A LAGRANGIAN MEAN DESCRIPTION OF STRATOSPHERIC TRACER TRANSPORT
by
EDUARDO PANTIG OLAGUER
S.B., Massachusetts Institute of Technology
(1980)
SUBMITTED TO THE DEPARTMENT OF
METEOROLOGY AND PHYSICAL OCEANOGRAPHY
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 7, 1982
Massachusetts Institute of Technology 1982
-
-
4
./
Signature of Author
y & Physical Oceanography
(rol
Certified by
Thesis Supervisor
Accepted by
Chairman, Department Committee
DRAW
LiBRAR ES
A LAGRANGIAN MEAN DESCRIPTION OF STRATOSPHERIC TRACER TRANSPORT
by
EDUARDO PANTIG OLAGUER
Submitted to the Department of Meteorology and Physical Oceanography
on May 7, 1982 in partial fulfillment of the
requirements of the Degree of Master of Science in
Meteorology
ABSTRACT
Following the approach of Dunkerton, the mathematical properties
of the Generalized Lagrangian Mean, developed by Andrews and McIntyre,
are exploited in simplifying the thermodynamic and continuity equations,
which are then used to calculate Lagrangian mean velocities and parcel
trajectories in the stratosphere. The inclusion of meridional temperature
advection and a non-zero equinoctal circulation are two improvements
introduced in the calculation, in addition to the use of mutually consistent temperature and diabatic heating profiles generated by the MIT stratospheric model. The resulting motions are in qualitative agreement with
the observed poleward and downward spreading of tracers such as water
vapor and ozone. Transit times for both poleward and downward transport
are estimated and compared with observations.
Thesis Supervisor:
Title:
Dr. Ronald Prinn
Associate Professor of Meteorology
3
CONTENTS
Page
I.
INTRODUCTION
4
II.
THE GENERALIZED LAGRANGIAN MEAN
5
III.
THE LAGRANGIAN MEAN THERMODYNAMIC AND CONTINUITY
EQUATIONS
7
IV.
NUMERICAL SIMULATION OF PARCEL TRAJECTORIES
10
V.
INTERPRETATION OF RESULTS AND CONCLUSION
15
TABLES AND FIGURES
17
ACKNOWLEDGEMENTS
39
BIBLIOGRAPHY
40
I. INTRODUCTION
Dynamical processes play a crucial role in determining the overall distribution of ozone in the stratosphere. If the meridional transport of ozone were dominated by the longitudinally averaged circulation,
one would expect the behavior of ozone to be largely governed by the
pattern illustrated in Figure 1. This, however, is not the case.
Observations indicate that the overall mean mass flow in the
stratosphere is directed poleward and is downward outside the tropics.
Such a pattern is often referred to as the "Brewer-Dobson" circulation,
which is represented in Figure 2. If this flow is inconsistent with the
two-cell pattern associated with the hemispheric zonal mean circulation,
it does, however, account for the observed high concentration of ozone
in the polar winter. This apparent paradox may be resolved by noting that
the eddies, i.e., the deviations from the zonally averaged flow, also
contribute to the transport of trace species, especially in the winter
stratosphere, where planetary waves are intensely active.
To understand this tendency for the mean flow to be compensated
by the eddies, it is useful to consider the properties of a fluid by following the motions of individual parcels. To do this for each and every
parcel, as required by a strictly Lagrangian approach, would be costly
and time-consuming. It may, however, prove fruitful to use a hybrid description- one that allows us to, in some sense, follow the motion of a
typical air parcel, while retaining the mathematical convenience that
Eulerian averaging affords. In this paper, we use just such a description
(developed recently in efforts to understand wave-mean flow interactions)
in order to account for the observed pattern of tracer transport.
5
II. THE GENERALIZED LAGRANGIAN MEAN
Determining the average sense of meridional mass transport requires that we define a suitable mean velocity of particles associated
with a given latitude circle. The recent work of Andrews and McIntyre
(1978a),hereafter referred to as AM, provides a useful basis for such a
definition, which we may visualize as follows. Imagine a massless rod
constrained to lie parallel to the x-axis. Attached to this rod by "massless springs" are fluid parcels which initially lie on the rod, but are
subsequently displaced from their equilibrium positions, subject to some
linear restoring force. (See Figure 3.) The parcels will thus drag the
rod at some velocity, which we shall identify as the Lagrangian mean
velocity of the fluid parcels.
The above description may be made mathematically precise. Let x
represent the current displacement from a point P
rigid rod initially coincident with P
tesian coordinates (j',7[',f')
Let
I
such that P is the current location of the
fluid parcel originally at point P
(X, t) = If (x+
.
of a point P on the
S__R
be the vector PRP with Car-
.
Now let
(II-1)
(X_, t) , t)
whereqYis some property of the fluid, and let ()
denote an average in the
x-direction such that
(xt)=
0
(11-2)
We define the Lagrangian mean of the property
J
as an average over the
displaced particles, i.e.,
f(x,ty
1/(x,t)
=
(11-3)
Furthermore, if u(x,t) is an Eulerian velocity field, then there is some
-L
u , such that
dx/dt
and DL
where D
=
u
(x,t)
u
(xt)
(11-4)
= ur
= 4/at + u .V
(11-5)
and
V= x +
_(x,t).
-L
u is, of course, the Lagrangian mean velocity.
AM discuss in detail the mathematical properties of the Generalized Lagrangian Mean. Rather than duplicate a number of their derivations,
we shall instead quote some of the more important results which we shall
need in the proceding development. These are listed below. (Note that we
have used the Einstein summation convention.)
(i)
(dy/dtV = D'
(ii)
(a'). + b+) L= a
+b
where a and b are constants
(iii)
a=a
L
-L
-L
(iv)
-L
k-S(x,t) =
L
-L( (x,t)- 1(x,t)
af/lix
+ O(a )
where a is a small disturbance amplitude and
+
In
(iv),
'
=
I(xt).
is often referred to as the "Stokes correction".
To apply the above mathematical ideas to the atmosphere, we need
-L
only interpret ( ) as a zonal average, and u (p,z) as the Lagrangian mean
velocity of a ring of particles centered at latitude 97 and altitude z.
III. THE LAGRANGIAN MEAN THERMODYNAMIC AND CONTINUITY EQUATIONS
We are now in a position to exploit the mathematical properties
of the Generalized Lagrangian Mean in a manner parallel to that of Dunkerton (1978). Consider first an exact equation of the form
d6/dt =
where
Q
(11I-1)
Q = (#/T)(GT/*t)diabatic ,with T and 8 denoting absolute and poten-
tial temperatures, respectively. Using property (i), we obtain
-L -L
-7L
D 0 =L
(111-2)
We now consider the first-order Stokes corrections to 8 and
= d(f f9')x.-
Q:
(C(.40.)9j)
JJ
(III-3)
J
-S
and similarly for Q . AM have shown that in the case of an incompressible
Boussinesq fluid,V-V=O(a 2),
whereas in general, f.)e'/ix.
is O(a).
Although we are interested in compressible fluids, we take this as an indication that the second term on the RHS of (111-3) does not constitute
the leading approximation to 6S. An appropriate scale analysis (see AM,
1976) in fact suggests that the leading contribution is due to the term
Because 9 for each air parcel is a quasi-conservative property
as long as
Q is not too large, #' should be small since the parcel's
trajectory.must at some point coincide with the rod's position. Therefore,
unless wave transience and turbulent diffusion are important, we may
neglect19
in (111-2).
At the same time, the leading contribution to Q
involves
Jt'Q'.
At low latitudes in winter, and during the summer, this term is negligible
-S
-
since Q'<<Q. Dunkerton proposed that the approximation Q4<Q is also good
at mid-latitudes during the winter, arguing that
i'
is in general 90
out
of phase with v' (assuming a linear restoring force or, alternatively,
due to quasi-geostrophy), whereas
(See Figure 4.)
Q' and T' are strongly anticorrelated.
Since planetary wave activity leads to strong correlations
between v' and T', Dunkerton assumed a correspondingly weak correlation
between
'
and
Q'.
The correlation
Table 1.
between v' and T'
We observe that this correlation is
latitudes during the winter. How large should
is
demcastrated roughly in
indeed strongest at -mid-
Y
Q,2?/X
=)'(f2
be f
to be negligible? Let us suppose that
v'
= Acos(l)
T'
+d..)
Q' =-Dcos(lk +Vc)
1?' = Csin(lA)
where
= Bcos (1
represents longitude,
1 a wavenumber,
and o( some phase angle.
It
then straightforward to show that. T =cos( and cf = sin4. The maximum
is
value of V does not appear to be greater than roughly 0.5 (The original
data likewise indicate that on a day-to-day basis,
cantly exceed 0.5.),
hence <f is
idate Dunkerton's argument.
at least 0.8,
Moreover,
r
does not signifi-
which w%'-ould seem to invalthe magnitude of
we may estima>
which represents sample trajectcries of air parcels
'Q' from Figure 5,
computed directly from an Eulerian model.
waves,
For planery
6
the order of 10 m. At mid-latitudes during the winte:,
0
Q, which is roughly 1 K/day, hence y'Q' is
order of
0
10 K m s
it
is
, i.e.,
comparable to v'T'
under similar
j'
is of
Q' may be of the
the order of
:nditions. Although
not clear from the above discussion that we ma-- in
general neglect
-S
Q in equation (111-2) , we shall do so for the time Deing, in the hope
that we might still reproduce some of the qualitati--
features of stra-
tospheric behavior.
Lastly, we focus our attention on time-aver-aged solstice conditions, when Jl/at
is negligible. With this and the
aforementioned
approximations, our thermodynamic equation reduces
-
-L -
-.L4 -
v eO/dy + w 48/dz = Q
(111-4)
-L
-L
To fully determine v and w , we need one further restriction,
namely that mass be conserved. The Lagrangian mean
z:mntinuity equation
(see AM, 1978) is
-L
D
where
=
dntst
+
det(Sc
dety
=.u
(111-5)
0
+a./dx.),
1f
=1 if i=j, but is
pa rcel
denotes the density of a fluid parcel.
, otherwise, and
In the case of an incompressible Boussinesq fluid, AM have
shown that
2
=1 -
i.e.
,
-
3
(1/2) 2 ( jk )x
xk
+
O(a3
(111-6)
to O(a). 1 Once again, we shall use this to justify replacing
=
withf in (111-5). Moreover,we neglect time-dependence and assume stratification in the vertical, so that our continuity equation becomes, in
spherical coordinates:
L Cos
1v
r
f)+
rc C59
af
cosg99
_
l_
j~-/
L
(111-7)
=0
where r denotes the mean radius of the earth.
Combining (111-4) and (111-7) we obtain:
-L -
w
=
and
-)L
a(v cosf')
1
co sf
f
)z
z
-
-
+
bz
+
r
e
ez
I
Oz
z
0)
oz
=
0
(111-9)
z
Equation (111-9) may be integrated with suitable boundary con-L
ditions to obtain the profile of v
,
-L
whereas that of w
ly upon application of (111-8).
represents a hydrostatic basic state density.
follows immediate-
IV. NUMERICAL SIMULATION OF PARCEL TRAJECTORIES
The approach outlined in the preceding section varies somewhat
from that of Dunkerton, who ignored the meridional advection term in
(111-4) on the grounds that Vincent (1968) found it to be small in the
lower stratosphere. Moreover, Dunkerton used heating rates from Murgatroyd and Singleton (1961) and static stabilities (i.e., values of Q9/az)
inferred from the U.S. Standard Atmosphere Supplements (1966) in obtaining
-L
-L
his profiles of v and w . The results of his calculation are shown in
Figures 6-9. It should be noted that the representative trajectories in
Figure 8 are based on a "two-cycle" model, where quiet conditions are
assumed during the equinoxes, and equal but opposite circulations prevail
during the two solstice seasons.
Our own calculations are based on diabatic heating and temperature profiles obtained from runs of the MIT three-dimensional chemicaldynamical model. The data, originally in spectral coordinates, were converted to grid form and longitudinally averaged to give values of temperature and diabatic heating at 26 pressure levels (0-80 km) and 15 latitudes (80.5N-80.5S). The density profile was then obtained by applying
the ideal gas law at each vertical level. The advantage to be gained by
using model results is one of internal consistency, i.e., values of
are computed using values of
Q. Moreover, we have, unlike Dunkerton, re-
tained the meridional advection term in the thermodynamic equation. (Originally, we had ignored this ourselves, but integration of the continuity
equation led to values of v
which did not correspond with the assumption
that meridional advection was small throughout the stratosphere.)
Essentially, equation (111-8) was integrated separately for each
-L
hemisphere assuming v to be zero near the poles (at latitudes 1 and 15),
and ignoring any possible discontinuities at the equator. We also assumed
-L
w to be zero at the ground and at the top of the atmosphere, to be consistent with the rigid lid approximation employed in the model. The numerical scheme used in the calculation is outlined below.
The various terms in equation (111-8) were evaluated using centered differences for the vertical derivatives and forward (or more appropriately, "equatorward") differences for the horizontal derivatives.
11
In the Northern Hemisphere (latitudes 1-8) for instance, the finite difference equivalent of (111-8) becomes:
v
-L cos
i,j+l
-L!.cos
v.
-
j+1
E
ij
-L
i+1,j
A...
13
(
-.
j+1
)Cos
J
B. .v..
1J
A. .=
1J
i+1
-..
+1 -
j
9
B.A.
= A
C..
13
=
D.. =
1+1
~
r...-DD.rD+1,
Q.(z.
Dj 13
ilr
i
+1
i+1
i-l
ij
i-1
Note that the subscripts
"i"
-l'
i-1,
+1,j
z
i+l
)/($
-z.
i+l
i
A-
Di-1,j + Dij .
- z
(IV-l)
1J
1+1, j14-1,
i-1, j
z
C..
-
13
2-
z.
1
- A
i+,
13
i-l,j
z ++1- zi-
+
where
-L
_
i-1
)
-
i-1,
i+l,j
and "j"
i-l,j
- z
here denote height and latitude grid
indices such that i=1,2,...,26 and j=1,2,...,8.
Solving (IV-1) for -L
v.
v.
.
=Cos
1,J+1
cos
~L .
J
3
,
we obtain the relation:
1,j+1'
-L +
v.
(
ji+ 1
'P
-L
)
A..(v.
fj+1
13
1
+ B. .v..
1)
13
.
i+1,3
i+1-
-L
.
1-1,j
.
z
-L- C..
13
(IV-2)
Together with the aforementioned boundary conditions, equation (IV-2)
completely determines the Northern Hemisphere profile of -L
v and hence,
-L
that of w via the finite-difference equivalent of (111-8), namely:
-L
w..
1J
=
D..
13
-
-L
v. .A../r
1J j
-L
The same equations determine the profiles of -L
v and w
(IV-3)
in the Southern
Hemisphere, except that the subscripts "j" and "j+l" are now switched
for j=9,...,15.
Calculations were performed for each individual day in July
and January, with the velocity profiles being subsequently averaged over
one month.. The results of this procedure are displayed in Figures 10-13.
It should be noted that the profile of w generally follows that of Q,
i.e., regions of heating are associated with rising motions, whereas
regions of cooling imply subsidence. Anomalously large velocities at a
few gridpoints in the troposphere and near the top of the model were
most probably due to limited resolution in approximating large temperature
gradients by finite differences. This was painfully obvious when at first
. ..
.
-L
-L
we tried eliminating
v instead of w
in equations (111-4) and (111-7).
Integration of the resulting equation in the vertical resulted in unreasonably large values of w almost everywhere, due to our heavy reliance
on meridional derivatives which were calculated using grid points essentially thousands of kilometers apart. The more reasonable values obtained
in our final calculation are due to greater reliance on vertical derivatives, which are probably more accurately estimated since the relevant
spacing between grid points is only a few kilometers.
In obtaining parcel trajectories from the Lagrangian velocity
profiles, we have attempted to improve on Dunkerton's assumptions about
equinoctal conditions. Figure 14 shows the seasonal variation in the
Northern Hemisphere of stratospheric zonal mean and eddy kinetic energy.
Note that the maximum and minimum of the zonal mean circulation occur
roughly in January and July, respectively, and that the equinoxes are
periods of most rapid change. We have therefore modelled the time variation of the Lagrangian mean circulation in the following manner:
-L
v (y,z,t)
-L
-L
v + v
s
w
2
+
-L
-L
v - v
s
w
2
.
cos(2-1t/360),
(t
in days)
(IV-4)
where t=O on July 15 and vL and -L
v are the July and January average mers
w
idional velocities, respectively, measured in km/day. A similar equation
describes the seasonal variation of w.
Representative trajectories cal-
culated with these assumptions are shown in Figure 15.
We have also investigated the transit times of typical air parcels in six regions, corresponding to the tropical, northern mid-latitude,
and southern mid-latitude upper and lower stratospheres. A tabulation of
these average transit times is.given in Table 2, which also illustrates
the effect of including meridional temperature advection and an equinoctal circulation. The numbers were obtained by simulating parcel trajectories over a period of 10 years. Parcels which did not reach the pole
during the period of observation were nevertheless assumed to have arrived there in 10 years. Parcels which did not reach the ground during
this period, however, were simply ignored in the computation of transit
times for motion towards the ground, because of the possibility of being
trapped at the top of the atmosphere, where the vertical velocity was
assumed to be zero. Since we have also assumed v
to be zero near the
poles, air parcels reaching polar latitudes are effectively trapped. For
the purpose of computing transit time towards the ground, however, we have
reset the meridional velocities near the poles in such a way as to reverse
the motion towards the equator (i.e., the value of v
at latitude 1 in
the model was set equal to the negative of the absolute value of v
cal-
culated at latitude 2, and similarly for the south pole). Note that for
motion towards the pole, meridional advection has the effect of reducing
the transit time by half for parcels originating from the equatorial upper stratosphere, while an equinoctal circulation has a similar effect
for both the upper and lower stratospheres. In most cases, an equinoctal
circulation also has the effect of halving transit time towards the
ground.
As a further test of the sensitivity of our numerical results,
we have simulated parcel trajectories assuming a time-varying circulation
patterned more after the variation of eddy (as opposed to zonal mean) kinetic energy. (See Figure 16.) Transit times computed with this assumption
are also shown in Table 2. Note the comparatively minor differences between these transit times and those based on the variation of zonal mean
energy.
Lastly, we have attempted to model in a very crude way the effect
of eddy diabatic heating by assuming that -S
Q =Q
at latitudes above 20N in
January and below 20S in July, so that the forcing term on the RHS of
(111-4) is effectively doubled. (We have already seen that
f'Q'
may quite
possibly be of the order of v'T' at winter mid-latitudes. Examination of
data from Richards (1967) leads to the conclusion that 0 (v'T')/Jy is of
the order of
Q
in these regions, so that Q
Q would not be unreasonable.)
The resulting mean velocities are shown in Figures 17-20, while the associated transit times assuming a circulation based on K are displayed in
Table 2.
It is of interest to note that two recently written papers give
Q
supporting evidence that
is indeed significant. Schoeberl (1981) looked
at the effect of dissipating planetary waves on the Lagrangian mean flow
and arrived at a poleward and downward circulation that appeared.to be
twice as strong as Dunkerton's circulation in the lower stratosphere.
Kurzeja (1981) calculated
-s
Q assuming that the accelerations caused by
eddy radiative dissipation balance those caused by the zonal mean diabatic
heating and found that Q
compared with Q.
was generally smaller, but not negligible when
V. INTERPRETATION OF RESULTS AND CONCLUSION
The parcel trajectories represented in Figure 15 reveal that a
poleward-downward circulation of the Brewer-Dobson type is indeed the
predominant pattern in the stratosphere, and apparently, in the mesosphere as well. We hesitate, however, in giving undue credence to our results in the uppermost portions of the atmosphere because of the rigid
lid approximation employed in the calculation. Whereas the diabatic heating profile used by Dunkerton represents warming in the summer hemisphere
and cooling in the winter, especially at high altitudes, our diabatic
heating is necessarily consistent with the condition that vertical velocity vanish at the top, and because of mass continuity, this leads to
disagreement with the heating rates of Murgatroyd and Singleton, particularly in the mesosphere. Thus, the oscillating behavior noted by Dunkerton between 50 and 70 kms is not evident in our -calculated trajectories.
Much of the controversy concerning tracer transport has been over
whether the observed motions are due mainly to organized mean motions or
to random turbulent diffusion. Evidence for the latter mechanism has been
based on observations analyzed by Feely and Spar (1960), who looked at
185
W fallout from several bomb tests conducted at Bikini (12N)) between
May and July 1958. They claimed that if the Brewer-Dobson model were valid,
185W should have reached the lower polar stratosphere by moving upward
into the high tropical stratosphere, migrating poleward at high altitudes,
and then descending into the lower polar stratosphere. Feely and Spar,
however, saw no convincing evidence for either significant upward debris
motion at the equator or large scale subsidence at polar latitudes. The
observed distributions of 185W did suggest a lateral spreading that was
poleward and downward, wherein transfer across isentropes was significant
between 50 mb and 100 mb. (See Figure 21.) This observed lateral spreading,
however, is not necessarily inconsistent with our own results, which are
based entirely on non-turbulent transport, since there are some trajectories, particularly in the lower stratosphere, which do not exhibit large
upward tendencies near the equator. While downward diffusion may possibly
play a significant role in determining the residence times of tracers
within the stratosphere, lateral diffusion might not be as important.
It is interesting to compare our results with those of Dyer and
Hicks (1968) who charted the poleward progression of volcanic dust from
the Mt. Agung eruption of March 17, 1963. The initial injection of the
dust created an equatorial reservoir at 8S at a height of 22-23 kms,
whereafter a significant fraction entered the lower Northern Hemisphere
stratosphere. The observed dust amounts analyzed by Dyer and Hicks showed
winter maxima which progressed at a consistent rate of 9.4 degrees of
latitude per month, suggesting an equator-to-pole transit time of roughly
9 months.
(See Figure 22.) Compare this with the transit times which were
computed taking into account the possible effect of eddy diabatic heating.
It is therefore apparent that adequate parameterization of
Q
is
crucial if meridional transport is to be simulated properly. One possible
way of accounting for the Stokes correction is to assume that v'=ud4'/dx.
Since v'=,)'/4x, where+' represents the quasi-geostrophic perturbation
stream function, we may write the Stokes correction as
This method may be used to estimate
Q= &(''Q'/u)/ay.
as well, which would enable one to
make a sounder assessment of the relative importance of turbulent diffusion
in the meridional transport of trace substances.
T A.B L E S
A N D
F I G U R E S
55
I
I
I
I
.
I
I
I
I
I
MEAN STRATOPAUSE
50
I
--
I
I
I
-
-
-
-
-too...
45
-0.1
-0.2
......
40 -
to
-0.2 -0.1
-
0-....
35-
.oo
'-. -0.5
.--
.-
1
-1.5
-
-....
0.5
-1.5
--
- .
-..
J
30
o
2
-.90.5
-
km
25
o..
..
.
.. ~
--
2.
tMEAN'*%
*TROROPAUSE
20 F-
-o
.4
15
0~
10
5
\r -1
-o
5
K.40
I Io
n
90
80
70
60
50 40
* NORTH
30
20
10
0
10
20
30
40 50
.SOUTH
60
70
80 90
Fig. 1. Mean meridional circulation for December-February. Mass flow is given in units of 1012 gm/sec.
(After Louis, 1975.)
40
km
Winter
Hemisphere
40
km
Summer
Hemisphere
30[-
130
Stratospheric
Westerlies
tratospheric
Easterlies
20k
20
Tropopause
10-
j
~...----
j
10
Ozone Destruction Near Ground
90 0
600
300
S00
30 0
600
90*
Fig. 2. Schematic illustration showing sources, sinks, and mass
transport of ozone. (After Duftsch, 1971.)
y,z
X P
ax
PO
g-o+R ( rod )
RO (initial position of
rod and particles)
Fig.
3.
.
ITi
l iit
JANUARY 1964
JANUARY 1964
-50 -
30mb
30mb
70*N
700 N
0
-- 0.5
--
0
-60-65
-1.5
-70-7 .5
-80W
*W 180
140
100 80 60 40 20 0 20 40 60 80 100
WEST
140
3-2.0
180 *E
EAST
Fig. 4. Variation of temperature and cooling rate with longitude. (After Newell, et al.,
1974.)
Table 1.
VALUES OF
LATITUDE
( N)
85
80
75
70
65
60
55
50
45
40
35
30
25
20
r
=_T
FOR JANUARY, 1965
PRESSURE (mb)
100
0.27
0.12
0.05
0.06
0.15
0.27
0.39
0.47
0.44
0.33
0.15
0.
0.01
0.13
50
30
0.05
0.04
0.06
0.12
0.19
0.27
0.36
0.45
0.47
0.45
0.38
0.25
0.08
0.14
0.11
0.09
0.10
0.15
0.21
0.27
0.34
0.39
0.38
0.35
0.33
0.32
0.31
0.
10
-0.02
0.03
0.09
0.15
0.21
0.28
0.35
0.41
0.42
0.36
0.24
0.26
0.40
0.11
*
Computed using data from Richards (1967), The Energy Budget of the Stratosphere During 1965, Report No. 21, MIT Dept. of Meteorology, Cambridge,
Mass.
10
30
£0
E
ctr
:D
60
100
LI)
LU 200
w~
300
400
6008501000--
EQ
600
30*
POLE
LATITUD E
Fig. 5. Typical air parcel trajectories for various atmospheric
disturbances. (After Wallace, 1978.)
km
70
.0
0.5
..-
'-0.5
km
70
0040
60
~-
-100
50
60
-20-
50
40
00
40
30
0
.-'_--
20
S
60
30
EQ
30
60
20
S
W
Fig. 6. Lagrangian-mean vertical
velocities in km/day.
30
I
I
60
30
I
EQ
30
60
Fig. 7. Lagrangian-mean meridional velocities in km/day.
km
70
km
70
D
60
E
50
50
C
40
30
30
20
S
60
30
EQ
30
60
W
Fig. 8. Streamlines associated
with Lagrangian-mean
velocities.
10
S
EQ
V
Fig. 9. Representative trajectories. Circles indicate
positions after 6 months.
Figures 6-9. S=summer pole, W=winter pole. (After Dunkerton, 1978.)
0
0
0
0
B
B
km
70
60
50
hi
40
30
20
-N
10
0.
60N
40N
20N
0
20S
40S
Fig. 10. July Lagrangian-mean meridional velocities in km/day.
60S
0
0
0
0
.7
0
0
km
70
60
50
40O4
30
.20
-10
.0
Fig. 11. July Lagrangian-mean vertical velocities in km/day.
0
0
0~
0
0
0
4
km
70
60
50
40
30
20
10
0
60N
40N
20N
0
20S
40S
Fig. 12. January Lagrangian-mean meridional velocities in km/day.
60S
km
70
60
50
4O&
30
20
10
0
60N
40N
20 N
0
20S
40S
Fig. 13. January Lagrangian-mean vertical velocities in km/day.
60 S
20
I0
20
20
K
JAN FEB MAR APR MAY
JUN JUL AUG SEP OCT NOV DEC
1964
-2
7
Fig. 14. Daily variations of energy per unit area (10 erg cm )-in
the form of K' and K between the 10 and 100 mb levels,
after Dopplick (1971).
Fig.
80N
60N
40N
20N
15
0
20S
40S
60S
80S
0.05
-70
0.1
-60
0.2
0.5
-50
1.0
-Q
E
E2.0
-40
Uj5.0
Q'
U)
F-
l
W 20
0:
50
-20
100
200
-10
500
1000
0
80N
60N
40N
20N
0
LATITUDE
20S
40S
60S
80S
31
Table 2a.
TRANSIT TIME IN MONTHS FOR MOTION TOWARDS THE POLES
*
Method
*
*
20N-60N
*
*
20-20N
*
*
20S-608 *
*
Meridional Advection
Circulation Based on K
*
*
*
8
12
*
13
24
*
*
16
24
*
*
*
*
No Meridional Advection
Circulation Based on K
.25.
24
Meridional Advection
No Equinoctal Circulation
25'
53
No Meridional Advection
No Equinoctal Circulation
43
*
*
48
*
*
Meridional Advection
10
-
Crculation Based on KL-,)
*
-. S
.-
Q=Q in Winter
Extratropical Regions
Table 2b.
*
*
*
*
*
*
*
2)
*
*
*
5
6
*
*
*
*
30-50
10-30
30-50
10-30
30-50
10-30
30-50
1
*
9
9
30-50 km
10-30 km
A,
*
*
9
6
*
*
30-50 km
10-30 km
TRANSIT TIME IN MONTHS FOR MOTION TOWARDS THE GROUND
*
Method
* 20N-60N
*
*
*
20S-20N
*
23
15
Meridional Advection _
Circulation Based on K
No Meridional Advection
Circulation Based on K
*
*
*
29
16
No Meridional Advection
No Equinoctal Circulation
*
*
*
*
*
*
32
29
*
*
50
31
*
*
38
29
*
*
67
54
*
25
18
*
*
71
58
Q = -.Q in Winter
Extratropical Regions
11
7
*
*
*
*
*
*
*
*
*
*
34
38
*
30-50 km
10-30 km
*
*
30-50 km
10-30 km
*
*
58
47
*
30-50 km
*
10-30
km
*
*
61
49
34
29
*
17
13
*
*
*
Altitude
*
*
*
29
28
*
-s
28
28
*
*
Meridional Advection
Circulation Based on K'
*
*
*
48
37
*
*
*
*
Meridional Advection
No Equinoctal Circulation
*
*
*
20S-60S *
*
*
*
*
1
*
*
2
Altitude
**
*
*
*
*
*
*
30-50 km
10-30 km
30-50 km
10-30 km
*
*
*
14
*
30-50 km
*
11
*
10-30
This method is equivalent to that of Dunkerton.
2
Meridional advection and a circulation based on K included.
km
NORTHERN HEMISPHERE
LATITUD ES (1-8)
VL= L+ (L--
vL) cos 2Tt
L
90
,,+ ( L-L9) COS 27t
/
2VW-_SL
~~
ORIGINAL CIRCULATION
vw
VL
w
J
F
M
A
M
J
J
A
S O
N
D
e
SOUTHERN HEMISPHERE
LATITUDES (9-15)
L = L+ (VL~cw
9'=9)+
S
wvg-L) COS
VL=L+(LLco2w
27Ti
/SL
S L
^, 60
_VL)
S COS 2VTf
90
VSL
-L
w
J
F
M
A
M
J
J
A
S O
N
D
t
t = Time in days beginning January 1
vL = Average January Lograngian mean meridional velocity in km/day
VL =Average July Lograngian mean meridional velocity in km/day
Fig.
16.
Circulation based on variation of stratospheric eddy
kinetic energy.
00
0
,
km
70
60 N
40 N
20N
0
20S
40S
Fig. 17. July Lagrangian-mean meridiona.1 velocities in km/day.
60S
-1-1-1 0~0
0
0
S
S
0
0
km
70
60
50
40w
30
20
10
60 N
40 N
20.N
0
20S
40S
Fig. 18. July Lagrangian-mean vertical velocities in km/day.
60S
S
0
000SS
0
0
km
70
60
50
40
30
20
10
60N
40 N
20 N
0
20S
40S
Fig. 19. January Lagrangian-mean meridional velocities in km/day.
60S
0.
0
0
0
0
0
0
0
km
70
60
50
30
20
I0
60 N
40N
20.N
0
20S
40S
Fig. 20. January Lagrangian-mean vertical velocities in km/day.
60S
tj
3C
-----
(D
r0
V
-
5
---
5----
-
MD ti %.W
0
575-------------------
--
---
5
-
-
-5
-
-
-~~~7~----0
(D(
----
-00'
-
1000
-40000--00-~
-- ~
~~~~~
7-
------
-
-
...-.-.-
---
-
~~~-
- 0
- 00--
-
6
-
-
- -- -0
0
-40-
T-50
::; U
0
ho
20 0)
25C
800
0
N 70*
600
50*
200
300
400
00
10*
10 0
S
LATITUDE
Broken lines represent potential temperatures ( K) for Jul-Sep, and Nov-Dec, 1957.
3C
o~0
575--.,
-550- ...---
.
-
5C)
'50
300
-
-N
.
-- 1-
- -
50
0
-
-
- -
-
- -
600-
575----
--------
55
-
2000
-- 10,600
-11
(~t
~t
(nC
MD-1c
0J
'"
10c )
150200250.
80 0 N 704
--
~
Ht
%.0r(
'LDCD
~
~
600
50*
~
~
-
-
-
-3
0---5---
40
0
30
LATITUDE
0
J
LL
___-
325
200
10*
00
S
100
w
POSITION
OF SUN
-
-1965
N.
0
0
-1964
-----.
-1963
BALI
-. ERUPTtON
90 0
600
30*
NORTH
0
3
)0
600
90*
SOUTH
Fig. 22. Poleward progression of winter maxima of volcanic
dust concentrations from Mt. Agung eruption.
(After Dyer and Hicks, 1968.)
39
ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr. Ronald Prinn for suggesting
the topic of this thesis and for his timely and valuable suggestions that
helped me to complete my work.
BIBLIOGRAPHY
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,1978a: "An Exact Theory of Nonlinear Waves on a Lagrangian Mean
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,1978b: "Generalized Eliassen-Palm and Charney-Drazin Theorems for
Waves on Axisymmetric Mean Flows in Compressible Atmospheres",
Journal of the Atmospheric Sciences, 35, 175-185.
Cunnold, D., F. Alyea, N. Phillips, and R. Prinn, 1974: "A Three-Dimensional Dynamical-Chemical Model of Atmospheric Ozone", Journal
of the Atmospheric Sciences, 32, 170-194.
Dopplick, T.G., 1971: "The Energetics of the Lower Stratosphere Including
Radiative Effects", Quarterly Journal of the Royal Meteorological
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Dunkerton, T., 1978: "On the Mean Meridional Mass Motions of the Stratosphere and Mesosphere", Journal of the Atmospheric Sciences, 35,
2325-2333.
Dutsch, H., 1971: "Photochemistry of Atmospheric Ozone", Advances in Geophysics, 15, Academic Press, 219-322.
Dyer, A.J., and B.B. Hicks, 1968: "Global Spread of Volcanic Dust from the
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Feely, H., and J. Spar, 1960: "Tungsten-185 from Nuclear Bomb Tests as a
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Kurzeja, R.J., 1981: "The Transport of Trace Chemicals by Planetary Waves
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Murgatroyd, R.J., and F. Singleton, 1961: "Possible Meridional Circulations
in the Stratosphere and Mesosphere", Quarterly Journal of the
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Newell, R.E., 1961: Geofisica Pura e Applicata, 49, 137.
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Scoeberl, M.R., 1981: "A Simple Model of the Lagrangian-Mean Flow Produced
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Sciences, 38, 1841-1855.
Sheppard, P.A., 1963: "Atmospheric Tracers and the General Circulation of
the Atmosphere", Reports on the Progress of Physics, 26, 213-267.
Wallace, J.M., 1978: "Trajectory Slopes, Countergradient Heat Fluxes and
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