-1 1963
SEP
MESOSPHERIC HEATING
and
SIMPLE MODELS OF THERMALLY DRIVEN CIRCULATION
by
Conway Leovy
A.B.,
University of Southern California
(1954)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
July, 1963
.
Signature redacted
Signature of Author .
Department a
Meteoilogy, July 24, 1963
Accepted by
.
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.
...
.
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.
.
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. . . . . . . . . . . . . . . . . . . . . . . . . . .
Chairman, Departmental Committee on Graduate Students
I
.
Certified by .
.
Signature redacted
Mesospheric Heating
and
Simple models of Thermally Driven Circulation
by
Conway Leovy
Submitted to the Department of Meteorology on July 24,
in partial fulfillment of the requirement
for the degree of Doctor of Philosophy.
1963
ABSTRACT
Possible thermally driven symmetric circulations of the mesosphere
and upper stratosphere are considered, subject to the following major
assumptions: the motions are in hydrostatic and geostrophic balance, the
motions are small perturbations about a state of rest which is defined by
the horizontal average of the radiative equilibrium temperature, the eddy
fluxes of momentum and heat act only in a diffusive manner on the mean
zonal momentum and temperature fields.
The assumptions of geostrophy and
small perturbations are justified by means of a scale analysis and by a
detailed numerical investigation of photochemical and radiative processes.
The derived meridional circulations have a principle flow branch
from summer pole to winter pole with maximum amplitude less than a meter
per second.
Good agreement with the observed amplitude, distribution,
and seasonal phase of the mean zonal winds is obtained if eddy viscosity
and eddy conductivity parameters are assumed to have values between
500 m 2 /Is and 1000
m2s
Radiative and photochemical processes are found to have an important
damping effect on motions of this type; a radiative-photochemical damping
parameter can be derived having the order of magnitude 10-6 per second.
Thesis Supervisor:
Title:
Jule G. Charney
Professor of Meteorology
ACKNOWLEDGEMENTS
It is a pleasure to acknowledge the encouragement, support and
valuable advice given by Professor Jule G. Charney.
The author is indebted to Dr. Lewis D. Kaplan for providing a
copy of the Curtis CO 2 cooling matrices and to Dr. R. Tousey for providing the recent solar spectrum used.
He also wishes to thank Pro-
fessor R. E. Newell for making figure 13 on the Rocket Network data
available.
Valuable discussions with Dr. R. R. Rapp, Mr. E. S. Batten,
Dr. R. M. Goody, and Dr. H.
I. Schiff are also greatly appreciated.
Preparation of a manuscript under conditions requiring both speed
and precision is a trying task, and the author wishes to thank Miss
Marie Guillot, Mrs. Theresa Blumen, Mrs. Berit Larsen, and Miss Ann
Ordway for this work, particularly Miss Guillot on whom the brunt of
the task fell.
The numerous figures were drafted by Miss Isabel Cole
and Mrs. Janet Leovy.
Much of the numerical work on the radiative and photochemical
problem was done at the M.I.T. Computation Center, Cambridge, Massachusetts.
TABLE OF CONTENTS
Page
Chapter 1.
INTRODUCTION
1
Chapter 2.
BACKGROUND OF THE PROBLEM
4
Chapter 3.
FORMULATION OF THE PROBLEM
A.
The General Equations
B.
Scale Analysis and Formulation of the
Linearized Problem
C.
Energy Equations
11
11
RADIATIVE EQUILIBRIUM AND HEATING
A.
General Remarks
B.
Photochemical Considerations
C.
The Equilibrium Computations
D.
The Time Dependent Problem
E.
The External Heating
F.
Discussion
29
29
35
44
SOME LINEAR SOLUTIONS
69
69
72
Chapter 4.
Chapter 5.
A.
B.
C.
D.
E.
F.
G.
Chapter 6.
DISCUSSION OF RESULTS
A.
B.
C.
D.
Chapter 7.
The Equations
Boundary Conditions on the Stream Function
Solutions in the Case of Rayleigh Viscosity
and Conductivity
Solutions for Constant and Exponentially Increasing Viscosity and Conductivity
Boundary Conditions on
, and an Example
with Eddy Viscosity and Eddy Conductivity
Eddy Viscosity, Eddy Conductivity and Radiative Coupling
A Solution in the Case of a Bounded Atmosphere
48
60
63
75
98
101
110
124
133
Comparison with Observations
The Evidence from Radioactive Tracers
Eddy Viscosity and Eddy Conductivity
Some Implications of the Deduced Circulations
CONCLUSIONS AND SUGGESTIONS
15
25
FOR FURTHER RESEARCH
133
139
143
149
154
Appendix 1. DETAILS OF THE PHOTOCHEMICAL & RADIATIVE CALCULATIONS
A-1
A.
Photochemical Data
A-1
B.
Calculation of CO2 Emission
A-10
C.
REFERENCES
j
Calculation of Ozone Emission
A-12
U
LIST OF FIGURES
Page
Figure
Figure
Figure
Figure
Figure
1.
2.
3.
4.
5.
32
Characteristic times for equilibrium in the ozone
problem.
39
Equilibrium temperatures, *K, computed by matrix
method.
46
Log (base 10) ozone concentration, molecules per
cc, computed by matrix method.
47
Diurnal variation of 03 Lat. 0, Dec. 21.
tial conditions: equilibrium.
54
Ini-
Figure
6.
Time integrations of temperature at the equator.
55
Figure
7.
Variations of ozone concentration on the equator.
56a
Figure
8.
Equilibrium temperatures, *K, computed by marching
method.
57
Log (base 10) ozone concentration, molecules per cc,
computed by marching method.
58
Figure 10.
Diurnal temperature range, *K.
59
Figure 11.
The basic statical equilibrium temperature distribution.
61
The "external heating" of an atmosphere whose temperature is T0(z), *K/day.
62
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure 16.
Rocket network zonal wind observations, taken from
Newell (1963).
79
The idealized "external heating" used in the dynamical
computations.
88
Vertical velocity (cm/s) model I, Rayleigh friction
Newtonian conductivity.
89
-
9.
Meridional velocity (m/s), model I, Rayleigh friction
Newtonian conductivity.
-
Figure
ii
Cooling rate due to CO2 versus Planck function at
15 microns.
90
mm
Figure 18.
Figure 19.
Figure 20.
Figure 21.
Figure 22.
Figure 23.
Zonal velocity (m/s), model I, Rayleigh friction
Newtonian conductivity.
91
Vertical velocity (m/s), model II, Rayleigh friction - Newtonian conductivity.
93
Meridional circulation (m/s), model II, Rayleigh friction - Newtonian conductivity.
94
Zonal velocity (m/s), model II, Rayleigh friction
Newtonian conductivity.
95
Phase relationships between v&,
the external heating, P = 0.3.
-'
A
A
ur,
-
Figure 17.
-
Page
u ,T
and
96
Comparison of "well-behaved" solution, and solution
matched to molecular dissipation layer. No radiative
coupling.
106
Zonal wind (m/s), model I with eddy viscosity and conductivity.
107
A
Figure 24.
Figure 25.
Figure 26.
Figure 27.
Figure 28.
Phase lag of L
conductivity, P
,
=
model I with eddy viscosity and eddy
1.
109
Roots of the equation
A
values of GP in parentheses.
114
Well-behaved solutions (dashed), and solutions matched
to eddy dissipation layer (solid), radiative coupling
included.
116
Vertical velocity (cm/s), model III, eddy viscosity
and conductivity with radiative coupling.
119
Meridional velocity (m/s), model III, eddy viscosity
and eddy conductivity with radiative coupling.
120
Figure 29.
Zonal winds (m/s), model III, eddy viscosity, eddy conductivity and radiative coupling.
121
Figure 30.
Phase relationships between
model III.
Figure 31.
A
V,
/I
er ,
A
A
A
,
7
and
122
Heating used by Murgatroyd and Singleton (1961), *K/day.127
Page
Figure 32.
Truncated series approximation to the heating of
Figure 31, *K/day.
128
Figure 33.
Vertical velocity (cm/s), model IV.
129
Figure 34.
Meridional velocity (m/s), model IV.
130
Figure 35.
Zonal Winds (m/s), model IV.
131
Figure 36.
Observed zonal winds (m/s), after Batten (1961).
134
Figure 37.
Zonal wind (m/s) for model IV with eddy viscosity
and eddy conductivity, showing the effect of various
boundary conditions on
144
Temperature distribution corresponding to model I
(OK)-
153
Figure 38.
Appendix 1.
Figure Al.
Solar spectrum from 1750 to 2100A.
A- 5
Figure A2.
Cooling rate due to ozone infrared emission versus
the 9.6 micron Planck function.
A-13
LIST OF TABLES
Page
Table
Table
Table
Table
Table
1.
2.
3.
4.
5.
6.
Reactions determining ozone and atomic oxygen concentrations.
36
Errors due to overestimating the absorption at sunrise.
67
Zonal wind data deduced from rocket network observations.
80
Solutions and separation constants for the
equation.
86
Coefficients in representation of Murgatroyd
Singleton heating.
-
Table
Comparison of deduced and observed zonal winds.
132
135
Appendix 1.
Table Al.
Spectral data used in photochemical calculations.
A- 7
Table A2.
Calculation of time constants used in Figure 2.
A- 9
Table A3.
Constants used in calculating cooling due to ozone
infrared emission.
A-14
1.
INTRODUCTION
The possibility and desirability of understanding the dynamics of
large-scale motions in the upper atmosphere arise as detailed
information,
particularly information on variations in space and time becomes available.
In this regard, the upper stratosphere (which in this work is taken
to be the region from 20 kilometers height to the mesopeak)
and mesosphere,
where the absorption of solar radiation by ozone is
an important process,
play a unique role.
Most of this region lies beyond the range of balloons,
and it is too low for probing by meteor trail or radio reflection techniques.
The only direct observations from this region must be obtained
by means of rockets.
Above 55 kilometers,
temperature cannot be measured
directly except by acoustical methods (see for example Stroud
et al, 1956,
1960, Nordberg and Stroud,
1961), and the alternative procedure of meas-
uring the vertical distribution of density is more difficult than in
regions accessible to satellites.
Nevertheless, wind observations by the
rocket network since 1958 have given us a large store of information
about these regions, at least in the North American region.
The variation
of zonal wind with height, latitude and season is now fairly well known
(Murgatroyd, 1957; Batten, 1961), and it has even been possible to construct
crude synoptic charts at mesospheric levels
(Teweles and Finger, 1962).
An attempt has been made to assess the importance of eddy processes up to
60 kilometers
(Newell, 1963).
-1-
In addition, the mesosphere and the lower thermospheric region
just above it play unique roles as atmospheric chemical laboratories.
Nowhere else are large-scale dynamics and molecular scale processes so
directly and intricately related.
The detailed distributions of trace
constituents must be known before the dynamics can be well understood,
and conversely a knowledge of large-scale circulations is important for
the understanding of a variety of chemical processes (Kellogg, 1961;
Wallace, 1962).
In developing a theory of large-scale circulations, a good procedure is often to start with highly simplified models, and perhaps the
simplest model of all is one in which deviations from zonal symmetry are
either ignored or else are treated parametrically.
attempt to apply such a model to the mesosphere.
This paper is an
It is recognized that
this approach leaves much to be desired; it has proved wholly inadequate
to deal with the dynamics of the troposphere and lower stratosphere (see
for example Starr, 1954).
Nevertheless, there is no a priori theoretical
reason for rejecting such a dynamical scheme in the mesosphere.
The theory
of baroclinic instability which seems to be generally applicable to the
troposphere, as well as to the dishpan experiments, depends crucially on
the temperature gradient along the lower boundary, as Charney and Stern
(1962) have shown.
It is not reasonable to expect that the mesospheric
stability properties depend critically on the surface temperature gradient,
and the latter authors have thus far only been able to demonstrate a
necessary condition for instability in the absence of a boundary temperature
-2-
gradient,
regardless of temperature gradients or lateral wind shears
within a region.
This necessary condition for instability is probably
exceeded in the mesosphere during the winter,
and in fact there is
considerable observational evidence that instabilities do arise in the
winter mesosphere.
On the other hand at other seasons the mesosphere
may well be stable according to this criterion, and indeed, large scalelarge amplitude eddies are notably absent from the summer wind observations.
It is probable,
then,
that a symmetric circulation model may be
applicable to the summer mesosphere, and possibly to the transition
seasons as well.
It has also been shown (Charney and Drazin, 1960) that
it is unlikely that the large-scale quasi-geostrophic motions of the
troposphere penetrate into the mesosphere, except perhaps for brief
periods during the spring and fall.
Another reason for applying a symmetric model to the mesosphere
is that it may lead to conclusions regarding the zonal circulation components which are incompatible with observations.
For example, we are
able to derive a mean dissipation rate which would be required to give
the observed magnitude of the zonal circulation,
furthermore an estimate
of the dissipation rate can be obtained from the observed phase relation
between the zonal flow and the annual component of solar heating.
If
these prove to be incompatible with each other or with what we know about
the physical mechanism of dissipation, we may reject the symmetric circulation concept, at least in the particular latitude, height and season
range where the symmetrically derived zonal circulation is incompatible
with that observed.
-3-
2.
BACKGROUND OF THE PROBLEM
The idea of axially symmetric convection driven by latitudinal
variations in heating is perhaps the oldest in theoretical meteorology.
Attempts to apply this concept quantitatively have ranged from that of
Oberbeck (1888) to Holl (1961).
In another type of approach, Kuo
(1954)
has investigated the free axially symmetric motions which may arise in
a rotating fluid when a horizontal temperature gradient exists.
Such
motions are a form of instability and will not be treated here; instead
it will be assumed that horizontal temperature gradients always lie well
below this instability threshold.
Aside from the instability of the mean state of the troposphere
with respect to quasi-horizontal disturbances (baroclinic instability),
another difficulty, perhaps even more fundamental, attaches to quantitative attempts to deduce the basic state of the troposphere from theoretical considerationsalone.
This is the fact that the state of radiative
equilibrium is one of negative static stability (Manabe and M6hler,
1961);
the observed positive static stability of the troposphere must be a consequence of the motions, yet the static stability exerts a controlling
influence on both the magnitude and qualitative properties of the motions
themselves.
Hence the theoretical problem is an essentially non-linear
one in this sense.
This problem has been discussed in detail by Lorenz,
(1953).
-4-
The upper stratosphere on the other hand is statically stable in
the radiative equilibrium state, and the same may be true even up through
the meoodecline.
In Chapter 4 an attempt to ascertain the radiative
equilibrium state of the mesosphere will be described,
and it is found
that the radiative equilibrium state does indeed appear to be statically
stable.
This result is of fundamental importance, since it means that theoretical studies of these regions, whether based on symmetric or non-symmetric
models may be able, in a first approximation at least,
to treat the vertical
temperature distribution as given mainly by the radiation field with a
relatively small component superimposed by the motions themselves.
This
is the underlying philosophy of the present treatment.
Wilckens
(1962) has reviewed our knowledge of mesospheric circula-
tions emphasizing the attempts to explain the observations in terms of
axially symmetric models; these last may be divided into two classes:
those which emphasize an explanation of dynamical properties,
and those
which emphasize an explanation of the distribution of trace substances.
In the first group, Kellogg and Schilling (1951) proposed that a meridional cross-isobar flow occurs from summer pole to winter pole.
They
utilized wind and temperature measurements available at that time to
deduce the average slopes of pressure surfaces as a function of latitude
and height for the summer and winter seasons.
Assuming that the southern
Hemisphere resembles the northern hemisphere in the corresponding season,
they extended their isobaric surfaces from pole to pole.
Using these
cross-sections they surmised the existence of a frictionally induced
-5-
cross-isobar component from the summer high pressure region to the winter
low pressure region.
Such a pole to pole circulation would be thermally direct, and
would differ radically from the lower atmosphere circulation where transport across the equator is generally regarded as of less significance
than transfer within hemispheres.
Murgatroyd and Goody (1958) have made a calculation of the distribution of radiative heating in the mesosphere.
Their results clearly
show that the principle zonally averaged diabatic heating has a maximum
at the summer pole and a minimum at the winter pole.
The underlying
reason for this difference between the distribution of heating in the
mesosphere and in the troposphere can be explained by the difference
between volume and surface absorption of solar radiation.
The principle
heating agent for the lower atmosphere is absorption of solar radiation
at the ground.
This is roughly proportional to the cosine of the zenith
angle multiplied by the length of day.
In the summer these two factors
oppose and nearly compensate each other with increasing latitude,
normal pole to equator temperature gradient is never reversed.
and the
On the
other hand, the heating produced by absorption in a volume is proportional
only to the length of the day, at least at reasonably small optical depths.
This is why the heating gradient reverses between summer and winter within
a hemisphere.
Murgatroyd and Goody's estimates of the radiative heating were used
by Murgatroyd and Singleton (1961) to deduce the magnitude and distribution
-6-
of meridional wind components which would be required to balance the
heating.
They did not consider momentum balance requirements,
and in
fact their circulation implies a particular distribution of momentum
sources and sinks to give the observed zonal momentum distribution.
Their calculation agreed in essence with the proposal of Kellogg and
Schilling and gave meridional velocity components up to about four meters
per second in the upper mesosphere.
They also had a less intense direct
circulation from equator to pole in the stratosphere.
The complimentary point of view was taken by Haurwitz (1961), who
calculated the
meiional components which would be required to balance
the loss of relative angular momentum,
eddy viscosity.
assuming that this loss is due to
Implied by this circulation is a distribution of eddy
sources and sinks of heat.
Haurwitz's circulation applies only to middle
latitudes and is somewhat more complex appearing than that of Murgatroyd
and Singleton,
but in general shows a direct circulation in the mid-
mesosphere, with indirect circulations at higher and lower levels.
The dryness of mid-latitude stratospheric air led Brewer
(1949) to
postulate a direct stratospheric circulation with rising air over the
equator and descent at middle and high latitudes.
Dobson (1956) pointed
out that such a circulation model could also account for the observed
latitudinal distribution of ozone.
Machta (1959) also suggested that
such a circulation could account for the distribution of fallout of radioactive materials initially deposited in the stratosphere.
Libby and
Palmer (1960) invoked a somewhat similar scheme to account for the distri-
-7-
bution of radioactive material.
More recently, however, Newell
(1961)
has shown that the ozone and radioactivity observations can be explained
by quasi-horizontal eddies in which downward and northward velocities
are positively correlated.
An important paper by Eliassen (1950) has a very considerable
bearing on the present problem.
Eliassen showed that in a rotating
circular vortex which is in hydrostatic and geostrophic balance, every
distribution of momentum and heat sources taken together with the distributions of angular momentum and specific entropy in the vortex implies
a particular meridional circulation, and furthermore the meridional circulation responds instantaneously to changes in sources or in vortex
structure provided these changes are slow enough that the vortex always
remains balanced.
This is because the equations for a balanced symmetric
vortex are just a special case of the general quasi-geostrophic equations
discussed originally by Charney (1948) and Eliassen (1949).
The meridional
component in the symmetric model arises from the divergence required to
maintain the vortex in geostrophic and hydrostatic balance.
momentum sources and heat sources is quite analogous.
The role of
A point source of
heat produces a dipole-like flow toward higher values of specific entropy
at the source point.
A point source of angular momentum produces a dipole-
like flow toward values of higher specific angular momentum at the source
point, and away from such values at other points on the same line of
constant specific angular momentum.
Distortions of the dipole fields
arise from the vortex structure.
In the present problem,
the assumptions of hydrostatic and geostrophic
-8-
balance are made, however Eliassen's approach is more suited as a diagnostic tool when actual sources and sinks of momentum and heat as well
as the zonally averaged vortex structure are known.
To determine both
zonal and meridional components of motion using Eliassen's theory it is
necessary to use an iterative procedure in time and to adapt his equations
to realistic boundary conditions.
The present work is thus an attempt to derive meridional and
zonal circulation components under conditions resembling those in the
mesosphere*, under the following main assumptions:
)
The vortex is in hydrostatic and geostrophic balance.
ii)
The motions are a small perturbation on a basic state
of
statical equilibrium,
which is defined by the lat-
itudinal and zonal mean of the radiative equilibrium
temperature field.
iii)
Sources of angular momentum and entropy due to both
large and small scale eddies result only in effects
resembling eddy viscosity and eddy conductivity.
The validity of assumptions i) and ii) will be explored to some
extent in Chapters 3 and 4.
Assumption iii) can only be verified or
*
rejected after more observational and theoretical work has been done.
Hereafter "tmesosphere" will be used to denote the entire region from
25 to 85 kilometers height.
-9-
The approach to be followed is similar in some respects to that
of Kuo
(1956), but differs in the treatment of dissipation,
in the
consideration of time variations, and in the treatment of static stability as a constant or slowly varying function of height,
an exponentially increasing function.
-10-
rather than
3.
A.
FORMULATION OF THE PROBLEM
The General Equations
Since hydrostatic equilibrium will be assumed throughout,
pressure
but pressure varies by several orders
can be used as vertical coordinate,
of magnitude through the mesosphere so that it is more convenient to use
TF(P)
instead the quantity
H
IT(?) =-
P5
where
(3.1)
is the ratio of pressure to surface pressure,
H
and
H
is
can be expressed in terms of the average
~r.0
H
where
/p
n
the average scale height.
temperature
defined by
R
TOO
(3.2)
is the gas constant for dry air and
the acceleration of
gravity.
IT
where
T
has the following useful properties: in atmospheric regions
is close to
,
-F0
TT
differences will approximate height
differences; if temperature differences are small everywhere,
iT
resembles
the height, and the main advantage of pressure coordinates in such an
atmosphere -
the elimination of the need to deal with more than one inde-
pendent thermodynamic variable -
dP
dT
P
H
_
e
is preserved.
We also have
-r/
(3.3)
H
-11-
Dt
p
where
W
The
PDt
P
nt
--
mt
is the density, so that as long as
(3.4)
~T~
is close to
closely resembles the vertical particle velocity,
TT'
D 7/
system was introduced by Eliassen (1949).
The equations of motion,
hydrostatic equilibrium, continuity, and
energy may now be written
L
;T. "-sP
Dt
(z +
D t.r
-k
(3.5)
r cosCOS
)s LL51
-L
P
COS
d
_
(3.6)
RT
T
H
It
+
4F
-
(3.7)
W
--
re cos
a
(r cos cp) + e TT/H
.r/H
a-
(3.8)
(3.9)
-
c3u
r COS
T
with, as usual
3t
at-
Le
(3f
L4
recos?-
-12-
a
J
C9 Tr
(3.10)
i
In these equations
"
is time,
longitude,
and
its rotation rate, and
is the earth's radius, SL-
r.
X
f
latitude,
Cp
is the
,M
specific heat of dry air at constant pressure.
LL
dependent variables are the zonal velocity
Lr
,
the geopotential height of a
the temperature
T
.
Tr
-W~
In addition to
,
the
the meridional velocity
(or pressure) surface
,
and
is diabatic heating per unit mass and per
unit time.
We now take an ensemble average of each term in these equations
where the particular type of ensemble may be thought of as a large collection of observations taken at the same point in (
X ,
and at the same time of year but at all times of day.
f,
Tr
) space
In this way the
slow annual variation is retained while the diurnal, semidiurnal and
irregular variations all become fluctuations about the mean.
aging operation is identified by the over-bar (_),
from it by the prime (
)'.
This aver-
and fluctuations
We also introduce the zonal average, iden-
tified by the curly over-bar
f(
)dA
(3.11)
*
S-
and indicate its fluctuations by a star (
)
The successive application
of the ensemble and zonal averages will be denoted by means of the caret
(
).
The zonal average automatically satisfies the Reynolds postulates
(Kampe de Feriet, 1951);
the ensemble average is assumed to do so.
quently, if we first take the ensemble average of equations
(3.9), and then the zonal average, we obtain
-13-
Conse-
(3.5) through
r
A
A
u
~Jrr 4-
Ll
C)
AS
n
-
a
-* +CO
(3.12)
dH
e Tr/
A
(
4jw -*
GL A' ' )
A
+
cr
a
-
diV
4+ ( 2. n.+
R*
(3.13)
--
+-
A
Lt
r. Cas9
+o
Ctr
T
e
- F(p, rr ft)
S4-r 1-)CsJ
r-e c 0 4
-- e
J
wf
w tU
e -A/
e-r
.'
-
E~
7rr t
RH
~TT
H
r. Cos
-- ( U,cos T)
a<p
+ eTr/H
c) T
(
-
(3.14)
(3.15)
-14-
L
A
A
re C Cos
CP
(3.16)
A
J Le
CP
C
( P ) T.,)
The problem of the symmetric circulation is to regard these equations as a closed set to be solved with suitable boundary conditions for
the doubly averaged quantities appearing on the left-hand sides.
The
eddy fluxes of heat and momentum as well as the diabatic heating act as
forcing functions for the symmetric components (Eliassen and Kleinschmidt,
1957, p.
B.
144; Saltzman, 1961).
Scale Analysis and Formulation of the Linearized Problem.
We assume that the averaged temperature field can be decomposed in
the following way
-
)~, TT, --(3.17)
T~_
where
T
,
q
( Tr~
which is uniform in all pressure surfaces,
is the temperature
distribution of the basic statical equilibrium state.
time dependent heating which varies within
to (3.17), we split
into
-15-
1t~
surfaces.
arises from
Corresponding
A
+
pT.)
(3.18)
is defined by
where
RTO
__
(3.19)
H
c)Tr
and we assume that
(TT=o)
=
Const
It is convenient to introduce the following nondimensional
variables:
;z
7T/
Y~ SiCT
Cos
A
2 -ft 1-e
A
LUT-
HodL'
(3.20)
T-
R
D1e
-
-
7 P,
Cr
-16-
L
cos p
F
coscp
r. er
cr
and we also define the static stability by the relation
New quantities appearing on the right-hand sides of these equations are
(Y
,
the annual frequency,
of the heating and motions.
and
D
the characteristic vertical scale
Two height scales
D
have been
H
and
introduced in order to contrast certain features of mesospheric motions
with those of the troposphere.
in the mesosphere
D
Murgatroyd and Goody,
"
3H
1958).
In the latter region
D
'\
H
, whereas
(see, for example, Batten, 1961;
Equations
(3.12)-(3.16) may now be put
into the non-dimensional forms
C) T'
1
'/2.c)2(3.
--t +
n
L0
(3.22)
+
LD2_
ro
-17-
(3.23)
-fl
+ nr
-
R
where
7Z.
-(3.24)
*,
7-T
W--
(3.25)
/r-nJ(rje
can be assigned orders of mag-
and
The operators
nitude unity since we are dealing with processes of annual and global
scales;
also has order of magnitude one by definition.
D > H
,
Because
equation (3.24) then shows that
ja)
Ur
(3.26)
We note also that
7r
where
U
l(u
U
R
~~0
(3.27)
is a characteristic (dimensional) zonal velocity, and
is the Rossby number for the problem.
-18-
R.
It is possible to place upper limits on
and
45
if we
assume
i)
that deviations of zonal and meridional winds from the
Re
not greater than
ii)
that the ratio of vertical eddy to horizontal eddy velocity
R
is no larger than
iii)
,
means are no larger than the mean zonal wind itself, i.e.
that the scale of variations of the eddy fluxes is no
smaller than
r
in the horizontal and
H
in the vertical.
These assumptions are quite reasonable throughout the mesosphere, but may
be violated by both internal gravity waves and tides above about 80 kilometers (Hines, 1963).
r1_N oT
0
(R
<
When they are valid,
2
(
OV07
(V
Ra
o
(3.28)
and
(3.29)
If the Rossby number is of order 1/10,
may be a very important term
is dominated by
in equation (3.21), but
Furthermore,
.5
72
in
(3.22).
if the inertial terms in (3.22) were to be comparable with
the Coriolis term, the dimensional meridional velocity would have to be
about 60 m/s (for
R0
= 1/10).
tainly much less than this,
Since the meridional velocity is cer-
the only term which can balance the Coriolis
-19-
term is the term involving
.
In other words,
the mean zonal flow
must be in approximate geostrophic balance except perhaps very close to
the equator, provided the above assumptions are satisfied and the Rossby
number is 1/10 or less.
Equation (3.28) gives what is probably a high
upper limit to the ageostrophic force in this case,
since the limit
corresponds to perfect correlation between eddy velocity components.
We will assume that the zonal motion is in exact geostrophic
balance,
and take as the complete expression for this
Y(3.30)
yZ)
Combining this with equation (3.23) gives the thermal wind equation,
+
_Y_
L (-YZ)
I
a -2
D
_Y
d
H
Y
a
(3.31)
~
from which it follows that
H 7(3.32)
We now examine the validity of linearizing equations
(3.25)
Since the motion and perturbation temperature fields are
produced by differential heating,
proaches zero.
they will approach zero as
This suggests the introduction of a parameter,
%
ap-
,
and (3.31).
(3.21),
such that
I
-20-
.<
(3.33)
and expansion of the dependent variables in power series in
E
.
For
the order of magnitude analysis we assume that
-e
0-
(3.3 4)
(3.3:))
and
and
T
.
are characteristic dissipation times for
(
where
Then to first order in
Tn
6
*O
C4T
(3.36)
CY0
(3.37)
d n,
--- Y
- W-
= 0
Dc~
-+
-A
(3.38)
T
+
R,
r
(3.39)
Second order terms give
(3.40)
-21-
L
&
H
7
Y
ay9
g
(3.41)
-r/
=
-
D C)
-
(3.42)
1T,
+.~~~~=n,
(
+
(3.43)
Using the fact that all derivatives have order of magnitude unity and
assuming that neither (i
divided by
)
wr
(
-4)
are greater than one year
, we can derive orders of magnitude for the solutions
2 qT
of these equations.
We find
(3.44)
H
-I
.g
i: . H
H
(3.45)
The most important term on the left-side of the second order set is
IJ
6
Y
ru
nTYfn
I
rj
(3.46)
H
A*
-22-
I* H
It follows that
7.-
(3.47)
27.
(3.48)
Since
E + T).
+
'
-
-I
(3.49)
-
(3.50)
'
n,
71
+
-
n G + 7l i
7n
then
f
41
Ro i
D
-+I
SH
-
,Y)
(3.51)
+-
* --
The applicability of the expansion in
6
RS
4
-
(3.52)
and the detailed analysis
of the linearized equations clearly depends on the size of the second term
in the square brackets; more specifically,
-23-
it depends on whether
4* D
-
___
The parameters
or
radiative heating field.
D
field;
'
R5
The time scales
eddy source terms
,
(3.53)
can all be derived from the
is the frequency of oscillation of this
the vertical scale and
statically stable,
e
) , and
,
in detail in the next chapter.
bution.
IfJ
D
4;
C
the amplitude will be discussed
If the radiative equilibrium state is
also depends on the radiative heating distriand
depend largely on the
but if the effect of the eddy source terms is prima-
rily dissipative, these time scales can be derived from the observed
phase relation between zonal wind and radiative heating.
When these parameters are evaluated in this way, it will be shown
that the ratio of the left to right-hand sides of equation (3.53) is not
greater than about 1/3.
This is not usually regarded as sufficiently
small to justify linearization, but it does suggest convergence of the
series, and that the solution of the linearized equations should give
at least a rough approximation to the true solution.
The linearized
*
equations will thus form the major focus of the remainder of the thesis.
But it will be shown in the next chapter that damping by radiative and
photochemical effects may also be important.
-24-
C.
Energy Equations
For completeness,
the energy equations corresponding to a rotating
symmetric vortex in geostrophic and hydrostatic balance will be given.
These can be written down for both the non-linear and the linearized
equations.
For the non-linear case the energy equations have been
derived by Eliassen (1950), but for a different coordinate system.
We shall assume no energy flux through upper and lower bounding
pressure surfaces of the mesosphere, and make use of the dimensional
equations.
In the non-linear case these are
(3.12),
(3.14),
(3.15),
and (3.16) together with
U
AT
r,Cos~)L.L~
+
=0
(3.54)
-r/HA
-Tr/i f
Multiplying (3.12) and
e
(3.54) by
LL
,
e
and
L~
respec-
tively, adding, making use of the continuity equation (3.15) and integrating
over the entire mesosphere gives the kinetic energy equation
- e
-
dV
- JC/HA
^
dV
v
.
is the volume element in (
dV =
e
COS C> d /
cdTrW
-25-
(3.55)
0
~
f
X
.
where
FU
,
Tr
) - space:
(3.56)
After some further manipulation involving the continuity equation and
the hydrostatic equation (3.14),
c)J
(3.55) takes the form
.Tr/H ^ 2-
-e/H ^
2
fe-
Tr/)4
V
H(3.57)
A
FU.dV
The potential-internal energy equation is obtained by multiplying (3.16)
by
CP
and integrating,
H
V
.CP e
d
then
fHRT/
fcP e f~C dv *f e
MIN/
H
-Tr/)4
1 A
A
V
(3.58)
'dV
We note only two unusual features of these equations -
the quantity
plays the role of density, and the kinetic energy does not
A
include any contribution from the
Lr
A
or
W
components.
The latter
feature of the geostrophic approximation has been pointed out by Eliassen
(1950).
heating.
Omitted from equation (3.58) is a term arising from frictional
Although ordinarily negligible in the lower atmosphere,
term might be significant in the upper mesosphere (Hines, 1963).
The linearized dimensional geostrophic equations are
ir
Lr
F S Q(3.59)
-26-
this
A
JLL
Sir)
LP
=
C)
0
(3.60)
R
P
H
c Tr
-Tr
I
4
(3.61)
I1
rCPscj
_
-H
A
--- P
8f
1
) =
(3.62)
A
R7
C, H
C
CP
(3.63)
where
-(
S(7T)
+
P,
H.
o
-o
-T )r
in a statically stable atmosphere.
;> 0
(3.64)
The kinetic energy equation is
obtained by a procedure exactly analogous to the non-linear case.
The
equation is
TT/H ^
C et
2
L
f
v=-
A
-Tr/
r
r
v-
f
_7/Hdv
e
ua
(3.65)
On the other hand the potential-internal energy equation is obtained by
-27-
j
multiplying
C TP
(3.63) by
and integrating to give
70S
-
I T/HCTP
-WH
(3.66)
-
f&e-,r/H pC
S
,,
T
T/H
e
2Ta
volume in (
is the available potential energy per unit
-
Clearly
TO --
X
,
S
p
,
~1T
) space (Lorenz, 1955).
-28-
4.
A.
RADIATIVE EQUILIBRIUM AND HEATING
General Remarks
The preceeding analysis has raised the following two questions
which require an investigation of radiative heating in the mesosphere:
How should the basic state,
T(a), be defined,
*
i)
and is such a state statically stable?
ii)
What is the distribution of
,
and does its magnitude
satisfy the condition for linearization (3.53)?
To these should be added a third question:
iii)
How do the motions influence the net heating rate
through redistributing the temperature?
To make these questions more concrete, a somewhat simplified model
will be considered.
Murgatroyd and Goody (1958) have pointed out that
the principle radiative heat balance components between 30 and 90 kilometers are absorption of solar energy by ozone,
s
,
and emission of
infrared radiation in the 15 micron bands of carbon dioxide,
If
r-
is the net heating per unit mass,
4
7-)] 4-
*
[
Unless otherwise specified, all quantities in this chapter are
dimensional and
Z
denotes the geopotential height.
-
29--
(4.1)
where
over
symbolizes the linear operation of weighted integration
Z
,
and
Bv(T) is the Planck function corresponding to 15
.
T-
microns and the temperature
The heating
7,
is a function
of temperature as well as position, so that the equilibrium temperature
7-e
is defined by the implicit relation
4L B,(r )]
' (?e )
+
Now we suppose that the temperature
<f T
=
(4.2)
0
T
differs from
7-
by an amount
which is small in the sense that
s (Tr ) +-
s (T
(T~
(4.3)
and
B,()=B,
+
are valid approximations.
(4.4)
or)T
JT;
Under these conditions
d(1pW)
-][+js
)
9
(4.5)
7-
T=T7C,.
'n
if
T = T -+ -r
,
dT - -I)
Te- 7-
-30-
so that
+
In particular,
F-7-=
(T; ~Te) +T p
T
-
(4.6)
Clearly a good way of defining
7
e
.
-F
is as the horizontal average of
With this definition, the first curly-bracketed term in (4.6)
depends only on
7e
which in turn depends only on physical properties
of the atmospheric gases, the solar spectrum, the earth-sun geometry,
and temperatures outside of the region of interest.
This term can
therefore be called the 'texternal heating" and identified with the
forcing function
00e
in the analysis of Chapter 3.
term depends linearly on
Tp
The second bracketed
J it should therefore by incorporated
into the homogeneous part of the dynamic equations and may be called
the "internal heating.?
This formulation is convenient since it shows how the truly external
librium temperature and a statical
equilibrium temperature
1 (
)
forcing function is related to the difference between the radiative equi-
and it displays the motion-produced radiative heating as a separate term.
Its validity depends on both (
7
- T
) and
Tph
being small enough
that (4.3) and (4.4) are satisfied; it does not depend on the assumption
of only a single infrared band but does require that concentrations of
radiatively active gases are unchanged by the motions.
Examination of
the Planck function shows that (4.4) is a good approximation for
1 T
50 0 K when
T,
is between 200 and 3000K.
The valid range
of (4.3) will be considered in the next section.
Fortunately the internal heating can be easily incorporated into
the dynamical model.
imation to
Zf,
Murgatroyd and Goody have shown that a fair approxis
-31-
j
I
+4 H-
+ 90 km
-
50 km
X 80 km
'
45 km
ri 65 km
V
40 km
o 55 km
x
dT
1.010 - .02214
BL
dt
0
79.2 (omitting 90 km da ta)
r
x
Q
x
V
C2
ty
-4
+
0,
0A
.
IN,
.0
-8
-12 1-
+
+
0
-16 1-
erg/cm-sec
)
(C0 2
I
100
Fig. 1.
I
200
I
I
I
300
400
500
Cooling rate due to CO2 versus Planck function at 15 microns.
the parameters
6
and
-4
being functions of
0
only;
this is
because radiation to space is more important than exchange of radiation
between layers in the mesosphere, and is only valid for temperature
profiles of approximately similar shape.
results for all heights.
Figure 1 is a replot of their
Linear regression of all points except those
at 90 kilometers gives the relation
1.010 -
.02214 BW
*
=
with a correlation coefficient of 79.2
.
Combining this with the follow-
ing linear approximation to the Planck function for the temperature range
=
2.055
(-r
-
145.8)
,
200 to 3000 K
gives a "best" linear relation between the heating rate and temperature:
o
-*
*(4.8)
,
=
5
7.68xlO 5 , and
-7
= 5.26x10
.
with
With these ideas about the relationship between external heating,
internal heating and the radiative equilibrium temperature as a background,
we shall proceed to a detailed examination of radiative equilibrium and
*
heating rates in the mesosphere.
It should be noted that this high correlation arises to same extent from
clustering with respect to temperature of several calculations made at
. would be more
A considerably larger value of
the same height.
appropriate for altitudes above 55 kilometers.
-33-
Radiative equilibrium and heating rate studies have been carried
out previously by a number of authors for stratospheric levels.
works such as those of Karandikar
Early
(1946) and Gowan (1947) suffered from
inadequate knowledge of the solar spectrum but displayed the major
qualitative features of the stratospheric temperature distribution and
heating.
Brooks
(1958) has examined
in the lower stratosphere,
CO
cooling and ozone heating
but the most complete study of radiative heating
components for the region below 55 kilometers is that of Ohring (1958).
He found a net deficit in seasonal and latitudinal averages of heating
throughout the lower part of this region.
Recently Manabe and Mhler
(1961) have presented computations of heating rates as well as equilibrium
temperatures for the lower stratosphere.
agreement with those of Ohring.
Plass
Their findings are in general
(1956) has computed cooling rates
due to the 9.6 micron band of ozone from the ground to 65 kilometers.
The only attempt to calculate net heating rates due to all the
important components throughout the mesosphere is the work of Murgatroyd
and Goody.
In addition to ozone heating and
COm
cooling, they incor-
porated cooling by the 9.6 micron band of ozone by making use of Plass'
results.
All of these studies have one feature in common -
a distribution
of ozone based either on the assumption of photochemical equilibrium or
on observations,
or a combination of the two was used.
Conversely,
attempts to calculate ozone concentrations theoretically have relied on
observed temperatures and air densities
-34-
(Craig, 1950; Dittsch, 1961;
Johnson et al., 1951; Paetzold, 1961).
The two problems are not inde-
pendent, however; the dependence of heating rates and hence of equilibrium temperatures on ozone concentrations is obvious, but it is also
known that the ozone concentration depends on the temperature (Craig and
Ohring, 1958).
The photochemical and radiative aspects of the problem
will therefore be considered simultaneously.
B.
Photochemical Considerations
Only reactions involving oxygen allotropes and non-reacting third
bodies will be considered.
hydrogen, hydroxyl,
The neglect of reactions involving atomic
and various other compounds of hydrogen and oxygen
may be serious in the upper mesosphere, but the occurrence of OH nightglow should be a good indicator of these reactions.
There is evidence
(Wallace, 1962) that nearly all the OH night-glow originates above 75
kilometers.
If this is.true, it should be safe to assume that these
reactions do not exert a controlling influence on ozone concentrations
below that level.
With this restriction,
the well-known reactions determining the
ozone concentration are given in Table 1.
In these expressions
JJ
is
the dissociation rate of molecular oxygen per molecule per second when
c,
( 1j)
is the absorption cross section of molecular oxygen in
ej(
cm 2/molecule and
dissociation;
for ozone,
100
J3
i)
is the quantum yield of primary photo,
and
e3
are the corresponding quantities
is the photon flux outside the atmosphere per cm2 and
-35-
'A
Reaction
0, +
v
Rate
0
J2 4=ex o2
Iov
exP(-x-cX3
X3) J'
V >'H 300 A
o0÷+
X= .1
O+ M
0
+ 03 -+ 204 13 ~
(
Table 1.
-- o'
3
x3 )
, =M.0.7 X 10m/
-+Oz +M
0 +02+ M
e3 +XI 1,11 e=3 (exP (~o;x
v > 8GXO A
0 0, -+
30
.5-
33
c T's
exp (-30Z5/--r)
-x ?o c& M/s
)
03 +
Reactions determining ozone and atomic oxygen concentrations.
d
second per wave number,
L)
is the wave number (cm
1) and
X
)< and
are respectively the total numbers of oxygen and ozone molecules between
the point in question and the sun and depend on the zenith angle as well
as height.
Details of the constants used will be found in appendix 1;
about the values of the reaction coefficients
and
-
i
,
it should be particularly noted that there is considerable uncertainty
In the following analysis it will be assumed that photochemical
O.
changes in
concentration are completely negligible, so that
always remains perfectly mixed with neutral molecules
in the ratio 21:79.
0
(nitrogen and argon)
It will also be assumed that all molecules are
equally effective as third bodies.
For our purposes, the most convenient way of writing the rate equations which result from the reactions in Table 1 is the following
d
where
of
0,
7),
,
,
0,,,
J3
n.
,
O
13
+n
and
n
f,,,
n
(4.10)
are respectively the concentrations
and third bodies.
Equation
(4.9) gives the rate
of change of the total number of odd oxygen atoms, i.e.: the total number
of atoms in the form of ozone or available to form ozone, while equation
(4.10) may be thought of as expressing the variation of distribution of
-37-
odd oxygen atoms between ozone and atomic oxygen.
.J
useful because, in the daytime,
This point of view is
is so large that equilibrium in
equation (4.10) is always achieved in a short time,
i.e. from a few
minutes up to about an hour depending on height and zenith angle.
>> -
-11
Since
at all heights, this equilibrium can be expressed
,
by the relation
42
T)3
-n?n ,
.
J
(4.11)
3
to a high degree of accuracy.
In contrast the equilibrium of total odd
oxygen concentration may be achieved in a much longer time.
To show this
we assume that (4.11) is exactly satisfied in the daytime, then (4.9)
becomes
(-
'..
+
Z
where
Equilibrium will be achieved when
-n,
T
Letting
-38-
ie
-2 43(4.13)
(4.12)
/
I
)
//
Craig
75
x
zenith angle 67*
o
zenith angle 0*
Wallace
- --
J- 1
65
/
*
/
55
tim
(das)/
-
charctersti
45
zenith angle 0*
4
-4
35
25
10
I
Fig. 2.
102
I
10 1
I
1
10 1
10 2
I
Characteristic times for equilibrium in the ozone problem.
10 3
10
5
10.
--I
and
3
and assuming
J2
-
L
- ,
(4.14)
is independent of time leads to the solution
4:J(4.15)
is an initial value of
where
.
Clearly
A
is the appropriate
time constant for the odd oxygen atoms, and because changes in the odd
oxygen concentration are much slower than the adjustment between atomic
oxygen and ozone,
A
is the crucial time constant for restoration to
Wallace
equilibrium following any disturbance in concentrations.
(1962)
has previously derived a solution similar to (4.15) for conditions prevailing above 65 kilometers, while Craig
(1950) has obtained an approx-
imate solution resembling (4.15) which is valid below 45 kilometers.
Equations
(4.14) and (4.15) should be valid throughout the stratosphere
and mesosphere provided the reactions in Table 1 are the only important
ones.
-I
-j
Figure 2 shows the two time scales
X
and
J3
corresponding
to a zenith angle of 600 and standard atmosphere temperatures.
shown are time constants due to Craig and due to Wallace.
-40-
Also
The discrepancy
between the present result and that of Wallace is significant.
It is
believed to be due to the fact that Wallace's calculation was based on
the work of Bates and Nicolet
temperature of 45000K near
the Schumann-Runge bands.
(1950) who assumed a solar blackbody
1) = 50000 cm
1
and assumed
eL=
0 for
Recent evidence indicates a solar black-body
temperature near 5000 K at 50000 cm 1 (Detwiler et al, 1961) and significant predissociation in the Schumann-Runge bands (Wilkinson and Mulliken,
1957).
The reader is referred to Appendix 1 for details on the data used
in this calculation.
A particularly important disturbance of ozone and atomic oxygen
concentrations occurs daily, when the sun sets.
At this time atomic
oxygen disappears by recombination -
very rapidly below 70 kilometers,
and very slowly above 80 kilometers.
Some of this atomic oxygen forms
ozone via the three body reaction involving two oxygen atoms or via the
two-body reaction.
The latter process is greatly enhanced at upper levels
at night due to the increase in ozone, and may produce very large decreases
in the total concentration of odd oxygen molecules.
The degree to which
these losses can be restored during the day depends on
A .
Below
about 60 kilometers ozone concentrations can be expected to approach
equilibrium values rather early in the day, but a few kilometers higher
ozone may remain significantly below its equilibrium concentration throughout the day.
For this reason, it may be necessary to seek the equilibrium
temperature as the solution of a time dependent problem in which these
nighttime variations are specifically taken into account.
-41-
We are now in a position to discuss the dependence of
temperature.
on the
The solar energy absorbed in any thin layer may be affected
both through weak temperature dependence of the absorption coefficient,
and changes in the optical depth of ozone,
but the most important effect
of temperature is in changing the in situ ozone concentration through
.
the temperature dependence of
/
AL43 r'77
4ro
Below about 65 kilometers
so that the equilibrium ozone concentration is,
.J3
from (4.11) and (4.13),
~~-
71 2_
(4.16)
T
Neglecting the effect of temperature on
S',
n3,
Above 65 kilometers
hence of
7
dependence of
,
(
=
43n,
J3 , we then have
e.
(4.17)
becomes rapidly independent of
-3
and
diurnal variations may complicate the picture, but the
on temperature at the high levels will certainly
S
not be greater than that indicated by
It follows that
151.5
(4.18)
-
s
(c~ =
cFTTe
(4.17).
T
and the next largest term in the expansion of
-42-
)
713e
, (-Q-+ c)
is
PPP
FT
-fT
2.
d
Near the mesopeak, where these terms are largest,
(4.19)
CP
per second and
15/2.. T
.e_ 02
c-
(
may then be as high as 2x10
per second, which may
be several times as important a damping effect as infrared radiation.
Equation
(4.19) imposes another type of linearity condition on the
problem,
independent of
(3.53).
The validity of the separation of the
heating function into an inhomogeneous part plus a linear term depends
on the size of the factor multiplying
in (4.19).
Using typical mesopeak values, we obtain
01
(
-
-
(
Y2-
so that the error in linearizing the heating function will be of the order
of 50% for
J XT
P 500K.
The important stabilizing effect of
previously by Craig and Ohring (1958).
43
has been emphasized
The present discussion is intended
to clarify the role of this effect in the present dynamical framework,
-43-
but is subject to the following caution: the degree of temperature
dependence of
have used a
k3
43
is not well established.
Although most workers
close to that given by Benson and Axworthy (1957),
there is some evidence (Leighton et al,
1959) that this may not be
the value applicable to kinetic processes.
temperature dependence, however,
Even with a much weaker
it appears that this particular damping
effect would be an extremely important one.
C.
The Equilibrium Computations
The procedure used in these computations was to assume a temperature distribution,
then use the approximate equilibrium equations
(4.11)
and (4.13) with the reaction rates given in Table 1 to deduce the concentrations of atomic oxygen and ozone at each level starting from the
highest level
(85 kilometers) and working down.
Concentrations and
amounts of energy absorbed were computed in this way for various zenith
angles corresponding to different times of day for each latitude.
total energy absorbed during the day,
over the day using Simpson's Rule.
the 9.6 micron band of ozone,
13.s
The
, was obtained by integrating
To this was added the heating due t6
3
.
The latter quantity was computed
by application of an empirical linear relation, derived from
Plass'
work,
between in situ heating rate and the 9.6 micron band Planck function at
the assumed temperature; this method is the same as that used by
Murgatroyd and Goody (see Appendix 1).
-44-
Cooling due to carbon dioxide
was calculated using Curtis' matrix method
and Goody,
1958).
A new temperature was then computed by solving for
in the matrix equation
B2wj
R.
where the
L
(Curtis, 1956; Murgatroyd
and
Bv
RP.
.J
-
frSi
-
(4.20)
3L
are the Curtis matrix coefficients,
correspond to individual heights.
and the indices
After inverting the
resulting set of Planck functions to find the new temperature at each
level,
the process was repeated,
and the entire procedure continued
until subsequent temperatures converged to within 10K at all heights.
Apparently because of the very strong restoring effect of
3
this procedure led to an oscillation in succeeding temperatures unless
some damping was introduced.
This was done by choosing at each step
a new temperature which was between the computed new temperature and
the old temperature but slightly closer to the new temperature.
Calculations using this procedure were carried out at latitudes 00
0
o
30 , 60 ,
o
and 90
for December 21st, and 30 , 60
0
o
and 90
for June 21st.
In addition to the assumptions already mentioned, it was assumed that
the temperatures below 22 kilometers and above 85 kilometers as well as
the density at 22 kilometers corresponded to the 1959 ARDC Model Atmos1959).
It was also assumed that the total number
*
phere (Minzner, et al,
The author is indebted to Dr.
of the Curtis Matrices.
L.D. Kaplan for kindly providing a copy
-45-
-J
80
210
20
70
00
1801(6
24023
250
260
2900
b1-
90
70290
300J
L''
90
Fig. 3.
0
(summer)
50
Equilibrium temperatures,
JI
latitude
Iv
*K, computed by matrix method.
30
DU
/ U
(winter)
7v
8.o
8.0
7.5
7.5
7.o
.7.0
-80
9.5
-70
10.0
-60
10.5
4-i
-2J
tiD
-50
11.0..
11.5
-40
12.0
i
12.5
-30
13.0
I
20
90
I
70
50
(summer)
Fig. 4.
30
10
latitude
10
30
(winter)
Log (base 10) ozone concentration, molecules per cc, computed by matrix method.
50
70
of oxygen molecules above 85 kilometers was equal to the number density
multiplied by the scale height at 85 kilometers; the total number of
ozone molecules above 85 kilometers was assumed to be equal to the ozone
number density multiplied by one half of the density scale height, but
85 kilometers is essentially at zero optical depth for ozone in any case.
Results for the equilibrium temperature and midday ozone concentrations are given in Figures 3 and 4.
D.
The Time Dependent Problem
Since diurnal variations may have an important effect in the
mesosphere, the marching problem was also attacked in an attempt to
obtain mean daily temperatures which would be obtained
from arbitrary initial conditions.
asymptotically
For the reason discussed in Section B,
it is possible that this type of calculation could yield significantly
lower temperatures in the upper mesosphere.
In the notation of the previous section, the marching problem is
to solve numerically
df
where
C
C?=i1
(4.21)
is the heat of recombination at the iih level; it can be
either positive or negative depending on whether more molecules are
dissociating or recombining.
The solar energy absorbed,
-48-
,
is a
function of
equations,
and the latter is determined from the photochemical rate
-n3
(4.10) together with
dtt
-
, -
3z
3
z.r
T,+nJ4.
A problem arises in the numerical integration of
3
(4.22)
(4.10) and (4.22)
since some of the reactions are slow or nonexistent at certain altitudes
or times of day but are extremely rapid at other altitudes or times of
day.
At some altitudes the first order reactions are dominant, while at
other heights the second order (non-linear) reactions are comparable to
the first order reactions.
To solve these equations numerically, the
procedure of linearizing the equations about initial values at each time
step has been adopted.
The exact solution to the linearized equations is
then computed, and the solution for the next time step is obtained by
linearizing about the new initial values.
The time step chosen should
then be small enough that changes in
and
-n,
are small within
71,
each time step at all levels where the second order reactions are not
dominated by first order reactions.
We write the equations pertaining
to daytime variations in the linearized forms
/
-49-
2.
(4.24)
o
where the subscript
dominates
-03
refers to initial values.
all other bracketed terms in these equations at all levels even for low
zenith angle.
The large time increment of one hour has therefore been
used during the daytime, except during the periods within an hour of
sunrise and sunset when ten minutes was used.
compared with the smallest expected value of (
Ten minutes is short
-
responds to the most troublesome non-linear term.
7,o
)1
which cor-
The term arising
reaction is altogether negligible below 75 kilometers,
from the
and is only comparable to
OT)J
above about 83 kilometers.
At these
heights all recombination reactions have a time scale considerably
greater than ten minutes.
At night,
ten minute intervals were used
during the first and last few hours, but some preliminary experiments
indicated that a much larger time interval would be adequate during
most of the night,
i.e. after rapid changes taking place during the
first two hours or so have neared completion.
Consequently one hour
intervals were used during the remainder of the night.
the linearized equations are,
for daytime:
A
nJ.
~ e)f
2~~
~
-k
n
300-b -)O(.5
A2
2
-50-
(o
Solutions to
[(Az~
P Z+ 7., cL-n,. 'A
)
3 ( \-1 'I
-
_-n.l
X_
(AAz
At
(4.26)
L
where
( CL - 0" ) , -f-- 4
2-.
CL -
]
112-
(4.27)
(6~7~-3
4- 4
~C
*
-X
(4.28)
and
(4.29)
303+
(4.30)
-n30
-J3
In all the experiments,
and
expressions for
was large enough that the radical in the
N
was always real, even when the solar
0
zenith angle was greater than 90
T) =
) w e6
.
The nighttime linear solutions are
(4.31)
0
-51-
CL U(4.32)
where
(4.33)
and
(4.34)
0-V
Twilight radiation (direct solar radiation for zenith angles greater
than 90 ) was considered using the method of Pressman (1954).
The assump0
tions regarding temperatures below 22 kilometers and air masses of
and
aboye 85 kilometers were the same as those used in the equilibrium
03
The energy gained through dissociation in the time interval
calculations.
3t
was calculated from the formula
(
=-
34.35)
where
2
-
is Planck's constant,
C
the velocity of light,
V/
and
are the dissociation frequencies of oxygen and ozone respectively,
and
Jn, ,
and
cfT1. are the changes in r,
and
-n3
during
c5t
Integrations were carried out for December 21st at latitudes 00 300and
600
and for 450 on June 21 using the equilibrium results as initial
conditions.
Integrations were carried out for 10 days at 450,
-52-
12 days
0
,
and 18 days at 30
and 60
.
at
In each of these cases the 24 hour temperature changes on the last
day did not exceed 0.25 K at any height below 82 kilometers.
Above 82
kilometers 24 hour temperature changes up to 0.8 K occurred on the last
day indicating assymptotic limiting temperatures a few degrees lower
than were obtained at these highest levels.
Figure 5 shows the results of an experiment to test the adequacy
of the time intervals used.
It shows the variations of ozone concentra-
tion during a 24 hour period at various heights on the equator using
day,
night, and twilight time increments half as long as those used in the
other experiments.
The open circles are results for the longer time
increments and identical initial conditions.
The agreement is very good
except for some slight discrepancy at 49 kilometers.
Even at that level
the daytime concentrations are in very good agreement.
Some of the daily temperature values for the equator are shown in
Figure 6.
Higher temperatures occurred at the highest levels on the
first two days as a result of the release of recombination energy.
There-
after the temperature fell in a more or less exponentially decaying
fashion, apparently approaching an asymptotic value near that reached on
the last day.
check,
This behavior was characteristic of all latitudes.
As a
an experiment was run for ten days at the equator, starting with
the tropical standard atmosphere of Cole and Kantor (1963) as initial
temperature distribution.
The resulting temperatures,
day is also shown in Figure 6,
of which the 10th
appeared to approach the same asymptotic
values.
-53-
S85 kms.
85kms
$
79
-9
5 1
73
67
07
55
55
o0
55
0mo
0
6
D
61:
49
-73---
1
2
3
4
5
6
7
Fig. 5.
8
9
10
11
12
Hours
Diurnal Variation of 03 , Lot. 0 , Dec. 21 .
13
14
After Noon
Initial
15
6
Conditions t Equilibrium
0~
17
I8
I9
20
21
22
23
24
day 12
day 0
90
I
80
tropica
standar
atmosphe re
(day 10
/
day 1
d
tropical
standard
atmosphere (day 0)
70 1-
60 k
50
]
.-W
10
40 k
30
day I
day 12
20 1-
Sday 0
temperature,
180
Fig. 6.
200
I
.
-,
*K
I
220
240
260
Time integrations of temperature at the equator.
-55-
I
280
I
300
Some details of the ozone distribution are shown in Figure 7.
The
depleting effect of the nighttime losses shows up clearly above 65 kilometers.
The sunrise concentrations are down by a factor of about 6 from
the equilibrium values, while the sunset concentrations are down by as
much as a factor of 3.
As Figure 2 would suggest,
the final ozone con-
centrations are established in four days or less at all altitudes below
80 kilometers.
The slowness of the approach to the asymptotic steady
state temperatures is due to the slow radiative processes rather than
the photochemistry.
Figure 8 shows the noon temperatures obtained by the marching
process on the last day.
angle at the poles,
Since there are no diurnal variations in zenith
the equilibrium values were used at these points.
The differences between Figures 8 and 3 are small except above 65 kilometers where the marching problem temperatures are somewhat lower.
Of
more significance is the change in horizontal temperature gradient above
the mesopeak in the summer.
The gradient is nearly uniform in the equi-
librium computation, but shows a definite sharpening in high latitudes in
the marching problem.
Slightly higher temperatures at very low levels
were obtained in the marching problem.
This is a consequence of the
calculation of heating rates in the equilibrium problem using the equilibrium concentrations corresponding to the zenith angle at each time of
day.
This procedure evidently underestimates the energy actually absorbed
near sunrise and sunset.
-56-
I
.
80-
x
-.
before sunrise
1/2 hr before sunset
- -i/2hr after sunrise
.
/2 hr before sunset
day 12
day II
11/2 hr after sunrise
day 4
I hr
xxx
oo
x
0.
xx
--
-
x
70-
x
x
Equilibrium
day 1I
day 4
Concentration
(initial values)
A A Direct
Observation by
Johnson et al
'60-
50~
-
40
30-
10
10
Log
Fig. 7.
1012
10 l
10
base 10 ozone concentration ( Particles per cc
Variations of Ozone
Concentration
on
the
Equator
103
r-,- - '
f
80
180 160
10 200
25
260
270
280
60
300
290
4
90
50
70
(summer)
Fig. 8.
I
I
30
10
I
10
latitude
Equilibrium temperatures, *K, computed by marching method.
30
I
50
70
(vinter)
90
7.o
7.5
8.0
-80
-
8.5
9.0
-70
9.5
10.0
-60
10.5
I'
4J
-50
11.0
11.5
-40
12.0
12.5
-30
13.0
I
20
90
I
70
50
(sumer)
Fig. 9.
7
30
10
10
latitude
30
50
(winter)
Log (base 10) ozone concentration, molecules per cc, computed by marching method.
70
0.5
80
1.0
1.5
2.o
2.5
70
3.0
-60
4-)
I
'H
43.o
50
C,)
4..5
30.
4.o
- 40
1.5
- 30
1.0
0.5
90
I
70
I
50
I
30
(summer)
Fig. 10.
Diurnal temperature range, *K.
__
10
10
latitude
30
iI
50
70
(winter)
90
I
The noon ozone concentration distribution corresponding to Figure 8
is shown in Figure 9.
Again the only important differences with the equi-
librium results are the decrease at the highest levels, and the sharpened
horizontal gradient toward the summer pole in the upper mesosphere.
The diurnal temperature range as a function of latitude and height
is shown in Figure 10.
This is the range occurring on the last day of
the time integrations at each latitude.
Above 80 kilometers a correction
was made for the overall 24 hour temperature trend.
As one might expect,
the diurnal range is approximately equal to the absorption of solar energy
multiplied by the smaller of the illuminated or dark fractions of the
24 hour period.
E.
The External Heating
The temperatures of Figure 8 have been averaged with respect to area
on the earth's surface, and the resulting temperature,
has been identified with the basic
statical
equilibrium state,
( 7 ), which
is shown
in Figure 11 along with the 1962 U.S. standard atmosphere for comparison.
Since the external heating is the heating which the atmosphere would
undergo when in
statical equilibrium,
marching program with
each latitude.
7
it has been computed using the
as the initial temperature distribution at
The marching equilibrium distributions of atomic oxygen
and ozone were also used as initial conditions.
out at latitudes 90 , 45
and 0
Integrations were carried
for June 21st, and 30 , 600, and 90
-60-
for
901h
T.(Z)
80
1%
1962 U.S.
standarc d atmosphere
70
60 f
50 h
4.)
401-
30
20
aI
180
Fig. 11.
aA
200
220
Il_
240
temperature, *K
260
280
The basic statical equilibrium temperature distribution.
-61-
300
80
-70
J
+
+
+4
+3
+2
+1
0
--
5
2
/
+8
-
60
0
40
-30
I
90
Fig. 12.
i
70
50
30
10
I
10
M
70
I
30
(summer)
latitude
The "external heating" of an atmosphere whose temperature is T0 (z), 'K/day.
50
(winter)
90
December 21st.
The 24 hour temperature change computed in this way gives
an approximation to the external heating rate in 0K per day, and is shown
in Figure 12.
Although calculated quite differently,
the results are similar to
the net heating rates derived by Murgatroyd and Goody.
The largest
difference is the relative weakness and lower altitude of the arctic
winter heat sink in the present computation.
This can be attributed
primarily to the warmer temperatures actually occurring above 50 kilometers over the winter pole than occur in the dynamically stable basic
state defined here.
F.
Discussion
An indication of the accuracy of these results is given by the
comparison of the computed average radiative equilibrium temperature
with the U.S.
standard atmosphere in Figure 11.
no reason why the two should be identical,
There is,
of course,
since there might well be
a net import or export of energy across the upper or lower boundaries;
nevertheless the consistently higher computed temperatures can probably
be attributed largely to errors in computation.
Several important possible error sources are the following:
i)
Inadequacy of the infrared radiative transfer model,
particularly the neglect of water vapor emission and
very crude treatment of ozone emission.
-63-
Part of the
large discrepancy above 75 kilometers may be due to
failure to adequately account for vibrational relaxation of
COs
which begins at about 75 kilometers.
This effect has been discussed by Curtis and Goody
(1956) and was taken into account in an approximate
way by Murgatroyd and Goody.
In the present compu-
tation the Curtis matrix coefficients for computing
cooling at levels above 75 kilometers were corrected
to give cooling rates in agreement with those of
Murgatroyd and Goody at those levels, but the approximation may be too crude.
ii)
Neglect of additional reactions,
with hydrogen compounds.
particularly those
These are probably unimportant
below 75 kilometers, but would certainly give a net
decrease in ozone concentration at any levels where
they are operative.
This would alter the temperature
in the right direction.
iii)
Inaccuracies in the solar spectrum,
ficients, or reaction rates.
absorption coef-
Although there is some
uncertainty in the dissociation efficiencies of the
Schumann-Runge bands as discussed in Appendix 1,
by
far the largest uncertainty here is in the reaction
rates.
Reference has been made
(see page 44) to
evidence for a higher value and weaker temperature
-64-
dependence of
13
than were used here.
Although
no temperature dependence was determined, Phillips
and Schiff (1962) also report a larger value for
at room temperatures.
Larger
smaller ozone concentrations
-,
43
would lead to
(below 60 kilometers
the ozone concentration decreases approximately as
the square-root of
temperatures.
3
) and would cause lower
This possibility is particularly
appealing since the direct observations of mesospheric ozone concentration by Johnson et al
(1951)
show lower ozone concentrations than those deduced
here at all levels.
A particularly good indicator
is the region 40 to 60 kilometers, where because of
the very short characteristic equilibrium time,
day-
time ozone concentrations should be very close to
their equilibrium values.
In addition, below 35 kilometers the results suffer from errors
due to the assumption of latitude independent temperatures below 22 kilometers, and from a breakdown of the Curtis cooling scheme below about
25 kilometers.
The extremely long characteristic photochemical equilibrium
times below 35 kilometers casts doubt on the validity of any scheme such
as this much below that level.
The equilibrium temperatures over the
winter pole should be regarded as only roughly indicative, since the Curtis
cooling scheme has been applied here under the assumption that density
-65-
variations from the standard atmosphere are not too large.
Furthermore
the temperatures above and below the mesospheric region control the
arctic night mesosphere temperatures, and the mechanism determining the
temperature in these bounding regions has not been considered.
nately, because of the small area involved,
Fortu-
the arctic winter temperatures
7-.
have little effect on
The effect of one additional error source, a type of truncation
error, can be estimated.
Because of the large concentrations of ozone
present above 60 kilometers just at sunrise, there will be a spike in
the solar energy absorption until the excess ozone is dissociated.
The
differencing scheme used tends to overestimate this effect because of
truncation.
We can, however, obtain an estimate of the integrated energy
absorption in the spike, and compare this with the computed spike energy
of the marching program.
n
V
From equation (4.10), we have at sunrise
o e-
(4.36)
a
approximately valid down to about 60 kilometers.
absorption per unit mass in the spike,
dt
where
U
Idt
The rate of energy
,E
will then be
c J~t(4.37)
t
cJ.ne
is a frequency near the center of the Hartley band,
40000 cm~, and
n 3,
is the predawn ozone concentration.
the total energy in the spike,
& E
-66-
,
is
say
From this,
independent of
J3
= .35x101 1
..
30
ergs/gm.
(4.38)
71;t
.
73
Table 2 gives a comparison between spike size estimated from this formula
and computed by the marching program on the
llth day at the equator.
XE
from equation marching
(4.38) program
2130
(molecules/cm )
Hei ght
(km
S)
85
76
67
SE from
(10-6 ergs/gm)
.503x104
15
.204x10
.756
Table 2.
.655x1010
11
.552x10
.11Ox10
12
Error
in comp uted
heating rate
( K/d ay)
4.6
6.4
+.l 8
9.4
12.3
+.2 9
5.3
8.1
+.2 8
Errors due to overestimating the absorption at sunrise.
Below 67 kilometers, the error decreases due to the decrease in the excess
From these data one may estimate
of ozone concentration over equilibrium.
temperature errors in the marching computation to be as much as +5 K near
70 kilometers arising from this cause.
In future computations,
the
details of the sunrise variations should be treated more accurately.
In spite of these error sources, the results appear to be reasonably
reliable, as Figure 11 suggests, at least in so far as the major features
are concerned.
In particular,
they are probably adequate to answer the
two questions posed at the beginning of this chapter.
we have defined the basic state,
0 (a
rather large static stability everywhere.
-67-
In the first place,
), and it turns out to exhibit
In fact,
the maximum lapse rate
of 40 per kilometer agrees very closely with that of the standard atmosphere.
It is very unlikely that correction of the various error sources
mentioned above could reverse this conclusion.
In the second place, we
are now able to estimate the value of the parameter
amplitude of
Cp
e
is about 8 K per day.
terms of the non-dimensional heating
I11 I =
E
j
C
.
The maximum
Expressing this in
from equation
(3.20), we obtain
.15-
Figure 12 also indicates a characteristic vertical scale for the external
heating of about 25 kilometers, so that
Hvr
1/3.
It will be shown in
the next chapter that the evidence provided by the phase relation between
zonal wind and heating suggests a value of
the Prandtl number is of order unity.
WIru 0.2, provided that
Substituting these values into
equation (3.53), we find that the ratio of the second order terms to the
first order terms in the dynamical equations is about 1/3.
-68-
5.
A.
SOME LINEAR SOLUTIONS
The Equations
The basic equations are
(3.36) through (3.39), but we modify
these by setting
D
in the analysis.
We also reintroduce
H
to avoid the confusion of two height scales
the subscript ordering the
E
expansion.
'n
-n
+
and suppress
Then
(5.1)
(I
)-
y2
d/
Y
82
---
Gd
and
-
Ur
(5.2)
= 0
(5.3)
T
RS
(5.4)
It is convenient to introduce a stream-function by means of the
identities
and to make the substitution
-69-
We are concerned with periodic response to periodic forcing so
e
I = 1 ,
For the annual component,
has
=
the constant component
0 , while any semi-annual component would have
(annual mean)
Y = 2
.
that the time dependence may be represented by the factor
With these additional assumptions and definitions, equations
(5.1),
(5.3) and (5.4) reduce to
C) e
P r+Rs
(5.5)
'
-
with (5.2) remaining unchanged.
(5.6)
The dependent variables are now to be
interpreted as the complex amplitudes of the harmonic time factor.
fix the phase of these quantities, we specify that
To
is real.
-70-
j
In order to obtain solutions to these equations, it will be
assumed that
and
0
are related to the mean zonal flow and
temperature fields in one of the following ways:
0
=
)
'
friction coefficient, while
4.
~,~-
is a Rayleigh
,
i)
may be interpreted either as a
radiative damping factor, or as Newtonian conductivity, or a combination of the two.
e
=
a0--
and
are
4
r
4.
and they will be assumed to be constants,
Chapter 4,
+
4-r)
even though, as shown in
is a function of both
and
.
40
-
and
In terms of the dimensional constants
j2 T
ii)
A/
=-/
nondimensional eddy viscosity coefficient, and
eddy heat conductivity.
means of
- -k
*
L
a nondimensional
These are related to the dimensional eddy
viscosity and eddy conductivity coefficients,
L) HW
,
is a
)<
-71-
)
and
-K<
,
by
Solutions will be obtained both for constant
Pi
B.
and
K
A/
<
and
,
and for
increasing exponentially with height.
Boundary Conditions on the Stream Function
When any of the forms above are substituted into equations (5.2),
(5.5) and (5.6), an elliptic equation in a single dependent variable can
be derived.
This equation is of second order in the
and of either second or fourth order in the
on whether
Z
and
a'
_
/
derivatives,
derivatives depending
are represented by Rayleigh friction-Newtonian
conductivity, or by eddy viscosity and eddy conductivity terms.
When
the Rayleigh Friction-Newtonian conductivity assumption is made, the
solution is completely determined by specifying
V'
on all boundaries.
When eddy viscosity and eddy conductivity are included one additional
condition on
2(
boundaries .
These additional conditions on
(or 7
)
is required on each of two horizontal
)(
will be discussed in
Section E of this chapter.
From a physical point of view, the boundary conditions on
are that there be no mass flux across any boundaries.
pressure surface,
Z
=C,
The constant
is not the lower boundary, but we assume
that it approximates the lower boundary sufficiently well that we can
7=o
.
This approximation
can be
*
assume no mass flux across
The upper horizontal boundary may be at infinity.
-72-
j
justified by the smallness of the parameter
.
te
With this approximation, the lower boundary condition becomes
at
=O.
.
(5.7)
The side "walls" in the problem are the north and south poles and we
= 0
V
must certainly require that
The
along these boundaries.
however.
actual lateral boundary conditions are somewhat more stringent,
From the dimensional continuity equation (3.15), we see that the horizontal divergence of
A
A
ir
must remain finite at the poles if
ur
Now let
to remain finite there.
Lr
where
as
r
(5.8)
is a constant and
--.-
.
-
--
is
(CO
V
is a quantity which remains finite
Then the horizontal divergence of
S
-
COSp
(r+ ) (Cosy S&p'V
Cos
is
L-
(5.9)
This quantity goes to a finite value different from zero only if
We can restate equation (5.8) with
r=
1
in terms of
. -yfinite as
-73-
>'-*
4
1
r
1
in the form
(5.10)
It should be noted that as an alternative boundary condition we
may require that the vertical component of relative vorticity of the
zonal flow must remain finite at the poles.
LuCOS
f)
,
I
argu- quantity is
it follows from an this
d
-
Since, in dimensional terms,
'A
-
ment identical to the one above that
-Y
(5.11)
finite
-- Xy
/~
as
is required.
If the equations are separable
(5.11) is automatically
satisfied when (5.10) is satisfied.
The condition of no mass flux at the upper boundary is simply
that
0
the condition is
In the case of an atmosphere extending to infinity,
somewhat different.
infinity,
(5.12)
on a rigid horizontal boundary.
In this case
e ur
must vanish as
-z
approaches
so that
e
0
is a sufficient physical requirement.
as
- ~+
00
(5.13)
This will turn out to be suffi-
cient to specify the problem mathematically as well.
We note that all boundary conditions, for
for
1P
2X
as well as
, apply to both the real and imaginary parts of these quantities.
C.
Solutions in the Case of Rayleigh Viscosity and Conductivity
In the following development,
R,
will be assumed to be
constant; an example illustrating the effect of changes in
the solutions will be given at the end of this section.
the appropriate expressions for
'
and
Q
1R,
on
Substituting
into equations (5.5) and
(5.6), and making use of (5.2), we obtain
s
+-4
X
(5.14)
)
(1Y 2
y)
A+
+
(+-
(5.15)
?aYY
If the Prandtl Number,
P
these reduce to
-75-
,
is equal to unity,
)-'*X
YYZ
RS
Y2-
_
YxL
J
+~
li
(5.17)
x
(
1-Y 2.
Y3- j
d 2.*_
62_ 81/
It follows that in this case
47
tional to
or
is independent of
On the other hand,
.
is in phase with
T
(or
4,
)
and
is inversely propor-
'X
by the phase angle
and lags behind
4
4
K
There are two limiting cases:
.4
i)
, corresponding either to a steady
-
In this case, zonal
state or to very large friction.
as well as meridional components are in phase with
the heating.
1..j
ii)
-
00
, the case of vanishing friction
and conductivity.
The zonal component lags in phase
by one quarter period,
while the meridional component
remains in phase with the heating.
The agreement in phase between
value of
-4
The phase of
W
and
regardless of the
is a consequence of hydrostatic and geostrophic balance.
'2
then depends on the time integrated meridional circu-
lation modified by dissipation so that when dissipation vanishes,
-76-
X
is one quarter period behind the heating.
P
Now suppose
I I
We write
.
(5.19)
-X
=)-r
+ Z XL
and, separating the real and imaginary parts of (5.14), obtain
z
R
5 ?
+
I
*
)
+
(5.20)
=
5-
11 \ -YJ -! L4
2.
L)
Y
/
( -P)(I
'An
+
(
R~(W) 6z7- + (1+
(5.21)
= j
({\-
)
+ fI I)I
')
E
4?n~~~
y./,~ ~~~ , I(j-p q
Equation (5.21) indicates that
.
Separating
the real and imaginary parts of (5.15), one obtains
y+
(\ (I-
(5.22)
-
M
-- 7-
L--Y
YZ
62.x Z_
c3Y
L
4-
2 X,
az
-77-
I
d~
dX,
~P
Equation
(5.23) suggests,
d~(5.23)
P
even in the case when
is not unity,
that
-4
Since
-r
results from both radiative damping and damping by
some kind of eddy interaction, an upper
limit to the ratio
.2/.ir
A is due to radiative damping alone.
can be given by assuming that
Using the results of Chapter 4,
I.
may then range from about 2.5 up
to about 10, depending on the size of the
dT
term.
T=T.
For
the annually varying component of the motion this means that
-
o.4f
quantity is related to
OC =+Tn
O.
,
perhaps much less than this.
,
the phase of
X
,
Since this
by the equation
(X/'Xr)
it is possible to check this conclusion by examining the actual phase
lag of the mesospheric zonal motion.
Figure 13, taken from Newell (1963) is a plot of rocket network
zonal wind observations over a two year period.
-78-
The points at 60,
48,
I
IC
I
0
0
CM
IC
00
0%
I
0
F
0
0
%
0
*o.
..*4.,
q
*o
.:
0
0
0
\0
IC
.0.0
5
0
N
0-
00
J
0
0
0
0
4J o
-5
A
to
0
-5"
1960
Fig. 13.
1961
Rocket network zonal wind observations,
taken from Newell (1963).
1962
and 36 kilometers were fitted with sine curves in an attempt to determine the phase lag relative to the equinoxes; the results are given in
Table 3.
Phase Lag
Relative
to 9,-
Amplitude of
Annual
Oscillation
Annual
Average
(m/s)
Height
(kms.)
(m/s)
(days)
60
+17+ 5
+71
9+ 3
.15+ .05
48
+22+ 5
+66
17+ 3
.29+ .05
36
+13+ 5
+40
25+ 6
.43+ .10
Table 3.
Zonal wind data deduced from rocket network
observations.
Evidently the phase lag does lie within this upper limit; it appears
to range about .25 in the mid-mesosphere corresponding to a characteristic time of 15 days, and decreases with height.
The small observed value of
.
value of
'X'(
~j (i--I)
, and corresponding deduced
suggests an important simplification to the general
time dqpendent problem.
involving
oC
In equation (5.22), we may neglect the terms
with an error of order
is small,
O(
.
If we also assume
we may neglect the terms involving
in equation (5.20) with an error of order
[
.
7;.
This
implies a characteristic decay time for momentum of considerably less
than 60 days, and seems quite reasonable.
Both of these approximations
-80-
j
will be adopted in the analysis involving eddy viscosity terms, as well
as in the present case.
In the present problem,
the result is a simpli-
fication from the solution of a fourth order inhomogeneous equation,
to
the solution of two inhomogeneous second order equations with identical
homogeneous parts.
When eddy viscosity and eddy conductivity terms are
the simplification is from an eighth order equation to two
included,
fourth order equations with identical homogeneous parts.
When terms of order
neglected,
(
)
have been
and
equation (5.20) reduces to
RS(~
6
9
_CP2
Y(L
(5.24)
We assume that this equation is separable in the form
LP
,rm(Y) - '1,,7-
ZI
(5.25)
Tn= o
-rn =o
>
(5.26)
satisfies
Then if
Lin Y
(- y)
dY
and
(5.27)
can be written
.
2y
Fe
-F(Y)
at
(5.28)
-81-
I
the boundary conditions (5.10) will be satisfied,
and the equation for
can be written
d~,
p~-~~L~nP'm
provided also that the set of
~(5.29)
functions corresponding to different
2-
Li
values of the separation constant
are mutually orthogonal.
Ti
We first examine the equation for
,
but instead of
(5.27)
consider
YL
where
(5.30)
Y
n
is an arbitrarily small positive parameter.
Then if
7
?rn
is also required to satisfy a condition analogous to (5.28), we have a
Y
.
-
I Inparticular,
Sturm-Liouville problem in the range
-
I
-
the set of functions
are both mutually orthogonal and complete.
The solutions of (5.27) with
(5.28) are also mutually orthogonal; they
cannot be complete however.
throughout the interval
-/
All solutions to (5.27) which are finite
K
Y
!
+ /
are special solutions with
vanishing second derivatives at the origin.
The parameter
X
arises in a natural physical way if the effect
of viscosity is retained in the meridional equation of motion.
This
effect might be expected to be felt in the immediate vicinity of the
equator where geostrophic control is relatively weak,
-82-
but it will be
shown that if
is small enough it has little effect on
for the lowest values of
Lm
.
and
The lowest separation con-
stants and solutions therefore reduce to those of (5.27).
To solve
(5.30), substitute
so that
(
4T': dJY
,Lz
+l
Wjq
y+&)~7n
(5.32)
This closely resembles the spheroidal wave equation in form and has the
same singularities
(see, for example, Stratton et al,
1957).
A convenient
way of solving the latter equation is by substitution of a series of
Gegenbaur polynomials,
for equation (5.32).
and the similarity suggests the same approach
Let
CO
F
where
T_
n
is the
(5.33)
Gegenbaur polynomial of first order And is related
to Associated Legendre polynomials by
T
y)
(Morse and Feshbach,
Pn_+
1
(5.34)
1953, P. 782).
Using the recurrence formulas for
the Gegenbaur polymials we obtain the recursion relation
-83-
An
+B-'
+ C.,
C
0
;CT
(5.35)
where
An= ___~
(z-rl+i)(.n --I)
L(Zn
sj(n÷2)n +=('n
-.
C
+
)(r+1)(n+3)
+n
--
(5.36)
+
B
n (N+3) + 2
Ln
C
The set of equations
(z-n+5)(2.n+7)
(5.35) can be solved for the separation constant
_C7f
and the infinite solution vector
by rewriting them in
continued fraction form
cx~'
Lin
-n --+ oo
F,'
(5.37)
/=
be finite at
-1
means that
(5.38)
0,114
4.
)
The requirement that
~a7~
An_+
O~a.
so that the continued fraction must converge, and as
-84-
(Ur rLI
-
A,
-"-
(5.39)
The recursion formula yields a series which is either odd or even, and
since
AO= At
we
=
0
(5.40)
have
OL0
B
A.
A7
~
B* - Cz A 4
A3
B-
A,
'
(5.41)
(5.42)
B- C 3 A3r
which can be solved by successive approximation for the even or odd
separation constants.
This method is analogous to that used in tidal
problems (Lamb, 1932, P.
If
rapidity.
L?
330 to 355).
is not too large,
the method converges with reasonable
It follows that very small values of
S
cannot much affect
the solutions or separation constants in this case.
separation constants,
For the lowest
we may then write
,. 2..
L.
(5.43)
and use (5.41) and (5.42) in the calculation with
&
equal to zero.
The lowest two separation constants and the corresponding solutions,
except for an arbitrary multiplicative factor are given in Table 4.
-85-
M
0
1
8.1282
Table 4.
12.5439
T7 =
0
1.000000
1.000000
T
2
.134336
.156724
T7
4
.008010
.011125
-n
6
.000263
.000457
_n
8
.000005
.000012
=
Solutions and separation constants for the
Y
equation.
Turning our attention to the equation for the
-
dependence
(5.29), we obtain the general solution satisfying the boundary conditions on
at
=
0
and
=co
(equations (5.7) and (5.13))
in the form
' +
P EM
+
1
-
where
+(I 4RPL)
(5.46)
and
l
= RP &(5.47)
3 +-If
-86-
9
(5.45)
Solutions for all other dependent variables can easily be worked out
'P
from the solution for
and
'Y, L
It should be noted that
.
n
Xr
In = 0
(Y)
,
, '(r
for example
X M(Z)
while the real and imaginary parts of
,
Yr
all separate in the same way as
W,
(5.48)
T
separate in the same way as
for instance
-~
=0
"
(5.49)
)C
Several numerical examples have been worked out.
Each of these utilized
the heating function
,
S 00~~~~2~.~
~=
with
Z,= 4,
~=
~(5.50)
Sao
9, and the amplitude factor
chosen to correspond
to a maximum heating amplitude at the poles of 80K per day.
used in the first example were:
fle
=
8 .71x10
0
=
1.995x10
H
=
8x10 3 m
=
3
.1695x10- s-2
-87-
m2
2
s 1
Other parameters
180j
0
-70
v
-60
+7
-50
+6
+5
+4
+3
+2
+1
0
-1
-2
-3
-4
-5
-6
-7
*-
-40
0
30
I
90
I
70
50
(summer)
Fig. 14.
I
30
I
10
I
10
I
30
latitude
The idealized "external heating" used in the dynamical computations.
I
70
50
(winter)
90
I
N
0
f
I
(
I
70
I
--60
+0.3
+0.1
+0.2
I
1
-0.05
-0.1
-0.2
-0.3
I
+0.0
-50
/
xx
/
--40
-7,
-30
I
9U
Fig. 15.
I
(~~
7.
,'I
7U
DU
IF
I
30
10
10
30
50
70
(summer)
latitude
(winter)
Vertical velocity (cm/s) model I, Rayleigh friction - Newtonian
conductivity.
90
f~
')
0.
0.2 v.
u-20.5
0.4
0.2
6
80
70
60
-40
-30
I
U
90
I
I
50
7U
(summer)
Fig. 16.
I
30
a
10
10
30
latitude
Meridional velocity (m/s), model I, Rayleigh friction-Newtonian conductivity.
50
70
(winter)
90
-60
-60
-40
-20
0
+20
+40
+60
+L 40
+60
+
-2'0 - 40
IQ
80
-70
8
.+80
E
w
-60
-50
-40
)
C
-30
I
90
Fig. 17.
I
70
50
(summer)
I
30
10
10
30
latitude
Zonal velocity (m/s), model I, Rayleigh friction-Newtonian conductivity.
I
I
I
50
70
90
(winter)
P
=
I
10 6 s
This example will be referred to as Model I.
The field
of
C. fe
derived from equation (5.50) is shown in Figure 14, while the amplitudes
A
eA
of the annual oscillation of
15, 16,
and 17.
P
Since
U
and
are shown in Figures
A
A.
Ur
phase of
,the
I
Gr
,and
is
A
A
A
identical to that of
U
,
Ur
lags behind
(L
,while
Ore
by 11.5
days.
In the second example, Model II, all parameters are the same,
except that
P
= 0.3,
n*
4m= 0.3x10
19,
and 20*.
s
-l
Ur
,
V
The resulting real
.
A
A
A
parts of the amplitudes of
-6
and
U
are shown in Figures 18,
The meridional components of the circulation are decreased
relative to the case with
P
i
,while
the increase in the zonal
component is small, even though the momentum dissipation parameter has
decreased by a factor of 3.3, while the thermal dissipation has remained
unchanged.
Figure 21 shows the phase relationships in days between the
meridional and zonal components and the heating.
It is to be noted that
the meridional components lead the heating by very nearly
(i-'P)
A
or 29 days, while, although the phase lag of
L
has increased slightly,
it is still quite close to
; if
TThe figures are drawn in terms of the pressure coordinate,
in terms of geopotential height, the distortion due
interpreted
TT -surfaces, which in these examples amounts to height
to slopes of
as 8 kilometers, should be taken into account.
much
as
of
changes
-92-
p
-
-~
~------=
--
1
I
(80
/
I
I
N
70
/
-
4,
+.15
-60
+.10
+.05
-. 05
-. 10
.15
4-5
/
-50LA
\1
-40
-30
I
I
I
I
I
I
30
10
10
30
i
I
90
Fig. 18.
70
50
(summer)
latitude
Vertical velocity (cm/s), model II, Rayleigh friction - Newtonian conductivity.
70
50
(winter)
90
I
0.10
0.20
0. 0
0. 0
80
.30
30
- 70
- 60
- 50
- 40
0.0
0.0
- 30
4-
I
90
Fig. 19.
i
70
I
50
(summer)
a
30
--
L-
I
A
I
10
10
30
50
latitude
-I
70
(winter)
Meridional circulation (m/s), model II, Rayleigh friction - Newtonian conductivity.
I
90
80
-70
K
w
E
100
I 00
80
80
60
60
40
40
20
20
-60
cm
-50
F40
0
0
-30
9V
50
7U
(summer)
Fig. 20.
30
I
10
10
30
latitude
Zonal velocity (m/s), model II, Rayleigh friction - Newtonian conductivity.
70
50
(winter)
90
80I-
70Phase lead of
or
Phase lead
of
0~
AA
60F-
501I
401bO
30h
I
I
I
+30
+20
Fig. 21.
Phase lead, dgys
+10
Phase relationships between
heating, P = 0.3.
I
0
A
A
-10
A
u., T , kr
-96-
A
Ur
I
-20
and the external
These results,
as well as the differences in the amplitudes
P=
between this case and the case with
i
,
are a consequence of the
dominance of the dissipation term over the vertical velocity term in
It is this term which controls the amplitude
the thermodynamic equation.
of the zonal motion, while the control on the meridional motion by the
thermodynamic equation is relatively weak.
The major control on the
We can conclude
meridional motion is through the momentum equation.
that in this type of system, phase lag information,
Table 3,
such as that in
can be interpreted in terms of a characteristic time for thermal
damping rather than momentum damping.
The differences between the two examples also illustrate the role
RS
of the static stability parameter
portional to
P
Rs P i
,
and otherwise depends only on
Model II
by
Rs
multiplied by 0.3.
%
RSP Lf
P
real parts of the solution for the case
the old
is pro-
J
The solution for
otherwise it depends only on the combination
while the solution for
,
real solutions of
.
,
is proportional to
.
Hence,
if we multiply the
these can be interpreted as the
P
= 1 and a new
R,
equal to
With this interpretation, we see that
a rather large decrease in static stability has only a small effect on
the solutions; amplitudes of both zonal and meridional components are
increased slightly,
the height of the maximum rises, and the rate of decay
of the motions above the heat sources is diminished.
-97-
D.
Solutions for Constant and Exponentially Increasing Viscosity and
Conductivity.
Let
K = K
N = N. e
NO
where
Ko
,
P
and
are non-negative constants.
This is a
generalization which includes both the cases of constant eddy viscosity
and conductivity ( P = 0),
(
f=
1).
The equations to be solved are then
(~
Y~)X
together with
(5.2).
,
,
--
0(5.51)
a
(<0
iT
and molecular viscosity and conductivity
(5.52)
If one again assumes that the terms containing
can be neglected in the equations for the real
and
parts, solutions can be obtained with the aid of the substitutions
Xe- I Z
(5.53)
One obtains
Z
(5.54)
Variables can be separated in the same way as for equation (5.24),
and furthermore, since this equation is of second order in the vertical,
-98-
we need specify no new boundary conditions.
mass flux vanish at
2
0
=
and
Again we require that the
.
The solution for
satisfying these conditions is
ft fCO
oo [
?~
(5. 55)
eij
't~ie~K
with
L'~ '-
/U
f34~
When
=
(-3)÷41~L~j(5.56)
+.,(-)
PL
2,
(5.57)
0, equations (5.55) through (5.57) reduce to the
solutions for Rayleigh friction and Newtonian conductivity.
obtained from the solution for
the terms involving
',.
'Xi
q)
can be
by integrating (5.51) twice with
neglected, provided boundary conditions on
have been specified.
Although similar in form,
different from that when
the solution when
0=; when
-99-
L
')(,
3
=
=1, we have
1 is very
fR.,
,
-
3
+
(5.58)
R
-
(5.59)
Consider the motion which would occur above some pressure level
above all heat sources.
The solution for
Y!r,
Z.
which gives vanishing
mass flux at infinity (see equation (5.13)) is
O
e(5.60)
Since in the actual atmosphere for global scale motions
R.PL
<1
this solution gives increasing meridional components with height, even
though the mass flux vanishes at infinity.
geostrophic assumption when
Z
Such a flow must violate the
becomes large enough that both dissi-
pative and inertial terms become large in the meridional equation of
motion.
Nevertheless this type of solution may be applicable over a
height range of several scale-heights in the lower thermosphere where
molecular viscosity and conductivity first become important.
The separated momentum equation is
a7
z/
(5.61)
-100-
from which we obtain
X~
pEL(RsPL)Y-z)
~,zoex
X,
as the formal solution for
(5.62)
which vanishes at infinity.
if we are to have vanishing of the zonal velocity at infinity,
im
and
must be of opposite sign.
Thus,
Xr,
This is in contrast to the behavior
of these quantities in the case of Rayleigh friction and Newtonian conductivity.
In the latter case,
in a region above all heat sources,
Xrr
Since
E.
OA-i) <
.(5.63)
o
,
and
Boundary Conditions On
'X
W
are of the same sign.
and an Example With Eddy Viscosity and
Eddy Conductivity
has been obtained,
When the solution for
X,
can be
determined from equation (5.61) provided that we specify two additional
boundary conditions on
'
,
one at the lower boundary and one at the
upper boundary.
In a region bounded above and below by rigid horizontal surfaces,
the necessary conditions are the no-slip conditions:
)
=
on both rigid horizontal boundaries.
-101-
(5.64)
When the atmosphere extends from the earth's surface to infinity,
lower boundary condition is still
the
(5.64) and a suitable upper boundary
condition is that momentum flux vanishes as
3-
approaches infinity,
or
0
as
~
Condition (5.64) does not apply at
;F= 0
-4
,
W
(5.65)
since this is a constant
pressure surface rather than a rigid horizontal surface.
The actual
phenomena taking place near the lower boundary are extremely complex,
and cannot be considered in detail here.
ever,
We make the assumption, how-
that pressure fluctuations vanish as
Z
.
-+0
consistent with the assumption of vanishing mass flux at
This is
'= 0 , and
corresponds to the physical condition of no energy flux across that
pressure surface.
It is also consistent with the finding of Charney
and Drazin (1960) that the energy in tropospheric baroclinic waves is
normally trapped below the mesosphere
.
With this assumption,
the lower
boundary condition can be written
'X= C
=
at
.
Solutions satisfying the upper boundary condition
have
'X
approaching a constant as
2
goes to infinity.
(5.66)
(5.65) will
In general,
Assuming that pressure perturbations vanish at the ground is not necessarily the best lower boundary condition.
Solutions obtained by assuming
that vertical energy flux and pressure perturbations vanish in the lower
stratosphere will be considered in Section C of Chapter 6.
-102-
this constant will be different from zero, in particular with constant
eddy viscosity and conductivity coefficients,
different from zero.
this constant will be
This is an unsatisfactory state of affairs;
since
we know that dissipative effects will ultimately increase exponentially
at high levels, while differential heating does not, we would expect
oscillatory components of the zonal velocity to vanish at infinity.
This behavior of
2X
can be obtained by joining the mesospheric solution
for constant coefficients of eddy viscosity and conductivity to a solution for the thermosphere where dissipation is by exponentially increasing
molecular viscosity and conductivity.
We know that the latter solution
becomes invalid at great heights because of the breakdown of the geostrophic
approximation,
and the solutions in the joining region will probably not
adequately represent conditions there,
but if the resulting solutions do
not have much effect on the meridional components at levels below the
joining region while giving the correct behavior of the zonal component
above the mesospheric heat sources, this should be a valid procedure.
We assume that the solutions are to be joined at some height
above all mesospheric heat sources.
continuity of mass, or
Y?
continuous; because the model is a viscous
one, we must have continuity of
If
.Xm,
At the interface, we must have
]X
and
Tm.
or
dXm
as well.
and the mass flux are to vanish at infinity, we have from
equation (5.62), that
(5.67)
NIiRs Paln
-103-
furthermore,
dm
<
'"
,
From the continuity of
and
cixM
~&
'
in the upper layer.
(5.68)
equations (5.67) and (5.68) must also be satisfied by the lower layer
solution at the interface level,
Zo
.
To satisfy both of these con-
ditions we replace the previously obtained "well-behaved" solution for
Tn
by a new stream function,
__ 1
T =
where
%,
is the
to be determined.
"boundary"
it
well-behaved" solution, and
Because
solution,
eA3
-
-,
is quite large,
(5.67),
(5.68),
-
RPk -ZoA
in particular
ZO
,
using equations
and assuming equality of the dissipation
coefficients at the interface,
~ ~ Izo)-
or
of the lower boundary condition.
(5.61) twice from 0 to
and (5.69),
is a constant
the second term,
is important only near the interface,
does not affect the satisfaction
Integrating equation
Cb
R,PC,,.-
we obtain:
(t
3-') ( I--)C6
(5.70)
( z)
4RPL, (/U7I) +
C6
(5.71)
-104-
I
P
where
It (Ei)
and
A
for the unknowns
C,
and
are known, we can solve this system
-
I, (2)
Since
This
method has been applied, assuming
A
and the resulting
(AT
Cr
,
tAT
and
,
tr~
and
E
iLL
solutions are shown in Figure 22.
corresponding to
/
on both
=
108 kilometers,
together with the amplitudes of the "well-behaved"
-4 have been replaced by
LA.
4
dependence of the real parts of the amplitudes of
=
k
and
E
All parameters
AM 3
the dissipation coefficients
=
1200
m2/s .
=
Cr
Although
at distances greater than three or
4
is continuous across
Zo
,
tinuity, which leads to a discontinuity in
been assumed that
any
g
/
and
=
92.8
Figure 23 shows the dependence
It is clear that the "boundary" solution has little effect
and
in this
in the first example, except that
example are the same as those
of
n
viscosity does not act on
on W-r
four scale heights below
E,
its gradient suffers a disconIf
Uf
at
,
Eo
.
Since it has
this does not violate
features of the model, but is nevertheless an unrealistic situation.
The actual transition to
molecular dissipation is not sharp, so that the
-105-
I
well-behaved~
solutions
I
I
-100
1
I'
I
I
-
I
I
I
80
I
1
-
60
I-J
- 40
20
0
vertical velocity scm/s) meridional
-2
Fig. 22.
-1
0
+0.5
-1
elocity (m/s)
0
+1
zona; velojity (9/s)
+2
0
20
I
40
i
I
60
80
Comparison of "well-behaved" solution, and solution matched to molecular dissipation layer.
radiative coupling.
No
80
n
-20
70
-40
-60
-80
E
-80
-60
-40
I-I
-20
20
0
40
60
80
W
80
60
40 20
I
-60
-5(
I/
U)
-4(
I
2(
I
90
I
70
I
50
(summer)
Fig. 23.
I
I
30
10
latitude
I
10
Zonal wind (m/s), model I with eddy viscosity and conductivity,
30
I
(winter)50
IV
70
I
90
solution in the vicinity of
heights below
is not reliable, but since a few scale
7.o
-Z. the actual solution for the meridional components
does not depend strongly on the assumptions made about the transition,
the solutions in the mesosphere may be valid.
The equations for
RSpbY)
L';
and
i
can be written
V
(5.72)
and
RSP
7
e
-
2
Z
--
5.3
(5.73)
K
which is similar to the phase relation for
Newtonian conductivity case.
X
On the other hand,
in the Rayleigh frictionthe phase of
I
is
given approximately by
I -('-
(5.75)
-108-
90
80 1-
70
-
60
50
40 k
4-i
30 H
20 I-
phase lag
I
0
Fig. 24.
20
10
Phase lag of
A
U
(days)
I
30
40
model I with eddy viscosity and
eddy conductivity, P= 1.
-109-
In terms of the characteristic vertical scale,
_
z
H
D
P)
H..-
D
we have
,
so that we have the order
m
magnitude phase relation for
In the present example, we have taken
from equation
Wi M
o
can be calculated without additional approximation
'
and the phase of
P = 1, so that
,
(5.76)
(5.74).
The resulting phase lag of
(
(or
Ui
) is shown
in Figure 24.
F.
Eddy Viscosity, Eddy Conductivity and Radiative Coupling
In this case the momentum and thermodynamic equations are
(-
1
)X
-I-
--
_
~=
(5.77)
and
s(L
4K-
and the thermal wind equation
T
and
(5.2) remains unchanged.
(5.78)
Again neglecting
in the equations for the real parts, we
obtain
-110-
+GP
RSP(
c" 9r
__Y
Z
2-
az
(5.79)
- GP I,
Y
=P(9
where
No
(5.80)
Again variables separate as in equations
d'
d2
-'m
2
dependence of the stream function is obtained as
_
- (' P+-R ?M)d '
ez
d -a
=
(5.81)
xj
) , we obtain
For the imaginary part of
a3
d)W
3
+G
+
equation for the
(5.25) and (5.26), and the
-
4L
4
~
=
YP
7P(-~
& X,?Z
(5.82)
with the corresponding separated equation for the vertical dependence
d3
r
10P' -G+
(5.83)
-111-
L
.,
where the functions
correspond to the real part of
'x
The general solution of equation (5.81) can be written
A
Lc
and
IP3
-- Q,-if~e"dj
e"P,
C
where
/M
are the roots of the cubic equation
%
3 ~s
P, (5.85)
&4-&
C
and the constants
,
C,
and
+
C3
are to be determined from the
boundary conditions.
By considering the function
three roots of
(,), we will show that the
(5.85) are real for non-negative
Ps?
and
rP,
and they can be ordered as follows:
<
0
1
-112-
(5.86)
and
,
-
00
-F
,
also goes to plus infinity and minus infinity respectively.
A = + 1
4
,
(.)
to the right of
=
+
-
RsPLm
.
When
,so
,
that one real root lies
-F
,
0
so that there are two other real roots, one between
=
and the other to the left of
= 0
case there is a root at
roots of
.
o
When
I)
/=
GP = 0
, unless
=
0
CP
U=+I
and
,
--3
,
#
In the limits,
, in which
But in this case the other two
(5.85) reduce to (5.46) and (5.47) so that the ordering (5.86)
still holds.
Figure 25 illustrates the dependence of the roots of
R(PL
(5.85) on
6.P
for three values of
The three constants are determined by the boundary conditions:
vanishing mass flux at
flux as
in
r-O
and
R- = 00
, and vanishing momentum
If mass flux is to vanish as
.
D
,
we
must have
C3
=
3
CI
4 ) f S'e
(44-
(5.87)
and vanishing momentum flux requires that
co
e
JS,
o
C2
(
=-f
/3)
--
d
(5.88)
as well.
The vanishing of mass flux at
C,
=-
(
Z
=0
C )
-113-
then requires that
(5.89)
~jJ
(1.0,
(0.5
t
2)
(.0,
p
3
)
(0.5 P)
/43)
0
(0.I,(0)
I-J
(0-
,
I-J
/2)
(0.1,
-2.0
Fig. 25.
-1.0
Roots of the equation
0
+1.0
of Cin+Gthss
v3
4P in parentheses.
, values
3)
+2.0
+3.0
These constants together with the condition
=:-
0
at
completely specify the solution.
X
when
L
A convenient way of obtaining the
has been determined is by direct inte-
gration of the momentum equation.
are
,
(.=?o
(5.90)
The two arbitrary constants arising
, which is given by (5.90) and
which can be calculated from the thermodynamic equation.
At
F
=
,
solution for
= 0
.
the latter equation takes the form
GIPT
d
,T
or
e~o
dz
If we denote
f
1%!
A
-t +AM~
,+U}-(pVz
C; ,+,(p //
A =
d7
by
d4
=O.
A
,we
dz
e=0
find that
(5.91)
-
r
The solution determined in this way will be called the "well-behaved"
solution.
An example showing the vertical variation of the real parts of
the maximum amplitudes of
illustrated in Figure 26.
to
L)=
A
A
A
Ur
U
and
In this example
300 m2 /s, and
-115-
GP
=
Li
N
for such a solution is
=
a
= 23.5 corresponding
.21333 corresponding to
10-6
I
100
I
I
/
I
/
I
/
/
/
/
801L
/
/
I
60 I-.
7
Q)
I'
N
'I
/
40 L
7
/
/
,1
20r
/
I
0
vertical ve ocity (cm/sl
meridiona
velocity (qms
1
-0.2
Fig. 26.
0
+0.2
-20
0
zonql velqcity (m/s)
veoiy(ms
meiioa
+20
0
20
40
60
80
100
Well-behaved solutions (dashed), and solutions matched to eddy dissipation layer (solid),
radiative coupling included.
120
140
All other parameters and the heating are the same as in
per second.
the Rayleigh friction-Newtonian conductivity examples.
X
vanishing of
does not permit
W,
Again the "well-behaved" solution for
Such a solution may again be ob-
at high levels.
tained by joining the solution to an upper layer with molecular viscosity
The upper boundary conditions on the lower layer
and conductivity.
(5.67) and (5.68), and we fit these by writing
solution are
T
fe
_~
eJQ
4
C e/~(
is the "well-behaved" solution.
where
(5.92)
The term involving
Cb
will be negligible at low levels, so that the lower boundary condition
A
(/-)jJC'.
=1-f
,(
,-i)C,
+/
-QcaAZ
+) P,)C C3
+-
(5.93)
O
Introducing (5.92) into (5.67) and (5.68) and utilizing the zonal momentum equation, we obtain
-
)U
~
~
~
(frrI(e/Aa ,)itR)L~
("L
-
5L
, R=LJ;
_
(Ju-
-
-
Equation (5.91) now generalizes to
is satisfied by (5.92).
14rn
on
fr.3
+
fr-
=t
t
-117-
R5
(5.94)
Q#K-1
PL fR {(IA
JI)e/ULo
+
( R5PL~m)
(Aii
PL{
)
and
Auz7-0
(ze
Cb6
s(/"3-1)
RPL~ *I.(-
)
--
(5.95)
when terms of order
Cb
I)
df
(~)
and
/o3
are neglected.
o
Since
)
)
f
e
are known, equations
solved for the quantities
A
CO-
,
(5.93) through (5.95) can be
and
0
b
which determine the
solution.
An example of this type of solution, which will be referred to
as Model III, has been worked out, using the "well-behaved" solution as
4 R3 Pom
a basis and with
to be 108 kilometers.
AA
(Ar
again equal to .4296 and
taken
Z.e
The resulting real parts of the amplitudes of
A
,
,
and
LL
are shown in Figures 27, 28, and 29*.
*
A
Note the different isoline spacings used in this example for
than in Models I and II.
-118-
tY
and LP
80
70
+.125 +.1( )0 +.075 +.050
+.025
0
-.
025 5-. 0500
.075 -. 100 -. 125
60
-
-5(f
0
-
0
40
-30
-. 050
-. 025
I
90
70
50
+.025
I
I
30
10
(summer)
Fig. 27.
Vertical velocity
(cm/s),
I
10
latitude
model III,
I
30
50
+.050
I
70
(winter)
eddy viscosity and conductivity with radiative coupling.
I
90
(
80
70
[-
+0.2
60 |+0.1
0.0
50
4
bb
.H
fe
10
30
-
40
-0.1
90
70
50
30
(summer)
Fig. 28.
10
10
latitude
30
(winter)
Meridional velocity (m/s), model III, eddy viscosity and eddy conductivity
50
70
with radiative coupling.
90
I
I
(If
-
80
E
'I'
I
w
70 1
I
I
I I
I
+100
-100
60
-80
b1~
+80
50
-60
+60
40
-40
+40
30
-20
90
I
70
I
50
(summer)
Fig. 29.
Zonal
+20
30
10
I
10
latitude
I
30
50
(winter)
winds (m/s), model III, eddy viscosity, eddy conductivity and radiative coupling.
I
7'0
I
90
/
90
80
I
I
t-
70 r
60 r
50
N
40
A
.
30 1-
.
N
phase of
LL TF
phase of
IW
20 t-
L~
-100
~
-80
-
-
I
-60
-40
I-
II
I
-20
0
+20
I
i----------I
+40
+60
+100
+80
phase lead (days)
Phase relationships between
-122-
U.
T
tr
, model III.
and
,
Fig. 30.
Equations
P I
,
(5,82) and
(5.83) indicate that even in the case when
there is a phase lag of
which is proportional to GP
In the example described above the phases of
P
and
X
relative to
the heating have been obtained by solving (5.83) and the imaginary part
of the zonal momentum equation.
The results are shown in Figure 30.
The solutions for this case exhibit surprising differences from
either the Rayleigh-Newtonian or eddy dissipation models.
The most
important difference is the replacing of the single meridional circulation cell by a much weaker two-cell pattern.
The lower cell is closed
by a weak return circulation in the lowest levels; in the example chosen,
it reaches a maximum amplitude of four centimeters per second at the
ground.
The zonal circulation is much weaker in this case than in the
case where only eddy viscosity and eddy conductivity act.
These effects are again a consequence of the dominant control
exerted by the dissipation terms in the thermodynamic equation.
only eddy viscosity and eddy conductivity act,
When
the region of large and
nearly constant negative shear of the zonal wind below the heat source
leads to a large perturbation temperature, but negligible curvature of
the temperature.
Divergence of the diffusive flux of both momentum and
temperature is therefore small in this region.
When the simple linear
radiative coupling term is included, such large vertical shear would
lead to a substantial heat sink in the region below external heat sources,
and would drive strong downward motions in this region; upward motions
are still required in the external heating region just above.
-123-
The net
result of these effects is greatly reduced perturbation temperature and
zonal wind shear below the external heat sources, and weak downward
motion in the same region.
Although the solutions for this case reduce to those for eddy
viscosity and eddy conductivity when
0
P -
0)
, as they should,
the same qualitative features are obtained for much smaller values of
than that used in this example.
G.
A Solution in the Case of a Bounded Atmosphere
The solutions obtained in the previous sections are very incon-
venient for taking general
)
variations of the heating into account.
This is because of the slow convergence of the
series and the
difficulty involved in calculating each of these functions.
A simpler
solution can be obtained for the Rayleigh friction-Newtonian conductivity model when vertical velocity is required to vanish on two constant
We use equation (5.24), and introduce the substi-
pressure surfaces.
tutions
u
2(t
h.
equation (5.24) then takes the form
-124-
The vanishing of
Wi-
on upper and lower boundaries means that
vanishes on these surfaces.
a sine series in
(V
A general solution can then be obtained as
Z
q,./V~nY)
in (-f)i!-(5.97)
with
Z7m(Y)
=
where
D
gi
(5.98)
is now the distance between the constant pressure surfaces.
Y ) has again been introduced to insure satisfaction of
The factor (1-
the side boundary conditions.
Vm
must satisfy the equation
An
dY
d
(5.99)
and
(5.100)
The homogeneous part of equation (5.99) is again similar to the
spheroidal wave equation, except that no solutions to the wave equation
exist which satisfy the boundary conditions for the separation constant
equal to 2 as in this case.
We conclude that equation (5.99) has only
inhomogeneous solutions, as we would expect from the elliptic nature
of the problem.
To solve, we again assume a series of the form
-125-
Co
en
Vrn
In
n(5.101)
as well as
)(5.102)
"
-d
7-=
Instead of the infinite homogeneous set
of equations,
(5.35), we have to
solve the infinite inhomogeneous set
A'
where
A
Y1+2
and
C.
+n
C.3
O
=1
(5.103)
are defined as in (5.36), but now
Bn = [+
Z5.104)
(
This system can again be solved by the continued fraction method if it
is assumed that all of the
c4
are zero for
1
greater than some
finite value.
An example has been worked out for this case using essentially the
same heating function which Murgatroyd and Singleton (1961) used (Model IV).
This heating function in turn was based mainly on the work of Murgatroyd
and Goody (1958), and Ohring (1958).
series and five terms in the
Ti
Only the first four terms in the
series expansions of
The coefficients in this series are given in Table 5.
-126-
06-
in
were calculated.
i
I
+4
80 L.
+
+2.
+1
o
n
-i
-z
-3
-4-5--6 -7
-9
-10
70l.
-11
60 k
I-A
['3
-6
~B
.4'
+6
50
-3
-
40
+3
30L
+.
-i
0
10
10
~30
50
70
90
90
to (summer) -1)
JU1UI
3U
latitude
Fig. 31. Heating used by Murgatroyd and Singleton (1961),
0
JO
(winter5
0
K/day
9U
0
80
+1
+9
4
70
-10%
.1
00
50
*r4
G)
-6
+3
40
+1
301-.
4-2_.
0
a
90
Fig. 32.
70
50
(summer)
30
10
latitude
10
a
30 (winter),
Truncated series approximation to the heating of figure 31,
OK/day.
I
70
I
90
0.0
80
0+
004
+.*08
+.12
70
-0
.12
60
+.2
;50
ci)
+.08
40
+.04
30
-e2
l
0
90
Fig.
Ii
00
33.
70(winter)
50
30
Vertical velocity (cm/s), model IV.
10
II
latitude
r
f
10
t
I
I
I
30
I
I
i
I
(summer)50
I
I
%
70
I
I
I
I
90
O.4
0.2
0.6
1.0
1
.4
1.2.
1.0
0.8 0.6
.0
0.8
0.f
a.z
0
0
I
607
00
-50
-40
0.0
-30
N
90
I
70
(winter)
Fig.
34.
50I
30
1Ai
latitude
Meridional velocity (m/s), model IV.
'
-*1
/
I
(summer)
..
70
+20+4/
1.O
40
vI*.
+.
0
-0+ 0
+44o
+2.0
o 4 -4o
-
0
0
-80
_
40
-20
80
-130
--
70
+10
-10
-10/
40
-
0
0
30
-
-2O'~
I
I
VU
7U
Fig. 35.
-..
F
-10
I'-
(winter)
D0
Zonal winds (m/s), model IV.
30
I-f
1u
latitude
10
30
(summer)
50
70
90
,rl
1
2
3
4
0
+.0034
-.0041
-.0260
-.0151
1
-. 4886
-. 1512
-. 0417
-. 0979
2
+.1526
+.0787
+.0849
+.1307
3
-.0712
+.0307
-.0424
-.0349
4
-. 1041
-. 0938
-. 0370
-. 0534
17.3622
37.2338
70.3553
116.7206
Coefficients in representation of Murgatroyd
-
Table 5 -
Singleton heating.
The resulting heating field is given in Figure 32.
Because of the asym-
metry in the heating, which is due to difference between the winter and
summer heating in one hemisphere, the results should be interpreted on
the basis of a steady state approximation to winter and summer conditions
within one hemisphere.
Parameters used in this calculation were
4/-_(
t
8.71x105
D
=
60x103
H
=
7.5x10 3 m
4
r
Vt
=
.743x10- s-1
A
A
,
2
-3 -2
,361x10
s
No
Resulting fields of
2
=
,and
UA
are shown in Figures 33, 34
and 35.
-132-
j
6.
A.
DISCUSSION OF RESULTS
Comparison with Observations
Mean cross-sections of mesospheric zonal winds have been published
by Murgatroyd (1957), Batten (1961), and Newell
from Batten, is typical of these cross-sections.
(1963).
Figure 36 taken
These data can be
compared with the results of the preceding chapter, particularly with
reference to the following features of the zonal wind distribution:
amplitudes of the maxima, heights and latitudes of the maxima, phase lag
of the zonal winds relative to the external heating,
between summer and winter.
and differences
In Table 6, such comparisons are made between
the cross-sections of Batten and of Newell and the various models which
have been examined in Chapter 5.
It should be noted that the latitude dependence for Models I,
II
and III reflects the particular eigenfunction chosen to represent the
variation of the heating.
Comparison of Figures 12 and 14 suggests that
this function is more concentrated toward the poles than the actual
heating, consequently the deduced zonal winds are concentrated more
toward the poles than they would be if a more realistic representation
of the
)/
variation had been used.
Model IV, which is based on more
realistic latitudinal variations of the heating, gives better latitudinal
agreement between the deduced zonal winds and the observations.
-133-
+10
+20
0 -20 -20
+2
0
I'
80
/ /
/
+6
+80
I
- 70
4
7 60
-
I-1
w
/- 50N
E
50
bb
wr
I
I
I
- 40
/0
- 30
I
I
to U
I
I
70
(winter)
Fig.
36.
Observed zonal
E
I
50
30
10
latitude
winds (m/s), after Batten (1961).
10
30
(summer)
50
70
90
Dissipation
Model
Amplitude of
Zonal Wind
Summer
Winter
Parameters
Phase of
Zonal Wind
Height of
Latitude
Wind Max-
of Wind
Maximum
imum
I
= 10-6 s-
(Rayleigh friction,
Newtonian Conduc= 1)
tivity,
-6
10
V
=4
10
(Rayleigh friction,
Newtonian Conductivity,
= 0.3)
-6
0.3x10
= 10-6
III
81 m/s
80 M/s
80 m/s
1200 m2/s
*
II
m/s
81
-l
=X
2
1200 m /s
Ia
(Eddy Viscosity and
Conductivity)
5
-1
s
-6
s
-l
-13
days
66 kms
560
-14
to
68 kms
56
-19
days
112 m/s
112 m/s
-16
days
72 kms
56
118 m/s
118 m/s
-20
to
75 kms
560
75 kms
42
65 kms
300
65 kms
40
-1
r
(Radiative Coupling
included)
1J = 300 m 2/s
*=
IV
300 m 2 /s
.74x10-6 s
-6 s -l
-25 days
120 m/s
85 m/s
Observations (Batten)
85 m/s
60 m/s
(Newell)
70 m/s
(Bounded Atmosphere)
-15
days
.74x10
Observations
Table 6.
-10 to
-20 days
Comparison of deduced and observed zonal winds.
0
Another interesting feature of Model IV, which is not based on
strictly antisymmetric heating,
is the good correspondence of the rela-
tive amplitudes of the winter and summer wind maxima with those observed.
This suggests that the heating of Murgatroyd and Singleton,
which is not
anti-symmetric about the equator, may be a better representation to the
true driving force than the nearly anti-symmetric heating derived in
Chapter 4 (Figure 12).
This anti-symmetry of the heating in Figure 12
is rather surprising in view of the very definite deviations from antisymmetry of the equilibrium temperatures.
two effects.
This is mainly a result of
One is the photochemical effect of
4
.
Where solar
absorption is large, the tendency to return to equilibrium is faster
because of the temperature dependence of
13
.
Since solar absorption
is largest near the summer pole and vanishes at the winter pole,
this
effect will not be anti-symmetric, and will partially compensate for
the departure from anti-symmetry of the equilibrium temperature.
The
second effect arises from the non-linearities in the dependence of
heating on the temperature.
For example,
-
in equation (4.6),
d -L
is an increasing function of temperature; this will tend to decrease the
amount of cooling taking place in an atmosphere which is warmer than
equilibrium relative to the warming in an atmosphere which is cooler than
equilibrium.
To the extent that these non-linear effects are important,
Figure 12 does not represent the true driving force.
The heights of the maxima in the unbounded atmosphere models are
in reasonable agreement with the observations.
-136-
This height in the models
depends mainly on the height of the heating maximum, which was taken as
54 kilometers to agree with Figure 12.
In the cases where the thermal
dissipation dominates momentum dissipation, the heights of the maxima
appear somewhat too high; much better agreement is achieved when
P = 1.
The most striking feature of Table 6 is the joint agreement in
amplitude and phase of the wind maxima with the observations.
these two features depend on the single parameter
P
r
,
Since
at least when
is not too greatly different from unity, this result lends support
to the validity of the dynamical model.
Whether dissipation is given
the eddy viscosity-eddy conductivity interpretation or not,
models support a thermal damping constant,
all of the
-4v 10-6 per second.
This is also the right order of magnitude for the radiative and photochemical damping constant derived in Chapter 4.
Apparently radiative
and photochemical damping are very important factors in mesospheric
dynamics.
No direct observations of zonal mean meridional circulations in
the mesosphere have been made, nor can such observations be made with
present techniques.
However,
Oort (1962) has measured meridional circu-
lations in the stratosphere up to 30 millibars
(about 26 kilometers).
His results indicate motions in which the annual average dominates the
seasonal oscillation.
In middle and low latitudes there is equatorward
flow with amplitudes up to 40 centimeters per second, while north of
latitude 60
there is poleward flow.
In contrast the circulations deduced
in Chapter 5 all have flow away from the winter pole and toward the summer
-137-
PF
pole at these levels, with maximum amplitudes ranging from 2 to 10 centimeters per second.
There are several possible reasons for this disagreement.
In the
first place, the circulations observed by Oort appear to be driven by
momentum sources associated with large scale horizontal eddies of the
troposphere.
This is indicated by the qualitative agreement of Oort's
circulation with circulations deduced by Dickinson (1962) which are
needed to balance the momentum sources provided by these eddies.
is also suggested by the general decrease with height of
This
Oort's circu-
lation in the same region where eddy momentum flux is decreasing with
height.
It is also possible that an equator to pole thermal forcing component in the annual mean, which has not been considered in Chapter 5,
makes an important contribution at these levels.
Such a heating compo-
nent has been demonstrated by Ohring (1958) for the stratospheric region
below 21 kilometers.
It arises primarily from latitudinal variations in
tropopause height and lower stratosphere temperatures,
been considered here.
which have not
Although such a distribution of heat sources would
produce an annual mean component of meridional circulation, they would be
expected to produce a circulation of the type postulated by Brewer (1949)
i.e.,
of opposite sense to that found by Oort in middle and low latitudes.
If a circulation of the type deduced in this study exists, it would
certainly be masked in the lower stratosphere by the much larger amplitude
circulation found by Oort.
-138-
The only other attempt to deduce quantitative circulations in the
mesosphere itself is the study of Murgatroyd and Singleton.
They obtained
mesospheric meridional velocities up to 4 meters per second above 50 kilometers.
The lower values found in the present study are a consequence of
the control exerted by the damping terms,
equation.
In this model,
especially in the thermodynamic
these terms determine the amplitudes of meridional
as well as zonal components.
Since the thermal damping constant is inde-
pendently known to be at least of the order of 10-6 per second, this control
will be important even if non-radiative damping effects are very weak.
B.
The Evidence From Radioactive Tracers
During the first half of August,
1958,
two high altitude nuclear
detonations deposited substantial amounts of the tracer Rhodium 102 into
the high atmosphere at latitude 17
North.
These detonations were unique,
in that they were the only ones containing large amounts of Rhodium 102.
Most of the material was originally injected at mesospheric levels and
above (Kalkstein, 1962), and the subsequent distribution of Rhodium 102
was monitored by aircraft covering a wide range of latitude and height
in the stratosphere up to 27 kilometers.
These observations, which have
been reported on by Kalkstein (1962) and by List and Telegadas (1961),
may indicate some features of meridional transport processes above 27 kiloThe essential features of the observed stratospheric distribution
*
meters.
are the following
All concentrations of Rhodium 102 reported by Kalkstein and by List and
Telegadas were corrected for a 210 day half life.
-139-
i)
No Rhodium 102 attributable to the high level
injections was observed until the middle of 1959
when there was evidence of a rise in concentrations
in the southern hemisphere, particularly between 100
and 300.
After this initial rise, concentrations
remained essentially constant through the southern
hemisphere summer.
ii)
Northern hemisphere concentrations remained low until
the latter part of 1959 when sharp rises took place,
particularly north of 30
0
.
No important trends were
observed in the northern hemisphere stratosphere
between the spring of 1960 and the end of 1961.
iii)
No samples were taken in the southern hemisphere at
middle and high latitudes between the southern winter
of 1959 and the southern winter of 1960.
latitude sampling resumed in June 1960,
When higher
there had been
a very large increase in Rhodium 102 concentrations
to values comparable with those at high latitudes
(above 30 ) in the northern hemisphere.
Low latitude
concentrations also increased during the middle of 1960,
iv)
List and Telegadas have estimated that only 10 to 15%
of the total injection of Rhodium 102 had reached the
lower stratosphere by May,
-140-
1960.
.
0
but did not reach values as high as those south of 30
Friend, Feely,
Krey, Spar and Walton
(1961) have discussed this
evidence, and have compared it with the distribution of Strontium 90.
They have also attempted to deduce the origin of the Strontium 90 by
measuring the ratio of Cerium 144 to Strontium 90.
High altitude injec-
tions of Strontium 90 should have a higher Ce 144/Sr90 ratio than high
yield low altitude injections.
The evidence of the Ce 144/Sr90 distri-
bution supports the general features of the Rhodium 102 pattern, except
that there is evidence that the initial mid-1959 southern hemisphere
rise of
Strontium 90 of high altitude origin took place primarily south
of 300.
There is no unique interpretation for these tracer distributions;
the final concentrations will depend on the initial cloud size and distribution,
and on the size distribution of particles since relative fall
rates will be important,
although the lack of any early appearance of
Rhodium 102 near the latitude of injection at the sampling altitudes,
and
the small total fallout from injection until May, 1960 indicates that
fall rates cannot be very important,
greater than 27 kilometers.
except perhaps at heights much
Kalkstein has given one possible interpre-
tation in terms of upper atmosphere transport processes.
Since the initial particle distributions with height and size are
not known, no attempt will be made to assess the vertical transport
process, we will, however, examine the efficacy of the circulations
derived in Chapter 5 in producing the observed latitudinal distributions.
The strongest circulation derived is that of Figure 34.
-141-
If we assume
that the particles observed had the benefit of transport by the strongest
meridional winds in the circulation, these could not have traveled farther
south than latitude 10
South between the early August injection,
and the
circulation phase shift on September 21 which corresponds to a Prandtl
number of unity.
During the subsequent northern hemisphere winter,
the
debris could have been carried northward by the circulation to latitude
45 or 50
0
.
If the Prandtl number is actually less than unity the debris
would have begun returning north before September 21, and could not have
reached as far as 100 South.
If the Prandtl number were greater than
unity, the phase shift of the meridional circulation would have been
later, and the circulation itself would have been somewhat stronger,
allowing greater southward transport of debris, but it is difficult to
account for a Prandtl number much greater than unity, since this would
require prohibitively high eddy viscosity.
If only the meridional cir-
culation were acting, the debris would not be able to move beyond the
limits of 100 South and 50
North in subsequent seasons.
As a consequence,
the increase in Rhodium 102 observed near 300 South in the southern winter
of 1959, and the poleward gradient of Rhodium 102 in the southern hemisphere beginning with the winter of 1960 indicates the presence of lateral
mixing processes at levels above the highest sampling altitude (27 kilometers).
Although the distribution of tracers could be accounted for
entirely by mixing processes, a circulation of the type discussed here
may well be an additional important factor in producing the observed
distributions.
The tracer observations clearly indicate stronger vertical
mixing during the winters of both hemispheres.
-142-
C.
Eddy Viscosity and Eddy Conductivity
The theoretical model permits the interpretation of the zonal wind
data in terms of an eddy conductivity parameter.
basis of the zonal wind alone,
We are unable, on the
to deduce the Prandtl number, but the
actual height variations of the zonal wind suggest that this parameter
is of order unity.
In the following discussion,
it will be assumed that
eddy viscosity and eddy conductivity coefficients are equal.
Table 6 shows that agreement with the zonal wind observations can
*
be obtained using values of
1200
m2/s.
and
-K*
ranging
from 300 m2/s to
The higher values may be due in part to the use of unrealistic
boundary conditions on zonal wind above and below the
mesosphere.
Figure
37 is intended to illustrate the effect of varying the boundary conditions.
It shows wind profiles for latitude 30
in the winter corresponding to
the meridional circulation of Model IV and eddy dissipation parameters
=-
=300 m2 /s .
The boundary conditions used were
'X = 0
at 20 kms
and
L~ =-Xx
for three values of
Nb
at 82.5 kms
.
Since the smaller values of
are more
appropriate, if the upper boundary condition is determined by a molecular
dissipation layer,
of about two.
*
Under these boundary conditions, the most appropriate value
would be about 600 m 2 /s
.
of
the wind values derived are still too high by a factor
-143-
80
=3.0
1-
,X=.750
)g=,3ll
70 1
60 F
50
Wind Profiles at 300, winter
assuming that the boundary
condition at 82. 5 kms is
40
for three values of
30
I
20
Fig. 37.
40
60
80
100
120
140
I
I
I
160
180
200
Zonal wind (m/s) for model IV with eddy viscosityand eddy conductivity,
showing the effect of various boundary conditions on %
.
0
-144-
Considering all of the models of Chapter 5,
it appears that the
best agreement with zonal wind observations is obtained if the eddy
viscosity and eddy conductivity are in the range 500 m 2/s to 1000 m2 s.
Of course there is no reason to expect the actual eddy viscosity and
eddy conductivity to be independent of height.
These values should be
most representative of the region of maximum wind, i.e.,
from 60 to 75
kilometers.
if any, may be responsible for
It is not clear what processes,
effects resembling eddy viscosity.
It is well-known that large-scale
eddies, of the order of thousands of kilometers are quasi-horizontal and
act to transport momentum and heat in a highly systematic fashion,
against the gradients of momentum and heat
often
(see, for example, Starr, 1954).
It is therefore unlikely that these processes could be successfully parameterized by means of vertical eddy viscosity and conductivity coefficients.
at the very smallest scales -
of the order of meters
-
On the other hand,
it is probable that eddy processes would approach the idealized case of
homogeneous and isotropic turbulence,
and conditions may resemble the
inertial subrange of Kolmogoroff theory.
At slightly larger scales,
Bolgiano (1959) has predicted a "buoyancy subrange",
forces strongly modify shear turbulence.
where buoyancy
These small scale turbulent
motions are very likely to produce eddy viscosity and eddy conductivity
effects.
Over the very large intermediate range of scales, practically
nothing is known about mesospheric motions, although Mantis
-145-
(1963) has
reported observations indicating a maximum in the horizontal kinetic
energy of the upper stratospheric zonal easterlies at a time period
of 10 hours.
At levels above 80 kilometers, motions which appear to
be internal gravity waves are found (Hines, 1961; Witt,
1962).
If the
gravity wave interpretation is correct, they should also occur at lower
mesospheric heights, but with reduced amplitude.
amplitude,
Although large in
these motions would not transport momentum or heat unless
nonlinear interactions were important.
There is some observational evidence bearing on the intensity of
At stratospheric levels, up to 19 kilometers,
*
small scale turbulence.
Kellogg (1956) has measured the dispersion of smoke puffs.
He finds that
the dispersion cannot be well described by a constant eddy viscosity coefficient,
since the puffs appear to be continually acted on by larger and
larger eddies as they grow.
Root mean square velocities of these eddies
were deduced to be of the order of 0.1
m/s.
If it is assumed thatthis
is representative of the vertical velocities of turbulent eddies,
it is
possible to derive an upper limit on their vertical scale since the work
done by such an eddy against buoyancy forces cannot exceed its kinetic
energy.
If the logarithmic potential temperature difference between an
at its maximum vertical
eddy element and its environment is
displacement,
,
the work done per unit mass against gravity during
the displacement is approximately
The measured dispersions were mainly in the horizontal plane.
-146-
where
V
is the characteristic eddy speed.
and for an adiabatic process,
In an isothermal atmosphere
this leads to:
V
This is not in disagreement with the observed dispersions of the puffs
If we now insist on
which were generally between 100 and 200 meters.
interpreting these turbulent dimensions in terms of an eddy viscosity,
we obtain
\Ye
r0.5~
<T
nS
Kellogg notes that dispersions and derived eddy velocities increase with
height,
and the value given above can be taken as a plausible lower
limit on an eddy viscosity coefficient for the mesosphere, even though
the vertical scale of stratospheric turbulence may be somewhat less than
5 meters.
Above 80 kilometers information on turbulence has been derived by
Greenhow (1959) and Greenhow and Neufeld (1959) from observations of
radio-meteor trails.
Applying the Kolmogoroff theory to the observed
dispersion, they deduce a turbulence power,
The same value of
,
of 7x10-3 watts per
was deduced by Blamont and de Jager
*
kilogram.
,
(1962) from the observed dispersion of sodium-vapor trails.
They also
There is some question that the turbulence responsible for the dispersions of both meteor trails and sodium-vapor clouds may be produced
by the meteors or rockets involved.
-147-
report that the observed turbulent elements have diameters of 0.5 km.
If these eddies are in the inertial
subrange, we should have, approx-
imately
V
400 m2s.
V3
On the other hand if there were a constant coefficient of eddy viscosity
(independent of scale) the dispersions of the meteor and sodium clouds
should, after a sufficiently long time,
exhibit a linear dependence of
mean square radius on the time.
There is no evidence for such behavior
in the observations of Greenhow,
or in those of Blamont and de Jager.
This fact can be used to place a lower limit on a scale-independent viscosity coefficient from the relation
4-te
where
r;-
is the dispersion at the time of the final observation,
1.
On this basis, both the observations of Greenhow and those of Blamont and
de Jager give
700 mIs.
There is no evidence of the behavior forecast by Bolgiano (1959) for the
bouyancy subrange in either the experiments of Greenhow and Neufeld or
in those of Blamont and de Jager.
On the basis of a Richardson number argument, the eddy viscosity
in the middle-mesosphere should be considerably less than that above
-14S-
85 kilometers.
The turbulence producing shears probably arise from
gravity waves and tidal motions.
If this is so, the shear should be
only about 1/3 as large at 70 kilometers as at 85 kilometers, while
the static stability at 70 kilometers is about half of that at 85 kilometers.
These values imply a Richardson number which is twice as large
at 70 kilometers as at 85 kilometers and consequently turbulence should
be less intense.
The inference concerning relatively lower shears at
mid-mesospheric heights is borne out by the observations of aufm Kampe
et al
(1962).
To sum up: there is no evidence for a true scale-independent eddy
viscosity coefficient, but if one insists on interpreting observations
of turbulence in terms of an eddy viscosity, probable lower and upper
limits on a mid-mesospheric value are
0.5
<
V k
400 m2/s.
This range suggests that small-scale turbulence by itself may not be
sufficiently intense to produce the required diffusion of the momentum
and temperature fields.
It is possible, however,
that motions at some-
what larger scales may produce effects resembling dissipation by eddy
viscosity and eddy conductivity.
In particular, non-linear interactions
between gravity waves and the zonal flow might be such a mechanism.
D.
Some Implications of the Deduced Circulations
There are a number of implications which circulations of the type
deduced here would have if they actually exist in the mesosphere.
-149-
Only
a few will be discussed here.
Any type of convective transport in a stratified fluid will tend
to increase the static stability of the fluid.
In the present case it
is possible to estimate the static stability increase which would result
The vertical divergence of the heat flux
from the deduced circulations.
can be written
where
hand,
L
]
now indicates the latitudinal average.
On the other
the additional heat loss due to the vertical circulation is
approximately
Cpe LT*
where
is the temperature increase arising from a circulation pro-
duced convergence of heat flux.
[T*J
will be determined by the
requirement that these quantities balance, or
r*
Using
H
=
=
5x10
LLA
(6.1)
PJ
per second,
r
=
0.2 cm/sec,
7
=
30 K and
8 kilometers, we obtain
[T]
Nu
13 K.
To obtain the static stability change, we must divide this quantity by
the characteristic scale of the mesospheric circulations.
-150-
If we take
D
to be 15 kilometers,
to
we find that the static stability change corresponds
lo /km
or a percentage change in
of less
than
D
15%.
This result can be compared with a similar calculation for largeA
scale horizontal eddies.
We can again use equation (6.1), but
now correspond to the eddies.
TP
utr
and
As an example, we may take values
for these quantities corresponding to the polar night jet, or i.JrN I cm/s,
T
200 K.
Such values will lead to
[T
n
30 K/km, and to static
stability modifications of up to 50% of the observed values.
It is clear
that eddy convection is a much more effective process than the simple
cellular overturning considered here as far as static stability changes
are concerned.
Because of this the good agreement between the over-all
observed mesospheric stability and the mean static stability deduced from
radiative and photochemical considerations
(see Figure 12) supports the
hypothesis of a symmetric meridional circulation as the primary convective
element in much of the mesosphere.
Kellogg (1961) has suggested that an important heating agent in
the polar winter mesosphere is the recombination of atomic oxygen in
descending air.
He estimates that descending velocities as low as
.05
centimeters per second at 95 kilometers would be sufficient to produce
a heating rate of 100K per day.
Such velocities at 95 kilometers are
apparently quite feasible within the framework of the models studied here.
-151-
It should be noted however that if such a large heat source were present
at high mesospheric levels, it would have a considerable effect on the
circulation, and would in fact probably serve as an important damping
agent.
It has been pointed out by Newell (1963) that the anamolously warm
temperatures in the polar winter mesosphere need not be explained by
photochemical action, but may instead be a consequence of eddy motions
driven by instabilities at lower levels and characterized by a countergradient heat flux in the upper mesosphere much like the counter-gradient
heat flux which has been observed in the stratosphere
(White, 1954).
The
calculations of Chapter 5 indicate that such a temperature distribution
may also result from symmetric convection,
the motions at the higher
levels being simply driven by differential heating at lower levels.
Figure 38, showing the temperatures deduced from Model I,
this feature.
illustrates
Either of these two dynamical explanations of the polar
winter warmth has the advantage over the chemical heating hypothesis of
also explaining the extremely cold temperatures occurring in the upper
summer mesosphere.
-152-
180
80
200
220
O)A r%
70
1-
60 H
220
240
260
280
340
320
50
-i
40-
30
I
vu
Fig. 38.
-
7U (summer)
Temperature
OU
JU
-I latitude-"'
distribution corresponding to model I
(OK).
I
AI
I
(winter) 50
.3L
70
90
7.
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH
Following are the principle conclusions which the analysis presented
in the preceeding chapters suggests:
i)
Calculation of equilibrium temperatures on the basis
of presently available radiative and photochemical
data indicates that the radiative-photochemical equilibrium state of the mesosphere is everywhere statically
stable,
at least when possible reactions involving
hydrogen are neglected.
ii)
The relevant non-dimensional parameter determining
the linearity of a thermally driven symmetric circulation is proportional to the product of the Prandtl
number, the annual frequency,
the vertical scale of
heating and motions and the characteristic time for
thermal damping.
the scale height.
It is inversely proportional to
On the basis of radiative calcu-
lations and the observed phase lag of the mean zonal
wind, and assuming a Prandtl number of unity,
this
parameter, which approximates the ratio of the nonlinear to linear contributions to the solution, has
the value 1/3 for the mesosphere.
The remaining
conclusions are therefore based on the linearized
equations.
-154-
M
iii)
A thermal damping parameter arising from infrared
radiation and from temperature dependence of the
photochemical reaction rates can be derived,
and
it has the order of magnitude 10-6 per second.
This is large enough to have an important damping
effect on any symmetric thermally driven circulation in the mesosphere.
iv)
When damping is accomplished entirely by eddy viscosity and eddy conductivity,
the deduced mesospheric
heat sources produce a single cell meridional circulation from summer pole to winter pole at high levels
with a much weaker return circulation below 35 kilometers.
The meridional circulation is approximately
proportional to the Prandtl number,
Prandtl number equal to unity,
and for the
the maximum horizontal
velocity is about 80 cm/s occurring near 65 kilometers
and in middle and low latitudes.
The phase of the cir-
culation relative to the external heating is roughly
proportional to one minus the Prandtl number
P
i.e.,
the meridional circulation leads the heating when
(I-P)
>0
and lags behind the heating when
(I-P) < 0
v)
When linear thermal damping of Newtonian type
-155-
cor-
responding to radiative and photochemical effects
is included in a model with constant eddy viscosity,
the meridional circulation consists of two cells,
one above the other;
the strongest branch is from
summer pole to winter pole above 40 kilometers,
weaker flow from winter pole to summer pole between
8 and 40 kilometers, and much weaker return flow
below 8 kilometers.
An example of such a circula-
tion calculated with the radiative damping constant
= 10 6 sec
and eddy viscosity and eddy conductivity coefficients
V
=
-K
= 300
m2
gives maximum amplitudes of 30 cm/sec, 10 cm/sec
and 4 cm/sec for the three branches.
The circula-
tion amplitudes are roughly proportional to the
ratio
DX +X
where
V
is a suitable characteristic vertical
scale of the heating.
The radiative damping constant
introduces a phase difference between meridional
circulation and heating even in the case when
irv)
For the zonal circulation,
P
=
1.
the amplitude and phase
lag are both roughly proportional to the characteristic
-156-
time for thermal dissipation i.e.,
"'/
K-
when
Newtonian thermal damping is not included and
:D. (
N
when it is included.
+
Reasonable agreement with the observed zonal winds
in amplitude,
phase lag,
of the wind maxima
and height and latitude
can be obtained by assuming that
the upper boundary condition on the zonal wind is
determined by molecular viscosity and conductivity.
vii)
The best agreement with zonal wind observations is
obtained by assuming
These values of
t/
are somewhat higher than
estimates based on deductions about small-scale
turbulence from observations of the dispersion
of sodium-vapor clouds and meteor trails above
80 kilometers, and smoke puff diffusion in the
stratosphere would suggest.
viii)
In an atmosphere in which molecular viscosity and
conductivity are responsible for diffusion of the
zonal momentum and temperature fields,
and at
altitudes above all heat sources, the meridional
components of motion increase exponentially with
height while the vertical mass flux decreases exponentially with height.
The zonal velocity and
-157-
the temperature decrease exponentially with height.
Such a solution can only be valid in a limited
height range because it implies a breakdown of the
geostrophic approximation at very high levels.
ix)
The circulations deduced are not strong enough to
account for the observed latitudinal distribution
of the artificial tracer Rhodium 102 without additional dispersion by horizontal mixing processes.
The circulations would assist any lateral mixing
to produce the observed distributions, however.
Several avenues of research which are now being pursued may shed
additional light on the problems considered here.
Attempts to evaluate
the ozone reaction constants in the laboratory are going on, and the
results that have been obtained in this work suggest that the value of
together with its dependence on temperature is the most critical
quantity for the understanding of the dynamics of the mesosphere.
Another
important question to be answered is: how much water vapor is there in
the stratosphere and mesosphere? This is important,
both because water
vapor may make an important contribution to the heat balance through
infrared radiation,
and because this information is fundamental to the
photochemical problem of the reactions involving hydrogen at higher
levels.
Further refinement of the photochemical-radiative equilibrium
and heating rate calculations presented in Chapter 4 should probably
include water vapor radiation and photochemistry.
-158-
Further direct
observations of the ozone concentration in the mesosphere, such as that
of Johnson, Purcell, and Tousey (1951) would be valuable,
especially,
since the ozone concentration at lower mesospheric levels is a sensitive
indicator of the accuracy of the parameters in the photochemical theory.
On the theoretical side,
study of the non-linear interactions of
tidal and gravity waves with the zonal flow could shed light on what
may turn out to be important mechanisms for the transport of momentum
and heat in the mesosphere.
Measurements of the intensity of small scale turbulence by observing the dispersion of artificially produced clouds in the main part
of the mesosphere, analogous to the observations of sodium vapor clouds
at higher levels, would be valuable.
Finally,
it seems clear that no really conclusive statements as
to the character of the meridional circulations in the mesosphere can
be made until momentum and heat fluxes due to the large scale eddies is
known with some degree of confidence.
This knowledge will require a
good distribution of observations with longitude in order to eliminate
the effect of standing waves, and a good distribution with time of day
in order to allow separation of tidal effects.
When these fluxes and
their divergences become known, it should be possible to determine
whether the resulting sources are "small " in the sense of the analysis
of Chapter 3.
If they are small,
it should be possible to incorporate
their effect into the present theoretical framework to evaluate their
-159-
contribution to the meridional circulation in much the same way that
Kuo (1956) evaluated the effect of empirically determined momentum
fluxes on tropospheric mean meridional circulations.
-160-
Appendix 1.
A.
Details of the Photochemical & Radiative Calculations
Photochemical Data -
For the photochemical computations,
the follow-ing
information is necessary: the spectral distribution of solar energy outside
of the earth's atmosphere, the spectral distribution of the absorption
coefficients of ozone and molecular oxygen and the dependence of these
coefficients on temperature and pressure, the values of the reaction rate
coefficients
, 1-
, and
3
and their dependence on temperature,
and the spectral distribution of the quantum yield factors,
Dutsch
e..
and
e3
(1961) has reviewed out knowledge of these quantities, but more
recent data is now available in all but the last of these areas.
It is convenient to discuss the data according to spectral regions.
i)
From 13530 cm~
to 20700 cm~
(7392 to 4831 A),
Chapuis bands of ozone must be considered.
the
These are
relatively weak, but they are of importance in determining ozone concentrations below 35 kilometers.
Ozone
absorption coefficients in this spectral region were
taken from Vigroux (1953).
Absorption in these bands
is essentially independent of pressure,
and Vigroux's
work also indicates that they are quite insensitive
to temperature as well.
workers
In accordance with earlier
e.g.: Craig (1950), Diitsch (1961),
it has
been assumed that all solar absorption by ozone at
wave-lengths shorter than 11,000 A leads to direct
A-1
e
dissociation, so that
has been taken to be
unity for all spectral intervals considered.
The
solar spectral intensity for this region was taken
from Johnson (1954).
ii)
From 28750 cm-1 to 40000 cm 1 (3475 to 2500 A)
absorption is by the Hartley-Huggins bands of ozone.
Absorption coefficients for this region were also
Temperature dependence is
taken from Vigroux.
slightly more important in the long-wave portion
of this region than in the Chapuis bands, but this
effect has again been neglected.
Johnson's paper
was again used as the source for the solar spectral
intensity data.
iii)
-l
(2500 to 2000 A) ozone
From 40,000 to 50,000 cm
absorption coefficients were taken from Inn and Tanaka
(1953).
Again it is safe to neglect pressure and
temperature dependence.
Beginning at 41400 cm 1 (2720 A),
the Herzberg continuum of molecular oxygen produces
direct dissociation (
e4=
1).
This absorption is
pressure dependent, but the recent work of Ditchburn
and Young (1962) indicates that at pressures less
than about 30 millibars this effect can be safely
neglected, so that these coefficients have been
assumed independent of pressure in this work.
A-2
The absorption coefficients of Ditchburn and Young
have been used.
Solar spectral intensities were
taken from Detwiler, Garrett, Purcell,
and Tousey
(1961); their values overlap with those of Johnson
near 40,000 cm-1.
iv)
From 50,000 to 60,000 cm~ 1
(2000 to 1667 A) absorp-
tion is primarily by the Schumann-Runge bands,
although the Schumann-Runge continuum occupies
the short wave end of this region.
These are very
sharp bands and do not lead to direct dissociation.
Dissociation may take place, however, as a result
of the following processes: direct dissociation to
Z
the
state in the Herzberg continuum which
underlies the long wave end of the region, direct
dissociation in a continuum resulting from a transition from the ground state to the
3
T,
state,
and predissociation resulting from perturbation of
the upper state of the Schumann-Runge transition
3
(
E
) with the
-TT,
state.
The intensity
of the Herzberg continuum has been estimated theoretically by Ditchburn and Young for frequencies
greater than 50,000 cm 1,
while Wilkinson and Mulliken (1957)
have observed the intensity of an underlying continuum,
which they assumed was due to the 3
A-3
TT,
transition,
at 56,200 cm
1 and 55,600 cm 1.
The latter authors
also observed well-defined broadening attributable
to predissociation at the
Lr= 12 level.
Absorption
intensities in the Schumann-Runge bands have been
measured by A. Vassy (1941) and by Watanabe,
and Zelikoff (1953).
rapid;
Inn,
Intensity variations are very
in the present calculation a smoothed absorp-
tion spectrum was used based on the observations of
Watanabe et al for the short-wave end of the region
and on Vassy for the longer wave-lengths.
Pressure
and temperature dependence has again been neglected.
The quantum yield factor,
e
, . was estimated from
the results of Wilkinson and Mulliken, and Ditchburn
and Young.
The solar spectrum for this region was
based on recent data which were kindly provided by
Dr. R.
Tousey of the Naval Research Laboratory.
These data are shown in Figure Al and exhibit lower
intensities than those published by Detwiler et al
by as much as a factor of two near 1880 A.
Ozone
absorption for this region was taken from Tanaka,
Inn and Watanabe (1953), and was again assumed to
be independent of pressure and temperature.
Below 1667 A (60,000 cm
1) there is a strong absorption by molec-
ular oxygen in the Schumann-Runge continuum,
A-4
but this radiation does not
5500
---
-Si
-- -- ~~ ~
UI (1)
z U)500K1.0 -
Al M (1)
S i II (1)A
0*
---ll
-
-
Al I
Mg 1 (2)
--
)
10.0
SI (2
Ln
4500*K_~
J
a-
01
0
U)
1750
1800
1850
1900
1950
2000
WAVELENGTH (A)
Fig. Al.
Solar spectrum from 1750 to 2100 A.
Photgraph provided by
Dr. R. Tousey of the Naval Research Laboratory.
2050
2100
penetrate below 85 kilometers, and was not considered.
The spectral
data actually used in the computations are given in Table Al.
Integration over the spectrum was done by simply summing over
the spectral intervals.
The height increment used in the photochemical
computations was 1.5 kilometers.
Table A2 gives the data used in computing the time constants
illustrated in Figure 2.
Experimental data bearing on the reaction rates
,and
have recently been reviewed by Harteck and Reeves
Kaufman and Kelso (1961).
Attempts to measure
(1961), and by
and
by no less
than nine different groups have been reported since 1957, but there is
a wide divergence of results, especially for
only three recent attempts to measure
-
.
On the other hand,
have been reported (Benson
and Axworthy, 1957; Leighton et al, 1959; Phillips and Schiff, 1962),
and only two of these have measured the temperature dependence.
There
is considerable disagreement between these two (Benson and Axworthy and
Leighton et al),
For these reasons,
the rate constants
I
are the source of greatest uncertainty in the calculations.
A-6
and
I
sity (photons/cm -sec
per wave number multi-
plied by 10
13500.
.000 258
14500.
15500.
16500.
.000 243
.000 223
.000 205
17500.
18500.
.000 183
19500.
20500.
.000 130
.000 119
29375.
30625.
31875.
33125.
.000 022
.000 018
-17
Absorption Coefficients
2
(cm /molecule multiplied by 10
Molecular Oxygen
.000 160
Width of
Spectral
Interval
(cm
-1
.000 035
.000 130
1000.
.000 275
.000 460
1000.
.000
.000
.000
.000
440
1000.
290
1000.
175
1000.
085
1000.
.000 012
.000 008 5
.025
34375.
35625.
36875.
.000 007 1
38125.
.000 001 5
.11
.36
.65
.91
39375.
40625.
41875.
43125.
.000 000 79
.000 000 63
.000 000 45
.000 000 42
.000 000 03
.000 000 11
.93
.74
.000 000 25
.50
44375.
45625.
46875.
48125.
.000 000 38
.000 000 31
.000 000 20
.000 000 11
.000 000 37
.000 000 64
.000 000 86
.000 001 02
.305
Table Al.
efficiency
for oxygen
dissociation
Ozone
.000 12
.001 1
.005 2
.000 003 4
.000 002 4
Quantum
17
)
2
)
-1
)
(cm
Solar Spectral Inten-
)
Mean Wave
Number
1.00
.172
.082
.040
Spectral data used in photochemical calculations.
1000.
1000.
1250.
1250.
1250.
1250.
1250.
1250.
1250.
1250.
1.00
1.00
1250.
1250.
1.00
1.00
1250.
1250.
1.00
1.00
1.00
1.00
1250.
1250.
1250.
1250.
(Page 1 of 2)
I
)
(cm
Solar Spectral Intensity (photons/cm2-sec
per wave number multi)
plied by 1017
00
Absorption Coefficients
(cm2/molecule multiplied by 10 1)
Molecular Oxygen
49375.
.000 000 062
.000 001 16
50312.5
.000 000 040
.000 001 50
50937.5
51562.5
.000 000 035
.000 002 50
.000 000 023
.000 005
52187.5
.000 000 017
.000 013 5
52812.5
.000 000 014
.000 071
53437.5
54052.5
.000 000 012
.000 000 011
.000 26
54677.5
55312.5
55937.5
56875.
.000 000 012
.000 000 006 2
.000 000 003 2
.001
.002
.005
.016
58125.
59375.
.000 000 002 32
.052
.000 000 001 72
.1
.000 000 004 8
Table Al.
.000 62
39
65
4
8
Quantum
efficiency
for oxygen
dissociation
Ozone
.030
.034
.038
.042
1.00
.94
.60
.33
.049
.056
.061
.067
.057
.028
.014
.015
.072
.071
.074
.077
.22
.080
.25
1.00
.080
.080
1.00
1.00
Spectral data used in photochemical calculations.
Width of
Spectral
Interval
(cm -1
)
Mean Wave
Number
(Page 2 of 2)
1250.
625.
625.
625.
625.
625.
625.
625.
625.
625.
625.
1250.
1250.
1250.
Height
(kilometers)
Oxygen
Concentration
(partigles
per cm
85
.301x10 14
. 548x106
.169x10
7
.953x10-2
.183x10-
79
.820x1014
.406x10- 1 6
.722x10- 8
. 953x10- 2
.327x10- 5
73
.219x1015
.927x10-
16
.334x10-8
.952x10-
67
.548x1015
.278x10- 1 5
61
.127x10 1 6
.950x10-
(sec
-1
)
)
(sec
3
-1
)
(sec
-1
)
J
3
(cm /sec)
15
4
55
.274x1016
.213x10O'
49
.576x1016
4
.197x10 14
43
.124x10
7
.122x10~
14
2
.195x10-
8
.949x10-2
.146x10-
8
.928x10-
.122x10-
8
.100x10-8
.640x10~
9
9
5
.617x10-
5
.146x10~
4
2
.532x10~4
2
.762x10-
.167x10~
4
.638x10-
2
.184x10-
3
.282x10-
2
.657x10~0
.921x10-
3
.904x10-
5
.277x1017
.475x10-15
.196x10~
31
.668x10 7
.185x105
.205x10- 1 0
.530x10-3
.889x10-
6
25
.169x10 1 8
. lllxo- 15
.208x10-
12
.416x10- 3
.382x10
7
Table A2
-
37
Calculation of time constants used in figure 2.
A-9
I
We have used the values
-4
-32
= .27x10
6
cm /molecule in agree-
ment with Reeves, Mannella and Harteck (1960), and with Kaufman and
,the
For
Following Benson and Axworthy, we have used
3 =
5 X 10 exp (-3o25-/7)
value
1.lx1-
Kelso has been adopted.
minations of
cI/sec
.
Kelso (1961).
cm /sec which was found by Kaufman and
This is the lowest of recent experimental deter-
reported in the literature,
and was chosen primarily
because of the finding by Phillips and Schiff that the rate of the
reactions
Q + OH -+H +
H-+-02+M -+
Oz.
H02+M
is extremely fast, so that higher values of
obtained in other ex-
periments may have arisen because of contamination by a reaction chain
initiated by these reactions.
Temperature dependence of
should be unimportant on theoretical grounds
and
(Bates and Nicolet, 1950),
and it has been neglected.
B.
Calculation of C
O2
Emission -
We have attempted to adhere as closely
as possible to the procedure followed by Murgatroyd and Goody.
The Curtis
matrix method assumes non-overlapping lines, and the cooling at the
level resulting from each such non-overlapping band is given by
.
P.
A-10
t
th
T)
is the Lorentz half-width of the lines at the pressure
0C,
,
where
is the number of lines in the band, and
is the line strength parameter:
O
band (cm2 -wave number per gram),
IAL
K
has the value 106.
the mass mixing ratio of
T
If
the acceleration of gravity.
per day,
is the mean line strength in the
is in degrees Kelvin
Murgatroyd and Goody divided the lines
in the 15 micron bands into two groups: "tweak lines",
and "strong lines",
Y-
933,
Tis
= 108.
'Y
The value of
= 33,
C
= 372,
In
used was
The Curtis matrices have been
.064 cm-1 at one atmosphere pressure.
computed for
CO. and
Y = 5, 100, 560, and 950.
The matrix for 950 was used
, but
for the strong lines with a slight compensating correction to
T)l
a new matrix was interpolated graphically for the weak lines.
Graphical
interpolation of the elements on the principle diagonal of the latter
matrix was guided by the requirement that the radiation to space for
each row, i.e.:
5
, should also interpolate smoothly.
The
resulting weak-line matrix was compared with that of Murgatroyd and
Goody by doing several sample calculations using vertical temperature
distributions of Murgatroyd (1957).
The results were in very good
agreement with those of Murgatroyd and Goody suggesting that the cooling
rates are not very sensitive to the exact interpolation used.
The matrix elements were computed with pressure as vertical
coordinate.
This coordinate was converted to a height scale with
A-11
increments of 3 kilometers between adjacent layers.
The error arising
from this conversion amounts to a height difference of not more than
4 kilometers, except in the computation of equilibrium temperatures in
the polar night zone.
As discussed by Curtis and Goody
(1956), above 75 kilometers the
decrease in the ratio of collision frequency to vibrational emission
frequency causes deviations from local thermodynamic equilibrium.
This
effect was taken into account approximately by Murgatroyd and Goody, and
in the present calculation the matrix rows corresponding to levels above
75 kilometers were simply multiplied by constant factors to bring the
results into agreement with those of Murgatroyd and Goody.
C.
Calculation of Ozone Emission -
Figure A2 is a plot of the cooling
rates calculated by Plass (1956) versus the Planck function at the center
of the 9.6 micron band,
B
.
These data suggest the following
formula which was used for ozone infrared cooling:
CP
C + C B0
dT .at
where the value of
C,
and
C
in ergs/gm-sec and cm 2-wave number/gm
are given in Table A3.
A-12
ML
11
10
X
20 kms
o
30 kms
&
40 kms
+
50 kms
a
60 kms
t
9
8
7
6
IN
5
0
4-)
4
4-)
3
At
+
2
1
a
a
0
0
x
x
I
100
i
200
I
300
400
B 0 3 (ergs/cm
Fig. A2.
I
2
-
500
sec - wave -
600
number)
700
Cooling rate due to ozone infrared emission versus the 9.6 micron Planck function.
from Pless (1956).
800
900
Data taken
I.
Height,
Kilometers
C,
64
0.
0.
61
31.5
-0.69
58
52.0
-1.15
55
65.5
-1.46
52
70.0
-1.55
31-49
70.8
-1.571
28
82.0
-1.571
25
95.0
-1.571
22
110.0
-1.571
Table A3.
Constants used in calculating cooling due
to ozone infrared emission.
A-14
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.=a
Biographical Note
The author was born July 16, 1933 in Hermosa Beach, California.
He attended the University of Southern California from 1950 to 1954
when he received his A.B. in the Division of Physical Sciences and
Mathematics.
From 1954 to 1958, he served in the Air Force as a
weather forecaster in Korea and in Florida, and as a weather reconnaisance observer in the Pacific.
His forecaster training was at the
University of California at Los Angeles.
graduate student at
Since 1958, he has been a
Massachusetts Institute of Technology, and has
occasionally been a consultant to the Rand Corporation.
to the former Janet Seitz, and has two children.
He is married