The Theory of Long-lived Atmospheric Anomalies in the Midlatitudes
Walter A. Robinson
University of Illinois at Urbana-Champaign
These notes on long-lived anomalies are split into two parts. The first is a discussion of
the anomalies generated internally by atmospheric dynamics, while the second describes
the possible role of local boundary forcing in generating long-lived anomalies. These are
intimately related topics. In middle latitudes, the boundary forcing is weak, and is
probably effective only to the extent that it stimulates patterns similar to those that the
atmosphere generates on its own. This may also be the case for remote forcing from the
tropics, a topic covered by Grant Branstator elsewhere in this series. On the other hand,
only boundary forcing can produce long enough timescales for these phenomena to be
relevant to the Dec-Cen problem.
Internal dynamics
Starting with internally generated variability, the first important point, is that the “modes”
of variability are only very loosely such. The preferred spatial patterns of variability are
only weakly preferred, and temporal spectra are generally red, unpunctuated by
significant peaks. An example of the latter is shown in Figure 1, from Feldstein 2000.
Fig 1. Power spectra of the a), NAO, b), PNA, c) ENSO, and d) WP patterns (Feldstein 2000).
Power spectra are shown for two prominent modes of atmospheric variability, the North
Atlantic Oscillation (NAO), and the Pacific North American (PNA) patterns. The spectra
are computed separately for sub and super annual periods. These spectra are generally
consistent with that of a red noise process, though the PNA has a significant peak on El
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Niño timescales, consistent with its observed association with forcing from the tropical
Pacific.
In regard to spatial structures, while there are preferred spatial scales for long-lived
patterns, the locations of their nodes and antinodes are only weakly, if at all, fixed. This
is pointed out by Kushnir and Wallace (1989). If one is going to read just one paper about
the observed structures of low-frequency variability, it should be this one. Kushnir and
Wallace show that patterns of long-lived anomalies occupy a continuum of locations in
space, and are significantly localized only on the longest – interannual – timescales,
where, presumably, external forcing comes into play. Even interannually, however,
localization is weak, as is shown by Figure 2. This plot (produced using the Climate
Diagnostics Center online atlas of NCEP/NCAR reanalysis products) shows a decade of
anomalous, monthly mean, upper tropospheric zonal winds averaged over the Atlantic
sector. What is expected is an upper tropospheric manifestation of the NAO. What is
seen, aside from the fact that the variability is greatest in winter, are anomalies with a
rigidly fixed meridional scale but without any preferred latitude. For example, in
February 1993 the node is at 50 N, while it is close to 40 N during the winter of 1994.
Fig 2. Monthly averaged 200 mb zonal wind anomalies averaged from 0 to 90° W. (www.cdc.noaa.gov).
We turn now to the dynamics of long-lived anomalies. These anomalies fall, very
loosely, into two categories, Rossby wavetrains and annular modes. The archetype for
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Rossby wavetrains is the PNA pattern. The scale of wavetrain patterns is set by the
Rossby-wave dispersion relation – they have roughly the scale and structure of zerofrequency barotropic Rossby waves. Stationary Rossby waves have group velocities
significantly different from zero, so wavetrain patterns propagate wave activity or wave
energy away from their sources on times less than a week. Therefore, if these are to be
long-lived anomalies, they must be vigorously forced.
The simplest context for studying Rossby wavetrain patterns is a zonally homogeneous,
dry, two-level model on the sphere (Robinson 1991a). In such a model, these patterns
occur with no preferred locations, but can be isolated, by first low-pass filtering the
output, and then performing time-lagged one-point correlations. Figure 3 shows an
example. These panels clearly show the rapid dispersal of eddy activity into the
Fig 3. Time lagged one-point correlations of the low-pass upper-level streamfunction in a two level model.
The base point is indicated by the x. Lags are a) –6 days, b) –3 days, c) +3 days, and d) +6 days.
(Robinson 1991a)
tropics, while the phase of the wave is nearly stationary. These wavetrains are very
nearly equivalent barotropic and do not extract energy baroclinically from the
background flow. In the atmosphere, energy can be extracted barotropically from the
zonally asymmetric background flow, but not in this model, where the background flow
is zonally symmetric. Interactions with transient eddies remain as the only possible
source for the wave activity shown in Fig 3.
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Given that transient baroclinic eddies are the source of these wavetrains, this could either
be a one-way interaction, or the wavetrains could (partially) organize the transient eddies
so that a transient-eddy feedback operates. Model experiments, some with imposed lowfrequency anomalies, indicate that such a feedback loop is present. Figure 4, from Qin
and Robinson 1992, suggests how such a feedback might work. The distortion and
guiding of eddies by a wavy jet – the waviness coming from the wavetrains – causes the
eddies to systematically reinforce the wavy jet.
Fig 4. Eddy vorticity (shading) and streamfunction (thin contours) for linear barotropic eddies propagating
on a wavy basic state (thick contours). (Qin and Robinson 1992).
The other main type of low-frequency pattern is zonally symmetric. Disturbances
with zonal wavenumber zero trivially have zero frequency and group velocity, so that
their persistence is limited only by friction. Such modes are observed ubiquitously in
models and in observations, and have been assigned a variety of names: zonal index,
Arctic oscillation, annular modes, and even “wobblers”. Lorenz (1951) performed the
first quantitative analysis of zonally averaged variability in observations. He found a
strong negative correlation in sea-level pressure between subpolar and middle latitudes,
present in both the mean seasonal cycle and its deviations. Perhaps the most remarkable
thing is how little we have added to his understanding in the subsequent 50 years. A
modern picture, showing the vertical structure, is shown in Figure 5 (from Thompson and
Wallace 2000).
Annular modes are prominent in the same dry, global, two-level model mentioned earlier
(Robinson 1991b). Like the wavetrains in this simple model, the annular modes can be
forced only by transient-eddy momentum fluxes. The model can once again manipulated
to demonstrate that the transient eddy forcing is not entirely stochastic – a transient-eddy
feedback operates. Figure 6 suggests how such a feedback might work. Surface drag
makes an anomalously strong barotropic jet more baroclinic. Transient baroclinic eddy
generation is enhanced where the baroclinicity is stronger. As the baroclinic eddies
propagate away from their source region, their momentum fluxes reinforce the anomalous
jet (Robinson 2000).
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Fig 5. Zonally averaged zonal wind and low-level geopotential structure of the Southern (a and c) and
Northern (b and d) Hemisphere annular modes. (Thompson and Wallace 2000).
Fig 6. Schematic of a positive transient-eddy feedback loop for zonal flow variations, involving surface
drag and the generation of baroclinic eddies.
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In the real world, especially its northern half, there are stationary waves generated by
topography and zonally asymmetric heating. These stationary waves influence and are
influenced by variability in the zonally averaged flow. Taking the latter effect first, the
influence of changes in the zonal flow on the stationary waves, Figure 7 is an example
from the work of Ting et al. (1997). The top panel shows the variability in the stationary
eddies, as represented in the 500 mb height field, associated with interannual fluctuations
in the annular mode, while the bottom panel shows the stationary eddy variability
associated with tropical Pacific sea-surface temperatures (El Niño). Over most of the
hemisphere the former is more important than the latter, though the word “associated” is
key here. Unlike the effects of El Niño, we cannot say that the zonal flow variations
come first and then modify the stationary eddies. The two occur together in a way that is
dynamically self consistent.
Fig 7. Variability (standard deviation) of 500 mb eddy geopotential heights associated with interannual
variability in the zonally averaged flow (top) and tropical Pacific sea-surface temperatures (bottom). (Ting
et al., 1997)
In fact, a strong argument can be made for causality in the other direction. Limpasuvan
and Hartmann (2000) show that in the Northern Hemisphere (unlike the Southern
Hemisphere or a simple zonally homogeneous model) the stationary eddy momentum
fluxes drive variations in the annular mode. Because the refraction of the longest
stationary eddies – the planetary waves – is affected by the zonal flow through the
Rossby wave index of refraction, there is again a possibility of a positive feedback loop
that could make the annular modes more persistent than otherwise. In this case, the
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modified planetary waves accelerate or decelerate the zonal flow, and the resulting
modified zonal flow refracts the planetary wave activity towards those regions of reduced
zonal wind. The involvement of planetary waves could explain the observed extension of
the annular mode throughout the depth of the stratosphere (in winter).
Observations suggest that the dynamics of the annular modes in the Northern and
Southern Hemispheres are different. The Southern Hemisphere mode works essentially
like that in the two-level model, and interactions with transient baroclinic eddies
dominate (this is further supported by recent observational work, in press, by Lorenz and
Hartmann). In the Northern Hemisphere, on the other hand, the stationary eddies,
especially those on planetary scales, are key. Two questions are raised by this apparent
dichotomy. First, if the dynamics of the modes in the two hemispheres are so different,
why are there structures so similar? The structures of the annular modes are, in fact,
much more similar between the hemispheres than are the zonally averaged mean flows
they perturb. Secondly, Lau (1988) showed that the North Atlantic Oscillation (NAO) is
essentially a manifestation of variability in transient-eddy statistics (the storm track), and
it is known that the NAO is highly correlated with the Northern annular mode. If the
Northern annular mode is a planetary-wave phenomenon, how is it that it is so closely
associated with the transient-eddy driven NAO? One, as yet untested, possibility is that
the transient eddies are important for driving the anomalous planetary waves.
Boundary forcing
Consideration of the influence of local boundary forcing necessarily starts from linear
theory, though, as will be seen, modeled responses to midlatitude sea-surface temperature
(SST) anomalies only sometimes resemble the linear response. Moreover, the
atmospheric patterns that occur in association with SST anomalies never resemble the
linear response. These patterns, however, cannot be interpreted as the atmospheric
response to anomalous boundary forcing. In middle latitudes, the dominant flow of
information is from the atmosphere into the ocean, so that for the most part the observed
atmospheric patterns are those responsible for creating the SST anomalies.
The mutual influences of the midlatitude ocean and atmosphere are discussed in the
superb review by Frankignoul (1985). We should, perhaps, be embarrassed by how little
new understanding has been added in the past 15 years. Frankignoul emphasizes that the
ocean does not impose an atmospheric heating anomaly. Rather, ocean temperatures lead
to the development of heating anomalies through boundary-layer processes, typically
parameterized using bulk formulae. In linear models, however, we typically impose a
heating anomaly.
In quasi-geostrophic theory, clearly relevant to this problem, a heating anomaly acts as a
source of potential vorticity below the heating, where the static stability is being
increased, and a sink above the heating, where the static stability is being decreased.
Heating cannot contribute to the vertically integrated potential vorticity. If there is
surface heating, this is equivalent to a source of potential vorticity at the lower boundary,
but there is a compensating sink above where the heating decreases with height.
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Fig 8. Linear quasi-geostrophic responses to thermal forcing. Shading indicates perturbation temperatures
and contours show geopotential heights. The top panels show the response to deep (left) and shallow
(right) heating, and the bottom panel shows the response to an imposed temperature anomaly at the surface.
Figure 8 shows some typical linear responses to heating, of which the literature contains
numerous additional examples. The zonal flow is a westerly baroclinic jet. In the top
panels the heating, centered at longitude 180°, is strongest at the surface and decays
exponentially with height. Regardless of whether the heating is deep (left panel) or
shallow (right), the response is baroclinic, with a surface low east of the heating. In the
bottom panel there is no heating. Rather, it is imagined that the boundary layer has
equilibrated with an SST anomaly, and a surface temperature anomaly is imposed. In
this, contrived, case there is a ridge at the surface east of the anomaly. Given that some
boundary-layer compensation presumably does occur in the atmosphere, the most
relevant case for a midlatitude SST anomaly would be a compromise between the upper
right and lower panels. The key point, however, is that the linear response to imposed
heating is always baroclinic, and that this linear response is not the atmospheric pattern
associated with SST anomalies in observations. Once again, it must be emphasized that
the observed pattern is the source of, and not a response to, the SST anomaly.
What observational evidence is there that the midlatitude atmosphere cares about the
underlying SST field? None that is unambiguous. The NAO shows a small amount of
interannual persistence (one-year lag autocorrelation of 0.15), and it is unlikely that the
atmosphere alone can provide the implied memory. This year-to-year persistence is,
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however, present only in the more recent portion of the record (since ~1930), and may
represent a response to anthropogenic climate forcing.
Somewhat stronger evidence comes from examining time-lagged relationships between
the dominant patterns of SST and sea-level pressure variations, particularly in the North
Atlantic (Czaja and Frankignoul 1999). Figure 9, provided by Yochanan Kushnir
(unpublished) provides another example. The top panels show the leading patterns of
variability of SST and sea-level pressure in the North Atlantic, and the bottom panel
shows the lagged correlations between them. The horizontal axis shows the month of the
SST anomaly, while the vertical axis is the month of the pressure anomaly. The positive
correlations in the upper left of the diagram indicate an association of the wintertime
atmospheric pressure with ocean temperatures during the preceding summer.
Fig 9. Leading patterns of sea-level pressure (top left) and SST (top right) variability in the North Atlantic,
and the lagged correlations between these patterns (bottom). (Kushnir 2000).
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Given the difficulties of determining the atmospheric response to the ocean using
observations, we turn to models. Unfortunately, models – general circulation models
(GCMs) – have not given a clear answer to the question of whether the atmosphere cares
about the ocean. The overall impression from numerous model experiments is that the
atmospheric response to midlatitude SST anomalies is weak, perhaps 10 m of 500 mb
geopotential height for each Celsius degree of SST anomaly. The weakness of the
response magnifies the importance of a number of issues surrounding such experiments:




Is the response statistically significant? This question is complicated if, as is
often the case, the putative response shares the structure of the model’s internal
variability.
Is the model adequate? Is its internal variability realistic?
Is the location of the SST anomaly appropriate for the model’s climate, which
may deviate significantly from the reality?
What is the appropriate experimental design? If the response equilibrates slowly,
a large ensemble of short runs may yield a different answer from a single long
run. Which is more relevant to nature?
In interpreting the results of such GCM experiments, it must be remembered that one is
really considering only the potential for feedback onto the atmosphere of an SST signal
generated by the atmosphere. This is brought home whenever one looks at the air-sea
fluxes in these experiments, which almost always have precisely the wrong sign in
comparison with observations (Figure 10). In nature, a warm SST anomaly is created by
the atmosphere and is associated with an anomalous flux of heat from the atmosphere to
the ocean. In an imposed-SST GCM experiment, however, the anomalous flux of heat is
almost always out of the ocean into the atmosphere.
Fig 10. A cartoon showing the “wrong” sign of air-sea fluxes obtained in GCM experiments with imposed
SST anomalies.
Here are two examples of GCM responses to imposed SST anomalies. The first is a
roughly linear-looking response from a coarse resolution (rhomboidal 15 spherical
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harmonic truncation) version of the Geophysical Fluid Dynamics Laboratory (GFDL)
GCM (Kushnir and Held 1996). Figure 11 shows the equilibrated response to a 4 °C
warm SST anomaly. The upper panel is a 6000-day perpetual January run, while the
lower is a perpetual October run. Note in particular the weakness of the response, only a
few meters of 500 mb height per degree.
Figure 12, on the other hand, shows much more robust responses obtained using the
European Center for Medium Range Forecasting (ECMWF) model, at a significantly
finer resolution (triangular 63 truncation). These results are obtained from 5-member
ensembles of integrations, each over a single winter signal. Here the response is roughly
complementary for positive and negative SST anomalies, an expected result that is
obtained only rarely in such GCM experiments.
Fig 11. Geopotential response to a 4°C SST anomaly in the GFDL R15 GCM for perpetual January (top)
and October (bottom) conditions. (Kushnir and Held 1996).
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Fig 12. SST anomalies in the Pacific (top) and Atlantic (bottom) and the 500 mb height responses to warm
(left) and cold (right) anomalies. (Ferranti et al. 1994).
Two generalities apply to these and most of the many other GCM experiments that have
been performed using imposed midlatitude SST anomalies. First, only the relatively
high-resolution (triangular 42 or higher truncation) models seem capable of producing an
equivalent barotropic ridge downstream of the SST anomaly, though not all highresolution models do so. Secondly, the model response can be very sensitive to the
model’s climatological flow (see Peng et al. 1997). These “basic states” can vary
markedly from month to month of the model’s annual cycle, and are often very different
from observations.
The dynamics of these responses are related to the dynamics of internal variability
discussed earlier, both in the importance of transient eddies and the fact that modeled
responses project strongly on the model’s patterns of internal variability. Peng and
Whitaker (1999) demonstrate the importance of transient eddy momentum fluxes in
producing an equivalent barotropic ridge response. Figure 13 from their paper shows,
using a linear model, that the heating, as discussed earlier, produces only a baroclinic
response (left panels), while the anomalous transient-eddy vorticity fluxes (those
“observed” in a GCM experiment with a warm SST anomaly) produce the same
equivalent barotropic response as exhibited by the GCM. In other words, the equivalent
barotropic ridge response to a warm SST anomaly is primarily a response to induced
changes in the transient-eddy fluxes, rather than a direct response to the heating induced
by the SST anomaly. Similar results have, in fact, been found for the midlatitude
response to El Niño.
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Fig 13. Linear model responses to heating (left) and transient-eddy vorticity fluxes saved from a GCM run
with a warm Pacific SST anomaly (right). Geopotential response at 250 mb (a), 850 mb (b) and crosssection along 40°N (c). (Peng and Whitaker 1999).
Figure 14 is a cartoon showing how eddy vorticity fluxes can “barotropize” the baroclinic
ridge that is directly forced by anomalous heating. If, consistent with the arguments
about eddy feedback in the first section, transient-eddy vorticity fluxes reinforce the
upper-level ridge, this eddy forcing can be transmitted to the surface by the secondary
circulation – this is just an application of the quasi-geostrophic omega equation.
Fig 14. Cartoon showing how transient-eddy vorticity fluxes and the resulting quasi-geostrophic
secondary circulation (arrows) can lead to the development of a surface ridge in response to a warm SST
anomaly. The colored contours show the linear geopotential response to shallow surface heating.
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That the response to SST anomalies (at least in some GCM experiments) and internal
variability are both driven by transient eddies, suggests that the SST response may
resemble the patterns internal variability. This if found to be so, both in as yet
unpublished work by Deser et al., and in the example shown in Figure 15 (Peng and
Robinson 2000). The top panels show the 250 mb geopotential responses to a warm
Pacific SST anomaly (the same set of GCM experiments analyzed by Peng and
Whitaker). These equilibrated responses are strikingly different in January and in
February. The lower panel shows the leading EOFs of monthly mean 500 mb
geopotential heights for the same two months, calculated from control runs with no SST
anomaly. The similarity between the GCM responses and the leading patterns of internal
variability is unmistakable. Again, this is not surprising given that both are largely
maintained by interactions with the transient eddies.
Fig 15. Perpetual January and February GCM responses at 250 mb to a warm Pacific SST anomaly (top).
Leading EOFs of 500 mb geopotential heights for January and February control runs (bottom). (Peng and
Robinson 2000).
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This raises the question of whether an SST anomaly is really needed at all. Figure 16 is a
cartoon showing, in meridional cross-section, mutually reinforcing interactions between
the sea-surface, the large-scale atmospheric flow, and the transient baroclinic eddies. If
one covers up the ocean, one gets a completely reasonable picture of how anomalous jets
can be self maintaining through their organization of and forcing by transient eddies – the
idea behind the transient-eddy feedback for annular modes discussed in the first section.
The ocean, then, may be relegated to the role of “nudging” internal variability in a
direction consistent with the SST anomalies. This is a weak effect compared to the
robust internal variability within the midlatitude atmosphere, but it could be important for
decadal and longer timescales. If, when the previous winter’s SST anomalies reemerge
in the fall, they create a preference for one sign or another for the internal modes of
atmospheric variability, this sign might then be favored, if only slightly, throughout the
winter. The result is that without strongly forcing the atmosphere, the ocean provides
enough interannual memory to significantly redden the spectrum of atmospheric
variability.
Eddy Mediated SST- Atmosphere Feedback
Tropopause
z
SST anomaly
Surface
Key:
N
Eddy activity flux
Eddy mom entu m fl ux
Anoma lous westerly jet
Ano malo us surfa ce
easterlies
Ano malo us surfa ce
westerl ies
Fig 16. A cartoon of midlatitude atmosphere-ocean interaction, in meridional cross-section, including the
effects of transient eddies.
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