A STUDY OF TRACER TRANSPORTS BY PLANETARY SCALE WAVES
IN THE MIT STRATOSPHERIC GENERAL CIRCULATION MODEL
by
GARY EDWARD MOORE
B.S., Walla Walla College
(1973)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(AUGUST, 1977)
Signature of Author
[,.11-
epartment of Meteorology,
August 29,
1977
Certified by
Thesis Supervisor
Accepted by.............................................................
Chairman, Department Committee
WIT
Ial
A Study of Tracer Transports by Planetary Scale Waves
in the MIT Stratospheric General Circulation Model
by
Gary Edward Moore
Submitted to the Department of Meteorology on August 29, 1977, in partial
fulfillment of the requirements for the degree of Master of Science.
ABSTRACT
The quasi-geostrophic global stratospheric model at MIT developed by
D. Cunnold, F. Alyea, N. Phillips, and R. Prinn is used to study the zonal
and time-averaged transport of potential temperature and ozone by the planetary zonal wavenumbers 1-6. Several types of covariances between the winds,
geopotential, and the tracers are calculated in order to evaluate the directions and strength of the fluxes and the winds causing the fluxes.
The eddy covariances show that, in the lower stratosphere, ozone and
potential temperature b6th move polewards and downwards in longitudes of
low geopotential, in agreement with observations. In the upper stratosphere
and lower mesosphereozone and potential temperature are negatively correlated. In the midlatitudes of this region the eddy flux of potential temperature horizontally diverges; with downwards eddy fluxes occuring in low
latitudes and upwards fluxes occurring in high latitudes.
The meridional eddy flux of ozone is found to be only 1-3% efficient
since the meridional wind and ozone waves are nearly 900 out of phase.
Another contributing factor to the inefficiency of the eddy transport is
the tendency for the transient eddy flux to nearly cancel the standing eddy
flux in the lowest layers of the stratosphere. The total flux of ozone is
further diminished in some regions of the atmosphere by the opposition of
the eddy fluxes to the mean circulation flux.
In the final chapters, some of the calculated covariances are applied
in an attempt to calculate the eddy diffusivities. Because of the unsuitability of the mathematical form of the equation used to define the eddy
diffusivity, Kyy, and the uncertainties in the averaged covariances; Kyy
could not be estimated in a straightforward manner with any accuracy. The
eddy diffusivity Kyy is then assumed to be the product of the meridional
EKE and the timescale typical of the mixing processes taking place. The
timescale was calculated two ways. One used the LaGrangian integral timescale of Taylor's dispersion theorem. The other was an attempt to find the
timescale typical of the advective processes by using a mean length scale
and the mean zonal wind. Both results showed reasonable agreement of the
timescale with previously observed timescales in the atmosphere.
Thesis Supervisor: Ronald Prinn
Title: Associate Professor
TABLE OF CONTENTS
Page
INTRODUCTION
1.
The stratosphere:
2.
Representation of the eddy fluxes
3.
A consistent model of tracer transport in the stratosphere
4.
Introduction to the MIT stratospheric GCM
5.
A comparison of selected model covariances with observation
6.
The behavior of the meridional and vertical eddy wind
a descriptive review of its behavior
velocities
7.
A survey of the eddy fluxes of ozone and potential temperature
in the model
8.
A survey of the phase difference between the winds and the
tracers
9.
Introduction to the mixing length hypothesis
10. A first attempt at obtaining eddy diffusivities
11. The use of Taylor's dispersion theorem in determining eddy
diffusivities
12. Conclusions and summary
ACKNOWLEDGMENTS
APPENDIX
REFERENCES
TABLE I
FIGURE CAPTIONS
FIGURES
-1Introducion
A large numerical general circulation model(GCM) such as the
stratospheric model developed by Cunnold et al. (1975) allows a study
of the simulated atmosphere to an extent not achieved by observation.
Physical transport processes associated with certain mathematical terms
can easily be switched on and off for the purposes of study.
Furthermore,
the predicted variables are easily analyzed without having to be modified
as observations like radiosonde data are.
A diagnostic study of the eddy transport of tracers such as potential
temperature and ozone in the MIT Stratospheric GCM should be helpful
in determining whether the simulations of the atmosphere bear a resemblance
to observations.
Secondly;'the diagnostic study should be helpful in both
the interpretation and as a guide for future observations such as improved
remote sensing by satellite.
The average
transport by the eddies of such tracers as momentum, heat
and certain chemical species presently remains one of the least understood
of the processes occurring in the general circulation.
The actual eddy
transport of a tracer is the result of the little understood complex interactions
occurring between the zonal waves which shears, stretches, and deforms
parcels of air.
The wave interactions are reflected in the behavior of the
tracer and wind variation's amplitude, and in the variation of the phase
angle between the two.
By making a diagnostic study of the behavior of the
wind and tracer amplitudes and their phase differences, as a function of
zonal wavenumber, one might possibly find a simple means of roughly describing
the tracer transport done by the long planetary waves.
-21.
The Stratosphere:
A Descriptive Review of its Behavior
A great body of observational studies of the stratosphere have been
done by members of the Planetary Circulations Project at MIT.
Historically,
some of the first statistically significant studies were conducted on the
IGY (1957-1958) data collected in the region 10-100 mb.
The result of
these investigations are detailed quite thoroughly in Barnes' (1962) report.
Further data for the region 2-200 mb have been summed up quite neatly in a
report given by Newell et al. (1974).
Newell has also organized and summar-
ized the observations of tracer transports in the lower stratosphere in an
earlier paper (Newell, 1963).
The mean temperature field shows a surprising feature in that the 10200 mb layer generally has warm poles and a cold equator, despite radiational cooling at high and midlatitudes, and warming in low latitudes.
The
stratospheric profile exhibits its greatest complexity during the winter.
Above 100 mb during this period, the pole becomes cooler than midlatitudes.
Consequently one finds a CWC pattern on traveling from the pole to the
equator.
The observed zonal wind field is dominated by a winter polar jet which
increases at 40* from a minimum near 50 mb to what appears to be a maximum
at about .2 mb.
During the summer, easterlies extend upwards from around
30 mb at the midlatitudes to a broad core which Murgatroyd (1969a) shows to
close at .2 mb.
The zonal averaged meridional circulation was once a source of controversy until the eddy heat flux observations presented by Newell et al. (1974)
were used by Louis (1974) to deduce the circulation from the thermodynamic
equation.
The circulation shows a winter Hadley cell which extends into
-3the lower stratosphere.
the stratosphere.
The Ferrel cells of both hemispheres also penetrate
Thus a two-cell circulation operates in the stratosphere
with a direct cell in low and midlatitudes, and an indirect cell at high and
midlatitudes.
As one compares a 50 mb geopotential map with a 500 mb map, it becomes
evident that there is a qualitative difference between the stratospheric
and tropospheric eddies.
The smaller eddies of the troposphere disappear,
leaving only planetary scale waves 1-3 to dominate at 50 mb.
Sawyer (1965),
in an address to the Royal Meteorological Society, mentioned some work done
by G. R. Benwell that showed a phase relationship between 50 and 500 mb,
but there was no apparent correlation of the amplitudes of the waves between
the two levels.
White (1954) was the first to note that the horizontal heat flux in
the lower stratosphere was counter gradient.
Peng (1963) and Newell (1965,1972)
firmly established that the eddy heat flux is northwards in the lower
stratosphere
right up to high latitudes where a minor southward flux occurs.
A counter-
gradient heat transport by the eddies directly implies that EAPE is being
converted to ZAPE which maintains the positive zonally averaged temperature
gradient.
Unfortunately the vertical transient eddy heat flux cannot be
directly measured.
However, Oort and Rasmussen (1971) found that the stand-
ing eddies in the lower stratosphere transport heat downwards towards the
tropopause.
In the lower stratosphere the diabatic heating is sufficiently small
to allow the calculation of adiabatic vertical velocities from the thermodynamic equation.
Jensen (1961) made a pilot study and found a downward
motion of less dense (warm) parcels and an upward motion of denser (cooler)
parcels.
Oort (1964)
also found a convergence of the vertical transport of
-4geopotential.
Such a convergence means that pressure forces are doing work
on the stratospheric air mass.
One might draw the tentative conclusion
that the stratosphere appears to be driven from the tropospheric motions
in a fashion that Newell calls a "refrigerator" mode.
Loisel and Molla (1962) found that by using adiabatic vertical velocities the vertical and northward wind components of the transient eddies
are negatively correlated in the lower stratosphere, but positively correlated in the troposphere.
Further work by Newell et al. (1974) found that
negative values of the standing eddy covariance between V and W occur up to
10 mb.
Dickipson (1962) discussed the momentum budget of the IGY wind data.
The data show a horizontal convergence of angular momentum into the regions
of maximum zonal flow by the eddies.
Newell (1972) used five years of wind
data to study the northward eddy transport of angular momentum.
He also
found a maximum of northward eddy momentum flux just south of the polar
night jet which fluctuated as a function of the jet velocity.
The counter-
gradient flux-of momentum indicates that a conversion of EKE to ZKE is taking place.
Of all the tracers measured, the distribution of ozone in the lower
stratosphere has probably been studied the most.
It was once surmised that
ozone was in photochemical equilibrium; but the ozonesonde network observations show that the maximum total columnar amounts of ozone are found in
high latitudes during the spring.
Such observations are in direct contra-
diction to the photochemical equalibrium theory which implies an equatorial
maximum of ozone during the summer.
Martin (1956) found from observations of the horizontal eddy ozone
fluxes that ozone must be removed from the low latitude and moved northwards.
Newell (1963) used IGY data to find that the meridional flux by the transient
-5and standing eddies is predominantly polewards in midlatitudes.
The spread of radioactive tungsten 185 from equatorial nuclear tests
has been studies by Stebbins (1960), Feeley and Spar (1960), and Newell
(1963).
Feely and Spar (1960) used a Gaussian model to simulate the spread
of the debris in the lower stratosphere.
They found eddy diffusivities in
the horizontal of 1-106 m2 sec-1 and in the vertical of the magnitude
1-40 m2 sec~
. Newell (1963) published meridional-vertical distributions
of tungsten 185 which showed the northward and downward spread of the center
of gravity of the radioactive debris cloud in the lower stratosphere.
-6-
2.
Representation of the Eddy Fluxes
In the atmosphere there are four types of wave motion.
Waves are
classified as to whether they are standing or traveling waves or steady or
transient waves.
Standing waves are geostationary and are distinctly rela-
ted to forcing features of topography and heating at the ground.
Traveling
waves are generally will-o'-the-wisps since their source of energy fluctuates
as a function of longitude and time.
It is impossible to separate transient
standing wave amplitudes from steady traveling wave amplitudes; thus only
steady stationary waves can be separated from the total eddy quantity.
The model of Cunnold et al. (1975) that is used in this study specifies
the wave components as spherical harmonics at a given level.
The eddy
amplitudes at a given latitude, level, and instant of time can be written
as a zonal Fourier series, i.e.,
In this study six zonal wavenumbers plus the zonal mean are used to represent a variable in the model.
The averaging scheme of Starr and White (1951) will be applied in
describing the covariances of variables.
denoted by brackets, <( )>.
a star, ( )*.
The zonal averaging operator is
The zonal deviation from the mean is shown by
The time operator shall appear an an overbar, ( ),
time deviation is represented by a prime, ( )'.
while the
The order of applying the
averaging operators results in different kinds of eddy products.
If the
time operator is applied first, the product of two variables A and B is
written following Starr and White (1951) as:
-7-
)
3)
(2.2)
Term 1 of (2.2) is simple the product of the time-zonal means.
The second
term represents the eddy product due to standing steady waves.
The third
term represents the contribution due to the transient standing waves and
both. the transient and steady traveling waves.
If the zonal-averaging operator is used before the time-averaging
operator, the two-variable product, AB, becomes:
<AI3> =-<A> <0e> + <A><1'>
'(1
+,Af~t
Q)
(2.3)
(3)
The first term of (2.3) is the product of the zonal-time averaged terms
and it does not equal the first term in (2.2).
The second term is the pro-
duct resulting from correlations between transients in the zonally averaged
variables.
The last term contains the amplitude contributions from all
four types of waves.
The second term of (2.2) is calculated by taking time averages of the
B and C coefficients of (2.1).
Since the longitudinally dependent terms of
the Fourier series are orthogonal, the zonally-averaged product of two variables, P and
Q, can
be written as:
(pf >
6
s Z ( 60 %89Ph c-Qc)
nh
(2.4)
The third term of (2.2) is calculated as a residual by subtracting the first
two terms on the right-hand side of (2.2) from the total covariance.
-8The amplitude of a given zonal wavenumber is given by the relation:
W~I = -S BQ ,+ IN
)
16,(2.5) %
(2.5)
The phase angle of a given zonal wavenumber is found from the relation:
0 V,1
rt
(2.6)
gIC1
The amplitude and phase angle contain all of the information to be known
about the waves.
The cpvariance of two quantities A and B can also be found directly by
going from a spherical harmonic representation to a Gaussian-weighted grid
Once on the grid, the multiplication
space via a Fourier transformation.
of the two variables can be carried out.
An exponential filter is used to
smooth the grid field so that the truncation of the product will not result
The exponential filter has the added
in significant negative overshoot.
advantage of no phase shift and the conservation horizontally averaged AB
value.
After filtering,the reverse Fourier transformation is used to trans-
form back to spherical harmonics.
The transformation automatically truncates
the product and eliminates aliasing.
The zeroth order harmonic becomes the
zonal average of AB that is desired.
The spectral-grid-spectral method
(SPS) has the advantage of not specifying more information in the meridional
direction than the spherical harmonics allow, as is sometimes the case with
the zonal Fourier series technique (ZFS).
-9A Consistent Model of Tracer Transport in the Stratosphere
3.
After reviewing previous observations, Newell (1964) proposed a logi*
cally consistent model based on the phase relations between T
*
,
*
V , W
and
,
*
03
Newell spoke of the tracer transports in the lower stratosphere as
.
arising naturally from the motions and temperature patterns of the troposphere.
In a midlatitude amplifying disturbance the temperature wave lags the
geopotential wave by 0-ff radians.
The lag results in perturbation potential
energy generation and a westward slope of the lines of constant phase with
height.
The disturbance amplifies due to cold advection beneath the upper
tropospheric trough, and warm advection under the upper tropospheric ridge.
The vertical motions must be downwards in the air behind the midtropospheric
trough and upwards ahead of the trough so that the divergent motions can
make the necessary geostrophic vorticity change in the upper troposphere.
*
The results pattern of winds imply positive correlations between V
*
and T
.
*
,
W
Geopotential and meridional velocity are negatively correlated,
while the vertical velocity is positively correlated with geopotential.
Figure 1 shows the typical phase relations of a geostrophic disturbance in
the troposphere.
*
ratio ozone, X
,
Note that in mid-troposphere the perturbation mixing
is generally positively correlated with temperature, so
that its eddy flux is generally polewards and upwards.
In the lower stratosphere the vertically propagating waves phase -still slopes
westwards with height.
Temperature, ozone, and the meridional winds of the
disturbance are still positively correlated.
The crucial difference, shown
*
in Figure 1, is that the lines of constant phase for W
do not slope west-
*
wards at the same slope the V
lines do.
As a result, in the lower
-10stratosphere the vertical velocity advances in phase relative to V
*
T
,
*
*
and X
so that all of the correlations are negative.
The phase relation
between the geopotential and the winds remains of the same sign as in the
upper troposphere.
"refrigerator".
Newell (1963)
likened the lower stratosphere to a
In this region warm parcels are further adiabatically
warmed as they are forced to descend down the temperature gradient into the
upper troposphere, thus warming it.
Cooler upper tropospheric parcels are
forced up the temperature gradient, adiabatically cooling as they go, to
where they end up cooling the lower stratosphere.
An interesting question is:
why does the vertical velocity's phase
change relative to the cool trough as one goes from the troposphere to the
stratosphere?
It would seem that the air motions are trying to occur in
such a manner as to decrease the stability of the lower stratosphere.
The
decrease of stability implies an increase in potential energy due to the
tropospheric waves.
should result.
w
equation.
Thus an upwards eddy geopotential flux, <W 4>
>0,
Sawyer (1965) addressed this question with a comment on the
If diffusion is neglected, and there is to be a small merid-
ional circulation, then the differential vorticity advection term in the
w
equation has to oppose the thickness heating term in order for the
equation to be balanced.
For such an opposition to occur in a cooling layer,
there must be a negative correlation between vorticity and vertical velocity.
Thus rising motion occurs above the geopotential maxima and sinking motion
above the minima.
The result, again shown in Figure 1, is consistent for
a mechanically cooled lower stratosphere.
In the case at hand it is of interest to discover whether geostrophic
disturbances of limited zonal wavenumber can explain the observations presented.
Although the phase model proposed seems, by observation, to
-11physically hang together, it has by no means been proven to be the only consistent scheme.
A GCM might uncover different relationships.
The limited
set of physics in the GCM make is possible to deduce a physical mechanism
whereby a batter parameterization can be deduced.
The GCM can also produce
those correlations that are the "missing links" in the observations.
An
example would be the direct calculation of all covariances that contain
the vertical velocity.
The GCM has a final obvious advantage in that it
can be extended to regions that are not easily observed.
-124.
Introduction to the MIT Stratospheric GCM
The data used in this study come from various runs of the quasi-geostrophic model of Cunnold et al. (1975).
The reader is referred to the
referenced cited for a more detailed explanation of the dynamical and
chemical formulations in the model.
At best only a simple physical descrip-
tion of the model will be given in this section.
The model has 25 prediction levels in the vertical axis which is in
increments of log pressure.
The lowest level is the ground and the highest
level is centered at 71 km.
The model uses a global spectral technique and
represents each variable at a level with 79 spherical harmonics.
The
longitudinal truncation only allows zonal waves of up to wavenumber six.
The set of dynamical equations used are the energetically consistent
balance equation set described by Lorenz (1960).
The model does not pre-
dict the globally-averaged temperature, but rather it is externally specified.
Another simplification is that the horizontal advection due to the
nondivergent wind is neglected.
Vertical transport of ozone and momentum
by the subspectral scale eddies is simulated with a specified vertical
eddy diffusivity.
Frictional stress, topography, and all types of thermal
forcing are included as terms in the vorticity and thermodynamic equations.
The ozone mixing ratio is predicted from two equations which separately
predict the globally-averaged ozone mixing ratio and its deviation from
the global mean.
The coupling between both prediction equations comes
only through a single term in the global average prediction equation.
This
term is the vertical divergence of the globally-averaged flux due to the
product of the deviations of ozone mixing ratio and the vertical wind from
the global mean.
-13The photochemistry in the model includes reactions of odd nitrogen and
water vapor with odd oxygen.
The photochemistry and the dynamics are
coupled through both the temperature dependent reaction rates and the
diabatic heating due to solar radiation absorption by ozone.
-145.
A Comparison of Selected Model Covariances with Observation
* *
Of crucial importance to the phase model is the <V W > covariance
*
which sums up the phase relation between V
eddies.
*
and W
for all classes of
Because of equatorial symmetry, only the winter and spring northern
hemispheric seasonal averages are presented in Figures 2 and 3 respectively.
*
A positive average correlation exists between V
the summer and winter months in the troposphere.
*
and W
during both
The positive correlation
indicates that upward and poleward motions are taking place in the model as
they do in the observed atmosphere.
The winter hemisphere has correspond*
ingly larger covariances due to the increased magnitudes of the V
winds.
*
,
W
The region of 200-30 mb possesses a negative average correlation
*
between V
*
and W
for both summer and winter seasons.
The observed adia-
batic vertical wind correlations are in a sense justified since it seems
that long planetary waves with geostrophic winds can have velocity components that result in the negative correlations.
In the winter season a
maximum covariance of one-half the magnitude of the upper troposphere occurs
at about 90 mb at 40-50*.
In the summer the maximum still lies at 90 mb,
but it is shifted much closer to the equator, having a peak near 20-30*.
While the lower stratospheric maximum is about the same magnitude as the
peak in the winter, it is almost five times larger than the upper tropospheric covariances.
This fact indicates that the lower stratosphere has
much less seasonal variability of the mixing slopes than the troposphere.
Above 30 mb the covariance patterns again change.
The winter season
*
undergoes a shift back to positive average covariances between V
*
and W
The positive covariances are about as large as those in the lower stratosphere up at about .5 mb.
It is not apparent at this point whether the
-15vertical velocity's phase has slipped back to that of the troposphere, or
whether it has advanced in phase half a vertical wavelength.
The summer
**
hemisphere maintains the negative V ,W
at high to midlatitudes.
correlation through the stratosphere
At equatorial latitudes a slight positive corre-
lation eixsts up to .5 mb.
Above .5 mb, the low latitude winter season correlations again become
negative.
In the summer season the positive correlation in the equatorial
regions also go negative.
*
*
After this once over of the V ,W
average covariances, one sees several
unique regions in the vertical, namely 1000-200 mb, 200-30 mb, 30-.5 mb,
and
above
and fall.
,.5
mb.,
The
regions
are
quite
distinct
especially
in
the
spring
These two seasons have nearly symmetric covariances that are
almost identical to one another.
In the upper stratosphere there appears
to be two distinct latitudinal regions.
The regions in the vertical are
represented by dominantly positive (1000-200 mb), negative (200-30 mb), and
positive (30-.5 mb) covariances.
The +,-,+,-,
pattern looks suspiciously
as if it might have its roots in vertically periodic motions.
* *
The vertical flux of geopotential by the eddies, <W $ >, indicates
the upwards or downwards export of potential energy.
A vertical conver-
gence of this covariance indicates that a source of energy within the layer
exports eddy pressure work to other regions.
Figures 4 and 5 show the vertical eddy flux of geopotential.
During
the winter season, eddy potential energy propagates vertically upwards at
high to midlatitudes clear to the top of the model.
the eddy covariance occurs in the troposphere.
Vertical divergence of
This result is hardly sur-
prising in view of the present knowledge of the energy cycle of the troposphere.
Convergence occurs from 200 mb to about 25 mb.
Such a finding
-16reinforces Oort's (1964) analysis which found the lower stratosphere to be
driven by the troposphere and not by internal energy sources.
occurs above 25 mb up to about 2 mb at 300 and .3 mb at 70*.
occurs above the region of divergence.
Divergence
Convergence
One would expect to find conver-
gence near the upper boundary as pressure forces propagating from below do
work on the layer adjacent to the rigid lid.
The most peculiar feature in the winter hemisphere is the strong region
of divergence in the low latitude region .1-.5 mb.
In the lower mesosphere
the mean temperature has a maximum at midlatitudes and minima at the poles
and equator.
Although the eddy heat fluxes are directed along slopes going
northwards and downwards, the eddy heat flux at low latitudes is downgradient like the stratosphere. An examination of the mean temperature and zonal
wind gradients in both the vertical and horizontal did not seem to reveal
any favorable conditions for baroclinic instability as a source for the
downwards propagating eddy potential energy flux below this region.
How-
ever a growing disturbance might occur if the ozone radiative-dynamical
coupling described by Leovy (1964) occurs.
If significant coupling occurs,
EKE is generated when parcels carry excess ozone on the downward side of
their trajectory and undergo a net heating due to solar absorption.
Some-
times the EKE generation outstrips the radiational damping of the wave
motions.
Barotropic instability cannot be ruled out either.
Since the
region of the presumed instability occurs southwards of the polar night jet
where large positive horizontal shears can make a change of sign in the
potential vorticity gradient.
The question can be settled only by looking
at more covariances and their divergences.
The summer season looks much like the winter season and high to midlatitudes; the only difference is that the magnitudes of the convergences and
-17-
divergences are much smaller than in the winter.
The low latitudes of the
summer season lack the odd feature of the winter season in the upper stratosphere and lower mesosphere.
During the spring and fall seasons there is still an upwards eddy flux
of geopotential in midlatitudes.
Also there is still convergence in the
lower stratosphere, divergence from 20 mb to the stratopause, and convergence
above that.
At high latitudes there are downward eddy fluxes of geopotential
for both hemispheric seasons.
It is of interest to check on the sign of the average producE of the
vertical and horizontal eddy fluxes in order to determine the direction of
the slope 'along which the eddies are moving ozone.
*
*
*
Figures 6 and 7 show the
*
seasonally-averaged eddy covariance, <V W 03 03 >.
During the winter season ozone is generally transported downwards and
equatorwards in the troposphere.
In the lower stratosphere at midlatitudes,
ozone is transported downwards and polewards in the region up to 30 mb.
maximum covariance occurs near 100 mb at 50*.
The
Thus it is quite possible for
large amounts of ozone to be brought into the troposphere during the formation of an upper tropospheric thermally-indirect front.
Danielson (1968)
found from analyzed observations that ozone-rich air is brought down into
the troposphere during a frontal occurrence.
Above 30 mb, the midlatitude winter season shows a tendency to have
ozone brought upwards and polewards with a slope sign like that of the
troposphere.
In the low to midlatitude mesosphere, the slope of the ozone
fluxes appears to be directed downwards and polewards.
Ozone convergence
is taking place in this region.
In the summer hemisphere the tropospheric and lower stratospheric
covariances are smaller; however, the direction of the slope remains unchanged.
--18The only significant difference is that the polewards and downwards slopes
extend to the top of the model throughout almost all the hemisphere.
During the spring and fall seasons there are still polewards and upwards directed flux slopes in the troposphere, and poleward-downwards
slopes in the lower stratosphere.
In the region 1-20 mb, a polewards and
upwards slope occurs in midlatitudes for both the spring and fall seasons.
The sign of the average product of the horizontal and vertical eddy
fluxes of temperature are also calculated in order to note the thermal
behavior of certain regions.
of <V W T T >.
*
the <W
Figures 8 and 9 show the seasonal averages
A connection can be drawn between these covariances and
*
#
>
dovariance by remembering that Eliassen and Palm (1960 *
*
Equation 10.12) showed that positive <W $ > directly implies a poleward
heat flux for quasi-geostrophic waves.
The winter season shows the eddies in the troposphere to be dominated
by mainly polewards and upwards slopes along which heat is transferred.
In
the upper troposphere near the tropopause gap the direction of the flux
slopes changes from polewards and upwards to polewards and downwards.
change in
the direction of the slope is
lower than in
The
the case of ozone.
The lower stratosphere up to about 30 mb has a polewards and downwards
slope along which the eddies move heat.
This result is consistent with
Newell's (1964) concept of the behavior of the lower stratosphere.
Since
the horizontal eddy flux direction is counter to the average temperature
gradient the region sppears to be destroying EAPE and converting it to ZAPE.
At the same time the vertical eddy flux is downgradient which indicates a
conversion of EKE to EAPE.
At very high latitudes the polewards and downwards slope, along which
sensible heat is moved, persists up into the mesosphere.
The covariance
-19increases steadily with height from about 50 mb on up.
Above 20 mb in the high to midlatitudes, the direction of the potential
temperature fluxes tends upwards and polewards.
In the low to midlatitudes
an upwards and equatorwards direction of the heat flux is evident.
A
large heat source is implied at midlatitudes to balance the divergence of
heat.
The most interesting feature of the winter hemisphere is the strong
region of an equatorwards and downwards flux of heat centered near 3 mb and
200.
This feature
appears to be connected with the <W
*
*
#
> feature during
the winter season.
The summer
season shows the polewards and upwards flux of temperature
in the mid to lower troposphere.
While the stratosphere and the mesosphere
have the expected polewards and downwards flux slopes that are opposed to
those in the troposphere, the covariances in the stratosphere are miniscule
compared to the winter season as the observations dempnstate.
The spring and fall seasons present a confusing situation in regard to
the direction of the heat fluxes.
The lower stratosphere up to 30 mb of
both spring and fall have polewards and downwards eddy flux slopes.
Above
30 mb both seasons possess a polewards and downwards flux slope at low and
high latitudes which continues on up into the mesosphere.
The midlatitudes
for both seasons have a polewards and upwards flux slope above 30 mb.
This
behavior is similar to that in the troposphere.
It is interesting to see how the two tracers correlate.
A positive
correlation is expected in the lower stratosphere due to the similarities
of the source regions.
The covariance of ozone to potential temperature is
shown in Figures 10 and 11.
-20The winter season shows ozone to be dominantly positively correlated
to temperature below 3 mb.
Above 3 mb the only positive correlation that
extends into the mesosphere occurs only at very high latitudes.
The only
remaining region where a negative correlation exists is in the lower tropical
troposphere where ozone is quite scarce since there is no significant
local source, while the temperatures are quite large.
In the summer season the positive correlation extends up to 5 mb)and
above 5 mb a negative correlation exists.
It should be noted that for both
winter and summer seasons the changeover in the sign of the correlation
occurs at about 10 mb near the equator.
The spring season has a predominantly negative ozone-temperature corelation in the lower to midtroposphere.
tion exists up to about 5 mb.
Above 300 mb a positive correla-
Above 5 mb the correlation becomes negative.
The fall season has a negative ozone-temperature correlation only in the
low latitude troposphere. The region of positive correlation extends up to
5 mb.
The region above 5 mb and equatorwards of 600 has a negative ozone-
temperature correlation.
The negative correlation above 5 mb is probably due to the fact that
the rate constants for the production and destruction of ozone are dependent
on the exponential of the inverse temperature.
As temperature decreases,
the production rate increases slightly, while the destruction rate due to
NO,0 decreases.
In the region below 5 mb the bulk of the ozone is produced
near the tropics; however, dynamical motions move ozone into warmer regions,
thus making the correlation positive in the stratosphere.
-21-
6.
The Behavior of the Meridional and Vertical Eddy Wind Velocities
A study of the eddy winds themselves is of prime importance since the
winds make up one-half of the amplitude factor of a flux.
phase lag between a tracer,
For a given
*
Q
,
and the wind, the strength of the meridion-
al wind of the eddies determines in part the size of the poleward flux
which is of primary importance in this study.
Another reason for the
interest in the eddy winds stems from the need to compare the modeled eddy
winds with the observations in order to see if the winds in the model are
of sufficient accuracy so as to result in the proper eddy fluxes.
A final
reason for investigating the meridional wind amplitudes is that the decay
and growth rates of the amplitudes of zonally propagating waves can be related to eddy diffusivities in a crude fashion as Green (1970) has pointed
out.
The model makes use- of only the horizontally nondivergent wind, V$,
for the horizontal advection of heat or ozone.
Thus the advection by the
eddies will be considered to be done entirely by the nondivergent wind.
Under these circumstances the nondivergent meridional wind relations of the
form:
(6.1)
can be written.
Figures 12, 13, and 14 show the area weighted variances <V2 >
<(V
2 >9
and <(V')2> 22
respectively for the months of June and December of Run 26.
The <V
and <(V') 2 > are the steady standing wave, and residual transient
eddy components respectively.
-22The total meridional EKE, <V >, has a winter maximum at 200-500 mb and
500 which is about three times greater than the summer maximum at 500.
The
observed winter maximum according to Oort and Rasmussen (1971) is about
2
2
2
2
320 m /sec compared with the model's 700 m /sec2.
The summer maximum is
2
2
2
2
observed to be 195 m /sec compared to the model's 263 m /sec2.
One dis-
tinct difference between the model's tropospheric meridional wind variances
and the observed variances occurs in the equatorial regions.
At the equator
the observed total variance <V2> equals about 65 m2/sec2 compared with the
model's huge value of 250 m 2/sec2
The difference in the variance can be ascribed to the fact that the
model is a quasi-geostraphic model.
Near the equator, the assumptions in-
volved in the quasi-geostrophic approximation do not hold.
The Rossby
number near the equator becomes of order 1 or greater, therefore making it
impossible to balance the Coriolis term with the pressure gradient term.
The balance should occur between the pressure gradient term and the inertial
term.
By using the Coriolis force, the wind in the waves is strongly over-
estimated.
These spuriously large amplitude eddies make it quite difficult
to calculate the interhemispheric tropospheric transport.
The lower stratosphere of the model exhibits a strong dropoff in the
EKE as one progresses from the troposphere to the lower stratosphere.
At
650 and 50 mb, the winter observations show a variance of 210 m 2/sec 2, but
2
2
the model gives a value of only 150 m /sec2.
height during the summer is even greater.
The dropoff of EKE with
Summer observations at 55* and
2
2
2
2
50 mb show values of 26 m /sec while the model has a value of 50 m /sec .
The highest analyzed observations by Newell (1972) show the total
winter meridional wind variance to be 225 m 2/sec2 at 10 mb and 600 while
the model has a value of 240 m 2/sec2 at this location.
Thus in the winter
-23hemisphere the tropospheric eddies are two times too strong; while in the
lower stratosphere they are too weak.
By about the altitude of 10 mb the
eddies appear to be about the right strength.
The equatorial regions suffer from quasi-geostrophic theory breakdown;
but despite the fact that the tropospheric meridional EKE is five times
too strong, the variance at 10 mb of 10 m 2/sec2 agrees quite well with the
values found by Newell (1972).
On the average the summer hemisphere shows
a tendency for the modeled meridional EKE to be one to two times greater
than the observations.
By studying the analysis of observations made by Oort and Rasmussen
(1971), one finds that in the region 200-500 mb the ratio of the transient
wave EKE to the steady standing wave EKE is about six during the winter,
and it increases to about 25 during the summer.
The midlatitude ratio ob-
served in the model is slightly more erratic but it is generally in the
range of four to ten during the winter and increases in the summer midlatitude regions to about 25 to 35.
The tropospheric ratio of the standing EKE
to the transient EKE seems to agree with observations in a satisfactory
manner at midlatitudes.
In the summer hemisphere, at 50 mb, the midlatitude steady standing
wave EKE is greater by an order of magnitude than what is observed.
The
transient EKE appears to be twice the observed value as was the case in
the lower troposphere.
Above 50 mb both types of eddy variances fall off
rapidly and settle down to a transient to standing wave ratio of five to ten.
In the winter hemisphere the standing waves appear to be attenuated
much too rapidly in the high to midlatitudes.
At 50 mb and 600 the observa-
tions show an EKE of 50 m 2/sec2 while the model has an EKE of 60 m 2/sec2
yet the standing EKE in the troposphere is two
times too large.
It seems
-24-
that the standing waves are sufficiently trapped so as to allow only the
amount of variance that is observed.
The transient waves are strongly
trapped so that at 50 mb only the amount of variance comparable to the
observations exists; despite the fact that in the troposphere the transient variance is two times too large.
Above 50 mb the transient wave to
standing wave amplitude ratio is about five to ten in midlatitudes.
The standing wave spectra are plotted in Figures 15, 16, and 17 as
a function of zonal wavenumber.
Each spectrum has been plotted so that the
summation over the zonal wavenumbers of all the amplitudes equals one.
what is plotted is the fractional contribution to the total variance.
132, 40, 12,
Thus
The
1, pnd .089 mb levels are of particular interest since the
levels each occur in a distinctly differently behaving section of the atmosphere.
The horizontal resolution is limited to the equatorial region and
the midlatitudes at 40*.
The winter hemisphere standing wave spectrum exhibits a spectral gap
at wavenumber 4 in the lower stratosphere.
wavenumbers 3 and 5.
In this region, maxima occur at
In the upper stratosphere the gap disappears and wave-
number 5 strongly dominates.
As one progresses up into the mesosphere the
peak moves to wavenumber 3 at 1 mb and finally to wavenumber 1 at .089.
The shift to dominant lower wavenumbers is both observed and expected from
theory.
At the equator a spectral gap still appears at wavenumber 4 in the
lower stratosphere.
The maxima occur at wavenumbers 2 and 5, with wavenum-
ber 2 being very dominant by 40 mb.
It is not clear why a spectral gap
should appear at the stationary wavenumber 4 since sources for the forcing
of wavenumber 4 should be about the same as wavenumber 5.
There is a possi-
bility that wave energy is being reflected from the truncation at zonal
-25-
wavenumber 6, which makes wavenumbers 5 and 6 too large.
The summer hemisphere exhibits maxima at wavenumbers 2 and 4 in the
midlatitude lower stratosphere.
In the upper stratosphere the maxima
shifts to wavenumbers 1 and 3.
At the altitude of 1 mb wavenumber 3 is
very dominant.
Upon progressing upwards to .089 mb an even more unusual
spectrum exists where wavenumbers 1 and 6 dominate.
The seeming dominance
of wavenumber 6 may again be due to the spectral truncation.
Figure 18 shows the energy-weighted zonal wavenumber contours.
The
weighted wavenumber is defined through the relation:
L~~Yvxv*>''/
I
(6.2)
A comparison of Figure 18 with Figures 15, 16 and 17 shows that the addition
of the transient wave variance weights the zonal wavenumber center of moment
towards a higher wavenumber.
In figure 18 one can see the Eulerian spec-
trum characteristic baroclinic peak near zonal wavenumber 5.
The shift of
the spectrum towards lower zonal wavenumber shows up quite strongly at high
latitudes and altitudes.
The trapping of wavenumbers greater than 2 appears
to be stronger during summer than winter.
This fact is probably due to the
easterlies and the zero line in the zonal wind field.
The average phase change of the standing waves as a function of height
is also of interest.
Van Loon et al. (1973) show that the zonal wavenumbers
1 and 2 vary roughly as ffand n/2 radians per 30 km respectively.
At mid-
latitudes the observed phase of wavenumber 1 varies linearly with height.
Wavenumber 2 at midlatitudes shows a larger rate of phase variation near the
ground.
-26Figures 19, 20, and 21 show the phase in radians as a function of
height for wavenumbers 1, 2, and 3 respectively during the winter.
At lati-
tude 40 during winter, wavenumber 1 appears to vary 27r radians in 30 km.
Furthermore its phase becomes constant with height above 1 mb.
This con-
stancy of the phase with height could be due to a standing wave pattern
caused by upward propagating waves reflecting off the regions of large <U>
and the lid.
Possibly the Newtonian cooling is not large enough to prevent
the wave reflection off the jet.
Simmons (1974) found that the <U> and the
presence of Newtonian cooling are critical in avoiding the reflecting wave.
In winter, stationary wavenumber 2 calculated by the model also exhibits abouta 2n radians shift per 30 km and a constancy of phase above 5 mb.
During the summer, the phase shift is only ffradians per 30 km at midlatitudes, but reflection is still apparent above 5 mb.
Notice that in both
wavenumbers 1 and 2 the further polewards one goes, the lower the level
drops at which the constancy of phase appears.
This effect could be the
result of either ducting or the change in the resulting damping rate of the
wave.
Wavenumber 3 has again a winter midlatitude 2ff radian phase shift
per 30 km and again it exhibits a changeover to constant phase at 5 mb.
In
the summer hemisphere the phase shift is only ffradians per 30 km.
Matsuno (1970) used a quasi-geostrophic model in spherical coordinates
and found that the standing waves are attenuated much too strongly.
Second-
ly, he found that his model gave phase shifts of ffradians and nearly 1.5ff
radians per 30 km for wavenumbers 1 and 2 respectively.
Matsuno also found
very little southward tilt of the isophases which indicated little or no
ducting of waves towards to equatorial zero line.
This result could stem
from the fact he was using a hemispheric, not a global model.
-27The excessive phase tilt in the lower 30 km of the model in this study
can be due to either an excessive southward phase shift or too much vorticity advection which, when balanced by vortex stretching, leads to a reduced
vertical wavelength for the waves.
In the equatorial regions all wavenumbers show that most of the phase
shift takes place above 50 mb where the zero line for <U> reaches the equator.
This fact suggests that ducting of the wave phase from different lati-
tudes could be responsible for this apparent shift.
Figure 22 shows the total vertical velocity variance due to the eddies.
There is not much to say about the variance except that it has a very strong
similarity, to the <V 2>.
One other note of interest is that in the upper
atmosphere, the maximum in the variance occurs equatorward of the polar
night jet.
The summer hemispheric atmosphere above 30 mb has little or no
vertical eddy velocities to speak of.
The only way to get at what the transient eddies are doing is to plot
and study the time histories of the phase and the amplitude of the wave.
A
preliminary study was made by making a time history for the month of December at the 12 mb level and at latitude 40 for wavenumbers 1, 3, and 5.
These histories are shown in Figures 23, 24, and 25.
The purpose of such a plot was merely to look at the typical behavior
*
of the amplitude and phase of V
as a function of time.
From such a plot
the velocity of the wave with respect to a fixed point can be found.
Second-
ly, it was unknown whether or not discontinuities would occur in the wave
history due to the daily sampling.
The failure to sample often enough has
been thought to be a problem for the higher wave numbers in particular.
Wavenumber 1 seems well-behaved in both amplitude and phase.
30-day decay in the trend of the wave amplitude shows up.
A definite
The phase exhibits
-28-
a general westward motion with no great discontinuities.
The average
phase velocity is about -9 m/sec with respect to the ground.
Wavenumber 3 appears to hesitate between going from a decaying mode to
a growing mode near days 2-4.
in the wavenumber 1 case.
respect to the ground.
The variations in amplitude are greater than
The phase shifts very quickly westward with
It moves at an average velocity of about -30 m/sec.
From the Rossby wave dispersion relation, the latitudinal scale of this
wave must be quite large in order to produce the large retrogression.
Wavenumber 5 has very large amplitude changes with a period of roughly
7 days.
Note that near day 6 the sampling of the wave seems to catch a
wave in the.act of switching from a decaying to a growing mode, which causes
The phase motion of the wave is east-
a spurious "bump" in the amplitude.
wards with an average phase speed of about 4 m/sec.
A calculation was made of the average mixing slope defined by Reed and
Unfortunately the mixing slope covariance shows
German (1965) as <W /V >.
a large amount of point-to-point variation.
The problems appears to be too
short an averaging interval, and an improper grid and spectral representa*
tion which allowed small values of V
to dominate the covariance.
Another less "noisy", but possibly less legitimate estimate of the mixing slope can be formed by taking the ratio <VW>/<VV>.
assumes that
Kyy = T<V2>
*
relation
and
Kyz = T<VW>
This approximation
instead of Reed and German's
*
L(Y) = L(Z)(W /V ).
T is a time scale that is not necessarily
velocity independent, but is characteristic of the exchange of parcels.
A
more developed discussion of this matter will be given in a later chapter.
Figure 26.
Shows the mixing slope defined by <VW>/<VV>.
The dominant
feature below 20 mb is the downwards-polewards slopes in the lower stratosphere and the upwards-polewards slopes in the troposphere.
The typical
-29stratospheric slope is 2-4.E-04 radians,
slopes of the same magnitude.
Reed and German (1965) found
In the troposphere, Hantel and Baader (1976)
found slope maxima of 1.OE-3 radians which are also reproduced in Figure 26.
In the summer hemisphere, the rest of the atmosphere above 20 mb has a
downwards-polewards slope that has a maximum near midlatitudes and minima
at the poles and the equator as one would expect.
The winter hemisphere
shows downwards-equatorwards slopes in the upper stratosphere and lower
mesosphere of the same sign as the troposphere.
The maximum slope is cen-
* *
tered near the region where the <W $ > anomaly exists in the mesosphere.
Above 1 mb, in the low to midlatitudes the downwards-polewards slopes
again dominate.
-30-
7.
A Survey of the Eddy Fluxes of Ozone and Potential Temperature in the
Model
The poleward fluxes of ozone are of particular interest in this study.
These fluxes are responsible for the ozone number density maximum which
appear in the lower stratospheric high latitudes.
maxima in Figure 27 where <03> is shown.
One can observe these
The ozone number densities simu-
lated by the model appear to agree reasonably well with the observations
analyzed by Wilcox et al. (1977).
The total meridional ozone flux due to the eddies is shown in Figure 28.
All fluxesthat are presented are area-weighted to give an idea of their true
contribution to the global flux budget.
The winter hemisphere shows an
equatorward flux of ozone at high to midlatitudes in the lowest part of the
mesosphere near 1 mb.
This flux is downgradient according to Figure 27.
Everywhere else in the winter hemisphere, down to the 30 mb level, a poleward flux exists which comes to a maximum between 10 to 20 mb and 400.
Below 30 mb a confusing region occurs.
Here the eddies act to cause
alternating signs in the flux in going from level to level.
At midlatitudes
the vertical average of the fluxes between 50 and 150 mb in midlatitudes are
actually of the same size as those near 20 mb.
Proof of this is shown in
Figure 29 where the average flux of the 50-150 mb layer is shown.
that large interhemispheric transports appear to be occurring.
Notice
The direc-
tion of the flux is towards the winter hemisphere, and this kind of export
is expected due to the exchange of stratospheric air with the troposphere
at midlatitudes.
Below 150 mb, the eddy ozone flux decreases, but it re-
mains polewards at high to midlatitudes.
In the equatorial troposphere the
eddy flux is directed equatorwards, which implies downgradient transport
-31into the tropical ozone-poor air shown in Figure 27.
In the summer hemisphere, the mesospheric horizontal ozone eddy flux is
generally directed equatorwards.
tains its poleward flux.
The midlatitude upper stratosphere main-
The lower summer stratosphere in the 50 to 150 mb
region has poleward fluxes north of 500, and net equatorwards fluxes south
of 50*.
It will be shown later that the vertical transient eddy flux is
responsible for the change of direction of the flux from one season to
another since it changes the mixing surface's slope from horizontally upgradient to downgradient fluxes.
The upper troposphere has poleward fluxes
of ozone, but the lower troposphere has equatorward directed fluxes.
This
behavior is reflected in the changeover of the horizontal ozone gradient
shown in Figure 27.
A further breakdown of the eddy ozone flux was made.
The steady stand-
ing wave flux and the residual transient wave flux are shown in Figures 30
and 31 respectively.
In winter the region from 30 mb up to the stratopause
has a transient eddy flux that dominates over the standing eddy flux by
factors of five to ten.
Both the eddy flux components are directed polewards.
Below 50 mb in winter, the two eddy components are much larger than the
total eddy flux.
However, at many places they nearly cancel each other and
leave only a small residual.
This behavior is quite puzzling.
Somehow the
transient eddies are tied to the standing eddies in such a manner that
causes them to oDnose one another systematically over a large number of
points.
The interhemispheric flux in the 50 to 150 mb region seems control-
led by the transient eddies which win out in the tug-of-war going on between
the standing and the transient eddies.
In the summer hemisphere the opposition of the flux components occurs
from the troposphere clear up to 30 mb.
At high latitudes the cancellation
-32effect continues up to the stratopause.
At midlatitudes in the upper
stratosphere and lower mesosphere, the transient eddy flux dominates over
the standing eddies by at least a factor of two.
The total vertical flux of ozone, including the mean circulation, is
shown in Figure 32.
The winter hemisphere shows a strong downwards flux
of ozone throughout the whole vertical extent of the atmosphere from 10-50*.
The maximum in the flux occurs near the tropopause.
This result agrees
with the flux directions given by the tropopause gap mechanism of Danielsen (1968) and the downward Hadley-Ferrel cell branches of the mean circulation.
At very high latitudes ozone is moved upwards at all levels.
The summer lemisphere troposphere has a downwards total flux of ozone
in the 20-40* belt, and an upwards flux in the 50-60* belt.
Upon comparing
the total flux in Figure 32 with the standing and transient fluxes in
Figures 33 and 34, one finds that the mean circulation transport of ozone
tends to oppose the transient eddy fluxes, and actually dominates the vertical fluxes in some regions.
Mahlman (1973) showed that this opposition of
the mean circulation and transient eddy fluxes also occurs in a primitive
equation model.
In summer, the total vertical flux of ozone in the midlatitude lower
stratosphere is directed upwards.
The upwards flux is due to the action of
the eddies, mainly the transient eddies and not the mean circulation.
This
direction of eddy flux is completely opposite to the winter hemisphere and
its cause is not clearly understood at the present.
At low latitudes in
the summer mid-stratosphere the total vertical flux is directed upwards.
However in this case the eddies have a combined downwards flux, but the
Hadley circulation opposes and wins out over the eddies.
Higher up, near
5 mb, the summer season midlatitude transient eddies are upwards and
-33-
dominate the eddy flux budget over the standing eddies by as much as a factor of ten.
downward.
In the low and high latitude regions the eddy ozone flux is
The domination of the vertical eddy flux by the transient eddies
is probably due to the fact that tropospheric-forced steady stationary
waves are trapped by the easterlies and the zero wind line.
Again, when
looking at the zonally-averaged streamfunction of Cunnold et al. (1975)
above 5 mb, one sees that the eddy flux is in opposition to the mean
circulation flux.
In the winter hemisphere for the low to midlatitudes, the 5 to 20 mb
layer has upwards transient eddy fluxes.
show downward flpxes.
The high latitudes of this layer
In winter, the 50 to 200 mb region shows downwards
transient eddy fluxes at most latitudes.
In the troposphere the transient
eddy fluxes can best be described as everywhere opposing the mean circulation transport.
The standing eddies in the winter hemisphere compete with the transient
eddies up to the 1 mb level where they abruptly taper off due to the fact
that the lines of constant phase become vertical at this level.
The 5 to
30 mb region at midlatitudes exhibits an upwards eddy flux which generally
acts to aid the transient eddies in opposing the mean circulation.
In the
tropical lower stratosphere a curious standing eddy flux pattern appears.
A downwards flux of ozone due to the standing waves appear in the summer
hemisphere, and an upwards flux appears in the winter hemisphere.
The horizontal flux of potential temperature due to all the eddies is
shown in Figure 35.
The flux of temperature is polewards clear to the top
of the model in the winter hemisphere throughout the high and midlatitudes.
This type of flux is expected if the wave disturbances are to propagate
vertically.
-34In the mid-troposphere, there occurs a convergence of heat into the
equator coupled to a divergence at the tropopause.
This feature also
appeared in Manabe and Mahlman's (1976) primitive equation model.
They
suggested that the flux was due to the gravity-Rossby wave modes.
The
quasi-geostrophic model can roughly simulate such modes at the equator,
and if they are forced by the applied heating they could be responsible for
the feature.
The most exciting feature by far is the equatorward flux of heat near
.5 mb in the low latitude winter hemisphere.
This flux appears to be rela*
*
ted to the strong low latitude divergence of <W $ >.
bits a strpng divergence of heat.
The 20-40* belt exhi-
Solar heating by ozone appears to be the
only significant source of heat for this region.
The equatorwards flux of
heat according to Eliassen and Palm (1960) would cause a downwards eddy
geopotential flux.
The vertical flux of potential temperature by all the eddies is shown
in Figure 36.
The vertical potential temperature flux is directed upwards
in most of the troposphere for both summer and winter as Oort and Rasmussen
(1971) show.
Above 10 mb, the high to midlatitude winter upper stratosphere
and mesosphere exhibit an upwards eddy flux of potential temperature.
In
the low latitudes, the winter upper stratosphere and mesosphere exhibit a
downwards eddy flux of potential temperature.
The lower stratosphere at high to midlatitudes of both seasons shows a
downgradient downward flux of potential temperature.
The negative correla-
tion between potential temperature and vertical wind velocity comes from the
tendency for the vertical motions to respond to the diabatic cooling in a
manner that causes downward motions to occur in cyclonic regions.
-35The tropical tropopause region shows an eddy upwards flux of potential
temperature in the summer hemisphere and a downwards flux in the winter
hemisphere.
These features appear to be related to the strong horizontal
convergence of heat by the eddies into this region.
In order to determine which wavenumbers are doing most of the transport,
the total eddy fluxes of ozone were broken down by wavenumber for the levels
132, 40, 12 and 1 mb for the latitudes 80*N, 40*N, 0*, 40*S, and 80*S.
The
fraction of the total covariance as a function of wavenumber is plotted in
Figures 37 and 38.
At 80* both December and June show that the flux is
extremely dominated by wavenumbers 1 and 2 at all levels.
During the winter
the eddy ftux at 40* in the lower stratosphere is divided fairly evenly
between wavenumbers 1, 5, and 6.
During the winter, wavenumbers 5 and 6
contribute significantly to the eddy flux up to 12 mb.
In the lower meso-
sphere, however, only wavenumber 1 makes a significant contribution.
At the equator there appears to be a tendency for all the wavenumbers
to contribute a flux in the same direction.
This is in contrast to mid-
latitudes where various wavenumbers of significant contribution tend to
oppose one another.
The lower equatorial mesosphere curiously enough has
a strong contribution to the total flux made by wavenumber 4.
This contri-
bution probably comes from the mixed mode Rossby wave.
During the summer, the midlatitude fluxes are either dominated by or
strongly influenced by a strong wavenumber 4 component at all levels.
The
contribution to the flux by wavenumber 4 is in the same direction at all
levels.
Wavenumbers 1 and 2 become significant only above 12 mb.
In all the months the midlatitudes do not show a dominant wavenumber
that would be useful in aiding a parameterization of the flux.
Only at
high latitudes and mesospheric altitudes would such a scheme work.
The only
-36hope for a useful parameterization in terms of phase would occur only if
all the wavenumbers had nearly the same phase difference between ozone and
the wind.
However, this hope is also destroyed as one sees that there is
significant opposition in the fluxes as a function of wavenumber.
One
apparently cannot restrict the parameterization scheme to one zonal wavenumber.
-378.
A Survey of the Phase Difference Between the'Winds and the Tracers
The particular eddy flux of interest in this study is the poleward
flux of ozone.
Consequently the study of the phase lags for the time being
*
will be limited to V
*
and 03.
In order to ascertain the gross "efficiency" of the eddies in transporting ozone, the average phase angle defined by
was
calculated.
This
phase
angle
contains
not
only
average size of the phase difference between V
the
information
and 03
on
the
but also the corre-
lation between the amplitude size and the phase difference.
The expression
is unsuitable in trying to get an idea of the phase difference.
Neverthe-
less it does give one an idea of the average "efficiency" at which the
eddies transport ozone.
The phase difference, y, is shown in Figure 39.
the eddies are very poor transporters of ozone.
It can be seen that
Only about 1-3% of the
total possible transport is realized since the phase angle is nearly always
near 7r/2 radians on the average.
inefficient.
The 50 to 200 mb region is notoriously
This result is to be expected since the net flux in this region
is about the flux one finds for a layer one-third the thickness near 20 mb.
The most efficient regions of transport can muster only about a 5% efficiency.
The average phase difference between 03
*
*
and V
for the standing waves
at levels 132, 12, 1 and .089 mb are plotted as a function of zonal wavenumber in Figures 40 and 41.
The most obvious fact is that the standing
eddies have a phase difference much greater than the gross phase difference.
-38This result should be expected since the standing eddy wind variance is
much smaller than that of the transient eddies, yet the standing eddy
fluxes are nearly able to cancel the transient eddy fluxes in some places.
This kind of behavior then implies that the phase difference should be
*
fairly large.
The phase difference between V
*
and 03
is quite random in
the sense that the phase difference randomly fluctuates back and forth
across the 900 line as a function of wavenumber.
There doesn't appear to
be any particular "spectrum" of phase differences that can easily be fitted
by a simple relation.
*
Phase histories of V
*
and 03
for the first three zonal wavenumbers
at 12 mb and latitude 40 are shown in Figures 42, 43 and 44.
The time
period for the histories is set in the winter month of December.
Zonal wavenumber 1 shows a fairly constant leading of the ozone wave
which on the average has a phase difference of 1.86 radians.
Wavenumber 2
shows ozone still to be leading, however the phase difference shows a larger
amount of variance.
ans.
On the average the phase difference is about 1.71 radi-
Wavenumber 3 retrogrades rapidly with a westward motion, but the ozone
wave still leads the wind.
The variance of the phase difference grows still
larger, and it is found that this trend continues out to wavenumber 6.
The
average phase difference for wavenumber 3 is about 1.81 radians.
In summary one can see that the transient eddies have phase differences
which are very close to 1.57 radians on the average.
The meridional wind
and the ozone in these eddies appear to be almost completely out of phase.
The standing waves on the other hand exhibit larger phase differences, but
in many cases the different wavenumbers show a tendency to cancel one another.
The transient waves show a fair amount of constancy in the phase difference
as a function of zonal wavenumber, while the stationary waves do not.
At
-39-
this point it is not exactly clear what the transient wave phase difference
depends on so that one can write these phase differences in terms of averaged quantities.
-40-
9.
Introduction to the Mixing Length Hypothesis
The gradient law approximation to the mixing length hypothesis has
been reviewed elsewhere, i.e., see Corrsin (1969).
For any question about
the historical development and the validity of usage of the mixing length
hypothesis one is referred to the previous reference cited and to Launder
(1974).
The mixing length hypothesis is at best a physically meaningless
analog to molecular diffusion.
One of the main drawbacks in parameterizing
the fluxes by eddy diffusion is a lack of physical guidance in calculating
the eddy-diffusion coefficients
K(I,J) = <L(I) V"(J)>.
Stone (1973) found functional expressions for Kyy in terms of the mean
state variables.
However this case is misleading since the eddy diffusivi-
ties really result from the explicit expressions for the heat fluxes, with
the fluxes, who needs the eddy diffusivities?
The first is
There are two approaches to getting eddy diffusivities.
to simply fit
or "tune" the eddy diffusivities to the observed fluxes.
The second way is
to use statistical methods like Kao (1965)
to deduce the
mean behavior of an ensemble of parcels.
Reed and German (1965) were the first to discuss relations between the
different elements of the two-dimensional eddy diffusivity tensor.
the two-dimensional case, a perturbation in the tracer field,
can be written to first order terms as:
In
Q" = Q
- <Q>,
-41Reed and German assumed that the turbulence to be approximately isotropic
so that
V"L
= W"L
.
This assumption allowed the definition of a mixing
slope as being the wind ratio of the form:
S=P(9.2)
By multiplying (9.1) by V" and both zonally and time averaging one can
arrive at the relation:
(9.3)
where
%
, and where e equals the density.
The triple covariance in (9.3) can be expanded if V"L
single variable.
and <V"Ly>,
where
is treated as a
By ignoring the correlations between variations in <o>
the resulting form of (9.3) can be written as:
Kyy = <V"LY>.
The same kind of manipulation can be done for the vertical flux corre-
lation with the result:
Reed and German (1965) were the first to use a direct method of determining two-dimensional eddy diffusivities from the wind and tracer statistics.
The method in reality is just a direct way of tuning the eddy diffusivities
as opposed to tuning them in a model by trial and error.
-42The eddy diffusivity of the midlatitude mid-tropospheric disturbances
was found by using Eady's (1949) approximation that for heat, mixing tends
to occur along slopes one-half that of the mean isentropes.
This approxi-
mation plus the observed eddy heat flux allows the direct calculation of
Kyy at that point.
The eddy diffusivity, Kyy, is extrapolated to other regions by the
use of Prandtl's (1945) improved assumption that the eddy diffusivities
are proportional to the variance of the wind. By using observed horizontal
eddy heat fluxes, Reed and German were able to calculate <a>.
They also
assumed <a> to vanish at the equator so that the ratio Kzz/Kyy equals
<a'a'> which is then held constant everywhere.
Unfortunately Gudiksen et al. (1968) found that the eddy diffusivities
were much too large and had to be reduced by factors to 5 to 10 so that the
modeled distribution of tungsten 185 would fit the observations.
By using
five years of data on the heat fluxes and wind variances, Luther (1973)
made a set of eddy diffusivities.
Luther extrapolated the values of Kyy
above the region of observation by assuming that the upward propagating
wave motions have a constant energy density so
that the wind variances
increase by the inverse square root of the density.
Later)Louis (1974)
used a meridional circulation that he deduced and the observed ozone distribution to get improved estimates for Kyy.
-4310.
A First Attempt at Obtaining Eddy Diffusivities
The model of Cunnold et al. (1975) can be facilitated to calculate all
of the variables in (9.4) except Kyy for the cases of ozone and potential
temperature.
If one substitutes < > = -3<Q>/Y/3<Q>/DZ, and makes some
mathematical arrangements, Equation (9.4) can be solved by Kyy by the expression:
4KI> 4 V
(10.1)
Equatiqn (10.1) presents some problems mathematically since it becomes
singular as
<a>
=
<f>
.
In the case that the tracer is potential tempera-
ture the possibility of a singularity becomes inevitable.
sphere <at> < < >
nature.
In the tropo-
since the disturbances are generally of a baroclinic
<O> > <>
While in the stratosphere
a "refrigerator" and EKE is changed to EAPE.
troposphere and the lower stratosphere
<a>
since the motions behave like
Somewhere in between the mid-
=
.
In the equatorial
regions <a> and <f> become quite small and they tend to equal one another.
The smallest amount of error in either <ct> or <6> can cause a large spurious
Kyy.
Several calculations using both the ozone mixing ratio and potential
temperature showed singularity problems in both these regions, and also at
high latitudes in the stratosphere.
The vertical eddy flux relation (9.5) also has the same problem of
singularities when
<a><I
>
.
Computations again showed the same
problem in approximately the same regions.
The uncertainty in the average
values tend to make it impossible to exclude even the pseudo-singularities
from occurring in (10.1).
-44Another, more basic complication in the direct computation stems from
the uncertainty in the average values.
One cannot time-average over an
infinite number of winter seasons to get a true winter average.
age value has a finite level of uncertainty.
Each aver-
The standard error gives a
measure of the random "noise" that the average is buried in.
The standard
error is not really a true measure of the amount of variation which takes
place about the mean since the spectrum of the winds is not a constant as
a function of frequency.
A better indicator of the uncertainty might come
from using a high order Markov process instead of a random process.
However,
it is a line of analysis that would be complicated, and for the effort the
estimate of the uncertainty is provided well enough by a random model.
The standard error is given by
a//N, where a is the standard deviation
of the uncertain variable and N is the number of independent samples of the
variable.
Averaging over a season instead of a month decreases the error
only by a factor of .58, while averaging over a year reduces it by a factor
of .29.
The standard error strongly reflects the effect of beginning and ending
If one averages over 30 days and the devia-
in a small averaging interval.
tions from the average are on the average 30 times larger than the average,
then if there are one or two more positive deviations than negative ones,
due to the point of beginning and ending, large errors are possible because
the standard error is of the order of the average value.
Confidence limits can be established by using the T statistic defined
A confidence level of 95% can be set by using a T statistic
by <(>/a/Ar.
of two which is
simply twice the standard error.
*
*
were set for <V 03 >,
*f*
<V V >,
The 95% confidence limits
-
<ca>, and <03>.
The appropriate monthly
-45values and their uncertainties are shown in Table 1 for selected levels
and latitudes.
*
*
From looking at Table 1, one can note that for the covariance <V 03 >
even the sign cannot be accurately ascertained in many cases.
The uncer-
tainties in the covariance are particularly large at level 21 and at high
and low latitudes.
An even worse case is the mixing surface slopes, <a>.
Sometimes the uncertainties are nearly a factor of two larger than the average value.
Variances and single variables fared much better.
Note that for
* *
<V V > the uncertainty is at least a factor of three less than the average
value, while for<0 3> at high altitudes, the uncertainty is one-tenth the
average.
In the case of <0 3> one can see the distinct increase of the uncer-
tainty to average ratio as one progresses from level 15 to 21.
The increase
in the uncertainty is probably due to the increased importance of the divergence of the eddy fluxes in determining the mean value.
From the uncertainties in the mixing slopes and the fluxes, it looks
as though only monthly averages taken from nine years of data will significantly reduce the uncertainties anywhere near to the fractional level of un* *
certainty in <V V > and <03> .
per experiment.
Unfortunately only three years are simulated
The first year is generally evolutionary towards the steady
yearly oscillation.
Run
26, which appears to be the most successful to
date, has only two years of simulation, thus only one month at a time can
be simulated.
Nevertheless one can at least make a preliminary study from
the small amount of data at hand.
As an example to illustrate what sort of problem uncertainty is in the
calculation of Kyy from (10.1), a specific calculation of the Kyy uncertainty was made for Level 15 and 400 S.
The uncertainty relation for Kyy is
-46derived in Appendix 1.
Substituting in the values from Table 1 along with
Kyy = 2.45E+6 ± 6.63E+6.
the ozone gradients gives the result:
The uncer-
tainty can be seen to be so large as to make it difficult to determine even
the sign of Kyy.
In the past, the assumption that Kyy is proportional to
-*2
<(V ) >) has been used quite often.
L<(V') 2 > +
It is of interest to find out what the
size of the constant of proportionality is: whether it is 1 minute, 1 day,
or 1 week.
<(V)
Secondly, one might ask if perhaps a different power of
2> + <(V ) 2>
might work better.
A test was conducted to see what the variance-gradient produce relation
* *
was to the, fluxes.
V*
e > and X was set equal to
. A scatter plot of X versus Y was made and shown in
[7
Figure 45.
30-60*S
Y was set equal to -<V
The region chosen included the levels 15-19 and the latitudes
for the northern hemispheric month of June.
A polynomial regression was made of X on Y.
The largest reduction of
variance was only .15 which indicated a large amount of random scatter.
The result when the Y intercept was zeroed indicated that the reduction of
variance was accomplished mainly by a linear fit with the slope of 1.lE+5 sec.
The large uncertainties in the Y's are the cause of the scatter and the negative X values which tend to detract from the validity of the plot, however
not much can be done about it.
In the troposphere, Schneider and Dickinson (1976) interpreted the
<(V) 2> + <(V )2>
to be roughly the in-
verse of the growth rate of the most unstable wave.
The growth rate goes
proportionality factor in Kyy
approximately as f/v/R'I, where f is the Coriolis parameter and RI is the
Richardson number.
For the midlatitude troposphere the inverse of the
growth rate is roughly three days.
In comparing this result with the lower
-47stratosphere the proportionality coefficient does not seem to differ much
despite the different dynamics of the two regions.
since if the slope had been on the order of 10
6
This is encouraging,
sec, the proportionality
argument would be much less credible.
One could find the proportionality coefficient by using Reed and German's (1965) technique, however it would be better if the coefficient could
be calculated directly.
There are two ways of doing this.
One way is to
make use of Taylor's dispersion theory which is discussed in the next section.
The other way is to try to formalize the physical mechanism which
controls the proportionality coefficient of interest.
By taking a,cue from the growth of a tropospheric growing baroclinic
eddy, one can obtain some physical insight as to what the proportionality
time constant means.
perature advection.
The growth of the eddy basically comes from the temThe advection must in a sense be "localized" so that
the advective processes will preserve the scale of the disturbance and not
*
try to distort it beyond recognition.
If the phase difference between V
*
is close to 7r/2 radians, the advective scale can be thought to be
and T
of the order of one-half the wavelength.
If the phase relation holds for
most wavenumbers, then the distance would be about one-half the meridional
EKE weighted wavelength.
The rate at which parcels are brought into a disturbance and taken
out, in an average sense is roughly determined by the absolute value of the
average zonal wind.
The coefficient of proportionality, which shall be
henceforth called T, is regulated directly by the advective scale, and
inversely by the zonal wind.
The idea is simply that the larger the wind,
the shorter the time it takes for a parcel to move the advective length
scale.
The coefficient T can be tentatively written as
-48-
(10.2)
pd
The constant C
II|
is simply a small constant to avoid singularities near
= 0.
T tends to remain constant in the vertical.
As one goes to higher
levels in the stratosphere, the zonal wind velocity increases, but the
meridional EKE weighted zonal wavelength also increases.
As one goes from
the equator to the pole the zonal wind velocity reaches a maximum near 50*.
The weighted zonal wavenumber on the other hand tends to remain constant.
One expects -some,latitudinal variation with T reaching a minimum at midlatitudes as Murgatroyd (1969) found.
Several experiemtns were conducted with this formula.
One experiment
was conducted on Run 17 data with the intent of getting well-behaved Kyy
eddy diffusivities.
In this experiment the advection length was not calcu-
lated explicitly, but rather was artificially fixed by specifying the Kyy
value at 50*.
The value chosen was
Kyy = 2.6E+5 m2/sec
and spring northern hemispheric seasons.
for both winter
The Kyy eddy diffusivities calcu-
lated arehown in Figures 46 and 47 respectively for the case C
= 5 m/sec.
*
Run 17 variances of V
were found to be roughly four times larger than
observation due to the lack of significant subspectral diffusion.
Secondly,
a large computational two-level oscillation showed up.
Compared with those of Gudiksen et al. (1968), the Kyy's show a much
wider range of values.
While the Gudiksen et al. set show only variations
of a factor of five in the troposphere, the values in Figure 46 show the
tropospheric variation to be about a factor of ten.
several features are common to both sets.
In the troposphere
These are the tropospheric
-49inimum at 20-30*N and the maxima near the jet core and at the equator.
Also appearing in both sets is the minimum in the lower troposphere of the
equatorial regions.
The Gudiksen et al.
Kyy's decrease slightly in midlatitude from the
tropopause maximum to a minimum near 25 km of about 4.OE+5 m 2/sec.
The
model Kyy's also exhibit a minimum in this region of about the same size,
but the decrease is about a factor of ten.
In going upwards from 50 mb,
the Gudiksen et al. values of Kyy increase much more rapidly at high latitudes than the model Kyy's do.
The equatorial regions exhibit a rapid
decrease in Kyy in the lower stratosphere for both sets.
However, the
minimum of the model Kyy's is about 3.OE+5 while the Gudiksen et al. value
is more than three times smaller.
For a further comparison one is referred
to the CIAP Monograph Vol. 1 (1975) for other sets of Kyy eddy diffusivities.
Another experiment was run using the meridional EKE weighted zonal
wavenumber found from (6.1).
The northern hemispheric month of June from
Run 26 was used in order to calculate T.
The factor C
The result is shown in Figure 48.
was arbitrarily set to 2 m/sec.
T does not seem so constant as was anticipated.
order of magnitude at most.
T varies about an
Fortunately the only sharp gradients in T occur
near the equator, and in the lower troposphere.
Upon looking at the pattern
of T one finds a strong inverse proportionality of T with (<V'V'> + <V V >).
Minima occur in T where maxima occur in the variance.
One might be led to
redefine the formula for Kyy as:
k= C1
(V v.> +
<
(10.3)
where n falls in the range .5 to 1.0 and C2 is a true constant, say the
space-averaged T.
-50-
11.
The Use of Taylor's Dispersion Theorem in Determining Eddy Diffusivities
Taylor (1921) demonstrated the relation of the distance dispersion,
<fi2>, to22the velocity variance,
coefficient, R(T).
If <.
V 2and the velocity autocorrelation
2> is the square of the distance the particles
have diffused in a time period T, the dispersion rate can be expressed by
the now well-known Taylor relation:
<
< V>
>y-Z , S C)/e3 z
(1.1
As t,goes to infinity, the integral in (11.1) becomes what is known
as the LaGrangian integral time scale (LIT).
Generally the time period t
is taken as an averaging time period much greater than the lifetime of the
individual eddies so that in most cases the integral in (11.1) closely
approximates the LIT.
The most important eddy diffusivity component Kyy
can be expressed as
r:.v9(11.2)
where T is now used to denote the LIT.
The LaGrangian frame of reference has been used in the discussion.
In
many cases the LaGrangian variables are computationally awkward since one
must work with moving particle trajectories, rather than the handy instantaneous point sampling of radiosondes.
In order to make use of the Euler-
ian variables, Hay and Pasquill (1959) proposed the following EulerianLaGrangian transformation:
-51-
=4(1)(11.3)
Lpe)
R
where RL AND RE are the LaGrangian and Eulerian autocorrelation coefficients.
The parameter
S
is a scale dependent parameter linking the
integral timescales by the relation
L =
E'
Kao (1965) proposed a relation for the
waves.
associated with Rossby
The relation takes into account the Doppler shifting effect of
the zonal wasterly wind which tends to move the energy spectrum peak
towards lower frequencies.
The average
S
which is independent over wave-
number cap .be expressed by the relation:
p~
lxc~
2/~ic9)
-ai
(11.4)
where LW2 is the square of the meridional EKE weighted mean zonal wavenumber of the turbulence.
A plot of
of June.
3 is shown in Figure 49 for the northern hemispheric month
At midlatitudes 3 appears to be about 0.5.
This value agrees
quite well with the set of 1 made by Murgatroyd (1969b).
1
as a whole
tends to have an inverse proportionality with the zonal wind profile,
with strong westerlies appearing as low
lies show
0.5.
1 to
be one or larger.
The jets appear as minima.
1 regions,
and the strong easter-
In the troposphere
1
is generally about
While at high latitudes the cos$3 term
becomes quite small; giving values of unity or greater.
The summer hemi-
sphere stratosphere exhibits an almost uniform region of
1
of 1-2 equatorwards of 60*.
of the range
-52In order to calculate T , the Eulerian integral timescale TE is
calculated by taking the sum over all time lags of the autocorrelation
coefficient Rvv(t).
Unfortunately the accuracy of the estimate of the
*
Rvv(t) is limited by the length of the time record of V
scale of the dominant eddies.
,
and the time-
The length of the record must be at least
five times longer than the dominant eddy timescales.
be careful of how the longest time lag is ended.
Secondly, one must
Otherwise the ending of
the record on a large Rvv(t) will prevent accurate estimate of the LIT.
Figure 50 shows the autocorrelation coefficient at 12 mb and 40*
for the month of December.
The form of the autocorrelation coefficient
*
*
as a function of time is somewhat like R (t) = exp(-P t)cos(Q t).
vv
Murgatroyd (1969b) used this form for Rvv(t) in his study.
for R
vv
(t) is simply the Fourier transformation of a spectrum with a peak.
Q are
P and
This relation
tral peak.
parameters that adjust the frequency and strength of the specIn the autocorrelation coefficient case at hand, the dominant
period seems to be about 12 to 14 days.
The meridional winds that went to make up the autocorrelation coefficients were preprocessed to remove any linear trend that would bias the
autocorrelation coefficient.
The 32-day time period seems to be insuffi-
cient in many regions for estimating the LaGrangina integral timesacle.
At 12 mb the autocorrelation shows a dominant period of about 12 days.
Obviously a 32-day sample cannot represent this frequency well. Secondly,
at the
3 1 st
day lag, the autocorrelation coefficient is still quite large.
This indicates that a very long period is biasing the total area under the
Rvv(t) curve.
This problem is compounded by the fact that TL is quite
small compared with any given Rvv(t) that goes into making the sum for T .
-53The LaGrangian timescale was plotted for those regions in which it
was felt that there was some significance in the TL obtained.
The LIT,
The timescale is meant for the northern hemi-
TL, is shown in Figure 51.
spheric month of June.
The timescale seems to have a maximum near 80 mb and a minimum in
the mid-troposphere.
file.
Murgatroyd (1969a) also found the same kind of pro-
Murgatroyd's values for TL are about one-half the values calculated
from the model data.
The variation in TL calculated range by a factor of
three between 400 and 50 mb.
The comparison of the calculated TL with the T shown in Figure 48
shows a rough smilarity.
In both cases there is a mid-tropospheric
minimum, a 50 mb maximum, and a maximum in the lowest part of the troposphere.
-5412.
Conclusions and Summary
One of the basic purposes of this diagnostic study on the data from
the MIT stratospheric GCM was to compare the observations of the eddy fluxes
of ozone and potential temperature with the model simulation.
Another pur-
pose was to present findings from the model in regions where observations
have not yet been made so that the results can act as a guide to future observations.
The study conducted on Runs 17 and 26 of the model has several
significant conclusions which can be made.
An attempt shall be made to
present the more important points of the study by type of analysis and by
region of the atmosphere.
In themidlatitude troposphere the surfaces along which ozone and potential temperature mix are directed polewards and upwards.
The slopes of the
mixing surfaces are of the same size (5.OE-4 radians) as those observed in
the troposphere.
In the troposphere the mean circulation fluxes tend to
oppose the eddy fluxes of ozone and heat.
In the troposphere the ratio of the meridional transient EKE to that of
the standing eddies is about that observed by Oort and Rasmussen (1971).
The midlatitude meridional EKE for both the transient and the standing eddies
is about twice that observed.
In the mid-troposphere the equatorial region behaves quite differently
from midlatitudes in that it exhibits an equatorward eddy flux of heat converging into the euqator, while ozone appears to be diverging.
The merid-
ional EKE is spuriously large at the equator also.
The stratosphere in the 10 to 150 mb region has some similarities and
some differences from the troposphere.
In the midlatitudes, the stratospheric
temperature flux remains polewards, and the vertical flux of geopotential is
-55positive like the upper troposphere.
The convergence of the vertical eddy
flux of geopotential, along with the downwards vertical eddy flux of temperature agrees with Oort's (1964) analysis which found the stratosphere
was forced by the pressure forces from the troposphere, and that the EKE of
the wave motions is being converted to EAPE.
Both ozone and potential temperature in
the lower midlatitude winter
stratosphere have polewards and downwards fluxes that tend to occur in
longitudes of low geopotential. The slopes of the mixing surfaces are about
the right size (4.OE-4 radians) and exhibit a maximum at midlatitudes as
previous estimates by Reed and German (1965) have shown.
The horizontal steady stationary wave fluxes of ozone tend to cancel
the transient eddy ozone fluxes in the lower stratosphere.
ponent is much larger than the total eddy flux.
Each flux com-
In the 50 to 150 mb
region the large variability makes the total eddy flux of ozone undergo
changes from polewards to equator-wards directed fluxes from level to level,
although the total eddy flux in this region is directed polewards during
the winter.
The vertical eddy fluxes ofozone also show a tendency to oppose
the mean circulation fluxes.
The meridional EKE for both the stationary and transient waves decreases
to about the observed values at 30 mb for most latitudes.
The only excep-
tion occurs at high latitudes where the steady stationary wave variances
decrease instead of increase as the observations show to be the case.
The vertical scale of the nondivergent meridional standing wave wind
seems to be roughly twice as small as it should be in the lower stratosphere.
This standing wave also exhibits a constant phase with height above 10 mb.
As one goes polewards and upwards, both the transient and standing wave
motions show a spectral shift towards lower wavenumber.
The fluxes of ozone
-56also exhibit this spectral shift.
The only true exception is near the
equator where wavenumber 4, due possibly to the mixed-mode Rossby wave,
dominates the transient eddy fluxes.
The waves as a whole show ozone and the meridional wind to be nearly
900 out of phase so that only 1-3% of the total potential flux of ozone
is realized.
The traveling wave fluxes of ozone seem to have better
behaved phase differences between ozone and the meridional wind than the
steady stationary eddy fluxes.
The region from .5 to 10 mb shows some interesting behavior in the
winter hemisphere.
The mixing slopes change sign with respect to the rest
of the stratosphere, and possess a sign simialr to that of the troposphere.
In the lowest part of the mesosphere the potential temperature fluxes are
mainly equatorwards on the low latitude side of 400 and polewards on the
high latitude side.
Heat diverges out of the region about 40*.
The verti-
cal fluxes of potential temperature equatorwards of 40* are downwards,
while those polewards of 40* are upwards.
The winter transient vertical eddy fluxes of ozone equatorwards of 40* are
upwards, while those polewards of 40* are downwards.
The horizontal eddy
fluxes of ozone show a convergence into the midlatitudes.
The convergence
is even strong enough to produce equatorward eddy fluxes of ozone near 1 mb
in midlatitudes.
Because of the tropospherically similar mean temperature gradients in
the vertical and horizontal, the high latitude mesospheric regions could
quite possibly be baroclinically unstable.
The downwards directed eddy
geopotential flux in the low latitudes is quite interesting, for in this
region one cannot rule out barotropic instabilities due to the large horizontal zonal wind shears on the low latitude side of the polar night jet.
-57Of course, further studies on the energetics of both regions would have to
be done in order to show conclusively the energy conversions going on.
However, the general directions of the mean gradients of wind, temperature,
and the eddy fluxes suggest the possibilities for instability to the extent
that such further study is highly encouraged.
The region above .5 mb seems to behave much like the 20 to 150 mb
region.
The mixing slopes tilt so that the eddy fluxes of potential tem-
perature are directed downwards and polewards.
The only significant dif-
ference between the two regions is that ozone and temperature are negatively
correlated.
The
later part of the study which was directed towards obtaining eddy
diffusivities demonstrated several points.
The first one is that one can-
not use the eddy flux equation to get the eddy diffusivities without running
into serious difficulties with singularities.
Secondly, the daily varia-
tion of the fluxes is so large that one needs at least 300 samples in a
particular timespan of averaging in order to reduce the uncertainty of an
average to an acceptable level.
The uncertainty has a tendency to mask
any relation between Kyy and the meridional EKE.
Several approaches of finding the time scale relating the meridional
EKE and Kyy were tried.
The first method involved the finding of the
typical advective timescale due to all the waves.
The second method used
Taylor's dispersion theorem to calculate the LaGrangian integral timescale.
The model's Eulerian variables were transformed to LaGrangian ones via a
Hays and Pasquill (1959) type transformation specifically derived for
Rossby waves.
The first method gave Kyy eddy diffusivities that compared favorably
-58-
with the Gudiksen et al. (1968) values.
The timescales found from both
methods compared favorably with those of Murgatroyd (1969b).
A longer
averaging period is needed in order to make a good estimate of the
Lagrangian integral timescale for the whole atmosphere.
By using the
<a>, <a2 >, the <v2 > and the T from the model it seems quite possible for
one to calculate meaningful eddy diffusivities.
-59ACKNOWLEDGEMENTS
The author would like to acknowledge the helpful discussions
that he had with Derek Cunnold and Fred Alyea.
He would also like to acknow-
ledge the patient and helpful editing of this thesis by his advisor, Professor
Ron Prinn.
A special acknowledgement should go to the author's wife, Layla,
who put up with the long sessions he spent with the computer.
supported by a research assistantship under Grant No.
rQ
The author was
4
ASA
-60-
APPENDIX
Equation (10.1)
can be expressed in the form
k b - - A (1 -ic
(A. 1)
where
AY <V'*>
C'C CC(AL-
( V M
The uncertainty of (A.1) can be expressed by taking logarithms of
(A.1) and then differentiating, leaving
=
_
The uncertainty, 6(B + CD),
6B and 6CD.
t CD)
(A.2)
can further be simplified if broken up onto
The two uncertainties are simply additive.
6CD can be written as 6CD = CD(6C/C + 6D/D).
The uncertainty
By combining these results
and substituting into (A.2) one arrives at the result
I Sk~
(Ct~~Zf(t~t
ii
(A.3)
-61-
The uncertainty of the derivatives are calculated by assuming
a centered finite difference approximation that makes the uncertainty
6(
) = 2-6<Q>/DN, where DN = DPHI, DZ, etc.
Given A, B, C, D and the
uncertainties of the type in Table 1, the uncertainty 6Kyy can be
calculated.
-62References
Barnes, A. A., Jr., 1962: Kinetic and potential energy between 100 mb and
10 mb during the first six months of the IGY. Final Report, Department
of Meteorology, Massachusetts Institute of Technology, 8-131.
ClAP Monograph 1, 1974: The natural stratosphere of 1974.
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Department of
Corrsin, S., 1974: Limitations of gradient transport models in random
walks and in turbulence. Advances in Geophysics, 18a, 25-60.
Cunnold, D., F. Alyea, N. Phillips, and R. Prinn, 1975: A three-dimensional
dynamical-chemical model of atmospheric ozone. J. Atmos. Sci., 32,
170-194.
Danielsen, E. F., 1968: Stratospheric-tropospheric exchange based on
radioactivity, ozone and potential vorticity. J. Atmos. Sci., 25,
501-518.
Dickinson, R. E., 1962: Momentum balance of the stratosphere during the IGY.
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Technology, Cambridge.
Eady, E. T., 1949:
Long waves and cyclone waves.
Tellus, 1, 33-52.
Eliassen, A., and E. Palm, 1960: On the transfer of energy in stationary
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Feely, H. and J. Spar, 1960: Tungsten-185 from nuclear bomb test as a
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Hantel, M. and H.-R. Baader, 1976: On the vertical eddy heat flux in the
northern atmosphere. Beitrffge zur Physik der Atmosph~re, 49,
1-12.
Hay, J. S., and F. Pasquill, 1959: Diffusion from a continuous source in
relation to the spectrum and scale of turbulence. Advances in
Geophysics, 6, Academic Press, New York, 345-365.
Green, J. S. A., 1960: A problem in baroclinic stability.
Met. Soc., 86, 237-251.
Quart. J. Roy.
Green, J. S. A., 1970: Transfer properties of the large-scale eddies and
the general circulation of the stratosphere. Quart. J. Roy. Met. Soc.,
96, 157-185.
Gudiksen, P. H., A. W. Fairhall, and R. J. Reed, 1968: Roles of mean meridional circulation and eddy diffusion in the transports of trace substances in the lower stratosphere. J. Geophys. Res., 73, 4461-4473.
-63Jensen, C. E., 1961: Energy transformation and vertical flux processes over
the northern hemisphere. J. Geophys. Res., 66, 1145-1156.
Launder, B. E., and D. B. Spalding, 1972: Lecture in Mathematical Models
of Turbulence, Academic Press, London and New York.
Leovy, C., 1964: Simple models of thermally-driven mesospheric circulation.
J. Atmos. Sci., 21, 327-351.
Lorenz, E. N., 1960:
364-373.
Energy and numerical weather prediction.
Tellus, 12,
Louis, J. F., 1974:
Ph.D. Thesis, Department of Geoastrophysics, University of Colorado.
Luther, F. M., 1973: Monthly mean values of eddy diffusion coefficients in
lower stratosphere. AIAA Paper No. 73-498, Paper presented at the
AIAA/AMS International Conference on the Environmental Impact of
Aerospace Operations in the High Atmosphere, Denver, Colorado.
Kao, S. K., 1965: Some aspects of the large-scale turbulence and diffusion
in the atmosphere.' Quart. J. Roy. Met. Soc., 91, 10-17.
Mahlman, J. D., 1973: Preliminary results from a three-dimensional generalcirculation/tracer model. Proc. Second Conf. on the Climatic Impact
Assessment Program, Report No. DOT-TSC-OST-73-4, U.S. Department of
Transportation, 321-337.
Manabe, S., and J. D. Mahlman, 1976: Simulation of the seasonal and interhemispheric variations in the stratospheric circulation. J. Atmos.
Sci., 33, 2185, 2217.
Martin, D. W., 1956: Contribution to the study of the atmospheric ozone.
Sci. Rep. No. 6, Department of Meteorology, Massachusetts Institute of
Technology, 49 pp.
Matsuno, T., 1971: A dynamical model of the stratospheric sudden warming.
J. Atmos. Sci., 28, 1479-1493.
Molla, A. C., and C. J. Loisel, 1962: On the hemispheric correlations of
vertical and meridional wind components. Geof. Pura e Appl., 51,
166-170.
Murgatroyd, R. J., 1969a: The structure and dynamics of the stratosphere.
In The Global Circulation of the Stratosphere, edited by G. Corby,
pp. 154-195, Royal Meteorological Society, London.
Murgatroyd, R. J., 1969b: Estimates from geostrophic trajectories of horizontal diffusivity in the mid-latitude troposphere and lower stratosphere. Quart. J. Roy. Met. Soc., 95, 40-62.
-64Newell, R. E., 1963: Transfer through the tropopause and within the
stratosphere. Quart. J. Roy. Met. Soc., 89, 167-204.
Newell, R. E., 1965: A review of studies of eddy fluxes in the stratosphere
and mesosphere. Department of Meteorology, Planetary Circulations
Project, Report No. 12, Massachusetts Institute of Technology.
Newell, R. E., J. W. Kidson, D. G. Vincent, and G. J. Boer, 1972: The
General Circulation of the Tropical Atmosphere, Vol. 1, MIT Press,
Cambridge.
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-65-
S'tebbins, A. K., III, 1960: Progress report on the high altitude sampling
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White, R. M., 1954: The countergradient flux of sensible heat in the
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standing waves. J. Geophys. Res., 78, 4463-4471.
T A B L
80*N
V03
40*N
00
1.44E6±7.18E6
-8.29E7±4.86E7
-6.78E7±7.9.1E7
-2.71E7±2.23E8
-1.97E9±8.34E8
-1.03E9±1.07E9
400S
-5.53E7±5.6E7
80*S
-1.17E7±9.66E7
2.0E10±1.13E10
3.65E9±3.37E9
5.68E10±1.45E10
5.24E10±2.22E10
-1.21Ell±3.7E10
-2.85E11±1.04E11
3.91E9±3.19E10
-6.16E11±7.15Ell
3.74E12±8.36E12
-2.67E12±8.8E12
-3.43E12±8.60E12
-2.02E12±5.61E11
Vv
9.38±1.43
28.5±6.90
253±7.07
202±48. 7
60.6±20.4
4.37±.66
5.30±1.13
32.3±8.38
132±26.1
13.0±1.79
3.23±. 65
5. 96±.65
10.2±1.30
98. 8±17.0
35.3±5.33
11.9±2.11
221±29. 7
170±32.0
414±85.0
14.0±3.18
1.03E9±1.17E8
1.22E9±6.60E7
1.21E9±2.02E7
7.53E8±1.14E8
4.25E9±2.24E9
6.4E10±3.73E8
7.61E10±3.30E8
8.38E10±3.32E8
1.12E11±2.55E9
2.19E11±1.03E10
1.95E12±4.42E10
2.88E12±2.16E10
2.75E12 2.21E10
1.41E12±2.03E10
1.30E12±5.18E10
2.4E12±9.41E11
1.04E12±4.57E11
1.OE11±3.59Ell
1.41E12±3.15E11
4.18E12±1.28E12
03
-1.3E-4±1.2E-4
-6.2E-5±6.7E-5
-3.OE-6±4.6E-5
1.6E-4±1.6E-4
-8.9E-5±8.5E-5
-1.2E-4±1.3E-4
-1.2E-5±1.4E-4
1.1E-4±3.1E-4
1.lE-4±3.5E-4
-1.5E-4±1.8E-4
-7.7E-5±1.6E-4
-2.6E-5±1.lE-4
-5.8E-5±1.OE-4
-2.lE-5±3.9E-5
-4.8E-5±1.7E-4
-1.5E-4±2.7E-4
-6.6E-5±1.7E-4
1.OE-4±5.6E-5
-2.9E-5±1.lE-4
7.9E-4±1.7E-3
-67-
Figure Captions
1.
The phase relations as implied by observations.
2.
The <V W > covariance from run 17 for the northern hemispheric winter.
* *
Units are 10
-2
2
-2
.
m sec
* *
3.
The <V W > covariance from run 17 for the northern hemispheric spring.
Units are 10
* *
4.
#
The <W
-2
2
-2
m sec .
> covariance from run 17 for the northern hemispheric winter.
3
-3
Units are m sec .
* *
5.
The <W $ > covariance from run 17 for the northern hemispheric spring.
3
3
Units are m sec .
* *
6..
winter.
*
22
2
2 -6
-2
m molecules cm sec
.
*
0
> covariance from run 17 for the norther hemispheric
Units are 10
222
2 -6
-2
m molecules cm sec
The <W V T T > covariance from run 17 for the northern hemispheric
1 2 2
-2
Units are 10 m K sec
winter.
*
9.
*
The <W V 0
spring.
8.
*
Units are 10
*
7.
*
The <W V 03 03 > covariance from run 17 for the northern hemispheric
*
*
*
The <W V T T > covariance from run 17 for the northern hemispheric
-2
1l2 2
Units are 10 m *K sec .
spring.
* *
10. The <03 e > covariance from run 17 for the northern hemispheric winter.
Units are 10 13molecules cm- 3K.
*
*
11. The <03 0 > covariance from run 17 for the northern hemispheric spring.
Units are 101molecules cm
K.
12. The area-weighted <VV> covariance from run 26 for the northern hemispheric month of June.
2
-2
Units are m sec
13. The area-weighted <V V > covariance from run 26 for the northern hemispheric month of June.
2
-2.
Units are m sec
-6814.
The area-weighted
<V'V'>
covariance from run 26 for the northern hemispheric
2
-2
month of June; units are m -sec
15.
The normalized variance spectrum of <V*V*> from run 26 at 40*S for the
northern hemispheric month of June
16.
The normalized variance spectrum of <V*V*> from run 26 at the equator for
the northern hemispheric month of June
17.
The normalized variance spectrum of <V*V*> from run 26 at 40*N the northern
hemispheric month of June
18.
The <V*V*> weighted zonal wavenumber from run 26 for the northern hemispheric
month of June.
19.
The phaseof the steady stationary zonal wavenumber 1 V* amplitude from
run 26 for the northern hemispheric month of June
20.
The same as 19 for zonal wavenumber 2
21.
The same as 19 for zonal wavenumber 3
22.
The area-weighted <W*W*> covariance form run 26 for the northern hemisphric
month of June; units are 10
23.
-6 2
-2
m sec
The amplitude and phase history for days 450-480 of run 26 for the zonal
wavenumber 1 meridional wind V* at 40*S and 12 mb.
24.
The same as 23 for zonal wavenumber 3
25.
The same as 23 for zonal wavenumber 5
26.
The mixing slope <VW>/<VV> from run 26 for the northern hemispheric month
of June; units are 10 4radians
27.
The number density of ozone <0 3> from run 26 for the northern hemispheric
month of June; units are 1011 molecules cm-3
28.
The area-weighted <VO 3 > covariance from run 26 for the northern hemispheric
month
29.
of June; units are 10
11 m-molecules-cm-3sec-1
The total area weighted horizontal eddy flux of ozone for the 50-150 mb
layer from run 26 for the northern hemispheric month of June; units are
-6910
30.
11
-3sec -1
m-molecules-cm
The area-weighted <V'O'>
3
covariance from run 26 for the northern hemispheric
11
month of June; units are 10
31.
The area-weighted <V'0 3'>
11
34.
8
m molecules cm
-3
sec
-1
The area-weighted <W'0 3'>
covariance from run 26 for the northern hemispheric
month of June; units are 10
8 m molecules cm-3sec -1
The area-weighted
<W*0 3*> covariance from run 26 for the northern hemispheric
month of June; units are 10
35.
-3
-l
m molecules cm sec
The area-weighted <WO 3> covariance from run 26 for the northern hemispheric
month of June; units are 10
33.
-3 -1
sec
covariance from run 26 for the northern hemispheric
month of June; units are 10
32.
m-molecules-cm
8
m molecule cm
-3
-1
sec
The area-weighted <V*0*> covariance from run 26 for the northern hemispheric
month of June; units are 101 m *K sec~1
36.
The area-weighted <W*®*>
covariance from run 26 for the northern hemispheric
month of June; units are 10-2 m *K sec~1
37.
The fractional contribution of each zonal wavenumber to the total meridional
eddy
ozone flux <V*0 3*>
as a function of zonal wavenumber.
The covariznce
<V*0 3*> is from run 26 for the northern hemispheric month of June
38.
Same as 37
39.
The phase angle, y from run 26 for the northern hemispheric month of June,
the units are 10-2 radians.
40.
The phase difference in degrees between ozone and the meridional wind for
the standing waves as a function of zonal wavenumber.
The phase angles
are from run 26 for the northern hemispheric month of June
41.
Same as 40
42.
The phase angles of IV*I and 103*1 for wavenumber 1 at 400 S and 12 mb
-70-
during the days 450-480 from run 26 for the northern hemispheric month of
June; units are degrees
43.
The same for wavenumber 2
44.
The same for wavenumber 3
45.
A plot of -<V*O*> versus <V*V*> [3<>/aD# + <a>3<0>/3z]
46.
from
The eddy diffusivity K
yy
-l
2
.4
units are 10 m sec
47.
The eddy diffusivity K
run 17 for the northern hemispheric winter;
from run 17 for the northern hemispheric spring;
42
-l
units are 10 m sec
48.
The timescale T with C1 =2 m/sec from run 26 for the northern hemispheric
month of June; units are 10
49.
4
sec
the Eulerian-Langrangian conversion factor
from run 26 for the northern
hemispheric month of June, units-none.
50.
The auto correlation coefficient R (t)
vv
for days 450-480 from run 26 at
400 S and 12 mb for the northern hemispheric month of June.
51.
The Lagrangian integral time scale from run 26 for the northern hemispheric
month of June; units are 101days.
+1
50MB
-STRATOSPHERE
-1
P
+1
500 MB
-TROPOSPHERE
-I
WEST
EAST
PHASE ANGLE
T i.-ure
(RADIANS)
,. .
80N
WINTER
40N
60N
20N
0
20S
SUMMER
60S
40S
80S
0.05
70
0.1
60
0.2
0.5
50
1.0
E
E 2 .0
40
5.0
10
30
20
50
20
100
200
I0
500
000
80N
60N
40N
20N
20S
0
LATITUDE
figure 2.
40S
60S
80S
SPRING
60N
40N
80N
0.05-
'
20N
0
i
FA LL
40S
60S
20S
I
I
I
-
I
I
I
80S
I
70
-10
0.10.2-
60
0.550
1.02.0
E
-40-
L
5.0-
n
U
30
20-
bJ
50-
20
100200-10
5001000
u
80N
-
--
1....
0
/
60N
40N
~0
- -\
20S
20N
LATITUDE
f i;ure
3.
~
40S
60S
80S
0
80N
0.05
1
WINTER
40N
60N
1
20S
0
20N
SUMMER
60S
40S
80S
-70
I-
0.1-60
0.2
0.5-
-50
-5
E 2.0
0
-- 40
U5.0-0
-30 (
n10w
a_,
t
20-CLU
5U0-
2
500
1000
0I
80N
i
60N
40N
1 -J - 1
20N
1
0
LATITUDE
f
ure 4.
20S
40S
60S
80S
0
80N
SPRING
60N
40N
20N
0
20S
FALL
40S
60S
0.05
80S
-70
0.1
0.2
-60
0.5
-50
1.0.
E 2.0
40
5.0
10
-30
20
50
20
100
20010
5001000
80N
60N
40N
20N
0
20S
LATITUDE
figure
.
40S
60S
80S
0
WINTER
80N
60N
I
SUMMER
40N
I
,
20N
0
20S
40S
60S
-70
-60
-50
E
0
0J
0
20S
LATITUDE
figure
r
.
80N
0.05 -
SPRING
60N
40N
FALL
20N
0
20S
40S
60S
80S
-70
Q
0.1-
0.2
60
-
00
05
0
0
1.0
E 2.0 -
-40
u 5.0 cl)
30
J20-
0
50
0
--
uJ
20
dI
100
2000
010
5001000
80N
60N
40N
20N
-
0
20S
LATITUDE
f i t:u re 7.
40S
i
60S
80S
0
80N
WINTER
60N
40N
20N
20S
SUMMER
60S
40S
0.05-
-70
110
0.1-
(
0.2-
0.51.0-
80S
-00
100
5
5
-10
~60
-50
1
In5
E
E 2.040
i 5.0r
--30
S20-
a:
L
50~
20
100200-10
500~
Is,-
0
I
1000
80N
60N
40N
~~~
20N
~
0
20 S
LATITUDE
f i gure
.
1-
)
40S
60 S
80S
0
8ON
SPRING
40N
60N
20S
0
20N
FALL
60S
40S
80S
0.05-
-70
\Y t o5
-5
0.I-1
0.2-
100
0
5
50
-60
0.51.0-
50
E 2.0--
-10
-. 0
5.0
50
5
-
E
0
0
40-
-5)
cr-I
5 ~--
10O
0
Cf)
120
82
AI0T0
50-
20
100-
00
200-0-1
500-
0iiI
8ON
60N
40N
20N
20S
0
LATITUDE
f i u re '.
40S
60S
8S
0
WINTER
60N
40N
80N
I
0.05-
I
I
I.
SUMMER
20N
40S
20S
60S
80S
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I
I
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-60
0.5-50
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E
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0
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2050-
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100-
2005001000
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60N
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0
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I
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0
20S
LATITUDE
I
I
40S
I
I
60S
I
IL
80S
0
SPRING
60N
40N
80N
0.05-
I
I
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I
20N
FALL
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60S
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I
I
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40
0
Dio-30
J 2050-
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100-
7-044 *-
c
20010
0
500-
1'~ ___
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1000
--
I
80N
I
I
60N
I
I
40N
I
I
20N
I
I
I
I
0
20S
LATITUDE
figure 11.
I
I
40S
J~
60S
80S
0
'
O
SOP
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ajnl--,!-
0
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50-
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10020010
5001000
w
o
80N
60N
-I
40N
50I
I
20N
0
LATITUDE
figure 14.
20S
40S
I
I
60S
I
I
80S
0
40* De
-6.
.089mb
.4 -mb
40 mb
132mb
.2
S2
3
Zonol
4
Wavenumber
figure 15
5
2
3
Zonal Wavenumber
figure 1b.
4
5
.6-
I mb
.4-
13
12mb
2 -
40mb
2
3
Zonal
4
Wavenumber
figure 17.
u~j
80N
60N
40N
20N
JUNE
0
20S
40S
60S
80S
:D
-70
z
uJ
-60
z
0
-50
Lu
E
40
30
20
10
8ON
60N
40N
20N
0
20S
LATITUDE
figure
1..
40S
60S
80S
0
PHASE OF V* ZONAL WAVENUMBER I.
N.H.JUNE
.06
:1
.2
40N
:'
*%%
8 0N
EQ
N.
P(MB)
5 -.
\
40S
40N
40I..........0S....
200N
200-
\1
EQ
.- 40 N
-3
-2
-0
O
RADIANS
f ii.ure 12.
PHASE OF V
ZONAL WAVENUMBER
2.
N.H. JUNE
.06
.2
40N
40S
P (MB)
80S
5
80N
\EQ*
40 k
N
40N
40S
8SS
EQ
200 k
\ -.
B8N
40N
-3
-2
-1
figure 20.
0
RADIANS
I
2
3
PHASE OF V* ZONAL WAVENUMBER
.06
3.
N.H. JUNE
40N
64
-
-
I
-
.2
EQ
80N
80S
40S-
P(MB)
5
4
N
%
0N
EQ
80.S
40 H
80N
404oS
40N
200
80N
II.
-3
-2
-I
figure 21.
0
RADIANS
1
2
3
308e
309
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Wavenumber
I
5-
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4-
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m/sec
300
34
---
2-200*
%%-
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f i g-u re
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40* Winter
Wavenumber
3
400*
8-
6-
P
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<I V*l >
m/sec
4
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I
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lO
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Days
figure 24.
20
25
30
400 Winter
'Wawenumber
5
10
8
<1 V*I>
m/sec
6
4
2
5
15
Days
0
fii
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20
25
30
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60N
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E 2.0
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10
30
20
50
20
100
200
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500
1000
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I
I
;ure
~
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60 S
80S
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LATITUDE
fi
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.
I
I
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I
60S
I
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0
1.0
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CM SEC'
40S
40N
EQ
0
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f iure
2J.
0.05-
80N
80N
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I
60N
60N
I
I
40N
40N
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I
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100200500000
10
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a
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LATITUDE
fir-ure 3 .
I
40S
I
I
60S
I
I~
80S
0
80N
60N
40N
20N
20S
0
40S
60S
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0
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500-C
100000
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0
LATITUDE
f i ure
1.
40S
60S
80S
0
C
80N
60N
40N
20N
0
20S
40S
60S
80S
70
0.05-
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0
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0.5050
1.0-
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100-
10
200-
1000
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5
80N
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55
40N
20N
0
20S
LATITUDE
f i. ure
3
.
40S
60S
80S
0
60N
80N
40N
20N
0
20S
40S
80S
60S
0.05-
-70
0.160
0.20.5 -
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w 5.0D1
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3
5
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0
ct
20
50
00
20000
0
5001000
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0
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LATITUDE
i ure 33.
I
40S
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I
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0
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80S
E
80N
40N
60N
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40S
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60S
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o~~~
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100
200
10
--------- 0
500
1000
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f igu re 314.
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15
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figure 43.
12 ib.
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300-
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