INFARED COOLING OF THE
ATMOSPHERE BY THE 9.6
MICRON BAND OF OZONE
by
WALTER JOSEPH SLADE JR.
B.S., Texas Christian University
1969
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSAC USETTS TNSTITUTE OF
TECHNOLOGY
February, 1975
Signature of Author ..........
...---
Depar Znt of
Certified by
.-
eteorology
...
Thesis Supervisor
Accepted by ....................................
Chairman, Departmental
Committee
INFRARED COOLING OF THE
ATMOSPHERE BY THE 9.6
MICRON BAND OF OZONE
by
Walter Joseph Slade Jr.
Submitted to the Department of Meteorology on 22 January 1975
in partial fulfillment of the requirements for the degree
of
Master of Science
ABSTRACT
Monthly values of the radiative cooling of the atmosphere from
0-70 kilometers by the 9.6 am band of ozone are calculated using a
random band model to represent frequency integration combined with a
3-parameter inhomogeneous path approximation. Effects of line shape
variability above 30 km are handled by a correction factor calculated from
the Voigt integral.
Calculations were made on a global basis using data from a threedimensional 26 level general circulation model above 30 kilometers and
observational data below 30 kilometers.
Resylts indicate maximum cooling at 50 km with a maximum value of
at 60 0N in June and at 600 S in December. Slight radiative
5.40C day
heating occurs above the tropopause with a maximum value of 0.2 C day
Values approach zero near the ground and near 70 km. Ozone is concluded
to be a major contributor to the radiative budget of the middle and upper
stratosphere.
Thesis Supervisor:
Reginald E. Newell
Title:
Professor of Meteorology
Table of Contents
I.
Introduction.
7
II.
Radiative Transfer Equation.
8
III.
Previous Work.
14
IV.
Transmission Function.
16
A.
Band Model.
16
B.
InhomogeneoVs Path Approximation.
19
C. Line Shape.
D.
V.
Calculation of Cooling Rate.
A.
Physical model.
25
26
26
B. Temperature and Ozone Data.
26
C. Ozone Spectral Data.
27
D.
VI.
Zenith Angle.
19
Climatological Averaging.
28
E. Numerical Methods.
28
Results and Conclusions.
31
List of Symbols and Definitions.
71
Bibliography.
73
Acknowledgements.
List of Tables
Table
1
Previous Q0 Calculations
Table
2
Band Model Parameters For Test Atmosphere
Table
3
Inhomogeneous Path Parameters For Slabs N to N-1
Table
4
Line Shape Parameters
Table
5
Line Shape Correction Parameters for Slabs
N to N-1
Table
6
Cooling Rates For Test Atmosphere
Page
15
List of Figures
Page
Fig. 1
Line Intensity Distribution vs.
Band Model Prediction
34
Fig. 2
[T
35
Fig. 3
[T] for February
36
Fig. 4
[T] for March
37
Fig. 5
[T] for April
38
Fig. 6
[T] for May
39
Fig. 7
[T] for June
40
Fig. 8
[T] for July
41
Fig. 9
[TI for August
42
Fig. 10
[T] for September
43
Fig. 11
[T] for October
44
Fig. 12
[TI for November
45
Fig. 13
[TI for December
46
Fig. 14
Ozone Density For January
47
Fig. 15
Ozone Density For February
48
Fig. 16
Ozone Density For March
49
Fig. 17
Ozone Density For April
50
Fig. 18
Ozone Density For May
51
Fig. 19
Ozone Density For June
52
Fig. 20
Ozone Density For July
53
Fig. 21
Ozone Density For August
54
Fig. 22
Ozone Density For September
55
Fig. 23
Ozone Density For October
56
Fig. 24
Ozone Density For November
57
Fig. 25
Ozone Density For December
58
for January
List of Figures - Cont'd.
Page
Fig. 26
Cooling Rate For January
59
Fig. 27
Cooling Rate For February
60
Fig. 28
Cooling Rate For March
61
Fig. 29
Cooling Rate For April
62
Fig. 30
Cooling Rate For May
63
Fig. 31
Cooling Rate For June
64
Fig. 32
Cooling Rate For July_
65
Fig. 33
Cooling Rate For August
66
Fig. 34
Cooling Rate For September
67
Fig. 35
Cooling Rate For October
68
Fig. 36
Cooling Rate For November
69
Fig. 37
Cooling Rate For December
70
I.
Introduction.
General circulation studies of the energetics of the upper
stratosphere (above 10 mb.) have been hampered by a lack of knowledge
of the vertical motion field and the contribution by non-adiabatic
effects (Tahnk, 1973). These terms are needed to compute the generation
of zonal and eddy available potential energy and the conversion between
potential and kinetic energy for both the mean and eddy flows. The
vertical motion can be calculated from the First Law of Thermodynamics
(in spherical coordinates):
T~
u
WC0..dCOC
>T
v
T
RT .T
CP
(1)
Q is the non-adiabatic heating term and, the stratosphere is equal to
, which has three components:
radiative heating; Q
QRAD = QCO 2 + Q3 + QSOLAR
(2)
dioxide;
CO2 is the cooling due to the 15 micron band of carbon
QSOLAR is heating due to the ultraviolet and visible bands of ozone,
and Q
is the cooling due to the 9.6 micron band of ozone.
QCO2
has been computed by a number of investigators; Plass (1956), Murgatroyd
and Goody (1958), Kuhn and London (1969), and Herman (1972). QSOLAR
has also been calculated a number of times, the most recent being
Cunnold (1974). For a variety of reasons, namely lack of detailed spectra
data owing to the complexity of lines and the asymmetric nature of the
molecule, experimental difficulties in measuring emissivities, and lack
of detailed measurements of the vertical distribution of ozone, Q0
has not been so thoroughly investigated. This paper will present a method
to calculate Q from 0-70 kilometers and present results of such calculations
03
on a monthly basis.
II.
Radiative Transfer Equation
The following assumptions are made:
1. The atmosphere is plane-parallel and horizontally stratified.
2. Refraction and scattering are ignored.
3. Local Thermodynamic Equilibrium (LTE) is assumed.
The source function is therefore identical to the Planck
black-body Function.
The equation of radiative transfer may be written:
where -r, the optical thickness, is defined by:
cl.T = kdu
ko
,the
absorption coefficient is defined by:
Ia =
S F()-0)
(5)
where F(J - Vo ) is a line shape function and S is the line intensity.
Assuming the surface is a black body at the temperature of the atmosphere
next to the ground, "Tg, so that:
di T5(6)
T,
and the upper boundary is the top of the atmosphere, or more correctly
the altitude where the density of the absorbing gas is zero. At the
upper boundary incident infrared radiation is ignored, so that:
( )(0.)
0
7)
The solution of (3) with boundary conditions (6) and (7) is:
(8)
I (J~r,,) = B(O.,Tl
for'/> 0 and
I4
for
U<O.
-~
13 (AjT)
g,)=
(9)
The flux across a surface of constant pressure p is:
F(p) = f.[f I&(vj,),I.d c,
dA
(10)
Substituting (8) and (9) into (10) and integrating by parts yields:
T)b)L
[()TfE
3
rlp
aLtar
JJ
E.3 &LI-TX.1UI
oLT
(1i)
where E3 is the exponential integral used to represent integration over
zenith angle.
E5 (T
and
T,
f
C.T4
(12)
the flux transmission function is
T Zf
(+-T)
(13)
The cooling rate is:
by
(14)
10.
The above derivation is relatively straight forward and well-known.
The actual calculation of cooling rates involves therefore three
integrations; over frequency, over height, and over zenith angle.
The difficulty in evaluating (11) arises from the fact that pressure,
temperature, and absorber amount very continuously along the path
Kj , the absorption coefficient varies with frequency,
since it is a function of line shape, temperature and pressure,
taken.
and since it is a function of line intensity, temperature again.
The greatest difficulty arises from the integration over
frequency.
This can be overcome by experimental methods.
The absorption
of ozone can be measured directly with a spectrophotometer over a
The main difficulty
variety of pressures, temperatures, and path lengths.
arises at very low pressures and path lengths where the absorption
is very close to zero.
These low pressures and path lengths are also
very difficult to duplicate in the laboratory.
For pressures and path
lengths less than those obtainable experimentally, extrapolation
has been used.
Another method used is line by line integration over the spectrum
by direct numerical integration.
This method has the advantage of
using the actual distribution of lines and intensities.
from all lines are included.
computer time required.
model.
The contributions
The obvious disadvantage is the massive
The most frequently used method is the band
The distribution and intensity of lines are represented by a
tractable mathematical function.
simplicity.
The main advantage is mathamatical
The disadvantages of band models are that the complex
distribution of lines is only crudely approximated and that contributions
from the wings of distant lines are not included, although these
have been overcome somewhat by newer random models.
A newer method is the opacity distribution function where the
fact is used that for a homogeneous atmosphere (K,) is not a function
of temperature and pressure), the transmission within an interval
11.
is independent of the ordering of the Kg 's within the interval.
By ordering the Kj values in an ascending order using a random
selection of frequencies, a smooth opacity distribution results. The
main advantage is reduced computer time over line by line integration.
The disadvantage of this method is the difficulty in converting to an
inhomogeneous atmosphere (Arking and Grossman, 1972).
The above discussion involves only a homogeneous atmosphere.
The real atmosphere is inhomogeneous with temperature and pressure
varying continuously along the path taken. Since the absorption
coefficient is a function of temperature, pressure and frequency, it
is advantageous to separate the integration over frequency from that of
the path taken(temperature and pressure). This method involves finding
a homogeneous atmosphere whose transmission is the same as the inhomogeneous
atmosphere being considered.
The most widely known and frequently used method of finding
this homogeneous atmosphere is the so-called "Curtis-Godson approximation"
(Curtis, 1952, Godson, 1953). It is defined as follows:
S =.
SCd
----
SO
(15)
U(16)
S, the line intensity, and c< , the half-width of a Lorentz line, are
defined:
5 .x
T) 3
(17)
12.
(18)
[T
Cdo
where m = 5/2 for ozone and the subscript o denotes reference state
conditions.
If variations of S and o.
with temperature are ignored
(15) and (16) reduce to:
UJ =
I,
(19)
=
L4
A mean S and c
(or
AU
U
(20)
and p) is found which relates the inhomogeneous
atmosphere to a homogeneous atmosphere.
The absorption coefficient at
the center of a Lorentz line is defined:
X" SJ
0i
(21)
A line is a strong line when x>'Ol, a weak line when x <4 1, and is in
the intermediate region when x'- l.
The Curtis-Godson approximation has
been shown to be valid for weak and strong lines but not as good for
the intermediate region, especially for ozone (Walshaw and Rodgers,
1963; Clark and Hitschfield, 1964; and Armstrong, 1968).
calculated heating rates of up to 40% have been found.
Errors in
The general
conclusion is that the Curtis-Godson approximation is invalid for
ozone above 100 mb.
To overcome this difficulty higher parameter methods have been
developed (Goody, 1964; Rodgers, 1968).
These methods use 3 and 4
parameters respectively instead of the 2 used in Curtis-Godson.
13.
Goody's 3-parameter method reduces the errors by a factor of 3 to 12.
The penultimate obstacle is the angular integration over zenith
angle.
The flux transmission is:
2I
jf
3 ( U U') CIz
(22)
and the intensity transmission is
T eXp
A diffusivity term
U U)(23)
is introduced:
U
u(24)
Rodgers and Walshaw (1966) have determined that a value of
=
1.66
introduces an error of less than 2% into heating rate calculations.
The last difficulty occurs in the numerical integration.
At very low pressures, the net flux across a pressure surface is
very small.
Care must be taken that errors introduced by numerical
approximations do not have the same order of magnitude as the net flux.
14.
III.
Previous Work.
Previous investigations into Q
are summarized in Table I.
Plass (1956) using the 1941 laboratory measurements of Summerfield
with 3 different ozone and 3 different temperature profiles found
was on the order of a few tenths of a degree per day below
that Q
20 km.
3
The sign (either heating or cooling) depended on the ozone
distribution.
From 35 to 60 km there was cooling of 2 to 3 C day~-.
Murgatroyd and Goody (1958) using Plass's results computed the
cooling for ozone along with other gases for the region 30 to 90 km.
They found a maximum cooling of 30C day~
at 40ON in the summer with the.
maximum cooling at all latitudes near the stratopause. Hitschfield
and Houghton (1960) integrated the transmission line by line over a
narrow potion of the 9.6 muicron ozone band. fThe-results were
extrapolated to the rest of the band and used with ozone and temperature
data from a sounding from Liverpool.
0.2 to
0.30 C
day~ 1
They found radiative heating of
in the region 10-20 km.
Manabe and Moller (1961) using the laboratory measurements of
Walshaw and assuming radiative equilibrium found that ozone heated the
atmosphere from 6 to 27 km with a maximum value of 0.3 0C day~1 near
17 km.
Kuhn and London (1969) using a random band model along with the
Curtis-Godson approximation found a maximum cooling of 3.50C day~1 at
the stratopause in the sunmer near 600 N. The entire stratosphere was
cooled but values approached zero near 70 km.
As part of a study of global radiative heating Dopplick (1970)
using a random band model along with Rodgers new
3
4-parameter method. He found thermal heating by ozone above the tropical
calculated Q0
tropopause associated with large gradients of ozone concentration.
Values ranged to 0.7 0 C day
at 24 km near 20 0N in the winter season.
Table 1
Previous Q0
Calculations
Name
Year
Region
Frequency Integration
Temperature Data
Ozone Data
Plass
1956
0-70 km
Laboratory measurements
of Summerfield
Rocket Panel
(3 soundings)
Rocket Panel
(3 soundings)
Murgatroyd,
Goody
1958
20-85 km
Plass' results
Murgatroyd
(seasonal )
Rocket data
Hitschfield,
Houghton
1960
0-33 km
Line by line
integration with
Curtis-Godson
Liverpool sounding
Liverpool sounding
Manabe, Moller
1961
0-30 km
Laboratory data of
Walshaw
Radiative equilibrium
calculations
Tonsberg and Olson
Kuhn, London
1968
0-70 km
Random band model
of Wyatt with
Curtis-Godson
Kantor and Cole
(seasonal )
Hering and Borden
Dopplick
1970
0-30 km
Random band model
of Malkmus with
Rodgers 4-parameter
method
M.I.T. General
Circulation Data
Library
Hering
16.
IV.
Transmission Function.
A.
Band Model
It was decided to use a band model for reasons of economy.
An opacity distribution function was considered briefly, but dropped due
to a still enormous requirement for computer time.
The 9.6 micron
band of ozone consists of over 10,000 individual lines.
To construct an
opacity distribution function would require picking 500 random points along
the band and summing the contributions from all lines within + 75 cm~
of the point, which in the center of the band would involve over 5000 lines.
Because of the asymmetric nature of the ozone molecule, a statistical
The distribution of intensities most often
or random model is required.
used is:
P()Lexsj
(25)
However this function underestimates the number of low-intensity lines.
To correct this Malkmus (1967) has developed a random "exponentialtailed" band model.
If S
and SMIN are the maximum and minimum values
of S respectively, then:
R =
(26)
MIN
Malkmus then defines the probability distribution as:
P(S)= I
S iu R
[ ex-,
9
) _e
j,
.
I~
)J
(27)
This distribution function is nearly identical to (25) over a large
range of S values.
It gives a greater probability to low values of S.
17.
Figure I. shows the distribution of line intensities for ozone taken
from McClatchey's (1973) line parameter compilation compared to that
T was taken to be 250 0 K,
predicted by the Malkmus model.
and SMIN = 0.2127 x 10
cm
SMA
= 0.7009
(atm-cm~), and R = 0.3295 x 10
Malkmus assumes a Lorentz line shape.
This line shape is valid
Methods for overcoming the line shape
for the atmosphere below 30 km.
problem above 30 km will be discussed in a later section.
The Lorentz line
shape is defined as follows.
Oo=
77
(
L)-
(28)
-)+e
Malkmus found that for a spectrum of randomly distributed Lorentz lines
given by (27),
the mean transrnssivity is:
+(29)
-1
,j_
2
- ex
where SE and J are:
CJ=
d
-77-
)EcS (30)
6..g(31)
is the mean line spacing of the band.
For development and test
purposes, a "standard" temperature-ozone profile was adopted.
This profile
is shown in Table 2 along with the variation with height of SE' S
and SMIN'
18.
Table 2
Band Model Parameters for Test Atmosphere
Level
T
K)
P
(mb)
0
10-3 atm-cm/km
S
SMIN
cm
SE
/atm-cm~1
-5
x 10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
211
219
226
234
242
250
258
267
268
261
254
249
243
237
231
225
220
215
211
210
213
222
234
248
266
287
0.0396
0.0594
0.0891
0.1340
0.2000
0.3010
0.4510
0.6770
1.0100
1.5200
2.2800
3.4300
5.1400
7.7100
11.6000
17.3000
26.0000
39.0000
58.5000
87.8000
132.0000
198.0000
296.0000
444.0000
667.0000
1000.0000
0.001
0.003
0.005
0.011
0.022
0.046
0.093
0.177
0.376
0.848
1.650
2.584
4.123
6.520
9.790
12.510
15.600
18.400
20.150
16.500
4.620
2.179
1.460
1.266
1.318
1.415
0.495
0.535
0.573
0.613
0.653
0.698
0.746
0.798
0.801
0.766
0.726
0.692
0.659
0.629
0.597
0.565
0.537
0.512
0.497
0.492
0.505
0.550
0.613
0.689
0.793
0.919
0.469
0.666
0.906
1.210
1.590
2.092
2.753
3.618
3.672
3.068
2.464
2.023
1.645
1.335
1.008
0.853
0.681
0.550
0.480
0.459
0.514
0.756
1.121
1.198
3.513
6.282
0.157
0.170
0.183
0.195
0.208
0.222
0.238
0.255
0.256
0.244
0.232
0.221
0.210
0.200
0.190
0.180
0.171
0.163
0.158
0.157
0.161
0.175
0.196
0.220
0.253
0.294
19.
B. Inhomogeneous Path Approximation
A three-parameter method was used to correct for the inhomogeneous
The method, an improvement over Curtis-Godson, was developed
atmosphere.
by Yamamoto, et al. (1972).
range of path lengths, the equivalent half-width
in the intermediate
is reduced;
S
respectively.
, R.
To correct for the deficiency of Curtis-Godson
,
and
x are calculated from (15), (16), and (21)
An empirical value
6
is calculated:
(32)
The reduced half-width o>.
is given by:
(r>Sae
(33)
Yamamoto has tested this method for a single Lorentz line and found that
the error in calculating the transmissivity for x = 1 is less than 4%
compared to 8% for Goody's 3-parameter and 32% for the Curtis-Godson
method.
The rate of error also depends on the thickness of the atmospheric
layer being considered.
The thinner the layer the smaller the error.
Since band models treat the mean line intensities and half-widths as
if they applied to a single line, this method is assumed to be valid for
band models although it has not been specifically tested as such.
Values for the test atmosphere of A/ ,
in
,
S, X,
and & are given
Table 3.
C. Line Shape
The shape of a spectral line at tropospheric and stratospheric
pressures and temperatures is affected by collisional and thermal
- processes.
Collisional-broadening results from the perturbation of
energy levels due to molecular collisions.
The resulting profile is
w
20.
Table 3
Inhomogeneous Parameters for Slabs N to N-
x
N
cm ~/atm
0.000006
0.000009
0.000014
0.000021
0.000031
0.000045
0.000066
0.000098
0.000148
0.000224
0.000339
0.000516
0.000782
0.001187
0.001790
0.002712
0.004112
0.006211
0.009275
0.013495
0.021681
0.027994
0.043946
0.064493
0.093593
cm ~(atm-cm)~i
0.165
0.177
0.190
0.203
0.217
0.232
0.248
0.256
0.250
0.238
0.226
0.215
0.205
0.195
0.185
0.175
0.167
0.161
0.158
0.159
0.170
0.182
0.208
0.237
0.274
0.0356
0.0523
0.0797
0.1208
0.1905
0.2848
0.4057
0.6139
0.8627
1.0705
1.1207
1.0764
1.0504
0.9916
0.8129
0.6372
0.4700
0.3331
0.2063
0.0828
0.0705
0.0522
0.0048
0.0038
0.0034
0.0045
0.0082
0.0154
0.0283
0.0533
0.0897
0.1369
0.2130
0.2918
0.3480
0.3604
0.3495
0.3429
0.3276
0.2771
0.2209
0.1613
0.1087
0.0592
0.01630.0128
0.0081
0.0002
0.0001
0.0001
NAW,
21.
Lorentzian and is given by (28).
Broadening of a line by the thermal
motion of the radiating molecules is known as Doppler broadening.
The resulting profile is essentially Gaussian.
-0.)
-
C<DVW
ex
OLD]
/4/49
(34)
where O(D , the Doppler half-width is given by:
e/gD
=
/kTt-2)
2
M/
C (
Y
(35)
Values for the Doppler and Lorentz half-widths are given in Table 4
for the temperature and pressures of the "test" atmosphere. Below
30 km the Lorentz profile dominates, while above 70 km the Doppler
profile is valid. Between 30 and 70 km both profiles must be considered.
The resulting profile is the Voigt line shape,
F(-0.) =
O/C
2
1
rx ..
(36)
77 '-0.
F
where
CL-D
~01 0* / .
(37)
22.
Table 4
Line Shape Parameters
cm~1/atm
cm~/atm
0.00078
0.00080
0.00081
0.00082
0.00084
0.00085
0.00086
0.00088
0.00088
0.00087
0.00086
0.00085
0.00084
0.00083
0.00082
0.00081
0.00080
0.00079
0.00078
0.00078
0.00079
0.00080
0.00082
0.00085
0.00088
0.00091
0.000005
0.000008
0.000011
0.000016
0.000024
0.000036
0.000053
0.000080
0.000120
0.000177
0.000270
0.000420
0.000624
0.000950
0.001444
0.002182
0.003321
0.005039
0.007611
0.011452
0.017120
0.025150
0.036620
0.053360
0.077396
0.111711
23.
V
is the Lorentz line center corresponding to each position on
the Doppler line.
One procedure is to try to adopt the Voigt profile to the
Malkmus model.
This proves to be exceedingly difficult mathematically.
A second method is that of Gille and Ellingson (1967).
the transmissivity of a Lorentz line shape then
If
1. is
V,the transmissivity
of a Voigt line is:
SC
T 7 V= TL
where C
(38)
is a correction factor given by
X~l
a
5(39)
7TX
where V
a, and
are:
1+.
a~ F
where
of
o.a=
X (TrY)
b
2
~;-
FQ-))
C
-vL.)
(40)
(41)
-(42)
is the Voigt line profile.
Details of the calculations
are in Gille and Ellingson and Young (1965).
Values of a,b, and c
for the 25 slabs of the test profile are given in Table 5.
24.
Table 5
Line Shape Correction Parameters for Slabs N to N-1
b
N
0.01
0.02
0.001
0.002
0.005
0,011
0,024
0.053
0.112
0.236
0.519
0.988
1.583
2.340
3.506
5.085
6.371
7.667
8.680
19.382
8.719
5.080
1.738
1.043
0.909
1.019
36.30
51.38
71.92
100.10
1.286
140.00
0.02
0.03
0.05
0.07
0.10
0.15
0.22
0.34
0.52
0.81
1.24
1.91
2.97
4.53
6.99
10.70
16.34
25.16
c
1.4455
1.6029
1.8142
2.0867
2.2909
2.3372
2.2270
1.9070
1.5569
1.3285
1.1441
1.0748
1.0365
1.0140
1.0008
1.0004
1.0002
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
25.
D.
Zenith Angle.
As discussed previously, the integration over zenith is
replaced by multiplying the absorber amount by 1.6667.
26.
V.
Calculation of Cooling Rate.
A.
Physical model.
To evaluate (11) the atmosphere from 0-70 km was divided into
26 levels identical to those of the General Circulation Model of Cunnold,
et al. (1975).
The pressure in atmospheres at level A/
I) = e xP E2(O.5X.2 4-A/V
A/
(43)
This particular method was chosen so that the Q
be compatible to calculations of Q
B.
calculations would
3
and QCO
performed using these same levels. Each slab
is approximately 3 kilometers in thickness.
is:
2
currently being
formed by the levels
Temperature and Ozone Data.
The monthly values of temperature and ozone concentration
which were used in these calculations are shown in Figures 2-25.
The
temperature data was supplied by Drs. Derek Cunnold and Fred Alyea of
M.I.T. and come from Run 17 of their model.
(Level 26) come, however,
Values for the surface level
from Oort and Rasmussen (1971).
This data compares
favorably with the observed data compiled by Newell, et al. (1974), with
the exception of the top levels (Levels 1 and 2) where the model predicts
slowly decreasing temperatures from the summer pole to the equator while
the actual observations show a gradual increase.
Above 30 km (Levels 1-14) the ozone concentrations used are those
of Run 17 of the model.
Comparisons with the observations of Krueger and
Minzner (1973) show that the model's ozone concentrations differ only
only by 2% or less.
Below 24 km (Levels 18-26) the ozone data is from
27.
Dopplicks compilation of
Hering (1967).
This data is a collection of
5 years (1963-1967) from the North American Ozone Network.
collected by balloon, went up to 10 mb.
This data,
Values from Levels 15-17 were
obtained as follows:
3e (
5(17)L=
~
~
~
LD
)+
0ca ( 17
eD(7=[36:p(7
1 FA(7]I
C
(44)
(45)
(46)
where
are the values of ozone concentration of Dopplick and
o and e
Cunnold and Alyea respectively.
C.
Ozone Spectral Data
Data used in the band model is that of McClatchey, et al. (1973).
Their compilation includes 10,722 lines ranging from 900.009 to 1180.522 cm1 .
Reference state conditions are as follows:
= 296'K
= 1000 mb.
C40
= 0.11 cm~
/atm.
So mn(
= 0.*399 x 10~19
(
(So
mAx)
E3 o mAX)
= 1052.860 cm-1
0 M1,V
0 (So
JI)
E (So
mP)
= 87.148
cm1
= 0.322 x 10-24
=
cm 1/mol-cm-2
1010.767 cm-1
= 1180.340 cm-1
cm
/
mol-cm 2
28.
Climatological Averaging.
D.
QO values are actually [Q
]
values where [
zonal averaging3 andC) denotes a time average.
values of Q
] denote
Strictly speaking the
should be calculated for each individual sounding and then
averaged over space and time.
values are used.
This is impractical, so [T] and [
3o
Rodgers (1967), however, has estimated that the errors
[T]4 instead of [TI4] is less than 0.05%.
in using
E.
Numerical Methods
The Planck function is assumed to be linear with pressure
between adjacent pressure levels (Drayson and Epstein, 1967).
Using
this assumption (11) can be differentiated and written:
SF-
Fv - Fm-I
*d( --
-T I
/:/)
-
+
(NIT1
1(8
(48 )
where
and
d [;-' T - -.;
(50)
29.
I
and
is the transmission between the ith and the jth levels.
Since there are 25 slabs for each sounding, 25 x 25 or 625 different
must be calculated.
values of
Q
~~
03 Ar
x
The cooling in degrees per day is
F.31 X1-3
(51)
B; the Planck black-body function is:
13
where x =
C c/XkT
I''
fAdx
_____
(52)
- I
and
are the limits of 9.6pam.band.
in Table 6 for the "test" dase.
Values of Q0
and B
3
are given
30.
Table 6
Cooling Rates For Test Atmosphere
w
B
deg/day
A0.00
-0.06
-0.10
-0.30
-0.61
-1.16
-2.03
-2.96
-3.22
-2.55
-1.71
-1.10
-0.77
-0.53
-0.32
-0.14
-0.04
+0.01
+0.01
0.0
-0.02
-0.01
-0.01
-0.01
-0.02
-0.01
ergs/cm2/sec
3429.56
4185.18
5221.92
6425.26
7806.73
9487.96
11521.70
13965.66
14110.58
12485.94
10652.60
9265.69
8004.89
6955.90
5923.22
5001.60
4249.36
3638.25
3303.20
3196.56
3467.80
4580.62
6425.26
9155.87
13679.06
20523.79
31.
VI.
Results and conclusions.
Monthly values of Q
are shown in Figures 26-37.
the heavy 0.0 line are positive indicating a net heating.
Values inside
Points to
note are that maximum cooling occurs at the stratopause, near 50 km,
rather than at 20-30 km where the ozone concentration is at a maximum.
Values closely follow the temperature distribution with maximum values
occuring in the summer hemisphere at the levels of maximum temperature.
Net heating occurs primarily above the tropical and subtropical tropopause
where there is a large vertical gradient of ozone concentration.
in the tropopause are less than 0.05 0 C day~1.
Values
Above the stratopause values
approach zero; however, ther consistently appears a net heating in the
top two levels in the summer mesosphere near the poles.
These values are
suspect, however, due to the very small net fluxes across pressure levels,
and possible errors in temperature and ozone data.
Comparision with other calculations shows agreement in major details.
Kuhn and London in their calculations from 30-70 km also found maximum
cooling near the summer stratopause.
Their maximum values however were
from 3 to 40C day~1 compared to 5.4 0C day~1 for this calculation.
An
explanation for this is that since they used seasonal summer and winter
values of temperatures instead of monthly values as in this investigation,
they smoothed the peaks in temperature values.
the stratopause used by Kuhn and London was
The maximum temperature at
277 0 K
at 600 at 50 km compared
to 2910K for the same point during July in this calculation.
rate is a function of the Planck function,
to the fourth power.
a maximum at
302 0 K.
itself
The cooling
a function of temperature
For the 9.6 ozone band the Planck function reaches
Briggs (1965) using rocket measurements at McMurdo Station
(Latitude = 780S) found temperature values at 50 km of over 3000K during the
first two weeks of January, 1963 indicating that daily values may be
even higher than the monthly values presented here.
January (Figure 26)- Maximum cooling occurs at 600 S at 50 kilometers
with a value of 5.4 0C day~1.
a value of 4.2 0C day~1.
Another maximum occurs at the equator with
Maximum cooling occurs at the stratopause for all
mom
32.
latitudes except near the North Pole where it occurs above the tropopause.
Heating occurs from 200N to 400S from 13 to 25 km with a minimum value
of 0.10C day ~1.
February (Figure 27) -
Maximum cooling of 4.20C day~1 occurs again
The secondary maximum at the equator remains with a
at 60 S at 50 km.
Maximum values occur at the stratopause for all
value of 4.0 0C/day.
latitudes including the North Pole.
Heating occurs from 400N to 4 0 0S
from 13 to 25 km again with a maximum value of 0.1 0C day~1
March (Figure 28) a value of
3.70 C
day
.
The maximum cooling occurs at the equator with
Cooling is generally symmetric about the equator
however values are slightly higher in the Southern Hemisphere.
cooling of over
10 C
day~1 at all latitudes at the stratopause.
There is
Heating
is reduced somewhat extending from 400 N at 200S from 15 to 23 km with
a maximum value of 0.05 0C day~1.
April (Figure 29) -
Maximum cooling is now centered in the Northern
Hemisphere for the first time with a maximum of 3.90 C day~1 at 50-53
kilometers at 600N,
an increase of 140c day~1 over the previous month
for this latitude.
A second maximum of 3.80C occurs at the equator.
Maximum cooling occurs at the stratopause for all latitudes except the
South Pole where it occurs at 27 km with a value of 0.6 0C day-1.
Heating
occurs from the North Pole at 20 S. The maximum value is 0.1 0C day~1 at
500 N at 25 km.
May (Figure 30) - Cooling Values reach over 4.6 0C day~1 for the
first time in the Northern Hemisphere with a maximum of 4.90C day~1
at 600 N.
The secondary maximum that has been present over the equator
has shifted to 10 N with a value of 3.90C day~.
Cooling is a maximum
33.
at the stratopause except for the South Pole where a-value of 0.30 C day~1
occurs at 31 km.
Heating occurs from 700N to 200 S from 15 to 27 km with
a maximum value of 0.20 C day~1 at 50 0N at 25 km.
This is the maximum
value of heating obtained all year.
June (Figure 31) -
Cooling is concentrated in the Northern Hemisphere
with a maximum value of 5.4 0C day~1 at 60 N at 50 km.
maximum of
4.00 C
day~
has shifted back to the equator.
The secondary
Maximum values of
0.40C day~ 1 occur at 20 km in the south polar regions; everywhere else
maximum values occur at 50 km.
Heating resembles April's profile extending
from the North Pole to 30 0S with a maximum value of 0.1 C day~
at 50 N at
23 km.
July-December (Figures 32-37)
-
These months are the opposite of
January through June with maximum cooling shifting
back to 600 S in December.
This paper has presented a method for calculating Q0
from
0-70 km: and has given results on a monthly basis of such calculations.
o
-1
Values obtained are approximately 2.0 C day
higher than previously
calculated.
It appears that Q
is a major component of Q in the
stratosphere especially in the middle and upper regions. Maximum values
of Q
occur in the same region as maximum values of QSOLAR indicating
that Zhe absolute value of Q may not be as large as previously believed.
Further work is needed above 60 kilometers to determine if Q0
zero or contributes heating in the mesosphere.
approaches
-
-2
-3
-5
-4
log
S
Figure 1.
-6
-7
00
11
0
[T]
60N
80N
I
~5
JANUARY
(*K)
40N
20N
26
27
0
20S
60S
40S
80S
2
- 50 -0.05
p
(mb)
230
S-0.5
1.0
040
2100
5.0
- 10
15 -30
- 20
2 0-03
- 20 - 50 0
2200
25 -
0
-1000
.0
80N
00
60N
40N
20N
)
LATITUDE
Figure 2.
20S
40S
60S
80S
44#
V
0
80N
60N
(T]
40N
(*K)
20N
0
4
0
FEBRUARY
40S
20S
60S
80S
0.05
LEV
0.1
0.2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
SON
60N
40N
20N
0
LATIT UDE
Figure 3.
20S
40S
60S
80S
1000
[T]
40N
(*K)
20N
0
20S
MARCH
60S
40S
70
-
0.05
HT
(kin) 0.I
LEV
0.2
5
p
(m b)
0.5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
0
L AT IT UDE
Figure 4.
-1000
SON
60N
T
(*K)
40N
20N
APRIL
0
20S
40S
60S
80S
0.05
LEV
0.1
0.2
p
5
(m b)
0.5
10
15
20
100
200
300
500
25
SON
60N
40N
20N
0
LATITUDE
Figure 5.
20S
40S
60S
80S
1000
[T]
80N
60N
(*K)
20N
40N
MAY
80S
1
60S
40S
20S
0
HT
210
LEV
~70 -0.05
-60 -0.2
30
5
(p
(m b)
240
250
-0.5
260
-50
-. 0
70
-2903280
10
C1
(kin)
220
-
2.0
210
4 0
-5 .
-5.0
- 30
15
-20
- 20
220
220202- 220
50
t100
20
-200
230 20 250-10O 300
260 270
25~.2
80N
60N
40N
500
40S_____________
20N
0
,
LATITUDE
Figure2 6.
20S
405
605
_
80S
00
0
1000
0T00 (*K)0J
[t OCK)
JUNE
LEV
5
10
15
20
25
8ON
60N
40N
20N
0
LATITUDE
Figure 7.
20S
40S
60S
80S
0
0
[I]
JULY
(*K)
0.05
LEV
0.1
5
0.2
P
(mb)
0.5
10
15
20
25
LATITUDE
Figure 8.
(i]
AUGUST
(*K)
40N
10.05
LEV
km)
0.1
0.2
p
5
(m b)
0.5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
8ON
60N
40N
20N
0
LATITUDE
Figure 9.
20S
40S
60S
80S
1000
[T]
8ON
60N
40N
(*K)
20N
0
20S
SEPTEMBER
40S
60S
S0
I
S
U'I
0
S
80S
0.05
LEV
0.1
0.2
p
(mb)
0.5
5
10
15
20
25
8ON
60N
40N
20N
0
LATITUDE
Figure 10.
20S
40S
60S
80S
p
80 N
60N
IT]
40N
(*K)
20N
0
20S
OCTOBER
40S
60S
80S
0.05
LEV
0.I
0.2
5
p
(mb)
10
15
20
25
80N
60N
40N
20N
0
LATITUDE
Figure 11.
20S
40S
60S
80S
4.)
I
IT]
40N
a
NOVEMBER
(*K)
20N
0
20S
40S
0.05
LEV
0.I
0.2
p
(mb)
0.5
230
240
..
...........
20
25
280
________________________9__..........___
BON
60N
40N
20N
0
LATITUDE
Figure 12.
20S
40S
60S
80S
iooO
U
U
[T]
80N
60N
40N
ID
(*K)
20N
0
20S
DECEMBER
40S
60S
80S
0.05
LEV
0.1
0.2
P
(mb)
5
10
15
20
25
80N
60N
40N
20N
0
LATITUDE
Figure 13.
20S
40S
60S
80S
0
BON
I
1)t
OZONE DENSITY
60N
40N
I
I
I
I
(10 1 MOLECULES/ CM )
20N
0
20S
I
I
I
61
f
I
I
40S
I
JANUARY
60S
I
I
80S
I
I
70
-
HT
LEV
0.1
(km)
60
50
0.05
-
0.2
p
(mb)
0.5
1.0
2.0
50
--
too0
40
,,
5.0
200
30
300
400 -
10
20
20
150
1OO
50
t0
200
300
500
100
80N
60N
40N
20N
0
LATITUDE
Figure 14.
20S
40S
60S
80S
0-
1000
di
OZONE DENSITY
40N
80N
60N
(10'0 MOLECULES /CM 3 )
0
20S
20N
40S
FEBRUARY
60S
80S
0.05
LEV
0.I
0.2
p
(mb)
5
10
15
20
25
80N
60N
40N
20N
0 LATITUDE
Figure 15.
20S
40S
60S
80S
OZONE DENSITY
40N
60N
80N
(1010 MOLECULES /CM
20S
0
20N
)
40S
MARCH
60S
80S
-70 -0.05
LEV
HT-0.
(kin)0.
60
-_---_
5
__
__ __
_
__ __
100
2.0
-4
_
__50__
P
(mb)
-0.5
~50-.
10
-1.
10
-0.2
-5.0
00
30
30 0
15
6
-20
00
0
507
00
30-
000
20 -
-200
- 300
-500
80N
250,
60N
1
40N
20N
0
LATITUDE
Figure 16.
20S
40S
60S
80S
0
- 1000
0
OZONE
8ON
a
(1010 molecules/ cm3 )
DENSITY
60N
40N
20N
0
t4
20S
ei
40S
*
APRIL
60S
80S
10.05
LEV
0.1
0.2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
80N
60N
40N
20N
0
LATITUDE
Figure 17.
20S
40S
60S
80S
1000
OZONE
80N
DENSITY
(1010 molecules/cm3 )
MAY
10.05
LEV
km)
0.1
0.2
P
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
8ON
60N
40N
20N
0
LATITUDE
Figure 18.
20S
40S
60S
80S
1000
0
OZONE
80N
DENSITY (1010
40N
60N
molecules /cm 3 )
0
20N
20S
JUNE
40S
60S
80S
10.05
LEV
,km)
0.l
0,2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
-20
50
20
100
200
300
500
25
80N
60N
40N
20N
0
LATITUDE
20S
Figure 19.
40S
60S
80S
1000
0
S
0II
0
OZONE DENSITY
40N
60N
80N
(IO'0 MOLECULES/ CM 3 )
20S
0
20N
40S
JULY
60S
80S
1
-70
LEV
:0.05
-0.1
5.
-60
5p
-0.2
(mb)
-0.5
0-50
10
--
2.0
50
-40
00
-5.0
30
300
15
400-
5
10
-20
5000
200 4200
60020
25
-
000
-300
- 1000
103000
500
2250080N
60N
40N
20N
0
LATITUDE
Figure 20.
20S
40S
60S
80S
OZONE
80N
60N
DENSITY
40N
(100 MOLECULES /CM 3 )
20S
0
20N
LEV
40S
AUGUST
60S
80S
70 0.05
HT
0.I
(kin)
------
0.2
60
5
p
(mb)
0.5
-10
1.0
10
2.0
50
100
5.0
20 0
15
400
o- 20
20
20
- 10
25
-
50
30
100
40
200
300
500
100550
... ..........
30
30
300
.1000
80N
60N
40N
20N
0
LATITUDE
Figure 21.
20S
40S
60S
80S
OZONE DENSITY (10'0 MOLECULES /CM 3 )
20S
0
20N
40N
60N
40S
SEPTEMBER
80S
60S
0.05
LEV
0.1
0.2
p
(mb)
0.5
80N
60N
40N
20N
0
LATITUDE
Figure 22.
20S
40S
60S
80S
'p
0.05
LEV
0.I
5
0.2
p
(mb)
0.5
10
15
20
25
I
80N
60N
I
I
40N
I- I
I
I
20N
I
L
I
0
LATITUDE
Figure 23.
I
20S
I
I
40S
I
I
60S
I
I
80S
1000
0
0
OZONE DENSITY ( Q10 molecules/cm 3 )
8ON
GON
40N
20N
0
0
0
1)
20S
NOVEMBER
40S
60S
80S
0.05
LEV
0.I
5
0.2
p
(mb)
0.5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
8ON
60N
40N
20N
0
LATITUDE
Figure 24.
20S
40S
60S
80S
1000
DENSITY
OZONE
6N
8N
(1010 molecules/cm3 )
40Nh
20N
20S
O0
4)
1'
4)
4)
DECEMBER
60S
40S
80S
70 0.05
HT 0.1
(kin)
LE V
0.2
p
(mb)
0.5
60
5
50
10
1.0
10
2.0
50
40
5.0
100
_ __i
__NI
l'
4 )
'l
O
200
15
-
30
300-
5400
500
000
50
50
20
20
-0
500
050
25
10
40N
20N
100
200
300
100
80N
10
20
400
60N
-
0
LATITUDE
Figure 25.
20S
40S
60S
80S
- 1000
COOLING
RATE
(*C/DAY)
JANUARY
0.05
LEV
0.I
0.2
5
p
(mb)
0.5
10
15
20
25
0
LATITUDE
Figure 26.
0
I)
COOLING RATE (*C/DAY)
60N
40N
20N
0
80N
FEBRUARY
40S
60S
20S
80S
70
-
HT
LEV
0.1
(kin)
5
-2.0
60
0.05
-
0.2
p
(m b)
,:
..
0.5
- 2.0
-340-
--
2.0
50
0 -.
1.0
10
2.0
i
s
-4.0
40
--.
5.0
-3.0
15
-0.3--
-
30
20
-
20
20 150
20
100
-0.3
0.0-
10
200
1300
500
25
J0
80N
60N
dI
40N
20N
0
LATITUDE
Figure 27.
20S
40S
60S
80S
1000
(*C/DAY)
COOLING RATE
ION
40N
20N
0
MARCH
0.05
LEV
0.1
0.2
p
(mb)
0.5
5
10
15
20
25
8ON
60N
40N
20N
0
LATITUDE
Figure 28.
20S
40S
60S
80S
mu
S
COOLING
RATE
U
(C /DAY)
S
ID
1D
APRIL
-0.05
LE V
- 0.1
-0.2
P
(mb)
-0.5
5
-1.0
10
-2.0
-5.0
-I0
15
- 20
-50
20
-100
-200
-300
-500
25
-1000
LATITUDE
Figure 29.
0
0
0
COOLING
RATE
MAY
(*C/DAY)
0.05
LEV
0.1
0.2
p
(mb)
5
10
15
20
25
80N
60N
40N
20N
0
LATITUDE
Figure 30.
20S
40S
60S
80S
COOLING RATE
40N
60N
80N
I
I
(oC/DAY)
0
20N
I
I
----__,,.
LE V
..
.-
JUNE
60S
80S
40S
20S
I
I
I
I
-
-- 0.0
0.05
70
HT
0.I
(kin)
o0. o
5
60
-
0.2
P
(m b)
0.5
10.3
-N
-40~4.0
10
50
-3.0-20)
1.0
~
--------- ~
2.0
--.---40
5.0
15
I0
30
20
-
-0.3
0
20
- 150
20
100
10
-
200
300
500
25
I
80N
I
I
60N
I
I
40N
I
20N
I
I
0
LATITUDE
Figure 31.
I
20S
I
I
40S
I
I
60S
I
I
80S
0-
1000
0
0
U
V
V
U
v
00
U
JULY
(*C/DAY)
COOLING RATE
0
20N
40N
60N
10.05
HT 0.1
LEV
(km)
0.2
P
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
80N
60N
40N
20N
)
LATITUDE
Figure 32.
20S
40S
60S
80S
1000
COOLING
RATE
(*C/ DAY)
0
20N
20S
0
0
00
00
AUGUST
E
40S
10.05
LEV
km)
0.
0.2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
80N
60N
40N
20N
0
LATITUDE
Figure 33.
20S
40S
60S
80S
1000
4
U
0
0
COOLING RATE (*C/DAY)
20N
0
40N
60N
80N
SEPTEMBER
40S
60S
20S
80S
0.05
0.0
LEV
.....
--
0. 3 - -
5
-
-- --
0.0 ,
0.1
0.2
p
(mb)
0.5
-1.0
-2 .0
-3.0
1.0
10
2.0
-2.0
l
-
5.0
-1.0
10
15
0.3
-........----------
-
~--.
20
0.0
- 0 _. _ ...
20
j20
.. .
150
100
I
-
I
I
I
I
I
I
I -
-
-
200
300
500
80N
60N
40N
20N
0.
L AT ITUDE
Figure 34.
20S
40S
60S
80S
1000
0
80 N
1U
COOLING RATE (*C/DAY)
0
40N
20N
60N
0
p
20S
OCTOBER
40S
60S
80S
0.05
LEV
0.1
0.2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
'5
20
50
20
100
200
300
500
25
80N
60N
40N
20N
0
LATITUDE
Figure :35.
20S
40S
60S
80S
1000
80N
COOLING RATE (*C/DAY)
20N
40N
60N
0
20S
0
V
0
0
NOVEMBER
40S
60S
80S
0.05
LEV
0.1
0.2
p
(mb)
5
10
15
20
25
80N
60N
40N
20N
0
LATITUDE
Figure 36.
20S
40S
60S
80S
0
0
0
0
COOLING
RATE
DECEMBER
(*C/DAY)
0.05
LEV
0.1
0.2
p
(mb)
0.5
5
1.0
10
2.0
5.0
10
15
20
50
20
100
200
300
500
25
80N
60N
40N
20N
0
LATITUDE
Figure 37.
20S
40S
60S
80S
1000
71.
List of Symbols and Definitions
a
=
radius of the earth.
=
half-width of a Lorentz line.
=
half-width of a Lorentz line at reference conditions.
D=
half width of a Doppler line.
Planck black-body function
3=
C
=
speed of light.
C
=
adjustment constant for line shape.
C
=
c
=
specific heat of air at constant pressure.
mean line spacing.
=
diffusivity factor.
=
Energy of the lower state.
E
E3=
F
h
=
Exponential integral of the third order.
adjustment constant for intermediate lines.
=
flux.
=
line shape function.
=
acceleration of gravity.
=
Planck constant.
=
intensity.
Boltzmann constant.
=
absorption coefficient.
=
=
wavelength.
level number.
O
P
=
zenith angle.
=
probability
F
=
pressure.
A
function.
72.
reference pressure 1000 mb.
fO
9Q
S
T
=
ozone density.
=
azimuth angle.
=
rate of non-adiabatic heating (per unit mass)
=
gas constant for dry air
=
line intensity.
=
line intensity at reference condition.
=
temperature.
=
temperature at reference state.
=
time.
=
flux transmission function.
intensity transmission function.
T
=
optical thickness.
U
=
=
zonal wind velocity.
absorber amount
f
=
cos
V
=
meriodional wind component.
O
=
frequency.
e
06 =
frequency of line center.
J
vertical wind velocity.
=
X=
absorption coefficient at line center.
73.
REFERENCES
Arking, A. and K. Grossman, 1972: The influence of line shape and band
structure on temperatures in planetary atmospheres. J. Atmos.
Sci., 29, pp. 937-949.
Armstrong, B.H., 1968: Analysis of the Curtis-Godson approximation and
radiation transmission through inhomogeneous atmospheres. J. Atmos.
Sci. 25, pp. 312-322.
Briggs R.S., 1965: Meteorological rocket data at McMurdo Station, Anarctica:
1962-1963. J. App. Met. L, pp. 238-245.
Clark, J. and W. Hitschfield, 1964: Heating rates due to ozone computed
by the Curtis-Godson approximation. Quart J. R. Met. Soc., 90,
pp. 96-100.
Cunnold, D., F. Alyea, N. Phillips, and R. Prinn, 1975: A three-dimensional
dynamical-chemical model of atmospheric ozone, -J.Atmos. Sci.
(To be published Jan., 1975).
Craig, R.A., 1965: The Upper Atmosphere:
Academic Press, 509 pp.
veteorology and Physics.
New York:
Curtis, A.R. 1952: Discussion of a statistical model for water vapor absorption.
Quart. J. R. Met. Soc. 78, pp. 638-640.
Dopplick, T.G. 1970: Global radiative heating of the earth's atmosphere.
Report -No. 24, Planetary Circulations Project; Department of
Meteorology, Massachusetts Institute of Technology, 128 pp.
Drayson, S.R. and E.S. Epstein, 1967: The calculation of long-wave radiative
transfer in planetary atmospheres. Tech Report, Department of
Meteorology and Oceanography, University of Michigan, 110 pp.
Fang, T.-M., S.C. Wofsy, and A. Dalgarno, 1973: Opacity distribution functions
and absorption in Schumann-Range bands of molecular oxygen. Preprint
Series No. 26, Center for Astrophysics, Harvard College Observatory
and Smithsonian Astrophysical Observatory, 15 pp.
Gille, J.C. and R.G. Ellingson, 1968: Correction of random exponential
band transmissions for Doppler effect. Applied Optics, 7(3),
pp. 471-474.
Godson, W.L. 1953: The evaluation of infrared radiative fluxes due to
atmospheric water vapor. Quart. J. R. Met. Soc., 79, pp. 367379.
74.
Goody, R., 1964: The transmission of radiation through an inhomogeneous
atmosphere. J. Atmos Sci. 21 pp. 575-581.
Gradshteyn, I.S. and I.M. Ryzik, 1965: Table of Integrals Series and Products.
New York: Academic Press, 1086 pp.
Herman, G.F., 1972: Radiative effects of carbon dioxide in the upper
stratosphere. S.M. Thesis, Department of Meteorology, Massachusetts
Institute of Technology, 46 pp.
Hitschfeld, W. and J.T. Houghton, 1961': Radiative transfer in the lower
stratosphere due to the 9.6 micron band of ozone. Quart J.R. Met.
Soc., 87, pp. 562-577.
Krueger A. and R. Minzner, 1973: A proposed mid-latitude model of atmospheric
ozone. Goddard Space Flight Center X-651-73-72.
Kuhn, W.R., 1966: Infrared radiative transfer in the upper stratosphere and
mesosphere. Ph. D. Dissertation, Department of Astro-Geophysics,
University of Colorado.
Kuhn, W.R. and J. London, 1969: Infrared radiative cooling in the middle
atmosphere (30-110 km). J.Atmos Sci. 26 pp. 189-204.
~
Malkmus, W., 1967: Random Lorentz band model with exponential taile
line-intensity distribution. J. Opt. Soc. Am., 57, pp 323-329.
Manabe, S. and F. Moller, 1961: On the radiative equilibrium and heat
balance of the atmosphere. Month Wea Rev. 89, pp. 503-532.
McClatchey, R.A., W.S. Benedict, S.A. Clough, D.E. Burch, R.F. Calfee,
K. Fox, L.S. Rothman, and J.S. Garing. 1973: AFCRL Atmospheric
Absorption Line Parameters Compilation, AFCRL-TR-73-0096,
Air Force Cambridge Research Laboratories, Air Force Systems Command.
78 pp.
Murgatroyd, R.J., and R.M. Goody, 1958: Sources and sinks of radiative
energy from 30 to 90 kn. Quart. J.R. Met. Soc., 84, pp. 225-234.
Newell, R.E., G.F. Herman, J.W. Fullmer, W.R. Tahnk, and M. Tanaka, 1974:
Diagnostic studies of the general circulation of the stratosphere.
Proceedings of the International Conference on Structure, Composition
and General Circulation of the Upper and Lower Atmospheres and Possible
Anthropogenic Perturbations, Vol. I, pp. 17-82.
Oort, A.H. and E.M. Rasmussen, 1971: Atmospheric Circulation Statistics,
NOAA Profesional Paper 5, National Oceanic and Atmospheric Administration,
Department of Commerce.
Plass, G.N. 1956: The influence of the 9.6 micron ozone band on the atmospheric
infrared cooling rate. Quart. J.R. Met. Soc. 82, pp. 30-44.
75.
Plass, G.N., 1960: Useful representations for measurements of spectral
band absorption. J. Opt. Soc. Am., 50(9) pp. 868-875.
Rodgers, C.D., 1968: Some extensions and applications of the new random
model for molecular band transmission. Quart. J. R. Met. Soc.,
94 pp, 99-102.
Rodgers, C.D., 1967a: The radiative heat budget of the troposphere and
lower stratosphere. Report A2, Planetary Circulations Project,
Department of Meteorology, Massachusetts Institute of Technology,
99 pp.
Rodgers, C.D., 1967b: The use of emissivity in atmospheric radiation calculations.
Quart. J. Roy. Met. Soc., 93, pp. 43-54.
Rodgers, C.D. and C.D. Walshaw, 1966: The computation of infra-red cooling
rate in planetary atmospheres. Quart. J. Roy. Met. Soc. 92 pp. 67-92.
Tahnk, W.R., 1973: The energy budget of the middle and upper stratosphere.
S.M. Thesis, Department of Meteorology, Massachusetts Institute of
Technology, 117 pp.
Walshaw, C.D., 1957: Integrated absorption by the 9.6/m. band of ozone.
Quart J. Roy. Met. Soc., 83, pp. 315-321.
Yamamoto, G., M. Aida, and S. Yamamoto, 1972: Improved Curtis-Godson
approximation in a non-homogeneous atmosphere. J. Atmos. Sci.,
29, pp. 150-1155.
76.
ACKNOWLEDGEMENTS
The author would like to thank Professor Reginald E. Newell
who suggested the topic and provided counsel and inspiration.
Special
thanks are due to Gerald Herman for many enlightening discussions of
the problems of radiative transfer.
Thanks also go to Dr. Fred Alyea
who provided the ozone and temperature data, to Isabele Kole who drafted
the charts, and Beth Prolman who typed the manuscript.
This research was conducted while the author attended the
Massachusetts Institute of Technology under the auspices of the Air Force
Institute of Technology and was supported by the United States Atomic
Energy Commission under Contract
No.
AT(ll-1)-2195.