Saltzman B. (editor) Anomalous Atmospheric Flows and Blocking
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184
J.
EGGER
ETAL.
Charney and DeVore
(
1979),
prescribed the interaction of large-scale modes
with shorter waves as a white-noise forcing and studied the impact of this
forcing on the
flow
climatology (see also Benzi
et
al.,
1984).
Shutts
(1983)
was able to show that dipole blocking patterns can
be
generated in barotropic
flow, where smaller eddies are excited by a wave generator (see also Kalnay-
Rivas and Merkine,
198
1).
However, it is a shortcoming of these theoretical
efforts that the choice of the forcing is largely intuitive and not based on
observations. The synoptic-scale forcing has been determined by Egger and
Schilling
(
1983)
on
a
day-byday basis from data. Egger and Schilling used
the resulting time series of the forcing as input in a model. They were not
concerned with blocking but with the variability of the atmosphere and
found that a large part of the observed long-term variability of the atmo-
sphere can be explained
as
a response of the planetary-scale modes to this
forcing. Since blocking is a phenomenon with relatively large time scales one
might expect that the origin and maintenance
of
blocking patterns can be
explained at least partly by the same mechanism. Here we want to find out
if
this view is correct. We prescribe the observed forcing in a barotropic model
of planetary flow. Numerical experiments are carried out in order to see if
blocking can
be
generated in this model.
2.
STOCHASTICALLY FORCED PLANETARY MODES
Our modeling strategy is fairly similar to that described in Egger and
Schilling
(
1983,
1984)
so
that only a brief outline is given here. We assume
that the barotropic component of atmospheric large-scale flow can be de-
scribed by the vorticity equation
(1)
where
v/
is the stream function,
rp
is latitude,
1-l
is
a radius
of
deformation,
a
the earth’s radius, and
R
the earth’s rotation rate.
A
damping term is in-
cluded on the right-hand side of Eq.
(1
).
We partition the streamfunction in a
planetary part
vr
to be resolved and treated explicitly and an unresolved
synoptic-scale part
vu
:
v
=
vr
+
vu
(2)
(d/dt)(V2
-
A2)tyr
+
Jr(cy,,
V2yr
+
2Ra-I
sin
rp)
-
kV4t,vr
(d/dt)(V2
-
A2)y
+
J(v,
V2ty
+
2l2a-l
sin
9)
=
kV4y
The equation for the planetary part is
=
-Jr(vry
V2y/,)
-
Jr(~ut
V2vr)
-
Jr(v/,t
V2y/,)
(3)
The subscript rat the symbol Jdenotes the projection of this Jacobian on the
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
185
resolved flow. The terms on the right-hand side of Eq. (3) describe the
interaction of the planetary-scale modes and the synoptic-scale modes. Of
course, there exists a similar prognostic equation for
vu.
This latter equation
will not
be
solved, however. Instead we lump all the three Jacobians together
and prescribe them as forcing. We split the streamfunction into a mean flow
and a deviation thereof:
vr=
u/+
v’
(4)
where
u/
=
-
Uoa
sin
a,
is the superrotational mode with a zonal wind profile
u
=
u,
cos
a,.
It is convenient to project
v/’
onto spherical harmonics
lMN
W‘
=
5
x
x
WmnP:+zn--I(sin
P)
ex~(imi’
(5)
where
P
is
the corresponding associated Legendre function and is longi-
tude. It is seen that we admit only those modes which are antisymmetric with
respect to the equator. We select the modes with
m
5
5,
n
5
5
as
planetary
modes
so that
M
=
N
=
5.
This choice is somewhat arbitrary and the results
depend on it.
We shall mainly present results obtained with a linearized version of Eq.
(3),
where the linearization is made with respect to the mean state described
by
p.
After performing the linearization, Eq. (3) can
be
transformed into a set
of equations for the expansion coefficients
(6)
where
a
contains the effects of damping and wave propagation andX,, is the
forcing by synoptic-scale eddies. Eleven years of geopotential height data
have been used to construct forcing time series
for
each of the planetary
modes. The statistical characteristics of these time series are discussed in
Egger and Schilling (1983, 1984). We quote here the main result that the
autocovariance of the real (and imaginary) part of the forcing can
be
approx-
imated by
(7)
m--M
n-1
(dIdt)vmn
+
avmn
=
xmn
Rxx(z)
=
Z
exp(-slzl) cos(p7)
where
Z,
s,
and
p
are the fitting parameters to be determined from data for
each mode and
7
is
the lag. Typically,
s
-
0.3
-
1.2 day-l;p
-
0.5
-
1.4
day-’.
Given Eq.
(7)
we can solve Eq.
(6)
for the covariances
RXy(r)
of the response
v/
and the forcingX(Egger and Schilling, 1984). For intercomparison R,,(z)
must be determined from data
as
well. The result is displayed in Fig.
1
for two
modes. The mode with
m
=
3,
n
=
2
is a low retrograde mode with an
Rossby frequency of
o
=
-0.7
X
10-l
sec-l.
186
J.
EWER
ETAL.
-4
-3
-2
-1
FIG.
I,
Covariance
of
forcing
and
response
RXv(?)
as
observed (solid lines) and
as
determined
theoretically (dashed lines)
for
two
modes
of
the planetary flow;
7
in
days,
R
in
lo6
m4
~(3~~.
Adapted
from
Egger and Schilling
(1984).
The agreement of the theoretical curve and the observations is quite good.
We have negative values of the covariance for
7
<
0
and a sharp increase of
R, when
7
approaches the origin. There is a maximum ofR,, after about
1
day and a slow decay afterwards. It
is
easy to understand the curve for
7
>
0.
A
positive forcing
X
at
7
=
0
creates a positive response, of course. Friction
and movement with the Rossby phase speed limit the increase of the re-
sponse with increasing lag. Furthermore, the forcing changes rapidly accord-
ing to Eq.
(7).
It is not
so
easy to understand the negative values of
R
for
7
<
0
and the reader is referred to Egger and Schilling
(
1984)
for a discussion ofthis
phenomenon. The mode with
m
=
4,
n
=
4
(Fig.
1)
is progressive and
is
close to the boundaries of the planetary wave group. There the agreement of
observations and theory is less satisfactory.
A
more exhaustive comparison
led Egger and Schilling
(
1984)
to conclude that the agreement is particularly
good if we look at the
slow
quasi-stationary modes. These modes experience
the fluctuations of the vorticity transfer from synoptic-scale modes as an
independent stochastic forcing. Since blocking
is
a relatively slow quasi-sta-
tionary process and since the planetary-scale response to stochastic forcing is
particularly realistic for the slow modes, we feel confident that the forcing of
blocks by synoptic-scale modes can be studied by aid of Eq.
(6).
Moreover,
Schilling
(1
986)
demonstrates that the kinetic energy of waves with
m
5
5
is
increased before the onset of blocking by this forcing.
3.
RESULTS
In almost all runs the observed winter mean value
U,,
=
12
m sec-l is
prescribed for the mean flow. The forcing
X,,
for each mode
of
the resolved
flow is inserted
as
observed
at every day ofthe numerical integration
(m
5
5,
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
187
n
5
5). Most of the results have been obtained for winter cases. These runs
are started with the undisturbed superrotation
@
=
-
Uoa
sin
a,
as initial state
and the flow evolves as a response to the forcing. The forcing
X,,
is pre-
scribed for 11 winters according to the observations. The experiments are
started on
20
November. The evaluation of the flow patterns begins on
1
December and the runs are terminated after
3
more months. The integra-
tions have been carried out first with the linear version of Eq.
(3).
They have
been repeated with all nonlinear terms included. In addition there has been a
nonlinear experiment where the integration has been extended over 2 years
and where an annual cycle was prescribed for
U,
(Muller, 1984).
The evaluation of the numerical experiments has been done mainly in
terms of blocking events. Blocking in the model is defined as follows: one
considers the streamfunction of the resolved flow in the latitude band 40”N
to 75 ON. Maps of the resolved flow are prepared every day and the extrema
of the deviation of the streamfunction from climatology are searched for.
Such an extremum is called a blocking anticyclone when the difference
A
=
~(x,
y,
t)
-
Fr(x,
y)
(-t,
time mean) satisfies the amplitude criterion
A
>
1.5
X
lo7
mz sec-’ and if the “anomaly” can be found for more than 4
consecutive days. In addition, slow anomalies are selected by requiring that
the shift of the location of the anomalies
is
less than 5
’
longitude per day. If
the blocking definition
is
applied to data we use a vertically averaged stream-
function. More details of the selection criteria can be found in Metz
(1
986).
There are two problems with such blocking definitions. First, it is not clear
if an observed block still looks like a block when only resolved modes are
admitted to describe the flow. In Fig.
2
we show an example of an observed
block (Fig. 2a) and its counterpart in the space of resolved modes. It is seen
that much detail is lost in the truncated version but the overall flow pattern is
fairly similar in both representations. In particular, the blocking patterns in
the Pacific and Atlantic are resistent with respect to this truncation. Second,
one cannot be sure if our definition of blocking yields about the same results
as have been obtained by other authors (e.g., Treidl
et
al.,
198
1).
To check
this possibility we took the flow observations for the
1
1 winters and searched
for blocks using the criterion. The result is displayed in Fig.
3.
There is a
pronounced maximum of blocking activity in the Atlantic and Pacific. This
result is in satisfactory agreement with the findings
of
others (e.g., Treidl
et
al.,
198
1).
The latitudinal distribution peaks at 55
O,
but there are blocks well
within the whole latitudinal band under search. These features also come out
clearly in Fig. 4, where the mean locations of blocking highs are shown. The
majority of blocks is situated in the Atlantic and Pacific. Note the almost
complete absence of blocks over North America.
Let us now have a look at the modeling results. Although it is straightfor-
ward to apply the blocking definition to the model results this need not be a
meaningful procedure. The only source of wave energy in our model is the
a
I
0-
FIG.
2.
500-mbar analysis
as
provided
by
the German Weather Service
on
14
May,
1972;
a,
original analysis redrawn;
b,
resolved modes
only.
Contour
interval,
8
gdam. Adapted
from
Miiller
(1
984).
188
20
18
16
>
l4
u
z
12
W
3
10
a
E8
LL
6
4
2
0
>
0
W
3
(3-
W
m
LL
z
20
10
0.
180WI
40WI
OOW
60W
20W
20-5
60E1 OOEl40E
LONGITUDE
7
-
-
-
-
-
'
7,
30
t
40
>-
30
0
z
W
3
W
&
LL
0
20
10
0
1
I
L
4
6 8
10
12
14 16
18
20
22
24
DAYS
FIG.
3.
Frequency distribution
of
blocking
highs
with
longitude, latitude, and duration.
Adapted
from
Metz
(1986).
189
190
J.
EWER
ETAL.
O0
FIG.
4.
Mean positions
of
blocking highs.
Adapted
from
Metz
(1986).
synoptic-scale forcing. In the atmosphere topographic effects will generate
planetary-scale waves. Moreover, the planetary modes can extract energy
from the mean flow.
So
the wave amplitudes
in
the model should be smaller
than those observed. In these circumstances blocks that can satisfy the am-
plitude criterion will
be
rare. We circumvent this problem by tuning the
damping in the model
so
as to have wave amplitudes that are close
to
those
observed. The value adopted is
k
=
6.0
X
lo5
m2 sec-'. This value is cer-
tainly within the range of what one would consider acceptable for barotropic
large-scale flow.
In Fig.
5
we show the distribution of the linear model blocks over the
Northern Hemisphere for comparison with Fig.
3.
The model
is
correct in
predicting the maximum position density over the Atlantic but tends to
predict blocking action in areas where none is observed. There
is
also agood
agreement
of
the duration
of
the blocks, although the model blocks tend
to
be longer lived.
It is not
only
in the gross statistical characteristics that the model comes
reasonably close
to
reality. If we
look
at the spatial structure of the simulated
blocks we also see quite a good agreement. We show first the evolution
of
a
blocking ridge at the Greenwich Meridian that recedes westward at the end
of
the blocking period (Fig.
6),
as obtained
by
MiiIler (1984) in the
run
with
>-
0
z
W
3
c3
W
m
LL
l8OW
12OW
60W OE 60E 120E
LONGITUDE
I
30
40
50
60
70
80
LATITUDE
I
W
w
m
LL
::
0
l-
4
6
8
10
12
14
16
18
20
22
24
DAYS
FIG.
5.
Frequency distribution
of
blocking highs with longitude, latitude, and duration as
obtained in the stochastically forced linear model. Adapted
from
Metz (1986).
a
P-
o,
f
I
OC
DAY
347
0:
DAY
349
FIG.
6.
Blocking event as obtained in a nonlinear integration
of
the stochastically forced
model.
Adapted from Miiller (1984).
192
c
00
DAY
355