AN OBSERVATIONAL STUDY OF BLOCKING
WITH REGARD TO THE THEORY OF
COHERENT STRUCTURES IN A BAROCLINIC ATMOSPHERE
by
PAUL JAMES HANCOCK
B.A., University of Chicago
(1983)
Submitted to the Department of Earth, Atmospheric and
Planetary Sciences in Partial Fulfillment of the Requirements
for the Degree of
MASTER OF SCIENCE
in
METEOROLOGY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1986
©Massachusetts Institute of Technology, 1986
/2
Signature of Author
p
I/
Center for 6 eteorology and Physical Oceanography
Department of Earth, Atmospheric and Planetary Sciences
June, 1986
Certified by
x
Paola Malanotte-Rizzoli
Thesis Supervisor
Accepted by
Chairman, Department Committee on Graduate Students
I~k Fj7Abf
AN OBSERVATIONAL STUDY OF BLOCKING
WITH REGARD TO THE THEORY OF
COHERENT STRUCTURES IN A BAROCLINIC ATMOSPHERE
by
PAUL JAMES HANCOCK
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences in partial fulfillment of the requirements for the
Degree of Master of Science in Meteorology
ABSTRACT
Considerable theoretical work has been done attempting to model
persistent atmospheric flow patterns (blocking) with localized, coherent
structures superimposed on a mean westerly wind, in particular by
In Malguzzi's theory, which assumes a baroclinic
Malguzzi (1984).
atmosphere, the existence of the coherent structure solution requires
that certain constraints be satisfied by the mean zonal wind field,
which is allowed to have both vertical and meridional shear. These
constraints imply the existence of a quantum mechanical potential well
embedded in the structure of a potential function V(y,z) which Malguzzi
derives. In dimensionless form, V(y,z) =
1
4
1
=
- u
- Uzz
+
Uz
-
where y is the meridional gradient of the quasi-geostrophic potential
2
vorticity, u is the mean zonal wind, and the buoyancy frequency (N ) has
been assumed constant.
The present investigation consists of calculating various profiles of V
from real atmospheric data and analyzing their structure, in an attempt
to determine the degree of applicability of the coherent structure model
to the real atmosphere. Numerous cases are considered. A composite of
twelve cases of Atlantic (positive) blocking is looked at, first for a
time of five days after onset, and subsequently during an entire time
series from 5 days prior to onset until 5 days after the onset of
blocking. Also, a composite of twelve cases of Atlantic "negative"
blocking, consisting actually of a strong "high index" zonal flow, and
the wintertime climatological mean flow are considered, for comparison
Finally, several cases of Atlantic blocking are investigated
purposes.
on an individual basis.
For the Atlantic positive blocking composite case at day +5 lag, the
results show undeniable evidence supporting the existence of the
potential well structure in the V field, although the available data
does not extend far enough to the south to capture the entire well.
There are some gaps observed in the positive V regions forming the rim
of the potential well, however, indicating that the meridional
confinement of the waves might not be as complete as assumed in the
The negative blocking composite results provide substantially
theory.
-2-
less support for a potential well in V, and the climatology results no
support. For the individual cases, defining U becomes a serious
problem; however in some of the cases where a representative U field
can be identified, very clearly defined potential wells are observed in
the V profiles, which practically match the theoretical ideal. Finally,
in the time series, evidence supporting the existence of a potential
well in V is minimal prior to the onset of blocking, but quite strong
after onset.
Finally, the magnitudes of N2 values and variations are estimated using
2 =
N 2 (z). The
observed data, and the V function is rederived assuming N
modifications to the V field are significant, but the overall structure
remains primarily determined by the structure of the mean zonal wind
field, except in isolated regions such as the tropopause where
temperature effects may dominate.
Thesis Supervisor:
Title:
Dr. Paola Malanotte-Rizzoli
Associate Professor of Oceanography
-3-
TABLE OF CONTENTS
Abstract
............................................................2
Table of Contents..................................................4
Acknowledgement.....................................................5
I.
Introduction..................................................6
II.
Theoretical background and motivation........................12
III.
Data and numerical procedure.................................25
IV.
Results....................................................32
A. Composite of twelve cases of Atlantic positive
blocking...............................................33
i.
The cases.......................................33
ii.
Some observations of the 500mb geopotential
height fields...................................33
iii. Observations of the calculated V function
profiles, for day +5....o......................38
B. Composite of twelve cases of Atlantic negative
blocking............................................... 48
C. The V field calculated from the climatological mean
............................. 51
geostrophic wind profile..
D. Individual cases of Atlantic (positive) blocking ....... 55
12Z, December 31,
i.
The case of
ii.
The case of OZ, December 23,
1971..............56
1976......... ..... 59
iii. Some further individual cases of blocking.
E. Time evolution of the V profile for the Atlantic
positive blocking composite.........
F. A few considerations
V.
regarding N 2 ...
Conclusions................................
.
62
..... 64
...
69
..... 82
References ...................................... *000000*60*60
..... 89
Figure Captions.................................
..... 91
Figures.........................................
..... 95
-4-
ACKNOWLEDGEMENT
First, I would like to thank my advisor and thesis supervisor,
professor Paola Malanotte-Rizzoli, for her suggestion of the topic of
this thesis, her continued support and encouragement of the work, and
her many useful comments and criticisms.
greatly appreciated.
Her enthusiasm was also
In addition, professors Randy Dole and Peter Stone
are thanked for numerous helpful comments and suggestions.
Finally,
thanks is due to Laura Werkheiser for her help in preparing the figures,
and to the 16th floor crew for making life there more enjoyable.
-5-
I.
Introduction.
Atmospheric blocking is a phenomenon which has been recognized by
meteorologists for quite some time.
Many observational and theoretical
studies have been conducted on blocking, yet there is still no
completely satisfactory explanation for this challenging feature of the
atmospheric circulation.
The observational studies provide much
information on the climatology and structure of blocks (e.g. Rex, 1950;
Dole, 1982);
some attempt to explain the dynamics which characterize
the development and maintenance of blocking (e.g. Hartmann and Ghan,
1980;
Illari, 1984).
Inherent in most of the theoretical studies is the notion that
blocking displays unusual persistence characteristics and therefore
requires a special explanation.
Whether blocking actually shows
unusually long persistence or not is still a matter open to debate;
however, in light of what Leith (1983) describes as "classical
predictability analysis" there does seem to be some basis for regarding
blocking as persistent.
The classical predictability theory assumes the
atmosphere to be an unstable, nonlinear, turbulent system in which any
minute perturbation will eventually grow to deflect the system from its
original predicted course, and lead to a completely different sequence
of weather events.
The main goal of the theory is to estimate the
growth rate of small errors.
A general consensus now exists that a
small error doubling time is on the order of 2.5 days (synoptic time
scale), and this figure will limit the general predictability times
which can be achieved.
Blocking events, however, often persist for two
to three weeks or more, with a major effect on the weather patterns in
-6-
their locality.
This observation suggests that some measure of
atmospheric predictability can be attained on time scales longer than
the classical predictability times, and raises the notion of weather
regimes which show greater persistence than individual weather events.
The identification of weather regimes has led to the idea that
there are certain preferred quasi-equilibrium states into which the
atmospheric flow organizes itself.
Numerous theoretical studies have
been conducted on these so called multiple equilibria, a notable example
being that of Charney and DeVore (1979).
As with many theoretical
studies relating to blocking, wave resonance is a central theme in their
theory.
These authors consider a few externally forced large scale
waves in a nonlinear barotropic model, and find that two distinct stable
flow states are possible --
one far from linear resonance displaying
small amplitude waves and strong zonal flow (high index flow), and the
other close to linear resonance with large amplitude waves and weak
zonal flow (low index flow).
This low index state is characteristic of
blocking (later referred to as "positive blocking").
The multiple
equilibria in these theoretical studies are brought about by retaining
nonlinearity in the governing equations of motion, and solving by
expanding the equations in Fourier series and then severely truncating
them.
Malguzzi and Speranza (1981) also find multiple equilibria in a
model which uses a weak forcing, resulting in only a weakly nonlinear
flow.
This possibility that the atmospheric flow is organized into
certain identifiable quasi-equilibrium states, between which it makes
relatively quick transitions, has led Leith (1983) to describe what he
calls a quantum theory of atmospheric dynamics and predictability.
-7-
-'-~LIBr Il~~rr.r
-I-- )-----r~L~L
According to Leith, there have been two approaches to the quantum
dynamics, namely a wave approach and a particle approach.
The wave
approach represents a global theory, in that large scale waves are
considered over the entire hemisphere, and stems from the study of
Charney and DeVore cited above.
A significant problem with this
approach concerns the severe truncation necessary to solve the
equations, which eliminates many smaller modes which could possibly
cause more numerous transitions between equilibrium states -
and
thereby reduce the persistence times of the states.
The particle approach to an atmospheric quantum dynamics consists,
according to Leith, of an attempt to model certain flow regimes (e.g.
blocking) with stable coherent structures, such as solitons or modons.
This theory represents a local approach to the problem, which emphasizes
the regional aspects of the atmospheric flow patterns, and is the
subject of further focus in the present study.
The coherent structures of this "particle" theory are special
localized solutions of the nonlinear dynamics equations, and are assumed
to be superimposed on a mean westerly current.
The two key aspects of
the model equations for this theory are nonlinear interactions and
linear dispersion of atmospheric waves.
Nonlinearity arises as the
result of advection, while a mean vorticity gradient (B-effect) gives
rise to linearly dispersive Rossby waves.
Either of these two effects
(nonlinear interactions or linear dispersion) alone could destroy any
local coherent structure, however together they can balance in such a
way as to preserve it.
Theoretical studies of the type described above, which seek
coherent structure solutions to the quasigeostrophic equations, have
-8-
r--n~-_
_I~U;L~
fallen into two basic groups.
The first group deals with solutions
known as modons, which consist of a vortex pair translating with
constant phase speed.
Modons are exact solutions to the inviscid
barotropic potential vorticity conservation equation, but have a
discontinuity in the streamlines at a particular radius from the modon
center.
McWilliams (1980) has shown a modon solution to resemble a
particular atmospheric blocking event quite well.
The second group of
solutions, known as solitons, are characterized by the results of Long
(1964).
Long studied waves in a a-plane channel, in which the mean
westerly current varied with latitude.
The resulting shear led to the
functional relationship between potential vorticity (q) and
streamfunction (*) being quadratic in *, and therefore necessitated
solving a Korteweg-deVries (KdV) equation.
Solitons are the exact
solutions to this equation, however it must be remembered that the KdV
equation itself was derived through a perturbation calculation from the
barotropic potential vorticity equation.
Many observed blocking patterns do have structures which resemble
the solitons and modons of these theoretical models;
not.
although many do
It is always of fundamental importance to maintain a keen
appreciation of the similarities and differences between theory and
observations, for such differences often lead to new developments.
The
present study represents an attempt to test a certain theoretical
prediction by conducting an observational investigation.
The extent to
which the observations confirm the theoretical prediction, and to which
they differ from the theory, are sought.
The theoretical work upon which this observational analysis is
based is the study of coherent structures in a baroclinic atmosphere by
-9-
Malguzzi (1984).
Malguzzi's work represents another attempt to model
persistent atmospheric flow patterns by localized, coherent structures.
The theory assumes a mean zonal wind profile which contains both
meridional and vertical shear, and has two latitudinal turning points,
one on each side of the supposed mid-latitude jet stream (two vertical
turning points as well).
These turning points define the edges of the
horizontal wave guide, in which the waves are confined.
This
confinement is manifested in the theory by the existence of a quantum
mechanical potential well embedded in the structure of a potential
function V(y,z) which Malguzzi derives.
That such a potential well
exists is crucial for the wave trapping upon which the theory relies,
and represents the particular theoretical prediction to be tested here.
The observational investigation consists of calculating various
profiles of V from real atmospheric data, and analyzing their
structure.
Most of the data is taken from actual cases of blocking, and
evidence of the characteristic signature of the potential well is
sought.
Much of this analysis amounts to a study of the mean zonal wind
field (averaged over a suitable local longitude band) upstream from a
blocking pattern.
In addition, a brief look is given to data taken from
cases of particularly strong zonal flow, the opposite phase of the
so-called "index cycle" from blocking.
The purpose here is to see the
structure of the V function during an unblocked flow regime, and compare
it to the patterns obtained from the blocking cases.
Hopefully, in
addition to providing observational evidence either for or against
Malguzzi's hypothesis, something may be learned about the nature of
blocking as well.
In section II, a review of Malguzzi's theory is provided,
-10-
__ ~I-~YI-LVI~
_I~I_1LL_.II
furnishing the motivation for what follows in the remaining sections.
Section III deals with the analysis procedure and the available data.
In section IV the results of the calculations are presented, along with
discussions of their significance.
Finally, in section V, conclusions
are drawn and an assessment is made as to the degree to which the
observational findings support the theory.
-11-
II.
Theoretical Background and Motivation.1
Log pressure coordinates provide the most clear and concise system
of notation, and are thus best suited for working with the equations to
follow.
The log pressure system is defined with vertical coordinate
z
-H01ln(p/p
0
) ,
where P0 = 1000mb is the standard reference pressure.
The density scale
height, HO = RTO/g, is assigned a typical mid-latitude value of 8km.
(Holton, 1979, chapter 11 contains a good review of log pressure
coordinates.)
In addition, the B-plane approximation will be made, allowing for a
Cartesian geometry, but with the dynamical impact of the earth's
f will
rotation taken into account by defining a Coriolis parameter f;
have a constant value, f0 , everywhere except where it appears
differentiated with respect to latitude, in which case 3f/ay = B
constant.
=
The other standard approximations used in quasi-geostrophic
theory will also be applied, and a review of these can be found in
Holton (1979), chapter 6.
With the goal of explaining some of the "coherent structures" which
are often observed during atmospheric blocking episodes, Malguzzi
chooses to develop a model with potential vorticity as the principle
dynamical quantity of interest, as opposed to many other quantities,
1
Most of the theory presented here follows the model developed in
Malguzzi (1984, PhD. thesis, chapter 1).
-12-
such as momentum, which could be used to describe the dynamics.
Potential vorticity is indeed the logical choice, if one adopts the
fundamental notion, as Malguzzi does, that the structures under
investigation are the result of some combination of Rossby waves
superimposed upon a mean zonal flow.
For this perspective leads one to
the natural separation of the zonal mean flow, and the perturbation,
which is simply the remainder after the zonal mean flow has been
subtracted from the total flow.
As Illari (1984) points out, the single
most powerful dynamical constraint on both the mean flow and eddies is
the approximate conservation of potential vorticity.
Having established the motivation for working with potential
vorticity, the starting point of the theory is the quasi-geostrophic
potential vorticity conservation equation (in log pressure coordinates)
for inviscid, stationary flows.
The assumption of no friction is a
reasonable approximation above the planetary boundary layer;
the
restriction to stationary flows is based on the assumption that the
coherent structures change little with time, and that the eastward
propagation speed of the pattern (group of waves) is very nearly zero.
Nonetheless, considering only stationary flows prevents one from
modelling the time evolution of the patterns obtained.
The equation is:
J(4,q) = 0.
Here, J represents the Jacobian of
The variable
p represents
i and
q (J(y,q)=4jxqy -
(1)
yqx)"
the geostrophic streamfunction,
(x,y,z) = gh(x,y,z)
fo
-13-
(2)
2
where g is the acceleration due to gravity (g = 9.806m/sec );
f 0 is the
standard mid-latitude value of the Coriolis parameter (f0 = 10- sec );
and h(x,y,z) is the geopotential height at a particular point (x,y,z) in
The geostrophic zonal wind is then given by the familiar
meters.
relation
u = -9
,
(3)
where the subscript denotes a partial derivative.
The variable q represents the quasi-geostrophic potential
vorticity, and is given (in log pressure coordinates) by the expression:
q = V
on the
(4)
(e-z/H
+ f + e /HO0
B-plane, with constant N2 . Here, V represents the two
2
dimensional gradient operator, with V the horizontal Laplacian;
1
1
the Coriolis parameter, f = By (j = 1.5 x 10-11m- sec- ).
f is
Additionally,
the Brunt-Vaisala frequency, a measure of the static stability of the
atmosphere, is given in the present coordinate system by the relation
N2
where,
T
KT
HO az
HO
R
(5)
R = gas constant for dry air = 287 JoK-lkg-I
T = absolute temperature (OK) = T(x,y,z)
K
= R/cp = 0.286.
At this juncture an important restriction is made by assuming for the
2
purposes of the model that N is constant throughout the domain under
-14-
I_____lml__C__Y~__III___I_~_
consideration.
Later, the validity of this assumption will be examined,
along with the possible effects that allowing a variable N
on the results.
could have
N is given a typical average value for the mid-latitude
troposphere, N = 1.25 x 10-2sec The variables
1
=
(N2
1.56 x 10-
sec-2).
and q can now be decomposed into zonal mean plus
i
deviation components:
(x,y,z) = 4(y,z) + *'(x,y,z)
(6a)
,
with the condition
=0
4'
lim
(6b)
+
xI
stating that the flow far upstream or downstream from the region of
interest is just the zonal mean flow.
In addition,
q(x,y,z) = q(y,z) + q'(x,y,z).
Now, the mean zonal wind 7(y,z), the wind upstream from the blocking
region to which q will correspond, is given by:
u(y,z)
(yz)
ay
= -
It is considered as having vertical and meridional shear, thus allowing
for a realistic mid-latitude jet as observed in the atmosphere.
The expression for q is then
2
+ f + ez/H
q =
ay
N
2
-15-
(ez/HO
~(z/H
.
__Y__n_______l___lllII -L-~X~X~X~X~X~X~X~X~X~L~L~L~L~L~L~L~L~L~L
Expanding the third term on the right and simplifying, one obtains
+ f +
-
N
y
f02
2-
a 22
q
az
z
2
2
HN
2
Taking the meridional gradient (a/ay):
q
This quantity
-
f02
yy2 + B
N2 =z
fo2
2
2
+ HON
a
2
az
y, the basic state quasi-geostrophic potential
vorticity gradient, is of considerable dynamical interest, due to its
implications for the stability properties of the atmosphere.
As Fullmer
(1982) notes, in the absence of boundary temperature gradients, a zero
in the qy profile is a necessary, and in at least a few special cases
sufficient, condition for baroclinic instability to occur.
Zeros of
qiy usually have associated baroclinically unstable modes of
oscillation.
At this point it is useful to non-dimensionalize the equation.
The
following scales are taken as representative of the motion systems under
consideration:
Lr = horizontal length scale =
Rossby radius of deformation = 1000km
U
= characteristic velocity scale (= peak jet velocity) = 45m/sec
H0 = vertical length scale = scale height = 8km.
Letting the star (*) variables be the old dimensional variables, while
the unstarred variables are non-dimensional, one has:
z* = zH0
y* = yLr
u* = uU
-16-
U
8* = B
2
Lr
Substituting into the equation for qy, one obtains:
U
82
y2
8y
2
y
Lr
+
f02 U a2
2
2
N2
N
H
az
U
2
Lr
+
02
2
az
N2
N
HO
Since
2
U f
4.5 x 10- 11 m-sec- 1
U
Lr
2
N
2
H02
the expression for the non-dimensional qy is:
2
2.
qy
y2
Y
+
8z2
+
_9(7)
Dz '
with
*
=
(4.5 x 10-
11
m-sec-1
)
The following non-dimensional equations can
the characteristic scale.
also be obtained in a similar manner:
yy + *zz
q=
- "z
+
BY
and
q
=
2
+
'zz -
z
*
(8)
Having established a few relations with which to work, solutions
of
the original equation (1) can be sought, along the lines of Malguzzi's
modelling techniques.
The general solution of equation (1) can be
written
q = F(9),
-17-
(9)
where F is an analytic function.
The model assumes weak nonlinearity,
taken to be a higher order effect, balanced by linear dispersion.
With
these two assumptions, there are only two cases which will lead to
coherent, localized analytic solutions:
a)
the deviation l' from the mean flow is weak, or
b)
the mean flow has weak meridional shear.
In light of observations, neither of these cases is particularly
realistic.
In blocking patterns, for instance, the deviation i' is
often strong enough to offset or even reverse a mean westerly flow (')
at mid-latitudes.
Also,
the observed mean flow certainly does not have
weak meridional shear, as a strong peak of westerlies occurs in midlatitudes flanked by substantial horizontal gradients (M/ay).
In order
to derive a solution with the model, however, one of these cases must be
selected, and case a) is given consideration as the one allowing for the
use of realistic mean wind profiles containing significant meridional
(and vertical) shear.
The fact that 4' must therefore be assumed to be
weak is a limitation of the model.
In any case, the next step is the expansion of the functional
F = F(P +
p')in a power series, and the cases a) and b) above require
different expansions.
Details of these techniques can be found in
Malguzzi's thesis (1984);
expanded around
7.
for the case of a weak deviation
', F is
Malguzzi then uses this expansion, along with the
upstream boundary condition (6b), and equation (8) for q',
from the relation q = F()
to derive
an expression which can be solved for the
perturbation streamfunction, 4', alone.
For this derivation the reader
is once again referred to Malguzzi (1984).
The upstream boundary
condition implies that 7 = F(T), and the resulting equation for -p'is
-18-
then:
P xx
+
P'z
zz -
+
*1yy
Y(
u -
u
y-( )+
-
u
uay
o(
' 3)
(10)
assuming certain fixed properties of the mean flow.
p'is
But, noting that
assumed to be small, one would like to
transform this equation into one for an 0(1) variable.
Therefore, the
following substitution is introduced by Malguzzi:
Let
4' =
so that
EI*ez/2
e < 1
* = 0(1).
and
The ez/2 factor allows elimination of the term containing the vertical
first derivative.
Substitution into (10) yields:
(I
+
$*xx
+
9*yy
+
#*zz
_Y__"p*
-
z/2
=
e
u+
-
_l
i
2
_Y
II
2 +O(g 2 )
.
(
II
Therefore,
*xx
+
'*yy
+
*zz - V*
=
-- ez/2
Q4*
2
(11)
+ 0( 2)
with
(12)
Equation (11) shows clearly that nonlinearity is O(e).
It is equation
(12) which will be focused on in this study.
In his thesis, Malguzzi specifies a particular mean zonal wind
profile, T(y,z), which is intended to represent the mid-latitude jet
stream in some gross average sense.
The profile has meridional and
-19-
vertical structure, and the variables are assumed to be separable:
7(y,z) = U(y)Z(z) .
In the north/south direction, the profile has two turning points at
latitudes ±Y 0 , where the wind speed reaches a minimum U(y) = U 0 > 0;
jet maximum occurs half way between these turning points.
a
These two
latitudes ±yg represent the edges of the jet, and define the horizontal
wave guide within which all of the solutions apply.
The actual latitude
values of ±yO are left unstated by Malguzzi, as they are not necessary
for the theory.
The theoretical solutions are assumed to be waves which
are meridionally trapped in the zonal wave guide, and thus they will not
be effected by the structure of the zonal wind outside this region.
In
the vertical, a similar situation applies, with minimum values of Z(z)
(turning points) occurring at the ground and at a height of roughly
19km, while a jet maximum is at roughly 10-11km.
wavelike solutions are also vertically trapped.
Analogously, then, the
It is essential to
remember that this zonal wind structure was chosen, and while it is
quite representative of the mid-latitude jet stream in a large scale
long-term average sense, it was designed to be compatible with the
assumption that there exists a region of horizontal and vertical finite
extent out of which no energy radiation occurs.
Indeed the entire
theory rests upon this assumption of energy trapping, which depends upon
the meridional and vertical structure of the mean zonal wind.
This brings up the V function of equation (12)
once again.
Using
Malguzzi's specified wind profile, the structure of the V function is
that of a potential well of finite depth.
If
p*of
equation (11)
broken down by orders of e, and a long scale is introduced in the
-20-
is
x-direction, then the leading order [0(1)] eigenvalue problem
(corresponding to eq.(11)) is the equation for the wave function of a
quantum mechanical particle in the potential well V.
The significance
of this statement will be clarified below through a brief review of the
theory leading to the solutions for the perturbation streamfunction.
It
is clear, though, how the theory relies so crucially on the structure of
V actually being that of a potential well.
the variable 4* is expanded as
In order to solve equation (11),
* = *(0) +
)
+ ...
Now, it is assumed that the nonlinearity, an 0(s) effect, is balanced by
dispersion in the zonal direction.
Hence, a long scale in x is
introduced:
x =
X
-X
,
X = 0(1)
with
Substituting these expressions into equation (11)
.
and separating terms
in powers of e, at lowest order one obtains
*(0)yy +
W(0)zz - V*(0)
=
0
In addition, due to the long x-scale, the zero order solution is
separable, allowing the following expression to be written:
9(0)
= A(X) (y,z).
Malguzzi's theory then proceeds through the steps leading to the
-21-
following zero order eigenvalue problem:
yy +
= -K4
zz - V
with
Zz
1
z = 0
,
2)
z
and
+ 0
y
as
+
, z
0.
The eigenvalue, K, is assumed to be of order e.
(13)
For further details of
these steps, as well as a derivation of the lower boundary condition,
one is referred to Malguzzi's thesis (1984, chapter 1 and appendix A).
This eigenvalue problem is used to determine solutions for p(y,z).
One still needs to solve for A(X), however, and for this the order
6 equation (derived from eq.(11)) must be looked at.
This order
E
expression can be written as a Korteweg-deVries equation after some
manipulation, the specific steps of which are again provided in
Malguzzi's thesis (1984).
The solution for A(X), localized in x, then
turns out to be
A(X) =
3K( 1 )
_K(1
sech2
2
with K ( 1 ) > 0, and 6 as defined in the thesis;
quantity given by K = sK( 1 ).
X)
,
(14)
K( 1) is an order one
These solutions represent long waves.
The coherent, localized solution for the perturbation
streamfunction is then
= 3
p(y,z) ez/ 2 sech2 (
-22-
x)
+ 0()
,
(15)
(using the definition
' = e*eZ/2).
In this equation, e represents
the meridional and vertical structure function of the coherent solution,
and is obtained as the eigensolution of equations (13);
corresponding eigenvalue, with K > 0 and K = O(e).
K is the
Equation (13) is the
equation for the wave function of a quantum mechanical particle in the
potential well V.
Thus, there will be an infinite number of bound
states corresponding to eigenvalues K which are smaller than the value
of the rim of the potential well.
If there were no potential well,
there would be no bound states, and no region of trapping, as the theory
presumes.
With Malguzzi's zonal wind profile, the rim of the potential
well has a value of approximately 4 (dimensionless units).
Malguzzi's calculations, with the model parameters assigned typical
mid-latitude values, lead to some interesting results.
The
eigenfunction p1 , corresponding to the lowest bound state KI, is not an
This solution does not
acceptable solution since in this case K 1 < 0.
lead to a coherent structure.
The second eigenmode
acceptable and has a corresponding K 2 = 0.22.
anomaly pattern
#2, however, is
For this case, the
4' associated with the function 42 is an antisymmetric
dipole, with the anticyclone centered north of the cyclone.
The
solution has an equivalent barotropic vertical structure, and is
meridionally and vertically trapped as well.
This is the structure
Malguzzi sought to duplicate with the model, and his solution bears a
good amount of resemblance to some observed blocking patterns,
particularly in the eastern Atlantic ocean.
The solution for blocking proposed in Malguzzi's thesis, then, -that of "stationary nonlinear Rossby waves of localized character
superimposed on a mean zonal wind with meridional and vertical shear",
-23-
with weak nonlinearity balancing weak wave dispersion --
appears to be
able to at least partially describe some of the blocking patterns
observed on synoptic maps.
It should be kept in mind, however, that the
entire theory relies on this V function derived by Malguzzi having the
structure of a potential well.
V is a function of the mean zonal wind
profile only, assuming constant N;
it is not clear whether its desired
structure will be duplicated if real atmospheric data taken during
blocking events is used to obtain a mean zonal wind profile rather than
using the hypothetical wind profile proposed by Malguzzi.
will be examined in the following sections.
-24-
This question
Data and Numerical Procedure.
III.
As with most any observational study in which real atmospheric data
is used as the basis for making calculations, the derived results will
not be nearly as smooth and clean cut as if a nice analytic function had
been used to approximate the data.
Such is the nature of observational
studies, given the inaccuracy in making measurements and the inherent
diversity and turbulence of the atmospheric motions themselves.
However, it is hoped that from such analyses some kernel can be
identified and associated with either some known physical process or
hypothesized theoretical behavior.
The data used in this study consists of geopotential heights and
temperatures at six levels (isobaric surfaces) through the atmosphere:
850mb,700,500,300,200 and 100mb.
The data was taken from the final
corrected version of the NCAR data sets, which cover the 15 year period
January 1963 through December 1977 (the exception is the 100mb data,
which only covers the period 4/21/65 through December 1977).
At each
level, data is given on a 40 latitude by 50 longitude grid, extending
0
0
from 22' to 900 N, and around the entire longitude circle, 0 ,5 E, ...
,50 W.
The data is given twice daily at 00 and 1200 Greenwich Mean
Time.
With the exception of the 100mb data which was read directly from
tape, the remaining data was accessed through Dave Gutzler's climate
data files, which have been placed on disk for easy accessibility.
Three preferred regions for blocking to occur in the northern
hemisphere have been identified by Dole (1982).
These are, broadly
speaking, the eastern Pacific ocean, the eastern Atlantic ocean, and the
northern Soviet Union.
Observations indicate that flow patterns
-25-
resembling the dipole structure of Malguzzi's theory, with the
anticyclone north of the cyclone, occur in all three blocking regions,
but with a slight preference for the eastern Atlantic.
For this reason,
cases of Atlantic blocking will be looked at in this study, but this by
In
no means rules out the applicability of the theory to other regions.
addition, only winter blocking events are considered, although blocking
occurs in all seasons.
A brief review of the calculational procedure is now provided.
is calculated according to equation (12).
V
Since the theory looks at
blocking from the perspective of a localized, rather than a global
phenomenon, only data within a specified longitude band in the vicinity
of the block will be considered.
The mean zonal wind, Z, is the
quantity of interest, and this is calculated geostrophically from the
geopotential height field;
the zonal wind upstream from the block is
The first step then is the
taken to represent the mean zonal wind.
calculation of the geostrophic streamfunction,
gridpoint.
Next the three dimensional
4
= gh/f 0 , at each
p field is transformed to two
dimensions by computing the zonal average
y field.
The actual choice of
the longitude band over which to average is somewhat arbitrary, and four
different choices are tried to see how sensitive the results are to this
criterion.
The narrowest band contains none of the more meridionally
oriented flow associated with the blocked region, while the wider bands
extend progressively further into the transition zone between the strong
zonal flow upstream and what might be called the blocking region (see
later figures).
Once this zonal average is obtained, 7 is calculated as
u= -a/ay.
Due to the small number of vertical grid points, and the high order
-26-
of differentiation required in the calculations, a finite difference
approach to computing derivatives is not satisfactory here.
Instead,
bicubic splines are used, which simultaneously fit a natural cubic
The
spline to the data in the meridional and vertical directions.
bicubic spline assures that second derivatives are continuous and
eliminates spurious values of the derivatives.
The field of u is
calculated in this manner, and the vertical resolution is artificially
expanded by computing T at five intermediate levels:
and 150mb.
800mb,600,400,250,
This does not actually increase the vertical resolution, as
these intermediate data points are not the result of independent
measurement, but of interpolation from surrounding data points.
Once the T field is determined, the quantities Uyy, uzz, and
Uz must be calculated in order to solve for V according to equation
(12).
The vertical derivatives are the most sensitive calculations due
to the sparsity of vertical gridpoints.
While natural cubic splines are
0
acceptable for the meridional derivatives, as the end gridpoints at 90 N
and 220 N can be ignored, this luxury is not affordable in the vertical
with only six independent data levels.
The cause can be aided by
considering Malguzzi's theoretical wind profile and some observations of
The idea
winter mean zonal wind profiles, and making some assumptions.
is just to get some feel for the 7 profile above 100mb and below 850mb
so that the vertical derivatives at these levels, which represent the
edges of the domain, can be determined more realistically.
Malguzzi's vertical profile of 7
In
there are turning points at the ground
and at roughly 19km, where the zonal wind reaches a minimum.
This
implies that 37/z = 0 at the ground and at some level above 100mb.
addition, it is assumed that 7 = 0 at the ground (z = 0, or p = p0 =
-27-
In
1000mb), which is a reasonable approximation in light of mean meridional
It
cross sections of 7 such as that found in Holton (1979, chapter 6).
is also assumed that 4j = 0 at some level above 100mb, coinciding with
the level at which 87/az = 0.
From the winter mean meridional cross
section of 7, it is apparent that there is a broad minimum in the wind
field between about 30-50mb (20-23km) in mid-latitudes.
The minimum
values of U range from near zero at low latitudes, to 10-15m/sec between
~45 0 N - 600 N.
Still further north, the wind field becomes quite flat
on the average, and 7 continues to gradually increase vertically into
the stratosphere.
For simplicity, the boundary condition u = Z/i
= 0
at 30mb (~23-24km) is chosen.
What was found when actually doing the calculations was that
significant differences do exist between the
calculated in the following manners:
Tz (and 7zz ) fields
a) with a natural cubic spline fit
to the data ranging from 850mb to 100mb, ignoring all aspects of the u
field outside of this range, and
b) by fitting a cubic spline to the
extended vertical range (including 7 = 0 values at the ground and at
30mb) with the specified condition
and 30mb).
aT/az = 0 at the endpoints (ground
With the natural cubic splines of method a), the 7z and
izz fields were found to vary erratically in space, and to be
particularly rough near the edges of the domain.
Furthermore, the
derived fields were not too stable in the sense that slight changes in
the 7 profile could render very large changes in the 7z and Zzz
fields.
The sensitivity of the results to the exact choice of boundary
conditions in method b) above, however, was small provided the choice
was reasonably realistic.
For example, applying the upper boundary
condition at 10mb and at 50mb was tried, with only a very minor change
-28-
in the
zGand tzz profiles at 100mb -- decreasing to no discernible
change in the mid and lower troposphere.
Also, applying the conditions
3/az = 0 ; T = 10m/sec at 30mb was tested, again with no significant
effect on the results below 100mb.
Of the results obtained by methods a) and b) above, those of method
b), with the assumed behavior above 100mb and below 850mb, are clearly
preferable.
A crude estimate of the extension of the U field above and
below the domain covered by the data has been made in an attempt to
preserve the accuracy of the derived fields near the edges of the
domain.
A crude estimate is all that is really possible since
geostrophy is a poor approximation in the planetary boundary layer, and
N is no longer nearly constant above the tropopause;
in other words, a
more precise extension would be pointless and irrelevant due to the
already existing model limitations.
With the vertical first and second derivatives of the mean zonal
wind field calculated with the boundary conditions as above, the only
remaining task is to calculate the meridional shear term, uyy.
For
this calculation, bicubic splines are once again applied, however this
time the extended 7 profile (U = 0 at the ground and 30mb) is assumed
for consistency.
The values of
iyy, and of the vertical derivatives,
at 220 N and 900 N are ignored due to their spurious behavior as
endpoints.
Thus the final domain for which the potential function V is
calculated ranges between 100mb and 850mb in the vertical, and between
latitudes 260 N and 860 N.
It should also be noted that the wind field u
is nondimensionalized according to the scales described in section II
before any of the derived fields required for the determination of V are
computed;
the characteristic length scales are also applied, and all of
-29-
the calculations are carried out in dimensionless units.
One final point to be made about the procedure of the calculations
concerns zeros in the 7 profile.
Since the equation for V contains a
fraction with 7 as the denominator, zeros of 7 are a potential source of
trouble.
In Malguzzi's theoretical wind profile T > 0 is assumed
throughout the domain, however when real data is used to obtain the wind
field, zeros sometimes arise.
As the domain here extends to 860 N, well
north of the mid-latitude jet stream and the prevailing westerlies,
zeros in 7 (in the lower part of the domain at least) can be anticipated
in at least some cases as the westerlies give way at high latitudes to
the polar easterlies.
Such a zero line in the ' profile represents a
discontinuity in the V field, where V is undefined.
In addition, in the
neighborhood of G = 0, where T is very small, V will be extraordinarily
large in magnitude.
1iI is
applied,
In order to smooth the V field a threshold value of
such that at any gridpoint where
15
is less than this
threshold, u is assigned the threshold value, while maintaining its sign
(+ or -).
Various experimentations proved that IGi = 4m/sec is a good
threshold value;
it provides enough smoothing to the V field, without
being too severe and causing an adjustment to the value of 0 at too many
So, for example, a gridpoint having 7 = im/sec (before
gridpoints.
nondimensionalizing) would be assigned 7 = 4m/sec, while if 7 = -Im/sec,
a new value of 7 = -4m/sec would be assigned.
It is important to note
that this adjusted 7 field is used only in the denominator of the
fraction in equation (12), and not in determining uz, uzz, or uyy.
Also, it does not remove the discontinuity in the V field associated
with the
= 0 line;
term
yy
zz
there will still be an abrupt sign change in the
z
there.
Furthermore, this smoothing
-30-
technique is only applied in the cases where it is necessary.
In some
of the cases, where 7 > 0 throughout the domain, or where the zeros are
limited to a small portion of the domain at high latitudes, the field of
V is calculated without using the adjusted u profile.
Malguzzi has a
similar notion in mind as he assigns U(y) = U 0 both poleward and
equatorward of his theoretical mid-latitude jet, as a threshold minimum
value.
Thus, with the method of calculating V having been described here,
in the next section the results are looked at for the various cases
considered.
The structures of the calculated V fields are compared and
contrasted to the potential well structure of Malguzzi's idealized V
function.
Also, the behavior of the V function is observed as the zonal
average wind profile is varied in space (different longitude bands over
which u is averaged) and time
(t profile for different observation
times).
-31-
IV.
Results.
The potential function V is calculated for numerous cases of
Atlantic blocking using observed atmospheric data.
These cases fall
into three basic categories:
a) A composite of twelve cases in which a blocking ridge is the
dominant feature.
These cases are characterized by a split flow at
500mb, with one branch of the westerlies being diverted well to the
north of its average position, and one branch to the south.
Anomalously weak zonal winds prevail in the "blocked" region
between the split flow.
As Dole (1982) has identified this pattern
as one phase of the "primary regional pattern of low frequency
variability",
it
will henceforth be referred to as the positive
blocking pattern.
b) A composite of twelve cases of what will henceforth be called
Atlantic negative blocking.
This is the opposite phase of the
primary regional pattern of low frequency variability from pattern
a).
It consists of an unusually strong zonal flow most often
displaced somewhat to the south of the climatological mean jet
position.
c) Individual cases of Atlantic blocking which will lack the benefit
of the smoothing provided by the compositing procedure.
In
particular, cases are chosen which display a well defined
anticyclone/cyclone vortex pair in the 500mb flow, resembling the
theoretical flow pattern obtained by Malguzzi.
-32-
A.
Composite of twelve cases of Atlantic positive blocking.
Composite fields have the advantage of being smoother than those of
individual cases, yet they still possess the primary features common to
the cases from which they are derived.
Here the compositing procedure
consists merely of taking the average of the twelve observed values at
each gridpoint.
The criteria used to define blocking, and to identify
the onset of blocking, is a matter still open to some debate.
A good
set of criteria for determining what constitutes a blocking event and
what does not can be found in Dole (PhD. thesis, 1982, pp. 43-44).
The
first time in a sequence of observations at which all of the criteria
are satisfied represents the onset of the blocking event, and is defined
as day zero.
Time is then measured relative to this day zero;
for
example, one might wish to make a calculation at "day -2", which
represents a time of two days prior to the onset of blocking.
i. The cases.
The cases used in this study all display well defined blocking
patterns.
They are the cases of Atlantic positive blocking identified
by Dole (1982), and the patterns observed are quite closely
geographically colocated.
The twelve day zeros used in the composite
are given below:
Case #
1
2
3
4
5
6
Day Zero
1/ 2/67
11/28/67
12/24/67
1/20/68
12/25/68
1/30/69
Case #
7
8
9
10
11
12
12Z
12Z
12Z
0Z
0Z
12Z
Day Zero
OZ
11/28/69
12Z
2/24/70
12Z
1/23/73
0Z
11/23/73
12Z
12/29/74
OZ
12/ 3/75
ii. Some observations of the 500mb geopotential height fields.
Data was gathered for eleven lag times, once every 24 hours from
-33-
five days prior to the onset of blocking until five days after onset.
Maps of the 500mb geopotential height field for each lag time are shown
The composite 500mb height map for day -5 shows a
in figures la-k.
rather zonal flow regime, with no outstanding troughs or ridges.
There
is certainly no apparent precursor at this point to the blocking event
which is to ensue.
As one examines the series of maps, it is not until
day -2 that the ridge which will amplify into the block becomes
0
0
distinctly evident, in the region around 20 -30 W.
On day -4 there is a
slight deepening and sharpening of some of the weak troughs seen on the
day -5 map, particularly those near 120'W and 75°W.
This trend
continues on the day -3 map with an eastward progression, so that there
0
are significant troughs at about 110W and 65 W at this time.
Zonal
flow is still the rule across the Atlantic.
A favored location for cyclogenesis at the surface is at the
Thus
inflection point in the 500mb streamlines to the east of a trough.
on the day -3 map there is quite likely a significant mid-latitude
cyclone in the western Atlantic (in the average of the 12 cases).
Such
a feature would produce significant warm advection to its east, which
would in turn tend to build a ridge at 500mb there.
As has been
suggested by numerous other authors (e.g. Illari, Collucci), the impact
of travelling mid-latitude cyclones on the development of blocking may
be a major one.
And, by day -2 a broad ridge is apparent in the eastern
0
Atlantic, with the upstream trough now centered near 50-55 W.
On the
subsequent maps this ridge continues to build while remaining nearly
0
stationary, centered at about 20 W.
The onset of blocking is declared
on day zero, although the pattern looks more fully developed on day +1.
By day +1 the ridge at 20'W has clearly become the most striking feature
-34-
of the northern hemispheric flow pattern, and the bulk of the prevailing
westerlies have been displaced far to the north of their average
0
0
location -- with the jet centered near 60 N at 20 W longitude.
So the
entire development of the blocking pattern spans only a few days, while
the pattern itself often persists for several weeks.
developing at
At the same time that the large scale 500mb ridge is
mid and into high latitudes, it is apparent that a trough develops in
the same longitude region, to the south of roughly 300 N.
A second,
weaker branch of the westerlies passes through the base of this trough,
forming the characteristic split flow of the blocking pattern.
0
Unfortunately data is not available south of 22 N, so part of this
southern feature remains obscured.
Once the pattern is fully developed
on day +1, it remains nearly stationary with some further
intensification of the ridge/trough system through day +5.
For comparison purposes the climatological mean 500mb geopotential
height map is presented in figure 2.
This is a winter seasonal mean,
and was computed by averaging at each gridpoint all the data from the 15
year data set between November 15 and the end of February, inclusive.
As expected, it is a very smooth field, showing the zonal asymmetry
arising due to east/west geographical contrasts.
Further insight can be gained concerning the development of
blocking by examining the fields of 500mb geopotential height anomalies
at each of the lag days.
The height anomaly at each gridpoint is the
difference between the height value at that point and the climatological
mean height value there.
3a-k.
These anomaly fields are shown in figures
The maximum height anomaly to be found anywhere on the day -5 map
is roughly 100m in magnitude, reinforcing the notion of a lack of strong
-35-
upper level features at this time.
On the subsequent maps, the maximum
height anomalies generally increase in magnitude each day, with the
extreme positive anomaly associated with the blocking ridge approaching
a value of 300m by day +5.
All of the figures reveal an abundance of alternating positive and
negative anomaly regions, having the appearance of wavetrains.
this behavior may be attributable to Rossby waves.
Much of
A glance at the day
-5 map (fig. 3a), for example, shows a typical pattern which can result
due to Rossby waves;
it is the series of alternating high and low
0
0
centers stretching from the low at (38 N,122 W) to the high at roughly
(420 N,20 W).
There is evidence of another weaker amplitude wavetrain
extending eastward from this final high center.
Some of the wavetrains
are observed to be stationary, occurring roughly in the same place on a
number of subsequent maps, while others are seen to propagate.
Of particular interest to the study at hand is the wavetrain seen
in the Atlantic/European sector on many of the maps, which is especially
clear on the day 0 profile (fig. 3f).
This wavetrain extends from the
low center at (300 N,45 0 W) northeastwards, and then eastwards, to the
high center at (640 N,69 0 E);
it is virtually stationary, and appears
well developed on all of the maps following the onset of blocking.
The
wavetrain becomes increasingly vague the farther in advance of day zero
one looks.
The major positive anomaly center and the more modest
negative center to its south which are associated with the blocking
pattern comprise a portion of this wavetrain.
The continued
amplification of this positive anomaly near 200-30W between day +1 and
day +5 is most extraordinary;
it is the only center in the wavetrain
which amplifies, the other anomalies maintaining a nearly constant
-36-
magnitude after day +1.
By day +5, the positive center has developed
into an impressive bulls eye, dwarfing all of the other features on the
height anomaly map.
These observations are consistent with the blocking spectra
calculated by Dole (1982).
Dole constructed composite vector time
series of zonal Fourier components k = 0-6 of the anomalies at certain
"key latitudes" (Dole, 1982, pp. 146-149).
He found no dominance by one
component, but rather that there are important contributions to the
positive anomaly made by a broad band of wavenumbers, mainly k = 0-4.
He also found that wavenumbers 6 and greater "do not contribute
significantly to the representation of the stationary pattern" (although
these propagating short waves may still be important to the underlying
dynamics).
The longer wavelengths are seen to amplify simultaneously
and in phase beginning around day zero, while remaining stationary.
This dominant impact of the low wavenumbers on the observed anomaly
pattern, Dole suggests, is consistent with the notion of a localized
Rossby wave source in the tropics.
Models of Rossby wave propagation
away from a localized source in the tropics indicate that only the long
wavelengths (k < 4) will appear with significant amplitudes at high
latitudes, the shorter wavelengths being trapped near the poleward side
of the mid-latitude jet.
The poleward side of the jet is also an
assumed region of wave trapping in Malguzzi's theory, although no
dependence on wavelength is mentioned there.
The observed wavetrain in
the Atlantic/European sector, described above, appears as if it could
possibly have a source in the tropics, thus accomodating these
theoretical ideas.
This discussion of the observed 500mb geopotential height and
-37-
height anomaly fields for the composite Atlantic positive blocking case
provides a good background on the typical characteristics and behavior
of blocking patterns -- the phenomenon which Malguzzi's model attempts
to describe.
The main objective to assess the structure of the
potential function V is now addressed.
iii. Observations of the calculated V function profiles, for day +5.
Fields of the V function are calculated according to the methods
described in section II.
For the moment only the day +5 fields are
considered, when the blocking pattern is highly developed.
A discussion
of the evolution of the V field in time from day -5 through day +5 will
be given later, in subsection E.
Equation (12), used to calculate V, states that
1
V(y,z) =
4 -
- Uyy - Uzz + u z
u
In doing the calculations, it was found that the four quantities
Uyy,
Uzz,
B,
and uz are all of the same order of magnitude at many
gridpoints.
In the dimensionless units,
at every gridpoint.
B has a constant value of 1/3
Thus, V is determined as the result of four,
roughly equally matched, competing effects.
gridpoints uyy and
At the majority of
zz, are negative, while uz is positive;
since
u is almost always positive, the result is that V tends to be negative
throughout the large part of the domain.
The interesting features arise
in those areas which display exceptions to these general trends.
As mentioned earlier, four different longitude bands are chosen
over which to average the zonal wind.
The central axis of the blocking
pattern lies between 200 -300 W (see figure 1k).
-38-
An appropriate band over
which to average, that includes only the upstream zonal wind with none
of the more meridionally oriented flow associated with the blocking, is
the band 900 W< x <600 W.
This longitude band is also an area of
relatively weak height anomalies (fig. 3k), making it a fairly
representative zonal average.
Wider bands, extending successively
farther east to 500 W, 400 W, and 30*W, are also used, to determine the
sensitivity of the calculated fields to the exact nature of the zonal
wind profile.
The band 90°W < x < 300 W actually includes a fair portion
of the blocking pattern.
Figures 4a-d show the meridional cross sections of the mean zonal
wind averaged over each of these longitude bands, in dimensional units.
The 90W < x < 30°W profile of G clearly displays the effect of the
split flow in the blocking region, featuring twin jet maxima centered
near 310 N and 610 N;
at all levels.
a relative minimum of 7 occurs between these maxima
Each of the peak wind values on this plot, at around
25m/sec, is relatively weak compared to typical magnitudes of
tropospheric jet maxima.
The " profiles corresponding to the narrower
longitude bands show a progressive deterioration of the split flow, and
development of a single, stronger jet maximum.
This single maximum is
centered near 380 N on the 90W < x < 60'W profile, and has a magnitude
of 35.3m/sec.
The southern portion of the jet is not visible in figure 4 since it
extends south of 220 N, where data is not available.
However, one can
expect based upon climatological average conditions, that the 1 field
continues to decrease south of the edge of the map;
easterly winds can
be expected to prevail south of about 20°N in the low levels, and extend
upward to roughly 150 - 200mb near the equator.
-39-
On the northern side,
it is not very clear where one might draw the line representing the edge
of the jet -
the "turning point at +Y0" in Malguzzi's theory.
This
turning point represents the edge of the horizontal wave guide in the
theory.
On the 900 W < x < 50,60 0 W maps the northern limit of the jet
might be thought to lie in
the vicinity of 66-70 0 N, as the T values
temporarily cease to decrease north of this region.
However, in the
twin jet profile (fig. 4a) there is no clear northern edge;
the edge of
the horizontal wave guide might be called at near 70-72 0 N below 400 500mb, but at higher levels it appears to extend considerably farther
north.
The 90W< x <60 0 W
U profile is in reasonably good agreement
with the theoretical r profile, then, while the wind field computed by
averaging over the wider band shows considerable differences.
Meridional cross sections of the potential function V, calculated
from each of the 7 profiles described above, are presented in figures
The smoothing technique of setting a minimum threshold value of
5a-d.
I,
for the denominator of the fraction in equation (12) is not applied
in these figures.
The key feature to note on these maps is the high
center ranging between 64-70 0 N at tropospheric levels (below -200mb).
This "wall" corresponds to the northern rim of Malguzzi's potential
Comparing the V and I profiles, one sees that this region of
well.
positive V corresponds to an area where U-yy and izz are
simultaneously positive, the opposite of their typical sign in this
domain.
This wall of positive V values becomes more dominant, and shifts
slightly southward, as one looks at the progressively narrower longitude
bands.
In the 900 W < x < 300 W profile, this northern barrier of
positive V values covers only a limited vertical range, extending
-40-
roughly between 570 and 310mb.
While it is not a negligible feature, it
is also not the sweeping barrier present in Malguzzi's theoretical V
profile;
is absent.
there are large gaps in the vertical where the northern wall
However, the northern rim of the potential well looks
considerably more impressive in the 90W < x < 600 W profile.
The
vertical range of the V > 0 region is markedly wider in this profile
than in figure 5a, extending here between 600 and 230mb.
While in
figure 5a the barrier resembles more of an island, here it has the
distinct character of a sustained wall, extending vertically through the
majority of the troposphere.
The shift of the vertical axis of the wall
from near 690 N in figure 5a to about 640 N in figure 5d is the result of
the exclusion of the northern branch of the jet in the split flow region
from the narrower ! field -- effectively moving the northern "edge" of
the jet slightly south.
Thus, as should be expected, the wind profile
which agrees more closely with that used in the theory yields the V
function profile more nearly resembling the desired theoretical
potential well.
As one is trying to identify a potential well, it seems appropriate
to look for such V>0 barriers as was shown to exist on the northern
flank on the other three sides of the supposed wave trapping region
also.
It is first noted that the V function field calculated by
Malguzzi from the assumed theoretical wind profile contains a potential
well with an upper rim at near 15km (~150mb) and a lower rim at near
1.5km (~850mb).
The level of this upper rim is near the top of the
present domain, while the level of the lower rim is below the region in
which V was able to be calculated here.
These observations suggest that
any upper or lower potential well rims may be difficult to detect in the
-41-
present study if for no other reason than the limited area of data
coverage.
However, in the cases of both the upper and lower rims, there are
additional complicating factors.
Before getting into these a brief
discussion of the actual observed behavior is presented.
In the upper
part of the domain, all four profiles (figs. 5a-d) show a region of
positive V values extending meridionally from ~45 0 N up to between
South of 450 N there is a zero line
620 -67 0 N, centered at about 130mb.
of V right at the top of the domain (100mb), extending all the way to
the southern edge at 260 N, with positive values above it.
These regions
of positive V values are characterized by negative values of uz , and
sometimes also by positive values of uzz, rare occurrences in the
lower parts of the domain.
These regions of V>0 in the top portion of the domain could
conceivably be thought to represent an upper barrier to Rossby wave
propagation -- the upper rim of the potential well;
their location is
quite consistent with that of the upper rim of Malguzzi's theoretical
well.
However, the present calculations, as well as the theoretical
model itself, disregard the important dynamical consequence of the
tropopause by assuming N 2 to be constant (and thereby eliminating all
temperature dependence of q and V).
Considerable theoretical work
(e.g. Held, 1983) suggests that the tropopause acts like a rigid lid,
arresting the vertical propagation of many Rossby wave modes.
The
height of the tropopause, for a winter seasonal mean, ranges from near
200mb at 260 N down to just below 300mb in the polar regions;
is below the V>0 regions discussed above.
this level
Therefore, it appears likely
that the tropopause may play the leading role as the upper barrier to
-42-
wave propagation, rather than the positive V regions in the lower
stratosphere which are calculated exclusively from the mean zonal
winds.
Accordingly, less significance can be attached to that portion
of the V profiles above the tropopause relative to the tropospheric
portion.
The ramifications of allowing N2 to vary will be discussed
more fully in subsection F below.
Now considering the lowermost portion of the domain, it is observed
that the V field is quite erratic in this area, displaying very strong
gradients.
There is evidence of a region of positive V values below
750mb, between roughly 270N and 420 N, and also between 51-58 0 N, on all
four profiles (figs. 5a-d).
However, the V values in this part of the
domain are heavily influenced by the lower boundary condition, to which
their widely fluctuating behavior is attributed.
It is suggested that
if a lower boundary layer were allowed for near the ground, then the
large fluctuations and possibly also the positive regions of V along the
lower edge of the domain could be made to disappear.
values are the result of large positive values of
assumed lower boundary conditions
Tzz.
These positive V
With the
that 7 = u z = 0 at the ground, there
must in fact be some region in the lower troposphere in which
izz > 0.
If this region were confined to a narrow lower boundary layer, which was
matched to the "interior" solution above, the large fluctuations of V
below 700mb on the plots would disappear.
Thus any association of these
low level positive V regions with a lower potential well rim seems
dubious.
However, the physical barrier of the ground will still act as
the lower edge of the waveguide.
It is interesting to note, as an aside, that calculations made by
Fullmer (1982) also show that "there is a tendency towards negative
-43-
These
values of 7y at low levels (below 750mb) for most latitudes".
negative regions "are caused by large values of the term containing the
vertical shear".
Since V = 1/4 - 7y/u, and ' is predominantly
positive, negative regions of qy are positive regions of V.
Fullmer's
calculations also show much stronger gradients of qy at 800mb and
below than at higher levels in the atmosphere.
Finally, regarding the southern rim of the potential well, one is
left only with speculation, given the available data.
do not extend south of 220 N, either.
Fullmer's results
However, a good case can be argued
for the possibility of a southern barrier existing in the V field, based
upon considerations of the climatological wind field, and subsequent
analogies to the character of the wind field in the region of the
northern barrier.
It is clear from the profiles of 7 given in figure 4 that the
southern "edge" of the jet, as mentioned earlier, lies somewhat to the
south of 260N. 1
At 260 N,
decreasing latitude.
ST is observed to be decreasing steadily with
Now, in order to guarantee a positive value of V,
in a region where 7 > 0, the quantity (3 - iyy - uzz + Tz) must be
less than zero.
The following arguments apply to the troposphere, below
roughly 200mb, and to the north of the critical latitude where 7
changes
sign.
To start with, 0 is constant with
= 1/3 in dimensionless units.
0
Based upon climatology, the zonal wind south of 22 N continues to
increase with height, but to a lesser extent than it does in mid-
1
The term "edge" of the jet is used in the sense of Malguzzi's theory,
i.e. that point where d falls below some value U0 , where U0 is
small compared to typical within jet magnitudes.
-44-
latitudes.
The conclusion is that while 7z remains positive south of
the domain, it is smaller in magnitude than at latitudes between
22-45 0 N, say.
Hence, the sum (8 +
Ez)
must be outweighed by the sum
(Myy+ Uzz), to accomplish a positive value of V.
Climatological
profiles indicate (see, e.g., Holton 1979, chapt. 6) that while T
0
continues to decrease to the south, equatorward of 22 N, it does so less
rapidly -- i.e. there is an inflection point where jyy = 0 somewhere
(between 15 -200 N on average, at 500mb).
point, -yy > 0.
South of this inflection
The situation is the mirror image of that near the
northern rim of the potential well, where another inflection point
occurs with 'yy > 0 to the north of it.
In addition, climatological
profiles indicate that on the average uzz will be positive between
10-200 N, below roughly 300mb;
again, this is an analogous situation to
the one in the northern rim region.
The fields of 7iyy and 'zz
are, however, quite sensitive to the
profile, and hence these fields most likely display a good deal of
case to case variability.
So, while the region just to the south of the
jet appears favorable for a southern rim of positive V values to occur,
the actual existence of this rim probably varies widely.
The regions
around the northern and southern "edges" of the jetstream represent
transition zones from strong gradients of 7 associated with the
mid-latitude jet, to much weaker 7 gradients away from the jet.
In
general, these transition zones are regions of positive Uyy and u,but in order for a "wall" of positive V values to exist, the rather
severe condition that (uyy + uzz
)
> (8 + 7z) must hold.
This
condition will certainly not hold in general, but as in the observed
northern rim region, it can be expected to hold sometimes to form a
-45-
southern rim.
It would be interesting to obtain tropical data with
which to test this hypothesis in a real situation.
All of the above discussion about the southern rim of the potential
In an actual blocking
well, it must be remembered, is pure speculation.
case there is in fact no reason to expect the observed i field to
resemble the climatological mean field, the basis of much of the above
reasoning.
In any event, even if nothing at all is suggested about the
V field south of 260 N, there is nothing in the observations to disprove
the existence of a southern rim;
the data simply does not extend far
enough to the south to prove the point either way.
A few more general comments about the V function profiles in figure
5 will be made.
The first concerns the potential well itself -- the
region inside of all the barriers discussed above.
The ideal well in
Malguzzi's thesis consists of a broad, flat bowl-shaped region, with V
values uniformly between -1 and -3 in dimensionless units, surrounded on
all four sides by the rim of positive V values.
The observed V profiles
are also relatively flat inside the rim, with values ranging for the
most part between -1 and -4.
There are a few isolated regions where the
well is deeper than a value of -4, most notably just to the south of the
northern rim below about 300mb, and at around 700mb at most latitudes.
There is nowhere, in the domain, that V becomes positive in the interior
of the well.
In addition, comparing the four profiles in figure 5 it is seen
that they are all quite similar in nature, in a broad sense.
They all
display erratic fluctuations and large gradients of V near the lower
edge of the domain, positive V regions in the lower stratosphere, and
evidence of a rim of positive V values extending vertically near
-46-
64-70'N;
these regions surround a potential well consisting of a
relatively flat V profile with dimensionless values generally between -1
and -4.
The major significant difference among the profiles is the
varying vertical extent of the area of positive V associated with the
northern rim.
Thus, varying the width of the longitude band over which
the zonal averaging is carried out, to the degree done here, does not
alter the general nature of the V profile in any cataclysmic way;
however, it does effect the character of individual features to an
extent which is significant for the theory at hand.
While all four
profiles show gaps in the rim, where V fails to exceed zero, these gaps
are considerably narrower in figure 5d than in 5a, due to the greater
development of the northern wall in the former case.
The potential well
in figure 5d exhibits a strong likeness to the theoretical potential
well, while in the 90 0W< x <30 0 W case the gaps in the rim raise doubts
about the structure representing a potential well at all.
Therefore,
the 90*W< x <60°W zonal average, more accurately representing
the upstream mean conditions, is clearly preferable;
the V field cross
section based on this narrower longitude average lends appreciable
support to the assumptions going into the theory described in section
II.
A few words concerning the smoothing procedure for the V profiles
are now noted.
As the threshold value for 1u1 is set at 4m/sec in the
procedure, only those gridpoints at which 17
< 4m/sec will have V
values which are modified from the unsmoothed field.
In the present
case of the Atlantic positive blocking composite field, for day +5 lag,
there are very few gridpoints which feature I1
< 4m/sec.
Therefore,
the smoothed and unsmoothed V cross sections look almost identical.
-47-
The
smoothed V field profile for the 90°W< x <60'W longitude average is
presented in figure 6, to be compared with figure 5d.
The only
differences occur at the extreme high latitude edge of the domain, and
0
in a small region between 600 N and 72 N extending upward to a maximum
level of close to 400mb at 660 N.
The rather discontinuous looking
contours north of 80'N in figure 5d appear much smoother in figure 6;
also, the deep low level breach in the northern rim at 64*N below 600mb
is considerably shallower in the smoothed profile.
Although the impact
of the smoothing procedure is modest in this particular case, it will
play an important role in some of the other cases to be looked at, which
include more substantial regions of light mean zonal winds particularly the individual cases of blocking, which have not had the
previous smoothing afforded by the compositing procedure.
Indeed, the
smoothing technique is essential in many of the V profiles, the general
character of which would otherwise be effectually obscured due to the
huge magnitude of V at a few points with small u.
In the following subsections, V function profiles calculated from
other flow patterns will be considered, in an effort to learn more about
the structure of V in various situations, and its relationship, if any,
to blocking.
B. Composite of twelve cases of Atlantic negative blocking.
Cases of "negative blocking" in the eastern Atlantic region,
consisting actually of an anomalously strong zonal flow, have been
composited in a completely analogous manner to the positive blocking
cases.
The selected cases are again those identified by Dole (1982).
-48-
The case dates of the twelve day zeros are given below:
Case #
Case #
Day Zero
12/31/65
1/17/66
1/18/67
2/13/67
1/ 8/69
1/ 8/70
1
2
3
4
5
6
7
8
9
10
11
12
12Z
0Z
0Z
OZ
0Z
OZ
Day Zero
1/ 4/71
1/ 5/72
1/30/72
11/30/72
1/ 1/74
1/24/74
OZ
12Z
12Z
12Z
OZ
0Z
The history of the evolution of the pattern from day -5 through day
+5 is not presented here.
Rather, only the day +5 results are
considered, representing the fully developed pattern.
Figure 7 shows
the composite 500mb height field for day +5, and figure 8 the
corresponding height anomaly field.
The height anomaly map displays a
pattern in the Atlantic-European sector which is strikingly similar to
the pattern on the day +5 height anomaly map for the positive blocking
composite, with the centers having opposite polarity however (see fig.
3k).
This observation is what led Dole (1982) to characterize these two
situations as opposite phases of the primary regional pattern of low
frequency variability.
The strong geostrophic zonal flow, peaking near
300 W longitude, is clearly seen in figure 7.
The upstream mean zonal wind is again considered as the zonal
0
average of u over the four longitude bands 90 W < x < 60,50,40,30*W.
Unlike in the positive blocking composite, here the
! field does not
appear to show any major differences depending upon the width of the
longitude band.
Therefore, only the meridional cross section of u
corresponding to the 90W < x < 60°W longitude band is presented here,
in figure 9.
As in the positive blocking composite there is a jet
centered near 40*N, however here the magnitude of the jet is
considerably stronger, with 7 peaking at 48m/sec as opposed to 35.3m/sec
-49-
in the positive blocking case.
Correspondingly, the meridional and
vertical gradients of 1 are stronger in the present case.
Finally, the computed V function cross sections for all four
longitude bands are shown in figures 10a-d.
These profiles incorporate
the smoothing technique, which is made necessary due to the zero line of
i near 73-76 0 N.
In these figures, evidence supporting the existence of
a potential well is much more tentative than in the positive composite V
profiles (fig. 5).
In all four of the figures, the lower edge of the
domain is characterized by the same strong V gradients as were observed
in the positive composite figures.
The major differences between this
case and that of the positive composite are with respect to the northern
wall of positive V values.
Significant regions of positive V values, resembling a northern
wall, are observed in all four of the present figures between -60° 700 N, above about 370mb.
However, assuming, as suggested previously,
that the tropopause plays a major role as an upper barrier to wave
propagation, one should focus attention mainly on the tropospheric
portion of the domain.
Below ~300mb there is evidence of a narrow
area of weakly positive V values near 520 N on some of the plots.
This
area, which conspicuously changes shape as a function of the width of
the longitude band over which u is zonally averaged, cannot readily be
considered as the northern rim of a potential well, however.
This
behavior is in contrast with the positive composite V profiles (fig. 5),
which show a pronounced tropospheric vertical wall of positive values.
The problem here is that there is no standard against which the
magnitude and size of positive V areas can be measured to determine
their significance as barriers, in the sense of the theory.
-50-
Thus,
although much weaker than the northern wall in the positive composite
case, the area of positive V in the present figures near 520 N could be
important.
What can be said, at least, is that the V profiles for the
positive blocking composite case (fig. 5) provide better support for the
model (i.e. for the existence of a well) than do those for the negative
composite case, where the northern wall of V > 0 is concentrated in the
upper layers of the domain.
Having looked at the V profiles associated with two very different
flow patterns, and found significant differences, it now seems relevant
to examine the V function calculated from the climatological mean
geostrophic zonal wind field, and compare that to what has been seen
already.
Such is the topic of the next subsection.
C. The V field calculated from the climatological mean geostrophic wind
profile.
Recall the winter climatological mean 500mb geopotential height
field shown in figure 2.
The V function profiles calculated from this
climatology, for the four longitude bands over which u is averaged, are
presented in figures lla-d.
The figures have been smoothed to remove
spurious values from the northern edge of the domain.
These results are
presented to help determine the fundamental nature of the V field, based
on average conditions.
The interesting features which arise in the V
function cross sections corresponding to particular flow patterns, such
as blocking, can then be identified as those regions where V differs
significantly from this climatology field.
It is noted that a different
climatological mean V field could be constructed by first computing the
V field for every observation time during the winter season, and then
-51-
averaging these together.
This method will not, in general, yield the
same fields as those presented in figure 11, which are calculated by
first averaging together all the height fields, and then computing V.
The first observations to come to mind upon looking at figures 11
are the relative flatness of the profiles over the bulk of the domain,
and the absence of a vertically oriented region of positive V values
which might represent the northern rim of a potential well.
The
positive values along the northern edge of the domain, confined almost
entirely to latitudes above 800 N, are too far poleward to have any
significance with regard to Malguzzi's theory;
negative values of U in this region.
they are caused by
The high degree of similarity
among the four figures is also noted, which is expected based on the
smoothness of the climatological wind field, which varies only slowly in
the zonal direction.
Another characteristic of these four cross sections is the
existence of strong gradients of V along the extreme lower edge of the
domain, below roughly 750mb, at most latitudes.
This characteristic was
also noted in all of the earlier V profiles, and owes its existence to
the physical lower boundary at the ground.
A final feature to note on
these figures is the area of positive V values in the upper portion of
the domain, between 110-150mb, to the north of 45'N.
This area
displayed a tendency for positive V values on the earlier profiles too.
Hence there can be assumed a climatological bias in favor of positive V
values in this region.
In a sense, then, the vertical component of the energy trapping
assumed in the theory appears to be more climatologically determined,
while the meridional component of the trapping depends more upon the
-52-
detailed structure of the mean zonal flow.
The lower boundary of the
Earth's surface is always present, while the tropopause exists regularly
and is well defined in climatological averages;
in the meridional
direction, on the other hand, the current "climatological" V profiles
display very little variation, and show no evidence whatsoever of a
vertically oriented wall of positive values which would represent a
barrier to wave propagation.
The absence of meridional V gradients, and
in particular of a northern wall, represents the greatest disparity
between this current V profile and the earlier V profiles associated
with blocking patterns.
In order to more quantitatively assess the variations between the V
profiles shown thus far, difference fields of V are calculated for the
following cases:
: Zonal averages
:over long. bands
Difference Field
Figure
IV( + blocking composite) - V(climatology)}
: 90ow< x <
:
30,40,50,60 0 W
{V( - blocking composite) - V(climatology)}
: 900 W< x <60 0 W
13a
0
0
{V( + blocking composite) - V( - composite)} : 90 W< x <60 W
13b
12a-d
The fields of V corresponding to the positive and negative blocking
situations, from which the climatology V value has been subtracted out
at each gridpoint, pinpoint those areas displaying anomalous V values.
Figures 12 and 13a can be thought of as anomaly maps for the V
function.
These fields, on the whole, are quite flat, displaying very
few contours other than the zero contour.
Most of the exceptions occur
along the lower edge of the domain, where the large gradients of V
-53-
naturally make the difference fields rougher, and in the very high
latitudes, north of around 750 N, where sign changes and generally weak
values of ' add to the roughness.
These areas aside, the only region in
figures 12a-d showing large magnitude anomalies, say in excess of 4
dimensionless units, is the region where the northern potential well rim
is located in the positive blocking profile.
The northern rim of the
potential well, and the enhanced pit just to its south, are therefore
the main anomalous features of the V field associated with the positive
blocking composite.
Figure 13a shows significant positive anomalies associated with the
positive region near 60-68 0 N, above ~370mb, in the negative blocking
composite V field (fig. 10d).
In addition, somewhat lesser positive
anomalies are seen to be associated with the weak tropospheric positive
V region near 52-540 N.
The zero line of U near 740 N for this case makes
the profile quite rough there.
Figure 13b assesses the differences between the V fields of the
positive and negative blocking composites.
The strong tropospheric
positive center near 62-68 0 N and negative center near 52-56 0 N reveal the
locations of the most significant differences.
The positive center is
due mainly to the large positive anomalies in the positive blocking
composite case, while the negative center has important contributions
from both positive anomalies in the negative composite case and negative
anomalies in the positive composite case.
The flat profile south of
500 N in figure 13b indicates the similarity of the fields in the two
cases there.
Finally, one further comparison of the V fields for the positive
blocking composite, negative blocking composite and climatology cases is
-54-
made.
Plots of V versus latitude for a slice at the 500mb level, for
the zonal averages of
figures 14a,b.
7
taken between 90 0W< x <50,60°W, are shown in
These figures clearly show the dominant northern wall of
V > 0 in the positive composite case, at least at 500mb;
the negative
composite case appears to mirror climatology much more closely,
suggesting the absence of a potential well.
Now that the structure of the V function has been examined in some
detail for these composite fields, it is worthwhile to take a look at
its behavior during some individual cases of Atlantic blocking.
D. Individual cases of Atlantic (positive) blocking.
The V function fields are calculated for a number of individual
cases of Atlantic blocking, each displaying its own uniqueness while
satisfying the set of criteria used to define blocking.
Many of the
cases which form the composite are looked at on an individual basis.
As
the geopotential height fields are not smoothed by any compositing
process, they display much more small scale variability than the
previous fields considered.
In particular, there is a much larger eddy
component in the geostrophic wind field, and the jetstream tends to be
more concentrated and much more meandering than in the composites.
The application of these individual case fields to the purpose at
hand has its pros and cons.
On the positive side, these represent the
real events which are being modelled by the theory, uncontaminated by
any averaging or approximation scheme.
In addition, single cases whose
blocking structure strongly resembles the dipole of Malguzzi's solution
can be selectively picked out for analysis.
Such a good duplicate of
the theoretical dipole structure would be extremely difficult to
-55-
reproduce in a composite field of many cases.
On the negative side, the
amount of detail and small scale structure present in these individual
case fields blurs the large scale features which an attempt is being
made to model.
If the large scale flow pattern is thought of as the
desired signal, then the small scale structure present on these single
case maps can be considered noise.
In this light, the compositing
procedure is beneficial in that it filters out the noise while
maintaining the signal.
In some of the cases it will be seen that the
zonal flow upstream from the block resembles the mean zonal flow about
as little as the flow in the blocking region itself.
It is very
difficult to obtain a regional mean zonal wind field to use in the
calculations.
The flow upstream from the block in the composite field,
on the other hand, is relatively representative of a regional mean zonal
flow.
Nonetheless, a few cases are found which seem reasonably well
suited to the purposes of this study.
i. The case of 12Z, December 31, 1971.
This case combines a number of attractive features, fitting the
design of Malguzzi's theory very closely.
field for December 31,
in figure 15.
The 500mb geopotential height
1971, at 1200 Greenwich Mean Time (GMT) is shown
A strong blocking pattern is observed over the eastern
Atlantic and western Europe (300 W - 30 0 E), with a well developed vortex
pair evident in the geostrophic flow.
The double vortex nature of this
flow regime is much stronger than that of the positive blocking
composite, where the high latitude anticyclonic vortex is much more
highly developed than the lower latitude cyclonic circulation.
Hence
this current flow pattern is in closer agreement with Malguzzi's
modelling results.
The split westerly jet in the blocking region is
-56-
very unambiguous.
In addition, the zonal flow upstream of the block is
quite smooth, and appears to be representative of a mean zonal flow;
it
is relatively free of the small scale distortions which reach
significant magnitudes in many of the individual cases.
Another attractive aspect of this particular situation has to do
with the strong wave number one component observed in the hemispheric
flow pattern.
The magnitude of the wave number one component is
determined by the degree to which the flow is not centered around the
north pole, but rather has one broad ridge and one trough around the
hemisphere.
In the present case, it can be seen merely by inspection
that a large amount of the total wave energy associated with the flow
will be in the wave number 1 component;
the strong ridge is centered
near 00 longitude, with the trough near the dateline.
The advantage of
this asymmetric configuration, from the perspective of the present
study, is that the entire blocking pattern is located to the north of
its average position, as determined by the composite 500mb height field
(the northern 500mb anticyclone is centered at 610 N here, as opposed to
490 N in the composite, while the cyclonic vortex center is near 450 N
here and near 350 N in the composite).
Therefore, the southern portion
of the pattern, which was too far south in the composite field, is now
within the domain covered by the data sets.
In a sense, one can now
"see" the entire blocking pattern.
The central axis of this blocking pattern lies near 0' longitude at
500mb, as opposed to the axis near 25*W in the composite pattern.
Accordingly, the longitude band over which the upstream zonal wind is
averaged to obtain the *
field is shifted eastwards to 650 W < x < 350 W.
This Ij field is presented in figure 16.
-57-
Another particularly attractive
feature of this case is noted at once upon examining the figure -namely, the clearly defined edges of the mid-latitude jet near 340 N and
A fairly strong jet maximum is located at (490 N,290mb).
600 N.
This iT
field displays considerably more detailed structure, and stronger
gradients than the 7 fields in the composite cases.
Figure 17 shows the V field profile for this case, calculated from
the zonal wind field in figure 16.
As expected, the V values fluctuate
much more widely here than in the previous composite cases, even with
the smoothing procedure.
The sharp oval shaped closed low center
between 60-640 N, and some of the large and abrupt fluctuations in V at
high latitudes are caused by sign changes of T.
Other large
fluctuations in V are caused by the strong gradients of u.
However, a
0
very clearly defined potential well is apparent between 36-56 N, with
all four edges of the rim visible.
Indeed, this feature comes
remarkably close to matching the theoretical ideal, with the northern
and southern rims stretching virtually across the domain.
Gaps in the
rim are practically nonexistent, the one main exception being the narrow
gap in the southern wall between 600-750mb.
The interior of the well is
the flattest region in the entire domain, with values for the most part
near -2 in dimensionless units.
The northern wall rises abruptly out of
the well and contains some large values of V.
A major difference between this V field and those of the composite
cases is observed in the meridional width of the potential well.
Here,
the well is narrow and sharply defined, spanning only 200 of latitude,
while in the composite cases the well stretches over a much greater
latitude band.
the 5 fields.
This difference is due to the differing characters of
In the present case, there is a concentrated mid-latitude
-58-
jet in the i field, relatively narrow in meridional extent, and with
clearly defined edges.
of the jet as the
Strong wind shears are observed near the edges
ETvalues
drop off rapidly.
In the composite cases, on
the other hand, the jet is observed to be very broad, with
7 values
decreasing much more gradually away from the jet maximum (compare figs.
4d and 16).
This case, then, is particularly well suited to looking at the V
function.
A strong blocking pattern is evident in the eastern Atlantic,
with a well developed vortex pair in the 500mb flow.
The upstream mid
latitude jet is located far enough to the north so that its southern
edge is within the domain of the available data, and is relatively
smooth and zonal.
The V profile contains a potential well which is much
narrower meridionally than those observed in the composite cases of
blocking, but with a structure which matches that of Malguzzi's
theoretical well remarkably closely.
This case stands as solid
observational evidence in support of Malguzzi's theory.
However, this
case seems to be the exception, rather than the rule, among the
individual case dates examined, with regard to its accomodation of the
theory in section II.
Some of the difficulties which arise in
individual case analyses will be seen below.
ii. The case of OZ, December 23, 1976.
This case was chosen due to the well developed anticyclone/cyclone
dipole in the 500mb flow in the region around 20-30 0W longitude.
This
structure can be seen on the 500mb geopotential height map for this
time, presented in figure 18, and strongly resembles the structures of
Malguzzi's theoretical solutions.
It is noted that the dipole here
occurs at even higher latitudes than in the previous case of 12/31/71,
-59-
with the high center near 670 N, and the low center near 510 N.
The problem of this case arises in determining 7, the upstream,
For upstream from the dipole is a massive
zonal average wind field.
trough, with the mid-latitude westerly jet centered near 400 N -the south of even the low center of the dipole.
well to
This massive trough was
an unusually strong and persistent feature through a good part of the
1976-77 winter, associated with record breaking cold over much of
eastern North America.
If the zonal wind is averaged over the standard
longitude band, 900 W< x <60'W, it is found that on the northern side of
this major trough (north of ~56 0 N) the geostrophic zonal flow is
easterly.
This
7 profile is shown in figure 19, and the easterly
maximum at 640 N is seen to be quite striking.
Strong meridional
gradients of 7 occur almost everywhere in the domain, and a second
strong westerly maximum occurs at 800 N.
The question arises, however, of what is the best upstream
longitude band over which to average the geostrophic zonal wind to
obtain 7.
As opposed to the earlier cases looked at, here the band
90*W< x <60 0 W is a region of strong height anomalies.
There is so much
smaller scale structure superposed on the mean zonal flow (in a time
averaged sense) in this particular case that different choices of the
longitude band will lead to greatly differing u profiles, and hence
differing V structures.
The jetstream displays large amplitude meanders
near and upstream of the blocking region (fig. 18), and thus there seems
to be no single longitude band over which a zonal average of u leads to
a representative ~ profile.
In other words, there is no nearby upstream
longitude band in which the height anomalies are small.
These large
upstream height anomalies, and the corresponding sensitivity of the
-60-
results to the exact choice of the longitude band on which to work, is a
real problem and a serious drawback to applying the present analysis
technique to single case data sets.
The problem is greatly remedied by
compositing a number of geographically colocated blocking events to
obtain much smoother wind fields.
In any event, the V function is calculated based on the U field
averaged over 900 W< x <60 0 W (fig. 19).
This V profile, with the
smoothing procedure applied, is presented in figure 20.
expected in this case, the results are not good.
As might be
The dashed in lines
show the positions of the U = 0 lines in fig. 19, and represent
discontinuities in V.
The plotting routine, using data from limited
gridpoints, attempts to smooth over these discontinuities.
There is no
northern rim of positive V values along the northern edge of the mid
latitude jet (~50-550 N), although the region corresponding to the jet
maximum does look like the interior of the theoretical potential well.
The upstream westerly jet in this case is positioned so far south
relative to the dipole as to seem almost decoupled from it.
Perhaps an
average over a different longitude band would yield a better potential
well in V;
or, possibly, formulating the problem in terms of a local
"zonal" wave guide which is actually oriented slightly southwest to
northeast rather than purely east-west (this would reduce or eliminate
the easterly jet in the M profile, and better line up the dipole with
the upstream jet) would give better results.
Several of the cases used in the blocking composite are next looked
at on an individual basis, in order to get a better feeling for the
behavior of V in single cases.
-61-
iii. Some further individual cases of blocking.
Four cases from the positive blocking composite are examined here
on an individual basis.
The cases displaying the most well defined
dipole structures in the blocking region are chosen.
time +5 days lag, of these four cases are:
3/1/70 12Z;
and 12/8/75 OZ.
The case dates, at
1/25/68 OZ;
12/30/68 OZ;
For each case, the 500mb geopotential
height field, and the V function profile based on a longitude average
900 W< x <60°W, are shown in figures 21-24.
All of the V profiles are
smoothed, but in figures 22b and 23b the threshold U value has been
increased to 8m/sec to provide some additional refinement of a rough
field.
The most successful of these cases, from the standpoint of the
theory, are those of 3/1/70 and 12/8/75, in figures 23 and 24.
In the
3/1/70 case, the dipole structure is particularly strong, as evidenced
in the 500mb height field, and the upstream flow is reasonably smooth
and zonal.
The V profile shows a very strong northern rim of positive
values near 520 N, stretching vertically across the domain.
In figure
24, the blocking dipole is much less dramatic than in some of the other
cases.
However, the V profile shows evidence of a potential well along
with both northern and southern rims;
the interior is not as flat as
theory would predict, with two fairly deep lobes split by a relative
maximum of V, approaching zero.
The other two V profiles (figs. 21,22) show the contaminating
effects of anomalies and small scale structure in the wind field
upstream of the block.
900 and 60°W;
Figure 21 has a "mini-split" of the jet between
so, while a northern potential well rim is seen near
520 N, the "interior" region of the well contains another maximum of
-62-
positive V values, rather than being flat.
Figure 22 shows an
exceptional dipole blocking structure, but the upstream flow is plagued
by a similar easterly wind between ~55-70 0 N as was seen in the
12/23/76 case.
The corresponding V profile is erratic, with no well
defined northern rim.
When working with individual case data sets, zeros and negative
(easterly) regions of T! seem to arise quite often, and are not confined
exclusively to the very high latitudes.
Malguzzi assumes a theoretical
1 field which is positive everywhere, and thus his theory does not
address the problem of u zeros;
in the composite fields, zeros of u are
confined to the polar latitudes, and can therefore more easily be
disregarded.
Perhaps, then, due to the special problems that arise with
T zeros, a distinction should be made between two classes of individual
cases:
those that have significant regions of easterlies in the u
field, and those that do not.
An excellent example of a case containing
easterlies is that of 12/30/68, shown in figure 22, where the easterlies
prevail between ~55-70*N at 500mb.
Another example is the 7 field in
figure 19.
Consider the zero wind line near 550 N in the 12/30/68 case, along
which V is undefined.
If qy < 0, then as this zero line is approached
from the south, V goes to positive c, while as it is approached from the
north, V goes to negative
will hold.
=. If 4y > 0, then the opposite situation
In either case, there will be a very narrow zone of positive
V values with magnitudes approaching infinity on one side of the u = 0
line or the other.
rim?
Does this zone represent a northern potential well
The answer to this question is uncertain, and no attempt will be
made to answer it here.
It may depend upon the meridional width of the
-63-
positive V zone and/or the wavelength of the waves.
In any case, it
will be stated only that the possibility exists for a barrier to wave
propagation along a ,u= 0 line.
Theoretical work (e.g. Held 1983)
suggests that at a critical latitude where i = 0, an incident Rossby
wavetrain will be at least partially absorbed, and perhaps partially
reflected.
E. Time evolution of the V profile for the Atlantic positive blocking
composite.
This final set of results is perhaps the most striking, and the
most supportive of the theory.
Returning to the Atlantic positive
blocking composite, seen in section A, the V function is now computed at
each of the eleven lag times from day -5 to day +5, to get some idea of
how the potential well evolves in time.
Almost coincidentally with the
establishment of the eastern Atlantic blocking pattern, a northern
potential well rim is seen to arise in the V profiles, which persists
through day +5.
Such a rim is absent at all times prior to the onset of
blocking.
The theory presented in section II was for stationary solutions,
and the technique cannot be generalized to include time dependent
behavior.
However, in his thesis Malguzzi does look at the full time
dependent problem using a different technique, namely that of truncated
orthonormal projection.
It is there found that the coherent (dipole)
structure obtained previously (in the long wave limit) is recovered by
solving a severely truncated dynamical system, and that the dynamics of
the coherent structure can be studied by analyzing this system.
This
technique is rendered more powerful than the previous one due to greater
-64-
mathematical approximations being made.
It is encouraging to know that
the coherent structure solutions are recovered in a time dependent
problem, and this fact suggests the relevance of determining the
structure of the V function at each time during a sequence of
observations.
The behavior of the V fields can then be compared with
the occurrence of blocking.
Recall that the time series of 500mb geopotential heights, from day
-5 to day +5, is shown in figures la-k.
The corresponding V function
profiles, with the smoothing procedure applied, and for a zonal average
of the upstream zonal wind taken over the longitude band 90'W< x <60 0 W,
are presented in figures 25a-k.
This time series exhibits some very
interesting behavior.
Looking first at the day -5 V profile (fig. 25a), there is no
evidence of a potential well structure.
In fact, this profile looks
very similar to the profile calculated from the climatological wind
field (compare with figure lid), featuring a relatively flat topography
over most of the domain, and a broad high latitude minimum of V.
The
small region of positive V values near 620 N is one of the few departures
from the climatological profile.
This insipid nature is consistent with
the earlier observation of a lack of significant features on the day -5
height map.
The day -4 profile of V is slightly more interesting,
displaying the most significant vertically elongated band of positive V
values seen prior to the onset of blocking.
This band is centered near
48 0 N, but is still unimpressive compared to the vertical bands of
positive V which are noted after onset, and questionable in its capacity
to represent the northern rim of a potential well.
By day -3, this area
of positive V is greatly reduced, and the V profile overall closely
-65-
resembles that based on the climatological winds once again.
patch of V > 0 near 60°N continues to persist.
The small
The day -2 profile of V
is astoundingly similar to figure lid (climatology).
Up to this point,
the V profiles have been fairly flat and quite similar to the V profile
based on climatological winds, with little or no evidence of any
embedded potential well structure.
Simultaneously, there has been no
observed blocking pattern in the eastern Atlantic.
As the blocking pattern becomes established in the eastern
Atlantic, a major change occurs in the nature of the V profiles.
On day
-1, with the blocking pattern seen to be in its incipient stage of
development (fig. le), a major vertical ridge of positive V values is
noted near 70°N.
Although quite far to the north, this feature is
substantial enough to bear watching, and is in sharp contrast with the
climatology profile of figure lid.
The small area of V > 0 near 480 N is
observed to be somewhat rejuvenated at this time as well.
The changes
in the V profiles between day -1 and day 0 are somewhat uninteresting.
The ridge near 700 N is considerably reduced in vertical extent on day 0,
but still undeniably present in the middle troposphere;
V > 0 band at 500 N is substantially enlarged.
meanwhile, the
It is still the case that
any potential well is, at best, loosely defined.
The changes which occur in the V profiles between day 0 and day +1
seem much more consequential.
Before describing these changes, however,
it is interesting to note that the development of the blocking pattern
at 500mb also seems much more substantial between day 0 and day +1 than
during the previous day.
The blocking ridge appears rather flat on day
zero, while it extrudes decidedly northward on day +1.
For example at
200 W, the ridge axis at this time, the 500mb height rises during these
-66-
two days can be deduced from the following table at three different
latitudes:
500mb heights in decameters
50°N
600 N
700 N
Day -1
563
533
515
Day
0
566
535
516
Day +1
570
546
521
It is seen that the ridge strengthens little between day -1 and day 0,
while it builds northward significantly between day 0 and day +1.
Getting back now to the V profiles, on day +1 (fig. 25g) there is the
first evidence of a sustained vertical wall of positive values, which
can demonstrably be said to represent the northern rim of a potential
well.
This wall seems to be the result of a northward displacement of
the upper tropospheric day 0 ridge near 500 N to about 580 N, and a
simultaneous merging with the mid and lower tropospheric ridge near
700 N.
By day +2, the wall of positive V values has developed into a
strong and dominant feature in the V profile, and become better aligned
in the vertical than on the day +1 profile.
which V = 0, is seen to lie near 620 N.
The edge of the well, along
This strong feature is highly
anomalous with respect to the climatology profile of figure 11d, and a
fundamental link in completing the potential well structure.
In the
remaining three figures, this vertical rim of V > 0 values is noted to
persist, with some diminution on the day +5 profile, which was looked at
earlier.
The feature is so prominent on the day +2 through day +4
profiles as to leave little doubt about the existence of a potential
-67-
well in the V field, assuming a southern rim exists also.
A further look is taken at the time evolution of the V profiles by
plotting, for each observation time, the value of V at 500mb versus
latitude.
The first major
These plots are presented in figures 26a-k.
spike of positive V values occurs on the day 0 profile at 700 N,
representing the northern rim of the potential well.
If a southern rim
exists, there will be another major spike somewhere off the left edge of
the graph.
The major spike of V > 0 is present on all of the plots
later than day 0, but on none of the earlier plots.
This point is
further illustrated in figures 27a,b which show the curves corresponding
to days -5 through 0, and days 0 through +5, plotted together.
The pre-
blocking picture, in figure 27a, shows an assortment of patterns, but
with no major positive regions of V apart from day zero;
figure 27b, on
the other hand, displays a major wall of positive V values between
~60-72 0 N.
In addition, the various curves behave more uniformly, and
with smaller amplitude oscillations, in the "interior" of the well (the
region south of 580 N) in figure 27b than in 27a.
Finally, in figure 28, a cross section of the 500mb V values, with
time on the vertical axis and latitude on the horizontal, is presented.
The northern barrier is seen to arise near day -1 at 70'N, and persist
through day +5.
This figure provides a particularly clear vision of the
time evolution of V, though admittedly only at 500mb.
Judging from
figures 25a-k, however, the behavior at most tropospheric levels would
be quite similar.
So, in the case of the composite fields at least, it is found that
prior to the onset of blocking no potential well exists in the V
profiles, while approximately simultaneously to the establishment of
-68-
blocking a potential well forms, and persists during the course of the
blocking event.
Thus, while this behavior implies that the formation of
a potential well in the V profile may be of little use as a predictor of
blocking, it is extremely consistent with Malguzzi's theoretical
reasoning.
However, although no cause and effect relationship is being
rigorously argued here, and many other processes most likely also
contribute to the development of a blocking event, it is possible that a
predictor for a potential well in the V function profile (calculated
from a zonal average of the zonal winds over certain preferred
geographical regions) may be at least partially a predictor for
blocking.
In other words, a process which will tend to create a
potential well in the V profile may merit further investigation into its
possible significance as a contributing mechanism to the development of
blocking.
This notion is consistent with the idea, for example, that
blocking in an eastern ocean (i.e. Atlantic or Pacific) often follows a
major cold air outbreak off a continent and into the western part of the
ocean;
such a cold air outbreak tends to create a strong baroclinic
zone over the western ocean, which is accompanied typically by a
concentrated jetstream -- a feature which is quite often manifested on
the corresponding V profile by the presence of a potential well
structure.
F. A few considerations regarding N 2
It is recalled that the theory presented in section II, where the
expression used to calculate V was derived, assumed N
throughout the domain covered by the data.
2
to be constant
This assumption eliminates
the temperature dependence of the quasi-geostrophic potential vorticity
-69-
(q), and therefore also makes V a function of the upstream mean zonal
In section II, the constant N 2
geostrophic wind (t) field only.
assumption was made in order to follow the approach of Malguzzi's
theory, however no observational justification was provided.
It is now
the task to try and assess how realistic this approximation is, and what
significance a variable N 2 might have on the results presented.
One can begin by considering once again the expression for N 2 in
log-pressure coordinates:
N2 = R
Hg
Since R, H 0 , and
K
z
+
.
(5)
Hg
are constant, spatial temperature changes must be
balanced by spatial changes in the vertical gradient of temperature in
order for N 2 to remain constant.
Observations (e.g. Holton 1979) show
that in much of the interior of the troposphere vertical temperature
changes and vertical changes of aT/az are indeed of opposite sign, but
it is not so clear to what extent they actually offset each other.
However, near the tropopause and in the lower stratosphere both T and
aT/az increase with height, indicating that N 2 will certainly not be
constant in these regions.
Therefore, it might be anticipated that any
errors arising due to the constant N 2 assumption will be particularly
serious near the tropopause.
The average tropopause level can be quite clearly determined by
plotting the zonal and wintertime average field of equivalent potential
temperature (6e) as a function of latitude and height.
This field,
calculated from a December, January, February three month average over
many seasons, was provided in Oort (1983). The plot, shown in figure 29,
-70-
indicates that in the long term wintertime average, the tropopause
extends from just above 200mb at 260 N, gradually downward to around
300mb at 800 N.
In an attempt to determine the degree of variability of the N 2
2
field, equation (5) was used to calculate the value of N at each
gridpoint using the data from the composite of twelve cases of Atlantic
positive blocking at day +5 lag, and for a zonal average of T between
900 W< x <30 0 W.
Bicubic splines were used to calculate the vertical
The results of this calculation are shown in
temperature gradient.
figure 30a, where the contour N 2 = 1.56 x 10
4
sec -
2
has been drawn in,
corresponding to the assumed constant value in the theory.
In addition,
contours corresponding to half this value and to twice the value are
also drawn.
It is seen that there is a fair amount of vertical
variability, with a considerably lesser latitudinal variability.
However, the value N 2
=
representative average.
1.56 x 10-4 sec - 2 does seem to be a fairly
These observations provide the justification
for allowing N2 to vary vertically but not horizontally, and for calling
the value 1.56 x 104 sec - 2 the characteristic N2 scale, in the
calculations presented below.
For comparison, the results of an N 2 calculation by Youngblut and
Sasamori (1980) are also shown (figure 30b).
These authors assumed N 2
to be a function of height only, and computed it using the hydrostatic
approximation and hemispheric mean data for January 1963.
The two
independent calculations show quite good agreement, in general.
For the
2
Youngblut and Sasamori results, the change in the value of N between
each successively higher level is also shown, to provide some feel for
the vertical gradient of N 2 at the various levels.
-71-
0
Finally, in figure 30c a meridional average between 22-70 N, at
each level, of the values shown in figure 30a is presented.
corresponding nondimensional values are also given.
The
The data were
averaged between 22-70*N because this is the region where the potential
well is found in the V profiles.
This data is used in later
calculations.
Returning to figure 30a, it is seen that N2 is rather large in the
lower troposphere, and after a sudden initial drop in the lowest two
kilometers, it gradually decreases with height up to around 350-400mb,
where it reaches a minimum.
Above this level, there is seen to be a
sharp increase in the N 2 values near the tropopause, and then a
In fact, the sharp vertical
continued more gradual increase higher up.
increase in N 2 can be used to determine the general location of the
tropopause.
It is also seen that there are very few regions where N
is
less than half the assumed constant value, or more than double this
value.
Thus, while the observed N 2 variability is not negligible, there
is also no order of magnitude variability.
In order to get some idea of the possible modifications that
allowing a variable N 2 would have on the results presented earlier,
specifically on the structure of the calculated V fields, a few of the
2
calculations are redone here assuming N2 = N (z).
One may begin again
with the expression
q = V2
+ f +
2ez/HO~
(e-z/H
3oZ
N2
az
in which N 2 was previously assumed to be constant (see eq. 4).
Expanding the vertical derivative in the third term on the right,
simplifying, and taking the zonal average, as was done before, yields
-72-
=
02 i
+ N2
z2 -
T(new)
yy +
where the subscript "(new)"
2
1
a(N 2
4
N
a-z
1
HN
) A
z
distinguishes this quantity from that
This equation is now nondimensionalized;
derived in section II.
the
assumed scales are as before, with the addition:
N2 E
N 2 scale = 1.56 x 10- sec -
with ~
n ,
2.
Substituting, one obtains the dimensionless expression
q(new)
=
(16)
1
Tz
z
:
Y
Tyy +
Similarly,
q'(new)
9
=
'
1
1
1
y(new)
8
yy -
z
(17)
,
Therefore, taking the
which is the counterpart of equation (8).
meridional gradient of (16)
an
1n
zz
+
and rearranging terms one also has
Uzz +
+
-
1)(
-
UI)zz
+
a
In this equation, the first four terms on the right are just
.u
(18)
the
previous expression for qy, with the final two terms representing the
modification arising due to a variable N
2
Continuing in a completely parallel derivation to that in section
II, the expansion of the functional F leads to the previously
encountered expression:
-73-
qy I
Y
q
u
((p,)
2
2
+ O(w')12
"
u
i
for q'(new) must be used, and equation
Now, however, equation (17)
(18) for qy(new)
LI
qY)
to give
1
1
+xx 'yy + -n VPzz
' zz - n
+
'z
-
1
n
an
22
(new)
-
, + 0(
' 2)
@z
(19)
4' = e *ez / 2
The substitution
9*xx
+
%*yy
1
,
+
1 an
n2 '
*zz
is again introduced, leading to
4Y(new) +1
1
j
-z 4n
2n
2
an
za
=
O()
V(new)
4* is not eliminated
It is noted that the vertical first derivative of
this time; hence the subsequent steps taken previously (and by Malguzzi)
I/n 2
will not lead to a Korteweg-deVries equation, unless
an/az
is
In any event, the new V function is given by
very small.
qY(new)
1
V(new)
=
4n
1
an
(20)
+ 2n 2 az
Substituting eq. (18) into eq. (20) for qY(new) gives
Y(old)
1
V(new)
=4
-
S1
n -
U
-
z
zz
1
n
i
1
41++ 4~-
-74-
ann
+n
2n 2
az
where the subscript "(old)" refers to the section II definition of a
quantity.
This equation can be rewritten in the more revealing form:
1(
1
V(new)
=
V(old)+
1-7
u-
1
uzz
u
an
1
uz
n2u
(21)
8
A
2
From equation (21), the modification arising due to a variable N
is clearly evident.
"B" vanish.
It is noted that if n ' 1, as before, terms "A" and
Term A represents the modification due to the actual
departure of N 2 from its assumed constant value, and term B the
modification due to the vertical gradient of N2 .
So, what effect will these modifying terms have on the potential
well structures found in many of the earlier V profiles?
will be considered while referring to figure 30.
This question
As the northern rim of
the potential well, in the positive composite cases, was usually found
between 62-70*N, the modifications will only be considered south of
700 N.
Term A will be most important in those regions where n differs
the most from unity.
In particular, near n = 1/2, the factor (1/n - 1)
= 1, and term A will make an order one contribution to the value of V.
The narrow area near 400mb is the only region in the domain where n
1/2;
=
this region is in the interior of the potential well, and is
characterized by positive uz and negative uzz.
Hence the uz and
"zz effects in term A will reinforce each other in this region, and a
significant negative modification to V will result;
term A will thus
tend to deepen the interior of the potential well near 400mb.
The
magnitude of term B depends upon both the difference of n from unity and
the vertical gradient of n;
often a tradeoff is found between these two
-75-
effects.
One might expect term B to be important near the tropopause,
where sharp vertical gradients of n occur.
In order to provide some quantitative feeling for the modifications
arising due to the inclusion of terms A and B in the expression for V,
and for the behaviors and relative importance of A and B over the entire
domain, the tables in figure 31 have been prepared.
The data in these
tables pertain to the positive blocking composite case, at day +5 lag,
and for a zonal average over 90*W< x <30 0 W.
The tables summarize
calculations made at the four latitudes 340 N, 46*N, 580 N, 700 N.
In deriving equation (21), N2 was allowed to vary with height only;
therefore, it seems appropriate to use the n values given in figure 30c
in the calculations at all latitudes.
However, figure 30a shows that
there is actually some meridional N2 variation, which becomes
particularly important with regard to the height of the tropopause.
Therefore, for each of the four latitudes, two tables are presented;
the first is based on the meridionally averaged n values given in figure
30c, and the second on the actual n values at that latitude, which can
be gotten from figure 30a.
Comparing the two tables for each latitude,
the final V(new) values are seen to be, for the most part, not greatly
different from each other.
Some general observations from the tables are now noted.
First of
all, it is seen that the modification produced by allowing a variable N
tends to increase V in the lower levels of the domain, between
500-700mb.
Between roughly 450mb and the tropopause the modification
acts in the opposite sense, tending to decrease V potential well.
increase V.
and deepen the
Near the tropopause the modification is again seen to
These trends are fairly uniform at the four latitudes, and
-76-
the contributions of terms "A" and "B" are seen to be of comparable
magnitudes.
At 70*N it is seen that the modifications tend to shift the
region of positive V values which represent an ill defined northern rim
slightly lower in the atmosphere;
positive values.
they do not remove the region of
At the other three latitudes, which lie in the
interior region of the supposed potential well, the generally negative
tropospheric values of V continue to prevail in the modified field.
The
0
notable exception is the positive value at 46 N and 500mb, which was
only a very weakly negative point previously however.
All in all, the modifications to the V field indicated in the
tables of figure 31 do not seem too impressive.
The inclusion of
temperature effects (i.e. variable N2) seems unable to significantly
change the basic topography of the previously calculated field.
The
considerably smaller magnitudes (with some exceptions) of the values in
the "A" and "B" columns of the tables than the values in the "V(old)"
column testifies further to this inability.
While it is encouraging
that the temperature effects do not seem to destroy the potential well,
particularly the northern wall, it is surprising that they do not have a
more significant impact near the tropopause.
As mentioned earlier, theoretical work (e.g. Held, 1983) suggests
that the tropopause may act as a rigid lid, halting many vertically
propagating wave modes, and might therefore be considered as the top of
the vertical wave guide.
2
In Malguzzi's model, however, due to N being
considered constant, there is no reason to expect the V profiles to
display any important characteristics (e.g. sharp gradients or positive
values) at the tropopause level.
By allowing N 2 to vary with height,
one might hope to see a positive upper wall of V values arise near the
-77-
tropopause level.
The tables of figure 31 show that while the
modifications do act to increase V near the tropopause, they are not
sufficient to overcome the negative V(old) values and create a
positive wall region.
At best, the V gradients are sharpened near the
tropopause, resulting from the negative modification in the upper
troposphere and the positive modification in the lower stratosphere.
However, it is suggested that at the tropopause there is actually a
2
narrow region of considerably more concentrated vertical N gradients
than the current data is able to capture.
The present vertical
resolution of the data is simply not fine enough to reveal such a narrow
zone, but rather the N 2 gradient becomes smeared and diluted across the
wide gap between data levels (recall, data is available only at 100,
200, 300, 500mb).
A hint of this sharp increase in N2 at the tropopause can be seen
in figure 29.
It is noted that an alternative definition for the
buoyancy frequency is:
N2
=
g de
Sdz
g
d(lnO)
dz
Thus, the vertical gradient of N2 will be proportional to the vertical
second derivative of 6.
(Figure 29 is actually a plot of equivalent
potential temperature, 8e , but for the present purposes of eyeballing
vertical gradients near the tropopause either 0 or
looked at.)
0
e plots can be
As a rough estimate, it is seen that the Oe field is
relatively flat in the troposphere, and then suddenly begins to ascend
sharply with height near the tropopause.
The curvature of the field
will have a sharp maximum at the tropopause level, as the following
-78-
sketches schematically illustrate:
4oo
£00
L
The field of 3n/az will thus apparently have a similar sharp peak near
the tropopause.
A crude estimate of the effect of this sharp maximum of an/az on
the modification to the calculated V profiles can now be made.
Any
significant modification will lie in term B, which contains 3n/az as a
factor.
A few assumptions will be made.
First it is noted that the
line of n = 1 in figure 30a very closely follows the level of the
tropopause;
this line will probably be embedded somewhere in the sharp
gradient of n at the tropopause.
Therefore, for the purposes of the
current crude estimate, n = 1 will be assumed at the tropopause implying that term A will be zero, making no contribution to the
modification of V.
Furthermore, it is noted in figure 4 that at mid-latitudes (south
of -60*N), the vertical derivative of u becomes very small near the
tropopause level.
In fact, the maximum of the mid-latitude jet occurs
-79-
very near the tropopause, and a second assumption that the corresponding
This assumption
= 0 occurs at the tropopause will be made.
line of '
is important because an/az is multiplied by the factor (1/2 - iz/U) to
obtain term B.
(1/2 - iz/Tf)
The behavior of an/az is linked to the behavior of
in determining a modifying term for V.
Thus, in order
for the sharp peak of an/az at the tropopause to have a significant
impact, (1/2 - Uz/Z) must not be too small there.
Above the
mid-latitude jet, i, is negative, tending to increase term B.
the jet, 7z is positive, and there is a level where ~,z/
term B is therefore zero.
Below
= 1/2, and
The assumption of uz = 0 at the tropopause
seems fairly realistic for mid-latitudes.
However, it is noted that at
high latitudes (e.g. 70°N), Tz remains positive well above the
tropopause, and (1/2 -Tz/f)
negative.
Thus, at high latitudes term B
makes a negative modification on V near the tropopause, and no positive
wall is anticipated.
To make a crude estimate of the magnitude of term B near the
tropopause (it has already been assumed that A = 0 there), an estimate
of
an/az
must be made.
the largest
an/az
From the table in figure 30c, it is noted that
occurs between 250 and 300mb, and has a value of
0.32/1.458km = 1.76 in dimensionless units.
Assuming that this value
has been cut at least in half due to the poor vertical data resolution,
it is conjectured that the actual peak of 3n/az has a value of 3.52 at
the tropopause.
With 'z = 0, and n = 1, term B therefore has a value
of +1.76 at the tropopause (mid-latitudes).
Assuming, based on the
figure 31 tables, that the tropopause occurs at -190mb at 340 N,
~230mb at 46*N, and ~255mb at 580 N, it is seen that a net
modification of +1.76 to the V profile at these levels will be
-80-
sufficient to create a wall of positive V values.
maximum of V will remain slightly negative,
At 58*N, the
as the wall begins to vanish
with increasing latitude.
The above estimates are very crude, and are not intended to be
highly accurate.
They are intended to point out that a narrow zone of
strong an/az near the tropopause, which has been attenuated due to poor
data resolution in the previous calculations, can act to create a narrow
wall of positive V values there, when a variable N2 is allowed in the
derivation of V.
As seen previously, the basic shape of the V profiles
in the troposphere, including the northern wall, remains largely
unchanged after allowing for a variable N 2 .
Therefore, it appears
likely that the major impact of allowing a variable N2 on Malguzzi's
theory will be in the emergence of the tropopause as the level of the
top of the potential well.
-81-
V.
Conclusions.
This study set out, as its primary objective, to determine the
extent to which the profile of Malguzzi's V function, calculated from
actual atmospheric data taken during observed blocking events, contained
a structure resembling a quantum mechanical potential well, in midlatitudes.
The real question, and the essential goal of the present
study, however, is to try and determine the degree of applicability of
Malguzzi's theoretical model to the real atmosphere.
In its idealized
setting, the model seems to explain certain observed atmospheric flow
patterns quite well.
In actuality, the atmosphere does not behave
ideally, and the model will undoubtedly bear more relevance in some
situations than in others -- perhaps in some instances of blocking it
may not be applicable at all.
In his development of the model, Malguzzi assumes the buoyancy
frequency, N 2 , to be constant, and therefore his V function, given in
equation (12), turns out to be a function only of the mean zonal wind
field, averaged over some suitable local longitude band upstream of a
blocking pattern, or coherent structure.
Therefore, requiring a
potential well structure in the V field for the model to be applicable
is equivalent to saying that "an isolated coherent structure exists only
if the mean zonal wind, upon which the structure is superimposed,
satisfies certain constraints" (Malguzzi, 1984).
Malguzzi sums up with the statement that only sufficient conditions
have been found which confirm the meaningfulness of the model, as
opposed to necessary conditions.
These are that, given a u(y,z)
profile, if there is a bound state of the linear problem given in
-82-
equation (13), which is asymmetric in y, and if the corresponding
eigenvalue, K, is sufficiently small, then the Korteweg-deVries dynamics
described by the model should be observed.
Hence the applicability of the model depends, in a large part, on
the structure of the mean zonal wind field, i(y,z), which justifies the
To
present investigation of the V function under various conditions.
calculate V in practice, however, the question of the proper definition
of
i arises,
to which there is no clear cut answer.
Also, as the number
of data levels is limited (850, 700, 500, 300, 200, 100mb), and the
calculation requires differentiating the raw geopotential height data
twice in the vertical and three times with respect to latitude, a fairly
smooth field of ;
is desirable.
Therefore, a composite of twelve
blocking cases is used.
The results, which were looked at in section IV,A, proved to be
quite supportive of the existence of a potential well.
The longitudinal
averaging of i was carried out over four bands of varying width, and the
best results came from the narrowest band, which included no part of the
downstream blocking pattern.
Clearly visible was a wall of positive V
values along the northern edge of the potential well, lending support to
the idea of meridional trapping of the waves.
A similar southern wall
was not observed, but this was judged to be the result of a lack of data
south of 220 N, where the southern wall is expected to occur, if present.
As for the upper and lower edges of the potential well, sustained
regions of positive V values were not observed;
however, the vertical
wave trapping assumed in the model is thought to be accomplished by the
presence of the earth's surface below and the tropopause above.
2
section IV,F, it was suggested that if N is allowed to vary with
-83-
In
height, a narrow upper wall of positive V values will likely arise near
the tropopause.
These observations agree quite well with Malguzzi's
assumptions regarding the structure of the V function, and imply that
his model is applicable, at least partially, to the composite of twelve
cases of Atlantic positive blocking, at day +5 lag.
The word partially
is used because the observed potential well structure was not perfect.
There were numerous gaps in the "walls", for instance, which might act
as leaks, allowing certain waves to propagate through.
A similar calculation of the V field was carried out for a
composite of twelve cases of Atlantic negative blocking, which consists
actually of an abnormally strong zonal flow in the "key region" (Dole,
1982), and on the wintertime climatological mean data set.
The negative
blocking case displayed considerably less support for the existence of a
potential well in V than the positive blocking case, and the climatology
case virtually no support.
These "negative results" were also
encouraging, suggesting, as one would expect, that Malguzzi's solution
is not applicable to explaining the negative blocking or climatological
mean atmospheric flow patterns.
With the encouraging results of the positive blocking composite
case in mind, the question of determining the structure of V for
individual cases of blocking arises.
Cases which strongly resemble
Malguzzi's dipole coherent structure can be selected for analysis, and
one would surely hope to find a potential well in the V profile for
these.
However, the matter of defining
(y,z) for the individual cases
turns out, more often than not, to be a difficult task.
The large
meanders of the jetstream, both upstream of, and associated with the
blocking pattern, often make a representative mean zonal wind profile
-84-
impossible to identify;
the calculated V profile often becomes
extremely sensitive to the exact choice of longitude band over which
is averaged.
T
Therefore, not a great deal can be concluded as to the
applicability of Malguzzi's model to actual individual cases of
blocking.
Some individual case situations were identified, in which the
zonal wind field upstream of the blocking pattern was not wavy, which
displayed practically ideal potential well structures in their
corresponding V profiles.
Thus it can be said, at least, that the model
is applicable to some individual cases of Atlantic blocking.
Finally, it proved interesting to return to the positive blocking
composite case, and compute a time series of V profiles from day -5 lag
through day +5 lag, to examine the evolution of the V fields as the
The most notable result was the
blocking pattern formed and grew.
emergence, approximately simultaneously with the establishment of the
blocking pattern, of a significant "northern wall" of positive V values,
which persisted thereafter through day +5 lag.
This observation
supports the existence of a potential well in V, and thus the
applicability of Malguzzi's solution, while the blocking pattern is
present, but not prior to its formation.
There are numerous parameters associated with the Z(y,z) profile,
which change with time.
Certain combinations of these parameters will
allow a potential well structure to exist in the corresponding V
function profile, at mid-latitudes.
This list of parameters includes
the maximum wind speed of the mid-latitude jet (Umax), the minimum
wind speed (which was found sometimes to be negative, causing additional
problems) at the turning points on either side (north or south) of the
jet (Umin), and the width of the jet (distance between these turning
-85-
points), yg.
of u -
It is assumed that the fundamental tropospheric behavior
increasing with latitude up to a maximum at mid-latitudes, and
then decreasing to the north -
will be the rule in general, however the
precise shape of the profile can vary.
In other words, the exact
curvature of the Ti(y,z) profile can vary, both meridionally and
vertically.
If a typical curvature is chosen, and regarded as fixed,
Malguzzi (1984) points out that large values of Umin and y 0 tend to
reduce the possibilities for bound states of the linear problem of
equation (13).
Thus, narrow mid-latitude jets (meridionally),
surrounded by regions of weak mean zonal winds to the north and south,
will favor the existence of a potential well in the V profile.
Furthermore, Malguzzi points out that weaker jets are less favorable to
support a potential well structure in the V profile, reducing the number
of possible combinations of parameters, which is consistent with the
fact that fewer blocking events are observed during the summer than
during the winter.
The parameters associated with the *(y,z)
profile discussed above
are continually changing in response to various atmospheric motions, for
a zonal average taken over any fixed local longitude band.
Certain
processes in the atmosphere may contribute to focusing the mid-latitude
jet into a relatively narrow, concentrated westerly stream, surrounded
by regions of substantially weaker zonal winds to the north and south.
Such processes will likely, therefore, also cause a mid-latitude
potential well to form in the V profile calculated from this wind field,
and may thus have important implications for the downstream development
of a coherent structure (blocking pattern).
Other studies have also suggested the possible importance of the
-86-
structure of the zonal wind field for the behavior of atmospheric
blocking.
As Dole (1982, Ph.D. thesis, p. 203) states:
"The evolution analyses suggest, however, that the explanation for
the initial developments [of the blocking pattern] may be rather
subtle. In the PAC [Pacific blocking] cases, an intriguing clue is
provided by the significant pattern located upstream over Asia and
the extreme western Pacific preceding the development. The
structure of this pattern suggests that the associated wind
anomalies are primarily in the zonal flow over both the Himalayas
and the southwestern North Pacific."
Dole also points out that typical values for the decay rates of blocking
patterns are comparable with estimates of the time scales for variations
in the zonal flow.
The explanation of the preference of certain geographic locations
for persistent blocking patterns to develop remains somewhat of a
mystery (Dole, 1982).
Geographically fixed forcings could play a large
role in determining where blocking occurs.
With regard to the current
theory, it is suggested that 7(y,z) profiles which favor the existence
of a potential well in their corresponding V fields do not occur
randomly around the globe, but rather are found more frequently in
certain geographic regions than in others -
namely, those regions
upstream of the preferred blocking regions.
Consider the western
Atlantic in winter, for example;
at mid-latitudes there is typically a
highly baroclinic zone associated with the Atlantic storm track in this
region, created by the regular fluxes of cold continental polar air off
the North American continent running up against the much warmer maritime
air found near the Gulf Stream.
A strong, concentrated jetstream is
often associated with this baroclinic zone.
One of the main problems
with this notion, which was encountered while looking at some of the
-87-
individual blocking cases in section IV,D, is that frequently this
concentrated jetstream is not oriented roughly east-west.
It often has
a significant meridional component, and sometimes is not even aligned
with the downstream (dipole) blocking pattern.
In any event, this investigation, which has amounted in large part
to a detailed study of the zonal flow fields upstream of blocking
patterns, has shown that the assumptions made in Malguzzi's coherent
structure model are approximately valid in many observed cases of
blocking.
In particular, Malguzzi's potential function V does indeed
contain a well-like structure at mid-latitudes, for a variety of actual
(positive blocking) data sets tested.
Although this potential well does
not have a flawless structure, often containing gaps in the walls for
instance, it does approximate the ideal much more closely than any
structures found in the V profiles corresponding to other atmospheric
flow regimes, such as "negative" blocking, climatology, or the flow
pattern prior to the onset of blocking in the positive composite case.
Even when the definition of V was modified to include temperature
effects, by allowing N 2 to vary with height, the potential well
structure remained;
in this case it was found that the shape of the V
profile was still primarily determined by the structure of the mean
zonal wind field, except in isolated regions such as the tropopause
where the temperature effects could dominate.
Thus it is concluded,
based on observational findings, that the theory of the self-interaction
of Rossby waves superimposed on a meridionally varying westerly current,
leading to the formation of a localized coherent structure, can be used
effectively to model certain atmospheric blocking patterns.
-88-
REFERENCES
Charney, J.G., and J.G. DeVore, 1979: Multiple flow equilibria in the
atmosphere and blocking. J. Atmos. Sci., 36, 1205-1216.
Colucci, S.J., 1985: Explosive cyclogenesis and large-scale circulation
changes: implications for atmospheric blocking. J. Atmos. Sci.,
42, 2701-2717.
Persistent anomalies of the extratropical Northern
Dole, R.M., 1982:
Hemisphere wintertime circulation. Ph.D. thesis, Massachusetts
Institute of Technology, Cambridge, MA.
, 1983:
Persistent anomalies of the extratropical Northern
Hemisphere wintertime circulation. In Large-Scale Dynamical
Processes in the Atmosphere, B. Hoskins and R. Pearce, eds.,
Academic Press, 95-109.
Fullmer, J.W.A., 1982: Calculations of the quasi-geostrophic potential
vorticity gradient from climatological data. J. Atmos. Sci., 39,
1873-1877.
Hartmann, D.L., and S.J. Ghan, 1980: A statistical study of the
dynamics of blocking. Mon. Wea. Rev., 108, 1144-1159.
Stationary and quasi-stationary eddies in the
Held, I.M., 1983:
extratropical troposphere: theory. In Large-Scale Dynamical
Processes in the Atmosphere, B. Hoskins and R. Pearce, eds.,
Academic Press, 127-168.
,
and D.G. Andrews, 1983:
On the direction of the eddy momentum
flux in baroclinic instability.
Holton, J.R., 1979:
Press, 391pp.
J. Atmos. Sci., 40, 2220-2231.
An Introduction to Dynamic Meteorology.
Academic
Illari, L., 1984: A diagnostic study of the potential vorticity in a
warm blocking anticyclone. J. Atmos. Sci., 41, 3518-3526.
Predictability in theory and practice. In LargeScale Dynamical Processes in the Atmosphere, B. Hoskins and R.
Pearce, eds., Academic Press, 365-383.
Leith, C.E.,
1983:
-89-
Long, R.R., 1964:
21,
Solitary waves in the westerlies.
J. Atmos. Sci.,
197-200.
Malguzzi, P., 1984: Coherent structures in a baroclinic atmosphere.
Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
, and P. Malanotte-Rizzoli,
1984:
Nonlinear stationary Rossby
waves on nonuniform zonal winds and atmospheric blocking. Part I:
the analytical theory. J. Atmos. Sci., 41, 2620-2628.
, and A. Speranza, 1981:
atmospheric blocking.
Local multiple equilibria and regional
J. Atmos. Sci., 38, 1939-1948.
McWilliams, J.C., 1980:
An application of equivalent modons to
atmospheric blocking. Dyn. Atmos. Oceans, 5, 43-66.
Mitchell, H.L., and J. Derome, 1983:
Blocking-like solutions of the
potential vorticity equation: their stability at equilibrium and
growth at resonance. J. Atmos. Sci., 40, 2522-2536.
Oort, A.H., 1983:
Global Atmospheric Circulation Statistics, 1958-1973.
NOAA prof. paper #14, U.S. govt. printing office, Washington, D.C.
Rex, D.F., 1950:
Blocking action in the middle troposphere and its
effect upon regional climate. Part I: an aerological study of
blocking action. Tellus, 2, 196-211.
Youngblut, C., and T. Sasamori, 1980: The nonlinear effects of
transient and stationary eddies on the winter mean circulation.
Part I: diagnostic analysis. J. Atmos. Sci., 37, 1944-1957.
-90-
FIGURE CAPTIONS
Figures la-k. 500mb geopotential height fields in decameters, for the
composite of twelve cases of Atlantic positive blocking, from day
-5 lag to day +5 lag. Covers 22-90*N. Negative longitudes are
degrees west, positive longitudes, degrees east.
Figure 2. Wintertime climatological mean 500mb geopotential height
field in decameters. Data averaged between November 15 and the end
of February, for the years 1963-77.
Figures 3a-k. 500mb geopotential height anomalies (meters), for the
composite of twelve cases of Atlantic positive blocking, from day
-5 lag to day +5 lag.
Figures 4a-d. Latitude-height cross sections of the zonal mean
geostrophic wind (i) upstream from the "block" (m/sec). Data for
the composite of twelve cases of Atlantic positive blocking at day
+5 lag.
Zonal averages taken between 90*W< x <30,40,50,60*W.
Figures 5a-d. V function profiles (dimensionless) calculated from the
zonal wind fields in figures 4a-d.
Figure 6. V function profile (dimensionless) for the Atlantic positive
blocking composite case at day +5 lag, based on a zonal average of
5 over 90°W< x < 60°W. Field has been smoothed by setting a
threshold minimum for lul of 4m/sec. (Same as figure 5d., but with
smoothing.)
500mb geopotential height field in decameters, for the
composite of twelve cases of Atlantic negative blocking, at day +5
Figure 7.
lag.
Figure 8. 500mb geopotential height anomalies (meters), for the
composite of twelve cases of Atlantic negative blocking, at day +5
lag.
Latitude-height cross section of the zonal mean geostrophic
0
0
wind (5), averaged over the longitude band 90 W< x <60 W (m/sec).
Data for the composite of twelve cases of Atlantic negative
Figure 9.
blocking, at day +5 lag.
-91-
Figures 10a-d. V function profiles (dimensionless) for the Atlantic
negative blocking composite case at day +5 lag, based on zonal
averages of U over 90*W< x <30,40,50,60*W. Fields have been
smoothed with a threshold minimum
'
= 4m/sec.
Figures lla-d. V function profiles (dimensionless) based on the
wintertime climatological mean U fields averaged over longitude
bands 900 W< x <30,40,50,60*W. Fields have been smoothed with a
=
4m/sec.
threshold minimum Ii
Difference V profiles (dimensionless) for the smoothed
positive blocking composite V values at day +5 lag minus the
smoothed climatology V values, for longitude averages 90*W< x <30,
Figures 12a-d.
40,50,600W.
Figure 13a. Difference V profile (dimensionless) for the smoothed
negative blocking composite V values at day +5 lag minus the
smoothed climatology V values, for a longitude average
900 W< x <60 0 W.
Difference V profile (dimensionless) for the smoothed
positive blocking composite V values at day +5 lag minus the
smoothed negative blocking composite V values at day +5 lag, for a
Figure 13b.
longitude average 90*W< x <600W.
Figures 14ab. Cross sections of the dimensionless V function values at
the 500mb level for the cases of the Atlantic positive and negative
blocking composites at day +5 lag and the wintertime climatological
mean, versus latitude. No smoothing of the V fields is employed.
0
a) based on a zonal average of 9 over 900W< x <50 W.
b)
based on a zonal average of 9 over 900W< x <600W.
500mb geopotential height field in decameters for 12Z,
Figure 15.
Data covers 22-90*N.
December 31, 1971.
Figure 16.
Latitude-height cross section of the zonal mean geostrophic
wind (5), averaged over the longitude band 65°W< x <350W (m/sec).
Data for 12Z, December 31,
1971.
V function profile (dimensionless) for the case of 12Z,
December 31, 1971, based on a zonal average of U between
650W< x <35*W. Field has been smoothed with a threshold minimum
Figure 17.
ll = 4m/sec.
-92-
500mb geopotential height field in decameters for OZ,
December 23, 1976.
Figure 18.
Latitude-height cross section of the zonal mean geostrophic
0
wind (5), averaged over the longitude band 90 W< x <60°W (m/sec).
Data for OZ, December 23, 1976.
Figure 19.
V function profile (dimensionless) for the case of OZ,
December 23, 1976, based on a zonal average of 5 between
90*W< x <60*W. Field has been smoothed with a threshold minimum
Figure 20.
5 = 4m/sec.
Figures 21-24.
Individual blocking cases at +5 days lag:
Figures 21ab. OZ, January 25, 1968.
a) 500mb geopotential height field in decameters.
b) V function profile (dimensionless), based on a zonal average of
U between 90*W< x <600 W. Field has been smoothed with a
threshold minimum Ju = 4m/sec.
Figures 22a,b. OZ, December 30, 1968.
a) 500mb geopotential height field in decameters.
b) V function profile (dimensionless), based on a zonal average of
U between 90*W< x <60OW. Field has been smoothed with a
threshold minimum
5
=
8m/sec.
Figures 23a,b. 12Z, March 1, 1970.
a) 500mb geopotential height field in decameters.
b) V function profile (dimensionless), based on a zonal average of
9 between 90*W< x <600 W. Field has been smoothed with a
threshold minimum fij = 8m/sec.
Figures 24a,b. OZ, December 8, 1975.
a) 500mb geopotential height field in decameters.
b) V function profile (dimensionless), based on a zonal average of
U between 90'W< x <60'W. Field has been smoothed with a
threshold minimum J
= 4m/sec.
V function profiles (dimensionless) for the Atlantic
positive blocking composite case, days -5 lag through +5 lag. All
0
0
based on a zonal average of 5 over 90 W< x <60 W. Fields have been
= 4m/sec.
smoothed with a threshold minimum I
Figures 25a-k.
-93-
Cross sections of the 500mb V function values from
figure 25, versus latitude, for days -5 lag through +5 lag.
Figures 26a-k.
Figures 27a,b.
a) The curves of figure 26 corresponding to days -5 lag through 0
b)
lag, plotted together.
The curves of figure 26 corresponding to days 0 lag through +5
lag, plotted together.
Figure 28.
Profile of the dimensionless V function values at the 500mb
level, with latitude on the horizontal axis and time (days -5 lag
through +5 lag) on the vertical axis. Values are for the Atlantic
positive blocking composite case, and based on a zonal average of 5
smoothing is applied with a threshold minimum
over 900 W< x <60*W;
IuI
= 4m/sec.
Plot of the wintertime and zonal average field of equivalent
Figure 29.
potential temperature ( e) as a function of latitude and height,
in degrees Kelvin. Data for a December, January, February three
month average during the years 1958-73 (see Oort, 1983).
N 2 profile, as a function of latitude and height,
calculated from the data for the Atlantic positive blocking
Figure 30a.
composite case at day +5 lag, and based on a zonal average of T
between 90*W< x <30*W. Units are 10- sec - .
Figure 30b. Table showing the results of an N 2 calculation by Youngblut
and Sasamori (1980). Assumed N 2 to be a function of height only,
and computed it using the hydrostatic approximation and hemispheric
mean data for the month of January, 1963.
Figure 30c. Table showing the meridionally averaged N2 values from
figure 30a between 22-70°N, at each data level. Values given in
dimensional (10-
Figures 31a-h.
sec - 2 ) and dimensionless units.
Tables showing the various
contributions to the
modification of the V profile which arises by allowing N
with height (see equation 21).
to vary
Data for the Atlantic positive
blocking composite case at day +5 lag, and based on zonal averages
of T and u over 90 0 W< x <30=W. Calculations made at four latitudes:
34, 46,
58 and 70 0N.
2
At each latitude, first table based on the N
values in figure 30c, and second table based on the N
figure 30a for that latitude.
values in
All units except pressures (mb) are
dimensionless.
-94-
o
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Figure 5a
90' w < x < 30 w
Z(km)
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28
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Figure 5b
90* W < x < 40*W
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Figure 5c
90'w < x < 50*w
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Figure 6
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9
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pf(,)
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Figure 10a
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< 30 W
p( 4 )
l ru)
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(latitude)
Figure 10b
90 W < x < 40W
,IA)
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Figure 10c
90W < x < 50 0 W
/
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52
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Figure 10d
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(mi
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72
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(latitude)
Figure 11a
900 W < x < 30W
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Figure 11b
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2(Kx)
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(latitude)
Figure 11c
90 0 W < x < 50W
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72
76
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(latitude)
Figure 11d
900 W < x < 6 0 *W
Ist
16 +
11 I
24 "
2e
32
35
40
44
48
52
55
60
64
68
72
76
80
64
(latitude)
Figure 12a
90°W < x < 300 W
I
a
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Figure 12b
52
36
40
44
46
52
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90 W < x < 40*W
64
66
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76
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94
(latitude)
/8
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56
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72
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(latitude)
Figure 12c
90 W < x
<
50'W
84
LH
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Figure 12d
90W 4< x < 600W
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32
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64
68
72
76
80
84
(latitude)
Figure 13a
90ow < x < 600 W
NKSO
HMO~
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IL
+
(latitude)
Figure 13b
90W < x < 60W
a)
3.1
J
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7
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Figure 15
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12Z
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I 100---
150-
2po -
2.50
rjo
300'
600,
700-
(latitude)
Figure 16
65W < x < 35'W
iz(ki.
I'
2
92
L
21
-19.
76
(latitude)
Figure 17
65 w < x < 350 W
100
90
80
-100
12/23/76
Figure 18
138-
OZ
'5.00F7
b+
i +
iot
24
28
2
40
44
48
52
56
55
60
64
68
72
76
80
84
(latitude)
Figure 19
90W < x < 60ow
10 0P(14)
loo
-?6
16-
i0
t
24
I
2A4
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I
2
32
35
40
44
48
52
56
60
64
68
72
76
80
64
(latitude)
Figure 20
90 0 W 4 x < 6 0 6W
-100
-90
1/25/68
Figure 21a
-141-
-80
OZ
Z,
i (o,)
100
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(latitude)
Figure 21b
90oW < x < 60 W
100
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12/30/68
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Figure 22a
20
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Figure 22b
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sa
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Figure 23a
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(latitude)
Figure 23b
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Figure 24b
90 W < x < 60*W
/I
16
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24 - 28
32
36
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Figure 25a
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(latitude)
Figure 25b
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Figure 25d
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84
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on
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Figure 25e
8
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7 (4latitude
(latitude)
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P(Mo
100
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(latitude)
Figure 25f
Day 0
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4
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6
72
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(latitude)
Figure 25g
Day +1
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(latitude)
Figure 25h
Day +2
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Figure 25i
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36
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o
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Figure 25k
Day +5
a)
9
7
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fi
pi
V f,.,d;oI
(
Cots
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Figures 26a,b
-16o-
Lu
(N)
Lkilluo (r.
16
310
30
Se4,t
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Figures 26gph
isavture ('N)
I)
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Figures 26i,j
i
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k)
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it
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Figure 26k
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Figures 27a,b
-166-
M
+2
+4
+24
24
23
32
36
40
44
48
52
56
60a
4
68
72
76
80
LATITUD
Figure 28
-167-
84
-
t&mb)
'I
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LIII
Iso
I
3 * *
G
7
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T_
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Figure 29
a)
2.08
2.22
2.64
2.99
2.99
2.84
2.63
2.30
2.05
2.04
1.85
47
1.16
0.99
1.14
1.441.61
z(F^)
N (10-
20
2.69
19
2.50
18
2.33
sec
-
)
A
*0.17
1.64
*0.11
2.22
*0.11
2.11
+0.09
1.4
1.68
1.89
2.11
2.33
2.52
2.68
2.83
2.93
2.93
2.94
2.98
3.03
3.04
2.95
2.82
2.75
2.75
2.02
+0.06
i
1.96
0.04
1.09
1.29
1.40
j6
1.82
2.06
2.30
2.56
2.75
2.81
2.92
3.08
.22
3.27
3.16
2.96
2.85
2.83
1.92
+0.11
1.81
+0.32
0.84
0.98
1.05
1.18
1.39
0
1.80
2.01
2.21
2.33
0.65
o
.72
9
0.89
1.04
1.15
1.25
1.36
1.
1.69
2.49
2.66
2.76
2.80
2.75
2.63
2.56
2.53
1.49
+0 34
1.88
2.00
2.03
2.03
2.07
2.09
2.09
2.06
1.15
*0.14
1.01
-0.04
0.97
0.67
0.64
0.68
0.73
0.
0.82
1.01
1.17
1.20
1.13
1.09
1.19
1.36
1.47
-0.04
1.46
1.01
-0.09
1.06
1.01o 0.99
0.96
0.95
0.94
0.95
1.00
1.04
1.12
1.18
1.17
1.12
1.06
1.07
1.22
1.36
1.39
-0.15
1.25
-0.18
1.46
1.42
1.37
1.31
1.28
1.30
1.37
1.47
.
1.46
1.43
1.41
1.41
1.37
1.32
1.40
1. 2
1.73
1.72
1.70
1.6
.
1.69
1.83
1.89
1.88
1.81
1.73
1.70
1.75
1.78
1.75
1.79
1.84
1.81
1.62
1.92
1.93
1.98
1.99
1.97
2.10
2.29
2.28
2.21
2.15
2.05
2.02
2.11
2.23
2.28
2.30
2.26
2.14
1.82
1.99
2.02
2.11
2.15
2.15
2.31
2.51
2.46
2.37
2.32
2.21
2.19
2.30
2.47
2.57
2.59
2.49
2.31
1.43
-0.19
-0.20
-0.76
2.58
21-.2
-30 -3---3 8---3
50
5-6-~
62LAT I
Figures 30a,b
t4 DE
66
70
74
78
82
90
N
= N (z)
Values
Meridionally Averaged,
22-70 N
Al /isec.L)
n (0 imenSio4ss)
100
2.25
1.44
150
200
2.49
2.22
1.60
1.42
250
1.79
1.30
1.15
0.83
0.85
0.54
0.67
0.90
p(mb)
300
400
500
1.04
600
1.40
700
800
1.75
2.08
850
2.24
Figure 30c
-170-
1.12
1.33
1.44
346N
A
.i
100
150
200
250
-0.31
-0.38B
-0.30
I
1474a~
"j
0.85
0.49
500
600
0.11
-0.11
-0.25
700
800
(it-7d
-L
2 ?a
n&
I-L
- ,,=
-0.13
0.20
300
400
8
1.27
-1.61
-0.78
-0.98
-0.86
-1.25
-1.93
-0.79
1.10
1.88
1.07
-1.98
-1.65
-3.86
18.54
-1.19
-0.93
-0.19
0.01
0.44
' /,,
--
1.70
{,
IVU /
-0.39
0.60
-0.32
0.01
0.23
0.13
-0.18
-1.06
0.19
0.22
0.06
-0.29
-0.95
-0.09
0.41
-4.60
1.32
1.77
q.00
1.19
1.70
0.87
0.43
0.20
-o.61
0.42
0.65
0.63
0.50
-0.65
0.06
-1.43
-1.34
-0.99
0.03
-0.27
-0.67
-1.07
-1.68
-1.27
0.11
1.62
2.02
2.25
1.26
-0.19
0.01
0.44
1.25
0.86
0.68
1.10
1.88
0.39
0.08
1.07
-1.98
-1.65
-1.19
-0.42
-0.60
-0.93
-5.01
-0.03
0.10
1.25
0.86
0.68
0.87
0.43
0.20
0.03
-0.27
-0.67
-1.07
-1.68
-1.27
0.19
-2.54
-1.71
-1.95
-1.97
-2.74
-3.74
-2.77
-5.99
16.2
I
-0.52
-1.93
-1.29
-1.60
-2.09
-4.09
-3.37
-1.09
-3.58
12.8
F"' 3ib (bp)
100
-0.48
1.27
150
-0.26
-1.61
200
250
0
0.32
-0.78
-0.98
300
400
0.75
1.14
500
600
700
800
0.63
0.19
-0.06
-0.22
-0.86
-1.25
-1.93
-0.79
-3.86
18.54
100
-0.31
-0.64
150
200
250
-0.38
-0.30
-0.13
0.71
-0.73
-1.26
300
400
0.20
0.85
-2.07
-1.17
500
600
0.49
0.11
0.29
-2.84
700
800
-0.11
-0.25
-7.23
-21.7
0.38
0.72
1.16
1.85
1.89
-0.40
-2.42
-1.89
0
-0.31
-1.22
-0.15
0.23
-4.08
0.37
0.19
0.23
-2.54
-1.71
-1.95
-1.97
-2.74
-1.49
-3.74
-2.77
-5.99
16.2
-1.21
-1.89
-2.56
-4.06
-3.34
-0.90
-3.51
13.4
46 N
-0.89
-1.90
-0.27
0.22
0.16
-0.41
-1.00
0.14
-0.31
0.79
5.42
-0.24
0.01
-1.14
1.54
-1.18
1.28
0.30
0.43
-0.35
-1.34
0.15
-0.44
-2.42
-1.55
-0.24
-4.00
-8.92
-23.0
0.17
-0.75
-2.68
-2.99
1.08
-2.84
-5.87
-12.9
-1.14
-0.92
1.54
-0.35
1.33
0.23
-0.69
1.18
1.47
2.26
4.66
31c ( top)
31
100
~ [ir
-0.41
-0.42
-0.64
150
200
-0.32
-0.13
0.25
-0.73
-1.26
-2.07
0.51
250
1.05
-1.17
0.64
0.29
-2.84
1.59
-2.60
0.39
0.08
-0.42
-0.60
-2.17
-1.49
-0.89
-1.90
I -1.04
I -5.01
300
400
500
600
700
800
0.14
-0.15
-0.32 [
0.71
-7.23
-21.7
1.25
2.21
All unis ex.epf prefsafes are JlmensionlO
Figures 31a,b,c,d
-171-
s.
0.26
-0.30
0.23
0.16
-0.04
0.09
0.35
0.49
-0.51
-1.23
0.19
-0.67
-0.39
1.07
1I 6.92
1.93
2.83
5.22
0.18
1.56
-1.34
-2.42
-1.55
-0.24
-4.00
-8.92
-2.75
-3.45
1.51
-2.46
-5.02
-!0.9
580N
A
B
e)
IIn
100
500
600
-1.91
-2.42
-1.89
1.07
-1.98
700
800
-0.11
-0.25
-3.59
10.69
-1.19
150
200
250
300
400
F;
f)
11-_--
-0.31
-0.38
-0.30
-0.13
0.20
0.85
0.49
0.11
-2.62
2.44
-0.36
-0.67
-1.64
-0.19
0.80
0.59
0.73
0.53
0.29
0.01
0.44
1.10
1.88
-0.21
-1.65
-0.93
A
-0.15
-0.93
0.11
0.09
0.01
-0.33
-1.62
-0.73
-1.62
-2.24
-2.84
-1.19
-0.21
0.39
-2.67
0.32
0.59
0.55
-0.23
1.45
2.67
2.67
2.64
-3.85
1.81
-1.32
-3.19
0.89
-0.89
-2.03
-3-31
-3.99
-4.92
-4.90
-1.35
-3.09
-5.84
-7.37
-4.31
5.66
5.63
-4.66
-2.44
31f (bf,)
1oo
-0.24
-2.62
-0.47
-0.44
2.44
-0.82
-0.20
0.80
150
200
0.23
0.79
0.73
0.53
0.63
-1.15
0.16
0.22
0.29
-0.21
0.13
-1.03
-0.94
-0.89
-0.73
-1.62
-2.24
-2.84
0.01
-0.02
0.44
1.10
1.88
-0.09
250
-0.33
-0.36
-0.67
300
-0.08
-1.64
1.53
400
500
0.54
0.39
-1.91
-2.42
-1.38
600
0.07
-1.89
-1.50
700
-0.14
-0.27
-3.59
10.69
-1.14
800
B
0.81
1.71
0.59
-0.13
0.50
-2.89
-2.8
-2.84
-0.66
-3.85
-3.88
-0.12
0.17
1.81
0.54
-0.99
0.42
0.44
-0.36
1.01
2.43
2.55
2.53
-1.32
-2.03
-3.31
-3.99
-4.92
-4.90
-7.37
5.66
-1.39
-2.74
-5.38
-4.85
-2.60
-4.32
5.30
70N
g)
100oo -.
150
200
250
300
400
500
600
700
800
31
-3.31
0.53
-1.11
-0.77
-1.14
-0.44
-0.38
-0.30
-o0.13
0.20
0.85
0.49
0.11
-0.11
-0.25
1.03
-1.88
-5.24
-29.3
F,3 31 . (b01
m).
0.34
-0.49
-0.52
-0.43
-0.23
-3.31
0.53
-1.11
-0.77
-1.14
400
0.38
-o.44
500
600
700
0.39
0.11
1.03
-1.88
-0.11
-0.26
-5.24
-2q9.
100
150
200
250
300
800
All un +s except pra rures
1.02
-0.20
-0.12
-0.00
-0.32
0.33
0.10
-0.50
-0.23
1.07
-1.98
-0.93
-1.12
-1.65
-1.19
-0.93
-1.45
-2.44
-0.37
0.50
-0.21
-0.04
-0.36
-0.95
-0.99
2.22
2.40
0.58
7.32
2.90
6.24
-1.13
-0.26
-6.71
-5-35
0.64
-0.51
0.08
0.60
-0.02
-0.09
-0.32
0.58
0.33
1.31
2.18
-0.86
-0.50
0.26
-0.93
-1.12
-0.17
0.40
-0.21
-1.47
-1.45
-1.24
-2.44
-0.99
-6.71
-0.99 -6.7
are dAmen~oleft~S
Figures 31e,f,g,h
-172-
0.58
7.62 -L
I
7.62
-4.54
-1.20
-2.24
-3.64
-1.40
-1.07
-0.66
1.14
-1.33
-1.84
2.57
-1.89
-8.05
-36.7
5.29
0.30
-4.57
-23.1
-3.42
0.01
-0.01
-4.54
-1.20
-2.24
-9.09
-1.45
-1.67
-0.19
-0.66
-1.07
-0.66
1.14
-0.93
-1 .06
2.57
-1.89
3.93
-2.03
0.96
2.13
3.03
-8.05
6.64 I -36.7
-36.7
6.64
-1.95
-0.22
-1.06
0.03
-4.44
-22.4
-22.4