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Forcing of Planetary-Scale Blocking Anticyclones by Synoptic-Scale Eddies
FORCING OF PLANETARY-SCALE BLOCKING ANTICYCLONES BY SYNOPTICSCALE EDDIES J. EGGER,w. METZ,AND G. M ~ L L E R Meteorology Institute University of”Munich D-8000Munich 2 Federal Republic of Germany
1. INTRODUCTION
The flow patterns typical of blocking are of large scale. Correspondingly, most theories of blocking concentrate on the large-scale aspect of blocking. The impact of synoptic-scale waves is excluded. For example, Egger (1978) used a truncated semispectralmodel of barotropic channel flow where only the largest modes where retained to demonstrate that blocking-type flow patterns can be created through a nonlinearly modified resonance mechanism. Tung and Lindzen (1979) advanced the view that blocking can be explained through linear resonance of forced planetary-scale waves. Schilling ( 1982) provided evidence that baroclinic energy conversion between large-scale flow modes can lead to blocking. These theoretical studies found some support from data analyses (e.g., Colucci et al., 1981) and from the analysis of global circulation model (GCM) data (Chen and Shukla, 1983). However, there is ample evidence that this basic view of blocking as a phenomenon which is almost completelydominated by the dynamics of the largest atmosphericmodes is not adequate. It has been realized for some time that traveling weather systems interact with the block. Green (1977), for example, pointed out that the transfer of momentum by the synoptic-scale eddies played an important role in maintaining the block linked to the British drought of 1976 (see also Illari and Marshall, 1983). Hansen and Chen (1982), when studyinga case ofblocking in the Atlantic, found that this block was forced by the nonlinear interaction of intense baroclinic synopticscalewaves with barotropic ultralongwaves. Fischer (1984)arrived at similar conclusions through an analysis of data obtained in numerical experiments with a GCM. Hansen and Sutera (1984) provided evidence that the transfer of enstrophy between these two wave regimes was different for blocking and nonblocking situations. These findings have been taken into account by modelers. Egger ( 198I), using a simplified version of the model proposed by 183 ADVANCES IN GEOPHYSICS, VOLUME
29
Copyright 0 1986 by Academic Ress, Inc. AU rights of reproduction in any form reserved.
184
J. EGGER E T A L .
Charney and DeVore ( 1979), prescribed the interaction of large-scalemodes 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 KalnayRivas and Merkine, 198 1). However, it is a shortcomingof these theoretical efforts that the choice of the forcing is largely intuitive and not based on observations. The synoptic-scaleforcing 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 atmosphere 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 FORCEDPLANETARY 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 described by the vorticity equation
+
(d/dt)(V2 - A2)y J(v, V2ty
+ 2l2a-l sin 9)= k V 4 y
(1)
where v/ is the streamfunction, 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 included on the right-hand side of Eq. (1 ). We partition the streamfunctionin a planetary part vrto be resolved and treated explicitly and an unresolved synoptic-scale part vu:
v = vr + v u
(2)
The equation for the planetary part is (d/dt)(V2 - A2)tyr = -Jr(vry
+ Jr(cy,,V2yr+ 2Ra-I sin rp) - kV4t,vr
V2y/,) - J r ( ~ u t V2vr) - Jr(v/,t V2y/,)
(3)
The subscript rat the symbol Jdenotes the projection of this Jacobian on the
PLANETARY-SCALEBLOCKING 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 Jacobianstogether and prescribe them as forcing. We split the streamfunction into a mean flow and a deviation thereof:
vr= u/+ v’
(4)
where u/ = - Uoasin a, is the superrotational mode with a zonal wind profile u = u, cos a,. It is convenient to project v/’ onto spherical harmonics l
W‘ = 5
xx M
m--M
N
n-1
WmnP:+zn--I(sin P) ex~(imi’
(5)
where P is the corresponding associated Legendre function and is longitude. It is seen that we admit only those modes which are antisymmetricwith 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 (dIdt)vmn + a v m n = x m n
(6)
where a contains the effects of damping and wave propagationandX,, 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 forcingcan be approximated by Rxx(z)= Z exp(-slzl) cos(p7)
(7)
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(Eggerand Schilling, 1984). For intercomparisonR,,(z) must be determined from data as well. The result is displayedin 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. E W E R E T A L .
-4
-3
-2
-1
FIG. I , Covariance of forcing and responseRXv(?)as observed(solidlines) 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 response with increasing lag. Furthermore, the forcing changes rapidly according 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 discussionofthis 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-stationarymodes. 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-stationary process and since the planetary-scaleresponse 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 (m5 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 @ = - Uoasin a, as initial state and the flow evolves as a response to the forcing. The forcing X,, is prescribed 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 integrations 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 = ~ ( xy,, 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 streamfunction. 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., 1981). To check this possibility we took the flow observationsfor 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 satisfactoryagreement with the findings of others (e.g., Treidl et al., 1981). The latitudinal distribution peaks at 55 but there are blocks well within the whole latitudinalband under search. These featuresalso 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 straightforward 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 O,
a
I
0FIG.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 W
12
3 10
a
E 8 LL
6 4 2 0
180WI 40WI OOW 60W 20W 20-5 60E1 OOEl40E
LONGITUDE
30
>
0
z
W
t
20
-
3
-
10
-
-
3
(
W
m
LL
-
7
-
'
0.
7
,
40
1
>- 30 0
z W
W &
LL
10
0
L I
3 0 20
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. E W E R 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 amplitude 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 certainly 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
: l30
40
50
60
70
80
LATITUDE
I
W
w
m
LL
0
4
6
8 10 1 2 14 16 18 20 22 2 4
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
D A Y 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
0 0
DAY 355
194
J. E W E R ET AL.
the annual cycle. To establish the typical blocking flow pattern in the model Metz (1 986) has computed the mean maps of the streamfunctionanomalies averaged over all Atlantic and all Pacific blocking periods separately. The result for the Atlantic case and for the linear run is shown in Fig. 7 and is to be compared to the corresponding map derived from the I 1 winters. The agreement is quite good, although the amplitudes are somewhat too low in the modelderived map. The correspondingmap for the nonlinear run is similar. The agreement for the Pacific cases is not so good but is still satisfactory (Metz, 1986). All this demonstrates that a majority of the most prominent characteristics of atmospheric blocking are captured by the model. Strictly speaking, this does not prove that atmospheric blocks can be caused by the synoptic-scale forcing. After all, the flow evolution in the model is not the same as that in the atmosphere, and the forcing acts in flow fields which are not those observed,However, if we can demonstrate that the model flowsare
FIG.7. Mean streamfunction ( lo6 m 2sec-I) anomaly for wintertime blocking in the Atlantic (40"W to 1O"W); a, observations; b, linear model result. Stippling: 95% confidence area. Adapted from Metz ( 1986).
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
195
similar to the observed flow over time, this will be a strong argument in favor of the forcing hypothesis. Metz (1986) has computed mean model flow maps averaged over all those days wherein Atlantic (Pacific) blocks have been observed in the atmosphere. For the Pacific blocks the result was negative. The mean flow obtained hardly resembles the observed mean flow for Pacific blocking. However, the agreement for the Atlantic cases is surprisingly good both for the linear and the nonlinear run (Fig. 8). We obtain the familiarhigh over the Atlantic almost at the correct position. This means that many of the Atlantic model blocks occur on the same days as in the atmosphere. As for the Pacific, possibly a data problem exists since the forcing fields are subject to large errors in data-sparse areas. It is also conceivable that synoptic-scale forcing is not so important for the Pacific blocks. The results presented lead to tentative conclusions: 1. The barotropic vorticity equation can be used even in a linearized form to study the impact of synoptic-scale forcing on the planetary flow. 2. The stochasticforcing creates blocking patterns in linear and nonlinear
196
J . EGGER E T A L
FIG.8. Mean streamfunction anomaly (lo6m 2sec-') in the model when the model flow is averaged over episodes of observed blocking. Stippling: 95% confidence areas. Adapted from Metz (1986).
planetary-scale flow. Location and duration of these enforced blocking ridges show reasonably good agreement with the observations, although local preferences are not as pronounced in the model results. This suggests that the forcing of planetary-scale modes by the nonlinear interaction with synoptic-scale weather systems contributes considerably to the creation and maintenance of blocking in the Northern Hemisphere. This is not to say that other mechanisms are unimportant. After all, it has been shown in case studies that this nonlinear transfer is not the only mechanism. Moreover, we have to keep in mind that the response of the model to the forcing may be too iarge since we chose relatively small damping rates. With stronger damping only a few of the blocking situations in the model would satisfy the amplitude criterion. Additional mechanisms would be required in order to create strong blocking anticyclones. Let us close with a comment on the forcing problem. The stochastic forcing in the numerical experiment is prescribed and, therefore, not in-
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
197
fluenced at all by the planetary flow in the model. In particular,the blocks in the model are enforced by the synoptic-scale eddies but these eddies do not “feel” the block in the model. Nevertheless, the blocking climatology of the model has been found to be reasonably realistic. This suggests that the influence of the planetary flow on the synoptic-scale forcing may be minor. However, it would be premature to draw such a conclusion. A significant interdependency of forcing and large-scale flow may be found in reality despite the fact that the model produces reasonably realistic results with independent forcing. This is suggested by the observed strong local preference of the eddy forcing and by instability calculations like those of Frederiksen (1983).
REFERENCES Benzi, R., Hansen, A., and Sutera, A. (1984). On stochastic perturbation of simple blocking models. Q. J. R. Meteorol. Soc. 110, 393-409. Charney, J., and DeVore, J. (1 979). Multiple flow equilibria in the atmosphere and blocking. J. Atrnos. Sci. 37, 1 157 - 1 176 Chen, T.-C., and Shukla, J. ( 1 983). Diagnostic analysis and spectral energetics of a blocking event in the GLAS climate model simulation. Mon. WeatherRev.111, 3-22. Colucci, S., Loesch, A., and Bosart, L. (1981). Spectral evolution of a blocking episode and comparison with wave interaction theory. J. Atmos. Sci.38,2092 -2 1 1 1. Egger, J. (1978). Dynamics of blocking highs. J. Atmos. Scz. 35, 1788- 1801. Egger, J. (1 98 1). Stochasticallydriven large-scale circulationswith multiple equilibria.J. Atmos. Sci. 38,2606 -26 18. Egger, J., and Schilling, H.-D. (1983). On the theory of the long-term variability of the atmosphere. J. Atmos. Sci. 40, 1073- 1085. Egger, J., and Schilling,H.-D. (1984). Stochasticforcing of planetary scale flow. J. Atmos. Sci. 41,779-788. Fischer, G. (1984). Spectral energetics analyses of blocking events in a general circulation model. Cont. Atmos. Phys. 57, 183-200. Frederiksen, J. ( 1983). Disturbances and eddy fluxes in Northern Hemisphere flows in stability of three-dimensional January and July flows. J. Atmos. Sci.40, 836-855. Green, J. ( 1 977). The weather duringJuly 1976:Some dynamical considerationsofthe drought. Weather 32, 120- 128. Hansen, A., and Chen, T.-C. (1982). A spectral energetics analysis of atmospheric blocking. Mon. Weather Rev. 110, 1146- 1159. Hansen, A., and Sutera, A. (1 984). A comparison of the spectral energy and enstrophy budgets of blocking compared to non-blocking periods. Tellus 36A, 52-63. Illari, L., and Marshall, L. (1983). On the interpretation of maps of eddy fluxes. J. Atmos. Sci. 40,2232-2242. Kalnay-Rivas,E., and Merkine, L. 0.(198 1). A simple mechanism for blocking. J. Atmos. Sci. 38,2077 -209 1. Metz, W. (1986). Transient cyclone-scale vorticity forcing of blocking highs. J. Atmos. Sci., submitted.
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Muller, G. (1984). Blockierungen in einem Modell mit stochastischer Forcierung. Diplomarbeit, Meteorol Inst. Univ. Miinchen. Schilling, H.-D. (1982). A numerical investigation of the dynamics of blocking waves in a simple two-level m d e l . J. Atmos. Sci. 39,998- 1017. Schilling, H.-D. (1986). On atmospheric blocking types and blocking numbers. Adv. Geophys. 29,71-99. Shutts, G. (1983). The propagation of eddies in diffluentjet-streams: eddy vorticity forcing of “blocking” flow fields. Q.J. R.Meieorol. SOC.109, 737-761. Treidl, R. A., Birch, E., and Sajecki, P. (198 1). Blocking action in the northern hemisphere: A climatological study. Atmos. Oceun 19, 1-23. Tung, K.-K, and Lindzen, R.(1979). A theory of stationary long waves. Part I: A simple theory of bloclung. Mon. Weather Rev. 107,714-734.
1. INTRODUCTION
The flow patterns typical of blocking are of large scale. Correspondingly, most theories of blocking concentrate on the large-scale aspect of blocking. The impact of synoptic-scale waves is excluded. For example, Egger (1978) used a truncated semispectralmodel of barotropic channel flow where only the largest modes where retained to demonstrate that blocking-type flow patterns can be created through a nonlinearly modified resonance mechanism. Tung and Lindzen (1979) advanced the view that blocking can be explained through linear resonance of forced planetary-scale waves. Schilling ( 1982) provided evidence that baroclinic energy conversion between large-scale flow modes can lead to blocking. These theoretical studies found some support from data analyses (e.g., Colucci et al., 1981) and from the analysis of global circulation model (GCM) data (Chen and Shukla, 1983). However, there is ample evidence that this basic view of blocking as a phenomenon which is almost completelydominated by the dynamics of the largest atmosphericmodes is not adequate. It has been realized for some time that traveling weather systems interact with the block. Green (1977), for example, pointed out that the transfer of momentum by the synoptic-scale eddies played an important role in maintaining the block linked to the British drought of 1976 (see also Illari and Marshall, 1983). Hansen and Chen (1982), when studyinga case ofblocking in the Atlantic, found that this block was forced by the nonlinear interaction of intense baroclinic synopticscalewaves with barotropic ultralongwaves. Fischer (1984)arrived at similar conclusions through an analysis of data obtained in numerical experiments with a GCM. Hansen and Sutera (1984) provided evidence that the transfer of enstrophy between these two wave regimes was different for blocking and nonblocking situations. These findings have been taken into account by modelers. Egger ( 198I), using a simplified version of the model proposed by 183 ADVANCES IN GEOPHYSICS, VOLUME
29
Copyright 0 1986 by Academic Ress, Inc. AU rights of reproduction in any form reserved.
184
J. EGGER E T A L .
Charney and DeVore ( 1979), prescribed the interaction of large-scalemodes 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 KalnayRivas and Merkine, 198 1). However, it is a shortcomingof these theoretical efforts that the choice of the forcing is largely intuitive and not based on observations. The synoptic-scaleforcing 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 atmosphere 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 FORCEDPLANETARY 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 described by the vorticity equation
+
(d/dt)(V2 - A2)y J(v, V2ty
+ 2l2a-l sin 9)= k V 4 y
(1)
where v/ is the streamfunction, 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 included on the right-hand side of Eq. (1 ). We partition the streamfunctionin a planetary part vrto be resolved and treated explicitly and an unresolved synoptic-scale part vu:
v = vr + v u
(2)
The equation for the planetary part is (d/dt)(V2 - A2)tyr = -Jr(vry
+ Jr(cy,,V2yr+ 2Ra-I sin rp) - kV4t,vr
V2y/,) - J r ( ~ u t V2vr) - Jr(v/,t V2y/,)
(3)
The subscript rat the symbol Jdenotes the projection of this Jacobian on the
PLANETARY-SCALEBLOCKING 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 Jacobianstogether and prescribe them as forcing. We split the streamfunction into a mean flow and a deviation thereof:
vr= u/+ v’
(4)
where u/ = - Uoasin a, is the superrotational mode with a zonal wind profile u = u, cos a,. It is convenient to project v/’ onto spherical harmonics l
W‘ = 5
xx M
m--M
N
n-1
WmnP:+zn--I(sin P) ex~(imi’
(5)
where P is the corresponding associated Legendre function and is longitude. It is seen that we admit only those modes which are antisymmetricwith 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 (dIdt)vmn + a v m n = x m n
(6)
where a contains the effects of damping and wave propagationandX,, 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 forcingcan be approximated by Rxx(z)= Z exp(-slzl) cos(p7)
(7)
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(Eggerand Schilling, 1984). For intercomparisonR,,(z) must be determined from data as well. The result is displayedin 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. E W E R E T A L .
-4
-3
-2
-1
FIG. I , Covariance of forcing and responseRXv(?)as observed(solidlines) 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 response with increasing lag. Furthermore, the forcing changes rapidly according 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 discussionofthis 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-stationarymodes. 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-stationary process and since the planetary-scaleresponse 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 (m5 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 @ = - Uoasin a, as initial state and the flow evolves as a response to the forcing. The forcing X,, is prescribed 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 integrations 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 = ~ ( xy,, 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 streamfunction. 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., 1981). To check this possibility we took the flow observationsfor 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 satisfactoryagreement with the findings of others (e.g., Treidl et al., 1981). The latitudinal distribution peaks at 55 but there are blocks well within the whole latitudinalband under search. These featuresalso 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 straightforward 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 O,
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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 amplitude 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 certainly 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
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the annual cycle. To establish the typical blocking flow pattern in the model Metz (1 986) has computed the mean maps of the streamfunctionanomalies averaged over all Atlantic and all Pacific blocking periods separately. The result for the Atlantic case and for the linear run is shown in Fig. 7 and is to be compared to the corresponding map derived from the I 1 winters. The agreement is quite good, although the amplitudes are somewhat too low in the modelderived map. The correspondingmap for the nonlinear run is similar. The agreement for the Pacific cases is not so good but is still satisfactory (Metz, 1986). All this demonstrates that a majority of the most prominent characteristics of atmospheric blocking are captured by the model. Strictly speaking, this does not prove that atmospheric blocks can be caused by the synoptic-scale forcing. After all, the flow evolution in the model is not the same as that in the atmosphere, and the forcing acts in flow fields which are not those observed,However, if we can demonstrate that the model flowsare
FIG.7. Mean streamfunction ( lo6 m 2sec-I) anomaly for wintertime blocking in the Atlantic (40"W to 1O"W); a, observations; b, linear model result. Stippling: 95% confidence area. Adapted from Metz ( 1986).
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
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similar to the observed flow over time, this will be a strong argument in favor of the forcing hypothesis. Metz (1986) has computed mean model flow maps averaged over all those days wherein Atlantic (Pacific) blocks have been observed in the atmosphere. For the Pacific blocks the result was negative. The mean flow obtained hardly resembles the observed mean flow for Pacific blocking. However, the agreement for the Atlantic cases is surprisingly good both for the linear and the nonlinear run (Fig. 8). We obtain the familiarhigh over the Atlantic almost at the correct position. This means that many of the Atlantic model blocks occur on the same days as in the atmosphere. As for the Pacific, possibly a data problem exists since the forcing fields are subject to large errors in data-sparse areas. It is also conceivable that synoptic-scale forcing is not so important for the Pacific blocks. The results presented lead to tentative conclusions: 1. The barotropic vorticity equation can be used even in a linearized form to study the impact of synoptic-scale forcing on the planetary flow. 2. The stochasticforcing creates blocking patterns in linear and nonlinear
196
J . EGGER E T A L
FIG.8. Mean streamfunction anomaly (lo6m 2sec-') in the model when the model flow is averaged over episodes of observed blocking. Stippling: 95% confidence areas. Adapted from Metz (1986).
planetary-scale flow. Location and duration of these enforced blocking ridges show reasonably good agreement with the observations, although local preferences are not as pronounced in the model results. This suggests that the forcing of planetary-scale modes by the nonlinear interaction with synoptic-scale weather systems contributes considerably to the creation and maintenance of blocking in the Northern Hemisphere. This is not to say that other mechanisms are unimportant. After all, it has been shown in case studies that this nonlinear transfer is not the only mechanism. Moreover, we have to keep in mind that the response of the model to the forcing may be too iarge since we chose relatively small damping rates. With stronger damping only a few of the blocking situations in the model would satisfy the amplitude criterion. Additional mechanisms would be required in order to create strong blocking anticyclones. Let us close with a comment on the forcing problem. The stochastic forcing in the numerical experiment is prescribed and, therefore, not in-
PLANETARY-SCALE BLOCKING ANTICYCLONES FORCING
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fluenced at all by the planetary flow in the model. In particular,the blocks in the model are enforced by the synoptic-scale eddies but these eddies do not “feel” the block in the model. Nevertheless, the blocking climatology of the model has been found to be reasonably realistic. This suggests that the influence of the planetary flow on the synoptic-scale forcing may be minor. However, it would be premature to draw such a conclusion. A significant interdependency of forcing and large-scale flow may be found in reality despite the fact that the model produces reasonably realistic results with independent forcing. This is suggested by the observed strong local preference of the eddy forcing and by instability calculations like those of Frederiksen (1983).
REFERENCES Benzi, R., Hansen, A., and Sutera, A. (1984). On stochastic perturbation of simple blocking models. Q. J. R. Meteorol. Soc. 110, 393-409. Charney, J., and DeVore, J. (1 979). Multiple flow equilibria in the atmosphere and blocking. J. Atrnos. Sci. 37, 1 157 - 1 176 Chen, T.-C., and Shukla, J. ( 1 983). Diagnostic analysis and spectral energetics of a blocking event in the GLAS climate model simulation. Mon. WeatherRev.111, 3-22. Colucci, S., Loesch, A., and Bosart, L. (1981). Spectral evolution of a blocking episode and comparison with wave interaction theory. J. Atmos. Sci.38,2092 -2 1 1 1. Egger, J. (1978). Dynamics of blocking highs. J. Atmos. Scz. 35, 1788- 1801. Egger, J. (1 98 1). Stochasticallydriven large-scale circulationswith multiple equilibria.J. Atmos. Sci. 38,2606 -26 18. Egger, J., and Schilling, H.-D. (1983). On the theory of the long-term variability of the atmosphere. J. Atmos. Sci. 40, 1073- 1085. Egger, J., and Schilling,H.-D. (1984). Stochasticforcing of planetary scale flow. J. Atmos. Sci. 41,779-788. Fischer, G. (1984). Spectral energetics analyses of blocking events in a general circulation model. Cont. Atmos. Phys. 57, 183-200. Frederiksen, J. ( 1983). Disturbances and eddy fluxes in Northern Hemisphere flows in stability of three-dimensional January and July flows. J. Atmos. Sci.40, 836-855. Green, J. ( 1 977). The weather duringJuly 1976:Some dynamical considerationsofthe drought. Weather 32, 120- 128. Hansen, A., and Chen, T.-C. (1982). A spectral energetics analysis of atmospheric blocking. Mon. Weather Rev. 110, 1146- 1159. Hansen, A., and Sutera, A. (1 984). A comparison of the spectral energy and enstrophy budgets of blocking compared to non-blocking periods. Tellus 36A, 52-63. Illari, L., and Marshall, L. (1983). On the interpretation of maps of eddy fluxes. J. Atmos. Sci. 40,2232-2242. Kalnay-Rivas,E., and Merkine, L. 0.(198 1). A simple mechanism for blocking. J. Atmos. Sci. 38,2077 -209 1. Metz, W. (1986). Transient cyclone-scale vorticity forcing of blocking highs. J. Atmos. Sci., submitted.
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Muller, G. (1984). Blockierungen in einem Modell mit stochastischer Forcierung. Diplomarbeit, Meteorol Inst. Univ. Miinchen. Schilling, H.-D. (1982). A numerical investigation of the dynamics of blocking waves in a simple two-level m d e l . J. Atmos. Sci. 39,998- 1017. Schilling, H.-D. (1986). On atmospheric blocking types and blocking numbers. Adv. Geophys. 29,71-99. Shutts, G. (1983). The propagation of eddies in diffluentjet-streams: eddy vorticity forcing of “blocking” flow fields. Q.J. R.Meieorol. SOC.109, 737-761. Treidl, R. A., Birch, E., and Sajecki, P. (198 1). Blocking action in the northern hemisphere: A climatological study. Atmos. Oceun 19, 1-23. Tung, K.-K, and Lindzen, R.(1979). A theory of stationary long waves. Part I: A simple theory of bloclung. Mon. Weather Rev. 107,714-734.