United States
Environmental Protection
Agency
Environmental Monitoring
Systems Laboratory
PO Box 15027
Las Vegas NV 89114
EPA-600/4-79-066
October 1979
Research and Development
xvEPA
Modeling Wind
Distributions Over PROPERTY OF
Complex Terrain
DIVISION
OF
METEOROLOGY
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U S Environmental
Protection Agency, have been grouped into nine series These nine broad categories
were established to facilitate further development and application of environmental
technology Elimination of traditional grouping was consciously planned to foster
technology transfer and a maximum interface in related fields The nine series are
1. Environmental Health Effects Research
2 Environmental Protection Technology
3 Ecological Research
4 Environmental Monitoring
5 Socioeconomic Environmental Studies
6 Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9 Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL MONITORING series Thissenes
describes research conducted to develop new or improved methods and instrumentation
for the identification and quantification of environmental pollutants at the lowest
conceivably significant concentrations It also includes studies to determine the ambient
concentrations of pollutants m the environment and/or the variance of pollutants as a
function of time or meteorological factors
This document is available to the public through the National Technical Information
Service, Springfield, Virginia 22161
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EPA-600/4-79-066
October 1979
MODELING WIND DISTRIBUTIONS
OVER COMPLEX TERRAIN
by
Mark A. Yocke
Mei-Kao Liu
Systems Applications, Incorporated
950 Northgate Drive
San Rafael, California 94903
Contract No. 68-02-2446
Project Officer
James L. McElroy
Monitoring Systems Research and Development Divisio
Environmental Monitoring Systems Laboratory
U.S. Environmental Protection Agency
Las Vegas, Nevada 89114
ENVIRONMENTAL MONITORING SYSTEMS LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
LAS VEGAS, NEVADA 89114
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DISCLAIMER
This report has been reviewed by the Environmental Monitoring
Systems Laboratory-Las Vegas, U.S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the U.S. Environmental
Protection Agency, nor does mention of trade names or commercial products
constitute endorsement or recommendation for use.
11
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FOREWORD
Protection of the environment requires effective regulatory actions
that are based on sound technical and scientific data. This information
must include the quantitative description and linking of pollutant sources,
transport mechanisms, interactions, and resulting effects on man and his
environment. Because of the complexities involved, assessment of specific
pollutants in the environment requires a total systems approach that tran-
scends the media of air, water, and land. The Environmental Monitoring
Systems Laboratory-Las Vegas contributes to the formation and enhancement of
a sound monitoring data base for exposure assessment through programs designed
to:
develop and optimize systems and strategies for moni-
toring pollutants and their impact on the environment
demonstrate new monitoring systems and technologies by
applying them to fulfill special monitoring needs of
the Agency's operating programs
This report discusses the development of an air flow model for urban
areas in complex terrain and the application of the model in Phoenix, Arizona.
The model will be incorporated into an existing method for the design of am-
bient air quality monitoring networks (see EPA-600/4-77-019 and EPA-600/4-
78-053). The method may be useful for regional or local agencies who have
a need to plan new or modify existing networks. The Monitoring Systems Design
and Analysis Staff may be contacted for further information on this subject.
j^* -'- ''-",," '-/ '- ^ '<' ~x"~y "v-
George- 'B. Morgan s
Director
Environmental Monitoring Systems Laboratory
Las Vegas
iii
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ABSTRACT
Accurate determination of wind fields is a prerequisite for success-
ful air quality modeling. Thus, there is an increasing demand for objective
techniques for analyzing and predicting wind distributions, particularly
over rugged terrain, where the wind patterns are not only more complex, but
also more difficult to characterize experimentally. This report describes
the development of a three-dimensional wind model for rugged terrain based
on mass continuity. The model is composed of several horizontal layers of
variable thicknesses. For each layer, a Poisson equation is written with
the wind convergence as the forcing function. Many types of wind pertur-
bations over rugged terrain are considered in this model, including diver-
sion of the flow due to topographic effects, modification of wind profiles
due to frictional effects in the planetary boundary layer, convergence of
the flow due to urban heat island effects, and mountain and valley winds
due to thermal effects. Wind data collected during a comprehensive field
measurement program at Phoenix, Arizona, were used to test the model. The
model was shown both qualitatively and quantitatively to perform reasonably
well in the application to Phoenix, and its utility was demonstrated by the
relatively modest computing and data requirements in this application.
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CONTENTS
FOREWORD ............................. iii
ABSTRACT ............................. jv
LIST OF ILLUSTRATIONS ...................... vi1
LIST OF TABLES .......................... viii
LIST OF ABBREVIATIONS AND SYMBOLS ................ ix
ACKNOWLEDGEMENT ......................... xi
I INTRODUCTION ........................ !
II WIND FLOWS OVER CITIES LOCATED IN COMPLEX TERRAIN ...... 3
A. Geostrophic Flow and Its Modifications ......... 3
B. Thermally Induced Local Circulations .......... 4
C. Modification of Wind Flows Over Mountains ........ 5
1. Airflow Over a Mountain Range ............ 5
2. Airflow Over a Solitary Hill ............ 8
III REVIEW OF PREVIOUS MODELING STUDIES
A. General Classification of Wind Models
1. Interpolation Techniques .............. n
2. Objective Techniques ................ 12
3. Diagnostic Models .................. 13
4. Dynamic Models ... ................ 13
B. Pertinent Existing Models ................ 14
1. Objective Techniques Based on Variational
Principles . . ................... 15
2. Diagnostic Models Based on Mass Continuity ..... 15
IV DEVELOPMENT OF A WIND MODEL FOR COMPLEX TERRAIN ....... 18
A. The Model Equation . .................. 18
B. Parameterization of the Vertical Fluxes ......... 22
1. Topographic Effects ................. 22
2. Boundary Layer Effects ............... 23
3. Thermal Effects ................... 27
C. Numerical Solution Procedure .............. 30
v
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V APPLICATION OF THE MODEL TO THE PHOENIX AREA . 34
A. Data Base 34
B. Discussion of the Results 39
C. Statistical Analysis of the Results 41
VI CONCLUSIONS AND RECOMMENDATIONS 47
APPENDIX A: DECOMPOSITION OF A VELOCITY VECTOR 48
APPENDIX B: COMPARISON OF PREDICTED AND MEASURED SURFACE
WINDS IN PHOENIX 5;
REFERENCES 105
VI
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ILLUSTRATIONS
1 Formation of the Lip-Valley Mountain Wind 6
2 Formation of the Down-Valley Mountain Wind 7
3 Classification of Types of Airflow Over Ridges with
Typically Associated Wind Speed Profiles and Streamlines ... 9
4 General Classification of Types of Airflow Over a
Solitary Hill 10
5 Cross-Sectional View of the Intersection of a
Hypothetical Terrain with a Three-Dimensional
Modeling Grid 20
6 Schematic Diagram of a Flow Contacting the Slope of a Hill . . 23
7 Parameters Used in Defining the Diversion Effect 24
8 Sketch Showing a Grid Cell Along the Left Boundary
of the Modeling Region 32
9 Modeling Grid Indicated on a Topographic Map of
the Phoenix Area 37
10 Frequency Distributions of Wind Speed and Wind Direction
Deviations for 15 February 1977 42
11 Frequency Distributions of Wind Speed and Wind Direction
Deviations for 16 February 1977 43
12 Frequency Distributions of Wind Speed and Wind Direction
Deviations for 7 March 1977 44
13 Frequency Distributions of Wind Speed and Wind Direction
Deviations for 10 March 1977 45
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TABLES
Phoenix Area Wind Measurement Station Names
and Coordinates 36
Mean Flow Imposed at Model Boundaries and Mountain-
Valley Wind Coefficients Used for Phoenix Wind Simulations . . 38
VI 11
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LIST OF ABBREVIATIONS AND SYMBOLS
ABBREVIATIONS
km kilometer(s)
m tneter(s)
m/sec meter(s) per second
MST mountain standard time
UTM Universal Transverse Mercator
SYMBOLS
A
a
A
B
b
B"
a.
B
d
e
f
F
f(x,y)
g
H
slope vector
slope scalar
potential velocity vector
multiplicative factor
intercept
solenoidal velocity vector
solenoidal potential
function
cutoff coefficient
drag coefficient
derivative symbol
base of natural logarithm
weighting function;
dimensionless velocity
Froude number
forcing function
gravitation constant
average surface elevation
in the grid cell; depth of
model region
max
h(x,y)
k
L
an
M(x,y)
N
r
Re
ri
(k)
highest terrain eleva-
tion
terrain height as a func-
tion of location
grid cell position
indexes
von Karman's constant
Monin-Obukhov length
natural logarithm
"equivalent mountain"
function
number of measurements;
number of vertical layers
number of parameterized
fluxes
distance
Reynolds number
current residual of the
i-th equation
free-stream (unperturbed)
temperature
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SYMBOLS (concluded)
'E
T(x,y)
U
U
oo
u,v,w
V
w
x"
x,y,z
temperature of the
current
temperature of the
environment
spatial distribution
of surface temperature
wind speed
free-stream (unperturbed)
velocity
wind components corres-
ponding to orthogonal
Cartesian coordinates
frictional velocity
interpolated wind speed in
x-direction
interpolated wind speed in
y-direction
vector velocity
weighting function;
vertical velocity
position vector
orthogonal Cartesian
coordinates
aerodynamic surface rough-
ness length
empirical coefficient
difference operator
mean square error
dimension!ess parameter
defined as
Re
AX
arctan (vh)
potential function
scalar functions
wind divergence
vorticity of flow; com-
ponent of total divergence
partial derivative
divergence operator
r
Y
adiabatic lapse rate
environmental temperature
lapse rate
dimensionless parameter
defined as u(z.)/U
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ACKNOWLEDGMENT
We wish to express our sincere graditude to Dr. Thomas W. Tesche, who
has made significant contributions to the formulation of the complex-
terrain wind model discussed in this report.
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I INTRODUCTION
Efforts to establish air quality monitoring networks for urban areas
have led investigators to develop various objective methods for selecting
optimum measurement sites. One promising approach developed by Systems
Applications, Incorporated (SAI) and the Las Vegas Environmental Monitoring
Systems Laboratory (EMSL-LV) of the U.S. Environmental Protection Agency (EPA)
entails the use of an air quality simulation model for the prediction of
pollutant concentration distributions (Liu et a!., 1977 and McElroy et al.,
1978). When this method was applied to the metropolitan Las Vegas area,
the wind speed and direction were found to be the environmental parameters
that potentially can most significantly affect the network configurations
(McElroy et al., 1978). In general, the pollutant concentration is approxi-
mately proportional to the inverse of the wind speed. Although a wind
direction that is invariable during the averaging period of the pollutant
measurements cannot change pollutant concentration levels, it can shift the
locations of concentration maxima and thus affect the selection of monitor-
ing sites.
In the past, meteorological stations have been deployed to measure wind
distributions, and the data have subsequently been interpreted to provide
the necessary input to an air quality simulation model. This approach suffers
from several deficiencies. First, characterizing the spatial variations for
a large urban complex may require many meteorological stations and, thus,
great expense. Second, reliable and objective interpolation schemes are
not available for reproducing acceptable wind fields from a finite number
of measurements, especially in complex terrain. Third, the historical data
used for monitor siting purposes generally lack detailed wind measurements,
except for synoptic-scale climatological information. For these reasons,
further studies on the modeling of wind fields were recommended.
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The objective of this project was to develop physically based wind
models that can be coupled with an air quality simulation model. The
ultimate goal was to apply the coupled wind-air quality model in the design
of air quality monitoring networks (McElroy et al., 1978). It was hoped
that injecting atmospheric dynamics into the wind analysis would improve
the accuracy of the wind predictions. Two types of special situations were
specifically considered. These were (1) a coastal city and (2) an inland
city in complex terrain. Whereas wind flows in coastal cities are typically
dominated by land and sea breezes, particularly those in the low-latitude
areas, the meteorological settings of inland cities are often characterized
by the presence of complex terrain. Since coastal areas will be addressed
in a separate document, this report focuses only on the problem of modeling
the wind fields over an inland city located in complex terrain.
The next chapter describes the general features of the wind patterns
over complex terrain. Chapter III discusses previous modeling studies.
Chapter IV proposes a modeling approach capable of predicting three-dimensional
wind distributions. Application of the model to the Phoenix, Arizona area
is described in Chapter V, Finally, conclusions and recommendations are given
in Chapter VI.
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II WIND FLOWS OVER CITIES
LOCATED IN COMPLEX TERRAIN
Several excellent reviews on wind flows over mountainous regions have
been made (Alaka, 1960; Reiter and Rasmussen, 1967; Flohn, 1969; Nicholls,
1973). They cover a wide range of topics related to the low-level airflows
over complex terrain and provide an extensive list of references on the
subject. Therefore, only a cursory description of the wind flows to be
modeled is included here.
The distribution of winds in the atmospheric boundary layer is affected
by air motions on all scales, ranging from the large-scale Rossby waves to
the smallest turbulent eddies. Depending on a variety of factors and condi-
tions, they all contribute in varying degrees of importance to the resultant
wind observed at any geographical location. Because of the large spread
in the spatial scales and the widely different mechanisms responsible for
their occurrences, the following discussion is limited to those wind phenomena
with spatial and temporal scales of interest to the dispersion of air pollu-
tants from an urban area.
A. GEOSTROPHIC FLOW AND ITS MODIFICATIONS
Above the surface boundary layer, air motion in the lower atmosphere,
particularly in mid latitudes, is generally determined by a balance between
the horizontal gradient of atmospheric pressure and the Coriolis force. The
resulting flow, which is parallel to the constant pressure contours or
isobars, has been termed the geostrophic wind. For situations where the
isobars are curved, the wind flow is further modified by the centrifugal
force and is known as the gradient flow. Deviations from the geostrophic
motion can also be caused by the frictional force and horizontal tempera-
ture gradients. For example, modification of the geostrophic flow by the
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ffictional force near the earth's surface leads to the well-known Ekman
spiral in the planetary boundary layerthe decrease in wind speed and
the counterclockwise turning (in the Northern Hemisphere) of the wind
direction as it approaches the surface.
On the other hand, if a significant horizontal gradient is present in
the mean air temperature, the geostrophic wind is further altered, with
the resultant flow called the thermal wind. The thermal wind is generally
directed along the tangent to the isotherms in such a way that the area
of lower temperatures is to the left in the Northern Hemisphere. Over
mountainous terrain, the horizontal temperature gradient may be generated
by the uneven heating of the slope.
B. THERMALLY INDUCED LOCAL CIRCULATIONS
Convective circulations in the atmospheric boundary layer are induced
by buoyancy as a result of inhomogeneities in the surface temperatures.
These thermal anomalies are, in turn, generated by differential heating
and cooling of the land and water masses, the mountains, and the valleys.
These diurnally varying flows, which include the land and sea breezes, the
valley winds, and the slope winds, can extend over areas of tens of kilo-
meters in extent. The inertia! interaction between these local winds and
the synoptic-scale flow is, therefore, one of the most influential factors
in determining the near-surface flow pattern in low-altitude coastal areas
and in mountainous regions. For example, in Los Angeles, the land and
sea breezes are typically the dominating winds near the surface during the
summer, when a strong, clockwise-turning sea breeze occurs during the day
and a weak, counterclockwise-turning land breeze arises during the night.
As solar heating is weakened in the winter, the interaction between the
sea breeze and the synoptic-scale flow becomes more apparent.
The thermally-induced flow of greatest consequence to mountainous
regions is the mountain-valley wind, which is also a result of uneven
cooling and heating. In the evening, radiational cooling of the upper
mountain slopes results in a thin wedge of cool air that descends the
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slopes and begins to move down the canyons. This valley or canyon flow
generally persists all night. After sunrise, the upper slopes heat up
more rapidly than the valley floor, which is shielded by the overlying
cool air mass. Gradually, the canyon wind changes direction and begins
to move up the canyon and over the heated upper slopes. The gross aspects
of the mountain-valley winds are now reasonably well understood. The
formation of up-valley and down-valley (drainage) mountain winds is illus-
trated in Figures 1 and 2.
C. MODIFICATION OF WIND FLOWS OVER MOUNTAINS
The presence of natural obstructions, such as mountains, obviously
alters the wind distribution near the surface. As discussed earlier,
changes in the atmospheric momentum and heat budgets also affect the air-
flow over mountainous regions.
1 Airflow Over a Mountain Range
According to Forchtgott's classification scheme (Forchtgott, 1949),
four types of airflows over a two-dimensional mountain range can be
identified:
> Laminar flow. Under light winds, flow over the ridges
forms a smooth, shallow wave; close to the surface,
vertical currents exist as a result of orographic lifting,
and downstream phenomena do not occur.
> Standing eddy. With stronger winds, a large, semi-
permanent eddy forms to the lee of the mountain, creat-
ing a larger effective shape of the mountain with respect
to the flow aloft.
> Lee wave. With stable stratification and even stronger
winds increasing with height, a lee wave system develops
downwind of the mountain ridge. The amplitude of the
waves is primarily determined by the shape of the mountain,
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(a) Formation of up-slope winds
shortly after sunrise when
the down-valley wind is
still blowing
(b) Predominance of up-slope winds
as the down-valley wind dies
in mid-morning
(c) Enhancement of up-slope winds
by the onset of up-valley
winds toward midday
(d) Maintenance of up-valley
winds as the up-slope winds
cease in late afternoon
Source: Defant (1951).
Figure 1. Formation of the up-valley mountain wind
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(a) Beginning of down-slope wind
shortly after sunset before
up-valley mountain wind dies
i(b) Down-slope wind and return flow
at center of valley after
up-valley wind dies in late
afternoon
(c) Down-valley mountain wind
as return flow at center
of valley ceases
(d) Down-valley mountain wind
as the down-slope wind
ceases at night
Source: Defant (1951).
Figure 2. Formation of the down-valley mountain wind
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but the wavelength depends on the atmospheric strati-
fication and the wind profile.
> Rotor flow. Under very strong winds limited to a
restricted vertical depth (on the order of the mountain's
height), severe turbulence and quasi-stationary rotary
vortices occur in the lee of the mountain ridge.
These flow patterns are presented schematically in Figure 3.
2, Airflow Over a Solitary Hill
Like wind flows over two-dimensional mountain ranges, those over a
single, isolated, round hill can be classified into four types:
> Bifurcation flow. Under light winds and stable strati-
fications, the airflow simply bifurcates at the base
of the hill. No vertical currents exist; the flow is
strictly horizontal.
> Laminar flow. Under moderate winds and neutral or
unstable stratifications, the airstream follows the
geometric shape of the hill.
> Lee wave. With stable stratifications and strong winds,
three-dimensional lee waves form.
> Turbulent flow. With very strong winds, turbulent motions
prevail.
These flow patterns are illustrated in Figure 4.
As commented by many previous investigators, such an ideal mountain
or valley probably does not exist in nature. Under most circumstances,
the terrain to be modeled is complex or featureless. In these situa-
tions, the eventual wind field may be the result of superpositions of
several of the flow patterns discussed above.
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01
f
o>
Wind speed
(a) Laminar flow
en
'i
0)
Wind speed
(b) Standing eddy
Wind speed
(c) Lee wave
CD
CD
1C
Wind speed
(d) Rotor flow
Source: Forchtgott (1949).
Figure 3. Classification of types of airflow over ridges
with typically associated wind speed profiles
and streamlines
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(a) Bifurcation fl
ow
(c) Three-dimensional lee wave
(d) Turbulent flow
Figure 4. General classification of types of airflow
over a solitary hill
10
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Ill REVIEW OF PREVIOUS MODELING STUDIES
A variety of wind models of interest to the present project have been
developed. They range from simple, straightforward interpolations of
wind measurements to complex, sophisticated, dynamic models. In the first
section of this chapter, general classifications of these models are des-
cribed, and advantages and limitations associated with each class are
discussed. The second section focuses on those recently developed models
that are particularly attractive and pertinent to the goal of this study.
A. GENERAL CLASSIFICATION OF WIND MODELS
The objective of any wind modeling effort is to develop a capability
for either forecasting or projecting the wind distributions as a function
of time or space. These models can be generally divided into four cate-
gories: interpolation techniques, objective techniques, diagnostic models,
and dynamic models. Each is discussed below.
1 . Interpolation Techniques
The simplest interpolation scheme conceivable is the weighting of
measurements by an influence factor. For example, in the spatial inter-
polation of wind speed, the interpolated value at location j can be written
as
(1)
11
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where v. and v. are the observed wind speed at location i and the inter-
polated wind velocity at location j, respectively, and r. . is the distance
' J
between locations i and j. The weighting function, f, for obvious reasons,
has generally been assumed to be a function of r... The inverse of the
' J
distance squared was the value chosen by Wendell (1970), Strand (1971), and
Weisburd, Wayne, and Kokin (1972), whereas the inverse of the distance
was selected by Eschenroeder and Martinez (1972) and Liu et al. (1973).
Despite its simplicity, at least conceptually, this approach has been plagued
by many fundamental difficulties. Among the problems associated with the
use of this simple interpolation technique are:
> The interpolated wind field is very sensitive to the choice
of the weighting function and other similar artificial para-
meters.
> The interpolated wind field is also sensitive to the number
of wind measurement stations and the network configurations.
> The interpolation scheme obviously lacks any basis in physics,
and does not take into account the presence of the terrain
unless reflected in the wind measurements.
As a result of these deficiencies, it was found that applications of this
simple interpolation scheme are almost always less than acceptable (Liu
et al., 1973).
2. Objective Techniques
Ideally, the ultimate goal of interpolation is to obtain a "best" fit
of the observational data. A logical approach, based on mathematical
optimization theory, is to minimize the deviations between the interpolated
value and the real value at a grid point.
Assume that observational data of a scalar quantity u are available
at N locations. The best fit, u(x), can be constructed from a linear
combination of the observed data, u(x.)> J - 1, 2, ..., N using
J
12
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N
u(x) =T w(x,7.)u(x..) . (2)
The weighting functions, w(x,x.), are determined by minimizing the mean
J
square error:
= [u(x) - u(x)]2 (3)
The mathematical basis for this approach has been developed by Gandin
in the U.S.S.R. (Gandin, 1965) and by Panofsky (1949) and Sasaki (1970) in
this country. The success of this method hinges on the assumption that the
density of the meteorological stations is adequate for deriving the degree
of details desired in the wind analysis. In other words, all necessary
information to describe the variation of the interpolated quantity must be
contained in the observational data. Unfortunately, this condition is
seldom if ever met in air pollution studies. Models falling into this cate-
gory are discussed in more detail in Section B.
3. Diagnostic Models
A common feature of diagnostic models is that they do not invoke the
full set of equations describing the continuity of mass, heat, and momentum.
In most cases, they are based simply on the equation of mass conservation.
Important dynamic processes that govern the distribution of the winds are
often parameterized using relationships established either theoretically
or empirically. A distinct advantage of this approach is that it bypasses
the need for numerically solving the dynamic equations, thus drastically
reducing the computational burden. The success of this approach depends on
the ingenuity in devising the model and the parameterization schemes. Such
models are discussed in Section B.
4. Dynamic Models
Dynamic models are generally based on the numerical solutions of all
pertinent equations expressing the conservation laws. Thus, these models
13
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can simulate in a predictive mode the complicated, nonlinear interactions
between the large-scale synoptic air motions and local circulations induced
by topographic and/or thermal perturbations. The selection of suitable
numerical techniques, the parameterization of the subgrid processes, and
the initialization of the model are the major problems confronting the
application of dynamic models. The use of dynamic models is further plagued
by the requirement of a comprehensive data base for the exercise and veri-
fication of the model as well as the need for a potentially large computing
budget to implement and operate the model in an explicit fashion.
Many dynamic models have been developed to simulate mesoscale atmos-
pheric flows. They range from relatively simple one-dimensional models
to complex three-dimensional ones. Examples of these models include:
> Land and sea breeze models--Estoque (1963), Pielke (1973).
> Urban heat island models--Myrup (1969), McElroy (1971).
> Planetary boundary layer models Estoque (1963), Deardorff
(1970).
> Mountain and valley wind modelsOrville (1965), Hovermale
(1965), Thyer (1966).
B. PERTINENT EXISTING MODELS
Of the four general categories of wind models discussed above, the
following two approaches appear to offer the most promise to the present
study:
> Objective techniques based on the variational principle
> Diagnostic models based on mass continuity.
These two approaches seem to be particularly attractive because:
> They invoke certain first principles such as mass continuity
to supplement the wind measurements.
14
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> They are relatively simple to use and inexpensive
to apply.
Considerable work relevant to these two modeling approaches has been carried
out. These studies are summarized below.
1. Objective Techniques Based on Variational Principles
Dickerson (1973) appears to have been the first to apply this well-
known technique from synoptic-scale meteorological analysis to air pollu-
tion problems. In an attempt to adjust the wind field in the San Francisco
Bay Area using sparse and irregularly spaced measurements, he adopted a
variational formalism similar to that of Sasaki (1970). Simply stated,
his algorithm, using an iterative procedure, allows the measured wind
field to be adjusted in such a way that the difference between the diver-
gence of the adjusted wind field and that of the measured wind field is
minimized in a least-squares sense. The requirement that the adjusted wind
field be divergence-free was called the "strong constraint" by Sasaki (1970).*
As a result, Dickerson's model can provide a smooth, mass-consistent, two-
dimensional wind field if enough wind measurements are available.
Dickerson's approach was extended to three-dimensional wind flows by
Sherman (1975) in a model known as MATHEW. Although the overall conserva-
tion of mass is strictly imposed in this model, Sherman introduced different
Gauss precision moduli for the horizontal and vertical directions. When
the lower boundary conditions were properly adjusted according to local
topography, significant improvements were reported in the computed wind
field (Sherman, 1978).
More recently, Liu and Goodin (1976) examined three different methods
for objective analysis, comparing the characteristics of each in the reduc-
tion of wind divergence and the rate of convergence of the iterative scheme.
Each of the three methods is characterized by one of the following constraints:
* In contrast, the requirement that the difference in the divergence, as
measured by the residue in the continuity equation be only approx-
imately equal to zero, is called the "weak constraint."
15
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> The divergence is minimized
> The vorticity is fixed
> The station measurements are fixed.
Liu and Goodin concluded that the most suitable method is the one in which
the measured winds are held fixed while wind vectors at adjacent points
are adjusted in order to reduce the divergence.
2. Diagnostic Models Based on Mass Continuity
To simulate the wind field over irregular terrain characterized by
inhomogeneous surface temperatures, Anderson (1971) devised a simple
diagnostic model that was essentially based on the vertically integrated
mass conservation equation in steady state. The resulting model
equation is a two-dimensional Poisson expression in which the forcing
terms are perturbations of the free stream due to topographical and thermal
anomalies. In this model, topographical and thermal perturbations are para-
meterized in terms of the slopes of the local topography and the temperature
differences at the ground, respectively. The parameterization scheme con-
tains empirical coefficients that must be determined through the use of
observational data. Anderson applied his model to the State of Connecticut
(Anderson, 1971) and to the Los Angeles air basin (Anderson, 1972). For
both cases, he reported reasonable success.
Liu, Mundkur, and Yocke (1974) applied this technique to the San
Bernardino Mountains to determine the feasibility of modeling the surface
wind fields for simulating the spread of wind-driven brush fires. The
predicted flow pattern reproduced the expected behavior of topographic
and thermal perturbations. The computed wind distributions compared reason-
ably well with observational data collected by the Forest Fire Laboratory
of the U.S. Forest Service and with values for the empirical coefficients
chosen from a sensitivity analysis of the model.
16
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In a similar study carried out in Norway, Gr0nskei (1972) also used
the same procedures to compute the wind field in Oslo. The only difference
between his model and Anderson's model is that Grtfnskei parameterized the
thermal anomalies in terms of the sulfur dioxide emissions over the center
of the city and the temperature differences between air and water over the
open Oslo fjord. The sulfur dioxide emissions are presumably related to
the strength of the urban heat island. More recently, Basso, Robinson,
and Thuillier (1974) applied Anderson's model to study the flow patterns
in the San Francisco Bay Area. The results of this effort also appear to
show reasonably accurate comparisons of simulations with observations.
A slightly more sophisticated version of this approach is a model pro-
posed by Fosberg and his colleagues (Fosberg, Marlatt, and Krupnak, 1976).
In their model, equations governing the changes of divergence and vorticity
were derived from the primitive equations. They assumed in the derivation
that the inertia! terms can be neglected and that dynamic changes in the
flow field due to disturbances are transmitted by impulses. The first
assumption should work fairly well when applied to the surface layer; the
second is equivalent to the quasi-steady-state assumption. Under these
assumptions, the divergence and vorticity equations are reduced to two-
dimensional Poisson formulations. Fosberg, Marlatt, and Krupnak (1976)
applied this model to the rugged area in northwest Oregon, and the resultant
predicted wind field was deemed to be reasonable when compared with the
observed surface winds.
17
-------
IV DEVELOPMENT OF A WIND MODEL
FOR COMPLEX TERRAIN
Of the four wind modeling approaches identified in the previous chapter,
the interpolation techniques, which are grossly simple and therefore often
produce unacceptable results, were dismissed outright. At the other extreme,
the application of the dynamic modeling approach to complex terrain did not
appear to be feasible at this time because of many fundamental problems
related to model formulation and excessive computation burden. As mentioned
in the previous chapter, both the objective techniques and the diagnostic models
appear to offer suitable alternatives for computing three-dimensional wind
fields over complex terrain.
In an exploratory study carried out for the U.S. Forest Service, a two-
dimensional wind model of the diagnostic type was developed for application
to complex terrain (Liu, Mundkur and Yocke, 1974). This model was based
on the solution of a two-dimensional Poisson equation expressing the con-
servation of mass. Perturbations in the prevailing wind field due to local
topographic or thermal variations were treated as forcing functions. This
model was applied to the Devil Canyon in the San Bernardino Mountains. The
predicted winds compared favorably with the observational data collected
by the Forest Fire Laboratory. The success of this undertaking has encour-
aged us to continue further development of this modeling approach.
A. THE MODEL EQUATION
Modeling the wind field in the lower atmosphere is essentially tanta-
mount to simulating the interactions between the free atmosphere and the
surface boundary layer of the atmosphere. Depending on the characteristic
spatial and temporal scales and other environmental parameters, such as
the prevailing wind, the thermal stability, and the topography, these inter-
actions take place at different levels of significance. For example, for a
18
-------
o
region with a characteristic horizontal dimension on the order of 10 km
or larger and a characteristic time scale on the order of days, the surface
layer can be viewed as a layer feeding energy to the free atmosphere. As
a result, any successful model on this scale must include the dynamic
changes in the large-scale motion that are due to the surface layer. In
contrast, on the scale of interest to the present project, with a horizontal
2
dimension of 10 km and a time scale of a few hours, the synoptic-scale air
motion can be viewed as nearly steady state. Consequently, the surface layer
can be regarded as a passive system driven by the synoptic-scale flow and
surface perturbations. This is the approach adopted in the present study.
The model equation is based on the three-dimensional steady-state
equation expressing the conservation of mass for an incompressible fluid:
3U.+ av. 3w = o
9x ay 9z u '
where x, y, and z are the orthogonal Cartesian coordinates and u, v, and
w are the corresponding wind components. As shown in Figure 5, the modeling
region is first divided into vertical layers. Note that the terrain is
allowed to intersect the modeling region; consequently, portions of the
modeling region (shaded in Figure 5) must be excluded in the calculations.
Note also that it is not necessary to assume that the vertical layers are
equally divided. By integrating Eq. (4) over each vertical slab, one can
obtain the following set of equations:
9u. 9v.
+ = -x>^ ' i = 1» 2, ..., N , (5)
where N is the total number of vertical layers and u. and v. are the verti-
cally averaged wind in the i-th layer, defined as follows:
19
-------
l\ \ \ \ \ \ \ \ \ \ [\
\N \ \ \ \ \ \ \ \ \\
> r \ --v Y \ y \ \ \ [V^17^
\\\ \\V\\ \V
\\K\\\\\ \\
O) -<3
-C !-
CO S-
re en
-a
en
QJ I
-a cu
re "a
-C O
re re
c
c o
2 -r-
O co
JZL C
CO
s_
cu re
ai +->
i
> i
re
r O
re -r-
c: -M
o cu
o o
a> co-
co >,
I -C
CO
co re
o
S- 4-
o o
O)
en
20
-------
u- = ^ f ] u dz , (6)
i J-,
1 rZ\*t , (7)
«i 7Z
l
AZ. = Zj - Z.
and ft. is the wind divergence in the i-th layer:
w(z.) - w(z. ,)
n(x,y)= ^ ' (8)
By defining the following two-dimensional potential functions* for each
of the vertical layers,
(9)
(10)
Eq. (5) can be cast into the conventional Poisson form:
2 9 0-j 9 4>-j
v *. = Y + ~- = -a. . (11)
9X sy
Once the distribution of the wind convergence is specified, solutions to
these equations with appropriate boundary conditions can be computed
readily using numerical techniques.
* The question regarding the possibility of expressing a velocity vector as
a potential function is addressed in Appendix A.
21
-------
B. PARAMETERIZATION OF THE VERTICAL FLUXES
The specification of wind convergence is the key feature of this model
It is proposed that the overall wind convergence is the sum of many com-
ponents, a).., as weighted by empirically determined coefficients, a.,
* J J
ft. =
In Chapter II, a discussion of the major physical processes that affect
the wind distributions in the planetary boundary layer was presented. Only
the following perturbations to the wind field over rugged terrain are, how-
ever, considered in the model:
> Lifting and diversion of the flow due to topographic effects.
> Wind profile modification due to frictional effects in the
planetary boundary layer.
> Convergence of the flow due to thermal effects.
- Urban heat island
- Mountain and valley winds.
These perturbations are treated through parameterization of the pertinent
processes as follows.
1. Topographic Effects
Because the terrain is part of the modeling region in the present model
formulation, certain aspects of the topographic effects are included indirect-
ly as boundary conditions. For example, the no-slip condition is imposed
whenever the flow encounters a solid surface. The often-observed lifting
and diversion of the flow is, however, handled by parameterizing the verti-
cal fluxes.
22
-------
For high wind speeds and neutral and unstable conditions, it is perhaps
logical to view an airflow contacting the slope of a hill to be perfectly
elastic. Based on kinematic considerations, the vertical velocity can be
expressed as (see Figure 6):
w = U sin (arctan vh)
e"klz
(13)
where h(x,y) is the terrain height as a function of location. The exponen-
tial term has been added to allow for the decay of the topographic influ-
ence away from the surface.
e = arctan (vh)
Figure 6. Schematic diagram of a flow
contacting the slope of a hill
For low winds and stable conditions, the kinetic energy of the air-
stream approaching a hill may be too small to overcome the potential energy
required to lift it over the obstacle. As a result, the flow is diverted
around the hill.* The physical processes governing the occurrence of these
phenomena are rather complex, and they have only recently received the
attention of air pollution researchers. As a first attempt to characterize
this effect, Eq. (13) was further modified by a multiplicative factor
(see Figure 7),
* The impingement of plumes upon cold mountain slopes, a phenomenon
that is well known in recent air pollution studies, is related to
this situation.
23
-------
r ~ -rc/ioo m
Unstable
(Y - r < 0)
Stable
(Y - r > 0)
B = 1
B = 0
Y - r
Y - r
Figure 7. Parameters used in defining the diversion effect
24
-------
B(FJ =
r
1
F,./Fr
r r
c
if ^
if ^
> - r c
(14)
where
Y - environmental temperature lapse rate,
r = adiabatic lapse rate,
F - - (Froude number),
Ah = hmax - h(x'y> '
Tm = free-stream (unperturbed) temperature
g = acceleration of gravity
U2
c = - (a cut_0-ff constant),
F = a critical Froude number where no flow diversion
c
takes place.
Modification of the temperature lapse rate, Y» as air flows over a hill has
been neglected as a first approximation herein. According to an analysis
by Lilly (1973), the critical Froude number can be taken as 0.5 for an
ellipsoidal mountain and 1.0 for a conical mountain.
2. Boundary Layer Effects
It is well known that surface friction plays an important role in
determining the distribution of the horizontal wind, particularly the
vertical wind profile in the atmospheric boundary layer. According to
Blasius (1908), the vertical velocity in the boundary layer can be obtained
from the similarity solution as follows:
25
-------
where
w(z.)
= n.
Re.
(15)
U^ = the free-stream (unperturbed) velocity,
Re = the Reynolds number,
[z.) = the vertical velocity at height z.,
AX
T~
Y'i -"(z^/U.:
Equation (15) is used to parameterize the frictional effect in the
atmosphere, from which the following equation is derived:
k AX
kU(zi)
U AZ.
u a^.. r
'TAX1
(16)
where u* is the friction velocity, k is von Karman's constant, and the
dimensionless f(z) can be computed from relationships obtained empirically
by Businger et al. (1973).
26
-------
for the stable case,
f(z) = Wf-\
\ O/
for the unstable case,
f(z) = in
'(r)
+ 2 tan
-1
(17)
- in
- 2 tan
-1
and
(18)
-1/4
where z~ is the aerodynamic surface roughness length and L is the Monin-
Obukhov length.
3. Thermal Effects
It is also known that flow can be induced by the conditions of uneven
surface heating. On the scale of interest to this study, over complex
terrain in urban settings, two types of atmospheric circulation were deemed
important:
> Flows induced by an urban heat island
> Mountain and valley winds (upslope and downslope flows).
27
-------
a. Urban Heat Island
Airflow over a heated island has been shown to have an appearance
similar to that over a mountain (Stern and Malkus, 1953). According to
Stern and Malkus, an "equivalent mountain" function is defined as follows:
if y - r > 0
(19)
0 i f Y - r <. 0
where T(x,y) is the spatial distribution of surface temperature. In parallel
to the discussions that led to Eq. (13), it can be assumed that the verti-
cal motion generated by heat island effects is
w = U sin (arctan vM) e~k2z (20)
The reader should note that in the absence of a driving wind, U, vertical
fluxes due to the urban heat island are zero. Furthermore, the equivalent
mountain formation was derived for flat terrain situations; therefore, this
parameterization is used only for flat portions of the modeling grid. In
sloping terrain, the treatment described in the following section is used.
b. Mountain-Valley Winds
Slope winds are micro/mesoscale breezes that blow normal to the topo-
graphic gradients due to temperature-dependent density differences. During
the night, radiative cooling of the slope cools the air just above it.
This process causes the air close to the ground to become denser than the
air at the same altitude but farther above the sloping surface. As a
result, the cold air slides down the slope. The reverse process occurs
during the day when the sun's radiation warms the slope and the air just
above the slope.
28
-------
Defant (1933) proposed the following expression to describe the
steady-state, average speed of a cold current gliding down a slope of
average elevation angle a:
slope
gh(T~ - TV)sin a
CDTC
1/2
(21)
where
V
h =
g =
TC'TE
a drag coefficient
the mean height of the cold current,
gravitational acceleration,
the temperatures of the current and environment,
respectively.
The reader will note that a driving wind is not necessary to induce down-
slope flow with Eq. (21).
Adapting Eq. (21) to our grid system but retaining the important func-
tional dependences, one can obtain the following expression for both upslope
and downslope flow:
slope
1/2
(22)
where H is the average surface elevation in the grid cell and H is the
max
highest terrain elevation affecting local flow. The terrain involving the
heights is substituted for the "sin a" term in Eq. (21) because the "sin a"
term is qualitatively correct only for estimating average slope velocity
and only if the elevation angle is uniform along the entire slope. In the
modeling grid, however, the velocity is computed at any point along
29
-------
a slope, not just the average slope velocity, and elevation angles are
rarely uniform along a slope. The substitution allows qualitatively
realistic spatially resolved estimates of upslope and downslope flow
behavior. With a proper selection of the constant in Eq. (22), quantita-
tively correct estimates should also be possible.
As in Defant's algorithm, Eq. (22) contains no dependence on driving
wind. In our model, therefore, the vertical fluxes for the mountain-valley
winds are self-generating. Since down- and up-slope flows have both hori-
zontal and vertical components, the model must account for the self-generating
horizontal component as well as the vertical component to maintain consis-
tency. Therefore, the parameterization of mountain-valley winds is entered
through the vertical flux term and through the horizontal boundary condi-
tions. All previous parameterizations of vertical fluxes described are
dependent upon the horizontal wind and are intrinsically consistent with
the horizontal flow component; thus, no adjustments to boundary conditions
are necessary for them.
C. NUMERICAL SOLUTION PROCEDURE
To obtain a solution to Eq. (11) with complex boundary conditions,
a numerical technique is required. Several direct Poisson solvers are
available that are based on block-cyclic reduction of a set of finite
difference equations (Buzbee, Golub, and Nielson, 1970; Swarztrauber and
Sweet, 1975). Although these solvers are convenient and inexpensive to use,
their application is restricted to simple rectangular modeling regions.
This presents a rather serious drawback for the present formulation because,
as described earlier, the terrain may intersect the modeling region.
Thus, an alternative solution technique was chosen that can accom-
modate the exclusion of portions of the modeling region with irregular
shapes and yet incur only a modest increase in computation time over
the direct solution techniques. The general form of the Poisson
equation is
30
-------
72+ = 14+14 = f(x,y) . (23)
ax 3y6
and the five-point difference approximation to Eq. (23) is
_ f f (24)
Ux? (Ay)2 i>j
i = 1 , .... M and j = 1 , . . . , N ,
where
= the potential function,
f(x»y) = the forcing function,
x, y = orthogonal Cartesian coordinates,
i, j = the grid cell indices of a grid system that
is superimposed on the modeling region and that
has M grid cells in thex-(i-) direction and N
grid cells in the y- (j-) direction.
Equation (24) is valid for all points within the modeling region, but
for those along the modeling boundary some additional computations are
required. Along the boundaries, the normal derivatives of (i.e.,
d/dx or d$/dy--often referred to as Neumann-type boundary conditions)
are specified, and these are used to compute values of $ at fictitious
grid points outside the modeling region. For example, consider a
grid cell (i»j) that forms part of the left boundary of the modeling
region as shown in Figure 8. To solve Eq. (24) for this point, one
must specify some value for . -, ., which exists at a point outside
i ~ > » j
the modeling region. The study team simply computed A. , . using
' ~ i > J
(25)
and substituted this expression in Eq. (24).
-------
i , j+1
ij
i+'UJ
/
Figure 8. Sketch showing a grid cell (i,j) along the
left boundary of the modeling region
The commonly used modified Gauss-Seidel iterative solution method
was selected because it is very well suited to the solution of the
five-point operator [Eq. (24)] for the Poisson equation (Dahlquist and
Bjb'rck, 1974). Like all iterative techniques, it starts from a first
approximation, which is successively improved until a sufficiently
accurate solution is obtained. To simplify the description of this
method, consider the linear system of equations
Ax = b
(26)
instead of the system described in Eq. (24). Equation (26) can be
written as
x .
xi
a..
i = 1, 2, ..., n and 3.. ? 0 . (27 )
In Gauss-Seidel's method, a sequence of approximations x^ , x ,
^
, ... is computed by
32
-------
aii
i = 1, 2, ..., n
Note that Eq. (28) can also be written as
x^k+1) = xjk) + r\k] , (29)
(k)
where r. is the current residual of the i-th equation and
i-1 /1., -M n
.
U J U J l
r = -= -^ . (30)
Now, to improve the rate of convergence, one can slightly modify Eq. (29)
to give
x. = x1 + ur. , (31)
where u is called the relaxation parameter, which is chosen so that
the rate of convergence is maximized. This improved iteration procedure
is called the "successive overtaxation method" (Dahlquist and Bjbrck,
1974).
In the present study, u> = 1.4 was found to give the best rate of
convergence in tests of the modified Gauss-Seidel solution to Eq. (24).
In these tests, the criterion for complete convergence was defined
so that
max r ±0.01 x . (32)
33
-------
V APPLICATION OF THE MODEL TO THE PHOENIX AREA
The Phoenix metropolitan area is located in central Arizona along the
channel and flood plain of the Salt River. It is surrounded by flat desert
wastelands and by barren mountain ridges ranging widely in elevation. The
river level is about 350 m above mean sea level near downtown Phoenix, and
mountain peaks rise as high as 1400 m above mean sea level within 20 km of
the downtown area. Since Phoenix is located in the great southwest desert
of the United States, it typically experiences clear skies and intense
surface heating and radiational cooling during the days and nights, respec-
tively. In light of the above comments, the Phoenix urban area is an ideal
location in which to test the performance of the three-dimensional wind model
described earlier, because all the following phenomena appear to occur there:
topographic obstructions to wind flow, urban heat island effects, surface
roughness effects, and mountain-valley winds (up- and down-slope flows).
Therefore, the Phoenix area was chosen as the initial test location for
the model.
In the next section, the data base used to carry out wind simulations
for four days in Phoenix is described. Section B discusses the results
of those simulations. Finally, Section C presents statistical analyses
of the results.
A. DATA BASE
Four days were selected for testing the wind model described in the
previous chapter. On three of these days low wind speeds prevailed, and
on one the wind speeds were high. The dates chosen were 15 and 16 February
1977 and 7 and 10 March 1977; these were chosen for completeness of data
and phenomenological interest. On these days, a maximum of 15 surface
wind stations collected data; however, for some hours a significant number
of these stations reported missing or invalid data. Names and coordinates
34
-------
of each of the 15 stations are shown in Table 1. No upper level wind or
temperature measurements are made near the Phoenix area, and National
Weather Service synoptic scale maps provided the only information about
upper level flows on these days. The model was exercised for all 24 hours
on each of the days.
The Phoenix metropolitan area and surrounding environs were divided
into a 40- by F>0-qrid of 2- by 2-km squares as shown in Figure 9. Five
vertical levels of 200 m each were chosen for the present application.
For each grid element, the average surface elevation, elevation of most
prominent terrain feature in the vicinity of the cell, and the aerodynamic
roughness length were determined. Terrain height information was obtained
from topographic maps of the area. Values of the roughness length were
estimated using the bulk aerodynamic method devised by Lettau (1969). For
this purpose, representative land use categories were established and
percentages of them within each horizontal grid element determined using
aerial photographs, results of visual ground surveys, street maps, and real
estate maps and charts. A composite value for each of the elements was
then derived by linear weighting of the percent coverage of each land use
category existing within the element.
For all days simulated, the values of the coefficients used in the
parameterization of heat island, frictional, and topographic vertical fluxes
were identical: 1.0, 0.01, and 0.3, respectively. These were derived
from previous model applications and sensitivity tests for other areas.
Only the boundary condition flows a
-------
TABLE 1. PHOENIX AREA WIND MEASUREMENT STATION NAMES AND COORDINATES
Station Name
Central Phoenix
South Phoenix
Glendale
West Phoenix
North Phoenix
North Scottsdale/Paradise Valley
Scottsdale
Mesa
Mesa Wind
Fire Station 13
Fire Station 17
Orange Grove
Williams AFB
Luke AFB
Taylor
UTM*
Coordinates
East
404.
401.
390.
395.
402.
414.
415.
423.
431.
409.
402.
385.
438.
372.
391.
21
21
57
04
10
27
83
85
69
13
71
61
00
00
09
North
3702
3696
3714
3708
3713
3719
3704
3698
3697
3704
3708
3725
3685
3711
3687
.44
.36
.82
.31
.71
.34
.66
.12
.50
.68
.81
.60
.00
.00
.72
Grid Cell Model
Coordinates
East
32
31
25
27
31
37
37
41
46
35
31
22
49
16
25
North
21
18
27
24
27
30
22
19
19
23
24
33
13
26
14
* Universal Transverse Mercator
36
-------
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37
-------
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o o o o
oooooooo
o o o o o o
0000000
o o o o o o o
o o o o o o
~ ~
CNJ fNJ CNJ CNI
CNJ CNI CNI
co co ro
o o o o o o
CM co *r *t *r
ro ro co co ro
I
^
II
Q,
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3 00 -!
Q O
O 31
^1 CL
i- o:
< o
O O O O
O O O O O
000
UJ Q
OO UJ
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Q- =>
CNJ r- CNI
o o
CNJ CNJ
ooo
CNJ CNJ CM CNJ CNJ
ro ro CNJ rxj
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38
-------
B. DISCUSSION OF THE RESULTS
The measured surface winds are plotted along with the corresponding
predictions for the lowest layer of the modeling grid in Appendix B for
every second hour on the four days. The predictions for all five layers
not presented herein are contained on microfiche.* The contours shown on
the predicted wind field plots represent terrain that is above the top
elevation of that particular grid layer; they are included so that the
reader may more easily elucidate the impact of elevated terrain and terrain
slope. Computations are not made by the model for grid cells within these
contours.
A glance at the measured winds on the days selected for application of
the wind model to Phoenix should convince most readers that complex flow
situations existed on three of the four days (15 and 16 February and 7 March
1977). Generally, chaotic wind patterns were observed on these three days
owing to weak synoptic flow and the dominance of local effects. Stronger
synoptic flow existed on the fourth day (10 March 1977), resulting in more
regular flow patterns. Certainly, the main goal of this study is to deter-
mine whether the wind model developed here can at least qualitatively match
such diverse wind patterns as were observed on these four days in the Phoenix
area. Success in this task would indicate that most of the important
pheonomena have been treated and have -been parameterized reasonably well.
Beyond this study, continued efforts to achieve a better quantitative
match of predicted and measured winds are suggested that would focus on
systematic adjustment of empirical parameters for a given application site.
For the present application, such adjustments have not been carried out.
Analyses of quantitative results presented later in this section are intended
to provide a benchmark against which to compare future improvements in model
performance, rather than an estimate of the model's ultimate prediction
accuracy; optimization of empirical coefficients in the model are certain
to improve its performance.
* The microfiche format was selected for compact presentation of all of the
576 plots generated in this study. The microfiche are available on request
from the Project Officer.
39
-------
On the basis of a qualitative comparison of the wind model predictions
with wind measurements, the results of this application of the model to the
Phoenix area are encouraging. The model reproduced the wind field on the
high wind speed day (10 March 1977) very well and required no mountain-
valley wind contribution. This finding is consistent with the intuitive
notion that these effects would be relatively insignificant under strong
synoptic flow conditions. Furthermore, this result indicates that the model's
treatment of topographic blocking, frictional, and urban heat island effects
is adequate.
In the early hours of 10 March the large scale flows appear to have been
generally out of the southwest at about 15 to 23 mph. Flows gradually began
to shift to a more westerly direction by 0200. By 0500 they were northwesterly,
and they became almost northerly in the early afternoon. Flow velocities
began to diminish somewhat at about 1800, slowing to about 10 mph by 2300.
Throughout the day, significant amounts of diversion can be seen in the
model predictions around the Sierra Estrella, South, and Maricopa mountain
features. Some minor flow disturbances can also be seen in the vicinity of
the mountain just north of Phoenix (e.g., Camelback, Phoenix Mountain). The
presence of these topographic features frequently tends to produce localized
areas of convergence and divergence in the Avondale and Tempe-Mesa vicinities
and along the Gila River. There is some indication that these are consistent
with the location of pollutant "hot-spots" in the Phoenix area (Berman, 1978).
On the low wind speed days, it appears that microscale (less than 2 km)
and mesoscale (larger than 2 km) drainage and upslope flows dominate the
chaotic flow situation. Although the model did not reproduce the flow char-
acteristics exactly for these days, it appears to have predicted the most
important mesoscale drainage and upslope influences. On these days, 15 and
16 February and 7 March 1977, large scale flows were light, ranging from 2
to 10 mph, and generally out of the west (varying between southwest and
northeast). As can be seen in the model predictions, local effects essentially
mask the large scale flow characteristics, particularly in the lowest levels.
A significant amount of downslope flow can be seen to persist around all
40
-------
significant terrain features. For example, up- and down-slope flows
associated with the gradually steepening terrain toward the McDowell
Mountains to the northwest dominate over a large part of the modeling
region. Up- and down-slope flows from the South Mountains have a signifi-
cant impact on the wind in Phoenix proper. Down-slope flows persist
during the hours of darkness on all three days. The down-slope flows
desist just after dawn and a few hours later the effects of up-slope
flow on major terrain features can be seen. These diminish in the early
evening, followed by the onset of down-slope flow a few hours later. Areas
of convergence and divergence caused by opposing flows up and down adja-
cent terrain features can be observed in the vicinity of Phoenix, especially
in tne Tempe, Mesa and Gila River areas. These patterns are fairly con-
sistent with the distribution of pollutants observed in those areas. A
specific example is the large area of convergence at night in West Phoenix.
This area typically has high concentrations of inert pollutants like carbon
monoxide during nighttime hours in light wind and clear sky situations
(Berman, 1978).
Some of the measured wind values on these days appear to reflect strong
local (microscale) effects (Pitchford, 1976). Although we cannot state
categorically that an aberrant wind speed or direction is the result of these
influences, this possibility should be kept in mind in evaluating the model's
performance. Some of the measurement stations appear to have poor exposure,
e.g., the south Phoenix station is immediately adjacent to a large tree.
Our analysis of prediction statistics was performed using all available data,
although some may be representative of local or short-time scale phenomena.
C. STATISTICAL ANALYSIS OF THE RESULTS
The wind model predictions at the surface were compared with the corres-
ponding surface-level measurements taken each hour throughout the four test
days. Based on these data, frequency distributions of the deviation in
predicted wind speed and wind direction as a function of the measured values
were constructed. Also, estimates of the mean (or expected value of)
deviations in these predicted parameters were computed. These frequency
distributions and means of deviations in wind speed and wind direction are
shown in Figures 10 through 13 for the four test days: 15 February,
41
-------
50
40 -
co
OJ
O
c:
O)
O
O
O
S-
O)
-Q
E
30 -
10 -
O WIND DIRECTION DEVIATION
D WIND SPEED DEVIATION
-180° -90° 0° 90°
Wind direction deviation (degrees)
180C
-8-404
Wind speed deviation (m/sec)
Note: The mean absolute wind speed deviation is 1.58 m/sec, and
the mean absolute wind direction deviation is 58.98°.
Figure 10. Frequency distributions of wind speed and wind direction
deviations for 15 February 1977
42
-------
50
40 '
en
-------
60
50
40
co
OJ
o
C
(LI
S-
S-
3
O
o
o
S-
OJ
-O
rs
30 -
20 -
10
O WIND DIRECTION DEVIATION
D WIND SPEED DEVIATION
-180C
-90
Wind direction deviation (degrees)
I I i
180°
-8
-40 4
Wind speed deviation (m/sec)
Note:
The mean absolute wind speed deviation is 1.9 m/sec, and
the mean absolute wind direction deviation is 40.58°.
Figure 12. Frequency distributions of wind speed and wind direction
deviations for 7 March 1977
44
-------
70
60
50
O)
o
c:
01
S-
S-
13
o
o
o
S-
-------
16 February, 7 March, and 10 March 1977. In these figures, wind direction
data points are indicated with circles, and wind speed data points with
squares. All measured wind data reported on these days were used to con-
struct these distributions and means. As noted earlier, many stations in
the Phoenix area may be influenced by microscale effects; consequently,
this statistical evaluation of the wind model's performance is quite
stringent, yet the performance of the model for these days in Phoenix is
quite encouraging.
These extensive statistical analyses of the predicted and measured
wind speeds and wind directions were also carried out to detect possible
systematic biases and random errors in the model. The figures illustrate
that the model generally predicts winds that are higher in magnitude (by
about 1.5 m/sec) and more clockwise in direction (by about 1 compass point
on a 16-point scale) than are the corresponding surface measurements. This
finding is not surprising because the predictions are averages for a layer
between the ground and 200 m above the ground, whereas the measurements
were obtained by sensors situated between about 10 and 20 m above ground.
The mean absolute deviation in wind speed predictions for all four days
combined is 1.96 m/sec, based on 831 hourly averaged wind measurements. The
four-day mean of absolute wind direction deviations is 49.8°. Both of these
values indicate that the performance of the model is reasonable when applied
to a complex and chaotic flow situation. Notice that the lower wind speed
days (15 and 16 February and 7 March) have lower mean wind speed deviations
than the higher wind speed day (10 March). In contrast, the mean direction
deviation is smaller for the high wind speed day, on which an organized
synoptic flow existed over the area. These results basically agree with
the observations in the preceding paragraph. Over 64 percent of all wind
direction deviations are within ±45° or +1 compass point for all four days.
46
-------
VI CONCLUSIONS AND RECOMMENDATIONS
A three-dimensional diagnostic wind model has been developed for rugged
terrain based on mass continuity. The model is composed of several hori-
zontal layers of variable thicknesses. For each layer, Poisson equation is
written with the wind convergence as the forcing function. Many types of
wind perturbations over rugged terrain are considered in this model, including
diversion of the flow due to topographical effects, modification of wind
profiles due to boundary layer frictional effects, convergence of the flow
due to urban heat island effects, and mountain and valley winds due to thermal
effects. Wind data collected during a comprehensive field measurement pro-
gram at Phoenix, Arizona, were used to test the model. The average deviation
between the predicted and observed wind speeds, based on 831 hourly measure-
ments, is 1.96 m/sec. The corresponding average wind direction deviation
is 49.8°. These gross statistics seem to indicate that the performance of
the model is reasonable when applied to a complex and chaotic flow situation.
Further improvement of the model is certainly possible. Among the
areas for which continued developmental effort would be most fruitful are:
> A better, more objective procedure to prescribe the boundary
conditions for initiating the model calculations.
> A systematic analysis of the model responses to optimize the
empirical coefficients in the parameterization schemes.
> Further refinement of the model by including other physical
processes, such as the wind direction shear, if proven important.
47
-------
APPENDIX A
DECOMPOSITION OF A VELOCITY VECTOR
48
-------
APPENDIX A
DECOMPOSITION OF A VELOCITY VECTOR
In its most general form, any velocity vector can be decomposed
into two parts, one of which represents a potential flow:
7 = A~ + B"
= V$ + B" . (A-l)
This decomposition is obviously not unique, because any potential flow
can be used for A. By imposing the additional condition,
v B" = 0 , (A-2)
one can show that a unique decomposition can be constructed, and the
second vector, B, can be given by a vector potential, B, (Aris, 1962):
F = V x § . (A-3)
This vector potential will satisfy Poisson's equation,
V26 = -u , (A-4)
where u> is the vorticity of the flow. This is the well-known Helmholtz's
decomposition for incompressible flow.
Alternatively, one can write
7 = A + B~ , (A-5)
A = v$ , (A-6)
6~ = v x B . (A-7)
49
-------
If the vortical component § is restricted to be in the plane perpendic-
ular to the vorticity vector itself, namely,
§ v x B = 0 , (A-8)
then it can be represented by two scalar functions,
§ =
Such a vector was termed complex lamellar by Lord Kelvin (Truesdell,
1954).
50
-------
APPENDIX B
COMPARISON OF PREDICTED AND MEASURED
SURFACE WINDS IN PHOENIX
51
-------
APPENDIX B
COMPARISON OF PREDICTED AND MEASURED
SURFACE WINDS IN PHOENIX
This appendix presents a comparison of the predicted and measured
surface winds in Phoenix every second hour for the four test days in 1977:
15 and 16 February and 7 and 10 March 1977. Each pair of comparisons
shows the predictions on the top of the page and the corresponding measure-
ments on the bottom labeled in mountain standard time (MST).
52
-------
PART 1-15 FEBRUARY 1977
53
-------
40
ID
20
30
40
SOUTH
WIND SPEED IM/5)
0 10
10
20
30
90
1C
:;/.:.;
:/;-
V -
/.
/
r i i i I i I I i I I I i I i i i i I I I i i i I I I I i i I i i I i I I i i i I i i i I I I i i i ln
C 10 20 30 40 5IT
30
to
0000 MST
54
-------
*L --j--^--1 ^ f * T T T
10
20
30
10
20
30
*L
r
,
cc
i i i i i i i i i
- , , ,
L_ ' '
.
'
' .
1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1
'
; *
; ; ; -! | ' ;
: r -
! - ' < -."
. '
:
i i 1 > i i i i i
0 2
i i i i i i > i t
"I1 "" ; * : " '-
! ' : :
' ;:...:
......
, ; . , . i>*;. i .
,'. ., i .[ ( , i-'i-"
i < i i i > >» -
, ;..,., , , ... , .:.
i i i i i i i i i
0 3
i i i i i i i i i
< * i i i i
.
/
*
,
*
»
;;. r, :;
f »
.
.,,,,,,
iiiiii.ii
0 4
:
.
< < t i t > ,
< < < i i
-:.:/: . ..
*
,..,.... ^ . -
,.,,.,,
* ' ' '
1 1 1 1 1 1 1 1 1
0 S
30
20
1C
0200 MST
55
-------
ww/niiri ' * < UWM
;;^|j)|jJll|!r:::::U::^
f\:'' 'Jx^HSjff^ -/:'C,
Uc~r777
WIND SPEED IM/S)
0 10
10
20
30
li.
Lt
- , , , ,
..--.-.
- , , , . ,
k.
* '
1 I 1 1 1 1 1 1 1
1
1 - -'
M < 1 1
( < -1 --I -
\ -
. - -
' ' ' '
* < i i ...
1 i i i j f > i i i
0 2
: I .;::..! .;...[ .1.4
:.::.: :;.;.;:
*< -i 1 1 > .., -i
, '
; y
*
:. . . .y. .-. _
,,,.,,, ,.,.-
: .. | : r*T
; :
. ' ' '
1 1 1 1 1 1 1 1 1
0 3
* < < i i i
\. .
' \
.
t "
,.; ;,: ;.t
.
:;:: : :/
1
1 1 1 1 1 1 1 1 1
0 4
..:..:,....
;;.:;;:::;
. «<
/
* * '
1 1 1 1 1 § 1 1 1 1
0 5
in
*>n
&
0400 MST
56
-------
4CC '1°,
20
30
40
3C
1C
/. . .: .. ..
* <
I i
i;:;;/:
30
20
10
10
20
30
40
0600 MST
57
-------
S0UTH
10
20
30
40
3C
2C
1C
:../.:;::
I i i
HIND SPEED IM/S)
0 10
30
10
10
20
30
4(1
0800 MST
58
-------
M I 4 » t I I I I I II 4 ' » t I I I I- M 4 I > t t I II I I Jf 4 I T I
WIND SPEED IM/5)
0 10
10
20
30
40
3C
20
1C
\
3C
20
10
10
20
30
40
1000 MST
59
-------
ID
20
3D
ltll«*(Jlll!*-*'4'Jt1lltt !**»(. fill
' ' ( ' ' vS'jpraf / v
\ Hfflfe/ / *
U "
WIND SPEED IM/5)
Q 10
10
20
3D
40
I-.
\- '
3C
1CU
_*r
30
V
zo
10
JO
20
30
40
1200 MST
60
-------
10
ninnu
\ \ \\ 11 f t f //////..,.
\ \ M i 1 f 11 // r //////..,...,.,
^^^^^» i t t i t t t f t f f t f f f fj.i.
\\\\\ \ * < < i i * * 1111 f r f 11 f
\\ \
it\U\\\\\\V\\.\
WIND SPEED IM/S)
o 10
icc
?r
1C
wc
10 2
;
1 , i , , (<
j *
! yX
I .S
s
-,,,,,, ii«,,.
.,,,,-,, i,.,,.
-, t t i . i > y f <-(«(... t >
I" !
!
'.
* I
i i i . i , i ] i 1 i i i i i i i i i
1C 2
0 3
' '
«,
< < i i i , i > *
< < -i i i , > »
< * i i i . -> >
i i i i i i i i i
0 3
0 4
! I ! ''!'.'. 1
'(till'
x
/"
k,- -
»
; -;j.-; ; \ -
\. , , , ,
i 1 1 1 i 1 1 i i
0 4
0 5
i i i i i i i i i
>-
, . .< i i . , -
/":"". ""."".
/ .
... . f .
1 1 1 1 1 1 1 ] ]
0 5
"fo
on
1400 MST
61
-------
WIND SPEED IM/5)
0 10
irc
c
i
1
- < i i i i .
. .
-
.
, , , ,
: i
D 2
< <(!-
...;,- .
^^
^^
i i . t \ *
<. < -i »- * >
'-(!)-' »
i i i i i i > i i
0 2
D 3
>>
\>r
^^^
M_
' < 1 I 1 > t -» '
< < 1 1 1 ' i ! 't
' i~l
1 i I 1 1 1 i 1 1
0 3
D 4
< < { 1 i i
,,.;,,,
v*'" ""
v -
... ,f , ,1,
{ 1 4 1 1 1
i i i i i i i i i
0 4
3 5
-
. - ,
j '' y ''
IS
' «^
'
1 1 I 1 1 1 I 1 1 1
0 5
In
7(-l
i n
r»
t?
1600 MST
62
-------
P \WJh] /*'*-'
K \i«/ /,,***
**** >»^i»*
s ** * * *s±+
Jf * # » *s»sVs*J*S*^^^jr**^**J**SI *^'
*l**^M*J++'sv****rf<*^*+*~*~j+J»s*j+*+ '*C
-
10
10
20
20
SBUTH
30
30
JO
40
WIND SPEED IM/S1
1 10
ar
r
cc
f
, >
i i i i i ' i i i i
1
**"
i i i i i , i i i
0 2
i 'i
0
' '
,* :
*^ : .
<* .
">
'
1 1- , !>
1 | 1 1 1 1
a
1 1 1 1 ! 1 1 1 1
* t , 1 .
^
X
^
»-
\
< v « * « » »
'. > 1 i i 1 1 1 1
0 4
i l i t i i l i r
< t i
* '
.
'**''*
' '
^ « <". * «.- » » t
"*"**. ">
"
*
"
"
l l l l 1 l i ' i
Q 5
40
i?
1800 MST
63
-------
*cc
10
20
30
40
M »?
'i iV.I 0 Yl \i ii ULi i u 17 :; ?i III!
mfZ
wk
W/M\ rt "*
f--<-« /, ,O ,,,
10
S8UTM
40
"""so1
WIND SPEED (M/S)
0 10
3D
20
30
40
L.
3c'r-
1C
30
20
10
10
20
30
40
2000 MST
64
-------
10
WIND SPEED IM/S)
Q 10
10
20
30
40
I i i i i i | i i i i i i i i i | i i I i I I i I I | I I I I i i i i i | i
iii i i i i i
:\L...: :.....:..
i_
t: ;:
L.
1C
tr
\
30
20
ID
lu
20
30
2200 MST
40
65
-------
PART 2--16 FEBRUARY 1977
66
-------
10
20
30
SBUTH
WIND SPEED IH/S)
0 10
10
20
30
40
ti. i i i i i i i i i r~
*- .....
!
r r
r "" " "
c ' ; '7 : ; "" ~
3C| - -
1_
[_
'
*
?r L_ .. .... ... , .,....,... ...
h
^
*"
f
'
i
_^
; i
r - r
"- -
0 2
-
: -
. .: / ....; 4
/
^^^
^*^*
.
111(11111
0 3
0000 MST
67
7 -
, ~\
r;... . . . _ i
:/ 'f\ ;- " .. :;;; .." ;j
/<<> ~ ->
^
'
* i
J 1 1 | | 1 1 ! 1 1 1 I 1 t 1 1 ' 1 1
0 40 5
*>n
n
S
-------
4C1-
10
4
\\\\^ > 1
V ^ * * 4
-U //''
?],//<'
t yyxv-'
'yy/^
I'r/i r
l tl 411.1 i|f l.n !*l I I I I I I
20 ' 30
SBUTH
WIND SPEED IM/S)
a 10
20
30
3Ch-
I: :;::.;
i....
h
30
20
10
10
20
30
' i ' ' i ' i i * ' ' In
40 5ff
0200 MST
68
-------
M I I » M I I I I M. I > > I I I M I 4 4 .1 * * .* I I I I I J I 4 I
WIND SPEED IM/S)
Q 10
ID
20
30
L
i
i '
i i
,
^~
.
t
I
t ,"'.. ' .
-c h-- - *
i
'
'
1
1-1 , ...
1 >- 1
t-
r
r
r
L.
r" ' ' '
-c i
i i i i iii
0 2
iii
- - : -7 ^
/
j.
"/
i I i i 1 i i i i"
0 3
0400 MST
69
^ j.
^
>
\< -: -'*
! !"! i ! i I.
0 4
........ ..., ,
"^ <
i i i i i i i i
0 5
-30
- 10
-------
10
20
4C>
I--
3Ct
r
L'.
f:
ich
10
10
10
~*^r ±\ ^«*
H | * '*'""'
yj * ' ' """
i ! i i i L/ t^ i» i* I* t* >* t* rr'*'-*
±3
<^
20
20
SBUTH
30
30
WIND SPEED IM/51
1 10
90
/:.
/
30
10
10
20
30
40
0600 MST
70
-------
10
WIND SPEED IM/S)
0 10
irc
or
1C
cc
i
i i i i i i i i
~
L ' ' '
1
.
' ' '
'
'
~
'
1
0 2
i i i i i i i i i
\
'
i i i ii , ,1 I
0 2
D 3
i i i i i i i i i
«-
S
*^V-
. , r«^.
i i 1 1 i i i i 1
0 3
0 4
111111111
. , yr.(* ', ,
0 4
a 5
"_ . . . if
*
t i i i i i ii
0 S
a0
TO
^r\
i n
,T
tf
0800 MST
71
-------
SBJTH
H1HD SPEED 1M/5)
0 ID
trc
ar
cc
1
.
.
L
1 1 1 1 , 1 1 \ 1
1
d 2
111111111
0 2
0 3
i i i i i i i i i
""» ' '"" ' ' ' '
: "*
\'
" ' :""'.::
.
f ...-
y .
'
1 1 1 i i i i , i
0 3
0 4
. . .*-.../.
7
I
^
.. ^^.*
<\ < ' tf^-
i i < i i i
0 4
3 5
1
' '
1
< < . i >
»
^
< f 1 ! -
0 S
ln
T
i?
1000 MST
72
-------
t » t f » M
10
10
SOUTH
20
30
30
WIND SPEED IM/S)
0 10
40
h-
:c|-
1C -
/.
:i:iii"in/..j:.
x-
30
10
10
70
30
40
1200 MST
73
-------
10
10
10
20
20
SBUTM
30
30
U1NO SPEED IM/SJ
0 10
40
'| 1 1 1 i ! 1 1 1 1
P
'
u - .
ta "
J_.
1
-
. *~
" ' ; : .:.: :*.
i
L , ,
-. .
- h""" TT r
r
i '
i - -
'
r
*~ " ' ' '
t- - - ...
h "
i
r' ~
-c i
i i i i i i i i i i
<-
' /
.
< »
I i i i i i i j i i
0 2
<*< t i ,-
\ \' \ \
0
. . , ...
**
.
/
i i i
-
*
/'
,
jr
'
1 1 1 1 1 1 1
0
1 I 1 1 1
,
^^
'
1 , I 1 1
40
-
*~
i i t i
i
X-
-
5
30
4-U
i n
ri
s
1400 MST
74
-------
SPEED IM/S)
10
1CC
tc
T
r
cc
i
i i i i .1 i i i
r
^
*~
L . .
t. . -. . _ _ _
-. :
.. . . .
1
0 2
1 1 1 T 1 1 1 1 1
/. .- _ J
....
1 ] 1 1 i , 1 1 1
0 2
0 3
i i i i i i i i i
*
< . i
" "
-»
i i i i i i j , i
0 3
0 40 5
:
.
i f - . . ,
' '
» :
'jf - -
*^ :
'
-» -^* :
V
«f
_ ',.>». >» . . .
; .
: '
:
'
:
;
ii , i , i , , , , ,
0 40 5
an
?n
8
1600 MST
75
-------
SOUTH
WIND SPEED (M/SJ
0 10
10
20
30
40
^
L .
'
r " -
3C
'
r
. .
r ( """ "' " '
r *
tC f * * * > » >
L . .. .
r
r
t :
r :
u
r
t ' ; ; ;
(-'''
r
t 1
X-...
0 2
\
»
-^
] ', ', I ', I ', : 1
0 3
i i i i i i i i i | i i i i i i i i i
:
.
V .
;
*
*
' : ^
: : '
0 40 5
*1U
o1
1800 MST
76
-------
10
20
WIND SPEED IH/S)
0 10
13
20
30
40
1C
i.
1 1 1 1 1 1 1 1
. . , _ _
-,,,.,. .. .
- < 1 1 1.
. 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1
<>
11-1) It
i i I ! , . i i i
0 2
/ <- -
- :
^
1 t 1 < | . i >
^^
i \ i 't \ i 7 j i "
a ^
2000 MST
11
<-< < > r
t T
:
i ' i _ . . ^
r . :
'. < f i * 4> .. .;.- i- < -<-, -ii . >
ttt»»i .>>> ji» »
: ^^
Jt '
\
! i i 1 i ! i i i ! i i i i ! i > . i
n 40 5
i n
,T
i?
-------
10
-« >»*»» »«.«.«.. » «.«-«.V
10
SBUTH
WIND SPEED IM/S1
3 10
10
20
30
40
,
i
i
i- i ,,(.,- -.., ,.| . i
. , r _ _ _ .
y
*** ' s
; ,
i
r ^
r
- i ...... .,....., --
,
; - ;
1 ;
- ' ' - ' ' ' ' '
C 10 20 3
< ' . . ,
1 '
.
/*
... .._._... ..._
«
t 0>^
1
. . -^T - . . _/ _
*** i
( , < , . , i. . - . .
. i . i I . < 1 t
0 40 5
*(j
2200 MST
78
-------
PART 3-7 MARCH 1977
79
-------
5BUTM
WIND SPEED IM/S)
0 10
cc l
L: : .: .:.:.. :..:..: :
t :
r
r~ " . .
t ' . '.. .:: .:..::': .:.
1
c .
r
|~
i
t"
" i__
1
*" - .
r
L -i j
r ' '
0 2
I 1 1 1 1 1 1 1 1
0
1
30
1 1 I 1 I 1 1 I 1 1 1 1
T :
T
1 :t
'1 ' : : : . i .:"
i: '
.
:
^*^Ss- T
\ : :
\ :
- :
:
|
T
40
1 1 ! | 1
*
;
'
*
:
< T
:
:
;
1
< i .
/. .
.
^
5
In
"»n
cU
_J.
1J
20
30
0000 MST
-------
"C 4/Ss
WIND SPEED (M/S)
0 10
10
20
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-------
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Dickerson, M. H. (1973), "A Mass-Consistent Wind Field Model for the
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108
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-79-066
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE ANDSUBTITLE
MODELING WIND DISTRIBUTIONS OVER COMPLEX TERRAIN
5. REPORT DATE
October 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Mark A. Yocke and Mei-Kao Liu
8. PERFORMING ORGANIZATION REPORT NO,
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Systems Applications, Incorporated
950 Northgate Drive
San Rafael, California 94903
10. PROGRAM ELEMENT NO.
1HE775
11. CONTRACT/GRANT NO.
68-03-2446
12. SPONSORING AGENCY NAME AND ADDRESS
U.S. Environmental Protection Agency-Las Vegas, NV
Office of Research and Development
Environmental Monitoring Systems Laboratory
Las Vegas, NV 89114
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
EPA/600/7
15. SUPPLEMENTARY NOTES
This report is the first in a series. For further information, contact
J.L. McElroy, Project Officer, (702) 736-2969 X241, Las Vegas, NV
16. ABSTRACT
Accurate determination of wind fields is a prerequisite for successful air
quality modeling. Thus, there is an increasing demand for objective techniques
for analyzing and predicting wind distribution, particularly over rugged terrain,
where the wind patterns are not only more complex, but also more difficult to
characterize experimentally. This report describes the development of a three-
dimensional wind model for rugged terrain based on mass continuity. The model
is composed of several horizontal layers of variable thicknesses. For each layer,;
Poisson equation is written with the wind convergence as the forcing function.
Many types of wind perturbations over rugged terrain are considered in this model,
including diversion of the flow due to topographical effects, modification of wind
profiles due to boundary layer frictional effects, convergence of the flow due to
urban heat island effects, and mountain and valley winds due to thermal effects.
Wind data collected during a comprehensive field measurement program at Phoenix,
Arizona, were used to test the model.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Mathematical models
Environmental models
Atmospheric models
Wind fields
Predicting wind distribut
Complex terrain
Three-dimensional wind mo
Wind perturbations
Testing mathematical mode
Phoenix, Arizona
ion
iel
L
143
55A
68A
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
21 NO. OF PAGES
122
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION is OBSOLETE
U S GOVERNMENT PRINTING OFFICE 1 979-683-091/22O1
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