UDC 551.510.53.534:551.513:551.509.313
Numerical Experiments on the Steady-State
Meridional Structure and Ozone Distribution
in the Stratosphere
V. R. KRISHNA RAO-Atmospheric Environment Service of Canada, Toronto, Ontario
ABnRACT-A two-dimensional circulation model of the
stratosphere, incorporating the mutual interrelationships
between radiation, photochemistry, and seasonal trans-
port processes, was run under steady-state assumptions
to study seasonal variations. The large-scale, quasi-
horizontal eddy processes were parameterized in terms of
time-zonal mean temperature and ozone mixing ratio
using the generalized diffusion formulation on a sloping
surface.
The computed distributions of temperature and ozone
mixing ratio in the meridional plane show satisfactory
agreement with the observations in different seasons,
thus accounting for the considerable discrepancies between
the radintive-photochemical equilibrium state and obser-
vations. The significant jetlike features such as westerlies
in winter and tropical easterlies in summer are well-
reproduced in the upper stratosphere.
As a consequence of the generalized diffusion, the mean
meridional motions developed the two-cell structure
typical of. that found in recent observational studies of
the winter lower stratosphere.
Quasi-horizontal eddies and mean meridional motions
contributed significantly to heat and ozone budgets,
whereas the vertical eddies had little effect except to trans-
port ozone downward in the winter lower stratospher?.
1. INTRODUCTION
Since the publication of Chapmans (1930) paper on the
photochemical formation of ozone, there have been
numerous attempts (Wulf and Deming 1936, Craig 1950,
Diitsch 1946, and others) to explain the discrepancies
between photochemical theory and the observations as
revealed by the classical ozone measurements by Dobson
et al. (1927). It is now generally accepted that much of
the discrepancy is due to atmospheric transport processes.
One of the first attempts to combine the nonphoto-
chemical processes with the photochemistry of ozone in a
numerical model was that of Prabhakara (1963). He
obtained the steady-state solution for the ozone equation
in the meridional plane by balancing the photochemical
effects with the ozone transports by the motion terms.
The specified mean meridional motions were based on
calculations by Murgatroyd and Singleton (1961)) and
the large-scale eddy effects were parameterized using the
Fickian-type diffusion formulation. Although moderate
success was achieved in simulating the gross featuresof
the observed ozone distribution, little insight was obtained
on the operation of the transport and physical processes
because of over-simplification of the complex interactions
among ozone, temperature, and circulation.
Recently, there have been attempts to model the winter
stratosphere with a more realistic approach. Joint radia-
tive, photochemical, and dynamical time-integration
studies of the winter stratosphere by Byron-Scott (1967)
and Clark (1970) reveal the importance of the large-scale
wave disturbances for the poleward transport of heat and
ozone. The primary purpose of these studies, however, was
to simulate the stratospheric winter warming phenomenon.
The most de tailed investigations of the diffusion of
tracer material in the winter stratosphere were those of
Hunt and Manabe (1968) and Hunt (1969). The model
used for those investigations was an 18-level verison of
the general circulation model of Smagorinsky et al.
(1965). Despite the omission of the thermal coupling
between the radiative and photochemical sources and
sinks, they found that both large-scale eddies and mean
meridional motions were important to the poleward
transports of the trace substances such as ozone. However,
these authors themselves agree that their studies provide
no information on the seasonal variations of ozone and
other such properties in the stratosphere.
I n this study, the major interactions between radiative,
photochemical, and transport processes are, retained in the
meridional plane, and the necessary modeling simplifica-
tion is achieved by parameterizing the eddy circulation in
terms of generalized diffusion related to time-zonal mean
properties (Reed and German 1965). The relative roles of
the mean meridional motions and the parameterized
eddies in maintaining the seasonal distributions of
temperature, winds, and ozone in the presence of radiative
and photochemical sources and sinks are the major focus
of this paper.
510 1 Vol. 101, NO. 6 1 Monthly Weather Review
2. THE MODEL and
Vertical Coordinate
To retain the advantage of pressure as a vertical co-
ordinate and to achieve adequate vertical resolution in the
stratosphere, we defined the coordinate for this experi-
ment, by the parameter 2 [following Eliassen (1949)],
where
(1) z= -In--. P
PS
In eq (l), p is the pressure,, and p , is the reference pressure
taken arbitrarily at 70 mb.
Averaged Equations
The atmospheric evolutions in this experiment are
represented by the two horizontal equations of motion,
and the hydrostatic, continuity, thermodynamic and
ozone equations. Following the method of Reynolds
(1894), these are averaged with respect to time, t, and
longitude, A, using the definitions
(-)=LJ7J2-( 2rr ) dxdt
and
( )=(->+( )
where (-) denotes the time-zonal average of any quantity
( ) over a period T and ( ) represents the departure from
the time-zonal average. The averaging period chosen here
is comparable to the duration of a season, and this removes
tidal and short-period fluctuations.
In the second equation of motion, the advection and
local acceleration terms are assumed to be small compared
to the Coriolis and pressure gradient forces; thus the
flow remains approximately in geostrophic balance ex-
cept near the Equator. Using these approximations and
neglecting viscous effects, we can write the final averaged
equations of zonal momentum, continuity, thermo-
dynamics, thermal wind, and ozone under steady-state
conditions as
u a Taii 1 a - -(E cos 4)+Z--fi+- - (vu cos2 4) a cos 4 a4 az ucos 4 %
a - - +-(Z u )--iu=O, (2) az
(3)
(5)
vajg Tax i a - a - - -+z--+- -((ox cos 4)+-(2x) a d 4 d Z acos$&p az --
--ZX--P=O, (6)
where,
a=radius of the earth,
c,=specific heat a t constant pressure,
f= Coriolis parameter,
po=pressure a t 1000 mb,
P=rate a t which ozone is produced from photochemical processes,
G=rate of heat addition per unit mass due to radiation,
R=gas constant for dry air,
T= temperature,
u=eastward wind component,
u=northward wind component,
Z=upward wind oompownt
e =potential temperature,
K = Rfc,,,
+=latitude, and
x=ozone mixing ratio.
Equations (2)-(6) constitute a system of - five equations
in five dependent variables; namely C, V, 2, i, and T, and
are sufficient to close the system mathematically with
proper boundary conditions, provided Q, p , and the eddy
fluxes of heat, momentum, and ozone can be expressed as
functions of the dependent variables.
A parameterization suitable for heat and ozone trans-
ports occurring on a meridionally sloping surface (Reed
and German 1965) was used to express the fluxes in terms
of the averaged dependent variables. This representation
is capable of producing meridional counter-gradient heat
fluxes in the lower stratosphere. As an example, the
expression given by Reed and German for the eddy flux
of heat in the meridional direction can be written as
--
(7)
where Cr represents the slope of the mixing path, 3
represents the slope of the e surface, and K,, is the coeffi-
cient of eddy diffusion in the y direction.
One can see from eq (7) that the flux is countergradient
if i; > 3. If on the other hand, Cr < 5, the heat flow is
down the gradient. Finally, when Cr = 0, eq (7) reduces to
a familiar expression for Fickian diffusion.
Using the definition for 0,
ae
- -& ae
a2
@%tan @= --7
-
we can write the horizontal and vertical heat fluxes as
June1973 1 Rao 1 511
and (8)
- where K,, = a K,,, the cross-component eddy diffusion
coefficient, H is the - scale height, z is the geometric height,
and K,, = (2 + afp)Ku,, the vertical eddy diffusion co-
e5cient. Assuming the large-scale eddy mixing of ozone
takes place in the same manner as heat, we have
and (9)
The seasonal and, latitudinal distributions of K,,, K,,,
and K,, are taken from Reed and German (1965) for the
lower stratosphere. Lacking such information at higher
levels in the stratosphere, we used these values as the
basis for the flux calculation in the model. However, an
attempt has been made to derive K,, at higher levels
using the wind statistics information given by Newell
(1963) with the assumption that K,, is proportional to
the variance of the meridional wind component.
Reed and German's choice of the diffusion coefficients
to calculate the heat transports in the model is justified
by the fact that the coefficients have been derived based
on heat flux and temperature data. There appears to be
some question, however, whether one could use the same
form of diffusion and the coefficients to describe the ozone
transports in the model. Preliminary calculations show
that the assumption on the form of diffusion of ozone is
satisfactory, although the coefficients based on heat flux
data may have the effect of slightly overestimating the
ozone transports in the meridional direction.
Momentum fluxes are parameterized as related to the
meridional thermal gradients (Williams and Davies 1965)
consistent with the thesis that the meridional momentum
flux is mainly controlled by the baroclinic processes.
Thus, following Williams and Davies we have
and
where
and
K,,(~m~.deg-'.s-~) =0.65x 1078K,
Kz,(cmz.s-l)=Kz,.
The symbol 8 represents the angular velocity of the earth.
Although the Williams and Davies choice is more ap-
propriate for a baroclinically active layer (a/p<1), one
can extend it to a passive system (a//3>1) by introducing
the concept of negative eddy viscosity (-Kvm) and see
whether the comparison of the model results with the
observations may shed some light on the physical justifi-
cation of the original assumption.
512 / Vol. 101, No. 6 / Monthly Weather Review
Substituting eq (8), (9), and (10) for eddy fluxes in
eq (4), (6), and (2), respectively, we obtain
- -
Since the calculations of 4 and will be made using
the actual temperature, it is more convenient to transform
eq (12) in terms of the actual temperature using the
following relationships between 8 and T:
and
When eq (14) and (15) are substituted in eq (12),
the thermodynamic energy equation becomes
-
t a n 4 a?i' K,,@ K,, - Q
+a ( K~~ + z+ K T ) - ,= 0.
Discrete Vertical and Horizontal
Structure of the Model
The two-dimensional grid used for the numerical
computations is shown in figure 1. It runs from 10 km to
55 km in the vertical and from 85's (negative) to 85'N
(positive) in the horizontal, and the poles represent the
lateral boundaries. In the vertical, the finite-difference
net consists of 16 discrete levels separated by a constant
A2=0.4236. This corresponds to a height increment of
'33 i
z 3 2 1
'31 I
' 3 0
z 2 9 I
TOP BOUNDARY
f 2 8 ;
' 2 7 ;
'26
- ' 2 5
I
I
I
1
I
I I
li j
'18 i
I 1 5 1
141
'13 I
' 1 7 ;
' 1 6 :
3. PHOTOCHEMICAL AND RADIATIVE
CONSIDERATIONS
The photochemical source term, @, in the ozone equa-
tion and the radiative heating term, Qlc,, in the thermo-
dynamic energy equation are computed by methods
similar to those of Leovy (1964) and Byron-Scott (1967).
The methods are reviewed briefly in the context of the
present experiment.
I
;; 1 . 1 1 1 1 1 1 1 1 1 .
,:. ................. .................... l l .. ................
,I ..................... i.38.5
.................. ; .................... ..................
' 9 ; 8 1 .. ................ lr
2 1 2 1 I(
' 1 1 ; 7 ..
' l o ;,,
' 6 1
z s :l I ....,.............
2 ;
2 1 ;
...................
' 81 ,: .................... 7 1 ,I ..
4 1 ,,
................. i ................
' z =" :: = = = = ~-" "&;; 3 ' LOWER BOUNDARY
-05 .-6S ,-45 ,-25 . -5 5 , 25 , 45 . 65 . 8,s
FIGURE 1.-Two dimensional grid used for numerical calculations in
the model. The boundaries of the model region are marked by
zs. The vertical dashed lines are the imaginary arrays outside
the poles.
-35.3
1.32.5
1.29.5 5
1-26.7 c where 1.23.95
!.2 1 .2 ~
-1 8.5 I
1-15.8
1-13.1
1-10.2
i; 4.3 1.1
(17)
I
j - 7.5
about 3 km in the US. Standard Atmosphere. In the
meridional direction, the grid spacing is 10' of latitude.
Boundary Conditions
The Ozone Photochemical Source Term, P
Assuming that only reactions involving oxygen allo-
tropes and a chemically inert background gas are signifi-
cant in the atmospheric ozone balance, one can show
(Lindzen and Goody 1965, Brewer 1966) that the ozone
source term can be written in the form
and
In the above expressions, M, M2, and Ma denote the
molecular masses of air, oxygen, and ozone, respectively,
and p z is the density of oxygen. Although the Chapman
theory can be questioned on a number of points based
on recent developments in ozone photochemistry (Hunt
1966, Johnston 1971), it is still the dominant photo-
chemical mechanism in the stratosphere and is compatible
with the nature of the experiment in this study.
The ratio of the reaction rate coefficients (K13/&)
is strongly temperature dependent and is generally ex-
pressed as
- K 1 3 -~1 exp (-$) K12-
specified at 10 km and the 2 values based on the thermal
equilibrium (Murgatroyd and Singleton 1961) are used
a t the top boundary.
For the purpose of solving the zonal momentum equa-
tion in conjunction with the equation of continuity,
we assume that both u and are antisymmetric with
respect to the poles. Unless this condition is ma.intained,
there would be infinite divergence and vorticity a t the
poles due to the spherical geometry of the earth. The
specification based on seasonal climatological values lacks
the rigor of physical argument. However, the sensitivity
of the solution in the interior of each of these boundary
conditions has been tested by varying the values at the
boundary.
source term as
If Ioy is the intensity of the direct solar beam at the outer
limit of the atmosphere [given in photons. .s-I. (wave
no.)-'], cy2(v)+pcy4(v), cy3(v), and 0, are the pressure-de-
pendent absorption cross section of molecular oxygen, the
absorption cross section for ozone, and the scattering cross
section for air, respectively; x2, x3, and x, are the total
numbers of oxygen, ozone, and air molecules above a
horizontal square centimeter situated at level 2. Finally,
if e2 (v ) and e3 (v) are the quantum yields or efficiencies
for oxygen and ozone dissociation, then the dissociation
June1973 / Rao 1 513
rates p2 and qa (Byron-Scott 1967) are given by
n
+$(z, >~4 (~);c 2 (Z >+~3 (~)2 3 (2 )
+P,(YMZ)I sec S -M ~ (21)
where sec 7 is the mean zenith angle between sunrise and
sunset a t a given latitude and season, d is the fraction of
the day that the sun spends above the astronomical
horizon, and (2, 00 ) represents the oxygen-weighted
mean pressure for the layer between level 2 and 00.
For the details on calculating sec F, d , 5(2, a), 2,, 5, and
z,, the reader is referred to Rao (1970).
The spectral data on I ,,, a 2 (v ), a 3 (v ), Pm(v), and e2(v)
required to evaluate p2 and q3 in the above equations were
taken from Byron-Scott (1967). The spectral domain con-
sidered covers the wave number range from 13,500 cm-'
to 59,375 cm-'. Contained in the range are the Hartley
(2000-3000 A), Huggins (3100-3400 A) , and Chappius
(4500-7500 8) bands of ozone and Herzberg continuum
and Shumann-Runge of molecular oxygen.
The Radiative Heating Term, &cP
The main components of heating contributing to the
sources and sinks in the thermodynamic energy equation
in the stratosphere are the heating due to absorption of
short-wave radiation by ozone and heating or cooling due
to exchange of long-wave terrestrial radiation in the
9.6-pm band of ozone and by the 15-pm band of CO,.
Thus,
,
If x (2 ) is the mixing ratio of ozone a t a level 2 and hv
is the average energy of a photon, then, assuming that the
quantum efficiency for ozone is unity, the heating rate
due to absorption of short-wave radiation by ozone in
OK/s a t any specific level is given by
where h is Plank's constant.'
The method employed in the present calculations to
obtain Qo3/cp was developed by Clark (1963); it is based
on the laboratory measurements of mean absorptivities
for the 9.6-pm band of ozone by Walshaw (1957). The
cooling rates (i.e., QCo2/c,> for various levels in the model
as we used in the case of the 9.6-pm ozone band, except
that the absorptivity matrix is computed using the empir-
ical formulas given by Howard et al. (1955) for the in-
tegrated absorption of the 15-pm C02 band (Rao 1970).
4. METHOD OF NUMERICAL SOLUTION
To describe the numerical procedure using the method of
successive approximations, we let u t f , 2tj, T?,, u t j ,
and x?, represent the nth-order approximation. Sub-
stituting and T:,j in the spatial finite-difference
analog of the zonal momentum equation [eq (ll)], one
can obtain ~7,:'. Using the corrected values of v:,:' in the
equation of continuity, we can write the (n+ 1) th approx-
imation for i a s
where (2,) are the nondimensional vertical motions
specified at the upper boundary, w k =l if k #j or 19, and
wk=d/:! if k=j or 19.
The corrected fields, v:,;l and Z:,$l, are now used to
solve the heat equation [eq (IS)] for the (n+l)th approxi-
mation of temperature. The well-known Liebmann relaxa-
tion technique is employed, and the computational pro-
cedure is basically similar to that described in Thompson
(1961), though not in all details. From the sequential
method of relaxation, the new approximation for the
temperature field is given by
where a, is the relaxation coescient. With respect to the
eq (16), R:,;.n+l represents residues at each gridpoint, and
Z(WT) i,, denotes the sum total of the weights given to the
central point (i, j) in various terms of the equation.
The finite-difference scheme adopted here to obtain
the residues from eq (16) involves central differencing
for both second- and first-order derivatives of diffusion
terms and the Lelivier (Richtmyer 1957) method of
windward-side difference approximation for the advection
terms using noncentered differences. The upstream
differencing introduces additional diffusion into the
system, which lacks physical argument. However, the
scheme was found essential to control the numerical
calculations in the model. After applying a similar pro-
cedure to the ozone equation [eq (13)], we can write
the (n+ 1) th approximation for the ozone mixing ratio as
Finally, from the thermal wind equation [en (5)l we details of the method are given by Rao (1970.)
Taking into account the fixed distribution of the
absorber, we used the same procedure to compute C02 have
f +l k *-l ,k )A 2 (26) R k =4
(T"+l, -P+' Un+l-
2AY
i . j -(uLB) f+- w k
The multiplication of qs(Z) by Y should be achieved within the integral of eq (21). fi k =j
514 / Vol. 101, No. 6 / Monthly Weather Review
where, (uLB)* are the specified values of the zonal wind
a t the lower boundary, wk=l if k #j or 4, and wk=1/2
if k=j or 4.
Until now, we have described the procedure to compute
the new approximations for all the variables once for the
entire grid. After computing the new radiative and photo-
chemical source terms from T?$l and x;;l at the
beginning of each iteration, the entire grid is scanned
repeatedly on all the variables and the successive approxi-
mations are computed until the following convergence
criteria are satisfied:
and
To obtain a convergent solution, one must satisfy the
ellipticity criterion for the system of equations. This prob-
lem has been overcome by ellipticizing the system. By
combining the zonal momentum, thermodynamic, thermal
wind, and continuity equations, we derived the necessary
(not sufficient) condition for thetsystem to be elliptic. This
can be written in the following form:
5. FINAL DESIGN OF THE
NUMERICAL EXPERIMENT
Four experiments were performed for the summer and
winter seasons. In the fist experiment (case l), the trans-
port processes were completely suppressed and a simple
radiative-photo chemical equilibrium solution for tempera-
ture and ozone mixing ratio was obtained. I n case 2, both
large-scale eddy effects and mean meridional motions were
incorporated, but the linear temperature approximation
was used to compute long-wave heating components in
the heat equation (Leovy 1964, Lindzen and Goody 1965).
With this approximation we can write
where T, and c denote the standard temperature and the
relaxation time for infrared cooling, respectively. The
values of T, and c were taken from Hering et al. (1967).
Case 3 was the same as case 2 except that the long-wave
heating components were computed in a detailed manner
from the physical considerations. In cases 2 and 3, the
eddy transfer coefficients, K,,, K,,, and K,,, were varied
only with latitude and season as shown in table 1. Case 4
is the same as case 2 except that K,, and K,, were varied
(Rao 1970) with respect to height as well as latitude and
season.
Case 2- and 3-type experiments were also extended to
the spring and fall seasons in a straightforward manner,
by changing the declination of the sun and by choosing
diffusion coefficients and boundary fields for these seasons.
The diffusion coefficients were again taken from Reed and
German (table 1) ; as mentioned earlier, they were derived
based on heat flux and temperature data in the strato-
sphere.
6. DISCUSSION OF THE NUMERICAL RESULTS
Rad ia t ive-P hotoc hem ica I Eq u i I i br iu rn
The meridional cross sections of the radiative-photo-
chemical equilibrium temperature and ozone mixing
ratio appaar in figures 2 and 3 for the summer and winter
seasons. Since the solar radiation does not enter into the
short-wave heating or the photochemical calculations
beyond 65N in the winter hemisphere, it was not pos-
sible to compute the equilibrium distribution of ozone
north of 65ON. The solar radiation has, of course, played
an important role in determining the temperature increase
with height and in the stratopause formation in the
summer hemisphere.
Comparison of figure 2 with observations by Murga-
troyd (1957) shows that there are significant differences
between the equilibrium and the observed distribution of
temperature in the meridional plane, particularly in the
winter season and also in the lower stratosphere during
the summer. For example, the computed equilibrium
distribution does not indicate a temperature increase from
the Tropics to higher latitudes in the lower stratosphere as
shown by the observations in both hemispheres (Murga-
troyd 1957). In fact, the computed variation in the lower
stratosphere is in the opposite direction. Furthermore,
the equilibrium temperatures in the winter stratosphere
are unrealistic when compared with the observations.
Figure 4 represents a meridional cross section of ozone
mixing ratio for the summer and winter seasons as deter-
mined from observations obtained by the North American
ozonesonde network (Hering and Borden 1964). Compari-
son of the equilibrium distribution of ozone mixing ratio
(fig. 3) with the observations reveals that the observed dis-
tribution cannot, be explained by radiative-photochemical
calculations alone. NoGe, for example, that the computed
values of ozone mixing ratio in the lower stratosphere are
substantially lower than those observed, particularly in
the high latitudes in both hemispheres, resulting in the
opposite latitudinal variation of ozone. This strongly
suggests the importance of meteorological transports in
determining the ozone distribution in the lower
stratosphere.
In the upper stratosphere, the ozone measurements are
les5 reliable; however, the computed values in summer
seem to agree with the observations, suggesting that the
June1973 1 Rao J 515
TABLE 1.-Latitudinal distribution of eddy diffusion coejicients K,,, K,,, and K,, for different seasons
Latitude (degree)
5 15 25 35 46 66 66 76 86
Summer
Kuu(10~ocm2/s)
KU,(1O%m2/s)
K,, (103cm2/s)
~
0. 90 1. 16 1. 41 1. 45 1. 25 0. 91 0. 61 0. 48 0. 45
-1.21 -1.78 -3.73 -7.33 -8.41 -5.36 -2.48 -1. 13 -0.68
1. 58 2. 00 3. 21 6. 51 7. 73 5. 38 2. 83 1. 16 0. 80
Winter
K,, (lO%m2/s)
K (1 O@cmz/s)
K,, (103cmZ/s)
1. 17 1. 46 1. 86 2. 35 3. 33 4. 46 5. 50 6. 25 6. 20
2. 68 3. 96 8.96 12. 53 12.41 11.51 10. 20 8. 56 7. 08
Spring
-3.48 -5.28 -10.39 -14.36 -15.91 -16.75 -15.33 -11.31 -8.05
K,, (lO~Ocrnz/s)
K,,(lO@cm2/s)
K,, (103cm2/s)
0. 77 0. 90 1. 02 1. 23 1. 35 1. 28 1. 17 1. 02 0. 87
1. 73 2. 22 4. 18 7. 25 7. 67 6. 27 4. 50 2. 23 1. 47
-2. 13 -2.80 -5.00 -7.85 -8.38 -7.38 -5.62 "1.32 1. 47
Fall
1. 43 1. 67 2. 15 2. 57 2. 93 3. 68 4. 12 3. 00 1. 96
-2.83 -3.78 -7.92 -12.98 -14.72 -11. 52 2. 35 12. 87 12. 60
2. 57 3. 22 6. 73 11.80 13.03 10. 07 8. 83 10. 57 11. 50
meteorological transports may not play a significant role
in determining the ozone distribution in those regions due
to short photochemical regeneration times. Such trans-
ports could be important at high latitudes in the winter,
however.
Tirne-Zonal Mean State of the Model Stratosphere
Under the Influence of Transport and
Radiative-Photochemical Effects
The meridional cross sections for summer and winter
seasons representing the zonal mean temperature, ozone
mixing ratio, and zonal wind as computed for the case 2
experiment are shown in figures 5 , 6, and 7, respectively.
The flow pattern resulting from the individual velocity
components of the mean meridional motions is illustrated
in figure 8. The flow pattern was obtained by computing
the stream function that satisfies the equation of con-
tinuity in the meridional plane.
The computed distributions show several features
more representative of the observed distributions than
the radiative-photochemical equilibrium model. Let us
consider the changes that took place in the temperature
structure and in the distributions of the ozone mixing
ratio from case 1.
We note that in t.he lower stratosphere the incorporation
of the transport processes resulted in a dramatic reversal
of the latitudinal temperature and ozone mixing ratio
gradient in the ca.se 2 experiment, with temperature and
ozone mixing ratio values increasing from low to high
latitudes (figs. 5, 6). In the winter lower strato-
516 / Vol. 101, No. 6 / Monthly Weather Review
sphere, for example, the temperature has increased about
1OoK from the Tropics to high latitudes, and the ozone
mixing ratio has increased by2a factor of 2 on the average.
Both features are consistent with the measured values in
the lower stratosphere, and they represent a significant
improvement on the results of the case 1 experiment.
We also note in figure 5 that the tropopause was indicated
at about 16 km in the Tropics and around 13 km in polar
latitudes, which is also consistent with the observations.
Note, however, that the temperature structure in the
winter lower stratosphere (fig. 5 ) does not indicate the
observed midlatitude warm belt with cold temperatures
at the pole. Calculations on a gridpoint basis, however,
show the highest temperatures around 50N and a slight
decrease of temperature toward the pole. As can be seen
later, this feature is more evident in case 4 results, when
vertical variation is introduced in the diffusion coefficients.
I t is important to mention at this point that the pre-
liminary experiments with Fickian diffusion have not
resulted in any sigdlcant increase of temperature and
ozone with latitude in the lower stratocphere, showing that
the large-scaJe, quasi-horizontal eddy mixing on a sloping
surface is mainly responsible for the above events.
The temperature and ozone changes in the middle and
upper stratosphere are small at all latitudes in summer
and small in the Tropics during the winter season. The
close comparison between cases 1 and 2 experimental
results in the summer upper stratosphere shows that the
transport processes are less important. In the winter high
latitudes, however, the changes from case 1 are quite
51.5
48.0
44.9
41.5
38.5
-35.3 E
-32.5
5 29.5
2 !? 26.7
23.9
21.2
18.5
-*
~
. 0.5 
15.8 1
*. -.-.-.-a 
significant, and values in case 2 compare favorably with
observations. For example, the computed temperatures
around 250K a t 50 km in the subpolar regions during
the winter season correspond well with the observed
values given by Murgatroyd et al. (1965). Furthermore,
the northward transports of heat against the dissipative
effects of long-wave radiation in winter high latitudes
have resulted in the formation of the stratopause (fig. 5)
although not well marked. We also note rather high values
of ozone mixing ratio in the winter polar regions (fig. S ),
which may account for the observations in those regions.
In view of the fact that there was no production of ozone
in the winter polar regions, these high values of ozone
mixing ratio can only be brought about by the meteor-
ological transports in our model from the production
regions.
Despite the good overall agreement in the main features
between computed temperature and ozone distributions
in case 2 and observations, there are differences in detail.
In the case of ozcne, we will highlight these differences,
which are not readily discernable in the meridional cross-
section presentation, by comparison of vertical profiles a t
specific latitudes for winter in figure 9. We note that the
computed values of ozone mixing ratio were consistently
overestimated a t all latitudes near the level of ozone
maximum. This is also true to a lesser extent in the lower
stratrosphere except in the high latitudes where the com-
puted values were slightly less than the observations.
In figure 10, the latitudinal distributions of total ozone
computed in cases 1 and 2 experiments are compared
50
1.90 2.30 BOO 350 6.00 6.00 600 6.W 590 590 590 4.00 3.00 3.W 2.00
OBSERVED OZONEMWING RATIO Lus/s)
40
-35
E
-30
-
2
-
25-
c
p-
915- 10 -
5 -
0 I I I I I I I I I I I . 3 I * 0  2
$, 80 70 60 50 40 30 20 10 EP 10 20 30 40 50 60 70 80 90 SUMMER LATITUDE (de@ WINTER
FIGURE 4.-Meridional cross section of observed ozone mixing
ratio in summer and winter. The distribution is based on bi-
monthly means of North America,n ozone-sonde data compiled
by Hering and Borden (1964).
with the observed amounts for the summer and winter
seasons. The latitudinal distribution of total ozone com-
puted from the case 3 experiment, which will be discussed
later, is also shown in figure 10. It is clear from figure 10
that the modeled transports and their interactions with
the radiative-photochemical effects in the case 2 experi-
ment brought the latitudinal distribution of total ozone
into good agreement with the observed total amounts.
However, as indicated by the vertical profiles (fig. 9),
the computed total amounts in the case 2 experiment
were overestimated in both hemispheres, except in high
latitudes during the summer. The transport processes,
June1973 1 Rao I 517
-35.3 -
51.5
48.0
44.9
41.5
38.5
-35.3
E
-32.5
g 2 9 .5
s
$26.7
23.9
21.2
18.5
15.8
-*
21.2
18.5
15.8
13.1
-
/ \
' -I c ' ' I " ' ,' ' ' 1( ' I /
85" 7965' 5s" 493s" 25" 15' SEa9 15" 25" 3945" SP 65" 75" 85"
SUMMER LAT ITUDE WINTER
FIGURE 5.-Steady-state distribution of temperature (OK) in the
meridional plane for summer and winter from the case 2
experiment. The tropopause and stratopause are indicated by
dashed lines.
5 ,
i, I 1
5
-
-
- .
-
-
-
-
-
-
-
as modeled, appear to result in excessive flow of ozone
northward near the level of ozone maximum, especially
in winter, which in turn produces higher values of ozone
mixing ratio and consequently higher total amounts
than observed. We note from the zonal wind cross section
(fig. 7) that the significant observed features such as the
jetlike strong westerlies in the subtropical winter season
and the strong easterlies in the tropical regions in summer
are reproduced reasonably well by the model computations.
Incursion of the summer easterlies into the tropical winter
below 35 km is consistent with the observations and is
probably caused by the interhemispheric transports. The
weak midlatitude westerlies in the summer upper strato-
sphere do not agree with the observations. The lack of
vertical structure in the diffusion coefficients in case 2
apparently contributed to the poor horizontal temperature
structure, resulting in the unrealistic zonal winds in that
region.
Another feature that is not consistent with the obser-
vations is the marked discontinuity in the zonal wind
field across the Equator in the upper stratosphere. This is
a result of the thermal wind approximation, which is not
a good one in the Tropics.
The mean meridional circulation developed in the model
has a two-cell structure in the winter lower stratosphere
(fig. 8) with, generally, a rising motion in the Tropics
and in the subpolar and polar regions with subsidence
flow in the subtropics and middle latitudes. The flow
51.5
48.0
44.9
4 1 .5
38.5
-35.3
E
-32.5
g 2 9 .5
s 2 26.7
23.9
21.2
18.5
-L
I -2 1 u
1 9 .1
8 5 "s 65"5$45"3$25"15" 56% 15" 2 9 35'4955" 65" 7'3 85" SUMMER LATITUDE WINTER
15.8 :
13.1 ' ' ' ' ' I c
8 5 "s 65"5$45"3$25"15" 56% 15" 2 9 35'4955" 65" 7'3 85" SUMMER LATITUDE WINTER
FIGURE 6.-Steady-state distribution of ozone mixing ratio in the
meridional plane for summer and winter from case 2 experiment.
13 1 L
-W T
I- -. .
85" 75" 65' 55" 45" 35" 25" 1s" 5=5" 19 25" 35" 45" 55" 65" 75'85"
SUMMER LAT I T UDE WINTER
FIGURE 7.--M_eridonal cross section of the time-zonally averaged
east-west, TI, wind component for summer and winter from
case 2 experiment. Negative values represent easterlies.
518 1 Vol. 101, No. 6 f Monthly Weather Review
FLOW PATTERN (ARROWS INDICATE DIRECTION) TROPICS(5'N) SUBTROPICS(35'N) SUBPOLAR(65'N) 51.5
48.0
44.9
41.5
38.5 -40
-
-\ 50- I
E
-35 \ I -35.3
I E t
I J -32.5
2 2 9 .5
I 26.7
23.9
21.2
18.5
15.8
13.1 0 5-
X
-
x
8 w
loo ; lb 1; 0 ; i o 1; 0 ; ;o ;5
OZONE MIXING RATIO (fiq/g )
FIGURE 9.-Vertical distribution of ozone mixing ratio in case 2
a t selected latitudes compared with the mean observations of
Hering and Borden (1964) for winter season.
85"75" 65" 55" 45" 35" 25"15" 'ifa? 15" 25" 35" 45" 55" 65" 75" 85" SUMMER LATITUDE WINTER
---- -- ---
4 -
FIGURE 8.-The flow pattern of the mean meridional circulation for
summer and winter from the case 2 experiment.
pattern for the summer lower stratosphere is similar to
that during the winter season except that the whole
The computed circulation pattern in the lower strato-
sphere, shown in figure 8, corresponds closely with the
on observed information (Teweles 1964, Vincent 1968).
Also, the magnitudes of the vertical motions in the lower
pattern is shifted toward the Equator. ( London 1967 )
1 -
indirect computation of the mean meridionial flow based o , , , , , , I , I , , , , , , , >. ,
90' Bo" 70" M" 40" 'O" 20' 'O" lo' 20" 'O" 40" 60" '05 90"
FIGURE 10.-Comparison of the equilibrium distribution of total
tribution of total ozone computed from case 2 (heavy solid line)
with the observed total amounts (dashed line) as given by
Prabhakara (1963) for summer and winter seasons. Also shown
for comparison are the pure photochemical equilibrium results
of London (1967) and find equilibrium results of case 3 (dotted
line).
SUMMR LATITUDE WINTER
stratosphere Of Our are generally within the range from 1 (&&-dot line) and the final equilibrium &s-
of the observed values.
Manabe and Hunt (1968) also calculated mean merid-
ional motions for the winter stratosphere. Their circula-
tions are similar to that shown in figure g for the lower
stratosphere except the vertical extent of their winter
polar cell was greater. Differences in the results can be
attributed to the seasonal lag and the lack of vertical
structure in the diffusion coefficients in the case 2 results,
in addition to the basic differences in the two models.
during the \,,inter indicates upward motion in the
sub tropical regions with the compensating downjvard
motions on either side of it. I n summer, hopTever, the flow
is generally upward.
Seasonal Transport and
evaluated the budgets Of heat and Ozone using the case
experimental results.
balance components with latitude are shown at two levels
representing the lower and upper stratosphere. The
Principal components contributing to the heat budget are the large-scale quasi-horizontal eddy effects, the vertical
component of the mean meridional motion, and the net
radiation in both the lower and upper stratosphere, and
for the summer and winter seasons. Note that in the lower
stratosphere the large-scale eddy processes are transport-
ing more heat from the Tropics and subtropics to middle
and high latitudes than is required to compensate for the
The flow pattern in the middle and upper stratosphere Reat budget. In figure 11, the variation of various heat
Physical Mechanisms
To assess the relative importance of various physical
and transport processes operating in the stratosphere, we
June1973 J Rao J 519
CI LOWER STRATOSPHERE ( 21 km 1 UPPER STRATOSPHERE ( 45 krn 1
l *~~~*l ~l l ~l ~l ~l s l 1 ~1 1 1 ~1 ~1 1 ~1 .1 1 1 1 1
85" 65" 45" 25" 5" 5" 25" 45" G5' 85" 85" 65" 45" 25" 5" 5" 25" 45" 65" 85"
SUMMER LATITUDE WINTER SUMMER LATl TUDE WINTER
FIGURE 11.-The latitudinal variation of the various heat balance components (deg/day) in the lower and upper stratosphere for
summer and winter from case 2.
radiational heat deficit. On the other hand, the excess heat
from the Tropics (generated by radiation) and from high
latitudes was removed by the vertical component of the
meridional circulation, which is simultaneously supplying
heat to the subtropics. Although there is negligible contri-
bution from the north-south component of the mean
meridional motion, it is not appropriate to separate its
budget effects from the vertical component because of the
complete balance between two components in the equation
of continuity.
Relatively little information is available on heat
transports from observed data, particularly in the middle
and upper stratosphere. However, the results of the model
with regard to heat transports by the large-scale, yuasi-
horizontal eddy effects in the lower stratosphere agree in
a qualitative sense with the observed estimates by Oort
(1963) of transient heat fluxes.
Manabe and Hunt (1968) have also made heat balance
calculations of the winter stratosphere from the general
circulation model. Exact comparison between the two
results is not possible. However, the heat balance
components they have computed at 25 km agree approxi-
mately in both magnitude and sign with the results shown
in figure 11 for the winter stratosphere except that the
strength of the mean meridional circulation, and therefore
the magnitude of its contribution to the heat balance, was
underestimated in the case 2 results.
Ozone budget. Although the predominant balance in
ozone is between the large-scale, yuasi-horizontal eddy
effects and the contribution from the vertical component
of the mean meridional motions, the contributions of
vertical eddy effects and the north-south component of
the mean motion are important for complete balance.
The relative roles of various mechanisms in the main-
tenance of ozone are illustrated in figure 12, as a function
520 / Vol. 101, No. 6 / Monthly Weather Review
of latitude, a t two representative levels of the lower and
upper stratosphere, respectively. Contrary to the classi-
cal view (Brewer 1949) that the late winter and spring
build-up in the high latitude lower stratosphere is attribu-
table to transport by the mean meridional cell, this model
demonstrates that the ozone transports to high latitudes
are mainly due to large-scale eddy fluxes. The vertical
eddy effects played a substantial role in the winter high
latitudes in balancing the northward transports of ozone
by the quasi-horizontal eddies. The main function of the
mean meridional motions in the two hemispheres is appar-
ently to remove the ozone tracer from the Tropics and
high latitudes, and deposit it in the subtropical regions.
On the other hand, the quasi-horizontal eddies transport
ozone northward into high latitudes mainly from this
sub tropical area of concentration. However, more ozone
was transported to high latitudes by the horizontal eddy
effects than was removed by the vertical, component of
the mean motion, and the excess was balanced by down-
ward transports due to vertical eddies in winter and by
the photochemical sink in summer in high latitudes. I n
the Tropics, the ozone removed by the vertical component
of the mean motion was replaced partly by the photo-
chemistry and partly by ozone convergence due to hori-
zontal eddy effects, which might have been caused by the
interhemispheric transports.
In the winter upper stratosphere, the main source of
ozone production was located in middle latitudes, pro-
bably due to the inverse temperature effect on the photo-
chemical production of ozone that was so evident in the
case 1 results. From this source, the quasi-horizontal eddy
processes transported ozone both northward and south-
ward (predominantly northward) to fill the photochemical
sinks in those regions. Again, in the winter high latitudes,
the excess ozone transported by the horizontal eddy pro-
SUMMER LATITUDE- WINTER SUMMER LATITUDE W1 NTER
FIGURE 12.-The latitudinal variations of various balance contributions to the ozone budget (1O-lpg . g-l . day-1) in the lower
and upper stratosphere from case 2.
cesses over the photochemical sink was balanced by ozone
removal due to the vertical component of the mean motion.
In the subtropical winter, apart from the mutual opposi-
tion between photochemistry and horizontal eddy pro-
cesses, the vertical eddy effects and the contribution from
the north-south component of the mean motion roughly
balance each other. In the summer upper stratosphere,
however, both horizontal and vertical eddy effects were
weak in the model; therefore, the photochemical sinks in
that region were mainly balanced by the contributions
from the vertical component of the mean motion.
The relative importance of the radiative-photochemical
and transport processes depends on their relative time
scales (characteristic times). I t appears that, due to small
characteristic times for radiative-photochemical pro-
cesses, the transport processes have relatively small effect
on the distribution of ozone or temperature in the summer
upper stratosphere.
Results of the Case 3 Experiment
The linear constraint on the thermodynamic energy
source was removed in the case 3 experiment by incor-
porating detailed calculations for the long-wave radiation
transfer based on physical considerations. The diffusion
coefficients used in this experiment were exactly the same
as in case 2 for summer and winter (table 1). The results
show that the meridional cross sections of temperature,
mean circulation, and ozone are similar to those in case 2
except for small differences in detail. The case 3 experiment
for summer and winter produced better temperature
gradients than case 2 did in the upper stratosphere but
poorer ones in the lower stratosphere. As shown in figure
10, however, improved latitudinal distribution of total
ozone occurred in the case 3 experiment. A detailed
discussion on case 3 results is given in Rao (1970).
Results of the Case 4 Experiment
One unsatisfactory feature we have noticed in the case 2
(also in case 3) experiment, particularly in the middle
latitude summer, was the lack 2f a realistic vertical
structure in the distribution of zonal winds. This de-
ficiency appears to result from a lack of vertical structure
in the diffusion coefficients, K,, and K,,, which contributed
to insufficient heat transports and horizontal temperature
gradients. Therefore, we decided to v$,ry the diffusion
coefficients, K,, and K,,, in the vertical dimension as
wel,l. The diffusion coefficients for the lower stratosphere
were taken as reported by Reed and German; those for
the higher levels were derived from Newell's wind statistics.
Except for the vertical variation of diffusion coefficients
in this experiment, conditions were the same as in case 2.
The temperature and zonal wind distributions resulting
from this experiment are shown in figures 13 and 14, and
the circulation pattern of the mean meridional motions is
illustrated in figure 15. Immediately apparent from figure
13 is the appreciable improvement in the horizontal
temperature structure in both hemispheres in the lower
stratosphere over that derived from the case 2 experiment.
For instance, the temperature gradient from the Equator
to the pole in summer in the lowest three levels of the
model is quite close to the observations. Note also the
15K increase of temperature from the Equator to a
maximum around 55"N with lower temperatures farther
north in the winter lower stratosphere, roughly in agree-
ment with the observations. In the middle and upper
stratosphere, however, the changes are slight; there is,
however, a better indication of the stratopause formation
in the winter high latitudes.
Associated with the improved horizontal temperature
structure in the lower stratosphere, is a much improved
vertical structure of the zonal wind distribution, especially
June1973 / Rao / 521
TEMPERATURE 1%)
23.9
SUMMER LATITUDE WINTER
FIQURE 13.-The steady state distribution of temperature in the
meridional plane for summer and winter in case 4. The stratopause
is indicated by the dashed line.
in the middle latitude summer (fig. 14). For example, the
specified zonal winds of 17-18 m/s in the middle latitude
summer were reduced to zero and slight easterlies in the
upper stratosphere. This confirms the idea that the poor
zonal wind structure in the middle latitude summer was
partly due to lack of vertical structure in the diffusion
'coefficients in case 2; the results, however, still do not
agree with the observed easterlies, which require even
stronger horizontal temperature gradients.
The circulation pattern of the mean meridional motions
for case 4 (fig. 15) shows a much greater resolution in the
two-cell structure in the summer lower stratosphere than
that for case 2. In addition, the vertical extent and the
strength of the high latitude cell in the winter lower
stratosphere are in better agreement with the results of
Manabe and Hunt (1968).
The meridional cross section of ozone in case 4 was
similar to that in case 2 and, therefore, it is not necessary
to present it here. However, it should be mentioned that
the total ozone amounts were higher in case 4 because of
stronger northward ozone transports by the quasi-
horizontal eddy processes in the lower stratosphere. I t
appears that the diffusion coefficients based on heat
transport data were a little too strong for the ozone
transports in the model.
522 / Vol. 101, No. 6 / Monthly Weather Review
' \
-10
1
I
I I
- 20
EAST-WEST WIND (m/s)
30
15.8
13.1
85" 75" 6C 55' 45' 35" 25' 15' 5'EQ5' 15" 25' 35" 45' E
SUMMER LATITUDE WINTER
u 5" 75' 65"
FIQURE 14.-The meridional- cross section of time-zonal mean
east-west wind component, u, for summer and winter in case 4.
The Boundary Conditions
To test the effect of lower boundary condition on
temperature, we reduced the lower boundary tempera-
tures for summer and winter by 5 percent, a change of
10-120K at each latitude, and the case 2 experiment was
repeated with the new conditions. The results show that
modification of lower boundary temperatures has no
influence on the distribution of temperature within the
model region except for some minor changes at the lowest
two levels near the lower boundary. When the upper
boundary temperatures were also changed by 5 percent,
the changes that resulted near the upper boundary were
even less significant. The lack of response of the interior
temperature field to these boundary changes is probably
due in part to the lack of significant vertical heat trans-
ports within the model.
The effects of changes in the ozone mixing ratio at the
lower boundary on the interior distribution of ozone were
also tested by deliberately minimizing ozone mixing ratio
values at the lower boundary. Altering the lower boundary
mixing ratio values by a factor of 2-3, depending upon
the latitude, alters the total ozone amounts by about 10
percent; there is an indication that this change is due to
downward transport of ozone by vertical eddy processes
FLOW PATTERN IARROWS INDICATE DlRECTlONl SUMMER 165ON) SUMMER 15ON) WINTER (65")
SUMMER LATITUDE WINTER
FIGURE 15.-The flow pattern of the mean meridional circulation
for summer and winter in case 4.
in the model when strong vertical gradients of the ozone
mixing ratio were imposed at the lower boundary.
The vertical velocities specified at the top boundary
of the model for this study are based on the ap-
proximate balance between the static stability term
and net radiation in the thermodynamic energy equation
and were taken from Murgatroyd and Singleton (1961).
Because there is some doubt as to the validity of these
values in the winter high latitudes, we decided to test a
specification - of zero vertical velocities at the top by making
Z =O everywhere at the top boundary. This is an extreme
condition, and it is unlikely to represent a true situation
in the real atmosphere at that level. The vertical profiles
of temperature and ozone mixing ratio corresponding to
the two sets of boundary conditions are compared in
figures 16 and 17 at 65"N and 5"N in summer and at
65"N in winter. Comparison of the two temperature
profiles at 65"N in winter with the observations shows
that the zero vertical velocity condition at the upper
boundary is quite unrealistic. For instance, the temperrt-
tures - corresponding to the dashed curve (representing
i=O at the top) are 10-13"K lower than the observed,
while the solid curve (with nonzero vertical velocities at
the top) agrees well with observations. This suggests
that the top boundary specification of the vertical veloci-
205 215 225 235 245 255 265 200 210 220 230 240 250 260 210 220 230 240 250
TEMPERATURE (OK)
FIGURE 16.-Influence of the modified vertical velocities a t the
upper boundary on the vertical distribution of temperature a t
selected latitudes. The dashed curves represent the zero vertical
velocity specification a t the top (Le., the modified boundary).
SUMMER 165"N) SUMMERISON) WINTER 165-M
55r I I
-40
E
- 35
%30-
0
W 125-
20 -
15 -
0 5 10 0 5 IO 15 0 5 10 15
OZONE MIXING RATIO ( u g l g )
FIGURE 17.--Influence of the modified vertical velocities at the
upper boundary on the vertical distribution of ozone mixing
ratio a t selected latitudes. The dashed curves represent the zero
vertical velocity specification a t the top (i.e., the modified
boundary).
ties in winter high latitudes is a reasonable condition.
The ozone changes caused by the variation of vertical
velocities at the top are not particularly significant,
although they are noticeable (fig. 17).
Case 2 and 3 Experiments
For Spring and Fall Seasons
For a detailed discussion of the results (including case
3), the reader is referred to Rao (1970). Here we present
and discuss briefly only the results of the case 2 experi-
ment. The results are shown in figures 18-20 with the flow
pattern for the mean meridional motions illustrated in
figure 21.
The temperature distribution in the lower stratosphere
fall (fig. 18) shows a pronounced temperature maximum
around 55ON from which the temperature decreases
toward both the Equator and the pole. In the spring
lower stratosphere, however, the temperature maximum
has shifted farther to the north and is located around 75"
to 80"N.
I n the middle and upper stratosphere (fig. 18), the
strongest horizontal temperature gradients are found in
June1973 Rao 523
85" 75' 65' 55' 450 35O 2 5 O 15' 5O 5O 15O 250 35O 45O 55O 65O 7 5 O 85O
SPRING LATITUDE FALL
FIQURE 18.-The steady-state distribution of temperature in the
meridional plane for spring and fall in case 2. The stratopause is
indicated by the dashed line.
middle and high latitudes in both hemispheres, in contrast
to the strong gradients that were shown in tropical and
subtropical regions in summer and winter in case 2. Note
that the temperatures are higher in the Tropics, steadily
decreasing toward both p,oles except a t about 45 km in
spring, where the temperature gradient is reveresd.
The zonal wind distributions for spring and fall (fig.
19) generally follow the horizontal temperature structure
except for the lower boundary effects. Therefore, com-
ments are not necessary except to mention that the double
wind maximum of westerlies in the upper stratosphere and
the tropical easterlies in the lower levels during the fall
season seem to correspond to the early winter conditions
in the real atmosphere.
In the cross section of ozone mixing ratio in the lower
stratosphere (fig. 20), we see that the model distribution
is roughly symmetrical with respect to the two seasons.
The stronger diffusion coefficients that correspond to the
early winter conditions appear to be too strong for the
fall season. The distribution of ozone near the level of
ozone maximum and above is similar to that in summer
and winter in case 2.
Figure 22 shows the comparison between the computed
latitudinal distribution of total ozone and the observed
amounts, with the solid and dashed curves representing
524 / Vol. 101, No. 6 / Monthly Weather Review
21 2
I
13.1 EO
85" 75O 65' 55' 450 350 250 15' 5' 15' 25" 35" 45' 65O 75' 85'
SPRING LATITUDE FALL
FIGURE 19.-The meridional cross section of time-zonal mean
east-west component, u, for spring and fall in case 2. Negative
values represent easterly winds.
observed and computed amounts, respectively. Also shown
in figure 22 is the latitudinal distribution of total ozone
computed from the case 3 experiment for spring and fall,
which is represented by the dotted curve. The computed
amounts are obviously overestimated at most cf the lati-
tudes, although the spring maximum in the high latitudes
is computed correctly. As expected from the meridional
distribution of the ozone, mixing ratio in the lower strato-
sphere, the computed amounts are a t least 25 percent higher
than the observed in the fall season; in spring, the percent-
age error is much less, especially in the high latitudes.
From the flow pattern of mean meridional circulations
for spring and fall (fig. 21), we see that the three circula-
tion cells in the lower stratosphere during the fall season
look like the direct extension of the upper tropospheric
cells during the early winter conditions. As a result of
the shift of temperature maximum northward near the
pole in the spring lower stratosphere, the polar cell almost
disappears, resulting in a two-cell structure. During the
fall, the middle'and upper stratosphere flow is upward in
subtropical and middle latitudes with compensating down-
ward motion on either side in a manner similar to that
during the winter in case 2. In spring, however, the motion
is upward from the Tropics to 55'N in the middle strato-
sphere with downward motion in the .high latitudes; this
MERIDIONAL DISTRIBUTION OF OZONE (pg/q1
4 8 0 'i - - _-
/
FLOW PATTERN FOR MEAN MERIDIONAL CIRCULATION
850 75" 65' 55' 45' 35' 25" 15' 5" 5' I T 25' 35" 45' 55' 65' 75" 85"
SPRING LATITUDE FALL
tu
850 75' 65' 55' 45' 35' 25' 15" 5" 5' 15' 25' 35" 45' 55' 650 ?so 85'
SPRING LATITUDE FALL
FIGURE 20.-The steady-state distribution of ozone mixing ratio in
the meridional plane for spring and fall in case 2.
FIGURE 21.-The flow pattern of the mean meridional motions for
spring and fall in case 2. Arrows indicate direction.
flow pattern shifts towards the Equator in the upper
stratosphere.
To provide a general survey and to conclude the
discussion of experimental results, we compare (fig. 23)
the latitudinal distributions of total ozone resulting from
the various numerical experiments (including the present
experiment) with the observed total ozone curve from
Prabhakara (1963) for the winter season. Figure 23 shows
that the relative performance of various models, either
steady state or time-dependent, cannot be judged from
the latitudinal distributions of total ozone alone. A clear
example of this is given by the results of Hunt (1969),
which provide much information on the transport mech-
anisms involved in the maintenance of ozone distribution
in the winter lower stratosphere while having the least
agreement with the observed total amounts. Although the
agreement between the total ozone curve computed frcm
the present model and the observations is not satisfactory,
we have contributed significantly to the understanding of
transport mechanisms responsible for the maintenance of
the seasonal distributions of various properties in the
stratosphere.
7. CONCLUSIONS
Despite the limitations of the model with regard to the
eddy processes, a number of interesting conclusions can be
0.5 r
50.2 1
53.1
$1
FIGURE 22.-Comparison of the latitudinal distribution of total
ozone computed for spring and fall in case 2 (dashed line) with
the observed distribution (solid line) taken from Prabhakara
(1963). Also shown for comparison is the distribution of total
ozone for spring and fall in case 3 (dotted line).
drawn from the comparison of the experimental results
with the observations. By comparing case 1 and case 2
results, we find that the modeled transpcrts and their
interactions with the physical processes are sufficient t o
rectify much of the initial discrepancies between the
observed and the radiative-photochemical distributions
of temperature and ozone in the winter stratosphere and
the summer lower stratosphere. I n the lower stratosphere,
the required heat and ozone were supplied to the sub-
June1973 J Rao J 525
0.7 r ACKNOWLEDGMENTS
The author is grateful to B. W. Boville for his encouragement
and many helpful suggestions during the course of this investiga-
tion, A. D. Christie for critically reviewing the manuscript, and
the Atmospheric Environment Service of Canada for support
and the approval of this publication. CI
$0.5
v  \(CASE 3 EXPT PRESENT )
--*--- -. \ HUNT ( 02 -HYDROGEN ATMOSPHERE 1969 1
90 80 70 60 50 40 30 20 10 -0
LATITUDE
FIGURE 23.-Comparison of the observed latitudinal distribution
of total ozone in winter given by Prabhakara (1963) with the
results of various time-dependent and steady-state models,
including the present model.
tropical regions by the mean meridional motions and then
transported northward by the quasi-horizontal eddy
effects. At higher levels in the winter stratosphere, the
temperatures were maintained by both subsidence and
and northward heat transports against the strong long-
wave cooling. The high ozone mixing ratio values in the
polar regions were mainly due to northward ozone trans-
ports by the large-scale eddy processes from the middle
latitude photochemical source.
As a consequence of the generalized diffusion in case 2
and later, the mean meridional motions developed the
two-cell structure typical of recent observational studies
of the winter lower stratosphere. Both quasi-horizontal
eddies and mean meridional motions contributed sig-
nificantly to all three budgets, whereas the vertical eddies
had little effect except to transport ozone downward in
the winter lower stratosphere.
The results of these experiments reveal the fundamental
role played by the mean meridional motions and the large-
scnle, quasi-horizontal eddy effects, especially the latter,
in the maintenance of the seasonal distributions in the
stratosphere, in the presence of the physical processes, and
constitute a valuable contribution to the general under-
standing of the seasonal climatology of the stratosphere.
Future extensions of this work should give more em-
phasis to the two-dimensional distribution of the diffusion
coefficients and should try to avoid the thermal wind
constraint.
~
REFERENCES
Brewer, A. W., Evidence for a World Circulation Provided by
the Measurements of Helium and Water Vapour Distribution
in the Stratosphere, Quarterly Journal of the Royal Meteorological
Society, Vol. 75, No. 326, London, England, Oct. 1949, pp. 351-
363.
Brewer, A. W., The Rate of Production of Ozone in the Lower
Stratosphere, Publication in Meteorology No. 80, Arctic Metoro-
logy Research Group, Department of Meteorology, McGill
University, Montreal, Canada, May 1966, pp. 195-198.
Byron-Scott, R., A Stratospheric General Circulation Experiment
Incorporating Diabatic Heating and Ozone Photochemistry,
Publication in Meteorology No. 87, Arctic Meteorology Research
Group, Department of Meteorology, McGill University, Mon-
treal, Canada, Apr. 1967, 201 pp.
Chapman, Sydney, A Theory of Upper-Atmospheric Ozone,
Memozrs of the Royal Meteorological Society, Vol. 3, No. 26,
London, England, June 1930, pp. 103-125.
Clark, John H., Infrared Heating Due t o Ozone, M.Sc. thesis,
Department of Meteorology, McGill University, Montreal,
Canada, 1963, 55 pp.
Clark, John H., A Quasi-Geostrophic Model of the Winter Strat-
ospheric Circulation, Monthly Weather Review, Vol. 98, No. 6,
June 1970, pp. 443-461.
Craig, Richard A., The Observations and Photochemistry of
Atmospheric Ozone and Their Meteorological Significance,
Meteorological Monographs, Vol. 1, No. 2, American Meteorological
Society, Boston, Mass., Sept. 1950, 50 pp.
Dobson, G. M. B., Harrison, D. N., and Lawrence, J., Measure-
ments of the Amount of Ozone in the Earths Atmosphere and
Its Relation to Other Geophysical Conditions, Proceedings of
the Royal Society of London, Ser. A, Part 11, Vol. 114, No. 768,
England, Apr. 1927, pp. 521-541.
Diitsch, H. U., Photochemische Theorie der Atmospharischen
Ozons unter Berucksichtigung von Nichtgleishchgewichtzustan-
den and Luftbewegungen, Doctoral Dissertation, University
of Zurich, Switzerland, 1946, 113 pp.
Eliassen, A., The Quasi-Static Equations of Motion With Pressure
as Independent Variable, Geojysiske Publikagoner, Vol. 17 NO. 3,
Norske Videnskaps-Akademi i Oslo, Universitetsforlaget, Oslo,
Norway, 1949, 43 pp.
Hering, W. S., and Borden, T. R., Jr., Ozonesonde Observations
Over North America, Val. 2, Environmental Research Papers
No. 38, U.S. Air Force Cambridge Research Laboratories,
Hanscom Field, Mass., July 1964, 280 pp.
Hering, W. S., Touart, C. N., and Borden, T. R., Jr., Ozone
Heating and Radiative Equilibrium in the Lower Stratosphere,
Journal of the Atmospheric Sciences, Vol. 24, No. 4, July 1967,
Howard, John N., Burch, Darrell E., and Williatns, Dudley,
Near Infrared Transmission Through Synthetic Atmospheres,
Geophysical Research Paper No. 40, U.S. Air Force Cambridge
Research Center, Hanscom Field, Mass., Nov. 1955, 244 pp.
Hunt, Barrie G., Photochemistry of Ozone in a Moist Atmosphere, Journal of Geophysical Research, Vol. 71, No. 5, Mar. 1966,
Hunt, Barrie G., Experiments With a Stratospheric General
Circulation Model: 111. Large-Scale Diffusion of Ozone Including
Photochemistry, Monthly Weather Review, Vol. 97, NO. 4,
Apr. 1969, pp. 287-306.
pp. 402-413.
pp. 1385-1398.
526 1 Vol. 101, No. 6 1 Monthly Weather Review
6
Hunt, Barrie G., and Manabe, Syukuro, Experiments With a Planetary Circulations Project, Department of Meteorology,
Stratospheric General Circulation Model: 11. Large-Scale 34 machusetts Institute of Technology, 1963, 122 pp.
Diffusion of Tracers in the Stratosphere, Monthly Weather Prabhakara, Cuddapah, Effects of Non-Photochemical Processes
Review, Vol. 96, NO. 8, Aug. 1968, pp. 503-539. on the Meridional Distribution and Total Amount of Ozone in
the Atmosphere, Monthly Weather Review, Vol. 91, No. 9, Sept. .Johnston. H. S.. Reduction of Stratospheric Ozone by Nitrogen . .
Oxide Catalysts From Supersonic Transport Exhaust,  Science,
Leovy, Conway, Radiative Equilibrium of the Mesosphere,
Journal of. the Atmospheric Sciences, Vol. 21, No. 3, May 1964,
pp. 238-248.
Leovy, Conway, Simple Models of Thermally Driven Meso-
spheric Circulations, Journal of the Atmospheric Sciences, Vol. 21,
NO. 4, July- 1964, pp. 327-341.
Lindzen, Richard S., and Goody, Richard, Radiative and Photo-
chemical Processes in Mesospheric Dynamics: Part I, Models for
Radiative and Photochemical Processes, Journal of the A t -
spheric Sciences, Vol. 22, No. 4, July 1965, pp. 341-348.
London, Julius, The Average Distribution and Time Variation of
Ozone in the Stratosphere and Mesosphere, Space Research Vol.
7: Proceedings of the 7th International Space Science Symposium,
Vienna, Austria, M a y 10-18, 1966, North Holland Publishing
Company, Amsterdam, 1967, pp. 172-185.
Manabe, Syukuro, and Hunt, Barrie G., Experiments With a
Stratospheric General Circulation Model: I. Radiative and
Dynamic Aspects, Monthly Weather Review, Vol. 96, No. 8,
Murgatroyd, R. J., Winds and Temperatures Between 20 km and
100 km-a Review, Quarterly Journal of the Royal Meteorological
Society, Vol. 83, No. 358, London, England, Oct. 1957, pp. 417-
458.
Murgatroyd, R. J., Hare, F. K., Boville, B. W., Teweles, S., and
Kochanski, A,, The Circulation in the Stratosphere, Meso-
sphere and Lower Thermosphere, Technical h o l e No. 70, World
Meteorological Organization, Geneva, Switzerland, 1965, 206 pp.
Murgatroyd, R. J., and Singleton, F., Possible Meridional Cir-
culations in the Stratosphere and Mesosphere, Quarterly Journal
of the Royal Meteorological Society, Vol. 87, No. 372, London,
England, Apr. 1961, pp. 125-135.
Newell, Reginald E., Preliminary Study. of Quasi-Horizontal Eddy
Fluxes From Meteorological Rocket Network Data, Journal
of the Atmospheric Sciences, Vol. 20, No. 3, May 1963, pp. 213-225.
Oort, Abraham H., On the Energy Cycle in the Lower Strato-
sphere, Scientific Report No. 9, Contract No. AT(30-1)2241,
Val. 173, NO. 3996, Aug. 6, 1971, pp. 517-522.
Aug. 1968, pp. 477-502.
.-
1963, pp. 411-431.
Rao, V. R. Krishna, A Numerical Experiment on the Steady State
Meridional Structure of the Stratosphere, Ph.D thesis, Depart-
ment of Meteorology, McGill University, 1970, 229 pp.
Reed, Richard J., and German, Kenneth E., A Contribution to the
Problem of Stratospheric Diffusion by Large-Scale Mixing,
Monthly Weather Review, Vol. 93, No. 5, May 1965, pp. 313-321.
Reynolds, O., On the Dynamical Theory of Incompressible Viscous
Fluids and the Determination of the Criterion, Philosophical
Transactions of the Royal Society of London, Ser. A., Vol. 136,
Richtmger, R. D., Difference Methods for Initial-Value Problems,
New York, N. Y., Znterscience Publishers, 1957, 238 pp.
Smagorinsky, Joseph, Manabe, Syukuro, and Hollon~ay, J. Leith,
Jr., Numerical Results From a Nine-Level General Circulation
Model of the Atmosphere, Monthly Weather Review, Vol. 93, No.
12, Dec. 1965, pp. 727-768.
Teweles, Sidney, Stratospheric-Mesospheric Circulation, Re-
search in Geophyszcs, Vol. 2, No. 19, Solid Earth and Interface
Phenomena, M.I.T. Press, Cambridge, Mass., 1964, p. 509.
Thompson, Philip D., h umerical Weather Analysis and Prediction,
The MacMillan Company, New York, N. Y., 1961 170 pp.
Vincent, D. G., Mean Meridional Circulations in the Northern
Hemisphere Lower Stratosphere During 1964 and 1965, Quar-
terly Journal of the Royal Meteorological Society, Vol. 94, No. 401,
London, England, July 1968, pp. 333-349.
Walshaw, C.D., Integrated Absorption by the 9.6 Band of Ozone,
Quarterly Journal of the Royal Meteorological Society, Vol. 83, No.
357, London, England, July 1957, pp. 315-321.
Williams, G. P., and Davies, D. R., A Mean Motion Model of the
General Circulation, Quarterly Journal of the Royal Meteoro-
logical Society, Vol. 91, No. 390, London, England, Oct. 1965,
Wulf, Oliver R., and Deming, L. S., The Theoretical Calculation
of the Distribution of the Photochemically-Formed Ozone in the
Atmosphere, Journal of Terrestrial Magnetism and Atmospheric
Electricity, Vol. 41, No. 3, The Johns Hopkins Press, Baltimore,
Md., Sept. 1936, pp. 299-310.
1894, pp. 123-164.
pp. 471-489.
[Received March 2.4, 1972; revised March 2, 19731
June1973 1 Rao I 527