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THE BAROCLINIC INSTABILITY OF SIMPLE AND HIGHLY STRUCTURED
ONE-DIMENSIONAL BASIC STATES
by
JAMES WILLIAM ANTHONY FULLMER
B.S., Drexel University
(1972)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MAY 1979
) James William Anthony Fullmer
Signature of Author ..........................................................
Department of Meteorology, May 1, 1979
Certified by ..............................................................
Thesis Supervisor
Accepted by
..............................................................
Chairman, Departmental Committee
on Graduate Students
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by
JAMES WILLIAM ANTHONY FULLMER
Submitted to the Department of Meteorology on
1 May
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy
1979
ABSTRACT
The baroclinic instability of simple and highly structured onedimensional basic states is studied using a frictionless, adiabatic, quasigeostrophic model on a s-plane. Square-root-pressure coordinates are used
at 48 levels in the vertical and calculations are made for the upper boundary conditions 4' = 0 and w' = 0. Many properties of the unstable waves
are considered: instability source, wavelength, growthrate, phase velocity,
steering levels, and the vertical structure of their amplitude, phase,
meridional entropy transport, and potential to kinetic energy conversion.
First, simple basic states are assumed. Their shear and static stability
each have a single value from 1000 to 250 mb, and another, independent value
from 250 to 0 mb. These simple basic states are determined by 3 parameters:
a shear ratio, a static stability ratio, and a generalized 8 parameter. A
wide expanse of parameter space is considered. The highly structured basic
states have zonal velocities and static stabilities obtained from one month
averaged data for latitudes 250 N to 650 N and for months (January, April,
July, and October), which represent seasonal extremes and transitions.
The longwave modes discovered by Green (1960) are shown to have
several interesting properties. To exist, they require a non-zero value of
qy and are destabilized by the presence of a Stratosphere or rigid lid.
Doubling times are moderately short (about 1 week) and their meridional circulation in the lower stratosphere agrees with observations. Pressure amplitudes, kinetic energy destruction, and meridional entropy transport are particularly strong in the lower stratosphere (relative to other levels).
Their kinetic energy is generated in the troposphere. Their entropy transports are countergradient in the lower stratosphere when a reversed shear
exists in that region.
Quasi-geostrophic potential vorticity meridional gradient profiles
(qy) for the one month averaged data possess a considerable number of zeros.
These zeros and their associated negative qy regions have a substantial effect on the unstable mode spectrum._ Some modes' growthrates are drastically
reduced when a particular negative qy region is removed. New modes (distinct from those discovered by Eady, Charney, and Green) exist only when
certain negative qy regions are present. Some of the new modes are potentially important for the general circulation of specific regions of the
atmosphere. In particular, longwave modes, in addition to those discovered
by Green, could be important for the stratospheric circulation, and some are
examples of in situ stratospheric baroclinic instability.
5
The unstable mode spectrum is also shown to be sensitive to small
changes in the unperturbed state. It is shown that only those changes
which drastically alter the negative regions and associated zeros of the qy
profile result in a substantial change in the unstable mode spectrum.
TABLE
OF
CONTENTS
ABSTRACT ..................................................................... 4
TABLE
I.
II.
OF
CONTENTS .............................................. 6
INTRODUCTION ................................................. 8
1.
Background and problems to be addressed
2.
Outline of the thesis
THE MODEL ..............
.... ............................... 16
1.
The model eigenvalue equation
16
2.
Boundary conditions
18
3.
4.
Fluxes and energetics
Numerical Accuracy
Figures
20
22
26
III. TWO-LAYER MULTI-LEVEL MODEL RESULTS ...
IV.
8
13
.........
....... 32
1.
Introduction and overview
32
2.
3.
4.
5.
Necessary conditions for baroclinic instablity
Nominal parameter values
Results for special and nominal parameter values
Results for the variation of one parameter with
33
35
36
6.
7.
others fixed at nominal values
Results for constant ^(T planes in parameter space
Calculations for w'=0 at p=0
Figures
40
45
49
55
RESULTS FOR THE HIGHLY STRUCrTURED BASIC STATES............114
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Introduction
Criteria for choosing input data for the model
The data sets
Calculation of q1
Processing the dAta for input to the model
Model results-classification of modes
The Eady modes
The Green modes
The Es and Sm modes
The S modes
The L modes
The necessary conditions for instability and the
existance of unstable modes
Importance of the unstable modes to the general
114
115
116
117
120
122
125
127
130
131
133
circulation of specifi.c regions
139
135
14.
V.
Calculations for w'=O at p=O
Figures
Tables
143
145
194
SUMMARY OF IMPORTANT RESULTS ............................. 217
APPENDIX ........................ .......
ACKNOWLEDGEMENTS ...................
........................ 220
............ ..........
. ....229
REFERENCES .................................................................... 231
INDEX TO CITED WORKS ....................
......................
235
-------
CHAPTER
--
I
INTRODUCTION
1.1
Background and Problems to be Addressed
One of the oldest and most interesting problems of dynamic meteor-
ology is the stability of a horizontally uniform steady flow.
The flow is
said to be baroclinically unstable when there exist perturbations which
grow exponentially with time, and which derive their energy from the available
potential energy of the mean zonal flow. Investigation of the baroclinic instability problem began with the works of Charney (1947) and Eady (1949).
Eady's model assumed a linear velocity profile, no variation of the coriolis parameter (P3= 0),
and rigid lids at the top and bottom. Unstable modes
were found above a certain wavelength.
rigid bottom but an unbounded top.
Charney's model atmosphere had a
A linear velocity profile was assumed
(with the velocity constant above a certain height) and variation of the
coriolis parameter was permitted.
Charney found instability only for waves
shorter than a certain wavelength which depended on the value of P (or on
the value of the wind shear).
Burger (1962) established for a continuous model with constant p 3 0
that all wavelengths have an exponentially unstable mode associated with
them except for certain isolated wavelengths.
Modes on the longwave side
of these isolated wavelengths always grew more slowly than those on the
shortwave side.
Green (1960) in a numerical model had also found that all resolved
wavelengths were unstable (with the exception of isolated wavelengths) when
P
#
0.
Green also extended Eady's and Charney's results to include a semi-
infinite "stratosphere" of increased constant static stability.
however, was kept uniform throughout his atmosphere.
"stratosphere" was to make longer waves more unstable.
The shears
The effect of the
His growth rate
spectrum is similar to that for SR = 1 in Figure 3.4, given below.
The
shorter modes to the right of the cusp point in Figure 3.4 will be referred
to as Charney or Eady modes.
Those on the other, longwave side will be
called Green modes.
Green showed that the fastest growing Eady mode did not have much
vertical structure.
However, Geisler and Garcia's (1977) numerical calcula-
tions found the Green modes to have streamfunction amplitudes in the stratosphere
an order of magnitude larger than those in the troposphere.
Geisler and
Garcia assumed a reference state temperature characteristic of a mid- latitude winter, and both a linear and a mid-latitude winter mean zonal velocity profile.
Both Green and Geisler and Garcia found growth rates of the
Green modes to be much less than those of the Eady modes.
However, there
is no certainty that modes of largest growth rate must dominate.
In fact,
Gall (1976) and Staley and Gall (1977) have shown that the longer unstable
waves can have a longer time to grow than the shorter waves because they
extend higher into the atmosphere to regions where their growth is not halted as quickly by interactions with the mean flow (see also the discussion
in Section IV-13).
The Green modes, therefore, could be of great importance
to the upper atmospheric circulation.
Many of their properties remain un-
investigated; in particular, their kinetic energy release and entropy transport.
Also uninvestigated is the dependence of the vertical structure of these
waves on
6, the shear of the mean zonal flow, and the static stability. The
study of these properties of the Green modes is one major concern in this thesis.
One-of the properties of the baroclinic modes which is essential to
the atmospheric general circulation is their transport of heat.
The trans-
port is generally poleward and upwards, down the horizontal gradient of the
mean temperature (Oort and Rasmusson, 1971), and accounts for a substantial
part of the total heat flux at mid and high latitudes (Oort, 1971; Palmen
and Newton, 1969, pp. 62-63).
The eddy heat fluxes prevent the occurrence
of very large meridional temperature gradients and associated intense westerlies.
The theoretical study of Stone (1972) shows that the baroclinic
eddies are essential to the determination of the atmospheric temperature
structure.
The studies of Phillips (1954) and Kuo (1956) show that the
eddies help drive the mean meridional circulation.
Models of the atmosphere,
consequently, must include the effects of these baroclinic eddies.
The lower stratosphere (from the tropopause to 30 mb) presents a curious exception to the general description of atmospheric heat flux given
above.
Murgatroyd (1969) points out that the latitudinal heating gradient
opposes the latitudinal temperature gradient in winter and summer and this
requires a net countergradient meridional heat flux.
The contribution of
the mean meridional circulation to the horizontal flux has been calculated
from observations of Oort and Rasmusson (1971) at 100 mb.
At this level,
the mean flux tends to be weaker (by a factor of 4 or more) than either the
transient or stationary eddy flux and is generally equatorward.
the observation of Oort and Rasmusson (1971) and Newell et al.
However,
(1974) show
the eddy heat fluxes to be poleward up to 10 mb and countergradient between
the tropopause and 30 mb.
The transient eddy fluxes are about 1/2 to 2/3
as strong as those of the stationary eddies.
Newell (1964a) has described
the lower stratosphere as a refrigerator which maintains the temperature
minimum at the equator through countergradient eddy heat fluxes.
The origin of the observed lower stratosphere net countergradient
heat flux has not been determined.
The lone observations of Oort and
Rasmusson at 100 mb are not strong enough to discount the mean meridional
circulation from playing an important role in determining the net heat flux.
The non-interaction theory begun by Charney and Drazin (1961) and extended by
Holton (1974) , Boyd (1976) and Andrews and McIntyre (1976) tequires that, in the
absence of diabatic effects and dissipation, the heat (and momentum) fluxes of
stationary eddies (even if these eddies have harmonic time variations) are
exactly balanced by transports of an induced mean meridional circulation.
Thus, only transient eddy motion or a circulation in which diabatic or
dissipative effects are important remain as candidates for producing the
net countergradient heat flux in the lower stratosphere.
Diabatic heatinq
rates are small in the lower stratosphere (< .50 K/day; Newell et al.
(1969),
Murgatroyd (1969), Houghton (1978)) but still may be important to the heat
budget (Vincent, 1968).
Viscous dissipation is likely to be small compared
to radiative damping (Holton, 1975).
Although a circulation with diabati-
city and possibly dissipation is a possible mechanism for generating the
net countergradient heat flux, this mechanism is outside the scope of the
current study.
The transient eddy component o
the net countergradient heat flux is
unlikely to come from in situ stratospheric baroclinic instability (Holton
(1975), Simmons (1972)).
Most of the transient eddy activity in the lower
stratosphere, Holton claims, is probably due to the remnants of baroclinically unstable modes which originate in the troposphere.
Green (1960) and
_~
~_ ~I
_ __
Peng (1965) have shown by numerical studies that the heat fluxes of the dominant
Eady mode are very weak in the stratosphere. Green's (1960) fluxes for the dominant Eady mode were down gradient as he did not consider the reversed stratospheric temperature gradient. The strength of the Green mode amplitude in the
stratosphere and the results of Gall (1976) mentioned above, indicate that these
modes may possess strong stratospheric heat fluxes. The possibility that these
fluxes may be significant and countergradient will be investigated, and implications for the parameterization of stratospheric eddy heat fluxes will be
discussed.
Necessary conditions for baroclinic instability were derived by Charney
and Stern (1962) . They found that, in the absence of boundary temperature gradients, the quasi-geostrophic potential vorticity gradient (qy) profile must
possess a zero for baroclinic instability to occur (for a more detailed discussion of this result, see section 111-2).
The calculations of Leovy in Charney
and Stern (1962) show zeros in q above 100 mb for data of 14 October, 1957.
Leovy
and Webster (1976) also suggest that qy may also have zeros on the tropical side
of the stratospheric jet.
Furthermore, Charney and Stern (1962) showed that
such zeros are necessary conditions for an internal instability; they, as well
as Green (1972) , McIntyre (1972) , Simmons (1972, 1974) and Dickinson (1973) have
shown that a zero in qy is a sufficient condition for instability in a number of
special cases.
Therefore, detailed profiles of u and T, closer to those found
in the atmosphere than the simple linear profiles studied by previous authors,
may possess zeros in
which have associated unstable modes.
Detailed profiles
of u and T are also of interest because small variations in their detail have been
shown to alter significantly the growth rate spectrum of the Eady modes
JGall and Blakeslee, 1977; Staley and Gall, 1977).
In this thesis, the,
effect of the structure of the u and T profiles on the baroclinic instability problem will be considered along with the sensitivity of the results
to small variations in these profiles.
1.2
Outline of the thesis
The problems raised in the discussion above will be addressed through a
comprehensive study of the baroclinic instability of both simple and highly
structured one-dimensional basic states. Many properties of the unstable
waves are considered:
instability source, wavelength, growth rate, phase
velocity, steering levels, and the vertical structure of their pressure
amplitude, phase, meridional entropy transport and kinetic energy generation.
The model used is the quasi-geostrophic s-plane model of Green (1960)
written in square-root-pressure coordinates.
This choice of vertical coor-
dinate gives an equal number of levels in the upper layer (stratosphere)
and lower layer (troposphere). The upper layer is thereby given sufficient
resolution without having an unnecessary number of levels in regions of
very little mass.
In Chapter II the mathematics of the model are devel-
oped and the numerical accuracy of the integration scheme is checked.
This study first extends Green's results by a "two-layer multi-level
model" (Chapter III).
ity,
P
Within each layer, the shear, u
,
and static stabil-
1 d In 8
P
dp , are constant but can have independent values.
parameter family results.
In addition to Green's generalized
A 3-
8-parameter,
up (upper layer)
- 8o Poo 2
layer) and static sta, there is a shear ratio SRer
T
f2 u0
u
(lower layer)
r (upper layer)
Green's assumption of the same shear in
(lower layer)
bility ratio a
Y
both layers has been relaxed.
A wide expanse of the three-dimensional
* Klein (1974) addressed the same baroclinic stability problem, but confined
himself to profiles of zonal wind and temperature which are characteristic
of a mid-latitude winter. In addition, he was restricted by his method
of analysis to only the most unstable mode at each wavelength.
parameter space is explored.
The dependence of the Eady and Green mode's
properties on the parameter values is studied in detail.
In particular,
it is shown that the Green modes do have strong countergradient heat fluxes
in the lower stratosphere although nearly all of their kinetic energy is
drawn from the troposphere.
Chapter IV presents a further extension of Green's model to more
realistic u and T profiles for the basic state.
The profiles are obtained
from monthly averaged data at different latitudes and are one-dimensional.
One cannot rigorously justify using monthly averaged data for the unperturbed
state, since the u and T profiles from monthly averaged data do contain the
effects of atmospheric eddies, and the unperturbed flow does not vary on a
much slower time scale than the eddies.
Nevertheless, the use of monthly
averaged data for the unperturbed state can be viewed as a reasonable way
of relaxing the simplified representation of u and T in previous baroclinic
instability studies.
It is a reasonable "next step" which allows one to,
see how the unstable modes are affected by detail in the structure of the
basic state.
The one-dimensionality of the basic state is justified partly by the
work of Moura and Stone (1976).
They found that if the unperturbed state
is locally unstable, it is globally unstable.
Local conditions (e.g. at a
single latitude) are important in determining instability even if meridional
variations are included and if the unstable modes have large meridional
scales.
The one-dimensional basic state is also justified by the secondary
role that the meridional variations are found to have in determining the
depth and location of'regions of negative qy. Calculations in Chapter IV
will show that vertical variations alone represent well the features of the
qy
profile which are important in determining stability properties.
As in Chapter III, the Green and Eady modes are studied in detail,
and the Green mode is again shown to possess strong countergradient heat
fluxes in the lower stratosphere.
In addition to the Green and Eady modes,
other long and short wave modes are shown to exist for the more detailed u
and T profiles.
This reveals an advantage of the eigenvalue approach over
the numerical time integration approach.
The latter gives only the most
unstable eigenvalue at each wavelength; the former gives all the unstable
modes.
Many of the new long and short wave modes are shown to exist only
when certain negative qy regions exist.
The existence of negative qy regions
is shown to be very sensitive to small variations in the basic state.
In-
deed, small variations in the basic state are shown to affect the unstable
mode spectra significantly only when they result in changes in the depth
and location of negative qy regions.
Most of the model calculations were done with the upper boundary condition
9' =
0 at p = 0 (?' is the perturbation stream function).
However,
to allow for a possible source of instability at p = 0, results assuming
W' = 0 at p = O0are presented and explained whenever these differ significantly from those assuming
Y' = 0 at p = 0.
Chapter II discusses the
choice of the upper boundary condition in more detail.
CHAPTER II
THE MODEL
The Model Eigenvalue Equation
2.1
The quasi-geostrophic model on a
purposes of this study.
a-plane is well-suited for the
It is simple enough to allow a detailed parameter
study of large-scale motions and it still provides a good approximation to
the properties of such motion.
The equations for such a model, written in
(p) coordinates, n being an arbitrary function of pressure, are:
7 =
--
(f
+
)
=
Dt
Tr' D
Dt
f
-
w'
o
(1
cr
f
-
(1)
(2)
0
where x, y, t are the eastward, northward, and time coordinates,
4
is the
horizontal streamfunction, f = fo + By is the Coriolis parameter, a =
1 d in 6s
dp
p
is the reference state static stability,
7'
d
i d
y dx
x dy
-
velocity,
dvelocity, i
=
x2
dp ,
+
2
y2
dp is the vertical
dt
D
is the vorticity, and
t
Dt
One next non-dimentionalizes with scales p
t
L
-
uo
, w
PooUo
L
.
Then, following Phillips
S=
poo x,y -
L,
, ~ Lu0
(1954), one defines:
+ qp'
' = Y(T) cos
x
-
y e ik (x-ct)
yy = 0
Stime
variation
is
hen
of theof
meano flw,
The time variation of the mean flow, -- is then of order
i'
.
With the
17
above substitutions, the differential equations become:
(T
T
-
=
(u-c) p2)
(u-c) T
T
- uf
7T
TP
TP2~
=
i
7
fL 2 u
ik
(3)
dTr
TPoo
[ mi-
1
(4)
fL2UO
where all quantities except those in brackets [ ] are non-dimensional and
Y
-
B aTPoo
T -
2
2
(k2+p2)
is the generalized P parameter, and P -
f2Uo
f
TPoo2
2
is
Using Phillips' 1954 grid, one writes the
the non-dimensional wavenumber.
vorticity equation at odd levels and the thermodynamic equation at even
levels.
Combining these finite difference equations as Phillips did results
i
in:
(u -c2)P2 ) _T
T-n
--
0 =
H
n n
(u
n-2
-c)
-
n
-c
(u
n
n+2
T
n
(5)
+
Z
n
T
(u -c)T
+
n-2
n
n
+
n
+ (u -c)
n
n
where n is odd and runs from 1 to N-1 = 97
n+2
(N is defined as the value of n at
the ground level and equals twice the number of levels at which
n
~
is written),
means the value of ( ) at level n,
Sr
+
n
n
(
I
n+l
)2
pn
+
7TI
1i
n
n-l
(A) 2
and
En
T/nl
nl
With appropriate boundary conditions (discussed below), eigenvalues
of this system of equations are found by the OR algorithm.
The algorithm
is one of the most efficient methods known for solving the complete eigenvalue problem for symmetric or non-symmetric matrices (Dahlquistand Bjbrck,
1969).
It is most efficient if the matrix value whose eigenvalues are
desired is in Hessenberg form.
The eigenvalue matrix is balanced, and put
into Hessenberg form and solved for its eigenvalues by EISPACK subroutines
*In the equations and symbols below aT is the nominal tropospheric value
of c given in Section 3.3.
_____
____
.....
provided by the MIT Information Processing Service
(IPS).
_IIn
ImrI
iIl_____
The IPS publica-
tion AP-42 describes these subroutines.
Eigenfunctions are obtained by assuming Real[j3 = Real
2l
the boundary condition is t' = 0 at the top, or by
assuming RealY
Im3
= 1 if the condition is
P'
= 0 at the top.
is then used to integrate downwards.
= 1 if
3
=
The eigenvalue equation
The homogeneity of the eigenvalue
equation justifies the arbitrary
choice forY1 orY3.
2.2
Boundary Conditions
At the bottom
0.
This is justified because in the first approxi-
mation for quasi-geostrophic motions
) 2 -gpw.
In the atmosphere for motions
of synoptic or larger scale, the other terms in the & - w equation are smaller by an order of magnitude or more.
The
&.=
0 lower boundary condition
also filters out horizontally propagating acoustic waves.
dary condition is applied at level N.
- tofN-3
N-1
N-3
This relation between
The lower boun-
It results in an equation relating
N-l1
andN
N-33
the eigenvalue equation (Eq. 5) at level N with
is obtained by writing
TT+N-l =0.
At p = 0, the top of the model atmosphere, both the author and
Holton (1975) have found that only the W = 0 boundary condition has been
used in the literature for quasi-geostrophic models.
theory, this condition can act as a rigid lid.
While being exact in
It can introduce spurious
free oscillations which may be unstable, and cause reflections of vertically
propagating modes
(Lindzen,A1968; Cardelino, 1978).
Geisler and Garcia (1977) assumed 4)'= 0 at a level "(typically 100
km) where solutions of interest have already decayed to a negligible value",
their model is in geometric coordinates.
Indeed, asymptotic solutions of
the quasi-geostrophic equations for U profiles linear with height show that
I
grows or decays exponentially.
'
Assuming
i'
=0 ina geometric coordi-
nate model retains only the realistic decaying solution.
Charney inMorel
et al. (1973) requires that the upward energy flux ps 4'w' is zero at z =
C.
This is satisfied if the decaying solution is selected because lim
ps*$z = lim p s
lim I
j=
Z-)c
lim p
k'w'.
The assumption behind the upper boundary condition
P'w' = 0 is that one's solution should decay away from
Zo
energy sources which are at finite heights.
For the model used here, problems with up constant have uz^u
z ->,
so that the available potential energy goes to zero as z-.o.
--
0 as
For this
case, which has no energy source near the top, Charney and Pedlosky (1963)
have shown that the energy flux of the disturbance will decay at least exponentially with height.
at the top.
The disturbance will then decay to zero amplitude
Thus, T' = 0 is a realistic boundary condition.
Consideration of energy sources above the stratosphere is not within the scope of the current study.
Calculations, therefore, are presented
primarily with Y' = 0 as the upper boundary condition.
However, comparisons
will be made with calculations which have c' = 0 as the upper boundary condition.
Important differences in the calculations resulting from the differ-
ent boundary conditions will be pointed out and their physics will be discussed.
In particular, it will be seen that the long wave Green mode experiences
a behavior analogous to reflection when t' = 0 is used.
Implementation of upper boundary conditions in the reduction of the numberofmodel levels from 49 to 48
sure of model levels).
(Figure 2.1 shows the height and pres-
The condition w0
=
0 is replaced by l' = 0 and
the remaining 48 equations at levels 3 to 97 are solved for their eigenvalues.
2.3
Fluxes and Energetics
To obtain the horizontal entropy fluxes, one first uses the hydroa
ap
static and Poisson relations:
ideal gas law P = pRT
1
fo p
Poo K
6 = (°)
T
(6);
to relate 8 to the streamfunction:
(8);
fo
R
PooK a _
p
P
Ox'
The meridional velocity is simply v' =
(9)
and the vertical velocity is
related to ) through the thermodynamic equation.
sumption that
v and
(7); and the
P
=even =
(m+
+ m- ),
m-even
2 m+l
m-l
0, w and 0, and 0 and 4.
With the additional as-
one multiplies the expressions for
Then averaging zonally and simplifying
results in:
(v'8')
('8')
'
aT
(--)
=
P
-K
2
1-K
P
kmE
1
((L)2
*
Real
(i
-'
+T-1 i
+
c.
I
P
(u
1
-
'
k,
9
-1
+ u1
+l
1
kme
Real
K-l
-1
A
-
*
1
++1
+
+
-1
-
2c.
' £+i
y +
i
*
-
£+
-u
*
Real {(i
1
£+i
*
T t+l
+l
- 2cr ) Real {i
r
U1
- IF
tan
tan
8
N
Lc
4kn
*
i
-i
E-1
]
1
+
£+1
where m = cos
py,
and c. are the real and imaginary parts
eklit , C
=
r
1
Given eigenvalues and eigenfunctions,
of the phase speed and ) is even.
The last cor-
the above non-dimensional correlations are then computed.
relation is often called the "pressure work term" for reasons given below.
The quasi-geostrophic energy equations have been derived by Charney
(in Morel, 1973) for geometric corrdinates.
Their counterparts for pres-
sure coordinates are:
dX
d
u''
u
dy dp
y
dt
Ro
''dy
dE
F
=
(
U v' u dy dp
u
y d
dt
K 1
Poo
'O' (
)
dy dp
PO 0
p
-
I
CK + CE + BE
v
2K
-
y (o)
I
dy dp
P
Jdy
R- f
o fp
Poo
where,
6'
=
L /L
=
CK + CA + BE
v''_
P dy dp -
r
,
(L
r
=
fZ.
y
Poo )
dy dp = -CE +CA
is the radius of deformation)
1
= i-1 ---C
+C
+B
2
*Note:
Once r(p)is known, writing the energy equations in Tr coordinates
requires only a very simple transformation.
"
- --
MwlIftiIIh
"~ -NIIWI
31,E , andA are the Eddy Kinetic Energy, Total Eddy Energy, and Eddy Available Potential Energy, respectively (Lorenz, 1967).
The power integral CK,
which represents transfer of kinetic energy between zonal mean and eddy flow,
is not present for this one-dimensional model since U
= 0.
The power inte-
gral CA, which represents the transfer of available potential energy between
zonal mean and eddy flows is studied by presenting profiles of the horizontal heat flux.
temperatures.
The horizontal heat flux has the more direct effect on
Profiles of the full power integral CE are presented because
CE describes the source regions for Eddy Kinetic Energy.
The last term,
BE, describes the transfer of eddy kinetic energy into or out of a given
pressure layer P1 to P2.
Since p is always -90 ° out of phase with v, the
M't' correlation reveals the relation of poleward to downward velocities
throughout the atmosphere.
It will be used to relate model eddy motions to
observations.
2.4
Numerical Accuracy
The numerical method described above is a second order scheme, and
numerical experiments didshow the error in growth rates varying inversely
with the square of the number of levels when7 =
P
is the coordinate.
For nominal values of the stratosphere/troposphere shear ratio, SR = -1.5,
and static stability ratio a p = 50 (see Chapter III for the calculation 6f these
values), Figure 2.2 illustrates the Eady mode's growth rate accuracy as a function
of YT (YT = 2 is nominal).
Note that many more levels are needed to obtain
accurate growth rates for higher 'T because the most unstable wavelength
becomes shorter, and the associated vertical structure becomes finer.
The model resolution of 48 levels is of good accuracy at least to wavenumber
P = 6 or wavelength = 1.8 x 103 km.
In low latitudes yT can be large and
T
one would need 15 or more levels to
even detect instability if YT = 10.
T
The results in Figure 2.2 are identical for both upper boundary conditions.
The Green mode, however, which has the longer wavelength of 8.3
x 103 km (P = 1.3)' for nominal parameter values, is resolved much better
when Y'P
= 0 is the upper boundary condition.
In this case, the
doubling time is determined to within 2% of its 48 level value (6.1 days)
when only 13 levels are used.
However, when w'
is the upper boun-
dary condition, 37 levels are required for the doubling time to be within
2% of its 49 level value (14.7 days) (the marked differences between these
estimates of the Green mode doubling time is discussed with the Chapter III
&)I
0 calculations).
=
The above and additional calculations showed that
when doubling times of long-wave modes were calculated, calculations using
(P'.O
= 0 converged more rapidly than those using the more common Oj('p=
= O upper boundary condition.
The ability of the model to converge over the entire range of the
vertical coordinate is illustrated in Figure 2.3, which shows results valid
for both upper boundary conditions.
For the meridional entropy flux of the
Eady mode, good convergence is obtained for the model resolution of 48
levels
(49 levels, for the w'
(p=0 )
= 0 boundary condition).
For entropy fluxes of the Eady mode at higher YT values and shorter
wave lengths, Figures 2.4 and 2.5 show model convergence (their results are
identical for both upper boundary conditions).
v'e'
s
The quantities v'8'
T
and
are the mass weighted zonally averaged meridional entropy fluxes for
the troposphere and stratosphere, respectively.
Each profile of v'9' has
been normalized, i.e. the maximum value for all levels in the profiles has
been set equal to 1.
Good convergence on values of v'e' T and v'e's exists
for modes with zonal wavelengths as small as 1.8 x 103 km.
Three additional checks were made on model accuracy.
The first is
a check for satisfaction (numerically) of the lower boundary condition.
Both upper and lower boundary conditions were, of course, used to determine
the eigenvalues.
Determination of the eigenfunctions, however, started
with the upper boundary equationY
1
= 0 and assumed an amplitude and phase
forY3 (see the earlier discussion in this chapter for a justification of
this assumption).
Then the model eigenvalue equation (Eq. 5) is used to
determine values of Y at lower levels.
The value of Yat the lowest level,
YN-1' is obtained from the eigenvalue equation written at level N-3, which
to
relatesL
N-l
.
-3 and y
N-5
N-3
The remaining Yequation, written at level
N-1 contains the lower boundary condition.
NN-l and
It must be satisfied by the
determined by the eigenvalue equation
N-3
N-3 values, which were
written at higher levels.
Satisfaction of this lower boundary equation was
checked for all model results.
Asecond check is satisfaction of Howard's Semicircle Theorem Test
(see Pedlosky (1964) for application of this theorem to quasi-geostrophic
models).
All values of the eigenvalues were found to be within the theo-
retical limits of the test.
A final check on the model's validity is its ability to reproduce
previous results.
2
A plot of growth rate contours vs
-1
rT-1
and wavelength
7r
P 2for Green's linear u profile reproduces his Figure 3 exactly (see Figure
2.6 and Green (1960),
(w'
or w' = 0)
Figure 3).
at the top.
Both calculations used a rigid lid
I1
--------- -
-
-
-
----
--
--
111 111
100
1
IIIYYIYIYIIIIYIYIYII
1Yi
1i11hi I IIAtIIId
U iIl
hIi
26
0
0--
80-
-200
60
- 400
E
E
mb
w
cE
w
w
0.
40-
-600
20-
-800
km
I
O0
0
Figure 2.1.
I
20
I
I
I
I
I
40
60
LEVEL NUMBER
I
80
Height and pressure of model levels.
I
1000
98
*I
27
04
48
o
442
030
36
24
C:
.0
In
oC
02
4
610
8
Figure 2.2. Growthrate vs.
T of the fastest growing Eady mode for
models with indicated numbers of levels for which 4 is written; SR = -1.5,
op= 50.
E
484
2
6
4
CL
o
600-
800
1000I
Figure 2.3.
I
v
1
.01
I
1.0
Vertical structure of the meridional heat flux of the fastest
growing Eady mode for YT = 2, SR = -1.5,
=
50.
29
i
1.0
I
I
.8
6
.6
8
.10
14
0
.2-
48
0
2
6
4
8
10
6T
Figure 2.4. As in Figure 2.2, except for the tropospherically integrated
meridional heat flux.
~-~I------
111 -.lliil li lllliiiii
30
4
IIII
6
4
4
10
0
2
4
6
8
10
tT
Figure 2.5. As in Figure 2.2, except for the stratospherically integrated
meridional heat flux.
2.0
1.5
1.0
0
2
4
WAVELENGTH (nondimensional)
6
PcI
Figure 2.6. Non-dimensional growthrate 1
divided by Y
as a function
of Y
and wavelength
p = 1, SR = 1, u(p=0) = 1, and W' = 0 at p = 0.
Neutral modes are shown by the heavy zero curve.
Tha
o
Neutral
-*
-------
-M-
u~~l
11I~~.
Y,
32
CHAPTER III
TWO-LAYER MULTI-LEVEL MODEL RESULTS
3.1
Introduction and Overview
This chapter treats the atmosphere as a 2-layer system having a
troposphere and stratosphere.
static stability,
within each layer.
and these
Each layer has an independent shear and
unperturbed flow parameters are constant
This approach simplifies the parametric study of how
the unperturbed state affects the perturbations.
Within each layer are
24 levels so that the vertical structure of the perturbations can be studied
in fine detail.
Three parameters completely determine the unperturbed state: Y',
the
generalized n-parameter, defined in chapter II; SR, the shear ratio
du)
du stratosphere / du
(dptropospher); and
dp stratosphere
stratosphere
/-
dp troposphere
troposphere
.
, the static stability ratio
The velocity field, u
n
=
n
ranges from 0 to
uo
(
1 between ground and tropopause
p Poo
1 ) and from 1 to 1 + 1
SR between
3
4
tropopause and p = 0:
1 U
4
The static stability parameter,
m
$
stratosphere
=
4
4 (l-)
(note:
1
SR p + 3 SR
49,as m is even).
m
troposphere
= 1 for m > 49 and
=
mo
1-
,
for m <49
'o
When values for YT, SR, and 69 are chosen,
the model eigenvalue equation is ready to be solved to any specified nondimensional wavenumber P.
Solutions will first be obtained for combinations of special and
nominal values of the parameters.
"Nominal" values are defined as those
appropriate for mid-latitude winter conditions:
0-= 50 (these values are calculated below).
YT=0 (no
YT = 2, SR = -1.5, and
Special values used are
-effect),SR = 1 (uniform shear throughout the atmosphere), SR = 0
= 1/4), and
(no shear above P/
stability throughout the atmosphere).
values will be studied.
= 1 (no stratosphere, uniform static
Every combination of these parameter
This will enable one to see the effects of: uni-
form shear, zero shear above p/P o o = 1/4, and nominal shear above P/Poo =
1/4, both with and without a stratosphere, and with and without the ,-effect.
The second part of this parameter study will use the nominal parameter values to define an origin in parameter space.
The study will include
parameter values on axes through the origin along each of which only one
parameter varies.
Both the first and second parts of the parameter study
will treat in detail mode growth rates and phase speeds, and the vertical
structures of streamfunction amplitude and phase, of meridional heat transport and of eddy kinetic energy generation.
This will be done for both the
Eady and Green modes.
The third part of the parameter study will discuss growth rates of
the Eady and Green modes on YT = constant planes in parameter space.
Inter-
esting vertical structure properties will be noted.
Finally, calculations will be presented to point out how changing
the upper boundary condition from )' = 0 to o'
3.2
= 0 affects the perturbations.
Necessary Conditions for Baroclinic Instability
An important part of this study will be the relation of unstable
modes to the necessary conditions for baroclinic instability, given by the
Charney-Stern theorem.
Charney (in Morel, 1973) derives the theorem for
geometric coordinates with a zero energy flux condition at the upper boun-
9'w' =
dary (lim
z-3c
lim
z.- oo
0), z =oo.
'I'=
7
The derivation in pressure coor-
dinates for W' = 0 at p = 0 is similar and results in the equation (analogous to Charney's equation 9.18):
2
Sf
f 2
( d-U)
,,2
C
i
p)
(1)
d
0 dy -
dy + JYdy
-c
1c
SE
-u
f
yy
2 1o dpp
u ) - f
25
o
(1
)
p-o
P
= 0.
This (for any U andX) results in
i.e. no energy flux at p = 0.
( 2)
p
For the profiles of this section (linear u, constant
forces lim
0
-c I
pP=0
p1
Y'
= 0
0),
at p = 0
lim
pe- o
P'cO'
= 0,
Also, ~y'=0atp=0results (for any u and C)
in the disappearance of the first integral in the above equation.
The
upper boundary condition >' = 0 would allow this integral to remain.
the upper boundary condition
Thus
V' = 0 has as necessary conditions for baro-
clinic instability:
= 0 in the fluid in the absence of a temperature gradient 6
Sy
at the lower boundary, or
(1)
(2)
q
au
p
a balance between the lower boundary integral and the qY integral.
The w' = 0 upper boundary condition would have as necessary conditions for
baroclinic instability:
(1)
q
= 0 in the fluid in the absence of temperature gradients at both
boundaries, or
(2)
a balance between upper boundary, lower boundary, and qy integrals.
* It is not consistent to use the a term in quasi-geostrophic models.
Calculations show that this term is y small. It is used only in the
calculations of Tables 4.1 to 4.4.
o' = 0 case has the distinction that a temperature gradient at the upper
The
boundary as well as a temperature gradient at the lower boundary and an internal zero of q
is a potential source for baroclinic instability.
Indeed,
later calculations show that there are modes which depend on a non-zero value
of
&
at p = 0 for their existence, that there are other modes which depend
on specific zeros of
Ei,
modes which exist when
for their existence, and that there are still other
/ 0 at the ground.
y
The modes which exist only
0 at p = 0 may not be realistic for the atmosphere; however, they
fore
5
y
may play
a role in laboratory experiments or oceanic flows.
will be discussed with the W'p
3.3
They
= 0 calculations.
Nominal Parameter Values
The origin or "nominal" point is chosen to approximate a winter
stratosphere and troposphere at 45 0 N.
Using U.S. Standard Atmospheric
values for January 45°N at 1000, 250, and 28 mb one has:
AZ
T
2
AZn
gAp
S
9
m
AZ
m
sec
14000 m
23200 nt/m
2
10200 m
secc
75000 nt/mn2
-5
n 327-In 273
4
10200 m
2 In 599-in 327
14000 mm
=
sec 2
.169 x 10 - 3
2
4
-2
m sec 2 kg - 2
thus:
a
= 52.5;
P
give
u 28
13 m/sec,
281000
SR = -1.55
for the shear ratio:
S= 1.64 x 10
-11
from above give:
m
y
-1
sec
u2 50 = 24 m/sec,
-1
,
= 2.19.
P
and U000 = 0
and 24 m/sec for U .
5
m
=10 nt/m,
f =10
-4
sec
-1
-2
and u 0 and CT
-
--
-- -
111111~~
III, ,
I'
llk,
IY IIY
~
I
II
I
IIIIIIIIYIY
36
The winter values of yT = 2.19, SR = -1.55, 6=
nominal values Y
= 2, SR = -1.5,
= 50.
52.5 are rounded to the
It should be noted that YT and
P are analogous to the Y and P parameters of Green (1960), and, as in Green,
0 for all calculations.
Since
2
2
pappears only in the combination k +P,
the results for any value of . can be recovered by a reinterpretation of
the value of k.
In this chapter, k represents the number of waves in a
latitude circle at 450
It-is also interesting to note that the value of
'T
corresponding to
the critical shear of the Phillips (1954) 2-level model is:
ST=
T
Stone
2F'
2.06 where u
Uc
c
(1978).
critical value.
= A=
fl
(R is the gas constant, see
The winter value of YT = 2.19 is very close to Phillips'
This shows again that the mid-latitude troposphere is
remarkably close to
two-level neutral stability for time averages of over
a few (3) months (see also Moura and Stone (1974) and Stone (1978)).
3.4
Results for Special and Nominal Parameter Values
Figures 3.1 to 3.4 present the doubling times and phase velocities
of the unstable modes for special and nominal parameter values.
It is
immediately apparent that the presence or absence of a stratosphere or of
the #-effect affects the results much more than variation of the upper
layer's shear.
The presence of a greater static stability in the upper
layer, not surnrisingly, lessens further the influence of upper layer shear
Eady modes grow faster for a reversed upper layer shear if no
variation.
stratosphere is present and very slightly more slowly if there is a stratosphere.
The overall shape of the doubling time curves is little affected
~ ~
'
by variation of SR.
This is not the case for YT.
(1960), zero A-effect stabilizes short waves.
As pointed out by Green
Also, when SR = 1 and there
is no stratosphere in addition to no A-effect, all modes are stable; this
differs from Green's results for a rigid lid
(see discussion of W' = 0 atp=0
'T
increases, short waves become more unstable (see Fig-
ures 3.17 and 3.18).
The presence of a stratosphere lessens doubling times
calculations).
As
of the Eady mode for all values of SR and
days.
"T from about 2 1/2 to 1 1/2
When both the stratosphere and a-effect are present, the long wave
"Green Mode appears and has its fastest doubling time (6 days) for the nominal,
reversed stratospheric shear.
The doubling time and wave number of the most
unstable Green and Eady modes for the parameter values
CT', SR, Op
= 2, 1,
50 compare well with those calculated by Geisler and Garcia (1977) for a
constant shear and more realistic temperature profile.
Differences are
less than 10%.
Phase velocities calculated by Geisler and Garcia are almost twice
as large as this model's values for
T, SR, f
= 2, i, 50.
is due to the much larger u values in the former model.
The difference
A fairer compari-
son between this model and that of Geisler and Garcia will be made in the
next chapter.
wavelengths
For zero
6-effect, phase velocities are very large at long
(note the change of scale in Figures 3.1 and 3.2).
When there
is both a 4-effect and a stratosphere present, the phase velocity goes
to zero with the growth rate at a "critical" wavelength which defines the
separation between the Eady and Green modes. The existence of neutral waves
stationary relative to the surface wind at this critical wavelength was discovered by Green (1960).
Steering levels, as in Green's study are below
the mid level (p/poo = 1/2) for all wavelengths.
-Iu,
Streamfunction amplitudes and phases are presented for the most unstable Eady and Green Modes in Figures 3.5 to 3.8.
presence of a stratosphere causes a sharp peak in
independent of
8-effect and shear variation.
equal to the peak at the ground.
also was observed by Green (1960).
when
YI
at the tropopause
This peak is approximately
The latter is narrower for YT
Variation in SR has negligible effect
pause as SR goes to negative values (Figures 3.6 and 3.7).
existing only for
T
0,
0; this
larger values of MIP occur around the tropo-
=
hen
50, but when
For the Eady mode, the
o
The Green mode,
1 has a strong peak in the stratosphere
which moves up from 180 mb to 120 mb as the stratospheric shear reverses.
Streamfunction phases are constant throughout the upper layer when
YT = 0 or o
= i1. When there is both a p-effect and a stratosphere, the
pressure wave tilts westward with height in the stratosphere for both Eady
and Green modes, the Green mode having a much greater tilt.
In the tropo-
sphere the Green mode also has a greater westward tilt.
For quasi-geostrophic disturbances, a westward tilt of the wave with
height means entropy is transported poleward (Lorenz, 1951).
The model
calculations of v'&' , Figures 3.9 to 3.12, illustrate this property for
the fastest growing Eady and Green modes:
v' ' values are positive.
of 0
all tilts are westward and all
When there is no
d-effect v'9' is independent
and SR values and is zero in the upper layer, reflecting the con-
stancy of phase there.
In the presence of both a stratosphere and a B-
effect, entropy transports are significant in the stratosphere, for both
modes.
The Green mode has a much stronger stratospheric transport.
peaks at the tropopause and remains large in the lower stratosphere.
It
The
Thus, nominal conditions
flux is strongest for the case of reversed shear.
produce modes which have larger stratospheric heat transports than all the
other cases considered.
A local maximum in the entropy flux in the lower stratosphere is observed
(Oort and Rasmusson, 1971; Newell et al., 1974).
Thus, the Green mode could
be very important.for lower stratospheric dynamics.
This problem will be
addressed in more detail in the next chapter.
Except for the Green mode with uniform shear (SR = 1),
no modes have
kinetic energy generated in the upper layer (see Figures 3.13 to 3.16).
Only this Green mode and an Eady mode with no stratosphere and SR = 1
have their kinetic energy destroyed in any region of the lower layer.
In
all cases there is, of course, a net generation of mode kinetic energy
And in all cases, there
when the entire model atmosphere is considered.
is a net mode
kinetic energy generation in the troposphere and net loss in
the stratosphere.
The maximum mode kinetic energy generation is in the mid
to low troposphere for all cases except the Green mode with reversed stratospheric shear.
This latter mode has its strongest kinetic energy genera-
tion in the upper troposphere.
Earlier discussion pointed out that for no stratosphere and uniform
shear there was apparently no Green mode for YT = 0 or 2, and there were no
unstable modes for SR = 1, (7 = 1 and Y T = 0.
stronger
Is there a Green mode for a
effect? What happens to it for small #?
These questions are
answered by a graph (Figure 3.17) similar to Green's (1960) figure 3 with
the difference that Green's rigid lid is replaced byf' = 0 at p = 0*
* Differences between this graph and Green's Figure 3 are discussed with
the
= 0
0'at p = 0 calculation later in this chapter.
40
The Green mode is seen to exist for large 8-effect (or small tropospheric
shear, u ) but is very weak, and its wavelength becomes infinite as yT or
q
= 8 goes to zero.
Also, as yT or q
goes to zero, growth rate per unit
YT remains constant so that the growth rate approaches zero with yT or q
.
When SR and
have their nominal values, and
ation is much different
(see Figure 3.18).
growth rate changes little as
rT
varies, the situ-
The fastest growing Eady mode
ycE~r~
to zero, and the Green mode is much
stronger and is at a shorter wavelength for IT < 10.
3.5 Results for variation of one parameter with others fixed at nominal
values
In this section two parameters are held at their nominal values and
the third varies over a wide range of parameter space.
Doubling times and
phase velocities are obtained for zonal wave numbers 0 to 15, and vertical
structures of wave properties are presented for the fastest growing Green
and Eady modes.
Shorter modes are trapped near the ground and are outside
the stratospheric focus of this thesis (Chapter IV illustrates properties
of these waves for realistic T and U profiles).
Figures 3.19 to 3.21 show that the doubling time of the fastest
growing Green mode is much more sensitive to parameter variation than is
the doubling time of the fastest growing Eady mode.
The wave numbers of
the most unstable Green and Eady modes are not very sensitive to the variation of SR and
0
but increase markedly as
T increases.
This increase of
most unstable mode wavenumber with YT was pointed out earlier (see Figures
3.17 and 3.18).
As SR takes on values from -12 to 4, the Green mode re-
mains constant in strength for reversed shear in the stratosphere but be-
The Eady mode, however, becomes
comes much weaker for large positive SR.
slightly stronger and the wavelength for which it is most unstable slightly
increases with SR throughout its range.
For values of
0- between 1 and
1000, the Green mode has its fastest doubling time, 6 days, near the nominal value,
value of a
= 50 (intermediate calculations, not shown, refines this
to be 60).
As yT varies from 0 to 10 there is a dramatic de-
crease in the Green mode's fastest doubling time - from 12 days (IT = 1)
to 3 days (YT = 10).
The wavenumber of the fastest growing Green mode
increases from 2.0 to 7.5.
long-wave.
Thus, at high
YT
the Green mode is no longer
Green (1960) has shown that longer waves are expected to have
longer doubling times because they only weakly satisfy the necessary condition for instability, viz., that particle paths have an average slope
between that of the isentropes and the horizontal.
ment was for rigid lids at top and bottom.
-Green's heuristic argu-
To the extent that a 50-fold
increase in the static stability approximates a rigid lid, one can say
that the Green mode grows slower at lower
(T when ~'= 0 in the upper boun-
dary conditions because it is at a longer wavelength.
The phase velocities, shown in Figures 3.22 to 3.24, all haveasharp
minimum at .the wavenumber which, by definition, separates the Green and
Eady modes.
The Green mode is defined to include all wavenumbers less than
that of the sharp minimum in the phase velocity.
minima in some of the phase velocity curves.
There are other, weaker
These are found to be asso-
ciated with weak minima (which may not be resolved) in the doubling time
curves.
Phase velocities are most strongly influenced by changes in 'fT
For all wavelengths, phase velocities decrease with increasing
T.
And,
for all wavenumbers and parameter combinations of this section (except SR
-- --- --' --
< -2),
sphere.
--
--
the unstable modes have a single steering level in the lower tropoFor a strongly reversed stratospheric shear (SR < -2) some modes
have an additional steering level in the stratosphere.
Streamfunction amplitudes and phases for the fastest growing Eady
and Green modes are presented in Figures 3.25 to 3.36.
The structure of
the Eady mode amplitude in the troposphere is little affected by varying
SR and a- (except when
.' = 1).
the ground and tropopause.
Peaks of the pressure amplitude occur at
The peaks are of approximately equal value and
the one at the tropopause decays less rapidly into the stratosphere for
small
- , when y
= 10.
When )T begins to increase beyond 3, the tropo-
pause peak begins to weaken substantially.
Stratospheric Eady mode pres-
sure amplitudes are largest when yT is near its nominal value.
The Green mode has larger streamfunction amplitudes in the stratosphere than the Eady mode for all parameter combinations of this section.
Figures 3.8 and 3.28 through 3.30 show that the Green mode's amplitude peaks
higher in the stratosphere, and has larger values there (relative to those
of other levels) when the parameters are near their nominal values.
The
peak, for near nominal parameter values, occurs at 120 mb, and the amplitude remains greater than half the peak value from below the troponause to
30 mb.
For YT
A
7, there are pressure amplitude peaks at the tropopause
and 75 mb which are separated by a sharp minimum at 140 mb (see Figure 3.30).
The upper peak is weaker and has a value of .7 times the tropopause peak
value.
The Green mode also has a peak at the ground which is as strong as
the peak in the stratosphere when
4 10 or SR
-
-4.
In comparison with
that of the Eady mode, the vertical structure of the Green mode's pressure
.
amplitude is much more sensitive to variations of SR orO
The same was
seen to be true regarding the doubling times of the Eady and Green modes,
and, as subsequent figures illustrate, the same is also true for the vertical structure of mode phase, meridional entropy transport and kinetic
energy generation.
The larger amplitude of the Green mode at stratospheric
levels renders it more sensitive to variations of SR and
0
, which are
really variations in the stratospheric shear and static stability because
YT is held constant.
The phase of the Eady mode streamfunction varies much less with
height than does the phase of the Green mode
(Figures 3.31 to 3.36).
cept for the case of a very stable stratosphere,
Ex-
> 1000, the Eady mode
pressure wave has only a slight westward tilt with height.
The tilt of the
Green mode is very strongly westward with height and is most strongly westward when YT or C
are large or when SR = -4.
The Green mode has its most
rapid vertical phase variation concentrated in a region just above the
tropopause (100-250 mb) and in the lower troposphere (550-750 mb).
phase can change 1800 in these regions.
The
The Green mode and, at a slower
rate, the Eady mode vertical wavelength approaches zero in the stratosphere
as
---30 0.
sphere as
Amplitudes of both modes also approach zero in the strato-
--
0>cand are negligible for 0
> 1000.
The meridional entropy transports, v'O', of both modes (Figures 3.37
to 3.42) are all poleward, as expected for quasi-geostrophic waves which
tilt westward with height.
The stratospheric transports are generally
very weak for the Eady mode (in comparison to the tropospheric transports)
except when the stratospheric shear is easterly (SR < 0),
but even these
-
-
____ __IIIIIu
_____ I
l III WIIIII1W
161
transports are far shallower than those of the Green mode in the lower
stratosphere.
Tropospheric Eady mode v'&' values are little affected by
but as YT increases, they decrease rapidly at all
changes in SR and 6,
levels except near the ground.
The entropy transports of the Green mode
in the lower stratosphere are generally stronger than those in the troposphere except for transports near the ground when SR is positive,@, < 10,
or y
7.
The stratospheric Green mode fluxes are countergradient and are
T
strongest when the parameters are near their normal values.
They have
their strongest peak at the tropopause and remain at greater than half the
peak value up to 130 mb.
for SR positive or
0
Stratospheric Green mode v'O' values are weakest
5< 10.
The generation of mode kinetic energy for both Eady and Green modes
occurs only in the troposphere except for the Green mode when SR > 0 (see
Figures 3.43 to 3.48).
In this unique case there is a weak, shallow region
above the tropopause where Green mode kinetic energy is generated.
The
generation of Eady mode kinetic energy is stronger in the lower and midtroposphere (except for YT = 10; in this case alone there is a weak destruction of mode kinetic energy in a region of the troposphere).
The generation
of Eady mode kinetic energy is affected only slightly by variations of SR
or 9,; however, the generation decreases rapidly as (T increases.
generation of Green mode kinetic energy when SR4 0,
5
The
Z 10, or 'T < 7
is nearly constant throughout the troposphere except near the ground where
it goes to zero with o; when SR> 0, p
= 10, or (T
>- 7, there is a mid-
tropospheric destruction of Green mode kinetic energy.
The strongest Green
mode kinetic energy generation is always in the lower troposphere.
In
summary, the strongest generation of both Eady and Green mode kinetic energy
is in the troposphere.
Therefore, both modes are tropospheric in origin.
They drive the stratosphere, i.e. they lose kinetic energy in that region
and build up zonal available potential energy there with counter-gradient
entropy fluxes when SR< 0.
The Green mode has much stronger and deeper
stratospheric entropy transports and kinetic energy losses in comparison
to its tropospheric values of these quantities than does the Eady mode.
It is also a (zonally) longer wave mode and can, according to Gall (1976)
more easily penetrate into the stratosphere.
Therefore, the Green mode
could be of much greater importance to lower stratospheric dynamics than
the Eady mode.
3.6
Results for constant YT planes in parameter snace
This section explores a much wider expanse of parameter space.
It
focuses on the growth rate of the most unstable Eadv and Green modes and
discusses properties of their vertical structure which may be of importance
to the general circulation at specific levels of the atmosphere.
It also
ascertains whether any new modes appear in the range of parameter space
Parameter space of this model has 4 dimensions represen-
which it covers.
ted by
(T, SR,
Gp,
and P.
The value of the wavenumber P is always chosen
to be that of a local maximum in the curve of growth rate vs P.
The re-
maining 3 dimensions are sampled by examining planes of constant
YT at YT
= .5, 2, and 6.
On these planes,
' varies from 1 to 1000 and SR varies
from 14 or 0 to -14, depending on the value of YT and the mode presented.
The doubling time of the fastest growing Eady mode is contoured on
the constant YT planes in Figures 3.49 to 3.51.
On all planes there is
only a slight variation of doubling time with SR-and CT when r
jP
f
> 20; and
-~~
..-
AUNM
W
00111WI
111011111MIN
1
,1
1,
II
I
IM
M
M
I
I
IIll
iII
46
the variation is even slighter for larger Y
*
tween 1.5 and 2.0 days.
days for small
.
Most doubling times are be-
There is a lengthening of doubling time to 3 or 4
a and ISRI > 10.
As seen in the previous section, wavenum-
bers of the most unstable Eady mode increase with Y .
*
ap approaches 1 and SR becomes large and negative.
they also increase as
When 0
> 20 they vary little with SR and
double from SR=-2 to SR=-10 (p
a
a
P
=50to
r =2 (SR=-14,
p
smallest values of
a
When YT is not large,
.
= 2, y
a
.
The wavenumber will typically
= .5or 2) ; it will also double from
y = .5 or 2) with the most rapid increase at the
T
On the YT= 2 plane, the most unstable wavenumber
is between 6 and 7 for a
> 20 and is near 15 for SR < -8 and
a < 2.
The fastest-growing Green mode undergoes a much less smooth variation
of doubling time with SR and
a (Figures 3.52 to 3.54).
for higher YT , independent of the values of SR and Cp.
the fastest doubling times are when
a
On all the
The longest doubling times
For YT = .5 or 2 there is a region of
and SR going positive.
shorter doubling times for nominal and larger values of a
The nominal point (SR= -1.5,
9p = 50,
YT planes,
ap is small and SR is large and negative;
this is the same region where the Eady mode is weakest.
occur for small
It is much shorter
and SR near -2.
YT = 2) occurs in a local region of shorter
doubling times whose values are about 6 days.
The YT = 2 plane shows the most
complicated variation of Green mode doubling time with SR and
a .
This, in
part, could be due to the difficulty in defining a fastest growing Green
mode when there is no minimum of growth rate between the Green and Eady modes
(see the SR = -4 curve in Figure 3.19).
In this case, a point .6 zonal wave-
numbers to the left of the point of greatest curvature was chosen to be the
wavenumber of the fastest growing Green mode (e.g. 3.6 was chosen as the
wavenumber of the fastest growing Green mode on the SR= -4 curveof Figure
eNo
lil llil
IIll
i
i
l
m
3.19).
In all cases there was a minimum in the phase velocity curve which
separated the Green and Eady modes, and the point of greatest curvature of the
doubling time curves was always on the long-wave (Green mode) side of this minimum. The wavenumber of the most unstable Green mode varies with the parameters
in a manner similar to that of the Eady mode.
On the
T = 2 plane the wave-
number is between 3 and 4 when a > 20 and is near 8 when SR <-10 and a < 5.
p
p
Much of what was discussed in the previous section about the Eady and Green
mode vertical structure also holds true for the much larger expanse of parameter
space considered in this section.
Stratospheric Green mode l$i
mode peak values of
[I
Exceptions are presented below.
<
values are weaker when yT
2. The Eady
at the ground and tropopause are equal except when
P < 10 or YT > 3; then the tropopause peak becomes weaker.
The peaks of
the Green mode streamfunction amplitude are at the qround and in the lower
stratosphere with the latter being much stronger except
and SR < 0, and
(2) when Y
(1) when a
= .5 and SR is large and negative.
< 15
The strato-
spheric peak of the Green mode amplitude is strongest when parameters are
all near their nominal values, and when y
T
= 6 with SR and a
p
near their
nominal values.
The streamfunction phases of the Eady mode and Green mode behave as
described in the previous section, except when y
= 6 and -8 < SR < -2.
In this case the phase varies about twice as rapidly for both modes.
The stratospheric v'8' peak of the Green modes is weak when a
< 20.
It is strongest when parameters are near their nominal values and also when
Y = .5 or 6; SR is near its nominal value, and a > 30.
PT
-
-
--
~~"~~UNIYINIYNIN
The previous discussion of this section has centered on the Eady and
Green modes.
For the parameter value combinations considered in earlier
sections of this chapter, there was only one conjugate unstable eigenvalue
for each wavenumber; this contained the Eady and Green modes.
The mean
Therefore, the un-
flow in all previous sections possessed no zeros of qc.
stable eigenvalue containing the Eady and Green modes owed its existence
to a temperature gradient at the ground.
In this section one has the possibility of an internal zero of qy
helping to satisfy the necessary condition for instability.
occur only at the boundary between the two layers (P = -).
This zero may
The value of
qy for the simple profiles of this chapter is easily obtained, as upp,
T
p
and
-
q
y
are all zero except at the internal boundary where there is a jump in u
p
0.
at
whereL
Thus, everywhere except at p=,
=
1
-
, so qy
31/4
1
q
=
There is a S-function in
.
goes through a zero at P =~
1
1denotes the jump in Fupacross
if
=
1
In terms of the para-
meters describing the profiles, this means:
SR <
Jp
)>
-1
u
0 for p )
p
p-p4
SR
if
Sp
4
for p) 4
1
if u < 0 for p
>
4
The case with u = 0 in the lower layer was applied to the underside of the
P
polar night jet by Murray (1960) and Simmons (1972).
is applicable to the profiles of the present model.
The condition SR
> C
Since the upper layer
of the present model is assumed to be more stable than the lower layer,
(T
> 1), for there to be an internal zero of q
, the shear in the upper
layer must therefore be more westerly than that of the lower layer (SR > 1).
This does not represent the situation at the earth's tropopause but does
approximate conditions just beneath the polar night jet.
Both Simmons and the author find that an unstable conjugate eigenvalue is associated with the zero of a
and O.
It occurs only when SR
>0
caused by discontinuities in u
for all values of yT.
For the present
model the eigenvalue is not well determined, even for 48 levels.
The dif-
ficulty is in resolving a mode whose strongest amolitude and v'O' values are
concentrated in a very shallow region around the internal boundary.
for 48 levels when
T
=
2, SR = 10, bp
Results
= 5 give doubling times of about
one week for wavenumber 8, and a vertical decay scale (from the internal
boundary) of about 40 mb.
The phase speed is westerly at 30 m/sec which
places the steering level at 225 mb.
Simmons' doubling times are slightly
longer than a week and his vertical decay scale is only a few kilometers.
The pressure wave of the mode tilts very slightly westward with height;
the total phase change from p = 0 to the ground is .1 radians.
Essentially
all of the mode's kinetic energy is generated at the internal boundary.
This mode will be referred to as the "internal boundary mode".
3.7
Calculations for a' = 0 at P = 0
In the discussion of numerical accuracy (Chapter II),
it was pointed
out that the fastest growing Eady mode growth rate and vertical structure
was not affected by changing the upper boundary condition from
,=
0 to
' = 0 if a nominal stratosphere was present.
A comparison of Figures 2.6
and 3.17 shows that this is far from the case if there is no stratosphere
(SR = 1, O, = 1).
These two figures' results are calculated from identical
assumptions, except for the upper boundary condition.
stead of changing little with yT (for o' = 0 at p = 0),
YT --- 0 when
Y'=
0 at p = 0.
Also, when
The Eady mode, inbecomes stable as
i' = 0 at p = 0, the Green mode's
wavenumber approaches zero as YT decreases below 3, and the mode becomes
very weak (the doubling time is longer than 60 days).
This compares with
a zonal wavenumber of 2.5 for small YT and a doubling time of 20 days for
the
W' = 0 at p = 0 calculations.
When nominal values of the parameters
are assumed, the fastest growing Green mode's doubling time is twice as
long when W' = 0 at the top.
This contrasts with the small effect on the
Eady mode when the upper boundary condition is changed and a nominal stratosphere is present.
The streamfunction amplitude of the Eady and Green mode for the case
of no stratosphere, SR = 1, and w'Ip 0 = 0, both had peak values at the
bottom and top of the model atmosphere; the peaks were of equal value (see
Green, 1960, Figures 5 and 6).
Forcing
9' = 0 at the top placed a severe
constraint on these modes, causing their growth rates to be much smaller.
On the other hand, for nominal values of 0- and
YT, and SR = -1.5,
0, or 1, the fastest growing Eady mode streamfunction amplitude went to zero
at the top for both upper boundary conditions, and the mode's growth rate
was unchanged (to within 2%).
For the same parameter values, however, the
fastest growing Green mode amplitude peaks, which occur in the lower stratosphere
(Figure 3.8),
are replaced by a node in that region and 2 peaks near
the top and at the tropopause when the upper boundary condition is changed
to
o' = 0 (see Figure 3.55).
The effect is analogous to what happens when
one gets spurious reflections; the effect results in the doubling of the
For the case using o' = 0 at P = 0, the
Green modes' doubling time (sic.).
fastest growing Green mode has a weak region of kinetic energy generation
within 15 mb of the top.
When c' = 0 at p = 0 and one parameter value is varied and the others
are held at their nominal value, one finds that for XT = 3, 4p >
40, or
-3< SR < -1 the same behavior analogous to reflection and destructive
interference occurs.
For the Green modes at
other parameter values, the
change in the upper boundary condition from W' = 0 to
of the
V' =
0 lowers the level
(II peak from the top down to the mid stratosphere; the growth rate
of these Green modes are substantially reduced.
For all parameter combinations where Y
o and
p > 10, the meri-
dional entropy transport for the Green and Eady modes is not strongly affected by changing the upper boundary condition.
The transports are very
weak near p = 0 for both modes and both boundary conditions.
The kinetic energy loss in the upper stratosphere for the Green mode
is smaller when LO' = 0 at p = 0.
Green mode kinetic energy loss or genera-
tion is little affected at other levels.
The kinetic energy loss or gene-
ration for the Eady mode is unaffected by changing the upper boundary condition.
A new type of mode appears for w' p=0 when there is a stratosphere
and when SR > the nominal value (compare Figures 3.19 and 3.56).
number of its growth 'rate peak for
The wave-
YT and 0? nominal varies from zonal wave-
number 4 to wavenumber 1 as SR increases from -1 to 30; the value of the
.Inllri
1
fastest doubling time remains constant at 4
days, as does the steering
-
The streamfunction amplitude peaks at the top and decreases
level (800 mb).
as a linear function of pressure to about 10% as much at the tropopause.
It remains close to this value below the topopause.
ward with height.
The wave tilts west-
Entropy transports are everywhere poleward and are es-
pecially strong in the lower stratosphere; the peak flux is at the tropopause.
Mode kinetic energy is destroyed in the lower stratosphere but is
generated everywhere else with greatest generation in the troposphere.
The
mode is only present when the temperature gradient at the upper boundary
is able to play a role in fulfilling the necessary condition for instability.
A second conjugate pair of eigenvalues (one eigenvalue of each pair
represents an unstable mode) exists when SR
Figure 3.57).
<
0 and C' = 0 at P = 0 (see
When 4' = 0 was the upper boundary condition, there was only
one conjugate unstable eigenvalue solution to the model's differential equation except for the internal boundary mode which exists when SR
) Cp . This
new solution for W' = 0 at P = 0 exists only when the temperature gradient
at the top boundary is positive (warmer toward the poles).
The new solu-
tion has growth rates which exceed those of the other solution (which contains the Eady and Green modes) when the Richardson number of the upper
layer is less than that of the lower layer.
son numbers (upper/lower) equals
O7/
SR2 .
The ratio of the two RichardThus, y < SR2 is the condition
for dominance of the new solution; as illustrated by Figure 3.58.
It can
also be seen from Figure 3.58 that the doubling times of the fastest growing mode of the new solution decreases quickly with R. of the upper layer.
The zonal wavenumber :-the fastest growing mode ranges from 3 when SR = -14,
0 > 40,
and 'T = 2 to 9 when SR < -3,
p = 2, YT = 2.
Phase velocities
are large and easterly; at
T = 2, SR = -8 and P = 50, the phase velocity
of the most unstable mode is -28 m/sec.
The phase velocity becomes more
negative as SR becomes more negative so that
around 50 mb.
steering levels remain
Streamfunction amplitudes peak at the top and decrease rapid-
ly toward the tropopause where they
are approximately 30% as large.
In
the troposphere, the amplitude gradually decreases to a value of 20% (of
the peak value) at the ground.
The phases show a strong eastward tilt of
the pressure wave with height in the stratosphere, a very slight eastward
tilt just below the tropopause and a slight westward tilt in the troposphere.
The phase change in the stratosphere is about
:/2 radians.
Correspondingly
the entropy transports are strong and equatorward in the stratosphere, weak
and equatorward in the upper troposphere, and weak and polewardin the lower
troposphere.
There is a strong generation of mode kinetic energy in the
stratosphere with the greatest generation being at the steering level. Mode
kinetic energy is lost in the troposphere but in a smaller amount.
The picture of this new conjugate eigenvalue that emerges is one of
a mode which is generated in the stratosphere and "drives" the upper troposphere
(through a loss of its kinetic energy) with weak counter-gradient
entropy fluxes.
Eady mode:
In a general sense, its properties mirror those of the
It is destabilized by a temperature gradient at the opposite
boundary and drives the upper troposphere instead of the lower stratosphere.
The internal boundary mode described earlier undergoes substantial
changes in growth rate and vertical structure when the upper boundary condition changes to c' = 0.
extremely short when
When Y , SR,
Its doubling time for the most unstable mode is
' = 0 at p = 0, and becomes much shorter when SR>>
, .
7 = 2, 10, 5, the doubling time is only 3/4 of a day and the
-
zonal wavenumber is 5.
' ...
.
.
. ..
I
, ,
t
ffL
lMm-
Unlike the situation when (' = 0 at P = 0, the
eigenvalue is well resolved for the N' = 0 upper boundary condition.
The
difficulty of resolving a mode whose pressure amplitude has a single very
sharp peak has been removed; the pressure amplitude now peaks at the top
and decreases only gradually to a value of 30%
250 mb internal boundary.
(of the peak value) at the
The amplitude in the lower layer gradually de-
creases to a value of .15 (of the peak value) near the ground.
speed and steering level are about the same as for the
dary condition:
30 m/sec and 225 mb, respectively.
The phase
= 0
0'upper boun-
The entropy transports
are strongest in the bottom of the upper layer, peaking at the internal
boundary; they are poleward (down gradient) and do not decrease to half the
peak value until 120 mb.
There are weak equatorward transports in the upper
part of the lower layer, and weak poleward transports near the ground.
The
eddy kinetic energy is generated in a region just above the internal boundary and destroyed in a region just below it, just like the new conjugate
mode described above.
25
.6 -
-22
T
E
0
\L
0
z
-f
/2
\
I
I
/ 13
/
vs. wavenumber forYT = 0,
I
= 1, and SR = 0 (dash), and SR = -1.5
(solid).
_I~
I_
_
--
"'
_go
28
25
22
16
I
,)
U
U
Cn
1
>-
z
w
H
I0
(9
Z
J
I
0
13
0
3
6
9.
PLANETARY WAVE NUMBER
12
Figure 3.2. As in Figure 3.1, foryT = 0, Cp = 50, and SR = -1.5
0 (dash), and 1 (dot).
(solid),
.6
12
I
U
O
9 -J
p-
00
I
L.
-6
3
3
6
PLANETARY
9
12
WAVE NUMBER
Figure 3.3. As in Figure 3.1, except:
(solid), 0 (dash), and 1 (dot).
YT = 2, aj
P
05
15
= 1, and SR = -1.5
-iIIrn
iii
18
15
12
E
In
0r
C0
>-
T
9
I
a-
6
0
3
6
PLANETARY
12
9
WAVE NUMBER
As in Figure 3.1, except: yT
Figure 3.4.
(solid), 0 (dash) , and 1 (dot) .
=
2
,Y
= 50, and SR = -1.5
PHASE
(4)
(radians)
200
400
CL
i,
600
800
1000
.2
.4
.6
.8
Iy I
Figure 3.5.
Vertical structure of the streamfunction amplitude (thick
lines) and phase (thin lines) for the fastest growing Eady mode, YT = 0,
and:
-, = 1, SR = -1.5 (solid); op = 1, SR = 0 (dot); CF = 50, SR = -1.5
(dash dot); cD = 50, SR = 0 (dash); and 0 p = 50, SR = 1 (cross).
- -
PHASE. (h)
pianoI
-
(radians)
200
400
600
800
1000
.2
.4
.6
.8
Iql
Figure 3.6. As in Figure 3.5, for yT = 2,
and 1 (dash)
and SR = -1.5 (solid), 0 (dot),
PHASE (N)
2
(radians)
4
.2
.4
0
200
400
600
800
1000
Figure 3.7.
0
As in
Figure
(dot), and 1 (dash).
3.5,
for y
14/1
= 2,
.6
.8
cy = 50,- and SR = -1.5 (solid),
Mfl
62
(radians)
PHASE (4,)
0
2
200-
1.
I0.
4
27r
-.
/..
I:
.
400-
/e
*
600
800 -
0
.2
.4
.6
.8
1.0
Figure 3.8. Vertical structure of the streamfunction amplitude (thick
lines) and phase (thin lines) for the fastest growing Green mode, YT = 2,
= 50, and SR = -1.5 (solid), 0 (dot), and 1 (dash).
200 -
400-
600 -
800 -
I000 I
-1.
0
I
I
-. 8
I
l
-.6
I
I
-. 4
I
I
-. 2
I
I
0
VI
.2
I
I
.4
I
I
.6
I
I
.8
I
iI
1.0
e'
Figure 3.9. Vertical structure of the meridional entropy transport for the
fastest growing Eady mode for the same cases as in Figure 3.5. Note, all
curves are identical.
It
_
I-
-
'
=1111,11
0-
200-
400 -
600
800
1000
-1.0
Figure 3.10.
-1.5 (solid),
-.
8
-.
6
-.
4
-.2
0
.2
.4
.6
.8
1.0
As in Figure 3.9, for the cases: YT = 2,
= 1i,and SR =
0 (dot), and 1 (dash). All curves are zero Pabove 250 mb.
200 I-
t~
\I
\
\ .
400
E
Li
D
VI)
wn
cr
0_
600
800 -
1000I
-1.0
I.
Il
-.
6
Ii
-.
4
II
-.2
I
.2
I
0
Figure 3.11. As in Figure 3.9, for YT = 2,
0 (dot), and 1 (dash) .
a
I
.4
I
.6
I
.8
I
1.0
= 50, and SR = -1.5 (solid),
--------
------
-- -----
----------
S
Iu
llui
200 I
t
I
I
1
I
I
I
I
I
I
I
I
I
I
400 -
I
I
t
t
I
I
600 -
800 -
1000 L
-1.0
II
Im
I
-.
8
-.6
-.4
•
I
I
.2
|
-.2
0
I
V
I
.4
I
.6
I
.8
1.0
'
Figure 3.12. Vertical structure of the meridional entropy transport for
the fastest growing Green mode for same cases as in Figure 3.8.
67
0
200-
a-.--
400 '
"
I.
a
S
1/.
C)
600
\',
800 -
1000
-1.0
I
-.
8
I
-.
6
I
-.
4
-.2
I
.2
0
CE (KE ,-
PE
I
.4
I
.6
.8
1.0
)
Figure 3.13. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE > PE is positive), for the fastest growing
Wady mode for the same cases as in Figure 3.5.
W
--"--I^
1--
1111111 11111
1111
1I1II
IYI
68
0I
I
I
200-
S
II
I
(solid),
O(I)°.|(dot),
400-
and 1
(dash) .
LU
/
*
800i
-1.
-.8I
1000
-1.
Asi
Figur3.4
(sld,0
-.6I
-~-6
(dtad
Figure 3.14.
As in
09
-4
-
-.2
0
- CE
(EK,
II -P
CE K#
iue31,frteCSCy
.
2I
)
.4I
4
.6I
.6
.8I
.
1
,a
n
R=-.
p
#ds)
Figure 3.13, for the casesy
1.0
.
= 2,
= 1,
and SR = -.
5
E
Z)I
1.
800
1000
-1.0
-.8
I
-.6
I
-.
4
0
-2
CE (K EPE
Figure 3.15.
-1.5
I
.2
I
.4
I
.6
I
.8
)
As in Figure 3.13, for the casesyT = 2, O
(solid), 0
(dot), and 1
(dash).
I
1.0
P
= 50, and SR =
---- --.-=mill
.4/
V/
200 -
400
6001
/I
%
I
800 r
N
1000
-1.0
1
-.8
,..
II
II
-.6
-.4
I
I
I
li
i
I
l
.2
.4
.6
.8
Ii
-. 2
0
I
I
I
i
1.0
CE ( KE-,-PE )
Figure 3.16.
for cases
y
T
As in Figure 3.13, except for the fastest growing Green mode
a = 50, and SR = -1.5 (solid) , 0 (dot), and 1 (dash).
D
= 2,
2.0
I
I
1.5
I
I
.5
1.0
.5/
6
4
2
0
WAVELENGTH (nondimensional)
Figure 3.17.
length'c
Non-dimensional growthrate
= 1, SR = 1, u(p-
) = i,
and
(_q divided by YT vs yT - 1 and wave=
00' at
p = 0.
Compare Figure 2.6.
- --
--
'
IrlllYII
2.0.
I
1 .6
1.5-
1.0
I,
1
l
.5.0
O-
0
pcz4
--
/O
0
2
WAVELENGTH
4
(nondimensional)
pa1
by YT vs YT
_
divided
growthrate
Non-dimensional
Figure 3.18.
p,= 0.
0
at
4'
=
= 50. SR = -1.5, u(p-) = 1, and
length:c
6
and wave-
co
o
'i-
LId
0
z
00
C
0
3
6
9
12
PLANETARY WAVE NUMBER
= 2, T = 50, and SR
Doubling time vs. wavenumber for Y
, and 8 (long dash)
(cross)
4
(dash),
0
,
-4
(solid)
(dot), -8 (dash dot),
Figure
-12
3.19.
11111110111w,
*'i*
*
I
i
l
*
i
Si
'L
"
I
=i
Figure 3.20.
As in Figuret 3.19, for the cases: y
aigure
=I1 (dot),
.
10 (dash do ), 40 (solid), 100 (d
(dspsh,
2
SR =-1.5,
and
and
-
I
'
z
I
i
m
o
.
!
".
-I
--
I
I
I
1
I
I
0
PLANETARY WAVE NUMBER
Figure 3.21.
As in Figure 3.19, for the cases SR = -1.5,
50, and
= 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
=1
(dash dot),
3 (solid) ,
5 (dash),
7 (cross)
,
and 10 (o
dash).
Y
T
L
-
111
ii1
111611
, , j
76
I
18
I
15
12
E
9I
0
I
-9
I Ii
0
Figure 3.22.
3.22.
Figure
Figure 3.19.
3
"I
6
PLANETARY
I
I
9
WAVE
12
o
15
NUMBER
for the same cases as in
Phase velocity
velocity vs.
vs. wavenumber
wavenumber for the same cases as in
15
12
U
',
E
95
90
JwO
7-
I
6
6
PLANETARY
Figure 3.23.
Figure 3.20.
9
WAVE NUMBER
Phase velocity vZ. wavenumber for the same cases is in
I_-~1-.^i~_-I_- 111
Will ll ill,
11,01
1
78
18
15
12
-
E
/
\
\9
/
9
J
"
\I
x
6•
I
x
3
I
-
II
0
Figure 3.24.
Figure 3.21.
3
6
PLANETARY
9
WAVE NUMBER
12
15
Phase velocity vs. wavenumber for the same cases as in
0
200
400
E
w
6
.0, f,
a
600
800
1000
0
III1i
.2
.4
.6
8
1.0
Figure 3.25. Vertical structure of the streamfunction amplitude for the
fastest growing Eady mode for the same cases as in Figure 3.19.
~
-m____-_rrt~cP1-s_
-1-1-1
i 1Ill.
OII
No MIII
80
0
.--
200
400E
SIU
,I
I'. I
.
8000\
0
.2
.4
.6
.8
1.0
2, SR = -1.5, and g
Figure 3.26. As in Figure 3.25, for the cases Y=
P
= 1 (dot), 10 (dash dot), 40 (solid), 100 (dash), and 1000 (cross).
81
0'
v.
200 /
."
/
1I
E\'
r600
•
\\\
I
1000
0
I
.2
Il
.4
6
.8
1.0
Figure 3.27. As in Figure 3.25, for the cases SR = -1.5, a = 50, and
YT = 0 (dot), 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10
(long dash).
_
/
er
600-
800-
1000
0
A
.2
.4
.6
.8
1.0
Figure 3.28. Vertical structure of the streamfunction amplitude for the
fastest growing Green mode, for the same cases as in Figure 3.19.
83
200-
400 -
/
i"i
'
II
800
000 -
0
Figure 3.29.
IIII
.2
.4
.6
.8
1.0
As in Figure 3.28, for the cases YT= 2, SR = -1.5, and c
= 10 (dash dot), 40 (solid), 100 (dash), and 1000 (cross).
P
~---
---
IIIIYlii
N11111
l9111111
iII
A-
200 -
2
//
,
/
/
400k
/
/
-o
E
./
/
/
$7
w
C,
,.
U
,
'7/
a600
1,
/
N'
800
N
Nr
1000
I
0
I
.2
I
I
I
.4
I
.6
1
I
.8
I
1.
1"11
Figure 3.30. As in Figure 3.28, for the cases SR = -1.5, Qp = 50, and
YT = 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
85
0-
*I
.I
I': \
400-
Il,Si
0
I.L
I
800 -
\,.: \
1000
II
O
2
PHASE
(
)
4
(radians)
2-rr
Figure 3.31. Vertical structure of the streamfunction phase for the fastest growing Eady mode, for the same cases as in Figure 3.19.
86
O-
:
*
4
m
"
200 ;I
400 LI
•
a
,11
o
a
400 -
10001
0
2
PHASE
Figure 3.32.
1 (dot),
As in
Figure 3.31,
1 (dot),
dot),
800
4
(N)
(radians)
for the cases y
(dash (solid),
,10 40
10 (dash dot), 40 (solid),
27r
100
'
100
2,
SR
1000
-1.5,
(dash),
and
(cross).
(dash),
and 1000 (cross).
and cr
I:
200 -
1'!
1I1
It'
.I'
:
:I
4001-
\I
A.;
600 -
I
I*
1
,I
I
rl
I
I g\
IN
\
800 -
r
\
\*
1000
I
I
|
PHASE
'
2'~
I
|
(4)
II
-
_
(radians)
Figure 3.33. As in Figure 3.31, for the cases SR = -1.5, Op = 50, and y
= 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
~_
~_____
^_~1_1~__ X__1
I ~I__
~_
~Y~___
~I_ _I_
_
__
_III,
88
400-
200
r
'
I
400
E
It I
0r
I
600
\
N,
,
800
,
'\
1000
1
O
1
2
PHASE (4)
4
(radians)
27r
Figure 3.34. Vertical structure of the streamfunction phases for the fastest growing Green mode, for the same cases as in Figure 3.19.
Or
*
Mr
i
200
x
I
X
400 -
I
600 -
1
800
1000
'\
1
"
-
I
I
2
PHASE
I
(4)
I
I
I I
4
(radians)
Figure 3.35. As in Figure 3.34, for the cases Y = 2, SR = -1.5, and a
10 (dash dot), 40 (solid), 100 (dash), and 1000 (cross).
=
------
whilim, 11, 1
-'IIIYIIYIYlliil
IA
I
\
\
200 I
aI
pI
It
o
I
I,
I
,I
I
400 9
I
E
I
I
-I
a\"t
LU
w
vD
C,
U)
w
a_
600
\L
i
I
k
\
\X
\
800
1000
\
Ii
N
I
I
II
PHASE
I
(4I)
I
4
(radians)
,
,I
2rr
Figure 3.36. As in Figure 3.34, for the cases SR = -1.5, p = 50, and yT
= 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
91
0-
io
400 -
uj
D
600-
800 -
000
-1.0
I
-.8
I
-.6
-. 4
-.2
0
I
.2
I
.4
I
.6
I
.8
I
1.0
v, el
Figure 3.37. Vertical structure of the meridional entropy transport for
the fastest growing Eady mode, for the same cases as in ifigure 3.19. Where
cases are not shown, they coincide with those presented: the curve for SR
= -8 coincides with that of SR = -12 at somelevels; the curves of SR = 0
and SR = +4 coincide with that of SR = -4 at some levels.
"'
-
-"YY 1III11
92
0-
200 -
400
I
,1
w
600
800
1000
-1.0
-.8
-6
-.4
-.2
I
0
I
.2
I
.4
.6
.8
1.0
v'8
Figure 3.38. As in Figure 3.37, except for the cases YT = 2, SR = -1.5,
and a = 1 (dot), 10 (dash dot),40 (solid), 100 (dash), and 1000 (cross).
Where cases are not shown, they coincide with those presented: below
250 mb, the Up = 40. 100, and 1000 coincide.
93
O-
200 -
400 E
I
A'
800 S\
*
1000J
-1.0
-.8
-.6
-.4
-. 2
O
.2
.4
.6
.8
1.0
Figure 3.39. As in Figure 3.37, exceptfor tie cases SR= -1.5, p = 50,
and y = 0 (dot), 1 (dash dot), 3 (solid), 5 ((Ia7,'hi), 7 (cross), and 10
(long dash).
(long dash) .
0
200
I
II,
400 -
I.
\"
\*
600
800
p\
N,
'1
\;.
I.:
\.\
1000
IR
- I.(
0 -.8
IB
nn I
I
-.6
-.4
IN
-. 2
0
V'
I8
II
I
m
I
m
.2
.4
.6
.8
gI
1.0
e'
Figure 3.40. Vertical structure of the meridional entropy transport for
the fastest growing Green mode, for the same cases as in Figure 3.19.
Or
200
~
fII
400 -
II
600 -
'I
*I
'
800 I-
A'
SI
Il
~1
1000
-1.0
Figure 3.41.
and cr
= 10
I
I
I
I
-.
8
-.6
-.4
-. 2
0
.2
.4
.6
.8
1.0
As in Figure 3.40, except for the cases Y. = 2, SR = -1.5,
40 (solid), 100 (dash), and 1000 (cross).
(dash dot),
~~~~__I__
____~~__
I It,
ill
11A,
0-
I
00 -
400 -
tU 0
I,I
0
v'i
600
I''II
800
1000
-1.0
-8
6
-.
4
-.
2
-.
0
.2
.4
.6
.8
1.0
Figure 3.42. As in Figure 3.40, except for the cases SR = -1.5, ap = 50,
and y = 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
0-
200
/
,,
400 -/
E
K
600
,
cI/00w
600
-1.0
-.
8
-.
6
-.
4
-.
2
0
CE ( KE -
.2
.4
.6
.8
1.0
P )
Figure 3.43. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE
PE is positive) for the fastest growing
Eady mode, for the same cases as in Figure 3.19.
1
-
~lIIi
-----
0
200
.///.
/ ,,
400 -
.-
0-8
E
/
.
-4
I
600
800
1000
-1.0
-.8
I
I
-.
6
-.4
I
-.2
I
0
I
.2
.4
.6
I
I
.8
1.0
CE (KE--PE)
Figure 3.44. As in Figure 3.43, except for the cases T = 2, SR = -1.5,
Op = 1 (dot), 10 (dash dot), 40 (solid), 100 (dash), and 1000 (cross).
0-
200-
400
Figure
CL
3.45.
SIr
•
As
in
600 '7(dot),
/
Figure
3.43,
except
for
the
cases
1 (dash dot),
/
800
(long
dash)
/
SR
7
-1.5,p
(cross)
4
/
-1.0 SCE -8
800
."
-
1000II
-I.0
-8
-.6
-. 4 .2 -. 2
/(KEPE
.4
.6
.8
1.0
.4l1
.6
.8
1.0
i
-.6
-. 4
-. 2
0
.2
=50,
100
O
200 -
.*-'. _--
-
x
"r~
400 -
I r.
I,;
2.
'I:
I'
600 -
F
:
-
I.,
I.
-
I.I
I:
r
I
~
S.
800 -\.
- I
-
\
.'
\\
\
\
S..
~\
,
t-..
1000
-1.0
I
-.8
I
-.6
I
-.4
I
-.2
I
I
I
I
I
I
0
2
.4
.6
.8
1.0
CE ( K E
PE)
Figure 3.46. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE + PE is positive) for the fastest growing
Green mode, for the same cases as in Figure 3.19.
101
I_.--'
*
Y
/.d
4-fv
0
*-<
I
200
l
I
a
l
I
l
w
400-
I It £
I
K
Ia
600 aI
\
4I.
.I
800
a:
10001
- I.(
0
-.
8
'***
% .
I
II
II
I
.2
.4
.6
.8
_
-.
6
-.
4
-.2
0
I
1.0
CE (KE,-PE )
Figure 3.47. As in Figure 3.46, except for the cases YT = 2, SR = -1.5,
and 0 = 10 (dash dot), 40 (solid), 100 (dash), and 1000 (cross).
11111101
102
44
200 h
I
I:
I
r
r
r
r
400
-o
E
S
v
t
91
\'
\;
r
r
r
r
r
r
800
V
\,'
I
I
I
I
I
r
600 k
1000
-1.0
1
Is
r
r
r
r
r
I
I
r
I
16
I
I
I
I
I
I
I
I
I
-.8
-.6
-.
4
-.2
0
.2
.4
.6
.8
CE (KE-
I
1.0
PE )
Figure 3.48. As in Figure 3.46, except for the cases SR = -1.5, a = 50,
and yT = 1 (dash dot), 3 (solid), 5 (dash), 7 (cross), and 10 (long dash).
0
80
60
40
26
0
100
1.5
-
-4
a:
1.6
cn
-8
1.7
18
2.0
2.5
3.0
-12
Doubling time (days) vs. SR and
Figure 3.49.
Eady mode for Y = 5.
p for the fastest growing
0
T
9
9
o
0
20
40
60
-4
1.7
-8
2.0
25
3.0
-12
Figure 3.50a. As in Figure 3.49, except for yT = 2.
80
8
cr
4
0
20
40
80
60
100
,-p
Figure 3.50b. As in Figure 3.49, except for yT = 2 and SR > 0.
Cn
9
9
S
o0
20
40
60
80
100
1.58
-4
-8
1.6
-12
I.?
1.8
.
Figure 3.51.
As in Figure 3.49, except for
= 6.
!
I
0
4
80
60
40
20
0
020
25
I0
Q2
30
-8
15
10
-12
Figure
3.52.
Green mode
Doubling time
for y
=
T4
(days) vs.
SR and a
for the fastest growing
5.
9
9
9
oo00
07,
0
0
r
-8
-12
Figure 3.53.
As in Figure 3.52, except for yT = 2.
0
ab
20
.0
3.0
-43. 7
,(n
-8
3.5
3.0
2.0
-12
Figure 3.54.
As in Figure 3.52, except for Y
= 6.
o
\o
il,
PHASE (4)
110
(radians)
200
400
600
800
1000
.2
.4
.6
.8
IkI
Figure 3.55. Vertical structure of the streamfunction amplitude (thick
lines) and phase (thin lines) for the fastest growing Green mode for the
same cases as in Figure 3.8. The boundary condition w' = 0 at p = 0 is
assumed.
111
8
6-
Uj
z
OJ
0
0
3
6
PLANETARY
12
9
WAVE
15
NUMBER
Figure 3.56. Doubling time vs. wavenumber for y
and the upper boundary condition w' = 0 at p = 0.
= 2, SR = 8,
= 50,
-
0111,
112
.6
0
.0
0
O
O
2
0
Figure 3.57.
3
9
6
PLANETARY WAVE NUMBER
As in Figure 3.56, except for SR = -8.
12
15
1.55
-4
-
1.7
1.6
,
C,
-8
1.5
.5
10
-12-
I!.
I
I
-
Figure 3.58.
Doubling
est growing mode.
The
are to the upper right
ating modes are to the
I
I
I
I
time (days) vs. SR and Gp for Y = 2, for the fastupper boundary condition is w'
0.
Eady modes
of tihe dashed line; new, stratospherically origin-Ilower left.
W
114
CHAPTER IV
RESULTS FOR THE HIGHLY STRUCTURED BASIC STATES
4.1
Introduction
The simplified temperature and zonal velocity profiles of the pre-
vious section do not contain the detail found in true atmospheric profiles.
The goal of this chapter is to include such detail.
The static stability
and unperturbed state of the model atmosphere will be chosen to include
seasonal variations.
These variations will be represented by monthly aver-
aged u and T data for the months January, April, July, and October.
Lati-
tudes from 200 N to 700 N in 50 intervals will be included.
These months represent seasonal transitions and extremes.
The data
available allow fine detail to be included in the vertical structure of
the unperturbed state.
The basic state is kept one-dimensional; longitudi-
nal and meridional variations are excluded.
In calculating the stability characteristics of these detailed basic
states, it will be of particular interest to determine how the modes of the
simplified profiles are changed; what new modes, if any, appear, whether the
modes grow fast enough to reach significant amplitude; and whether these
modes could play a significant role in the general circulation of specific
regions of the troposphere and stratosphere.
These questions are addressed
in this chapter by discussing the longitudinal scale, growth rates and phase
speeds of the unstable modes, and by examining the vertical structure of
their pressure amplitude, their meridional entropy transport, their meridional-vertical velocity correlations, and their conversion of available
potential energy to eddy kinetic energy.
115
4.2
Criteria for Choosing Input Data for the Model
There are 44 pairs of u and T profiles from the four months and 11
latitudes selected.
The pairs used in the calculations will be chosen from
these 44 pairs according to two criteria:
First, low, middle, and high
latitudes should be represented in each seasonal month; the latitudes 250 N,
450 N, and 650 N are selected to meet this criterion.
ion is more complicated and requires discussion.
The second criterThe Charney-Stern theorem
states that in the absence of boundary temperature gradients a zero in the
potential vorticity meridional gradient (q ) is a necessary condition for
instability.
Furthermore, studies of Green (1972), Murray (1960) as cor-
rected by McIntyre (1972), Simmons (1972, 1974), and Dickinson (1973) have
shown that a zero in qY has associated unstable disturbances.
The studies
of Simmons (1974, 1972) also show that pressure amplitudes and meridional
heat fluxes of the unstable modes are confined to a region which extends
only a few kilometers above or below the height of the zero in q
.
These
results indicate that a zero in the q, profile may result in unstable modes
and the location of the zero may be important in determining the vertical
structure of those modes.
Therefore, when choosing u and T profiles for
the model, this second criterion will be observed:
Profiles will be selec-
ted which represent all the qy zeros present in the larger data set of 11
latitudes and 4 months.
The second criterion applies, of course, to the one-dimensional calculations of q,; because the basic states are one-dimensional; meridional variations inu and a (T) are ignored. However, calculations of qy, including both
vertical and meridional variability of -u and 6 (T) are made for all data
116
sets and are compared with calculations which exclude meridional variations.
Regions where the two methods of q
gions where u
yy
is large.
calculation differ are found to be re-
However, the zeros of qy are found to be little
y
affected by the exclusion of meridional variations; thus, the one-dimensional calculation of q
is qualitatively realistic.
The second criterion,
therefore, allows for an adequate representation of potential source regions
for baroclinic instability.
4.3
The Data Sets
The data sets are chosen to cover more than one year where possible,
so that the study is climatologically representative.
Data from the north-
ern hemisphere is more plentiful and thus better suited for a climatological
study.
Four data sets are used and are described below.
They will be re-
ferred to as Data Sources 1 to 4, respectively.
1)
Oort and Rasmusson (1971) provide 5-year means (May 1958 to April
1963) for all months, for latitudes 100 S to 750 N in 50 intervals and for
levels at 1000, 950, 900, 850, 700, 500, 300, 200, 100, and 50 millibars.
2)
R. Newell has provided the tables of u and T (for the years 1964
to 1968) from which the plots in Newell et al. (1974) were made; this
data covered all months and latitudes 200 N, 250 N, 300 N, ...
850 N, and
is for levels at 200, 150, 100, 50, 30, and 10 millibars.
3)
Tahnk and Newell (1975) supply u and T values at 5 mb and 2 mb;
these data were available only in 100 intervals from 150 N to 750 N,
and the data set is only for one year, 1972.
4)
Finally, for levels between 1 and .1 mb, data from Newell and Tahnk
(in preparation) are used; the data are from Meteorological Rocket
Network observations (MRN); the data cover an 11-year period (1961 to
1971), but are only for North America.
117
Because the last two data sets are very limited as to time and/or spatial
coverage, they are used only as guides to extrapolate u and T profiles
time periods,
100 mb, and
1) the data sets are from different
It should be noted that
above 10 mb.
2) they suffer from poor station coverage at levels above
3) they employ different analysis techniques.
These three
the data sets' values of u, T,
factors account for the differences among
and hence, values of qY in regions where they overlap.
The u and T values
of these data sets are presented in the appendix.
4.4
Calculation of qy
The calculation of qy is done for all data sets, with and without
meridional variations in u and 0 (T).
First, a cubic spline was applied to
the u and T profiles within each data set to find values at model levels.
The cubic spline assures that vertical second derivatives are continuous;
this eliminates the possibility of spurious ;
file yields p
and
definition of 0,
values.
The splined T pro-
-profiles from Equations 6, 8, and 9 in Chapter II, and the
and these profiles are used with that of _ in the exact
definitions, Equation 2 of Chapter III, and in the model approximation:
q
- f
to obtain the two sets of qy values.
2
(
1
u )
(1)
The results are in Tables 1 to 8.
In the _q calculations which include U
and 0 terms, there is a
I
yy
y
dominance of positive values which is due to the P effect and vertical shear
variation.
There is a tendency towards negative values of q
(below 750 mb) for most latitudes.
at low levels
These negative regions start at levels
118
ranging from the surface to 900 mb, and in some cases are associated with
two zeros.
They are caused by large values of the term
f
2
(
1 -
U )p.
r
This
same term results in the high latitude 600-750 mb negative -y regions in
January, April, and October, and the July high latitude 375-500 mb negative
q
region.
There is little seasonal continuity of qy at higher levels for
the 5 year data sets; only July has significant negative q, values above
The other months have a few weakly negative -
600 mb.
regions in levels
covered by both the Oort and Rasmusson (1971) and Newell et al. (1974) data
sets, but none of these appears in both sets.
q
In July, the only negative
region below 5 mb in which the horizontal shear term is comparable to
the vertical shear term is that at low latitudes between 125 and 175 mb.
u
, although relatively small in the regions at 550 and 650, 90 to 150 mb,
and in the region at 650, 30 to 60 mb, is just enough to tip the balance
from positive to negative q
values.
These negative q
regions will not be
present in the model calculations of q.
The Tahnk and Newell (1975) data set has q
tudes 450 and 500 N for all seasons.
zeros at 30 mb for lati-
At 650 N there is a zero at 25 mb for
April and a zero at 50 mb for Ocrober. These zeros are due to large values
2 1
u ) . They are not present in the 5-year Newell et al. (1974)
of f C
data set.
Also, the zeros occur in a region where data points are sparse
(there is no data in the Tahnk and Newell (1975) data set between 10 and
100 mb).
Therefore, their climatological importance is questionable and
they will not be included in U and T profiles used in the model.
The MNR data have zeros at high and low latitudes between 1 mb and
5 mb.
These are due to large u
and may result in instabilities which are
barotropic in character (Holton (1975), p. 88).
119
The calculations of Leovy in Charney and Stern (1972) show zeros in
qy above 100 mb for 14 October 1957 at 150W.
in the October monthly averaged data.
These zeros are not present
This is not surprising as one data
set is only for a specific day and longitude and the other is averaged over
one month and all longitudes.
Leovy and Webster (1976) suggest (without
presenting calculations) that
I may have zeros in winter on the tropical
side of the stratospheric jet (they used hemispheric daily u data to calculate l ).
The MRN data for January have similar zeros but this data is only
for North America.
set.
The zeros are not in the Tahnk and Newell (1975) data
zeros are due
Both the Leovy and Webster (1976) and the MRN data q
to large horizontal curvature, and instabilities associated with these zeros
would be expected to be barotropic in nature (Leovy and Webster, 1976).
The values of qy calculated for the model (Eq.1 of Ch. IV) when compared with q
values calculated from Eq. 2 ofCh. III) reveal the contribution
of meridional shear variation.
70is
The term involving meridional variation of
a factor of 5 or more smaller than
u
YY
.
The u
YY
term is large and
negative between 75 and 450 mb south of 350 N in April and January and October.
July values are strongly negative (nearly as large as January values)
at the same levels but for latitudes 400N and 450 N.
In all data sets ex-
cept the MRN, July is the only month with large positive u
values; these
yy
are at 250 N and 300 N between 175 and 350 mb; this, of course, reflects the
more northerly position of the summer zonal velocity jet.
juyy 1
The region where
is large are those regions where the model values of q
from the observed values.
differ most
There is a negligible effect on the zeros of
below 175 mb for all months.
In January and October there are some differ-
ences above 175 mb, but the zeros here in the observed _q
do not appear in
120
both data sets.
In July there is a greater difference, as noted above, and,
0
in addition, in the region 400 N to 50 N, 50 mb to 120 mb.
has q
The MRN data set
zeros resulting from large uyy ; there are almost no zeros among the
corresponding model values.
and a do not have a
Overall, however, meridional variations in u
zeros calculated from
substantial effect on the number and location of q
zonally and monthly averaged data.
Therefore, one-dimensional _4
tions are qualitatively representative of the full two-dimensional
calculay field.
A one-dimensional basic state thus includes essentially all the potential
sources of baroclinic instability that would be included in a two-dimensional basic state.
This gives further support to the qualitative realism of
studying the baroclinic instability of one-dimensional unperturbed flow.
4.5
Processing the Data for Input to the Model
The profiles of u and T for the four months, January, April, July
0
0
and October and latitudes 250 N, 45 N, 65 N have already been chosen to be
used in the model.
Examination of the q
fields
(model values) reveals
that January 350 N and October 500N merit study for associated unstable modes.
January 350 N provides q
zeros at high (50, 100 mb ) and mid (480, 600 mb)
0
October 500 N provides a close comparison to October 45 N with the
levels.
major difference being in strength of the negative q
of the q
region or closeness
zeros.
None of the data sets includes all model levels.
In order to obtain
a single u or T profile for all levels, the datawere plotted and smoothed
by hand.
The Oort and Rasmusson data were followed closely to about 100 mb;
the Newell et al. (1974) data from 100 mb to 10 mb.
Between 10 and 2 mb,
121
the Tahnk and Newell
(1975) data were used.
Above 2 mb, the Newell and
Care was taken so that
Tahnk (in preparation) data were used as a guide.
zeros of qY were affected minimally.
The smoothed profiles appear in Fig-
ures 4.1 to 4.8.
To minimize error due to interpolation, points at levels for which
the data sets have values are then read from the smoothed u and T profiles .
Next, a cubic spline is applied to each set of u or T values to yield values
The cubic spline again assures that vertical second deri-
at model levels.
vatives are continuous, and thereby avoids spurious q
small scale variations in model results.
zeros and erroneous
The splined u and T values consti-
tute the input i and T data for the model.
The
19 to+12.
q values associated with the model input profiles are in Tables
A comparison of these values with those q
values from the model
calculations for individual data sets shows that joining the data sets
affected qy zeros only within the overlap region of the data sets.
negative q
Three
regions which appear in only one of the data sets do not appear
**
in the model q
profiles
.
mains in the model profile,
The final model q
One negative q
region (for January 350 N) re-
although it is present in only one data set.
profiles have a weak negative _q region for April 250 N
at 400 mb which is not present in observed q
profiles.
Two negative qY
regions, July 250N, 125-175 mb and July 650 N, 30-60 mb, are missing from
*
The levels, 19 in number, are: .1, .2, .5, 1, 2, 5, 10, 30, 50, 100,
200, 300, 400, 500, 700, 850, 900, 950, and 10M0 millibars.
**
These 3 negative q regions are for the Newell et al. (1974) data set.
They are: April 6 oN, levels 37-41; July 250 N, levels 39-41; and October 450 N, levels 27-31.
_ . _____I___
___
111
122
the final profiles; in the observations the latter region is weak and the
Upper
former is strong because of large meridional shear variation in u.
level zeros in q
are still present in the input data set of the model:
0
at January 350 N (65-85 mb), July 45 N (85-125 mb), and July 65°N (140-175
Therefore, the u and T profiles chosen for this instability study
mb).
on the whole give a reasonably accurate representation of mean atmospheric
qy zeros, and thereby also give a good representation of some potential
sources of baroclinic instability in the real atmosphere which were not
studied before.
S
In addition to the complete E and T profiles, the reference values
-1
-5
4
-2
-2
+5
S
are
and u = 0 m sec
nt m-,
= 10
,
.321 x 10 -5 m sec-2
o
oo
From these
used in the numerical calculations.
+
of
n
-
C-7
n l
is calculated from r
T
and C=
The parameter YT and the fields of u and
YT is obtained.
Rd
[
n
(
P
)
(
P- a
The field
k-
T)] .
are all now determined and the
model set of finite difference equations is ready to be solved for eigenvalues and eigenfunctions for any given wavenumber P.
4.6
Model Results- Classification of Modes
The doubling times and phase speeds of unstable modes associated with
the - and T profiles are presented in Figures 4.9 to 4.21.
assume
' = 0 as the upper boundary condition.
All calculations
There is no figure for July
at 250 N because no unstable modes were found to exist in that case.
This is
because there is no temperature gradient at the lower boundary nor any in0
ternal zero of q, associated with the July 25 N data; the necessary condi-
tions for instability are thereby not met (the relation of the modes presented in Figures 4.9 to 4.21 to the necessary conditions for instability
123
will be discussed at the end of this chapter).
planetary wavenumbers .3 to 15
Modes are presented for
(when vertical resolution permits), except
for October at 250 N where the range is
3 to 18 (there are no unstable modes
less than wavenumber 3 for this case).
Note that the physical wavelength
associated with the wavenumbers changes with latitude.
All cases except July 250 N, October 250 N show instability at all zonal
wavelengths resolved by the model except isolated wavelengths.
Doubling times
tend to be longer for 650 N. Modes also tend to peak at shorter physical wavelengths
for higher latitudes, which may be due to the increase of 3 with latitude.
Within any given season, there is a large difference between the doubling
time and phase velocity curves at different latitudes.
As one changes
season at the same latitude, there is somewhat less of adifference in these
curves only for 650 N.
The modes presented in the 14 cases under discussion are divided
into 6 categories:
1)
The Eady modes, whose fastest growing mode has the shortest doubling
time of all modes and usually occurs between wavenumbers 5 and 8.
2)
The Green modes, which are on the long wave side of a cusp which sepa-
rates them from the Eady modes.
Exceptions to this are the cases of April
and July at 650 N, where the Green and Eady modes merge.
There is still a
minimum in the phase velocity curve and the wavenumber of this minimum is
defined as the division between Eady and Green modes in these cases.
The
Eady and Green modes are part of a single conjugate solution to the eigenvalue problem which are separated by a cusp point.
11111116
_ ____
I__
124
3)
The "shortwave Eady modes" which have a local growth rate maximum on
the shortwave side of the fastest growing Eady mode.
0
only for January, April, and October at 65 N.
These modes occur
They are defined to exist
for wavenumbers greater than that of the local growtbrate
modes' curve.
minimum in the Eady
This minimum is at wavenumbers 6.4, 7.2, and 6.8, respectively,
for the 3 cases cited above.
4)
The shortwave modes which have their shortest doubling time at the
wavenumbers of the local growthrate
tioned immediately above.
minimum of the Eady mode curve men-
They have a cutoff on both sides of the peak
value, and are always weaker than the Eady modes at the same wavenumbers.
These modes form a solution to the eigenvalue problem which is distinct
from that containing the Eady and Green modes.
5)
Shortwave modes other than those of category 4, whose fastest growing
mode has a shorter wavelength than that of the fastest growing Eady mode.
These modes are a conjugate solution to the eigenvalue problem which is
separate from the solution containing the Green and Eady modes.
Their
doubling times are generally much longer than that of the fastest growing
Eady mode.
6)
Longwave modes, other than the Green modes.
The fastest 4rowing long-
wave mode is defined to have a wavelength longer than that of the fastest
growing Eady mode.. These modes may occur either on the longwave side of a
0
cusp point which separates them from the Green modes (e.g. January, 45 N),
or they may form a separate conjugate solution to the eigenvalue problem.
For ease of reference, the modes in the six respective categories are given
the symbols E, G, Es , Sm, S, and L.
125
Figures 4.9 to 4.21 give several examples of each mode category defined
above.
Table 4.13 presents properties of the fastest growing mode of all
of these examples .
Subsequent discussion of the unstable modes will,
like Table 4.13, be organized according to mode category.
4.7
The Eady Modes
The Eady modes are present for all cases except July at 250 N.
The
fastest growing E modes are usually between 3000 and 5500 km in wavelength
(planetary wavenumbers 5 to 8).
has
A
A notable exception is October 250 N, which
= 2000 km and pwn = 18 (see Table 4.13).
Doubling times of the fast-
est growing E modes range between 1.5 and 4.3 days and average about 2 days;
they are shortest for 450 N.
The case of January 450 N has pwn = 7.7 and
doubling time (D.T.) = 1.7 days; this compares well with pwn = 6.6 and D.T.
= 1.6 days for the fastest growing Eady mode, when nominal winter parameters
are used in constructing the u and Oprofiles.
The fastest growing E mode
in Geisler and Garcia's (1977) calculation which used the Lindzen-Hong profile (to simulate winter at 450 N) has a planetary wavenumber greater than 9
and a doubling time of 1.7 days.
The doubling time agrees well with this
model's value for January 450 N.
The wavenumber is difficult to define be-
cause the E mode growth rate peak is very flat in the Geisler and Garcia.
calculation.
With the again notable exception of October 250 N, phase velocities
* Modes with doubling times greater than 20 days are not included. The
fastest growing Green modes for April and July at 650 N are chosen to be
those with their wavenumber just on the longwave side of the point of
maximum curvature in the Eady-Green mode growth rate curve.
MWM
M11
126
of all the E modes are positive except when the growth rate goes to zero.
The value of the phase velocity is less than 7 m/sec, so there are steering
u = c
levels for all the E modes in the lower troposphere.
upper stratosphere for some cases (see Table 4.13).
r
also in the
The fastest growing
E mode has an easterly phase velocity only for October 250 N, but the u profile is such that in this case, as in all the others, there is a steering
level in the lower troposphere.
The phase velocity curve for January 450 N
is similar to that for the simple u and Oprofiles constructed from nominal winter parameters.
Geisler and Garcia's (1977) Lindzen-Hong profile
phase velocities are higher (around 14 m/sec for the Eady modes), but the
Lindzen-Hong profile also has higher u values near the ground than does
the January 450 N u profile, so that the steering levels forboth these cases
are at nearly the same height (2 km or 800 mb).
Some properties of the vertical structure of the fastest growing
Eady modes are presented in Table 4.13 for all the cases for which the mode
exists.
The complete
jWI, Phase (p), vO , and eddy energy transfer profiles
are presented in Fiqures 4.22 to 4.29 for 8 of the 13 cases.
The cases of
January 450 N, April 450 N, and April 650 N are similar in vertical structure
0
to that of January 650 N, and the cases of October 450 N and October 65 N are
similar in vertical structure to that of October 500 N.
The streamfunction amplitudes show a peak at the ground and near the
tropopause
for all cases except April and October 250 N.
There is gener-
ally a strong resemblance to the case of simple profiles constructed from
* The tropopause is defined to be the pressure level where the temperature stops
decreasing as one ascends from the ground. It is between 200 and 100 mb
for the profiles of this section and is at lesser pressure for lower
latitudes.
127
nominal winter parameters.
The two exceptions decay rapidly in the upper
troposphere to extremely small values throughout the stratosphere; they also
have the highest planetary wave numbers of all the cases, as one would expect.
The phase profiles show that the pressure wave tilts gradually westward with height except in the upper stratosphere where the westward tilt
is very strong.
Calculations of the phase difference between v' and W'
show that the meridional circulation associated with this wave is strongly
poleward and upward throughout the troposphere and lower stratosphere and
changes over to poleward and downward circulation in the upper stratosphere
where [((
is small.
The meridional entropy transports are poleward throughout the model
atmosphere for nearly all cases.
Weak shallow equatorward fluxes at the
ground for October 250 N and January 650 N are the only exceptions.
The
transports are strong only in the lower troposphere and peaks there or at
the ground.
Mode kinetic energy release is strongest in the mid and/or lower
troposphere (depending on the input a and T profiles) and goes to zero with
w at the ground.
The greatest destruction of mode kinetic energy is in the
lower stratosphere, but the amount of destruction is very small compared
to the amount of kinetic energy generated in the troposphere.
4.8
The Green Modes
The Green mode spectrum joins that of the Eady mode at a cusp point.
The Green modes are present with the Eady modes for all cases except July
250 N.
The fastest growing G modes have wavelengths between 5000 and 10000
km (planetary wavenumbers 2.5 to 5, except for a value of 7 for October 250 N).
.
I_
128
Doubling times for the fastest growing G modes very from 4.6 to 19 days,
0
and average 8 1/2 days; the longest doubling times are for profiles at 65 N.
The case of January 45*N has the fastest growing G mode at pwn = 4.8 with
D.T. = 6.9 days.
This compares very well with Geisler and Garcia's values
of pwn = 5 and D.T. = 7.5 days for the Lindzen-Hong progile.
Calculations
using simple ii and 7 profiles constructed from nominal winter parameters
give pwn = 3.3 and D.T. = 6 days for the fastest growing G mode.
Phase velocities for the G modes are positive except for April and
October at 250 N.
The range of phase velocity values is like that of the
E modes and steering levels are similar to those for the E modes.
The G
0
modes for the case of January 45 N shows the same disagreement with the
phase velocity calculations of Geisler and Garcia as did the E modes; the
reason for this was discussed above.
The phase velocities of the G modes
for January 450 N agree well with those for the case using nominal winter
parameters.
The vertical structures of the fastest growing G modes for 8 of the
13 cases are presented in Figures 4.30 to 4.37.
0
The cases of April at 25 N,
0
0
450 N, 650 N are similar to that of January 25 N and the cases of July 65 N
0
and October 450 N are similar to that of October 25 N, and thus these five
cases are not illustrated; all cases, however, are included in Table 4.13.
The streamfunction amplitudes show strong peaks in the stratosphere
and secondary peaks near the ground.
0
The January and October 65 N fastest
growing G modes have strong amplitudes throughout the stratosphere.
Those
0
of January 450 N and, especially, October 50 N have very narrow peaks near
the top (p = 0).
Stratospheric amplitudes are much-stronger than tropo-
spheric amplitudes for the fastest growing G modes; the opposite is true
129
for the dominant Eadymodes.
Geisler and Garcia's stratospheric amplitudes
for the G mode in some cases are not much stronger than tropospheric values.
They attribute this to a very small difference between the minimum zonal
wind in the stratosphere and the mode phase speed.
has a large value for this difference: 6 m/sec.
The case of January 450 N
However, for the cases of
October 450 N and 500 N the difference is small (K 1 m/sec) yet the fastest
growing G modes for these cases have very strong peaks in the stratosphere.
Results from this model are inconclusive regarding the dependence of stratospheric G mode amplitudes on the difference between the stratospheric zonal
velocity minimum and mode phase speed.
The streamfunction phases indicate a westward tilt with height of the
fastest growing G mode pressure wave throughout the atmosphere for all cases
except for January 650 N and October 250 N, which have a shallow region of
eastward tilt at the ground.
The westward tilt is strongest in the strato-
sphere and mid-troposphere for most cases.
Calculations of the v' and
W'
phase difference show that the meridional circulation of the fastest growing G modes is poleward and upward in the lower troposphere and poleward
and downward in the upper troposphere and stratosphere.
October 650N are exceptions.
In these cases, the circulation in the stra-
tosphere is poleward and upward.
The stratospheric circulation of the
Green modes agrees well with the v''
(1964b).
January 65N and
covariance data presented in Newell
Newell's 18-month data set shows poleward and downward transient
eddy motion at 70 and 45 mb in all seasons and latitudes 200 N to 700 N
except for the case of winter and spring above 550N.
In this case, the
circulation is poleward and upward.
* Molla and Loisel's
(1962) v'w' covariance data (for only 2 months) pre-
9111111111Y
--1
130
The meridional entropy transports are poleward with strong peaks in
0
0
0
the stratosphere for all cases except January 45 N and 65 N and July 45 N.
The pressure wave' for the exceptional cases also hada weakwestward tilt
with height in the stratosphere.
0
January 450 N and October 50 N are the
only cases which have peaks of v'S' near the top, at 20 and 10 mb, respectively.
The G modes in general have much stronger stratospheric v'8' rela-
tive to tropospheric values than do the E modes.
The release of mode kinetic energy is strongest in the lower troposphere with some modes also having strong values in the upper troposphere.
The destruction of mode kinetic energy is very strong in the stratosphere,
and it is much stronger (relative to tropospheric values) for the G modes
in comparison with the E modes.
0
Exceotions are the cases of January 65 N,
October 650 N, and July 450 N; the first 2 of these exceptions actually have a
weak generation of mode kinetic energy in the stratosphere.
4.9
The E s and Sm Modes
The E
s
and S modes always occur together.
m
and are present in all seasons except summer.
They exist only at 650 N
As shown by Table 4.13 and
Figures 4.13, 4.16, and 4.21, all the Es modes are remarkably similar to
each other
as are all the S
m
modes.
The most unstable E
s
and S modes are
m
(cont'd from previous page)
sents a similar picture for the lower stratosphere. However, Wallace (1977)
states that: "Calculations of the temporal correlation between vertical velocity and temperature by Molla and Loisel (1962) ... indicates that mean air
parcel trajectories in the waves slope upward toward the pole". No vertical
velocity-temperature correlations are presented in Molla and Loisel (1962).
Their v 'w covariances do show poleward-upward transient eddy motion in a
0
100-50 mb layer, but only above 55 N for only the 2 months January and April 1958.
131
planetary wavenumbers 13 and 6 1/2 and have doubling times of 3 and 5 days,
respectively.
The phase speed and steering level height of the most unsta-
ble Es mode (2 m/sec, 1.5 km) are half of the respective values
for the Sm
modes.
The vertical structures of the most unstable E
s
and S
m
modes are
presented for January 650 N (Figures 4.38 to 4.41); the structures of the
modes for April and October are very similar.
The streamfunction amplitude
of the Sm mode peaks in the upper troposphere and secondly, near the ground.
The much shorter zonal wavelength Es mode peaks near the ground and is
trapped in the lower troposphere.
The pressure waves of both modes tilt
gradually westward with height except for shallow regions of weak eastward
tilt near 750 mb and the ground.
The meridional entropy transports in
these shallow regions are correspondingly weak and equatorward.
transports are primarily poleward for both modes.
The entropy
They are strong in the
mid and lower troposphere for the Sm mode and strong in the lower troposphere for the E
s
mode.
The meridional circulation for both modes shows a
strong poleward and upward correlation of v' and U' at all levels.
The
generation of eddy kinetic energy is strongest in the mid troposphere for
the S mode and in the lower troposphere for the E mode.
m
s
The destruction
of eddy kinetic energy is very weak and occurs in shallow regions.
The Sm
and especially the Es modes are very short-wave modes and are strongly active
only in the lower or mid troposphere.
They would be of little importance
to the stratospheric general circulation, because of their shallow depth.
4.10
S Modes
The short wave modes have zonal wavelengths less than 3700 km.
They
~ IYYIIIUI
--
132
occur at a wide range of mid latitudes and, except for an Sm mode at 65*N,
they are not present in October.
S modes varies from 4 to 15 days.
The doubling times of the most unstable
The most unstable S mode for two cases,
April and July at 45N was not resolved by the model; they are both shorter
than 1500 km.
These
Modes at pwn = 13 were taken to represent these cases.
two cases also have a cusp point at wavenumber 5.7.
At wavenumbers less
than 5.7 there are weaker longwave modes which form with the S modes, a
distinct unstable solution to the eigenvalue problem at all wavelengths
except that of
the cusp point.
Phase velocities are high (4,-15 m/sec) for the S modes of low latitudes (250 N, 350 N) and are low (Z 3 m/sec) for the S modes of higher latitudes (450 N, 650 N).
The S modes whose fastest growing mode is unresolved
and their longwave counterparts have a remarkably constant phase velocity
for all wavenumbers except very near the cusp point.
is
~-
3 m/sec, and steering levels are at 890 mb.
The phase velocity
The other S modes have
steering levels higher up in the mid troposphere and in the stratosphere.
Figures 4.42 to 4.45 present the vertical structures of 4 of the 6
most unstable S modes.
0
The two S modes for April 25 N are similar in ver-
0
tical structure, as are the S modes for April and July at 45 N.
The streamfunction amplitudes for all the fastest growing S modes
except those which are unresolved are strong only in the upper troposphere
and peak just below the tropopause.
0
The April and July 45 N S modes for
pwn = 13 are trapped in the lower troposphere and peak at the ground.
The streamfunction phases vary very little with height except near
the ground and in the lower stratosphere.
In the latter regions, the wave
0
tilts westward except for July 65 N which has a substantial eastward tilt
133
between 140 and 180 mb.
a westward tilt.
v'I'
Near the ground the April and July 450 N modes have
For all seasons, the wave has a strong poleward-upward
correlation in the troposphere which becomes poleward-downward in the
stratosphere..
The meridional entropy transports are strongly poleward in the midand/or upper troposphere for all cases except April and July at 450 N.
The
only appreciable transports for these two cases are poleward and near the
Only July 650 N has a significant region of equatorward transports
ground.
and these are near the tropopause.
Eddy kinetic energy is generated only in the upper troposphere for
all cases except April and July at 450 N.
These two cases have all their
kinetic energy generated near the ground; kinetic energy for the S modes of
these cases is destroyed between 830 and 900 mb.
The S modes for April
250 N lose their kinetic energy in the upper troposphere, just below the
tropopause and, to a lesser extent, in the mid-troposphere.
January 350 N
and July 650 N have shallow regions in the mid and upper troposphere where
there is a small loss of mode kinetic energy.
4.11
The L Modes
There are no L modes at 250 N for all four months, nor are there any
L modes with doubling times shorter than 25 days for April.
The wavelength
of the fastest growing L modes varies between 3.5 and 13.5 x 10 3 km.
are only 2 L modes of wavelength less than 8 x 10
longest doubling times (13 to 14 days).
There
km, and these have the
The doubling times for the most
unstable modes of wavelength 8 x 103 km or greater varies between 6 and 11
days.
The phase velocity of the L modes varies between 2.5 and 7 m/sec
----
-- --
--IIY IYIIIHY
U
III1
134
except for the case of January 350 N, whose L modes have much faster phase
The steering levels in the troposphere for
velocities of 18 to 20 m/sec.
the L modes ranges from 920 to 390 mb.
January 351N, July 45*N, and July
65N also have steering levels in the stratosphere (see Table 4.13).
The vertical structures of 4 of the 7 fastest growing L modes are presented in Figures 4.46 to 4.49.
The omitted cases:
January 450 N (pwn= 2.3),
July 650 N, and October 650 N are similar in vertical structure to those of
0
January 45*N (pwn = 3.6), January 35 N and January 65*N, respectively.
The streamfunction amplitude of all the fastest growing L modes peaks
at the tropopause or higher.
0
The fastest growing L modes for January45 N,
January 650 N and October 650 N peak just below the top of the model atmosphere;
their structure would be resolved better if there were levels above 1 mb
Only
(47 km), the highest level for which streamfunctions are calculated.
0
the L modes for July 450 N and January 35 N have appreciable streamfunction
amplitudes in the troposphere; these occur in the upper troposphere.
The most unstable L mode's phases show that their pressure waves
have a zero or westward tilt with height throughout the model atmosphere
0
for all cases except January 350 N and July 65 N.
eastward tilt between 350 and 75 mb.
These cases have a strong
For all cases, the v'w' correlation
indicates a poleward upward circulation at lower levels which becomes poleward and downward in the upper troposphere and stratosphere.
The meridional entropy transports for the fastest growing L modes
0
0
are everywhere zero or poleward except for January 35 N and July 65 N.
For
these cases, there are strong equatorward transports in the region of the eastward
tilt
of their pressure wave.
The poleward transports of the fastest growing L modes
135
all peak in the stratosphere, only July 450 N has a secondary peak at the
ground.
The most unstable L modes whose streamfunction amplitude peaks
near the top also have their meridional entropy flux
and eddy energy
transfer peaks at or near the top.
The fastest growing L.modes for July and January at 45 0 N have most
of their kinetic energy generated in the troposphere between 850 and 250
mb.
These modes also have a substantial amount of their kinetic energy
destroyed in the stratosphere.
The remaining four fastest growing L modes
have most of their kinetic energy generation in the stratosphere and lose
their kinetic energy in the lower stratosphere or upper troposphere.
These
four modes are examples of in situ baroclinic instability of the stratosphere.
Two of these modes (those for January and October at 650 N) have
their peak v'&' and
JIJ
values at or near the top.
An accurate represen-
tation of their structure above 10 mb would require more levels in that
region.
The other two L modes (January 350 N and July 650 N) are well re-
solved.
Their significance is discussed in the next two sections.
4.12 The Necessary Conditions for Instability and the Existence of Unstable
Modes
In the above discussion, it has been shown that there are long and
short wave modes (the L, S, and S modes) associated with the u and T prom
files obtained from monthly averaged data.
Figures 4.9 to 4.21 show that
these modes exist at the same wavenumbers as the Green and Eady modes.
In
one case (April 250 N) 2 S modes exist at the same wavenumbers as an E mode.
These long and short wave modes were not present for the parametric u and T profiles of the previous chapter.
For 3 u and T profile
pairs of this chapter,
~~ ~
--
~
~
~
mimia~-~~YIYYIY
IIII~Y
136
the Eady modes had a strong shortwave local maximum at approximately waveWhy do
number 13; this also did not occur for the parametric profiles.
these new long and shortwave modes exist?
Why did the Eady modes' growthThe answers
rates in some cases acquire a local maximum near wavenumber 13?
to these questions come from an examination of the relevant quasi-geostrophic potential vorticity profiles.
The Eady and Green modes exist for the
chapter if
i and T profiles of this
1) there is a temperature gradient at the lower boundary, or
2) there is a zero in the qy profile.
The case of January 250 N had a sur-
face temperature gradient and E and G modes, but no qy zeros.
The case of
April 450 N was altered without affecting the qy zeros so that it had no
surface temperature gradient; its E and G modes were only slightly affected.
The case of July 250 N shows that indeed no modes exist if the necessary conditions for instability are not met.
Removal of internal qy zeros or changing the surface temperature
gradient does substantially alter the gorwth rate spectra of the G and shortwave E modes.
Doubling the surface temperature gradient for January 250N
decreased the fastest G mode doubling time from 5.3 to 3.3 days.
of the 600-730 mb negative q
Removal
region for the cases of January, April, and
October 650 N resulted in removing the local maximum of E mode growth rates
near wavenumber 13.
(By definition, this means removing the Es modes).
The simple parametric u and Tprofiles do not have this negative
q
region.
Removal of the 730-800 mb negative qy region for the cases of April and
July at 450 N increased the fastest growing G mode doubling time from about
8 1/2 to 17 days and .increased the doubling times of E modes between wavenumbers 11 and 15 from about 2 1/2 to 10 days.
The removal of the negative
137
y region for the cases at 650 N required increasing u by .3 m/sec at 700 mb
0
and by .1 m/sec at 500 mb; for the cases at 45 N, the removal was accom-
plished merely by increasing U at 850 mb by .3 m/sec.
These and other cal-
culations show that a very small change in the U profile can result in a
drastic change in the zeros of the
(
The curvature of the u profile, u
, is what has an important role in de-
PP
termining y.
This need not be the case.
profile.
y
Substantial changes in the _ profile could be made without
significantly affecting the zeros of q
if Upp is not appreciably affected.
The above calculations have shown that small changes in the
result in the removal of q
spectra.
E profile can
zeros which drastically affects the growth rate
However, the cases of January, April and October at 650 N have
widely differing u profiles, but very similar q
y
at 600, 730, and 880 mb).
profiles (all have zeros
Their growth rate spectra (and even the phase
velocities and mode structure) are remarkably similar.
Thus, a substantial
change in the U profile need not have an important effect on the growth rate
spectra unless is it accompanied by a substantial change in the zeros of
the
y profile.
Calculations of Gall and Blakeslee (1977) and Staley and
Gall (1977) have also shown that the growth rate spectrum at short wavelengths is very sensitive to low level change in the G and static stability
profiles.
They claimed that this was because the short waves "felt" only
the part of the 5 profile below 500 mb.
This explanation could not apply
to the longwave Green modes of this study, which were affected by low level
changes in u.
Gall and Blakeslee (1977) and Staley and Gall (1977) did not
discuss the effects of their low level changes in u and O on qy.
The S modes, like the E modes, do not exist if the 600-730 mb negas
m
tive qy region is removed.
This is more than just removing a local maximum
138
in a growth rate curve.
An entire distinct conjugate solution to the eigen-
value problem exists only when a specific negative qy region (with its associated zeros) also exists.
The profile of qy requires negative values for
it to allow, in conjunction with the surface temperature gradient, the
bracketed term in Eq. 1 of Ch. III to be zero so that C. need not be zero for
1
the S
m
of '
modes.
The E and G modes in these cases do not need negative values
to meet the necessary conditions for instability.
The L modes (except for January 450 N) and S modes exist only when
certain negative qy regions are present; these regions are specified in
Table 4.13.
These modes
(with the exclusion of January 450 N) are part of
a conjugate solution to the eigenvalue problem which is distinct from that
containing the Green and Eady modes.
The L modes for January 350 N and July 650 N
are the only modes associated with negative q
modes are
regions in the stratosphere. These
an example of in situ stratospheric . baroclinic instability.
They
have downgradient meridional entropy transports and have their kinetic energy generated at the levels of their associated negative qy regions.
The S , S, and L modes were not present when the simple parametric i
m
and T profiles were used.
These modes are seen to exist for the more com-
plicated u and T profiles of this chapter because these profiles have
zeros not present in the simple parametric profiles.
modes which exist only when a specific negative q
4
The S , S, L, and E
m
s
regions exists always
have downgradient meridional entropy fluxes at the levels of that region.
In about half the cases, the fluxes peak at one of the q
the negative
y region (See Table 4.13).
zeros bounding
Modes which are associated with
the surface temperature gradient have downgradient fluxes at the surface.
Thus all modes have downgradient meridional entropy fluxes in the region of
139
their source of instability, and their maximum entropy transport is generally
above or at the top level of their destabilizing negative qy regions.
Several cases indicate that the amount of negative
is important in
determining the strength of the growth rate of the associated unstable mode or in
determining whether it exists at all.
October 50 0 N, July 250N and January 350 N
each have a negative qyvalue at only one level, and none have an unstable mode
associated with this negative qy region (to within model resolution).
and October 450 N each have negative q
April 250 N
regions 2 levels deep and do have associated
unstable modes which have small growth rates and/or have small extent in wavenumber space.
July 450 N, April 450 N, January 350 N, January 65 0 N, April 650N,
and October 650 N all have much larger negative q regions and all have associated
unstable modes of much larger growthrates.
4.13 Importance of the Unstable Modes to the General Circulation of Specific Regions
Foz modes of different wavelengths, one cannot conclude that the modes
with larger growth rates will dominate the general circulation; their growth may
cease sooner than those which have slower growth rates. This was shown by Gall
(1976) and
Staley and Gall (1977) for the case of long vs. short waves. They found
that quasi-geostrophic unstable modes of wavelength < 3000 km were trapped below
500 mb whereas modes of wavelength > 4000 km extended through the depth of the troposphere. They also showed that modes which were trapped in the lower troposphere
(the shortwave modes) stopped growing sooner than modes which extended to upper
levels.
Therefore, the longwave G and L modes of this model, although having sig-
nificantly smaller growthrates than the E and Es modes, may be very important to
the general circulation in the upper troposphere and stratosphere.
All the fastest growing G modes
(those discussed in this chapter,
and those associated with parametric u and T profiles of Chapter III) have
_~
~~
~CL"L~L
^^
~ -
-
IIIIIIIIYYYLIYY
I1I~I
i
140
their strongest streamfunction amplitudes inthe stratosphere and/or the
upper troposphere just below the tropopause. The peak value of the streamfunction amplitude always occurs near the tropopause.
The meridional entropy
transports of the fastest growing G modes in all but four cases (see Table 4.13)
are strongest in the stratosphere with most cases having their strongest
transports in the lower stratosphere.
The transports are poleward and may
be the cause of the observed secondary maximum of poleward transports
in the lower stratosphere (Oort and Rasmusson, 1971; Newell et al., 1974).
All but 3 of the fastest growing G modes have most of their kinetic energy
depleted
in the stratosphere.
The Green modes, then, are potentially very
important to the general circulation of the lower stratosphere and may also
be important for the general circulation of the upper stratosphere and the
region just below the tropopause.
The L modes, in addition to the G modes, have strong () .5 of their
peak value) amplitudes, meridional entropy transports, and kinetic energy
losses in the stratosphere.
Four of the fastest growing L modes may be
important to the general circulation above the lower stratosphere.
However,
these modes have large amplitudes and entropy fluxes very near or at the top of
the model atmosphere and may require more levels in this region for an accurate determination of their vertical structure.
The fastest growing L mode
for July 450 N is the only L mode which has strong poleward entropy fluxes
in the lower stratosphere.
The L modes for January 350 N and July 650 N are
well resolved and are examples of stratospheric in situ baroclinic instability.
They both have strong equatorward entropy transport in the lower
stratosphere.
Observations of Oort and Rasmusson (1971) and Newell et al.
(1974) disagree as to the direction of the transport in the lower stratosphere
141
0
for January 35 N, but they agree that the transports are weak.
650 N, they'agree that the transports are weak and poleward.
For July
The G modes,
which have growth rates comparable to those of the L modes in these cases,
show weak poleward fluxes for January 350 N and strong poleward fluxes for
July 65 0 N.
Thus, the G modes and L modes arising from in situ baroclinic
instability may both be important
to
the general circulation of the lower
stratosphere for January 350 N and July 650 N.
Most S and Sm modes have strong streamfunction amplitudes throughout
the troposphere.
Three cases, the S modes for April 250 N and July 650 N have strong
fluxes in the upper troposphere; all fluxes are poleward except for July 650 N
which has a region of equatorward fluxes between 160 and 180 mb. All the other S
and S modes have poleward v's' values which are strongest in the mid or lower
m
troposphere.
The S and S modes could be important for the general circulation
m
of the upper troposphere.
In the mid and lower troposphere they are likely
to be dominated by the E and Es modes which have strong amplitude and entropy
transports in these regions and which, in all cases except January 350 N and
January 650 N, have much larger growth rates at the same wavenumbers.
The Eady modes, like the S and Sm modes, have strong streamfunction
amplitudes throughout the troposphere.
In some cases the E modes have
strong amplitudes for about 20 mb above the tropopause.
The meridional en-
tropy transports of the E modes is strong only in the lower troposphere as
is the generation of their kinetic energy (there are no regions where E
mode kinetic energy loss is large relative to the amount generated).
The
above description of the Eady modes associated with the u and T profiles
of this chapter also applies to the Eady modes associated with u and T profiles generated from the parameter values:
yT = 2, SR = 0, 1, or -1.5,
142
and tr
= 50.
Thus, the general circulation of the troposphere, especially
the lower troposphere, is expected to be very strongly affected by the Eady
modes.
The Es modes and E modes of similar wavelength have strong amplitudes, v'S' values, and eddy.energy transfer values only in the lower troposphere.
These modes are therefore likely to be important to the general
circulation of only the lower troposphere, with E modes being dominant.
The most unstable G modes have been shown to be potentially important to the general circulation of the lower stratosphere, in contrast to
the most unstable E modes which have very weak amplitudes and fluxes in the
stratosphere (relative to those in the troposphere).
The longest E modes
have wavelengths comparable to those of the Green modes.
The question
arises as to whether they share some of the G mode properties and also are
potentially important to the stratospheric general circulation.
Calcula-
tions for 10 of the basic states of this chapter show that the long wave
E modes, which are
within half a planetary wave number of the cusp point
dividing them from the Green modes, do share many of the properties of the
Green modes.
In particular, they have much stronger stratospheric ampli-
tudes and fluxes than those of the most unstable Eady mode.
at a maximum usually at the tropopause.
Amplitudes are
The v'8' values are greater than
1/2'the nominal value generally between 90 and 40 mb and sometimes between
the tropopause and 40 mb.
poleward and downward.
The circulation-of these longwave E modes is
Nearly all of their kinetic energy is generated in
the troposphere, and they have a much greater dstruction of kinetic energy
in the stratosphere than does the most unstable Eady mode.
Thus, in addi-
tion to the G (and possibly L) modes, the longest wave E modes may be im-
143
portant to the general circulation in the stratosphere.
4.14
Calculations for W' = 0 at p = 0
Changing the upper boundary condition in the parametric study of
Chapter III resulted in large changes, for some parameter combinations, in
the Eady and especially in the Green modes' growth rates, phase velocity,
and vertical structure.
stratosphere).
The changes were drastic when
Oj
equaled 1 (no
In addition, new modes appeared which were associated
with a temperature gradient at the upper boundary when t' = 0 was the upper
boundary condition.
The calculations of this chapter are virtually unaffected by a change
in the upper boundary condition from Y' = 0 to
static stability,
0
= 0.
0'
This is because the
, becomes very large near p = 0 and would damp out the
effects of the upper boundary condition.
It would also not allow a temper-
ature gradient at the upper boundary to be of any importance in affecting
the necessary conditions for instability.
From Eq. 1 ofCh. III one sees that
large 0 at the top allows one to neglect the integral whose integrand is
evaluated at the upper boundary.
bility for
Thus, the necessary conditions for insta-
' = 0 at p = 0 become, for very large
-at p = 0, nearly iden-
Thus, modes which are present for Y'
tical to those for Y' = 0 at p = 0.
at p = 0 will also be present when N' = 0 at p = 0 and vice versa.
Since O at p = 0 is large but finite, there is a small difference
between calculations assuming different upper boundary conditions.
When w'
= 0 at p = 0 there are for January 650 N and October 650 N additional longwave modes of planetary wavenumber
( 1.5 and doubling time > 2 weeks.
144
These modes exist only when the temperature gradient at the upper boundary
is non-zero and only when
= 0
0'is the upper boundary condition.
kinetic energy is generated mostly in the stratosphere.
Their
145
1
I
I
.'1
/
• *
."/
/
,
oIl
*
250 N
E
IO
U)r*.
/
/ '35 0 N
/
/45ON
650N
./
10 ~-
w
\
-\
/1/
*..'/
JANUARY
/
..
JANUARY
100-
.000"
1000
0
Figure 4.1.
I
20
I
I
I
I
40
60
80
MEAN ZONAL VELOCITY (m/sec)
Zonal velocity profiles for the basic state for January.
100
146
*
I
AP
\
100
-1000
0
-
,
.
0
0/C,
20
\45N
MEAN
20
VELOCITY
5ZONAL40
40
MEAN ZONAL VELOCITY
Figure 4.2.
As in Figure 4.1, for April.
60
( m/sec)
80
60
80
(m/sec)
147
/o
*
00C
I
\
JULY
1000
cre
-60
Figure 4.3.
•
°
\
-40
-20
MEAN ZONAL VELOCITY
As in Figure 4.1, for July .
0
(m/sec)
20
148
E.0e
S
.
,"4
0//
E
//
•
IGOl*
I
250N
I
//
45N
65
MI0
..
D
OCTOBER
N
.
I000
0
Figure 4.4.
20
40
MEAN ZONAL VELOCITY
As in Figure 4.1, for October.
60
(m/sec)
80
149
JANUARY
*0\\0
I\
,/
65N/
1
0
/ /*
/
1000."
*
/
.26
1 .*
100
35N
*
0/"
/450N
0
I
1000
200
*
220
240
260
TEMPERATURE
Figure 4.5.
280
300
(oK)
Temperature profiles for the basic state,
for January.
150
_.
-\
APR IL
S/*
/*C
2
A
-
C//*
U)J
I
100
1000 -
I000200
Figure 4.6.
". i
I II
220
I
240
260
TEMPERATURE (oK)
As in Figure 4.5, for April.
280
300
151
I'.
I
I'
..
I
I
I
I
i
JULY
o
*
\
**\
.
c9
E
*
0n
/
, *o
0*_/
.*
100 -. '25 0 N
1
65 0N
450N
S.
1000"
200
.
220
240
TEMPERATURE
Figure 4.7.
--
As in Figure 4.5, for July.
260
(oK)
280
300
152
-o
100
1000
200
220
240
260
TEMPERATURE
Figure 4.8.
As in Figure 4.5, for October.
(0 K)
280
300
153
18
.8
15
.6
12 o
T
4-
9
1
ZU)
0
-
o
0
o
I
3
1
6
PLANETARY
I
I
9
WAVE
12
0
15
NUMBER
Figure 4.9. Doubling time (thick lines) and phase velocity (thin lines)
vs. wavenumber of unstable modes for January, 250 N. Solid curves represent
the Green and Eady modes.
154
23
I
.8
-20
6
17;
U,
-14
4
/
2-
L.
I
/"
0
0
3
6
PLANETARY
-
/0
I
I
9
12
WAVE NUMBER
PLANETARY WAVE NUMBER
Figure 4.10.
As in Figure 4.9, for January, 350 N.
5
15
.
8
155
18
I
15
6
12 -
o
.2-
00
9
0 .0
-Ju.
0
3
6
PLANETARY
Figure 4.11.
9
WAVE NUMBER
As in Figure 4.9, for January, 45 N.
12
15
I-
Illilaillil
156
I
.8
1
18
15
6
12
-i
-
E
oo
0
uJ
'I\-
m
-
z
cn
o
>
()
6 I
D
d
I
o0
0
Figure 4.12.
3
6
PLANETARY
\
9
WAVE
12
NUMBER
As in Figure 4.9, for January, 65'N.
15
157
.8
I
21
- 18
.6
-
-
15
u,
-12
o
4-F
0
0
-j
4-
9
-
1233
0
Z
6
2-
S /I
0
3
6
9
PLANETARY WAVE NUMBER
Figure 4.13.
As in Figure 4.9, for April, 250 N.
O00
12
15
U6
158
18
-15
.6
12 o
u,
H
-
00
9-
Li
- 6
0
0
Figure 4.14.
l
I
3
12
9
6
PLANETARY WAVE NUMBER
As in Figure 4.9, for April 45*N.
15
159
18
12
(n
0
E
Li,
>u
o
o
0
0
9
H
o-
3
Figure 4.15.
6
PLANETARY
9
WAVE NUMBER
As in Figure 4.9, for April 650 N.
12
-j
w
160
18
.8
15
6
E
-----
o
-----
O
t OO
PLANETARY
z
WAVE
NUMBER
c
-a
m
9
0
12
159
a
.-
r-a
0
Figure 4.16.
3
6
PLANETARY
9
WAVE NUMBER
As in Figure 4.9, for July, 450 N.
12
15
J
161
_1
I-
0
z
D
0
0
o(1
-16
3
6,.
I
3
I
6
PLANETARY
Figure 4.17.
I
9
.
2
1
. I
12
I
WAVE NUMBER
As in Figure 4.9, for July 650 N.
162
II3
8
0O
.6
-r
E
U,
0
T
5
<
0
.
2- -12
-15
0
3
6
9
PLANETARY
Figure 4.18.
12
WAVE NUMBER
As in Figure 4.9, for October 250 N.
15
18
163
.8
18
15
6-
12 j
7
E
0
S90
i-
2 -
I
6
.2-
0
o.
0
Figure 4.19.
3
13
I
6
PLANETARY
9
WAVE NUMBER
As in Figure 4.9, for October 450 N.
12
15
Y
Y
YIIIIY
YIIIIIYII
lii~
'~^IIYYYI
-
164
18
.8
15
6
-o
0
9
Li
J0
S-i
z ----
I
I
SI
- -
0
m.
0
6
.2 -
0
0
Figure 4.20.
3-
I
3
6
PLANETARY
I
9
WAVE NUMBER
As in Figure 4.9, for October 500 N.
I
0
12
15
165
18
15
12 Z
E
>-
7o
o
90
Uj
Ho
I
6
3
0
0
Figure 4.21.
3
6
PLANETARY
9
WAVE
NUMBER
As in Figure 4.9, for October 650 N.
_________________________________________
IIYI
IImuYi lYYi
IIIYIII.,
166
400-
uJ
I*I
800-
1000I
0
.2
.4
II
.6
.8
1.0
Figure 4.22. Vertical structure of the streamfunction amplitude for the
fastest growing Eady mode. Cases: January 25,N (dot), January 350 N (dash
dot), January 650 N (solid), and April 25N (dash).
167
''N
0
2
4
.6
.8
Figure 4.23. As in Figure 4.22, for cases: July 450 N (dot), July 65 0 N
(dash dot), October 25 0 N (dash), and October 500 N (solid).
----
~
wlli
168
N
200
'N
I'
400
*
I
I
I
I I
I I
I
I
*
I\
I
II
II
600
800
Irnrn
2
4
PHASE I/i} (radians)
2rr
Figure 4.24. Vertical structure of the streamfunction phase for the fastest growing Eady mode. Cases are those in Figure 4.22.
169
0\1
"
I
II
600
400 -
1000I
U
aI
0
,
I
2
I
4
2
IYIIYYYII
I
170
200
I
I
I
I
400r
i
600-
'.I
800-
1000-
-1.0
-.8
-. 6
-. 4
-. 2
0
.2
.4
.6
.8
1.0
Figure 4.26. Vertical structure of the meridional entropy transport for
the fastest growing Eady mode. Cases are those in Figure 4.22.
171
0
*I
200
V.
1**
I.
400
.0
E
w
n
09
w
600
800
"
IY
1000 L
-1.0 -. 8
-.6
-.4
-.2
0
.2
.4
.6
.8
1.0
Figure 4.27. As in Figure 4.26, except for the cases July 450 N (dot), July
65N (dash dot), October 250 N (dash), and October 500 N (solid).
milli
172
0
a
K*
200
*g**
400
J
r
r
600
r
r
r
I
800
1000
-LO1
-.
8
-.
6
-.
4
-.2
0
.2
.4
.6
.8
1.0
CE (KE-,-PE
Figure 4.28. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE - PE is positive) for the fastest growing
Eady mode. Cases are those in Figure 4.22.
173
0-
200-
I
400
E
Cr
600
800
.-
1000
-1.0 '-.8
-.6
-.4
-2
0
.2
.4
.6
.8
1.0
CE (KE- PE)
Figure 4.29. As in Figure 4.28, except for the cases July 450 N (dot), July
650 N (dash dot), October 250 N (dash), and October 500 N (solid).
...
-U
h
h
h
W11
jINIh
174
""
I
-
400
/
y/
/.
600
00
1000
.2
.4
.6
.8
I1
Figure 4.30. Vertical structure of the streamfunction amplitude of the
0
0
fastest growing Green mode. Cases: January 25 N (solid), January 35 N
0
0
(dash dot), January 45 N (dot), and January 65 N (dash).
175
0
.4
200
I...
1**
600
800
r\
.2
.4
'4*'
Figure 4.31. As in Figure 4.30, except for the cases: July 450 N (dot),
October 250 N (solid), October 500 N (dash dot), and October 650 N (dash).
II
~"
I
III~I'll,flINI
bI
176
0
200
600
PHASE {4i}
4
(radions)
Figure 4.32. Vertical structure of the streamfunction phase of the fastest
growing Green mode. Cases are those in Figure 4.30.
177
0--
.a
200400-
U,
w
I
Cr
a-
600 -
"
\
800 -
1000
I
0
2
I
4
I
27r
PHASE {4,} (radians)
Figure 4.33. As in Figure 4.32, except for the cases July 450 N (dot),
October 250 N (solid), October 500 N (dash dot), and October 650 N (dash).
11~111~ ~~_~_
Y
178
"""""""'
""'
I\
i "
200 -
400-
600 -
800 -
1000
-1.0
I
I
I
I
-.8
-.
6
-.
4
-.2
I
I
I
,
I
I
I
I
.2
.4
.6
.8
I
0
I
I
I
I
1.0
Viet
Figure 4.34. Vertical structure of the meridional entropy transport of
the fastest growing Green mode. Cases are those in Figure 4.30.
179
'Z
200
/I
I
I
I
I
I
I
I
400I
I
I
I
I
I
I
J
I
J
I
1
600
r
I
I
I
r
I
r
I
800 -
...
r
r
1000
-1.0
-.8
-.6
-.4
-. 2
0
.2
.4
.6
.8
1.0
v'8'
Figure 4.35. As in Figure 4.34, except for the cases July 450 N (dot),
October 250 N (solid), October 500 N (dash dot), and October 650 N (dash).
-101IN11141
180
I
/
II
200
E
(n
U)
II
0
"
\
.
I
CL 600
•
/
800
1000
-1.0
I
,
-.8
I
-.6
-
I
-.4
I
-.2
I
.2
0
CE (KE--P
E
I
.4
I
.6
I
.8
1.0
)
Figure 4.36. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE-* PE is positive) for the fastest growing
Green mode. Cases are those in Figure 4.30.
181
00
I
-f
.
200
I
400 E 800o
\,
w
D
600
CE
(KE
E
Figure 4.37. As in Figure 4.36, except for the cases July 450 N
October 250 N (solid), October 500 N (dash dot), and October 650 N
(dot),
(dash).
-
-- 11hi
182
E
w
Cr,
D
Cn
a
600 -
800 -
I
100
0
I
.2
I
I
.4
.6
.8
1.0
Figure 4.38. Vertical structure of the streamfunction amplitude of the
fastest growing E s and Sm modes. Cases: E s, January 650 N (dash), and
Sm, January 650 N (solid).
183
200 -
4001-
600 I
I
I
r
800
I
r
1000
n
I
I
h,
I
,|
I I
I
I
2
4
PHASE {\} (rodians)
Figure 4.39. Vertical structure of the streamfunction phase of the fastest
growing Es and Sm
m modes. Cases are those in Figure 4.38.
_
._._I
_____
184
0-
200-
E
.i
400
600-
800
1000
-1.0
000
-. 8II
-.6
-.
-.2
0
.2
.4
.6
.8
1.0
v'8
Figure 4.40. Vertical structure of the meridional entropy transport of the
fastest growing E s and Sm
m"modes. Cases are those in Figure 4.38."
185
0
200
400
600
800
I000
-1.0
-.8
I
-.6
I
-.4
-. 2
I
0
CE ( KE
I
.2
.4
.6
.8
1.0
PE )
Figure 4.41. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE-3 PE is positive) for the fastest growing
ES and Sm modes. Cases are those in Figure 4.38.
111'
186
200 -
I
-
0
400 -"
E
I
WI
I0
" "
"
I
I
/
/
. .
.
_.i..'
"
.
10001
0
.2
.4
.6
.8
1.0
Figure 4.42. Vertical structure of the streamfunction amplitude of the
fastest growing S modes. Cases:
January 350 N (dot), April 250 N (solid),
0
0
July 45 N (dash dot), and July 65 N (dash).
187
0-
200 --
1 II
400 E
D
n
Cln
U)
I
:
600
800 -
1000
I
O
2
4
27r
PHASE {4} (radians)
Figure 4.43. Vertical structure of the streamfunction phase of the fastest
growing S modes. Cases are those in Figure 4.42.
Y1
-
188
0-
-
200 -
.
400E0
fastest
growing
S
modes.
Casse
are
-4
-4
-. 2
v, e,
those
in
Figure
4.42.
600
600-
800
800-
-1.0
-1.0
-_8
-8
-.
-. 66
-.2
0
0
.2
.2
.4
.4
.6
.6
.8
.8
1.0
1.0
VI 8'
Figure 4.44. Vertical structure of the meridional entropy transport of the
fastest growing S modes. Casse are those in Figure 4.42.
189
0
p
200O
'a-
400
-3
600
800
1000
-
-1.0
-8
-. 6
-4
-. 2
0
CE (KE--
.2
.4
.6
.8
1.0
PE
Figure 4.45. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE--PE is positive) for the fastest growing
S modes. Cases are those in Figure
-ImII
190
/
*.
..
.
I
I
200
...
.
-
400
\
(I)
S."
a.
600 -
0
.2
.4
.6
.8
1.0
Figure 4.46. Vertical structure of the streamfunction amplitude of the
fastest growing L modes. Cases are: January 350 N (dash dot), January 450 N
(dash), January 65 ON (solid), and July 450 N (dot).
191
..-----,.
-
_-
I
I
200 *
I
II
400-
\
L
Icr
"
I
600-
.
800 -
I
i
I
I
1000
0
II
I
2
4
PHASE {4'} (radians)
I
2rr
Figure 4.47. Vertical structure of the streamfunction phase of the fastest
growing L modes. Cases are those in Figure 4.46.
- In
--
-
Nl,,
192
..
(U
200
- 400-
600
800-
1000 -1.0 -.8
-.
6
-.
4
-.2
0
.2
.4
.6
.8
1.0
Figure 4.48. Vertical structure of the meridional entropy transport of the
fastest growing L modes. Cases are those in Figure 4.46.
193
0
200
400
600
800
1000 L
-1.0
-.6
-.4
-.2
.2
0
CE (KE---
.4
.6
.8
1.0
PE)
Figure 4.49. Vertical structure of the transfer of eddy kinetic to eddy
available potential energy (KE -> PE is positive) for the fastest growing
L modes. Cases are those in Figure 4.4E
QUASIGEOSTROPHIC P)TENTIAL VORTICITY MIRIDIONAL GRADIENT (UJ)
TABLE 4.1.
LEVELS 7-29,
LEVELS 13-41. 1?
SOURCES: LEVELS 25-97, #1
JANUARY*
SEE SECTION IV-5 FOR A ODSCRIPTIQN OF THE DATA SOURCES.
LATITUDE()=
25
50
35
-531.
12677.
5818.
10913.
-1141.
4385.
3258.
4426.
-4111.
40
45
50
FOR
DATA (10I/(MSEC))
LEVELS 3-71 44o
03,
5b
60
65
1064.
-5937.
-8508.
14153.
2782,
2740.
5772.
5447.
14149.
LEVEL
1933.
1678.
1369.
1161.
990.
882.
816.
706.
445.
31.
915.
3913.
7893.
P487*
6809.
1318.
1101.
729.
511.
369.
257.
140.
30.
-86.
-41.
1316.
3925.
6898.
7426.
5999*
4118.
4681.
3630.
3836.
4038.
2030.
-105.
-591.
-1049.
-1406.
-1558.
-1394.
933.
705.
254.
37.
-27.
-25.
-17.
80.
304.
766.
1439.
2356.
3277.
3376.
2658,
1221.
892.
317.
79.
95.
208.
312.
542.
937.
1507.
1568.
1436.
1164.
945.
b47.
1831.
1539.
992.
825.
954.
1109.
1128.
1216.
1408.
1690.
1386.
834.
134.
-118.
138.
n
*Level n is related to pressure level p by:
5636.
5643.
4173.
37184.
3382.
2957.
2508.
2049.
1623.
1313.
1267.
1732.
4387.
4264.
3098.
2576,
1989.
1308.
524.
-355.
-1272.
-2088.
-2550.
-2369.
1986.
1823.
1589.
1217.
810.
368.
713.
716.
1267.
1324.
1354.
1365.
1359.
1341.
1320,
1303.
1303.
1342.
1195.
1134.
1615.
1778.
1935.
2084.
2224.
2349.
2455.
2537.
2589.
2609.
2330.
2126.
1894.
1611.
1298.
1138,
1152.
1149.
981.
479.
323.
2077.
5423.
5688.
4306.
8944.
7507.
9718.
?449.
2997.
2758.
2326.
2165.
2178.
2155.
1629.
1472.
1593.
1727.
1724.
1772,
1894.
2046.
1577.
2212.
21P3.
2210.
2305.
2407.
1873.
1053.
827.
-94.
-181.
327.
-417.
-38.
3309.
3173.
2909.
2785.
2699.
2593.
2448.
2320*
2224.
213b.
1790.
1344.
874.
807.
1189.
/2
where N=98.
,a
40
C2
C.4
-C
VS
N0 a0%inC4y
Ucm 81NonO..4VCO
f
In4-Q aC*
.a v1.In.4
0
*
N'
r
.
a
-4
n
lcma'
1'4-,
0
woo
4
On
a .4
p
.
C
IV CYn 41
*I'
a, -r fn~on r.
c W N . Go
-rc
1 N
.4.4
.4NV)~
-.4~if
o -a
=- s P- wl
W
.
~vacur
W .4 cc
.
--
-A &on
I
4045-.41mWe't44n
p n .ccW
v VS
N C V
,
)C
N N
- 4
.
L 4
~ ~
N
VS .
-V
?1- a, .- fn
c*~
-
0S4 'nN
f--
-44aW4aN
'~
z
n-n
I or-
I.'
~
)
%W- M4P-4r
a'
C%0
-.
0r 4n W, t- W
.444,..4.*5
'cd44a
I.-
v
cc~
a, a'-P.'W
Y
1-. -
Wpof C
o%
P.
~
~
0 Fn4rN, 0
N I- 'a a W
~
ao-
W W
a V
on0 N c a f CC
r ,NC
7
4
~
4n
Sa %C4
4, 0 4P
NI
4 .4r4Zc,4'VSS0
N
C2
4 NO Of" P P-4
04VS
~~
~
5P.-
a
a0%
a a'
10
n
0 P- '1
N-4 'D -N
N44aW4-4.4r4
N
4
N
NW)
~.4f-m
~~~~~~~~
r~
~
-4~ "~ r
4 V
V
4 04 f- 4n
1.~
P. f. %&W) # CY .0 0' NO%a a, n
f
V4
.4~~~5-
PVS a'
n
N a,
4 0
.4NS4N'4.a'0
195
TAdLE
0
0
0
4.2.
AS
IN 4.1
APHIL.
FOR
40
35
30
LATITU'r ):
45
65
60
55
50
L EVEL
569.
779.
1559.
3044
1
1395.
14"p.
31
3684.
4473.
2279.
b42.
5701.
21 1.
1977.
168 .*
1476.
242F.
2473.
2516.
2551.
256P.
2556.
2504.
18229
2172.
1157.
*LT
-4
312.
215.
50.
-187*
-494.
-863.
-1269.
-1663.
-1966.
-2122.
271.
1741.
?4'5.
2282.
2024.
1715.
1371.
114".
10P4.
1046.
967.
813.
1083.
24~4.
491P.
5879.
4990.
2411*
2104.
171P.
1 3 5.
042?
6P1.
5'(0.
2215.
1908.
1447.
1029.
63q,
400.
324
4.
346.
2174.
1716.
1186.
719.
242*
220.
321.
"77.
723.
14(107
507!.
5746.
4(4',.
1372.
2171.
3018 4.
3242.
2697.
1307.
1504.
167.
15h?.
1321.
1 08.
1392.
770.
478.
4 A8.
4 74.
431.
460.
641.
933.
1380.
1849.
1570.
1370.
1130;
472.
773.
549.
674.
790.
717.
630.
568.
579.
747.
1041.
13i'1.
1394.
11 54.
(7
-2395.
-663.
648.
275.
26'.
-267.
-792.
-1409.
-2122.
-2'.01.
-3650.
-4173.
-"93.
-1377.
-1s88.
2.
5542.
4626.
691.
185.
e92.
501
37.
-4R2.
1300.
1150.
1031.
1343.
4550.
1503.
1299.
Q09.
585.
1 07.
1 t29.
1659.
1462.
1209.
2659.
2564.
2064.
2176s
2334.
2357.
2463.
1590.
2464.
.
876.
906.
075o
1100.
1057.
7?24.
567.
479.
S24.
658.
666.
786.
922 .
1059.
12071.
1448.
1756.
155?.
106.
361.
163.
639.
668.
717*
762.
820.
940.
1173.
1502.
1949.
2447.
1908.
A61.
-628.
-1325.
- 32.
'0
Ob
TABLE 4.?
CONTINUfr).
LATITUEEPE):
25
30
35
40
45
so50
55
60
65
877.
739.
597.
462.
449.
4923
592.
75?.
600.
635.
679.
725.
754.
782.
817.
8620
728.
791.
R77o
978.
920.
993.
113t.
1387.
17q3.
2513o
3975.
5925.
7717.
8433o.
6117.
371P.
36190
3547.
345.
281b.
2098'
13917
731.
17
-4750
1091.
4694.
8506.
12805.
93b0.
-113,9.
-29604.
-12071.
620.
736.
851.
llP6,
173.
24553
3880.
5b46.
7U90.
331.
8214.
7317.
6622.
5173.
3385.
2115.
1165.
241.
-654.
-b15130
-2351.
-687.
3329.
6q549
9970.
v201e
-10305.
-16781o
-(206.
LEVEL
25
27
29
31
33
3
37
169.
1773.
1802.
1788o
2041.
2501.
3221.
4377.
5204.
8472.
10664.
11229.
109610
9546.
6789.
4652.
30p0.
1~2?
21.
306.
1413.
2497.
3510.
354?.
3298.
305.3
2798.
2560.
2330.
2592.
3324.
4034.
4758.
28 3.
-4344.
1279.
2844.
6218.
918F.
11 8a.
13293.
1361o
11762.
6782.
3596.
2522.
16481.
1147.
1177.
1470.
1811.
2166.
242f.
2662.
2884.
3103.
3303.
4 1
2N84.
1624.
613.
-2..
1Pebl.
245P.
1937.
2291.
S3097.
41
43
45
47
49
51
53
55
57
5
61
6'
61
67
69
71
73
75
77
79
81
F
8,
87
89
1
93
95
97
179a?.
1811.
1722.
1484.
1426o
1556o
2010.
1284.
13140
13480
1405.
1641.
2000.
2515.
32f9.
4520.
6636
920
10612.
11813.
11427.
80p6.
5764.
5205.
4414.
3572.
2703.
1809.
1060.
!65.
11 62.
lqq5*
2710.
34ql.
4229.
4h93.
'411.
-110.
-3188.
-530
-2674.
106.
10822.
7339o
867.
847.
044.
85h.
51.
1094.
1323.16q7
1242.
979.
730.
510.
4580
488.
610.
846.
774.
668;
553.
441o
477.
577.
740.
9 3.
231V
3395.
4677.
5919.
7312.
8635.
9409.
9869.
9638.
772m.
4233.
2C69.
2005.
2114.
2366.
2415.
2458.
?459.
249.
24q2
7 540
1241.
-12S7.
-37!6.
-F117.
-33)3.
926!0
15)857.
9196.
1241.
1886.
2661.
3534.
4723.
6397.
8855.
11123.
11852.
9901.
4580.
1202.
1449.
151.
2595.
2834.
3035.
3238.
3414.
3574.
3744.
2136.
-1191.
-4510.
-7824.
-4223.
11822.
9101.
5167.
1330.
18?50
2302.
2776.
3460.
46009
6701.
9344.
11087.
9950.
4278.
851.
1679.
2642.
35bsq9.
3655
3532.
3401.
3301.
3172.
3082.
27b?.
2204.
1561.
770.
-851.
-30190
8075.
6024.
965.
1309.
1590.
1852o
2258.
29559
4288.
6061.
16000
7978.
5673.
3519.
3462.
3432.
3423.
3038.
2(56*
22b5o
1922.
1012500
2048
4002.
6140.
L98
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-15937.
-4v45.
990.
974
940.
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4.3*
LATITUDEtN)=
AS
IN 4.1*
25
FOR JULy.
30
40
35
45
50
55
60
65
LEVEL
-657o
253
3428.
-220.
1902.
2078.
-2514.
-3737.
1480.
1587.
2507.
582.
2255.
93.
2019.
1427.
1786*
1409.
1674.
2107.
2405.
2620.
2763.
2837.
2846.
2791.
2676.
24qA .
2271.
1298.
1791.
1630.
1575.
1521.
1470.
1420s
1371.
1321.
1271.
1218.
1170.
2594.
2454.
2359.
2249.
2096.
1946.
1804.
1623.
1442.
1328.
935.
296.
-888.
-17.
2061.
I9400
2362.
2298.
2220.
2107.
1945.
1798,
1669.
1503.
1305.
110'..
826.
562.
425.
221.
-172.
1632.
4445.
357
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6367.
5769e
-182.
-328.
1143*
841.
545.
277.
3 .
-176.
-375.
176.
282e
473.
746.
1091.
-560.
1482e
k90.
1437.
1399.
1309.
1015.
588*
259.
82.
-94.
-310.
-500.
226.
1866.
4320.
5489,
4674.
1877.
2202 ...
-730s
-872.
-120.
-92.
61.
S102.
2449.
3755.
-148.
-207.
2374.
275.
1086.
189.
41.
-67.
1690.
1670.
1618.
1376.
1006.
709.
530.
321.
14.
-389.
110.
523.
2211.
3096.
1758.
1673.
586.
370.
2011.
1999.
1954.
1753.
1454.
1226.
1102.
960.
750.
45. *
306.
1102o
2356
2231
-962.
-968.
1168.
1071.
913,
669.
187.
72.
-29.
-110.
-73.
475*
1508.
2773.
3061.
2378.
924.
771.
538.
332.
194e
1490
1630
236.
692.
488.
209.
42.
22.
104.
4735
348.
143.
-31.
-106.
-42.
135.
427.
400.
719.
790.
792o
688.
222.
424.
758.
1251.
1043.
4H3.
-326.
1057.
99.
-1182.
471.
182.
-716o
-663.
-1656.
-1306.
877.
1484.
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35
30
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45
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3540.
1406.
1443.
3774.
2101,
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2389.
2440.
4962.
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-666.
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1509.
2663.
3694.
3369.
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2048.
1764.
1469.
1162.
942.
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1459.
1156.
706.
587.
491.
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2000.
3944.
5145.
4574.
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574.
1038.
2121.
3717.
4670.
4111.
1929.
1726.
1318.
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741.
559.
9245.
6400.
5022.
18T0.
1601.
1096.
821.
730.
660.
406.
307.
339.
270.
358.
866.
1781o
3065.
3640.
3162.
247.
265.
615.
1232.
2064.
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596.
512.
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656.
722.
t29.
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716.
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7533.
2096.
2072.
2240.
2094.
1863.
1558.
1191.
777.
343.
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-405.
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2011.
19 6.
1926.
1686.
1375.
1106.
602.
190.
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1337.
902.
1552.
2424.
2234.
1988.
1750.
1517.
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1052.
5443.
1596.
3141.
2447.
2156.
1780.
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1952.
2025.
2041.
2006.
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1718.
1612.
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2831.
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2368.
2100.
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2064.
1883.
1704.
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1518.
1434.
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1651.
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684.
677.
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QUASIGEOSTROPHIC POTENTIAL VORTICITY MERIDIONAL GRADIENT
TABLE 4.5.
LEVELS 13-41, 82,
SOURCES: LEVELS 25-97, #1o
MODEL APPROXIMATION, FOR JANUARY.
SOURCES.
#3, LEVELS 3-7. #4. SEE SECTION IV-3 FOR A DESCRIPTION OF THE DATA
LATITUDEfN)z
25
30
35
40
45
50
55
THE
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LEVELS 7-299
60
.
65
LEVEL
3
5
7
948.
2995.
1995.
7
9
11
13
15
17
19
21
23
25
27
29
1700.
1594.
1886,
1918.
1961.
2009.
2059.
2106.
2142.
2162*
2157,
2121.
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
2134.
2051.
1903.
1649.
1304.
1063.
945.
785,
513.
94.
164.
1862.
4544.
4628.
3438.
2313.
1806.
2418.
1845.
2970.
1731.
2312.
2328.
2503.
1894.
1592.
1102.
737.
431.
180.
-56.
-286.
-524.
-621.
436.
2615.
5161.
5516.
4146.
2891.
2704.
1998.
987.
485.
-63.
-609.
-1072.
-1337.
-1297,
1790.
1426.
831.
476.
280.
173.
104.
157.
382.
883.
1565.
2461.
3345.
338R0
2702.
1789.
1409.
776.
488.
461.
533.
606.
834.
1260.
e199.
2039.
1985.
1769.
15740
14590
3619.
1856.
1184.
4118.
802.
2231.
1882.
1547.
966.
797.
957.
1168.
1271.
1467.
1780.
2178.
1959.
1461.
786.
534.
766.
2063.
1811.
1328.
1220.
1402.
1613.
1702.
1855.
2088.
2348.
1975.
1307.
454.
184.
603.
1790.
-295.
1875.
3337.
3787.
2841.
2775.
2630.
2404.
2104.
1752.
1397.
1133.
1119.
1602.
3310.
3406.
2567.
2273.
18350
1242.
492.
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-2254.
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-2720.
2274.
2183.
2023.
1763.
1416.
1598.
1432.
1735.
1667.
1588.
1501.
1407.
1311.
1219.
1140.
1085.
1069.
2030.
1826.
1493.
1168.
820.
541.
314.
30.
-396.
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2232.
5553.
5892.
4316.
2446.
2586.
2327.
2244,
2099.
1754.
1665.
1740.
1844.
1892.
1998.
2176.
2369.
1933.
1210.
331.
109.
638.
2339.
2311.
2135.
2067.
2025.
1986.
1931.
1896.
1888.
1874.
1577.
1166.
729.
703.
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TABLE 4.6.
LATITUOFtI=
AS IN 4.5, FOR
25
APRIL.
30
35
40
45
50
55
60
65
LEVEL
2248.
1924.
2131.
2169.
1876.
2210.
1869.
1846.
1960.
1992e
2034.
2085.
2141.
2198.
2250,
2287.
2298.
2271.
2104.
2016.
1864.
16289
1317.
1092.
975.
841.
651.
431.
635.
1832.
3942.
4645.
3811.
1820.
2098.
2496.
1817.
1983.
2399.
1975.
1838.
1752*
1641.
1503.
1339.
1155.
9f3.
789.
1312.
920.
540.
588.
431.
304.
188.
343.
208.
254.
780*
7H2
1291.
2093.
3918.
4426.
3554.
2075.
292?2.
3012.
2410o
313.
612.
11 68*
1487.
1770.
1964.
1872.
1536.
1336.
742.
448.
448.
476.
571.
784.
1056.
14417.
1848.
1858.
1495,
1400.
11721.
656.
488.
618.
765.
744.
735.
743.
801,
970.
1234.
1518.
1536.
1282.
1099.
935.
680.
601.
709e
840.
895.
980.
1106.
1285.
1277.
1178.
998.
b88.
858.
1082.
1226.
1047.
609.
392
612.
428.
197.
-85.
-420.
-803.
-1212.
-1604.
-1906.
-2205.
-3050.
-3691,
-3926,
-1048,
-1435.
1680.
1523.
2488.
865.
39.
-593.
-1352.
463.
-5.
-529.
1877.
1569.
1031.
633.
361.
233.
205.
1437.
2770.
1363.
1495e
1346.
12180
930.
542.
1575,
1399.
1161.
R52.
673.
663.
19900
1742.
1737.
2277.
1748.
1801.
1730.
lqq7,
2065.
1896.
1605.
1272.
881.
1866*
1922.
2199.
-2062.
830.
737.
613.
592.
688.
824.
962.
1159.
1433.
1766*
1547.
1038.
317.
20,
228.
582.
568.
587.
598.
625.
734.
973.
1312.
1774.
2292.
1771.
740.
-738.
-1432.
-942.
a
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TABLE 4*7.
LATITUDE&)=
AS IN
*5,
25
FOR
JULY.
30
35
40
45
50
65
60
55
LEVEL
1529.
1900.
2170.
1706.
1719.
2114.
2069.
2122.
1947.
1935.
1Q24o
1913.
1901.
1889.
1876.
1860.
1843.
1079.
1972.
1951.
1923o
1828.
1687.
1607.
1613.
1646.
1725.
1905.
1833e
1538.
766.
-354.
-1150.
1831.
1791.
1735.
1621.
146R.
1370.
1341.
1317.
1297.
1307.
12'~?.
1254.
1375.
1475.
1320,
1828.
1627.
1943.
1692.
1675.
1782.
1563.
1552.
1613,
1554.
1530
1625.
1608H
1518.
1533.
1558.
1590.
1626.
1657.
1677.
1677.
1649.
1590.
1257.
1109.
934.
731.
1656.
1613.
1553.
1368.
1096.
1.87.
763.
605.
373.
60.
83.
662,
220.
3698.
3632.
1557.
1356.
1012.
798.
997.
1723.
1290,
1226.
1845.
566.
1671.
283.
87.
-226.
-675.
-1296.
-971.
350.
30lf *
482106
4 106.
516.
928.
339e
1290.
1687
2079.
2395.
2536.
2394.
-437.
-528.
-416.
1044.
657.
222.
388.
625.
141.
-69.
-271.
-204.
-565.
1248.
1156.
1024.
717.
282.
-3271.
799.
b69.
505.
260.
lb.
1457.
139 4.
-186.
1139.
1079.
908.
1045.
905.
700.
427.
131.
-330.
-610.
-954.
-1283.
-705.
-211,
-314.
-389.
-337.
232.
782.
3076.
4058.
3054.
1275.
2525.
2770.
1994.
869.
698.
712.
528R
444.
231o
253e
101.
88.
780
72.
169.
116.
244.
492.
311.
559.
957.
1528.
1400.
916.
177.
-140.
-40.
923.
1120.
1244.
1240.
1093.
844.
554.
408.
1950
44.
4.
85.
266.
565.
1037.
1685.
1321.
4300
-797.
-1243.
-900.
Li
j.
4j
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TABLE 4o8.
LATITUDE &)=
AS IN 4.5,
FOR OCTOBFH.
35
25
40
45
50
55
60
65
LEVEL
2439,
2256.
2423.
2309.
2191.
2460.
2207.
1943.
2822.
1791,
1635.
1843.
1751.
1641.
1519.
1 '8o9.
12c1.
1974.
1894.
2052.
2009.
1957o
1897.
1831.
1763.
1699.
1647.
1615.
1622.
2103.
2039o
1929.
1778.
1604.
1506.
1473.
1433.
1371.
1302.
1244.
1460.
2207.
2859.
2730.
2095.
2115.
2670.
1011.
1973.
1 10.
1528.
1880.
1661.
1280.
1250.
932.
604.
36P.
205.
694.
556.
.18.
261.
134.
448.
143,.
3236.
4302.
3830.
1847.
1992.
1677.
-21.
-|q.
-1 .
444.
1444.
3P29.
3211.
2008.
2529.
646.
2133.
2004.
1871.
1611.
1965.
18640
1763o
1!21*
1295.
936.
554.
1217.
P62.
1798.
1538.
10810
730.
461.
102.
-150
-72.
-6.
4 54.
1304.
2510.
3(j00.
241?7o
2228.
2121.
1700.
1k15.
16060
1403.
1130.
796.
417.
20.
-355.
-645.
-775.
-678.
-257.
-520.
-6317.
-5560
1750.
1442.
895.
591.
491.
436.
299.
206o
167o
226.
581.
1182.
1q90.
?271.
18370
1747o
1412.
0oo.
018.
524.
583.
526.
532o
615.
784.
876.
992.
1134.
i 156.
1015.
2132.
b1502.
1528.
1829.
477.
177.
-160.
-414o
-5370
-471,
1144.
1052.
1000.
2050.
1936.
1740.
1515,
1275.
1128.
1053.
964.
835.
683.
714.
1283.
2638.
3564o
3212.
1950.
1720.
2680.
17470
1424.
808.
539.
571.
685.
668.
1697.
1418.
859.
629.
684.
791.
177.
713.
957.
823.
939.
1211.
1102.
921.
621.
249o
4930
1 4.3.
0 q 0*
354.
454.
1293~.
295.
C
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POTtNTIAL VORTICITY MERIDIONAL
OUASIG[,OfGPI'HI(
TABLf 4.'0.
MODFL APPROXIMATION AND
(.ef)),
(tOI
4(5y)t
.T
GAl[
F(ORJANUARY.
SORCESt
)ATA
ALL
FOR
L
CO POIT
E
LA.ITITOLL
.5
45
2
65
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2097,
S19 8.
2228.
2224.
2161.
20 5.
119.
1742.
1549.
1303.
1 (Ib6.
C. C,0
2.
1411.
2219.
37811,
5781.
611h3.
7040.
C613.
5553.
15 .
4 , 7..
273A.
1473.
2101.
2164.
1871.
1672.
1920.
147P.
1063.
332.
-304.
-478.
-3,1.
63.
124.
14,25.
232b.
2841,
3926.
5346.
7185.
8444.
8532.
7244.
7066.
A491
727 .
4205.r
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464•9.
14,
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5 ''. .?
419h.
5396,
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744.
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1502.
1374.
-143.
-1874.
-1052.
-271
4O'0.
1 ?, .
1q24.
1 '74.
2213.
2472.
2422.
2004.
1421.
1024.
903.
1212.
1256.
1317.
1357
1315.
1253.
1170.
1093.
1171,
1322.
1541.
1853.
2302.
?" 65.
3562.
4028.
4617 .
5416.
61,75,.
7812.
8423.
7705.
4'66.
28339
2654.
26352
2759.
2565.
2391
2246.
2127.
2041.
2 1'?.
500.
2102.
3265.
3310.
2357.
2159.
2188.
2639.
2450.
2057.
1629.
1368.
1140.
970.
881.
779.
711.
(179.
689.
740
876.
1052.
1306.
1717.
239b.
3562.
4642.
5292.
5476.
4777.
4140.
4199.
4127.
3915.
2706.
1316.
-22.
-1243.
-2311.
-3230.
-1664.
1943,
470~
10
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1481.
79.
2312.
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4163.
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214
Table 4.13. Summary of unstable mode properties. Explanation of columns:
2) MO-monthof profile.
1) MODE-mode types as defined in sections IV-6.
5) PWN - planetary
wavelength.
WL
4)
(ON).
profile
of
3) LAT - latitude
the phase spe4d.
of
part
real
CP
7)
time.
doubling
DT
6)
wavenumber.
level(s).
steering
S.L.
9)
speed.
phase
the
of
part
8) CI - imaginary
amplitude
streamfunction
the
of
value
10) 1IPK - pressure level of the maximum
11) FIPK - pressure level of the maximum absolute value of the meridional
entropy flux. The v'e' peaks are .positive unless a "-" preceeds the entry.
12) 17I >.5 - the region where the streamfunction amplitude is greater, than
1/2 the maximum absolute value.
13)
Iv'e'j
> .5 - the region(s) where the
absolute value of the meridional entropy flux is greater than 1/2 the maximum absolute value. The regions contain positive v'O' values unless other14) ICEI > .5 - the regions where the absolute
wise indicated by (*).
value of the eddy kinetic to eddy available potential energy transfer is
greater than half the maximum absolute value. The regions are for kinetic
15) Qy < 0 - regions
energy generation unless otherwise indicated by (*).
(1) where the quasi-geostrophic potential energy vorticity meridional gradient is less than zero and (2) which are associated with the given unstable mode. See section IV-12. Each mode described is the fastest growing mode for the mode type given.
MODE MO
AT
YL
(K(M)
PwN
T
(DAYS)
CI
Ce
(M/SEC) (M/SEC)
TABLE
4.13
I-i l
I
S.L.
(H)
t(8E)
(PR)
II>.5
9
(8B)
Iv-j >.5
IMn)
ICI>.5
4402505004008008RO0-
6040
2.20
3.59
4.53
2.71
1.49
843
755
770
780
362
230
1000
338
1000
1000
920
880
115-1000
110-1000
150-1000
140-1000
00-1000
900-1000
720-1000
820- 940
2.2
0.39
2.32
1000
880
430-1000
820-1000
1.7
6.12
2.62
600- 830
920-1000
810- 950
E
E
E
E
1
1
1
1
25
35
45
65
5400
5500
3700
3500
6.8
6.6
7.7
4.8
1.9
1.5
1.7
300
E
4 25
3900
9.4
E
4 45
3500
8.2
1.72
8.17
E
7 65
2700
6.2
4.0
1.32
0.89
E
10 ?5
2000
18.2
4.3
-2.21
0.60
E
E
E
10 45
10 50
10 65
3700
3200
3200
7.7
8.1
5.2
1.6
1.b
2.5
6.89
7.45
3044
2.84
2.70
1.66
843,
60*15,1
730,
50.15
752.
20
735
76
770
88
8709
50.30
750
770
752
6
1 25
10000
3.6
5.3
2.84
2.46
806
175
60
160
1000
E
4 65
3100
5.4
2.3
3000
1.71
E
7 45
3700
7.7
2.0
4.92
2.33
735
150-1000
880
180-1000
230
1000
130-1000
270
840
170-1000
620- 820
910-1000
690-1000
820
880
630- 980
810-
1000
1000
1000
1000
1000
80
130-1000
170-1000
160-1000
680-1000
790-1000
810- 960
500540400770-
840
920
700
920
60- 400
860-1000
70- 310
900-1000
60
10-
100
508304572020300820-
140*
880
65*
840
150*
390
920
558105555- 170
7002525- 90
800680240- 300
660-820.910-1000
8585- 200
140W
900
180*
750
120
910
800
1 35
10000
3o3
9.6
9.31
1.37
6
1 45
5900
4.8
6.9
5.95
1.06
806
8
20
6
1 65
6500
2.6
19.0
1.96
0.43
820
211
880
175
70.
193
70
42
1000
230
107
211
70
80- 450
60- 100
143
60
60- 320
820-1000
5- 20
130- 200
10- 510
920-1000
35- 200
5200
700
1007
-2.32
0.63
G
10 45
9300
3.1
7.7
8.10
1.50
G
10 t0
6300
4.1
7.4
7.84
1.16
750
5
10
6
10 65
5600
3.0
15.0
2.95
0.48
790
230
880
0.72
2.31
6
4 45
8000
3.6
8.1
6.83
1.19
G
4 65
7400
2.6
13.0
2.96
0.64
G
7 45
8000
3.6
8.6
5.75
1.16
G
7 65
5000
3.4
17.1
1.36
0.39
830- 950
130
10 25
4.6
4555
900-1000
10- 70
193
G
4.4
50-
900
10- 500
900-1000
60- 350
970-1000
60- 400
820-1000
30- 440
850-1000
90- 350
940-1000
150- 430
862,
50910o2
700,
5010
752(
20
7001
80
770S
88
8805
6015
700
8300
530- 810
315
G
4 25
940
830
940
670
920
20
1000
735
G
tIp)
55-
10-
430- 690
770- 920
300- 810
330- 900
820- 900
110
60
810- 960
608304568050750820-
200
150 R
880
180
810
180*
900
920
TABLE
MODE MO LAT
WL
PWN
(k)
ES
C9
1
4
ES
10
OT
(DAY.)
.5.
CP
(V/SEC)
CI
(M/SEC)
4.13 CONTINUED
S.L.
(m*)
IvpI
( Mr)
843
840
940
1000
17
I
IYI>.5
(MB)
'I ).5
(MB)
880
880
800-1000
790-1000
(Mp
C*I.S5
tI
(MB)
(M)
840- 930
830- 930
840- 920
840- 920
600- 730
600- 730
1400
12.'4
1200
14.2
2.r
1.5F
1.9b
0.50
0.01
1200
13.1
2.8
2.38
0.56
840
1000
880
800-1000
940- 950
830- 920
600-
430830400820400R20-
670
930
670
920
670
960
430- 660
600- 730
430- 660
600- 730
430- 660
600-
360- 530
350- 520
480-
95- 150 *
240- 410
240- 410
380-
18
SM
I1
2600
6.4
4.3
3.17
0.77
650
360
600
190-1000
SM
4
2300
7.2
5.0
3.9P
0.60
360
600
210-1000
SM
10
2500
6.P
4.9
4.32
0.65
650
30
650
340
600
190-1000
S
S
1
4
3100
10.0
3.6
16.90
1.30
290
480
140-
590
3200
11.6
10.0
16.00
0.40
230
110
140-
380
95-
150
S
4
S
4
3700
10.0
15.4
14.60
0.31
210
110
110-
350
100-
420
2200
13.0
11.0
3.52
0.24
0.36
510
7', 4
380
96
413
94
880
1000
1000
930-1000
qP0-1000
9$0-1000
920-1000
S
7
S
7
P200
13.0
8.0
?.oO
1000
7
1500
11.4
15.6
3.90
0.12
900
67
435
155
1000
s
290
400
190-
L
I
13500
2.2
11.3
19.10
1.52
491
110
-80
40-
L
LL
L
L
1
II
1
7
8000
12400
12000
5000
3.6
2.3
1.4
5.4
6.03
8.9
11.00
14.0
6.94
7.40
2.69
P065
1.56
1.75
1.33
0.48
3
1
b
210
L
7
3500
5.0
13.1
4.45
0.34
L
10
8400
2.0
11o.0
3.54
1.02
752
735
717
920
65
386
165
740
160
3
10
0
10
70
-170
20
223
1001103-
460
160310-
180*
440
65-
110*
5- 20
0- 20
5- 60
65- 110
940-1000
160- 200*
5-
60
860- 890 4
910-1000
860- 890 *6
910-1000
J30- 440
71-
120
210
010
565250- 400
160- 200
10-
30
730
730
730730400-
50-
E0
600- 730
F80
140- 1) 0
600-
730
217
CHAPTER V
SUIMMARY OF IMPORTANT RESULTS
The climatological profiles are shown to be unstable to severallong
and short wave modes.
This may due in part to the absence of thermal and
frictional dissipation.
Newtonian cooling would lessen the growth rate of
long wave modes (Geisler and Garcia, 1977), and surface frictional dissipation would tend to stabilize the short wave modes which are confined to
the ground.
In the more realistic case, which includes thermal driving
and dissipation, the unstable mean flow supplies potential energy to the
perturbations to balance the loss of perturbation potential energy through
thermal dissipation, and the loss of perturbation kinetic energy through
frictional dissipation (Charney, 1959; Pedlosky, 1970, 1971; Stone, 1972).
The long wave modes discovered by Green (1960) are shown to have
several interesting properties.
To exist, they require a non-zero valueof
y and are made -muchmore unstable by a stratosphere or rigid lid.
Their
wavelength becomes c as yT (or 8) goes to zero, and the Green modes become
T
shortwave modes as yT becomes large.
ing Green modes are moderately short:
Doubling times of the fastest growabout 6 days in the case of a nomi-
nal winter stratosphere with linear u and T profiles, and less than 10 days
for most of the climatological profiles.
In the parameter study, the Green
mode amplitudes are particularly strong in the lower stratosphere (relative
to other levels) when nominal winter conditions are assumed.
Amplitudes
are. also strong in the lower stratosphere for the cases in the climatological study.
For both the parametric and climatological studies, the stream-
function phase of the Green modes varied much more than that of the Eady
modes, and the variation was usually confined to small regions in the lower
stratosphere and mid troposphere.
The phase difference between the meri-
218
dional and vertical velocities indicates that the Green modes have a meridional
circulation in the lower stratosphere like that observed. The meridional entropy transport for the Green modes in the parameter study is countergradient
and particularly strong in the lower stratosphere when nominal winter conditions are assumed.
In the climatological study, transports are strong and
polewards in the stratosphere in nearly all cases; they are countergradient
for the cases with reversed shear. The Green modes' amplitude, meridional
heat flux, and kinetic energy destruction are much larger in the stratosphere than are the same quantities for the Eady mode; this is true for both the
parametric and climatological studies. The Green modes are thereby more
sensitive to changes in the unperturbed state of the upper layer.
Quasi-geostrophic potential vorticity meridional gradient profiles
calculated from monthly averaged u and Y values possess a considerable number
of zeros.
The zeros and their associated negative regions of qy have a sub-
stantial effect on the spectra of the unstable modes. Some modes had their
growth rate drastically reduced when a negative y region was removed.
New
modes, distinct from the Eady and Green modes, were shown to exist because certain negative q, regions are present, and these modes tend to have a larger growth
rate when their associated q regions extend over greater depths. Their fluxes
y
at the levels of those regions are always downgradient. Certain of the new modes
are potentially important for the general circulation of specific regions
of the atmosphere.
Some S and S modes could be important for the circulam
tion of the upper troposphere.
The E 'and some S modes could influence
s
middle and lower tropospheric circulation.
In addition to the G modes,
some L modes could be important for the stratospheric circulation.
these L modes are destabilized by negative q
Two of
regions in the stratosphere
219
and are examples of in situ stratospheric baroclinic instability.
A very small change in the u profile (.3 m/sec or less) can have a
drastic effect on the unstable mode spectra by reducing growthrates or by
entirely eliminating certain modes.
affected.
Both long and short wave modes are
Staley and Gall (1977) and Gall and Blakeslee (1977) reported
extreme sensitivity of short wave mode growth rates to small low level
changes in U and the static stability.
They attributed this to the short-
wave vertical structure which they claim causes the mode to "feel" only
low level parts of the basic state profiles.
This explanation cannot hold
for the long wave modes of this study, which were affected by low level
changes in W.
Rather, it was shown that only the small changes in 1 which
drastically alter the stability properties of the qy profiles (by removing
negative qy regions and their associated zeros) result in a substantial
change in the unstable mode spectra.
In the stratosphere, long wave modes have been shown to be potentially important in transporting most of the heat, even when the flux is
countergradient (see Section III-4 and -5, and IV-8, -11, and -13).
Most
parameterizations of the eddy heat flux (e.g. Green, 1970; Stone, 1972;
and Held, 1978) have assumed that the most unstable Eady mode is respon- .
sible for the eddy heat flux.
These parameterizations are therefore not
likely to be capable of representing the stratospheric heat flux.
Most
of the long wave modes which are potentially important for stratospheric
heat transport have their kinetic energy generated in the troposphere.
Therefore, a successful parameterization of the stratospheric heat fluxes
is likely to have to rconsider these long wave modes and to treat the
stratosphere and troposphere together.
220
APPENDIX
Zonal Velocity and Temperature Data
-a
%
00>4
3
3'~NJ 3'
I
3
000
0'
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228
ACKNOWLEDGEMENTS
229
Support for this thesis was in part from the'National Aeronautics
and Space administration under grants NSG 5113 and NGR 22-009-727, and
in part from the United states Department of Transportation under Grant
AT(11-1)-2249.
0
The list of those who helped and inspired me in the course of my
education is indeed very long, and, for that reason, could not be
comprehensive. Nevertheless, I want to thank those to whom I am most
indebted.
First, my advisor Professor Peter Stone has always been a source
of inspiration and encouragement. He has been very understanding in
times of slow progress, and always constructive in his criticism.
I also want to thank Professor Edward Lorenz for his good friendship
and deep inspiration both as a scientist and a true gentleman.
I acknowledge Professor Reginald Newell's guidance in the early years
of my study at M.I.T.. Miss Isabelle Kole and Mr. Sam Ricci are to
be thanked for their friendship and excellent work of drafting the
figures. I am indebted to Ginny (Powell) Siggia for typing the text,
and wish her and her husband love in their marriage which took place
during the time in which she typed my thesis. I am grateful to Diana
Lees Spiegel for her many kind words and competent assistance in
coding and running my programs. To Professor Eugenia Rivas I give
thanks for her warm friendship and helpful discussions. Revisions
and extensions of the thesis were suggested by Professor Jule Charney.
I am thankful for the interesting discussions I have had with my
fellow students, in particular: Brett Mullan, Gary Moore, Lin Ho,
Ron Errico, Minoru Tanaka, Long Chiu, and Bijoy Misra. Administrative
assistance was cheerfully given by Jane McNabb and Virginia Mills.
I wish to thank Susan Ary, David Nergaard, and Kathy Huber of the
Department for their good friendship. And I wish many broken (meteorological) records to the weather station at Saugus Mass. in return for
the countless bits of practical advice and ceaseless wit of its
operator, Eddie Nelson.
My gratitude for my educational experience extends beyond the
Department of Meteorology first of all to my parents, my earliest
educators, James Fullmer and Antonine Fullmer. The instilled in me
the importance of a good education and made that education available.
To them this thesis is lovingly dedicated. My gratitude in addition
extends to Sr. Patricia Margaret who also helped start me out on my
academic journey, to Mr. Philip. Hart who gave me excellent and
inspirational instruction in mathematics, to all my ment6os at St.
Joseph's Prep from whom I received the priceless gift of a broad
education, to Professor Leonard Cohen for his friendship and guidance
at my undergraduate school, Drexel University, and to Wade Rockwood
and Fr. Bob Moran for their kind and good advice.
g
g
S
230
Very special thanks go to my longest and closest friend,
Ted Kuklinski. Ted has helped me in every way a good friend could.
His caring and encouragement have always been there to sustain me.
I have had the rare and wounderful experience of being in school
with this kind and selfless man for no less than SIXTEEN years.
I also want to express my deep gratitude to the many members
of the Paulist Center Community who have touched my life in a myriad
of different and beautiful ways. They have supported me in rough
times and shared the joy of good times. From them I have learned
much about the meaning and beauty of life. Just a small fraction
of these good people do I mention here: Bill Dougherty,-Loretta
Jordan, Mary Lou Doran, Blair, Joe, James, and Jennifer Mergel,
Bob Paine, Vinny and Elaine Adams, Jim McCarthy, Elanor Morrissey,
Linda Jeffrey, Jane Vincent, Nancy O'Brien, Bev Eder, Paula Ripple,
Rich Zablocki, Dick and Kathy Noonan, David and Kathy Konkel,
Kathie Voigt, Kathi Connors, Cathy Wright, Mary Hardigan, Janet Gawle,
and especially Glen Hoffman.
Finally, I want to give my greatest thanks to God who has given
life, my talents, and my friends, and without whom this thesis
my
me
could not have been written.
231
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INDEX TO CITED WORKS
(Chapter-Section)
Andrews & McIntyre
III-1
1976
Houghton, 1978
I-i
Boyd, 1976
I-i
Klein, 1974
IV-13
Burger, 1962
I--I
Cardelino,
Cardelino, 1978
1978
11-2
Charney, 1947
I-i
Charney, 1959
V
Charney & Drazin, 1961
III-1
Kuo, 1956
I-1
Leovy & Webster, 1976
I-1, IV-4
Lindzen, Batten & Kim, 1968
II-2
Lorenz, 1951
III-4
Charney & Pedlosky, 1963
II-2
Lorenz, 1967
II-3
Charney & Stern, 1962
I-1, IV-4
McIntyre, 1972
I-1, IV-2
Dahlquist & Bj6rck, 1974
II-1
M.I.T. I.P.S., 1973
II-1
Dickinson, 1973
I-1, IV-2
Molla & Loisel, 1962
IV-8
Eady, 1949
I-i
Morel (Ed.), 1973
11-2, 3; III-2
Gall, 1976
I-1, III-5,
IV-13
235
Moura & Stone, 1976
i-2, III-3
Gall and Blakeslee, 1977
I-1, IV-12, V
Murgatroyd, 1969
I-1
Geisler and Garcia, 1977
I-1, 11-2, 111-4, IV-7, 8
Murray, 1960
111-6, IV-2
Green, 1960
I-1, 2; 111-3, 4, 5, 7; V
Newell, 1964a
I-i
Green, 1970
V
Newell, 1964b
IV-8
Green, 1972
I-1, IV-2
Newell & Tahnk (in prep.)
IV-3, 4, 5
Held, 1978
V
Newell et al., 1969
I-i
Holton, 1974
I-i
Newell et al., 1974
I-1, III-4, IV-3, 4, 5,
Holton, 1975
I-1, 11-2, IV-4
Oort, 1971
I-1
13
1I1111
236
Oort & Rasmusson, 1971
I-1, 111-4, IV-3, 4, 5, 13
Palmen & Newton, 1969
I-1
Pedlosky, 1964
II-4
Pedlosky, 1970
V
Pedlosky, 1971
V
Peng, 1965a
I-1
Peng, 1965b
I-1
Phillips, 1954
I-1. II-1, III-3
Simmons, 1972
I-1, III-6, IV-2
Simmons, 1974
I-1, IV-2
Staley & Gall, 1977
I-1, IV-12, 13; V
Stone, 1972
I-i, V
Stone, 1978
III-3
Tahnk & Newell, 1975
IV-3, 4, 5
U.S. COESA, 1962
III-3
Vincent, 1968
I-1
Wallace, 1978
IV-8