RESEARCH ON DIFFUSION IN ATMOSPHERIC BOUNDARY LAYERS:
A POSITION PAPER ON STATUS AND NEEDS
by
Gary A. Briggs
and
Francis S. Binkowski
Meteorology and Assessment Division
Atmospheric Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
ATMOSPHERIC SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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NOTICE
The information in this document has been funded by the United States
Environmental Protection Agency. It has been subject to the Agency's
peer and administrative review, and it has been approved for publication
as aln EPA document. Mention of trade names or commercial products does
not constitute endorsement or recommendation for use.
The authors, Gary A. Rriggs and Francis S. Binkowski, are on assignment
to the Atmospheric Sciences Research Laboratory, U.S. Environmental
Protection Agency, from the National Oceanic and Atmospheric Administration
(NOAA), U.S. Department of Commerce.
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ABSTRACT
The introduction of a new understanding of atmospheric boundary
layers (ABLs) has caused a major change in the view of the diffusion of
pollutants. The turbulence parameters now standard in ABL work, have
provided a method for systematically organizing diffusion parameters.
Concurrently with these advances, alternatives to the operational models
have emerged, but existing experimental data sets are inadequate for
model comparisons and evaluations. The most important knowledge gap is
the lack of an adequate specification of the relevant meteorology both at
the point of release and downwind. A second major inadequacy is experimental
measurements of plume characteristics up to 100 km from the release point.
There is also a great need for formulating new operational models based
upon this newly acquired experimental data and the new alternative
approaches. Finally, it is recognized that a modest but steady effort
is necessary.
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EXECUTIVE SUMMARY
INTRODUCTION
The purpose of this study is to review the current state of science
concerning diffusion within the atmospheric boundary layer (ABL) and to
identify major issues and research needs. The study is limited to
considering only the diffusion of non-buoyant, conservative substances
over homogeneous and level or moderately hilly terrain. Thus, chemical
transformation, wet and dry deposition, buoyancy effects, and diffusion
in complex terrain are all topics which are excluded. Certainly these
are very important topics both from the point of view of the research
scientist and from that of a regulator; however, each of these topics
deserve a review study of the same scope as attempted here. The topics
that will be included here are: a review of diffusion experiments and
models, and a discussion of research needs. We begin, however, with a
highly simplified description of the major meteorological factors affecting
diffusion. The main report contains the necessary details and references.
METEOROLOGICAL FACTORS CONTROLLING DIFFUSION
Material released from the ground or from a stack is mixed and
transported by air motion. This process dilutes the material and carries
it horizontally away from the source. This mixing process is called
turbulent diffusion, or simply diffusion. The turbulence responsible
for this mixing may result from the wind blowing over the ground, generating
turbulent eddies, or it may result from heated parcels of air or thermals
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that are generated in air heated by contact with ground which has been
warmed by the sun. Thus, we distinguish between "mechanical" and
"convective" turbulence, because in addition to having different generative
mechanisms, they have considerably different diffusive properties.
The current view of the ABL divides it into three conceptual states,
the neutral boundary layer (NBL), the convective boundary layer (CBL), and
the stable boundary layer (SBL). The NBL occurs when there is a negligible
transfer of heat from the ground surface to the air. This occurs with
strong winds and usually when most of the sky is covered with clouds. Only
mechanical turbulence is generated in the NBL which extends up to or even
into the cloud bases. The SBL grows when the ground is substantially
cooler than the air above it causing a stable profile to develop. This
means that an upward moving parcel of air, for example, experiences a
downward restoring force because it is colder, therefore denser, than its
surroundings. Mechanical turbulence is generated as the wind blows over
the ground, but vertical motions are very much restricted by the stability.
Vertical diffusion, consequently, is restricted and plumes of effluent
dilute slowly during transport downwind. The depth of turbulent mixing
is also very restricted in the SBL, being of the order of 100 m or
less. It is under these conditions that some of the most serious air
pollution episodes can occur. The CBL occurs in the daytime, with low to
moderate wind speeds and clear to partly cloudy conditions. Then the
ground is warmed and thermals rise from the surface to generate convective
turbulence. The diffusion process is dominated by the vertical transfer
of material in the thermals and downdrafts. The height to which
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these thermals rise is called the mixing depth. As a result of this
efficient type of mixing, the CBL often has a "well mixed" condition in
which the vertical profile of a pollutant becomes constant from a few
meters above the surface to the mixing depth.
On a day without many clouds, and with neither fronts nor shorelines
nearby, the diurnal course of the ABL is as follows: starting about an
hour before sunrise, the ground has cooled to its minimum temperature.
The temperature profile is stable but there is some vertically restricted
mechanical turbulence from the surface to the top of the SBL. Any plumes
within this SBL are very well defined. Any plumes above the SBL have
virtually no vertical dilution but may be spread horizontally by meanders
caused by large scale horizontal eddies. As the sun comes up, thermals
begin to rise from the ground but must work against the stable temperature
profile until all the air in the developing CBL is warmed and mixed. The
CBL, thus formed, grows until the diminished surface heating in the late
afternoon can no longer overturn stable air above the CBL. Typically,
the maximum mixing depth is of the order of a kilometer. Any plumes
beneath this height released into the previously stable air are mixed
downward to the ground, a process referred to as fumigation. Late in the
afternoon, the strength of the thermals diminishes. Soon a new stable
layer is established at the surface and a new SBL forms. The transitional
times around sunrise and sunset are as yet poorly understood. Little is
known about the structure of the ABL and even less about diffusion during
these periods.
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In a typical diurnal cycle, the large changes in stability correspond
to large changes in diffusion rates. Some method of indicating the
ambient stability is necessary and there are several indices of stability
currently used. The index most indicative of the physical processes at
the surface is the Obukhov length, L. By convention L is defined to be
negative with an upward flux of heat at the surface (unstable conditions).
In principle, this length is defined such that at height z = L above the
ground, the buoyant production (or destruction) and the mechanical
production of turbulent kinetic energy are comparable. Hence, above z = L,
the turbulent state of the ABL is dominated by buoyancy effects and
below z = L, the turbulent state of the ARL is dominated by the mechanical
overturning caused by the wind blowing over the surface. Profiles of
mean wind speed and other meteorological quantities are functions of z/L
near the surface.
Besides the Obukhov length, other quantities of importance in describing
the turbulence in the ABL are the surface heat flux, the surface friction
velocity scale u*, and the mixing depth. A velocity scale analogous to
u* has been developed for the CBL. Called the convective velocity scale,
w*, it is essentially defined by the surface heat (or buoyancy) flux the
depth of the CRL; w* has proven to characterize the turbulent velocities
within the CBL very well. The depth of the CRL divided by L serves as a
useful indicator of the relative importance of the thermals as a major
transfer mechanism; the larger this ratio is, the more important thermals
are to the description of diffusion. As fundamental as these variables
are to our understanding of ABL processes, u* and the surface heat flux
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are not easily measured; both are needed to determine L using its
definition. An alternative index related to L, called the Richardson
number, Ri, uses a temperature difference and velocity differences between
two heights: in another form known as the bulk Richardson number, only
one wind measurement is used. There are simple methods to convert from
Ri to Obukhov length. An estimate of the site roughness length, z0, is
necessary with the bulk Ri approach, while the velocity differences
approach strongly demands wind sensors that are properly calibrated. With
the temperature differences, and at least one wind velocity, the same
methods that yield the Obukhov length can be used to obtain u* and the
heat flux. The only other essential measurement needed for characterizing
ABL turbulence and diffusion is the mixing depth. This can be obtained
from good profiles of temperature and humidity. In the case of the SBL,
a profile of wind speed is also required. In recent years it has become
possible to estimate this depth by using remote sensors such as lidars and
acoustic sounders. Lateral diffusion at large distances is enhanced by
wind direction changes with height, which are best determined from wind
velocity soundings.
EXPERIMENTAL AND THEORETICAL BASES FOR OUR UNDERSTANDING OF DIFFUSION
The current understanding of diffusion processes is based upon
results from field experiments, laboratory experiments and theoretical
models. Field experiments may be conveniently grouped into two classes,
those in which the tracer material is released from very near the surface
and those in which the release is elevated. The data from previous
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experiments has been very useful in formulating our current understanding
of the diffusion process but there are major inadequacies. First, the
tracers used in most early field studies were not conservative. This means
that as a cloud or plume of tracer was transported downwind, the material
stuck to the ground or deposited, thus reducing the concentration further
downwind by an uncertain amount. The theoretical view of diffusion
prevelant at the time of the earlier experiments was that the vertical
profile of concentration was Gaussian. If this were true and the tracer
were conservative, the standard deviation of the concentration, Sigma-z,
is easily obtained from ground samplers alone. When deposition occurs,
however, the Sigma-z estimate is biased by an unknown amount to be an
overestimate. A second difficulty with these data is that with one
notable exception, the Prairie Grass experiment done in 1956, the
experiments did not take sufficient meteorological data for either
interpretation using contemporary concept or for validation of current
diffusion models. Two recent experiments, one in Copenhagen, Denmark
and one under EPA sponsorship in Boulder, CO, had extensive meteorological
data. The Danish experiment used a conservative tracer, while the EPA
sponsored experiment used remote sensing to get concentration profiles at
several locations downwind of the source. This experiment, named Convective
Diffusion Observed with Remote Sensors (CONDORS), was designed to verify
some new insights into CBL diffusion that were gained from numerical
experiments and laboratory studies.
Laboratory studies of ABL diffusion are gaining wider acceptance as
the scientific community has learned how to simulate ABL processes more
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faithfully. The major problem faced in a laboratory simulation is to
maintain the proper balance of forces and yet attain the required
reduction in scale. In strictly neutral conditions, this is not too
difficult. One problem is that if the scale reduction factor is very
large, viscous effects smooth out turbulent eddies in the laboratory
which would not be smoothed in the atmosphere. This is usually overcome
by making the bottom of the wind tunnel have a rougher texture than strict
scaling would predict. A major difficulty is properly simulating buoyancy
effects. Since gravity cannot be easily scaled down on the earth's
surface, density differences must be exaggerated. This distorts other
aspects of the simulation so that technical trade-offs are necessary.
The most effective use of laboratory work then is to examine a particular,
but important aspect of the diffusion problem. One such application
which illustrates this point is the work done at the EPA Fluid Modeling
Facility that showed that a tall slender building required a shorter
stack for good plume dilution than would have been predicted by using
building height alone.
Experimental work alone is insufficient for an understanding of
diffusion. A proper conceptual framework is also necessary. Most current
applications of diffusion models are based upon the Gaussian distribution
in both vertical and horizontal directions in order to characterize the
diffusion. The fiaussian distributions are based upon empirical data
collected primarily during near surface releases over relatively flat
terrain. Unfortunately, most of the model applications are for situations
that were not considered in the experimental situations which provided
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the empirical data for the formulation of these models. Finally, for
diffusion of non-buoyant releases within a CBL, the Gaussian assumption
for the vertical distribution of material was recently shown to be
incorrect. Current practice has adjusted the necessary parameters so
that surface concentrations are nearly correct for the CBL case even
though the vertical profile is not. There are however, several modeling
methods that have been demonstrated to be superior for particular applications,
In the interest of brevity, only three examples will be mentioned. The
first is the statistical method which relates the standard deviations of
the concentration distribution directly to the observed turbulent
intensities of the wind field. Since the statistics of the wind field tend
to be fairly uniform in the horizontal direction, this method has the
greatest success in characterizing the horizontal concentration distribution
for transport over an area of homogeneous land-use. A second approach is
to simulate the diffusion as a random walk or "Monte Carlo" process.
Recent work in this area has allowed for vertical and horizontal
inhomogeneities in the flow field. A disadvantage to this approach is
that it generates individual material trajectories and then constructs
the distributions directly. For realistic cases, a very large number of
trajectories must be calculated. Finally, a third method is numerical
simulation of the diffusion process by solution of the relevant governing
partial differential equations in an Eulerian framework. There has been
great progress made in formulating these models which require a number of
simplifying assumptions. For example the laboratory work on surface
releases in the CBL inspired a set of numerical simulations that in turn
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included a prediction about unexpected behavior of an elevated release.
The numerical work predicted that the plume centerline for a non-buoyant
release would descend to the surface, then rise again. This prediction
inspired further laboratory work that verified the finding. The CONDORS
experiment was conducted specifically to confirm the numerical and
laboratory findings for diffusion in the CBL. Here is an example of how
the various methods for studying diffusion interact in the discovery and
confirmation of major new findings in diffusion theory. All of the work
mentioned in this example was supported by EPA.
NEEDS FOR FUTURE WORK
Before listing the future needs it is imperative to state at the
outset that the human and fiscal resources for these endeavors are not
trivial. A modest, steady commitment of personnel and funds for an extended
period is necessary, as the work described requires a sustained scientific
effort.
The first major need is for field data for selected meteorological
scenarios for the development, testing and evaluation of diffusion
meteorology models. The characterization of those meteorological variables
important to the diffusion process is a major impediment to the implementation
of new and emerging approaches to diffusion characterization. To meet
these needs requires detailed profiles of the relevant variables within
the first kilometer or so of the atmosphere. In addition, sufficient
measurements are needed to characterize the surface energy balance, the
forcing of the wind fields by the large scale migratory pressure systems,
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the horizontal temperature field and the cloud conditions. These studies
should be conducted over a variety of terrain types using a minimum of
preparation and personnel. This would allow the study of the meteorological
events of interest at the locations of interest. A portable instrumented
tower, a tethersonde (a tethered balloon carrying an instrument package),
and an acoustic sounder would be the basic tools for such studies. The
development of such meteorological field studies would greatly facilitate
the transfer of theoretical results into practical operational models
useful for decision making.
A second major need is for the comparison, evaluation and construction
of new operational models. An interdisciplinary team with skills in
meteorology, chemistry, numerical modeling and statistics is required to
meet this need. Such a team would evaluate the models, not only with
respect to experimental data, but also with respect to the expected
uncertainty in the model inputs. These type of comparisons are necessary
to make improvements in the operational models.
A third major need is for diffusion experiments, both in the field
and in laboratory settings. The laboratory studies are needed to test
theoretical results in specific simplified situations that are free of
confounding influences. The field data are needed because there is a
wide disparity between the flat homogeneous land where the best of the
previous field studies have been conducted and the complex woodlands and
urban-suburban developments typical of the American landscape. One
example of important information to be gained from such field studies is
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the characterization of the concentration field at 100 km downwind from
the source. Current models extrapolate to this distance based upon
measurements to only 3 km. Vertical and lateral profiles of concentration
even at widely spaced plume transects would improve the current situation
considerably. Other examples include the characterization of releases
made during the hours of transition, that is near sunrise and sunset,
as the behavior of the atmosphere during these periods is very poorly
understood. For any of these examples, a good diffusion experiment is of
necessity a good meteorological experiment, since a full complement of
meteorological measurements is essential to the success of the experiment
and to the continued usefulness of the data set.
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CONTENTS
Abstract iii
Executive Summary v
Table and Figures xix
Abbreviations and Symbols xx
Acknowledgements xxiii
1. Introduction, Purpose and Scope 1
2. Atmospheric Boundary Layers 3
Important Meteorological Variables and Scales
Affecting Diffusion 5
The Diurnal Cycle and Transitional Periods 13
Neutral Boundary Layers 26
Convective Boundary Layers 29
Stable Boundary Layers 35
3. Diffusion Experiments 47
Field Experiments 49
Laboratory Experiments 74
4. Diffusion Modeling as Practiced 87
The Gaussian Plume - Its Utility and Limitations 87
Empirical Diffusion Coefficients and Stability Typing . . 91
Evaluations of Sigma Curves and Typing Schemes 103
5. Diffusion Modeling Alternatives 109
Surface Layer Similarity Models Ill
Gradient Transfer and Higher Order Closure Models .... 115
Convective Scaling Models 109
Large Eddy Simulation Models 130
Statistical Models 134
Random Perturbation Models 143
6. What Are the Needs? 149
Experimental Versus Modeling Needs 149
What Kind of Experiments are Needed? 152
Meteorological Measurements 156
Evaluation and Concentration Fluctuation Issues 160
Stable Boundary Layer Issues 163
Convective Boundary Layer Issues 168
Calms and Other Inconvenient Events 171
Resource Issues 172
7. Implementation of Improved Modeling Practices 175
Estimating Needed Parameters from More
Obtainable Measurements 176
Building Better Operational Models 180
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8. Assessment 183
Atmospheric Boundary Layers 184
Diffusion Experiments 186
Diffusion Modeling as Practiced 190
Alternative Modeling Approaches 194
Needs for Diffusion Experiments 200
Research Modeling Needs 206
Technical Transfer Needs 207
References 210
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TABLES
Number Page
1 Summary of Tracer Experiments in Homogeneous
Terrains 52-53
FIGURES
Temporal behavior of measurements in a CBL at the
Boulder Atmospheric Observatory, Colorado, USA
13 September 1983 20
Temporal behavior of measurements in a SBL at
Cabauw, The Netherlands, 30-31 May 1978 23
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LIST OF ABBREVIATIONS AND SYMBOLS
ABL Atmospheric Boundary Layer
b Oimensionless constant in Eq. 2.5 for h, SBL
C Concentration (generalized)
CBL Convective Boundary Layer
D Wind direction
f Coriolis parameter
g Gravitational acceleration
h Mixed depth height for NBL or SBL
H* Turbulent flux of buoyant acceleration at the surface:
H* = (g/V) ^TT1"
i Subscript, "inversion"
k von Karman's constant, = 0.4
K Eddy diffusivity (generalized) (K^, heat) (Km, momentum)
L Obukhov length: L = -u*3/(kH*)
n A generalized exponent
N Brunt-Vaisala frequency: N? = (g/9)d9/az
NBL Neutral Boundary Layer
o Subscript refers to surface value
p Atmospheric pressure
P/G/T Pasquill/Gifford/Turner
P0 Standard atmospheric pressure (1000 mb)
0 Source strength (rate of release)
r Correlation coefficient
Re Reynolds number
Rj Gradient Richardson number
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% Bulk Richardson number
SBL Stable Boundary Layer
t Time
T Absolute temperature
Te Eulerian time scale for ABL turbulence
T] Lagrangian time scale for ABL turbulence
Tv Virtual absolute temperature
u Wind speed in the downwind direction (usually defined by
mean wind direction at source height).
TJ" Mean wind speed in the downwind direction
Ug Geostrophic wind speed - the forcing formation for the wind
O ^"
u* Friction velocity: u**- = -w'u1 = TO
U Average "u through mixed depth
v Wind speed in the lateral direction
w Wind speed in the vertical direction
w* Convective scale velocity: w* = (H*z^)^'^
x Distance downwind of source
y Lateral distance from plume center!ine
z Height above surface
ZT Mixing depth in CBL, usually capped by an inversion
Z0 Roughness length
Zs Source height
p Average slope of a site
9 Potential temperature, T(pQ/p)2/7
A0 Difference in 9
aa Standard deviation of azimuth angle of wind
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ae Standard deviation of elevation angle of wind
au, av, aw Standard deviation of wind components u, v, or w
av, ay, az Standard deviation of concentration in lateral (y)
and vertical (z) direction
p Density of air or fluid
T Horizontal kinematic stress, time scale
X Concentration
v Kinematic viscosity
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ACKNOWLEDGEMENTS
The authors wish to express their deep appreciation to John S. Irwin
of the Meteorology and Assessment Division for his penetrating review and
constructive suggestions. We also express our profound gratitude to
Barbara Hinton and Brian Eder for their Herculean labors in typing this
manuscript and preparing the figures, respectively. Finally, since no
task can be finished without supervisory support and guidance, we wish to
take this opportunity to thank John F. Clarke and Francis A. Schiermeier
for their patience and support, especially during the final phase of
delivery of this document.
Further we wish to thank Evelyn Poole-Kober for her work in obtaining
the references and for proofreading the manuscript.
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SECTION 1
INTRODUCTION, PURPOSE AND SCOPE
The basis of most applied diffusion modeling today, excluding buoyancy
effects, was developed in the late 1950s and early 1960s. These Gaussian-
empirical models were built on the results of a few field experiments
that were quite limited in scope and that had problems with the tracers
employed and instrumental accuracy, although they represented the state of
the art at the time. Those results have been extrapolated to distances 15
to 125 times as large as the range of measurements and to more extreme
stabilities; furthermore, they have been applied to much rougher or much
more hilly sites and to much higher sources than used in the original
experiments. Some of these extrapolations were guided by statistical
theory, but most were pencil-and-straightedge or freehand extrapolations.
On this basis, hundreds of multi-million dollar decisions were made and
continue to be made. .
This extrapolative situation, based on such limited experimental
data, did not come about for lack of progress in diffusion science since
1960. In Section 3, over 20 major field experiments and numerous labora-
tory experiment are reviewed that were made after the two experiments that
support most applied diffusion modeling today. In Section 2, large advances
in knowledge about atmospheric boundary layers that have been made since
1970 are reviewed; these advances provide much support for better diffusion
modeling. In Section 5, it is seen that classical diffusion theories have
been further extended and that many new theoretical techniques with
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considerable validation have been developed as a result of availability
of faster and cheaper computers. Hundreds of new papers on diffusion
experiments, theory, and model validation appear every year in conference
volumes or in journals. Thus, much progress has occurred since 1960 at
the research level.
There is a perception in the scientific community that the best of
the newest work is not being assimilated into the regulatory process in a
rapid enough manner. A counterview is that to effect the transfer of new
technology into the regulatory process requires more than a demonstration
of a new idea with a single data set; a proof of superiority over existing
techniques in wide variety of cases is also necessary. Historically,
only a small fraction of the resources for research has been committed
for this transfer.
The purpose of this paper is to provide an evaluation of what has
been accomplished by past diffusion research and what most needs to be
done. The scope is purposely limited to "basic" atmospheric boundary-layer
diffusion, leaving out the complications introduced by source buoyancy,
plume deposition, chemical reactivity, and complex flows due to structures,
large terrain features, and land-water boundaries. This limit is imposed
because basic boundary layer diffusion, in itself, involves complex
phenomena and a large body of literature (the few hundred references used
in this report represent only the most classic or most recent research
advances). Also, a good understanding of basic diffusion must be the
starting point for advances in modeling the more complex diffusion problems,
Each of these problems can best be addressed separately, by specialists,
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as some already have. Most basic and specialized areas in diffusion research
were reviewed in Atmospheric Science and Power Production (Randerson,
1984), and complex terrain models were recently assessed by Schiermeier (1984)
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SECTION 2
ATMOSPHERIC BOUNDARY LAYERS
What have we learned about their dispersive properties?
The atmospheric boundary layer (ABL) refers to that part of the
atmosphere directly affected by turbulent mixing near the surface. Heat,
water vapor, pollutants, and momentum mix through this layer much more
readily than through the air above it, which is only sporadically turbulent
(in clouds and in "clear air turbulence"). This layer of continuous
turbulent mixing, the ABL, is virtually always present, although its depth
ranges widely from day to night, season to season, and place to place (it
is absent only on windless nights in topographic "bowls" of cold, dense
air). It is also the part of the atmosphere in which nearly all human
activity takes place, including release of pollutants. Fortunately for us,
turbulence is effective at dispersing and diluting pollutants; without it,
molecular diffusion would diffuse a point-source plume to a height and
width of only 4 m in 24 hr (edges defined by 50% of center-line concentration),
Unfortunately for those people trying to predict turbulent diffusion, the
rate at which it proceeds varies quite widely, depending on wind, atmospheric
stability, the height of the ABL, the height of release of pollutants and
surface characteristics in a rather complex way.
The ABL may be viewed as the place where the large scale pressure
systems constituting "weather" interface with the planet's surface. The
geostropic wind, ug, is a measure of forcing by the pressure field, which
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changes as the large migratory systems ("highs" and "lows") move across the
land. The surface fluxes of heat, moisture, momentum, and pollutants into
or out of the ABL are all strongly influenced by surface cover and local
land-use (a pasture has very different texture and thermal properties from
those of an industrial area, for example). It is in the ABL that the
influences of large scale forcing and local surface fluxes are recounciled;
this process accounts for the great variability in wind, temperature, and
diffusion rates which we observe.
The discussion that follows begins with definitions and explanations
of all meteorological variables important to diffusion in ARLs. Then the
typical diurnal cycle of meteoroloigcal change and some of its effects on
diffusion is described, with particular attention to transitional periods.
Finally, current understanding of ABLs, their turbulence structure and
their diffusive abilities are described. Because stability has the greatest
effect on diffusion rates and because there are basic structural differences
in the turbulence of stable boundary layers (SBLs), neutral boundary layers
(NBLs), and unstable or "convective" boundary layers (CBLs), it is convenient
to consider these separately.
IMPORTANT METEOROLOGICAL VARIABLES AND SCALES AFFECTING DIFFUSION
Regardless of the stability, the same basic collection of meteorological
variables are of prime concern for diffusion modeling. They will be
introduced in this section. Overbars indicate a time average, and primes
indicate deviations from the time average. Subscript "o" refers to surface
values.
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There are several ways to measure stability. One measure is the
static stability, which in an incompressible fluid is -dp/dz, where p
is the fluid density and z is height above the surface. This can be used as
a stability measure in air, but the more easily measured quantity is dT/az,
where T is absolute temperature (air is nearly an ideal gas, so at constant
pressure, p « 1/T). Because air is compressible, a better stability
measure is 38/dz, where 9 is the "potential" temperature defined
by 9 = T(p0/p)2/f7 (p is ambient pressure, and p0 is a standard, sea level
value, usually 1000 mb). This quantity compensates for heating and cooling
due to pressure changes, so that if air rises or falls without radiating or
absorbing heat its 9 is constant, i.e., 9 is a conservative property (this
type of process is called "adiabatic"). If 59/az = 0, the stability
is "neutral", meaning that if air is displaced vertically, it remains at
the same 9, T, and p as ambient air on that constant pressure surface (p
is primarily a function of z); thus, it develops no buoyancy. However,
if 59/dz > 0, upwardly displaced air will find itself surrounded by
higher 9, higher T, and lower density air; therefore it will be heavier
than the ambient air. This displaced air acquires buoyancy, or negative
buoyancy, which tends to return it to its original height; hence, the air
is "stable". If d9/az < 0, the opposite happens, and displaced air
acquires buoyancy acting in the same direction as the displacement. This
displaced air tends to keep rising or falling, even accelerating, and the
air is called "unstable". Turbulence develops spontaneously in unstable
air; this is called "convective" turbulence.
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There is a simple relationship between the vertical gradients of 9 and
T, derivable from the definition of 9, the ideal gas equation of state,
and the hydrostatic equation, ap/az = - pg, where g is gravitational
acceleration (this equation follows from the fact that static pressure is
just the weight per unit area of the air overhead). For the earth's
atmosphere, the result is
as/az = (aT/az + o.ni°c/m) (p0/p)2/7 (2.1)
Naturally, the larger the 39/az, the more stable the air is;
0 < 39/az < 0.01°C/m is considered slightly stable; 39/az > 0.01°C/m
is usually considered moderately stable, unless a9/az > 0.03°C/m,
which is roughly the dividing line for "very" stable. However, static
stability is not the only consideration. A vertical gradient of mean wind
speed, aU/az, tends to encourage overturning and the development of
turbulence. If equal amounts of air from two layers mix, one layer with a
mean wind speed TT = uj and the other with U = u;?, the conservation of momentum
principle gives u" = (u^ + U2)/2 for the mixture. However, the kinetic energy
(KE) of mean motion, U^/2, is less than the original amount, (u^/2 + U2^/2)/2
by the amount (uj> - u^)^/4. This surplus KE feeds turbulent motion, i.e., it
becomes turbulent KE. This source of turbulent KE is present whenever aU/az
is not zero, regardless of sign. These considerations lead to a very
useful stability index, the gradient Richardson number:
Ri = (g/9)(ae/az)/(au/az)2 (2.2)
This number is dimension!ess; it compares the potential energy increase needed
to mix stable air with the surplus KE that would be available after mixing
-------
due to au/az. When Ri > 1, turbulence is extinguished rapidly, as
there is not enough surplus KE to drive air mixing "uphill" against the
positive a0/az. At Ri = 0.5 there is an approximate energy balance,
and at Ri < 0.25, turbulence develops spontaneously from the slightest
perturbation, even if ae/az is positive. Thus, Ri is a physically
meaningful stability index; it also highly correlates with diffusion rates
in certain cases, which are discussed in Section 4.
Wind speed and its fluctuations are very important quantities in
diffusion modeling because they have four effects: transport, axial dilution
(for continuous releases), shearing of the plume or puff, and turbulence-
induced growth and dilution. The usual meteorological convention is to
define x as the longitudinal horizontal spacial coordinate (extending in
the direction of the mean wind), y as the lateral horizontal coordinate
(perpendicular to the mean wind), and z as height above the surface. For a
point source, we define x = 0 and y = 0 at the source. The corresponding
components of velocity are u, v, and w, respectively. The direction of x
is generally defined by the mean wind direction at the source height, so
that v" = 0 at that height. However, if the wind direction turns with
height, a7/az * 0; this tends to increase the plume width with distance,
as the plume top is transported in one direction, and the plume bottom is
transported in another. Turbulent velocities are defined by deviations
from the time average, e.g., v1 = v - 7. Averaged over the same time span,
or "sampling time", we must have U7", "v"1", and w1" = 0. The mean motion in
the atmosphere closely parallels the ground, so over flat ground we usually
can assume that ^7=0 also. Turbulence velocity standard deviations are
8
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defined by av = (v'v1 j, etc. These are very important quantities in
diffusion modeling, as the classical statistical theory prediction (Taylor,
1921) for diffusion close to the source is well proven:
-------
friction effects, the roughness length, z0. It can be determined experimen-
tally from wind speed profiles near the ground, and it scales roughly as
about 1/10 the height of most "drag elements" on the surface (trees, grass,
crops, buildings, etc.); z0 can be crudely estimated by inspection of a
site (Weiringa, 1980). The velocity scales Ug and u* are roughly proportional
but Ug/u* is influenced by z0 and by the stability.
Turbulent velocities and diffusion near the surface, especially their
vertical components, are strongly correlated with u* and the flux of buoyancy
due to heating or cooling of the surface. This flux by (g/Tv)w'T'v = H*,
where Tv is the virtual absolute temperature, which includes the effect
of water vapor on air density (Tv = T/(l - 0.6q), where q is the specific
humidity). Except over very moist surfaces, the humidity effect is small
and H* may be determined by using only the "dry" temperature, T, in place of
Tv. H* represents the turbulent flux of positive buoyant acceleration
caused by heating of the air (or vaporizing of water) next to the ground;
it is proportional to the upward sensible heat flux plus about 1/7 of the
latent heat flux (Briggs, 1985a). When the surface is colder than the
surface air, the heat flux is downward, and H* is negative. H* is also
equal to the rate per unit mass at which turbulent energy is created by
buoyancy; it is the rate of potential energy release in vertical, turbulent
motions. When H* is positive, 99/5z becomes negative due to stronger
heating near the surface, and convective turbulence is produced. When H*
is negative, 99/az becomes positive due to stronger cooling near the
surface, and H* represents a rate of loss of turbulent energy. When
turbulence is produced by wind shear, it is called mechanical. The
10
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rate of mechanical production of turbulent energy is tbu/az, so
u*^ = TO also relates to turbulence production. Both u* and H* can
be measured using fast-response measurements of u1, w1, and T' (or Tv').
Together, u* and H* form an important length scale, the Obukhov length
(Monin, 1959), L = -u*3/(kH*), where k = 0.4, the von Karman constant
(Hogstrom, 1985); this length is defined negative when H* is positive
(unstable conditions). Near the surface, it is well established that
aU/az = k-iuVz, so the ratio of buoyant to mechanical production of
turbulent energy is H*/(u*2dTT/az = -z/L. Thus, z = |L| is a dividing
line between dominance of buoyancy effects (above |L|) and mechanical
effects (below |L|). According to surface-layer similarity theory, dis-
cussed here in Section 5, aU/az = u*/(kz) times a function of z/L and
(g/9)ae/dz = H*/(ku*z) times another function of z/L; these hypotheses
have been well supported by field measurements (e.g., Businger et al.,
1971). It follows that in the surface layer (the lowest 10% or so of the
ABL), R-j is a function of z/L. It is also possible to relate either
Ri or L to the Bulk Richardson number, R^ « AG/TT, by using surface-layer
similarity equations for IT and 9 (Irwin and Binkowski, 1981). Both L and
Ri have been shown to be highly correlated with vertical diffusion from
surface sources (e.g., Weber et al., 1977). This is consistent with the
fact that vertical turbulence quantities, e.g., aw/u*, have been shown to
be functions of z/L.
The wind speed gradient in the surface layer, kzau/az = u*m,
where m is a function of z/L, can be integrated to give
11
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u = (u*/k)[ln(z/z0) - <|,m], (2.3)
where <|>m is a function of z/L, given by the integral of (1 - m)d1n(z/L).
This equation provides a way to calculate u for an elevated release using
U measured closer to the surface, if we can measure or estimate L.
The remaining parameter that is of great importance in describing the
diffusion potential is the height of turbulence mixing in the ABL. For
CBLs this is usually designated as zj (most CRLs are capped by a temperature
inversion, i.e., positive aT/az, hence the subscript "i"). For SBLs this
is usually designated as h, which will be used here for NBLs also. First of
all, it is important to know whether an elevated source is within or above the
mixing layer, as there is normally no turbulent diffusion above this layer.
Except in clouds and in occasional patches of "clear air turbulence", the
atmosphere above the ABL is always stable and non-turbulent, regardless of
stability swings at the surface (the upper air is not subject to intense
heating and cooling like the air next to the surface). Especially on clear
nights with low Ug, when h can be quite small, elevated and even medium-level
sources may be above h. Second, turbulence intensity and structure is a
function of z/z^ or z/h; the intensity falls off rapidly with height near
z-j or h. Third, in the CBL it has been found that the size of the largest
turbulent eddies approximates zj and that cru and av are proportional to
the scale velocity w* = (H*z1-)^' throughout the mixing layer.
In summary, in the order of discussion, the meteorological variables
and scales most relevant to diffusion are aT/az, 39/az, Ri, TJ", V,
aw» f» ug» u*» zo» H*» L» and zi or n-
12
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THE DIURNAL CYCLE AND TRANSITIONAL PERIODS
Before discussing neutral, convective, and stable boundary layers
considered separately, it is worthwhile describing what is known about the
normal diurnal sequence of stability changes in the ABL. This description
excludes overcast, windy conditions, which tend towards neutral regardless
of the time of day. In the course of a typical day, the ABL goes through a
wide range of structure, being stable at night and convective during most
of the daylight hours. During the transitional periods around sunrise and
sunset, rapid changes occur. Rapid changes also occur during thunderstorms,
at the onset of sea breezes, and with frontal passages. Because these
latter events are accomplished by brisk wind speeds, they are not likely to
be causes of serious pollution episodes. The steady state assumptions
customarily made in diffusion modeling can be more or less justified for
1-hr averaging during the major phases of the 24-hr cycle, but have little
validity during the transitional periods, as will be seen. These
transitional periods have received less study than the major ABL states,
but they can be described at least qualitatively.
A good time to begin this description is the hour before sunrise,
because this is ordinarily a time of steady state. Radiative cooling has
created stable stratification in the lowest 300 to 1000 m, and there may be
a remnant of the inversion above the previous day's maximum z-j (radiative
cooling can intensify the inversion if there was much haze below that ZT).
There is stronger stability near the surface and a shallow mixed layer to
h produced either by (1) wind shear-generated turbulence, especially
13
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when Ug is large, (2) slope flows generated by cooling of the surface air
and gravity, especially when Ug is small, or (3) convective mixing produced
by more intense radiational cooling at the top of a fog extending to depth
h, especially when Ug is small in humid conditions on clear nights.
When sunlight first reaches the surface, at low angle, most of its
energy goes at first into evaporating dewfall or fog; this latent heat
produces some increase in buoyancy, as water vapor is lighter than air, but
only about 1/7 the buoyancy produced by the sensible heating of air.
Consequently, there is not much convective activity until most of the
liquid water or frost has evaporated. This may take no time at all in a
desert, about an hour for typical dewfall or frost in a humid climate, or
half a day for a deep fog. After this evaporation is accomplished, a much
larger fraction of solar energy ends up as sensible heat in the air next to
the surface. The air becomes lighter and rises; small thermals converge to
produce larger ones, and these impinge against the top of the mixed layer,
overshoot a little into the stable air above, mix with some of it, and
bring it back down as the mixture subsides and feeds into downdrafts. This
is the beginning of the development of the daytime CBL. The thermals cause
large undulations in Zj, especially in the first hours of CBL development.
If there happens to be an elevated layer containing pollutants, as occurs
when a buoyant plume rises and stratifies in non-turbulent air at night,
the pollutants, too, are entrained by invading thermals and are mixed down
to the ground in the "inversion breakup fumigation" process. The depth z-j
continues to grow until early to midafternoon when solar heating and H*
begin to diminish. Sometimes during the day ZT appears to "jump", as
14
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thermals can traverse a near-neutral layer aloft quite easily once they
reach nearly the same potential temperature as the neutral layer. The
average 9 is nearly uniform through the bulk of the mixed layer, and it
increases steadily through the day as long as H* is positive (until about
1 h before sunset). The heating rate diminishes through midday and afternoon
as H* decreases and the depth being heated, z-j, maximizes. The negative
39/az layer near the ground reaches maximum in intensity near noon,
when H* is at a maximum. As H* decreases in midafternoon, this negative
a9/az decreases, so the surface temperature may begin to decrease even
though H* is still positive and 9 is still slowly increasing above 0.1 z-j.
Exactly what happens in the CBL in the very late afternoon is still an
area of speculation. In clear conditions, H* goes to zero about an hour
before sunset, as outgoing thermal radiation from the warmed surface nearly
balances incoming solar radiation. The CBL may begin to collapse before
H* goes to zero as the thermals become weak. Convective downdrafts outside
the thermals have a slight stable stratification due to positive
39/at = -wa9/az. When new thermals are very weak, they cannot pene-
trate through this stable stratification all the way to the previous z-j, so
the mixing depth descends. It is not known how rapidly this occurs. The
continuation of positive moisture flux from the surface during z-j collapse
may produce a negative humidity gradient, which leads to more rapid radiative
cooling at lower heights, and further stable stratification that will
hasten the demise of turbulence aloft. In dry conditions, the CRL collapse
process may go slower. Without cooling due to radiative flux divergence
(due mostly to humidity), the stratification remains close to neutral to z-j
15
-------
and the turbulence gradually "runs down", with a time scale ~ ZT/CJW, on
the order of 1 hr in the late day.
While what remains of the C8L aloft is running down or collapsing due
to stratification developing from radiation flux divergence, the SRL begins
development before sunset, as soon as H* turns from positive to negative.
As the sun sinks, incoming solar radiation falls to zero, while outgoing
long-wave radiation from the surface continues all night. Because the surface
emits almost like a black body, this radiation flux is « Ts4, where TS is
the absolute surface temperature. In spite of nocturnal cooling, absolute
temperature does not change much, so the radiation balance at the surface is
rather constant on clear nights (if a cloud deck moves over the site, it
radiates back like a black body at slightly smaller T^, so the outgoing
surface radiation is almost balanced and the net radiation is reduced
drastically). The net radiative cooling at the surface is divided between
cooling the ground slab, condensing or freezing water in any dew, fog, or
frost, and cooling the air next to the ground. If it is very humid, as
after a summer shower, fog or dew can form right away, but usually this
does not occur until later at night as the surface temperature approaches
the dew point; of course, this does not happen at all in dry conditions.
The soil heat flux to the surface is fairly constant through the night, but
the surface cooling rate decreases as the heat conducts out of an increasingly
deeper ground slab (Garratt and Brost, 1981).
Surface parameters as well as SRL structure evolve rapidly in the
first few hours after transition. This is evident from the results obtained
16
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with second-order closure ABL models (Wyngaard, 1975; Brost and Wyngaard,
1978), which appear to give good simulations of observations from the 1973
Minnesota experiment (Rao and Snodgrass, 1978; Wyngaard, 1975). The heat
flux becomes increasingly negative up to a point, about 1 hr after transition,
and then gradually diminishes in magnitude in spite of the relative constancy
of the net radiation flux. At the same time, u*, L, and h rapidly reduce
from their values at t = 0 (transition), but they may approach a rough
steady state in 3 or 4 h, if not interrupted by a turbulent burst (a sporadic
episode of turbulent mixing due to changes in the structure of Ri). This
probably happens because the increasing stability near the surface restricts
turbulent motions, so that u* drops, and as turbulence at the surface
diminishes, the heat transfer from surface to air is restricted, which
causes H* to drop. (Heat must first pass through a "laminar sublayer"
whose thickness is proportional to v/u*; this is the same insulating
layer that our bodies lose at higher wind speeds, hence, the "wind chill"
factor.) The bulk Richardson number for the whole layer gives some insight
into the behavior of h. Wind shear can support turbulence working against
stable stratification only as long as Rb = (g/9)(9^ - e^n/u^ is
less than 0.3 or so. The surface air cools more rapidly than air higher
up, so if ITh remains constant, the increasing (eh - QO)» which increases
the "heaviness" of the surface air, pulls h downward. Of course, tTn can
also change as h moves downward, or as momentum is transferred in gravity
waves or due to changes in driving forces like Ugf. Thus, h can behave
rather irregularly through a night (e.g., see Fig. 2). In some of the
Minnesota runs, Ug dropped during the night and turbulence became immeasurable
17
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after about midnight. In such cases, we might as well forget about turbulent
diffusion until sunrise, and just try to determine in which direction a
plume of pollutants is heading and whether it is spreading due to horizontal
eddies or meandering. In this situation, there is no substitute for wind
speed and direction measured near the point of release.
Several efforts have been made to mathematically simulate the complete
diurnal behavior of the ARL, e.g., Yamada and Mel lor (1975) and Binkowski
(1983). Roth of these simulations show a rapid decay of CRL turbulence in
the late afternoon, and predict a maximum in the negative surface heat flux
in the SBL about 2 hr after transition to negative values.
The temporal evolution of a fairly typical CBL, as shown by actual
measurements, is plotted in Fig. 1. The measurements are from the Boulder
Atmospheric Observatory of NOAA's Wave Propagation Laboratory during the
CONDORS experiment (Moninger et al., 1983). The day, 13 Sept. 1983, was
cloudless and very convective; u" dropped as low as 2 m/s during the midday,
giving Tj/w* = 1 and |z-j/l | exceeding 100. Fig. la shows the development of
z-j and T at 10 m. For about an hour after sunrise there is evidence of
continuation of a drainage wind from the west up to a height of about 100 m
(this flow is peculiar at this site and probably originates from the east
slope of the Rocky Mountains, which begin 25 km to the west). This flow
breaks up soon after the heat flux turns positive (Fig. Ib) and ZT climbs
more or less steadily until noon, at which time it seems to reach its limit
of = 1000 m for that day (z] sometimes leveled off for several hours during
18
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19
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20
-------
this experiment, and on some days it appeared to grow in stair-step fashion).
The scatter in the lidar measurements, which were made at various azimuths
and shot elevations, give an idea of the magnitude of undulations in local
z-j, perhaps ± 100 m. In Fig. Ib we see that the sensible heat flux (« w'T1)
is very much controlled by the insolation, although it lags behind insolation
in the first hours of daylight; this lag may be due to both outgoing thermal
radiation from the surface to the clear sky and channeling of solar energy
into evaporation of wetness in the ground cover (predominantly deep grass).
Figure Ic shows turbulence velocities at 10 and 200 m together with w*
and u*. .Although there is much scatter in the av values and the aw
values at 200 m, after ZT passes through 200 m at about 8 AM, they appear
to be essentially controlled by w*. The 20-min average u* values are much
smaller and fluctuate widely. Note that w* is nearly constant from 10:10
to 14:10. In contrast, av and
-------
hours. Before CBL development, the wind was from the north at ~5.6 m/s
at 200 m and from the west at ~2.1 m/s at 10 m, in the presumed drainage flow.
The wind directions become easterly (from the east) at midday, probably due
to forcing from the upslope flow on the east slope of the Rockies to the
west, and phenomenon absent at sites far from mountains. The wind speed at
200 m is quite steady until 2-5 approaches that height at 07:20; then it
drops and fluctuates widely during the most convective hours until it
regains some of its steadiness at w'T' and w* become negligble about 1 hr
before sunset.
The temporal evolution of a SBL at a flat site (Cabauw, the Netherlands),
as shown by measurements, is plotted in Fig. 2. The night, 30-31 May 1978,
was clear and the large scale driving force was fairly constant, as is
shown by the geostrophic wind speed and direction in Fig. 2d. In spite of
this we see in Fig. 2a large oscillations in h, as determined by the top of
the layer with significant downward heat flux. Note that the acoustic
sounder or "sodar" representation of h is in general agreement, but does
not fluctuate as widely as h determined by w'T1 < 0.001°C-m/s (this
determination was made by extrapolation in the range 200-270 m). The
temperature at 10 m falls almost steadily until 45 min after sunrise, and
then rises rapidly and steadily. Various energy fluxes near the surface
are plotted in Fig. 2b. The net (incoming minus outgoing) radiative flux
crosses to negative about 1 1/4 hr before sunset and then becomes practically
constant through this clear night; it crosses to positive about 1 1/4 hr
after sunrise, rapidly increasing. The tranpirative response of the grass
and trees may account for the continued upward flux of water vapor (positive
22
-------
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23
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latent heat flux) to 1 hr after sunset and its quick reversal back to upward
flux minutes after sunrise. The early transition from positive to negative
heat flux, about 3 hr before sundown, is surprising to us (the authors);
perhaps the grass surface is cooling due to transpiration more rapidly than
it absorbs solar heat (late afternoons in summer, grass often feels cool to
bare feet). The delay in positive sensible heat flux until 2 1/2 hr after
sunrise mirrors the evening transition behavior; this could be due to wind-
induced evaporation of dew. During the SBL hours, Fig. 2b shows a striking
correspondence between sensible heat flux and u*, supporting the constant
T* = w'Tr/u* approximation suggested by van Olden and Holtslag (1983).
In Fig. 2c we see that variations through the night in turbulent
velocities also strongly follow variations in u*, especially the velocity
measurements at z = 20 m. However, aw at 200 m tends to not respond to
u* and remains at a near constant low value during periods of h < 200 m
(roughly 1900-2300 and 0200-0700 GMT). These small
-------
10 m rises with u*); less stability causes less suppression of vertical
turbulence, which tends to increase surface friction. Immediately after
transition to positive heat flux at 0600 (Fig. 2b) we see the rapid develop-
ment of a CRL (Fig. 2a); in its early development, it is capped by a thick
layer of negative heat flux.
In Fig. 2d we see relatively small differences between 20- and 200-m
values of wind speed and wind direction just before evening transition and
increasingly independent behavior afterward. This characterizes a principle
difference between CBLs and SBLs. As h drops below 200 m, 71200 approaches
and oscillates around ug, the geostrophic wind speed. (Some of these
oscillations might be due to variations in the local Ug or pressure field;
the measured values represent Ug on a large scale.) In constrast, ^20 changes
little during this night and is seemingly indifferent to changes in other
variables. After evening transition the wind direction at 20 m begins
turning to the left of the geostrophic wind, which means that it turns toward
low pressure; this is normal behavior near the surface in a SBL. This wind
direction difference begins to diminish in the morning at the same time
that TIQ and u* begin to rise, indicating increased coupling with upper
flow at this time. The wind direction at 200 m approaches the large scale
geostrophic wind closely until 2300, then turns to the right nearly 50°, an
event for which we have no explanation.
Figures 1 and 2 illustrate dramatically the many complex changes that
occur in ABLs around the times of morning and evening transition. They are
also illustrative of many of the understood features of quasi-steady state
25
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CBLs and SBLs, so will be useful references for the C8L and SBL subsections
to follow. In addition to the understood features, measurement records
like these often reveal events we cannot readily explain, such as the brief,
marked increases in magnitudes of turbulence variables at Ifi30 MST in Fig. 1
and at 0345 GMT in Fig. 2.
NEUTRAL BOUNDARY LAYERS
A neutral boundary layer is one in which buoyancy forces are negligible.
Thus, 56/az = 0 throughout a NBL and temperature drops 1°C per 100 m
increase in elevation. For this condition to hold near the surface, H* must
be zero or very small because at small z/L, 59/5Z = (9/g)H*/(ku*z).
Also, the production (or dissipation) of turbulent energy by buoyant forces
must be small compared to the production by mechanical forces (wind shear);
this requires that h < |L|. This condition is met rather infrequently,
which is perhaps why there has not been as much research on NBLs as on CBLs
and SBLs. As L « u*3/H*, a combination of high winds and negligible heating
or cooling of the ground is required to produce a neutral boundary layer.
This latter requirement is most likely to be met during overcast conditions,
when very little solar radiation reaches the ground and most upward long-
wave radiation from the surface is balanced by downward radiation from the
cloud base (clouds and most surfaces radiate approximately as black bodies).
There are also moments of H* = 0 just after sunrise and just before sunset
in the normal diurnal cycle when it is not overcast, as H* changes sign from
positive in the day to negative at night. This condition is so transient
that the ARL does not have time to adjust to any kind of neutral structure.
26
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The NBL is driven by the pressure gradient; the magnitude of the
acceleration in the direction of the gradient is Ugf, which follows from
the definition of ug. Balancing this when the NBL is in a steady state
(no acceleration) are the Coriolis accelerations TTf, directed to the right
of the wind direction in the northern hemisphere, and fractional drag.
This drag is given by a-t/az, where T is the horizontal stress; T = u*2 at
the surface and goes to zero at the top of the NBL. Near the surface, U
is small, so the Coriolis acceleration is weak and aT/az nearly
balances the pressure gradient acceleration, Ugf; higher up, the Coriolis
acceleration is stronger and a-t/az becomes weaker, so the wind turns
more to the right and approaches the geostropic wind near z = h. The two
velocity scales ug and u* are more or less proportional, but ratio is
influenced by the surface Rossby number, u*/(fz0). The ratio Ug/u* ranges
from about 20 over rough surfaces to about 50 over very smooth surfaces
(water and snow) in neutral conditions (Counihan, 1975).
Turbulence velocities in the NBL are most closely related to u*. The
vertical velocity standard deviation, given by crw = (w'w1 J1/2, is about
1.3u* near the surface (Hanna, 1981). The lateral (across the wind) and
longitudinal (along the wind) components of velocity variance, ov and au,
respectively, also scale to u*, but observations show some variability in
the ratios crv/u* and au/u* at different sites. This may be caused by
surface inhomogeneities or topographic influences; av/u* ranges from
about 1.3 to 2.6 and au/u* ranges from about 2.1 to 2.9 near the surface
(Lumley and Panofsky, 1964). Observations of these quantities high up in
the NBL are lacking, but Hanna (1981) suggested exponentially decreasing
27
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values using the scaling height u*/f; this is as an approximate fit to a
second-order closure numerical model of the ABL (Wyngaard et al., 1974).
Hanna also suggested an explicit form for the Lagrangian time scale; this
is roughly the time it takes diffusing particles to "forget" the turbulent
velocity they experienced at release, and it is an important time scale in
diffusion theory (especially statistical theories.)
The height of the NBL is of concern, since it is where turbulence
velocities drop off and where the wind speed approaches the geostrophic
value. It is usually assumed that h « u*/f, but this can only be true if
99/dz = 0 through a deeper layer, so that h is not limited by stable capping.
In this case, Deardorff's (1972) large eddy numerical modeling indicates
that H = Ug at a height 0.2u*/f, but turbulence velocities fall off gradu-
ally, rather than abruptly. His results are in rough agreement with Hanna's
(1981) approximations, av/u* = aw/u* = 1.3 exp(-2zf/u*), except that
Deardorff's av does not fall off significantly from its surface value
until z > n,17u*/f. Typically, u* = 0.5 m/s with IT = 5 m/s at z = 10 m, and
f = 10~4 s-1 at midlatitudes, so 0.2u*/f is ~ 1000 m. Obviously, u*/f cannot
be the correct form of h near the equator, because there f = 0. NBLs can
also be limited by stable capping at z = z-j, just like CBLs, especially when
z-jf/u* is small. Krishna and Arya (1981) investigated this case with an
eddy diffusivity model. They concluded that there is no significant
entrainment at z-j when Zi > 0.2u*/f, but there is significant entrainment
when Zj = 0.05u*/f, along with other changes in boundary layer structure.
Yokoyama et al., (1977) analyzed aircraft turbulence measurements in five
early-morning to mid-morning NBLs capped by an inversion, for which hf/u*
28
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ranged from 0.09 to 0.16 (h was defined by the zero intercept of measured
turbulence energy dissipation rate profiles; h * zj within +150 m and -50 m).
For these boundary layers, aw was observed to be proportional to (1 - z/h),
a linear dropoff, in contrast to the exponential dropoff predicted by NBL
models with z^f/u* = 0.45 or larger.
CONVECTIVE BOUNDARY LAYERS
Our understanding of CBLs, which prevail during the daytime, has
greatly improved since convective scaling with z^ and w* = (H*z1-) '^ was
proposed by Deardorff (1970a, b). Experiments in CBLs in Minnesota (Izumi
and Caughey, 1976; Kaimal et al., 1976), England (Caughey and Palmer, 1979),
France and Spain (Druilhet et al., 1983), and other places, including over-
sea measurements, have confirmed the utility of this scaling for most
turbulence-related quantities, including diffusion rates (for a review of
the development of convective scaling and its application to diffusion, see
Briggs, 1985a). The associated temperature scale, 9* = w'T'/w*, has also
proven useful for describing the temperature profile and temperature
variances in the lower half of CBLs. For a comprehensive review of knowledge
of CBLs as of 1981, see Caughey (1981). Unreferenced information in the
discussion below can be found in Caughey's review.
An ABL with turbulence driven mainly by buoyancy, rather than by wind
shear, is considered convective. Usually, the buoyancy arises from the
surface due to heating by solar radiation absorbed at the surface. However,
occasionally an "inverted" CBL develops due to cooling at the top of a
cloud or a fog layer produced by outgoing long-wave radiation; because the
29
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magnitude of this heat flux is considerably smaller than the typical heat
flux produced at the ground by solar heating, wind speed must be fairly low
for convective effects to dominate. Cloud-topped CBLs and convective mixing
in fogs have received very little attention.
For the more usual CBL, with turbulence driven by heating of the
surface, highly efficient vertical mixing is observed to develop; major
updrafts and downdrafts extend through nearly the whole layer (Willis and
Deardorff, 1976b). The following are some consequences of this vigorous
vertical mixing: (1) horizontal momentum becomes well-mixed, so that there
normally is negligible wind shear from z/zf = 0.1 to 0.9, (2) potential
temperature becomes well-mixed, but there is an increasingly negative d0/5z
towards the surface, and (3) passive material becomes almost uniformly
mixed in the vertical in a time = 4 z-j/w* after release, typically about a
half hour. The CBL is capped by stable air; waves in this air and undulations
in Zi, are produced by the impingement of stronger updrafts against the
stable capping layer. These strong updrafts also entrain some of the
warmer, stable air aloft and bring it back down into the CBL, creating a
negative heat flux at the top of the CBL; this usually ranges from 0.1 to
0.4 times the surface heat flux in magnitude. Another consequence of this
entrainment is that significant horizontal momentum is brought into the
CBL; if there is a large difference between the wind in the CBL and the
wind just above z-j, this entrainment can cause considerable wind shear in
the upper part of the CBL.
30
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In CBLs, velocity variances are most closely related to w*, at least
when Zj > 20|L|. (The ratio z-j/|L| is considered the index characterizing
the degree of convectiveness. A z-j/|L| > 50 can be considered "strongly"
convective; however, using a numerical model, Deardorff (1972) found
significant changes from NBL turbulence structure at z-j/|L| = 1.5, which is
considered only weakly convective.) Panofsky et al. (1977) fit measured
values of both |L| but < 0.1 zi (provided
that z-j/|L| » 10), and aw = w* times a function of z/z\ at z > 0.1 z-j.
For the lowest 10% of the CBL, Panofsky et al. (1977) suggest
aw = (2.2u*3 + 2.3 H*z)1/3. Empirical fits to field observations for the
W
upper CBL aw have been suggested by Irwin (1979) and Hicks (1981).
31
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One aspect of vertical velocities in the CBL that has considerable
consequences for diffusion is that they are strongly non-Gaussian in
distribution, at least through the bulk of the mid-CBL. Lamb (1981) has
shown this from his numerical modeling, and field measurements also show
highly skewed w distributions. For z/z-j = 1/4 and 1/2, the mode of Lamb's
distributions is at w « -0.5w*. This agrees well with the observed descent
rate of average plume centerlines from elevated sources, which will be
discussed later in this report. Lamb's interpretation of this, which has
wide support in field observations, is that downdrafts occupy a larger
percent of the horizontal area than do updrafts (thermals); the updraft/
downdraft area ratio is about 0.4/0.6. Updrafts tend to have about 1.5
times as large absolute velocity, so the condition w" = 0 is satisfied. The
significant skewness in w1 distributions makes conventional Gaussian plume
modeling unrealistic for CBLs.
Hanna (1981) has given relationships for Lagrangian time scales in the
CBL; basically they are ZT/W* times functions of z/z-j. Because updraft and
downdraft velocities average about 0.6w* and 0.4w*, respectively, the time
it takes for a passive substance to traverse z-j once up and once down is
approximately 4z-j/w*, which is about how long it takes concentrations to
become almost uniformly mixed. For turbulence measurements from fixed
sensors (Eulerian measurements), the peak energy content is observed at a
frequency = 1.3z-j/IT for the horizontal components and approaches 1.5 Zi/U
for w1 in the upper part of the CBL (Kaimal et al., 1976). This is in
rough agreement with Deardorff and Willis1 (1976b) laboratory observations
that typical downdraft "cell" sizes were » 1.5 z-j. One implication of
3?
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these observations is that for good ensemble averaging of any quantity in
the CBL, including concentration, one should average over a period many
times 1.5 z-j/IT, to average out the effects of individual eddies; 1.5 z-j/IT
typically ranges from 1 to 15 min.
The height T.\ begins at about the height of the nocturnal boundary
layer at sunrise and increases rapidly through the morning once heat flux,
or H*, becomes significantly positive (this may be delayed until most dewfall
or surface wetness from precipitation has been evaporated). As a first
approximation, the growth of z-j can be predicted by using an early morning
temperature sounding and the time-integrated surface heat flux and by assuming
that all the heat flux is accounted for in warming of the air beneath z-j from
its initial temperature to a uniform potential temperature, 9s(z-j)
(subscript "s" refers to the sunrise potential temperature profile). Such
a model is called an "encroachment" model. In differential form, it can be
written simply as
Zi(d9s/az)i dzi = (vTHdt, (2.4)
where w'T1 is proportional to the sensible heat flux at the surface and
(59s/az)j is the morning profile value at z = Zj . According to Tennekes
(1984), the encroachment method handles about 80% of the Zi prediction
problem; adding an additional 20% heat flux as a crude approximation of the
negative heat flux at z\ adds another 10% accuracy, and further refinements
in the prediction method "tend to get lost in the unavoidable inaccuracies
of most experiments."
33
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Cumulus clouds often develop just above the mean mixing depth, especially
in the afternoon on humid days. These are caused by the transport by thermals
of humid air from near the surface to its lifting condensation level (LCL).
Vertical motion in CBLs is nearly adiabatic, so air cools nearly 1°C for
every 100 m of rise. At first, only the most vigorous thermals containing
the largest absolute humidities reach the LCL to form small scattered
cumuli. As z-j grows, more and more thermal s reach the LCL and the percentage
of sky cover by cumuli increases; this phenomenum has been discussed and
modeled by Wilde et al., 1985. One consequence of this mechanism in very
humid weather is that, at some point during the day, nearly the whole sky
becomes covered with cumulus clouds, which limits the solar radiation
reaching the ground and, in turn, limits further growth of z-j.
Depending on conditions above z-j, "cloud venting" (Ching et al., 1983)
may occur, causing transport of air and pollutants out of the mixed layer
into the overlying stable air. This venting occurs when "forced" clouds
(Stull, 1985) are carried by their vertical momentum past z-j to a level of
free convection (LFC), at which point the latent heat released by condensation
of water vapor is sufficient to cause the clouds to develop positive buoyancy
relative to the surrounding dry air. (The existence of a LFC depends on
the surface humidity and the mean temperature profile.) When this happens,
the clouds become "active" (Stull, 1985) and develop vertically into cumu-
lonimlus clouds. The rate at which CBL air can be vented aloft by these
clouds has not yet been quantified, but Issac et al. (1984) have estimated
that up to 90% per hour of air below cumulus cloud bases enters the clouds
in some conditions; much of this air subsides and returns to the CBL,
34
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but some significant chemical changes can occur during the wet phase.
The CBL tends to "collapse" in the late afternoon as its driving force,
the positive surface heat flux, is removed. Little is known about how this
occurs. (This phenomenon was discussed in the subsection on "The Diurnal
Cycle and Transitional Periods.")
STABLE BOUNDARY LAYERS
Research on SBLs has lagged behind the advances made for CBLs, but
much progress has been made in the last seven years. Part of the reason
for delayed progress was the difficulty of developing instrumentation that
responded adequately to the very low turbulence levels that occur in SBLs.
In addition, the structure of SBLs is inherently more variable than that of
CBLs and cannot be so easily generalized; it is not known whether the
relationships that have been developed on the basis of "ideal" terrain
experiments will transfer to more typical sites. Terrain slope is critical
because even extremely slight slopes, on the order of 0.1°, lead to downslope
gravitational forces that are as important as the large scale pressure
gradient in driving the flow (Caughey et al., 1979). The angle between
downslope direction and pressure gradient force also can have a critical
effect. Besides these effects, the temperature and stability structure of
SBLs can be considerably affected by radiational cooling, and the wind
velocity profile evolves slowly through the night, rarely achieving a
steady state. These effects cause the Ri profile to evolve also, so that
"turbulent bursts" may occur. Short of that, large amplitude wave motion
may develop at the top of a CBL, causing its height to fluctuate (Lu et
al., 1983).
35
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There has been much improvement in knowledge of SBL turbulence
structure for flat sites in undisturbed conditions (no discernable wave
motion). High quality experiments in Minnesota (Caughey et al., 1979)
and in the Netherlands at Cabauw (Driedonks, van Dop, and Kohsiek, 1978) have
made this possible. Nieuwstadt (1984a,b) has shown that many relationships
that are valid for the surface layer, the lowest 10% or so of the SBL, are
appxoximately valid for the whole SBL if jocal scaling is used. Instead
of scaling with u*, the surface value of -c1/2, we scale with the value
of tl/2 at the height in question. Likewise, we use the local value of
heat flux and a local value of the Obukhov length, A = -t 3/2[k(g/T)wTTr].
The reason that local scaling works is probably because eddy sizes are
limited in stable conditions to a small fraction of the total SBL thickness
(this is not true for CBLs or NBLs). Thus, each layer responds only to the
immediately adjacent layers and not to surface fluxes. One of the useful
results of Nieuwstadt's data analysis is that crw = 1.4T*'2 throughout
the CBL. For the eddy viscosity of momentum, Km, he found that
Km = t-^AfjU/A) for the whole layer, in contrast to the expression
usually assumed for just the surface layer, Km = u*Lfg(z/L) (Km is defined
as T * a~u/az and fj and f2 are functions).
Nieuwstadt (1984a) also developed a simple prediction for the profiles of
the local fluxes by using the equations for 39/at, all/at, and aV/at
in horizontally homogeneous boundary layers. To do this, h assumed
(1) stationarity, with time derivatives of a0/az, u, and v set equal to zero,
(2) a constant flux Richardson number, defined by Rf = (g/T)w'T'/(i: aii/az)
(vector values, rather than scalar values, are used for T and aTT/az), and
36
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(3) h = 3l/4(kRifU*L/f)*/ ^. Assumption 1 is not really valid for the mean
quantities like Hand 9, but it seems to give good predictions for turbulence
quantities (i.e., turbulence may adjust quickly to changes in gradients of
mean quantities). The value of Rf, like Ri, increases from zero at z = 0
to a nearly constant value « 0.2 through most of the SBL. Assumption 3
agrees with earlier arguments for the functional form of h, to be discussed
later; however, a simple solution for t is obtained only with this particular
choice of constants, for which h = 0.35 (u^/f)1/^. The solution obtained
is T/u*2 = (1 - z/h)3/2. This was shown to be in good agreement with the
Minnesota and Cabauw observations. The result for heat flux,
w'T'/fw'T1)0 = (1 - z/h), follows more readily from just the stationarity
assumption a(99/9z)/3t = 0. The Cabauw observations fit this well
enough, but it is seen in Nieuwstadt's (1984a) Figure 10 that the Minnesota
observation would better fit a higher power in (1 - z/h)n, with n = 2
instead of 1 (this difference in sites could be due to differences in
radiation flux divergence). Some consequences of Nieuwstadt's forms for \
and trr are A/L » (1 - z/h)5/4 and aw/u* = 1.4(1 - z/h)3/4.
Lateral variances for Cabauw were not reported, but it might be noted
that the au/u* values for Minnesota reported by Caughey et al., (1979)
give a fairly good fit to 2.3(1 - z/h)3/4. Hanna (1981) suggested
a linear decrease with height in aw, au, and av as empirical fits to
the Minnesota data, with, for instance, crv/u* = 1.3(1 - z/h). However,
expressions like these with au and av of the same order as aw can only
represent SBLs in their most ideal state, with only small-scale, nearly
isotropic turbulence and no large horizontal eddies or meanders. Thus,
37
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they should be interpreted as minimum values; on some nights, at some sites
0 and 39/3z > 0.01°C/m at z < h-f; however, in the
upper part of a SBL and above, ae/az is strongly influenced by radiational
cooling, which depends on the water vapor and C02 profiles among other things.
Also, turbulent exchange tends to reduce 30/dz, not increase it, so
39/az greater than some threshold value makes no sense as an indicator of
the top of the turbulent boundary layer, h. Other doubtful measures of h are
the height of significant cooling, h9, and the lowest height of a maximum
38
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in the wind speed profile, hu (it is true that a "jet", a wind speed maximum,
often develops above the SBL, in response to inertia! forces free of ffictional
drag). However, there is one mean profile quantity that is a good indicator
of the height of turbulent mixing, and that is the Richardson number, Ri
(Eq. 2.2), or the Bulk Richardson number, Rb = (g/9)(9 - 90)z/Tf2, where 9
and "u are measured at the height z (Rb does not require the precision measure-
ments that Ri does, so it is often the more practical choice). By using data
from 25 nights of the Wangara meteorological experiment in Australia, Mahrt -
et al. (1982) showed that the height at which Ri = 0.5 correlated highly
with u* and, by implication, with L, as u* and L were also highly correlated
(Mahrt and Heald, 1979). In contrast, the correlations between hu or hg
and u* were about half as large, about 0.42, and the correlation between h-j
and u* was even less. They noted that Ri was normally small and relatively
constant with height in a layer near the ground, but that this layer
was capped by a sharp increase in Ri. In the same journal issue, another
paper using the Wangara nocturnal data appeared by Wetzel (1982). He
showed that the height at which Rb = 0.33 correlated well with u* and L,
with poorer in correlations for hg, hu, and h-j in place of Rb. He also
found another profile height index that correlated with u* and L almost as
well as the Rb index, namely the top of the linear part of the 9 profile
(these profiles were unusually regular, however, which may be more charac-
teristic of ideal, flat sites like Wangara). Arya (1981) showed that u*
and L correlated fairly well (r = 0.63) with h estimated from acoustic
sounder returns at Cabauw, whereas the correlation of this h with he was
only 0.28 and with hu was not even statistically significant.
39
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Ideally, we would like to be able to estimate h on the basis of surface
parameters, so that something simpler than a well instrumented tall
meteorological tower would do the job. Zilitinkevich (1972) developed a
simple SBL model with
h * b(u*L/f)l/2 (2.5)
This equation has received much testing in recent years. While this h
correlates significantly with measured h, it is far short of perfect. For
acoustic sounder measurements at Cabauw, Arya (1981) found a correlation
of r = 0.67, while Nieuwstadt (1984a), who excluded weakly stable
cases, found r = n.77. Nieuwstadt and Tennekes (1981) found better
correlation with a prognostic model; h never reached its equilibrium
value because the relaxation time was too long (this time depended on the
surface cooling rate, a90/at, and became ~ 10 hr late at night). However,
according to Nieuwstadt (1984a), part of the success of this model was due to
the fact that the initial h was specified; because time scales are long,
predicted and observed h only slowly diverge. Because there is no ready
way to specify the initial h, he concludes that diagnostic models like Eq.
2.5 are more practical for now. However, Eq. 25 is a steady-state solution,
and the SBL is so slow to adjust to a balance of forces that a steady state
may not often be achieved. A second-order closure numerical model developed
by Rrost and Wyngaard (1978) gave predictions that fit Eq. 2.5 well, but
the steady state, with b = 0.4, was not approached until t = 5 hr after the
onset of surface cooling. At t = 2 hr, b = 0.7 was the best fit. The
measured values of h from the Minnesota experiment runs, which were centered
40
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at about 1 hr after transition to negative H*, fit Eq. 2.5 with b = 0.7 to
0.9 when h was defined as the height at which heat flux extrapolated to
only 5% of its surface value (Caughey et al., 1979).
Garratt (1982) compared Eq. 2.5 with h data from four different sites,
Koorin and Wangara in Australia (both using the Ri method), Cabauw (acoustic
sounder), and Minnesota (heat flux profile). By using a different criterion
for h than did Caughey et al., (1979), he calculated a best fit of b = 0.37
for Minnesota; for Cabauw and Wangara, he calculated b = 0.42, and 0.39,
respectively, which supports the generally accepted value of b = 0.4. However,
for the only low-latitude site, Koorin, at 16°S, the best fit was b = 0.13.
This is three times smaller than the other sites; the f-1/2 term in Eq. 2.5,
even if incorrect, does not account for this difference (remember that f
contains the sine of latitude). Garratt accounted for this difference by
running Brost and Wyngaard's model for each site while including the signi-
ficant terrain slopes at Koorin (0.11°) and Minnesota (0.08°) and the
angular difference between the fall line and mean surface wind direction;
the results of these model runs gave very good agreement with the best-fit
values of b observed at each site. This result should make us cautious
about the general use of a surface parameter approach like Eq. 2.5 until it
is validated for sites with more typical slopes; certainly, these often
considerably exceed 0.1°.
There are diagnostic alternatives to Eq. 2.5. For sites where there
are significant slope flows, gravitational acceleration may be much stronger
than Coriolis acceleration. (This only requires that g(A0/9)p » uf,
41
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where A9 is the potential temperature difference between the drainage flow
and "free" air and p is the terrain slope, in radians. If A9 = 5°C, 9 = 300°C,
u = 3 m/s, and f-1 = 104 s, p » 0.1° implies that gravitational force
dominates.) In such cases, u*/f should not be a significant length parameter.
Instead, it may be found that h « L, as Kantha and Long (1980) found to be
true in mechanically-stirred stratified fluid in a laboratory tank. For the
simple two-dimensional slope flow model that Briggs (1981) fit to Ellison
and Turner's (1959) laboratory slope experiments, h « x, the distance down
the slope, and the mean flow speed U « (H*x)l/3> this gives h « iP/H* (the
dependency of h and U on p cancel out in this expression). If U = u*,
then h « L for slope flow; we might expect something like surface-layer
similarity profile assumptions to apply to a slope flow, in which case
constant h/L would imply a nearly constant U/u* (influenced slightly by a
small increase in In(h/z0) down the slope), so that there is an internally
consistent argument for h <* L. Idealized as this model may be, it is
adequate warning that L may be highly variable across rolling or hilly
terrain at night.
In sorting through these alternatives, one should be aware of strong
correlations between u*, L, and other SBL parameters; these correlations
also provide possibilities for useful simplifications. Van Ulden and Holtslag
(1983) showed that the surface-layer temperature scale, T* = w'T'/u*, is
nearly constant over a wide range of u* at Cabauw (see Fig. 2b), becoming
much smaller only at very small u*, < 0.1 m/s; this constant T* does depend
somewhat on cloudiness. Some consequences of constant T* are H* « u*,
L « u*2, and, if Eq. 2.5 holds, h « u*3/2. Venkatram (1980) showed that
42
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L = 1100 u*2 (mks units) gives a good fit to field experiment data at three
relatively smooth and flat sites; furthermore, h = 2400 u*3/2 (mks units)
gives a good fit to the Minnesota measurements of h, which ranged from 30 m
to 400 m. Nieuwstadt (1984a) carried this simplification one step further
to show that h = 28 IP/2 (mks units), where IT is measured at 10m, worked
about as well as Eq. 2.5 for Cabauw observations. However, the constant in
this relationship probably depends on site parameters like z0; furthermore,
in drainage flows, the relationship may be more like h « Tj"3, as already
discussed. Briggs (1982a) suggested an easily measured replacement for L,
Ln = u"3/Rn*, where Rn* is defined like H* but with net long-wave radiation at
the surface replacing heat flux. In doing this, he showed that
H*/Rn* = 0.5(1 + S/L)'1 for the Project Prairie Grass measurements (Barad,
1958), where S = 15 m, a site-specific length constant. This implies two
regimes. One is a near-neutral regime with large L and H* « Rn*, limited
only by the radiation balance; this gives L * u*3 and Eq. 2.5 reduces to
h « u*2 or h « L2/3. For a very stable regime, with L « S, H* « L « u*3/2 and
Eq. 2.5 reduces to h <* u*5/4 « 1.5/6. jn between these asymptotes, there is a
broad range that approximates the constant T* solutions, as suggested by
Venkatram (1980). Note that this intermediate regime gives h <* L3/4, and
for all cases we have discussed, including slope flows, a strong correspond-
ence between h and L is predicted; namely, h « |_n with n ranging from 2/3
(flat, near neutral) to 1 (slope flow). Therefore, h = function(L) might
be a versatile approximation, even with L replaced by an easily measured
quantity like Ln = "u3/Rn*.
43
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Besides slope effects, in SBLs we must be aware of radiative cooling
effects on boundary layer structure. Little work has been done on this
but Garratt and Brost (1981) made a theoretical investigation by using the
same second-order closure model that so successfully modeled the effects of
slight terrain slopes (Brost and Wyngaard, 1978; Garratt, 1982). To do this,
they had to assume some typical profiles for water vapor, Cf)2, and temperature
in the troposphere plus a value for the surface emissivity (percent of black
body radiation emitted). Six different cases were run, including two for which
radiation flux divergence (radiative cooling) was left out of the calculation.
The results showed that radiative cooling has large effects on the profiles
of heat flux (w'T1) below O.lh and on 9 and Ri profiles near and above
z = h. Radiative cooling also has significant effects on the profiles of "u
and 9 in terms of surface-layer similarity theory largely because of the
development of an above-surface minimum in w'T1; this affects H* and L,
which are defined at the surface. Without radiative cooling, crw above
z = h does not extinguish nearly as rapidly after transition to night.
On the other hand, the inclusion of radiative cooling in the model has
relatively little effect on TT, T, 99/dz, and Ri below z = h, or on
the evolution of u*, Rh at z = h, or b = h/(u*L/f)1/2 (from Eq. 2.5). At
z = h, Rb increased from approximately 0.1 at t = 1 hr to 0.25 at t = 2.5
hr regardless of initial conditions or inclusion of radiation; b increased
slowly from about 0.32 at t = 1 hr to 0.4 at t = 12 hr, except that it was
of the order of 10% larger in the first few hours when radiative cooling
was omitted.
44
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At present, there are no models that could possibly predict occurrences
of large amplitude waves or large time-scale horizontal eddies (meanders)
in the SBL. It is understood that the possibility of large gravity waves,
which may "break" into turbulence or initiate a turbulent burst, exists
when Ri is near a critical value (~l/4) through a substantial layer.
However, it is difficult to predict the evolution of Ri through a night, as
it depends on far upwind initial conditions and the structure of the humidity
profile that develops as the CBL shuts down and surface evaporation and
plant transpiration continue around the time of transition. DeBaas and
Driedonks (1985) made a case study of a wave occurrence measured at the
Cabauw meteorological tower; Ri = 0.2 to 0.3 was measured for certain wind
directions on the tower at z = 30 to 160 m. Large waves were present, as
could be seen in T versus time traces from the instruments at z = 120 and
160 m; also, aw more than doubled between 80 and 160 m, a reverse of its
usual behavior. The authors developed a linearized wave model which
successfully predicted the frequency range of the waves that developed,
especially as seen in the turbulence spectra for w1. However, the wave
magnitude could not be predicted with this model. Another episode of large
wave motions near the top of a SBL was described in detail by Lu et al.
(1983); this episode was apparently caused by the passage of a shallow front.
Large time-period wind direction shifts and meanders occur in SBLs;
these can cause 1-hr average lateral diffusion to be many times as large as
would be predicted using boundary-layer turbulence models, which describe
only three-dimensional eddies are restricted in size by the ground and by
stable stratification. Large two-dimensional horizontal eddies, with no such
45
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constraints on size, often exist and tend to be site specific. Hanna
(1983) has summarized empirical information from a number of sites. At
every site, the magnitude of the lateral wind direction variance, which
directly relates to initial lateral dispersion, tends to increase approxi-
mately as u~l in stable conditions. However, av tends to be nearly constant
(within a wide range) with "u at all sites. Its average value ranges from
0.3 m/s at Porton (England) to 1 m/s at a complex terrain site in California
Hanna also presents some wind velocity observations for a site on the Snake
River Plain in Idaho, a broad plain bounded by mountains. For this site
av = 0.5 m/s for most 1-hr average values; occasionally values are much
larger, but they are rarely less than 1/2 of this. Superimposed on these
fluctuations were wind direction shifts with time periods ranging from
about 1 to 4 h. Such large time-scale oscillations could be due to seiche
effects peculiar to that kind of topography, with cold air trapped in a
large basin.
46
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SECTION 3
DIFFUSION EXPERIMENTS
How far have they done, for what sources, under what conditions?
Good experimental information is essential for building confidence
in the abilities of modeling techniques to work under all proposed
conditions of application. Some partial confidence develops from experience
with model applications, provided that gross model failures are rare,
non-critical, or simply not publicized. However, each user is concerned
with only a limited range of source and meteorological parameters. Since
there is no collective archive of experiences with model failures and
successes, information of this nature available to a model user tends to
be anecdotal and patchy. It sheds little light on the causes of model
failures and gives little basis for confidence in the model for application
to radically different source types or meteorological situations.
It is far preferable to test and compare model performances against
data from carefully executed diffusion experiments. Such experiments
should include trustworthy measurements of the diffusion pattern of a
conservative tracer (one that does not lose or gain mass flux with distance)
and every source and meteorological variable that might relate to variations
in the diffusion pattern (variables such as wind direction, stability,
source buoyancy, etc.) If these measurements are complete, then the data
set is "universal" in that it provides a standard of comparison for any
model, past, present, or future. Unfortunately, such complete experiments
are rarely accomplished. Also, there is a seemingly limitless number of
47
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combinations of source, terrain and surface configurations, effluent
characteristics, and meteorological conditions for which dependable
models are desired. Each field or laboratory experiment covers only a
very limited range of these combinations. Usually, only one source type
is utilized, e.g., it may be a passive source of conservative tracer, a
buoyant source with no stack, building, or terrain downwash conditions
complicating the flow, or a source of uniform glass beads to study deposi-
tion with known source characteristics. There are always constraints on
the range of meteorological variables, due to the location, the limited
goals and resources of the experimenters, or factors like the undesir-
ability of allowing the equipment to get wet. Furthermore, the diffusion
measurements themselves have many limits. The sensitivity and number of
sensors used puts limits on the range and density of measurement locations,
and the great majority of field measurements have been confined to near-
surface concentration values due to the expense and logistical difficulties
of measurement in the vertical dimension.
For some of the same reasons that diffusion experiments are limited,
e.g., manageability, this paper reviews only the state of the science
for continuous, passive, point sources of (presumedly) conservative
tracers with no complex terrain or large surface inhomogeneities, such as
land/water boundaries, within the range of diffusion of concern. These
limitations nevertheless permit inclusion of most earlier diffusion
experiments and quite a few recent ones.
48
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FIELD EXPERIMENTS
There have been a few dozen field experiments on ABL diffusion that
can be considered of research grade. While this seems a substantial number,
they all have many limitations, and many "data gaps" continue to exist.
A few of the earlier experiments, such as Hbgstrbm's (1964), used
plume photography to determine plume geometric variables, such as the
vertical standard deviation, cjz, and the lateral standard deviation, cry.
This is a valid and relatively inexpensive method for characterizing
observed plume geometry, for both "instantaneous" or "relative" plume
diffusion (i.e., diffusion relative to the instantaneous plume centerline)
and for "total" or "time-averaged" diffusion (Nappo, 1984). This method
is limited to the range that the plume can be seen against the background;
depending on the plume opacity and the background conditions, this may be
tens of kilometers in very stable conditions or only a few tenths of a
kilometer in convective (unstable) conditions.
The vast majority of the field experiments use a metered release of a
specific gas or some type of small particles as a "tracer". This is
sampled using aspirated boxes known as "samplers" or, in the case of some
fluorescent particles, sticky surfaces which catch the particles by
impaction. A considerable number of these receptors are set out in an
array intended to define the plume. Usually, they are set out in near-
surface "arcs" of constant or approximately constant distance from the
source with only a few degrees of azimuth between them. In a few of these
experiments, attempts were made to define the vertical concentration
49
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profile of the plume by setting up a vertical array of samplers. This is
always a more difficult and expensive feat, and is often frustrating.
Because the towers or tethered balloon cables supporting the vertical
array of samplers are very sparse in the horizontal plane, they often
fail to define the plume's vertical dimension adequately, or they even
miss the plume altogether. Rarely has such an array extended through the
whole mixing layer, so its usefulness is confined to near the source
where vertical diffusion is still within its bounds. Very recently,
diffusion experiments have been made that use remote sensors, especially
lidar, that effectively solve the problem of measuring concentrations in
the vertical dimension; two of these experiments will be described at the
end of this section.
There have been several good surveys of diffusion field experiments
up to about 1978, which this discussion will exploit. Horst et al.
(1979) selected ten field experiments that they considered adequate
for testing predictions of 0y and
-------
somewhat wider than that of the present report, and it included diffusion
experiments in complex terrain, near shorelines, and within forests.
Draxler's survey includes 20 "classical" diffusion experiments, made over
homogeneous terrain, and 32 experiments in other categories. Another
survey of some relevance was made by Sklarew and Joncich (1979); it
covered field programs plus monitoring networks at power plants. Although
these sources were all buoyant, buoyancy effects dominate lateral diffusion
only at small to moderate distances. At tens of kilometers, plumes from
buoyant sources diffuse much like passive plumes released from the same
effective source height or from any height within a well-developed convec-
tive mixing layer, which tends to mix all material nearly uniformly
throughout its depth.
Classic Tracer Experiments
It is not possible in a few pages to discuss individually each field
experiment falling within the scope of this paper. Instead, information
from Horst, et al . (1979), Draxler (1984), and also from tables published
in Draxler (1976) and Irwin (1983) have been used to construct the summary
of experiments shown here in Table 1. Included are four experiments
that Draxler (1984) considered non-classical because of rough (but not
"complex") terrain, urban siting, or low wind speed conditions; these are
the Julich and Karlshruhe experiments, the St. Louis experiment, and the
INEL, Idaho experiment, respectively. For these experiments, the sites
seem reasonably homogeneous, at least among typical "real world" sites,
as opposed to ideally flat and uniform sites like that of the Prairie
51
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Table 1. Summary of Tracer
Site or
Program
Surface Source
Round H111
Prairie Grass
NRTS, Idaho
Ocean Breeze
Dry Gulch
Green Glow
Hanford-30
St. Louis
INEL, Idaho
Hanford-67
Elevated Source
Rrookhaven
Porton, ENR
Han ford
NRTS, Idaho
Agesta, SUED
Suf field, CAN
Julich, GER
Karlsruhe, GER
Hanford-67
Cabauw, NETH
Copenhagen, DEN
Terrain/Roughness
Length (cm)
Low hills, <30m/>10
Flat prairie/0.6
Flat, sagebrush/1.5
Low dunes, <6m
Sloping mesa, ravines
Flat, sagebrush/3
Flat, sagebrush/3
Urban, <30m relief
Flat, sagebrush/1.5
Flat, sagebrush/3
Fields, woods, hills/100
Rolling downs/5
Flat, sagebrush/3
Flat, sagebrush/1.5
Wooded hills <50m/60
Rolling prairie/2
Forest, farm/40 to 80
Forest, farm/110
Flat, sagebrush, 3
Flat meadows, trees/10-2f)
Urban resident ial/60
Stab. a
U-S
U-S
U-S
U-S
U-S
U-S
U-S
U-S
vS
N-S
U-S
sU-sS
s
u
N-vS
U-S
U-S
N
N-S
SU-N
II-N
Release/
Receptor
Heights (m)
0.3/2
0.5/1.5*
-1/-1.5*
2-3/1.5-4.5
2-3/1.5
2.5/1.5*
2.5/1.5*
sfc, bldg"Vsfc
1.5/0.8*
2/1.5*
108/sfc
140-2551/cable
56/1.5*
46/1*
50/photos
7 & 15/0
50 S 100/sfc
100/sfc 20-30
26 S, 56/1.5*
80 4 200/sfc
115/sfc
nur.b
(min)
10
10
60
30
30
30
60
60
60
10-30
60
30
15-60
30
60
30-60
60
10-30
fiO
60
Footnotes: * Plus vertical profiles, a Stability range, U = unstable, N - neutral,
S » stable, s ' slightly, v * very. b Duration of release or of sampling. c All
include u" or u profile. d Deposition may significantly reduce concentrations at
the larger arc distances (Gryning et al. 1983). e Six 17.5-m towers at x = 100 m.
f Wind and temperature, surface to specified height. 9 Profiles of u and T to 16 m,
aircraft soundings to 3 km, solar radiation, soil temperature profiles and more.
n Significant deposition noted. 1 Five 30-m towers at x = 400 m. J Significant
deposition loss (Simpson, 1961). k Reliability questioned by Horst et al. (1979).
References for Table 1: Round Hill, Cramer, Record and Vaughan (1958). Prairie
Grass, Barad (1958) and Haugen (1959). NRTS, Islitzer and Dumbauld (1963). Ocean
Breeze and Dry Gulch, Haugen and Fuquay (1963). Green Glow and Hanford-30, Fuquay,
Simpson, and Hinds (1964). St. Louis, Pooler (1966) and McElroy (1969). INEL,
Sagendorf and Dickson (1974). Hanford-67, Nickola (1977). Brookhaven, Smith (195fi),
Porton, Hay and Pasquill (1957). Hanford, Hilst and Simpson (1958) and Elderkin
52
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Experiments in Homogeneous Terrain
Tracer
so2d
S02d
Uranine dye"
Diffusion Data
3 arcs
5 arcs, towers6
6 arcs, towers1
Maximum
Range
(km)
0.2
0.8
3.2
Meteorological
Datac
oa, ae, 2m; AT, 12-1. 5m
aa, ae, 2m; profiles1' 16m,
aa,
-------
Grass experiment (Barad, 1958). A scan of column 2 of Table 1 shows experi-
ments made in a wide variety of site types. For completeness, the more
recent experiments made in Cabauw (Nieuwstadt and Van Duuren, 1979) and
in Copenhagen (Gryning and Lyck, 1984) are included; both experiments
used elevated releases of S?Q, which has been favored in recent years as a
very conservative tracer that can be accurately measured.
Table 1 cannot be considered an adequate, or even absolutely accurate,
description of each field experiment; it is very compact and omits many
details. Yet it does serve to give an overall view of what has and has
not been studied in the field in scientific diffusion experiments. Within
the surface source and elevated source categories, the listings are
approximately chronological. This ordering shows a trend toward more
mass-conservative tracers such as SF^ in more recent experiments.
There is evidence of deposition losses for almost all the tracers
used in experiments before the early 1970s. Certainly, this can be
suspected for any of the particle or dye tracers, for surface releases and
for elevated releases after substantial surface impact occurs. Deposition
may not affect lateral diffusion, or ay, very much, but it can greatly
distort vertical diffusion at distances of about 1 km or more. For two
of the surface source experiments, it was possible to estimate the decrease
in mass flux of tracer using an arc of samplers on towers. In the Green
Glow experiment, which used zinc sulfide particles, the loss of source
material at x = 3.2 km was estimated by Simpson (1961) to range from 78
to 97%, with 18 to 45% loss even at x = 0.2 km. The greatest loss was
for the most stable run. Similarly, 10 to 54% losses of uranine dye at
54
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x = 0.4 km were estimated for the NRTS experiments (Islitzer and Dumbauld,
1963). Significant deposition was also suspected for the multiple-tracer
releases of the Hanford-67 series beyond x = 0.2 km (Horst et al . 1979).
Even S02 gas is known to deposit on the ground; Gryning et al . (1983)
estimated a 20 to 25% reduction in surface concentration at x = 0.2 km
for the Prairie Grass experiment. At the farthest arc, x = 0.8 km, they
estimated a 25 to 45% reduction, the greatest reduction being in stable
conditions. The significance of these deposition losses is that we must
regard all estimated vertical dispersion results from the surface source
experiments with suspicion or at least as in need of careful interpretation.
The reinterpretation of the Prairie Grass experiment by Gryning et al.,
if correct, implies that all previous determinations of a2 from this
experiment at x = 0.8 km in stable conditions may be about a factor of
1.8 too large, since they were based only on surface samples and the
assumption of a Gaussian, conservative plume. Hence, for surface releases,
only the INEL experiment using SFs as a tracer should be free of this
interpretive problem (this was a rather specialized experiment, run under
extremely stable conditions, with sampling to only 0.4 km).
Another tracer that should not be considered conservative is oil
fog, which was used in several of the elevated source experiments. Oil
fog consists of micron-sized droplets formed by condensation of oil heated
to a vaporous state and mixed with air; it is produced much like steam.
It contains some volatile substances, and these slowly evaporate after
the tiny droplets have formed, as they have a very large surface area-to-
volume ratio. Lidar scans of oil fog plumes have indicated a loss of
55
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total reflectivity with distance. Scans from the 1983 CONDORS experiment,
described at the end of this section, indicate a 50% loss in the total
return signal (after correction for attenuation and background haze)
from the plume in travel distances of 1.4 to 2.8 km for most cases
(Eberhard, 1985: personal communication). The rate of evaporation depends
en the oil type, and the oil chosen for this experiment was relatively
low in volatile substances; most oils would evaporate faster. Presumedly,
this decrease of droplet size, mass flux, and reflectivity with distance
would not much affect plume width determinations based on a certain
percentage of the center!ine concentration, as was used at Brookhaven
(the data base for the Brookhaven ay and az curves). However, the
concentration-to-source strength ratio, X/0, of oil fog will become
increasingly lower than that for a conservative tracer; when az is
inferred from surface measurements by assuming a Gaussian, conservative
plume, as was done at Brookhaven, increasing overestimates of az with
distance would result.
The Agesta experiments of Hb'gstrb'm (1964) also used oil fog, but
plume parameters were determined by photographing puffs of oil fog from
near the releasing point; each puff was photographed for about 15 min,
until it became difficult to define, and then another puff was released.
The meandering component of diffusion was determined from the vertical and
lateral standard deviations of puff center!ines as the succession of puffs
passed designated travel distances (estimated from travel time and wind
speed measurements); this determination should not be affected by droplet
evaporation. However, the relative diffusion estimates, based on the
56
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average puff size relative to the puff center, could become increasingly
too small with distance, as evaporation could cause the apparent edges of
the oil fog puff to shrink somewhat. At the furthest distances, the
opposite effect could occur, as the more rapid disappearance of the
apparent puff could lead one to believe that it became more rapidly
diluted by a more rapid increase in ay and az.
With the possibility of "disappearing tracers" in mind, consider the
limitations of the diffusion data in experiments listed in Table 1.
First of all, for about half of the experiments, tracer measurements were
made only within a few meters of the surface, by using arcs of samplers.
In such cases, only the lateral dispersion near the surface, such as ay,
and the crosswind-integrated concentration, /Xdy, are measured directly.
The /Xdy measurements are universally meaningful only if little of the
tracer is lost at the distance of the arc, i.e., if the integrated mass
flux overhead = Q, the rate of release. If significant deposition has
occurred, this measurement is relevant only for material with the same
deposition rate under the same conditions. Very often, az has been
inferred from /Xdy measured on surface arcs by assuming (1) a conservative
tracer and (2) a Gaussian form of vertical concentration distribution
about the release height. We have reason enough to believe that assumption
1 is very risky after about 0.1 to 0.4 km of plume contact with the
surface for the majority of the tracers (some have yet to be proven or
disproven as approximately conservative tracers). Assumption 2 has been
shown to be roughly correct in neutral and stable conditions, and in
57
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unstable conditions for near-surface sources at distances less than
0.5 ziTT/w* (Pasquill , 1974; Willis and Deardorff, 1976a). At larger
distances and for elevated sources in unstable conditions, this assumption
may be grossly incorrect (Willis and Deardorff, 1976a, 1978, and 1981).
The az curves most used in current practice, Pasquill's and
Brookhaven's (Section 4), were derived using surface sampling data with
assumptions 1 and 2, above; therefore, these curves should be considered
very doubtful for the unstable cases, unless they are applied to the same
type (surface or elevated) of source using the same assumptions. Pasquill's
non-neutral az curves are based mostly on the Prairie Grass experiment,
so they apply best to near-surface sources. Brookhaven's curves are
based mostly on their 108-m oil fog releases, so are more appropriate for
elevated sources. In both data sets, it seems likely that the mass flux
of the tracers decreased significantly at the larger distances, violating
assumption 1. This would have the effect of causing overestimates of
-------
this deposition would tend to deplete the lower part of the plume more
than the upper part of the plume and thus distort the profile shape
compared to that of a conservative tracer plume. However, surface deposi-
tion should not reduce the maximum height obtained by the tracer at any
given distance; the settling velocities, or fall speeds, of the particulate
tracers were negligibly small. Therefore, the az values computed from
tower profiles for the surface plumes should be roughly similar to those
of truly conservative tracers, although they will be affected somewhat by
the profile shape distortion. To take an extreme example, if the Gaussian
curve centered at z = 0 is distorted by a factor <= z, the computed second
moment around z = 0 (i.e., az) is increased by a factor 21/2, or by
41%. The increase is probably considerably less for actual cases affected
by surface deposition, perhaps around 20% at most; thus, tower-derived az
values may be acceptable in spite of surface deposition. The Table 1
footnotes on tower distances then suggest that usable az information
exists for surface sources from x = 0.1 to 3.2 km. However, in every
case the site is flat and smooth; z0 ranges from 0.6 to 3 cm. There are
further restrictions regarding stability. For very stable conditions, we
have only the INEL experiment with towers only at x = 0.2 km (in the most
stable runs the plumes may have reached their maximum vertical growth,
with az only = 4 m, as close in as 0.1 km; the surface arcs indicated
no trend in /*dy from x = 0.1 to 0.4 km, and SFg is considered a conser-
vative tracer). Moderately stable and neutral cases are well covered,
but the towers are too short to encompass the rapid vertical development
of plumes in unstable conditions.
59
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For elevated sources, surface deposition should have negligible
effect when az < zs, the source height, because surface contact is
minimal up to this point. Deposition should also be less serious than for
surface sources at all distances, since concentrations near the ground
are nearly always lower for an elevated release. Towers were used in two
experiments, but at NRTS they were only 30 m high. The source was at 46 m,
so only a small portion of the plume, not enough to determine az, could
be measured. The towers for the Hanford-67 experiment (Nickola, 1977)
ranged from 27 to 62 m in height, so they could have substantially defined
the vertical profile only for the 26 m-high release at the farther arcs
(with the taller towers) in the more stable cases. Nonetheless, Horst et
al. (1979) were able to determine best-fit az values for a Gaussian
plume on the basis of the partial profiles at x = 0.2 km, with good
agreement among results for the different tracers used. They considered
this procedure inadequate at the larger distances, due to profile distortion
by deposition. Two of the elevated-source experiments used impaction-type
collectors mounted on tethering cables with much less restrictive heights
than tower networks, but there were no elevated, lateral plume measurements
(it would be unwieldy to construct an arc of tethered balloons, and they
would tend to get tangled in turbulent winds). The Porton experiments
(Hay and Pasquill, 1957) defined
-------
Crosswind-integrated surface concentration is directly related
to az if the profile shape (e.g., Gaussian) and the overhead mass flux
are known. Most historic az estimates were made by using surface
sampling arc /Xdy measurements and assuming a Gaussian plume shape and
a conservative tracer, but one or both of these assumptions are question-
able in the majority of cases, as discussed earlier. However, /Xdy
is useful in its own right, mainly for computing long-term averages of
surface concentrations. We can consider it to be reliably determined
only for those experiments using a conservative tracer, which would only
be the SFs experiments in Table 1. This includes only the measurements
to x = 0.4 km for a surface release in very stable conditions at INEL and
two elevated-source experiments, both relatively recent and both European.
(It is curious that, in Table 1, all the surface-release experiments were
made in the U.S.A., while most elevated-source experiments were made in
Europe.) At Cabauw, there was only one "arc", really lines of samplers
following roads. In Copenhagen, three arcs were established over a
useful range of x, 2 to 6 km. Neither experiment included stable cases,
nor very convective cases, rare in these northern, coastal regions. It
is encouraging that both of these experiments were made in "real-world"
locations with surfaces typical of many industrial and power production
sites.
Because ay is not so affected by tracer evaporation or deposition,
Table 1 indicates that there is a wide variety of useful ay information,
particularly as derived from sampling arcs. Many of these determinations
extend to 4 to 6 km downwind, for both source types. For surface sources,
61
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the Hanford experiments listed in Table 1 (including project Green Glow)
extend cjy information to 13 or 26 km, for all stabilities. However,
keep in mind the fact that in stable conditions, ay is very site dependent;
gravitational flows of cool, denser air near the surface develop even on
extremely slight slopes, and these collide and interact with each other,
causing horizontal meander. The investigators for the INEL experiment,
which was performed during very low wind speed and very stable conditions,
concluded that the best "model" for horizontal diffusion at short distances
was the frequency distribution of observed wind directions (Sagendorf and
Dickson, 1974). One of the unique aspects of this experiment was that
the arcs extended over a full 360°; during some of the one-hour runs, the
plume meandered through the full 360° of arcs. Experts have been saying
for years that the only reliable indicator of ay averages of more than
a few minutes duration in stable conditions is wind variance measurements
at the site at the height of release (Hanna et a!., 1977); no amount of
experimental information on
-------
Copenhagen included z-j estimates from soundings. Of the historic experi-
ments, only the Prairie Grass data tabulation of 1956 included temperature
and humidity soundings to such a depth that Zi could be determined at a
later date, as was done by Nieuwstadt (1980) and Briggs and McDonald
(1978). Both these references showed that the Prairie Grass unstable
data compared well with predictions that use convective scaling, which
requires z-j, hut scatter was somewhat large owing to the rather short
release time of 10 min; the bulk of the experiments had 30- to 60-min
releases or sampling durations.
From the point of view of the statistical theories of diffusion, the
most important measurement besides wind speed is wind direction variance.
The lateral direction variance, aa, is needed for correlations with ay,
and the vertical directional variance, ae, is needed for correlations
with az (the subscripts "a" and "e" stand for azimuth and elevation).
Note that aa or "gustiness" of lateral wind direction was measured in
almost every experiment in Table 1. However, most older wind vanes were
poorly damped and had too much inertia, so they did not yield accurate
cra measurements. Better instruments were available by the late
1960s. In about half of the experiments, but dominately the elevated
source experiments, ae was measured; of these, the only ones with
direct tower- or cable-measured vertical dispersion were Prairie Grass at
0.1 km, NRTS at 0.2, 0.8, 1.6, and 3.2 km (surface source) and 0.4 km
(elevated source), and Porton, with measurements to 0.5 km. Thus, there
is a rather limited experimental basis at present for comparing vertical
diffusion with statistical modeling predictions.
63
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Surface-layer similarity theory, favored by some modelers, especially
for vertical diffusion, requires some estimate of the Obukhov length, L.
This parameter is very difficult to measure directly, as it requires high
quality turbulent flux measurements. This was done only in the Cabauw
experiment and in the more recent CONDORS experiment to be discussed
shortly. However, the Richardson number, Ri, is a function of 7/L and
can be used to determine L; Ri is defined by temperature difference and
wind speed difference between two levels. It was measured and tabulated
in four of the experiments, but could be derived for any of the experiments
having high quality profiles of temperature and wind. In addition, L can
be inferred from surface roughness and the bulk Richardson number by
using known flux/profile relationships from surface-layer similarity
theory. This number requires only AT between the two levels and IT at
one level to be determined; such measurements were made for every experiment
in Table 1, so it is possible in principal to make some estimate of L.
However, precisely calibrated measurements of T are required to accurately
measure AT; often these measurements are not sufficiently reliable.
Another limitation on this approach is that the surface layer may be quite
shallow in moderately to very stable conditions, so for the flux/profile
relationships to be valid, AT and IT must be measured rather low, preferably
below z = L; this condition was not met for many of the elevated source
experiments.
It is also desirable to compare diffusion observations with the more
empirical prediction schemes that use Pasquill/Turner categories or AT.
While every experiment included either temperature profiles or AT, any
64
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comparison using AT would be confounded by the great variety of height
pairs used to define it; experiments including detailed temperature
profiles provide the possibility of standardization. Either cloudiness or
solar insolation measurements are necessary to determine Pasquill or
Turner categories for each run. This information was provided in only five
of the experiments; this seems a serious oversight in the other experiments,
considering the ease of these particular measurements and the popularity
of the Pasquill/Turner method. It may be possible to assign a Pasquill
category to these experiments by working backwards through other schemes
that relate L, Ri, R& to these categories (e.g., Colder, 1972). Such
indirectness should not be necessary. Unfortunately, too many past
diffusion experiments seem designed to test only one or two modeling
approaches rather than all of them.
Long-Range Measurements
As might be expected from a simple extrapolation of trends in Table 1,
as needs for diffusion data extend out to 100 km or more, the experimental
information tapers off rapidly. This is especially so for vertical
diffusion data, which are virtually nonexistent beyond x = 3 km. In
convective conditions, which normally prevail during the daytime, this
is not a great lack because it is well established that tracers and
light particulates become nearly uniformly mixed from z = 0 to z-j by
vigorous vertical turbulence. Above zj concentrations fall off rapidly
to near zero at z = 1.1 Z}. For passive tracers, this vertical concentration
distribution is approximately established at a time of travel = 3 z-j/w*,
65
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typically 20 to 40 min; x is then typically 2 to 10 km. In neutral and
stable conditions, theoretical models suggest that complete vertical
mixing through the turbulent boundary layer takes a time of the order
of f~l, where f is the Coriolis parameter. This is about 3 h at middle
latitudes, far beyond the usual travel times associated with vertical
diffusion measurements.
When one considers diffusion over days of travel, daytime mixing to
the height z-j is so rapid compared to vertical mixing in stable conditions
that it is usually adequate to assume that vertical diffusion is "frozen"
from about sunset until the convective mixing layer develops through each
layer the following morning, as long as there is no appreciable settling
velocity or scavenging by precipitation. When cumulus clouds develop in
daytime with bottoms near Zj, material can be drawn up into them by the
updrafts supporting the cumuli, a process known as "cloud venting" (Ching
et.al., 1983). This can take material out of the boundary layer; if the
material has a very long residence time, over a period of many weeks it
will mix through the whole troposphere, where the earth's weather takes
place. On these time scales, we inevitably are faced with processes that
are beyond the scope of this paper, which is limited to "simple" boundary-
layer diffusion. To stay within this scope, we shall limit the travel
times under consideration to about 1/2 a diurnal cycle (12 hours), or
travel distances to a few hundred kilometers.
For lateral dispersion over such distances, somewhat more than a
dozen sets of observations have been summarized by Pasquill and Smith
66
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(1983) and by Draxler (1984). These observations are nowhere near as
substantial and well-documented as those in Table 1 for short ranges.
Some long-range observations are from a classified project (Slade, 1968)
and some are unpublished. Most of the published observations have very
little meteorological information other than broad designator terms like
stable and neutral. Because many of these observations were made by
traversing a plume at a series of distances and heights with a single
aircraft, and there was inadequate time for repeated traverses over such
distances, they were really quasi-instantaneous measurements of plume
width or of 0y, rather than a time average. Traditional statistical
theory predicts that instantaneous plume widths approach time average
widths at large travel times compared to the Lagrangian turbulent time
scale, but this notion has been increasingly questioned simply because
horizontal eddies in the atmosphere at any time scale up to days can be
found (synoptic "highs" and "lows"). In one of the unpublished experi-
ments (Ferber and List, 1973) referenced by Draxler (1984), both instan-
taneous aircraft measurements and time-averaged surface sampling of a
3-hr release of SFs were made for the same 76-m high release in windy,
neutral conditions; the time-averaged surface-measured ay values were
3 1/2 times as large as the aircraft-measured ay values in the range of
6 to 18 km downwind. For 1-hr averages at x ~ 100 km, the differences
are probably much less, but there appear to be no data for testing this
assumption.
The ay versus x data for the x = 2- to 100-km range summarized by
Draxler (1984) all show similar power laws of growth, with n in ay « xn
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ranging from about 0.7 to 0.9. This seems to hold regardless of stability
or of whether ay was from instantaneous aircraft traverses or from
surface sampler measurements of a 3- or 4-hr tracer release. Similarly,
Pasquill and Smith (1983) fit a diverse collection of data in the range
x = 3 to 1000 km with n = 0.875. These are significantly larger values
of n than the value 0.5 predicted by statistical or eddy diffusivity
theories at large x. Draxler (1984), Pasquill and Smith (1983), and many
others have suggested that wind direction shear may be responsible for
the more rapid growth of ay at large x. However, the large-x experiments
so far seem to lack wind profiles and other supporting meteorological
measurements for testing models of plume growth, so we cannot progress
much beyond empirical power laws.
Useful Buoyant Plume Observations
Although this paper does not cover buoyancy effects and will pass
over diffusion data from buoyant sources, such as those surveyed by
Sklarew and Joncich (1979), diffusion of plumes from buoyant sources is
not always dominated by buoyant effects. Thus, sometimes observations
from buoyant plumes can be used to test aspects of passive diffusion
models, especially when atmospheric diffusion is rapid, as in unstable
conditions. For instance, Carras and Williams (1983) determined vertical
plume growth versus distance by using time lapse photography at four
stack sites in Australia during convective conditions. For the three
sources of lesser buoyancy, good agreement was found with Lamb's (1979)
convective scaling predictions for passive diffusion; the most buoyant
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source had enhanced growth at small x, but its plume growth also fit the
passive plume asymptote at larger distances (>1 km). For ay, Briggs
(1985b) demonstrated that buoyancy has little effect on plume growth in
convective conditions if the dimensionless parameter F* < 0.06; F* can be
defined as (w*/U) times the ratio of the stack buoyancy flux to the
buoyancy flux from a surface area n Zj2, owing to surface heating and
evaporation. Thus, F* is a measure of the relative buoyancy contained in
a segment of bent-over plume compared to that in atmospheric thermals.
Except for large sources of buoyancy in low wind speed conditions, the
F* < 0.06 criterion is usually met. Briggs (1985b) also presents evidence
that suggests that buoyant plumes trapped within a convective mixing
layer become uniformly mixed in the vertical, just as passive plumes do,
when X = (w*/U)x/zi > 30 F* (or, equivalently, x > 30 Fb/w*3), where Fb
is the stack buoyancy parameter. Thus at these large distances, the
diffusion of buoyant plumes released into and trapped with a convective
mixing layer will be essentially the same as passive source diffusion
from any zs < Zi. In such cases, plume width measurements at large
downwind distances are useful even if the plume is initially buoyant; a
good example of such measurements are those reported by Carras and
Williams (1981), which extend from 15 to 1000 km downwind of an isolated
stack, a considerable extension beyond the observation distances in
Table 1.
Remote Sensing Experiments
Another type of experiment not included in Table 1 is remote sensing
experiments, which are very recent. These overcome the limitations of
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surface-based sampling by measuring plume concentrations with reflected
electromagnetic radiation. Lidars have actually been used since the
early 1960s to define the rise and other geometric properties of buoyant
plumes. They send out narrow pulses of monochromatic light from a laser
and then record the time lag and intensity of the reflected return. Present
operational lidars work best with plumes composed of ~1 urn-sized particles,
which are easily produced by using an oil fog generator (other, more
conservative tracers are available, but these are also more expensive).
A lidar "scan" consists of a series of pulses made at a fixed azimuth
with a small increment in elevation angle made between each pulse. Up to
10 pulses per second have been obtained; a mapping of plume concentrations
at a half-dozen azimuth angles can be completed in 2 or 3 min with such
instruments; repetition of such rapid scanning for 30 min or more produces
an adequate ensemble of individual scans for defining an average plume.
To infer concentrations of tracer, or rather of tracer reflectivity,
requires careful processing of the return signal to compensate for range,
background haze, and attenuation of the reflected light returning through
the plume and background haze. This degree of sophistication is a recent
achievement (Eberhard et al., 1985); earlier experiments used lidar
mainly to define the edges of buoyant plumes. One limitation of lidar is
that for eye safety reasons it should not scan within 15 m or so of the
surface. Thus, it is best used for elevated sources or for surface
releases at distances where they are diffused through several hundred
meters or more. Lidar's great advantage is that it can detect sufficiently
dense plumes at a 5-km or more range. This allows the mapping of diffusion
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through a mixing depth of 2 km or more. Thus, lidar measurements comple-
ment observations that can be made by using samplers on surface arcs and
towers.
Lidar has been used in two recent passive plume experiments, one
carried out in stable conditions and the other carried out in unstable
conditions. Both use oil fog as the plume tracer for the lidar, which
was provided and operated by the National Oceanic and Atmospheric Admini-
stration's Wave Propagation Laboratory (WPL). The experiment made in
stable conditions at Cinder Cone Rutte, Idaho, was primarily a complex
terrain experiment, but ay and az determinations were made by lidar
at small distances, 0.1 to 0.9 km, upwind of the isolated hill (Strimaitis
et al., 1983). Source heights ranged from 20 to 60 m, and meteorological
measurements on a nearby 150 m tower included temperature at eight levels,
three-dimensional wind speed and turbulence intensity at five levels,
insolation, and net radiation. These measurements are sufficient for
testing any presently known modeling approach and have been used to test
a combination statistical/similarity equation for az (Venkatram et al.,
1984).
The CONDORS experiment (CONDORS = Convective Diffusion Observed with
Remote Sensors) made at WPL's Boulder Atmospheric Observatory was designed
to define three-dimensional plume concentrations in highly convective
conditions by using both 1idar-detected oil fog and doppler radar-detected
"chaff". This tracer consists of bundles of aluminum-coated thread which
are chopped into 1.5-cm lengths (half wavelengths) and ejected in an air
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jet. The radar can map the chaff plume for tens of kilometers and can
detect concentrations as low as 10~5 threads/m^ with almost no background
or attenuation problems except for "ground clutter" presented by metallic
objects such as power lines. The radar's lateral and vertical resolu-
tion is limited by its 0.8° beam width. The main drawback of this remote
sensor is that the lightest chaff presently available has a settling velocity
of about 0.3 m/s. Used in vigorous convection with vertical turbulence
velocities of about 1 m/s, the chaff plume does mix up to z-j, however, its
mode was observed to be displaced downward compared to the oil fog plume,
with this displacement increasing with distance (Moninger et al., 1983).
In half of the CONDORS runs the chaff and oil fog were released side-by-side
at heights ranging from 167 to 280 m, using the 300-m meteorological
tower. For other runs, except for one with both sources on the ground,
the chaff was elevated and the oil fog was on the ground. In addition,
the main part of the experiment in 1983 had one sampling arc at x = 1.2
km, and SF^ was released from the elevated release point; this made
possible some comparison of X/Q inferred from the remote sensors and
surface tracer measurements (Eberhard et al., 1985). Because some of the
chaff deposits and the oil fog slowly evaporates, a constant value of 0
cannot be assumed, as for a conservative tracer; instead, the total flux
at each distance, IT //Xdydz, was substituted for Q. (This is far superior
to experiments with non-conservative tracers and only surface samplers,
because there is then no direct measurement of the total flux of tracer
versus distance or of the extent of vertical diffusion.) Meteorological
measurements in the CONDORS experiment included temperature, three components
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of wind speed measured with sonic anenometers, turbulent variances, and
turbulent fluxes measured at eight levels on the tower. In addition,
solar radiation was measured and three or four rawinsonde releases were
made during each of the 11 two-hour runs; z-j was determined from the
rawinsonde temperature and humidity soundings, the top of the haze layer
as seen by lidar, and the top of the chaff as seen by radar.
New Experimental Capabilities
It is essential that the most useful experiments have been those
with a full complement of meteorological measurements. Of the historic
experiments, Prairie Grass stands out, even though it was one of the
first, made in 1956. This data set has been used much more than any
other, largely because almost every needed meteorological parameter was
measured or could be estimated from the available measurements. Some of
the recent experiments like Cabauw, Cinder Cone Butte, and CONDORS have
returned to this philosophy of measurement; they, too, should prove to
be very versatile bases for model testing and will raise the standards
for future diffusion experiments. All of the earlier experiments suffered
to some degree from tracer deposition, unreliable turbulence measurements,
and physical/economical limits on vertical sampling. Recent experiments
have benefited from the use of more conservative tracers, much improved
turbulence instrumentation, and remote sensors. These capabilities
present many opportunities for dependable measurements over much greater
ranges of distance, height, and stability conditions than was previously
possible.
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LABORATORY EXPERIMENTS
Fluid modeling experiments in wind tunnels and water channels have
long been used as tools for simulating large-scale turbulent flows over
and around obstacles. Confidence in these tools has increased to the
point that they have been used extensively to investigate diffusion from
releases on and near buildings and terrain features (Hosker, 1984). They
are increasingly used to model diffusion in complex terrain, including
stability effects, primarily because they are much cheaper to operate
than large-scale experiments, particularly in difficult terrain. In the
last few years, the U.S. EPA has supported several paired field/fluid
modeling experiments on diffusion in complex terrain; these were directed
towards improving modeling of diffusion in complex terrain and testing
the reliability of fluid modeling in such applications (Schiermeier,
1984). With some limitations, the fluid modeling simulations have been
quite successful.
In addition to being less expensive than field experiments, laboratory
modeling offers control over the meteorological variables, so that both
the flow and surface characteristics can be idealized. Therefore, it is
an attractive alternative for investigation of "pure" turbulent diffusion,
free of the complexities found in the field due to uneven terrain, uneven
surface heating or cooling, uneven surface roughness, and variable winds
that cause part or all of the plume to miss the samplers. In bypassing
these complexities, fluid modeling is naturally unable to simulate some
features of the real atmosphere. The most conspicuous missing element is
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plume meander due to large-scale horizontal flow structures many times
larger than the mixing depth. Such eddies, mini-fronts, or lines of
convergence may be caused by large-scale topographic variations, surface
inhomogeneities, wind direction shears, or even cloud shadows in the real
atmosphere. Straight-sided wind tunnels and water channels cannot simulate
wind direction shear effects, and their walls prevent the development of
any horizontal eddies larger than their width. Thus, laboratory lateral
diffusion measurements correspond to the minimum lateral diffusion that
occurs under the same stability conditions in the atmosphere during
periods free of these larger scale effects. Laboratory ay values
probably represent 10- to 20-min averages in the real atmosphere better
than 60-min averages.
The great reduction of scales in laboratory modeling facilities also
causes a great reduction in the range of turbulent eddy sizes compared
to the atmosphere. The smallest eddy sizes are limited by fluid viscosity
and the turbulent energy dissipation rate. In the atmosphere, the diffusive
ability of very small eddies (~1 mm) is usually inconsequential, except
for the mixing of chemically reactive species. The problem with very
large scale reductions is that viscosity begins to smooth and distort even
the largest eddies, which are the most effective agents of diffusion, so
that none of the full-scale turbulent diffusion is well simulated. In
the atmosphere, only the motion of eddies smaller than about 1 cm are
effectively smoothed and dissipated by the action of molecular viscosity.
The degree of scale reduction that avoids viscosity-induced distortions
is most limited when modeling plume buoyancy or ambient stratification
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(stable or unstable atmospheres). This is because gravity then plays a
role, and gravity cannot be adjusted. Gravitational acceleration has the
dimensions of velocity squared divided by length; if lengths are reduced
by a factor S, velocities such as wind speed must be reduced by a factor
S1/2 to maintain the ratio of inertia! forces to gravitational forces (this
assumes that full-scale density differences are maintained). The importance
of viscosity is measured by the inverse of the Reynolds number, Re, which
is a length scale times a velocity scale divided by the (kinematic)
viscosity. Thus, Re must be reduced by the factor S3/2.
To illustrate, suppose we wish to model a 1 km width full-scale terrain
segment in a 2-m wide wind tunnel, so S = 500. A full-scale wind of 10 m/s
blowing over roughness elements of size 1.5 m has Re = 10^. If we model
neutral flow, we can maintain the 10 m/s flow speed and reduce the roughness
size by the factor of 500, to 3 mm; this results in Re = 106 * 500 = 2000,
which is large enough for good turbulence simulation. However, if we
wish to model a stratified atmosphere, we must also reduce the wind speed
by a factor of (500)1/2 to 0.45 m/s, which reduces the Re to 90; this is
submarginal , so that viscosity smoothes the flow around the roughness
elements and they appear smoother to the wind than they should be.
A non-practical solution to this problem is to put the whole wind tunnel
in a giant centrifuge in order to increase the effective gravitational
acceleration by substituting centripetal acceleration. More practical
solutions are (1) to use less scale reduction, thus modeling less area,
(2) to use increased density or temperature differences to increase buoyant
accelerations, which is equivalent to increasing gravity, or (3) to
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experiment with exaggerated roughness elements to achieve a good simulation
of full-scale wind speed and turbulence profiles. All of these strategies
have their limits, and the bottom line is that there are limits to the
scale reduction possible to successfully simulate atmospheric turbulence or
turbulent diffusion. Several other scaling limitations of fluid modeling
have been discussed by Snyder (1972 and 1981).
Laboratory diffusion experiments have been made in wind tunnels,
water channels, and water tanks. The latter have been used to study
diffusion by purely convective turbulence. Since tank walls prohibit
crossflow, there can be no turbulence generated by boundary-layer wind
shear. Water channels are relatively easy to stratify and are most often
used as towing channels for buoyant sources or for elevated releases
upwind of a terrain feature; that is, the model is towed along the length
of the channel. This simulates the effect of a uniform crossflow velocity.
This technique allows easy maintenance of strong, stable stratification,
which is not easily maintained in a return flow mode (model stationary,
water moving through channel and recirculating). It is used to study
only the effects of crossflow and stable stratification, however, it does
not permit simulation of ambient shear-generated turbulence. Such simu-
lation requires a flow of fluid across a long fetch because the turbulent
boundary layer that develops increases in depth slowly and is only a
small fraction of the fetch. The boundary layer depth usually determines
the scale-down factor. Only wind tunnels have been used to study diffusion
in shear-generated turbulence. Large facilities with long fetches are
cheaper to build and to operate using air as the operating fluid rather
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than water because water is so heavy; it is also easier to measure flow
velocities and turbulence in air. Large wind tunnels used to simulate
atmospheric boundary layers are generally about 2 m high, 2 to 4 m wide,
and 30 m long.
Neutral Boundary Layers
Simulation of diffusion in NBLs is relatively easy to perform in
wind tunnels because no heating or cooling is required and large flow
speeds can be used. This helps keep model Reynold's numbers high enough
to create well-developed turbulence that closely resembles full-scale
turbulence. The main requirement for good boundary layer simulation is a
long fetch over a rough surface. The boundary layer develops to a reason-
able depth, e.g., 1 m, more quickly if turbulence is initiated through
the whole depth of the tunnel at the upwind end by using an array of drag
elements known as "vortex generators". Surface-release diffusion experi-
ments in neutral flow were performed by Robins (1978) using such a confi-
guration. The wind tunnel was unusually wide, 9.1 m, and 2.7 m high;
this aspect ratio should allow freer development of horizontal eddies and
lateral diffusion. Experiments were performed with two different floor
roughnesses, one rather smooth to simulate "rural" diffusion (z0 = 3 cm
full-scale) and one rough to simulate "urban" diffusion (z0 = 130 cm
full-scale). The depth of the developed boundary layers at the release
point was 0.6 m and 2 m, with scale-reduction factors of 1000 and 300,
respectively, for a full-scale boundary layer thickness of 600 m. Measured
turbulence intensities, when scaled with the friction velocity, u*, were
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within 5 to 10% of best estimates from field measurements. The mean
plume measurements revealed nearly Gaussian lateral plume profiles and
quasi-Gaussian profiles in the vertical, fitting X <* exp(-azn) better
with n = 1.7 than with n = 2, the Gaussian value. The growth of the
vertical depth of the plumes fit quite well a prediction based on surface-
layer similarity theory, 6 In(0.176/z0) = 0.22x, where 6 is the
height at which the concentration falls to half its surface value. The
lateral spread averaged approximately 20% less than a prediction based on
statistical theory. Overall, Robins' experiment showed that quite good
simulation of turbulence and diffusion, especially for vertical diffusion,
in a neutral ABL can be achieved in a large wind tunnel. For lateral
diffusion, growth due to wind directional shear cannot be simulated in a
parallel-walled tunnel. In the atmosphere, wind shear effects may
contribute substantially to
-------
of wind and wind shear, but the tank has proven to provide a reasonably
good simulation of purely convective turbulence; turbulent velocity
variances, nondimensional ized by the convective velocity scale w* and
plotted against z/z-j, were in good agreement with aircraft measurements
for the vertical component, aw» and were about 30% lower than the
atmospheric measurements for horizontal components (Willis and Deardorff,
1974). This may be due partly to the horizontal motion-inhibiting effect
of the tank walls.
Diffusion experiments were made in this tank by instantly releasing
over 1000 tiny (~1 mm), neutrally-buoyant oil droplets from a line
source crossing the tank at various fixed source heights, zs. These were
photographed looking down the plume axis, and inventories of particle
positions were made at a series of times, t, after release. To interpret
these results for vertical and lateral diffusion in terms of distance
downwind of a continuous, point source in a crossflow, it is simply
assumed that x = ~ut. This assumption appears reasonable, considering the
fact that "u is nearly constant with height in a very convective ABL,
except close to the surface. Since the tank diffusion is, in effect,
integrated along the longitudinal coordinate, x, in the photographs, this
technique fails to account for longitudinal diffusion; Deardorff and
Willis (1975) estimate this phenomenon to be significant for point sources
only when U < 1.5w*, rather low wind speeds (a typical summer midday
value of w* is 2 m/s).
The results of the Willis and Deardorff convective tank diffusion
experiments were surprising and somewhat troubling to the diffusion
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modeling community because they revealed features of vertical diffusion
that were at odds with assumptions common to all Gaussian models, so
widely accepted and used. The near-surface releases, with zs/z-j = 1/15,
diffused in a Gaussian manner at first, up until t = 0.5 z-j/w*, but
shortly thereafter the locus of maximum concentration lifted off the surface
and lofted into the upper half of the mixed layer. This was accompanied
by a rapid decrease in surface concentrations until nearly uniform vertical
mixing from z = 0 to zj was achieved at t = 3 z-j/w*, the limit of the
experiment. (In convective ABLs, updrafts and downdrafts extend through
nearly the whole layer with mean up and down speeds of 0.6w* and 0.4w*,
respectively (Lamb, 1982); therefore, the time needed for one complete
"stir" is approximately Zj/(0.6w*) + Zi/(0.4w*) = 4zj/w*.) The elevated
releases, at zs/Zi = 1/4 and 1/2, exhibited non-Gaussian vertical behaviors
from the outset; the locus of maximum concentrations descended from the
release heights until they reached the surface at t « 2 zs/w*. Naturally,
this produced higher surface concentrations than predicted by Gaussian
models, which assume that the centerline of the plume remains at z = zs.
At larger t, the locus of maximum concentration lifted off the surface and
lofted into the upper part of the mixing layer, just as had occurred for
the surface releases.
These results were unexpected and stirred some controversy, particularly
because convective tank experiments are a relatively new tool for simulating
atmospheric dispersion. Some modelers questioned whether the observed
diffusion patterns occur in the real atmosphere. Unfortunately, the
answers could not immediately be found in existing field data due to many
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deficiencies already discussed, especially the lack of vertical concentra-
tion profiles to any substantive height. The same phenomena were observed
in the numerical modeling results of Lamb (1979), which provided some
confirmation. (Lamb's work followed the surface-release tank experiments
and actually preceded and inspired the elevated-release tank experiments.
For a history of developments in convective diffusion, see Bn'ggs, 1985a.)
Papers by Bn'ggs and McDonald (1978) and Nieuwstadt (1980) confirmed the
fact that surface concentration results from the Prairie Grass field
experiments were consistent with the results of Willis and Deardorff
(1976a) and of Lamb (1979), when convectively scaled, except that ay
values in the tank were about 30% smaller than the field values (this
seems consistent with the smaller horizontal velocity variances observed
in the tank). Preliminary results from the CONDORS field experiment,
which was largely motivated by the results of Lamb's and Willis and
Deardorff's experiments, did show the descending plume behavior for a
zs/z-j = 1/2 case, with a remarkably good comparison with the convectively-
scaled tank results for crosswind-integrated concentrations (Moninger et
al., 1983). (Convectively-scaled means that heights are scaled with z-j,
time after release or x/TF is scaled with ZT/W*, and concentrations
_ O
are scaled with 0/(uzj ).) Further good comparisons of the tank experiment
results with some power plant plume data, the Cabauw field experiment,
and CONDORS data were shown in Rriggs (1985a). The convective tank
experiment results are now being taken quite seriously, and many researchers
have been developing mathematical diffusion models that fit these results
(such models are surveyed in Briggs, 1985b).
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More recently, Poreh and Cermak (1984) demonstrated that convective
ABL simulations producing results similar to those of the tank experiments
could be performed in a large boundary-layer wind tunnel. This has the
advantage of including some of the crossflow and wind speed shear effects
found in the real atmosphere. They ran these experiments using both a
smooth floor consisting of a 12 m long heated aluminum plate and a rough
floor, with twisted-link chains stretched across the plate. The simulation
had some shortcomings, e.g., the fetch of heated plate upwind of the
source was only about 1 TT(zi/w*), which is not sufficient for full convective
turbulence development (Briggs, 1985a), and the convectively-scaled vertical
velocity variances were only about 2/3 of values observed in the atmosphere.
Nonetheless, plume behaviors similar to those observed in the tank experiments
were observed in the tunnel. The lateral diffusion rate was larger, even
somewhat larger than that observed for the Prairie Grass field experiments
(Nieuwstadt, 1980), in spite of the lateral width restriction of about
2.4 z-j in the tunnel. An improved convective diffusion experiment was
recently completed in the same tunnel; stronger stable capping and
increased dimensionless upwind fetch over the heated plate were achieved
(Poreh, 1985: personal communication).
Stable Boundary Layers
Some basic diffusion experiments in SBLs layers have been done in
the same wind tunnel facility at Colorado State University. In fact, a
recent series of baseline diffusion measurements were made in this tunnel
for every combination of (1) stable, neutral, or convective boundary
layer, (2) smooth or rough floor and (3) surface source or elevated
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source (one or two different heights) (Cermak et al., 1983). The smooth
floor configuration still developed turbulence (a tripping device in
these experiments helped maintain turbulence throughout the test section
of the tunnel). The effective roughness length of a smooth surface is « v/u*,
where v is the kinematic viscosity of air (0.15 cm2/s at room temperature);
in these experiments it corresponded to approximately 1 cm at full scale
(similar to the Prairie Grass field site). The rough floor configuration
gave z0 = 16 cm, full scale, typical of mixed surface cover. To achieve
the SBL, incoming air was preheated to 78°C before entering the test
section, which had its floor temperature cooled to about 0°C. This large
temperature difference helped to compensate for scale-down effects by
increasing buoyancy forces. Unlike earlier experiments that achieved
only slightly stable boundary layers, with diffusion slightly reduced
from the neutral case (Chaudhry and Meroney, 1973), the recent experiments
achieved substantial stability. Vertical diffusion at the larger distances
was reduced by factors up to 5.5 compared to neutral conditions with the
same tunnel configuration. The estimated values of the Obukhov length,
16 cm and 25 cm for the smooth and rough floors, respectively, are frac-
tions of the boundary layer depth = 1 m), that correspond to moderately
stable conditions in the atmosphere (Pasquill category "F"). Measurements
from this series of experiments were included in a comparison of wind
tunnel ay measurements versus field data and statistical-theory predic-
tions (Li and Meroney, 1984).
A recent diffusion experiment in a boundary-layer wind tunnel in Japan
(Ogawa et al., 1985) achieved even stronger SBLs, with Obukhov lengths
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as small as 1/13 times the boundary layer depth. Vertical diffusion was
sharply reduced from its neutral growth at the greater stabilities, results
comparable to those of Cermak et al. (1983). A somewhat surprising result
was that lateral diffusion was reduced from its neutral growth by about
20% in the moderately stable SBLs, while it was actually somewhat larger
than its neutral growth in the strongly stable SBLs. Ogawa et al.
attributed this growth to the development of horizontal meanders in the
very stable flow, a phenomenon that field observations of lateral turbulence
velocities tend to support (Hanna, 1983). The Ogawa et al. (1985)
observations also showed significant departures from Gaussian vertical
profile shapes for the neutral and very stable cases. These experiments
were conducted only for a surface source and a smooth floor configuration.
Potential Further Uses of Laboratory Simulations
The results reviewed in this section support the idea that fluid
modeling, in the laboratory, is a valid way to improve our understanding
and modeling of diffusion. However, it is obvious that this tool has not
been fully exploited. Considering the fact that it is much cheaper to
run diffusion experiments in the laboratory, under controlled conditions,
than in the field, it makes sense to use laboratory facilities as much as
possible. These laboratory experiments should be supplemented by selected
field experiments to provide confidence-building and to measure features
that cannot be produced in the laboratory, like crosswind shear-induced
diffusion. For neutral conditions, elevated releases in a well-developed
boundary layer could be run by using several contrasting floor roughness.
In a wide tunnel it would be informative to run experiments with a patchwork
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pattern of contrasting surface roughnesses (on a scale small compared to
the tunnel width); this would be much more like the real world and might
reveal effects of the surface inhomogeneities on lateral diffusion (surface
drag inhomogeneities would cause vertical torques that might produce
larger and more vigorous horizontal eddies). For convective conditions,
a similar exploration could be made of the effects of surface heat flux
inhomogeneities, which would provide forcing for the turbulent mixing
much more like the real world than a uniformly heated aluminum plate
(most places of interest have a complex pattern of different types of
vegetation, different radiation balances, and different degrees of moistness
and thus, large contrasts in surface heat flux). For stable conditions,
it now appears possible to simulate fairly strong stable ABLs. There is
a need for much more work on the effect of height of release relative to
the boundary layer height for all stabilities.
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SECTION 4
DIFFUSION MODELING AS PRACTICED
How well does it reflect the current state of knowledge?
THE GAUSSIAN PLUME - ITS UTILITY AND LIMITATIONS
The vast majority of "production line" diffusion models are Gaussian,
i.e., they make the basic assumption that concentration distributions
relative to the center of a puff or plume have a Gaussian shape. In a
recent summary of scientific reviews of eight models (Fox et al., 1983),
most of the reviewers "felt that they were reviewing a single, physically
incomplete, Gaussian model with slight variations, rather than eight
substantially different models". They also felt that these models do not
represent the state of the science. The reason that Gaussian plumes have
come to dominate the model market is a matter of precedence and convenience.
Early diffusion models were naturally influenced by much earlier treat-
ments of molecular diffusion. Molecular diffusion in a fluid is caused
by countless collisions and rebounds among the molecules, which can be
treated statistically as a "random walk" process. When concentration and
diffusivity vary on scales that are large compared to the mean molecular path
length, the diffusion process can be described by the "parabolic equation",
5 ax
dx/dt = (K — ) , (4.1)
sx-j ax-j
where x-j (i = 1, 2 or 3) represents the three orthoganal spacial coordi-
nates. In this context, K, the diffusivity, is directly related to the
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product of mean molecular path length and speed. In the absence of shear
and when K is constant, the solution of this equation for a tracer released
at a point at t = 0 is Gaussian; the standard deviations about the center
are
a = (2Kt)l/2 .
This approach was adapted to modeling diffusion in the atmosphere by
replacing molecular diffusivity with "eddy diffusivity", which in analogy
to random molecular motions is supposed to be proportional to the product
of mean turbulent eddy size (or spectral peak) and turbulent velocity
variance. Whenever the atmosphere is turbulent, this K is orders of
magnitude larger than the molecular diffusivity, so the latter is thought
to be negligible. The classical treatment for diffusion from a ground
source using the parobolic equation with u" « zm and Kz « zn was first
published by Sutton (1953), where Kz is the vertical component of K. The
resulting distribution in the vertical is Gaussian only for the special
case m = n. In general, x * exp (-constant • zP) with p = 2 + m-n. If
n = 1, as it is observed to be in the neutral surface layer, and if wind
shear is neglected (m = 0), an exponential distribution results. Values
of p fitting observed surface-source concentration distributions have ranged
at least from 1.15 to 3 (Pasquill, 1974, p. 205; Ogawa et al., 1985).
However, Pasquill demonstrated that the practical effect on the ratios
of various profile constants is very small for 1.25 < p < 2.5, which suggests
that there is no harm done in assuming a Gaussian distribution (p = 2)
for these cases. Large errors result if a Gaussian distribution in the
vertical is assumed when the locus of maximum concentration from a ground
88
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source becomes elevated; no value of p fits this case. This phenomenum
has been observed in convective conditions, and has undoubtedly resulted
in some erroneous diffusion modeling as a consequence of assuming Gaussian
plume distributions.
There are many in which a Gaussian plume shape is an acceptable
approximation. For a sufficient sampling duration, lateral diffusion
near the source is Gaussian if the distribution of lateral turbulent
velocities is Gaussian. This is normally the case, except when the sampling
period encompasses a substantial wind shift. The same applies to vertical
diffusion from an elevated source, almost to the point that significant
concentrations reach the ground, but this is not valid in the convective
mixing layer. For this case, it has been amply demonstrated that vertical
turbulent velocities have a skewed distribution favoring downdrafts. The
simple Gaussian assumption is wrong here and underpredicts maximum ground
concentrations (Briggs, 1985a). The vertical velocity distribution can be
fit with a double-Gaussian shape, however, and this has been assumed in
some recent dispersion models. A near-Gaussian shape is often observed
in lateral concentration distributions at distances of many kilometers.
This is to be expected when the plume width far exceeds the width of the
largest horizontal eddies because then horizontal diffusion becomes akin
to a random walk process. At moderate to large distances in unstable
conditions, after a travel distance of about 3(U/w*)zj, passive material
becomes almost uniformly distributed from the surface to the top of the
mixed layer. Gaussian models can approximate this by assuming reflection
at the surface and at z = Zj (see Pasquill, 1976b).
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Some further discussion of when plumes are and are not Gaussian is
given by Pasquill (1978, pp. 6-9). He notes that power station plumes
have been observed to be Gaussian only in their lower half. Weil (1983)
speculates that the Gaussian assumption may perform satisfactorily in
convective conditions for vertical distributions of strongly buoyant
plumes, citing numerical simulations of Lamb (1982); however, the assump-
tion works poorly for passive or weakly buoyant plumes. Strengths and
weaknesses of Gaussian plume modeling have been discussed by Smith (1981).
The conditions of applicability he lists are somewhat more restrictive
than the ones stated here. Strengths that he lists are ease of computation,
familiarity, and adaptability; this means that ay and az can be altered
to better fit new observations without fundamentally changing the model.
Some of the limitations of Gaussian diffusion models that he disccusses
are the following: they do not account for wind direction shear, they
cannot describe near-calm situations, and they do not represent diffusion
during stable conditions very well, when vertical diffusion may be controlled
by source properties (for buoyant plumes) and lateral diffusion is due
mostly to wind direction shear and meandering. Gaussian models have also
been extrapolated to long range diffusion predictions over curved trajec-
tories and changing meteorological conditions, which has no basis.
To a considerable extent the shortcomings of Gaussian models can be
overcome by adjustment of ay and az values to account for complicating
factors which distort plumes from this ideal. At least this is usually
possible for making practical estimates of surface concentrations. Lateral
surface concentration distributions can almost always be fit reasonably
90
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well with a Gaussian curve. However, manipulations of
-------
are certain to serve as metersticks for progress in developing improved
models. A fairly comprehensive discussion of existing sigma curves and
stability typing schemes, including some that will he skipped here, has
been given by Weber (1976). The discussion here is limited to four generic
schemes. Gifford (1975, 1976) also offers an overview and considers
schemes for exceptional flows, including near calm, very stable condi-
tions, urban diffusion, diffusion over water, diffusion downwind of
buildings, and diffusion in rugged terrain.
Brookhaven Curves and qa Typing
Stability categorization methods based on wind direction fluctuations
go back at least 50 years (Giblett et al., 1932). The meteorology group at
Brookhaven National Laboratory (BNL) developed a system of five stability
categories based on inspection of one-hour segments of strip chart recordings
of lateral wind direction fluctuations (Singer and Smith, 1966); this is
a quick way to estimate aa. They related these categories to measurements
of tracers released over about 1-hr periods at the top of a 108-m tower
at BNL, which is in rolling, forested terrain of relatively large roughness.
The tracers were uranine dye, oil fog, and argon 41. The lateral plume
dispersion, ay, was measured at the surface directly, but az was inferred
indirectly by assuming conservation of the tracer fluxes and a Gaussian
plume concentration distribution centered at the release height in the
vertical (these were shown in Sec. 3 to be somewhat risky assumptions).
It was also assumed that ay and az have the same power law dependences
on x, and optimum fit power laws were determined for each category.
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This typing scheme has a very direct physical basis for expecting
good correlation with lateral diffusion because near the source passive
material goes as the wind blows, i.e., it tends to travel in straight
lines at first, so that the wind direction distribution also describes
the concentration distribution. In terms of Taylor's (1921) statistical
theory, ay = aax at small x. As material travels far enough to be
influenced by a number of turbulent eddies of various scales, ay/(aax)
tends to get smaller, but this dropoff is gradual and the rate of dropoff
is not very different in different stabilities. Hence, aa or the wind
direction tracer measured at the height of release over the desired sampling
period correlates well with ay at any distance, provided that the
stability has not undergone large changes since the time of release.
To address vertical diffusion, statistical theory gives us az = aex
near an elevated source, where ae is the standard deviation of vertical
wind angle. This measurement is somewhat more difficult to obtain than aa
and is often not made. However, aa can substitute for ae as a measure
of vertical diffusion if atmospheric turbulence is nearly isotropic, so
that ay and az are of the same order. This is true in NBLs and CBLs
but not in SRLs, because there horizontal motion can far exceed the scale
of vertical motions, which are restricted by vertical stability. Cramer
(1957) noted that aa correlated well with ay measurements made day
and night and with daytime values of az, but the correlation with
nighttime az was nil. Briggs (1985a) argued on the basis of convective
scaling theory that aa should also correlate well with vertical diffusion
from a ground source while it is in the "free convective regime", with
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az <«: H*1/^3/2. This follows from the approximate relationships t =* x/u
and aa a av/u = 0.6w*/u; substituting the definition w* = (H*z-j)1//3,
the free convective az « (aax)3/2z.j~1/2. Because a-,3'2 ranges far more
widely than z-j'1/2, the correlation between az and aa is strong.
Pasquill/Gifford/Turner Curves and Pasquill Typing
Pasquill's typing scheme, based on wind speed, insolation, and
cloudiness, has been very widely adapted, sometimes with minor modifications,
The scheme is more or less intuitive and was suggested as a way to relate
diffusion rates to easily obtainable measurements. It was in use at the
British Meteorological Office in 1958 and became widely used after
Gifford's (1961) adaptation of the scheme for Gaussian plume model ing.
Turner (1961) elaborated on the scheme somewhat by replacing subjec-
tive solar insolation categories with solar elevation and cloud cover
estimates. Smith (1972) proposed a scheme using wind speed, insolation
and cloudiness, but with a continuously variable numerical stability
index. His az curves were based on an eddy diffusivity model and
included corrections for surface roughness;
-------
the worst hours of the year; this approach would tend to overestimate
typical 1-hr concentrations, however.) They are based primarily on appli-
cation of the statistical theory of Hay and Pasquill (1959) to wind
direction fluctuations observed at a 16 m height over relatively smooth
terrain with z0 = 3 cm. There were some very limited tracer measurements
supporting the
-------
1977 or Briggs, 1982b); this correlation also applies to the Richardson
numbers, Ri and Rfo, since they relate strongly and monotonically to L.
Golder (1972) used five sets of meteorological data to develop nomo-
grams relating R-j and L to Pasquill categories and surface roughness.
Briggs (1982a) showed that L relates well to "psuedo" L's defined by
Ls = -u"3/R| and Ln = -u"3/Rj!;, where R* and R* are g/(cppT) times
insolation and net radiation, respectively. It was pointed out that Ls
and Ln simply amount to quantified forms of Pasquill's categories.
The moistness of the surface affects the ratios H*/RS* and H*/Rn*, so also
affects the relationship between Pasquill categories, Ls, or Ln and L;
however, this is secondary compared to the role of Tf3 in establishing this
relationship.
Pasquill's categorization method is less effective for ay because
scales other than the height above the ground can have more influence on
lateral turbulent motions. In convective conditions, z-j affects lateral
diffusion even near the ground, since av = 0.6 w*. At small x,
ay/x = aa = av/U <= (u*/u) |z.j/L| '•', and z^ has as much influence
on ay as L. In stable conditions, large horizontal meanders are very
terrain sensitive and are not controlled at all by L, which is the scale
limiting vertical turbulent motions. However, the Pasquill's categorization
method has validity for 3-min sampling times, as represented by Pasquill's
cfy curves, since large eddies have little effect at this time scale;
the small eddies are more isotropic, even in stable conditions.
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A very dubious, but frequent, practice is the application of Pasquill/
Gifford/Turner (P/G/T) az values to pollutants from tall stacks. This
was a major point of criticism in recent rural model reviews by a committee
of atmospheric diffusion specialists (Fox et al ., 1983). They felt that
the system is based too much on surface data and that the classification
system has a strong bias towards the neutral stability category. We now
know that, in convective conditions, the locus of maximum concentrations
from near-surface sources lift off the ground at x = 0.5 ("u/w*)z-j, and
thereafter the surface concentrations reduce very rapidly; this has a
strong bearing on the interpretation of Pasquill's oz curves for "A"
and "B" conditions (Briggs, 1985a). These curves are based on observed
jurface concentrations from Prairie Grass surface releases, and they
depend on the assumption that the vertical concentration distribution is
Gaussian with its maximum at the surface. Combined with the conservative
tracer assumption, this gives az <* (/xdy)~*. However, if the locus of
maximum X elevates, causing /Xdy to reduce rapidly at the surface,
the above assumptions lead to an exaggerated growth in the calculated
cfz. Reanalyses of the Prairie Grass data using convective scaling
suggest that this is what happened, since the surface concentrations, at
least, behaved very much like those in laboratory tank experiments (Briggs
and McDonald, 1978; Nieuwstadt, 1980; Briggs, 1985a). The Pasquill "A" az
curve in x expands much faster than "very unstable" curves derived from
elevated sources, such as the BNL curves. Little harm is done if the
P/G/T curves are applied to low sources using a Gaussian model, since
that is how they were derived, but large distortions can result if they
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are applied to very elevated sources. For instance at about 1.8 km in
"A" conditions, a plume is predicted to be four times as high as it is
wide. This seems counterintuitive given the current understanding of
diffusion under convective conditions (Lamb, 1979).
Hogstrom's Sigmas and (d9/az)/Uf2 Typing
Remarkably little attention has been given in diffusion literature
to Hb'gstrb'm's experimental study of diffusion from elevated sources made
two decades ago (Hdgstrbm, 1964). The data are very useful and his analysis
was detailed, scientific, and full of significant conclusions, especially
regarding diffusion from elevated sources in stable conditions.
The experiments were unique in that lateral and vertical diffusion
were measured optically using photographs of puffs shot from near the
release point, rather than from the usual lateral camera position.
Each puff consisted of a 30 s oil fog release. The puffs were assumed to
advect at the rate of the mean wind speed measured at source height.
Total diffusion was measured by releasing a succession of puffs over
about 1 h, each one released as the former one faded. For each experiment,
the standard deviations of the puff center!ines around their mean position,
(jyC and ozc, were determined from the photographs. In addition, the
mean relative diffusion around each center-line, ayr and azr was estimated
by assuming a bi-Gaussian X distribution and that the visible edge of the
puff represents a constant value of /Xdx (Roberts, 1923); cases of
uneven or variable background were omitted. Total diffusion for each
experiment was calculated by assuming vectorial addition of center!ine and
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relative diffusion, e.g., - jr i
were performed at Agesta with zs = 50 m and at Studsvik with zs = 24 m
and 87 m (both sites are in Sweden). Conditions were mostly neutral to
very stable, with only a few experiments made in slightly unstable condi-
tions (plumes disappear more rapidly in unstable conditions).
Five different stability indices were tested for their ability to
order data consisting of the most stability-dependent measure,
ayr ' Ozr at x = 2000 m, for the largest group of experiments, Studsvik
at 87 m. Plots were made for each index and a line of best fit was drawn
by eye; correlation coefficients were computed using this function. The
best correlation, 0.95, was obtained with s = 105(m3/C°-s2)(d9/dz)/Uf2,
where Uf is the speed of the "free wind" determined from routine, twice-
daily rawinsondes released nearly 100 km away from this site averaged
through a layer near 500 m. Uf actually worked better than u measured at
at the site itself. The lowest correlations were with d"u~/5z instead
of IT, and with d9/3z alone. In all cases, ae/az was determined only
from the 30-m and 122-m levels of the meteorological mast, so was the same
as A9/(92m). Thus, Hogstrom's stability index is a form of the Bulk
Richardson number, with s <= A9/Uf2; however, it represents stability near
the height of release better than surface layer stability, so is not
directly related to L or Pasquill categories.
The components of relative diffusion for the Studsvik 87 m releases
were highly correlated with s. At distances between 1 and 4 km, a2r
approximated its neutral-condition value divided by (1 + 0.022s). Similarly,
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cfyr a^ 1 km approximated its neutral value divided by (1 + 0.012s).
A very significant conclusion about aye, which can be regarded as
the "meandering" component of diffusion, is that it showed no dependence
on stability. It grew roughly linearly with distance in the range 1/4 to
4 km. Another interesting conclusion made by Hogstrom is that the wind
direction shear effect on aye was also almost independent of stability;
the larger shear with increasing stability was compensated by smaller crz.
(He estimated this effects contibution to
-------
It is of interest to note the relative importance of azr and
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approximated if the buoyancy flux F^ = 300 rrr/s^ (typical for these
sources) and u = 6 m/sec. Although ambient turbulence or stable stratifi-
cation are bound to be important in this distance range, it seems likely
that growth induced by buoyant rise is also a significant component of
the TVA
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AT/Az tends to be a function of heat flux only, while u has much
stronger influence on dispersion; thus, no correlation at all is seen
between AT/Az and measured az or f^dy in unstable conditions
(Weber et al., 1977; Briggs and McDonald, 1978). During stable conditions,
AT/Az, Weber et al. noted that u*, H*, and L are all well correlated
with each other in the surface layer and AT/AZ correlates fairly
well with near-surface vertical diffusion, provided it is measured through
a substantial layer roughly coinciding with the plume layer. (The best
correlations with az were obtained with (22 - zj) > 5m and zz/zj > 2,
where Z2 and zj are the upper and lower heights of temperature measurement,
respectively).
EVALUATIONS OF SIGMA CURVES AND TYPING SCHEMES
We find surprisingly few comparisons of diffusion data with rival
schemes for ay, az, and stability classification. There have been
dozens of papers demonstrating that different stability schemes disagree
with each other, but only a few papers address the question of which one
is best. Weil (1977) compared the performance of different Gaussian
model schemes applied to three power plants, which involves the additional
uncertainties associated with buoyant plume rise. The Brookhaven ay
and az values worked best, but they were chosen by an algorithm of
Weil's own devising based on wind speed and A9 (this most closely relates
to bulk Richardson number classification). The TVA scheme was the poorest
performer. Pasquill's scheme was intermediate in performance, even though
its basis was in surface source data; its performance was greatly improved
by displacing the stability category towards the unstable direction by one
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category. Similar results were obtained at all three power plants, except
for a few exceptional cases at Chalk Point; this was the only plant having
cra measurements, so it seems that aa classification schemes were
not adequately tested.
Quantitative tests of the correlation between various stability
indices and vertical diffusion from surface sources were made by Weber et
al., (1977). Only linear correlations were tried, and az was mostly
inferred from surface-arc /Xdy measurements and the Gaussian plume
assumption, using the Prairie Grass, Green Glow, and NRTS data sets. The
indices tested were AT, net radiation, aa, ae, IT, the gradient
Richardson number (Ri), and z/L. For unstable cases, AT and net
radiation did not correlate with az at all. The wind direction variances
had mixed results with mostly rather weak correlations, but the wind
vanes used in these early experiments may have been poor. The best
correlations were with Ri and z/L, which are closely related to each
other. For stable conditions, ae gave the most consistently good
correlations with az, but Ri, z/L, and AT also gave good corneldtions.
At the farthest distance, 3.2 km, all correlations fell to 0.5 or less.
The Prairie Grass data set provided the best correlations; these data
were further used by Rriggs and McDonald (1978), who compared (/Xdy)~l
with various indices and nondimensionalizations. The single best stability
index was L, but U^/A9 worked almost as well and it is much easier to measure;
1J2, by itself, applied separately to stable and unstable data categories,
worked slightly less well but still provided good correlation with /Xdy;
this supports Pasquill's scheme for ground-level sources, since it depends
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most strongly on u. Measured between 1 and 8 m, A0 also provided good
correlation in stable conditions, but none at all in unstable conditions.
Rood ordering of the data was obtained with nondimensional plots involving
_o
L, uVAS, or, for unstable cases, convective scaling with z^ and w*;
L-scaling had a slight advantage in stable conditions, as did convective
scaling in unstable conditions.
Comparisons of ay and az schemes against data from a large number
of both surface- and elevated-source experiments were made by Irwin
(1983). He tested the P/G/T scheme, with classification by Turner's
(1970) method, along with five schemes based on wind direction fluctuations,
with
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must remember that Pasquill's
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of the atmosphere itself; for these 60-min averages in neutral to
moderately unstable conditions, then, 50% of the /Xdy fell within about
a factor of 1.?. of the mean and 75% of the observations fell within about
a factor of 1.5 of the mean prediction; even the "perfect model" nay
produce little improvement in this scatter. Three models had insignificant
bias: Hay and Pasquill (1959), Draxler (1976), and the P/G/T method
(Turner, 1970). The az enhancement due to roughness and urban-enhanced
heat flux recommended by Pasquill (1976a) resulted in underestimates of /Xdy .
for this suburban area about a factor of 1.4. The largest mean underestimate,
of about a factor of ?., was by a model taken from Hanna et al . (1982);
although Gryning and Lyck did not state the model specifically, it probably
is the Briggs, "urban" ay and ax curves, which are based on curves
from the St. Louis diffusion experiment (McElroy and Pooler, 1968). This
suggests that urban cry and az curves give too fast a growth rate for
suburban locations. R)r ratios of predicted-to-observed maximum
concentrations for Copenhagen, all four methods using aa to predict ay
gave the least scatter; three of these also gave insignificant bias: Hay
and Pasquill (1959), Hraxler (1976), and Pasquill (1976). For overall
performance in predicting ay, /^dy, and *max, it appears that the Hay
and Pasquill and the nraxl er methods were clear winners in this contest.
To summarize these evaluations, it appears that models based on
ay = aax F(t), with a moderate degree of sophistication in the
specification of f(t), like Draxler (1976), give the best results so far
for lateral dispersion. The nraxl er (1976) formula for az, based on ae
measurements, also appears to be the most dependable method for predicting
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vertical dispersion, although several similar methods are competitive
with it. This supports the recommendations of scientists at the various
workshops since 1977 that favor characterization of
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SECTION 5
DIFFUSION MODELING ALTERNATIVES
The "conventional" models discussed in Section 4 all assume a Gaussian
plume and all are mostly empirical. However, we find that most of the
stability classification schemes have some physical basis. The Brookhaven
03 classification relates directly to ay at short distances, with
the relationship degrading very slowly with distance. It relates well
to az when conditions are nearly isotropic, i.e., lateral and vertical
turbulence velocities and length scales are about the same; this condition
holds best for elevated sources in neutal and unstable conditions.
Pasquill's classification based on IT and insolation or cloudiness is
crudely related to the Obukhov length, L, the most important length scale
affecting vertical turbulence in the lower part of the ABL. Thus, it
correlates best with az from surface or low sources. It correlates
fairly well with short-term averages (3 min) of ay, since the small
eddies affecting this time scale are more isotropic, but considerably
underestimates typical 1-hr averages of ay. Hogstrom's (ae/az/Uf2)
classification is a bulk Richardson number, which is strongly related to
L if ae/az is measured near the surface. Measured near 100 m, it
is a good index of the relative vertical stability at heights of elevated
releases. For neutral to stable conditions it correlated very well
with az, both-relative and total, and with ayr, the relative part of
lateral diffusion (this was accomplished by smaller eddies which are more
isotropic); however, it did not correlate with the total (time-averaged)
10Q
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cry. TVA's index AT/Az and variations on it like AT, A9, or a0/az
are very weak indicators of the atmosphere's diffusive capacity, because
wind speed has a stronger effect than the static stability.
Thus, these classification schemes, except for AT/AZ, each have
some areas of success as diffusion indicators, and none is "universal".
The Gaussian plume assumption has been a useful workhorse, but we have
identified some diffusion conditions during which it leads us astray, as
the atmosphere takes on more complex behavior. The worst aspect of the
models reviewed in the previous section is that each was built on a
very small set of diffusion measurements, measurements that had quite a
few shortcomings; since the curves are empirical, there is simply no
basis for extrapolating them to much larger x or using them for all types
of sources and terrains. Here is where more scientific models become
practical tools. If they can be validated for enough parameter combinations
within their range of applicability, there is some basis for confidence
in extrapolations and interpolations.
Over homogeneous terrain and under near steady conditions, typical
Gaussian models predict peak concentrations within about a factor of 2.
There is widespread belief that this kind of performance can be
improved upon by models which incorporate adequate physics, if they
can be supported by adequate meteorological measurements. This section
describes six general methods of approach that have had proven success in
some areas of testing; none of them appear to he the universal solution
for all diffusion problems, but a combination of approaches should be able
to provide the large step in modeling improvement that is greatly needed.
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SURFACE LAYER SIMILARITY MODELS
In the early 1960's, there were a number of attempts to rationalize
atmospheric dispersion rates in terms of fundamental boundary layer
parameters that were discussed in Sec. 2. The inspiration for this work
was the so-called "similarity theory" of Monin and Obukhov (1954), which
was becoming a successful tool for describing profiles in the surface
layer of wind speed, temperature, humidity, and vertical turbulence
quantities. This theory assumes that vertical turbulence near the surface
is a function of height and velocity and length scales based on surface
fluxes of heat (or buoyancy) and momentum (these scales were defined in
Sec. 2). Thus aw is assumed to be u* times a function of normalized
height, z/L, on dimensional grounds. Vertical gradients of u", T, water
vapor, etc., are assumed equal to their fluxes at the surface divided by
(u*z), times some function of z/L. These functions must be determined
empirically and may be different for different quantities. The usefulness
of this approach, essentially a dimensional analysis, was shown decisively
by the results of the 1968 Kansas field experiment (Businger et al.,
1971; Kaimal et al., 1972).
If the theory describes the structure of turbulence near the ground,
then it should also be capable of describing turbulent diffusion of
passive tracers released at the surface; this was tried in 1957 by Kazansky
and Monin for describing the shape of smoke plumes (see Monin, 1959).
Rifford (1962) reviewed progress with similarity theory up to that time
and developed predictions of diffusion from the surface starting with the
111
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equation
cfz/dt = cu*<|)(I/L), (5.1)
where T is the mean height of the concentration, c is a dimension!ess
constant, and is a universal function of z/L. Other early examples of
the application of Monin-Obukhov similarity theory to diffusion are the
papers by Cermak (1963), which used mostly wind tunnel diffusion observa-
tions, and by Panofsky and Prasad (1965). The latter authors noted that
lateral dispersion does not fit the theory as well as vertical dispersion;
this observation has been reconfirmed by many others since then, and
seems to follow from the fact that lateral turbulence is influenced by z-j
at day and by topographically influenced meanders at night. Thus, surface
layer similarity theory omits significant factors affecting lateral
dispersion.
The subject lay in a relatively dormant state, stymied in part by
the difficulty of measuring L, until after analyses of the 1968 Kansas
wind and temperature profile data were published. Chaudhry and Meroney
(1973) made use of the Rusinger et al., (1971) relationship between
Richardson number, R^ = (g/9) (ae/az)/(a"u7az)2, and L, to make improved
estimates of L for the Prairie Grass data. They also advanced
the theoretical argument that $ = ^~^, where ^ is the dimension!ess
temperature gradient given by ku*ae~/az = (w1 e'/z)h, and 9 is potential
temperature. The argument was made using an eddy diffusivity approach.
The simplest way to do this is to recast Gifford's equation as
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dz2/dt = 2bu*z = 2K , (5.2)
where K is an eddy diffusivity for passive material diffusing from the
surface. Chaudry and Meroney argued that this diffusion mechanism is
similar to that for heat flux from the surface, so K « Kn, the eddy
diffusivity for heat; this is defined by w'e' = Khd9/dz. Substitution
of the profile law gives K^ = ku^z/^ and, therefore, <{> = n •
These authors were able to demonstrate that this hypothesis leads to a
much better fit with the Prairie Grass data than does Monin's (1959)
hypothesis for <|>, which is too weakly dependent on L. They used the
observations at x = 100 m, the only arc having samplers on towers; this
permitted direct determination of "z (no Gaussian assumption required).
Horst (1979) carried this analysis further using all the arc distances
of the Prairie Grass experiment, ranging from 50 to 800 m, but he had to
assume a vertical distribution function to relate 7 to /xdy at the
surface, since only near-surface samplers were used. Based on Prairie
Grass profiles at x = 100 m and other evidence, he chose a modified Gaussian
profile assumption: X-l « exp(z/b"z")r with r = 1.5. He then showed that
K = Kn leads to a satisfactory fit with observations, much better than
K = Km, which had been suggested by Pasquill. He also compared the predic-
tion with /Xdy observed at x = 3.2 km at the National Reactor Testing
Station (Islitzer and Dumbauld, 1963), finding more scatter but definite
skill using K = Kn. Van Ulden (1978) also used K = Kn in a similar analysis
for arc distances x = 50, 200, and 800 m; further, he made an allowance
for the non-zero height of the Prairie Grass samplers, z = 1.5 m. In a
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recent re-analysis of the Prairie Grass data, Gryning, van 111 den and
Larsen (1983) again demonstrate a good fit of the observations using
surface layer similarity and K = K^, with poorer results using Km. They
find improved fits when they include an estimate of deposition in their
model, as S02 is now known to deposit on vegetation.
Two recent papers attempt to simplify application of surface layer
similarity to vertical diffusion from the surface. Both of these utilize
the Prairie Grass data. Venkatram (1982) argues that both the neutral
and very stable asymptotes for /Xdy at the surface are independent of
u". (To simplify discussion, Twill be used in place of az and cloud
height h, since they are all closely related.) For surface, concentrations
from surfaces sources f^dy « 0/(TT"z). As T « u*t « (u*/u)x for the neutral
case, the U dependence cancels out: f-^dy « 0/(u*x). For extreme stability,
with ~z~ » L, Venkatram assumed u" « u*z/L and a constant eddy diffusivity,
K « u*L. Substituting this into 7« (Kt)1/2, with t * x/u" and the above
IT, we get T« L2/3xl/3 an(^ j-^y « 0/(u*L^/^x^/3). (jne existence of this
asymptote in the ARL can be questioned in light of new knowledge about SBLs,
because h/L never gets large enough.) Venkatram used profile determinations
of L and u* to nondimensionalize the data, showing that the two asymptotes
fit the neutral-stable Prairie Grass data rather well, except for some
widely scattered points at larger x/L values. Briggs (1982b) derived the
same asymptotic relationships, arguing that there is a very slight dependence
on ~z/z0 at intermediate values of x/L, which can be neglected. He also
extended this approach, using cruder approximations, into the convective
regime, and suggested analytical expressions for Q/(u*x f-^dy) versus
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x/L. The numerical constants in his analysis differ from Venkatram's
asymptotes by about 20%, partly because Briggs adjusted for the non-equality
of the source and receptor heights in the experiment. Venkatram (1980)
also demonstrated a strong correspondence between u*, L and 7j in stable
conditions, which could be used to simplify application of these results.
It can be concluded that surface layer similarity models can give
very good results for vertical diffusion from near-surface sources at
distances up to 1 or 2 km. The theory does not contain enough information
about the variation of turbulence with height to give good results for
sources at more than 1/10 the ABL height. Except for averaging times of
less than 10 min, for which diffusion is accomplished mostly by small,
isotropic eddies, this approach can only predict the minimal (rather than
typical) lateral diffusion. This is because it contains no physics relating
to large horizontal eddies.
GRADIENT TRANSFER AMD HIGHER ORDER CLOSURE MODELS
One of the earliest and most frequently used approaches to prediction
of diffusion in the atmosphere is the gradient transfer assumption, or
"eddy diffusivity" theory, or "K theory". It assumes that the local
turbulent flux of a material is directly proportional to the mean local
gradient of that material, and is aligned into that gradient (towards
lower concentration). The constant of proportionality, K, is called the
eddy diffusivity and is supposed to be a property of the flow, rather
than of the substance. Thus, in a steady state situation, K should be
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only a function of position (x, y, z) and boundary conditions. For just
the vertical component of turbulent flux of substance C, we can write the
gradient transfer assumption quite simply
= -K aC/az . (5.3)
If there is no mean advection, then the rate of change of concentration
at each point is given by
aiU/at = - owrc"I7oz = o(KoC/az)/dz. (5.4)
This approach is tempting in its simplicity, and builds on a long
history of mathematical treatments of molecular diffusion. In that
context, K is just 1/3 of the mean molecular speed times the mean free
path length between molecular collisions. The most direct analog for
turbulent diffusion is K proportional to the mean eddy velocity times a
mean eddy diameter, or turbulence length scale (such as aw times the
Lagrangian time scale). However, this analogy is often inappropriate,
and there are only certain situations in atmospheric diffusion for which
K theory is approximately valid. Pasquill (1974, section 3.1) has discussed
this at some length. The question of validity has to do with the difference
between diffusion and dispersion, terms that usually are used as if they
were interchangable. "Diffuse" in its Latin root means to pour or to
spread out, and seems to connote smearing. Thus, local flux proportional
to local gradient seems a good description of "diffusion". "Disperse" in
its Latin root means to scatter, which better describes what big eddies
do to small plumes - wave them this way and that. The relative size of
the eddies and the plume or puff is the key. The idea that local mean
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concentration gradients have anything to do with "dispersion" of small
plumes by large eddies seems absurd, if we try to visualize this process.
Yet K theory has been indiscriminately applied to dispersive situations.
On the other hand, it is intuitively a good assumption to apply to "diffu-
sion" of large plumes by small eddies. Pushed to its limit, we might
expect this assumption to hold as long as mean concentration gradients do
not change drastically on the scale of the most effective eddies.
There are three cases of atmospheric diffusion which might be
describable using ordinary K theory, as described above. The diffusion
of material released near the surface is one. This application actually
occupies the fuzzy region between validity and non-validity, because the
most effective eddies doing the upward mixing are those of the order of
the plume thickness or slightly larger; it can be viewed as an approach
that might work, especially when the eddy structure maintains similarity
relative to the height above the ground. This condition is most closely
met in the lower part of the NRL. Here, K « u*z has proven validity, as
shown by many of the papers referenced in the last subsection. The SBL
is also a reasonable candidate for K theory, because the eddy size becomes
gradually restricted by the scale L as z/L becomes larger and vertical
diffusion becomes more of a diffusive process (eddies small compared to
scale of variations in mean gradients). The unstable case is just the
opposite; the higher the material mixes, the larger the dominate eddy
size until the dispersant is drawn into thermals which lift the material
all the way to z-j. Here, the eddy diffusivity assumption is inappropriate,
At best, it might apply as long as the maximum mean concentration remains
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on the ground and the vertical distribution is quasi-Gaussian; there is
growing evidence that this condition holds only for t < 0.5 z-j/w*, or
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only after the plume thickness is of the order of L or A, the "local"
Obukhov length defined by Nieuwstadt (1984a,b). The stable stratification
limits vertical mixing motions to a depth comparable to L or A, except
near the ground where z is a more limiting factor. Observations of
layers of near-constant a¥/az and aU/az in stable, dry boundary layers
above z = L indicate that Km = Kn = 0.08u*L, where Km and K^ are eddy
diffusivities for momentum and heat, respectively. We would expect K for
the diffusion of passive matter to be about the same. Ordinary K theory
has no validity for elevated plumes in neutral and unstable conditions,
because the plume thickness is then smaller than the most active eddies,
which are dispersive rather than diffusive.
The third possible case for limited validity for K theory is that of
lateral diffusion in convective conditions at distances beyond several
(U/w*)zj. By this time the plume width is several z-j, larger than the
dominant cell size (about 1.5 z-j). The asymptote for oy suggested by
Deardorff and Willis (1975), based on their laboratory tank experiments,
is 0.53 (w*zit)1/2, which would result from K = 0.14 w*z^ (note that, in
this case, convective scaling with w* and z-j applies, rather than surface
layer scaling with u* and L). However, in the atmosphere, where wind
shear plays a role and there are no horizontal restrictions like the tank
walls, there are horizontal eddies or rolls of much larger scale than zj
in the line of mean wind direction. Some indication of this is given
in the report on the Minnesota convective boundary layer experiment
(Kaimal, et. al., 1976). Very frequently, wind direction shifts are
observed to occur on time scales of about one hour on convective days
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(personal observation). Then continuous plumes are still "dispersing" on
these time scales, and a ay « x*'^ or « t*'* regime may be short-lived,
about 1 hr at most. Lateral dispersion observations in neutral and
unstable conditions have mostly yielded best fit power laws between x^-?
and xO*9, in contrast to the x^«5 asymptote given by K theory. This suggests
that plumes are usually influenced by larger and larger horizontal eddies,
contrary to K theory assumptions.
One unrealistic consequence of any diffusivity model is that non-
zero concentrations are predicted out to infinity, implying an infinite
velocity of propagation. An example is the Gaussian distribution, which
results when K is assumed constant. Although these far-flung concentra-
tions are quite insignificant in magnitude, the physical unreality of
this aspect of such models is discomforting. An alternative approach,
called the telegraph equation, has been developed by Monin and Yaglom
(1965). Essentially, it restricts the rate of change of flux by adding
the term t dw'C'/at to w'C1 in the flux-gradient equation, where T is
a limiting time scale. The result is that there is a limited velocity of
propagation proportional to (K/-c)l/2, and the flanks of the concentration
distribution have a cutoff. Closer to the center of the distribution
there is a slight change, but not enough to have any practical consequence
(see Pasquill 1974, Fig. 3.2).
The most serious 1 imitation'of simple eddy diffusivity approaches is
that non-local effects, such as those brought about by large turbulent
eddies, are neglected. The local turbulent flux of concentration is
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assumed to be a function of only the local gradient of concentration.
When large eddies bring in fluid containing zero concentration, which
therefore comes from regions of zero gradient, this assumption of depend-
ence only on local gradient is obviously wrong. Several attempts have
been made to construct models which avoid this deficiency. One involves
a more sophisticated version of K-theory called spectral diffusivity
(Christensen and Prahm, 197fi; Rerkowicz and Prahm, 1980). The concentra-
tion field is described in terms of Fourier components, with c(k) being
the amplitude of the Fourier mode of concentration with wavenumber k
(waves per unit length). The eddy diffusivity is assumed to be a function
of k; thus, K(k) can account for the variability of turbulent energy in
each range of eddy size. The eddy diffusivity equation for a Fourier
mode becomes ac(k)/5t = -k^K(k)c(k). In a sense, this model is
non-local , since each Fourier component represents concentration spanning
the whole dominion under consideration. A particular form that has been
proposed for K is KQ/(1 + Bk^'^). This gives a constant K, with the same
result as ordinary diffusivity theory, for very large wavelengths (small
k), i.e., for variations in concentration on a scale large compared to
energetic turbulent eddies. For small wavelengths K <* e^/^k-^/^,
where e is the turbulent energy dissipation rate; this is the asymptote
for the "inertia! subrange" of turbulent eddies driven by the larger
eddies. Prahm, Rerkowicz and Christensen (1979) applied this theory to
time averaged plume concentrations, adding the effect of "meandering" of
the plume center! ine by means of a "spectral phase diffusivity coefficient'
that is made to fit statistical theory predictions for large and small
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asymptotes of k and t. The resulting plume growth is, not surprisingly,
in agreement with the main results of statistical theory, but the shape
of a narrow plume is different from Gaussian. There is no experimental
confirmation of this shape as yet, and its practical significance seems
doubtful. The theory has primarily been applied to two dimensional global
transport models, with eddy sizes ranging up to several thousand kilometers.
Another type of non-local K theory was suggested by Voloshchuk
(1976). In this model, K is both a function of z and of a displacement
distance, 1. If K(z,l) has a non-zero value only for 1 = 0, the equation
reduces to the telegraph equation for the local case, like that developed
by Monin and Yaglom (1965). While this method appears to have potential
for accounting for non-local concentration gradients, the actual specifi-
cation of K(z,l) for various stabilities might be quite difficult, con-
sidering the effort that has been spent on trying to specify K for a
single variable, z.
A new mechanism for turbulent transfer has been developed by Stull
(1984) and applied in one form by Stull and Hasegawa (1984). This mechanism
is called transilient turbulence theory, from the Latin verb "transilire"
meaning to jump over or leap across. .Stull introduces the mechanism in a
finite difference framework, then goes to a continuous representation.
In either form, the mechanism adjusts the concentration at a point by
considering the concentrations at all neighboring points which can be
influenced by the flow. For example, consider only vertical transfer,
for which the concentration at a point is influenced by all of the points
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in the vertical. The formal expressions for the continuous case are
similar to the equations for radiative transfer in an absorbing, but
nonscattering medium. Fiedler (1984) independently developed a similar
concept, but started from a different perspective, namely the spectral
diffusivity concept of Berkowicz and Prahm (1980). Stull's approach was
to visualize turbulence as consisting of a spectrum of eddy sizes each
contributing to the turbulent transfer. The interesting feature of the
mechanism is that the coefficients defining the transfer process can be
specified ahead of time, or can be made responsive to changing flow
conditions. The coefficients can be made to represent any type turbulent
flow. The same model with different coefficients can represent diffusion
in a CRL or in a SRL. Further, Stull points out that the transilient
approach can also be used in combination with a second order closure
model to close the higher moments. This new development has much promise.
K theory is considered "first-order" closure, meaning that closure
assumptions are applied to the second moments of turbulent quantities,
such as w'C1 and w'u1, that relate them to first order quantities like (T
and "u. About 10 years ago, work accelerated on "second-order" closure
models for turbulent boundary layers and diffusion, carrying this approach
to a new level. These models begin with the exact equations for second
moment terms like concentration flux or momentum flux, which are needed
to solve for a"C/at or 5~u7dt. As in all turbulence models, the exact
equations contain terms with new unknowns, and always more unknowns than
equations, so closure assumptions are still necessary. In this case, the
new unknowns are third moments, such as u1w'Cr , and assumptions must be
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made relating them to terms already being computed, i.e., second moments
or mean quantities. These assumptions may be just as questionable as
assumptions made about K in first order theory, and the third moment
terms may be modeled quite crudely; the advantage is that second moment
terms like turbulent fluxes do not depend on these assumptions alone, but
also depend on exact terms; thus, they are less susceptible to error on
account of the failure of a closure assumption. Nevertheless, the output
of second-order closure models depends on the performance of the closure
assumptions, and these should be scrutinized for proper physical charac-
teristics and tested against data in a variety of flow situations. One
shortcoming that second order closure models so far share with K theory
is that the closures are local; thus, the effect of large eddies on the
third moment terms cannot be handled properly.
One of the advantages of second-order closure modeling of diffusion
is that buoyancy-driven vertical mixing is included; this is done with
the term (g/e")9'C' in the equation for aw'C'/5t. Whereas gradient
transfer theory fails to model the lifting of the mode of maximum
concentration from a surface source in convective conditions, Lewellen
and Teske (1976) demonstrated a second-order closure model that does
this. The qualitative features of /Xdy that were observed by Willis
and Deardorff (1976) in the laboratory are reproduced quite well, but the
surface concentrations do not reduce quite as fast in this numerical
model. Pasquill (1978) also reports some success with second-order
models in the case of vertical diffusion from an elevated source, another
case which cannot be handled by conventional K theory. Second-order
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closure models have also been used to study the SBL, as reported in Sec. 2
Wyngaard (1975) and Rao and Snodgrass (1978) were able to reproduce in
their models the approximate behavior with time of quantities like heat
flux, friction velocity, and boundary layer height, h, which is of vital
relevance to diffusion modeling. It evolves rapidly after sunset, and a
low or medium level source could be below or above h; this determines
whether or not vertical diffusion occurs at all.
CONVECTIVE SCALING MODELS
A very active area of research at present is the application of
convective scaling to daytime diffusion models. The idea was first
applied to atmospheric boundary layer turbulence by Deardorff (1970b),
who first tried this scaling in the context of a large eddy simulation
model of neutral and unstable boundary layers; this model will be discussed
in the next subsection. Subsequent successes using this scaling in the
context of CBL turbulence measurements were reviewed in Sec. 2.
For |zj/L| equal to 4.5 and larger, Deardorff (1972) found that w* scales
the magnitude of turbulent velocities better than u*. Typical daytime
values of convective and surface layer scaling quantities are: z-j ~ 1000 m,
u* ~ 0.4 m/s, and H* ~ 0.004 m2/s3. This gives L 40 m and |z.j/L| ~ 25,
which is quite convective according to Deardorff's results. The criterion
|z-j/L| > 4.5 would be met even with l/5th the above values of z-j or H*,
or with these values unchanged and u* = 0.7 m/s; this typically corresponds
to wind speeds of about 6 m/s near the ground; U = 6 m/s with moderate
insolation corresponds with stability class "C-D" in Pasquill's table,
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between "neutral" and "slightly unstable". We can conclude that convective
scaling would be appropriate for most of the daytime on most days at most
sites on land. One condition that produces sustained maximum average
ground concentrations near large power plants is a sunny summer day with
low wind speed; these are very convective conditions, with |zj/L| much
larger than 100. This is a case which should certainly benefit from the
convective scaling models recently under development.
Application of the convective scaling idea to diffusion began with
the now-famous convective tank experiments of Willis and Deardorff (1974,
1976, 1978, 1981, 1983). These experiments and their controversial results
were discussed in Section 3 under "Laboratory Experiments"„ These results
were unexpected and cast much doubt on the ability of conventional Gaussian
models to accurately model daytime diffusion. Naturally, there was much
interest in verifying these results outside the modeling tank. The
numerical experiments of Lamb, using Deardorff's (1972) model, was a
first attempt. The results were qualitatively quite similar to those of
the tank experiments, with nearly the same descent rate of the maximum
concentrations from elevated releases, but with slightly less decrease in
surface concentrations at the larger distances. Utilizing field data
that were already available, both Briggs and McOonald (1978) and Nieuwstadt
(1980) demonstrated that the "Prairie Grass" surface values of /Xdy
were quite similar to those for the lowest release in the tank when
nondimensionalized using convective scaling. The field experiments
showed more scatter; this was attributed to the short release time, only
10 minutes, which is inadequate to sample many eddies (a typical "eddy
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passage time" is 1.5 z-j/u, which can range from 2 to 30 minutes or more).
The Prairie Grass ay/Zi values were about 40% larger than those in
the tank; part of this may be due to the absence of horizontal motion
restriction in the atmosphere.
It is ironic that field experiments done in the two decades following
Prairie Grass did not include sufficient measurements to determine z-j, a
parameter that turns out to be indispensible for modeling diffusion in
convective conditions (the 1956 experiment included aircraft soundings of
temperature and humidity, as well as rawinsondes, so z-j was easy to
determine in retrospect). A recent field experiment designed to test
convective scaling on diffusion in the atmosphere, especially for compari-
son with the tank experiments, was carried out at the Roulder Atmospheric
Observatory in August and September of 1982 and 1983; this was a coopera-
tive experiment between the U.S. EPA and NOAA's Wave Propagation Laboratory.
It is described in Section 3. Some preliminary results have been reported
by Moninger et. al., (1983), Eberhard et al. (1985) and by Briggs (1985a).
In the first paper, a striking resemblence was seen for nondimensionalized
j"*dy between a 29-min period with z^/z^ - 0.45 and the tank experiment
for zs/Zi = 0.49. However, it is not yet known how much the 0.3 m/s fall
speed of the chaff tracer distorts the distributions (this concern motivated
the simultaneous side-by-side releases with oil fog). In Rriggs (1985a),
five periods of chaff distribution were analyzed in comparisons with the
tank and numerical experiments and three other field experiments including:
the most convective cases from Project Prairie Grass, the five most
convective 30-min periods of SF^ releases at Cabauw, the Netherlands,
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which really were only marginally convective (van Duran and Nieuwstadt,
1980), and, finally, some SO;? measurements at the Morgantown power plant
(Weil, 1979). The latter observations were screened to remove buoyancy-
dominated cases. The results of these comparisons show a good deal of
consistency, in convective scaling. All experiments support the tank-
numerical cjy/z-j curves, except for two BAD periods with very large wind
direction shear; these had about double the ay. The near-surface z-jIT/xdy/Q
for one of the surface chaff releases dropped to a value about 30% lower
than the tank minimum, but all the remaining data were in better agreement.
The most significant result was for the maximum surface value of /xdy
from elevated sources; there was a definite trend with increasing zs/z-j ,
with Cy1 = zs"u/xdy/0 = 0.48(1 + 2zs/Zi). Conventional Gaussian models
give Cy1 = 0.48, regardless of source height. These maxima were all
found in the neighborhood of x = 2 u" zs/w*.
Another recent confirmation of convective scaling validity comes
from photographic thickness versus distance of plumes from three power plants
in Australia (Carras and Williams, 1983). Good agreement with az/z-j in
Lamb's numerical experiments is found for the less buoyant plumes
(F^ < 300 nr/s^). Further testing of Willis and Deardorff's convective
scaling results was recently accomplished in a wind tunnel, at Colorado
State University (Poreh and Cermak, 1984). This was described in Section
3, under "Laboratory Experiments." Rather good agreement with the tank
results was obtained, in spite of lack of complete development of convective
turbulence intensity and other restrictions that accompany seal ing-down
to wind tunnel size.
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At present most theoreticians seem convinced that convective scaling
is the best way to go for modeling daytime, non-neutral diffusion ("the best
thing since sliced bread" was heard in this context at a recent conference).
Many modelers have been fitting the experimental results with analytical
formulas for
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LARGE EDDY SIMULATION MODELS
Another numerical modeling approach to boundary layer turbulence
that is really quite different from second order closure models has been
called the "subgrid closure" or large eddy simulation (LES) model. This
is the approach first applied long ago to prediction of long-range global
weather patterns (Smagorinski, 1963), then further developed for the CBL
by Deardorff (1970). The equations for the motions of fluids are believed
to be complete and can be solved exactly for many non-turbulent flows.
The "closure problem" arises with turbulent flows because the equations
can be solved exactly only if initial and boundary conditions are known
exactly and we attempt to calculate every detail of the motion, down to
the smallest eddy. This, of course, exceeds the computational capacity
of any computer so far envisioned (unless we regard wind tunnels and
water channels as analog computers!). For tractability, we have to
introduce some kind of averaging, and this always introduces some new
unknown, such as entrainment velocity or shear stress. Some assumption
must be made about how the new unknown(s) relate to quantities already
computed to "close" the problem and make a solution possible. In the LES
modeling approach the domain of computation is divided up into grid cells
and the exact, or "primitive", equations are used to solve for the mean
velocity, temperature, water vapor content, etc., of each grid cell for
each time step. Thus, the details of the large eddy turbulent motions
are actually computed in time and space; this gives a "picture" of the
turbulence, if desired. Patterns of mean quantities are obtained by
applying time averaging, just as would be done in a field or laboratory
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experiment. This contrasts to higher order closure models, which predict
mean patterns but none of the details of motion. What is lacking, however,
is exact computation of the way neighboring grid cells interact with each
other through turbulent exchange of momentum, heat, water vapor, etc., by
subgrid scale turbulent eddies. The closure problem is still there, but
in this approach it has been pushed down to the scale of the grid cells
used; this follows because bulk averaging is done on this scale with
nothing known about the details of motion within the grid cells.
One advantage of this approach is that the structure of fine scale
turbulence is simpler than that of the large scales where turbulent energy
is produced. The small eddies are more isotropic and are driven only by
the "inertial" forces induced by the larger eddies. This is called the
"inertia! subrange" of turbulence, and turbulent velocity spectra follow
a simple -5/3 power law with frequency or wave number in this regime.
This ought to make the closure problem simpler on this scale, with greater
universality of successful solutions. The closure that has been used is
a rather primitive eddy diffusivity assumption; the rate of subgrid
turbulent exchange between cells is assumed proportional to the difference
between the cells of the quantity under consideration. This seems a
"safe" use of K theory because if the grid cells are small enough, the
gradients of mean quantities do not vary much on that scale. K theory is
used only to account for exchange done by eddies of the grid cell size
and smaller.
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The eddy diffusivity for subgrid exchange used by Smagorinski (1963),
Lilly (1967), and Deardorff (1972) is based on the cell size and velocity
differences between adjacent cells. In effect, it is assumed that the
subgrid turbulence reacts instantly to changes in local velocity gradi-
ents. More specifically, the subgrid K is taken to be proportional to
the square of the grid size times the local deformation rate, which is
always positive. The constant of proportionality is chosen to produce
flux-gradient relationships in agreement with those derived from careful
field experiments, like the 1968 Kansas program. Deardorff (197?) uses
an eddy diffusivity for heat, Kh, three times as large as the one for momen-
tum, Km (except for the layer of grid cells adjacent to the surface); K^
and Km are different because of the pressure-velocity correlation. Pres-
sure forces affect momentum exchange but not heat exchange. Presumedly,
this applies to material exchange also, so on a subgrid scale the turbu-
lent exchange of concentrations between cells should be described by Kn.
Deardorff (1972, 1974) first applied this type of model to ARLs for
neutral and convective cases. Many of these results were mentioned in
the previous subsection on convective scaling models, since they led to
his convective scaling suggestion. One very instructive product of this
model, not available with other models, is the mapping of fields of turbulent
quantities computed for any given grid plane and time step. You can see
the eddies, so to speak. It must be remembered that these are only model
predictions, however, and it must be asked how well the model simulates
what is known about the ARLs. Heardorff's model results have compared
very well with aircraft and tethered balloon measurements for convective
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cases. The greatest shortfall has been in the magnitude of computed
lateral turbulent velocities. This may be partly due to the limited
width of the domain of computation, 2 Zj (the edges of the domain are not
treated like solid walls; cyclic boundary conditions are used, so whatever
advects in or out of one wall is balanced by advection through the opposite
wall). The model is not capable of computing subgrid details like entrain-
ment at the top of the mixing layer; instead, a "solid" lid at z = z-j was
used, with w1 = 0 there.
Deardorff's model output for velocity fields was used directly by
Lamb (1978, 1979) for numerical diffusion experiments. In effect, Lamb
released thousands of numerical "particles" into the already-computed
turbulent velocity fields, computed their trajectories, and counted them
over grid volumes to get average concentration fields. He also added to
the particle velocities a random subgrid scale velocity component, scaled
by subgrid-scale turbulent energy. The results for release heights near
the surface, at z « z-j/4, and at z = z^/2 in convective cases are in
satisfactory agreement with Willis and Deardorff's modeling tank results,
as was discussed in the previous convective scaling subsection. They agree
especially on the "unorthodox" qualitative behaviors of the surface and
elevated plumes.
Large eddy models are at their best in large eddy turbulence, which
is why the convective case has been the most explored. Deardorff also
modeled the NBL. In principal, this type of model can also be applied to
SBLs. A much larger computational capacity will be needed, however,
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because the largest eddies (for vertical turbulence, at least) are much
smaller. The nocturnal mixing depth typically goes to 10 L, and eddy
size is limited by L, rather than z-,-. Instead of a computational domain
20 cells high, as Deardorff used, at least 100 are needed. Much smaller
time steps would be needed also because the rotation period for the most
energetic eddies scales to L/u* instead of z-j/w*, about an order of
magnitude shorter time period. If Brost and Wyngaard (1978) are correct,
the model would have to be run a much longer time because even on slight
terrain slopes a steady state is not achieved during a night. In contrast,
the CRL approaches steady state in its turbulent properties after a few
z-j/w*, typically 0.5 to 1 hr.
STATISTICAL MODELS
It was realized long ago that if enough were known about the
statistics of turbulence and the mean flow field, it should be possible
to calculate the dispersion of passive substances resulting from the
turbulence. The formal foundation for this calculation was laid by G. I.
Taylor in 1921 and is summarized by Pasquill (1974, see section 3.4). Its
main limitation is that it applies only to homogeneous turbulence, i.e.,
the dispersant is assumed not to spread into a region where the statistics
of turbulence are different from the origin. Thus, it applies best to
horizontal diffusion over terrain flat and homogeneous enough to not
cause large variations in the lateral turbulent velocity statistics. It
can be applied to vertical diffusion close to an elevated source, but it
becomes mathematically unsound as the material spreads and approaches the
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ground, since vertical turbulence statistics vary strongly with height
near the ground. This is not to say that this application should not be
tried; with appropriate modifications, useful results might be obtained,
as is proven by data comparisons with the "bouncing ball" models mentioned
in the subsection on convective scaling models (these models are based on
turbulence statistics at just the source height). The statistical
approach of Taylor has been extended and adapted for the case of vertical
diffusion from a ground source (Hunt and Weber, 1979) by separating
the motion into mean rise velocity, dT/dt, and fluctuations about this
mean and using the fact that aw does not vary much with height.
The statistical theory of Taylor and the simplified approach to it
developed by Hay and Pasquill (1959) need a capsule review here because
they are so fundamental to all subsequent work with statistical models.
Taylor's theory, as applied to turbulent diffusion, begins by describing
the displacement of a single particle in terms of its time history of
velocity, assuming that it follows the motion of a single parcel of fluid.
This is the Lagrangian velocity, and all statistics of such velocities
are called Lagrangian. A second particle can be added and the relative
displacement can be described in terms of the Lagrangian statistics of
the two particles to compute the relative diffusion. The most frequent
application of this theory is to total diffusion, i.e., the variance of
displacement of a large number of particles released steadily from the
same point. One fundamental result, written here for only the lateral
component of diffusion, is
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da2/dt = 2 / v'(t)v"(t + g) d£
y o (5.5)
- 2 a/ /
o
where t is the travel time, av is the total velocity variance, and R(£)
is the Lagrangian correlation coefficient of velocity separated by a time
lag C« Since R(£) * 1 at small £, i.e., there is no significant
change in parcel velocity over a small time, it follows from the above that
ay = avt. For very large travel times, if R(£) approaches zero faster
than 5"1 at large 5, ay = (Za^t^1/2, where tL is the limit of
the integral of R(£) over a large time; t|_ is the "Lagrangian time
scale". This is the same result given by constant eddy diffusivity, with
K = av2t|_. Correlation coefficients have a direct mathematical relation-
ship with spectra, so the basic theory of Taylor was later developed in
terms of integrals of Lagrangian velocity spectra by Batchelor (1949) and
others.
These results are exact and relatively simple, but true Lagrangian
velocity statistics are rarely obtainable. In an approximate way, large
scale horizontal velocities can be deduced from the trajectories of
tracked balloons, but these never follow the vertical air motions perfectly,
What is needed are cheap, small, neutrally buoyant air motion "trackers".
Because it is much easier to obtain Eulerian velocity statistics, which
are measured at a fixed point such as a meteorological tower, Hay and
Pasquill (1959) tried a different approach. They assumed that the
Lagrangian velocity correlation, R|_(£), is similar in shape to the
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Eulerian velocity correlation, Rj:(t), except for an expansion of the time
scale by a factor p; that is, R|_(£) = RgU/p) . Pasquill (1974) has
argued that the exact shape of the velocity correlation is not critical
anyway. The value chosen for p is more important. One simple estimate for
p is based on a scaling approximation for the two time scales. The
Eulerian time scale, t£, is proportional to the time it takes an eddy to
pass a fixed point, which is A/If, where a is the size of the dominant
eddies. The Lagrangian time scale is proportional to the time it takes to
traverse an eddy with a turbulent motion, JL/0V, if lateral diffusion
is being considered. Then p « t^/t^ « "u/crv = i~ , where i is called
the turbulence intensity. Pasquill (1974) gives estimates of (pi)
ranging from 0.35 to 0.8.
The most useful result so far from the Hay-Pasquill hypothesis has
been the following relationship, written here for only the lateral component
of spread,
Here, we use the approximations aa - crv/u" for the azimuth angle variance
and t = x/U for the time of travel; T is the sampling duration; and
t/p is the averaging time, which acts like a filter. By smoothing
over an averaging time t/p before calculating variances, the contribution
to c?v of high frequency eddies that only move the particles back and
forth a number of times by the time they reach x is suppressed. Using
the above, all that is needed to calculate dispersion in homogeneous
turbulence is a record of the velocity or azimuth angle fluctuations at
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the release point plus a good estimate of p; this method can be applied
to different sampling durations in a straightforward manner. This approach
is, in fact, the main basis for PasquilTs curves for ay, rather than
direct diffusion measurements. Pasquill (1974) has accumulated much
experimental support for this approach, especially for ay.
Statistical theory goes back over 60 years, yet, since 1976, there
has been a marked increase of interest in it as a practical tool for
diffusion modeling. The first of these recent attempts to make practical
simplifications in statistical theory was by Draxler (1976), who used the
forms suggested by Pasquill (1971)
ay = avt fi(t/tj_) and (5.7)
az = awt f2(t/tL) , (5.8)
where av or aw are measured over the same sampling duration as ay or az,
with an averaging time approaching zero (i.e., unfiltered fast-response
measurements). Draxler attempted empirical evaluations of f^, fp, and t\_
separately for elevated and surface sources, horizontal and vertical
diffusion, and stable and unstable conditions. Altogether, 11 different
sets of field experiment observations were used. Analytical best fits
for ?i and f% were suggested as were best values of Ty? for each category,
where T^/2 is the time that f^ or f? is observed to drop to 1/2 (the derived
value of t|_ was 1.64 T^/2 ^or most cases). ^1/2 ranged from 50 s for az of
ground sources in stable conditions to 1000 s for ay of elevated sources
for all conditions. However, the scatter in individual run best fits to
was very large, over several orders of magnitude. The scatter in the
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logarithms of the ratio of computed to observed ay and az, using the
suggested Ty2 values, ranged from a standard deviation of 1.29 to 2.05.
These were somewhat better than the comparison with the Pasquill-Gifford
predictions, but still represent errors of factors of 4 to a - not a very
comfortable place to rest the "state-of-the-art". Nevertheless, in 1977
this approach was recommended by an AMS workshop as a step toward improved
modeling (Hanna et al., 1977). For ay, the simple form suggested by
Pasquill (1976a) was also recommended, with 0y/(aax) a function of
only x. It should be recognized that much of this scatter is due to
poorer instrumentation for measuring aa and ae in the older experiments.
Draxler's method performed much better when compared to recent experiments
(Gryning and Lyck, 1984).
Subsequently, there have been attempts to reduce the amount of pure
empiricism in this approach, to better model the variability of quantities
like 0y/(aax). Doran et al . (1978) showed how this quantity,
plotted against x, systematically depends on the sampling duration and
the averaging time used in different experiments. They qualitatively
related this dependence to velocity spectra considerations and suggested
that wind speed and stability will also have important effects. The same
authors studied additional data bases and concluded that aa measured at
elevated release points is not a good indicator of surface level ay;
near-surface aa orders the diffusion data far better (Horst et al.,
1979). Examining data for rough terrain experiments, they found
indications of problems with cra measurements; in the older experiments,
wind vanes may have been either sluggish or insufficiently damped.
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Irwin (1979) suggested some forms for crz/(aex) in which ae is the
variance of the elevation angle. For neutral and stable conditions he
presented them as empirical functions of travel time, with one curve for
zs > 100 m and another for zs < 50 m, and a suggested interpolation;
these were based on averages from five experiments. For unstable conditions,
he made the physically reasonable assumption that t[_ is proportional to
z-j/w* and presented the Willis and neardorff tank data, Lamh's numerical
results, and some limited field data as functions of tw*/Zi, the convective
scaling dimension!ess time scale. Separate curves were fit to ground
source and elevated source data. This approach does appear to give
considerably less scatter than the purely empirical analyses.
A somewhat similar scientific approach was recently used for elevated
source values measured in a SBL (Venkatram et al., 1983). The data were
lidar measurements of
-------
curve az = awt(l + t/2t|_)~1//2. However, In the upper half-decade of
their t/t|_ values, some tendency towards a -1 slope in the plot for
cjz/(awt) can be seen. This suggests az « aw t|_ as a possible limit,
a limit that has been suggested by Hunt (1982) for the case tL « N-1.
The effect of sampling duration on plume width and average concentra-
tion can be readily determined by using statistical theory, a capability
that escapes most other modeling approaches. All that is needed is a set of
velocity spectra or a time series of fast-response wind speed or directions
from which quantities like (
-------
"Cluster growth", or relative diffusion, is another problem area
that, in principle, is amenable to solutions using statistical theory.
In practice, this is a much more difficult problem to solve because one
must know or assume properties of relative Lagrangian velocity statistics,
which depend on the spacial separation of particles as well as time
lags. Taylor first addressed the problem (Batchelor, 195?), and Smith and
Hay (1961) attempted to simplify it by replacing Lagrangian velocity
correlations with Eulerian ones, in the manner of Hay and Pasquill (1959).
This theoretical matter then seemed to get little attention until Sawford
(1982a) discussed and compared both approaches. Data for testing predictions
for relative diffusion were limited and contained much scatter, but
Sawford believed that Taylor's approximation gives more realistic predic-
tions.
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RANDOM PERTURBATION MODELS
In the last few years there has been a resurgence of interest among
theoreticians in mathematical simulations of turbulence diffusion that
use randomness as one element of the simulation. These are variously
called random walk, Markovian random walk, Monte Carlo, random force, and
Langevin equation methods. Much of the mathematics of these methods has
been borrowed from treatments of Brownian motion, which were made much
earlier in the century by a number of prominent scientists, e.g., Einstein
(1905).
Most of these methods also lean heavily on statistical approaches to
diffusion modeling, since they depend on velocity correlation functions
for the non-random part of the modeling. An exception is Lamb's (1978)
modeling of the CBL, which uses Oeardorff's (1973) large-eddy, subgrid-
closure model to determine motions caused by large eddies and randomized
velocities for diffusion caused by eddies on the subgn'd scale. Undoubtedly,
computational advances account for some of the renewed interest in these
methods, especially the Monte Carlo techniques that "compute" the trajec-
tories of thousands of particles using random number generators.
The most common starting point is the assumption that
v(t + At) = R(At) v(t) + v'(t) (5.9)
(Smith, 1968), where v is the velocity of a small volume of air, At is
a small increment of time, and v' is a random change in the velocity,
with V1" = 0 and v1 v = 0. It follows that for stationary, homogeneous flow,
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R(At) must be the Lagrangian autocorrelation function (Gifford, 1982).
Another form of the above uses the series expansion of v(t + At) in terms
of At, making At small enough that higher order terms can be neglected,
so that v(t + At) = v(t) + Atdv/dt. Substitution into the above yields
dv/dt + [1 - R(At)]At-!v(t) = v'/At. (5.10)
This is a form of Langevin's equation, which was widely utilized in
theories of Brownian motions; some of these theories are referenced by
Gifford (1982), who also mentions several applications of the equation to
turbulent diffusion made in the early 1960s.
The above equations, while simple, have many interesting properties.
If the process is stationary, v7 must be constant in time; this imposes
the condition that V'z = v7!!! - (R(At))2], so that the random velocity
changes do not cause an increase or decrease in the total averaged kinetic
energy. Quite a few investigators have assumed that R(At) has an
exponential form, as a matter of convenience; Gifford (1982, 1983) claims
that this form of R can be derived from the fundamental equation. On the
other hand, Pasquill and Smith (1983) claim consistency of this equation
with any form of R(At) that decreases linearly in At for small At; they
also fit nonlinear forms of R, R = 1 - 9Atn for small At with n * 1, to
more complex velocity change assumptions involving v at time steps prior to
t (double regression and triple regression models). However, the exponential
form is the simplest, and it has been claimed that diffusion predictions
are not sensitive to the exact form of R (e.g., Pasquill, 1974). Smith
(1968) notes that the exponential form for R implies impulsive forces acting
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on the parcel, as the analogy to Brownian motion would suggest, but this
may be of little consequence to the practical outcome, the diffusion
prediction.
Some controversy exists over the valid applications of this
assumption to diffusion prediction. Gifford (1982) applied the Langevin
equation, with R = exp-(pt), to derive analytical predictions of the mean
displacement of particles with an initial velocity v0 and the mean-square
displacement of the same particles, ay2. Applied over all initial
values of v0, with v0^ = "v2" = av2, he derived the same equation for
total variance that Taylor (1921) derived from statistical theory (Taylor
also assumed the exponential form of R). Applied to small initial values
of v0, Gifford showed that the results approximate ay2 = v02t2 at small
t, (?/3)av2pt3 at intermediate t, and 2 av2p~l at large t (p'1
is the Lagrangian time scale of the turbulence and
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that the ay equation applies to the averaged diffusion of all puffs
leaving the source with the velocity v0, not to the average relative
of an isolated puff diffusion.
Although we must be careful about the interpretation of various
results, the Langevin equation with an exponential form of R is very
mathematically malleable. It was applied to relative and total diffusion
from a finite-sized, finite-duration source by Lee and Stone (1983a).
They also computed peak to mean concentration for a point source. A
comparison paper explores relationships between Eulerian and Lagrangian
time scales, so important in statistical diffusion theory, using the
Langevin equation with Monte Carlo simulations (Lee and Stone, 1983a).
Sawford (1982) used a similar method, except that he applied it to the
relative motion between a pair of particles, a more fundamental approach
to the problem of relative diffusion.
Most of the above results apply to horizontal diffusion because of
the assumption of homogeneous turbulence; this assumption is seldom
justifiable in the vertical dimension. Ways to extend the application of
the method to the case of a gradient in turbulent kinetic energy were
developed by Wilson et al. (1982) and Ley and Thomson (1983); this requires
compensation for a bias velocity that develops in the direction of the
gradient. Some conclusions from these studies are given by Pasquill
and Smith (1983); one is that about 5000 random-walk trajectories must be
calculated to get smooth profiles. Davis (1983) used these simulation
methods to obtain a number of practical results for elevated-source
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diffusion in neutral conditions. He explored the effects of varying the
release height, the surface roughness, and the friction velocity by making
Monte Carlo experiments for a number of parameter combinations.
Very recent work on random walk modeling in inhomogeneous flows
(Thomson, 1984) and inhomogeneous and unsteady flows (van Dop, Nieuwstadt,
and Hunt, 1985) has shown that the Langevin equation can be used under
these conditions, but a careful analysis is necessary near boundaries.
A more elaborate Monte Carlo scheme using Eq. 5.9 was developed by
Baerentsen and Rerkowicz (19R4) for vertical diffusion in convective
conditions. Recognizing the highly skewed nature of the vertical velocity
distribution owing to the larger fraction of horizontal area occupied by
downdrafts, they assumed separate statistics for updrafts and downdrafts,
in effect, a double Gaussian distribution of w. Particles in an updraft
or downdraft may stay in it until reaching z = zi or z = 0; then,
perfect reflection is assumed, with a switch in the updraft/downdraft
category that selects the statistical parameters. A particle traveling
in an updraft, for instance, rises with the mean velocity of updrafts at
that height plus a deviation velocity based on the Monto Carlo method and
the mean variance of updraft velocities at the height being traversed.
All velocity statistics were estimated, in terms of convective scaling,
on the basis of field data, water tank data (Willis and Deardorff, 1974),
and Lamb's (198?) numerical modeling. The Monte Carlo scheme provided a
very good duplication of the vertical diffusion results from the tank
experiments (Willis and Deardorff, 1976, 1978, 1Q81).
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The computation of thousands of particle trajectories of the Monte
Carlo method can be avoided by means of the "integral equation" approach,
which integrates all probabilities mathematically by considering the
probability of a "collision" (change in v) in each grid square (Smith and
Thomson, 1984). This requires much less computation than the Monte Carlo
method. With either approach, it is relatively easy for today's computers
to use the random perturbation methods to calculate diffusion; once
confidence in the assumed turbulence parameterizations is established,
these methods can he used to explore the effects of release height,
roughness, stability, etc, much more easily than by doing field experiments
for every combination of circumstances.
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SECTION 6
WHAT ARE THE NEEDS?
Perceived data and modeling deficiencies
EXPERIMENTAL VERSUS MODELING NEEDS
If there is a overwhelming scientific consensus on anything, it is
that we need well-conceived new experiments much more than we need more
models. This is not to say that there are no needs for new models or model
development. There have been frequent calls to replace the P/R/T curves,
especially for elevated-source modeling, and to replace the Gaussian plume
assumption in situations in which it seems dubious, especially for vertical
diffusion in convective conditions. However, alternative models are
available, as Section 5 of this paper amply shows. The real need is to
prove that, at least, some of them work better than models in use, and
this requires a broader and more reliable experimental base than we have
at present. Comparative studies like Irwin's (1983) are handicapped by
the doubtful quality of wind variance measurements in pre-1970s experiments,
the likelihood of significant deposition losses in the same, the sparsity
of direct vertical diffusion measurements, and the rapid decline in data
for distances greater than 1 or 2 km downwind. In addition, scientists
have often expressed alarm and amazement at how far accepted modeling
schemes extrapolate beyond the data bases supporting them; Smith (1981)
compares this practice with extrapolating the parameters for a two-
story house to construct a building the size of the World Trade Center.
This is hardly an exaggeration; P/G/T az values based on Prairie Grass
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measurements to 800 m have been extrapolated to 100,000 m. More extended
experiments are needed both to support better models and to evaluate the
adequacy of models that have been used for over 20 years.
A glance at the reference list for Table 1 supports the claim of
Gifford (p. 4 of Smith, 1981) that experimental efforts dried up after
1968, at least for field experiments (the Hanford-67 series is an exception,
spanning 1967 to 1973). He and Smith (1981) attributed this to a misplaced
faith in numbers produced by computers, a commodity that became ever more
abundant and cheap. In the period following 1968, considerably better
tracers, sampling equipment, remote sensing techniques, and turbulence
instrumentation were developed. So we are now in a position (if we had
the money) to do much higher quality diffusion experiments. The recent
elevated-source experiments in Europe, in Copenhagen and Cabauw, give us
some encouragement, as does the CONDORS experiment in Colorado (Agterberg
et a!., 1983; firyning, 1981; Moninger et al., 1983). These were all
limited objective, but high quality, experiments. There has been moderate,
continuing support for laboratory diffusion experiments through the 1970s
up to the present. This support, however, has shifted toward studies of
complex terrain, acidic deposition and buoyancy effects; there is little
work at present on basic diffusion phenomena.
On the model development side, the opinion expressed by Smith (1981)
that it needs no new support seems extreme; nor do we share his
confidence that "new sources of field data will be devoured by the modelers
as quickly as they become available." Although computer power becomes
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steadily cheaper, it takes considerable human effort to assemble all the
new and old pieces of information, process it effectively using computers,
and intelligently evaluate the results, including needs for model
revisions. Comparison of one model with one experiment is a relatively
modest task, but much more is learned, and more effort is required, when
many models are compared with that experiment (e.g., Gryning and Lyck,
1984). Best yet are efforts to compare many models with many experiments
(Turner and Irwin, 1982; Irwin, 1984). These efforts require at least
the current level of support for model evaluation, as most scientists
engaged in this activity feel that it will take a long time to meet
current needs at the current rate of commitment. Furthermore, it takes
much well thought out effort to translate research models, once validated,
into appliable models, given the fact that we do not normally have high-
grade meteorological measurements available. This need will be
addressed at length in Section 7. The need for better applied models is
indisputable. The scientists participating in the Rural Model Review
were critical of the eight models reviewed as being too similar and
showing too little skill (Fox et al., 1983). The AMS committee con-
ducting this review strongly urged that "the scientific community submit
models that it considers technically better than those available today."
Because modeling improvements require suitable data bases for
validation, specific modeling needs and experimental needs tend to parallel
each other. Specific areas needing attention will be addressed in the
following subsections.
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WHAT KIND OF EXPERIMENTS ARE NEEDED?
Suggestions for the kind of experiments that are needed for improvements
in diffusion modeling are numerous. This section will discuss generalities,
leaving specific phenomena needing investigation to the sections on the
SRL, the CBL, and on "calms and other inconvenient events".
One thing is clear: scientists want diffusion experiments with a
complete complement of measurements. It is frustrating to be unable to
use diffusion measurements to test a model because some critical meteoro-
logical variable was not measured. Sometimes a climatological or other
estimate of the missing variable can be substituted, but this muddies
the analysis, making the results less certain. There are many past
experiments that give only aa and/or cre; there is no hint of even the
P/fi/T stability. In either case, no matter how high the quality of the
diffusion measurements, the data are useful only for testing a certain
subset of diffusion models. Considering the trivial added cost and
bother of adding "cloud cover" to the list of measured variables, this is
ridiculous. A number of other meteorological parameters are more difficult
to obtain, but nevertheless are quite cost-effective in maximizing the
usefulness of the experiment. The goal should be to produce a set of
measurements that can be used to test any physically reasonable diffusion
model, present or future. The Prairie Grass experiment is an outstanding,
solitary example of what can be done; every possibly relevant meteorological
variable that could be measured in 1956 was measured. Who envisioned
then that the aircraft soundings to 3000 m would be useful for determining
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z-j and w*, when convective scaling theory was developed in 1970? This
vintage experiment has been more useful than any other, even in model
evaluations of the most recent decade.
There is also a strong consensus that diffusion measurements are
needed at much larger x than in past experiments, e.g., see Ching et al.
(1983); no one is comfortable with extrapolations to 100 km based on data
to about 3 km. Many scientists mention 100 km as a goal distance, as
recommended by the AMS Review Panel on Sigma Computations (Randerson,
1979). Especially needed are direct measurements of az or °^ vertical
concentration profiles; in the CBLs, this need extends only to distances
where the material becomes well-mixed from z = 0 to z\ , say to
x = 4 "u z-j/w*. In neutral and stable conditions, there is little idea of
the distances at which vertical growth terminates. There is also uncer-
tainty about dy at large x; does it approach an asymptote <* x^, <= x*,
or something in between? Extrapolating from 3 to 100 km, the choice of
asymptote can make quite a difference. There are many who believe that
wind direction shear is an important factor in ay growth beyond x = 10
km; thus, 9a(z) should be measured through the depth of the boundary
layer containing the plume.
There have also been a number of calls for experiments over some
variety of terrain (e.g., Randerson, 1979). The Prairie firass experiment,
while very valuable, represented an extremely flat, smooth terrain. The
terrains in which most people live and most pollution problems occur
lie broadly between the prairie field of wheat stubble and "complex"
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terrain. There is a need for experiments in areas with typical surface
roughness, with trees, buildings, and scattered clearings (the Copenhagen
and Cabauw experiments, Table 1, are good examples). There is also a need
for experiments in rolling or moderately hilly areas, especially in stable
conditions, when drainage flow effects develop even on the slightest of
slopes.
The need for plume measurements at larger distances parallels a
need for more complete plume characterization. A sparse sampler network
merely results in questions about representativeness, whether the peak
concentration was measured, and where the centerline of the plume was
located. Where it is permissible to create a highly visible plume, even
old-fashioned smoke photography can produce more useful results, giving
both instantaneous and time-averaged plume growth (Nappo, 1984). This is
not feasible for large x, however, except in stable conditions at a site
remote from development. Remote sensors, especially lidar, are seen by
many as the most satisfactory way to get three-dimensional plume measure-
ments. Lidar can either be surface-based or mounted on a plane traversing
a plume from overhead (Johnson, 1982). Lidar signal processing techniques
have now developed to the point that X/Q isopleths can be mapped (Eberhard
et al., 1985). The overhead "burden" (f^-dz) of SC^ and other gases can
be measured by van mounted correlation spectrometers, and the ground
level concentrations can be measured with fast-response gas analyzers by
using four to six traverses per hour across a plume on a highway; this
technique has been used to define power plant plumes at distances up to
25 km (Weil, 1977). For much larger distances, several excellent tracers
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have been developed that do not significantly deposit and that can be
detected at great distances, up to x = 1000 km (Johnson, 1982). The
logistics of operating and maintaining a sampler network on such scales
is difficult and considerably increases the cost of an experiment.
A few other experimental needs have been mentioned. One is for more
elevated-release experiments, with larger zs, typical of the taller sources
we are modeling. The recent experiments discussed in Section 3 had source
heights ranging from 115 to 285 m, so there has been some progress for
more elevated release experiments. The majority of releases from taller
sources are also quite buoyant, which involves a new range of phenomena
that is best studied with buoyant plumes as the objects of measurements,
rather than passive releases. One very good idea, at least if kept on a
scale that makes it not too expensive, is a multiple day around-the-clock
experiment such as done in the Tennessee Plume Study (Schiermeier et al.,
1979). Past experimenters have sought short periods of steady-state
meteorology in order to get data in the ideal conditions assumed by
modelers. The atmosphere is never in a steady state for long, and many
modelers and users wonder if transition periods, turbulent "bursts" in
the SBL, and other transitory phenomena are being mismodeled.
The above discussion has concerned needs for field studies. While
some phenomena, like wind direction shear effects and large-x diffusion,
may not be practical objects of laboratory modeling studies, wind tunnels,
water channels, and tanks are very useful for studying short-range phenomena,
Since they are cheaper to operate than field studies, they ought to be
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more fully exploited. A few suggestions regarding these studies were
made at the end of Section 3.
METEOROLOGICAL MEASUREMENTS
As mentioned in the previous subsection, diffusion field experiments
cannot be fully utilized for testing models unless they are accompanied
by complete meteorological measurements. There are also practical needs
concerning meteorological measurements to be used in applied modeling.
This subsection addresses both of these concerns.
The meteorological parameters needed for testing current research
models are aa and ae at source height, u*, L, and z0 and, for daytime,
z.j and w* (there is one redundancy among these, because w*^/z^- = u*^/kL).
For large-x, lateral diffusion experiments it is important to measure
mean wind direction, D, through the layer containing the plume, so that
the rate of change of wind direction with height can be correlated with
ay growth. In stable conditions, potential temperature profiles, 9(z),
should be measured through the layer containing the plume because an
important time scale affecting turbulence and vertical diffusion is N~l,
where N
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at some height representative of the plume; better, u(z) profiles should
be measured through the layer containing the plume. It is also desirable
to have more than a minimal number of U, "ea, and T measurement levels
in case an instrument becomes impaired during the course of the experiment,
which is a strong possibility.
These measurements can be made roughly or with high quality instrumen-
tation, depending on the overall level of effort in the experimental
program. The surface roughness, z0, can be crudely estimated by eye, or it
can be more properly determined from wind profiles taken near the ground
in near-neutral conditions. An even better method is given by Wieringa
(1976, 1980). This estimate of roughness uses the longitudinal turbulence
intensity and gives an effective roughness. The surface flux quantities
u* and L require fast response measurements of u, w, and T to be measured
directly, and such measurements require some experience in quality assurance.
The next step down is to estimate u* and L from established flux-profile
relationships and profiles of U and 9; this approach has often been
used in analyses of the Prairie Grass experiments, as excellent profile
measurements were made on a 16-m tower, but fast-response turbulence
instrumentation did not exist in 1956. The minimum measurements for
estimating L and u* are the bulk Richardson number, IT^/A9, and ZQ.
Particular care should be taken in making these measurements; U should be
measured near z = 10 m, and measurement between about 2 m and 10 m has
been recommended for A9 (Hoffnagle et al., 1981). Potential temperature
is then calculated as a direct function of temperature and ambient pressure.
The lateral and vertical wind direction variances can be replaced by the
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velocity variances av and aw, respectively, by using aa » av/u and
ae = aw/TJ. If u*, L, and Zi or h are measured, it is now possible to
estimate ov and ow from these parameters on the basis of boundary layer
research, with the exception of av in stable conditions (large horizontal
eddies may depend more on the terrain than on surface fluxes in stable
conditions). However, the atmosphere does vary from its own norm, so it
is much better to measure av or aa and
-------
In making the sophisticated measurements, one should not overlook
the simple ones. To determine the P/G/T category, all that is needed
besides IT is a measurement of insolation, net radiation, or a cloudiness
estimate. For slightly more refined systems, some notation of surface
moisture (dry, lush plant cover, dew, soil set from recent rain, etc.)
and a relative humidity measurement are needed.
This relates to another experimental need. Little work has been
done on how to get the best estimates of difficult to measure parameters
that are from relatively simple measurements. This must be done if research
diffusion models are to be usable in improving operational models. Some
work along this line will be discussed in Section 7; more work is needed.
It is especially valuable to try out the simpler measurement schemes
in the context of a diffusion experiment, so that we can discover how
much degradation of diffusion model performance occurs when we use simple
meteorological measurements as proxies for the difficult ones. Can a
research model applied in such a way beat out the P/G/T or Brookhaven
systems without a large increase in measurement effort? One step in this
direction was made by Weil and Brower (1984) in the context of power
plant plume modeling. They demonstrated that a Gaussian model using
improved dispersion parameters outperforms a current operational model.
Briggs (1982) showed that the Prairie Grass values of u*/*dy/0 correlate
about as well with simple substitutes for L as with L itself, the best
measure in previous comparisons (Briggs and McDonald, 1978). The substi-
o o
tutes were u /Rs* and u /Rn*, where Rs* and Rn* are proportional to solar
insolation rate and net radiation, respectively, rather easily measured
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quantities. Even IP alone provided useful correlation, but with increased
scatter.
Many scientists have been thinking about the need to get the best
estimates of quantities like u* and L for operational modeling purposes.
Another concern is to get representativeness, because L can vary by a
factor of 3 or more from field to forest (in this regard, mean quantities
likeU3 have more spatial averaging in them than quantities like the
turbulent heat flux. The AMS Workshop on Stability Classification Schemes
and Sigma Curves made recommendations on instrumentation for diffusion
modeling (Hanna et al., 1977). More detailed recommendations resulted
from the U.S. EPA Workshop on On-Site Meteorological Instrumentation
Requirements to Characterize Diffusion from Point Sources (Strimaitis et
al., 1981); this report included recommendations on instrument response
specifications. Recent comparisons of turbulence velocity responsiveness
and accuracy of six commercial systems were made by Kaimal et al . (1984a);
comparisons were also made of four Floppier acoustic sounding systems
capable of remote measurements of wind speed, direction, vertical velocity,
and vertical velocity variance, aw (Kaimal et al., 1984b).
EVALUATION AND CONCENTRATION FLUCTUATION ISSUES
A surprising number of scientists are calling for more research on
fluctuations of concentrations about their mean value in steady conditions,
x" (defined x' = X - X~ )• It is the nature of turbulence to be very
variable even when the conditions producing it are steady (for example,
note the gustiness a few feet in front of a steadily rotating electric
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fan). Given this natural variability, the question exists of how to
know when x" , or If, or v'w', etc, has been determined. The answer is
that you cannot measure a "true" average quantity, but the longer you
average in steady conditions the better the precision of the measurement.
The problem in the atmosphere is that "steady" driving forces do not
remain so for very long, so there is a limit to how precisely x" and
and mean quantities can be measured no matter how good the instrumentation
(in a wind tunnel, you can maintain steady flow conditions as long as
your budget allows, so you can get much closer to true averages). However,
if one can measure the statistics of the fluctuating quantities, it is
possible to state the probability of the resulting finite-time average
being within a percentage of the true average. This should be a major
consideration in the evaluation of the worth of experiments. (For instance,
each Prairie Grass run consisted of only a 10-min release of tracer;
because large convective eddies take several minutes to advect past a
source, individual daytime trials showed large scatter with regard to
one another, regardless of the parameterization scheme being tested.
Scatter can be reduced by combining runs with similar parameter values.)
At the other end of the modeling process, the consequence of turbulent
variability is that even with the perfect model and perfect input para-
meters, a perfect prediction of the results that will occur for a particular
hour cannot be made. Just as the weather is rarely "average" for the date,
time, and place, atmospheric diffusion routinely deviates about its own
mean, even given the same set of driving forces.
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Thus, knowledge of concentration variability is essential to rational
model evaluation. Given that even the perfect model cannot be in perfect
agreement with data, except occasionally on chance, what degree of agreement
should be our goal, and what degree of disagreement really indicates
modeling deficiencies? This is the main motivation for recent interest
in X' measurements and models, as there has been much interest in how
to properly evaluate diffusion models and how to specify error bounds in
models (Fox, 1981; Fox et al., 1983). Other practical needs for X1
include prediction of flamability of accidental gas releases, the
response of insect pests to smells locating their host plants, and
prediction of lethal doses of fast-acting toxic gases. The U.S. EPA-AMS
Workshop on Updating Applied Diffusion Models (Weil, 1985) focused on
this issue. This workshop recommended pursuit of several X1 modeling
approaches that appear promising, including second order closure models,
large eddy models, and random perturbation (Monte Carlo) models (these
are described in Section 5). The greatest needs are for X1 data.
Laboratory simulations were recommended as valuable, as X1 measurements
are easier to make in a wind tunnel or water tank than in the field,
and X" can be determined more accurately by extending the "steady" conditions,
Recent work by Willis and Deardorff (1984) in their modeling tank began
exploration on X'/X for buoyant plumes in a CBL.
A related need is to find the most effective ways to evaluate
diffusion models with field data, which are always flawed and have unknown
deviations from the true x" . The AMS committee overseeing the Rural
Model Reviews was obviously quite disappointed in the results of extensive
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statistical comparisons between eight models and sampler data from one
power plant; scatter of predicted-to-observed X ratios was so large that
none of the sophisticated statistical tests detected significant differences
in model performance (Fox et al., 1983). Using a much simpler statistic,
the fractional error, and stratifying the same data by distance and
stability, Irwin and Smith (1984) were able to identify differences in
model performance*and when and where they occurred. A number of scientists
have stated a need for more studies like this, i.e., evaluations of
evaluation methods for diffusion models.
There is sometimes a need to adjust diffusion models for different
sampling times. A number of authors have suggested power laws in sampling
time to correct ay (Gifford, 1975). These are mostly derived from appli-
cation of statistical theory to wind direction fluctuation measurements,
but such corrections should be functions of stability and distance, i.e.,
existing methods tend to be overly generalized. Measurements and models
appropriate to the X1 question will also probably serve to improve our
handling of the sampling time question.
On a final note, a need frequently voiced that has much to do with
research is to convincingly demonstrate how uncertainty in predictions of
X can be usefully incorporated in routine regulatory decision making.
STABLE BOUNDARY LAYER ISSUES
Next to X1 and model evaluation needs, the most frequently voiced
research needs have to do with SRLs. Perhaps this is in part because
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there has been so much progress in the last decade on understanding
diffusion in CRLs. The SBL has been relatively neglected partly because
it is more difficult to make turbulence measurements in stable conditions,
when turbulence intensities are quite low, and partly because SBLs are
more complex than CRLs. They may have very large wind direction shears
and are much more terrain sensitive. This is because downslope gravita-
tional forces on cooled air near the surface compete with large scale
pressure forces and momentum transferred from above in forcing the flow,
even on very slight slopes. At the AMS workshop in 1977 it was flatly
admitted that "in very stable conditions our present knowledge is in a
very poor state" (Hanna et al., 1977). Since the participants summarized
the problem so well, here is the rest of the quote:
"Our observations refer almost totally to "ideal" (flat, smooth)
sites, and we have good evidence that roughness and terrain
slopes greatly affect diffusion in the stable boundary layer.
It is recognized that turbulent mixing normally occurs only in
a shallow layer near the ground in these conditions, but the
thickness of this layer varies over a very wide range, from
meters to hundreds of meters, depending strongly on the upper wind
and on site topography. Whether this layer encompasses or falls
short of a given source height obviously will have a critical
effect on diffusion from the source. We also need to know much
more about the frequency of occurrence of "turbulence episodes"
that have been observed at night. These greatly increase the
mixing height, evidently for significant periods of time such as
several hours. Except for the urban nocturnal boundary layer,
which is convective and somewhat more akin to the daytime
boundary layer, we have only begun to formulate possible ways
to predict the nocturnal mixing height. Measurements of this
height, for instance using a remote sensor such as an acoustic
sounder, at a variety of sites and under a variety of wind and
cloudiness conditions would greatly improve our knowledge."
There has been some progress in understanding the turbulent structure
of momentum-driven SBLs since 1977 (Nieuwstadt, 1984a, b); this was greatly
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helped by high quality meteorological measurements made on a 200-m
mast at Cabauw, the Netherlands, and an acoustic sounder to measure
the height of turbulent mixing, h (Driedonks et al., 1978). However,
this was a very flat site, with no significant driving of the boundary
layer by downslope forces (Nieuwstadt claims that, in contrast to "the"
CRL, there are many different SBLs, depending on the balance of driving
forces). Even with this ideal site, the investigators decided to exclude
data from some time periods because of evidence of gravity waves; these
are long-period waves (several minutes or more) that cause h to fluctuate
up and down and also cause quantities like aw to increase with height.
Normally ow decreases with height and near zero values as z approaches h.
The waves are not "turbulence", cause no mixing, and could perhaps be
filtered out, but because they muddy the analysis of an ideal SBL, these
periods were discarded. The effects of gravity waves and ways to predict
their occurrence in SBLs are other areas completely void of information.
The need to study SRLs in rolling or hilly terrains is also very great,
as no work has yet been done in this area.
The most urgent need is for observations of h in a variety of terrains
under different meteorological conditions, so that ways to model it using
near-surface measurements may he tested; h has a rough correlation with
u* and L for flat sites, but we have no appropriate measurements for
sloped sites and no theoretical reason to expect the same correlations
to apply. This is a most important parameter because turbulence drops
to zero at z = h; h acts as a "lid" on diffusion from sources below h,
and there is no vertical diffusion for sources above h.
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Lateral diffusion over averaging periods like an hour remains an
enigma for stable conditions. Not only is it very sensitive to terrain,
it is sensitive to the history of the flow for hours upwind. This is
because when turbulence is suppressed due to static stability there is
almost no dissipation of energy in large horizontal eddies, which are not
suppressed by static stability. It is like someone let the brake off.
This is especially true above h, where there is no turbulence. The
•?•
horizontal eddies that cause plumes to meander may originate from flow
around a mountain, or a thunderstorm a few 100 km upwind many hours
earlier or from the remnants of convective eddies from the previous
afternoon. Horizontal eddies may be present in a wide range of degrees
on different nights at the same site, depending on the history of the
flow. For a high source above h at night, there is no consequence at
night because there is no downward diffusion. However, when the mixing
depth builds up to the plume level in the morning, fumigation occurs;
whether this causes high concentrations or not depends on the lateral
diffusion that occurred during the night. Relow h, large horizontal
eddies are very sensitive to terrain. For this reason, Pasquill
recommended no cry curves except for 3-min averages, and the 1977 AMS
committee strongly urged aa measurements at the site in question (Hanna
et al., 1977). Recently it was found that nighttime av fluctuations on
a tower correlated with minute pressure fluctuations at the ground (Zhou
and Panofsky, 1983). With more investigation, this might turn out to be
an effective basis for categorizing ay at night by using a surface
measurement.
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The need for testing the effects of wind direction shear on ay was
already mentioned; these are especially large effects in SBLs. Furthermore,
we have almost no grasp on the prediction of the rate of change of wind
direction with height at night, even in idealized terrain. This is an
area that might be helped by second-order closure boundary layer models,
which have some demonstrated success at simulating SBL flow on a slope
(Brost and Wyngaard, 1978).
Turbulence episodes or bursts have been mentioned as a phenomenon
requiring attention for nocturnal diffusion. These are sudden jumps in
h, accompanied by increased turbulence and downward mixing of warmer
temperature, larger "u, and any pollutants that were "captured" by the
increase in h. These jumps occur when the stability in the layer above h
becomes marginal, with a Richardson number Ri = (g/e)(d9/az)/(alf/az)2
near a value of 0.5 or so. This can happen as the wind profile
above h evolves slowly due to the turning of the earth (before the
turbulent burst the layer is practically frictionless, so U responds only
to inertia and the horizontal pressure gradient); it can also be caused by
weakening of 59/az by more rapid radiative cooling from above.
When Ri falls below the critical value 0.25 somewhere in the layer,
turbulence develops spontaneously and can spread at least through all
layers where Ri < 0.5. These events are extremely difficult to predict.
For the source below the original height of h, more vertical dilution
will occur when h jumps, so such events would serve to reduce surface
concentrations. However, for the source above the original h, accumulated
pollutant may suddenly mix to the surface in such an episode, somewhat
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like morning fumigation, (Evidence of such events have been seen in
nighttime SO;? monitoring records from near a power plant, but high concen-
trations occurring during such an event have yet to be observed.)
CONVECTIVE BOUNDARY LAYER ISSUES
Although knowledge of diffusion characteristics of CRLs has advanced
more than that of SBLs, important research needs remain. These needs are
primarily for more extended, three-dimensional diffusion measurements,
studies of late afternoon transitions, and studies of cloud effects.
Except for surface measurements near a surface source, measurements
of diffusion in the CBL to date are overwhelmingly from laboratory studies.
There were surface arc tracer measurements at one distance at Cabauw and
at three distances in Copenhagen for elevated SF^ releases in rather weakly
convective conditions. These satisfy only a small part of the need for
field data for comparison with the laboratory measurements, which showed
such unexpected vertical dispersion behaviors. Some CRL experimental
needs put forward by Ching et al. (1983) were (1) surface X and x(z)
profiles to x = 6 km for both elevated and surface-source releases, (2)
complete concurrent meteorological measurements including z-j, (3) long
averaging or release times of at least several hours, (4) early morning
and late afternoon releases to study transition, and (5) studies of cloud
effects. Participants at the U.S. EPA-AMS Workshop on Updating Applied
Diffusion Models (Weil, 1985) also expressed the desirability of including
a dense surface sampling network and using remote sensing for three-dimen-
sional plume measurements. Some of these experimental needs are addressed
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by the CONDORS experiment in Section 3. But, the CONDORS experiment did
not address the issues of diurnal transitions and cloud effects. This
experiment was limited by (1) only a single surface sampling arc at
x a 1.2 km, (2) settling and deposition of some of the chaff plume, and
(3) oil fog detection and mapping by lidar to only x = 2 km or dimensionless
X = 1.5. The critical area of maximum surface impact for elevated oil fog
releases was well covered by lidar, but it could not distinguish oil fog
from background haze at much larger distances due to highly diffusive condi-
tions and limits on the amount of oil fog that could be released (this
technique might be doubly effective at a remote, very low turbidity site).
The midday CBL usually approximates a steady state, with slowly
increasing z-j and nearly constant w*, so it is rather well understood.
Many modelers feel uneasy about the application of steady-state models to
transition periods early and late in the daytime. The growth of z-f in
the early morning, with entrainment of initially stable air aloft, has
been measured and modeled with some success; inversion breakup fumigation
caused by z-j growth has been studied in the laboratory (Deardorff and
Willis, 1982) and in a field experiment (TVA, 1970); this study predates
convective scaling, but only crude estimates of w* and z-j could be made
from the tabulated measurements). What needs study the most is the
dying-out of convective turbulence in the late afternoon and the concurrent
collapse of the mixing depth, which tends to leave material "frozen" at
heights that it mixed up to early in the afternoon (Ching et al., 1983).
Kaimal et al. (1982) suggest that z-j may collapse rather suddenly in the
late day, but evidence is sketchy.
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Cloud effects on turbulence and diffusion have received very little
attention in spite of the fact that clouds occur rather commonly. There
are many degrees and types of cloudiness and many questions. Scattered
cumulus clouds often form at the top of vigorous thermals; these have
ground shadows that create cool spots of low or reversed surface heat
flux. Does this phenomenon alter the balance of updrafts and downdrafts
and affect vertical diffusion? When these clouds develop significantly
into stable air above Zj, they may "vent" pollutants that were drawn into
the supporting thermals; this phenomenon of cloud venting needs more
study (Ching et al., 1983).
When the sky becomes very overcast, solar heating of the ground is
cut off and H* becomes very small, zero, or slightly negative, cutting
off the buoyant forces driving thermals from the surface. However, cooling
can occur at the top of the boundary layer due to radiative cooling of
cloud tops and cool outflows; cooling from above can cause an "upside-down"
CBL driven from above. This may sustain convective turbulence late into
the afternoon on days with afternoon cumulus development. There is a
basic need to study CBL turbulence in cloudy and partly cloudy conditions,
which is probably done best with turbulence-instrumented aircraft. Heat
and moisture fluxes need to be measured just below the cloud bases, as
well as close to the ground, so that we can find out if current CBL models
will work for top-driven CBLs merely by turning them upside down and
using -H* at z « Zi in place of H* at the surface. Very likely, further
model modifications will be needed.
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A good start in this direction has been made by the University of
Wisconsin Boundary Layer Research Team (Stull and Eloranta, 1984; Wilde
et al., 1985; and Stull, 1985) who studied the CBL with cumulus over
Oklahoma. Albrecht et al. (1985) have studied the marine ARL topped with
stratocumulus over the eastern Pacific. This is an example of the ABL being
strongly cooled from the top.
CALMS AND OTHER INCONVENIENT EVENTS
A perennially dodged question is, "how do you model calms?" Most
models have X « Tj-1, so their predictions for U = 0 are alarming.
One psuedo-answer to the question is that the wind never really stops
blowing; it is just too slow for instruments to respond. When this
happens, use the Gaussian model with u" = 1 m/s. A second such answer is
just like the first, except that "U = 0.5 m/s is the magic minimum wind
speed that makes the model work (this answer doubles X, of course).
These responses seem less than scientific, at the least. Plumes have been
observed to rise and fan out in all directions in stable, nearly calm
situations and have been observed to loop in all directions in nearly
calm, convective conditions; in fact, this condition sometimes gives the
highest observed mean surface concentrations at power plants (Bowne,
1984: personal communication). A few physically-based models for diffusion
in If = 0 conditions do exist, but very little has been published. Recently,
a model for zero and very low u" diffusion in the CBL was proposed (Deardorff,
1984). Adequate testing of such models requires a dense sampler network
close to the source occupying all compass directions or remote sensing of
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a plume near the source and some sensitive wind speed measurements (e.g.,
sonic anemometers).
Another "inconvenient event" for which no special modeling techniques
exist is rain. No diffusion experiments have been carried out in rain,
and some instruments perform well only in dry weather. Continuous S&2
monitoring has been made at many power plants, but accompanied by only
standard, rather crude meteorological measurements. Furthermore, S02
is scavenged by rain and snow, which adds a large complicative factor to
the analysis. Examination of such monitoring records should help answer
the question of whether diffusion during precipitation events is a worst
case. Scavenging and wet deposition are very important questions here,
but are beyond the scope of the present paper. When precipitation comes
from stratus clouds, as is usual for mid-latitude winter storms, there is
almost no positive or negative heat flux at the surface, so we usually
assume that "neutral" diffusion applies in these conditions. To my
knowledge, no one has tested this assumption. When precipitation is
convective, as in thunderstorms and on-again-off-again showers, large,
cool downdrafts are generated by evaporative cooling in the rain cells.
This condition undoubtedly affects diffusion in a profound way, but how?
RESOURCE ISSUES
Efficient use of research funding is an issue that is raised frequently
in the context of research needs. As this section has shown, it is easy
to assemble a rather long list of areas in which our knowledge is deficient.
These areas are not "ivory tower" concerns, but affect many practical
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decisions involving investments that, collectively, total billions of
dollars. Too much of the diffusion modeling supporting such decisions
has been ad hoc or highly extrapolative, lacking appropriate validation
for the application. Usually this reflects lack of appropriate data, but
too often insufficient use is made of existing data that could provide
partial validation, at least. More encouragement of evaluation efforts
is needed, either through allocation of funds for contracted work or
through allocation of scientific staff time. These efforts must also
include a careful analysis of the range of applicability of any new models
as well as outlining the details of model operation.
The long list of research needs necessitates prioritization, as
there are not enough resources (including scientists) to address all the
needs. However, prioritization is not enough; there is also the strategic
problem of how to make the most of resources. Given a certain level
of funding, one could choose to direct it all towards the top priority
need, or divide it among the top 10 or 20 priorities. The consensus
among diffusion research scientists seems to be that more is accomplished
by supporting a multiplicity of small research projects with diverse
goals, carried out by individuals or small teams, than by supporting a
large project with narrow goals.
There are needs for occasional "big experiments", but these need
not be budget busters. The Prairie Grass experiment of 1956, the most
classic example, was a multi-agency effort; four universities and 60
scientists and technicians participated. The recent CONDORS convective
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diffusion experiment was more modest, involving about 20 people during
data collection and costing about 1/4 million dollars (a large fraction
of the cost was for computer processing of the remote sensing signals);
yet, it was a well-instrumented experiment that accomplished its basic
goal: to define diffusion in three dimensions for elevated and surface
sources for highly convective conditions. Similar experiments on about
this scale or a little larger (with more surface sampling) run in neutral
and stable conditions as well as convective conditions at several diverse
sites would give a far better experimental base for testing models and
for checking against laboratory modeling. Laboratory modeling is a very
cost-effective tool for basic diffusion research as well as for complex
flow situations. It cannot provide all the answers, but it can be used
to explore many fundamental phenomena in a very controlled and systematic
way. Numerical modeling, such as second-order closure, LES, and Monte
Carlo models also can be very cost-effective tools for basic explorations,
if the models are reasonably efficient. Furthermore, it would also be
cost-effective to support analyses of many under-used past meteorological
and diffusion measurements in the light of contemporary boundary layer
concepts.
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SECTION 7
IMPLEMENTATION OF IMPROVED MODELING PRACTICES
The technical transfer problem
There is a very large gap between diffusion modeling as practiced
and the state of the science at the research level. The Pasquill curves
are still the basis of most diffusion modeling in the United States; they
were developed around 1958 for short-term averages from low sources,
contained many ad hoc assumptions, and were supported in a rather spotty
way by diffusion and wind direction variance measurements. More informed
modelers prefer the Brookhaven curves for elevated sources, but these too
were developed in the same era with limited diffusion measurements at one
field site and with many questionable assumptions (like Gaussian vertical
distribution and non-evaporation of oil fog droplets). Obviously, more
field and laboratory measurements on diffusion have been made since then.
Tremendous progress has also been made in understanding turbulence in the
ABL and in successful mathematical modeling of many aspects of atmospheric
diffusion. At any conference or workshop concerning diffusion issues,
however, one can sense much frustration over this situation.
It seems obvious that not enough effort is being made to translate
research into better practice. What are needed are major efforts to
simplify research models sufficiently to make them viable as practical
tools and to find adequate substitutes for the parameters that are difficult
to measure.
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The frequent lack of effective measurements is a real handicap in
this endeavor. If we must restrict our input variables to the type of
measurements routinely made at airports up to now, we can do little more
than define Pasquill categories and make crude estimates of L on the
basis of wind speed sampled for 1 min every 3 hr at an unusually smooth
site. Of course, there is little possibility of significantly improved
diffusion modeling using such restricted measurements. For this reason,
diffusion scientists have repeatedly recommended effective on-site
instrumentation and upgraded measurements at NWS stations (Hanna et al.
1977; Strimaitis et al., 1981).
ESTIMATING NEEDED PARAMETERS FROM MORE OBTAINABLE MEASUREMENTS
Except for modeling diffusion at short distances, < 1 km, from sources
low in the ABL, the mixing height, zj or h, must be measured or estimated
to do any kind of diffusion modeling. One cannot model at a distance
pretending that there is no lid on diffusion. For elevated sources at
night, there is a good chance that there is no vertical diffusion at all
if zs is above the nocturnal mixing depth, h; however, if zs < h, vertical
mixing down to the ground occurs and the whereabouts of h becomes a
critical question. Other quantities required depend on the modeling
approach; these include D, ae, u", u*, L, and w*, discussed in Section 6.
All of these should be measured in any diffusion research experiment for
maximum utility of the results, but direct measurement of many of these
parameters is too difficult to be recommended in applied modeling.
Scientists are aware of the problem and have proposed a number of schemes
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for estimating the difficult parameters from those more easily measured.
There are many ways to estimate the surface-layer parameters u* and L
(or H* = u*3/(0.4L), which also appears in the definition of w*). The
profile method, using IT(z) and 9(z) profiles from a small tower, is
most often suggested (Rriggs and McDonald, 1978; Nieuwstadt, 1978;
Irwin and Binkowski, 1981; Berkowicz and Prahm, 1982). However, site
uniformity and height of vegetation are major factors to be considered
when placing instruments. Without profiles, a crude estimate of the
surface roughness is needed with any of the methods; this estimate can be
made from site inspection (Wieringa, 1980). Pasquill and Smith (1983)
give nomograms for estimating u*, L, and H* from wind at a single height,
"uj, and potential temperature difference (A9) between 0.2 z\ and zj.
In convective conditions, H* is highly correlated with A9 alone (Dyer,
1965). In general, u* and L can be determined from IT at a single height
if an estimate of H* is available. Methods for doing this by using
cloudiness and some site-dependent parameters for surface moisture have
been developed (van Ulden and Holtslag, 1983; Holtslag and van Ulden,
1983). An alternative method developed by Berkowicz and Prahm (1982)
requires net radiation and humidity deficit near the surface to estimate
H*; both are relatively easy measurements to obtain (humidity deficit
relates directly to temperature and relative humidity). A similar method
for estimating L by using IF and net radiation or insolation was given by
Briggs (1982a); it is very simple, but lacks consideration of the surface
moisture.
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The variances of lateral and vertical wind directions are not as
difficult to measure as turbulent fluxes, and a good choice of
instrumentation is available (Kaimal et al., 1984a). Lacking measure-
ment, they can be estimated from aa = av/u~ and
-------
observations from 10 days at Cabauw in the Netherlands (Driedonks, 1982);
excellent results were obtained from models that included a u* term in
addition to the above inputs.
For the nocturnal h, Nieuwstadt and Tennekes (1981) developed a
prognostic equation that tested well against observations at Cabauw, but
it requires many input parameters (including the geostropic wind).
Nieuwstadt (1984b) also compared some simpler schemes for h against
Cabauw observations; he found fairly good correlations with Zilitinkevich's
(1972) steady-state prediction, h = 0.4 (u*L/f)1/2, and with a semi-empirical
equation, h = 28 i^n, ' (mks units). The constant in the latter equation is
dimensional, and should be regarded as site-dependent. It is very similar
to the semi-empirical equation of Venkatram (1980), h = 2400 u*3/2 (mks
units), which gave a good fit to Minnesota measurements (Caughey et al.,
1979).
Except for a» in stable conditions, for which there is no substitute
for measurements of aa, it appears that all meteorological parameters
for modeling diffusion can be estimated from a morning rawinsonde sounding,
U at a height of 10 m or so, and some type of sensible heating/cooling
estimate; the latter requires solar radiation, net radiation, or cloudiness
and solar elevation and possibly a humidity measurement.
What is greatly lacking at present is independent testing of these
parameter substitutions at different sites. For almost every technique
outlined above, testing was conducted at only a single site. There is no
way of knowing whether the methods transfer to sites of different character.
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Furthermore, in cases in which there are many approaches, as for the
estimation of u* or L, we would like to know which one is best and which
is the simplest technique that is "good enough."
All of these questions could be settled nicely with comparative
meteorological measurement experiments carried out during some daytime
and some nighttime periods at three or four contrasting sites, for instance,
flat and rather smooth; suburban or forested; gently rolling; and hilly.
Full meteorological measurements should be made concurrently, so that any
simplified scheme can be tested against the direct measurements. Such a
program would accelerate improved modeling efforts tremendously and save
years of guesswork, non-optimum on-site measurements, and modeling attempts
with the wrong input parameters.
BUILDING BETTER OPERATIONAL MODELS
Implementing improved models requires research and commitment.
Adding an occasional experiment with partially complete meteorological
measurements to the potpourri of past experiments and the continued
development of scientific, albeit esoteric, models is not enough for
progress in a practical sense if the last 20 years' experience is an
indicator. Making the technical transfer happen requires a large effort
to sort through the numerous scientific models (Section 5), determine
the major features having the most practical significance, search for
effective substitutes for the difficult-to-measure meteorological inputs,
sift through the available diffusion data for experiments with the required
measurements and no debilitating uncertainties, perform extensive
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comparisons between the data and several promising models plus the old
models (both with and without various substitute meteorological parameters),
evaluate the results for the most effective models given several different
quality levels of input parameters, and then cast the favored model(s)
into operational form with computer programs and/or nomograms. Most of
this hard work would be classified as applied research and a prime research
priority (the only research deserving higher priority is conducting
experiments that are designed to facilitate meteorological and diffusion
model comparisons and that make complete and trustworthy measurements).
The task so briefly stated above is so large, especially when the
variety of operational modeling needs is considered, that no individual
or small group could carry it out in just a few years, even if they were
relieved of all other responsibilities. A few scientists have made
efforts in this direction, but their time is divided among many competing
tasks, so that they cannot tackle the whole of the technical transfer
problem. Another factor for research scientists to consider is that
there is more recognition for original research than for midwifing technical
transfer. However, many scientists achieve a great sense of satisfaction
when their research results are translated into something practical, and
they are willing to assist in the process.
A few of the efforts towards better operational models should be
mentioned. Pasquill's (1961) original fiaussian scheme was such an effort,
on the part of a research scientist, and was a great step forward at
the time. Smith (1973) refined this system for vertical diffusion from a
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source near the ground by introducing continuous, rather than discrete,
stability categories and providing convenient nomograms, including one
for a correction for surface roughness (his curves were based on the
results of an eddy-diffusivity model). Irwin (1979) first advanced a
generalized scheme that used convective scaling for daytime diffusion and
on-site a a and ae data to characterize ay and az anc' provided meteorological
substitutes for nonavailable ora and ae measurements. This has evolved
into a more complete approach that makes use of the latest advances in
knowledge of ABL structure, especially for stable conditions (Sivertsen
et al., 1985; Irwin et al., 1985). Weil and Rrower (1982) developed a model
for power plant applications that uses u* and w* to characterize av and aw;
these determine short-range dispersion in the usual way, based on statis-
tical theory (e.g., ay = crvt). These forms for cjy and az are then modified
by simple, empirical functions of x.
The willingness of scientists to seek model simplifications in order
to encourage practical applications was amply demonstrated at the 1984
Workshop on Updating Applied diffusion Models, cosponsored by the U.S. EPA
and the AMS (Weil, 1985; Briggs, 1985b). This willingness should be
further encouraged by a program of sustained and systematic efforts to
translate research results into improved modeling practice. This task
requires a thorough familiarity with models and the available data sets
for validation.
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SECTION 8
ASSESSMENT
The purpose of this paper is to review the current state of science
concerning basic diffusion in the ABL and identify major research needs
in this area. Basic diffusion in this paper means diffusion of conserva-
tive, non-buoyant substances over reasonably homogeneous, ordinary terrain,
Thus, the scope here excludes consideration of buoyancy, deposition,
chemical transformation, complex terrain, and shoreline flow issues.
Even so, a vast number of research papers concern basic diffusion, and
there are many complexities to consider as various forces and surface
conditions come into play in the ABL. This assessment section will
summarize the state of knowledge concerning the mechanisms driving turbu-
lent diffusion in the ABL, the scope of experimental information that has
been obtained, the theoretical and experimental basis for diffusion
modeling as currently practiced, and the strengths and weaknesses of a
half-dozen approaches to improved diffusion modeling. It will then set
forth what most needs to be done to expand the experimental base for
model validation and further develop the most useful modeling approaches.
Finally, it is suggested that more effort is needed to translate research
advances into modeling practice. Model simplifications should be guided
by comprehensive evaluations using all reliable experimental evidence.
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ATMOSPHERIC BOUNDARY LAYERS
Progress toward the understanding of turbulent and mean structures of
ABLs accelerated after the 1968 Kansas surface-layer field experiment
(Izumi, 1971), in which accurate profiles of potential temperature and
wind speed components, including fast-response turbulence measurements,
were made on a 32-m tower. This experiment, among others, provided
verification of the Monin-Obukhov surface-layer similarity theory, (which
is discussed in Section 2). This theory has been very successful in
predicting profiles of mean quantities and vertical turbulence in the
surface layer; it also works fairly well for the horizontal turbulence
velocities in neutral conditions and the high-frequency portion of the
horizontal turbulence velocity in stable conditions. It fails to correlate
with low-frequency horizontal velocity changes in stable conditions,
which have periods ranging from about ~ 1/4 to 4 hr, if they are present
at all; it also does not correlate well with horizontal turbulent velocities
in unstable conditions.
The situation with respect to CBLs vastly improved after the convec-
tive similarity scaling was introduced by Deardorff (1970a,b) and was
tested in the 1973 Minnesota experiment (Izumi and Caughey, 1976), which
included tower measurements to 3?. m and extended turbulence and mean
measurements to 1200 m by mounting instruments on a cable tethering a
large balloon-. Convective scaling is discussed in Section 2. Following
the analysis of the Minnesota experiment in terms of convective scaling,
for the first time, the behavior of the major parameters of turbulent
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structure was understood for the whole of the ABL. Further results and
confirmation of earlier results for CRLs have been obtained at many sites
now, mostly by research aircraft. One very significant finding that
affects diffusion is that the mode of vertical velocity is negative,
about -0.5w*, throughout the mid-mixing layer; this explains why the
center!ines of elevated plumes have been observed to descend in laboratory,
numerical, and some field experiments. Little work has been done on the
effects of partial cloudiness on CRL structure, and no work has been done
on "upside-down" CBLs driven by radiative cooling at the top of a fog or
heavy overcast of clouds.
Progress has been made in understanding NRLs, but mostly by means of
numerical models rather than by experiments. Perhaps this is because
truly neutral (H*/u*3 * 0) conditions are rather infrequent, and overcast,
windy conditions are not the best flying weather. There is ample infor-
mation for the lower part of the NBL, for which it appears that surface-
layer similarity works very well for the vertical component of turbulence
and fairly well for the horizontal components; these may be influenced by
surface roughness or slope inhomogeneities. Higher up, Coriolis accelera-
tion affects NBL structure, including turbulence, and the Coriolis parameter,
f, is important. Further details of NBLs are contained in Section 2. There
is need for more experimental measurements in the upper part of the NBL.
Much progress has been made in understanding SBLs since the Cabauw
experiments begun in 1977, but understanding is far from complete because
of the varied nature of SBLs. They are strongly influenced by the slightest
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terrain slopes and are also affected by radiative cooling, which depends
on humidity and CO;? profiles. Also, they are subject to disruptive
events like gravity waves and turbulent bursts, which are hard to forecast.
Much has been learned about undisrupted, pressure gradient-driven SRLs in
flat terrain. This work is discussed in Section ?..
The diurnal cycle of ARLs is basically well understood, except that
there is not much experimental information on the transitional periods
near sunrise and sunset. The least understood phenomenon is the collapse
or running down of CBL turbulence in the late afternoon, which may even
overlap the beginning of S8L development at the surface; this collapse
begins as soon as H* turns negative, about 1 hr before sunset. The rate
of CBL collapse may depend on humidity, which increases radiation flux
divergence and hastens the development of stable stratification aloft.
DIFFUSION EXPERIMENTS
Several dozen field experiments have been conducted using tracers or
oil fog for basic diffusion studies, but only a few of them included a
full range of desirable meteorological measurements, and these experiments
are quite limited in terms of distance and plume sampling. Until recent
experiments, the tracers released were not very conservative, which
confounds interpretation of their results for vertical diffusion and
surface concentrations. Laboratory modeling using tanks, water channels,
or wind tunnels has been used with success in modeling diffusion in all
stabilities, but its potential has not been exploited very much. In
total, we have a patchwork quilt of experimental information that contains
18fi
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many blank squares and many uncertain ones, especially for vertical
concentration distributions and diffusion at distances greater than 3 km.
Almost all field experiment estimates of oz, the vertical diffusion
parameter, were made using only surface concentration measurements by assuming
a Gaussian vertical distribution of concentration, X, and a conservative
tracer, i.e., //(Xu")dydz = 0, the source strength. It has gradually
come to light that all the tracers used, except for recent gaseous tracers
like SFfi, were not conservative; S02 gas and particulate tracers deposit
significantly, and oil fog evaporates substantially within a travel distance
of 2 km or less. Unless reasonably good estimates can be made of the
deposition or tracer loss rates and the data can be appropriately reanalyzed
(e.g., Gryning et a!., 1983), the az results from these experiments
must be held questionable. The same is true of the crosswind-integrated
concentration, /Xdy. We have reliable surface X measurements for SFfi,
but only for neutral and slightly unstable conditions at x = 2 to fi km
for elevated releases and only for very stable conditions at x < 0.4 km
for surface releases. In a few of the earlier experiments, tower or
balloon tethering cable measurements were obtained for vertical distribu-
tions from elevated releases before significant deposition could occur,
but these were complete only for 26-m releases in neutral to stable
conditions and ~ 200-m releases in near-neutral conditions.
For lateral diffusion, ay, deposition is less of a problem and
there is a more complete collection of reliable tracer measurements;
these include some to 26 km for surface releases and some to 10 km for
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elevated releases. The meteorological measurements accompanying the
longer range experiments was sufficient for testing only certain classes
of models. As we look at available plume width measurements at greater
distances, up to 100 km or so, we find rather scant meteorological
information. Many of these measurements are quasi-instantaneous plume
widths obtained by a single aircraft traverse at a given distance.
Under some circumstances, observations of buoyant plume parameters can be
used to approximate passive plume dispersion, especially for ay after
vertical mixing is complete, as in the CBL; this gives useable data in
the 15- 30-km range for ay in unstable conditions.
Most field experiments suffered from meteorological instrumentation
deficiencies, so they cannot be used to test all types of models.
The earlier turbulence instrumentation was not accurate, and many
experiments lacked sufficient wind and temperature profile measurements
and even simple measures like solar insolation and cloudiness. Only one
of the earliest experiments included soundings adequate for obtaining z-j,
a key CRL parameter. The most recent field programs have included measure-
ments of Zi and a complete array of good meteorological measurements.
Hopefully, this is indicative of better experiments to come. Not only do
we now have good turbulence instrumentation available, but we have much
better tracers and remote plume sensors like lidar, which has been used
in two very recent basic plume diffusion experiments.
Fluid modeling has proven a reliable tool for studying many aspects
of diffusion in ARLs. It is cheaper than field experiments and is far
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more controllable, although the drastic reduction in scale does impose
some limits on what can be simulated, particularly when buoyancy is
involved. For instance, it is not possible in present facilities to
simulate the effects of large horizontal eddies or wind direction shear
on ay, but otherwise fluid modeling has proven quite versatile in
simulating vertical turbulent diffusion and the minimum cry in the
absence of the above effects.
A recent series of wind tunnel experiments at Colorado State University
simulated basic diffusion for an elevated release and a surface release
over a smooth surface and over a rough surface for CBLs, NBLs, and SRLs
(Cermak et al., 1983). For the unstable runs, fairly good agreement was
obtained with other CRL diffusion experiments, in spite of short dimen-
sionless fetch and weak capping. Moderately stable ARLs (h/L = 4 to fi)
were obtained by using a strong temperature differential, 78°C, between
cooled floor and heated inflow, which is necessitated by the scale-down.
Other wind tunnels have given excellent comparisons with field measurements
in neutral conditions for az as a function of x and z0, the roughness
length, but with ay about 20% smaller than in the field under similar
conditions (this is perhaps due to the constraining effect of the tunnel
walls). A similar decrease in ay, compared to field experiments inter-
preted in terms of convective scaling, has been noted for the convective
tank experiments of Willis and Deardorff (197fi, 1978, 1981). Their unique
facility, basically just a water tank with a uniformly heated bottom,
provided the pioneering results for diffusion in very convective conditions.
The somewhat controversial results for vertical diffusion from this
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series of experiments have been essentially duplicated, by using convec-
tive scaling, in numerical experiments in the CS1I wind tunnel and in the
few field observations for which adequate measurements exist.
Both wind tunnels and convective water tanks could be put to further
use to make systematic studies of source height effects, surface roughness
effects, surface roughness and surface heat flux inhomogeneity effects,
and concentration fluctuation statistics, a matter of great current
interest. Also, as wind tunnels and water channels have been successfully
used to simulate diffusion in complex terrains, it seems that they could
also be used to study the effects of typical rolling or hilly terrains on
diffusion.
DIFFUSION MODELING AS PRACTICED
The basic diffusion models used in standard practice today were
developed 25 to 30 years ago. The stability classification schemes that
they use, although crude, are not "unscientific". They have some basis
in the physics relevant to ABL diffusion, but this basis is quite incomplete
in the light of what has been learned about ABLs since 1968. The
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This assumption, while far from rigorous, has proven amazingly resiliant.
For sufficient averaging times, 1 hr or so, it works as well as any shape
for lateral diffusion. Because vertical turbulence is not homogeneous,
systematic deviations from the Gaussian shape have been observed for K.(z)
profiles. In most cases, the consequences of using Gaussian X(z) for
ground-level concentration predictions are not great and the prevailing
mood seems to be in favor of not changing. However, for vertical diffusion
in unstable conditions in CBLs, the consequences appear to be considerable.
The Brookhaven stability classification scheme (Singer and Smith,
1966), based on wind direction fluctuations, was an adaptation of a scheme
proposed at least as early as 1932 (Hiblett, 1932) and founded on statistical
theory (Taylor, 1921). The basic result for short-range diffusion,
cjy = crax, where oa is the lateral wind direction variance, can be
stated succinctly as "the plume goes as the wind blows" and is so direct
as to defy any contrary argument (as long as the wind vane faithfully
responds to the wind). The correlation between ay and aa becomes
less direct in a few tens of meters of travel, and it was an intuitive
assumption that some empirical correlation would remain at distances of
tens of kilometers. This assumption has held up, although it can be
improved upon. A more risky assumption was that vertical diffusion would
correlate with cra also; this has proven true at short ranges in unstable
and neutral conditions when turbulence eddies are more or less isotropic
(high as the they are wide); however, it is definitely risky for stable
conditions when site- and time-specific large horizontal eddies, or
meanders, can make lateral wind fluctuations and dispersion much larger
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than vertical, at least over 1-hr averages. The Rrookhaven curves were
based mostly on surface measurements of oil fog droplets from a plume
released at zs = 108 m in rolling, forested terrain. The usual assumptions
of Gaussian *(z) and a conservative tracer were applied to infer az
values from surface concentrations. As even heavy-oil fogs can lose 50%
of their lidar cross-section in about 2 km of travel, there is reason to
doubt the conservative-tracer assumption and the resulting az values.
The surface ay values should not have been greatly affected by oil fog
evaporation.
Pasquill's (1961) stability classification scheme, which was adopted
by fiifford (19fil) and adapted by Turner (1961) for use with Pasquill's
curves, is based upon measurements usually available at airports, wind
speed at z = 10 m and insolation or cloudiness. It is an intuitive scheme,
but as was shown later, it correlates significantly with the physically
very important nbukhov length, L (Bolder, 1972; Rriggs, 198?). It only
lacks adjustment for the effect of local surface roughness. Like L, it
gives good correlation with vertical turbulence and diffusion near the
surface. It correlates with lateral turbulence and ay only when the
turbulence is nearly isotropic, as for 3-min averages in neutral and
stable conditions. In unstable conditions, turbulence is somewhat
isotropic at an elevation of 0.1 zj or more, but not at the ground; z-j
has as much influence on ay as does L, and Pasquill categories have
little to do with zj. Pasquill's ay and az curves were intended for
use with near-surface sources, as they are based on near-surface data
(also, the categorization scheme, like L, has decreasing relevance higher
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up in the ARL). The 400 m, although the rapid drop in surface concentrations was quite
consistent with Willis and Deardorff's results (Nieuwstadt, 1980).
Some brief comments are due on two other stability classification
f\
schemes. Hogstrom's (1964) suggestion of s = A9/Uf as an index in which
Uf is the "free wind" velocity at about 500 m was based on testing of five
different indices for correlation with observed ay and az values from
his oil fog puff experiments. For his site, uf worked better than IT at
source height z = 87 m. The potential temperature difference was measured
between 30 and 122 m, bracketing the source height. At least in the surface
layer, A9/u? is a bulk Richardson number that does relate directly to L.
Using uf and A9 from higher up, there would not be a direct correspondence,
but s would broadly indicate dynamic stability in the elevated layer
encompassed by A9. For Hogstrom's experiments, which were made only in
neutral to very stable conditions, s correlated very well with changes in
relative diffusion,
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center!ines in the vertical, 0ZC. There was no discernible trend with
stability in ayC, the meandering part of lateral diffusion. The suc-
cesses, and failure, of this index for Hogstrom's elevated releases
roughly parallel those of Pasquill's scheme, which is likewise loosely
related to L. However, if Hogstrom had been able to use his puff technique
in unstable conditions, he probably would not have found much correlation
with s, as A9 is very small in CBLs above 30 m (for good correlation with
L, A9/u"2 should be measured below 20 m; the bottom level should be as
low as 1 m, if practical). For the same reason, the use of AT at consider-
able elevation as a stability index could not distinguish at all between
unstable and neutral conditions. Measured near the surface, AT or A6
correlates well with H* in unstable conditions, hut not with IT or u*,
which have much more effect on diffusion. Thus, AT alone is inferior
as a stability index; even u" alone broken into a "day" and a "night"
category works much better (Briggs and McDonald, 1978).
ALTERNATIVE MODELING APPROACHES
Many papers have been published concerning more scientific approaches
to basic diffusion modeling. Thankfully, they sometimes include testing
of a model against some kind of diffusion observations. However, compre-
hensive testing of a variety of alternative approaches has not been done;
meteorological measurements in data sets make such comparisons very diffi-
cult, if not impossible. This paper has divided the approaches into six
generic categories for discussion and attempts to identify the strengths,
weaknesses, and potentials of each method.
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Surface-layer similarity theory expresses gradients of mean quantities
in terms of (u*z)~l times their surface fluxes times functions of z/L
and expresses turbulence variances as u* times functions of z/L. This
theory has been quite successful at predicting profiles of IT, 9, and other
mean quantities in the lower ABL. Near the surface, aw/u* - 1«3; 1 15 or so,
there is little doubt that convective scaling is a better approach. This
replaces u* and L with w* and Zj as principal scales. Convective scaling
has been demonstrated in numerous measurements programs to correlate well
with turbulent velocities throughout CBLs (surface-layer similarity is
superior only for aw below z = |L|). Experience has shown that approaches
that order turbulence measurements well also order diffusion well. The
fact that similar results are obtained for diffusion in such diverse
media as water tanks, wind tunnels, numerical experiments, and field
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experiments when the results are nondimensionalized with convective
scaling does much to instill confidence in the method. The correct
scaling, in itself, does not constitute a diffusion model; it merely
provides a way to summarize data efficiently and in a more universal
format to allow comparisons of data from diverse experiments. It also
suggests a simplified format for diffusion models. A number of equations
for ay and az and non-Gaussian vertical diffusion models using w* and
z-j scaling have been proposed (Briggs, 1985b). Some of these also include
neutral asymptotes that use u* as a second velocity scale.
Gradient-transfer, or "eddy diffusivity", models assume that the flux
of a quantity is proportional to its gradient, and that the quantity moves
toward the region of lower concentrations. The constant of proportionality,
K, is called the eddy diffusivity in turbulent flow and has units of
Iength2/time. This theory is built on the same analogy to molecular
diffusion as the Gaussian plume assumption and shares some of its virtues
and weaknesses. It is rarely valid for lateral diffusion, as ay must
be larger than the largest horizontal eddies for the assumption to hold
true. In the CBL, Ky = O.lw*z-j does provide a supportable asymptotic
prediction for the minimim a =* 0.5 (w*z1-x/tl)1/2, but usually horizontal
motions on a larger scale than z^ are present and an x*'2 asymptote is
not observed (Briggs, 1985b). Gradient transfer theory has the most validity
for vertical diffusion from a surface source because the largest eddies
affecting this diffusion are not much larger than az (except in CRLs at
x > 0.5 "u" z.j/w* when the mode of *(z) lifts off the ground against
the predictions of gradient-transfer models). It has been a useful tool
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for producing practical az predictions when good estimates of K(z) are
made by using surface-layer similarity scaling (Smith, 1973). Some more
advanced models, like spectral diffusivity theory, attempt to account for
non-local effects on fluxes of concentration; these have not been developed
enough to encourage practical application.
A more advanced technique is second-order closure modeling, which
assumes that third-moment quantities like u'w'x' are functions of
local second-order or mean quantities or their gradients. Any local
closure technique works best for stable conditions when turbulent eddies
are small, and second-order closure models have been highly successful at
predicting the behaviors of SRLs, including the effects of terrain slope
and radiative cooling. They have hardly begun to be applied to turbulent
diffusion in SBLs, but it seems that they would provide an excellent
exploratory tool, especially for the effects of zs/h and terrain slopes.
Large eddy simulation (LES) models were first applied to prediction of
global weather patterns and have provided very good results for turbulent
structure in CBLs and NRLs; both of these contain large eddies, of the order
of their height. Unlike the second-order closure models, which predict
time-averaged properties, the big-eddy models compute the instantaneous
structure of eddies larger than the computational grid size. This provides
actual "pictures" of the turbulence structure. To get averaged quantities,
one computes the time averages by "sampling" over many time steps, much
as in a field experiment. To account for the interactions-between adjacent
cells, an eddy-diffusivity assumption is applied only for small scales,
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the "subgrid" scales. This permits counter-gradient diffusion to occur
on the large scales, which does occur in simulated CBLs. These models
have given very good results for diffusion in CBLs. They have further
potential for this case and probably would also be a good tool to explore
the effects of zs/z-j and z-jf/u* on diffusion in NRLs.
Statistical models in various forms have long been workhorses of
diffusion modeling. They are based on the idea that if enough is known
about the statistics of turbulent motions, both mean instantaneous and
time-averaged diffusion of passive substances following those motions can
be predicted. The most immediate results, like
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sources in CBLs have used observed vertical velocity statistics at source
heights to predict *(z) versus distance essentially by assuming that tL
is large.
Since 197fi, there has been a move to improve ay and 0Z predictions
by going back to the basic statistical theory prediction that cjy/(avt)
and crz/(awt) are functions of t/t|_. The motivations are that
(1) this is surely correct at small t/t|_, (?) this is simple, (3) the
functions of t/tj_ are weak, not strong, so their form and the estimates
of t|_ are not critical , and (4) we now know enough about ARL structures
to be able to estimate av and 0W at source height in many ways, if
they are not measured directly. A number of functional forms and ways to
approximate t|_ have been suggested, and much testing against data sets
has been performed for this method. It seems a desirable approach for
generalized applied diffusion modeling and, once optimized, should give
adequate predictions except when the fiaussian plume assumption itself is
inadequate, i.e., for vertical diffusion in the CRL.
A further development along the statistical line of approach, but
with "artificial" statistics, has been the random perturbation/random
force/Markovian random walk/Monte Carlo/Langevin equation methods that
have been revived and extended in recent years. The basic idea is to
compute the velocity of particles in a turbulent flow assuming for each
particle the velocity of the previous time step reduced by whatever is the
drop in the autocorrelation function for the time step At plus a random
velocity perturbation. This is provided by a random number generator
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in a computer, of course, and is distributed in a way that maintains the
proper average velocity statistics for the flow. Many results can be
produced by mathematical manipulation of the basic equations. Monte Carlo
diffusion simulations, using thousands of computed particle trajectories,
have produced further results, such as peak-to-mean concentrations, relative
diffusion, and relationships between t(_ and tp. Methods have been developed
to allow application of these techniques to vertical diffusion i.e., with
inhomogeneous turbulence. For instance, the technique has been applied
to vertical diffusion in the CRL with good simulations of experimental
results. However, such applications require quite a few assumptions
about the turbulence statistics and particle behaviors at boundaries.
It would seem that these methodologies need considerably more testing
against research-grade observations, perhaps wind tunnel observations,
before they can be relied upon as general purpose research tools.
NEEDS FOR DIFFUSION EXPERIMENTS
Without question, there is a broad consensus that we must broaden
the experimental base if we are to make progress towards better diffusion
modeling. The many flaws and omitted measurements in most past experiments
make them impossible to use for even-handed comparisons with both old and
new modeling approaches, although they are useful as auxiliary tests of
certain models. The field experiments with adequate, or nearly adequate,
meteorological measurements were too limited in distance range or stability
range to provide for most of our model validation needs and usually had
only surface concentration measurements. Loss of tracer by deposition was
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a problem in the great majority of experiments. Laboratory modeling of
diffusion appears to be capable of fairly accurate simulations of most
ABL diffusion aspects and could be used to a much greater degree to
systematically explore the effects of source height and changed surface
conditions.
The usefulness of any diffusion field experiment is determined by
the completeness of the accompanying meteorological measurements as
much as by the scope of concentration measurements. Most past experiments
included only a selection of meteorological parameters, so they cannot be
tested against most modeling approaches. A complete set of meteorological
parameters includes (1) variances of vertical and lateral velocities or
wind directions,
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For experiments in stable conditions, the slope of the site is important,
and in uneven areas the topography surrounding the site may be important.
The site description should include inhomogeneities such as tree lines
and varied surface cover. For the majority of past experiments, the
Pasquill category cannot be determined because solar insolation was not
measured and no notation was made of cloudiness. Such simple measurements
should not be neglected. For alternative schemes, these measurements of
net radiation and some notation or measure of surface humidity conditions
should also be included.
In all conditions, plume concentration measurements are needed to
larger distances. Some vertical profiles of concentration, even if very
sparse horizontally, are greatly needed to at least the distance where
the plume fills the boundary layer (from about 10 km in very convective
conditions to perhaps 100 km in neutral conditions). Lateral plume
width, or ay, measurements are needed to distances of the order of 100 km;
even if measured at only a few distances, it seems that interpolation
through large distance factors would be preferable to the present practice
of extrapolating from 3 to 100 km. Three-dimensional plume concentrations
can now be obtained by lidar within a few kilometers of the source, and
excellent tracers with negligible deposition loss now exist that can be
distinguished from background hundreds of kilometers away.
For stable conditions, in SBLs, the greatest need at present is for
both boundary layer measurements and a few diffusion experiments at a
variety of site types. Almost all that we know now pertains to flat
20?
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sites or to sites with very slight slopes, ~0.1°; even this much slope
has been observed to cause quite significant effects on h and on the
turning of wind with height and time. Because of the great significance
of h to diffusion modeling, it would be very helpful to have a "traveling
boundary-layer experiment" (even without diffusion measurements) to
measure h at sites of differing character - broad slopes, short slopes,
hilly, rolling, etc. To correlate measured h with parameters measureable
at the surface, we would also want u*, L, IT, z0, net radiation, cloudiness,
and humidity. A portable instrumented tower, a tethersonde, and an
acoustic sounder would be adequate basic tools for such experiments. We
also need to develop some sort of climatology of nighttime
-------
extend to about 25 km. There is especially a need for turbulence
measurements aloft in the late afternoon, as we have almost no informa-
tion on CRL collapse; in addition to the usual CRL parameters, these
measurements aloft should be accompanied by humidity profiles, as radiation
flux divergence may greatly affect the process. Finally, as clouds are
rather common in afternoon CRLs, we need to stop avoiding them and make
turbulence measurements when clouds are present in varying degrees to see
how correlations with CRL parameters are affected. In overcast, daytime
conditions, there may be more cooling at the top of cloud layers than
heating at the surface, and heat fluxes at the surface and at the cloud
base may be in delicate balance. These conditions need to be measured
at the same time as other CRL turbulence measurements. It is likely that
some of these ARLs would turn out to be "neutral", which is good because
there is so little information on the upper part of NRLs and the effect
of Zif/u* on their turbulent structure. These latter tasks could be
performed by research aircraft.
There have been many calls for studies of concentration fluctuations,
X1, and their statistics. These are needed for rational evaluation
of modeling uncertainties, for determining sampling time effects, and for
flamability and toxicity predictions for some gases. "Raseline" experiments
on X'/X" could most easily be performed in a wind tunnel, e.g., for an
elevated and a surface source, over a rough and a smooth floor, and in
unstable, neutral, and moderately stable conditions. For CRLs, such
measurements could also be made in convective tanks, as some have already.
In addition, because of uncertainties introduced by laboratory scale-down
204
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and the finite sample and probe sizes, some X1 measurements should be
made in conjunction with full-scale diffusion experiments, even if only
at a few locations per experiment.
Large voids exist in our understanding of and appropriate modeling
of diffusion during stability transition, calms, and rain. It would be
helpful, in the course of diffusion experiments in progress, to include a
few runs that span the transition from unstable to stable conditions
near sunset. Inversion breakup fumigation has received some laboratory
tank study, but no full-scale experiment has been attempted with full CBL
turbulence and profile development measurements. Calms are difficult
to find in the atmosphere, especially during short periods of readiness
to launch a field experiment, but they do occur. It seems likely that
some case-study calm periods could be found among existing power plant
sn£ and meteorological monitoring archives; this, at least, is a place to
begin. There is a real need to study calms, as near-zero wind periods
during unstable conditions have produced "worst case" surface concentrations
at some power plants. A study of this phenomenon could be made in a
convective laboratory tank, but it probably would have to be two or three
times wider than the one used up to now. As for rain, we have usually
assumed that passive diffusion proceeds as it does in dry, neutral condi-
tions, but there is no evidence that this is so; often, showery conditions
are accompanied by convective downdrafts produced by evaporative cooling.
205
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RESEARCH MODELING NEEDS
Because research diffusion models are numerous and seem somewhat
ahead of the game at present in that there are generally inadequate
diffusion observations for proper model validations, we have chosen to
somewhat emphasize the experimental needs. We believe that this reflects
the prevailing opinion in the scientific community at present. However,
we do this with expectations that research model development will continue
and will improve as better experimental data become available. Perhaps
some approaches will stand out as more clearly rewarding when this happens.
In the meantime, there are some areas in which we see more immediate
benefits possible from some of the models.
We are beginning to understand the rudiments of SRL turbulence
structure, but have not yet performed diffusion experiments concurrently
with needed basic measurements like h. It seems that second-order closure
modeling of SRLs so far has produced very good simulations of SRL evolution
in time and of slope effects, at least on broad slopes. Surely these models
could also be used to advantage to explore vertical diffusion in such
boundary layers (these models presently only compute the vertical dimension)
Slope effects, radiative cooling effects (probably small), and zs/h
effects could be readily explored, given adequate computer time. These
models will also give the wind direction shear, but they cannot be used
to predict lateral diffusion. For ay, further use could be made of
statistical theories, including random perturbation modeling, especially
with good wind direction fluctuation measurements.
20fi
-------
For CBLs, it seems possible that radiative cooling could be added to
a LES model. In conjunction with diminishing H* at the lower boundary,
such a model could simulate CBL collapse, at least until the big eddies
are no longer present. The overcast boundary layer, with various balances
between surface H* and cloud base H*, could also be simulated with LES
models. One advantage of these models is that they can be readily used
to predict vertical and lateral diffusion from any height within the
resolution of the grid cells; all it takes is sufficient computer time to
compute numerous particle trajectories. A more speculative possibility
for these models is simulation of turbulence and diffusion in CBLs topped
by scattered clouds, which form at the top of more vigorous thermals;
this would probably require some form of cloud updraft parameterization
and some accounting of subsidence outside the clouds.
TECHNICAL TRANSFER NEEDS
Even without further research advances, we have gained so much new
knowledge about boundary layer turbulence structure and diffusion since
1968 that it is greatly perplexing to many scientists that so little of
this has filtered down to applied modeling. Of course, this is not an
automatic process. Research models contain some parameters that are
difficult to measure. The models are often complex entities that do not
lend themselves well to routine usage, at least not in their original
format. Furthermore, many operational modelers do not feel inclined to
switch from the old schemes until they are proven inferior to an alterna-
tive scheme. This requires, first of all, support of a scientific effort
207
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to validate, simplify, revalidate, and then package preferable models in
a way that simplifies application and requires only minimal meteorological
and source inputs.
In Section 7, we reviewed efforts that have been made to find adequate
ways to estimate the required, difficult meteorological parameters by using
simpler meteorological measurements. A rather large array of methods has
been suggested, but, in most cases each method has only been tested with
data from one site. This is inadequate, especially in stable conditions.
One reason that full-meteorology diffusion experiments have been recommended
here is that they allow any number of schemes for input simplification to
be tested both against the fundamental ABL parameters, such as L, and
diffusion data in a modeling context. Furthermore, any applicable model
or model simplification can be tested against the same diffusion data.
This is rarely achievable with past experiments, except for very recent
ones. Only testing of old and new models side by side against "universal"
data sets like the above will settle the question of how much modeling
improvement, if any, can be obtained for various levels of effort.
Apart from diffusion experiments, traveling boundary-layer experiments
like the one suggested here for SBLs would settle the question of which
input simplification methods work best at sites of various characteristics.
Any modeling approach, old or new, will work vastly better if we can make
a better estimate of the turbulent mixing depth, h or z-j.
208
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Such efforts will bear fruit, to the satisfaction of research
scientists and model-users alike, only after some period of sustained
support for researchers who can concentrate on the job of transforming
validated research diffusion models into efficient and more accurate
operational models than those in common use.
209
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