoyancy; such models do not transport the
variances and fluxes in the right direction or to the right place, where they
are needed dynamically. As an example, Lewellen (1975) uses transport models
for u.u. and u.Q which are proportional to the gradients of these quantities.
Such models are appropriate to turbulence which is nearly isotropic and in
which buoyancy effects are negligible. To our knowledge, there is only one
model for the turbulent transport terms which will work when buoyancy is domi-
nant (Zeman, 1975).
The moral to be drawn from all of these examples is that modeling of the
third-order terms is a very sophisticated business. It is not enough to
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devise a model that is dimensionally and tensorially correct; it is necessary
to consider the detailed dynamical behavior of the model vis-a-vis real tur-
bulence, under a range of parameter values, for all physical phenomena present
one at a time and in various combinations. Turbulence is, after all, a com-
plicated phenomenon capable of behavior of almost infinite variety; it is too
much to expect to construct in a naive way a model that behaves even approxi-
mately like it.
VERIFICATION OF SECOND-ORDER MODELS
In continuum mechanics, the behavior of a material is determined by the
partial differential equation describing the conservation of momentum. This
equation relates the variables describing the motion in the neighborhood of
any point. In order to determine the flow field in a particular situation,
this equation must be solved together with a set of boundary and initial condi-
tions. Similar to the situation in turbulence, the differential equation for
conservation of momentum is not closed, but contains the stress, which must
be related to the other flow variables. Such a relation is termed a consti-
tutive relation. When dealing with a new material, one formulates a constitu-
tive relation connecting stress and deformation history in a properly invariant
way, identifies in this general relationship certain invariant functions which
define its behavior, and devises experiments which will permit measurements
of these invariant functions. An invariant function is a function of tensorial
invariants which is consequently the same in every coordinate system. The type
of experiment usually devised is the simplest flow which will permit the func-
tion in question to be measured without the interference of other phenomena
extraneous to the measurement. These flows are usually called viscometric,
and great pains are taken to eliminate secondary flows, lack of two-dimension-
ality, nonuniform temperature, and so forth, so that one effect can be examined
at a time.
We feel it is appropriate to do the same thing with the second-order
models. The sets of differential equations for the second-order quantities
play the role of the momentum equation for a material. Like that equation,
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they are not closed, and relations must be found connecting the unknown func-
tions appearing in the equations to the second-order quantities; these rela-
tions play the role of the constitutive relations. When these relations have
been found, the equations can be solved in a particular domain with boundary
and initial conditions to predict the behavior of the turbulence. In order
to determine the constitutive relations, sets of viscometric experiments must
be devised. In other words, we feel that no effort should be made to repro-
duce field data until each physical phenomenon has been examined alone and in
combination with others in simple, well documented (laboratory) flow situa-
tions, over a range of parameter values. It is clear that any laboratory
experiment that elucidates a phenomenon which occurs in the atmosphere is
relevant to meteorology. One should not think only in terms of laboratory
experiments which reproduce at once ail major phenomena which occur in the
atmosphere. Such experiments are not only difficult (Synder, 1972) (perhaps
impossible) to perform, but they are seldom as useful as carefully defined
experiments which display in isolated form a single phenomenon at a time.
Calibration of a model from an experiment in which several phenomena are
present at once is dangerous; the modeling of several terms will be in question,
corresponding to the several phenonmena, and the wrong term may be adjusted
to bring the results of the model into agreement with the data. For example,
suppose that, with a given set of coefficients, the model predictions do not
match the measurements. Is this because of the effect of buoyancy on the trans-
port, or anisotropy? Or has the passive scalar transport (as distinct from
the effect of buoyancy) been modeled incorrectly? Suppose, by fiddling, we
manage to adjust the coefficients so that the model predictions match the
measurements; we still have no assurance that the values obtained are universal,
and will work under different circumstances. We may have matched the data by
putting in too much buoyant transport and not enough passive transport. In a
situation in which these are in different ratio, the model will no longer match
the data.
It is also important to calibrate the model against several similar flows
that differ only in the various parameter values. Consider, for example, the
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Reynolds number. In the atmosphere, the Reynolds number may be considered to
be infinite; in the laboratory, it is finite. For this reason, any laboratory
flow must be carried out for at least two values of the Reynolds number so that,
with the help of theoretical insight on the behavior of the various terms with
increasing Reynolds number (usually not too difficult to acquire), one may extra-
polate to infinite Reynolds number. If an external influence is applied to a
flow (such as the application of mean velocity gradient), the external influence
may introduce a new scale (a time scale, in the case of the mean velocity gra-
dient) . The ratio of this scale to that of the turbulence, say, becomes a new
variable, and it is important that several experiments be considered in order
to determine the influence of this variable, if possible, it is also desir-
able, after considering each phenomenon separately, to consider the phenomena
together in various combinations, since turbulence is far from linear and the
various phenomena may combine in unforeseen ways.
Thoughout all of this, we are assuming that turbulence can be adequately
described by a finite set of differential equations plus boundary and initial
conditions. That is to say, the behavior of turbulence can be described in
terms of relations among several variables in a neighborhood, plus conditions
on the boundaries in space and time. If this were not true, there would be
nothing universal about information obtained from a given experiment; i.e.,
the relations among the variables in a neighborhood (as opposed to the details
of the flow field) would depend on the boundary conditions. Certainly this
assumption is not exactly true for turbulence anywhere, and in particular near
a boundary in space or time (Lumley, 1970). That it is true to a sufficient
approximation away from boundaries in flows that do not vary too abruptly in
space or time is the central assumption of second-order modeling, and this has
been borne out by experience in modeling the various flows. In other words,
in the progress made thus far, relations determined from one viscometric flow
have been satisfactory in another with different boundary conditions. There
is no reason to expect this assumption to be violated in the future,- hence,
we may have a certain amount of confidence that, if a set of viscometric flows
exemplifying the various phenomena can be devised, the relations we find will
be universal.
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Let us consider the phenomena occurring in turbulent flows for which we
need calibrating experiments (see Table 1). Turbulence decays; anisotropic
turbulence becomes more isotropic; scalar fluctuations are produced from
gradients; turbulence can be stretched, which alters its dynamics; the stretch-
ing can be combined with rotation, which is even more complicated; in inhomo-
geneous situations, transport occurs; the transport may be influenced by
buoyancy. This listing is not yet sufficiently precise; for example, the
separate effects of temperature fluctuations or concentration fluctuations
must be examined. Thus, the observation that "turbulence decays" requires
further investigation: turbulent veloo-ity fluctuations decay, and turbulent
temperature fluctuations decay. Do they decay at the same rate in a non-
buoyant isotropic situation? How is the decay rate of the temperature fluctua-
tions influenced by anisotropy of the velocity field? A time scale can be
defined for the velocity field, and a similar one for the temperature field.
Do these time scales tend to relax toward each other?
In this way, each entry in the list of phenomena can be broken down,
forming, finally, a collection of definitive experiments. It is important to
remember that each experiment is not of interest for its own sake, but rather
as a paradigm of a particular phenomenon which occurs in the atmosphere (but
in combination with so many other phenomena that we cannot separate effects).
In just this way, a laboratory experiment designed to measure normal stresses
in polyisobutylene may seem rather esoteric, but is essential to predict the
behavior of STP in an engine bearing. Table 1 presents a final list of the
various phenomena and some experiments to elucidate them. An X in the left-
hand column denotes that the phenomenon is displayed in the experiment listed
at the right. These experiments have, for the most part, been carried out
over the last 20 years with the purpose of shedding light on turbulence dynam-
ics (not for the calibration of any particular model).
It may be asked whether the phenomena listed are exhaustive. Is turbulence
capable of other types of behavior overlooked here? There is no way to guar-
antee that none have been overlooked; the phenomena listed represent the con-
sensus of the turbulence community at the present time. However, it is not
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TABLE 1. EXPERIMENTS THAT ELUCIDATE PHENOMENA OF TURBULENCE
ty fluctuations
ature fluctuations
0 Ot
> H
X
X X
X
X X
X X
X X
X
X X
X X
X
X X
X X
Phenomenon
I
0)
« >i 3
in 4-i JJ
C O k
>i -H -H V ^
l-l rH JJ p (!) Q CJ
8 *J C C C C E
(i)
(2)
X (3)
X (4)
X X <5>
X <6>
X X <7'
XXX (8)
XXX (9)
X XX X (10)
XX XXX (11)
X X X X X X (12)
Experiment and Investigator (s)
Decay of isothermal, isotropic turbulence - Comte-
Bellot and Corrsin (1966)
Decay of temperature fluctuations in isotropic turbu-
lence - Mills et al. (1958); Lin and Lin (1973); Yeh
and Van Atta (1973)
Return to isotropy - Uberoi (1956, 1957); Mills and
Corrsin (1959)
Decay of temperature fluctuations in anisotropic
turbulence - Mills and Corrsin (1959)
Decay of heat flux - No experiment exists
Production of temperature fluctuations from a uniform
gradient by isotropic turbulence - Wiskind (1962);
Alexopoulos and Keffer (1971)
Homogeneous strain of isothermal turbulence - Marechal
(1972); Tucker and Reynolds (1968)
Homogeneous strain of turbulence with a mean tempera-
ture gradient - No experiment exists
Homogeneous shear - Rose (1966); Champagne et al.
(1970)
Two-dimensional wake - Townsend (1956)
Convectively driven mixed layer - Willis and Deardorff
(1973); Lenschow and Johnson (1968); Lenschow (1970,
1974); Telford and Warner (1964); Rowland (1973)
Two-dimensional hot non-buoyant wake - Mimaud-Lacoste
(1972); Freymuth and Uberoi (1971)
x x x x ^ x (13) IVo-dimensional thermal plume - W- K. George (experi-
ment presently under way)
161
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likely that any more will be found, or — if some are — that they will be of
great importance.
Experiment 5 requires some explanation. In order to measure the strength
of the tendency to return to isotropy, we can produce an anisotropy of the
velocity field (say, by stretching) and permit this to decay. This is equiv-
alent to inducing a Reynolds stress and permitting it to decay, since the
Reynolds stress tensor may be diagonalized. (That is to say, if at a point
we have a Reynolds stress and various intensities, there is always another set
of orthogonal axes through the same point in which the cross-correlations
vanish, leaving only unequal intensities.) In exactly the same way, we should
produce a heat flux representing an anisotropy of the combined fields and allow
it to decay without the presence of a temperature gradient. One of the in-
teresting and unanswered questions is whether the heat flux and the Reynolds
stress would decay at the same rate. The problem lies in creating a heat flux
without a temperature gradient or in removing the gradient after the heat flux
has been set up. This could be done with a grid of small heated jets, so that
an excess velocity would be associated with an excess temperature (initially).
Such questions are not academic; in this case, the answer determines the turbu-
lent Prandtl number (the ratio K /IC ) in homogeneous situations, and hence
equilibrium levels of heat flux.
Experiment 8 also has not been accomplished. When turbulence is stretched,
one must consider whether the stretching would change the value of the heat
flux if a mean temperature gradient were present. If it changed, how would the
change depend on the rate of stretching? This experiment would be a very simple
modification of Experiment 6. One would simply add a mean temperature gradient
(by heating the grid bars), first in the direction of positive strain rate and
then in the direction of negative strain rate.
Where existing experiments have been suggested in Table 1, they have
usually been picked for several values of the relevant parameters. In Experi-
ment 7, for example, there are two quite different values of the ratio of
turbulent time scale to time scale of the strain rate.
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In several cases, the experiments mentioned are, in one respect or another,
far from definitive. For example, the investigations listed under Experiment
2 display three separate evolutions of the ratio of thermal to mechanical time
scale. We are currently carrying out this experiment in an effort to obtain
more precise values. The two investigations mentioned under Experiment 3 show
two opposite responses of the decay of dissipation to anisotropy; presumably,
both are not correct. A third experiment is needed to settle the matter. Of
course, the reason for these conflicting results is the fact that experiments
are difficult to accomplish and, despite the greatest care, still retain a
certain imprecision. This serves to underline why it is of the greatest
importance that the model be calibrated against laboratory data, and not against
field data. If laboratory data can still be ambiguous after the greatest care
has been taken to define experimental situations, consider the ambiguity in
field data, where so many variables are not under control! Unfortunately, in
field data the ambiguity is not evident even when the data of different experi-
ments are compared, since the experiments are typically not even nominally
comparable.
There are several experiments not appearing in Table 1 which might have
been listed under "Hard to Classify." First, there is the vital question of
the response of anisotropic turbulence to changes in mean conditions. Experi-
ments such as 6 and 9 always start with a grid-produced turbulence, which is
approximately isotropic, and then apply a distorting mean velocity or tempera-
ture field. It is equally important to determine how a turbulence which has
been subject to a certain mean field (and has, in consequence, acquired a cer-
tain structure) reacts to the sudden application of a quite different distorting
mean field. This corresponds much more closely to nature, and it is essential
to know if a model reproduces this behavior properly. Examples abound, the
most common being the response of the surface mixed layer to a sudden change of
terrain: the structure has come to equilibrium with particular conditions of
surface roughness and heating, producing characteristic values of heat flux
and anisotropy; following a change of terrain, the layer must accommodate it-
self to new values of roughness and heating. An experiment suitable for the
nonthermal aspects of this might begin like Experiment 6, with a constant area
163
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duct exponentially growing in one cross-stream direction and exponentially
ri~
shrinking in the other. The duct would be elliptical at first and would
become circular, and — in the classical experiment — would become elliptical
in the orthogonal direction. By a suggestion of J. Mathieu (private communi-
cation) , however, the duct could be cut at the circular section and rotated
45° so that the strain rate field applied in the second half of the duct would
be at 45° to that applied during the first half. The ability to predict such
a flow would be a very critical test of a model.
Second under "Hard to Classify" is the problem of the presence of a sur-
face. It is not difficult to show (Lumley, 1970) that what we are trying to
do with second-order modeling is only possible sufficiently far from a sur-
face, or sufficiently long after initiation of a flow. That is, we are assum-
ing that the distributions of the mean quantities through second order in a
region uniquely determine the values of the mean quantities of third order.
We can demonstrate that this is not the case one order lower by considering
the relation of Reynolds stress to mean velocity profile; it is possible to
have the same mean velocity profile with and without the presence of a wall,
with two quite different Reynolds stress distributions. Presumably the same
thing is true at all levels, so that we may expect our modeling of various
terms to break down in the vicinity of a surface. The efforts of various
workers to take account of this are described in Reynolds (1976); the matter
is far from settled. Most of the work to date has been applied to the Rey-
nolds stress (in particular, the pressure-strain correlation) and to the
equation for the dissipation of turbulent kinetic energy, with very little
applied to the thermal equations. That there is a problem here is evident,
since in the neighborhood of a wall one customarily measures the ratio of
streamwise to vertical heat flux near -3 (Monin and Yaglom, 1971); nearly the
same ratio is measured for the ratio of streamwise to vertical turbulent
energy flux (Schon, 1974). On the other hand, in homogeneous situations
(Webster, 1964) one measures a value much nearer to unity, and this is approxi-
mately what the various current models predict with nearly any closure assump-
tions. Hence, it is clear that — after all relevant physical phenomena have
been satisfactorily modeled away from a surface — some major modifications
to the models will be required to account for the presence of a surface.
164
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It seems likely that the majority of these modifications will consist of cor-
rectly taking into account the effect of the reduced Reynolds number on the
return to isotropy near the wall, as well as including in the pressure gradient
correlations terms corresponding to the surface integrals (resulting from the
solution of the Poisson equation for the pressure). There may, however, be
other, more subtle, effects.
Third under "Hard to Classify" are inhomogeneity and three-dimensionality.
At this point (after all the flows of Table 1 have been modeled), the model
will have been calibrated against a collection of flows which represent (at
most) inhomogeneity in one direction at a time, and parallel flow. In every
case, the modeling of the various terms is inherently three-dimensional; never-
theless, there will be constants that cannot be unambiguously determined by
calibration against such simple flows. We will need well documented laboratory
flows embodying change of direction with altitude and streamwise inhomogeneity.
Snyder (1972) has discussed the modeling of the former. For this purpose, a
flow such as the mixing layer formed by the interaction of two orthogonal
streams of uniform mean velocity profile would be suitable. Such a flow was
set up by Bradshaw a number of years ago (private communication) but the re-
sults have not appeared in the literature to the best of my knowledge. Stream-
wise inhomogeneity might be examined by attempting to predict the flows dis-
cussed by Tani (1969) for the isothermal case (corresponding to a change of
surface roughness), or the flow of Johnson (1959) corresponding to a sudden
change of surface heating.
At the present time, nine of the flows in Table 1 have been more or less
satisfactorily matched; it is anticipated that substantial progress on the
remaining four (plus the "Hard to Classify" problems) will be made in the next
several years. This should not by any means imply that all problems will be
resolved. In many cases, it will be necessary to bypass a problem, temporarily
adopting a theoretical estimate, to return later to resolve the issue defini-
tively. This is certainly the case with the resolution of the third problem
(above), Experiments 7 and 8, and the question of the response of anisotropic
turbulence to a distorting field, all of which require the results of difficult,
carefully done experiments.
165
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Of course, we cannot wait even a few years for a substantially resolved
model. There is a present and continuing need for practical models, and it is
possible to meet this need within the framework outlined in this paper. That
is, so long as one is aware of the basic problem (of simulating a particular
type of behavior), one can adopt engineering approximations appropriate to
particular situations, while awaiting better models. This is the usual approach
of all workers in the field. The Rotta (1951) model for the return to isotropy
is such an approximation, for example; it has been used with more or less satis-
factory results since the beginning of modeling efforts, although users have
been aware of limitations in various circumstances. Many of the simulations
used by Zeman (1975) are such engineering approximations. Only recently have
more evolved models been suggested (Lumley, 1976). Even relatively simple
models, if applied judiciously, can provide very satisfactory results in many
situations (Lewellen and Teske, 1975).
We have made no mention of contaminant; however, temperature in a non-
buoyant situation is the same as a passive contaminant. Chemically active
contaminants are another matter, and a very complicated one. In principle,
the behavior of chemically reacting species can be modeled using the same
concepts that we have described above. However, there is relatively little
good experimental information for calibration of the model, and only sparse
theoretical guidance. The development of satisfactory models for chemically
reacting species will be a long struggle. Incorporation of particles with
finite terminal velocity, radiation, absorbing boundary conditions, and the
like do not represent a problem, although there is again little experimental
information, and time will be required.
WHAT CAN WE EXPECT FROM SECOND-ORDER MODELS?
Second-order models have certain basic limitations — certain things that
cannot be expected of the models no matter how much time and effort are
expended. First, models of this sort presume certain things — principally,
that the length and time scales of the turbulence are small relative to the
length and time scales characterizing the distributions of the mean quantities
(Lumley, 1970); that is, that the situation is quasi-steady and quasi-homogeneous.
166
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This is not as serious a limitation as it sounds; while the equations are an
expansion (keeping first-order terms only) about the homogeneous, steady
state, we can actually go fairly far from this state before serious error
develops. Thus, in modeling decaying flows, or wakes, very satisfactory
results are achieved when the scale ratios are between 1/2 and 1. This covers
very many situations of practical interest, but probably not all. We must
always bear in mind that the errors are likely to get worse with more abrupt
changes or larger spatial gradients.
The model is restricted by the length and time scales of the region in
which it is applied. That is, we envision applying the model in a region 20 km
on a side, and 2 km high; it is presumed that the boundary conditions on the
sides and top of the box are given by a mesoscale prediction program, of which
this is one grid square (and in which all these complex effects are parameter-
ized) . Certainly the model cannot predict diffusivities corresponding to length
scales larger than the box or to time scales longer than the transit time
across the box. This is a limitation on the application of any model in this
situation. Mesoscale variability must be introduced by variation of the bound-
ary conditions. The second-order model can respond realistically to time-varying
boundary conditions so long as the time scale of their variation is not less
than three to five times the turbulence time scales (the situation is essentially
the same as that discussed in connection with the time-to-recover from a poor
initial condition).
Treating mesoscale variability by variation of boundary conditions is
essentially the same as arbitrarily dividing atmospheric motions into "weather"
and "turbulence," and suffers from some of the same problems. This can be
unambiguously done when there is a spectral gap (Lumley and Panofsky, 1964),
and corresponds to the time scale restriction just given. If there is signi-
ficant energy near the spectral cut (within a factor of 2 on either side),
then there will be additional interaction terms, and the effect of the "tur-
bulence" on the "weather" will not be simply a Reynolds stress. However, it
must be admitted that other approximations which (supposedly) are subject to
this restriction behave well even when the restriction is fairly seriously
violated. For example, the Navier-Stokes equations are based on the assumption
167
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that the time scale of the motion described is long compared to the time Vr
between molecular collisions, so that the molecular velocity distribution will
have time to come to equilibrium. For this reason, these equations should
break down above a Mach number of the order of 2. Nevertheless, they work
well to Mach numbers of the order of 10. The Heisenberg approximation for the
turbulent spectrum (Monin and Yaglom, 1971) is based on cutting the spectrum
at an arbitrary wave number, where no gap at all exists, and representing the
effect of the small scales on the large by a Reynolds stress; despite the ab-
sence of a gap, the results are excellent in most respects. Hence, it is
quite possible that handling mesoscale variability by variation of boundary
conditions will be satisfactory to a value of the time scale ratio consider-
ably smaller than the suggested safe value of 3 to 5. It is important to
remember, however, that there is no fundamental reason to expect this to be" so.
Within these very general limitations, a good second-order model is capable
of very realistic predictions. In a homogeneous turbulence, without wind shear
or buoyancy, a second-order model predicts that a continuous point source pro-
duces a Gaussian plume far downstream; the dispersion of the plume corresponds
to an exponential Lagrangian correlation. The initial spread is linear and
the later spread parabolic (as it should be). As time grows and the integral
takes in more of the correlation, longer and longer time scales are responsible
for the spread, corresponding to the classical description of dispersion (Corr-
sin, 1962). In the early stages of dispersion, the plume cross section is not
Gaussian, but goes to 0 faster at the edges. If this were all the second-
order model were capable of, it would hardly be an improvement on the Gaussian
plume model, or the other gradient-transport models. The true merits of the
second-order models only become evident when one considers an inhomogeneous
multiple source situation, with buoyancy and wind shear; then diffusion
coefficients and length scales are all automatically predicted. In fact, the
entire turbulent field is predicted, together with the distribution of wind
and temperature, as well as mean concentrations and fluxes and variances of
concentration, and all from first principles (in the sense that the behavior
of the model has been calibrated against the behavior of real turbulence in
prototypical situations) without the necessity of intervention by the operator.
168
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ACKNOWLEDGMENTS
This research was supported by the U.S. Environmental Protection Agency
under Grant No. R800397 with The Pennsylvania State University, and by the
Atmospheric Sciences Section of the U.S. National Science Foundation through
Grant DES75-13357.
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NATURAL REMOVAL OF GASEOUS POLLUTANTS
Robert L. Kabel
Department of Chemical Engineering
Center for Air Environment Studies
INTRODUCTION
Prom 1973 to the present there has been an increasing awareness of the
importance of incorporation of natural processes for pollutant removal in air-
quality simulation models. This awareness has developed as air pollution con-
siderations have expanded from local to regional and even global scales. An-
other major stimulus came with the recognition that transport and deposition
of atmospheric constituents could significantly pollute bodies of water. A
corollary is that natural sources of atmospheric pollutants (and potential
water pollutants) can be expected to be included in quantitative models along
with man-made emissions in the near future.
For perspective, we might consider the estimated emissions from natural
and anthropogenic sources for various gaseous pollutants. It is known that
the combustion of fossil fuels produces far more SO than the natural volcanic
source. On the other hand, natural sources (especially biological decay) of a
reduced form of sulfur, H S, far outweigh the anthropogenic sources. Of
course, the H S is readily oxidized to SO« in the atmosphere, giving an
approximate balance between natural and anthropogenic sulfur in the atmo-
sphere. We might also consider the background concentrations and major iden-
tified sinks of the various pollutants. The largest background concentrations
occur for those constituents which are chemically most inert and least water
soluble. The importance of these concepts will become more evident as this
paper progresses. Detailed discussion of this background material is given
by Rasmussen et al. (1974, 1975).
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NATURAL REMOVAL PROCESSES *'r"
In this section, the natural processes by which tropospheric pollutants
are eliminated or converted are considered. Discussion of each kind of removal
process is given and the limitations of the paper are more fully defined.
Table 1 indicates the various removal processes:
TABLE 1. REMOVAL PROCESSES
Interfacial Bulk
Vegetation Atmospheric Reactions
Soil Precipitation Scavenging
Water Bodies Dry Deposition
Stone Aerosol Scavenging
Tropopause
There are two main categories of removal processes. Interfacial processes
occur at a boundary of the troposphere. By contrast, the bulk processes occur
within the troposphere and have a more volumetric character.
Bulk Processes
Atmospheric Reactions—
Probably the best known removal process, chemical reaction in the atmo-
sphere, is also a generation process. The oxidation of H S to SO mentioned
earlier is a simple case. Because of the photochemical smog problem, an entire
literature has evolved from 25 years of intense research. Altshuller and
Bufalini reviewed the literature in 1971. Hecht and Seinfeld (1972) have pro-
posed an 81-step reaction mechanism for photochemical smog formation as well
as a simplified 15-step version. Smog is by no means the only important con-
sequence of atmospheric reactions. Each case is rather special, and no attempt
will be made here to deal with this massive topic.
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Precipitation Scavenging—
Two mechanisms of precipitation scavenging exist. In the first, the pol-
lutant is incorporated into the droplets and/or particles during the nucleation
and growth phases of cloud formation. The pollutant may even contribute to
this process; for example, see Twomey (1971). Second, when the solid or liquid
precipitation falls, gases and particles are collected during vertical passage
through the polluted region. The relative importance of these two scavenging
mechanisms is not well defined. But the phenomenon of acid rain is well known.
Cogbill and Likens (1974) provide excellent perspective on the causes, nature,
and impact of this phenomenon. At present, the pollution of surface regions
(especially water bodies) via precipitation is receiving extensive attention.
A case in point is Murphy's (1975) report on measured phosphorous input by
rainfall into Lake Michigan. Thus, precipitation scavenging is observed to
be an effective cleanser of the air and polluter of the water, at least for
certain contaminants.
Predictive tools are also becoming available. A comprehensive analysis
of reversible interaction of raindrops with atmospheric contaminants has been
presented by Hales (1972). Further study by Hales et al. (1973) has led to
the development of a mathematical model for predicting ground level washout
fluxes and average concentrations in the rain as a function of location beneath
a plume. The above references deal with liquid precipitation, which is surely
the more important case. However, some serious attention has been given to
scavenging by snow (Forland and Gjessing, 1975). The theory of precipitation
scavenging is perhaps rather well advanced in comparison to its practical appli-
cation. Measurement programs are now being conducted in all parts of the world.
In this paper, we will not be concerned further with the subject.
Dry Deposition—
The term "dry deposition" has been used mainly to imply all removal pro-
cesses at the earth's surface except precipitation scavenging. Such a charac-
terization is so general as to be almost useless for modeling purposes. Accord-
ingly, several processes usually lumped under this heading will be treated in
detail elsewhere in this paper. There is, however, one bulk process which
might be accurately called dry deposition. Consider the following simple but
177
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important mechanistic illustration. A common air pollutant, sulfur dioxide,
is oxidized and absorbed in a water droplet to give sulfuric acid. The oxida-
tion can and does occur either before or after the absorption. The resulting
acidic droplet is a natural site for the absorption of ammonia, another common
and inherently basic pollutant. Since ammonia is very soluble in water, its
absorption can also precede that of the sulfur dioxide. In either case, the
result is that the uptake of one component is enhanced by the uptake of the
other. If the water in the droplet now evaporates (ammonium sulfate is not
particularly hygroscopic), a dry particle of ammonium sulfate remains and can
eventually work its way to the ground. The chemistry and importance of such
processes for the above case are well presented by Miller and de Pena (1972).
Aerosol Scavenging—
The previous illustration shows how ammonia can be scavenged by a sulfuric
acid aerosol. Of course, the aerosol need not be an aqueous solution of an
absorbed gas. The potential variety of particles is staggering. A good review
of aerosol scavenging as a removal process is given by Hidy (1973). Because
particle deposition will not be examined further in this paper, attention is
called to some recent and valuable research by Sehmel (1975) on experimental
measurements and predictions of particle deposition and resuspension rates.
Interfacial Processes
Tropopause—
The tropopause can be thought of as the upper edge of the troposphere
and hence as an interface. At the present time, this interface is involved
in a dramatic controversy. Man-made chlorofluorocarbons are introduced into
the troposphere in a variety of ways. Because of their extreme chemical in-
ertness, these pollutants eventually diffuse through the troposphere into the
stratosphere. There, it is postulated, they participate in a series of reac-
tions which lead to ozone depletion (Molina and Rowland, 1974). The ozone
layer may also be affected by pollutants of natural origin. Nitrous oxide is
produced in tremendous amounts by biological decay and is also quite inert in
the near-earth region. Accordingly, it too may reach the stratosphere with
results similar to those postulated for the chlorofluorocarbons (Friend, 1976) .
178
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Thus pollutant removal from the troposphere at its upper "boundary" can take
on real importance in special cases. Although the tropopause can be thought of
as a boundary of the troposphere, the passage of material through it is really
a matter of atmospheric diffusion. This is the subject of a completely separate
literature.
The greatest attention in this paper will be given to the earth interface
processes. First, some illustrations of the removal mechanisms discussed
earlier will be presented. Then the matter of quantitative modeling of such
processes will be taken up. It should be realized that the matter addressed
here is the effect of things (vegetation, soil, water, and stone) on pollution,
and not the effect of pollutants on things (which has received so much atten-
tion) .
Vegetation—
The ability of plants to exchange large quantities of gaseous materials
with their surroundings is dramatically evidenced by the photosynthetic process
involving carbon dioxide and oxygen and by evapotranspiration of water. A
whole body of literature exists with respect to these compounds. The most
prominent research on uptake of ordinary pollutants has been conducted by
Hill (1971) and continued in collaboration with other workers. Some of Hill's
early results are shown in Table 2. A striking trend of increasing uptake
rate with increasing pollutant solubility in water is evident. Bennett et al.
(1973) have developed a quantitative model for gaseous pollutant sorption by
leaves. They emphasize the importance of estimating the internal solute con-
centration. The prediction of pollutant uptake by vegetation is still in an
uncertain state; however, a good beginning has been made. The extensive litera-
ture on plant physiology offers promise for rapid progress in this area.
Soil—
Soil can be a very effective sink for atmospheric pollutants, especially
if the soil is moist and the pollutants are water soluble. However, carbon
monoxide is virtually insoluble in water and yet is taken up in huge quantities
by the soil as a result of microbiological activity. The discovery of this
179
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TABLE 2. SOLUBILITY IN WATER AND UPTAKE RATE OF POLLUTANTS
Pollutant
CO
NO
°3
N02
so2
Uptake Rate
in Alfalfa*
(mol/m2 s) x 109
0.0
2.1
34.7
39.6
59.0
Solubility
at 20° C
g/100 g
0.00234
0.00625
0.052
decomposes
10.8
*Concentration of the gas in the chamber was 2 x 10 mol/m .
massive sink for CO only a few years ago resulted in a reduction in the esti-
mate of the residence time of carbon monoxide in the atmosphere from 3 to 0.1
yr. This illustrative case of the importance of an overlooked natural re-
moval process in estimating global budgets is considered in detail by Ingersoll
(1972). Quite a number of experimental studies on pollutant uptake by soils
have been conducted. But the variety and complexity of soils, complicated
especially by their biologic character, makes controlled experimentation dif-
ficult and quantitative modeling an awesome objective.
Water—
Water is surely a critical factor in almost all pollutant removal pro-
cesses. It also plays a major role via the direct absorption of water soluble
pollutants in the three-quarters of the earth's surface which is covered by
water. Ammonia is chosen to illustrate this removal mechanism because of its
high solubility (about six times that of SO ; see Table 2). Cattle feedlots
put large amounts of ammonia into the air. Of this, Hutchinson and Viets (1969)
showed that the amount of NH removed by the atmosphere by precipitation was
"insignificant compared to the amount absorbed directly from the air by aqueous
surfaces in the vicinity of cattle feedlots."
In an attempt to quantify the removal of ammonia released from a point
source (such as from a plume produced by a sewage treatment plant) by an aqueous
180
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surface, Calder (1972) developed a simple mathematical model which takes into
consideration the atmospheric transport and diffusion of the plume as well as
a characterization of the removal process. His conclusion was that 20% of the
ammonia could be removed from the plume in passing over a 30-km-long lake at a
wind velocity of 5 m/s. Because of the importance of aqueous sinks (large
lakes, oceans), this paper will deal with this interfacial process in much
greater detail at a later point.
Stone—
Sulfur dioxide in the atmospheric environment has caused inestimable
damage to frescoes, monuments, and other edifices throughout the world. The
damage is a result of enhanced weathering rates caused by the attack of sul-
furic acid (SO + 1/2 o + HO ->• H»SO ) on the carbonate matrix of limestone
and sandstone. Spedding (1969) showed that the SO uptake rate is dependent
upon the moisture level in the atmosphere. The H SO. reacts with the carbonate
matrix to form gypsum as follows:
CaCO + H SO + HO -»• CaSO • 2H 0 + CO
Because the resulting salt is more soluble in water than the carbonate, the
gypsum would be more readily leached out from the stone. The stone would also
be subject to physical disintegration because of the volume expansion that
accompanies the mineral change (Luckat, 1973).
Luckat and Spedding provide values for SO,, removal by stone of 6 to 200
and 50 to 200 mg/m day, respectively. Luckat"s data were obtained in highly
industrial sections of Germany and Spedding had a concentration of 360 yg
SO /m (100 times the worldwide background level) in his experiments.
Taking 5 mg/m day as the lower limit of these data, the total earth's
14 2
surface of 5 x 10 m , and an estimate that 1% of the earth's surface is stone
9
capable of removing SO , the annual removal rate is calculated to be 4.5 x 10
^ <
kg SO /yr. This rate is 5 to 20 times smaller than any of the estimated
rates for other natural SO2 sinks. More details on this calculation and com-
parison are given by Rasmussen et al. (1974). Clearly this process could be
181
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modeled by applying the theory of adsorption and reaction rates to this hetero-
geneous reaction system. However, detailed analysis is not warranted because
of the minor impact of this removal process on global budgets. Indeed, it is
doubtful if it could even be of much consequence on a local scale.
PRINCIPLES
The remainder of this paper will be dedicated to describing principles
and methods for quantitative modeling of the transfer of mass at the atmosphere-
earth interface. The existence of atmospheric transport models which provide
information on temperature, pressure, humidity, winds, pollutant concentrations,
etc. is presumed. Of course, many models exist; see Anthes (1978). However,
most do not incorporate removal at the earth's surface. Consider a recent model
by Shir and Shieh (1975) which has been prepared to be compatible with the on-
going Regional Air Pollution Studies in the St. Louis area: "Many model options
are provided which enable users to study conveniently the significant effects
which these options have on the final concentration distributions." Unfortu-
nately, the options do not include allowance for a finite pollutant flux at
.the surface. Shir and Shieh explain that "absorption of the SO by the ground
surface is neglected because it requires data describing the surface properties
and their absorption rate." They call for investigation of the surface absorp-
tion rate. Allowance for a finite surface flux manifests itself simply as a
boundary condition on any properly constructed atmospheric transport model.
Interfacial Flux Calculation
The calculation of the downward flux of a pollutant at a point on the
earth's surface can be accomplished if the concentrations and eddy diffusivities
are known as functions of height either by prediction or measurement. The pro-
duct of the vertical concentration gradient and the eddy diffusivity is the
flux at any height. Such fluxes can be extrapolated to the interface to find
the desired surface flux. This is easier said than done, because eddy trans-
port coefficients are not easily determined and the vapor phase concentrations
(especially near the interface) are seldom accurately known. Thus this micro-
scopic approach, even when used with parameterized turbulent transport co-
efficients, has not proved very useful in practical calculations. A macroscopic
182
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approach which has found widespread application in chemical engineering, mete-
orology, and other fields is described below.
The vertical flux of a material in a gas phase, F (0), can be represented
by the equation
Fg(0) = kg(z) [Cg(2) - Cg(0) ] {Eq. 1}
The flux is proportional to a concentration driving force [C (z) - C (0)] in
g g
which C (z) is the pollutant concentration at some height z above the sur-
g
face; and C (0) is the concentration in the gas phase right at the interface.
The coefficient of proportionality, k (z), is called the gas phase mass trans-
g
fer coefficient. The indicated z dependence implies that the coefficient must
be selected to be consistent with the height at which the pollutant concentra-
tion is to be measured. The choice is quite arbitrary, although 1 and 10 m
are commonly used in presenting meteorological data. If the gas phase concen-
tration is known by measurement or model prediction at some height and the
gas phase mass transfer coefficient is obtained for the same height from a
correlation, the surface flux can be calculated if the interfacial concentra-
tion is known. Sometimes C (0) is taken to be 0. However, this implies that
g
the nonatmospheric side of the interface has an infinite capacity for the
material and offers no resistance to the rate of mass transfer. In the case
of uptake by rock, soil, and/or vegetation, neither of these assumptions is
likely to be valid. Indeed, it may be the atmospheric phase resistance which
is negligible. These solid phases may have to be characterized by a complex
series of resistances according to their individual peculiarities.
By contrast, an aqueous phase (such as a lake or ocean) can be dealt with
similarly to the gas phase. The flux equation is
F£(0) = k£(z') [C£(0) - C£(z')] (Eq. 2)
where C (0) is the pollutant concentration in the liquid phase right at the
interface; C (z1) is the concentration at some depth z' ; and k (z1) is the
J6 Jo
liquid phase mass transfer coefficient corresponding to the depth at which the
183
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pollutant concentration is known. Again, C (z1) is obtained by measurement
or prediction and k (z1) from a correlation. Correlations for the gas and
liquid phase mass transfer coefficients are the subject of later parts of this
paper. In general, the interfacial concentration C (0) is not easily obtained.
x/
At steady state, there is continuity in the transfer of mass across the
interface. That is, the flux of pollutant through the gas must be equal to
the flux of pollutant through the liquid:
Fg(0) = F£(0) (Eq. 3)
Equating Equations 1 and 2 leaves only two unknowns, the gas and liquid inter-
facial concentrations. A relationship between them is supplied by the postfu-
late that phase equilibrium exists right at the interface. The validity of
this postulate of interfacial equilibrium is well documented for numerous cases.
For any particular case solution, thermodynamics can be used to obtain a quanti-
tative expression such as
C CO) = f [C4CO)1 (Eq. 4)
g
An example of a rigorous calculation is given for SO in equilibrium with
fresh water and with seawater (Rasmussen et al., 1974). A commonly seen form
of Equation 4 is C (0) = H C0 (0), known as Henry's Law.
g x
The calculative procedure is as follows:
a. C (z) and C (z1) must be known or specified,
g x,
b. k (z) and K (z1) must be predicted, and
g J6
c. Equations 1 through 4 are solved simultaneously for C (0), C (0),
g x,
and F (0), the interfacial compositions and the desired absorption
flux.
184
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Deposition Velocity
Equation 1, which effectively defines the gas phase mass transfer co-
efficient, will be recognized as being conceptually identical to Chamberlain's
(1953) original formulation of the now-familiar deposition velocity. Because
the interfacial concentration was usually not known, researchers over the years
have made the perfect sink, C (0) = 0, assumption. Thus, the deposition velo-
city has come to be defined as
> 5)
often without attention to its height dependence. Although (obviously) one
can calculate a deposition velocity for any flux and concentration measurement,
the values obtained have not been successfully correlated in a general sense
(Chamberlain, 1975) . The principal flaw has surely been in the ignoring of
the role of the nonatmospheric phase.
Part of the charm of the deposition velocity is that it has units of
velocity and can be applied to solid particles as well as gases. The easy
visualization of falling particles encourages one to conceive of gases moving
downward with a velocity v . Indeed, diffusional theory (Bird et al., 1960)
gives
nA = °A VA (Eq. 6)
where n is the mass flux of A with respect to stationary coordinates, C is
the concentration of component A, and v is the velocity of species A with
.A
respect to stationary coordinates. Comparing Equations 5 and 6, it appears
that v, and v could be identical if C, and C were taken at the same location.
A g A g
However, all terms in Equation 6 have point values only. Thus n would be the
A
vertical flux at the point where C was measured, whereas F(0) is the vertical
f\
flux at the surface. Hence v is not the same as v and is, in fact, only a '
ratio between two measured quantities, without physical significance except as
a coefficient of proportionality.
185
-------
Although the deposition velocity is firmly entrenched in the meteorology
literature, use of the mass transfer coefficient formalism is strongly urged
for the following reasons:
a. Equation 1 will surely be more successful in correlating surface
fluxes (deposition) than Equation 5.
b. Misleading physical significance is less likely to be attached to
k (z) than to v .
g g
c. Mass transfer coefficients have been shown to be useful in charac-
terizing nonatmospheric phases. The nomenclature given in Equation
2 for the aqueous phase is in widespread use in the oceanography
literature.
Need for Data
Equations 1 and 2 are guides to the experimental determination of the gas
and liquid phase mass transfer coefficients, k (z) and k (z1). For complete
g *
specification of k (z), the flux through the interface and the gas phase con-
centrations at the interface and at some known height must be measured. Better
still, vertical profiles of concentration could be measured. This provides one
way of determining the interfacial concentration. For carefully specified
circumstances, the interfacial gas phase concentration can be obtained from
knowledge of the liquid phase characteristics. An analogous procedure will
suffice for finding k (z1). Much of the data presently available are incomplete
J6
in some way. The height or depth of concentration measurement is unspecified,
an interfacial concentration (usually 0) has been chosen without verification,
and/or the flux has been obtained from an eddy diffusion model rather than
measured directly. Unless all factors are measured, it is like using one
model to develop another. As adequate data become available, correlations
based on fluid-mechanical and physical-chemical considerations (such as those
described below) will provide reliable estimates of the mass transfer coeffi-
cients for use in the boundary conditions on air-quality simulation models.
GAS PHASE MASS TRANSFER COEFFICIENT
In the calculation of deposition of a pollutant from the atmosphere to
any medium at the earth's surface, it is necessary to quantify the mass transfer
186
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through the air near the surface. As described earlier, this can be achieved
if the gas phase mass transfer coefficient can be predicted.
Conceptually, the starting point is the Reynolds analogy, which expresses
a similarity among mass, heat, and momentum transfer. One expression for the
Reynolds analogy is
2
kg(z) = CD u(z) = u.,./u(z) (Eq. 7)
This expression says that the mass transfer coefficient, k (z), is linearly
g
proportional to the horizontal velocity, u(z), which in turn is a function of
height, z, and that the coefficient of proportionality is the familiar drag
2
coefficient, C . Noting that C = [u./u(z)] , Equation 7 can be expressed
D D *
as shown in terms of the friction velocity, u^. The chemical engineering
literature contains many modifications of the Reynolds analogy. One of the
best known is the Chilton-Colburn (1934) equation, which allows for variation
among chemical species by inclusion of the Schmidt number (Sc = u/pD) :
*) ="
" 2/3
S u(z) Sc /J
Owen and Thomson's (1963) research on heat transfer provides a somewhat dif-
ferent correction to allow for the fact that different species may diffuse at
different rates, and (especially) to allow for the fact that surfaces may ex-
hibit a bluff body character not accounted for by the Reynolds analogy. Their
equation is
1
kg(z)
Uj,
(Eq. 9)
In all of these equations, u^ is the familiar friction velocity, which is re-
lated to the shear stress at the surface by u,. = /T /p .
* o
Owen and Thomson's Equation 9 has been accepted by many investigators
who have then tried to obtain correlations for B . One such correlation is
suggested by Dipprey and Sabersky (1963):
187
-------
- 10.25
(Eq. 10)
The Reynolds number in Equation 10 is the roughness Reynolds number,
Re = u. z /v, where v is the kinematic viscosity and z is the so-called
0*0 o
roughness height. Button (1953) has tabulated values for z for various types
of underlying surfaces. Charnock (1955) has developed the following correla-
tion for a water surface:
z = b u/g
o *
(Eq. 11)
where g is the gravitational acceleration and b is a constant for which various
values have been given. One more relationship is required and is provided for
neutral conditions by the logarithmic velocity profile:
u(z) = I
*
(Eq. 12)
In this equation, k is the von Karman constant, usually given a value of 0.4.
For nonneutral conditions, corrections can be made to Equation 12 as required
(Panofsky, 1963).
To obtain a value of the gas phase mass transfer coefficient, the calcula-
tion proceeds as follows." From a value of wind speed u(z) at a single height
z, Equations 11 or 13 and 12 can be solved for z and u,. Note that this allows
o x
specification of the vertical velocity profile via Equation 12. A method for
obtaining u^ directly for airflow over water is provided by Hicks (1973) :
CD =
u(10)
= [0.65 + 0.07 u(10)] x 10
-3
(Eq. 13)
Knowing the appropriate physical properties for the air and the pollutant of
interest, the correction factor B can be obtained from Equation 10. From
the known velocity profile, Equation 12, and the calculated values of u^ and
188
-------
B , the mass transfer coefficient can be determined from Equation 9 for what-
ever height the pollutant concentration is known.
The above development is only illustrative, as many of the ideas have
reached far more sophisticated levels and almost classic form in a variety of
special applications. On the other hand, generalization is lacking when it
_ -i
comes to predicting B and hence k . Figure 1 shows the mass transfer coef-
ficient as a function of the roughness height for a variety of correlations,
including the Reynolds analogy. A neutral atmosphere with a wind velocity at
10 m of 10 m/s was chosen, with SO as the diffusing gas. For smooth surfaces
(low z ) the correlations are in rather good agreement. However, they diverge
o
sharply as roughness increases. To date, proper interpretation of adequate
data has not occurred to reconcile the discrepancies.
LIQUID PHASE MASS TRANSFER COEFFICIENT
The kind of data analysis and correlation evaluation recommended in the
previous section has recently been completed by Brtko (1976) for the liquid
phase. This paper presents only the major features of his work.
Researchers in chemical engineering, oceanography, and other fields have
measured liquid phase mass transfer coefficients. Most often the data have
been interpreted in terms of the thickness of a hypothetical stagnant film
and thus have resisted successful correlation. Danckwerts (1951) proposed a
surface renewal theory which is conceptually attractive but contains a surface
renewal rate parameter which has proved hard to predict a pr-iori-. Recently,
Fortescue and Pearson (1967) and Lament and Scott (1970) have proposed roll
cell mass transfer models based on different characterizations of the turbulence
in the liquid phase.
Brtko and Kabel (1976) have adapted the two models to the situation where
(
the roll cells are the result of wind-induced turbulence in the liquid phase.
The result for the more successful of the two models is
k£(z') = 0.4 Sc
1/2
3
Vu.
kz1
a
w
3/21
1/4
(Eq. 14)
189
-------
M
C
O
•H
0
O
+J
C
0)
-H
U
o
o
s-i
-------
To use this model, only the friction velocity in the air at the free liquid
surface, the depth z1 in the liquid phase at which the concentration is
known, and the temperatures of both phases are required. The friction velocity
is obtained as described earlier. The temperatures enable the estimation of
the physical properties, and k is the von Karman constant.
Data with which to compare Equation 14 are few, and most of those that
do exist are inadequate for this purpose in one way or another. Comparisons
with some of the better defined data are given below. Liss (1973), measuring
the desorption of oxygen in a wind-water tunnel, obtained values of about a
factor of 2 lower than those predicted by Equation 14. Probably the most
reliable field data are those of Broecker and Peng (1971) and Peng et al.
(1974), who measured the efflux of radon 222 from the Atlantic and Pacific
oceans. The field data fall about a factor of 3 above the predictions of
Equation 14. Work is continuing to reconcile the remaining discrepancies.
Nevertheless, the agreement appears satisfactory inasmuch as the models men-
tioned here are the only predictive models known to the author.
OVERVIEW
The importance of accounting for natural sources and sinks of atmospheric
pollutants in air-quality simulation models was discussed. Several removal
processes were described and illustrated with significant examples from the
literature. Some indication of the state of development of quantitative
understanding of each process was given. Emphasis was then placed on those
mechanisms active at the earth's surface. The principles of modeling mass
transfer at interfaces were presented and discussed in the context of common
practice. In particular, for the important case of pollutant transport across
an atmosphere-water interface, a brief evaluation of methods of predicting gas
and liquid phase mass transfer coefficients was attempted.
191
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ACKNOWLEDGMENTS
The author wishes to acknowledge continuing technical input from the
members of the Select Research Group in Air Pollution Meteorology. In parti-
cular, recognition of the contributions to parts of this paper by graduate
students Karen H. Rasmussen, Lawrence B. Hausheer, and Wayne J. Brtko is made.
Special appreciation is extended to the U.S. Environmental Protection
Agency for its financial support of this project via Grant No. 800397, ad-
ministered through the Center for Air Environment Studies and the Department
of Meteorology of The Pennsylvania State University.
NOMENCLATURE
B Measure of deviation from Reynolds analogy, dimensionless
b Proportionality coefficient, dimensionless
C Point solute concentration, yg/m
n
C Drag coefficient, dimensionless
C Atmospheric pollutant concentration, yg/m
C (0) Atmospheric pollutant concentration at gas-liquid interface,
g 3
yg/m
C (z) Atmospheric pollutant concentration at height z, yg/m
g
C (0) Pollutant concentration in liquid phase at gas-liquid interface,
yg/m
C (z1) Pollutant concentration in liquid phase at depth z1 , ug/m
JC
D Molecular diffusivity of pollutant through medium of interest,
m /s
2
F(0) Interfacial flux of pollutant, yg/m s
2
F (0) Flux of pollutant leaving gas phase at interface, yg/m s
g 2
F (0) Flux of pollutant entering liquid phase at interface, yg/m s
X
g Acceleration of gravity, m/s
H Henry's Law constant, yg/m of gas per yg/m of liquid
k von Karman constant equal to 0.4, dimensionless
k (10) Gas phase mass transfer coefficient for 10 m, m/s
g
192
-------
k (z) Gas phase mass transfer coefficient for height z, m/s
k (z1) Liquid phase mass transfer coefficient for depth z1 , m/s
2
n Mass flux of A with respect to stationary coordinates, yg/m s
f\
Re Roughness Reynolds number u. z /v, dimensionless
o * o
S Schmidt number v/D, dimensionless
c
u(10) Gas phase velocity at 10 m, m/s
u(z) Gas phase velocity at height z, m/s
u^ Gas phase friction velocity vr/p, m/s
v Velocity of species A with respect to stationary coordinates,
/\
m/s
v Deposition velocity, m/s
z Height above earth's surface, m
z' Depth beneath water surface, m
z Roughness height, m
o
Greek Letters
y Absolute viscosity of phase of interest, kg/m s
ite
3
2
v Kinematic viscosity of phase of interest vi/p, m /s
p Density of phase of interest, kg/nf
p Density of gas phase, kg/m
a 3
p Density of liquid phase, kg/m
W 2
T Gas phase shear stress at surface, kg/m s
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OBSERVATIONAL REQUIREMENTS FOR VALIDATION OF
AIR POLLUTION METEOROLOGY MODELS
Dennis W. Thomson
Department of Meteorology
INTRODUCTION
Significant advances have been made during the past several years in the
development of three-dimensional regional and finer-scale meteorological models.
One-dimensional models for both the daytime and nighttime planetary boundary
layer may now be run on desk-top calculators and employed routinely for some
air-quality predictions (such as passive contaminant concentrations). We now
also understand in some detail both the practical potential and limitations of
a variety of simplified transport and diffusion models, including box, plume,
puff, and particle-in-cell types. Presently, although several extensive (but
rather specialized) meteorological and air pollution measurement programs are
either in progress or in advanced planning stages, one of the major problems
prolonging the development of comprehensive, predictive meteorological models
for air pollution applications is the lack of measurements and analyzed data
which can be readily used to test and evaluate the models (Anthes, 1978).
Unfortunately, in this case recognizing the need for validation data does
not "half solve the problem." The application of conventional "synoptic" data
systems and analysis techniques on a regional (or smaller) scale is prohibi-
tively expensive. This is a consequence of the required density, both spa-
tially and temporally, of in-sitw surface and aerological measurements.
It appears that many of the problems (including fiscal problems) asso-
ciated with the use of existing operational systems for fine-scale experiments
could be solved through appropriate application of sophisticated, continuously-
operating remote sounding and — perhaps, on an intermittent basis — instrumented
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aircraft observations. With the exceptions of sodar (Sound Detection and Rang-
ing) and path-integrating optical anemometers, however, most of the contemporary
remote-probing systems of potential interest are still in the research and
developmental stages. Neither these systems nor their users are probably pre-
pared to cope with the operational demands of assembling a set of atmospheric
observations of the type which could be used for definitive model validation
and intercomparison experiments.
Several excellent review papers and publications exist which summarize
the capabilities of many direct sensors (Kaimal, 1975; Johnson and Ruff, 1975)
and indirect sounding systems (Derr, 1972; Little, 1972). This paper differs
in that I have attempted to recommend the use of specific meteorological ob-
servations or measurement systems on the basis of model data requirements
rather than system capabilities. Classification of the various models
generally follows that of Anthes (1978). No attempt has been made here to
review the nature and quality of existing climatological and synoptic observa-
tions, for they would presumably be available for use regardless of any parti-
cular local (or regional) air pollution meteorology requirements.
GENERAL CONSIDERATIONS
The meteorological measurements required to determine the time rate of
change of a passive contaminant are determined, either explicitly or impli-
citly, by the concentration equation
i£ -_""*"
3t H dz
+ source and removal terms,
->-
in which the mean and turbulent horizontal transport terms are V*VQ and
n
V «V Q1, respectively; and the mean and turbulent vertical transport terms
H H
are w -^ and W " . The remaining term, QV-V, is the divergence of the
dZ dZ
velocity field. Suitable observations of the space- and time-dependent velocity
198
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field are, obviously, of paramount importance. Viewed in this way, the role
of other basic meteorological variables (e.g., temperature, pressure, radia-
tion, and moisture) is that of completing the basic set of equations which
must be used to determine or predict the mean wind field and turbulent velo-
city fluctuations. The scalar variables appearing in that set of basic
equations, the so-called "primitive equations," include: pressure, tempera-
ture, specific humidity, and the latent heats of evaporation and condensation.
Ideally, in a complete model all of these variables are specified everywhere
within its domain, and on its upper, surface, and four lateral boundaries.
In fact, however, a complete set of observations of all the apparently important
variables will probably never be realized. The task before us, then, is to
determine for a variety of less-than-complete models the minimum number of
measurements which can be used and still enable us to make predictions of
atmospheric structure and behavior that will aid an air pollution meteorologist.
ONE-DIMENSIONAL MIXING-LAYER MODELS
For most daytime conditions, in the absence of significant advection or
subsidence, one-dimensional mixing-layer models are now so refined as to tax
the state-of-the-art measurement systems which may be used to acquire model
validation data (Tennekes, 1978).
Daytime mixing-layer growth, H(t), may be essentially treated as the
result of the time-dependent sensible heat transport, S (t), into the lower
d.
atmosphere. The initial condition of the lower atmosphere is defined by an
early morning temperature profile, T(z,t ). To a lesser extent, the growth
rate also depends upon the variable entrainment rate at the top of the mixing
layer. The temperature profile may be adequately determined using either a
conventional radiosonde (if the trace is carefully analyzed) or — better yet —
one of several new commercial balloon- or kytoon-borne "mini-sondes" (Morris
et al., 1975). Either will provide T(p), from which (using the hypsometric
relation) T(z) may be calculated.
For studies of mixing-layer climatology, S (t) may be sufficiently well
cl
specified by a "mean" surface energy balance (Tennekes and van Ulden, 1974).
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However, prediction of H(t) on a given day requires precise specification of
S (t) . Recall that '"""
GL
S (t) = (l-a)R(t) - eaT 4(t) - E(t) - S (t)
ci S S
At a given site the albedo may be estimated or measured (Sellers, 1965;
Kondratyev, 1969), and the global radiation then monitored using a pyrometer.
Since
R (t) = (l-a)R(t) - eaT 4(t)
LI S
use of a ventilated or dome-type net radiometer to remove uncertainties asso-
ciated with the albedo and outgoing infrared terms would be preferable. A
problem remains, of course, in that local radiation (or energy budget) measure-
ments may not be representative of those for the surrounding area (Dabberdt and
Davis, 1974). The relationship between an areal average of R (t), which will
determine H(t), and observations at a given site can probably be best deter-
mined by experimental comparison of observed and predicted H(t) values.
Since the soil heat flux is normally small [S (t) ^ 0.1 R (t)] and well
behaved, it is possible to parameterize it in terms of R (t) (Blackadar, 1978).
But when a model such as Blackadar1s is used, certain corrections should be
made (for example, for recent rainfall in a normally dry area).
The most difficult term to specify is E(t). It can easily be fourfold
S (t), and strongly depends upon factors such as surface and subsurface soil
cL
characteristics, vegetative cover, and recent rainfall history. Probably
operational measurements of E(t) or S (t) by the "eddy correlation" technique
3.
(w'q1, q'T1) would, due to the delicacy of available sensors, be prohibitively
expensive. For inversion-rise predictions, measurement of the mean vertical
dry and wet bulb temperature gradient so that a Penman-type (Sellers, 1965)
or Bowen ratio technique may be employed is the best apparent solution. The
limitations in these techniques are not likely to give problems during inver-
sion-rise conditions. Highly reliable and sufficiently accurate electronic
bridges for thermistor or resistance elements may be built or purchased for
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a few hundred dollars — a fraction of the cost of many air-quality monitoring
instruments. Given the initial temperature profile and the time-dependent
sensible heat flux, an inversion-rise model such as Tennekes1 (1973) may then
easily be run on a desk-top programmable calculator (Benkley, 1976). Table 1
summarizes data requirements for mixing-layer models and some suggested
measurement techniques.
Measurement of H(t), at least to heights of the order of 1 km, may now
be routinely performed using the sodar acoustic sounding technique (Russel et
al., 1974; Aerovironment, 1976). Several different commercial systems are
now available throughout the world. It is only a matter of time before sodar
will be as familiar a working tool to the air pollution meteorologist as radar
presently is to the severe storms forecaster.
In principle, an additional application of sodar for inversion-rise
predictions exists. Using the relationship derived by Wyngaard et al. (1971)
2
for estimating the vertical sensible heat flux from C (sensed by a mono-
static sodar), it may be possible to obtain S (t) as well as H(t) from a
3.
sodar system. The two are independent measurements in that S (t) is based on
a
signal amplitude at a given range, whereas H(t) is derived from the charac-
teristic rising inversion signature seen on the sodar height-time recordings
(Aerovironment, 1976) . Evaluation of this technique is presently underway at
The Pennsylvania State University.
Much research remains to be done on the modeling of nocturnal and stable
planetary boundary layers. Above homogeneous terrain, techniques such as those
employed by Blackadar (1978) and Wyngaard (1975) show promise. However, in
complex terrain even the simplest meteorological situations will probably
require two- or three-dimensional models.
Our qualitative understanding of stable boundary layers has been greatly
enhanced during the past 8 to 12 years as a consequence of the wealth of sodar,
lidar, and high-resolution radar measurements (e.g., see Ottersten et al.,
1973). The stable mixing layer appears characteristically to exhibit either
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highly erratic behavior or to regularly include wave-type features. Patches
of turbulence, probably generated by shear instability, can generate both
upward and downward momentum and heat fluxes. We do not yet know if the quasi-
periodic "bursts" of turbulence at arbitrary heights as a consequence of local
dynamics in stability are generated at the surface, or are the consequence of
(for example) wave motions aloft.
Blackadar (1978) has modeled the evolution of a nocturnal jet. His
sensitivity studies show that detailed vertical temperature and wind profile
and surface heat flux data would be required if a running model were to be
updated with actual observations. In the case of the nocturnal boundary layer,
the required sampling frequency at several height levels extending through the
jet could be of the order of several minutes. Either sodar or lidar might be
used to continuously monitor nocturnal mixing-layer depth (Russel et al., 1974;
Thomson, 1975). However, neither system can presently provide all the informa-
tion necessary to make predictions of its temporal evolution.
TABLE 1. DATA REQUIREMENTS FOR MIXING-LAYER MODELS
Variable
Sensor
Notable Advantages; Disadvantages
H(t)
T(z,t)
R(t)
Rn(t)
Sa(t), E(t)
S (t)
S
Sodar
Aircraft
Helicopter
Lidar
"Minisonde"
Radiosonde
Pyrometer
Net radiometer
AT, AT , T
w w
Heat flux plates
Continuous, 1-1.5 km max. height
Detailed 'in-s'itu meas. ,- cost, min.
altitude in urban areas
Maneuverability; cost
Outputs particulate loading;
maintenance cost
Optimal design
Reg. NWS obs.; limited resolution
Simple, reliable; solar radiation
only
Fragile
Bowen ratio assumption required
Subject to corrosion and frost action
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BOX MODELS
The concentration of a pollutant predicted by a box model may be written
S
Q =
ivl'D-H(t)
where S is the source strength and D the lateral dimension of the box. For
climatological predictions of average concentrations, box models do very well
(Gifford and Hanna, 1973). If the dynamic response of such a model is also
considered (Lettau, 1970; Tennekes, 1978), it can also be successfully applied
to a variety of more complicated situations. Obviously, however, a box model
breaks down in air stagnation situations when the wind speed falls to 0 or
in circumstances in which significant horizontal or vertical gradients are
present in any of the independent variables.
From an observational point of view (Table 2), a box model has the ad-
vantage that observations from only one location must be representative of
the box (typically, urban-scale). Since the flushing time constant is a few
hours, measurements on that time scale are sufficient. In the meteorological
i->-i
conditions in which a box model may be applied, |V| is a slowly varying param-
eter, and available surface or winds determined from synoptic analyses will
certainly be adequate. The height of the "lid" of the box H(t) may be contin-
uously monitored (using a sodar system) or predicted (using one of the afore-
mentioned models). If a tristatic Doppler sodar system is used, wind speed
and direction at, say, 300 m will be continuously available. In fact, since
many of the Doppler sodars now include minicomputer processors, if the source
term is known, the sodar computer can be programmed to output the model re-
sults as well as the meteorological data. Because the meteorological require-
ments for a box model are minimal, the largest uncertainty in its application
will probably be the time-dependent emissions inventory.
GAUSSIAN PLUME MODELS
In steady-state conditions, application of a Gaussian plume model requires
only a measurement (Table 3) of the mean wind speed and specification of the
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TABLE 2. DATA REQUIREMENTS FOR BOX MODELS
Variable
Sensor
Notable Advantages; Disadvantages
H(t)
|v) wind direction variations (Slade,
1968). The respective standard deviations, aQ and a,, may be either visually
y (p
estimated from a continuous strip chart record or electronically estimated in
a high-pass filter "sigma" meter circuit.
One reason for the success of Pasquill's classification scheme is that
estimates of a and a derived from local instrument measurements are highly
y z
sensitive to site-specific biases. In an urban environment, 100/1 or even
10/1 fetch (particularly independent of direction) rarely exists for wind
direction sensors situated within the surface layer.
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At the present time, one of the most serious measurement deficiencies
in the application of Gaussian plume models is that of poor (or nonexistent)
a and a values at altitudes of 150 to 600 m, the effective stack height for
y z y
the high stacks now in use at many power and large industrial plants. In
principle, there is no reason why such measurements could not be made using a
Doppler sodar system. Both wind speed (including vertical velocities) and
direction, and the variations thereof, may be continuously monitored to heights
in excess of 500-700 m with contemporary systems. Experiments for validating
sodar-derived a and a estimates for high stack prediction applications could
be easily conducted at a large power plant using extended exposure photography
of the plume as a diffusion rate reference (Hogstrbm, 1964; Slade, 1968). For
experimental purposes, it is also possible to obtain detailed plume cross-section
data using a lidar operating in a range-height-indicator mode (Johnson, 1971).
TABLE 3. DATA REQUIREMENTS FOR PLUME MODELS
Variable Sensor Notable Advantages; Disadvantages
(v(z ,t)| Cup anemometer Reliable, surface obs.
Insolation or m . , , _, -,-,,_-,. ^
Trained obs. Observer-dependent quality
Cloudiness
a0(zo,t),
. . Bivane In common use; fragile, limited to
(j> o' tower height
Doppler sodar Continuous; high initial cost
2
H(t) Sodar Bistatic, C sensing, system pref.
for interpretation of nocturnal layer
structure
LAGRANGIAN PUFF MODELS
The previously discussed models may all be applied using observations
from a single location as long as it has been established that the location
is representative of the larger region of interest. However, in the case of
more complicated models which may be applied to interpretation of regional-
scale transport and diffusion, it is necessary to discuss not only the di-
versity of measurements required at a single point but also the number and
siting of network stations or the use of mobile platforms.
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In a Lagrangian "puff" model, the center of mass of a puff of pollutants
is advected along a trajectory. Thus, determination of the three-dimensional
wind field from which trajectories may be computed is essential (Table 4).
TABLE 4. DATA REQUIREMENTS FOR TRAJECTORY AND PUFF MODELS
Variable Sensor Notable Advantages; Disadvantages
V(x,y,z,t) Radiosonde Data analyzed and available; sparse
network network and only 2 per day
NWS upper air Significant local error possible
analyses
Tetroon systems Outputs traj., cost
K(x,y,z,t) Tetroon systems Outputs K along traj., cost
In some situations, such as emission from a high stack — when neither
stagnation nor rapidly changing meteorological conditions are present — tra-
jectory estimates derived from synoptic meteorological analyses may be adequate.
But a puff model is most likely to be applied on a 5- to (perhaps) as much as
400-km scale. Within this domain, underlying terrain characteristics may
strongly influence both the direction of the mean flow, such as the veering
which is often observed east of the Appalachian mountains, or local diffusivity
variations which may result from, for example, orographic convection. At the
present time, the use of constant-level balloons (tetroons) is probably the
best in-situ technique for estimating trajectories on the urban scale (Dickson
and Angell, 1968; Angell, 1975; Gage and Jasperson, 1976). It is also possible
to infer diffusion coefficients from a constant-level balloon system.
The problem of determining regional-scale trajectory variations or local
diffusivity differences using a network of either in-situ or remote observa-
tions is analogous to that of evaluating the vertical temperature gradient on
the basis of two independent mercury-in-glass thermometer readings. In general,
in order to minimize systematic and random errors, measurements of any gradient
are best achieved by either employing a sensor which detects the gradient
directly or by physically moving a single instrument, in this case the tetroons,
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from one location to another. The rms errors of pibal or raob wind measure-
ments (World Meteorological Organization, 1971) are comparable to the magnitude
of urban- to regional-scale variations. Remote probing systems such as Doppler
sodar have, in this case, only the advantage of continuous operation. Varia-
tions in the vector wind must still be determined from the difference of two
"absolute" measurements.
In some cases, urban and mesoscale gradients also may be measured using
an instrumented research aircraft. The advantages and limitations of airborne
measurements are discussed in detail in the context of grid-point models.
GRID-POINT DIFFUSION MODELS
In an Eulerian grid-point diffusion model such as Shir and Shieh's (1974),
values of the required meteorological variables must be specified (at least
initially) at each grid point. Even if it were possible, in principle, to
obtain measurements of the required parameters at every grid point (16,800 in
Shir and Shieh's model), it would be pointless to do so. Most of the time the
mean gradient between adjacent grid points will be well within the noise level
of available instruments. Furthermore, a high spatial or temporal density of
observations for a region within the domain of the model in which "nothing is
happening" can in effect reduce the overall signal-to-noise ratio, since con-
centration of measurement resources there can divert attention from a feature
of interest. In fact, one of the principal advantages of an integrated modeling-
measurement program is that a "real-time" model can be used to guide, for ex-
ample, mobile platforms to specific areas of interest. Lacking a model for
guidance, it is generally advisable to concentrate observational systems in
those locations which are expected to clearly differ either geographically or
temporally.
We note also that, in general, application of a grid-point model does
not imply that the required observations must be conventional meteorological
parameters such as surface temperature and pressure (Table 5). The latter
are required only for truly dynamical models such as Anthes" (1978). Rather,
parameters such as stability, eddy diffusivity, and mixing height are the
important variables normally used for solving the concentration equation.
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Continuous measurement of the mixing height at a number of selected grid
points within a mesoscale region could be performed easily (and economically).
However, it is not at all clear that continuous, quantitative measurements of
eddy diffusivity and atmospheric stability over a significant depth of the mixing
layer can be obtained anywhere without costly instrumented tower or indirect
sounding installations. For urban- or regional-scale areas, there is presently
no alternative to model or statistical parameterization of the diffusion vari-
ables. In the event that not even local estimates of these are available, the
quantities will have to be approximated on the basis of accumulated experience
with surface measurements of pollutants dispersed from local sources or, perhaps,
some tracer experiments (Pasquill, 1974).
TABLE 5. DATA REQUIREMENTS FOR GRID-POINT DIFFUSION MODELS
Variable
Sensor
Notable Advantages; Disadvantages
H(t)
V(x,y,z,t)
K(x,y,z,t)
Sodar
Aircraft
Helicopter
Lidar
Radiosonde
network
NWS upper air
analyses
Tetroon systems
Parameterized in
terms of climatology
or submodel
Continuous, 1-1.5 km max. height
Detailed in-situ meas.; cost, min.
altitude in urban areas
Maneuverability; cost
Outputs particulate loading;
maintenance cost
Data analyzed and available; sparse
network and only 2 per day
Significant local error possible
Outputs traj., cost
Presently, an important deterrent to the wider application of remote
sounding systems is the difficulty of "inverting" the system outputs into the
familiar meteorological variables (such as the vertical temperature profile).
In the author's opinion, more attention should be given to seeking solutions to
the inversion problem which will result in quantitative estimates of selected
208
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air pollution parameters such as vertical diffusivity. For example, using
indirect sensing to estimate the vertical flux of a passive contaminant on the
basis of a derived vertical temperature profile could be both difficult and
highly prone to error. But the same might easily be done on the basis of a
2 2
sodar-derived C or C profile. Then, to obtain adequate flux estimates, a
regional-scale experiment might require only a few systems situated at loca-
tions representative of the major types of underlying terrain.
GRID-POINT DYNAMICAL MODELS
The logistical magnitude of the measurements problem (Table 6) for
initialization of dynamical grid-point models is such that, at least for any
"operational" applications, we have little choice other than to rely on auto-
mated objective synoptic analysis of surface and upper air network data. In
any case, the finer the scale, the less will be the sensitivity of a given
forecast to initial conditions (Anthes, 1978). That means, however, that
precise monitoring of the time-dependent lateral boundary conditions and adequate
measurements of critical parameters at selected interior grid points are
essential.
Determination of a state parameter (such as pressure) at every surface
boundary grid point is certainly not necessary. In fact, instrument-generated
noise or local-terrain-related or nonhydrostatic effects would probably generate
artificial, large-amplitude acoustic gravity waves in a running model. However,
spatially-representative pressure observations on the scale of terrain smoothing
(4 A x) could control the error growth rate of the mean geostrophic wind. Pres-
sure and other meteorological sensors suitable for such a network are now avail-
able in systems such as the Portable Automated Mesonet (PAM) developed at the
National Center for Atmospheric Research.
For regional model test and evaluation programs, lateral boundary measure-
ments of state parameters and winds might be obtained using instrumented air-
craft. Assuming a ground speed of 70 m/s, about 2 1/2 hr would be
required to make a traverse at a single level along one boundary (600 km).
Hence one disadvantage of aircraft for mesoscale observations is that the
209
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measurements time scale does not differ greatly from that of many of the
important processes. Use of an aircraft to obtain only vertical temperature
and humidity profiles is uneconomical, since the operating costs of a vertical
profile to 15,000 ft would be comparable to the launch cost of a conventional
radiosonde. If a high altitude (> 30,000-ft) jet research aircraft is avail-
able, a single pass along a lateral boundary would require less than 1 hr, and
dropsondes (Cole et al., 1973) might be cost-effectively launched to obtain
vertical profile measurements.
Still, of all the measurement systems which can be used for -in-s-itu
aerological observations, instrumented aircraft can provide the most diverse
and highest quality data set (Atmospheric Technology, 1973) . Temperatures to
better than 0.3 K, relative humidities to within better than 5%, and winds to
the order of 5 m/s are all well within the state of the art. By appropri-
ately designing flight patterns for an aircraft equipped with an inertial
guidance system and gust probe, one can also obtain estimates of derived param-
eters such as turbulent sensible and latent heat, and momentum fluxes, divergence
and vorticity estimates, and highly detailed vertical and horizontal gradients
of all the various measured parameters (Lenschow, 1975).
Using an instrumented aircraft, Redford (1976) has shown how the intensity
of turbulence at various levels changes upwind, above, and downwind of the St.
Louis metropolitan area. The primary limiting factor to, for example, air-
craft gradient estimates for studies of model initial-condition or validation,
is the natural small-scale heterogeneity of the atmosphere (Godowitch, 1976).
Many of the data-processing routines which are required for essentially
real-time processing of conventional surface and upper air synoptic observations
have been developed in conjunction with studies related to AFOS (Automation of
Field Operations and Services) (Cahir and Norman, 1975) and the NCAR PAM system.
However, much work (in both hardware and software) remains to be done before
it will be possible to process aircraft and indirect sensing observations in
real time for use in a running regional- or finer-scale model. For example,
consider just the problem of data transmission. On the urban scale, it is
possible to use VHF radio telemetry from an aircraft to a centrally located
210
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TABLE 6. DATA REQUIREMENTS FOR REGIONAL GRID-POINT DYNAMICAL MODELS
Variable
Sensor or System
a) Initial Conditions
T , p and q
(x,y,z,tQ)
V(x,y,z,tQ)
H(t)
b) Lateral Boundary Conditions
P(zo,t)
T, p and q
•
T, p and V
(x,y,z,t)
R(x,y,t)
Rn(x,y,t)
Clouds
d) Surface Boundary Conditions
T, p, q, and V
(x,y,z,t)
a(x,y,z)
Z
NWS surface + upper air analyses
Objective analyses of NWS network data
Raob and sodar network
Linear microbarometer array
Aircraft
Dropsondes
Raob and sodar (for PEL)
NWS upper air analyses
Satellite and surface obs
Satellite obs
Satellite obs
Mesoscale network such as NCAR PAM
Seasonal aircraft reconnaissance
Estimated from land-use survey
data receiver which is wired to a computer. On the regional scale, however,
use of a satellite relay link may be the only reasonable method for transmitting
aircraft observations from any location within the model domain.
Although satellite visible and infrared pictures are tremendously helpful
in interpreting meteorological phenomena and processes on all scales — including
211
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air-pollution-related studies (Lyons, 1975) — only one sophisticated interactive
satellite data-processing system which might be used for air-pollution-related
studies is presently under development. This is the so-called MCIDAS (Smith,
1975) system of the University of Wisconsin Space Science and Engineering
Center. The main difficulty in using quantitative satellite data in real
time is the problem of translating the photographic images into useful numbers
(such as plume trajectories or turbidity coefficients).
To my knowledge, neither air-quality variables nor specific pollutants have
yet been integrated into a three-dimensional dynamical meteorological model.
Consequently, model validation must be based on predictions of common meteoro-
logical variables. Of many possible candidates, the distribution and amount
of precipitation is one which is sensitive both to the parameterization of
physical processes which are also important to pollutant behavior, and to the
accuracy and reliability of the particular computational techniques used.
Consequently, quality rain data from either existing or new gauge networks or
from improved weather radar observations are essential to progress in air pollu-
tion meteorology research. The speed and direction of planetary boundary layer
winds may be a second useful verification variable. By using boundary layer
winds, perhaps measured continuously with Doppler sodar, verification would
be far less sensitive to site-specific observational anomalies and/or to
limitations of the particular surface layer parameterization scheme employed
in the model.
CONSIDERATIONS FOR THE ORGANIZATION OF MESOSCALE EXPERIMENTS
Many of the specific technical components which must be included in planning
a major field experiment are summarized by Johnson and Ruff (1975). But signi-
ficant progress towards an understanding of mesoscale meteorological and air
pollution phenomena does not depend, probably, so much today on organization of
a particular experimental program as on the relation of that program to other
efforts. The present evolutionary trend in experimental meteorology is toward
big programs such as the Global Atmospheric Research Program (GARP), the GARP
Atlantic Tropical Experiment, and the National Hail Research Experiment. Al-
though an urban-scale experiment of limited scope such as the St. Louis Regional
212
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Air Pollution Study (RAPS) can (just) be managed by a single government agency,
the logistics of future mesoscale experiments are such that multiagency support
is probably essential if the effort is not to be subcritical in one or more
areas of study.
Alternating hot and cold discussions are currently underway regarding
several mesoscale experiments. They include the NOAA Severe Environmental
Storms and Mesoscale Experiment; the Electric Power Research Institute Sul-
fate Regional Experiment; an ERDA multistate MAP3S program; and the EPA successor
to RAPS, which is also likely to be concerned with sulfate transportation and
transformation problems.
I would suggest that — while each of the above programs may succeed to
the extent that there will be limited, specific, useful results — conducting
them independently does not represent an optimal use of public resources. If
measurements and scientific resources were pooled, the coverage would not only
increase in proportion to the number of individual components, but also as a
consequence of the interactive (covariance) terms.
In the same sense that successful execution of the GATE field program
required the combined expertise and resources of many nations, successful com-
pletion of a comprehensive national mesoscale experiment will require the un-
selfish cooperation of every possible U.S. agency and institution.
NOMENCLATURE
C 2 temperature structure parameter
C 2 velocity structure parameter
D lateral dimension of box model
E atmospheric latent heat flux
H depth of mixing layer
n level index
p pressure
q specific humidity
Q contaminant concentration
213
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R global radiation
R net radiation
n
S source strength
S atmospheric sensible heat flux
3.
S soil sensible heat flux
s
t time
t initial time
o
T air temperature
T soil surface temperature
+s
V wind
-*•
V horizontal wind
H
w vertical veloctiy
z height
z aerodynamic roughness length
x albedo
e emissivity
9 direction of horizontal wind
a Stefan-Boltzmann constant
a horizontal dispersion coefficient
a vertical dispersion coefficient
Z
4> direction of vertical wind
REFERENCES
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Pasadena, California.
Angell, J. K. 1975. The Use of Tetroons for Probing the Atmospheric
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Anthes, R. A. 1978. Meteorological Aspects of Regional-Scale Air-Quality
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Third Annual Progress Report. U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina.
Atmospheric Technology. 1973. Issue on NCAR Research Aviation Facility.
Benkley, C. 1976. Studies of Planetary Boundary Layer Growth Using Sodar
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Park, Pennsylvania.
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Blackadar, A. K. 1978. High-Resolution Models of the Planetary Boundary
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Annual Progress Report. U.S. Environmental Protection Agency, Research
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Cahir, J. J. , and J. M. Norman. 1975. A Forecaster-Initiated Real-Time
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Cole, H. L., S. Rossby, and P. K. Govind. 1973. The NCAR Wind-Finding
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Dabberdt, W. F., and P. A. Davis. 1974. Determination of Energetic Charac-
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Derr, V. E. 1972. Remote Probing of the Troposphere. C55.602.T75, U.S.
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Dickson, C. R., and J. K. Angell. 1968. Eddy Velocities in the Planetary
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Appl. Meteor. 7:986-993.
Gage, K. S., and W. H. Jasperson. 1976. Diffusion Coefficients Estimated
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Gifford, F. A., and S. R. Hanna. 1973. Modeling Urban Air Pollution.
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Godowitch, J. M. 1976. Case Studies of Aircraft and Helicopter Temperature
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Hogstrom, U. 1964. An Experimental Study on Atmospheric Diffusion. Tellus
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Johnson, W. B. 1971. Lidar Measurements of Plume Diffusion and Aerosol
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American Meteorological Society, Raleigh, North Carolina, April.
Johnson, W. B., and R. E. Ruff. 1975. Observational Systems and Techniques
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Kaimal, J. C. 1975. Sensors and Techniques for Direct Measurement of
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Kondratyev, K. Y. 1969. Radiation in the Atmosphere. Academic Press, New
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217
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-78-049
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
SELECT RESEARCH GROUP IN AIR POLLUTION METEOROLOGY
Third Progress Report
5. REPORT DATE
August 1978
6. PERFORMING ORGANIZATION CODE
7. AUTHOH(S)
R. Anthes, A. Blackadar, R. Kabel, J. Lumley,
H. Tennekes and D. Thompson
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Dept. of Meteorology and Center for Air Environment
Studies.
The Pennsylvania State University
University Park, Pennsylvania 16802
10. PROGRAM ELEMENT NO.
1AA603 AB-02 (FY-78)
11. CONTRACT/GRANT NO.
R-800297
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory - RTF, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Interim 10/74 - 10/76
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT
Six individual investigators, who have conducted different but related meteor-
ological research, present in-depth technical reviews of their work. Prime con-
clusions are that (1) a scale analysis shows that different models are necessary
for meteorological processes on urban, regional and global scales; (2) for high
resolution models of the nocturnal planetary boundary layer, K theory models are
very efficient, realistic and useful; (3) the mixing height has a significant
effect on climatology models; (4) second moment closure methods are useful for
convective situations, properly testing counter gradient fluxes; (5) natural
sources and pollutants acting at the surface of the earth are important for air
quality simulation models; and (6) a combination of conventional micrometeorological
and acoustic sounder techniques are sufficient for verifying locally applied inver-
sion use, box, and regional scale modles, and urban and regional models require a
variety of in situ and remote observation.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
* Air pollution
* Meteorology
* Mathematical models
* Boundary layer
13B
04B
04A
12A
20D
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport}
UNCLASSIFIED
21. NO. OF PAGES
232
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
218
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