Saltzman B. (editor) Anomalous Atmospheric Flows and Blocking
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Part
IV
NUMERICAL EXPERIMENTS
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INSTABILITY THEORY AND NONLINEAR
EVOLUTION
OF
BLOCKS AND MATURE
ANOMALIES
J.
S.
FREDERIKSEN
Commonwealth Scientific and Industrial Research Organization
Division
of
Atmospheric Research
Aspendale, Victoria
31
95
A
ustralia
1.
INTRODUCTION
The results of studies which indicate that blocking and teleconnection
patterns may be generated through the three-dimensional instability of at-
mospheric flows are presented. The instability properties of climatological
flow for the Northern Hemisphere are examined. It is found that three-di-
mensional instability theory produces very low-frequency teleconnection
patterns such as the Pacific- North American and North Atlantic Oscillation
patterns, as well as medium-frequency onset of blocking modes upstream of
the mature patterns and higher frequency monopole cyclogenesis modes,
which have maximum amplitudes in the observed storm tracks. Thus, the
main observed modes of variability in the troposphere have analogs obtained
from instability theory.
The instability solutions are compared with the structural changes that
occur in the time evolution of observed anomalies such as blocks and with
nonlinear simulations of the development of mature anomalies.
Blocking, which was noted as long ago as 1904 by Garriott (1904), is a
primary example of anomalous circulation. It has subsequently stimulated
numerous synoptic studies because of its profound effect upon the weather
and climate (see, for example, Namias, 1947; Berggren
et
al.,
1949; Rex,
1950; OConnor, 1963). In the early tentative theories blocking was consid-
ered as a barotropic process (e.g., Yeh, 1949; Rossby, 1950; Rex, 1950;
Thompson, 1957). In particular, Rossby
(
1950) and Rex (1 950) suggested
that block development was analogous to a hydraulic jump in open-channel
flow.
In recent years there has been a resurgence of interest in studying anoma-
lous circulations based on observations, theories, and numerical simula-
tions. Many of these aspects are covered in this volume. Here, our purpose is
to provide a relatively self-contained presentation of three-dimensional in-
stability theory of the onset of blocking, large-scale teleconnection patterns,
211
Copyright
0
1986
by
Academic Press, Inc.
All
rights
of
reproduction
in
any
form
resewed.
ADVANCES
IN
GEOPHYSICS,
VOLUME
29
278
J.
S.
FREDERIKSEN
and cyclogenesis.
As
well we compare the results of this theory with studies
based on observations and numerical simulations.
2.
THREE-DIMENSIONAL INSTABILITY
THEORY
2.
I.
Model Details
The two-layer quasi-geostrophic P-model that is used for this study was
discussed in detail in Frederiksen (1982a) and was originally derived by
Lorenz
(
1960).
Here we briefly summarize the equations and describe how
the instability problem
is
formulated. In dimensionless form the P-model
equations take the form:
M/dt
=
-
J(ty,0)
+
OV
’x
+
Fe
(la)
(1b)
028=
V
-
2pVT
(14
dV’lyldt
=
-J(d(/,V’v
+
2p)
-
J(.s,V’T)
+
Fy
dV’T/dt
=
-J(v,V2z)
-
J(.s,V*~
+
2p)
+
V
*
2pVx
+
F,
(
Ic)
The original dimensional equations are made dimensionless by taking the
earth’s radius
a
and the inverse of the earth’s angular velocity
R-’
as length
and time scales, and
a2Q2/bCp
as
a temperature scale, where
Cp
is the specific
heat of air at constant pressure and
b
(-
0.124)
is a dimensionless constant.
In Eq. (l), the average
y
and shear
T
streamfunctions are taken to be
related to the streamfunctions in the upper
t,ul
and lower
ty3
layers through
the relations
v
=
t(v’
+
v3),
=
+(vl
-
w3)
(2)
Further,
0
is the average potential temperature,
x
is the velocity potential in
the lower layer, and
p
is sin
4,
where
4
is latitude. The two layers are
frequently taken to have streamfunctions representative of
250
and
750
mbar, although other levels are also commonly taken as representative. Here
we shall take the representative levels for the basic flow field in the upper and
lower layers as being
300
and
850
mbar. The dimensionless static stability
parameter
5
is given by
ii
=
bCp
~0/2a~R~
(3)
where
A0
is the mean potential temperature difference between the upper
and lower layers. The effects
of
diabatic heating, topography, dissipation,
etc. are taken to be included formally in the operators
Fyr, F,,
and
Fe.
INSTABILITY THEORY AND NONLINEAR EVOLUTION
279
2.2.
Basic
State
In this section, we use basic climatological Northern Hemisphere winter
300- and 850-mbar streamfunction fields derived from observations in the
manner described by Blackmon (1 976). The average over the eight winters
1963/ 1964 to 1970/ 197
1
was used. The basic streamfunctions (denoted
by
a
bar) were expanded in a general spherical harmonic decomposition.
1
(4)
j=
1,3
FIG.
1.
Geopotential height (meters)
of
300-mbar surface
for
Northern Hemisphere average
winter.
280
J.
S.
FREDERIKSEN
at each level. Here
1
is longitude,
PC(
I()
are orthonormal Legendre functions,
Yp
are spectral coefficients,
p
is zonal wavenumber, and
v
is
total wave-
number.
A
triangular truncation with truncation wavenumber
18
was used.
Figure
1
shows the corresponding geopotential height for
300
mbar.
Once
v/
has been specified through Eq.
(4)
the relations Eq. (Id) deter-
mines the basic potential temperature
e.
The basic state is taken to be
stationary and the operators
Fy
,
F,,
and
Fo
are supposed to balance the basic
state. For the sake of simplicity these operators are also supposed not to act
on the perturbations, i.e., they are inhomogeneous. The implications of this
prescription have been discussed in some detail in Frederiksen
(
1979).
In
this article we concentrate on the results for one value
of
the static
stability parameter, viz.,
Case
1.
=
0.0067; i.e.,
A8
=
23
K
(54
We also briefly mention some results for
two
other values of
5
also consid-
ered
in
Frederiksen (1982a, 1983b), viz.,
(5b)
Case 2a.
=
0.008; i.e.,
A8
=
28
K
where
A8
denotes the mean potential temperature difference between the
two layers. In addition we shall consider a barotropic model for which the
equation of motion is
Eq.
(
1
b) with
7
set to zero. Our barotropic results are
essentially a subset
of
those obtained by taking the static stability parameter
to infinity in the baroclinic model.
-
-
Case
3.
[T
=
0.01;
i.e.,
A8=
35
K
(5c)
2.3.
Disturbances
For the disturbances we take general perturbations [which for conve-
nience we denote by the original symbols in
Eq.
(l)] of the form
ma.
.m
P,
0
=
Re{
c
2
frnPn4
x
exp[iW
-
ool}
(6)
m--m
n-lml
wherefis any of the disturbance fields. Here
rn
is the zonal wavenumber,
n
is
the total wavenumber, and
o
=
or
+
ioi
is the complex angular frequency.
In practice only a finite number of nonzerof,, are retained in Eq.
(6).
Here a
parallelogrammic truncation is used in which
Iml=
0,
1,
. .
.
,
15;
we con-
sider only disturbance streamfunctions which are antisymmetric between
the two hemispheres and take
n
=
Irnl+
I,
.
. .
,
Iml+
I1
for the stream-
functions.
The method of linearizing
Eq.
(I)
about the basic states and solving the
INSTABILITY THEORY AND NONLINEAR EVOLUTION
28
1
consequent eigenvalue
-
eigenvector equations for the complex angular fre-
quency wand the amplitude coefficientsf,, is described in the appendices
of
Frederiksen (1982b).
2.4.
Growth Rates and Phase Frequencies
The nondimensional and dimensional growth rates
(mi
and
a$)
and phase
frequencies (or and 0:’) together with the periods are shown in Table
I,
for
Case
1
,
and for all modes from the fastest growing down to the first nonpro-
pagating but growing mode. We notice that there is quite a variety in the
phase speeds of the modes, from very rapidly propagating modes with pe-
riods of l
.8
days to modes that are nonpropagating.
2.5.
Disturbance Streamfunctions
To discuss individually each
of
the 22 modes in Table
I,
as
well as the
modes for
Cases
2a and
3,
would probably not be profitable. Rather, one can
make a rough classification of the modes. Any such classification will serve
only as a guide, and some modes will have characteristics of more than one
group.
2.5.1.
Monopole Cyclogenesis Modes (Class
A).
The typical member of
this class is Mode
1
of Case
1
for which the instantaneous disturbance
streamfunction’ in the upper layer is shown in Fig. 2a, while the amplitude of
the disturbance streamfundon in the upper layer is shown in Fig. 2b. The
instantaneous disturbance has an essentially monopole structure (with lati-
tude) except for a hint of secondary poles in the Atlantic in the upper layer.
Both
the instantaneous disturbance and the amplitude of the disturbance
have largest values in the observed positions
of
the storm tracks.
The regions of preferential development may be seen to
be
slightly down-
stream and poleward of the jetstream maxima and the reason for this may be
shown to be related to instability criteria as described in Frederiksen
(
1979).
The location of these regions is in quite good agreement with bandpass-fil-
tered observations (e.g., Blackmon, 1976, Fig. 5a), but the development
off
the east coast of North America is somewhat more pronounced than in the
Pacific compared with the observations. This, however, depends on the
particular mode examined as discussed below.
I
It is the same
as
that shown in Fig. 2a
of
Frederiksen
(1
982a) except that the arbitrary phase
and amplitude are different.
282
J.
S.
FREDERIKSEN
FIG.
2.
(a) Disturbance streamfunction and (b) amplitude ofdisturbance streamfunction
in
upper layer
for
fastest
growing
mode
in
Case
1
(of
class
A).
Similar modes, for which the amplitude in the North Atlantic ocean is
larger than in the Pacific, are Modes
2
and
4
of Case
I
as well as a number of
modes ofcases 2a and
3.
In each case, their periods are quite small. There are
also monopole cyclogenesis modes which are similar to that shown in Fig. 9b
of Frederiksen (1983a) for which the amplitudes in the Atlantic and Pacific
are nearly equal. Mode
6
of
Case
1
is an example. The modes with the very
shortest periods, such as
8
and
14
of
Case
1,
have considerably larger ampli-
tudes in the Pacific than in the Atlantic.
In
each case, the regions ofpreferen-
tial development occur in the same geographical locations as the observed
storm tracks (Blackmon,
1976;
Blackmon et
af.,
1977).
INSTABILITY THEORY AND NONLINEAR EVOLUTION
283
FIG.
2b.
2.5.2.
Pacific Onset-of-Blocking Dipole Modes (Class B).
Mode
3
of Case
1
,
for which the instantaneous disturbance streamfunction in the upper layer
is shown in Fig.
3,
is a typical member of this class;
so
is Mode
1
of
Case
2a
shown in Fig.
6
of Frederiksen (1982a). These modes have longer periods
than the monopole cyclogenesis modes, and become the fastest growing
modes when the static stability parameter is increased from
0.0067
to
0.008
so
that the importance of baroclinic processes is reduced. They have dipole
structures in the Pacific Ocean, with largest amplitudes upstream of the
observed regions of mature blocks. Moreover, the zonal scale of the distur-
bances is larger than for the monopole cyclogenesis modes. The fact that the
high
-low
vortex pairs have largest amplitude upstream
of
the main regions
ofthe occurrence of mature blocks in the Pacific would
seem
to be consistent