EPA-600/4-76-039
July 1976
Environmental Monitoring Series
RADIATIVE EFFECTS OF POLLUTANTS ON
THE PLANETARY BOUNDARY LAYER
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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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 five series. These five 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 five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations. It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-76-039
July 1976
RADIATIVE EFFECTS OF POLLUTANTS
ON THE PLANETARY BOUNDARY LAYER
by
A. Venkatram and R. Viskanta
School of Mechanical Engineering
Purdue University
West Lafayette, Indiana 47907
Grant No. R803514
Project Officer
James T. Peterson
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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11
DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for
publication. Approval does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection
Agency, nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
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in
ABSTRACT
The objective of this study was to gain better understanding of
the effects of pollutants on the thermal structure and pollutant
dispersal in the planetary boundary layer. To this end numerical
models of the boundary layer were constructed. Gaseous pollutants
in the boundary layer were considered to absorb and emit thermal
radiation, while aerosols were allowed to absorb and scatter solar
energy. A series of numerical experiments were conducted for a
variety of summer atmospheric conditions, and the modification of
the temperature and concentration distributions by the radiatively
participating pollutants was investigated.
The first part of the investigation consisted of the construction
of a one-dimensional numerical model of the boundary layer. The model
incorporated the two-stream method for the computation of radiative
fluxes in the solar spectrum, and the turbulent kinetic energy model
to account for turbulence. A series of numerical experiments were
performed to determine the role of pollutants in modifying thermal
structure and pollutant dispersal in the boundary layer. The results
of the experiments showed that the predominant influence of gaseous
and particulate pollutants on the surface temperature was one of
warming. Under the conditions investigated the reduction of solar
flux at the surface was not accompanied by a decrease in the surface
temperature. The maximum predicted temperature rise was 2.8 C over a
45 hour period. Radiative participation by pollutants increased the
stability of the surface layer during the day. This led to a decrease
in the surface pollutant concentrations. During the night, the warmer
surface temperatures caused the surface layer to become less stable
which in turn increased surface pollutant concentrations. Elevated
layers of pollutants were found to play an important role in modifying
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IV
mixed layer growth. Solar participation generally hindered mixed
layer growth while thermal participation helped the vertical expansion
"of the mixed layer.
The second phase of the study involved the construction of a two-
dimensional numerical model to study the effects of pollutants on
urban-rural differences in thermal structure and pollutant dispersal.
Results of numerical experiments showed that radiative participation
by pollutants increased the maximum urban heat island intensity by as
much as 50%. Pollutants also induced temperature "crossover," the""
maximum of which was 0.3 C. The effects of pollutants on pollutant
dispersal were found to be significant. At the source height (100 m)
in the urban area, the pollutant concentration was reduced by as much
as 13.5% jjuring the night.
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CONTENTS
LIST OF ILLUSTRATIONS viii
LIST OF TABLES xii
LIST OF ABBREVIATIONS AND SYMBOLS xv
ACKNOWLEDGMENT xviii
I. CONCLUSIONS 1
1.1 Conclusions of One-Dimensional Simulations 1
1.2 Conclusions of Two-Dimensional Simulations 3
II. RECOMMENDATIONS 5
2.1 Suggested Improvements in Modeling 5
2.2 Suggested Additional Simulations 6
III. DISCUSSION OF PROBLEM 8
3.1 Problem Area 8
3.2 Background 10
3.2.1 The Atmospheric Boundary Layer 10
3.2.2 Urban Heat Island Phenomenon 12
3.2.3 Effect of Pollutants on Thermal Structure .... 13
3.3 Objectives of This Study 15
3.4 Rationale 16
3.5 Scope of This Study 18
IV. FORMULATION OF PROBLEM 20
4.1 Physical Model 20
4.2 Formulation of Model Equations 22
4.3 Boundary Conditions 25
4.4 Parameterization of Urban-Rural Differences 30
4.5 One-Dimensional Model 31
4.6 Stability and Numerical Accuracy of the 1-D
Momentum Equations 35
4.7 Numerical Scheme for Two-Dimensional Model 40
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VI
V. TURBULENCE MODELING 46
5.1 Introduction 46
5.2 The Eddy-Diffusivity Model 48
5.3 Atmospheric Turbulence Modeling 52
5.4 Kinetic Energy Turbulence Model 54
5.5 Mixing Length Model 59
5.6 The Equilibrium Layer 64
5.7 Turbulent Diffusivity for Two-Dimensional Model .... 65
VI. RADIATION MODEL 70
6.1 Introduction 70
6.2 The Radiative Transfer Equation 71
6.3 Radiative Transfer in the Solar Spectrum 76
6.4 Radiative Properties of Aerosols and Gaseous
Absorbers in the Solar Spectrum 77
6.5 Two-Stream Equations 81
6.6 Band Models 86
6.7 Infrared Radiation Properties of Gases 89
6.8 Radiative Fluxes in the Thermal Spectrum 93
VII. RESULTS AND DISCUSSION: ONE-DIMENSIONAL MODEL 99
7.1 Introduction 99
7.2 Test Simulation 99
7.2.1 Initial Conditions and Parameters
Used in Test Simulation 99
7.2.2 Results of Test Simulations 102
7.3 Radiative Effects of Pollutants 114
7.3.1 Introduction 114
7.3.2 Effect of Pollutants on Thermal
Structure of Boundary Layer 116
7.3.3 Effect of Pollutants on the Earth-
Boundary Layer System 126
7.3.4 Effect of Pollutants on Pollutant Dispersal ... 131
7.4 Effect of Elevated Layers of Pollutants 134
7.5 Effect of Changing the Height of the Elevated
Polluted Layer 152
7.6 Effect of Changing Aerosol Properties 169
7.7 Effect of Choice of Gaseous Pollutant 176
VIII. RESULTS AND DISCUSSION: TWO-DIMENSIONAL MODEL 177
8.1 Introduction 177
8.2 Initial Conditions and Surface Parameters 179
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8.3 Effect pf Pollutants on Thermal and Solar Fluxes .... 182
8.4 Effect of Radiative Participation on the Surface
Temperatures of the Urban-Rural System 188
8.5 Effect of Radiative Participation by Pollutants on
the Vertical Potential Temperatures Profile 196
8.6 Effect of Radiative Participation on the
"Crossover" Effect 202
8.7 the Urban Heat Island and the Effects of
Radiativ^ Participation 208
8.8 Pollutanjt Distributions in the Urban-Rural System . . . 214
8.9 Effect of Radiative Participation by Pollutants on
Pollutant Dispersal 220
REFERENCES . . . ii 229
!
APPENDIX A: Calculation of Directly Transmitted and Diffuse
SolarjRadiation at the Top of the Atmospheric
Layer! 242
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xiii
Table Page
7.11 Summary of Simulations with Elevated Layers 135
7.12 Comparison of Thermal Fluxes (in W/m2) at the Surface
(Elevated layer at 300 m) . 141
7.13 Comparison of Solar Fluxes (in W/m2) at Surface
(Elevated layer at 300 m) 142
: 7.14 Comparison of Potential Temperatures (in K) at 1 m
1 (Elevated layer at 300 m) 144
7.15 Comparison of Surface Temperatures (in K) of Simulation
with Non-Participating (NP) Pollutants and Simulation
with Solar Participation (SP) only (Elevated layer at
300 m) 145
7.16 Comparison of Temperatures (in K) at 1 m (Elevated
Layer at 600 m) 161
7.17 Comparison of Solar Fluxes (in W/m2) at Surface
(Elevated Layer at 600 m) 163
7.18 Comparison of Thermal Fluxes (in W/m2) at Surface
(Elevated Layer at 600 m) 164
7.19 Comparison of Aerosol Concentrations (in ug/m3) at 1 m
(Elevated Layer at 600 m) 165
7.20 Comparison of Solar Fluxes (in W/m2) and Temperature
(in K) at 1 m (Elevated Layer at 1200 m) 168
7.21 Summary of Simulations Performed to Study the Effect
of Aerosol Property Variation; rs = 0.2, H = 01 170
7.22 Comparison of Temperatures at 1 m to Study the Effect
of Aerosol Parameter Variation 171
7.23 Comparison of Solar Fluxes at Surface to Study the
Effect of Aerosol Parameter Variation 172
8.1 List of Two-Dimensional Simulations 179
8.2 Grid Spacing, Surface Parameters and Pollutant
Parameters 181
8.3a Comparison of Incident Thermal and Solar Fluxes at the
Surface (in W/m2) for Simulations P and NP at x = 16 km
(urban area) ..... 183
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Figure Page
7.11 Nighttime Potential Temperature Excess of Participating
Simulation over Non-Participating Simulation 128
7.12 Effect of Radiative Participation on Earth-Atmosphere
Albedo 130
7,13 Aerosol Concentration Profiles for Time 05:00 to 20:00 . . 132
7.14 Potential Temperature Profiles for Time 05:00 to 08:00 . . 136
7.15 Potential Temperature Profiles for Time 09:00 to 12:00 . . 137
7.16 Potential Temperature Profiles for Time 13:00 to 16:00 . . 138
7.17 Potential Temperature Profiles for Time 17:00 to 20:00 . . 139
7.18 Aerosol Concentration Profiles for Time 05:00 to 08:00 . . 147
7.19 Aerosol Concentration Profiles for Time 09:00 to 12:00 . . 148
7.20 Aerosol Concentration Profiles for Time 13:00 to 16:00 . . 149
7.21 Aerosol Concentration Profiles for Time 17:00 to 20:00 . . 150
7.22 Variation of Mixed Layer Height with Time Elevated
Pollutant Layer at 300 m 151
7.23 Potential Temperature Profiles for Time 05:00 to 08:00 . . 153
7.24 Potential Temperature Profiles for Time 09:00 to 12:00 . . 154
7.25 Potential Temperature Profiles for Time 13:00 to 16:00 . . 155
7.26 Potential Temperature Profiles for Time 17:00 to 20:00 . . 156
7.27 Aerosol Concentration Profiles for Time 05:00 to 08:00 . . 157
7.28 Aerosol Concentration Profiles for Time 09:00 to 12:00 . . 158
7.29 Aerosol Concentration Profiles for Time 13:00 to 16:00 . . 159
7.30 Aerosol Concentration Profiles for Time 17:00 to 20:00 . . 160
7.31 Variation of Mixed Layer Height with Time Elevated
Pollutant Layer at 600 m 167
7.32 Effect of Forward Scattering Factor on Earth-Boundary
Layer Albedo of Second Day 174
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Figure Page
7.33 Effect of Single Scattering Albedo on Earth-Boundary
Layer Albedo of Second Day ...........
8.1 Potential Temperature Isopleths at Time = 11:30 Mrs . . . 189
8.2 Potential Temperature Isopleths at Time = 15:30 Mrs . . . 190
8.3 Potential Temperature Isopleths at Time = 21:30 Hrs . . . 191
8.4 Potential Temperature Isopleths at Time = 01:30 Hrs . . . 192
8.5 Comparison of Surface Temperature Difference Between
Participating and Non-Participating Simulations at
Center of City ............... 193
8.6 Surface Temperature Variation at x = 0 (Rural Beginning) . 195
8.7 Difference Between Temperature Profiles at Center of City
(x = 16 Km) for Participating and Non-Patticipating
Simulations (Time = 11:30, 1st Day) ........ 197
8.8 Difference Between Temperature Profiles at Center of City
(x = 16 Km) for Participating and Non-Participating
Simulations (Time = 21:30, 1st Day) ........ 198
8.9 Difference Between Temperature Profiles at Center of City
(x = 16 Km) for Participating and Non-Participating
Simulations (Time = 03:30, 2nd Day) ........ 199
8.10 Difference Between Temperature Profiles at x = 0 for
Participating and Non-Participating Simulations
(Time = 11:30, 1st Day) ............ 200
8.11 Difference Between Temperature Profiles at x = 0 for
Participating and Non-Participating Simulations
(Time = 21:30, 1st Day) ............ 201
8.12 Difference Between Temperature Profiles of Urban Center
and Rural Beginning for Participating and Non-Participating
Simulations (Time = 11:30, 1st Day) ........ 205
8.13 Difference Between Temperature Profiles of Urban Center
and Rural Beginning for Participating and Non-Participating
Simulations (Time = 21:30, 1st Day) ........ 206
8.14 Difference Between Temperature Profiles of Urban Center
and Rural Beginning for Participating and Non-Participating
Simulations (Time = 03:30, 2nd Day) ........ 207
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Figure Page
8.15 Effect of Radiative Participation on the Urban Heat
Island Effect 210
8.16 Negative Heat Island Created by City 212
8.17 Effect of Surface Parameters on the Heat Island
Effect 213
8.18 Aerosol Vertical Concentration Profiles at Center
of City (x = 16 km) for Simulation NP 215
8.19 Aerosol Vertical Concentration Profiles at Downwind
of City (x = 24 km) for Simulation NP 216
8.20 Variation of Aerosol Concentration with Time at
x = 16 km for Simulation NP 218
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xn
LIST OF TABLES
Table Page
4.1 Comparison of Numerical Solutions of the Momentum
Equations with Exact Solution 39
7.1 List of One-Dimensional Numerical Simulations 100
7.2 Vertical Grid Coordinates and Physical Properties and
Parameters Used in Test Simulation 101
7.3 Summary of Surface Parameters and Pollutant Properties
Used in Simulations of Section 7.3 115
7.4 Comparison of Thermal and SoTar Fluxes at the Surface
for Simulations with Participating (P) and Non-
Participating (NP) Pollutants 117
7.5 Comparison of Potential Temperatures (in K) at 1 m and
100 m for Simulations with Participating (P) and Non-
Participating (NP) Pollutants 119
7.6 Comparison of Surface Temperatures (in K) and Diffusivi-
ties (in m2/s) at 10 m for Simulations with Participating
(P) and Non-Participating (NP) Pollutants 120
7.7 Comparison of Terms in Energy Equation for Simulations
with Participating (P) and Non-Participating (NP)
Pollutants, Time is 12:00 hours of Second Day 123
7.8 Comparison of Energy Fluxes (in W/m2) at the Surface
for Simulations with Participating (P) and Non-
Participating (NP) Pollutants 124
7.9 Comparison of Energy Fluxes (in W/m2) at the Surface
for Simulations with Participating (P) and Non-
Participating (NP) Pollutants 125
7.10 Comparison of Aerosol Concentration (in yg/m3) for
Simulations with Participating (P) and Non-
Participating (NP) Pollutants 133
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xm
Table Page
7.11 Summary of Simulations with Elevated Layers 135
7.12 Comparison of Thermal Fluxes (in W/m2) at the Surface
(Elevated layer at 300 m) 141
7.13 Comparison of Solar Fluxes (in W/m2) at Surface
(Elevated layer at 300 m) 142
7.14 Comparison of Potential Temperatures (in K) at 1 m
(Elevated layer at 300 m) 144
7.15 Comparison of Surface Temperatures (in K) of Simulation
with Non-Participating (NP) Pollutants and Simulation
with Solar Participation (SP) only (Elevated layer at
300 m) 145
7.16 Comparison of Temperatures (in K) at 1 m (Elevated
Layer at 600 m) 161
7.17 Comparison of Solar Fluxes (in W/m2) at Surface
(Elevated Layer at 600 m) 163
7.18 Comparison of Thermal Fluxes (in W/m2) at Surface
(Elevated Layer at 600 m) 164
7.19 Comparison of Aerosol Concentrations (in ug/m3) at 1 m
(Elevated Layer at 600 m) 165
7.20 Comparison of Solar Fluxes (in W/m2) and Temperature
(in K) at 1 m (Elevated Layer at 1200 m) 168
7.21 Summary of Simulations Performed to Study the Effect
of Aerosol Property Variation; rs = 0.2, H = 01 170
7.22 Comparison of Temperatures at 1 m to Study the Effect
of Aerosol Parameter Variation 171
7.23 Comparison of Solar Fluxes at Surface to Study the
Effect of Aerosol Parameter Variation 172
8.1 List of Two-Dimensional Simulations 179
8.2 Grid Spacing, Surface Parameters and Pollutant
Parameters 181
8.3a Comparison of Incident Thermal and Solar Fluxes at the
Surface (in W/m2) for Simulations P and NP at x = 16 km
(urban area) 183
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XIV
Table Page
8.3b Comparison of Total Radiation (Solar + Thermal) at
x = 16 km for Simulations P and NP; Total Radiation
in (W/m2) 183
8.4 Comparison of Incident Thermal and Solar Fluxes at
the Surface (in W/m2) for Simulations P and NP at
x = 0 km (Upwind Rural Area) 185
8.5 Comparison of Incident Thermal and Solar Fluxes at
the Surface (in W/m2) in Urban and Upwind Rural Areas
for Simulation P 186
8.6 Comparison of Aeroxol Concentrations (in yg/m3) at
z = 1 m and z = 100 m at x = 16 km for Simulations
P and NP 222
8.7 Comparison of Aerosol Concentrations (in yg/m3) at
z = 100 m at x = 24 km for Simulations P and NP . . . . 223
8.8 Comparison of Eddy Diffusivities (in m2/s) at z = 1 m
and z = 100 m for Simulations P and NP at x = 16 km . . . 225
8.9 Comparison of Eddy Diffusivities (in m2/s) at x = 16 km
for Simulations NP and P 227
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LIST OF ABBREVIATIONS AND SYMBOLS
A wide band absorptance
A fractional water vapor absorption
b backward scattering factor
C concentration
Cp specific heat at constant pressure
cv specific heat at constant volume
Efc blackbody emitted flux defined as
En exponential integral of order n defined as
En(t) = f y"~2 exp(-t/y)dy
F radiative flux (solar or thermal) in +z or -z direction
F total radiative flux in +z direction
f Coreolis parameter or forward scattering factor
g acceleration due to gravity
I intensity of radiation
I. Planck's function
h thickness of equilibrium layer
H Hal stead moisture parameter or maximum model height
H* height of boundary layer in two-dimensional model
Hm height of boundary layer in one-dimensional model
H anthropogenic heat production rate
K turbulent eddy diffusivity
k thermal conductivity or turbulent kinetic energy
L horizontal extent of urban-rural system
LH latent heat of evaporation
L Monin-Obukov length
MC surface pollutant source
p pressure
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XVI
>y) scattering phase function
P.B.L. planetary boundary layer
p Legendre coefficient
X/
P Legendre polynomial of order a
X/
Ri gradient Richardson number, Eq. (5.3.1)
r reflectance
r(y'-ni) bidirectional reflectance
$cn volumetric pollutant source strength
T thermodynamic temperature
t time
u horizontal velocity along x-axis
u* surface friction velocity
v horizontal velocity along y-axis
w vertical velocity along z-axis
x horizontal coordinate or distance along urban-rural system
y horizontal coordinate perpendicular to x
y. water vapor path length
z vertical coordinate
z0 roughness length
Greek Letters
a thermal diffusivity,
3 extinction coefficient (K + a)
YC counter-gradient heat flux parameter
e emissivity, emittance
9 potential temperature or zenith angle
ic absorption coefficient or von Karman constant
A wavelength
y cosine of zenith angle of a beam of radiation
yfl cosine of solar zenith angle
v frequency
p density
a scattering coefficient or Stefan-Boltzmann constant
T shear stress or optical depth
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xv ii
TQ optical thickness of P.B.L.
<{> azimuthal angle
CD albedo for single scattering,a/3
Subscripts
aer aerosol
Cn species
v frequency
g geostrophic
H heat
i index denoting node along x-axis
j index denoting node along y-axis
k index denoting node along z-axis
M momentum
o surface value
s solar or soil
t thermal
w water vapor
Superscripts
+ positive z direction
negative z direction
1 turbulent fluctuation
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xvm
ACKNOWLEDGMENT
This research was supported by the Meteorology and Assessment
Division, Environmental Protection Agency, under Environmental Protec-
tion Agency Grant Nos. R801102 and R803514. Computer facilities were
made available by Purdue University Computing Center and the National
Center for Atmospheric Research which is supported by the National
Science Foundation.
The support of the project by the Environmental Protection
Agency and the help provided by Dr. James T. Peterson, The Grant
Project Officer, is acknowledged with sincere thanks.
Except for minor changes, this report constitutes the doctoral
dissertation of Mr. A. Venkatram which was submitted to the Faculty
of Purdue University in partial fulfillment of the degree of
Doctor of Philosophy.
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I. CONCLUSIONS
1.1 Conclusions of One-Dimensional Simulations
On the basis of the numerical simulations performed and the results
obtained using the one-dimensional transport model it is concluded that:
1. The turbulence model was capable of reproducing important
features of mixed layer formation. The counter-gradient heat
flux parameter was necessary to model the collapse of turbu-
lence during the night. The results of a test simulation
using the turbulence model compared favorably with the O'Neill
observations.
2. Pollutant aerosols reduced the solar flux at the surface by
15% on an average. Thermal participation by gaseous pollutants
lead to an increase of the downward thermal radiation at the
surface by about 10% at night. The magnitudes of the pollutant
caused changes in solar and thermal fluxes at the surface were
in good agreement with observations (Rouse, et al., 1973; Oke
and Fuggle, 1972).
3. Absorption of solar radiation by aerosols increased the tem-
perature of the boundary layer by as much as 0.5 C during
a day. The absorption and emission of thermal radiation by
gaseous pollutants lead to warming of the surface layer of
the boundary layer, and cooling at higher levels.
4. Radiative participation by aerosols and gaseous pollutants
lead to an increase in the surface temperature. The maximum
temperature increase was 0.35 C during the day and 2.8 C
during the night. The reduction of solar flux at the
surface decreased the surface temperature by a maximum of
0.31 C during the first 8 hours of simulation. However, no
temperature decrease was observed on the second day of
simulation.
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5. Solar energy absorption by aerosols increased the stability
of the surface layer. This led to a decrease of surface
pollutant concentrations during the day by a maximum of 1%.
The significantly warmer surface temperatures during the
night decreased the stability of the surface layer causing
higher surface pollutant concentrations. The maximum increase
was 18%. In presenting these conclusions it is important to
reiterate that the pollutant source was elevated at 100 m.
6. Elevated layers of pollutants played a significant role in
modifying mixed layer growth. Solar heating led to the
formation of sharp inversions which hindered the vertical
expansion of the mixed layer. Cooling induced by gaseous
pollutants helped the growth of the mixed layer.
7. The height of the elevated pollutant layer was an important
factor in determining the final height of the mixed layer.
When the pollutant layer was placed at 300 m the mixed layer
was able to penetrate the stable layer created by solar
heating. However, when the pollutant layer was located at
600 m the inversion strength became large enough to prevent
the mixed layer from penetrating the inversion.
8. In the process of creating a sharp inversion solar heating
also led to the formation of an unstable layer above the
inversion. Thus, when the mixed layer penetrated the
inversion its growth thereafter was relatively rapid in
the unstable layer above the inversion.
9. Elevated pollutant layers had a significant effect on
pollutant dispersal as they affected the growth of the mixed
layer. Dispersion was enhanced when the mixed layer height
was large. On the other hand, a small mixed layer thickness
was conducive to the buildup of pollutants near the surface.
10. Aerosol parameters were important in determining the effective
albedo of the earth-boundary layer system. In general, an
increase in single scattering albedo and backscattering
factor led to an increase of the effective reflectance. The
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effective albedo was most sensitive to the aerosol back-
scattering factor.
11. In most of the simulations, aerosols decreased the effective
reflectance of the earth-boundary layer system at small
zenith angles, and increased it at large zenith angles.
Thus, aerosols had a warming influence in the hours around
noon, and a net cooling effect around sunset and sunrise.
1.2 Conclusions of Two-Dimensional Simulations
On the basis of the preliminary results of the two-dimensional
simulations it is concluded that:
1. The predominant effect of radiative participation by
pollutants on the surface temperature is one of warming
both in the rural as well as in the urban areas. The
maximum pollutant induced temperature increase in the
urban center was 1.86 C. When only aerosols were allowed
to participate radiatively the reduction of solar flux at
the surface caused a maximum temperature decrease of 0.3 C.
2. Radiative participation increased the nighttime urban heat
island by a maximum of 50% in the early morning hours.
Solar participation decreased the urban temperature excess
during the day, while thermal participation increased the
heat island intensity throughout the day.
3. Radiative participation by pollutants was an important
contributing factor to the "crossover" effect. During the
night, when no temperature crossover occurred in the simula-
tion with non-participating pollutants, radiative cooling
in the simulation with participating pollutants induced a
temperature crossover of 0.3 C.
4. The predominant effect of radiative participation by
pollutants on pollutant dispersal was that of decreasing
pollutant concentrations in the boundary layer. In the
urban area (x = 16 km) the maximum pollutant induced
reduction occurred during the night and was 13.5% at the
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source height and 8.2% at the surface. The rural decrease
(x = 24 km) was 5.5% at 100 m and 7.5% at the surface.
It is necessary to emphasize that the magnitudes of the
pollutant induced concentration changes are largely
dependent on the particular turbulence model used in the
study. Definitive conclusions on the effects of pollutants
on pollutant dispersal cannot be drawn without examining
various turbulence models.
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II. RECOMMENDATIONS
2.1 Suggested Improvements in Modeling
This investigation involved the modeling of a large number of
physical processes. Although every attempt was made to model the
physical processes as realistically as possible, the wide scope
of this study gave rise to a number of suggestions for future
research. Some of the problem areas which should receive attention
are:
1. An improved mixing length model is needed. The self-similar
profile used in this study does not account for the effect
of local stability on the scale of turbulent eddies. This is
unrealistic, and an improved procedure to calculate the
mixing length is needed.
2. The solar radiation transfer model should be improved to
account for the selective absorption by gases and aerosols.
As a starting point the solar spectrum can be divided into
two regions taking advantage of the fact that water vapor
absorbs mainly in the region, 1-4 urn. This division of
the solar spectrum should be kept to the minimum to prevent
computational requirements from becoming excessive.
3. A procedure should be incorporated into the thermal radiation
model to account for the absorption and emission of a com-
bination of pollutant gases. A realistic pollutant model
should include the most important pollutants found in the
atmosphere. The incorporation of such a model necessarily
involves a procedure to account for the overlap between the
pollutant gas bands. As narrow band models are time
consuming, it is necessary to modify wide band models to
account for overlap.
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4. Radiative transfer in the polluted urban atmosphere is at
least two-dimensional and very complicated. Some simple
approximate yet mathematically and numerically tractable
analysis must be developed to account for two-dimensional
effects.
5. A simple model must be developed to handle the transport
of water in the soil layer. The currently available
(Eagleson, 1970) soil transport models involve considerable
data handling and cannot be readily incorporated into a
numerical model. A soil transport model in combination with
a physically realistic procedure to account for evapotrans-
piration at the air-soil interface is needed because latent
transport may be a significant fraction of the net energy
transfer at the earth's surface.
6. An improved procedure to solve the two-dimensional conserva-
tion equations must be developed. Possible improvements
are the use of a staggered grid system and the incorporation
of a conservative Arakawa scheme. This might solve the
problems encountered in the attempt to incorporate the
turbulent kinetic energy model into the two-dimensional
numerical model.
7. As the location of pollutant sources determines whether
radiative participation by pollutants increases or decreases
surface pollutant concentrations it is necessary to develop
a realistic pollutant emission model. The model should
account for the space (elevated or surface based) as well
as time (source strengths and time variation) distributions
of pollutant sources.
2.2 Suggested Additional Simulations
An analysis of the results of the numerical simulations showed
that a greater understanding of the problem could be obtained by
performing a number of additional simulations. Specifically, the
following numerical studies should be conducted:
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1. Sensitivity studies should focus attention on the role of
surface parameters in determining the effect of radiative
participation on the surface temperature. The results of
this study indicate that the effect of the reduction of
solar flux at the surface on the surface temperature is
dependent on the surface evaporation rate. Other studies
(Mitchell, 1971; Wang and Domoto, 1974) also show that
radiative effects are related to the surface albedo and
the surface moisture. It is clear that additional simula-
tions would provide valuable insight into these effects.
2. Numerical experiments must be conducted to investigate the
effect of various outflow boundary conditions (downstream)
on the temperature, velocity and species distributions.
3. Simulations must be performed to determine the relative
importance of urban-rural surface parameters in producing
the heat island and the crossover effect.
4. Simulations must be performed to examine the role of the
turbulence model in determining pollutant induced changes
of pollutant dispersal. This can be accomplished by per-
forming a series of simulations with identical initial
conditions but with different turbulence models. Some of
the suggested turbulence models are surface layer similarity
formulations (Businger, 1973), Richardson number correlations
(Pandolfo, et al., 1971), turbulent kinetic energy models
(Launder and Spalding, 1974), second-order models (Donaldson,
1973), and simplified sub-grid scale model (Orlanski, et al.,
1974). As atmospheric turbulence is still incompletely
understood, only conclusions based on results of simulations
which examine the effect of the choice of the turbulence
model will find acceptance.
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III. DISCUSSION OF PROBLEM
3.1 Problem Area
The rapid increase in industrialization and urbanization over
the past few decades has been accompanied by relatively significant
changes in the climate in and around urban locales. This inadvertent
climate modification has been evidenced by observations (Chandler,
1965; Summers, 1966; Oke and East, 1971; Oke, 1972) of urban-rural
differences of various aspects of climate such as temperature, humidity,
visiblity, radiation, wind and precipitation. One of the better known
features of the urban environment is the so-called heat island effect
which is associated with the existence of warmer surface temperatures
in the urban regions than in the surrounding rural environs. The
heat island effect is also related to the differences in the thermal
structures between urban and rural areas. Recent observations
(Bornstein, 1968; Clarke, 1969) show that while fewer surface inver-
sions occur over cities than over rural areas, weak elevated inver-
sions occur at a greater frequency over urban areas. It is generally
accepted that the heat island is primarily caused by the differences
in the urban-rural surface characteristics such as thermal properties,
surface moisture, surface albedo, roughness, and heat sources. Uhile
these differences can be used to explain thermal effects at the surface,
they cannot be directly related to the formation of elevated inversions.
It has been suggested (Bornstein, 1968) that these inversions could be
caused by the radiative effects of pollutant layers. Recent theoreti-
cal studies by Atwater (1970) and Bergstrom (1972) in addition to
lending some credence to the theory show that pollutants can contribute
significantly to the heat island effect.
A qualitative explanation of the process of weather modification
by air pollutants is fairly simple. Particulates in the atmosphere
absorb and scatter the incoming solar radiation leading to a reduction
-------
in the solar energy reaching the earth and thus a cooling. However,
Mitchell (1971) has recently shown that while stratospheric aerosols
do lead to a cooling, slightly-absorbing aerosols in the troposphere
may serve to increase surface temperatures. Other pollutants, such as
C02, which are relatively opaque to long-wave thermal radiation, can
produce a "greenhouse" effect leading to an increase in surface
temperatures. This simple explanation of the effect of anthropogenic
pollutants on the climate is clearly more relevant to the global scale
than to the small scales of an urban area. The microclimate of an
urban area is determined by the energetics and the dynamics of the
planetary boundary layer, the evolution of which is largely con-
trolled by turbulence. Thus, estimates of the effects of the
pollutants on the climate have to be based on studies which account
for the interaction of turbulence and radiation, both of which are
extremely complex phenomena. There have been relatively few studies
on the radiative effect of pollutants on a local scale (Atwater, 1970,
1972, 1974; Pandolfo et al., 1971; Zdunkowski and Mcquage, 1972;
Bergstrom and Viskanta, 1973), but the results indicate that the
effects can be substantial within a period of a few days. However,
the models used in these studies were relatively crude, and the
results obtained can at best be considered tentative. Considerably
more research effort is needed before short and long-term effects
of pollutants can be estimated.
Thermal modification by pollutants is just one of the many
effects of urbanization on the atmospheric structure of the city and
the rural surroundings. Although observational data can provide
information on the urban microclimate, it cannot be utilized in deter-
mining the causative factors of the atmospheric structure of the urban
area. An understanding of the various effects of urbanization can be
obtained only by isolating possible causative factors, and studying
their effects by varying them one at a time. Such a procedure is not
practical from the point of view of an observational program as it is
not possible to manipulate the urban environment. One technique
which can be employed to help fill this observational gap is numerical
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10
simulation. A numerical model describes the dynamics and thermo-
dynamics of a city in terms of differential equations (conservation
laws) and boundary conditions. It is through these boundary condi-
tions that city parameters such as tall buildings, heat production
by the city and pollutant emission are described. The main advantage
of numerical simulation lies in its ability to predict the effect of
changes of initial and boundary conditions on the planetary boundary
layer variables. Thus, a numerical model can serve as an invaluable
tool in determining the effects of urban changes such as urban renewal,
park development, industrial siting and freeway location. Numerical
simulation can also provide guidance for observational programs by
determining the relative importance of atmospheric variables. This
information can help to minimize redundancy in the field program.
Observational programs, in turn, can provide invaluable feedback for
improvement of the numerical model.
3.2 Background
3.2.1 The Atmospheric Boundary Layer
As the winds created by the rotation of the earth blow over the
surface of the globe, they maintain a layer of "frictional influence,"
or atmospheric boundary layer. The boundary layer is turbulent, and
extends to heights of the order of one kilometer. The large scale
flow or the free atmosphere receives much of its heat and virtually
all of its water vapor through the turbulent transfer processes in
the boundary layer. The turbulent activity in the boundary layer is
also responsible for the transfer of kinetic energy from the atmos-
phere to the boundary layer. Since, as much as one-half of the
atmosphere's loss of kinetic energy occurs in the boundary layer,
the turbulent motion in the boundary layer has an appreciable influ-
ence on the evolution of weather systems (Tennekes, 1974).
As man lives inside the boundary layer, his interaction with
the transport processes of the boundary layer determines the quality
of his daily environment. Pollutant and heat wastes injected into
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11
the boundary layer are dispersed vertically by turbulent mixing and
horizontally by the wind. Thus, the primary variables of micro-
meteorology are the wind velocity vector and the boundary layer height
which essentially determines the extent of vertical mixing.
The boundary-layer thickness varies from about a few tens of
meters on calm nights to about two thousands meters on sunny summer
afternoons (Tennekes, 1974). This variation in the boundary layer
thickness is caused primarily by the effects of vertical heat transfer
on turbulent motion. During the day, upward heat transfer enhances
turbulent activity which is responsible for the entrainment of the
stable nonturbulent layer which caps the boundary layer. During the
night, the downward heat transfer caused by the cooling of the earth's
surface, suppresses turbulence and the boundary layer rapidly
decreases in height. In relatively rare situations with negligible
heat transfer the boundary layer thickness is determined by the shear
stress at the surface and the local rate of the earth's rotation.
During the day, the thermally enhanced turbulent mixing of the
boundary layer is so effective that the vertical distribution of
boundary layer variables such as potential temperature, moisture,
pollutants and wind is more or less uniform through most of the
boundary layer. For this reason, meteorologists refer to the boundary
layer as the "mixed" layer. The distinctive structure of the mixed
layer has allowed the construction of simple models (Leahey and
Friend, 1971; Tennekes, 1973; Carson, 1973) to predict the variation
of the mixed layer height during the day. Considering their simplicity,
these models have had remarkable success (Tennekes and Ulden, 1974;
Leahey and Friend, 1971) in forecasting mixed layer heights. However,
these forecasts are necessarily short-term (12 hours), and consider-
ably more sophisticated models are required to predict the diurnal
variation of the boundary layer.
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12
3.2.2 Urban Heat Island Phenomenon
One of the better known features of urban climatology is the
existence of higher temperatures in the urban area than in the rural
surroundings. This phenomenon, known as the urban heat island effect,
has been documented for over a century (Howard, 1833). Howard's
measurements showed that the urban temperature excess was greatest
at night, when it amounted to 2.0 C. During the day, the heat island
effect was found to be negative with the city being 0.15 C cooler
than the surrounding countryside. But on the average, the city was
found to be warmer than the countryside by about 1 C. These results
are similar to those of more modern investigations (Chandler, 1962).
Several mechanisms have been suggested to explain the formation
of the heat island. Howard (1833) attributed the urban temperature
excess to anthropogenic heat production. Recent observations (SMIC,
1971; Bornstein, 1968) show that the anthropogenic heat production
term may dominate the other energy sources during the winter months,
and may be as large as 2% times the solar input. Bornstein (1968)
and Chandler (1961) have also emphasized the importance of reduced
evaporation rates in urban areas. They argue that as less energy is
required for evaporation, there is excess solar energy to heat up
the urban surface. It is also believed (Mitchell, 1961) that the
higher thermal conductivities and heat capacities of paving materials
and buildings of the city, allow more energy to be stored in the
"surface" layer of the city than in that of the countryside during
the day. This additional stored energy which is released during the
night creates the excess in the urban temperature. The smaller urban
albedos (Craig and Lowry, 1972) can contribute to the daytime heat
island effect. During the day, the higher eddy diffusivities (Bowne
and Ball, 1970) associated with the rough surface (tall buildings) of
the city tend to counteract heat island causative factors. However,
the higher levels of turbulence can lead to a reduction of the city's
ventilation by "breaking" the wind. Kratzer (1956) and more recently
Fuggle and Oke (1968) have attributed the heat island to the blanket-
ing effect of atmospheric pollutants. It is argued that layers of
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13
pollutants absorb radiation during the day, and re-emit thermal
radiation during the night. This increased downward thermal radia-
tion results in the observed nocturnal temperature excess.
From the preceding discussion, it is clear that the urban heat
island is the result of a complex set of interacting physical
processes. Over recent years, a better understanding of the heat
island phenomena has been obtained by supporting observations with
numerical modeling studies such as those of Myrup (1969), Tag (1969),
McElroy (1971), Pandolfo (1971), and Nappo (1972). As the physical
processes involved in the formation of the heat island are exceedingly
complex and incompletely understood, these models are necessarily
simple. However, they have provided valuable insight into the
relative significance of the factors responsible for the urban
temperature excess.
The magnitude of the heat island effect is important from the
point of view of practical applications such as the design of human
comfort systems and urban planning. Thus, the need for "quick"
estimates of the urban-rural temperature difference has given rise
to a number of empirical studies which attempt to correlate the
urban temperature excess with urban parameters such as population,
wind speed, cloud cover, etc. Some of the more successful correla-
tions are due to Ludwig (1970), and Oke (1972, 1973).
3.2.3 Effect of Pollutants on Thermal Structure
Observations show that there are distinct differences between the
vertical temperature profile of the city and that of the surrounding
rural area. The tower measurements of Munn and Stewart (1965) made
over southern Ontario indicated higher percentages of inversion-free
nights at the urban sites. Bornstein (1968) studying New York's
thermal structure, found that while the rural sites exhibited a
greater frequency of surface inversions, a greater number of elevated
inversions occurred over the city. He also observed that while city
temperatures were generally higher than those over the countryside
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14
up to heights around 300 meters, a temperature crossover effect
occurred around 400 meters where temperatures were generally lower
than those of the surrounding area. This finding is similar to those
of Duckworth and Sandberg (1954). More recently, Rouse, et al.
(1973) observed sharp elevated inversions over a polluted industrial
area in Hamilton, Ontario. These inversions were strongly developed
in the morning, weakened in the hours around noon, and strengthened
towards the evening.
The described features of the urban thermal structure are
believed to be caused by the radiative participation of pollutants
(Sheppard, 1958; Atwater, 1970; Bergstrom and Viskanta, 1972). It
is argued that the increased nocturnal thermal radiation associated
with the cooling of pollutant layers, heats up the earth's surface
thus preventing the formation of a surface radiation inversion. On
the other hand, the cooling of elevated pollutant layers may be
responsible for the temperature "crossover" effect and the formation
of elevated inversions.
There have been relatively few theoretical investigations on the
effect of pollutants on the thermal structure. Using a one-dimensional
radiative-conductive model Atwater (1970) showed that the cooling due
to layers of pollutants could cause elevated inversions. Bergstrom
and Viskanta (1972) studied the effect of pollutants using a one-
dimensional model which accounted for turbulence effects through semi-
empirical eddy diffusivity correlations. Their results indicated that
pollutants could be responsible for the destruction of the nocturnal
surface inversion. They also showed the formation of elevated
inversions due to pollutant induced cooling. Utilizing a boundary
layer model developed by Pandolfo (1971), Atwater has investigated
the thermal changes induced by pollutants. He concluded from his
results that the major effect of pollutants was to cause the delay of
the onset of unstable conditions after sunrise.
Most one-dimensional simulation (Atwater, 1970; Bergstrom and
Viskanta, 1972; Zdunkowski and McQuage, 1972) studies show that
pollutants may contribute significantly to the heat island effect.
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15
However, more realistic investigations based on two-dimensional
models seem to downplay the importance of pollutants as a causatory
factor of the urban temperature excess. Atwater's (1974) results
show that pollutants are only a minor factor in the formation of the
urban heat island. The more recent two-dimensional study of Viskanta,
et al. (1975) shows that under certain conditions radiative participa-
tion can in fact reduce the urban-rural temperature difference. This
result is significant in that it contradicts other studies. From the
preceding discussion it is evident that considerably more research is
required before a satisfactory understanding of the effects of
pollutants on the thermal structure is obtained.
3.3 Objectives of this Study
The primary objective of the study is to enhance understanding of
the effects of pollutants on the urban environment. Specifically, the
research program proposed to determine the role of pollutants in modify-
ing the thermal structure which, in turn, affects pollutant dispersal.
Particular attention will be paid to the interaction of pollutants with
the stable layer. To this end, the net effect of the most important
pollutants on radiative flux and its divergence, temperature, concen-
trations, and flow fields will be predicted.
The specific aims of this study are:
1) Construction of a physically realistic radiative transfer model
in the urban boundary layer in which gaseous and particulate
pollutants are present.
2) Construction of a turbulence model capable of representing
turbulent transport effects over a wide range of atmospheric
conditions.
3) Development of a one-dimensional transport model for simulating
the thermal structure and dispersion in an urban atmosphere.
4) Simulation of a variety of atmospheric situations using the one-
dimensional transport model to study the effect of pollutants on
the urban climate.
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it)
5) Development of an unsteady two-dimensional transport model to
study the effect of pollutants on urban-rural climatic differences.
Some of the questions this study will address itself to are:
1) Under what meteorological conditions is radiative transfer
important enough to compete with turbulent transport?
2) What should the pollutant concentration be in order to affect the
thermal structure and pollutant dispersion?
3) How do the radiatively participating pollutants affect the height
of the mixed layer during the day?
4) Does the variation of the pollutant concentration with height
(i.e., sharp concentration gradients) contribute to the formation
of an elevated stable layer?
5) How do pollutants affect the stability of the inversion? Do they
stabilize the inversion or do they accelerate the breakup of the
inversion?
6) What role do pollutants play during the formation of surface
inversions at night?
These questions relate to the effect of pollutants on the thermal
structure and pollutant dispersal. In addition to these questions, the
research program hopes to shed some understanding about the structure
of the mixed layer and its interaction with the inversion.
3.4 Rationale
After presenting the objectives of this study it is necessary
to answer the following question to justify the research effort:
Apart from the academic interest in the effects of pollutants, what is
the practical importance of the slight modification of thermal structure
the radiative participation of pollutants may be able to cause? The
key to answering this question can be found in the nature of atmospheric
turbulence. The planetary boundary layer is turbulent, and except for
a shallow layer next to the earth's surface, the turbulence in the
boundary layer is thermally controlled. During the day, the negative
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17
potential temperature gradients in the atmosphere are responsible for
the production of turbulent kinetic energy, and during the night the
slight positive potential temperature gradients inhibit turbulence.
These potential temperature gradients which are so effective in
promoting or suppressing turbulent activity are typically of the
order of a few degrees per kilometer. In view of this, it is easy
to see the significance of the small changes in temperature pollutants
can produce. These temperature changes become all the more important
when pollutant effects are localized, i.e. temperature changes over
small vertical distances producing large potential temperature
gradients.
With the background provided by the discussion of the preceding
paragraph it is easy to see the importance of the possible thermal
effects of pollutants in the practical tasks of (1) forecasting of
pollution episodes, (2) calculation of pollutant dispersion, (3)
prediction of thermal structure of a city. The turbulent planetary
boundary layer serves as a conduit through which the city flushes out
its pollutant and heat wastes. Since the atmospheric variables such
as temperature, wind speed and pollutant distribution are nearly
uniform (mixed) inside the boundary layer, fairly accurate estimates
of the pollutant concentration and the temperature can be obtained
from a knowledge of the wind velocity, boundary layer height, and the
sources of heat and pollutant emissions (Tennekes, 1974). The variables
which can be possibly affected by pollutants are the wind and the
boundary layer height. Pollutants may modify the wind by altering
the turbulent intensity. Clearly this effect will be important during
the night when turbulent transport does not overwhelmingly dominate
radiative transfer. Pollutant induced changes of a few degrees at the
earth's surface could also change the stability of the surface layer
quite drastically. As turbulent activity during the night is confined
to the surface layer, this alteration of stability could affect the
intensity of vertical mixing and thus pollutant dispersal. During
daytime, the vertical expansion of the boundary layer is limited by
a capping inversion. Clearly, the rate of growth of the boundary
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18
layer is dependent on the stability of the inversion. As there is no
turbulent activity inside the inversion, pollutant induced heating or
cooling can cause dramatic changes in the stability of the inversion.
Thus, elevated layers of pollutants inside the inversion can play a
significant role in controlling boundary layer height, one of the most
important variables of micrometeorology (Tennekes, 1974).
It is clear that the radiative participation of pollutants can
affect thermal structure and pollutant dispersal through the feedback
mechanisms described in the preceding paragraph. It is desirable
therefore to understand these feedback mechanisms and obtain estimates
of pollutant caused thermal changes. If numerical simulations show
that the radiative interaction of gaseous and particulate pollutants
with solar and atmospheric radiation is important, the inclusion of
radiative transfer in air quality simulation models (AWSM) may become
mandatory.
3.5 Scope of this Study
This section discusses the improvements made by this study over
previous investigations of the effect of radiative participation by
pollutants on local climate. The specific contributions of this
study are listed below:
1) A relatively sophisticated turbulence model was developed and
incorporated into the dynamic numerical model. All previous
studies (Atwater, 1970, 1972, 1975; Pandolfo et al., 1971;
Zdunkowski and McQuage, 1972; Bergstrom and Viskanta, 1973;
Viskanta, Bergstrom and Johnson, 1975) of the radiative effects
of pollutants have utilized relatively crude turbulence models.
As the thermal structure of the boundary layer is determined by
the interaction between radiation and turbulence, it is unrealistic
to emphasize the role of radiation at the cost of that of turbu-
lence. Bergstrom and Viskanta (1972) and Viskanta, Bergstrom
and Johnson (1975) have used sophisticated radiation models in
combination with relatively simple turbulence models to study
changes induced by pollutants. In view of the importance of
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19
turbulence, it is clear that the results of such studies have to
be viewed with a degree of skepticism. Even partial understanding
of the effects of pollutants can only be obtained by using radia-
tion and turbulence models which are representative of the relative
importance of the two physical processes.
2) A simple solar radiation model based on the two-flux method was
developed and incorporated into the numerical model. The radiation
model although relatively crude compared to that of Bergstrom and
Viskanta (1972) affords substantial savings in computer time
without introducing unacceptable idealizations. The model accounts
for absorption and scattering of solar radiation in a consistent
manner, and it represents an improvement over the radiation model
developed by Atwater (1971). Atwater's treatment of scattering
can at best be considered "fairly approximate." Furthermore, his
assumption of extinction coefficients being independent of
pollutant concentrations ignores the importance of the distribu-
tion of pollutants in determining the heating rates. The model
used in this study trades a degree of sophistication for con-
siderable flexibility in the variation of aerosol parameters
such as forward scattering factor and single scattering albedo.
3) As the turbulence model was able to reproduce important features
of mixed layer formation, it was possible to study the radiative
effects of pollutants on the growth of the mixed layer. A number
of simulations were performed to determine the role of elevated
layers of pollutants in modifying mixed layer growth. No such
studies appear to have been conducted to date.
4) The simulations have investigated the effects of radiative
participation by pollutants on the effective albedo of the earth-
boundary layer system. The effective albedo is an important param-
eter which determines the total energy absorbed in the system.
Workers (Atwater, 1970, 1972, 1974; Bergstrom and Viskanta, 1973;
Pandolfo, et al., 1971) who have coupled the dynamics of the
boundary layer with the influences of air pollutants have not
studied this important effect.
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20
IV. FORMULATION OF PROBLEM
4.1 Physical Model
A schematic diagram of the earth-atmosphere system being considered
in this study is shown in Figure 4.1. The physical model consists of
four layers: (1) the free atmosphere where the atmospheric variables
are determined by the large scale flow [as the simulation time scale
of the model is relatively small (48 hours), the free atmospheric
variables can be assumed to be unaffected by the flow in the boundary
layer]; (2) the "polluted" atmosphere in which the atmospheric variables
such as velocities, temperature and pollutant concentrations are
functions of space and time, and change in response to varying boundary
conditions primarily at the lower boundary (earth-air interface) and at
the upper boundary (free atmosphere); (3) the soil layer in which the
temperature is a function of depth and time only; and (4) the litho-
sphere in which the temperature is constant over the time scale of
simulation.
The urban parameters such as anthropogenic heat sources, surface
albedo and emittance, thermal diffusivity and conductivity of the soil,
roughness height, and surface and/or elevated pollutant sources are
prescribed functions of the horizontal distance along the urban-rural
complex. The variation in topography is not accounted for, i.e.,
except for the roughness variation the terrain is assumed to be uniform.
The forcing function of the model is the time dependent solar
irradiation. Incoming short-wave solar radiation heats up the earth's
surface during the daytime; a part of this radiation is absorbed and
scattered in the polluted atmosphere. The solar energy absorbed at
the air-soil interface is partly transferred to the atmosphere and
the soil by conduction. Another part of the energy is used for
evaporating water, and the remaining energy is reradiated as longwave
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FREE
ATMOSPHERE
V \ SOLAR RADIATION xTT>v SOLAR RADIATION
XV COLLIMATED A* f 'A DIFFUSE
MODEL
HEIGHT
POLLUTANTS AND
HEAT INJECTED
POTENTIAL
TEMPERATURE
LITHOSPHERE
Figure 4.1 Schematic of Earth-Atmosphere System
CONSTANT TEMPERATURE
REGION
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22
thermal radiation. The distribution of energy in the atmosphere is
accomplished through convection, radiation and turbulent diffusion.
The atmosphere also emits thermal radiation part of which is absorbed
at the air-soil interface. During the night, the earth's surface
cools in the absence of solar irradiation. Thus, the soil-atmosphere
system goes through a diurnal cycle of heating during the day, and
cooling during the night. These variations in air temperature and
thermal stratification, in turn, affect the velocity, moisture and
pollutant concentration fields through changes in intensity of turbu-
lent mixing.
4.2 Formulation of Model Equations
In deriving the conservation equations the following assumptions
are made:
1) The transport processes are two-dimensional, i.e., all the
variables are functions only of the vertical co-ordinate z and one
horizontal co-ordinate x, and are independent of the co-ordinate y.
Clearly, the assumption of two-dimensionality is not very realistic
in view of the fact that turbulence is a three dimensional phenomenon.
However, this model does not attempt to resolve turbulent motion;
turbulence effects are parameterized with the turbulent diffusivity.
Thus, this assumption depends on whether the urban-rural system can
be considered to be one-dimensional, i.e., the urban-rural character-
istics vary only along the x-co-ordinate. It is clear that realistic
city parameters would vary in two dimensions giving rise to three-
dimensional flow effects. Although it is not clear how important
three-dimensional circulation patterns are, it is felt that the two-
dimensional model is capable of providing understanding of the most
important urban effects. The inclusion of the third dimension in
addition to increasing the complexity of the problem would have
increased the computer time as well as storage by a few orders of
magnitude.
2) Atmosphere is hydrostatic. This approximation has been used
in several heat island studies (Estoque, 1963; Estoque and Bhumralkar,
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23
1969) and is valid when the horizontal scales are much larger than
the vertical scales. It allows the calculation of the pressure
distribution from a knowledge of the temperature distribution.
3) The Boussinesq approximations are applicable to the boundary
layer. The approximations can be summarized by the statements:
1) Density fluctuations result principally from thermal effects
rather than pressure effects, and 2) density variations in the mass
and momentum conservation equations may be neglected except when they
are coupled to the gravitational acceleration in the buoyancy force.
Spiegel and Veronis (1960) have shown that the Boussinesq approxima-
tions are valid when the fluid layer under consideration is confined
to a thickness which is much smaller than the minimum of the scale
heights of the atmospheric variables. Furthermore, it is necessary
that the motion-induced fluctuations do not exceed, in order of
magnitude, the static variations of the variables. The conditions
mentioned are satisfied in the atmospheric boundary layer. An
important consequence of the approximations is that the continuity
equation reduces to the incompressible form. This together with the
hydrostatic assumption, allows the calculation of the vertical
velocity from the continuity equation.
4) The viscous dissipation term in the energy balance can be
neglected. It can be easily shown (Spiegel and Veronis, 1960) that
this assumption is valid at the relatively low velocities found in
the atmosphere.
5) In applying the equation of radiative transfer to the
planetary boundary layer, the atmosphere is considered to be
horizontally homogeneous—the radiation model is one-dimensional.
As an accurate solution of the one-dimensional transfer equation
requires substantial computer time, it would be impractical and
presently unwarranted in terms of computer time and storage to use
a two-dimensional radiation model.
6) Energy transfer in the soil layer is one-dimensional. As
the horizontal temperature gradients are much smaller than the
vertical temperature gradients, conduction in the horizontal direction
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24
can be neglected in comparison to that in the vertical direction; the
energy equation reduces to the one-dimensional diffusion equation.
With the preceding assumptions the governing equations can be
written as (Estoque, 1973)
Planetary Boundary Layer
x-momentum:
y-momentum:
3v , 3v - 1 3p . 1 y . 9 L 3v ,n ,
3x~+w 3?- -fu - p plif + 35T Kx ax (4'2'
+|?=0 (4.2.ic)
Continuity:
3x ' 3z
Energy:
3tf+U3x"+W3z~= 3T + 3x~ [Kx 3x~J + Se '4'2
Water Vapor:
*\ f* t\ f* ^O f\ /*
O%/ r\ f 0 ^ "\ f\
W 3z 3z~ [^1 "3~r"j + 3x"
8C,, sc,
IT + " IS
Pollutant Concentration (Species):
8Cn 9Cn 9Cn ^n
Q. j. ,, D. j. u, 0. - Q. x i/ _ x c
u + w + K + s;
at u Civ " a-7 —^ T T3" "X, ^„ T Jr
3t 3x 3z 3z 3x [ x 3x j Cn
n = 1, 2, ..., N (4.2.If)
Hydrostatic Equilibrium (z-momentum):
|f = - P9 (4.2.1g)
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25
Equation of state:
p = pRT (4.2.1H)
Soil Layer
Energy equation:
3T 32T
5 - s (4.2.11)
3t "S
In Eqs. (4.2.la) and (4.2.1b) T and T are the turbulent momentum
x y
fluxes, FQ and Fc are the turbulent heat and pollutant fluxes,
respectively, SQ and Sc are the internal sources of heat and
pollutants, and Cw is the volumetric rate of water vapor generation.
The turbulent fluxes and the source term in the energy equation are
given by,
TX = pKM 3u/3z (4.2.2a)
T,, = pKM 3v/3z (4.2.2b)
= -KH(36/3Z - YC) (4.2. 2c)
=-Kc 3Cn/3z (4.2.2d)
S =-(Po/P)+H (4.2. 2e)
In Eq. (4.2.2c), yc is a small positive quantity which accounts for
the countergradient heat flow often observed in the atmosphere. On
the basis of Tel ford and Warner's observations (1966), Deardorff
recommends a value of 0.7 x 10~3 K/m for y .
\*
4.3 Boundary Conditions
The boundary conditions are specified at the top and bottom of
the atmospheric layer under consideration. At this point it is
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26
necessary to distinguish between the top of the atmospheric layer and
the top of the boundary layer. The top of the atmospheric layer is
the height of the last vertical grid point which is placed around 2 km.
The top of the planetary boundary layer, on the other hand, is a
function of space and time, and varies from a minimum of about 200
meters to a maximum of about 1500 meters. The definition of the
boundary layer height will be discussed in following sections. It
is appropriate to mention here that the top of the atmospheric layer
serves as a maximum bound on the height of the boundary layer.
At the earth's surface (z = z0) the boundary conditions are:
u = v = w = 0 (4.3.la)
T = T0(x,t) (4.3.1b)
Cw = Cwo(x,t) (4.3.1c)
Cn = Cno(x,t) (4.3.Id)
The ground temperature T0(x,t) is calculated from an energy balance
at the earth's surface,
F! - F2 + F3 - F, - F5 - F6 = 0 (4.3.2)
where
F! - thermal (longwave) emission from the ground
F2 - atmospheric eddy transfer of sensible and latent heat
F3 - soil heat flux
Flt - atmospheric thermal radiation flux due to water vapor,
carbon dioxide, and pollutant gases and aerosols
F5 - incident solar radiation flux absorbed at the surface
F6 - anthropogenic heat flux
-------
27
The expressions for the fluxes can be written as
Fj = etaTj (4.3.3a)
(4.3.3b)
F. = -k S
9T_
(4.3.3c)
o
3 "S 82
F,, = etF~ (0) (4.3.3d)
Fs = (1 - rs)'F-(0) ' (4.3.3e)
The surface water vapor concentration is calculated by postulating
that the ratio of the actual evaporation rate to the potential evapora-
tion rate is a constant (Halstead, et al . , 1957). Then, assuming that
the water vapor potential causing evaporation can be represented by
the difference in water vapor concentrations between the surface and
the first grid point one can write
H = [Cw(Zl) - Cwo]/[Cw(zJ - Cwsat] (3.3.5)
where Cw(z1) is the water vapor concentration at the first grid point,
and C . is the water vapor concentration at the surface for saturation
conditions. The ratio of the actual evaporate rate to the potential
evaporation rate, H, is called the Halstead moisture parameter and
ranges from unity for evaporation over water to zero for dry soil.
It should be pointed out that the modeling of evaporation by way
of the Halstead moisture parameter is at best a very rough approxima-
tion to the actual physical process of evapotranspiration. This
approach completely ignores the important role the soil plays in
controlling the rate of evaporation; it assumes that the soil offers
no resistance to the water which is transported through it and which
eventually evaporates at the soil-air interface. Also, the Halstead
moisture parameter approximation does not allow for the drying out
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28
of the soil. This model for evaporation was used only because other
simple models are not significantly better. Manabe's (1969) model
assumes that there is a finite amount of water in the soil. This
water is depleted as evaporation takes place at a rate proportional
to the amount of water left in the soil. Although this approach
allows for the drying out of the soil, it also ignores soil resistance.
Although more detailed models for predicting evaporation from the
earth's surface are available (Philips, 1957), their applicability to
the problem under consideration is limited because the heterogeneity
of the urban soil layer does not allow calculation of "average" soil
properties such as permeability, moisture potential, etc.
The surface pollutant concentration is given by specifying the
pollutant source Mcn as follows
Mf = ~Kr 3Cn/3z at z = 0 . (4.3.6)
n t, n
Elevated pollutant sources are modeled by source terms SQ in
the pollutant species equations.
At the bottom of the soil layer the temperature is taken to be
a constant
Ts(x,z,t) = constant at z = -ZA (4.3.7)
The boundary conditions at the top of the atmospheric layer (z = H)
are specified by requiring the values of the atmospheric variables to
remain constant, i.e.,
(4.3.8a)
(4.3.8b)
e = eg (4.3.80
cw = cwg (4.3.8d)
cn = cng (4.3.8e)
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29
Boundary conditions (4.3.8) are consistent with the notion the
variables u , v , 6 , Cwg and Cng associated with the large scale
weather system do not vary significantly over the time scale of
simulation.
In the free atmosphere the flow is assumed to be geostrophic
so that the pressure gradients at z = H can be replaced by
3x
H
= Pfvg (4.3.9a)
= -pfun (4.3.9b)
H g
As the boundary layer does not extend to heights greater than
2000 m, the geostrophic balance at z = H is not affected by boundary
layer turbulence.
Since [8p/3x]H is a constant, the pressure along x at z = H is
given by
p(x,H) = p(0,H) + HF x (4.3.10)
LdxJH
where p(0,H) is specified.
The lateral boundary conditions were assumed to be periodic in
the x-direction, i.e., the atmospheric variables have the same values
at the beginning and the end of the urban-rural system. Physically,
the boundary condition means that at distances far enough from the
urban area the atmospheric variables attain values which are not
significantly different from those at large distances upwind of the
urban complex. Clearly, the formulation of the periodic boundary
condition assumes that there are sinks of energy and pollutants between
the end of the urban area and the end of the rural area.
The imposition of periodic boundary conditions forces pollutant
concentrations to build up over the urban rural system. Although
this situation might be unrealistic in general, it represents
stagnating conditions during which pollution episodes occur. As
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30
the earth-atmosphere interface can absorb momentum as well as
energy, the velocity and temperature yields are not expected to
be affected by the conservation effects periodic boundary condi-
tions tend to introduce. Thus, the numerical scheme accounts for
advection effects on the velocity and temperature fields and at the
same time avoids the difficulties associated with inflow and outflow
boundary conditions (Roache, 1972). It should be mentioned that
Orlanski, et al. (1974) have studied heat island formation using a
model with periodic boundary conditions.
The upstream variables are calculated from the one-dimensional
conservation equations. Then, the downstream variables are updated
by
^d (L.z) = i|>u(0,z) (4.3.11)
where ij^ and $u represent the downstream and upstream variables
respectively, and x = L denotes the end of the urban-rural system.
The specification of the initial conditions will be discussed
in a later section. With suitable initial conditions and the boundary
conditions described in the preceding paragraphs, the formulation of
the problem will be complete when the eddy diffusivities and the
divergence of the radiative flux are known. The computation of eddy
diffusivities and the radiative fluxes will be discussed in detail
in chapters describing the turbulence and the radiation models used
in this study.
4.4 Parameterization of Urban-Rural Differences
From the point of view of the numerical model it is necessary to
describe the difference between the urban and rural areas in terms
of parameters. The roughness height is used to parameterize the
variation of the height of structures in the urban-rural system. The
tall buildings in the urban area are associated with relatively large
values of roughness length. Since the roughness length serves as the
mixing length for the calculation of the turbulent kinetic energy at
the surface, a larger roughness length would predict a larger generation
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31
of turbulent kinetic energy. This increased turbulent kinetic energy
would be reflected as larger diffusivities which would be responsible
for the slowing of the wind velocities over the urban area.
The dryness of the urban area relative to that of the rural area
is modeled in terms of smaller values of the Hal stead moisture param-
eter. As there were no studies to guide the selection of the moisture
parameters, values were fixed by numerical experimentation.
Although cities are constructed with materials which are lighter
(Bray, et al., 1966) in the solar spectrum than the foliage and soil
of the surrounding countryside, the trapping effect of buildings can
lead to an overall lower solar albedo for the urban area (Craig and
Lowry, 1972). This study assumed that the albedo of the urban area
was lower than that of the rural area.
As the urban area is the main source of pollutants and heat, it
was assumed that the heat and pollutant sources were distributed only
in the urban area. The pollutant sources were elevated and their
strength was allowed to vary sinusoidally, increasing during the day,
and decreasing after nightfall.
The values and the distribution of the urban-rural parameters
can be found in Chapter VIII.
4.5 One-Dimensional Model
An examination of the objectives of this investigation showed
that it was possible to achieve some of them with a one-dimensional
model. As the major emphasis of the one-dimensional simulations was
on the radiative effects of pollutants, there was some justification
in assuming that the atmosphere is horizontally homogeneous.
Pollutants are most likely to have the greatest effect on the thermal
structure through radiative participation when convective motions are
small--stagnating high pressure centers with light winds. Thus, the
simulations performed with a horizontally homogeneous (1-D) model
would represent situations in which pollutants would play a significant
role in determining the thermal structure.
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32
The highly nonlinear nature of the governing differential equa-
tions raised the possibility of errors being introduced by the
numerical scheme used to solve them. It is well known (Roache, 1972)
that the presence of the nonlinear advective terms in the two-
dimensional equations can lead to serious difficulties in the con-
struction of a satisfactory numerical method. Even if the method is
stable, there is always the possibility of numerical errors becoming
as large as physical effects. On the other hand, one-dimensional
numerical schemes are relatively simple, and can be "checked out"
with no difficulty. Thus, it was felt that the relatively error-free
one-dimensional numerical method was better suited to investigate
radiative effects whose magnitudes are usually very small.
In this section, the one-dimensional model equations will be
presented, and the special numerical techniques used in their solution
will be described.
The one-dimensional model equations are:
x-momentum:
8u 3 \v 9ul . cl x //i r -, \
8t= 3l(KM^j + f2)
Energy:
Species (Pollutants and Water vapor):
+ § • n = lf 2} ••" N (4>5'4)
The rest of the equations and the boundary conditions remain unchanged
from the two-dimensional model.
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33
Equations (4.2.1) and (4.2.2) can be combined into one equation
by defining a complex variable w*
w* = (u - u ) + i(v - v ) (4.5.5)
Multiplying Eq. (3.5.2) by i, and adding Eq. (3.5.2) one obtains a
single equation for w*
(4-5-6)
At first glance, it seems that the introduction of w* would
simplify the computational process considerably. However, it should
be noticed that the numerical solution of w* involves complex
arithmetic; the mathematical operations required are not much
different from that needed in the solution of the two Eqs. (4.5.1)
and (4.5.2). The main advantage of defining w* is the programming
simplicity it affords. Furthermore, it will be shown later that the
implicit finite difference equation for the linearized form of Eq.
(4.5.6) is unconditionally stable.
Further, programming facility can be obtained by combining the
aerosol and gas concentrations into a single complex variable. The
special form of the boundary conditions for the potential temperature
and the water vapor concentration did not allow a similar conversion
into a complex variable.
The finite difference approximations employed in the numerical
solution of the one-dimensional model equations can be illustrated by
considering the following equation
_
9t 3z 3z
where represents the variables of the model, and S, denotes terms
containing first order derivatives. Then, the finite difference
approximations for the terms in Eq. (3.5.7) can be written as
-------
34
If
.
9
3?
az
.n+l ,n+l
- z.J
(4.5.8a)
(4.5.8b)
In the above approximations j denotes the j-th grid point counted
vertically upwards, and n denotes n-th time step. Thus, 4>. represents
the value of the variable $ at the n-th time step and located at the
j-th vertical grid point. Further
and
K, =
K S
Kj>/2
(4.5.9a)
(4.5.9b)
The first derivative terms in 4> are represented as
where
(4.5.10)
a, =
32 =
(4.5.lib)
It is noted that the finite difference scheme is first order
accurate in time, and second order in space. A second order accurate
scheme in time was initially used but abandoned when it was found
that the solution oscillated. It can be easily shown (Roache, 1972)
-------
35
that the finite difference method used in this study is uncondi-
tionally stable when the eddy diffusivity is a constant. However,
it should be stated that no definite conclusions about the stability
of the system can be drawn when the eddy diffusivity is an arbitrary
function of space and time. The presence of nonlinear source terms
in the diffusion equation can also complicate convergence and
stability studies.
The application of the finite difference approximations at the
grid points results in a tridiagonal matrix which can be very
efficiently solved using the Thomas algorithm (Rosenberg, 1969).
4.6 Stability and Numerical Accuracy of the 1-D
Momentum Equations
An examination of the momentum equations (4.5.1) and (4.5.2)
shows that a finite difference representation of the equations would
contain the source terms, f(v - v ) and -f(u - Ug). As these source
terms have to be considered to remain constant over the time step of
integration, it is not possible to ensure stability for large time
steps. One solution to this problem of stability is to introduce
the complex variable w* as has been done in Eq. (4.5.6). It will be
theoretically shown in this section that the finite difference
representation of Eq. (4.5.6) is unconditionally stable. Also, as
a check on the stability and the convergence of the numerical scheme,
the classical Ekman layer solution will be compared against the
steady state numerical solution of the finite difference equation
for w*- This examination of the momentum equations will help to
validate the numerical schemes used in the solution of the other
model equations which are all similar in form to the momentum
equations.
The linearized form of Eq. (4.5.6) can be written as
(4.6.1)
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36
The insertion of the finite difference approximations of Section 4.1
into Eq. (4.2.1) results in
wn+1 - wn + ifw"+1At =
J J J
(4.6.2)
The stars (*) have been dropped for convenience, and a and b are
given by
(4.6.3a)
b E 2K/[(Zj - Zj_1)(z - z-.)] (4.6.3b)
Rearranging Eq. (3.2.2) one obtains
v/J+1(l + if At + aAt + bAt) - aAtw^J - bAtw^? = v/J (4.6.4)
J J""" J- J ~ •*• J
For simplicity it is assumed that the distances between the grid
points are equal. The stability analysis for unequal grid spacing
involves a great deal of algebra; but it can be proved that the system
is still unconditionally stable. The results of a numerical experi-
ment will lend additional justification to this statement.
Decomposing w into a Fourier series, one can write each mode of
the series as
_n+i ik JAz
= wn+1 e z (4.6.5a)
= w e (4.6.5b)
ik7(j-l)Az
n ~n ikzAz
= wn e (4.6. 5d)
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37
where Az is the vertical grid spacing, and w and wn are the complex
amplitudes corresponding to the wavenumber k at the (n+1) and
n time step respectively.
Substituting Eqs. (4.6.5) into Eq. (4.6.4) one obtains
wn+1[(l + ifAt + 2aAt) - aAt(ei6 + e"ie)] = wn (4.6.6)
where
6 E ikzAz (4.6.7)
wn+1/wn = i/[i + ifAt + 2aAt(l - cose)] (4.6.8)
Then,
The amp! i cation factor G is given by
G = 1/[1 + if At + 2aAt(l - cose)] (4.6.9)
Then, to prove that the system is unconditionally stable, it is
required to show that the amplitude of G is less than or equal to
unity for all values of 6. It is easily seen that this condition
is satisfied, as the minimum value (modulus) of the denominator of
the amplification factor is greater than unity.
As a final check on the stability of the system, a numerical
experiment was conducted. The analytical solution of the following
problem was used as a basis for comparison against the numerical
solution of Eq. (4.5.6).
The differential equation under consideration was
(4.6.6)
and the boundary conditions were chosen to be
w* = -(u + iv ); z=0 (4.6.11a)
s7 3
w* = 0 ; z = h (4. 6. lib)
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38
It is easy to see that in terms of the u and v velocities, boundary
conditions (4.6.11a) and (4.6.lib) reduce to
u = v = 0; z = 0 (4.6.lie)
u = u , v = v ; z = h (4.6.lid)
The solution of Eq. (4.5.6) with B.C's (4.6.11a) and (4.6.lib) is
straightforward and is given by
u - ug = |(-ug cosaz + vg sinaz)e"a(z+2h)
+ (u cosa(2h - z) - v sina(2h - z))e"a(z+2h)
j j
+ (u cosa(z - 2h) - v sina(z - 2h))ea(z~2h)
+y j
+ (-u cosaz - v sinaz)e"az|/D (4.6.12a)
v - vg = j(-ug sinaz - Vg cosaz)e~a(4h"z)
+ (u sina(2h-z) + v cosa(2h-z))e"a(z+2h)
j j
+ (u sina(z - 2h) + v cosa(z - 2h))ea(z"2h)
+ (ug sinaz - vg cosaz)e"azl/D (4.6.12b)
where
D E e"4ah - -2e"2ah cos2ah + 1 (4.6.13a)
a=/f/2K (4.6.13b)
The solution has been expressed in terms of u and v to facilitate
comparison with the numerical solution.
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39
Table 4.1 Comparison of Numerical Solutions of the Momentum Equations
with Exact Solution
Vertical
Height
z(m)
0
1.0
5.0
10.0
20.0
30.0
40.0
50.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
1100.0
1200.0
1300.0
1400.0
1500.0
1600.0
1700.0
1800.0
1900.0
2000.0
2100.0
2200.0
Exact
u
0
.00349
.01762
.03562
.07271
.11125
.15118
.19248
.41822
.95241
1.57119
2.24740
2.95785
3.68309
4.40706
5.11688
Solution
V
0
.01903
.09493
.18929
.37629
.56099
.74341
.92355
1.79043
3.35880
4.71573
5.87357
6.84579
7.64657
8.29039
8.79171
5.80248 9.16470
6.45640
7.07342
7.65041
8.18599
8.68035
9.13500
9.55261
9.93682
10.29205
10.62335
10.93629
11.23678
11.53100
9.42307
9.57989
9.64749
9.63741
9.56036
9.42623
9.24412
9.02238
8.76872
8.49023
8.19356
7.88498
7.57050
Numerical
At =
u
0
.00346
.01744
.03526
.07199
.11017
.14975
.19069
.41535
.95135
1.57158
2.24890
2.96017
3.68597
4.41028
5.12026
5.80588
6.45970
7.07655
7.65330
8.18859
8.68264
9.13698
9.55429
9.93819
10.29313
10.62415
10.93681
11.23704
11.53100
Solution
150 s
V
0
.01905
.09500
.18942
.37654
.56137
.74391
.92418
1.79150
3.35940
4.71572
5.87294
6.84432
7.64438
8.28755
8.78829
9.16083
9.41886
9.57546
9.64298
9.63294
9.55606
9.42221
9.24047
9.01921
8.76608
8.48820
8.19218
7.88428
7.57050
Numerical
At =
U
0
.00346
.01744
.03526
.07199
.11017
.14975
.19069
.41535
.95135
1.57158
2.24890
2.96017
3.68597
4.41028
5.12026
5.80588
6.45970
7.07655
7.65330
8.18859
8.68264
9.13698
9.55429
9.93819
10.29313
10.62415
10.93681
11.23704
11.53100
Solution
300 s
V
0
.01905
.09500
.18942
.37654
.56137
.74391
.92418
1.79150
3.35940
4.71572
5.87284
6.84432
7.64438
8.28755
8.78829
9.16083
9.41886
9.57546
9.64298
9.63294
9.55606
9.42221
9.24047
9.01921
8.76608
8.48820
8.19218
7.88428
7.57050
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40
The parameters chosen for the numerical experiment were,
f = 10"" sec"1; K = 50 m2/sec; h = 2200 m;
u = 11.531 m/sec; v = 7.5705 m/sec (4.6.14)
The velocities were initialized with arbitrary logarithmic
profiles, and the transient numerical solution was allowed to approach
steady state. Three time steps were used: At = 75 s, 150 s and
300 s. All the three solutions converged to the same steady state
solution (to the fifth decimal place) within fifty interations. The
results of the numerical experiments are presented in Table 4.1. In
addition to showing the stability of the system (nonlinear grid
spacing) the results also reveal the accuracy of the numerical scheme.
It is seen that the numerical scheme is accurate to within 1.1 percent
of the analytical solution. The maximum error occurs at the third
grid point (5 m) where the small velocities and the relatively large
gradients are most likely to give rise to errors. It is felt that
this error is well within the acceptable limits for this type of
study in which trends rather than numerically precise results are
more significant.
4.7 Numerical Scheme for Two-Dimensional Model
The two-dimensional model equations can be represented by the
following general form:
_ _ K
Dt ~ 3t 8x 3z 9z fz 3z 3x [x 3x
where cf> represents the model variables.
As the accuracy of the results of a numerical investigation are
determined by the numerical method used, the selection of a suitable
numerical technique is very important. A number of numerical methods
for solving equations of the type presented above are available in
the literature (Roache, 1972; Rosenberg, 1969). These methods can be
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41
classified into two broad categories, implicit and explicit. Explicit
methods are "one step" methods in which a variable is advanced to a
new time level using only information from the previous time step.
Thus, the updating of a variable involves the solution of a linear
algebraic equation with one unknown, and the computational requirement
is very small. Explicit methods can be made very accurate, and at the
same time they can be designed to obey important conservation laws
(Roache, 1972). It is clear that the utility of a numerical method
is limited if it does not conserve quantities like kinetic energy
and momentum. From a numerical viewpoint, it is important to use a
conservative method to avoid nonlinear advective instability (Lilly,
1965). In order to take advantage of these desirable features of
explicit methods, certain stringent restrictions have to be placed
on the time steps and the grid spacing to keep the solution stable.
These restrictions are functions of the nonlinear coefficients in
the differential equations, and although in principal it is possible,
it is not always practical to use time steps and grid spacing deter-
mined by the stability condition of the particular explicit numerical
method. In addition to being variable (as they are dependent on the
time varying coefficients), time steps can become so small that the
total number of computational steps required to integrate a variable
over a time interval can become large enough to make computer time
requirements prohibitive.
Implicit schemes approximate the advective and the diffusion
terms in the conservation equations in terms of the variables at the
new step. Thus, it is necessary to solve a system of algebraic
equations to obtain the values of the variables at the new time step.
It is clear that implicit methods are computationally more complex
than explicit methods; however, most implicit schemes are uncondi-
tionally stable and thus for a given grid size it is possible to use
time steps which are large enough to make the total computer time
smaller than that required for an explicit scheme. It should be
mentioned that implicit schemes are not conservative, and considerably
more numerical complexity is required to obtain the degree of accuracy
of an explicit scheme.
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42
Experimentation with several explicit methods showed that it was
impractical from the point of view of computer time to use the time
steps required by the stability conditions. Thus, implicit techniques
were examined, and the method selected is based on the time-splitting
method due to Marchuk (1965). The familiar alternating direction
finite-difference method (Roache, 1972) was not used because the
explicit differencing of the terms during each half time step gave
rise to oscillations with large time steps. On the other hand, the
implicit differencing of the time-split equations kept the solution
stable for relatively large time intervals. Essentially, time-
splitting consists of "splitting" the integration of a two-dimensional
(or three-dimensional) differential equation into one-dimensional
steps. Then, the steps involved in the integration of Eq. (4.4.1)
are represented by
(4-7-2a)
+ " • " * «+) <4-7-2b)
Equation (4.7.2a) is solved first, and the solution obtained serves
as an input to Eq. (4.7.2b). The result of the integration of Eqs.
(4.7.2a) and (4.7.2b) is the final solution; no meaning is attached
to the intermediate solution. The validity of these operations is
intuitively clear; theoretical justification can be found in the
paper by Marchuk (1965).
It is noticed that Eqs. (4.4.2a) and (4.4.2b) can be solved
using implicit finite-difference schemes. The finite-difference
representation of Eq. (4.4.2a) can be written as
n+1
(4.7.3)
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43
where ? . represents the value of $ located at (x., z., t ), and
i >3 i J ii
<4-7-4a)
"21 E ("1-l.J + "1 /2 (4-7'4c)
The finite-difference analog for the diffusion term, ^— [(Kx
r\
can be written in the same manner as that for the term ^— [(K,
oZ Z
which has already been described before.
The method used to finite-difference the advective term u
was motivated by a conservative explicit scheme for the advection
equation. It is noticed that the finite-difference method is second
order accurate in space, and first order in time.
With the exception of the term w 8/3z, the finite-difference
analogs of the terms of Eq. (4.7.2b) have already been discussed in
Section (4.5). The advective term w 3<)>/8z can be written as
(4.7.5)
where
f _ „ n \
3 (4.7.6a)
and
• s < VM'/K'J+I - zj-i»zj+i - zj» <4-7-6b)
b E (Z - 2J/[(z - ZJtZ - I.)] (4.7.6C)
It should be noted that the scheme given by Eq. (4.7.5) is second
order accurate in space.
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44
The set of finite-difference equations for Eqs. (4.7.2a) can
be represented by tridiagonal matrices which can very efficiently be
solved using the Thomas Algorithm (Rosenberg, 1969).
The finite-difference approximations used to calculate the
vertical velocity and the pressure are
w.
= w-
where
l - (u, ,
dXJ • • I ,J
' »J
0/Ax
- Zj)/2 (4.7.7)
(4.7.8a)
(4'7'8b)
and
R/C
R/CD / ir
9po.i72cp)K.j
cD/R
P
(4.7.9)
It is noted from Eq. (4.7.7) that the vertical velocity is computed
by integrating the continuity equation in the direction of increasing z
(increasing j). The condition that the vertical velocity is zero at
the air-soil interface translates into
w. , = 0;
1 J-1-
i = 1, 2,
N
(4.7.10)
where N is the number of grid points along the x-direction.
The pressure is computed by integrating the hydrostatic equation
in the direction of decreasing z. In Eq. (4.7.9) pn . is the pressure
u, i
at the top of the atmospheric layer corresponding to the location x..
Equidistant grid-spacing was used in the x-direction. As the
vertical gradients of the atmospheric variables are larger near the
air-soil interface thar
those at greater heights, it was necessary
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45
to use a non-linear grid-spacing in the vertical (z) direction to
provide good resolution; the grid-spacing was small near the inter-
face and became larger away from the ground. The details of the
grid system can be found in Chapter VII.
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46
V. TURBULENCE MODELING
5.1 Introduction
The most difficult problem associated with the numerical modeling
of the planetary boundary layer is finding a satisfactory method to
simulate the effects of turbulence. Over the past few years, there
has been considerable interest in atmospheric turbulence, and the
active research in the field has produced a multitude of turbulence
models. The state-of-the-art has been excellently surveyed in the
papers by Reynolds (1974a), Mellor and associates (1973), and Lumley
and Khajeh-Nouri (1974). The currently available turbulence models
are described in the following paragraphs.
Eddy-diffusivity or mixing length models express the trans-
port of a variable as the product of the gradient of the variable and
an eddy diffusivity. This model which was proposed independently by
G. I. Taylor and L. Prandtl is based upon kinetic theory concepts, and
draws an analogy between the irregular motion of molecules in a fluid
and the less uncorrelated motion of turbulent "eddies." Then, the
eddy diffusivity is expressed as the product of a velocity characteriz-
ing the turbulent motion and a mixing length which corresponds to the
mean free path in kinetic theory. Although the serious shortcomings
of the kinetic theory analogy have been recognized, mixing length
models have found widespread use mainly because of their simplicity
and their surprising success in treating a wide variety of turbulent
flows. It should be pointed out that in flows characterized by a single
length scale, the mathematical formulation (not the physical model) of
the mixing length model can even attain a certain degree of respect-
ability. This reinterpretation of the model under certain conditions
will be discussed in more detail in a later section.
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47
Higher order models bypass the concept of eddy diffusivity
and attempt to calculate the second moments of turbulent fluctuations
directly. Thus, there is no necessity to express the second moments
of fluctuating quantities (Reynolds stresses, etc.) in terms of the
mean field. Basically, the models attempt to formulate differential
equations for the moments. These differential equations, in addition
to containing the required moments contain moments of higher order.
It is possible to construct differential equations for each of the
higher order moments, but an attempt to continue this process of
formulating differential equations will result in an infinite set of
equations which cannot be solved. Thus, it is necessary to "truncate"
the differential equations at a certain stage and express the higher
order moments as functions of moments of lesser degree. For example,
the third order moments occurring in the equations for the Reynolds
stresses would be functions of second moments like the Reynolds
stresses, heat flux, etc. Thus, the classical closure problem
consists of constructing "suitable" functions for the higher order
moments. These functions (functionals to be exact) have to obey
certain physical and mathematical constraints. In fact, most of the
present research on higher order models (Lumley and Khajeh-Nouri,
1974; Reynolds, 1974b) has concentrated on methods to systematically
construct these functionals.
Most of the working higher order models (Donaldson, 1973;
Wyngaard, et al., 1974) are second order, and their success in modeling
turbulent flows seems to lend credence to the basic philosophy of
higher order modeling which states that the less critical is the
closure approximation the higher the degree of the turbulent moment
it is applied to.
Direct numerical simulation of turbulence utilizes the fact that
the Navier-Stokes equations can be used to simulate turbulence if the
numerical technique can resolve scales comparable to those of the
dissipative eddies. Most of the direct simulation studies (Orszag
and Patterson, 1972; Fox and Lilly, 1972) have used Fourier decomposi-
tion techniques and attempt to have their small wavelengths of the
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48
order of the turbulence microscale of the flow. The range of wave
numbers necessary to simulate the flow becomes larger as the Reynolds
number increases; thus, such an approach becomes rapidly impractical
from the point of view of computer core requirements as well as cost
as the Reynolds number increases (Fox and Lilly, 1972). An alternative
approach to direct simulation of turbulent is the so-called sub-grid
scale modeling (Deardorff, 1974) which takes advantage of the existence
of a wide inertia! subrange in high Reynolds number flows. Then, it
is possible to choose a "practical" grid size which lies within the
inertia! subrange without putting too much strain on computer core
requirements. The motion below the grid scale is modeled making use
of the fact that in the inertia! subrange the spectral energy flux is
equal to the dissipation, and the precise nature of the postulated
dissipative mechanism does not influence the large scales of the motion
(Tennekes and Lumley, 1972). Direct numerical simulation techniques
suffer from the disadvantage that one calculation is not sufficient;
each calculation is a realization in an ensemble and a sufficient
number of independent runs must be made to obtain stable statistics.
Thus, such simulations tend to use up a great deal of computer time,
and it is not practical to incorporate direct numerical simulation of
turbulence into numerical models of the planetary boundary layer.
However, it should be emphasized that direct numerical simulation of
turbulence has provided valuable insight into the nature of turbulence.
5.2 The Eddy-Diffusivity Model
Before describing the kinetic energy model for turbulence, the
concept of eddy diffusivity will be discussed in some detail to provide
justification for the utilization of a modified eddy diffusivity in
this study. The eddy diffusivity model can be best explained by con-
sidering the Reynolds stress tensor -u^-. The model would give,
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49
It is easy to see that the above equation is a direct counterpart
of the laminar Newtonian stress law with K the eddy diffusivity
replacing v the kinematic viscosity. The implications of the above
formulations are discussed in great detail in the book by Tennekes
and Lumley (1972) and in the review paper by Reynolds (1974a) and
need not be repeated here. It is sufficient to reiterate that the
physical basis for the eddy diffusivity concept is incorrect in most
of the details. In spite of its shortcomings, the model has been
widely used because of its simplicity and its surprising success in
treating a wide variety of turbulent flows.
From equation (5.2.1) it can be seen that K has to be of the
form,
K = cv£ (5.2.2)
where v is a velocity characterizing the turbulent motion, & is the
so-called mixing length and c is a constant. Substituting (5.2.2)
in (5.2.1) one obtains,
^uTLTT= cv£(u, . + u, .) (5.2.3)
i j i ,j j, i
To be specific, consider the case, i = 1, j = 2 and u =0. Then,
2,1
(5.2.3) becomes
-u'u' = cvA u (5.2.4)
1 ^ 1 5 2
By associating £ with the size of the large turbulent eddies which
cause most of the transport, it is possible to justify Eq. (5.2.4)
without resorting to kinetic theory concepts. If it is accepted that
the larger turbulent eddies derive their energy directly by the
straining action of the mean flow, one would expect the time scale of
the mean flow to be of the same order as the time scale of the large
eddies. Then,
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50
v/A = c.|u. J (5.2.5)
1 1 , 2
or
v = ClA|ii J (5.2.6)
In most turbulent flows uj and u2 are of the same order of magnitude:
Uj and u*2 = 0(v). Then, one can write,
::iF[u2~ = c12v2 (5.2.7)
where c12 is an undetermined constant. Substituting (5.2.6) into
(5.2.7) one gets
:l"X=ClC12*l"ll2l«lf2 t5-2'8)
Equation (5.2.8) is the one originally proposed by Prandtl. If the
correlation between u{ and u2 is good, and the straining of the eddies
is an effective mechanism, the coefficients c1 and c12 should be of
order 1. From (5.2.8) it is easy to see that the eddy diffusivity can
be expressed as
K = c£2|ui)2| (5.2.9)
and the fluctuating velocity v can be written as
v = &|u | (5.2.10)
The preceding arguments to derive (5.2.9) imply that the turbulence
is in equilibrium with the mean flow; the dynamics of larger eddies
are controlled by the mean flow. Thus, the model would predict small
eddy diffusivities in regions of small shear. However, there are
situations, especially in atmospheric flows, in which small velocity
gradients are associated with large momentum transport. Clearly, such
situations make the physical basis for the eddy diffusivity concept
even less firm. One partial solution to the problem is to use a
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51
turbulent velocity scale which is less sensitive to the mean gradients,
and the natural alternative velocity scale is the square root of the
turbulent kinetic energy k. Then, the eddy diffusivity becomes
K = ck^ (5.2.11)
By allowing the kinetic energy to evolve it is possible to have large
turbulent kinetic energies in regions of small shear. Thus, the above
formulation allows for large fluxes even when the mean gradients are
smal1.
The mixing length £ is the most elusive variable in the eddy
diffusivity model. For turbulent flows bounded by solid surfaces, the
mixing length is taken to be
I = KZ (5.2.12)
where K is the von Karman constant and z is the distance from the
solid surface. Equation (5.2.12) was the form suggested by Prandtl
and was used in derivation of the "law of the wall" (Cebeci and Smith,
1974) which has been experimentally verified. The basic idea of using
the distance from the solid surface as the mixing length has remained
unchanged ever since Prandtl first put it forward in 1926. Various
investigators have suggested improvements to the model mainly to
increase the accuracy of their predictions. Details of ingenious
mixing length models for a wide variety of engineering flows can be
found in the book by Cebeci and Smith (1974). The mixing length model
which has found widespread use in atmospheric modeling is that due to
Blackadar (1962). The model assumes that the mixing length increases
linearly with height near the earth's surface and reaches an asymptotic
value at greater heights. The equation for the model is
£ = KZ/(I + KZA) (5.2.13)
The asymptotic mixing length X is related to the height of the
planetary boundary layer. The formulation given by (5.2.13) is
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52
based on experimental findings (Rossby and Montgomery, 1935) which
indicates that a ceases to increase with height at upper levels.
Furthermore, it incorporates the physically realistic idea that I
should not exceed a fixed fraction of the boundary layer height.
Other mixing length models which are similar to Blackadar's have been
suggested by Ohmstede and Appleby (1964) and Lettau (1962). The more
recent turbulence models (Launder and Spalding, 1972) do not specify
the mixing length but allow it to evolve with the mean field through
a differential equation. This approach which is more physically
acceptable than the arbitrary specification of the mixing length has
certain disadvantages which will be discussed in a later section.
5.3 Atmospheric Turbulence Modeling
The language of atmospheric turbulence differs in many details
from that of engineering turbulence because the effects of thermal
stratification are considerably more important in the planetary
boundary layer. In contrast to engineering turbulent flows in which
turbulence is maintained by velocity gradients (eddy diffusivity ideas
are more applicable), atmospheric turbulence is mainly produced by
buoyancy. The effects of wind shear are felt only in a shallow layer
next to the earth's surface.
The effect of stratification on the turbulence in the boundary
layer is traditionally described with the aid of the Richardson number
which is defined as
Rl- _ £ 38/ff9u)2 , f3v
K I ~ TT -rTTV
I Oi.
The definition of the Richardson number assumes that only the vertical
velocity and temperature gradients are significant in determining
the turbulence characteristics. This assumption is valid in the atmos-
pheric boundary layer in which horizontal gradients are usually very
small compared to vertical gradients. A positive Richardson number
implies stable conditions in which turbulence produced by shear is
being converted into potential energy by the positive potential
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53
temperature gradient. A negative Richardson number, on the other hand,
would mean that buoyancy is aiding shear in producing turbulence. The
significance of the preceding statements will become clear when the
turbulent kinetic energy equations are examined in a later section.
The eddy diffusivity used in atmospheric turbulence modeling is
expressed as follows (Estoque, 1963),
K =
3u
f(Ri) (5.3.2)
where f is a function of the Richardson number. The function f has
been derived using semi-empirical arguments, and investigators
(Estoque, 1963; Pandolfo, 1966) have had success in deriving velocity
and temperature profiles in the surface layer using these functions.
An alternative approach to surface layer modeling is that, advanced
by the Soviet school (Moin and Yaglom, 1971). It postulates that
the wind and temperature profiles are universal when they are non-
dimensional ized using the proper scales. In the diabatic surface
layer, u* the friction velocity serves as the velocity scale, and
the temperature and the vertical distance are scaled by the parameters
defined below,
j*0 ~ ~v~ •• 'Q/*•«* (5.3.3)
and
(5.3.4)
where (9'w')0 is the surface heat flux, K is the von Karman constant,
T is the mean temperature of the surface layer and g is the accelera-
tion due to gravity. The temperature scale is 9*0, and L0, which is
the Monin-Obukhov length, serves as the length scale. The physical
interpretation of the Monin-Obukhov length will be discussed later.
Then, it is postulated that the temperature and wind profiles
can be expressed as,
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54
KZ_ du_ _ , (zj
u* dz ~ q>m(lQ
d0
Experimental observations (Businger, et al . , 1971) show that the non-
dimensional velocity and temperature profiles can indeed be described
by universal functions which depend upon the stability (the direction
of the heat flux) of the surface layer. A discussion of the stability
functions can be found in the book by Monin and Yaglom (1971). The
turbulent layer above the surface layer is not as well understood,
and there is considerable controversy surrounding the selection of
scaling variables. However, a series of numerical experiments by
Deardorff (1972; 1974) indicate that it is possible to find velocity
and length scales in the well mixed boundary layer. It is not yet
clear whether it is possible to find scaling parameters for the stable
boundary layer.
Over the past few years, several models (Carson, 1973; Deardorff,
1972; Tennekes, 1973) have been proposed to predict the evolution of
the well -mixed boundary layer. These models parameterize the boundary
layer and do not attempt to predict the detailed turbulence structure.
Although this approach is the most natural one to predict the important
boundary layer variables, such as mixed layer height and boundary layer
potential temperature, it suffers from the disadvantage that it can
predict only the gross features and cannot treat local temperature
changes caused by heat sources or radiative heating or cooling. As the
object of this investigation is to study the effect of pollutants on
the thermal structure, it is necessary to use a turbulence model which
will enable one to resolve localized cooling or heating. The model
adopted in this study is described in the next section.
5.4 Kinetic Energy Turbulence Model
Most of the models which have attempted to treat the important
physical processes in the boundary layer have used semi-empirical
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55
Richardson number correlations to calculate eddy diffusivities. These
models have had a degree of success in treating shear dominated
turbulence but predict imprecise eddy diffusivities in regions with
small wind shears. These inaccurate predictions can give rise to
incorrect physical effects. The limitations of the semi-empirical
eddy diffusivity correlations have been clearly recognized, and
recent papers by Shir and Bornstein (1974) and P. A. Taylor (1974)
discuss the need for improved models for turbulence. Recently,
Wyngaard et al. (1974) have demonstrated that for some special cases
it is computationally feasible in terms of time and money to
incorporate "higher" order turbulence models in numerical models of
the planetary boundary layer. Their 1-D model requires 10 minutes
on the CDC-6600 for a twenty-four hour simulation. It is suspected
that inclusion of radiation, surface energy balances, horizontal
variation of surface parameters advection and other effects which
would be required for a realistic boundary layer model would require
considerably more than ten minutes computer time (CDC-6600) if turbu-
lence models like that of Wyngaard, et al. are utilized.
It is believed that the kinetic energy turbulence model represents
the best compromise between the simplistic semiempirical Richardson
number correlations and the computationally cumbersome "higher" order
models. The kinetic energy model has been developed and utilized by
Spalding and his co-workers (1969) to model a wide variety of engineer-
ing flows. Peterson (1969) has applied the model to the problem of the
modification of planetary boundary layer by roughness changes under
neutral conditions. Del age (1974) has applied the model to the
nocturnal atmospheric boundary layer, and Lykosov (1972) has studied
the diurnal variation of the boundary layer using a kinetic energy
turbulence model. These and other studies indicate that the kinetic
energy model is indeed an improvement over the Richardson number
correlation ;approach. In view of this, it was decided to incorporate
the turbulent kinetic energy model in the present investigation.
The turbulent kinetic energy equation is presented below. The
details of the derivation can be found in books on turbulence (Monin
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56
and Yaglom, 1971; Cebeci and Smith, 1974). The equation after making
the usual closure assumptions is
where
k = ilTir (5.4.2)
is the mixing length, -u'w1 is the shear stress in the x-direction,
-v'w1 is the shear stress in the y-direction, and w'9' the upward heat
flux.
The term Dk/Dt represents the rate of change of kinetic energy
and in expanded form reads
l (5.4.3)
The term 3/8z(k'i £ k/9z) accounts for the diffusion of turbulent
kinetic energy. The terms -u'w1 3u/8z and -v'w1 9v/3z represent
production of turbulent energy by mean shear, and the term g w'G'/T
accounts for buoyancy production of turbulent kinetic energy. The
term e is the viscous dissipation of turbulent energy. As all of the
available experimental evidence indicated that the dissipation of the
large scale eddies e can be written as
e = Cpk^A (5.4.4)
where C~ has to be determined experimentally CD = 0.09 is the commonly
accepted value (Launder and Spalding, 1972).
Invoking the eddy diffusivity concept one can express the second
moments in Eq. (5.4.1) as
(5.4. 5a)
(5.4. 5b)
(5'4'5C)
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57
where K^ is the eddy diffusivity for momentum and KH is the eddy
diffusivity for heat. This study assumes that KH/KM is 1.35 which is
the value observed under neutral conditions (Businger, 1972). Although
1<^ is observed to be greater than 1.35 KM under unstable conditions,
no attempt was made to model the variation of eddy diffusivity ratio
with stability. It should be mentioned that, as yet, there is no
satisfactory theory to predict the diffusivity ratio.
Then, one can write
KM = k%£ (5.4.6)
and
KR = oA (5.4.7)
where a = 1.35 (a = KR/KM = 1.35). With Eqs. (5.4.4) through (5.4.7)
Eq. (5.4.1) can be rewritten as,
rx i It i "\ • I ' f "V O f \ 1 f \O
Dk = A l^o 1*1 k%, f[9u. 2 + [Ml2 + 9v
Dt 9z K * 9z K ^ 9z 9z 9z
oaf 39 _ Y
T 9z Yc
- CD k^/£ (5.4.8)
In the surface layer, experimental observations indicate that the
production of turbulence is matched by dissipation. If this is taken
into consideration Eq. (5.4.8) would read,
T a7 " Yr
I I QL U
j ^ j \ f
or
(5.4.10)
°D
where Ri, the Richardson number, is defined as
9u}2
fr*m <—)
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58
This definition is a slightly modified form of that expressed by
(5.3.1).
From Eq. (5.4.10) it is easy to see that k becomes negative if
Ri > I/a. A physical interpretation of the solution for k given by
Eq. (5.4.10) is that turbulence can exist only if the Richardson
number is less than a critical Richardson number given by
Ri = I/a (5.4.12)
t* I
There is still considerable controversy over the value of Rirr, and
\* I
it is not even entirely clear that a clearly defined critical
Richardson number even exists (Monin and Yaglom, 1971). However,
experimental evidence suggests that true turbulence cannot exist
in flows with Richardson number greater than 0.25 (Webb, 1970).
As the Monin-Obukhov length serves as a length scale under stable
conditions, it is useful to derive the important length from the turbu-
lent kinetic energy equation. Equating the shear production term to
the buoyancy production term, and using the experimentally verified
approximation of the constancy of fluxes in the surface layer, one
obtains from Eq. (5.4.1)
uj
(5.4.13)
where z^ is the height at which turbulence production is matched by
dissipation and where the x-axis is directed along the direction of
surface shear, and 9u/3z has been replaced by U*/KZ,.
The surface friction velocity u* is defined by
(5.4.14)
Then in terms of the definition given by (5.3.3) for 9*, one obtains
from (5.4.13)
z = L0/a (5.4.15)
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59
Thus, from Eq. (5.4.15) it is easy to see that L0, the Monin-Obukhov
length, represents the height below which one would expect turbulence
production to be dominated by shear effects. Above heights of the
order of L0 buoyancy would play the major role in producing turbulence.
Under stable conditions l_0 is a positive quantity and it can be shown
quite easily that the size of the dominant eddies scale with the
Monin-Obukhov length (Monin and Yaglom, 1971).
5.5 Mixing Length Model
Before Eq. (5.4.8) can be solved one has to have a formulation
for the mixing length £. Near the surface (z = 0) £ is known to
^,
behave like £ = Cn4 KZ. The calculation of the mixing length above
the surface layer is a problem still under consideration. The mixing
length can either be specified or calculated from a differential
equation for £ or a related quantity such as the dissipation rate.
The latter procedure has been used with a great deal of success by
Spalding and his co-workers at Imperial College (1969). More recently,
Wyngaard, et al. (1974) have used a dissipation rate equation to
compute the time scale of their turbulence equations. Lykosov (1972)
used a modified from of von Karman formulation for the mixing length.
Delage (1974) has used a prescription which is similar to that proposed
by Deardorff (1972). Delage's formulation, which is applicable only
to the nocturnal boundary layer, makes use of the fact that above the
surface layer the dominant eddies scale with the Monin-Obukhov length
under stable conditions. The expressions due to Blackadar (1962),
Appleby (1964) and Lettau (1962) have already been discussed. The
unknown which appears in most of the mixing length formulations is
the asymptotic mixing length which is a fraction of the boundary layer
height. Traditionally, the boundary layer height was taken to be a
fraction of the length scale formed with the friction velocity u*, and
the Coriolis parameter f. The asymptotic mixing length A would be
written as
X = C u*/f (5.5.16)
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60
The effects of stratification on X are taken into account by
writing
X = C(u*/f) g(y) (5.5.17)
where g is a function of the stratification parameter y given by
y = icu*/(fL0) (5.5.18)
Implicit in the formulation given by (5.5.17) is that surface fluxes
control the height of the boundary layer, and that the atmosphere is
neutrally stratified above the surface layer. Observations, however,
show the turbulent planetary boundary layer is capped by a stable layer.
Thus, the vertical expansion of the boundary layer is clearly con-
trolled by the stability of the air aloft. Then, the boundary layer
height can only be calculated from the internal structure of the
boundary layer. Numerical experiments by Deardorff (1974) show that
u*/f cannot be used as a scaling parameter under unstable conditions,
and the height of the mixed layer is the relevant scale. This seems
to indicate that, at least, under unstable conditions the maximum
mixing length should be related to the mixed layer height. However,
under stable conditions, formulae like (5.5.17) might be applicable
(Businger and Arya, 1974) because the turbulent region is confined to
the layer next to the surface making it possible for the surface
fluxes to control the boundary layer height.
For the purposes of this investigation, it was decided to
prescribe the mixing length rather than use the dissipation rate
equation which is still in the process of development. Initially,
Blackadar's (1963) expression for the mixing length was utilized.
The asymptotic mixing length was related to the mixed layer height
through the equation
X = 0.1 Hm (5.5.19a)
where Hm is the mixed layer height. Hm was defined as the height
at which the turbulent kinetic energy reached an arbitrarily small
value (= 10"6 m2/sec2). At the outset, it is clear that Blackadar's
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61
mixing length model has an undesirable feature in that the mixing
length increases monotonically throughout the boundary layer. Although
it allows for a gradual leveling off of the mixing length, the
relatively large mixing lengths tend to amplify small instabilities
in the capping inversion above the boundary layer. This can give rise
to two maxima in the eddy diffusivity profiles as illustrated in
Figure 5.1. As the mixing length is associated with the dominant
scale of turbulent motion, a model which predicts large mixing lengths
in regions in which there is little or no turbulent activity, is
physically unrealistic. Clearly, a mixing length formulation should
allow for a decrease of the mixing length in the upper part of the
boundary layer.
Recent experimental (Kalmykov, et al . , 1975; Deardorff, 1974)
studies show that the mixing length does indeed decrease towards
the edge of the boundary layer. Kalmykov, et al . (1975) conducted
an extensive study of the behavior of the mixing length in a variety
of shear dominated flows. From their large volume of experimental
data, they concluded that the mixing length profiles are self-similar
and can be well represented by the following expression:
1= CDKn (1 - n)/(l + n) (5.5.l9b)
where
I = £/Hm (5.5.20a)
(5.5.20b)
and CD is the dissipation constant, and K is the von Karman constant.
Equation (5. 5. 19b) predicts that H increases linearly near the surface,
reaches a maximum of about 0.04 at n = 0.5, and decreases to a value
of 0 at the edge of the boundary layer. This study does provide some
basis for the construction of the mixing length model. However, the
results cannot be directly applied to the study of this research
because of the significant differences between shear dominated flows
and those controlled by thermal stratification. Deardorff 's numerical
modeling study of the convective boundary layer shows a mixing length
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62
EDDY DIFRJ3IVITY F0R HERT
2500 ri 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
2000 •
1500 •
1000
500
0.00 .10 .20 .30 .40 .50 .60 .70 .80 .M
EDDY 01FFUS1VITY M2/S
Figure 5.1 Eddy Diffusivity Profiles with Blackadar's Formulation.
Simulation is for Time 19:00 to 22:00 Hours. A, B, C, D
Represent Profiles at 1 Hour Intervals.
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63
behavior similar to that of the investigation of Kalmykov, et al.
However, the mixing lengths are considerably larger as gravity
controlled convection tends to elongate turbulent eddies in the
vertical direction. The mixing length I reaches a maximum of about
0.072 at n = 0.5 and decreases to a value of around 0.0405 at n = 1-0-
The rather large value of 1 at n = 1.0 is associated with gravity
waves rather than true turbulence. It is clear that Deardorff's
results have greater applicability to the study under consideration.
The mixing length model used in this investigation was constructed
to incorporate the most important features of the results described in
the preceding paragraph. To summarize, the mixing length model
possessed the following characteristics:
£ = C^ KZ near z = 0 (5.5.21a)
£ = 0.8 CD Hm; z = Hm/2 (5.5.21b)
£ = 0.1 CD Hm; z = Hm (5.5.21c)
The value of 0.8 CD Hm for SL at Hm/2 was obtained from Deardorff's
study. Since Hm was taken as the height at which there was virtually
no turbulence, £ was assumed to have the small value of 0.1 CD H
The larger value obtained by Deardorff was associated with gravity
waves rather than turbulence. The magnitude of 0.1 CQ Hm used in
this study is based on the observation that the overshoot of con-
vective elements at the top of the boundary layer is approximately
ten percent of the boundary layer height. The mixing length at the
edge of the boundary layer is not critical as it is associated with
small values of turbulent kinetic energy.
A fourth order polynomial was utilized in the formulation of the
mixing length distribution. In addition to being able to satisfy the
required boundary conditions, a fourth order polynomial permits
flexibility in the choice of the magnitudes of the mixing lengths
at the middle and at the top of the boundary layer. The dimension!ess
mixing length £ can be then written as
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64
1= an + bn2 + en3 + dn* (5.5.22)
It should be mentioned that polynomials have been used before
(Fedyayevsky, 1936; Sherstyuk, 1969) to describe mixing length
behavior. The coefficients a, b, c, and d are determined by apply-
ing the boundary conditions (5.5.21). Implicit in Eq. (5.5.22) is
the assumption that the mixing length profile is self-similar.
Justification for this assumption is provided by Kalmykov, et al.
(1975). Furthermore, Deardorff (1972, 1974) in his investigations
shows that the height of the boundary layer serves as a scaling
height for atmospheric variables during convective conditions. Above
z = H the mixing length was assumed to decrease linearly to zero
at the maximum model height around 2000 meters.
5.6 The Equilibrium Layer
As the turbulent time scales are very small near the earth's
surface, very short time steps would have to be utilized in the
numerical solution of the turbulent kinetic energy equation. The
small length scales associated with the turbulent eddies give rise
to relatively large (compared to the other terms in the turbulent
kinetic energy equation) dissipation rates. Thus, the time step
used in the numerical solution of the kinetic energy equation had
to be made small enough to prevent the dissipation rate, which
remains constant over the time interval, from driving the turbulent
kinetic energy to a physically unrealistic negative value. However,
small time steps made the computational costs excessive; the way
around this problem was to take advantage of the near equality of
the production and dissipation of turbulent kinetic energy in the
surface layer (Tennekes and Lumley, 1972). Also, the time rate of
change of kinetic energy is almost two orders of magnitude smaller
than the other terms (Lenschow, 1974). Using these observations
about the surface layer the turbulent kinetic equation can be written
as,
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65
(5.6.1)
Then, the surface layer in which Eq. (5.6.1) is valid is called
the equilibrium layer in this study—the production and dissipation
of turbulent kinetic energy are in equilibrium. The equilibrium layer
is similar to the constant flux layer used by boundary layer modelers
(Estoque, 1963; Sasamori, 1970; Bornstein, 1972). Both the constant
flux layer and the equilibrium layer are based on the principle of
quasi-stationarity--negligible time lag between a physical change and
the forcing causing the change. In fact, one would expect the thick-
ness of the equilibrium layer to be of the same order of magnitude as
that of the constant flux layer. Based on Estoque's (1963) value for
the constant flux layer, the thickness of the equilibrium layer was
chosen to be 50 meters.
Equation (5.6.1) can be solved for the turbulent kinetic energy k,
k =
2
JL
CD
IM] 2 j. fly]2 ag_(3e_
^ 1 "r I _ 1 ^ _i. 1 f. ^
8z ' 3z T 3z " Yc
(5.6.2)
Once k is known from Eq. (5.6.2) KH and K.. can be computed from Eqs.
(5.4.6) and (5.4.7).
5.7 Turbulent Diffusivity for
Two-Dimensional Model
Initially, the turbulence model described in the preceding sections
Was incorporated into a one-dimensional transport model which was then
successfully utilized in simulating a variety of boundary layer events.
An attempt to incorporate the turbulence model based on the turbulent
kinetic energy equation into the two-dimensional urban transport model
gave rise to instability problems with the numerical scheme, and after
considerable experimentation it was decided to use a simpler scheme to
compute the turbulent diffusivities above the equilibrium layer.
The generally accepted behavior (Estoque, 1963; Deardorff, 1967;
Bornstein, 1972) of the eddy diffusivity can be summarized as follows.
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66
The eddy diffusivity increases near the ground reaching a maximum a
few hundred meters from the surface, and then decreases to a small
value at the top of the planetary boundary layer. Based on this
behavior numerical modelers (Estoque, 1963; Wu, 1965; Deardorff,
1967; O'Brien, 1970) have suggested a variety of eddy diffusivity
models. Estoque (1963) assumed that the eddy diffusivity decreased
linearly from a maximum at the top of the surface layer to zero at
the top of the boundary layer; Deardorff (1967) used an empirical
formulation but accounted for the experimentally observed (Telford
and Warner, 1964) counter gradient flux. Wu (1965) used the formu-
lation given by Eq. (5.4.9) but modified the mixing length to account
for its gradual leveling off at great heights. O'Brien (1970)
assumed that the eddy diffusivity profile could be fitted with a
cubic polynomial. Pandolfo, et al. (1963) extended surface layer
based Richardson number correlations for the eddy diffusivity to the
entire boundary layer. It should be pointed out that the described
methods to calculate the diffusivity profiles are semi-empirical in
nature, and are used only because better techniques such as higher-
order modeling (Donaldson, 1973) and subgrid scale modeling (Deardorff,
1974) are computationally cumbersome and at this stage cannot be
incorporated into practical boundary layer models.
Eddy diffusivities based on local Richardson numbers are accurate
only in regions of strong positive wind shear (Shir and Bornstein,
1974). Thus, in the region above the equilibrium layer, where the
velocity as well as the temperature gradients are small, it is neces-
sary to use a technique which is not dependent on the Richardson
number. The natural alternative is to specify the eddy diffusivity
profile as has been done by several investigators (Estoque, 1963;
O'Brien, 1970). It is not clear why an arbitrary specification of
the eddy diffusivity profile is preferable to imprecise eddy diffusivity
values obtained from Richardson number correlations until it is pointed
out that the numerical differentiation involved in the computation of
the Richardson number gives rise to large errors in regions in which
velocity and temperature gradients are small. These errors in turn
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67
lead to unrealistically jagged eddy diffusivity profiles as evidenced
by Pandolfo's (1971) results. Furthermore, Richardson number correla-
tions for the eddy diffusivity are based on the equilibrium assumption
(production equals dissipation of turbulence) which is clearly not
valid above the surface layer (Lenschow, 1973). From the preceding
discussion it should be clear that the specification of the eddy
diffusivity profile is not more empirical than the use of Richardson
number correlations; the main advantage of the specification of the
profile is the numerical facility it affords.
This study utilized the O'Brien (1970) cubic polynomial to specify
the vertical eddy diffusivity profile. The O'Brien model ensures the
continuity of the slope of the eddy diffusivity profile at the top
of the equilibrium layer, and it also allows the eddy diffusivity to
reach a maximum above the height of the equilibrium layer. These
physically realistic features of the O'Brien model are not found in
the models due to Estoque (1963) and McPherson (1968). It should be
mentioned that Sasamori (1970) and Bornstein (1972) have incorporated
the O'Brien diffusivity distribution into their "successful" boundary
layer models.
The O'Brien formulation assumes that the following are known:
K(H*), (8K/9z)*, K(h),
where H* is the height of the boundary layer, h is height of the
equilibrium layer, and K is the eddy diffusivity. If it is assumed
that the variation of K at H* is zero, then the eddy diffusivity
distribution can be written as,
K(z) = K(H*) + [2~ H!l2JK(h) - K(H*)
+ 2[K(h) - K(H*)
H* - h
Equation (5.7.1) yields parabolic curves, as predicted by Blackadar
(1962), Lettau (1962), and Lettau and Dabberdt (1970), and as observed
at night by Elliot (1964).
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68
Sasamori (1970) assumed that h was a function of the eddy
diffusivity near the ground, but used a fixed value for H*; Bornstein
(1972) fixed the values of both h and H*. In these above mentioned
studies h refers to the height of the layer near the surface in which
the fluxes (momentum, heat and mass) can be considered to be constant
with height. As has been discussed in the preceding section, this
investigation defines h as the thickness of the surface layer in which
it is assumed that the production and dissipation of turbulence are
in balance. It is clear from both definitions that h should be
a function of the turbulence characteristics of the surface, and a
realistic model should account for the variation of h with time.
Sasamori (1970) has suggested a method to compute h from the internal
characteristics of the surface layer; however, the method has not yet
been widely accepted, and most boundary layer modelers (Estoque,
1972; Wagner and Yu, 1972; Bornstein, 1972) have chosen to fix the
value of h. In view of these considerations it was decided to use
a constant value of 50 meters for h (Estoque, 1963).
The boundary layer thickness H* is a function of space and time
(Tennekes, 1974), and it can vary from a few tens of meters on clear
nights with light winds to about two thousand meters on sunny summer
afternoons. The boundary layer is identified with the turbulent region
next to the earth's surface, and the details of the formation of the
boundary layer or the mixed layer have already been discussed in a
preceding chapter. In the one-dimensional model, the turbulent kinetic
energy distribution was used to define the boundary layer thickness.
For the reasons given before the turbulent kinetic energy equation was
not solved in the two-dimensional model and it was necessary to use
an alternative definition for the boundary layer thickness. As the
mixed layer is usually capped by a stable layer in which there is
virtually no turbulence, the boundary layer thickness was defined as
the height at which the potential temperature gradient exceeded an
arbitrary positive value which was large enough to suppress turbulent
activity. Implicit in this definition of the boundary layer thickness
is the assumption that shear effects are of secondary importance
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69
compared to thermal effects and it is only necessary to have a large
enough potential temperature gradient to inhibit turbulent activity.
Clearly, this assumption cannot be justified when the boundary layer
is growing and is confined to the high shear region next to the
earth's surface, or when the upward heat flux is small (Monin-Obukov
length is large). Thus, it would seem that a better definition for
the boundary layer thickness would involve some sort of Richardson
number criterion. However, as has been discussed before, the numerical
computation of the Richardson number presents problems which have not
yet been resolved. Thus, for the purposes of this study it was
decided to use the simpler potential temperature gradient criterion
to define the boundary layer thickness. Then, the mixed layer height
was defined as the lowest level at which 30/8z exceeded 4 C Km" , a
value suggested by Deardorff (1967).
As the region above the boundary layer is not turbulent, the
eddy diffusivity is assumed to be zero above z = H*. The error
introduced by neglecting the molecular diffusivity is negligible over
the time scale of simulation.
The eddy diffusivity below z = h is computed from turbulence
equilibrium considerations, and the term (9K/9z)h is calculated using
a first order finite difference scheme.
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70
VI. RADIATION MODEL
6.1 Introduction
The spectrum of atmospheric radiation can be divided into two
fairly distinct regions. Most of the radiation emitted by the sun and
reaching the earth is confined to the spectral region between 0.3 ym
to 4 ym, while the energy emitted by the earth atmosphere system lies
between 4 ym to 100 ym. The spectrum of solar radiation is very close
to the spectral distribution of the energy emitted by a blackbody at
approximately 6000 K. Similarly, the earth atmosphere system can in
effect be replaced by a blackbody at 245 K.
The principal absorbers of solar radiation are water vapor in the
troposphere and ozone in the stratosphere. Water vapor absorbs pri-
marily in the near infrared region, 0.7 ym z. X <. 4 ym, while ozone is
effective in the ultraviolet (X <_0.35 ym) and in the visual range
(0.5 ym <_ X <. 0.7 ym). In addition to being absorbed, solar radiation
is scattered by gas molecules (Rayleigh scattered) and aerosols (Mie
scattered).
In the thermal part of the spectrum, the principal absorbers are
water vapor and carbon dioxide. Water vapor has strong bands centered
at 2.7 ym and 6.3 ym, and a rotational band extending from 20 ym to
100 ym under standard atmospheric conditions. Carbon dioxide absorbs
strongly in bands centered at 2.7 ym, 4.3 ym and 14.8 ym. Ozone has
a band centered at 9.6 ym. The 8 ym to 12 ym region, which is
referred to as the atmospheric window, is relatively transparent, and
it is in this region that the presence of pollutant absorption bands
is important.
Most of the ultraviolet region (X <. 0.35 ym) of the solar spectrum
is absorbed by stratospheric ozone when the solar energy reaches the
top of the atmospheric boundary layer which usually extends to heights
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71
around 2 kilometers. Inside the boundary layer, the short-wavelength
solar energy is scattered by gases and natural and man-made aerosols.
It is also absorbed by water vapor and pollutant gases such as N02.
Solar energy is also absorbed to a smaller extent by aerosols. Thermal
radiation is absorbed and emitted by water vapor, carbon dioxide,
and pollutant gases such as carbon monoxide and ammonia. Aerosols in
addition to absorbing can also scatter thermal radiation. At the
surface of the earth, solar radiation is absorbed and reflected
diffusely while thermal radiation is reflected, emitted, and absorbed
diffusely. Figure 6.1 schematically illustrates the interaction of
atmospheric radiation with the planetary boundary layer.
6.2 The Radiative Transfer Equation
The equation of radiative transfer expresses the conservation of
radiant energy in terms of the change of intensity of a pencil of
radiation traversing an absorbing, emitting, and scattering medium.
The derivation of the equation of transfer can be found in standard
works on radiative transfer (Chandrasekhar, 1960, for example) and it
is sufficient to present the equation,
31 (z,u,)
1 pvv*»n >) is the intensity of the pencil of radiation at a point
defined by the spherical coordinates z,y,. The equation then states
that the change of intensity (the term on the left) of the beam of
radiation is caused by the processes signified by the terms on the
right of the equation. The term -3v(z)Iv(z,y,c|>) accounts for the
extinction of radiation by scattering and absorption; 3v(z) is called
the extinction coefficient and is the sum of the absorption coefficient
K (z) and the scattering coefficient c?v(z). The term K (z)Ib (z)
accounts for the increase of the intensity by the emission of the
medium and the term
-------
Incident
Solar
Radiation
Natural
Atmosphere
Leaving Scattered
Solar Radiation
Leaving Thermal
Radiation
V
Scattering and Absorption
by Natural Gases and
Aerosols
Emission and
Absorption by
Natural Gases
. Scattering and
\ . Absorption by
Njf Gases and
fil Aerosols, Natural
Polluted and Polluted
Atmosphere
" Incident Transmitted
2 Scattered \
V f^ \ Reflected \ Reflected
/
Emission,
©Absorption and
Scattering by
Gases and
Aerosols, Natural
and Polluted
Incident
4
Emission Reflection
/ //*'/ ///V///A///
Absorption ' Absorption / Absorption
Absorption
Figure 6.1 Physical Model for Thermal and Solar Radiation
ro
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73
o (z)
f
p^z.u1,*' -> y,4>)Iv(z,u ',<
lJ1
accounts for the contribution of radiant energy to the beam by scatter-
ing. The scattering phase function p (z,y',' ->• y,cf>) represents the
fraction of energy scattered from a pencil of radiation propagating
in the direction y'.cj)1 into the solid angle centered around the
direction y,4>. In writing Eq. (6.2.1) is it assumed that the radiative
transfer is quasi steady, the index of refraction of the medium is unity,
and the atmosphere is in local thermodynamic equilibrium. The coordi-
nate system used in formulating Eq. (6.2.1) is illustrated in Figure 6.2
Equation (6.2.1) is a nonhomogeneous integro-differential equation
and its solution presents formidable mathematical difficulties. Addi-
tional complexities arise in attempting to characterize a medium which
scatters and absorbs radiation. A number of techniques have been
utilized to solve the equation of transfer, and descriptions of these
models can be found in books by Goody (1964) and Kondratyev (1969).
An excellent summary of methods applicable to a scattering atmosphere
can be found in a report published by the Radiation Commission of the
I. A.M. A. P- (1974).
In applying the equation of transfer to the atmosphere, it is
assumed that the atmosphere is horizontally homogeneous. This assump-
tion is based on two considerations which are:
1. Computational simplicity: A 2-D or 3-D radiative transfer
model would require excessive computational effort. The
significance of this statement becomes clear when one notes
that an accurate solution of the 1-D radiative transfer
equation requires substantial computer time.
2. The approximate horizontal homogeneity of the boundary layer:
Over the horizontal scales (-20 km) of interest the gradients
of the atmospheric variables are small compared to the
vertical gradients. Therefore, there is some justification
in considering the atmosphere to be horizontally homogeneous.
An approximate method to handle horizontal inhomogeneity
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74
Figure 6.2 Coordinate System for Radiative Transfer Equation
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75
would be to compute the 1-D radiative fluxes along hori-
zontal locations and use interpolation to compute fluxes
at intermediate points.
Using the assumption of horizontal homogeneity Eq. (6.2.1) can
be written as
%U) r+1
V ' - (Z,U' -> u)Iv(z,u')du' (6.2.2)
J-,
In writing Eq. (6.2.2) it has been assumed that the phase function as
well as the intensity has been averaged over the azimuthal angle c|>.
This equation can be more conveniently expressed by defining the
following quantities. The optical thickness T at the frequency v is
given by
rz
TV(Z) = Bv(x)dx (6.2.3a)
o
and the single scattering albedo to is defined by
u)v(z) = av(z)/3v(z) = av(z)/[Kv(z) + av(z)] (6.2.3b)
Using the definitions given by Eqs. (6.2.3a) and (6.2.3b), Eq. (6.2.2)
can be written as
dl (T ,y)
l Pv(u' * Vi)Iv(Tv,u')dy' (6.2.4)
The radiative energy flux F across a plane at height z is defined by
J
Iv(y)ydy (6.2.5)
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76
The fact that the atmospheric radiation spectrum can be divided
into two distinct regions can be advantageously utilized in solving
Eq. (6.2.4). In the solar spectrum, extending from 0.3 urn to 4.0 urn,
emission can be safely neglected, and in the thermal spectrum (4.0 urn
to 100 urn) scattering can be assumed to be negligible. The former
assumption is based on the fact that most of the emission occurs in
the thermal spectrum, and the latter on the fact that aerosols, which
cause most of the scattering, do not interact with long-wave
terrestrial radiation. Also, the effect of larger aerosol (~1 ym) ,
which can interact with thermal radiation, is negligible because
larger aerosols are found in small numbers (Peterson, et al . , 1969).
6.3 Radiative Transfer in the Solar Spectrum
In the solar spectrum, the emission term in the radiative transfer
equation can be neglected and Eq. (6.2.4) reduces to
dl oj
For convenience in writing the equation the dependence of I on y and
T has been dropped. Even in its relatively simplified form, Eq.
(6.3.1) presents mathematical difficulties, and except for the simplest
scattering phase functions, it does not permit an analytic solution
and numerical methods have to be resorted to. Any satisfactory solu-
tion of the equation would entail computation of the radiative fluxes
over small wavelength intervals in order to account for the strong
absorption in certain narrow wavelength bands. This computational
process is costly and cumbersome, and several approximate methods have
been suggested to reduce the numerical effort. The methods usually
involve one of the two approximations: (1) discretization of the
angular distribution of the radiation intensity examples of which are
provided by the six-stream method of Chu and Churchill (1955), the
Eddington approximation used by Pitts (1954), the Chandrasekhar
approximation (1960), modified two-stream method by Sagan and Pollack
(1967), and the spherical harmonic approximation used by Canosa and
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77
Panafiel (1973); (2) approximating the scattering phase function by a
simpler mathematically tractable function examples of which are
provided by the small angle approximation due to Romanova (1962) and
the delta function approximation due to Potter (1970).
The scope of this study did not permit detailed calculation of
the radiative fluxes; considerations of computational efficiency
played the major role in the choice of the technique to calculate the
fluxes. The detailed radiation model due to Bergstrom (1972) which
was available for use in this research program was not selected
because it required computer time which was excessive from the point
of view of this study. An examination of other available methods
showed that the so-called two-stream method met the requirement of
computational convenience without introducing unacceptable idealiza-
tions. As the name implies, the method approximates the angular
distribution of intensity by two intensities, one of which character-
izes the radiation field in the forward direction and the other, the
backward direction. Also, the phase function is parameterized by
introducing the forward and the backward scattering factors which are
the fractions of the radiative flux scattered in the forward and
backward directions, respectively. These statements will become clear
when the two-stream equations are derived. The two-stream method has
been used by Sagan and Pollack (1967) to study scattering in Venusian
clouds, and by Rasool and Schneider (1971) to study the effect of
aerosols on global climate. More recently, Wang and Domoto (1974)
have adapted the method to treat nongray gaseous absorption with
multiple scattering in the planetary atmosphere.
6.4 Radiative Properties of Aerosols and Gaseous Absorbers
in the Solar Spectrum
As radiative transfer was just one of the several physical
processes modeled in this investigation, it was necessary to use a
radiation model which would be simple and at the same time yield
relatively accurate estimates of the radiative fluxes. The adoption
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78
of the two-stream approximation leads to a reduction of computational
effort only if the properties of the radiative participants are
parameterized. From the point of view of storage as well as computer
time, it is necessary to avoid detailed integrations over wavelength.
For reasons which will be discussed later, aerosols allow parameteriza-
tion of properties. Gaseous absorbers such as N02 and S02, on the
other hand, have to be accounted on a wavelength basis because their
strong absorption precludes mean property representations. Thus, it
was found convenient to assume that aerosols and water vapor were the
only participants in the solar spectrum. This emphasis on aerosols is
not unrealistic especially in view of recent studies (Rasool and
Schneider, 1971; Mitchell, 1971; McCormick and Ludwig, 1967; Braslau
and Dave, 1973; Landsberg, 1970; Wang and Domoto, 1974; Kondratyev,
et al., 1974) which indicate that aerosols can play a major role in
determining the radiation balance of the earth-atmosphere system.
It is assumed that radiative properties of aerosols can be
represented by spectrally averaged quantities. This assumption is
based on a recent study by Bergstrom (1971) which shows that the
absorption and scattering coefficients of slightly absorbing aerosols
(10% carbon content) vary very slightly in the wavelength interval,
0.3 ym <_ X <_ 1.0 ym, a region which contains 70% of the solar radiation
Furthermore, as aerosols are predominantly scatterers, the spectral
distribution of the solar radiation is not as drastically altered as
it would have been by strong absorption bands, and there is justifica-
tion in defining mean radiative properties weighted with respect to
the solar spectral distribution. A number of studies (Rasool and
Schneider, 1971; Atwater, 1970) have also used representative values
for the radiative properties of aerosols in calculating the solar
fluxes.
In estimating the optical path due to the water vapor, the non-
linearity of the absorption of the water vapor bands has to be taken
into account. Relatively large amounts of water vapor inside the
boundary layer do not cause significant heating because most of the
energy in the important water vapor bands is absorbed in the region
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79
above the planetary boundary layer. Thus, a large increase in water
vapor path length is accompanied by a proportionately much smaller
increase in absorption.
In order to avoid lengthy computations involving wavelength
integration, a parameterized expression (Lacis and Hansen, 1974) for
water vapor absorption has been used to calculate the effective water
vapor optical thickness. For a cloud-free atmosphere, Lacis and
Hansen (1974) use the following expression to compute the absorption
by water vapor in the £-th layer of the atmosphere:
where irF is the solar constant, y0 is the cosine of the solar zenith
angle, Rg is the ground albedo, y^ is the effective water vapor amount
traversed by the direct solar beam in reaching the £-th layer, y* is
x/
the effective water vapor path traversed by the reflected radiation in
reaching the £-th layer from below, and A is an empirical function
of £-th; A (y,,) gives the fraction of energy absorbed in a layer of
effective water vapor thickness y,,. This study utilizes the function
A (yj suggested by Yamamoto (1962) which is given by
W V X»
Awv(y) = 2.9y/[(l + 141.5y)°-635 + 5.925y)] (6.4.2)
where y has the units of cm of preci pi table water vapor.
In order to utilize Eq. (6.4.2) to calculate an effective thick-
ness required in the two flux formulation, consider a beam of solar
radiation traversing a layer £. The energy absorbed in this layer Ar,
is then given by,
-------
30
where y£ is the effective path length measured vertically downwards
from the top of the atmosphere. Assuming that the layer can be
represented by an incremental optical thickness AT, one can write
- exp(-AT/y0)] =
Then,
AT = -y0 ln[l - (Awv(y£+1/u0) - Awv(y£/y0))/(l - Awv(y£/y0)]
(6.4.5)
There is some ambiguity in the definition of the incremental optical
thickness given by (4.4.5). Optical thicknesses can be computed for
the radiation reaching the layer from below, and for the scattered
radiation. Thus, it is possible to associate several optical thick-
nesses with the same layer. This is inconvenient and it is necessary
to choose a "dominant" optical thickness. Assuming that most of the
solar energy is contained in the direct beam, the optical thickness
was defined by considering the absorption of the direct beam. Equa-
tion (6.4.5) defines the "direct beam" optical' thikcness. Clearly,
it is desirable to have a definition based on scattered radiation to
handle cases when the zenith angle is large (early morning and evening)
and most of the solar radiation is diffuse. However, it is not
practical to use two different optical thicknesses. Also, as the
magnitude of the diffuse radiation is small, the computation of the
optical thickness at large zenith angles is not critical.
Empirical methods (Paily, et al . , 1974) were used to compute the
direct and diffuse radiation at the top of the boundary layer. Though
empirical, these methods have been tested thoroughly and given
accurate estimates of the total fluxes. Thus, the basic method to
compute the radiative fluxes in the solar spectrum consisted of two
steps: (1) compute scattered (diffuse) and directly transmitted solar
fluxes at the top of the planetary boundary layer using empirical
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81
methods, and (2) compute radiative fluxes in the boundary layer using
the more accurate two-flux method. The fluxes computed in step 1 serve
as boundary conditions for the calculation in step 2.
6.5 Two-Stream Equations
The two-stream equations can be derived by considering the radia-
tive transfer equation in the form,
u dT= -1 +f f ' p(y' -yJKT.vW (6.5.1)
The dependence of the intensity on the frequency v has been dropped
on the basis of the mean property assumption discussed in the
preceding section.
The boundary conditions for Eq. (6,5.1) can be written as,
I(0,y) = 2 [ r (-y1 +y)I(0,-y')y'dy' ; y>0 (6.5.2a)
' o
and
I(TO,U) = S6[y - (-y0)] + Id ; y > 0 (6.5.2b)
where r is the mean bi-directional reflectance of the earth's surface,
yQ is the optical thickness of the boundary layer, S is the direct
beam component (at zenith angle cos" y ) of the solar radiation at
the top of the boundary layer and Id is intensity of the diffuse
component. Then, Eq. (6.5.2a) states that the intensity of the radia-
tion traveling in +y direction is the sum of the contributions from
the beams of radiation reflected at the surface. Equation (6.5.2b)
states that the source of radiation at the top of the boundary layer
consists of the direct and the diffuse components of solar radiation.
To aid in the solution of Eq. (6.5.1) the scattering phase
function is expressed as a finite series of Legendre polynomials
N
p(T,y' +y) = I pj(T)P.(y)P(.(ljil) (6.5.3)
i=Q * * *
-------
82
where P0(y) is a Legendre polynomial of order JL Details of the
x/
derivation of Eq. (6.5.3) can be found in the book by Chandrasekhar
(1960).
Substitution of Eq. (6.5.3) into Eq. (6.5.1) yields
ydl=-I+^£ p4(r) p P£(y)P£(y')I(T,y')dy' (6.5.4)
Xf~~U "- *
To simplify the application of the boundary condition (6.5.2b)
the directly transmitted solar radiation is subtracted from the mean
intensity I which is then redefined by
(TO-T)/U
I(r,y) = I(T,U) - S6[y - (-yQ)]e (6.5.5)
Substitution of (6.5.5) in (6.5.4) results in
-' - U N
+ I p P (y) I(T,y')P£(y')dy'
il ,_-. X, X- X,
The two-stream approximation can be introduced by considering the
integral F given by
F = I(r,y)ydy (6.5.7a)
J-i
Using a two-point Gaussian quadrature formula, F can be expressed as
F = (I+ - I~)//3 (6.5.7b)
Thus, I represents the intensity radiation traveling in +y (= 1//T)
direction, and I", that of the radiation traveling in the -y (= -1//3)
direction.
The two-stream equations are obtained by multiplying Eq. (6.5.6)
by dy and ydy, respectively, and integrating over y from +1 to -1,
-------
83
dl! = (fu . i)r + bwl- + |S[1 - /3(1 - 2b)y0]e" T°"
(6.5.8a)
i HT- , „
- — 7T- = (fu) - 1)1" + bu)I+ + £S[1 + /T(1 - 2b)o) Je
~
(6.5.8b)
where I and I~ have been defined by Eq. (6.5.7b). The backscattering
factor, b, and the forward scattering factor, f, are defined by the
equations
i r+1
2b = 1 - i P(y)udy (6.5.9a)
J -i
f = 1 - b (6.5.9b)
Then, assuming that the earth's surface reflects diffusely, the
boundary conditions can be expressed in terms of I and I
I+(0) = r
/5 PnSe °'
0
(6.5.10a)
and
I"(TO) = Id (6.5.10b)
Even with the simplifications afforded by the two-flux approxima-
tion it is not possible to obtain analytic solutions for Eqs. (6.5.8)
because w, f, and b are functions of the optical depth. It should be
mentioned, however, that Wang and Domoto (1974) have obtained an
approximate solution to the two-flux equations using a perturbation
technique.
In order to facilitate an analytic solution of the two-stream
equations it is assumed that the radiative properties of the consti-
tuents of the planetary boundary layer can be represented by a mean
single scattering albedo, and a mean forward scattering factor which
are defined by the following equation
-------
84
ID = (TR + WTA)/(TR + TA + TQ) (6.5.11)
where w is the mean single scattering albedo, w is the aerosol single
scattering albedo, TA is the aerosol optical thickness, TR is the
Rayleigh optical thickness, and TG is the optical thickness of the
absorbing gas.
The mean forward scattering factor f is defined by
f - (ft, + fRTR)/(TA + TR) (6.5.12)
where f is aerosol forward scattering factor, and fR is the Rayleigh
forward scattering factor (fR = 0.5).
With the assumptions described in preceding paragraphs, Eqs.
(6.5.8) reduce to a pair of coupled linear differential equations,
and the solution is straightforward
nir
I(T) = a^e + a262e"aT + Cxe (6.5.13a)
«T rt-r "^n"1)/^
I"(T) = 3^ + B2e"aT + C2e ° ° (6.5.13b)
where
a = /3[(1 - u f)2 - (ub)2]% (6.5.14a)
a1 = v/3" u b/(/3"(l - w f) + a) (6.5.14b)
a2 = /3 uF/(/3~(l - w?) - a) (6.5.14c)
where
F = 1 - f (6.5.14d)
Before defining 3i , 32» C1 and C2 it is convenient to define the
intermediate quantities,
-------
85
b2 = /I o» F (6.5.15a)
S1 = /3(1 - /1(1 - 2b)yQ)wS/2 (6.5.15b)
S2 = /3(1 + /3(1 - 2F)y0)wS/2 (6.5.15c)
a = /!(! - w f) (6.5.15d)
dx = a - 1/y* - b2 (6.5.15e)
d2 = (ai ~ rs^e ° - (ct2 - r )e ° (6.5.15f)
n
t1 = (PC2 - C, + rs/TyoS)e ° ° - (Id - C2)(a2 - rs)
(6.5.15g)
tz = (a2 - rs)(Id - C2) - (rsC2 - C, + r$ /I y0S)e" aT°"T° "°
(6.5.15h)
Then,
Cx = [S^a - l/y0) + S2b2]/d1 (6.5.16a)
C2 = [S2(a + l/y0) + S1b2]/d1 (6.5.16b)
3X = t1/d2 (6.5.16c)
32 = t2/d2 (6.5.16d)
Then, in terms of F and F , the radiative flux F is given by
r+1 f+1f~ (T -T)/yi
F = 2rr Kr.yjydy = 2v I(T,y) + S6[y - (-y )]e ° lydy
J-l l _iv J
2lTrT+ T~1 O Cx.00 //-ri-7\
= — [I - I ] - 2iryQSe (6.5.17)
/3
-------
86
The divergence of the solar radiative flux can be written as
aS2e"aT(l - o2)
(C, - C2)/u0J//3 - ^
t
(6.5.18)
6.6 Band Models
Infrared absorption and emission of thermal radiation is a conse-
quence of coupled vibrational and rotational energy transitions. As
these transitions occur at discrete frequencies the absorption spectra
of polyatomic gases tend to be discontinuous. Figure 6.3 illustrates the
absorption spectrum of carbon dioxide, and it can be seen that absorp-
tion occurs in fairly distinct bands which are separated by regions in
which negligible absorption occurs. The location of a particular band
is determined by the associated vibrational transition, while the
"lines" in the band are governed by rotational transitions which
accompany the vibrational transition. Thus, a typical infrared
spectrum would consist of a few bands each of which would consist
of hundreds of lines.
Although modern spectroscopy can resolve detailed line structure,
it is impractical to do line by line calculations to compute radiative
fluxes routinely. The alternative to this type of exact calculation
is to utilize band models. Before proceeding on the subject of band
models, it is convenient to define the band absorptance or the
effective bandwidth
A = [ [1 - exp(-K L)]d(v - v ) (6.6.1)
JAv
where A is the band absorptance of a homogeneous layer of gas of
thickness L, and VQ is the center of the interval of integration Av,
-------
87
4.3
4.8-J
4 3 2.5
Wavelength, X, pm
2.0
1.67
I I 1 I I I I I I I I )
.5 LO 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6-OxlO3
Wave number, rt. cm"*
Figure 6.3 Low-Resolution Spectrum of Absorption Bands for C02 Gas at
830 K, 10 atm, and for Path Length Through Gas of 38.8 cm
(Siege!, R., and Howell, J. R.: 1972, Thermal Radiation
Heat Transfer, McGraw-Hill Book Company, New York,
814 pp, pg 406)
-------
88
and < is the monochromatic absorption coefficient. Physically, A
represents the effective bandwidth over which the emitted radiation
can be considered to be black. The integrand [1 - exp(-
-------
89
wide band model. The models mentioned in the previous paragraphs are
narrow band models while some of the wide band models are the box
model due to Penner (1959) and the exponential wide band model due
to Edwards (1964, 1967).
The preceding paragraphs provide a very brief description of
band models. More details can be found in excellent summary papers
by Tien (1968), Cess and Tiwari (1972), and Tiwari (1975).
6.7 Infrared Radiation Properties of Gases
It should be clear from the previous section that the definition
of band absorptance is strictly applicable only to a homogeneous gas
layer. Thus, in order to utilize the concept of band absorptance in
the computation of radiative fluxes in an inhomogeneous atmosphere,
it is necessary to use a method which "corrects" for the inhomogeneity.
One such method is the Curtis-Godson approximation which essentially
replaces an inhomogeneous layer by a "radiatively" equivalent
homogeneous layer enabling the use of narrow band models. An
alternative to this technique is to use emissivities. The concept
of emissivity as it is used by meteorologists (Brooks, 1950; Sasamori,
1968; Eliott and Stevens, 1961; Atwater, 1970) is empirical in nature;
it associates an emissivity with a gas layer, and the effects of
inhomogeneity are taken into account by using empirically determined
pressure and temperature correction factors. Rodgers (1966) has shown
that the emissivity approximation is capable of the same order of
accuracy as the more exact methods (for example, the diffuse approxi-
mate due to Rodgers and Walshaw, 1966), but only if used in an
empirical rather than a theoretical manner. Emissivities must be
fitted to calculated or accurately measured atmospheric fluxes rather
than directly to spectral data obtained in the laboratory.
As the principle behind the emissivity approximation is now well
enough established for most meteorological purposes, this study
utilized empirical emissivities wherever possible. For water vapor,
the emissivity correlations derived by Atwater (1970) from Kuhn's
(1963) emission data were used. It should be mentioned that Kuhn's
-------
90
data are based on measured atmospheric fluxes. Also, as the effects
of carbon dioxide overlap with water vapor at 15 ym have been sub-
tracted out, the use of Kuhn's data avoids the problem of overlap.
For carbon dioxide the emissivity correlation proposed by Shekter
(1950) and subsequently revised by Kondratyev (1969) was used. As
emissivity correlations are not available for common pollutant gases
such as ammonia, carbon monoxide and ethylene, it was necessary to
use wide band absorptances to define an effective emissivity given by
eef(u> = \ Ebv/W™- Tef Pef>/Tef (6-7-1}
where i refers to a band, Et,v. is the blackbody emitted flux at the
center of the band, u is the mass length, a is the Stefan-Boltzmann
constant, and r is the diffusivity factor, the meaning of which will
be explained later. The effective pressure P f is defined by
Pef = u f P(t)dt (6-7'2)
Jo
and T -r is the temperature corresponding to P ~. Equation (6.7.2)
has been used by Jurica (1970), and is based on the Curtis-Godson
approximation.
The technique implied by Eq. (6.7.2) to correct the temperature
and pressure variation is empirical, and has been used in this study
because it is simple compared to the more rigorous methods (Chan
and Tien, 1969; Edwards and Morizumi, 1970), and Jurica (1970) has
had some success in using the method. It should be pointed out that
the definition of emissivity given by Eq. (6.7.1) has meaning only
when applied to a homogeneous gas layer. Rodgers (1966) has shown
that it is not possible to define a unique emissivity when an
inhomogeneous gas is being considered--the upward and the downward
fluxes require different emissivities to make the definitions of the
emissivities compatible with the radiative transfer equation. It
should be reiterated that the concept of emissivity does not "fall
out" of the radiative transfer equations, but can be very useful when
-------
91
treated in an empirical manner. It is felt that in the absence of
empirically derived emissivities for the pollutant gases there is
justification in using Eq. (6.7.1) to provide estimates for the
emissivities.
The Edwards exponential wide band model was used to calculate
the effective emissivities of the pollutant gases. This choice of
a wide band model was based on two considerations. Firstly, the
Edwards model has enjoyed considerable success in correlating band
absorptance data; secondly, its utilization allows the incorporation
of band absorptance correlations which can be expressed as single
explicit functions of the path length (Tien and Lowder, 1966; Goody
and Bel ton, 1967; Cess and Tiwari, 1972).
The exponential wide band model assumes that the mean line intensity
is an exponentially decaying function of the wave number. Then, utiliz-
ing the narrow statistical band model, the expression for the band
absorptance can be integrated over the bandwidth. The band absorptance
can be then written as
A = A(C15 C2, C3, 3, X) (6.7.3a)
where
C2P
3 = 4c-f- (6.7.3b)
X = pL (6.7.3c)
and L is the thickness of the homogeneous gas layer, p is the density
and Pe is an effective pressure which takes pressure broadening by
other gases into account. Cl9 C2, and C3 are functions of temperature
and characterize the structure of the band. A summary of the functions
Cj, C2, and C3 for a number of gases can be found in papers by Edwards
(1964a, 1964b) and in the review paper by Tien (1968). In the original
formulation due to Edwards, different pathlength ranges involved the
utilization of different functional forms of A. As it is desirable
to have a single explicit expression for the band absorptance for the
-------
92
whole path length range, researchers (Tien and Lowder, 1966; Cess and
Tiwari, 1972) have suggested single continuous correlations for the
band absorptance. One such formulation is that due to Tien and Lowder
(1966) and has been used in this study. The band absorptance is
given by
A(u,t) = In{uf2(t)[(u + l)/(u + af2(t))] + 1} (6.7-4)
2
where
A E A/C3
u E (C/CJX (6.7.5)
t E (C^Cjp
and
f2(t) = 2.94[1 - exp(-2.60t)] (6.7.6)
As has been mentioned before, gaseous pollutants are important
only when they absorb in the atmospheric window extending from 8 ym
to 12 ym. Thus, a representative pollutant has to meet the requirement
of being a strong absorber in the atmospheric window region. An
examination of the absorption properties of a number of pollutant gases
showed that ammonia was the best candidate for a representative
pollutant. Ammonia has its strongest band (at 300 K) centered at
10.5 ym, and it absorbs very strongly in the entire window region.
Also, ammonia seems to be becoming an increasingly important air
pollutant since it is associated with the presence of a large popula-
tion (Ludwig et al., 1969). The importance of ammonia as a pollutant
has motivated a number of studies on the infrared properties of the
gas (France and Williams, 1966; Walsh, 1969). Recently (1973) Tien
has correlated the France-Williams (1966) data to yield the wide-band
parameters which were used in this study. In addition to ammonia,
this investigation has also used sulphur dioxide, carbon monoxide, and
methane as possible pollutants. The wide band parameters for carbon
monoxide and methane were taken from the review paper by Tien (1968),
and those for sulphur dioxide from a more recent paper by Chan and
Tien (1971).
-------
93
An alternative to using a single gas as a representative
pollutant is to use a combination of gases which together cover the
entire window region. One such combination could consist of S02 and
C^. C2Hn absorbs strongly in the 9 to 12 ym region and S02 absorbs
in the range 7 to 9 ym. The proportion of the gases in the mixture
can be adjusted to produce the desired absorption properties.
6.8 Radiative Fluxes in the Thermal Spectrum
In the thermal spectrum, scattering can be neglected and the
radiative transfer equation reduces to
dl (T ,y)
y — V^ - = -I (T ,y) + IK (T ) (6.8.1)
di v v bv v
The boundary conditions are
,
(6.8.2a)
I~(Tov,y) = 0 (6.8.2b)
where it has been assumed that the earth's surface is a diffuse
emitter. In this equation T is the effective optical thickness
of the earth's atmosphere, and
I+(Tv,y) E I(tv,y); y > 0 (6.8.3a)
I~(Tv,y) = I(Vy); y < 0 (6.8.3b)
Boundary condition (6.8.2a) states that at the earth's surface, the
source of radiation consists of the radiation emitted by the ground
and the reflected radiation. Boundary condition (6.8.2b) states that
there is no thermal source at the top of the atmosphere.
The solution to Eq. (6.8.1) with boundary conditions (6.8.2) can
be readily written as (Viskanta, 1966)
-------
94
, . -T /y fTv -(T -t)/y
I*(Tv,y) = I*(0,y)e v + Ibv(t)e dt/y, y > 0
o
(6.8.4a)
fTOV -(t-T
r(Tv,y) = j Ibv(t)e v dt/y , y < 0 (6.8.4b)
The radiative flux F (i ,) can be expressed as
Iv(Tv,y)ydy
,1 (-0
= 2u I Iv(Tv,y)ydy + 2TT I Iv(Tv,y)ydy
= ZTT I' I^(Tv,y)ydy + 2TT f l'(Tv,y)ydy = F^ - - F.
where
v
(6.8.5)
2ir I(T,y)ydy (6.8.6a)
F~ = -2TT T(Tv,y)ydy (6.8.6b)
o
Substituting Eqs. (6.8.4) into Eq. (6.8.6) yields
fTv d
+ 2\ Ebv(t) dt
o
CTov .
F'(T) = 2 E. (t) r Et - T ]dt (6.8.7b)
JT
•bvv"' dt "a1" v
"V
where En(t) is the exponential integral function, and Eb is the
blackbody emitted flux.
-------
95
Then, the total (integrated over the spectrum) fluxes can be
written as
F+(a) = {V(Tv)dv = 2e f Ebv(0)E,(Tv)dv
0 0
+ f 2(1 - e)Fv(0)E3(Tv)dv
o
2 Ebv(t) Ht
and
f°° f uv H
F"(z) =2 E, (t) S_ E Ft - T ]dtdv (6.8.8b)
I I DV Q u 3 v
Jo J T
T
r°° r ov
Recalling the definition of T ,
3 v
TV(Z) = f Kv(z)dz (6.8.8c)
0
it is noticed that it is not possible to interchange the order of
integration in Eqs. (6.8.8a) and (6.8.8b) unless the gaseous layer
under consideration is homogeneous, i.e., K / K (z). Thus, a
precise computation of fluxes in an inhomogeneous layer would involve
line by line integrations along the path followed by integration over
the wave numbers. Clearly, such a method is impractical, and it is
necessary to use an alternative procedure such as the Curtis-Goodson
approximation described in the preceding section. Before considering
an inhomogeneous layer, it is convenient to examine the computation of
fluxes in a homogeneous gas layer. The application of Eqs. (6.8.8a)
and (6.8.8b) to a homogeneous layer yields
F+(z) = j [eEbv(0) + (1 - e)F~(0)]2E3(Kvz)dv
o
d or r.. /_ ^^..^ (6.8.9a)
o o
-------
96
and
F-(z) = | f Ebv(0 ^ 2E3[KVU - z)]dvdS (6.8.%)
'Z 0
where h is the effective thickness of the atmosphere. In meteorological
application (Goody, 1964) E3(t) is approximated by
E3(t) = -jj-exp(-rt) with r = 1.66 (6.8.10)
Using Eq. (4.8.10) with the assumption that the widths of absorption
bands are small compared to the entire thermal spectrum (so that Eb
in a band can be replaced by the value at the center of the band) Eqs.
(6.8.9) can be written as
F+(z) = I [£Ebv + (1 - e)F-i(0)][Av1 - A^rz)]
and
where
+ IeEbv-Avj+| I EbvU)Ai[r(z • ?)]d? (6-8.Ha)
J J o ' *
fh
F"(Z) = J I Ebv.(?)Vr(? • Z)]d? (6.8.lib)
A(t) = [ [l - exp(-< t)]dv (6.8.12a)
JAM v
'Av
and
A'(t) = 3A(t)/3t (6.8.12b)
and i refers to the absorption bands of the gas, and j denotes the bands
in which no absorption occurs; Ebv. is the blackbody emitted flux at the
center of the i-th band.
-------
97
It is seen that the fluxes in a homogeneous layer can be expressed
in terms of band absorptances which have been experimentally correlated
For meteorological applications, the earth's surface is assumed to
be black (Goody, 1964), and Eqs. (6.8.11) reduce to
F+(z) = < - £ E A (rz) + f J E (W[r(z - ?)]dC
i vi JQ i i
(6.8.l3a)
and
rh
F'(z) = j I Ebv>(£)A,.[rU - z)]d? (6.8.13b)
The introduction of the emissivity given by Eq. (4.7.1) into
Eqs. (4.8.13a) and (4.18.13b) yields
F+(z) = al"[l - e(z)] + aT(g) 3e(*" U dg (6.8.14a)
o J oc,
0
and
F"(z) = j aTMC) °fc^r" Z) dg (6.8.14b)
where T is the temperature of the ground.
Equations (6.18.14a) and (6.8.14b) are the form of the radiative
transfer equations employed in meteorological investigations. It is
clear that this simple form results only when the gas layer under
consideration is homogeneous. However, the approach used by meteor-
ologists assumes that the same equations apply to an inhomogeneous
atmosphere if the emissivity is corrected for inhomogeneity. Although
this approach is empirical, its success in computing fluxes has proved
it to be a very useful alternative to laborious narrow band calcula-
tions (with the Curtis-Godson approximation).
This study used Eqs. (6.8.14a) and (6.8.14b) to compute the
infrared fluxes. The numerical scheme adopted was similar to the
ones used by Jurica (1970) and other meteorologists (e.g., Atwater,
1966). The thermal fluxes are evaluated by dividing the atmosphere
-------
98
into a number of homogeneous sublayers and the upward flux F and
the downward flux F are given by the equations
N
F'(z.) = I alV (6.8.15a)
1 n=i+l n n
F+(z.) = I aTVen + al"[l - e(z.)] (6.8.15b)
i n=2 n n s i
where N is the number of layers the atmosphere has been divided into
and the incremental emissivity Ae is defined by
Aen = e(lzn - zil} - e(lzn-l - zil> (6'8-16)
Then, the net thermal flux Ft is given by
F(zi) = F+(ZI) - F'(ZI) (6.8.17)
The emissivity e is evaluated at an effective temperature and pressure
defined by Eq. (6.7.2).
-------
99
VII. RESULTS AND DISCUSSION: ONE-DIMENSIONAL MODEL
7.1 Introduction
This chapter will present and discuss the numerical simulations
performed with the unsteady one-dimensional models. To gain a degree
of confidence in the turbulence model, the results of a test simulation
were compared against typical observations of the O'Neill Study
(Lettau and Davidson, 1957). The comparison emphasized the numerical
reproduction of important physical features rather than accurate
predictions. As the purpose of this study was to understand physical
trends, no attempt was made to "adjust" the parameters of the model to
actually reproduce observations. The relative success of a number of
simple boundary layer models (Estoque, 1963; Deardorff, 1967, Sasamori,
1970) in producing results which compare very well with observations
indicates that such an adjustment of parameters is possible.
The one-dimensional simulations emphasized the effects of pollutants
on thermal structure and pollutant dispersal. The role of elevated as
well as surface layers of aerosols and pollutants in modifying vertical
temperature and concentration profiles was investigated. The sensitivity
studies focused attention on the effect of varying aerosol parameters
and pollutant gases. A list of the one-dimensional simulations is
presented in Table 7.1.
7.2 Test Simulation
7.2.1 Initial Conditions and Parameters Used
in Test Simulation
A non-uniform grid system was used in the one-dimensional simula-
tions. The reasons for the smaller grid spacing near the earth's
surface have already been discussed. Table 7.2 presents the vertical
-------
Table 7.1 List of One-Dimensional Numerical Simulations
Section
7.3
7.3
7.4
7.4
7.4
7.4
7.4
7.4
7.4
7.4
7.4
7.5
7.5
7.5
7.5
7.5
Ref:
NP
P
NP
P
SP
TP
NP
P
SP
TP
SP
1
2
3
4
5
Radiative Gaseous
Participation Pollutant
None
Thermal & Solar NH3
None
Thermal & Solar NH3
Solar
Thermal NH3
None
Thermal & Solar NH3
Solar
Thermal NH3
Solar
None
Solar
Solar
Solar
Solar
Aerosol Properties
0)
0.90
--
0.90
0.90
—
—
0.90
0.90
—
0.90
—
0.80
0.99
0.90
0.90
f
__
0.85
—
0.85
0.85
—
—
0.85
0.85
—
0.85
—
0.85
0.85
0.85
0.50
Sex
m2/ug
10~6
—
10"6
10"6
—
—
10"6
10"6
—
10"6
—
5xlO"7
5xlO~7
5xlO"7
5xlO"7
Elevated
Layer
Height
—
--
300
300
300
300
600
600
600
600
1200
—
--
--
—
—
o
o
-------
Table 7.2 Vertical Grid Coordinates and Physical Properties and Parameters Used in Test Simulation
a) Atmospheric Coordinates
j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
z(m) 0 1.0 5.0 10.0 20.0 30.0 40.0 50.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
j 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
z(m) 800.0 900.0 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0 1900.0 2000.0 2100.0 2200.0
b) Soil Coordinates
j
z(m)
1
0
2
0,01
3
0.05
4
0.1
5
0.3
6
0.5
c) Physical Properties and Parameters
ks = 2.0 W/nTC Coreolis Parameter f = 10"1* sec"1
p = 1.5 x io3 Kg/m3 Latitude = 42.5 • N
c s = 1.0 x 103 J/Kg C u = 11.53 m/s
rs = 0.16 v = 7.52 m/s
e = 0.95 ZQ = 0.01 m
Solar Declination =11° H = 0.01
Surface temperature = 300-5 K
-------
102
grid coordinates and the physical properties and parameters utilized
in the study.
The test simulation was started at 05:00, and the initial condi-
tions were representative of early morning atmospheric conditions
during the period August-September, 1953 over the flat prairies at
O'Neill, Nebraska. Specifically, the potential temperature profile
was based on Carson's (1973) comments. Carson has made a detailed
study of the O'Neill profiles (Lettau and Davidson, 1957) and concludes
that the typical early morning profile is characterized by two gradients
The nocturnally established surface inversion is typically about 400 m
-,-3
deep and has a gradient of about 18x 10" K/m. The layer above it,
extending to about 2 Km, has a typical gradient of about 6x 10 K/m.
Thus, using this information, an initial potential temperature profile
was constructed after assuming an appropriate surface temperature. The
initial surface temperature and velocity profile corresponded to
August 25th of the O'Neill observations. The water vapor was assumed
to be uniformly distributed with a concentration of 1.5 gm/m3. The
soil temperature was taken to be uniform at the surface temperature
over the depth of the soil layer. The upper atmospheric data were
taken from McClatchey, et al. (1972). The time step used in the simu-
lation was 75 s.
7.2.2 Results of Test Simulations
In this section, the test simulation will be discussed in detail.
The results will be compared against observational data of the O'Neill
Study (Lettau and Davidson, 1957). To ascertain the validity of the
turbulence model, the results will be discussed with reference to other
recent boundary layer studies such as those of Orlanski, et al. (1974)
and Lykosov (1972). As Lykosov (1972) also used a turbulent kinetic
energy model, special emphasis will be placed on his results.
Figures 7.1 to 7.3 illustrate the evolution of the potential
temperature and eddy diffusivity over a twenty-four hour period.
They show that the model can reproduce important features of mixed
layer formation. The nocturnal surface based inversion is rapidly
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103
2S3«r-, -r -.
TEMPERRTURE
(500 •
5
5
MOO
2000
IMO
MO
EDDY DIFFUSIVITY F0R HEflT
"T r-f•-i i . | i | r | i | i | . -|- --t
500 502 5M JM 508 913 5<2 5I
-------
104
»00
2000
TEMPERflTURE
I , ,-,1T | I I I r |TT1
EDDY DIFFUSIVITY F0R HEflT
« ' • jV J~***~-*™ff-Ja'i.h^' ' ••••• . J..M j .. ... . • . •- - — - ~
319.0 311.0 312.0 313.C 314.0 315.0 316.0 317.0 311.0 519.0
40 60 10 fOt 120 140
COOT DIFFUSIVITY H2/S
EDDY 01FFUSIVITY F0R HEflT
304 306 301 319 312 3U 316 311 320
TEHPOtflTURE K
10 15 29
EDCY OIFFUSIV1TY HZ/S
Figure 7.2 Potential Temperature and Eddy Diffusivity Profiles for
Time 13:00 to 20:00 (See Figure 7.1 for arrangement).
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105
TEMPERRTURE
2500
IMO
1001
SOI
2500
1500
1MI
290) 500 502 5M 906 908 910 912 5U 516 911 920
TEHPERflTURE K
EDDY DIFFUSIVITY F0R HEflT
°t.O 1.0 2.0 9.0 4.0 5.0 6.0 7.0 1.0 ».0 10.1
EDOY 01FFUSIVITT HZ/S
TEMPERRTURE
IMO •
1000 •
MO •
EDDY DIFFUSIVITY F0R HEBT
2500 f—I—I—I—P—|—I—r—T—T—|—i—T—T-T—|—r—.—i—,—j-
2011
Ml
90S 310
TEHPERflTURE K
10 IS • 20
EOOY DIFFUSIVIU M2/S
Figure 7.3 Potential Temperature and Eddy Diffusivity Profiles for
Time 21:00 to 04:00 (See Figure 7.1 for arrangement).
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106
eroded away as the sun heats up the earth's surface. It is noted that,
in agreement with Carson's observations (1973), the stronger surface
inversion which is about 400 m deep is eroded from below in about 4
hours. The mixed layer grows more rapidly against the smaller resist-
ance of the stable layer above the nocturnally established surface
inversion, and reaches a maximum height of around 1300 meters in ten
hours (15:00). This agrees fairly well with the O'Neill observations
of August 25, 1953 as illustrated in Figure 7.4. The observed initial
inversion rise is more rapid than that predicted. This discrepancy
could be associated with the uncertainty in the initial conditions
and surface parameters.
The predicted temperature profiles are in qualitative agreement
with observations (Lettau and Davidson, 1957; Clarke, et al., 1971).
The daytime profile is characterized by a shallow superadiabatic layer
extending to less than 100 m, and the mixed layer above it has a slightly
stable gradient indicative of a countergradient heat flux. The turbu-
lent mixed layer is capped by a sharp inversion which has a typical
strength of 1.5 K.
The eddy diffusivity profiles show the increase in turbulent
activity as the mixed layer grows. The eddy diffusivity increases in
the lower portion of the mixed layer, reaches a maximum at heights
around half of the boundary layer thickness, and decreases to zero at
the edge of the boundary layer. The largest values of eddy diffusivity
occur during the day and are of the order of 100 m2/sec which is in
agreement with other investigations (Orlanski, et al., 1974; Deardorff,
1967).
The evening and nighttime potential temperature and eddy diffusivity
profiles dxhibit some interesting features. In about two hours after
the surface starts cooling at around 14:00 hours, turbulence is virtually
extinguished in the bulk of the boundary layer. This is indicated by
the rapid decrease in the mixed layer height and the collapse of the
eddy diffusivity profiles. This unusual behavior of the nocturnal
boundary layer has been observed experimentally by Kaimal, et al.
(1975) in Minnesota, and has also been theoretically predicted by
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107
1400
1200
1000
§
5
a
X
600
200
A O'NEILL OBSERVATIONS
AUGUST 25, 1953
0
PREDICTED THIS WORK
6 8 10
TIME AFTER SUNRISE (HR>
12
Figure 7.4 Comparison of Mixed Layer Heights
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108
Orlanski, et al. (1975). The typical boundary layer variation of the
O'Neill Study is illustrated in Figure 7.5. The discontinuities in
the boundary layer depth near sunrise and sunset are indicative of
the sharp changes in turbulent activity associated with high surface
cooling rates. It is interesting to note that the model used in this
study predicted this physical phenomenon only when the counter-gradient
parameter was included in the energy equation. Otherwise, the mixed
layer continued to grow even after sunset. The importance of the
counter-gradient flux seems to indicate that the upward transport
of heat during the day is caused by surface generated buoyant plumes
penetrating the weakly stable mixed layer. The turbulence in the
mixed layer being associated with buoyant parcels is extinguished as
soon as the surface cools and stops producing buoyant plumes. The
numerical model of this study predicted negligibly small nocturnal
boundary layers. Although there is not enough observational data to
disprove the validity of this result, it was necessary to set a minimum
value of 200 meters for the nighttime boundary layer to prevent the
formation of unrealistically large surface inversions. The value of
200 meters was based on the Wangara data (Clarke, et al., 1971) which
indicated that the nocturnal boundary layer varied from around 150
meters to 250 meters (Melgarejo and Deardorff, 1974) in height.
Admittedly, this treatment of the boundary layer is empirical; how-
ever, there is no satisfactory theory for the stable boundary layer
and the scope of this study did not permit a further investigation
of the problem.
Figure 7.6 shows the potential temperature variation at 50 m
over a simulation period of 12 hrs. It is seen that the temperature
variation shows reasonably good agreement with the O'Neill observations
Figures 7.7 and 7.8 reproduce some results obtained by Lykosov
(1972). It is evident that the results show some noticeable differ-
ences from those of this study. According to Lykosov's model, the
mixed layer height varies by a factor of about two, from 600 meters
in the night to about 1200 meters in the day. This study shows the
boundary layer height varying by a factor of about six, from 200
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109
A
2000 r-
1600 -
1200 -
800 -
Apparenf noon
Time of day
Figure 7.5 The Mean Boundary Layer Thickness , Deduced for the
O'Neill Data and Plotted with Standard Errors as Functions
of Time of Day, t, in Mean Solar Time (Carson, 1973).
-------
no
T(K)
312
310
308
306
304
302
A C^NEILL OBSERVATIONS
AUGUST 25, 1953
0
PREDICTED THIS WORK
2468
TIME AFTER SUNRISE (HR)
10 12
Figure 7.6 Comparison of Temperature Variation at 50 M
-------
Ill
meters in the night to about 1300 meters in the afternoon. This wider
variation of the mixed layer height agrees more closely with observa-
tions (Clarke.et al., 1971). Although the observations indicate that
weak turbulence may extend to greater heights than the nocturnal
boundary layer, turbulent activity above the boundary layer cannot
be as large as that predicted by Lykosov. It is clear from Figure
7.7 that the surface inversions extend to heights of 600 meters
indicating that the turbulence does not decay as rapidly as it should
above the typical nocturnal boundary layer height of 200 meters.
There is a significant difference between the eddy diffusivity
profiles predicted by Lykosov and those obtained in this study.
Figure 7.8 shows that the heights at which the turbulent exchange
coefficient attains a maximum are around 100 meters during the course
of a diurnal cycle. On the other hand, this study predicts a very
wide variation of the height at which the eddy diffusivity becomes a
maximum, and as the height is approximately half the mixed layer
thickness it is dependent on the variation of the mixed layer thick-
ness.
The test simulation has shown that the model predictions are
reasonably consistent with observational as well as theoretical
studies. It is felt that a detailed investigation of the effect
of initial conditions as well as surface parameters will help to
improve the capabilities of the model.
As an additional test of the turbulence model, a numerical
experiment was conducted to study stress profiles in a neutral
boundary layer. The potential temperature was assumed to be uniform
in the boundary layer, and the velocity field was initialized with
an arbitrary logarithmic profile. The unsteady momentum equations
were used to obtain the steady state stress profiles by allowing
the time dependent solutions to approach steady state. The absence
of thermal stratification made it possible to use a simple mixing
length model which was similar to that suggested by Blackadar (1962).
As the boundary layer height scales with u*/f under neutral conditions,
£ was assumed to be given by
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112
f, km
S.S
o.f
-*
ff.teg
Figure 7.7 Profiles of the Potential Temperature 0(z) = (ja - y)z +
v(z) for various hours of the day: (1) 2; (2) 6; (3) 10;
(4) 14; (5) 18; (6) 22 h (Lykosov, 1972). (v is Eddy
Diffusivity)
Figure 7.8 Isopleths of the Turbulent Exchange Coefficient v. The
Dashed Lines are Profiles of v for Three Times of Day
(0, 2, and 4 hrs) (Lykosov, 1972).
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113
1.0 i-
0.8
0.6
z/H
m
0.4
0.2
0
-0.8
WYNGAARD, ET AL
PRESENT MODEL
-0.4
0.4
U'W'/U
v'w/uS
Figure 7.9 Turbulent Shear Stress Profiles in a Neutrally Stratified
— - Atmosphere
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114
£ = C* K(Z + z0)/(l + C|* K(Z + z0)/*0) (7.3.1)
where
£0 = 0.009 C]* u*/f (7.3.2)
Figure 7.9 shows the stress profiles obtained. It is seen that they
compare remarkably well with the results from a considerably more
complicated turbulence model formulated by Wyngaard, et al. (1974).
However, the velocity defect profiles did not agree as well with
those of Wyngaard, et al. Further adjustment of the constant in the
formula for £0 will be necessary for improved agreement.
7.3 Radiative Effects of Pollutants
7.3.1 Introduction
This section will describe the numerical simulations performed
to investigate the radiative effects of gaseous and particulate
pollutants on the thermal structure and pollutant dispersal in the
planetary boundary layer. As pollution episodes occur when the winds
are low, the initial velocities were taken to be one fourth of those
used in the test simulation. The initial temperature structure
(temperature gradients) was identical to that of the test simulation.
Since the purpose of the simulations was to study the effects of
pollutants on the planetary boundary layer as a whole, it was decided
to use surface parameters which were not exclusive to the urban area.
The surface parameters used are based on those suggested by Pandolfo,
et al. (1971) and are presented in Table 7.3
The aerosol parameters are based on the calculations of Hansen
and Pollack (1970). The aerosol extinction coefficient was chosen
so that with the average mass loadings obtained in the simulations
the optical thickness in the solar spectrum would be in the range 0.1
to 0.2. These values of optical thickness are based on the measure-
ments of Herman, Browning and Curran (1971) which place the mean
atmospheric optical thickness in the visible spectrum around 0.1.
The single scattering albedo of 0.90 corresponds to a slightly
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115
Table 7.3 Summary of Surface Parameters and Pollutant Properties
Used in Simulations of Section 7.3
a) Surface Parameters
ks = 2.0 W/m/c Latitude = 42.5°N
p = 1.5 x 103 kg/m3 Solar Declination - 1TN
o
c = 1.0 x io3 J/kg/c
rc = 0.2
o
et= 1.0
H = 0.1
ZQ = 0.01 m
u = 2.88 m/s
v = 1.88 m/s
Surface Temperature = 285 K
Height of Elevated Source = 100 m
S = 0.05 |sin (irt/24)| ugm/m3/s (Aerosol and gas)
b) Aerosol Properties
3ex = 10"6 mVugm
f = 0.85
u = 0.90
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116
absorbing aerosol--the amounts of backscatter and absorption are
equal in a layer of an optical thickness of 0.5. As ammonia absorbs
strongly in the 8-12 ym region, it was used as a representative
pollutant. Aerosol properties are also listed in Table 7.3.
The pollutant source was assumed to be elevated in order to
model industrial emissions which are the most important pollutant
sources. It is noted that the source is allowed to vary during the
course of the day. The source strength increases during the day,
reaches a peak 12 hours after sunrise and decreases during the night.
This variation of the pollutant source is a simple representation
of industrial emission during the course of the day
(Roberts, et al., 1971). The lack of data and the objectives of
this study did not warrant a more detailed treatment of pollutant
emissions.
Two numerical simulations were conducted; in the first one the
pollutants were taken to be non-participating, and in the second one
the pollutants (aerosol and gas) were allowed to participate radia-
tively. As a large number of variables are involved in the simulations,
it is impractical to present all the numerical results of the experi-
ments. Only some selected results which are relevant to the under-
standing of the radiative effects of pollutants are presented. It
is noted that most of the results are given in tabular form in order
to emphasize the very small differences between the variables of the
non-participating simulation and those of participating simulation.
7.3.2 Effect of Pollutants on Thermal Structure
of Boundary Layer
Table 7.4 illustrates the effect of radiative participation of
pollutants on the thermal and solar fluxes at the surface. It is
seen that aerosols decrease the solar flux at the surface by as much
as 25% at large zenith angles (17:00 of second day). At small zenith
angles the attenuation of solar flux is around 5% (11:00 of second
day). The average attenuation of about 15% agrees well with recent
observations by Rouse, et al. (1973) over Hamilton, Ontario. Thermal
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117
Table 7.4 Comparison of Thermal and Solar Fluxes at the Surface for
Simulations with Participating (P) and Non-Participating
(NP) Pollutants
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
Solar Flux (W/mz)
NP P NP-P
237.60
550.4
738.5
734.9
543.0
233.5
0
0
0
0
0
0
233.5
540.0
725.2
722.2
534.1
229.8
0
0
0
0
0
216.2
532.8
726.4
720.5
516.6
198.4
0
0
0
0
0
0
184.9
491.8
687.9
682.0
477.2
172.6
0
0
0
0
0
21.4
17.6
12.1
14.4
26.4
35.1
0
0
0
0
0
0
48.6
48.2
37.3
40.2
56.9
57.2
0
0
0
0
0
Thermal Flux (W/mz)
NP P NP-P
294.0
305.7
320.3
331.9
337.2
336.0
328.0
321.1
316.5
313.3
310.6
308.2
311.6
329.0
344.3
351.6
354.7
352.3
343.3
336.2
331.2
327.7
324.6
294.3
308.6
327.6
344.1
353.8
356.0
351.3
348.2
346.7
345.5
344.3
342.8
345.2
362.4
379.7
388.4
393.7
392.7
386.6
382.4
380.2
378.6
377.1
- 0.3
- 2.9
- 7.3
-12.2
-16.6
-20.0
-23.3
-27.1
-30.2
-32.2
-33.7
-34.6
-33.6
-33.4
-35.4
-36.8
-39.0
-40.4
-43.3
-46.2
-49.0
-50.9
-52.5
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118
participation by ammonia increases the downward thermal flux by a
maximum of about 16% at 03:00 of the second day. The average
increase of about 10% is consistent with observations made by Oke
and Fuggle (1972) who measured an increase in the downward thermal
flux of approximately 10 percent in Montreal, Canada.
Table 7.5 lists the potential temperatures for the simulations.
It is seen that during the first 4 hours (7:00 to 11:00) the poten-
tial temperature at 1 m is slightly lower for the participating
simulation than for the non-participating simulation. An examination
of Table 7.6 shows that during the first day the surface temperature
for simulation P is cooler due to the decrease in solar flux. After
19:00 of the first day the surface temperature as well as the tempera-
ture at 1 m is warmer for the simulation with participating pollutants
for the remaining period of simulation. The temperature excess
reaches a maximum of 3.25 C at the surface at 03:00 of the second
day. During the night, the substantial increase of downward thermal
flux by the participating pollutants leads to an increase of the
surface temperature. However, during the day the increase in solar
attenuation is not accompanied by a decrease in surface temperature.
The surface temperature for the simulation with participating
pollutants is about 0.15 C on an average higher than that with non-
participating pollutants during the second day. An explanation for
this apparent anomaly will be provided in the next paragraph. It is
noted that while solar absorption causes a slight increase in poten-
tial temperature (-0.18 C) at 100 m, the large concentration of
pollutants at the source located at 100 m leads to cooling at night.
The temperatures in the boundary layer are higher on the second day
than those of the first day because the thermal structure at the
beginning of the second day is considerably different from that
initially. It is clear that periodicity in the thermal structure
variation can be obtained by adjusting the initial conditions.
However, there is no necessity to force the temperature (or velocity)
field to repeat itself 24 hours later because experimental observa-
tions (Clarke, et al., 1971) do not exhibit "precise" cyclic behavior.
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119
Table 7.5 Comparison of Potential Temperatures (in K) at 1 m and 100 m
for Simulations with Participating (P) and Non-participating
(NP) Pollutants
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
285.54
290.62
294.66
296.63
296.48
294.36
290.60
288.61
287-19
286.14
285.23
284.41
287.79
294.40
298.04
299.05
298.60
296.46
292.72
290.73
289.27
288.17
287.16
z = 1 m
P
285.23
290.37
294.59
296.69
296.54
294.31
291.02
289.28
288.25
287.46
286.82
286.27
288.82
294.88
298.30
299.39
298.88
296.66
293.72
292.14
291.19
290.46
289.87
NP-P
0.31
0.25
0.07
-0.06
-0.06
0.05
-0.42
-0.67
-1.06
-1.32
-1.59
-1.86
-1.03
-0.48
-0.26
-0.35
-0.28
-0.20
-1.00
-1.41
-1.92
-2.29
-2.71
NP
286.42
288.63
291.86
294.04
295.06
295.00
294.71
294.71
294.50
294.03
293.57
293.07
292.09
292.72
295.58
296.83
297.45
297.75
296.94
296.93
296.76
296.30
295.88
z = 100 m
P
286.50
288.48
291.83
294.16
295.24
295.13
294.76
294.76
294.41
293.91
293.38
292.85
291.70
293.33
296.03
297.35
298.02
297.75
297.34
297.31
296.92
296.41
295.87
NP-P
-0.08
0.15
0.03
-0.12
-0.18
-0.13
-0.05
-0.05
0.09
0.12
0.19
0.22
0.39
-0.61
-0.45
-0.52
-0.57
-0.03
-0.40
-0.38
-0.16
-0.11
0.01
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120
Table 7.6 Comparison of Surface Temperatures (in K) and Diffusivities
(in m2/s) at 10 m for Simulations with Participating (P)
and Non-Participating (NP) Pollutants
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
Surface Temperature
NP P
285.76
291.75
296.23
289.01
297.18
294.14
289.95
288.10
286.81
285.81
284.93
284.12
288.06
295.43
299.36
300.16
299.15
296.20
292.05
290.21
288.89
287.82
286.84
285.39
291.45
296.13
298.03
297.17
294.05
290.44
288.89
287.92
287.17
286.56
286.03
289.03
295.80
299.48
300.41
299.29
296.39
293.17
291.75
290.87
290.19
290.19
Diffusivity of Heat
NP P
0.90
1.48
1.72
1.75
1.58
1.10
0.43
0.59
0.55
0.56
0.56
0.58
0.89
1.34
1.70
1.77
1.59
1.06
0.40
0.58
0.55
0.55
0.56
0.83
1.46
1.70
1.73
1.56
1.01
0.48
0.60
0.59
0.59
0.60
0.61
0.91
1.33
1.71
1.74
1.54
0.83
0.48
0,52
0.60
0.60
0.60
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121
This study indicates that the predominant effect of aerosols
is that of warming of the earth-boundary layer system. Aerosols
decrease the solar flux reaching the earth's surface, an effect
which, at first glance, would tend to decrease the surface tempera-
ture. However, a more careful examination of the surface energy
balance shows that the reduction of solar flux need not be accom-
panied by a decrease in the surface temperature. It is seen from
Tables 7.4 and 7.8 that the evaporative flux is a large fraction of
the incoming solar plus thermal fluxes. As a relatively small
fraction of the solar radiation contributes to sensible heating,
the surface temperature is not very sensitive to changes in the
incident flux. Thus, if all other physical processes, such as
surface turbulence, were unaffected by aerosols, a decrease in the
solar flux would cause a relatively small decrease in the surface
temperature in order to bring the surface energy fluxes into balance.
However, the readjustment of the surface energy balance can be
accomplished by changes of the temperature of the surface air layer,
and these changes would be sufficient to offset the small decrease
in surface temperature which would be necessary otherwise. The
results of this study show that the aerosols by absorbing solar
radiation in the surface layer cause the necessary change in the
thermal structure of the layer next to the earth's surface to
prevent the decrease in the surface temperature. By direct absorp-
tion of solar radiation aerosols lead to an increase in the tempera-
ture of the surface air layer. This temperature increase causes two
effects which are equivalent to the reduction in surface temperature.
First, the higher temperature above the surface decreases the upward
turbulent heat flux by decreasing the temperature gradient causing
the heat flux. The reduction in the unstable temperature gradient
decreases the eddy diffusivity of heat of the surface layer which in
turn decreases the turbulent heat flux.
An examination of Table 7.8 shows that the total downward flux
(solar plus thermal) is altered very slightly by radiative participa-
tion of pollutants. The reduction in solar flux (see Table 7.4) is
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122
almost matched in magnitude by the increase in downward thermal flux.
This result is consistent with measurements by Rouse, et al. (1974)
over Hamilton, Ontario. Thus, the relative insensitivity of the
surface temperature to the reduction in the incident radiation
together with the small decrease in the total downward radiation
allows the surface temperature to be determined primarily by the
aerosol induced heating of the surface layer.
The effectiveness of the heating caused by absorption of solar
radiation by aerosols is evident from Table 7.7 which compares the
solar heating term against the turbulent heating term in the energy
equation. It is seen that radiative participation by aerosols
increases the solar heating term by a factor of three. Furthermore,
at some heights the magnitude of the solar heating term is almost
the same as that of the turbulent heating term.
The effect of radiative participation by aerosols on the stability
of the surface layer is evident from Table 7.6. During the day, the
diffusivity at 10 m is decreased by radiative participation. During
the night, the increase in the thermal radiation and the accompanying
increase in surface temperature causes the surface layer to be less
stable. This effect is reflected in the increase of turbulent
diffusivities during the night as shown in Table 7.6.
The modification of surface fluxes by radiative participation is
clearly indicated in Tables 7.8 and 7.9. It is seen that the daytime
turbulent fluxes are reduced due to radiative participation. The
reasons for this have already been considered. Although the stability
of the surface layer is decreased by radiative participation during
the night the substantially higher temperatures at the surface lead
to a reduction of the downward turbulent fluxes. The soil fluxes are
smaller with radiative participation than without because the higher
soil temperatures allow less energy to flow into the soil layer during
the day, and the higher surface temperatures during the night reduce
the energy transported towards the surface. The evaporative fluxes
are higher for simulation P than for simulation NP because the higher
surface temperatures increase the saturation water vapor pressure at
the surface and thus the concentration.
-------
123
Table 7.7 Comparison of Terms in Energy Equation for Simulations
with Participating (P) and Non-Participating (NP)
Pollutants, Time 1s 12:00 hours of Second Day.
z(ra)
1
5
10.0
20.0
30.0
50.0
100.0
200.0
300.0
400.0
- SFs 1
az Pcp
NP
5.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
x 105(K/s)
P
12.0
12.0
12.0
12.0
12.0
12.0
12.0
11.0
11.0
11.0
9<% n
[* if t O\J ^
3z LKH I9z " Yc'
NP
68.0
11.0
18,0
25.0
29.0
35.0
38.0
38.0
35.0
29,0
)] x 105(K/s)
P
62,0
10.0
15,0
22,0
27.0
32,0
35.0
35.0
33.0
27,0
-------
124
Table 7.8 Comparison of Energy Fluxes (in W/m2) at the Surface for
Simulations with Participating (P) and Non-participating
(NP) Pollutants
Time
07:00
09:00
11:00
13:00
15.: 00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
Solar Plus
NP
531.6
856.1
1058.7
1066.7
880.2
569.5
327.7
321.1
316.5
313.3
310.6
308.2
545.1
869.0
1069.5
1073.8
888.8
582.1
343.3
336.2
331.2
327.7
324.6
Thermal
P
510.5
841.4
1054.0
1064.6
870.4
554.5
361.3
348.2
346.7
345.5
344.3
342.8
530.1
854.2
1067.6
1070.4
870.9
565.3
386.6
382.1
380.2
378.6
377.1
NP
74.9
176.2
274.4
322.0
284.5
168.2
35.3
19.3
6.9
1.1
0
0
28.2
137.3
288.7
345.1
295.9
163.7
31.0
15.3
2,7
0
0
Latent
P
70.9
171.1
269.4
320.9
282.0
156.8
39.0
20.3
12.3
6.7
3.7
1.8
14.0
140.2
309.5
353.6
297.9
136.3
40.1
21.1
12.8
7.1
4.0
-------
125
Table 7.9 Comparison of Energy Fluxes (in W/m2) at the Surface for
Simulations with Participating (P) and Non-participating
(NP) Pollutants
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
- 64.0
-164.2
-182.8
-145.2
- 78.0
- 2.6
74.0
58.2
50.1
45.5
44.0
43.0
-107.0
-214.4
-185.4
-138.2
- 77.8
- 5.5
67.5
50.2
42.4
40.1
39.2
Soil Flux
P
- 52.9
-161.9
-187.7
-148.8
- 78.3
- 3.0
60.2
44.4
34.4
29.1
25.7
23.6
- 90.9
-202.8
-172.8
-137.0
- 73.8
- 11.9
43.2
27.4
18.8
13.7
10.5
Turbulent
NP
- 14.1
-104.0
-164.0
-151.8
- 75.7
20.0
33.0
30.4
23.8
20.5
18.9
18.1
- 18.8
- 84.0
-139.4
-130.0
- 61.2
23.1
32.2
30.7
23.8
21.0
19,8
Flux
P
- 10.3
- 98.3
-159.7
-147.0
- 68.1
22.8
30.5
22.4
20.6
17.5
15.8
14.8
- 14.0
- 75.8
-128.8
-117.5
- 44.6
20.4
28.7
21.9
19.4
16.7
15,2
-------
126
The effect of pollutant participation on the thermal structure
is evident from Figures 7.10 and 7.11. Direct solar energy absorption
by aerosols increases the temperature of the boundary layer by about
0.1 C on the first day, and about 0.55 C on the second day of simula-
tion. These higher temperatures persist during the night as seen in
Figure 7.11. The nocturnal surface temperature is about 2 C higher
at 05:00 of the second day and about 2.8 C higher at 21:00 on the
same day. This excess temperature is caused by the increase in
downward thermal radiation due to gaseous pollutants. It is seen
that in the vicinity of the elevated source at 100 m, the high
pollutant concentrations cause a small degree of cooling.
7.3.3 Effect of Pollutants on the Earth-Boundary
Layer System
The effective solar reflectance or albedo of the earth-boundary
layer system can be used to determine the overall effects of pollutants
participating in the solar spectrum. If the addition of pollutants is
accompanied by a decrease in albedo the resultant effect is one of
warming, and an increase in albedo caused by pollutants would lead to
a relative cooling of the earth-boundary layer system. It is clear
that while aerosol and water vapor absorption tend to decrease the
albedo, aerosol and Rayleigh scattering tend to increase the albedo.
Figure 7.12 illustrates the effect of aerosol participation on the
effective albedo of the earth-boundary layer system. It is seen that
the albedo reaches a minimum at solar noon and increases with zenith
angle. This result is consistent with the findings of Bergstrom
and Viskanta (1973). The albedo with non-participating pollutants
is lower than the assumed surface albedo of 0.2 because water vapor
absorption in the boundary layer dominates Rayleigh scattering (Lacis
and Hansen, 1974) under clear skies. It is noted that aerosol
participation increases the effective albedo at large zenith angles.
This increase is caused by two factors. As the aerosols used in
this study scatter predominantly in the forward direction (f = 0.85)
backscattering becomes important only when the optical path length is
-------
127
700
600
500
400
Z(M)
300
200
100
0
0
0,1
13:00
(Isi DAY)
13:00
(2ND DAY)
0,2
0,3
0,5
0,6
A9p.Np(C)
Figure 7.10 Daytime Potential Temperature Excess of Participating
Simulation over Non-Participating Simulation
-------
128
ouu
700
600
500
Z(M)
400
300
9m
M
M
»
•
••
•
>
^~
100 •
0
21:00 (2ND DAY)
05:00" QSTDAY)
0
1
2
3
A0p.Np(C)
Figure 7.11 Nighttime Potential Temperature Excess of Participating
Simulation over Non-Participating Simulation
-------
129
large. Furthermore, as the increase in the zenith angle is accom-
panied by an increase in the water vapor path length above the
boundary layer, the incremental water vapor absorption in the boundary
layer is decreased. At low zenith angles around noon aerosol absorp-
tion of solar radiation leads to a decrease in the effective reflect-
ance as seen in Figure 7.12. An important consequence of this
decrease in albedo is the net heating of the earth-boundary layer
system during the hours around noon. As 60% of the total solar
energy available during a day is incident on the earth-atmosphere
system during the period denoted by AB in Figure 7.12, the warming
effect of aerosols can be significant. The effect of this warming
on the surface temperature is dependent on the proportions of latent
and sensible heating at the surface as well as amount of convective
contact between the aerosol layer and the surface.
It is interesting to compare the results of this study with
those of similar investigations (Bergstrom and Viskanta, 1973;
Atwater, 1975). Bergstrom and Viskanta (1973) as well as Atwater
(1975) while predicting warmer atmospheric temperatures find that
solar attenuation caused by aerosols leads to a decrease in the
surface temperature. Bergstrom and Viskanta (1973) show that the
aerosol cooling effect on the surface can be as large as 2 C over
a period of two days. It is evident from the discussion of the
preceding paragraphs that the effect of aerosol induced boundary
layer warming on the surface temperature is dependent on the degree
of solar attenuation as well as the surface parameters. Bergstrom
and Viskanta used relatively high pollutant mass loadings which lead
to a greater reduction of the solar fluxes (12% at solar noon). As
they neglected evaporation, the surface temperature was sensitive
to this reduction in solar flux, and consequently their surface
temperature was decreased. Thus, it is clear that it is not possible
to draw any conclusions on the effect of aerosols on the surface
temperature without qualifying the conclusions with statements about
surface parameters, radiative properties of pollutants, concentra-
tion and distribution of pollutants, latitude and time of the season.
-------
130
0,1
06:00
SOLAR FLUX AT THE TOP OF PEL
-I 1000
10:00 12:00 14:00
LOCAL TIME (HR>
Figure 7.12 Effect of Radiative Participation on Earth-Atmosphere
Albedo
-------
131
7.3.4 Effect of Pollutants on Pollutant Dispersal
Figure 7.13 shows the evaluation of the aerosol concentration
profiles over a period of 16 hours. The figure has been presented
primarily to illustrate the effect of introducing pollutants into
the boundary layer from an elevated source. It can be seen that the
aerosol concentration reaches a maximum at the source and decreases
above and below the source height of 100 m. During the day the
aerosol concentration is almost uniform through the boundary layer.
The stability of the nocturnal boundary layer leads to large
(~700 pg/m3) aerosol concentrations at the source. As turbulent
activity is confined to the lowest 200 m, pollutants injected from
the source accumulate in the surface layer. Increased vertical
mixing in the nocturnal boundary layer only serves to transfer
pollutants from the source to the ground. The relevance of this
statement will become clear when the effects of pollutants on
pollutant dispersal are discussed in the next paragraph.
Solar heating due to aerosols during the day decreases the
instability of the surface layer, and the increase in the thermal
radiation during the night tends to decrease the stability of the
surface layer. Thus, radiative participation by pollutants decreases
the amount of pollutants transported from the source to the surface
during the day, and increases the amount transported during the
night. These effects are evident from an examination of Table 7.10.
With pollutant participation the aerosol concentration is lower by
a maximum of about 64 yg/m3 (7%) during the day. The effects of
pollutants on pollutant dispersal are more significant during the
nighttime. Radiative pollutant participation increases the aerosol
concentration at 1 m by as much as 90 ug / m3 (18%). At 200 m the
aerosol concentration is lower with pollutant participation than
without. The higher surface temperatures caused by pollutants lead
to increased vertical mixing and lower pollutant concentrations at
200 m.
Radiative participation by pollutants did not affect the growth
of the mixed layer to an appreciable extent in the simulations
-------
132
RER0S0L C0NCENTRRTI0N
RER0S0L C0NCENTRRTI0N
2090
1500
M « 70 75 «0 85
RER9SO. UNCCNTftnniN UG/N3
7i n M too «o
RER9S8L MNCENTWHIW IX/H3
AEROSOL C0NCENTRRTI0N
5
*
MO
2500
50
150
170
RER0S0L C0NCENTRRTI0N
i i i i i i i i i i i i i i i i i i i > i ' i
201 MO 410 MO
fCFWSBL CWCENTRfUim UC/H3
TOO
Figure 7.13 Aerosol Concentration Profiles for Time 05:00 to 20:00.
Top Row is for Time (Left to Right) 05:00 to 08:00 and
Bottom Row is for Time 09:00 to 12:00.
-------
133
Table 7-10 Comparison of Aerosol Concentration (in yg/m3) for
Simulations with Participating (P) and Non-participating
(NP) Pollutants
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
55.0
105.0
122.0
130.0
145.3
175.3
188.3
188.6
208.3
275.8
344.7
407.3
504.4
805.1
414.5
298.6
292.8
315.3
330.9
331.8
347.9
414.2
477.5
z = 1 m
P
53.4
106.4
124.6
131.4
146.0
177.5
188.4
188.7
238.8
322.4
407.5
487.3
593.5
780.6
350.1
280.5
277.6
302.5
322.5
323.7
385.9
476.6
568.8
NP-P
1.6
- 1.4
- 2.6
- 1.4
- 0.7
- 2.2
- 0.1
- 0.1
-30.5
-46.6
-62.8
-80.0
-89.1
24.5
64.4
18.1
15.2
12.8
8.4
8.1
-38.0
-62.4
-91.3
NP
50.3
95.0
115.0
123.5
138.3
163.7
165.7
165.7
165.7
165.7
165.7
165.7
167.5
487.0
377.0
289.4
284.5
301.2
300.5
300.5
300.5
300.5
300.5
z = 200 m
P
50.1
94.5
117.3
124.7
138.9
164.3
164.1
164.1
164.1
164.1
164.1
164.1
165.4
437.5
321.5
272.0
268.9
281.3
282.0
282.0
282.0
282.0
282.0
NP-P
0.2
0.5
- 2.3
- 1.2
- 0.6
- 0.6
1.6
1.6
1.6
1.6
1.6
1.6
1.1
49.5
55.5
17.4
15.6
19.9
18.5
18.5
18.5
18.5
18.5
-------
134
described in this section. However, radiative participation was
found to be very important in determining the mixed layer height
when pollutant layers were situated at heights above the surface.
The simulations investigating these effects are described in the
next section.
7.4 Effect of Elevated Layers of Pollutants
Pollutants which are vertically dispersed when the mixed layer
grows during the day, can form elevated layers when the boundary layer
contracts around sunset. Elevated pollutant layers could also result
from the emission of pollutants from tall smokestacks which can be
as high as 320 m (Cole and Lyons, 1972). As these elevated layers
are associated with stable regions (above the mixed layer) where
turbulence is inhibited, their radiative participation can lead to
a significant modification of the thermal structure of the stable
layer. Since the stability of the inversion capping the mixed layer
controls the growth of the mixed layer, elevated pollutant layers
can affect pollutant dispersal and fumigation as well as the tempera-
ture of the mixed layer.
In this section, the simulations performed to investigate the
effect of elevated pollutant layers will be discussed. The pollutant
parameters are based on those calculated by Hansen and Pollack (1970)
for typical aerosols. Table 7.11 summarizes the particulars of the
simulations and the pollutant parameters used. The pollutant layers
were chosen to be 300 m thick and the maximum pollutant concentration
was taken to be 400 ug/m3. The relatively large value of 400 ug/m3
was used to highlight radiative effects. The vertical distribution
of pollutants within the elevated layer is shown in the initial
profiles for the pollutant concentrations.
The first set of simulations were performed with the lower edge
of the pollutant layer located at 300 m. The results of four differ-
ent simulations are shown in Figures 7.14 to 7.17. These figures
illustrate the evolution of the potential temperature profiles over
a 16 hour period. The temperature profiles of simulation NP
-------
135
Table 7,11 Summary of Simulations with Elevated Layers
a) Aerosol Parameters
3 = io"6 Jn!
pex IU yg
f = 0.85
u = 0.90
b) Pollutant gas: NH3 (Ammonia)
c) Simulations:
NP - No participation by gaseous pollutants (thermal)
and aerosols (solar)
P - Participation by gaseous pollutants and
aerosols
SP - Participation by aerosols only
TP - Participation by gaseous pollutants only
-------
136
.•500
TEMPERflTURE
TEMPERflTURE
ZMO r- , i
be » » .1
TENPCRflTlAE K
a>2M2H2M2»2M2M2M3N»23M
roramucK
5
i
TEMPERRTURE
TEMPERflTURE
Ml
TEtfERRTUIE K
TCWERRTUC K
Figure 7.14 Potential Temperature Profiles for Time 05:00 to 08:00.
Top Left is Simulation NP, Top Right is Simulation P,
Bottom Left is Simulation SP and Bottom Right is Simula-
tion TP. Elevated Pollutant Layer at 300 m.
-------
137
2900
20M
TEMPERRTURE
TEMPERRTURE
3M 292 29a 296 298 XS 302 SW
2MO
TEMPERRTURE
TEMPERRTURE
2MO
2001
INI
2M
TQVQnTUKK
t S» 2M
torownitt K
M MI
Figure 7.15 Potential Temperature Profiles for Time 09:00 to 12:00.
Elevated Pollutant Layer at 300 m (See Figure 7.14 for
arrangement).
-------
138
TEMPERATURE
TEMPERflTURE
ISOO
SOI
2MI
2M 396 298 300
TEHPERflTUtE K
K.I Mt.l 2M.I JJ7.I 2M.I 2M.I Xt.t Ml.I 5U.O N3.I 5M.I
TOCCffiTURE K
TEMPERflTURE
TEMPERflTURE
. . .>^. . . 1 , f | !,...
a*.i tn.t at.i Si.t M.I
2MO
.i »«.i 9ti.« su.i JH.I JM.I
K
Figure 7.16 Potential Temperature Profiles for Time 13:00 to 16:00.
Elevated Pollutant Layer at 300 m (See Figure 7.14 for
arrangement).
-------
139
2900
2000
TEMPERATURE
TEMPERflTURE
:SC2922M2962»XJOJ
-------
140
(non-participating pollutants) show the usual development with the
nocturnally established inversion being eroded away as the mixed layer
grows during the day. The surface starts cooling in about 10 hours
and a surface inversion is re-established. The results of simulation
TP show the cooling effect of pollutants which participate only in
the thermal spectrum. The cooling is quite pronounced within 4 hours,
and the mixed layer grows rapidly as the stable layer above it is
cooled. At the same time the thermal participation of the pollutants
creates a sharp inversion (~2 K) at the top of the mixed layer. This
strong inversion tends to slow down the growth of the mixed layer.
The formation of this inversion has been predicted in theoretical
studies by Atwater (1970) and Bergstrom and Viskanta (1972). Observa-
tions (Bornstein, 1968; Rouse, et al., 1974) of elevated inversions
over industrial areas lend some validity to this theoretical prediction.
Simulation SP (aerosol participation only) shows the effects of
aerosol heating. In about 8 hours the heating leads to the formation
of a highly stable layer at a height of around 400 meters. The layer
above the stable layer is marked by a negative potential temperature
gradient. It is evident from Figure 7.15 that the sharp temperature
gradient caused by aerosol heating effectively impedes the growth
of the mixed layer. It is only after 9 hours that the mixed layer
is able to penetrate the inversion, and the growth thereafter is
relatively rapid as the layer above is unstable.
The results of simulation P (participation by aerosols and gases)
show the opposing effects of gases and aerosols. The heating caused
by aerosols is counteracted to a certain extent by the cooling induced
by the gaseous pollutants. Figure 7.15 shows that the pollutant
layer heating up at the bottom and cooling at the top. The stabilizing
effect of heating prevents the mixed layer from growing as rapidly
as in the case when there is no pollutant participation. Once the
mixed layer penetrates the inversion it grows at a rate comparable
with that of simulation NP.
Tables 7.12 and 7.13 compare the thermal and solar fluxes at
the surface for simulations P, NP, SP and TP. As expected, solar
-------
141
Table 7.12 Comparison of Thermal Fluxes (1n W/m2) at the Surface
(Elevated layer at 300 m)
Time
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
00:00
01:00
02:00
03:00
04:00
NP
292.8
293.9
299.1
305.7
312.9
320.2
326.8
331.8
335.4
337.2
337.4
336.0
332.7
327.7
324.0
321.1
318.5
316.5
314.8
313.3
311.9
310.6
309.4
Thermal
P
314.3
315.1
319.8
326.3
333.6
341.3
349.0
355.2
358.9
360.7
360.8
359.1
355.7
351.1
347.7
345.1
342.9
341.2
339.7
338.3
337.1
336.0
335.0
Fluxes
SP
292.8
293,3
297.9
304.2
311.4
319,0
326.5
332.7
337.0
338.5
338,4
336,7
333,2
328.3
324,5
321.5
318.9
316.8
315,1
313.6
312.2
310.9
309.6
TP
314.4
315.8
320.9
327.6
334.8
342.3
348.8
353.9
357.5
359.5
359.7
358.2
355.0
350.1
347.0
344.4
342.3
340.6
339.0
337.7
336.5
335.4
334.4
-------
142
Table 7.13 Comparison of Solar Fluxes (in W/m2) at Surface [Elevated
Layer at 300 m)
Solar Fluxes
Time
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
NP
70.5
257.6
402.9
550.4
665.9
738.5
762.1
734.9
659.5
543.0
396.3
233.5
69.7
P
56,3
200.6
365.5
518.7
640.6
717.8
742.9
713.9
634.0
511.3
359.3
197.2
55.9
SP
56.3
200.6
365.5
518.8
640.7
718,0
743.3
714.3
634.4
511.7
359.6
197.4
55.9
TP
70.5
257.6
402,8
550.3
665,7
738.3
761.8
734.5
659.1
542.7
396.0
233.3
69.7
-------
143
participation by pollutants decreases the solar flux at the surface
by about 20% at large zenith angles (06:00) and by about 4% around
solar noon. The average solar attenuation of about 12% is consistent
with measurements made by Rouse, et al. (1973) in Hamilton, Ontario.
Thermal participation by pollutants increases the downward thermal
flux by about 10%. It is noticed that around noon the solar attenua-
tion is of the same magnitude as that of the decrease in the thermal
flux; this indicates that cooling caused by pollutants might be com-
pensated to a degree by pollutant induced heating. The thermal fluxes
of simulation P (solar and thermal participation) are slightly larger
than those of simulation TP (thermal participation only). This is
to be expected as the increase in the atmospheric temperature caused
by solar heating is accompanied by an increase in thermal fluxes.
Table 7.14 presents the potential temperatures at 1 m for the
simulations. During the first 6 hours after sunset, the effects of
the reduction in the surface solar flux dominate over those of
absorption of solar radiation in the surface layer thus reducing the
temperature at 1 m for the simulations with solar participation. At
13:00 hours the effects of solar heating become evident with the
temperature at 1 m becoming higher in the simulations with solar
participation (P, SP) than that in the simulation with non-
participating pollutants. This temperature excess is about 0.1 C
on an average over the period of simulation. It is interesting to
note that the beginning of the solar heating is accompanied by the
increase in the surface aerosol concentration from the background
value of 50 yg / m3 to 130 ug/m3. It is seen from Table 7.15 that
absorption of solar radiation in the surface layer also affects the
surface temperature. Although the solar flux reaching the surface
is reduced by about 15% for simulation SP, the surface temperature
is 0.1 C higher during the major portion of the period of simulation.
The complex physical processes responsible for this seemingly
anomalous temperature excess have already been described in
Section 7.3.
-------
144
Table 7.14 Comparison of Potential Temperatures (1n K) at 1 m
(Elevated layer at 300 m)
Potential Temperature
Time
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
00:00
01:00
02:00
03:00
04:00
NP
283.57
285.55
288.10
290.62
292.85
294.66
295.93
296.63
296.80
296.48
295.66
294.36
292.47
290.60
289.50
288.61
287.80
287.19
286.64
286.14
285.68
285.23
284.80
P
283.75
285.43
287.89
290.44
292.73
294.67
296.08
296.84
297.00
296.64
295.76
294.41
292.69
291.08
290.10
289.28
288.68
288,16
287.69
287.28
286.90
286.55
286.23
SP
283.38
285.00
287.50
290.07
292.38
294.35
295.91
296.89
297.12
296.68
295.77
294,37
292.56
290,70
289.60
288,71
287,88
287.28
286.73
296.23
285.76
285.30
284.87
TP
283.92
285.97
288,49
290.97
293.14
294.82
295.93
296.60
296.78
296.47
295.66
294.40
292.62
290.95
289.99
289.11
288.55
288.03
287.57
287,15
286.78
286.43
286.11
-------
145
Table 7.15 Comparison of Surface Temperatures (1n K) of Simulation
with Non-participating (NP) Pollutants and Simulation
with Solar Participation (SP) only (Elevated Layer at
300 m)
Time
07:00
09:00
11:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
Temperature
NP
285.76
291.75
296.23
297.18
294.14
289.95
288.10
286.81
285.81
284.93
284.12
288.06
295.43
299,36
300.16
299.15
296.20
292.05
290.21
288.89
287.82
286.84
SP
285.12
291.09
295,88
297.17
294,10
290,02
288.19
286,90
285,89
284.99
284,17
287.44
294,93
299.53
300.39
299.24
296,40
292.22
290.38
289.05
287.96
286,95
-------
146
Figures 7.18 to 7.21 show the effect of radiative participation
of pollutants on pollutant dispersal. During the first four hours,
the pollutant layer is unaffected as the mixed layer has not yet
grown high enough to disperse the pollutants. During the following
4 hours, the effect of pollutants is evident from the different rates
of pollutant dispersal as shown in Figure 7.19. Thermal participation
by increasing the initial rate of mixed layer growth is conducive
to pollutant dispersal while solar participation decreases the rate
at which pollutants diffuse away from the pollutant layer. At the
end of 8 hours after the beginning of the simulation, the surface
pollutant concentration is 180 yg/m3 for no participation, 170
ygm/m3 for thermal and solar participation, 135 yg/m3 for solar
participation only, and 200 yg/m3 for thermal participation only.
From Figure 7.20, it can be seen that solar participation can slow
down pollutant dispersal even after 9 hours. Once the ririxed layer
penetrates the inversion created by solar heating, pollutants are
mixed rapidly, and radiative participation becomes unimportant.
Figure 7.22 shows the effect of radiative participation on the
growth of the mixed layer. It is seen that the largest mixed layer
heights (1300 m) occur for the simulations in which there is solar
participation. An explanation for this apparent anomaly is fairly
simple. Solar heating creates a sharp inversion at the height of
the aerosol layer; at the same time the layer above the inversion is
destabilized. Initially, the inversion created by solar heating
inhibits the growth of the mixed layer, but once the mixed layer
penetrates the inversion its growth is relatively more rapid than
for the other simulations as the layer above the inversion is unstable
The effects of the destabilizing influence of solar heating are
evident on the second day when the mixed layer growth is greatest
(1420 m) for the simulations with solar participation. Cooling
induced by gaseous pollutants helps the growth of the mixed layer
as evidenced by the larger mixed layer heights for the simulations
with thermal participation.
-------
147
a»ITTT
OER0S0L C0NCENTRRTI0N
2000
5
ICO 158 2t: 2GC N: 355
. CWCENTRflTlBN UG/H3
2MI
1MO
5
5
mi
RER0S0L CONCENTRATION
• ii
I M 111 15» Ml 34 Ml 554 411
, CMZNTMTIIN U&M
1MI
OER0S0L C0NCENTROTI0N
I M III IM Ml 2M Mi 3M 411
CMENTinriM
OER0S0L C0NCENTROTI0N
2MI
ISM
I M <» IM Ml » SI* SM 411
Figure 7.18 Aerosol Concentration Profiles for Time 05:00 to 08:00.
Top Left is Simulation NP, Top Right is Simulation P,
Bottom Left is Simulation SP, and Bottom Right is Simula-
tion TP. Elevated Pollutant Layer at 300 m.
-------
148
2500
2000
RER0S0L C0NCENTROT10N
1000
7-77
100 150 200 250 3C3 SO 400
RCMSa. MNCCNTRRTI8N UO/M3
2500
Mot
* ,
1001
Ml
flER0S0L C0NCENTRRTI0N
M 100 1M Ml Of Ml
UG/W
2SOO
s
*
MO
RER0S0L C0NCENTROTI0N
1M 211 2M XI
flERtS* CMXNTRRTIM UC/H3
410
2011
1HI
RER0S0L C0NCENTRflTI0N
III 1M 211 2M Ml JM 411
. CMXNTmnw UG/W
Figure 7.19 Aerosol Concentration Profiles for Time 09:00 to 12:00.
Elevated Pollutant Layer at 300 m (See Figure 7.18 for
arrangement).
-------
149
aw i—i—r
BER0S0L C0NCENTRflT!0N
80 100 120 140 160
CfNCENTRflTIW UG/H3
180 200
2)00
1MO
5
5
RER0S0L C0NCENTRRTI0N
M III 121
IM IN 211
2500
RER0S0L C0NCENTRflTI0N
1M IM Ml 2M SH
. WCEMTMIIIH UC/W
f€R0S0L CBNCENTRflTIBN
2111
IIM
M IM 121 Ml IM IM Ml
ue/n
Figure 7.20 Aerosol Concentration Profiles for Time 13:00 to 16:00.
Elevated Pollutant Layer at 300 m (See Figure 7.18 for
arrangement).
-------
150
RER0S0L C0NCENTRflTl0N
20M
ISM
1000
BOO
60 M 100 120 140 160
2MO
1500
5
*
RER0S0L C0NCENTRRT10N
N 111 lit 141
REMML CMXMTRflTIM UG/IQ
RER0S0L C0NCENTROTI0N
I MO
M III IM
C*CENTRf)TIW
Ml
2101
Illl
RER0S0L C0NCENTRRTI0N
III 121 Ul IM
Figure 7.21 Aerosol Concentration Profiles for Time 17:00 to 20:00.
Elevated Pollutant Layer at 300 m (See Figure 7.18 for
arrangement).
-------
151
MIXED LAYER GROWTH
MIXED LRYER GR0WTH
|
IMI
5 10 15 2J 25 JO JB 43 J5 50
TIME H»
MIXED LRYER GROWTH
IN
S II 15 21
TIME m
5
5'
'Ml
MIXED LfiYER GR0MTH
* ii « 21 a » » « «s s«
TIHEHR
Figure 7.22 Variation of Mixed Layer Height with Time Elevated
Pollutant Layer at 300 m. Top Left is Simulation NP,
Top Right is Simulation P, Bottom Left is Simulation SP,
and Bottom Right is Simulation TP.
-------
152
7.5 Effect of Changing the Height of the
Elevated Pollutant Layer
A second set of simulations was performed to determine the
effect of the height of elevated pollutant layer on thermal structure
and pollutant dispersal. The lower edge of the pollutant layer was
located at 600 m. Figures 7.23 to 7.30 illustrate the variation of
the temperature and concentration profiles over a period of 16 hours.
The results indicate the same trends as those of the first set of
simulations. However, as the greater height of the pollutant layer
allowed the pollutant induced heating and cooling to proceed over a
longer period of time, the inversions formed were considerably sharper
as indicated in Figure 7.24. It is seen from Figure 7.25 that the
inversion created by solar heating is large enough to prevent the
growth of the mixed layer beyond the height of the pollutant layer.
The effect of the creation of this "lid" on pollutant dispersal is
evident in Figures 7.29 and 7.30.
The simulations can be better understood by examining the
variables presented in Tables 7.16 to 7.19. Radiative participation
by aerosols and gases increases the temperature at 1 m. The tempera-
ture excess is about 0.2 C during the day and reaches a maximum of
1.5 C at 03:00 of the second day. It is noted that the temperature
excess produced by thermal participation alone is slightly higher
than that with thermal and solar participation during the first four
hours of simulation. This indicates that initially the effects of
the decrease in the solar flux at the surface offset those of solar
heating and the increase in downward thermal radiation. With solar
participation alone, the temperature at 1 m is slightly lower
(-0.6 C) than that with no participation during the first 16 hours
of simulation. An examination of Table 7.16 shows that the lower
surface concentrations (pollutant dispersal is impeded by solar
heating) allows the effects of surface solar flux reduction to
dominate over those of solar heating during the first 16 hours
leading to the decrease in the temperature at 1 m. However, as
-------
153
TEMPERATURE
2500
2MO
1900
»2»2902»22W29629»300S023W
TEHPEWTURE K
2500
Ml
TEMPERflTURE
at
MIMIIMJMM30I9U9M
K
VjMZMMIMIMiWlMMMtMIM
TEMPERRTURE
286 a» sic 292 aw 296 .>.« x:
: K
Figure 7.23 Potential Temperature Profiles for Time 05:00 to 08:00.
Top Left is Simulation NP, Top Right is Simulation P,
Bottom Left is Simulation SP, and Bottom Right is Simula
tion TP- Elevated Pollutant Layer at 600 m.
-------
154
TEMPERATURE
TEMPERflTURE
2MO
MOO
1900
1000
t ' '£»at sea
TEHPCRRTUK X
2Mt
5
5
Ml
Torcmnjc x
TEMPERfiTURE
TEMPERRTURE
2StO
rorcmruK K
TEWflWTURC K
Figure 7.24 Potential Temperature Profiles for Time 09:00 to 12:00.
Elevated Pollutant Layer at 600 m (See Figure 7.23 for
arrangement).
-------
155
ZBM
20H
1500
TEMPERRTURE
TEMPERflTURE
900
NJy^
296 2M J00~ 302 3M 306
1.1 IM.I m.i 2M.i m.t M.I m.i NI.I NI.I sw.i MM JM.I
roramruK K
TEMPERflTURE
sou
2000
TEMPERATURE
500
2SS 300
TDrCMTURE K
3M S5"
Figure 7.25 Potential Temperature Profiles for Time 13:00 to 16:00.
Elevated Pollutant Layer at 600 m (See Figure 7.23 for
arrangement).
-------
156
TEMPERflTURE
2900
TDVOWTUK K
TEMPERATURE
200)
1940
TO«I»TI« K
Figure 7.26 Potential Temperature Profiles for Time 17:00 to 20:00.
Elevated Pollutant Layer at 600 m (See Figure 7.23 for
arrangement).
-------
157
zaoo
AEROSOL CONCENTRATION
AEROSOL CONCENTRATION
1000
0 50 100 150 200 250 300 !BO 400
UC/MJ
2«00
1MO
s
V
1(11
HI
'l
•
.
'
,.
*
1 <« ISC Ml 29* Ml 9M 41
REMML CMCENTMITtM UG/W
RER0S0L CONCENTRATION
AEROSOL CONCENTRATION
2MO ri i i i ,
2011
2000
1500
i
y
1000
500
•
'
'
M 111 1M Ml IN Ml SH
-------
153
2300 i-T
r
s
5
RER0S0L C0NCENTRRTI0N
0 90 100 150 200 2BO 300
(CMS*. MNccNTMTim UG/M
480
RER0S0L C0NCENTRRTI0N
III
-------
159
2900
flER0Sffl_ CONCENTRATION
AER0S0L CONCENTRATION
100 iso 200 ao soo » wo
. UNCCNTRflTim U6/H3
2MO
(Ml
Ill 150 HO 2M Ml
fltWSJL CMCENTRRUIN I
RER0S0L C0NCENTROTI0N
ISM
'« III IM Ml 2M Ml
flEWSJL WCCMTMT1IN UC/H9
411
200
2000
RER0S0L CONCENTRATION
60 BO 100 120 140
AEMML MNCCNTMTIIN
IK KG
Figure 7.29 Aerosol Concentration Profiles for Time 13:00 to 16:00.
Elevated Pollutant Layer at 600 m (See Figure 7.27 for
arrangement).
-------
160
BER0S0L C0NCENTRflTI0N
2SOO
20M
IMC
1000
MO
2500
f€R0S0l C0NCENTRRTI0N
• 11111111111111 > 111111 ii 11111111"" i >
«0 100 120 140
CMCENTftRTIlN U6/MJ
* H 70 M M HI 111 121 IN 141
(WtSiL CWCENTmrilN UG/HS
5
flERBSBL C0NCENTRRTI0N
111 1M Ml at Ml 5M 411
(CXtML CMCCNTRnTIM UG/H3
S
*
RER0S0L C0NCENTRflT10N
"0 TO H; *o 100 no co isc u; IK
PCR030L CmCCNTRRTieN UC/M3
Figure 7.30 Aerosol Concentration Profiles for Time 17:00 to 20:00.
Elevated Pollutant Layer at 600 m (See Figure 7.27 for
arrangement).
-------
161
Table 7-16 Comparison of Temperatures (in K) at 1 m (Elevated Layer
at 600 m)
Temperature
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
02:00
NP
285.55
290.62
294.66
296.63
296.48
294.36
290.60
288.61
287.19
286.14
285.24
284.41
287.79
294.40
298.04
299.04
298.60
296.46
292.72
290.72
289.27
288.17
287.16
P
285,65
290.69
294.78
296.80
296.68
294.50
291.09
289.25
288.16
287.27
286.55
285.93
288.91
295.00
298.33
299.29
298.82
296.63
293.29
291.50
290.33
289.42
288.67
SP
285.00
290.07
294.23
296.36
296.37
294.18
290.60
288.53
287.28
286.23
285,27
284,43
287.25
293,92
298.04
299.49
299.29
297.10
293.44
291.47
290.00
288.74
287.66
TP
285.95
290.97
294.99
296.78
296.52
294.43
290.97
289.19
288.03
287.15
286.42
285.81
289.10
295,06
298.05
299.00
298.52
296.47
293.09
291.30
290.15
289,26
288.52
-------
162
soon as the surface concentration reaches 102.7 yg/m3 solar heating
becomes more significant and the temperature at 1 m is heated above
that of the simulation with no radiative participation. This tempera-
ture excess reaches a maximum of about 0.5 C at 03:00 of the second
day. The temperature increase is caused to a smaller extent by the
increased downward thermal radiation emitted by the atmosphere which
is warmer due to solar heating by aerosols.
Tables 7.17 and 7.18 list the thermal and solar fluxes absorbed
at the surface for the simulations. As expected, pollutant participa-
tion increases the downward thermal flux. Solar heating also
contributes to this increase by heating the atmosphere. The maximum
downward thermal radiation excess is around 25 W/m2 (7%) and occurs
at 21:00 of the second day. Solar fluxes are reduced by aerosol
participation. The maximum reduction occurs at large zenith angles
(17:00) and is about 90 W/m2 (30%). The small differences between
the solar fluxes of simulations P and SP are caused by variation of
the water vapor path length. It is noted that the reduction of
pollutant dispersal by solar participation alone leads to relatively
large local heating rates.
Table 7.19 shows the effect of radiative participation by
pollutants on pollutant dispersal. It is seen that initially solar
heating due to aerosols slows pollutant dispersal from the elevated
layer as indicated by the lower surface concentrations for the
simulations with solar participation. The concentrations of simula-
tion P show that cooling induced by the gaseous pollutant helps the
mixed layer penetrate the sharp inversion created by solar heating
due to aerosols. Once the mixed layer penetrates the inversion
its growth is more rapid than for the other simulations because the
region above the inversion is unstable. The more rapid growth and
consequently the higher final mixed layer thickness is indicated by
the lowest value (125.7 yg/m3) of surface concentration at 03:00
of the second day. It is interesting to note that the greatest
surface (1 m) aerosol concentration occurs for the simulation in
which aerosols are the only participants. An explanation for this
-------
163
Table 7.17 Comparison of Solar Fluxes (in W/m2) at Surface (Elevated
Layer at 600 m)
Solar Fluxes
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
237.6
550.4
738.5
734.9
543.0
233.5
0
0
0
0
0
0
233.5
540.0
725.2
722.2
534.1
229.8
0
0
0
0
0
P
200.6
518.7
717.9
714.2
511.5
197.3
0
0
0
0
0
0
197.6
509.0
704.8
701.6
502.9
194.0
0
0
0
0
0
SP
200.6
518.8
718.0
714.3
511.7
197.3
0
0
0
0
0
0
197.6
509.2
704,9
701.8
503.2
194.5
0
0
0
0
0
TP
237.6
550.3
738.3
734.5
542.7
233.3
0
0
0
0
0
0
233.3
539.3
724.0
721,0
533,2
229.5
0
0
0
0
0
-------
164
Table 7,18 Comparison of Thermal Fluxes (in W/nj2) at Surface
(Elevated Layer at 600 m)
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
294.0
305.7
320.3
331.8
337.2
336.0
327.7
321.1
316.5
313.3
310.6
308.2
311.6
329.0
344.3
351.6
354.7
352.3
343.3
336.2
331.2
327.7
324.6
Thermal
P
314.9
326.5
334.3
353.4
360.5
359.2
351.1
345.1
341,2
338.3
336.0
333.0
337.0
353.3
368.5
375.6
378.8
376.2
367.6
361.2
357.0
353,9
351.4
Fluxes
SP
293.3
304.1
318.7
330.7
337.1
336.0
327.6
321.0
316.5
313,4
310.5
308.0
310.6
327.4
343,8
353,3
358.1
355.8
346,3
338,9
334.0
330.0
326.6
TP
315.3
327.2
341.9
353.4
359.5
358.2
350.1
344,4
340.5
337.7
335,4
333.4
336.8
353.3
367,2
374.1
377.4
375.0
366.4
360,3
356.1
353.1
350.6
-------
165
Table 7.19 Comparison of Aerosol Concentrations On yg/m3) at 1 m
(Elevated Layer at 600 m)
Aerosol Concentrations
Time
07:00
09:00
11:00
13:00
15:00
17:00
18:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
50.0
50.0
50.0
81.7
149.7
152.0
151.7
151.7
151.7
151.7
151.7
151.7
151.7
141.4
150.8
144.9
134.4
130.2
130.1
130.1
130.1
130.1
130.1
P
50.0
50.0
50.0
76.3
147.7
149.3
149.0
149.0
149.0
149.0
149.0
149.0
149.0
149.0
147.9
138.6
129.3
125.8
125.7
125.7
125.7
125.7
125.7
SP
50.0
50.0
50.0
53.4
87.7
102.6
102.7
102.7
102.7
102.7
102.7
102.8
102.8
103.0
104.1
133.4
133.6
141.5
141.5
141.5
141.5
141.5
141.6
TP
50.0
50.0
50.0
128.4
149.0
143,8
143.3
143.3
143.3
143,3
143.3
143.3
143.3
143.1
142.2
136.8
129.9
126.6
136.4
136.4
136,4
136.4
136.4
-------
166
can be found by examining the effect of solar heating on the height
of the mixed layer. Solar heating does inhibit dispersal as indicated
by the lower surface pollutant concentrations till 11:00 of the
second day. At the same time the sharp inversion induced by solar
energy absorption prevents the vertical expansion of the mixed
layer. Thus, the pollutants which are allowed to disperse over a
smaller vertical extent reach higher concentrations at the surface.
Figure 7.31 illustrates the effect of radiative participation
on the growth of the mixed layer. It is evident that the height of
the elevated pollutant layer determines the effect of solar heating.
When the pollutant layer is at 600 m solar heating proceeds over a
longer period of time and the inversion formed is large enough to
limit the growth of the mixed layer. The mixed layer is not able
to penetrate the inversion and does not grow beyond 820 m. It can
be seen that cooling induced by gaseous pollutants helps the mixed
layer to grow. This effect is evident in the simulation in which
there is thermal and solar participation. While the mixed layer is
not able to penetrate the solar heating induced inversion when
aerosols are the only radiative participants, it does so with the
assistance of gaseous cooling. As the layer above the inversion
is unstable the mixed layer grows to a height of 1420 m, the largest
value for the simulations.
It is clear from the simulations described that aerosols lead
to a surface layer temperature increase by absorbing solar radiation
in the surface layer of the boundary layer. Thus, this warming
effect is dependent on the presence of fairly large aerosol concen-
trations near the surface. This line of reasoning indicates that
aerosols would cause cooling if they are distributed well above the
surface layer. To test this hypothesis a simulation was performed
with the aerosol layer located at 1200 m. Only solar participation
was allowed to prevent the dispersal of pollutants from the layer.
Table 7.20 compares the solar fluxes and the temperature at 1 m
against those for the simulation with no radiative participation.
The small reduction in the solar radiation during the second day
-------
167
I'M
1200
l no
MIXED LAYER GROWTH
8
200
0 8 1C 19 X a 30
TIW m
40 « M
MIXED LAYER GR0HTH
IMI
Mil
Ml
I I 11 II M
TIME Ml
MIXED LAYER GR0HTH
I,,
g
SMI
I I II IS 21 a M
T1K M
41 M
1400 nTTT
MIXED LAYER GTOHTH
1200
1000
r
i
Jj 800
5 60S
i
•
•at
1 B 10
is 20 a so
TINEM
Figure 7.31 Variation of Mixed Layer Height with Time.Elevated
Pollutant Layer at 600 m. Top Left is Simulation NP,
Top Right is Simulation P, Bottom Left is Simulation SP,
and Bottom Right is Simulation TP.
-------
168
Table 7.20 Comparison of Solar Fluxes (in W/m2) and Temperature
(in K) at 1 m (Elevated Layer at 1200 m)
Solar Flux
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
NP
374.6
550.4
738.5
734.9
543.0
233.5
0
0
0
0
0
0
233.5
540.0
725.2
722.2
534.1
229.8
0
0
0
0
0
P
200.3
515.8
712.3
708.4
508,3
196.9
0
0
0
0
0
0
197,6
505.6
698.7
695.5
499.6
193.9
0
0
0
0
0
Temperature
NP
285.54
290.62
294.66
296,63
296.48
294.36
290,60
288.61
287.19
286,14
285.23
284.41
287.79
294.40
298.04
299.04
298.60
296.46
292.72
290.73
289.27
288,17
287.16
P
284,70
289,80
293.95
296.01
295.76
293.42
289.71
287.68
286.21
285.10
284.08
283.18
285.98
292.79
296.97
298.09
297.63
295,32
291.64
289.64
288.15
286.93
285.84
-------
169
is caused by the increased amount of water vapor in the boundary
layer. It is interesting to note that solar attenuation leads to
a surface layer temperature decrease of about 1.2 C during the
second day.
7.6 Effect of Changing Aerosol Properties
Recent studies (Mitthefl, 1971; Russell and Grams, 1975) show
that tropospheric aerosols cause cooling or heating of the earth-
boundary layer system depending upon their properties. As the
properties of aerosols are not well known, it is not possible to
draw definite conclusions about their effects. In order to gain
some understanding of the importance of aerosol properties, aerosol
parameters such as single scattering albedo and forward scattering
factor were varied in a number of simulations. The aerosol extinction
coefficient was chosen so that the maximum aerosol optical thickness
of the boundary layer was around 0.14, a value which is consistent
with measurements made by Herman, et al. (1971). They placed the
mean aerosol optical thickness of the atmosphere around 0.1.
Some of the most important simulations are listed in Table 7.21
and some selected results are presented in Tables 7.22 and 7.23.
Aerosol participation decreases the temperature at 1 m in all the
simulations during the first 26 hours. This indicates that during
this period the absorption of solar radiation in the surface layer
is not significant enough to offset the effect of the reduction of
solar flux at the surface. Around 11:00 of the second day of simula-
tion the aerosol optical thickness is about 0.1 and the effects of
solar heating become evident. The results of simulations 2 and 4
show that the surface air layer is about 0.1 C warmer during the
remaining 20 hours. This result indicates that very slightly
absorbing aerosols (w = 0.9) can cause warming of the surface layer.
The predominantly scattering albedo of 0.99 of simulation 3 and the
small forward scattering factor of 0.5 (b = 0.5) of simulation 5
lead to a decrease in the temperature of the surface air layer. The
temperature deficit reaches a maximum of about 1.23 C at 07:00 of
-------
170
Table 7.21 Summary of Simulations Performed to Study the Effect of
Aerosol Property Variation; r = 0.2, H = 0.1
Simulation No. Parameters Characterizing Radiative Participation
No radiative participation
Aerosol participation; w = 0.8, f = 0,85
3ex = 5 x lo"7
Aerosol participation; u> = 0.99, f = 0.85,
3ex = 5 x io"7 m2/yg
Aerosol participation; w - 0,90, f = 0,85,
3ex = 5 x IO"7 mVyg
Aerosol participation; u> = 0,90, f = 0.5,
3ex = 5 x io"7 m2/yg
-------
171
Table 7.22 Comparison of Temperatures at 1 m to Study the Effect
of Aerosol Parameter Variation
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
22:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
1
285.54
290.62
294.66
296.63
296.48
294.36
290.60
288.61
287.19
286.14
285.23
284.41
287.79
294.40
298.04
299.04
298.60
296.46
292.72
290.73
289.27
288.17
287.16
Temperature (K)
2 3
285.33
290.46
294.59
296.63
296.48
294.27
290.59
288.60
287.19
286.15
285.23
284.41
287.27
294.29
298.23
299.28
298.82
296.65
-£92.91
290.92
289.48
J288.33
287.29
285.12
290.30
294.44
296.49
296.32
294.06
290.16
288.09
286.56
285.43
284.41
283.49
286.53
293.79
298.87
298,90
298.39
296.05
292.18
290.13
288.58
287.36
286.25
4
285.37
290.50
294,64
296.67
296.50
294,29
290.61
288.61
287,20
286.15
285.24
294.41
287,34
294.34
298.22
299.25
298.76
296.50
292.79
290.82
289.38
288.25
287.22
5
285.29
290.30
294.38
296.36
296.18
294.00
290.38
288.41
287.03
285.99
285.08
284.26
287.01
293.67
298.60
298.57
298.08
295.95
292.33
290.40
289,01
287.82
286.88
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172
Table 7.23 Comparison of Solar Fluxes at Surface to Study the Effect
of Aerosol Parameter Variation
Time
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
05:00
07:00
09:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
01:00
03:00
1
237.6
550.4
738.5
734.9
543.0
233.5
0
0
0
0
0
0
233.5
540.0
725.2
722.2
534.1
229.8
0
0
0
0
0
2
223.8
537.5
727.5
722.0
523.7
209.5
0
0
0
0
0
0
198.6
504.9
695.0
690.0
492.7
187.2
0
0
0
0
0
Solar Flux (W/m2)
3
228.4
542.6
732.7
728.5
532.1
217.5
0
0
0
0
0
0
210.9
522.9
715.4
711.7
514.0
202.4
0
0
0
0
0
4
226.2
541.7
733.0
728.3
529.8
214.0
0
0
0
0
0
0
205.3
516.1
708,9
704.6
505.7
195,4
0
0
0
0
0
5
220.7
527.7
712.2
704.8
510.1
214.6
0
0
0
0
0
0
191.8
482.5
661.6
655.3
467.7
179.2
0
0
0
0
0
-------
173
the second day for simulation 3. For simulation 5 the maximum
temperature decrease is about 0.8 C at 07:00 of the second day.
The lower temperature deficit of simulation 5 is associated with
the lower single scattering albedo of 0.9.
The effects of radiative participation of pollutant layers on
the total energy balance of the earth-boundary layer system is
reflected in the variation of the effective albedo of the system.
Figures 7.32 and 7.33 show the effect of aerosol property variation
on the system albedo. The effective albedo with no radiative
participation is lower than the surface albedo of 0.2 because of
the water vapor absorption of solar energy in the boundary layer.
Figure 7.32 illustrates the sensitivity of the earth-boundary layer
albedo to the forward scattering factor. Aerosols with forward
scattering factors of 0.5 and 0.7 lead to cooling of the earth
boundary layer system. Aerosols with a forward scattering factor
of 0.9 lower the albedo below the albedo with no participation during
the major portion of the period of solar irradiation. Thus, the
thermal effect is clearly one of warming. At this point it is
appropriate to recall the physical meaning of the forward scattering
factor. The forward scattering factor represents the fraction of
the incident solar energy scattered in the forward direction. Thus,
a small forward scattering factor is associated with large back-
scattering. This explains the results presented in Figure 7.32.
Figure 7.33 illustrates the effects of aerosol single scattering
albedo on the effective system reflectance. It is seen that aerosols
with a single scattering albedo as high as 0.99 can cause warming of
the earth-boundary layer system during a short period around noon.
This heating effect is associated with the large forward scattering
factor of 0.85. The radiation scattered predominantly in the forward
direction is absorbed by water vapor and to a much smaller extent by
aerosols, thus causing the warming. It is noted that water vapor
absorption becomes important at small zenith angles around noon when
the smaller water vapor path length above the boundary layer allows
greater solar absorption in the boundary layer. As water vapor
-------
174
0,3
3
cc:
LU
0,2
0,1
SIMULATION NP
06:00 08:00 10:00 12:00 14:00 16:00 18:00
LOCAL TIME (HR)
Figure 7.32 Effect of Forward Scattering Factor on Earth-Boundary
Layer Albedo Of Second Day. 3ex = 5* 10~7 m2/yg, CD = 0.9
-------
175
0,3 r
o:
!5
>-
0,2
0,1
CU =0,99^- u) =0,8
06:00 08:00
10:00 12:00
LOCAL TIME (HR)
14:00 16:00 18:00
Figure 7.33 Effect of Single Scattering Albedo on Earth-Boundary Layer
Albedo of Second Day. $.v = 5 x 10'7 mVug, f = 0.85
C A
-------
176
absorbs proportionately larger amounts of solar energy above the
surface than near the earth's surface, the overall warming effect is
not evident at the surface. This explains the temperature deficit
of the surface layer for simulation 3.
7.7 Effect of Choice of Gaseous Pollutant
As the choice of ammonia as a representative pollutant may be
criticized as unduly restrictive, a simulation was performed using
methane. Methane is produced by bacterial action in swamps, marshes
and sewage. It abosrbs strongly in a band centered at 7.6 urn, and
it is representative of a typical hydrocarbon pollutant. The
emissivity of methane was derived from the wide band correlations
presented by Tien (1968). The results of the simulation with methane
showed that the effects of radiative participation by methane were
extremely small at the concentrations (-200 ug/m3) predicted by
the numerical model. The temperature at 1 m was increased by a
maximum of 0.1 C, and the effect on pollutant dispersal was negligible.
-------
177
VIII. RESULTS AND DISCUSSION: TWO-DIMENSIONAL MODEL
8.1 Introduction
This chapter discusses the results of the simulations performed
with the two-dimensional model. At the outset it should be stated
that the model is in the process of development and as such the results
to be presented are to be considered preliminary. The construction
of the one-dimensional model and the simulations performed with it
represented the first stage of an ongoing research. The preliminary
one-dimensional simulations were conducted to test the radiation and
turbulence models developed in this study, and additional simulations
provided valuable understanding of the effects of radiative participa-
tion of pollutants on thermal structure and pollutant dispersal. The
second phase of the research study consisted of the incorporation of
the radiation and turbulence models into a two-dimensional transport
model. Several problems were encountered in the process and some of
them have not yet been resolved. An attempt to incorporate the
turbulent kinetic energy model into the transport model resulted in
the numerical scheme becoming unstable. Thus, it was necessary (for
the time being) to adopt a simpler model for turbulence. Specifically,
the O'Brien cubic profile (O'Brien, 1970) for diffusivity was utilized.
Sasamori (1970) and Bornstein (1972) have developed relatively success-
ful P.B.L. models based on the O'Brien profile. In using the O'Brien
diffusivity model, the height of the planetary boundary layer was
allowed to vary with time. This modification represents an improvement
over the versions of the O'Brien model used by Sasamori (1970) and
Bronstein (1972) both of whom fixed the height of the boundary layer.
As the mixed layer height varies considerably (by a factor of five)
over the course of a day, it is unrealistic to assume a constant
boundary layer height. It should be emphasized that the instability
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178
problem associated with the turbulent kinetic energy model is under
investigation. Possible solutions include an improved numerical
scheme and/or improved procedures to finite difference the non-linear
production and dissipation terms in the turbulent kinetic energy
equation.
The effects of radiative participation on vertical stability
have already been discussed in the preceding chapter. The complex
feedback mechanisms associated with the development of the thermal
structure in the boundary layer have been examined in some detail,
and will not be emphasized in the discussion of the results presented
in this chapter. As the two-dimensional model is still in the process
of development, the discussion of the results of the simulations will
be brief in comparison with that of the one-dimensional model. It is
felt that a more detailed examination is not warranted at this stage
of the research program.
The major emphasis of this chapter will be laid on the two
important effects of urbanization, the heat island effect and the
temperature "crossover" effect. These two effects associated with
the rural-urban system have been studied extensively (Bornstein, 1968,
1972; 01fe and Lee, 1971; Oke and East, 1971; Oke and Maxwell, 1975)
both theoretically and observationally. Although it has been sug-
gested (Bornstein, 1968; Rouse, et al., 1974) that radiative partici-
pation by pollutants could contribute to these effects, there have
been relatively few theoretical studies (Atwater, 1971, 1972, 1974,
1975; Bergstrom and Viskanta, 1973; Viskanta, Johnson and Bergstrom
1975) which have investigated t'n's possible role of pollutants.
Atwater (1974, 1975) using a two-dimensional model concludes that
pollutants are only a minor factor in the formation of the urban heat
island. His results also indicate that pollutants have minimal
influences on the vertical thermal structure. However, the more
recent results of Viskanta, Bergstrom and Johnson (1975) which are
based on a more sophisticated radiation model indicate that pollutants
can have a significant effect on the urban heat island. As there is
a general lack of consensus on the radiative effects of pollutants on
-------
179
urban-rural temperature differences, it is hoped that the results of
the simulations presented in this chapter will throw some light on
the controversial topic.
The secondary objective of the simulations was to investigate
the relative importance of urban parameters in determining the heat
island. As pollutants are responsible for radiative effects, it was
interesting to study the changes induced by pollutant participation
on their own dispersal. A list of the simulations performed is given
in Table 8.1.
Table 8.1 List of Two-Dimensional Simulations (see Table 8.2 for
other Urban-Rural Parameters)
Reference
N
P
SP
TP
1
2
3
4
Radiative Change in
Participation Urban Parameter
No participation
Thermal & Solar
Solar only
Thermal Only
Thermal & Solar
Thermal & Solar z0 = 2m, H = 0.01
Thermal & Solar H = 0.01
Thermal & Solar H = 0.01, u* = 0.2 ra/s
Change in
Rural
Parameter
H = 0.05
—
8.2 Initial Conditions and Surface Parameters
The urban-rural system was assumed to extend 30 km in the
horizontal direction and 2 km in the vertical direction. The city
itself was taken to be 15 km long and the adjoining rural areas on
each side of the city were assumed to be 7.5 km in horizontal extent
(Columbus, Ohio, is about 20 km square). As the planetary boundary
layer does not extend much beyond 1.5 km (Lettau and Davidson, 1957)
for the conditions chosen for the simulations, the value of 2 km for
-------
180
the vertical extent of the system serves as an upper limit for the
boundary layer height.
Sixteen uniformly spaced grid points were used in the horizontal
direction (x-axis). The first 5 grid points represented the rural
area, the next 6 represented the urban area, and the remaining 5 were
situated in the rural area. Variable grid spacing was used in the
vertical direction. In particular, the smallest spacing was used near
the surface to provide good resolution (see Table 8.2).
The surface parameters used in the simulations were based on
values suggested by Pandolfo, et al. (1971) and Oke (1975) and are
presented in Table 8.2. It was assumed that there was no variation
in soil properties but they were different in rural and urban areas.
Thus, heat island effects are produced primarily by the differences
in evaporation rates and heat production between the urban and rural
areas.
The temperature field was initialized with a profile typical of
early morning conditions in O'Neill, Nebraska (Lettau and Davidson,
1957). The nocturnally established inversion extended to 300 m and
had a gradient of 18 x 10"3 c/km. The stable layer above the inversion
had a gradient of 8 x io~3 c/km. The initial velocity field was
assumed to be uniform in the horizontal direction, and the vertical
profile was given by.
u(x,z) = v(x,z) = u*ln((z + z.)/z.)/K (8.2.1)
where
u* = 0.1 m/s, zi = 0.1 m (8.2.2)
and K is the von Karman constant. The maximum velocity predicted by
Eq. (8.2.1) was 2.48 m/s at the top of the atmospheric layer. As this
study was intended to be comparative, it was felt that there was no
necessity to initialize the temperature and velocity fields with field
data.
As in the one-dimensional simulations, the pollutant sources were
assumed to be elevated. They were placed at a height of 100 m along
the urban area. It is noted that the source strength is taken to be
-------
181
Table 8.2 Grid Spacing, Surface Parameters and Pollutant Parameters
a) Grid System
Horizontal grid spacing Ax = 2000 m
Beginning of rural area: x = 0
j
z(m)
1
0
2
1.0
3
5.0
4
15.0
5
30.0
6
50.0
7
100.0
8
200.0
9
300.0
10
400
.0
j
z(m)
11
500.0
12
600.0
13
700.0
14
800,0
15
1000.0
16
1200.0
17
1400.0
18
1600.0
19
1800.0
20
2000
.0
b) Surface Parameters for Urban Rural System
Parameter
Surface Albedo, r
Surface Emissivity, et
Hal stead Moisture Parameter, H
Soil Conductivity, ks, W/m C
Soil Diffusivity, as, m2/s
Surface Roughness, z m
Artificial Heat Production, Hp, W/m2s
Pollutant Source Strength, Scn, yg/m3s
Rural
0.25
1.00
0.10
2.0
1.28 x 10"6
0.1
0
0
Urban
0.15
1.00
0.05
2.0
1.28 x 10"6
1.0
40
0.5
c) Pollutant Parameters
Gas: NH3 (Ammonia)
Aerosol Properties:
o> = 0.9, f = 0.85, 3QV = 10"6 m2/yg
GX
-------
182
ten times that used in the one-dimensional simulations. This was
necessary because advection reduced the pollutant concentrations to
relatively small values (80 yg/m3). The source strength was increased
in order to generate pollutant path lengths large enough to produce
radiative effects.
The horizontal diffusivity Kx was taken to be 100 m2/s. The
main influence of the introduction of horizontal diffusion was that
of increasing the stability of the numerical scheme. Increasing the
value of the horizontal diffusivity did not affect the velocity and
temperature fields to an appreciable extent. This is to be expected
as the horizontal gradients of the variables are relatively small.
The pollutant concentration fields were affected to a greater extent
as the gradients near the boundaries of the urban-rural system were
quite large. It was felt that a detailed investigation of horizontal
diffusion effects was not warranted from the point of view of this
study. It should be mentioned that Lamb and Neiburger (1970) used hori-
zontal diffusivities as high as 800 m2/s without noticing significant
changes in their results. Lee and 01fe (1974) have investigated the
effects of very large horizontal diffusivities and conclude that the
results of calculations with diffusivities as large as 101* m2/s are
similar to those of zero prescribed horizontal diffusivities except
that vertical velocities and peak crossover temperatures are reduced
by about 40 percent.
The simulations were performed for summer conditions identical
to those of the one-dimensional simulations. The choice of summer
conditions was based on the observation that stagnating high pressure
centers which cause pollution episodes are associated more frequently
with summer.
8.3 Effect of Pollutants on Thermal and Solar Fluxes
As the solar and thermal fluxes incident on the air-soil interface
determine the surface tempeature, it is instructive to examine the
effects of radiative participation by pollutants on these fluxes.
Table 8.3 lists the solar and thermal fluxes at the surface in the
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183
Table 8.3a Comparison of Incident Thermal and Solar Fluxes at the
Surface (in W/m2) for Simulations P and NP at x = 16 km
(urban area)
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
Table 8.3b
Thermal Fluxes
NP
302.2
303.5
315.2
327.2
336.1
338,6
334.9
325.0
318.9
314.5
310.9
307.9
305.2
P
337.7
359.0
375.8
391.3
400.7
400.3
394.9
388.2
381,3
374.7
370.3
367.6
365.8
P-NP NP
35.5
55.5 421.8
60.6 784.8
64.1 963.8
64.6 899.0
61.7 613.6
60.0 207.9
63.2
62.4
59.9
59.4
59.7
60.6
Solar Fluxes
P P-NP
370.6 -51.2
742.4 -42.4
932.4 -31.4
863.6 -35.4
565.0 -48.6
165.2 -42.7
Comparison of Total Radiation (Solar + Thermal) at x = 16 km
for Simulations P and NP Total Radiation in (W/m2)
Time
07:30
09:30
11:30
13:30
15:30
17:30
NP
725.3
1064.0
1291.0
1235.1
952.2
542.8
P
729.6
1118.2
1323.7
1264.3
965.3
560.1
P-NP (%)
0.59
4.40
2.50
2,30
1.30
3.10
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184
urban area (x = 16 km) for simulations P and NP- It is seen that
radiative participation increases the downward thermal fluxes substan-
tially. The average increase is about 20%. Solar fluxes are reduced
by aerosols; the reduction is largest at large zenith angles in the
early morning and late evening. The maximum decrease is about 21%
at 17:30, and the reduction is only 3.2% at 11:30. It is interesting
to note that the increase in thermal fluxes is always greater than the
decrease in solar fluxes. This indicates that under the conditions
being investigated radiative participation increases the total radia-
tive energy incident on the surface. These results are consistent
with recent observations made by Rouse, et al. (1973). They made
simultaneous measurements of global solar and incoming thermal
(longwave) radiation at roof-top sites in a heavily polluted zone
and relatively clean control sites in and around Hamilton, Ontario.
Their measurements (27 March 1972) showed the solar flux at the
surface to be reduced by about 4% at noon and 21% in the early
morning (07:00) by pollutants. Thermal fluxes in the industrial
area were about 25% larger than those in the control sites. During
the day, the total radiation (solar plus thermal) received at the
surface was slightly larger for the industrial area. Specifically,
at noon the increase was about 2%, a value which agrees very,well
with that predicted in this study. It is realized that a meaningful
comparison between the fluxes predicted in this study and those
observed cannot be made without a knowledge of the pollutant concen-
trations in the area in which the observations were made. Solar flux
measurements in combination with information about the aerosol con-
centration profiles (and aerosol properties) are not readily available.
Thus, it is not possible to test the validity of the radiation model
by comparing predictions with field data. However, the type of com-
parisons made in this section do show that the predicted trends are
consistent with observations.
Table 8.4 lists the solar and thermal fluxes in the rural area
(x = 0). It is seen that the background pollutant concentration
(50 yg/m3) leads to an increase in the downward thermal flux by as
much as 13% at 05:30 of the second day. The solar fluxes are affected
-------
185
Table 8.4 Comparison of Incident Thermal and Solar Fluxes at the
Surface (in W/m2) for Simulations P and NP at x = 0 km
(Upwind Rural Area)
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
Thermal Fluxes
NP
302.2
301.9
313.3
324.7
333.6
336.6
333.2
323.5
317.3
312.9
309.3
306.2
303.4
P
337.7
338.0
349.4
361.4
370.7
373.7
370.2
361.0
355.7
351.5
348.1
345.1
342.5
P-NP
35.5
36.1
36.1
36.7
37.1
37.1
37.0
37.5
38.4
38.6
38.8
38.9
39.1
Solar Fluxes
NP P P-NP
422.6 396.6 -26.0
786.2 770.0 -16.2
965.6 956.2 - 9.4
900.8 888.8 -12.0
614.8 593.1 -21.7
208.3 184.5 -23.8
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186
Table 8.5 Comparison of Incident Thermal and Solar Fluxes at the
Surface (in W/m2) in Urban and Upwind Rural Areas for
Simulation P
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
Thermal Fluxes
x = 0 km
337.7
338.0
349.4
361.4
370.7
373.7
370.2
361.0
355.7
351.5
348.1
345.1
342.5
x = 16 km
337.7
359.0
375.8
391.3
400.7
400.3
394.9
388.2
381.3
374.7
370.3
367.6
365.8
AF" (0)
0
21.0
26.4
29.9
30.0
26,6
24.7
27.2
25.6
23.2
22,2
22,5
23.3
Solar Fluxes
x = 0 km x = 16 km AFS" (0)
396.6 370.6 -26.0
770.0 742.4 -27.6
956.2 932.4 -23.8
888.8 863.6 -25.2
593.1 565.0 -28.1
184.5 165.2 -19.3
-------
187
to a smaller extent. Aerosol participation leads to a reduction of
the surface solar flux by about 2% on an average. As in the urban
area, the increase in thermal flux more than compensates for the
reduction in solar flux at the surface.
Table 8.5 compares the solar and thermal fluxes at the surface
in the rural and urban areas for simulation P. It is seen that the
difference in the thermal fluxes between the rural and urban areas is
not as great as the increase in thermal radiation caused by the
background pollutant concentration of 50 yg/m3. This shows that a
large increase in pollutant concentration over the urban area is not
accompanied by a proportional increase in the downward thermal flux.
Thus, relatively small pollutant concentrations (50 yg/m3) can cause
thermal effects comparable in magnitude to those caused by much higher
pollutant concentrations (-300 yg/m3 over a height of 800 m in the
urban area). The average thermal flux excess of the urban area is
about 7%. The solar fluxes show a greater difference between the
urban and rural area than the decrease associated with the background
aerosol concentration of 50 yg/m3. The average difference is less
than 10%.
An analysis of the surface solar and thermal fluxes predicted in
this study shows that pollutants generally increase (2%) the total
radiation reaching the surface during the day. Thus, one can expect
radiative participation by pollutants to increase the surface
temperature even during daytime. At night, the radiative effect is
unequivocally that of warming of the surface temperature, as the
downward thermal fluxes are increased by as much as 20% by pollutant
participation. A secondary conclusion of the analyses is that an
increase in aerosol concentration (for the conditions investigated)
has a greater effect on the solar flux than that produced by an equal
increase in gaseous pollutants (thermally participating) on the thermal
flux.
-------
188
8.4 Effect of Radiative Participation on the Surface
Temperatures of the Urban-Rural System
The major emphasis of this section will be on pollutant induced
changes of surface temperature variation both in the rural as well as
the urban areas. Effects related to the differences in the thermal
structures of the rural and urban areas such as the heat island effect
and the crossover effect will be treated in later sections.
Figures 8.1 to 8.4 illustrate the potential temperature isopleths
at four selected times for the simulation with non-participating
pollutants as well as the simulation with participating (solar and
thermal) pollutants. Figure 8.2 shows the development of the mixed
layer during the daytime. From about 400 m at 11:30, the mixed layer
grows to a height of about 1000 m at 15:30. The formation of the
nocturnal surface based inversion is clear in Figure 8.3 which illus-
trates the evolution of the temperature field during the evening
and early morning hours. Although details of the differences between
the participating and non-participating simulations cannot be seen
very easily, overall differences are clearly discernible. It is noted
that participation by pollutants increases the temperature of the
surface layer. This warming effect is quite small during the day
while during the nighttime hours the temperature excess near the
surface is about 1 C as can be seen in Figure 8.4. The explanation
for this warming influence of pollutants has already been given in
Chapter 7 and need not be repeated here.
Figure 8.5 illustrates the effect of radiative participation by
pollutants on the surface temperature variation at the center of the
city (x = 16 km). It is seen that the results show trends similar
to those of the one-dimensional simulations. However, there is one
major difference. With solar participation only the surface tempera-
ture is lower than that for the simulation with no participation.
This finding is not consistent with that of the one-dimensional
simulations, and it illustrates the importance of surface parameters
in determining the effect of pollutants on the surface temperature.
-------
189
2000
1200
.--. -. .
15
0
X(KM)
Figure 8.1 Potential Temperature Isopleths at Time = 11:30 Mrs. Top is
Non-Participating, Botton is Participating; (for details of
grid system see Table 8.2)
-------
190
2000 s
1200
12 18
X(KM)
24
30
Figure 8.2 Potential Temperature Isopleths at Time = 15:30 Mrs;
(See Figure 8.1 for arrangement)
-------
191
2000 -
J200
600 F
Z(M)
200
15
0
2000 e
1200
600
Z(M)
200
15
0
r-.Ujj.J-:
>-2M.
-sir
-39tr-
-SOI.-H
-Ml.-
-501.-
-501.-
-2M.
-SOl.-
-SOI.-
-SOI.-
-19*.-
0
12 18
X(KM)
30
Figure 8.3 Potential Temperature Isopleths at Time = 21:30 Mrs; (See
Figure 8.1 for arrangement)
-------
192
2000 ESS
1200
600
Z(M)
200
15
0
2000
—aoo.-
1200 -
600
Z(M)
200
-JOO.
-JOO.
-100.-
-at.
-JOO.-
-SCO. —
-300.
-JOO.
-3DO.
Figure 8.4 Potential Temperature Isopleths at Time = 01:30 Hrs- (See
Figure 8.1 for arrangement)
-------
193
06:00 10:00
14:00 18:00
LOCAL TIME (HR)
22:00 02:00
Figure 8.5 Comparison of Surface Temperature Difference Between
Participating and Non-Participating Simulations at
Center of City
-------
194
As the urban moisture was relatively low (H = 0.05) the surface
temperature was sensitive to the reduction in solar flux at the
surface caused by aerosol participation. It is noted that the
temperature deficit is the greatest at large zenith angles and
reaches a minimum around noon. This is to be expected as the
largest solar attenuation occurs around sunrise and sunset. During
the daytime hours (05:00 to 17:00) the surface temperature for
simulation TP is higher than that of simulation P because the effect
of the increase in downward thermal radiation due to thermal partici-
pation by pollutants is reduced by the attenuation of solar flux at
the surface by aerosols in simulation P. The maximum daytime tempera-
ture increase caused by thermal participation is about 0.5 C at
07:30 while the maximum surface temperature difference between simu-
lations TP and P is about 0.3 C at 07:30. During the late evening
and early morning hours, the surface temperature of simulation P
becomes higher than that of simulation TP, and the temperature excess
reaches a value of about 0.1 C at 05:00 of the second day. It is
seen that radiative participation by pollutants causes a surface
temperature increase of about 1.8 C at 05:00 of the second day.
Figure 8.6 illustrates the effect of radiative participation on
the surface temperature variation at the beginning of the rural area
(x = 0). It is seen that the relatively small pollutant concentrations
of the rural area (50 pg/m3) can cause surface temperature changes
which are comparable in magnitude to those caused by much higher
concentrations (400 ug/m3) at the urban center. This underscores
the nature of radiative effects, in particular absorption which is
not proportional to the pollutant concentration (optical thickness)--
the incremental absorption of radiation becomes smaller as the optical
depth increases. It is noted from Figure 8.6 that the maximum
temperature excess caused by pollutants is about 1.0 C at 05:00 of
the second day.
-------
195
06:00 10:00
14:00 18:00
LOCAL TIME (HR>
SIMULATION P
SIMULATION NP
22:00 02:00
Figure 8.6 Surface Temperature Variation at x = 0 (Rural Beginning)
-------
196
8.5 Effect of Radiative Participation by Pollutants on
the Vertical Potential Temperature Profile
Figures 8.7 to 8.9 show the effect of radiative participation by
pollutants on the vertical potential temperature profiles at the urban
center. It is seen from Figure 8.7 that while the surface temperature
is reduced by solar participation, the temperature of the boundary
layer is increased by the heating associated with the absorption of
solar energy. Solar heating is also responsible for the higher
temperatures below 400 m for simulation P than for simulation TP.
Above 400 m, the effects of radiative cooling predominate as indicated
by the temperature deficits of the simulations with thermal participa-
tion.
The potential temperature difference profiles for 21:30 hours
and 03:30 hours (2nd day) show trends similar to those of 11:30 hours.
In the absence of solar irradiation, the effects of solar heating
of the daytime hours become very small as seen in Figures 8.8 and 8.9
(solar participation only). For simulation SP the temperatures are
smaller than those of simulation NP below 400 m and warmer above.
The effects of solar heating are confined to heights above 400 m
because at large zenith angles (around sunset) most of the solar
absorption occurs in the upper part of the boundary layer.
It is seen that thermal participation by pollutants causes
cooling of the boundary layer above 400 m, and the resultant tempera-
ture deficit increases during the night and early morning hours. At
the same time, the increase in downward thermal radiation caused by
radiative participation leads to warming at heights below 400 m.
This warming effect is greater for simulation P than for simulation TP
due to the higher temperatures caused by daytime solar heating.
The rural (x = 0) potential temperature difference profiles are
illustrated in Figures 8.10 and 8.11. An examination of Figure 8.10
shows that at 11:30 the predominant effect of solar participation is
one of reducing the solar flux at the surface (see Table 8.5). Back-
ground aerosols (50 yg/m3) do not cause sufficient solar heating to
-------
1000 r-
Z(M)
SIMULATION SP
SIMULATION P
SIMULATION TP
0
0,5
A9P-NP(C)
Figure 8.7 Difference Between Temperature Profiles at Center of City (x = 16 Km) for Participating
and Non-Participating Simulations (Time = 11:30, 1st Day)
-------
1000
Z(M)
-1,0
SIMULATION SP
SIMULATION P
2,0
00
Figure 8.8 Difference Between Temperature Profiles at Center of City (x = 16 Km) for Participating
and Non-Participating Simulations (Time = 21:30, 1st Day)
-------
1000 r-
-1,0
SIMULATION SP
SIMULATION P
SIMULATION TP
AGp.Np(C)
VO
Figure 8.9 Difference Between Temperature Profiles at Center of City (x = 16 Km) for Participating
and Non-Participating Simulations (Time = 03:30, 2nd Day)
-------
200
1000 r-
Z(M)
0
-0,4
SIMULATION SP
SIMULATION TP
SIMULATION P
'P-NP
Figure 8.10 Difference Between Temperature Profiles at x = 0 for
Participating and Non-Participating Simulations (Time
11:30, 1st Day)
-------
201
1000
800
600
Z(M)
400
200
0
-1,0 -0,5
0
SIMULATION SP
SIMULATION TP
SIMULATION P
0,5 1,0
Figure 8.11 Difference Between Temperature Profiles at x = 0 for
Participating and Non-Participating Simulations (Time
= 21:30, 1st Day)
-------
202
offset the effect of the reduction in solar flux at the surface.
This explains the cooler temperatures at heights below 400 m for
simulation SP. Solar heating in the upper part of the boundary layer
does lead to warming in simulation SP. The temperatures below 400 m
are higher for simulation TP than those of simulation P because of
the minor role played by solar heating. However, by 21:30 (see
Figure 8.9) the effects of solar heating become more significant as
indicated by the higher temperatures below 400 m for simulation P
than those of simulation TP. The higher temperatures are caused by
the increased downward thermal radiation from the atmosphere heated
by solar radiation during the day. The negative values of A6u_r
above 400 m for simulations P and TP are associated with radiative
cooling due to gaseous pollutants.
8.6 Effect of Radiative Participation on the
"Crossover" Effect
It is recalled that the temperature crossover effect is associated
with the existence of lower temperatures around 400 m over the city
(Bornstein, 1968; Davidson, 1967). Below 400 m city temperatures are
generally higher than those over the countryside. Two physical
mechanisms have been suggested to explain the "crossover" effect.
Lee and 01fe (1974) believe that the effect is caused by the lifting
of stable air over the city. This suggested mechanism can be better
understood by considering the energy equation. Since the maximum
temperatures occur around the center of the city, horizontal advection
of energy is relatively small in the region. As the "crossover"
effect occurs in regions where turbulent diffusion is negligible,
the energy equation can be written as
98 _ . . 99 (a r I N
It" ~W 91 (8>5-1)
It should be pointed out that the simplified form of the energy equa-
tion has been used only to emphasize the dominant energy transfer
processes in the region in which the "crossover" effect is expected
-------
203
to occur. It is seen from Eq. (8.5.1) that local cooling occurs
(36/at < 0) when the vertical velocity w is directed upwards in a
stably stratified region (86/8z > 0). It is clear that these condi-
tions are satisfied over a city. As stably stratified air is advected
over a city the wind velocity is decreased by the rough city and the
resulting upward motion causes cooling ("crossover" effect) in the
elevated stable region which has not yet been affected by surface
generated turbulence. From the preceding argument it is clear that
the postulated mechanism for the crossover effect is dependent on
two-dimensional (or three-dimensional) effects.
Lee and Olfe (1974) have conducted numerical experiments to test
the hypothesis presented in the previous paragraph. Using a constant
eddy diffusivity model they obtained crossover temperatures as large
as 0.7 C. However, their calculations using a formula relating the
eddy diffusivity to the local shear and stability did not yield much
temperature crossover because the eddy diffusivity oscillated rapidly
with altitude. Bornstein's (1972) results based on a more sophisticated
numerical model do not show the crossover effect.
The second suggested mechanism for the crossover effect is radia-
tive cooling due to pollutants. Using one-dimensional models, Atwater
(1971) and Bergstrom and Viskanta (1973) have shown that radiative
cooling by pollutants can induce the temperature crossover. However,
by definition, the temperature crossover is a two-dimensional effect
as it compares rural and urban temperature profiles. It is not
possible to explain the crossover effect in terms of the radiative
cooling in the urban area without considering the cooling induced by
the background pollutants in the rural area. In fact, recent two-
dimensional calculations by Atwater (1975) show that the crossover
effect cannot be explained in terms ofj^adiative cooling.
It should be pointed out that the prediction of the effect in a
numerical experiment is dependent on the particular turbulence model
used. It is probable that Bornstein (1972) did not obtain the
temperature crossover because he did not account for the variation
of the planetary boundary layer height. His assumed constant mixed
-------
204
layer height of 1050 m gave rise to relatively large diffusivities in
the upper part of the boundary layer. The turbulent heating associ-
ated with these diffusivities offset the cooling induced by the
lifting of stable air. It is also possible that Bornstein did not
predict the "crossover" effect because he did not account for radia-
tion transfer.
The urban temperature excess profiles obtained in this study are
shown in Figures 8.12 to 8.14. It is seen from Figure 8.12 that the
temperature crossover occurs even in the simulation with no radiative
participation. The maximum temperature deficit occurs at the top of
the mixed layer which is around 500 m thick at 11:30. The magnitude
of the temperature crossover is about 0.2 C which is smaller than that
predicted by Lee and Olfe (1974) but it agrees favorably with the
observations made by Bornstein (1968) over New York.
The effects of radiative participation on the daytime temperature
crossover are small. Thermal participation increases the crossover
effect while solar participation decreases it slightly. Radiative
participation has more noticeable effects during the late evening
and early morning hours. It is noted from Figure 8.11 that at 21:30
no temperature crossover occurs for the simulation with no radiative
participation. It is also interesting to notice that solar heating
during the day has a considerable influence on the magnitude of the
temperature crossover at night. The maximum crossover of 0.25 C
occurs for the simulation with solar as well as thermal participation.
This indicates that while solar participation (in combination with
thermal participation) heats up the urban and rural boundary layers
to the same extent (see Figure 8.11) during the day, the higher
pollutant concentrations over the urban area lead to greater cooling
over the urban area than over the rural area during the night. As
cooling is the only radiative effect in the simulation with thermal
participation only, the temperature crossover effect is smaller than
that of the simulation with solar and thermal participation. It is
seen that the maximum crossover occurs around 150 m.
-------
1000 r-
Z(M)
SIMULATION SP
SIMULATION P
SIMULATION TP
SIMULATION NP
no
O
01
Figure 8.12 Difference Between Temperature Profiles of Urban Center and Rural Beginning for
Participating and Non-Participating Simulations (Time = 11:30, 1st Day)
-------
1000 r
SIMULATION TP
SIMULATION NP
SIMULATION SP
ro
o
Figure 8.13 Difference Between Temperature Profiles of Urban Center and Rural Beginning for
Participating and Non-Participating Simulations (Time = 21:30, 1st Day)
-------
1000 i-
Z(M)
SIMULATION TP
SIMULATION P
200
0,2
0
SIMULATIONS NP AND SP
no
o
0,8
1,2
Figure 8.14 Difference Between Temperature Profiles of Urban Center and Rural Beginning for
Participating and Non-Participating Simulations (Time = 03:30, 2nd Day)
-------
208
The results presented in Figure 8.14 show trends similar to that
of Figure 8.13. At 03:30 (second day) the temperature excess profile
of simulation SP differs little from that of simulation NP. A very
small temperature crossover (-0.04 C) is noticed in the simulation
with no participation. The maximum temperature crossover occurs for
simulation P at a height of 300 m.
The temperature crossover of less than a degree predicted in
this study agrees fairly well with recent observations made by Clarke
and McElroy (1974) over Columbus, Ohio, and St. Louis, Missouri. The
measurements made over Columbus, Ohio, are of particular interest as
the city size assumed in this study is comparable to that of Columbus.
The evening temperature profiles (September 20, 1968; March 22, 1969)
show temperature crossovers of about 0.4 C which is consistent with
the predictions of this study.
8.7 The Urban Heat Island and the Effects
of Radiative Participation
It is recalled that theoretical studies (Myrup, 1969; McElroy,
1971; Atwater, 1972) show that the urban heat island is caused
primarily by the differences in the surface parameters between the
rural and urban areas. Also, it is possible to produce the heat
island in a numerical experiment by varying any one of the surface
parameters in isolation. In view of this it is necessary to discuss
the surface parameters used in this study in relation to their role
in creating the urban surface temperature excess. During the daytime
hours, the heat island is created primarily by the difference in
evaporation rates between the urban (H = 0.05) and rural (H = 0.1)
areas, while during the night the assumed anthropogenic heat pro-
duction (H = 40 W/m2) in the urban area is the major cause of the
urban temperature excess. The heat production term also contributes
to the daytime heat island. The higher roughness length (z0 = 1.0 m)
of the urban area tends to decrease the heat island effect during the
day by increasing the upward turbulent heat flux at the surface. At
-------
209
night, when the turbulent heat flux is directed downwards, the increased
urban roughness reenforces the heat island. The lower solar albedo
of the urban area (rs = 0.15) as compared to that of the rural area
(rs = 0.25) also contributes to the daytime heat island.
Figure 8.15 shows the variation of the urban surface temperature
excess as a function of time. It is seen that the temperature excess
is about 0.6 C during the day, and increases very sharply to about
1.1 C at 18:00. The heat island shows a small increase during the
night and reaches a maximum of 1.2 C at 06:00 of the second day (no
radiative participation). This variation of the heat island intensity
is consistent with experimental observations made by Oke and Maxwell
(1974) over Montreal and Vancouver, Canada. However, the nighttime
heat island intensity was considerably smaller than the values
observed typically. A possible explanation for this is the neglect
of soil property variation along the urban area in this study. The
difference in the soil heat capacities between the urban and rural
areas is believed to be the primary cause of the heat island during
the summer months (Peterson, 1969). Again, however, little observa-
tional evidence is available to substantiate or refute this proposition.
It is seen from Figure 8.15 that radiative participation by
pollutants affects the heat island intensity to an appreciable extent.
Thermal participation increases the daytime temperature excess by
about 0.1 C and the nighttime heat island intensity by as much as
0.3 C (25%}. Solar participation decreases the urban temperature
excess by less than 0.1 C during the day. These trends can be
explained in terms of pollutant induced changes of the thermal and
solar fluxes at the surface. The higher pollutant concentrations
over the urban area lead to a greater increase of downward thermal
radiation (see Table 8.5) as compared to that over the rural area.
This explains the increase in the heat island intensity due to thermal
participation. Solar participation on the other hand decreases the
solar flux reaching the surface (see Tables 8.3 and 8.4). This
reduction is accompanied by a decrease in the surface temperature.
It is noted that the urban temperature excess is the largest for the
-------
210
1,8 r
o
-------
211
simulation with both solar and thermal participation. This is to be
expected as the atmosphere is warmer in simulation P than in simula-
tion TP due to solar heating during the day.
Figures 8.16 and 8.17 show the results of a very limited number
of simulations performed to investigate the relative importance of
surface parameters in producing the heat island. It is evident from
Figure 8.14 that the daytime urban temperature excess is caused
primarily by the differences in the moisture parameters between the
rural and urban area. When the urban moisture parameter is increased
to that of the rural area, the daytime heat island intensity becomes
negative and reaches a minimum of -1.5 C at 12:30 (Figure 8.16). The
negative heat island is caused by the increased upward turbulent
fluxes in the city which leads to a cooling in the urban area relative
to the rural surroundings. This result has been observed experimentally
(Mitchell, 1961) and has also been predicted theoretically by Nappo
(1972). Nappo concludes from his study that negative heat islands
could occur when a city is surrounded by desert, or when a city with
much vegetation and water is surrounded by farmland. He offers
Albuquerque, New Mexico, and Minneapolis, Minnesota, as possible
examples of such situations.
Figure 8.17 shows the effect of decreasing the urban moisture
parameter. It is seen that the reduced evaporation in the urban area
leads to a heat island intensity of 4.5 C. It is also noted that the
maximum temperature excess occurs during the day instead of the night.
Although this situation has not been observed experimentally it has
been predicted in theoretical studies (Myrup, 1969; Tag, 1969).
Two numerical experiments were conducted to study the effect of
varying the urban roughness length and the initial velocity on the
heat island intensity. The results are presented in Figure 8.17. It
is seen that increasing the roughness to twice its previous value
decreases the maximum daytime heat island by about 1.5 C. This
reduction of heat island intensity is caused by the increase in the
upward turbulent flux at the surface. The second numerical experiment
was performed after increasing the initial velocity to twice its
-------
212
1,0
0,5
0
-1,0
-1,5
06:00 10:00 14:00 18:00 22:00
LOCAL TIME (HR)
02:00 06:00
Figure 8.16 Negative Heat Island Created by City. H (urban) = 0.05,
H (rural) = 0.05. No Moisture Parameter Variation
-------
213
O
4,5
4,0
3,5
3,0
2.5
2,0
1,5
1,0
II
Jl
\ V"- u40.
\ \
\
\ \
/— Z0« Im (urban)
\\
\
\
\
\
\
\
\
• I
06:00 10:00 14:00 18:00 22:00 02:00 06:00
LOCAL TIME
Figure 8.17 Effect of Surface Parameters on the Heat Island Effect.
H (urban) = 0.01, H (rural) = 0.1
-------
214
previous value (u* was increased from 0.1 to 0.2 m/s). The daytime
heat island is not reduced appreciably because the evaporation rates
in the rural and urban areas are increased to the same extent, and
their difference which determines the heat island intensity during
the day is not altered significantly. However, the nighttime heat
island is reduced by about 1.5 C at 06:00 of the second day. This
decrease in the urban temperature excess with the increase in the
wind velocity is consistent with trends of observations (Oke, 1973).
8.8 Pollutant Distributions in the Urban-Rural System
Before discussing the effects of pollutants on pollutant dis-
persal it is useful to examine the pollutant concentration profiles
in the urban and rural areas. Only aerosol concentrations are presented
as the concentrations of the gaseous pollutant were assumed to be
identical to those of aerosols. The lack of data on pollutant emission
did not warrant an accounting of the difference between the emission
rates of gases and aerosols. The assumption of equal aerosol and gas
concentrations allowed a reduction of the total numerical effort as it
was only necessary to calculate the aerosol concentrations. Figures
8.18 and 8.19 illustrate the development of vertical aerosol concen-
tration profiles in the urban region (x = 16 km) and in the downwind
rural area (x = 24 km). The effect of the strong elevated source at
100 m is evident in the urban concentration profiles. The concentration
reaches a maximum at the source height and decreases gradually to the
background value of 50 iag/m3 at the top of the mixed layer. The shape
of the concentration profile around the source is roughly Gaussian
during the day as well as the night. The daytime concentrations in
the rural area are almost uniform through most of the mixed layer and
decrease quite sharply to the background value at the top of the mixed
layer. This predicted shape of the rural concentration profiles is
consistent with observations (Edinger, 1973). However, the urban
concentration profiles (especially at 15:30, 1st Day) are not well
mixed. This indicates that the effects of the assumed source are
larger than those observed. It should be recalled, however, that a
-------
1000 r-
600
Z(M)
400
200
0
T = 11:30 (1ST DAY)
T = 15:30 QST DAY)
rv>
!—•
en
T = 03:30 (2ND DAY)
Figure 8.18 Aerosol Vertical Concentration Profiles at Center of City (x = 16 Km) for Simulation NP;
(The gradients at z = 0 are zero)
-------
1000 r-
600 -
Z(M)
400 -
200 -
0
T = 15:30 QST DAY)
T = 11:30 QsT DAY)
T = 03:30 (2ND DAY)
ro
i—>
01
Figure 8.19 Aerosol Vertical Concentration Profiles at Downwind of City (x = 24 Km) for Simulation NP;
(The gradients at z = 0 are zero)
-------
217
relatively large source strength was used to highlight radiative
effects.
It is seen from Figures 8.18 and 8.19 that the maximum pollutant
concentrations occur during the late evening (see Table 8.3 also) and
early morning hours. The best time for pollutant dispersion is in
the afternoon (-15:30). It is clear from the concentration profiles
that the mixed layer height is an important parameter in determining
the concentration levels. At 11:30 (1st Day) the mixed layer is
around 500 m and the average aerosol concentration in the rural
boundary layer (see Figure 8.17) is about 300 yg/m3. It is seen that
the subsequent expansion of the mixed layer is accompanied by a decrease
in the average aerosol concentration. At 16:00 (1st Day) the mixed
layer has grown to 800 m and the average aerosol concentration has
decreased to a value of around 200 yg/m3. During the night, the
mixed layer decreases in height, and pollutant concentrations increase
due to limited vertical mixing. The relatively large concentrations
at the source during the night are the result of pollutants being
injected into a stable atmosphere (see Tables 8.5 and 8.6).
The variation of aerosol concentrations at the surface and at
the source are illustrated in Figure 8.20. As the simulation is
started in the early morning (05:30) the pollutants build up rapidly
in the shallow boundary layer (400 m). By about 06:30 the concentra-
tion at the source has exceeded 700 yg/m3 while the concentration at
the surface (1 m) has reached a value of about 420 yg/m3. Soon after
sunrise, the mixed layer starts growing, and the aerosol concentration
at the source starts decreasing. Till about 10:00 the surface con-
centration increases as pollutants from the source diffuse towards
the surface. The height, h , at which the concentration reaches the
background value of 50 yg/m3 (+ 2 yg/m3) is also shown in Figure 8.20
in order to explain the concentration variation. The mixed layer
height defined with respect to the potential temperature gradient is
about 100 m less than h at any given time. It is seen from Figure
8.20 that the aerosol concentrations at the source as well as the
surface start decreasing at almost the same rate around 10:00. This
-------
218
1000
900
V.
s
700
600
500
40Q
300
I
I
I
CQer(Z = lOOm)
Caer(Z=lm)
I I
I
;:00 10:00 14:00 18:00 22:00 02:00 06:00
LOCAL TIME (HR>
Figure 8.20 Variation of Aerosol Concentration with Time at x = 16 Km
for Simulation NP
-------
219
indicates effective vertical mixing which accompanies the growth of
hc- At 16:00 hrs, hc reaches a maximum height of about 800 m, and
the surface and source concentrations reach their minimum values of
340 yg/m3 and 385 yg/m3, respectively. At 17:00 hrs hc starts
decreasing quite rapidly as the boundary layer becomes stable.
It is seen that the pollutants left in the atmosphere as the
mixed layer collapses are transported (diffused and advected) away
horizontally. This rapid dispersion of pollutants above the mixed
layer is caused by the imposition of periodic boundary conditions in
the horizontal direction. As there are no data on the residence
time of pollutants above the mixed layer, it is not possible to make
any statements about this seemingly unrealistic feature of the model.
However, a simple estimate based on the assumption that pollutants
above the mixed layer are removed only by horizontal advection indi-
cates that the residence time above the urban-rural system cannot
be much greater than 4 hours.
As expected, the collapse of hc around 17:00 hours is accompanied
by sharp increases in the aerosol concentrations at the source as
well as the surface. These increases are the result of pollutants
building up in the stable shallow evening boundary layer. It is
seen that the pollutant concentrations reach a maximum (local) around
19:00 hours, and then level off during the early morning hours. This
"overshooting" of the pollutant concentrations above the nocturnal
level is caused by the initial difference between the time scales of
pollutant buildup and pollutant dispersion by advection and diffusion.
The smallest difference between the source and surface concentrations
is about 45 yg/m3 and occurs around 14i:00 hours when vertical dis-
persion is most effective. The maximum difference of 300 yg/m3
occurs in the stable conditions of the nocturnal boundary layer.
It is interesting to note that the pollutant concentration variations
at the source and surface are similar except that there is a time lag
of about 1 hour between the changes at the source and those at the
surface. This time lag is to be expected as the distance between the
source and surface represents a capacitance for the flow of pollutants.
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220
The results of this section are in reasonable agreement with
observations (Edinger, 1973). However, they also illustrate the
limitations of the model. It is seen that the elevated source over
the urban area has an unrealistically large effect on the shape of
the concentration profiles. This feature is attributed to one or
both of the following reasons. The assumed source strength may
have been too large. Secondly, the O'Brien diffusivity model which
does not reflect local stability of the boundary layer cannot
represent the intense vertical mixing in the "mixed" layer. Conse-
quently, the daytime boundary layer is predicted to be less well
mixed than observations indicate it to be.
8.9 Effect of Radiative Participation by Pollutants
on Pollutant Dispersal
Before presenting the results of this section, it is instructive
to examine the physical processes through which radiative participa-
tion by pollutants affects pollutant dispersal. While radiative
participation by pollutants affects the potential temperature directly
through the radiative flux term in the energy equation, it modifies
the pollutant concentration only indirectly through the turbulent eddy
diffusivity (diffusion term in the species equation). The eddy
diffusivity is a function of the potential temperature and velocity
gradients as well as the mixing length. By modifying the potential
temperature and velocity (through eddy diffusivity) gradients,
radiative participation alters the eddy diffusivity which in turn
affects the pollutant concentration. Thus, it is clear that an
understanding of the effect of radiative participation on pollutant
dispersal is dependent on the modeling of turbulence. The implication
of this will be discussed in detail in a later paragraph.
In the light of the preceding discussion it is appropriate to
examine the eddy diffusivity formulation of the two-dimensional model
in relation to radiative effects. In the equilibrium layer, the eddy
diffusivity is a function of the local temperature and velocity
-------
221
gradients. Thus, it can be expected that pollutant induced changes
in these gradients will be accompanied by changes in pollutant concen-
trations in the equilibrium layer (50 m thick). In the transition
layer, eddy diffusivities are specified by the O'Brien cubic profile
which does not allow local stability to affect the eddy diffusivity.
As the shape of the O'Brien profile is determined by the turbulence
characteristics of the equilibrium layer, changes in the transition
layer eddy diffusivities only reflect changes in the eddy diffusivities
in the 50 m layer adjacent to the earth's surface. Thus, potential
temperature modification caused by pollutants does not affect the
pollutant concentration locally. This limitation of the model is
especially serious from the point of view of stability changes induced
by solar heating. As solar energy is absorbed mostly above the
equilibrium layer, this model cannot be expected to be sensitive to
the stabilizing action of solar heating. As the stability of the
equilibrium layer is controlled to a large extent by the surface
temperature, a high correlation between changes of surface tempera-
tures and changes of pollutant concentrations can be expected. A
higher surface temperature would imply a decrease in the stability of
the boundary layer. This would mean better vertical mixing which in
turn would lead to more effective horizontal advection. Thus, a
warming of the surface temperature can be expected to be accompanied
by a decrease in the vertical pollutant mass loading. It is felt
that this discussion will help to provide better understanding of the
results of this section.
This study assumed that the concentrations of aerosols were
identical to those of the gaseous pollutant (NH3). This allows the
combination of aerosol and gaseous pollutant to be treated as a
single pollutant which participates radiatively in the thermal as
well as the solar spectrum. It is recalled that aerosols participate
only in the solar spectrum, and the gaseous pollutant participates
only in the thermal spectrum.
Tables 8.6 and 8.7 show the effect of radiative participation
by pollutants on pollutant dispersal. As expected the warming effect
-------
222
Table 8.6 Comparison of Aerosol Concentrations C1n ug/m3) at z - 1 m
and z = 100 m at x = 16 km for Simulations P and NP
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
NP
50.0
420.2
455.3
446.1
392.7
337.9
398.4
472.3
524.6
502.5
491.7
497.6
514.0
z = 1 m
P
50.0
430.7
450.1
446.7
375.5
321.8
380.3
499.0
523.0
477.4
451.1
461.0
481,5
P-NP
0
10.5
- 5.2
- 0.6
-17,2
-16.1
-18.1
26.7
- 1.6
-25.1
-40.6
-36.6
-32.5
NP
50.0
732.4
518.6
503.6
433.3
384.3
537.9
932.2
839.5
793.9
780.6
785.6
793.6
z = 100 m
P
50.0
705.0
513,3
503,5
414,8
366.0
511.3
829.2
744.6
686.0
723,3
726.7
737.2
P-NP
0
-27.4
- 5.3
- 0.1
-18.5
-18.3
-26.6
-103,0
-94.9
-107,9
-57.3
-58.9
-56.4
-------
223
Table 8.7 Comparison of Aerosol Concentrations (in ug/m3) at z = 1 m
and z = 100 m at x = 24 km for Simulations P and NP
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
NP
50.0
318.5
334.5
319.4
290.2
262.6
300.4
373.6
427.5
417.0
407.2
404.2
406.5
z = 1 m
P
50.0
322.1
330.3
317.8
280.9
251.2
291.5
343.4
404.1
389.0
376.6
377.3
382.6
P-NP
0
3.6
- 4.2
- 1.6
- 9.3
-11.4
- 8.9
-30.2
-23,4
-28.0
-30.6
-26.9
-23.9
NP
50.0
475.8
346.5
329.5
295.1
269.0
363.3
609.0
596,4
579.2
569.4
563.4
555.6
z = 100 m
P
50,0
467,8
341,7
327.5
285.7
259.1
343.0
602,1
566.1
546.8
562.6
558.2
549.0
P-NP
0
- 8.0
- 4.8
- 2.0
- 9.4
- 9.9
-20.0
- 6,9
-30.3
-32.4
- 6.8
- 5.2
- 6.6
-------
224
of pollutants on the surface temperature leads to a reduction in the
pollutant concentrations in the boundary layer. As explained before,
this decrease of pollutant concentrations is the result of the
decrease in the surface layer stability (see Table 8.8) when pollutants
are radiatively participating. It is noted from Table 8.6 that the
reduction of pollutant concentrations is smaller during the day than
during the night. This is to be expected as pollutants cause a
greater surface temperature increase during the night than during
the day. At the surface, the daytime decrease of aerosol concentra-
tion is about 4.8% at 15:30, and the reduction during the night is as
large as 8.2% at 01:30. The effects of radiative participation on
concentrations in the vicinity of the source are almost of the same
magnitude (percentage) as those on surface concentrations. The
concentration reduction due to pollutants at 100 m is about 4.8%
at 15:30 and about 13.5% at 23:30. It is noted that pollutant
induced changes in the concentrations are smaller in the rural area
than in the urban area. The maximum reduction of surface concentra-
tion in the rural area is about 7.5% at 01:30 as compared to 8.2% in
the urban area. At the urban source height (no source in rural area)
of 100 m the reduction in pollutant concentration is only 5.5% at
23:30. Clearly the larger decrease of 13.5% (107.9 yg/m3) in the
urban area is caused by the presence of the pollutant source.
As changes in pollutant concentrations are a direct result of
changes in eddy diffusivities it is necessary to examine the effect
of radiative participation by pollutants on eddy diffusivity profiles.
Table 8.8 compares eddy diffusivities at selected heights for simula-
tions P and JUP. It is seen that radiative participation has a
noticeable influence on the eddy diffusivities at 1 m. The maximum
pollutant caused increase is about 14.8% (0.04 m2/s) at 23:30. At
the source height of 100 m the increase is as large as 63% at 19:30.
It is not surprising to note that the period (19:30 to 05:30) of the
largest changes in diffusivities is also the time interval during
which the pollutant concentration are altered the greatest.
-------
225
Table 8.8 Comparison of Eddy Diffuslvltles (in m2/s) at z =; 1 m
and z = 100 m for Simulations P and NP at x = 16 km
Time
05:30
07:30
09:30
11:30
13:30
15:30
17:30
19:30
21:30
23:30
01:30
03:30
05:30
NP
0.61
0.26
0.39
0.42
0.45
0.43
0.38
0.24
0.27
0.27
0.27
0.27
0.27
z = 1 m
P
0.61
0.27
0.39
0.42
0.45
0.43
0.34
0.28
0.31
0.31
0.28
0.28
0.28
P-NP
0
0.01
0
0
0
0
-0.04
0.04
0.04
0.04
0.01
0.01
0.01
NP
1.12
3.35
19.38
24.47
26.46
22.59
8.84
1.83
2.52
2.68
2.73
2,81
2.98
z = 100 m
P
1,12
3.91
20.05
25,04
27.32
23.35
9,48
2.99
3.65
3.73
2.87
3,00
3.25
P-NP
0
0.56
0.67
0.57
0.86
0.76
0.64
1.16
1.13
1,05
0.14
0.19
0.27
-------
226
As the shape of the O'Brien diffusivity profile in the transition
layer is determined by turbulence characteristics of the equilibrium
layer, it is interesting to examine the effects of radiative partici-
pation by pollutants on the stability of the surface layer as reflected
in the eddy diffusivities of the transition layer. Table 8.9 compares
the eddy diffusivity profiles at two selected times for simulations P
and NP- It is seen that the pollutant induced increase in eddy
diffusivity of about 1.6% at 30 m (13:30) is magnified to 8.3% at
200 m in the transition layer. The increase becomes as large as
103% at 500 m during daytime (13:30). At 21:30, the change of 39.0%
caused by pollutants at 30 m in the equilibrium layer is reflected as
a 39% increase at 200 m in the transition layer. The analysis of
this paragraph indicates that the effect of radiative participation
by pollutants on pollutant dispersal should be viewed in the light
of the limitations of the particular turbulence being used.
It is clear from the preceding discussion that in this study it
is not possible to draw any firm conclusions on the effects of
pollutants on pollutant dispersal in the transition layer. As the
stability of the surface layer is controlled to a large extent by
the surface temperature, the only important radiative effect of
pollutants from the point of view of pollutant dispersal is that of
warming or cooling of the surface temperature. It is obvious that
this statement has important implications. With a turbulence model
which does not predict the eddy diffusivities realistically in the
transition layer, it is not possible to make definitive recommendations
regarding the inclusion of radiation transfer in Air Quality Simulation
Models.
It is necessary to reemphasize the importance of the turbulence
model in determining the effects of radiative transfer on pollutant
dispersal. The effects of radiative participation are largely dependent
on the sensitivity of the eddy diffusivity formulation to local stabil-
ity changes caused by radiative heating or cooling. Thus, surface
layer based eddy diffusivity formulations such as those used by
Bornstein (1972) and Estoque (1973) cannot be expected to react to
-------
227
Table 8.9 Comparison of Eddy Diffusivitles (in m2/s) at x = 16 km
for Simulations NP and P
Height
(m)
1
5
15
30
50
100
200
300
400
500
600
NP
0.45
1.17
3.45
7.32
13.67
26.46
36.34
31.36
18.79
5.91
0.00
Time = 13:30
P
0.45
1.19
3.50
7.44
13.93
27.32
39.37
36.79
25.66
12.00
1.87
P-NP
0.00
0.02
0.05
0.12
0.26
0.86
3.03
5.43
6.87
6.09
1.87
NP
0.27
0.63
1.21
1.91
2.23
2.68
2.12
0.76
0.00
0.00
0.00
Time = 21:30
P
0,32
0,74
1,69
2.67
3.10
3.73
2.95
1.06
0,00
0.00
0,00
P-NP
0.05
0.11
0.48
0.76
0.87
1,05
0.83
0.30
0.00
0.00
0.00
-------
228
pollutant caused changes in the transition layer. On the other hand,
the model suggested by Pandolfo (1966) is extremely sensitive to local
stability. Bergstrom and Viskanta (1973) and Atwater (1974) using
Pandolfo's model conclude from their studies that radiative participa-
tion can change pollutant concentrations at the ground by as much as
20%. It is clear that this relatively large change is tied to the
turbulence model used. As atmospheric turbulence is still incom-
pletely understood, it is not always possible to examine critically
the validity of the assumptions on which a particular turbulence model
is based. Clearly, conclusions regarding the effects of pollutants
on pollutant dispersal have to be qualified with statements about the
turbulence model used.
-------
229
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APPENDIX A
Calculation of Directly Transmitted and Diffuse Solar
Radiation at the Top of the Atmospheric Layer
This study utilized a combination of empirical procedures sum-
marized by Paily, et al. (1974) to calculate the direct and diffuse
solar fluxes at the top of the atmospheric layer being studied. The
method is given below for convenience.
The solar altitude a is the angle in the vertical plane between
the sun's rays and the horizontal. The zenith angle u is the comple-
ment of a. Using spherical trigonometry y can be written as
u = cos"1 [sin sin6 + cost}) cos6 cosh] (A.I)
where is geographic latitude in radians, 6 is declination of the
sun in radians, and h is local hour-angle of the sun in radians.
An approximate relation to obtain 6 is given by
• <5 = (23.45 7T/180) cos[(2fr/365)(172 - D)] (A.2)
where 6 is in radians and D is the number of the day in a year with
D = 1 for January 1 and D = 365 for December 31.
The attenuation of solar radiation is represented by three
coefficients which are functions of the optical air mass, the moisture
content of the atmosphere and the dust content of the atmosphere.
The directly transmitted solar radiation St at an arbitrary height
in the atmosphere is given by
St = S0(a" - d) (A.3)
where S0 is the solar radiation at the top of atmosphere, a" is
the coefficient which accounts for attenuation for scattering and
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243
absorption by gases and d is the atmospheric dust depletion coeffici-
ent. These coefficients are expressed by the relations
a" = exp{m[-(0.465 + 0.130w)][0.179 + 0.421 exp(0.721 m)]}(A.4)
and
d = 1 - 0.99m (A.5)
The optical air mass m is given by the following equations:
m = m0(pa/p0) (A.6a)
mQ = [cosy + a(a + b)"0]"1 (A.6b)
where m0 is the optical air mass at sea level where the barometric
pressure is p , u is the zenith angle, and pa is the barometric
pressure at the altitude at which the solar fluxes are being computed.
The constants a, b, and c have the values:
a = 0.15 (A.7a)
b = 3.885 (A.7b)
c = 1.253 (A.7c)
The term w in Eq. (A.4) is the precipitable water vapor content in cm
above the altitude at which solar fluxes are being computed.
Assuming that half of the scattered radiation reaches the
altitude being considered, the diffuse solar radiation S^ can be
written as
Sd = SQ[0.5(1 - a1) + 0.5d] (A.8)
where a1 is a coefficient which accounts for attenuation by scattering
only and is given by
a1 = exp{m[-(0.465 + 0.134w)][0.129 + 0.171 exp(0.880m)]} (A.9)
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
REPORT NO.
EPA-600/4-76-039
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
RADIATIVE EFFECTS OF POLLUTANTS ON THE PLANETARY
BOUNDARY LAYER
5. REPORT DATE
Julv 1976
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT
A. Venkatram and R. Viskanta
9. PERFORMING ORGANIZATION NAME AND ADDRESS
School of Mechanical Engineering
Purdue University
West Lafayette, Indiana 47907
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
R803514
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
13. TYPE OF
Interim
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
The objective of this study was to gain a better understanding of the effects
of pollutants on the thermal structure and pollutant dispersal in the planetary
boundary layer. To this end numerical models of the boundary layer were constructed.
Gaseous pollutants in the boundary layer were considered to absorb and emit thermal
radiation, while aerosols were allowed to absorb and scatter solar energy.
First, a one-dimensional numerical model of the boundary layer was
constructed. The model used the two-stream method for the computation of radiative
fluxes, and a turbulent kinetic energy model to account for turbulence. A series of
numerical experiments were performed to determine the role of pollutants in modifying
thermal structure and pollutant dispersal in the boundary layer. The results showed
that the predominant influence of gaseous and particulate pollutants on surface
temperature was warming. Radiative participation by pollutants increased the
stability of the surface layer during the day. During the night, the warmer surface
temperatures caused the surface layer to become less stable.
The second phase of the study involved the construction of a two-dimensional
numerical model to study the effects of pollutants on urban-rural differences in
thermal structure and pollutant dispersal. The effects of pollutants on pollutant
dispersal were found to be significant. At the source height (100 m) in the urban
area, the pollutant concentration was reduced by as much as 13.5% during the night.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COS AT I Field/Group
*Air pollution
*Boundary layer
*Solar radiation
*Thermal radiation
*Mathematical models
13B
20D
03B
20M
12A
13. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
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