Saltzman B. (editor) Anomalous Atmospheric Flows and Blocking
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14
A.
C.
WIN-NIELSEN
c2ry2
Y;
l-----l
I
%C,
Y:
FIG.
4.
Energy relations
for
a wave-wave interaction barotropic
model.
obtained by a spherical harmonic decomposition
of
the real mountain
height;
m,
,
m,,
and
m3
are therefore realistic.
It may on the other hand
be
interesting to look at the results if the moun-
tains were excluded.
This
situation
is
depicted in Fig.
6.
It is seen that
multiple steady states can still exist without the mountains, that the transfor-
mations maintain the same directions as before although the energy
amounts
are
changed, and that the relative distributions of energy on the
components are maintained.
It
would thus appear that the mountain effects
GLOBAL SCALE CIRCULATIONS-A REVIEW
-
22.30
K,
15
13.20 -33.38 28.58
1
K,
1
c
-
1.95
3.18
0.61
0.10
K,
240’2
0.67
Y
1.79 0.88
0.79 0.26
K3
1.35 0.3;
2.0
0.84
FIG.
5.
A
specific
energy
diagram
for
the
model mentioned in Fig.
4.
Energy amounts in
the
unit in
kJ
m-2,
others in
102
W
m-*.
are not vital for bifurcation, but have modifying influence. Studies of this
kind are of course never conclusive, but
the
present study indicates that
wave- wave interactions provide
a
better mechanism for multiple steady
states because the “window” in parameter space is more open.
4.
BLOCKING
AS
A
STRUCTURAL
ENTITY
Time-averaged maps of the atmosphere at various pressure levels, the
so-called normal maps, where the averaging time is many years, show certain
16
A.
C.
WIN-NIELSEN
22.80
I
13.55 -33.07
20.02
K,
.
K,
-
235.7
*
110.1
'
+y$jp
0.64
FIG.
6.
The
same
situation
as
in
Fig.
5
but
without
the
mountain
effect.
I
T
characteristic features such
as
well-defined troughs and ridges. These fea-
tures are presumably
of
a stationary nature since otherwise they would have
been
eliminated in the averaging process. Indeed, it is these maps which are
used to represent the stationary flow (zonal flow and waves)
of
the atmo-
sphere.
Synoptic experience shows that the atmosphere additionally has other
structural entities which are of shorter duration, but which nevertheless are
of a quasi-stationary nature, and has a typical lifetime far exceeding that
of
a
traveling disturbance. Some of these quasi-stationary atmospheric structures
are seasonally dependent such
as
the continental wintertime anticyclone or
0
4.05
8.05
i
3.11
-
2.06
-
1.70 0.61
~
Kz
KZ
--
-
5.0
.
.
do
2.31
16.7
5.17
I
1.77
I
j
211
-
i
K3
14.4
1.18
0.56
K3
4.5
.
GLOBAL SCALE CIRCULATIONS- A REVIEW
17
the monsoon circulation. These structures will normally be seen on time-
averaged maps for the proper individual month or season because their
creation and maintenance and disappearance are strongly connected with
the seasonal cycle more than the general circulation in the annual sense.
On
a
still shorter time scale it is observed that the atmosphere from time to
time creates features which have
a
typical structure and exist for
a
certain
period in an essentially unchanged form. The blocking situation is an exam-
ple of such a structure. If the phenomenon of blocking as an atmospheric
structure is to
be
discussed from a theoretical or practical point of view it
would appear desirable to have a well-formulated definition of it. While such
definitions exist, it is nevertheless true that there is no general agreement on
them.
One
of
the first to take a theoretical interest in blocking was Rossby (1 950)
based on synoptic investigations by Berggren
et
al.
(
1949), Elliott and Smith
(
1949), and Namias and Clapp (1949).
A
more extensive “climatology”
of
blocking was carried out by Rex (1 950), who also provided a detailed de-
scription of some blocking cases from a synoptic point of view
as
well as an
analysis of the climatological aspects of blocking. The definition of blocking
action provided by Rex (1950) is that the blocking case must exhibit the
following characteristics:
1.
The basic westerly current must split into two branches.
2.
Each branch must transport an appreciable mass.
3. The double jet system must extend over at least 45
O
of latitude.
4.
A
sharp transition from zonal-type flow upstream to meridional-type
5.
The pattern must persist with recognizable continuity for at least
10
downstream must
be
observed across the current split.
days.
On the basis of this definition Rex (1950) was able to make a statistical
analysis of blocking using the years 1933- 1949 (inclusive).
A
main result
was that blocking seems to have preferred locations in the eastern Pacific and
eastern Atlantic regionswith mean values around
150”
Wand 10” W, respec-
tively. While Atlantic blocking cases may exist at
all
times of the year there is
a minimum around August and a maximum in April. Pacific blocking cases
are very rare in August and September, but show a broad maximum in the
period March-May. From the data it is also possible
to
compute the per-
centage of blocked days. For the Atlantic cases one gets
27.6%
and for the
Pacific case
12.2%.
Considering that the Atlantic cases are
68%
of all cases,
one gets as an average that
22.7%
of all days are blocked days. This result is
not inconsistent with a much later result obtained by
A.
Sutera (see this
volume), who, using a different criterion, found that about 30% of all days
18
A.
C.
WIIN-NIELSEN
were blocked days, but his results were obtained using a condition that the
block should exist for only
4
days.
Concerning the persistence of Atlantic
blocking cases it is found by Rex
(
1950) that the average duration is about
17
days while the highest frequency is found for blocking cases which last
14
days. These studies show also the typical anomalies in temperature and
precipitation which make the blocking action very important from a predic-
tive point of view.
Austin (1 980), in a study of blocking, is less stringent in the definition of
the phenomenon.
A
distinction is made between the synoptician's point of
view and that of the theoretical meteorologist. The first view is simply that
blocking is a spatially isolated phenomenon consisting of a stationary high-
pressure cell, warm in the troposphere and cold in the stratosphere, persist-
ing in a region where westerly winds are normally found. The requirement is
that the persistence
is
substantially longer than
3
days. On the other hand, it
is also stated that, to those who are engaged in spectral studies, blocking is
any stationary long wave of large amplitude, meaning that blocking is seen as
a global-scale phenomenon.
Although the split in the jetstream is not made part ofthe definition itselfit
is nevertheless mentioned that it occurs, particularly in those cases where a
high-pressure cell at about 60"N is located to the north of a low-pressure cell
at about
40
"N.
It
will
be noted that both of these definitions emphasize the connection
between the split of the jet upstream and the location of the blocking high-
pressure cell downstream. The definition
as
such does not imply any causal
relationship between the split of the jetstream and the blocking anticyclone.
On the other hand, the description of the irregular variation between the low-
and high-index zonal circulation and blocking, as, for example, described by
Namias and Clapp
(
195 I), could lead to the impression that the creation of
the blocking action follows the split in thejetstream in time. It
is,
indeed, this
idea which forms the basis for the theoretical considerations put forward by
Rossby (1950), who investigated the possibility that the blocking phenome-
non is analogous to a hydraulic jump. This theory is far from satisfactory.
Another attempt at understanding the behavior of the zonally averaged
flow and especially its interaction with the large-scale eddies was made by
Thompson
(1
957)
who developed a heuristic theory ofthe long-period varia-
tions in the zonal flow, i.e., a prediction equation for the time variation as
influenced by the statistics of the eddy behavior. The theory is for two-
dimensional barotropic
flow,
and it should be stressed that it is entirely
internal in the sense that Frictional and topographical effects are excluded. It
is obvious that in forming the prediction equation for the zonally averaged
flow
it
is
a necessity
to
parameterize the influence of the eddies
on
this flow.
Such a parameterization
is
possible only under a general hypothesis, which
in Thompson's
case
is
as
follows:
GLOBAL
SCALE
CIRCULATIONS
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19
1.
The most effective agencies of momentum transport are eddies of very
large scale, i.e., eddies with a meridional scale comparable with the width of
the channel.
2.
The statistical measures of the scale and intensity of turbulence at any
time may be identified with the statistics of a v-field whose amplitude is a
constant multiple of the amplitude of the initial v-field, and whose scale in
the x-direction is a constant multiple of the scale of the same initial field.
The general assumptions are equivalent to a closure, and they permit the
formulation of the prediction equation for the zonally averaged flow. This
equation has, after some simplification, the form
azu/at2
=
-kz~f[p
-
dzu/ayz
-
(2/~f)u]
(8)
where
U
is the zonally averaged wind,
vf
the initial variance of the meridio-
nal velocity,
p
the Rossby parameter,
Af
a
scale parameter, and
k
a constant.
Thompson (1 957) shows by numerical integrations ofthis and similar, but
more complicated, equations that a well-defined sharp jet in middle latitudes
will split into two jetstreams. The positions of the jet maxima travel to the
north and the south relative
to
the single original maximum. The typical
time scale of the phenomenon is several days, and good agreement is
ob-
tained between a theoretical calculation of the speed of the traveling maxima
and a determination of this speed from observational studies. The theory
permits
also
a calculation of the changes in the variance of the eddy vorticity
determined from the predicted changes in the zonal wind. The main con-
clusion from the study is that “a mechanism for the initiation and develop
ment of blocking is contained in the theory of barotropic flow.”
The mechanism for the splitting of the simple jetstream into two separate
jets was tested by Win-Nielsen
(1
96
1)
in a low-order, barotropic system
containing six components. The time scale as calculated by Thompson for
typical cases was confirmed by these experiments, which did not use any
other closure mechanism than the severe truncation in wavenumber space
inherent in a low-order model.
The studies which have been summarized very briefly above are in sharp
contrast to the blocking mechanism proposed by Charney and DeVore
(
1979) which was mentioned in an earlier section of this paper. We should
note that the zonal wind in the latter study is a measure of the total angular
momentum which in the model is influenced by the topography and the drag
effect from the planetary boundary layer. The main point is, however, that
the zonal wind in
this
model does not vary with latitude and, consequently,
any description of a split in the jetstream is thus excluded from the begin-
ning. This is not the case for the studies by Wiin-Nielsen (1 979) and Kiillen
(1 98 l), which at least contain the interaction between the eddies and the
zonal flow albeit in a low-order model.
20
A.
C. WIN-NELSEN
Most
of
the examples given by Thompson
(1957)
are related to a sharp
symmetrical jet. For these examples it can
be
shown that the general equa-
tion with good approximation can
be
reduced to Eq.
(8).
The numerical
integrations show that the
sharp
jet in general will split in
two
branches, one
traveling north and the other south. It may
be
of
some interest to make a
further analysis
of
Eq.
(8).
In the following discussion we shall consider a case
which assumes that the
initial
convergence of the momentum transport
vanishes, which in two-dimensional barotropic
flow
is
equivalent to
dU/
at
=
0
at
a
=
0.
In addition, in agreement with the previous study we shall
also assume that
u2
=
u;
=
const., where
U;
is
the initial value at the center
of
the channel.
If
we under these assumptions determine
Us(
y)
in such a way
that the right side of
Eq.
(8)
vanishes, we know that
a2U/dt2
=
0.
It is then
easy to
show
that the third and all other higher derivatives vanish
or,
in other
words,
U,(y)
is
a steady state.
To
obtain this steady state in a convenient
form
we nondimensionalize the independent variables
as
follows:
z
=
t/d,
q
=
y/
W,
d
=
I
day
=
0.864
X
lo5
sec
(9)
and the steady state equation
is
a2us/aq2
+
(2
w2/n;)uS
=
pw2
(10)
u,=v,=Ido-li,,
q=o;
u,=u,=u,-tl,,
q=l
(11)
Equation
(
10)
is solved
with
the boundary condition
where
Go
=
bution
Uo(q).
The solution to
Eq.
(10)
is
then
U
dq
is constant and can
be
calculated from the initial distri-
Using Thompson's theory we may adopt the
following
values
for
the various
parameters:
w2/A;
=
n2(
1
+
W2/L2)
p-
16
X
m-I
sec-'
W=5X106
m
L=28Xl@
m
GLOBAL
SCALE CIRCULATIONS
-A
REVIEW
21
a
b
t
Us,
rn
sec-1
FIG.
7.
Steady-state wind profiles for
various
boundary conditions.
(a)
Obtained by
solving
the finitedifference
equations;
(b)
based
on the analytical solution.
Figure
7
shows
U,(q)
for two values of
Uo
and
U,
. If
Uo
=
U,
it is seen that
Us(q)
is symmetric around the center of the channel. We return next to Eq.
(8).
From this equation and the equation
(13)
u,
=
u-
us
(14)
(15)
The solutions
of
Eq.
(1
5)
are
obtained by integration
of
the corresponding
finite-difference equation. For this purpose we introduce a grid size
of
Aq
and a time step of
AT.
Counting the time steps by
n
and the space variable by
m
we have
-
o
=
-u~(dZ/WZ)[)WZ
-
a2uS/aq2
-
(~W~/A;)U,]
we obtain by subtraction and denoting
--
-
g(d2/w2)[a2u1/aq2
+
(2
w2/,1;)u,
I
aT2
u,(n
+
1,
m)
=
2u,(n,
rn)
-
u1(n
-
1,
m)
+
?I;
-
-
r
-
Wz
dZ
Aq
x
u,(n,
m
-
1)
+
u,(n,
m
+
1)
1
As
mentioned earlier we shall
be
particularly interested in the possibility
that the initial jetstream may split in two branches. We may get a first idea
of
this possibility by investigating the initial tendency in the center of the
22
A.
C.
WIN-NIELSEN
channel if we restrict this
part
of the investigation to a jet which is symmetri-
cal around the center of the channel. Setting
n
=
1
and denoting
rn
=
Mfor
the center we note that
M)
=
u,(O,
M)
(17)
and we shall consider the quantity
u,
(2,
M)
-
u1
(
I,
M),
which, according to
Eq.
(
16)
and recalling the symmetry, may
be
written as follows:
UI(2,
M)
-
u1(1,
M)=
2~l(l,M+
1)-
(18)
The jet
will
have a tendency to split ifthe left-hand side of Eq.
(
18)
is negative
or
if
the bracket in the equation is negative. From this condition we get
(19)
Equation
(
19)
says
that the tendency for the splitting ofthe jet will occur if the
jet is sufficiently sharp. Using the same parameter values
as
before and
Aq
=
0.1
we find that
M+
1)/u1(1,
M)
5
1
-
(W2/Lf)(A7,lY
u,(l,M+ l)/ul(1,M)sO.9
(20)
Equation
(
19)
is an initial tendency only and does not necessarily result in
a division of the jet in two branches.
It
may, however, be used as a guide for
numerical experiments. For these experiments we have to select an initial
wind profile
u,
=
ul(q).
The following profile was seiected:
(21)
Since we consider a jet which is symmetrical around the middle we select
a
=
0.5
and continue to use
A1
=
0.1.
The center value
is
u,,,
while the value
one grid-size away is
(22)
The value
b
corresponding to Eq. (20) is then
b
=
0.3.
If
b
is smaller than this
value there is an initial tendency for a decrease
of
the center value. Such a
case is shown in Fig.
8
calculated forb
=
0.25. After
I
day
we observe that the
maximum value has decreased and that the jet has become wider. This
process goes on further and after
I
.5
days it is observed that the two branches
start
to appear.
A
case with
b
=
0.35
is
shown in Fig.
9.
In this case the jet
continues to increase and become broader. The cases illustrated in Figs.
8
and
9
are close to the dividing case where the equality sign applies in Eq.
UI
=
uo
exP{-
Ktl-
a)/bI2)
uo
exp[-
(
1
/
1
Ob)’]
GLOBAL SCALE CIRCULATIONS-
A
REVIEW
23
0.4.-
u/uo
FIG.
8.
A
time integration leading to a division
of
the jet after about
1.5
days. Parameters:
a
=
0.5.
b
=
0.25.
(19). More extreme cases of the type considered by Thompson (1957) are
easily computed. Figure 10 illustrates a case of a very sharp and narrow jet
which after
1
day has developed two branches. The two maxima move away
from the center.
The brief review
of
Thompson’s theory made above shows that in its
simplified form it has a steady state determined by the p-effect and the
boundary condition (Fig.
7).
The equation for the time development
of
the
zonally averaged flow contains the mechanism for the splitting of the jet if it
is sufficiently sharp, as shown by the examples (Figs.
8
and 9). In extreme
FIG.
9.
A
time integration
leading
to an increase and a broadening
of
the jet. Parameters:
a
=
0.5,
b
=
0.35.