ANNULAR MODES OF THE TROPOSPHERE AND ...pauluis/Olivier_Pauluis_Homepage...garding terminology, the term \Annular Modes" was rst used in Thompson and Wallace [2000] (which was received

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ANNULAR MODES OF THE TROPOSPHEREAND STRATOSPHERE

Paul J. Kushner1

Copyright 2010 by the American Geophysical Union. Reviews of Geophysics, ???, /pages 1??

8755-1209/10/£15.00 Paper number• 1 •

2 • KUSHNER: ANNULAR MODES OF THE TROPOSPHERE AND STRATOSPHERE

AbstractA pedagogical review of the phenomenology, dynamics and simulation of the Annular Modes is presented. First,

the manner in which the Annular Modes capture the principal features of variability in the zonal mean circulationof the extratropical stratosphere and troposphere is demonstrated. Next, elementary ideas about Annular Modedynamics, in particular the distinct eddy mean-flow interactions that govern the modes in the troposphere andstratosphere, are introduced. The review concludes with a discussion of the representation of the Annular Modesin simplified and comprehensive general circulation models, in the context of stratosphere-troposphere interactionsin intraseasonal variability and climate change.

KUSHNER: ANNULAR MODES OF THE TROPOSPHERE AND STRATOSPHERE • 3

1. INTRODUCTION

For the theory of the atmospheric general cir-culation it is important to not only account forthe circulation’s principal climatological features,such as the jet streams of the troposphere and thepolar night jet of the stratosphere, but also to de-velop a dynamical description of how the circula-tion varies about its climatological mean. In thischapter, we present a pedagogical review of the An-nular Modes [hereafter AMs; Thompson and Wal-lace, 2000; Thompson et al., 2000; Limpasuvan andHartmann, 1999, 2000], which are the principalmodes of variation of the extratropical circulationof the troposphere and stratosphere on timescalesgreater than a few weeks. Because much of theobserved extratropical variability is manifested inthe AMs, explaining their dynamics represents animportant theoretical task. Beyond this, the AMsare of applied importance because of their impactson many climate processes and because of their re-lationship to climate change signals [e.g. Shindellet al., 1999; Thompson et al., 2000; Thompson andSolomon, 2002; Kushner et al., 2001; Russell andWallace, 2004; Miller et al., 2006; Butler et al.,2007].

In this Chapter, we will discuss the observedAMs (Section 2), which consist of the SouthernAnnular Mode (SAM) in the Southern Hemisphereand the Northern Annular Mode (NAM) in theNorthern Hemisphere; present some elementaryideas on their dynamics and their representation inhighly simplified models (Section 3); and exploretheir central role in extratropical climate variabil-ity and change, with an emphasis on their be-haviour in more complex models (Section 4). Wewill adopt the viewpoint that the AMs representcoherent variations of the zonal mean circulationthat interact with synoptic-to-planetary scale at-mospheric waves and the zonal stresses they in-duce [e.g. Limpasuvan and Hartmann, 1999, 2000].We will find that, despite their importance, theAMs remain enigmatic because they are derivedempirically and because there is no simple first-principles theory describing their structure and dy-namics in the stratosphere and troposphere. Ourcurrent theoretical understanding is instead repre-

1Department of Physics, University of Toronto,Canada

sented in simplified nonlinear models in which AMstructures emerge as the leading modes of variabil-ity [e.g. Vallis et al., 2004; Holton and Mass, 1976;Scott and Polvani , 2006] and in detailed dynamicalanalyses of their temporal variability [e.g. Robin-son, 2000; Lorenz and Hartmann, 2001, 2003]. Inthe absence of a predictive theory, and in light ofthe many ways in which AM responses can be stim-ulated by climate forcings, we need to rely on com-prehensive climate models to quantitatively simu-late the AMs and predict their future behaviourunder climate change.

Before proceeding, we highlight some of the im-portant papers from the late 1990’s and early2000’s that defined the AMs and recognized theirsignificance. These papers represent a startingpoint for further exploration for the interested stu-dent.• The original description of the AMs as the

principal modes of variability of the extratropi-cal circulation, for the stratosphere and the tro-posphere of both the Northern Hemisphere andSouthern Hemisphere; their relationship to theNorth Atlantic Oscillation; and their impacts onweather variability is primarily due to Thomp-son and Wallace [see Thompson and Wallace,1998, 2000; Wallace, 2000; Thompson et al.,2000; Thompson and Wallace, 2001, among otherstudies]. The principal original work describ-ing the eddy mean-flow interactions that main-tain the AMs is that of Limpasuvan and Hart-mann [1999, 2000] and Lorenz and Hartmann[2001, 2003]. This set of studies revitalized a lineof work on variability of the zonal mean circulationthat dates back several decades. (The historical lit-erature is well referenced in the cited papers.) Re-garding terminology, the term “Annular Modes”was first used in Thompson and Wallace [2000](which was received by J. Climate in 1998) andLimpasuvan and Hartmann [1999] (which was re-ceived by Geophys. Res. Lett. in 1999). The NAMand SAM were originally named the Arctic Oscil-lation and the Antarctic Oscillation; these termsare still used in some papers. The relationship be-tween the AMs and the classical zonal index, whichis a dipolar mode of variability that corresponds tothe tropospheric AM in zonal wind, is detailed inWallace [2000].• The observational description of the time de-

pendent coupling between stratospheric and tropo-spheric variability through the daily evolution ofthe AMs, and its implications for prediction of cli-


mate variability on seasonal timescales, is primar-ily due to Baldwin, Dunkerton and collaborators[e.g. Baldwin and Dunkerton, 1999, 2001; Bald-win et al., 2003]. These papers followed a longline of inquiry that identified influences that thestate of the stratosphere might have on the tro-pospheric circulation, involving both zonal meanand regional circulation features [e.g. Boville, 1984;Kodera et al., 1990; Perlwitz and Graf , 1995].• The connection between the Annular Modes

and climate change in the extratropics and toAntarctic stratospheric ozone depletion is due toThompson and Wallace [1998]; Shindell et al.[1999]; Fyfe et al. [1999]; Thompson et al. [2000];Kushner et al. [2001]; Sexton [2001]; Thompsonand Solomon [2002]; Polvani and Kushner [2002];Gillett and Thompson [2003], among others.

The discovery of the AMs has spurred a large lit-erature that extends well outside the atmosphericand physical climate sciences. We do not providehere a comprehensive review of this literature, butaim to highlight a select set of points intended tointroduce the AMs to researchers new to the field.

2. OBSERVED AM STRUCTURE

The AMs are atmospheric teleconnection pat-terns — that is, continental-to-planetary scale pat-terns involving statistical coherence between spa-tially remote points — that are derived using a va-riety of methods [as recently reviewed in Baldwinand Thompson, 2009; Baldwin et al., 2009]. Typi-cally, the AMs are derived using empirical orthog-onal function (EOF) analysis of selected extratrop-ical meteorological fields. EOF analysis representsa straightforward multivariate statistical approach,but one that risks artificially conflating spatiallyand dynamically distinct phenomena. In this sec-tion, we will show that the AMs indeed representphysically meaningful and statistically robust pat-terns that connect the zonal mean variability of thetroposphere and stratosphere.

Using notation similar to Baldwin et al. [2009],EOF analysis decomposes an n× p data matrix Xconsisting of n temporal observations at p pointsinto a series of n× p matrices according to

X =r∑

i=1

yieTi = y1eT

1 + y2eT2 + . . .+ yreT

r , (1)

where r is the rank of X, which is never greaterthan the minimum of n and p; yi is a column vectorrepresenting the ith n-point principal component(PC) time series; and ei is a column vector rep-resenting the ith p-point spatial EOF. The EOFsare the eigenvectors of the p× p sample covariancematrix S = XTX, and are ranked in descending

order of the size of the positive quantity eTi Sei —

when divided by the total variance in the field, thisquantity is known as the “variance explained” or“represented” by EOF i. The r PC time seriesyi are mutually orthogonal, as are the r EOFs ei.A spatial pattern associated with another field Xthat is coherent with the jth EOF is found by re-gression according to

xj =XTyj

yTj yj

(2)

where X is an n × p data matrix consisting of pn-point time series. If we set X = X, which is theoriginal data matrix, then the spatial pattern thatresults from (2) is the jth the EOF: xj = ej . Theeffects of spatial weighting are straightforward tobring into this formalism [Baldwin et al., 2009].

In this notation, the AM EOF corresponds toe1 for an appropriately chosen X and the AM in-dex corresponds to the PC time series y1. To de-fine the AMs, one must choose the meteorologi-cal field X and the way to sample that field inspace and time. A review of various approaches[e.g. Gong and Wang , 1999; Thompson and Wal-lace, 2000; Wallace, 2000; Baldwin, 2001; Lorenzand Hartmann, 2001; Monahan and Fyfe, 2006;Kushner and Lee, 2007], not all of which use EOFanalysis as the basis for the AM index [e.g. Gongand Wang , 1999], shows that the different meth-ods yield similar results in the sense that the result-ing AM indexes are strongly temporally correlated.But this similarity can mask systematic underlyingdifferences: for example, Monahan and Fyfe [2006]show that the meridional structure of the AM de-fined using geopotential height differs from thatusing geostrophic wind because the loading pat-tern of variance for a field differs from the loadingpattern for the field’s spatial gradient. Focussingon the vertical structure, Baldwin and Thompson[2009] show that the choice of input field affectsthe manner in which stratosphere-troposphere cou-pling (see below) is manifested on short timescales.

To help understand the nature of the variancerepresented by the AMs, we examine the spa-tial and temporal structure of extratropical zonalmean variability to be captured by the EOF anal-ysis. We start with the zonal mean geopotentialZ = Z(θ, p, t), where the overbar indicates a zonalmean and (θ, p, t) are latitude, pressure and time.We then remove its climatological mean; call thisquantity Za, where the superscript a indicates ananomaly from the climatology. We then define

Z(θ, p, t) = Za(θ, p, t)√

cos θ, (3)

which, after some further processing, will providethe input to the EOF calculation. The geometricfactor

√cos θ relates to the cosine-latitude weight-


ing within the inner product for the EOF calcula-tion [Baldwin et al., 2009]; we will for this analysisretain it in the definition of Z and now examineZ as a measure of variability in the zonal meancirculation.

We first show the periodogram (i.e. the discreteFourier temporal spectrum) of Z as a function oflatitude from South Pole to North Pole and oftimescale from weeks to years for the lower tropo-sphere in Plate 1a and for the lower stratospherein Plate 1b. We also show the extratropical merid-ional mean of these spectra in Plate 1c. While thevariance of Z from (3), by construction, goes tozero at the poles, the variance of Za amplifies toa maximum at the poles (not shown). Because fortimescales shorter than annual geopotential vari-ance is generally stronger in the extratropics thanthe tropics (as expected from geostrophic balance),the definition of the AMs is relatively insensitive tothe equatorial extent of the domain of analysis [seee.g. Baldwin and Thompson, 2009]. In the tropo-sphere, there is a broad band of variance polewardof about 30 degrees latitude and a suggestion of aminimum in variance around 60 degrees latitude ineach hemisphere. Plate 1c shows that in the tro-posphere the variance has for the most part satu-rated on timescales longer than the intraseasonal(multiple-week) timescale. In the stratosphere, thevariance is relatively strong at higher latitudes, andshifted to longer timescales; the variance continuesto build up beyond intraseasonal timescales. Thestructure in latitude and frequency is similar inboth hemispheres, suggesting that the dynamics ofthe AMs in each hemisphere might be dynamicallysimilar. This hemispheric symmetry is less evidentfor zonal wind than for Z (not shown).

We now examine the seasonal dependence ofthe variability of the zonal mean circulation rep-resented by Z. We calculate the monthly meanZ, remove that part of the result that is linearlycoherent with the Monthly ENSO Index [MEI de-fined by Wolter and Timlin, 1993, see Lorenz andHartmann, 2003], linearly detrend this result, andfrom this last result create seasonal time series forthree sequential months each year. (For example,we create a time series of the months January-February-March for each year from 1979 to 2008,representing a three month “JFM” season, yieldinga 90-record time series.) We denote by Zmon theresult of this sequence of operations on Z. Thiscaptures seasonal and longer timescale variabilityand eliminates submonthly variability, for which,from Plate 1, the variability can be seen to be farfrom saturation.

We show the seasonal cycle of the standard de-viation of monthly mean Zmon in the extratrop-ics, as a function of center month (Februrary in

the previous example) in Plate 2, for lower tro-pospheric, upper tropospheric, and lower strato-spheric levels. The ENSO filtering removes vari-ance at low latitudes (not shown). In the tropo-sphere of both hemispheres, Zmon has a strongwintertime maximum in the higher extratropicsand a weaker maximum in the lower extratropics.These features are vertically aligned, suggestingthat the underlying variability might be equivalentbarotropic in the troposphere. The seasonal cycleis more pronounced in the Northern Hemispheretroposphere than in the Southern Hemisphere tro-posphere. The maximum in variance in the South-ern Hemisphere troposphere arrives in late fall, ear-lier in the season than the corresponding maximumin the Northern Hemisphere troposphere. In theNorthern Hemisphere stratosphere, the maximumvariance is found in the winter but in the South-ern Hemisphere stratosphere, the maximum vari-ance is delayed towards the spring. The period ofmaximum variance in the stratosphere representsthe active season of the stratospheric polar vortexin each hemisphere, when the polar stratosphericwinds are eastward and display considerable vari-ability; this season is delayed towards spring inthe Southern Hemisphere [Thompson and Wallace,2000; Hartmann et al., 2000].

Plates 1-2 represent the variability that the AMs,as leading modes, should capture. To calculate theAMs, we set X = Zmon(|θ| > 200, p = 850 hPa, t)in (1), i.e. we use extratropical lower troposphericZmon as input to the EOF analysis [Thompsonand Wallace, 2000]. The NAM and SAM arethe leading EOFs in the Southern Hemisphere andNorthern Hemisphere. We repeat this calculationfor each calendar month and for each hemisphere.Plate 3 plots the regression of the AMs on Zmon,using (2), for the same vertical levels as in Plate2. These are the patterns of variability coherentwith the lower-troposphere based AM in its posi-tive phase at each level, for each calendar monthand for each hemisphere.

Comparing Plates 2 and 3, which have equiva-lent contour intervals in each panel, we see thatthe leading EOF captures the principal featuresof the variance and its seasonal cycle throughoutthe troposphere and stratosphere, even though theAM definition involves only information from thelower troposphere. The tropospheric AM EOFconsists of a dipolar structure in Zmon that reflectsthe high-extratropical and low-extratropical max-ima in variance in the two hemispheres and thebroad band of variance seen in Plate 1a. Duringthe active season when the variance is large in thelower stratosphere, the AM anomaly pattern hasa consistent sign between the stratosphere and thetroposphere poleward of 600 latitude. This verti-


cal coherence indicates that the stratosphere andtroposphere are coupled in the active season.

Despite the evidence of coupling between thestratosphere and troposphere, Plate 3 also suggeststhat the main features of the tropospheric AMstructure appear to be independent of the strato-sphere. In particular, the dipolar tropospheric AMin Plate 3 persists throughout most of the year, es-pecially in the Southern Hemisphere. The strato-spheric AM, on the other hand, is strongly ampli-fied during the active stratospheric season and isstrongly attenuated in the inactive season when thestratospheric winds are easterly. This implies, aswill be discussed in Section 3, that to some extentthe dynamics of the stratospheric and troposphericAM are distinct and can be considered separately.

During the active season, when the variabil-ity of the stratosphere and troposphere is verti-cally coherent, the NAM and SAM are remarkablyhemispherically symmetric [Thompson and Wal-lace, 2000]. Plate 4 shows the 850 hPa SAM andNAM signatures in the zonal wind (that is, X cor-responds to u in (2), where u is the zonal meanzonal wind). In the troposphere, the AM in thewind in each hemisphere is strongly dipolar whilein the stratosphere the AM is more monopolar. Al-though the node of the SAM is located a few de-grees poleward of the node of the NAM, the pat-terns essentially represent a reflection of each otheracross the equator. To quantify the hemisphericsymmetry, we also show the associated hemispher-ically symmetric component of the patterns, usym,which is obtained by first reflecting SAM patternuSAM and then averaging it with the NAM patternuNAM:

usym(y) =12

[uSAM(−y) + uNAM(y)],

where y is the Cartesian meridional coordinate thatwe will use in the rest of the Chapter. The sym-metric component explains 96% of the variance inthe AMs (using a log-pressure weighting polewardof 10 degrees latitude), and the symmetric patternlooks quite similar to both the SAM and the NAM.Applying such a construction to the zonal meanwind in each hemisphere (bottom row of Plate 4)does not give the same qualitative sense of hemi-spheric symmetry: the Northern Hemisphere jet isdominated by the subtropical jet, while the South-ern Hemisphere jet structure is more split betweenits subtropical and extratropical parts; the North-ern Hemisphere surface jet is not as sharp, and thestratospheric jet is relatively weak. (The resultingsymmetric component [bottom right panel of Plate4] explains 88% of the variance of the total wind.)

The fact that the AMs are more hemisphericallysymmetric than the background circulation (as isalso found for the tropospheric AMs outside the ac-

tive season — not shown) suggests that the AMsare governed by dynamics that does not dependhighly on the meridional structure of the back-ground flow. It also means that the spatial rela-tionship between the jets and the AMs is distinctin each hemisphere. In the Southern Hemisphere,the tropospheric jet maximum at each altitude (redsolid line in the upper left panel of Plate 4) is lo-cated along the node of the SAM; the SAM in itspositive phase thus describes a poleward shift ofthe jet. Given a shift δy of the jet, we expect awind anomaly that is related to the vorticity ofthe jet according to

uSAM ∼ u(y − δy)− u(y) ≈ −δy uy = δy ζ,

where u(y) represents the tropospheric jet profileas a function of meridional coordinate y, wherecoordinate subscripts indicate partial derivatives,and where ζ = −uy is the zonal mean vorticityof the jet. This relationship suggests that spatialscale of the tropospheric SAM is linked to the spa-tial scale of the barotropic shear, i.e. of the vor-ticity of the jet, and the amplitude of the SAMto the degree of displacement of the jet. Tak-ing extratropical tropospheric vorticity values ofζ ∼ 10−5 s−1 and uSAM ∼ 1− 2 m s−1 for the SAMin the troposphere suggests that typical seasonal-to-interannual SAM fluctuations are related toseasonal-to-interannual variations in the positionof the jet of about δy ∼ ζ/uSAM ∼ 100 − 200 km;climate trends discussed in Section 4 correspond totropospheric jet shifts of this magnitude [Kushneret al., 2001]. The relationship between the jet po-sition and the SAM node holds at each longitude,when the regional structure of the SAM is analyzed[Codron, 2005]. In the stratosphere, the jet axis isaligned with the maximum of the SAM. Thus, inthe Southern Hemisphere active season, the SAMin its positive phase is described as a poleward shiftof the tropospheric jet and a strengthening of thestratospheric polar vortex.

This empirical description for the SouthernHemisphere does not generalize simply to theNorthern Hemisphere. In the Northern Hemi-sphere, the relationship between the stratosphericvortex and the NAM is similar to that for theSouthern Hemisphere (see red lines in Plate 4b).But in the troposphere, positive NAM conditionsconsist of a reduction in strength of the subtropi-cal jet and an intensification on the poleward sideof the jet, rather than a simple poleward shift ofthe jet. Thus we do not find a simple connectionbetween the barotropic shear of the zonal meanwind and the tropospheric NAM structure. Therelationship between the Northern Hemisphere jetstream and the NAM varies regionally: in theNorth Atlantic sector, the NAM corresponds toa poleward shifted North Atlantic jet, whereas in


the North Pacific sector, the NAM correspondsto a weaker subtropical jet [Ambaum et al., 2001;Eichelberger and Hartmann, 2007]. We will returnto this point in Section 3.

While our observational analysis has so far fo-cussed on monthly and longer timescales, the dy-namics of zonal mean extratropical variability, andof the AMs in particular, is controlled to a largedegree on shorter timescales, as we will discuss inSection 3. A daily index of the AM is found byprojecting the AM EOF e1 (which is typically cal-culated, as we have, from seasonal mean or lowpass filtered data) onto the daily circulation usingy1,daily = Zdailye1/eT

1 e1, where Zdaily representsthe daily time series of the geopotential anomalyat the p spatial points of the EOF. When AMvariability is analyzed on daily-to-multiple weektimescales, the vertically coherent static patternsin Plates 3-4 resolve into vertically propagatingsignals in which tropospheric AM anomalies lagstratospheric AM anomalies by periods of severalweeks [Baldwin and Dunkerton, 1999, 2001; Bald-win and Thompson, 2009]. This relationship isdiagnosed by calculating the AM and its princi-pal component time series at each vertical leveland then using lag-correlation or compositing tech-niques to determine the vertical variation of thetime lags. As an example, we show in the toppanel of Plate 5 a Baldwin and Dunkerton [2001]type composite over low NAM index events (weakor warm stratospheric vortex events). The figureshows a vertically coherent NAM signal that prop-agates down from the stratosphere into the tro-posphere and leads to like-signed NAM anoma-lies in the troposphere that persist for 40 days ormore. These downward propagating signals aretypically associated with extreme events in thestratosphere, such as stratospheric sudden warm-ings and vortex intensification events [Limpasuvanet al., 2004, 2005]. For example, during strato-spheric sudden warmings, a weak vortex anomaly(negative NAM event) is found to burrow downinto the stratosphere; the NAM signal then makesits way into the troposphere, where it is manifestedas an equatorward shift of the storm tracks andan equatorward intensification of the troposphericjet [Baldwin and Dunkerton, 2001]. The bottompanel of Plate 5 shows a similar calculation to thetop panel, but using the geopotential averaged overthe 600N-900N polar cap. The similarity betweenthe two panels suggests that we can use the polarcap geopotential as a simple proxy for the AM; wewill do so in Section 3.

The multiple week lag between stratospheric andtropospheric AM signals suggests that the strato-sphere can provide a source of predictability thatextends beyond the classical limit for troposphericweather predictability [Baldwin et al., 2003]. Bald-

win et al. [2003] argue that the tropospheric AMis observed to be more persistent in winter andthat this persistence results from stratosphere-troposphere coupling. A version of their analysisis presented in Plate 6, from Gerber et al. [2008a],which plots in the top row the seasonal cycle of theautocorrelation decay timescale of NAM and SAMvariability from the NCEP reanalysis. The char-acteristic timescales of the AM, consistently withPlate 1, is on the order of 1-2 weeks in the tropo-sphere and several weeks in the stratosphere, witha strong peak during the active season for the SAMand NAM, suggesting a stratospheric connectionfor tropospheric persistence of these modes [Bald-win et al., 2003]. The bottom row of Plate 6 showsthe same diagnostic for the set of climate simu-lations in the Intergovernmental Panel on ClimateChange’s Fourth Assessment Report, which we willcomment on in Sections 3 and 4.

We conclude this section by discussing the re-gional, i.e. the longitudinal, structure of the AMsin the troposphere, which has been the subjectof considerable debate [e.g. Deser , 2000; Ambaumet al., 2001; Wallace and Thompson, 2002; Val-lis et al., 2004; Kushner and Lee, 2007]. If weregress the NAM index on a longitudinally varyingfield such as the two dimensional sea-level pressure,we typically obtain a pattern consisting of a lowpressure anomaly over the pole and high pressureanomalies over the North Atlantic and North Pa-cific (see Figure 1a, from Deser [2000], which showsthe regression of the sea-level pressure on the NAMindex from Thompson and Wallace [1998]). Theregressions have a predominantly zonally symmet-ric or “annular” appearance, although the ampli-tude of the patterns is greater over the oceans thanover land. Similar predominantly zonally sym-metric structures are found in both hemisphereswhether or not the input to the EOF calculation[X in (1)] is zonally averaged [Thompson and Wal-lace, 2000; Baldwin and Thompson, 2009]. Thus,the zonal structure of the circulation that gener-ates the AM is not critical to the zonal structureof the circulation that is coherent with the AM.

The point of debate about the zonal structure ofthe AMs centers on the fact that the zonal struc-ture depends strongly on the analysis domain: forexample, when the domain of the EOF calculationis restricted to the North Atlantic basin, the Pa-cific centre of action is no longer present in theregression and only the North Atlantic Oscillationtype structure remains (Figure 1b); similarly whenthe domain is restricted to the North Pacific basin,only a North Pacific centre of action remains (Fig-ure 1c). While all three patterns share a simi-lar meridional structure, there is no coherence be-tween the basins [Deser , 2000]. Consistently, ondaily timescales it is found that high AM index


days are associated with regional (less than 90-degree scale) dipolar circulation anomalies [Cashet al., 2002; Vallis et al., 2004; Kushner and Lee,2007]. This is in contrast to what we found forthe vertical structure: for example the verticallycoherent structures in Plate 3 involve only lowertropospheric information, and the vertically prop-agating stratosphere-troposphere signal in Plate 5demonstrates a consistent statistical link betweenthe zonal mean tropospheric and stratospheric cir-culation.

Kushner and Lee [2007] explore the connectionbetween hemispheric AM variations and regional-scale variability. They find that day-to-day zonal-mean AM variability is consistent with the oc-curence of zonally localized dipolar patterns thathave a decoherence width of about 900 longitudeand that propagate eastward with a phase speedof about 80 longitude per day. In Figure 2,the regression pattern in surface circulation andthe longitude-time propagation structure for thesedipolar patterns in the Southern Hemisphere areplotted. Kushner and Lee argue that these pat-terns, which are similar in the Northern Hemi-sphere and Southern Hemisphere, are propagat-ing versions of NAO type variability; outside theNorth Atlantic sector they are not dominant butsecondary patterns of variability. These regressionpatterns have a westward phase tilt with height,suggesting that they are propagating and moder-ately baroclinic long waves. Kushner and Lee pro-pose that these waves arise spontaneously and ran-domly, and their projection onto the zonal meanproduces a zonal mean anomaly with the merid-ional structure of the zonal-mean AM. These ideasare not fully explored and have yet to be reconciledwith the better established viewpoint that AM dy-namics involves an interaction between synopticeddies and the zonal mean flow. This viewpointwill be discussed in the next section.

3. ELEMENTARY IDEAS ON ANNULARMODE DYNAMICS

If the AMs represent the dominant fluctuationsof the extratropical zonal mean circulation, theneddy mean flow interactions, in which the zonalflow controls the eddies and the eddies exert areturn control on the zonal mean flow, are re-quired to explain AM dynamics. To see why, con-sider the role of eddies in classical quasigeostrophic(QG) dynamics [e.g. Andrews et al., 1987]. For theQG scaling that the extratropical circulation is ob-served to satisfy, the zonal mean circulation evolvesaccording to (see the Appendix)

qt + (v′q′)y = S. (4)

where coordinate subscripts represent partialderivatives, overbars represent the zonal mean,primes represent eddy (zonally asymmetric) quan-tities, v is the meridional geostrophic velocity, qis the QG potential vorticity, and S is a forcingand dissipation operator that includes the effectsof momentum dissipation and diabatic heating. Inthis notation, v′q′ represents the meridional fluxof eddy potential vorticity. Consider how tem-poral anomalies in the zonal mean potential vor-ticity, which we denote, qa, are related to tempo-ral anomalies in the zonal mean potential vorticityflux, (v′q′)a. Using the principal of inversion ofpotential vorticity [Hoskins et al., 1985], from thezonal mean potential vorticity anomaly qa and re-lated surface information we can diagnose the zonalmean geopotential anomaly, and hence the zonalmean wind anomaly from geostrophic balance andthe zonal mean temperature anomaly from hydro-static balance. Thus (4) represents the evolutionof the entire zonal mean state.

If we remove the climatological mean from (4),multiply by qa, and take a density weighted volumeaverage (denoted by angle brackets), we obtain avariance equation for the zonal mean potential vor-ticity anomaly:

d

dt

⟨12

(qa)2⟩

= −⟨

(v′q′)ay · qa

⟩+⟨Sa · qa

⟩, (5)

where (v′q′)ay and Sa represent the eddy potential

vorticity flux divergence anomaly and the dissipa-tion anomaly. In the Appendix, we argue that⟨Sa · qa

⟩≤ 0 to within a boundary term of in-

determinate sign. If this boundary term can beneglected, this implies that Sa acts to attenuatethe amplitude of zonal mean potential vorticityanomalies. This in turn would mean that extrat-ropical dynamics cannot sustain fluctuations of thezonal mean unless, averaged over the spatial do-main, the eddy potential vorticity flux convergenceanomaly −(v′q′)a

y is positively correlated with thepotential vorticity anomaly qa. This suggests thatAM dynamics requires the eddy potential vorticityflux anomalies to amplify rather than attenuate lo-cal potential vorticity anomalies, over some part ofthe domain, in order to maintain AM variability.1We will argue that the means by which AM vari-ability is maintained by eddy forcing is distinct inthe stratosphere and troposphere.

In Plate 5 we showed that the AM is repre-sented well by the geopotential averaged over thepolar cap; [Baldwin and Thompson, 2009] discussthe strengths and weaknesses of using the polarcap as a proxy for the AM. For dynamical pur-poses the polar cap is a useful control volume towork with. In the context of QG dynamics, fol-lowing a similar approach to Hu and Tung [2002],we consider the integrated potential vorticity for


a polar cap bounded at the top and bottom byz = zT and z = zB and bounded meridionally bythe North Pole y = yN (900N) and by y = yS . Thiswill lead to relationships between circulation, polarcap isentropic thickness, eddy fluxes, and dissipa-tion. If we denote a density weighted average overthis polar cap by <>p, we find that the integratedpotential vorticity anomaly tendency satisfies (seeAppendix)

< qta >p =∫ zT

zB

dz (ρ0uta)∣∣∣∣y=yS

+∫ yN

yS

dy(f0ρ0θt

a/θ0z

)∣∣∣∣z=zT

z=zB

(6)

Here, the |y=yS notation represents the integralevaluated at y = yS , and the |z=zT

z=zBnotation rep-

resents the integral evaluated at z = zT minus theintegral evaluated at z = zB. Locally, the potentialvorticity, which is approximated by the QG poten-tial vorticity, represents the ratio of the absolutevorticity to the thickness between isentropic sur-faces, or the product of the absolute vorticity andstatic stability. A positive potential vorticity ten-dency requires either (or both) a positive tendencyin vorticity or a negative tendency in thickness(which is a positive tendency in static stability).Analogously, a positive polar-cap averaged poten-tial vorticity tendency involves either (or both) apositive tendency in circulation around the polarcap (acceleration of the zonal wind u around thepolar cap), as represented by the first term on theright hand side of (6); or a negative tendency inaverage thickness, as represented by an increasein the vertical difference of the term proportionalto θa on the right hand side of (6). These effectsare coupled via the residual circulation that is ex-tensively discussed in Andrews et al. [1987] andHaynes et al. [1991].

Polar cap potential vorticity anomalies can beinduced by anomalies in eddy fluxes and in dissi-pation. We can express the QG potential vorticityflux in terms of the QG Eliassen Palm (E-P) flux:

v′q′ = ρ−10

[−(ρ0v′u′)y + (ρ0f0v′θ′/θ0z)z

]= ρ−1

0 ∇ · ~F , (7)where

~F = (F(y), F(z)) = (−ρ0v′u′, ρ0f0v′θ′/θ0z)

is the E-P flux. We find that

< qta >p =∫ zT

zB

dzρ0(v′q′)a

∣∣∣∣y=yS

+ < Sa >p

=∫ zT

zB

dz(F a

(y)

)y

∣∣∣∣y=yS

+ F a(z)

∣∣∣z=zT ,y=yS

z=zB ,y=yS

+ < Sa >p (8)

The polar cap potential vorticity can thus be in-creased by an eddy flux of potential vorticity intothe cap. This eddy potential vorticity flux can in-volve either (or both) eddy momentum flux conver-gence at the cap boundary, which is equivalent to avorticity flux into the cap, since F(y) = (−u′v′)y =v′ζ ′; or a transport of thickness (vertically vary-ing meridional heat transport) into the meridionalboundary of the cap, via the terms involving thevertical difference in F(z) ∝ v′θ′ which are eval-uated only at the polar cap boundary. We notethat the vertical EP flux at a given latitude drivespotential vorticity anomalies poleward of that lat-itude. Through the EP flux divergence, both eddythickness and momentum fluxes can thus inducea PV tendency, which corresponds to a tendencyin circulation or in stratification. The connectionof the vertical EP flux (meridional heat flux) tostresses on the mean flow is related to the conceptof form stress, which involves the correlation be-tween fluctuations in pressure and in the height ofisentropic surfaces for a system in geostrophic andhydrostatic balance [e.g. Vallis, 2006].

We consider two special cases of the relation-ships in (8). If we take zB → 0, where z = 0 is theEarth’s surface (expressed approximately in log-pressure coordinates) and zT → ∞, we obtain thewell-known vertical mean momentum balance be-tween acceleration, mechanical forcing, and eddymomentum fluxes (see Appendix):∫ ∞

0dzρ0

[ua

t + (u′v′)ay −Xa

]= 0 at y = yS ,(9)

where X represents x-momentum dissipation. Thisexpression provides a mechanism for eddy momen-tum fluxes to produce the dipolar wind anomaliesthat characterize the tropospheric AM. Eddy mo-mentum flux divergence (or vorticity flux) out ofone latitude band into another will be balanced bydipolar wind structures via either the ut term orthe X term. This description suggests that simplelayer models that are dominated by vorticity fluxeswill be useful to understand the vertically coherentAM dynamics. The vertical integral averages outthe vertical coupling by the mean meridional cir-culation: when we look in more detail, we see thatthe eddy momentum transfer occurs mainly in theupper troposphere and is communicated verticallyto the surface via a thermally indirect eddy drivenmean meridional circulation, which is balanced byfrictional dissipation [Limpasuvan and Hartmann,2000]. The mean meridional circulation that is co-herent with eddy momentum fluxes is clearly seenin AM regressions [Thompson and Wallace, 2000].

As another special case, it is observed that forthe lower stratosphere the contribution from themeridional component of the E-P flux to the to-tal eddy driving is small compared to the vertical


contribution [Newman et al., 2001; Hu and Tung ,2002; Polvani and Waugh, 2004]. Thus, we expectstratospheric AM variability, which involves fluc-tuations in the strength of the polar vortex, to bedominated by variability in the vertical EP fluxcomponent and thus by the thickness flux. For ex-ample, on intraseasonal to interannual timescales,the vertical component of the EP flux correlatesvery well with polar cap stratospheric tempera-tures [Newman et al., 2001; Polvani and Waugh,2004]. This suggests that a useful simple model ofthe stratosphere will focus on the forcing of strato-spheric winds by vertical EP fluxes.

Before discussing simple theoretical models, wereturn to observations to examine the eddy fluxesthat are coherent with the AM. Limpasuvan andHartmann [1999, 2000] calculate the contributionsto ~F and ∇ · ~F (in primitive equation form, whichis approximated by (7)) that are coherent with theSAM and NAM index via a composite technique,using all-year data. The main focus of the paperis on the tropospheric AMs in observations andgeneral circulation models. Plate 7 shows com-posite EP flux cross sections for high AM indexconditions (top row), for low AM index conditions(middle row), and their difference (bottom row),adapted from Limpasuvan and Hartmann [2000].There is some degree of hemispheric symmetry inthese patterns. In both hemispheres, high AM in-dex conditions correspond to a deflection of EPflux away from the poles and towards the trop-ics. This figure also shows similar behaviour in thestratosphere in both hemispheres: high AM indexconditions correspond to positive EP flux diver-gence anomalies in the high latitude stratosphereof each hemisphere, indicating reduced wave driv-ing in the high latitude stratosphere. The patternof wave driving is consistent with the AM structurein the manner we just described: equatorward de-flection of wave activity, since F(y) ∝ −u′v′, corre-sponds to an increased poleward eddy flux of east-ward momentum, which exerts a westward stresson the equatorward side and an eastward stresson the poleward side. This is consistent with thedipolar structure of the tropospheric wind anomaly(e.g. Plate 4). Reduced wave driving in the strato-sphere allows polar stratospheric temperatures torelax closer to radiative equilibrium and hence al-lows the polar vortex to strengthen. A key differ-ence between the two hemispheres is the relativeimportance of stationary versus transient eddies inthe EP fluxes — stationary eddies are dominantin the eddy momentum flux driving in the North-ern Hemisphere, while transient eddies dominatethe eddy momentum flux driving in the SouthernHemisphere [Limpasuvan and Hartmann, 2000].

The balance between eddy fluxes, circulationanomalies, and dissipation does not address thefundamental questions of how the AMs arise, whythey are the dominant mode of variability, what de-termines their evolution and dynamics, and whatcouples their stratospheric and tropospheric parts.Much insight can be gained by careful observa-tional analysis. For example, Lorenz and Hart-mann [2001, 2003] investigate the role of the in-teractions between eddy momentum fluxes and theAM zonal mean wind anomalies in detail and quan-tify the positive feedback between the two. In thisfeedback, the zonal mean flow organizes the ed-dies in such a way as to enhance the momentumflux driving of the AM, and hence its persistence.But to answer our fundamental questions it is alsoimportant to consider simplified theoretical mod-els within a model hierarchy [Held , 2005]. We firstconsider the tropospheric AM, which, as we learnedfrom our observational analysis, is not critically de-pendent on coupling to the stratosphere (see thediscussion of Plate 3), and for which vorticity fluxesassociated with horizontal propagation of EP fluxwill play a prominent role. For the stratosphericAM, we will turn to models in which the dominanteddy driving comes from form stresses associatedwith the meridional eddy flux of heat; for thesemodels, vertical propagation of EP flux dominates.

Fundamentally, variability in the extratropicaltropospheric zonal mean circulation reflects eddymean-flow interactions involving baroclinic andstationary eddies; these give rise to considerablevariability even with time independent boundaryconditions [e.g. Robinson, 1991]. Coupling to theocean, to the land surface, to the stratosphere,etc., might modulate this dynamics but does notexert a leading order control. For example, oneof Limpasuvan and Hartmann’s [2000] main con-clusions is that an atmospheric general circula-tion model (AGCM) with a prescribed sea sur-face temperatures (SST) field that varies season-ally, but not from year to year, produces a realis-tic NAM and SAM structure with eddy fluxes thatclosely resemble the features seen in Plates 4 and 7.This also holds true for coupled ocean atmosphereGCMs in which SSTs are predicted, confirmingthat coupling to the ocean is not a critical issuefor AM variability [Miller et al., 2006]. An im-portant caveat, which we will return to in Section4, is that these GCMs produce AMs that are toopersistent. This is evident from the bias towardsthe long timescales for the climate model simula-tions seen in the bottom row of Plate 6 [adaptedfrom Gerber et al., 2008a]. AM variability ap-pears in AGCMs with simplified boundary condi-tions (such as an aquaplanet) [Cash et al., 2002]and in simplified GCMs, which will be discussed inSection 4.1, with linear representations of diabatic


heating and mechanical dissipation [Yu and Hart-mann, 1993; Held and Suarez , 1994; Polvani andKushner , 2002; Song and Robinson, 2004; Gerberand Polvani , 2009; Ring and Plumb, 2008]. Butan AGCM, in any configuration, still represents ahighly complex system, and it is useful to have inhand even simpler models with which to test ideas.

The simplest model in which realistic tropo-spheric AM variability is spontaneously generatedis a two-layer QG model with the density relaxed toa prescribed baroclinically unstable reference dis-tribution. The width of the region over which thisreference distribution is baroclinically unstable de-termines the width of the baroclinic zone. In onelimit the baroclinic zone is infinitely wide and zonalmean variability consists of multiple jets that ex-hibit a slow and unpredictable meridional wander[e.g. Panetta, 1993]; the jet and eddy scales are in-ternally determined by differential rotation (β) andthe eddy energy, which is related to the imposedbaroclinicity. Lee and Feldstein [1996] show thatthe emergence of more realistic AM type variabil-ity — which they describe in terms of zonal indexvariability — is linked to the width of the baro-clinic zone. Some of the possible regimes are seenin Figure 3: when the jet is narrow (Figure 3a) theleading mode is a monopolar “pulsing” mode andrepresents an alternate sharpening and flatteningof the jet profile. As the jet widens (Figure 3b),a dipolar AM EOF emerges that corresponds toa meridionally shifting jet. Once the barocliniczone is sufficiently wide, a double jet structureemerges and the leading EOF represents the degreeto which the two jets are merged together or sepa-rated (Figure 3c). Finally, for an even wider zone,the leading mode involves AM dipoles for both jets(Figure 3d). In all these regimes the zonal indexor AM EOF is present, but it is not always thedominant mode [see also Feldstein, 2000].

An arguably even simpler model of troposphericAM variability is the stirred barotropic model ofVallis et al. [2004], in which the forcing from baro-clinic eddies is not internally generated but is ex-ternally imposed as a stochastic forcing. The non-linear barotropic potential vorticity dynamics or-ganizes this forcing into a jet whose variability canbe tuned by adjusting the structure of the forcing.Similarly to Lee and Feldstein [1996], the transi-tion from a narrow-jet “pulsing” regime to a wide-jet “wobbling” regime is governed by the width ofthe stirring region (Figure 4). This model can alsobe used to investigate the effect of storm-track lo-calization on the formation of the North AtlanticOscilation, thus linking in a simple framework thetransition from hemispheric-scale to regional-scalevariability. Gerber and Vallis [2005] strip the dy-namics down even further by studying AMs us-

ing stochastic models in which angular momentumconservation is built in as a global meridional con-straint and coupling in longitude can be introducedin a controlled way.

We return to the observation in Section 2 thatthe NAM and SAM are more similar to each otherin the troposphere than the Northern and South-ern Hemisphere tropospheric jets. Classically, theobserved zonal mean zonal wind reflects two typesof jets: the subtropical jet for which axisymmet-ric angular momentum transport from the tropicsare important, and the higher latitude eddy-drivenjet which is dominated by eddy mean-flow inter-actions we have just described [e.g. Lee and Kim,2003]. (This classical viewpoint is evolving withthe realization that eddy potential vorticity fluxesare also important for the subtropical jet [e.g. Kimand Lee, 2001; Walker and Schneider , 2009].) TheSouthern Hemisphere November troposphere zonalwind displays aspects of both jets: there is an up-per tropospheric (baroclinic) subtropical jet corenear 300S in the upper troposphere and a verti-cally deep (equivalent barotropic) eddy-driven jetnear 500S (bottom row, left panel of Plate 4). TheSAM corresponds to meridional shifts of the latterfeature. In the Northern Hemisphere, by contrast,the zonal mean jet in JFM (bottom row, middlepanel of Plate 4) is dominated by the subtropicaljet near 300N. But for the tropospheric zonal meanwind sector averaged over the North Atlantic re-gion (not shown), a higher latitude eddy driven jetis seen and the NAM corresponds to poleward andequatorward shifts of this feature [Ambaum et al.,2001; Eichelberger and Hartmann, 2007]. Thus,a similar connection between the eddy driven jetand the AM can be found in both hemispheres,although the connection is more regional in theNorthern Hemisphere. Eichelberger and Hartmann[2007] use a simplified GCM (Section 4) to explaindynamically the local relationships between North-ern Hemisphere jet structure and NAM variability.

Unlike the tropospheric AM, which is manifestedas a jet meander, the stratospheric AM is man-ifested as a strengthening and weakening of thejet. Eddy forcing is important in stratospheric AMvariability but stratospheric waves are primarilygenerated from tropospheric sources that are ex-ternal to the stratosphere. The model of Holtonand Mass [1976] that has been further explored anddeveloped by many researchers [e.g. Yoden, 1987;Plumb and Semeniuk , 2003; Scott and Polvani ,2006] provides the simplest formulation for cap-turing stratospheric AM variability. In this model,the zonal mean stratospheric circulation is relaxedtowards a prescribed radiative equilibrium profile.Planetary Rossby waves are generated by prescrib-ing a zonally asymmetric lower boundary, conven-tionally chosen to represent the stationary wave


field in the upper troposphere. The upwelling waveactivity flux can be calculated in various ways; inthe simplest formulation it is parameterized us-ing WKB theory. In Figure 5, from Plumb andSemeniuk [2003], the wind and geopotential vari-ability resembles the observed downward propagat-ing AM signals in the stratosphere. The model inPlumb and Semeniuk [2003] is effectively one di-mensional in the vertical and represents in a sim-ple way the interaction between stationary Rossbywaves with upward group velocity and the zonalmean flow. The resulting downward propagat-ing signal in geopotential and winds is a conse-quence of enhanced wave activity absorption (EPflux convergence) where the winds are weaker, andthe reestablishment of the eastward winds at levelsshielded from upward propagating wave activity.

The Holton-Mass model exhibits distinct circula-tion regimes that depend mainly on the amplitudeof the zonally asymmetric lower boundary forc-ing. For example, in the weakly forced subcriticalregime that was explored in Plumb and Semeniuk[2003] and is illustrated in Figure 5, the period andevolution of the stratospheric flow is determinedentirely by the zonally asymmetric forcing of thelower boundary. Plumb and Semeniuk [2003] showthat downward propagation of Za does not implydownward propagation of stratospheric informa-tion, since the source of the wave driving and thewave activity absorption are determined from theprescribed boundary evolution and state informa-tion below a given level. This example shows howsome aspects of stratospheric variability might belargely determined by tropospheric dynamics. Inthe strongly forced supercritical regime [e.g Yoden,1987; Scott and Polvani , 2006], the zonal meanHolton-Mass model exhibits spontaneous vacilla-tions of the zonal wind even for time independentlower boundary forcing. This regime provides anexample of a more active stratosphere that mightnot depend on the detailed time evolving state ofthe troposphere, provided that the overall waveforcing from the troposphere is sufficiently strong.Thus within this simple model we see a wide rangeof possibilities for the way in which the strato-sphere might depend on the troposphere.

To summarize, we have learned that tropo-spheric and stratospheric AM variability can beisolated in truncated models whose dynamics arecomplementary. Tropospheric AM models requirerelatively fine horizontal resolution to capture thesynoptic scale eddies that are involved in eddy mo-mentum flux transport, but require only coarse ver-tical resolution involving even only one or two ver-tical layers. Stratospheric AM models require rela-tively fine vertical resolution to capture the down-ward propagating wave activity absorption region,

but require only coarse horizontal resolution tocapture planetary Rossby waves. But neither typeof simple model can realistically simulate the ob-served coupling, via the AMs, between the tropo-sphere and stratosphere that is depicted in Plate5. One might imagine constructing a model thatcouples the two types of truncated systems, butthis direction has not been explored to our knowl-edge. To simulate realistic AM variability, we turnin the next section to more realistic GCMs withsufficient vertical and horizontal resolution to cap-ture the dynamics as a whole.

4. ANNULAR MODES IN GENERAL CIRCULATIONMODELS

Given the limitations of the truncated modelsdiscussed in Section 3, we turn to models of greatercomplexity in the model hierarchy, and considerthe simulation of AM variability and AM responsesto climate forcings in simplified GCMs and com-prehensive climate models. We will also discussrecently developed ideas related to the fluctuationdissipation theorem, whose application to climatesimulation helps elucidate the simulated AM re-sponse to climate forcings. Along the way, we willconsider the general question of stratospheric in-fluence on the troposphere, in which the AMs arestrongly implicated.

4.1. Simulation of Intraseasonal VariabilityAs discussed in Section 3 modern climate mod-

els generally simulate the AM spatial structurecredibly, especially in the troposphere [Limpasu-van and Hartmann, 2000; Miller et al., 2006]. In-deed, realistic tropospheric AM structures appearto be a basic feature of any model that realis-tically represents the zonal mean circulation andtropospheric large-scale waves (baroclinic and sta-tionary waves). It was also established soon af-ter the original work of Baldwin and Dunkerton[1999, 2001] that comprehensive GCMs with suf-ficient stratospheric resolution are capable of cap-turing intraseasonal stratosphere-troposphere cou-pling in free-running climate [e.g. Christiansen,2001] and seasonal forecast [e.g. Charlton et al.,2004] settings. Christiansen’s [2001] simulationsof a free running climate model (with no infor-mation about the observed atmospheric state) areable to spontaneously produce realistic downwardpropagating NAM anomalies that progress fromthe stratosphere into the troposphere. In cur-rent models with good stratospheric representa-tion [e.g. Baldwin et al., 2010], downward propa-gating stratosphere-troposphere signals appear tobe a generic feature that is easy to capture in someform.


Charlton et al. [2004] demonstrate a practicalapplication of this capability by using a numericalweather prediction model to investigate whetherstratospheric information can provide useful pre-dictive information in the troposphere beyond thelimit of deterministic weather prediction. They ini-tialize ensembles of simulations with different po-lar vortex strengths and obtain a downward prop-agating NAM response which penetrates into thetroposphere over the course of several weeks (seeFigure 6). Practically, this study shows that accu-rate stratospheric information and representationcan potentially improve seasonal forecasts. Thequestion of stratospheric representation on captur-ing tropospheric AM variability arises in a varietyof contexts, including the response to stratosphericsudden warmings, ENSO-related variability, and tovariability in seasonal snow cover [e.g. Gong et al.,2002; Scaife and Knight , 2008; Fletcher et al., 2009;Ineson and Scaife, 2009; Bell et al., 2009; Cagnazzoand Manzini , 2009]. But these studies, and simi-lar ones which examine the downward influence ofaltered stratospheric initial states [e.g. Hardimanand Haynes, 2008], do not unambiguously addressthe question of whether the stratosphere exerts adownward influence on the troposphere. This isbecause the prescribed stratospheric state mightdepend on the prior state of the troposphere, inparticular the history of wave activity fluxes fromthe troposphere to the stratosphere [Plumb and Se-meniuk , 2003; Polvani and Waugh, 2004; Scott andPolvani , 2006]. We will show a more direct exam-ple of stratospheric influence in connection with cli-mate responses to stratospheric cooling and ozonedepletion below.

Having established that comprehensive climatemodels capture the downward propagating AM sig-nals, the dynamical question of what the mini-mum ingredients are to capture these signals re-mains open. To answer this, we turn to “simpli-fied GCMs”, by which we mean models that in-tegrate the primitive fluid equations of motion onthe sphere at relatively fine horizontal and verticalresolution but that include only the simplest rep-resentations of radiative transfer and subgrid scalemomentum dissipation, and that do not includemoisture [e.g. Scinocca and Haynes, 1997; Polvaniand Kushner , 2002; Taguchi and Yoden, 2002; Ger-ber and Polvani , 2009]. Simplified GCMs typi-cally represent diabatic heating by a term −(T −Teq)/τr in the thermodynamic equation, which cor-responds to Newtonian relaxation to a prescribedequilibrium temperature profile Teq on a radiativetimescale τr; and represent momentum dissipationby a simple mechanical drag −~u/τd, where ~u is thehorizontal velocity and τd is a timescale for mo-mentum dissipation. The reference temperatureprofile Teq is baroclinically unstable in the tro-

posphere and leads to an eddy-driven jet streamqualitatively similar to observed. In the strato-sphere, Teq is constructed to produce a polar nightjet. We emphasize that these models are only rel-atively simple compared to comprehensive GCMs;in absolute terms, they represent a highly complexdynamical system of partial differential equations.Still, compared to comprehensive GCMs, they havethe advantage of being relatively easy to reproducein a manner that allows them to provide the ba-sis for benchmark calculations across different re-search groups [Held and Suarez , 1994; Galewskyet al., 2004].

In simplified GCMs, stratospheric variability de-pends in a complicated way on tropospheric bound-ary conditions and dynamics. For example, mod-els with time independent and zonally symmetricboundary conditions can generate some realisticaspects of stratospheric variablity [Scinocca andHaynes, 1997], and the stratospheric mean stateand variability are sensitive to changes in topo-graphic amplitude and structure [Taguchi and Yo-den, 2002; Gerber and Polvani , 2009]. By simplyvarying the topographic forcing, these models canbe tuned to represent the stratospheric variabilityof either hemisphere [Taguchi and Yoden, 2002].

Our interest in this subsection is to deter-mine the minimum requirements for generating thedownward propagating AM signals. Gerber andPolvani [2009] addressed this question by varyingtopographic scale and amplitude. For a given timeindependent Teq they are able to tune their modelfrom a stronger polar vortex regime without to-pography in which the AMs are troposphericallytrapped, to a weaker polar vortex regime with to-pography in which the AMs have large amplitudein both the troposphere and stratosphere (Plate8). In this regime, the model is capable of pro-ducing realistic coupled stratosphere-troposphereevents (Plate 9a), including the build up of waveactivity flux into the stratosphere that precedes anegative stratospheric NAM event (Plate 9b). Thisregime is very sensitive: if the topographic am-plitude is increased, the vortex becomes relativelyweak and only produces small amplitude NAMevents; if the topographic amplitude is reduced,the vortex becomes too strong and robust to besignificantly weakened by wave activity pulses; ifthe wavenumber of the topography is changed, thecharacter of the variability of the polar vortex alsochanges qualitatively. As for the other exampleswe will describe in this section, none of these sen-sitivities were predicted by preceding theoreticalwork, which highlights how simplified GCMs re-main a primary theoretical tool to explore coupledstratosphere-troposphere variability.

To conclude this subsection on intraseasonalAM variability, we consider the tropospheric ori-


gin of stratospheric NAM anomalies in connec-tion with surface forcing, including forcing fromEl Nino-Southern Oscillation (ENSO) related seasurface temperatures in the tropical Pacific [Ine-son and Scaife, 2009; Bell et al., 2009; Cagnazzoand Manzini , 2009; Garfinkel and Hartmann, 2008]and from snow cover anomalies over Eurasia [e.g.Cohen and Entekhabi , 1999; Gong et al., 2002; Co-hen et al., 2007; Fletcher et al., 2009]. Althoughthe stratosphere is capable of generating internalvacillations independently of the state of the tro-posphere, as discussed in Section 3, it is likely thatsome portion of stratospheric variability shouldpotentially originate from tropospheric variabil-ity. Recent work by Garfinkel et al. [2010] showsthat this should occur, for ENSO and for Eurasiansnow forcing, when the wave component of cir-culation anomalies related to the surface forcingalign with the climatological stationary wave andthereby reinforce or attenuate the wave activityflux into the stratosphere. The lagged nature of theresponse implies that surface forcing can furtherextend the window of seasonal predictability viastratosphere-troposphere interactions. ENSO forc-ing provides a Rossby wave source from the tropicsto the extratropics, and the work by Ineson andScaife [2009] and Cagnazzo and Manzini [2009]shows that stratospheric representation makes adifference to how this wave activity drives zonalmean NAM anomalies. Surface forcing by Eurasiansnow cover anomalies in fall can similarly enhancethe tropospheric wave activity flux into the strato-sphere, thereby inducing a dynamical warming anda downward propagating NAM signal in NorthernHemisphere winter. This variability is simulatedin comprehensive GCMs in which the snow coverforcing is prescribed [Gong et al., 2002]. The ef-fect is sensitive to stratospheric details [Fletcheret al., 2009] and is not captured as an intrin-sic mode of variability in the CMIP3/IPCCAR4models [Hardiman et al., 2008]. This work sug-gests that obtaining improved predictability of AMevents of this kind will require considerable im-provements in the simulation of stationary wavesand land-surface processes.

4.2. AM Climate Trends and ClimateResponses

Much of the original interest in the AMsstemmed from the fact that circulation trendsthrough to the end of the 20th century projectedstrongly onto the AMs, in a manner correspondingto a positive NAM trend in the Northern Hemi-sphere and a positive SAM trend in the South-ern Hemisphere [Thompson and Wallace, 1998;Thompson et al., 2000; Hartmann et al., 2000;Thompson and Solomon, 2002]. For example,

Thompson et al. [2000] show that the pattern ofland-surface temperature trends in the NorthernHemisphere in the late 20th century are largelyconsistent with a positive NAM trend. In theSouthern Hemisphere, Thompson and Solomon[2002] reported that the seasonality and pattern ofSAM trends suggested a direct influence by ozonedepletion on the tropospheric circulation of theSouthern Hemisphere (see Figure 7a-b, from ?).Since the NAM’s peak in the late 1990’s, the NAMtrend has weakened and reversed [Overland andWang , 2005; Cohen and Barlow , 2005], but thelong term SAM trend remains robust [Fogt et al.,2009]. Although current trends in the AMs are rel-atively ambiguous, understanding the cause andexpected future of these trends remains a com-pelling research question. In this subsection, wewill investigate AM responses to climate change insimplified and comprehensive GCMs, and in Sub-section 4.3 highlight some theoretical issues relatedto these responses.

To see how circulation trends might relate to AMvariability, suppose that a circulation related fieldlike sea-level pressure exhibits a long-term trend.In the notation of Section 2, we can express a datamatrix F of n observations at p spatial points,which has zero time mean at each point, as thesum of a linear trend term and a residual term:

F = tfT∆ + R, (10)

where t is an n-point column of times of observa-tion and R is a residual term which has no lineartrend. The spatial vector f∆ is a p-point columnof trend coefficients, obtained by linear regression.(Alternatively, f∆ can represent simply the differ-ence in the field between two time periods, e.g. 21stand 20th centuries, in which case no linear regres-sion is carried out.) We are interested in what de-termines the spatial structure of this quantity andwhether it is related to natural patterns of vari-ability. For example, its projection on the AM isf∆,1 = f∆T f1/fT

1 f1, where f1 = FTy1/yT1 y1 is the

projection of the data matrix onto the PC timeseries of the AM. The projection coefficient f∆,1

provides the sign and degree of coherence betweenthe trend pattern and the AM pattern; the quan-tity

(f∆,1

T f∆,1

)/(f∆T f∆

)represents the fraction

of variance represented by the AM-related compo-nent of the trend.

Figure 8, from Miller et al. [2006], shows thefraction of variance represented by the projectionof the SLP response to climate change, diagnosedas the difference between the late 21st century andthe early 20th century, onto the AM and otherleading modes of extratropical variability, in thesimulations included in the Coupled Model In-tercomparison Project 3 prepared for the Inter-governmental Panel on Climate Change Report 4


(CMIP3/IPCC AR4). “Climate change” here in-cludes the effects of anthropogenic greenhouse gasloading, ozone depleting substances like chlorofluo-rocarbons implicated in photochemical ozone loss,and aerosol polution. In the Southern Hemisphere,the figure shows that the SAM is the dominant pat-tern of SLP response, and Miller et al. [2006] showthat the models produce a positive SAM responsethat agrees at least in sign with current trends.In the Northern Hemisphere the situation is muchmore complicated: the projection onto the NAMis weak, and Miller et al. [2006] show that thereis disagreement on the sign of the projection ontothe NAM as well. The situations in the two hemi-spheres are sufficiently different that they shouldbe considered separately.

The most robust and well characterized AM re-sponse to climate forcing occurs in the SouthernHemisphere. In particular, there is now exten-sive evidence from observations and models thatphotochemical Antarctic ozone loss has induced astrong positive tropospheric SAM response. This isborne out in several studies involving comprehen-sive GCMs [e.g. Sexton, 2001; Gillett and Thomp-son, 2003; Miller et al., 2006; Cai and Cowan,2007; Son et al., 2008; Shaw et al., 2009]. The mid-dle and bottom rows of Figure 7, from ?, illustratethis effect: they show the Antarctic temperatureand geopotential response for the CMIP3/IPCCAR4 models that include ozone depletion and thosethat do not. Models with ozone depletion consis-tently obtain a response in temperature and geopo-tential qualitatively similar to the observed trends,while those models without ozone depletion do notcapture this response. Although this is not a cleancomparison, because the sets of models representedin each row of Figure 7 are different, the mainconclusion that ozone depletion is required to ex-plain tropospheric circulation trends in the South-ern Hemisphere in the late 20th century has beenconfirmed in several studies. The seasonal cycle ofthe response to ozone forcing is a downward propa-gating signal reminiscent of transient intraseasonalvariability, and suggests a strong connection be-tween the two; this might be expected from theperspective of fluctuation dissipation theory dis-cussed in the next subsection, which connects inter-nal modes of variability to climate responses. Theeffects on the troposphere of ozone depletion areexpected to reverse as ozone depleting substanceemissions are reduced and the ozone layer recoversinto the mid 21st century [Son et al., 2008; Perl-witz et al., 2008]. This negative SAM response isexpected to be accompanied by an equatorwardshift of the tropospheric jet in Southern Hemi-sphere summer.

In simplified GCMs, the ozone-depletion effecton the tropospheric annular mode can be mod-

elled by examining the stratosphere-troposphereresponse to cooling imposed in the stratosphere[Polvani and Kushner , 2002; Kushner and Polvani ,2004; Gerber and Polvani , 2009; Chan and Plumb,2009]. Polvani and Kushner [2002]; Kushner andPolvani [2004] carry out such a simulation using asimplified GCM with zonally symmetric boundaryconditions. In the simplified GCM, for sufficientlystrong stratospheric cooling a dramatic polewardshift in the tropospheric jet occurs that is mani-fested as a positive tropospheric AM response thatis consistent in sign with the effects of ozone de-pletion. These papers provide an unambiguousexample of stratospheric influence on the tropo-spheric circulation in a simplified dynamical set-ting. As was the case for using simplified GCMsto understand intraseasonal variability, the modelsalso provide dynamical insight. These and subse-quent studies using simplified GCMs with variousthermal and mechanical forcings [Song and Robin-son, 2004; Ring and Plumb, 2007, 2008; Chan andPlumb, 2009] suggest that the same eddy mean-flow interactions that generate AM variability arealso responsible for amplifying the response to im-posed stratospheric forcing.

The simplified GCMs are sufficiently computa-tionally inexpensive that wide parameter regimescan be explored. Such explorations show thatmany potential sensitivities exist that can generatenon-robust responses to stratospheric cooling. Thesituation is nicely captured in Figure 9 [from Chanand Plumb, 2009], which shows that the strengthof the poleward shift of the tropospheric jet (thepositive SAM change) in response to stratosphericcooling (a strengthening polar vortex) is contingenton the existence of multiple regime behaviour forthe tropospheric jet. When the tropospheric equa-tor to pole temperature gradient is weak (in thefirst three columns of the figure), the troposphericjet can be found in a subtropical position or can bein a state where it jumps between subtropical andextratropical positions on a very long timescale.Under these conditions, as the stratospheric polarvortex is strengthened (proceeding down the rowsin the figure), the tropospheric jet position canbe shifted poleward. This behaviour characterizesthe regime found in Polvani and Kushner [2002];Kushner and Polvani [2004]. As the troposphericequator-to-pole temperature gradient is strength-ened, the subtropical jet regime vanishes, and thesensitivity to stratospheric cooling is greatly re-duced. Along the same lines, the tropospheric AMresponse to stratospheric cooling is greatly attenu-ated by the presence of zonal asymmetries [Gerberand Polvani , 2009]. Thus we see a connection be-tween AM responses to climate perturbations andthe ease with which the tropospheric jet can fluc-tuate between different regimes. The implications


of this behavior for present day, future, and pastclimates, has yet to be explored.

4.3. Perspective from Fluctuation DissipationTheory

Why should we expect climate trends to be re-lated to the AMs at all? The idea comes froma series of papers that have appealed to fluctu-ation dissipation theory in statistical mechanics[Leith, 1975; Palmer , 1999; Gritsun and Bransta-tor , 2007; Ring and Plumb, 2008]. The theory pos-tulates that, under general conditions, the prin-cipal pattern of response to climate forcings willproject strongly onto persistent modes of internalvariability of the system whose spatial structureis similar to that of the forcing. This means thatmany aspects of the response to climate changecan be predicted from knowledge of the system’sunforced climate variability. Formal techniques fordiagnosing fluctuation-dissipation are described inRing and Plumb [2008] and Gritsun and Branstator[2007]. Even without the formal approach, we canobtain useful information by empirically decom-posing a response pattern into parts coherent withand unrelated to the internal variability [Thomp-son et al., 2000; Kushner et al., 2001; Deser et al.,2004]. For example, if we consider the SouthernHemisphere response to greenhouse forcing [Kush-ner et al., 2001], we can decompose the zonal windresponse and the eddy momentum flux convergenceresponse (Figure 10, top row) into a part that iscoherent with the SAM (middle row) and a resid-ual (bottom row). The residual is interpreted asthe direct response to climate change, while theeddy driven AM response is interpreted as the in-direct response. Deser et al.’s [2004] analysis ofthe Northern Hemisphere circulation response tohigh-latitude surface forcing used this approach toshow that the direct response to sea ice and sea-surface temperature forcing in a GCM is localizedto the forcing region, and that the indirect responseis an eddy driven hemispheric teleconnection pat-tern. In a more controlled setting, Ring and Plumb[2007] show explicitly how the coherence of an ap-plied forcing — in this case, a zonally symmetricmomentum forcing applied in the vicinity of the jetin a simplified GCM — with the AM determinesthe sign and strength of the AM response.

Decomposing circulation responses into directand indirect parts is useful for a broad brush char-acterization but is qualitative and has little pre-dictive value. More quantitative analyses basedon fluctuation-dissipation theory have been carriedout in simplified GCMs [e.g. Ring and Plumb, 2008;Gerber et al., 2008b] or in very long runs of compre-hensive GCMs [e.g. Gritsun and Branstator , 2007,which deals with the global response to tropical

heating]. A key prediction of the theory is thatthe expected pattern of response f∆ ∼

∑iRifiτi,

where Ri is the projection of the forcing onto modefi and τi is the decorrelation timescale of the un-forced mode. Thus, we expect that more persistentmodes will dominate the response, other factors be-ing equal. Pursuing this idea, Gerber et al. [2008b]emphasize how AM timescales determine the am-plitude of the response to external forcings: mod-els with long AM timescales have an AM responsethat is proportionately greater, in agreement withthe predictions of the theory. Ring and Plumb[2008] used fluctuation dissipation theory in evenmore detail in a simplified GCM forced mechani-cally and thermally. The responses in these sim-ulations scaled linearly with the projection of theforcing onto the internal modes in general agree-ment with the theory, but the results did not agreequantitatively with its predictions.

There is thus fairly strong evidence to supportthe conclusion from fluctuation dissipation theorythat the AM timescale is a key parameter control-ling the strength of the indirect or free responseto climate forcings. But the AM timescale, fromthe current modelling evidence, turns out to behighly non-robust and difficult to simulate accu-rately. For example, Gerber et al. [2008b] showthat AM timescales in simplified GCMs are highlysensitive to horizontal resolution, vertical resolu-tion, numerical diffusion, and other factors. Incomprehensive GCMs, the IPCC AR4 models pro-duce AM timescales that are too long in both theNorthern Hemisphere and Southern Hemisphere,as seen in Plate 6. In simplified GCMs, we havealready seen how some aspects of the sensitivity ofthe AM timescales is related to the structure of thejets; in particular, the existence of multiple regimesfor the jet, in which the jet remains in a subtrop-ical or in an extratropical location for long peri-ods of time, greatly lengthens the timescale [Chanand Plumb, 2009]. In the IPCC AR4 models, Kid-ston and Gerber [2010] find a connection, similar toChan and Plumb in the simplified GCMs, betweenthe climatological bias, the AM timescale, and cli-mate responses: those models with an equatorwardbias in the Southern Hemisphere jet have a morepersistent SAM and shift poleward more under cli-mate change. From the analysis of simplified andcomprehensive GCMs, in light of fluctuation dis-sipation theory, we conclude that AM responsesmight be too strong in present day GCMs. In ad-dition, we conclude that the responses will be sen-sitive to many factors, even if the spatial structureand amplitude of the AMs are realistically repre-sented in the models.


4.4. Stratospheric Representation and AMResponses

With this background we are in a better posi-tion to address the question of stratospheric influ-ence on the AM response to tropospheric climatechange. The seminal paper in this area is the studyof Shindell et al. [1999], who argue that accuratestratospheric representation is necessary to repro-duce the NAM trend of the late 20th century. Theargument involves a comparison between two ver-sions of comprehensive GCMs forced by anthro-pogenic greenhouse gas loading: a GCM with veryfew levels in the stratosphere (which we will call“low top”), and a model with a relatively morestratospheric resolution (“high top”). Of the twomodels, only the high-top model produces a posi-tive NAM response similar to the observed trend.Shindell et al. argue that the only in the high-top model is the effect of planetary wave refractionproperly accounted for: the positive NAM responsein the stratosphere involves refraction of wave ac-tivity away from the polar lower stratosphere intolow latitudes [Shindell et al., 1999, 2001]. Thiswork emphasizes that if the climate response trulyprojects onto the coupled stratosphere-troposphereAM then the wave driving response is intrinsicto the AM response and must be properly repre-sented. While subsequent work has debated thefindings and interpretation of the Shindell et al.work, it remains highly significant for the subse-quent investigation it has stimulated.

The idea that stratospheric representation is im-portant to tropospheric climate dates back to thework of Boville [Boville, 1984; Boville and Cheng ,1988]. This work shows that the configuration ofthe model stratosphere (including factors influenc-ing the polar night jet like stratospheric diffusion,mechanical drag, and vertical resolution) has a sig-nificant influence on the tropospheric circulation.The patterns of sensitivity to stratospheric repre-sentation, in both the winds and the wave driving,resemble the AM structures. But the extension ofthis idea, that stratospheric representation is im-portant to tropospheric climate change, is not cur-rently straightforward to resolve. In apparent con-tradiction to Shindell et al. [1999], it is clear thatlow-top models are capable of obtaining AM re-sponses similar to the observed trends [Fyfe et al.,1999; Kushner et al., 2001; Miller et al., 2006]. Butseveral factors confound simple generalizations: aswe have said, the model responses are not robust(Figure 8); such responses might be exaggeratedbecause of the long AM-timescale bias; and someobserved AM trends are not robust. Thus, the is-sue of stratospheric representation on the response

to climate change needs to be addressed separatelyfor each climate response and for each model.

A necessary step to address the issue of strato-spheric representation is to systematically varystratospheric resolution while minimizing otherchanges. When model lid height is lowered andother changes are minimized, at least two mod-els show relatively little sensitivity in their AM re-sponses to climate forcing [Gillett et al., 2002; Sig-mond et al., 2008; Shaw et al., 2009]. For example,in Sigmond et al. [2008] and Shaw et al. [2009],a specialized low-top version of a high-top GCM(the Canadian Middle Atmosphere Model) is con-structed so as to minimize inconsistencies betweenthe two. (Plate 10 summarizes some of the resultsfrom these studies.) The lid of the low-top GCMis located at 10 hPa, which is lower than the lidof most present-day climate models. Yet both thelow- and high-top models exhibit a strong positiveNAM response to greenhouse warming [Sigmondet al., 2008] and a strong positive SAM responseto ozone depletion [Shaw et al., 2009] — see Plate10. Thus, stratospheric resolution by itself is nota key factor in controlling the AM response to cli-mate forcings, at least in this model.

While in clean tests, stratospheric resolutiondoes not necessarily influence tropospheric AM re-sponses, the responses can be surprisingly sensitiveto other factors. In particular, the models in Plate10 are sensitive to the details of the parameteri-zation of momentum dissipation by subgrid-scalegravity waves. In Shaw et al. [2009], for exam-ple, if the flux of momentum from parameterized“non-orographic” gravity waves is not deposited inthe top model level in a manner that satisfies mo-mentum conservation in the vertical, but insteadallowed to figuratively “escape to space”, the tro-pospheric SAM response to ozone depletion is su-pressed. In Sigmond et al. [2008], the positive tro-pospheric NAM response to greenhouse warmingcan be suppressed by increasing the strength of theorographic gravity wave flux from below. Thesesensitivities are relevant to current modelling prac-tice, in which an escape-to-space boundary condi-tion on the gravity wave fluxes near the model lidis often used, and in which a range of tunings of theorographic gravity wave flux are used. It would beinteresting to see how much of the spread in AM re-sponses in current GCMs could be attributed to in-consistencies in the implementation of gravity waveparameterizations, since these schemes are becom-ing better constrained observationally and bettercharacterized numerically [Alexander et al., 2009;McLandress and Scinocca, 2005].


5. CONCLUSION

The identification and characterization of theAMs, due in large part to the University of Wash-ington ‘school’ with its hallmark of dynamicallymotivated observational analysis, has representeda significant advance in atmospheric science thathas strengthened the link between our field andthe environmental sciences as a whole. This isbecause the AMs provide a unifying frameworkfor large-scale variability in the extratropics thatlinks stratospheric and tropospheric dynamics,that highlights common strands between Northernand Southern Hemisphere variability, and that con-nects intraseasonal variability to climate-timescalefluctuations. Even though it is incomplete, ourcurrent statistical and dynamical understanding ofthe AMs can be applied in a variety of environmen-tal contexts, including investigation of temperatureextremes [Thompson and Wallace, 2001], oceaniccirculation, the global carbon cycle in the North-ern and Southern Hemispheres [e.g. Russell andWallace, 2004; Butler et al., 2007], Antarctic sur-face climate [e.g. Thompson and Solomon, 2002],and many other diverse topics that are outside thescope of this Chapter. Because the AMs representa characteristic pattern of climate response as wellas climate variability, further attention will con-tinue to be warranted as we move forward withattempts to predict the future of evolution of theEarth system.

A key issue that remains unresolved in this dis-cussion is the mechanisms underlying stratosphere-troposphere AM coupling, and the coupling of AMsto the land and ocean surface. We have attemptedto clarify those aspects of tropospheric and strato-spheric AM variability that can be understood inisolation and those for which surface and strato-sphere coupling are more critical. The strato-spheric and tropospheric AMs can be regarded dy-namically distinct, but they are linked by eddydriven processes that we are only beginning tounderstand and quantify. Likewise, we are onlybeginning to understand the role of snow, oceansurface temperatures in the tropics and extratrop-ics, and other surface forcings in generating AMvariability. In the effort to understand the variousdrivers of AM variability, simplified general circu-lation models with sufficient horizontal and verti-cal resolution currently provide our main tool todevelop theoretical understanding. At each stage,both comprehensive and simplified GCMs have fur-nished surprising results, especially in relation toAM timescales and AM climate responses. Thesedevelopments are currently leading to fundamentalinsights that, we expect, will solidify our under-

standing of the observed AMs and their climateresponses.

Regarding the climate responses themselves, anintensive research effort on the Southern Hemi-sphere extratropical circulation response to climatechange is warranted at this point. A positiveSouthern Annular Mode (SAM) response to green-house gas increases and ozone depletion appears tobe a robust signal common to many climate mod-els, and seems to be in agreement with observa-tions in the second half of the 20th century. Butthere are reasons to expect, based on the long AMtimescale bias identified in Section 4, that the mod-els will tend to be correspondingly biased towardsoverly large amplitude AM responses. Accuratelyrepresenting the SAM response will become morechallenging and critical as the ozone layer recovers,since the climate and ozone forcing signals are ex-pected to produce opposite signed SAM responses[e.g. Son et al., 2010]. This issue will require asystematic attack using the full model hierarchybefore it can be fully resolved.

6. APPENDIX

We recall some of the main results of quasi-geostrophic (QG) theory [Andrews et al., 1987,Ch. 3]. The QG equation in log pressure and tan-gent plane linear Cartesian coordinates is

Dq = −Xy + Yx + f0ρ−10 (ρ0Q/θ0z)z , (11)

where the QG potential vorticity (also known asthe pseudo potential vorticity) q is defined by

q ≡ f0 + βy + ψxx + ψyy + ρ−10

(ρ0f

20N−2ψz

)z,

the QG approximation to the advective derivativeacting on the QG potential vorticity q is

D =∂

∂t+ u

∂

∂x+ v

∂

∂y,

where the static stability N2(z) is associated withthe reference potential temperature θ0(z). Some ofthe remaining notation is explained as follows:

ψ Geostrophic streamfunction(u, v) = (−ψy, ψx) Geostrophic velocityθe = f0θ0zψz/N

2 Potential temperatureperturbation from reference θ0z

(X,Y ) Momentum dissipationQ Diabatic heating

on potential temperature.

If we take the zonal mean of (11) and remove theclimatological mean of the result we obtain (4) with

S = −(Xa)y + f0ρ−10

(ρ0Qa/θ0z

)z. (12)

To show why we expect the density weighted vol-ume average of Saqa to be negative, we assume


that the momentum forcing is linear drag withtimescale τm and the diabatic heating is relax-ational on timescale τr. If the damping timescalesare equal, with τm = τr ≡ τ , then Sa = −τqa

and Sa · qa = −τ(qa)2 ≤ 0 at each point. Butif the momentum and thermal damping timescalesare different, it is more difficult to find a generalresult. We find

Sa = τ−1m (ua)y − f0τ

−1r ρ−1

0

(ρ0θa

e/θ0z

)z

= −τ−1m (ψa)yy − τ−1

r ρ−10

[ρ0f

20N−2(ψa)z

]z

and

qa · Sa

= −{τ−1m (ψa)yy + τ−1

r ρ−10

[ρ0f

20N−2(ψa)z

]z

}·{

(ψa)yy + ρ−10

[ρ0f

20N−2(ψa)z

]z

}= −

{τ−1m

[(ψa)yy

]2+(τ−1

m + τ−1r )ρ−1

0 (ψa)yy

[ρ0f

20N−2(ψa)z

]z

+ τ−1T

[ρ−1

0

(ρ0f

20N−2(ψa)z

)z

]2}Now the cross term in this expression is propor-tional to

(ψa)yy

(ρ0f

20N−2(ψa)z

)z

= ∇ ·G + ρ0f20N−2[(ψa)yz]2,

where

G=[−ρ0f

20N−2(ψa)y(ψa)zz, ρ0f

20N−2(ψa)yy(ψa)z

]Density weighted averaging corresponds to 〈f〉 =∫ρ0dV , and so⟨

qa · Sa⟩

= −⟨τ−1m

[(ψa)yy

]2+(τ−1

m + τ−1r )f2

0N−2[(ψa)yz

]2+τ−1

r

[ρ−1

0

(ρ0f

20N−2(ψa)z

)z

]2⟩+(τ−1

m + τ−1r )

∫∇ ·GdV.

Thus,⟨qa · Sa

⟩evaluates to a negative quadratic

term plus a flux divergence that integrates out tothe boundaries. We have not found a simple jus-tification for zeroing out this flux integral, partic-ularly its vertical contribution, and thus we mustallow that boundary terms might provide a sourceof variance of the zonal mean PV independent of

eddy feedbacks in (5). But if we choose boundaryconditions that zero out this flux, for example con-sidering the set of solutions for which (ψa) = 0 atthe boundary, then

⟨qa · Sa

⟩will be negative.

The polar cap density-weighted average poten-tial vorticity tendency in (6) is found from (4)

< qta >p

=∫ zT

zB

∫ yN

yS

dydz[−ρ0(ua)yt + f0ρ0(θa

e )t/θ0z

]using ua = 0 at y = yN , which corresponds to theNorth Pole. Equation (8) for the eddy fluxes anddissipation is derived similarly, provided we set themeridional component of the EP flux F(y) = 0 atthe North Pole. To obtain the vertically integratedmomentum balance (9) we take ρ0θe/θ0z → 0 aszT → ∞ and use the zonal mean of the standardQG lower-boundary condition

Dθe = Q at zB = 0;

topographic effects are neglected in this expression.

ACKNOWLEDGMENTS. The support of the Natu-ral Sciences and Engineering Research Council of Canada andthe Canadian Foundation for Climate and Atmospheric Sci-ences is gratefully acknowledged. Thanks go to R.A. Plumband E.P. Gerber for careful reviews; to K. Smith and D.W.J.Thompson for additional helpful comments; and to L. Mudrykfor providing Plate 5, V. Limapsuvan for providing Plate 7,and T. Shaw and M. Sigmond for providing the data for Plate10.

NOTES

1. A mathematically similar argument for the variance of the eddypotential vorticity leads to the classical result that the eddypotential vorticity flux must be down the meridional poten-tial vorticity gradient, i.e.

⟨v′q′ · qy

⟩< 0, in order to main-

tain eddy variance. The sum of the eddy and zonal meanvariance equations is related to conservation of total potentialenstrophy.

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Paul J. Kushner, Department of Physics, University ofToronto, 60 St. George St., Toronto, ON, M5S 1A7, Canada


EOF of SLP

Poleward of 20N

… for the North-Atlantic

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Pacific sector only.

Northern Annular Mode

(NAM)

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(NAO)

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(NPO)

Deser 2000

(a) (b) (c)

Figure 1. Regression of SLP on principal componenttime series from EOF analysis of SLP in three domains:(a) the domain poleward of 200N for the Northern Hemi-sphere. (b), as in (a), for the North Atlantic sector. (c),as in (a), for the North Pacific sector. Adapted fromDeser [2000].


a) ps regression on u1(t; = 100) 0o 30oE 60oE 90oE 120oE 150oE 180oW 150oW 120oW 90oW 60oW 30oW 0o

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(b) Time-longitude correlation for pattern (a)

Figure 2. (a) Composite regression pattern for localdipole index on daily surface pressure. The local dipoleindex represents the principal component time series forthe second EOF of surface pressure at a single longitude.The regression with two dimensional surface pressure iscalculated for each longitude, shifted to a common lon-gitude for plotting purposes, and averaged. Thus, thegeographic boundaries are provided for a sense of scaleonly. Contour interval: 1 hPa. (b) Longitude-time lagcorrelation plot of Southern Hemisphere dipole indexesassociated with the pattern in (a). The dipoles progateeastward as coherent packets in both hemispheres with acharacteristic speed of about 8 degrees longitude per day.Contour interval 0.1. From Kushner and Lee [2007].


Figure 3. Two-layer quasigeostrophic model simulationof dependence of leading EOF of zonal wind variability(solid) on width of baroclinic zone as represented by thezonal mean zonal wind (dotted) [Lee and Feldstein, 1996].The baroclinic zone is progressively widened proceedingfrom panel (a) to (d); between panels (b) and (c) thezonal mean wind transitions from a single-jet to a two-jet structure.


270 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 5. (a) Schematic of the leading EOFs associated with a pulsingjet. The solid line is the mean zonal wind itself, the dotted line isthe EOF of the zonal velocity, and the dashed line the EOF of thepressure or streamfunction field. (b) As in (a), but for the leadingEOF associated with a wobbling or oscillating jet.

FIG. 6. The mean value and the first EOF of the zonally averagedzonal wind simulations with a narrow stirring region (approximately3! half-width) and a broader stirring region (approximately 12! half-width). For the narrow forcing region, the first EOF is a almost a‘‘pulse,’’ with a structure similar to that of the jet itself. For a widerforcing region, the first EOF is closer to a ‘‘wobble.’’ Figures inparentheses indicate variance accounted for.

Both pulsing and wobbling behavior frequently occurin numerical simulations, and one factor determiningwhich is dominant is the width of the stirring region. Ifthe stirring region is narrow (narrower than the resultingjet), then the jet position is effectively fixed and the firstEOF resembles the jet itself. If the stirring region iswider than the natural width of a single jet, but notsufficiently wide to support two jets, the jet’s positioncan vary within the stirred region. The behavior in thesetwo cases is illustrated in Fig. 6. (One-dimensionalEOFs are constructed from daily fields; two-dimensionalEOFs are constructed from the fields after applying a10-day running average.)With a stirred region of similar meridional extent to

that of the baroclinic zone on earth a wobbling or a‘‘mixed’’ mode tends to prevail (Fig. 7), with the secondEOF looking more like a pulse. The first EOF of thestreamfunction is dipolar, with a node somewhat pole-ward of the mean position of the jet and the lower bandmore or less coincident with the mean position of the

jet. These structures are apparent in both the EOF ofthe zonally averaged fields, and in the zonally averagedEOF of the two-dimensional fields (not shown). Thefirst EOFs are typically well separated from the otherEOFs. Similar structures are seen in the observations(Feldstein and Lee 1998; Lorenz and Hartmann 2001)and in simulations with a general circulation model(Cash et al. 2002). Indeed Feldstein and Lee (1998)characterize the EOFs of the Northern and SouthernHemispheres as being either a strengthening and weak-ening of the jet or a latitudinal movement of the jet,although the interpretation is complicated by additionalvariations in the subtropical jet.Although useful as descriptive phrases, the pulsing

and wobbling modes are not necessarily distinct phys-ical modes. Recall that the mean wind is somewhat pole-ward of the center of the stirring, because of predom-inantly equatorward breaking of the Rossby waves. Ifthe stirring is stronger, the jet is not only stronger butis pushed slightly poleward, and the pulse and the wob-ble are synchronized. The EOFs are describing this inthe most economical way possible, subject to their or-thogonality.

c. Two-dimensional patterns

When one looks at the EOF of the two-dimensionalfields, the zonal average of the first EOF (of either ve-locity or streamfunction) is usually very similar to theEOF of the zonally average field, although the subse-quent EOFs are less distinct. The first EOF normally iswell separated from the others, although the varianceaccounted for is typically less than 20%. The first EOFof the two-dimensional streamfunction is illustrated in

Figure 4. Latitude dependence of the zonal mean zonalwind (“jet”) and the leading mode of variability of thezonal mean zonal wind (“EOF 1”) in a stochasticallystirred barotropic model [Vallis et al., 2004]. Resultsfrom simulations in which the stirring region is 3 degreeslatitude wide and 12 degrees latitude wide are shown.The narrow 3-degree jet has a “pulsing” EOF; the wider12-degree jet has “wobbling” EOF that corresponds tothe AM.


Figure 5. Response in geopotential and zonal wind ofa simple Holton-Mass β-channel stratospheric model toperiodic forcing from the troposphere. The model rep-resents extratropical eddy mean flow interactions fromupward propagating Rossby waves generated by a pre-scribed zonally asymmetric lower boundary. The pre-scribed forcing has a period of 200 d. From Plumb andSemeniuk [2003].


Figure 6. Ensemble mean NAM response to switch onstrengthening of stratospheric polar vortex, introducedby means of changing stratospheric initial state. Theinitial tropospheric response in the first two days arisesfrom an adjustment to a new balanced state in the strato-sphere. The positive NAM response, which occurs aftera delay of about two weeks, is the result of changes tothe stratospheric state. From Charlton et al. [2004].


shows the vertical level structure of the ozone ensemblemodels. Models are sorted from left to right in accordance totheir number of levels between 300 hPa and 10 hPa. It isseen that the CSIRO and GFDL models have the lowestresolution in this region.[12] The models are separated into a low vertical resolu-

tion ensemble (LOW) consisting of both CSIRO and GFDLmodels and a high vertical resolution ensemble (HIGH) thatconsists of the rest of the models from the ozone ensemble.Thus, the LOW models have 6 or fewer levels in the heightinterval between 300 hPa and 10 hPa and the HIGHmodels have 7 or more levels. The LOW ensemble consistsof 9 simulations while the HIGH ensemble consists of32 simulations. Subdivision of the HIGH ensemble intogroups of models with at least 12 levels and models withfewer than 12 levels between 300 hPa and 10 hPa revealedno significant differences in geopotential height trendsbetween these two groups.

[13] Figure 3 shows that in the stratosphere HIGH andLOW models perform rather similarly. In both ensembles,maximum cooling is simulated in November at 100 hPa andis of similar magnitude in each (�7.7 K in the HIGHensemble and �7.3 K in the LOW ensemble). Note thatcooling in the GFDL CM2.0 model is only �3.4 K per30-year meaning that this model does not capture themagnitude of cooling correctly. Excluding this model fromthe LOW ensemble produces results similar to those shown.In the HIGH ensemble (Figure 3c), cooling propagatesdown to 700 hPa in summer as in observations. TheLOW ensemble (Figure 3a) simulates tropospheric warmingwith a maximum in spring exceeding 0.5 K per 30-year.[14] In both ensembles, negative trends in the stratospheric

geopotential height are observed in spring although in theLOW ensemble they are of slightly smaller magnitude. Thedifference becomes clearer in the troposphere where no signof the downward trend propagation is evident in the LOW

Figure 1. Seasonal cycle of linear trends in (left) temperature (K per 30 years) and (right) geopotential height (m per30 years) over the Antarctic: (a)–(b) observations, (c)–(d) ozone ensemble average, and (e)–(f) no-ozone ensembleaverage. Shading denotes trends that exceed 1SD of the respective monthly time series.

L20806 KARPECHKO ET AL.: SH TROPOSPHERE RESPONSE TO O3 AND GHG L20806

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Figure 7. Antarctic trends for 1968-1999 in (left)temperature (K per 30 years) and (right) geopotentialheight (m per 30 years) as a function of calendar monthand pressure for observations (a)-(b), ensemble mean ofCMIP3/IPCC AR4 models that include the effects ofozone depletion (c)-(d), and ensemble mean of the modelsthat do not include the effects of ozone depletion (e)-(f),from Karpechko et al. [2009]. Averages use points corre-sponding to the stations from Thompson and Solomon[2002]. This and other studies (e.g. Cai and Cowan[2007]) confirm that ozone depletion is responsible forat least half of the 1968-1999 trend in the SAM in thetroposphere.


Figure 8. From Miller et al. [2006]: (a) Percentageof variance represented by projection of SLP trends inthe 20th century onto four leading modes of extratropi-cal variability in the CMIP3/IPCCAR4 models, for theNorthern Hemisphere. (b) As in (a), for the SouthernHemisphere.


a robust propensity for the midlatitude winds to bemorebarotropic and hence the existence of stratosphericwesterlies above the jet, but a more thorough explana-tion is beyond the scope of this study.

4. Discussion

Other simple models have demonstrated that gradu-ally changing a single parameter (e.g., increasing thediabatic heating in the tropics) can change the locationof peak eddy activity from one latitude to another (e.g.,Lee and Kim 2003). As discussed in Gerber andPolvani (2009), there is evidence that this system is ac-tually ‘‘teetering’’ between these two states. This is clearly

demonstrated in Fig. 3f by the bimodal distribution ofthe eddy-driven jet’s distribution in experiment 1b.To demonstrate fully that e 5 210 K lies in between

two regimes, we perform an additional experimentwith e 5 0 K, thus placing the tropospheric Te maxi-mum equatorward of that of the PK case. As shown inFigs. 3a, 3i, and 3m, the distribution of the eddy-drivenjet’s location is unimodal and is unambiguously locatedin the subtropics in experiment 0a or in the midlatitudesin cases 2a and 3a. A comparison with Fig. 3e shows thatthe control case of the PK setup indeed sat in betweenthese two states.There are other examples in the atmosphere that ex-

hibit ‘‘regime behavior.’’ For instance, using a primitive

FIG. 3. A relative histogram of the latitudinal location of the maximum daily averaged near-surface zonally averaged zonal winds for allmodel runs listed in Table 1. Histograms in bold are model runs for which the decorrelation time for the leading principal component isgreater than 200 days. Within each column, the same stratospheric equilibrium temperature profile is used, with polar vortex intensities(g) increasing to the right. Within each row, the same tropospheric equilibrium temperature profile is used, with the magnitude of theequator to pole temperature difference increasing downward.

JULY 2009 CHAN AND PLUMB 2111

Stro

nger

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ar N

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Stronger Tropospheric JetFigure 9. Frequency distribution of meridional locationof the tropospheric jet maximum in a simplified GCM forincreasingly cold polar stratosphere (increasingly strongpolar vortex), going left to right in each row, and forincreasingly strong tropospheric meridional temperaturegradient, going from top to bottom in each column. Go-ing from weak to strong tropospheric temperature gra-dient and stratospheric polar vortex, the jet variabil-ity passes from a subtropical regime to an extratropicalregime, through a mixed bimodal regime. In the mixedregime, the jet can be found in either an extratropicalor a subtropical location, and can persist in one of theselocations for a long period of time; the darker bars indi-cate high-persistence regimes with the marked timescalesτ . In addition, in the mixed regime, the jet is relativelysensitive to stratospheric cooling. In the extratropicallytrapped regime, the sensitivity to the stratosphere is at-tenuated or vanishes. Adapted from Chan and Plumb[2009].


U -∂[u’v’]/∂y

Response

SAM

Residual

Figure 10. Analysis of zonal wind response (left col-umn) and eddy momentum flux convergence response(right column) to greenhouse warming in the GFDL R30model, from Kushner et al. [2001]. The top row showsthe response, the middle row shows the part of the re-sponse that is coherent with SAM variability based onthe model’s lower tropospheric winds, and the bottomrow shows the residual (top row minus middle row). Theresidual is confined to the upper troposphere and lowerstratosphere, which Kushner et al. [2001] suggest is thelocation of the strongest forcing of the meridional tem-perature gradients in the response to climate change;they thus interpret this as the direct response to climatechange. The SAM related response, on the other hand,involves the eddy-driven SAM variability that is inter-preted to be stimulated by the direct forcing; they thuscall this the indirect response to climate change.


(a) (b)

(c)

Plate 1. (a) Log of the power spectrum of the zonal

mean geopotential anomaly Z from (3) at p = 850 hPa.Units are arbitrary. The data used is NCEP/NCAR re-analysis [Kalnay et al., 1996] daily averaged geopotentialfrom 1979 to 2008. A simple five point running meanacross frequencies has been applied to create smootherplots. (b) As in (a), for p = 50 hPa. (c) Meridional meanof the log-spectra in (a) and (b) poleward of 22 degreeslatitude, scaled to have equal area under each curve.


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ovD

ec

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20

JulAug

SepO

ctN

ovD

ecJanFebM

arApr

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90Std_Z50, c.int.=10.0

JanFebM

arApr

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ctN

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ec

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Plate 2. Seasonal cycle of the standard deviation ofthe seasonal time series of the zonal mean geopotentialanomaly Zmon at 850 hPa (left column, contour interval4 m), 250 hPa (middle column, contour interval 5 m) and50 hPa (right hand column, contour interval 10 m), forthe Northern Hemisphere and Southern Hemisphere ex-tratropics. Calendar months are arranged so that wintersolstice is at the center of each panel.


JulAug

SepO

ctN

ovD

ecJanFebM

arApr

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Jun

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50

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90Reg_Z850, c.int.=4.0

JanFebM

arApr

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Jun

JulAug

SepO

ctN

ovD

ec

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30

20

JulAug

SepO

ctN

ovD

ecJanFebM

arApr

May

Jun

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30

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70

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90Reg_Z250, c.int.=5.0

JanFebM

arApr

May

Jun

JulAug

SepO

ctN

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ec

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JulAug

SepO

ctN

ovD

ecJanFebM

arApr

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Jun

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90Reg_Z50, c.int.=10.0

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Regressions for Z850EOF1

Plate 3. Regressions on the 850 hPa NAM (top row)

and the 850 hPa SAM (bottom row) of X = Zmon at

850 hPa (left column), X = Zmon at 250 hPa (middle col-

umn), and X = Zmon at 50 hPa (right column). Contourintervals and format as in Plate 2


Plate 4. Top row: AM zonal wind regression patternsfor Southern Hemisphere November 850 hPa SAM (left),Northern Hemisphere JFM 850 hPa NAM (middle) andthe hemispherically symmetric component of these twopatterns (right). Bottom row: climatological mean zonalwind for November (left), JFM (middle), and the hemi-spherically symmetric component of these two patterns(right). The red curves in the top left panels indicate theapproximate location of the jet axis in each hemisphere.Contour interval in top row is 0.5 m/s and in bottom rowis 5 m/s.


Plate 5. Composites of 50 warm vortex events basedon NCEP reanalysis data from 1958-2007. For each win-ter season (NDJFM) the strongest geopotential heightanomaly at 10hPa was used. Contour intervals are spacedby 0.2. Index values between -0.1 and 0.1 are unshaded.The zero level is shown by the heavy black contour. Thetop panel shows a composite of nondimensional NAM in-dex (based on geopotential height anomalies, and usingthe zonal-mean EOF method described in Baldwin andThompson [2009]). The bottom panel shows a compos-ite for the time series of polar-cap averaged geopotentialheight (600N-900N), normalized by its standard deviationat each level. Figure courtesy of L. Mudryk.

SAMNAM

NCEP

203CM

Plate 6. The seasonal cycle of NAM and SAMtimescales, computed with the methods of Baldwin et al.[2003], for NCEP reanalysis (top row) and for 20th cen-tury integrations of coupled climate models in the FourthAssessment Report of the Intergovernmental Panel onClimate Change (bottom row). Adapted from Gerberet al. [2008a].


200

400

600

800

1000

PRES

SUR

E (h

Pa)

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1000

PRES

SUR

E (h

Pa)

80S 70S 60S 50S 40S 30S 20SLATITUDE

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Pa)

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)

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(km

)

NAMSAM

HIGH

LOW

HIGH-LOW

Plate 7. Composite E-P flux cross sections for highand low AM conditions in each hemisphere, and theirdifference. Contouring indicates regions of EP flux di-vergence/convergence, with shaded regions indicatingEP flux convergence. Contour interval for the toptwo rows is 1.5 m s−1 day−1, and for the bottom rowis 0.25 m s−1 day−1. Negative values (associated withwestward acceleration) are shaded and the zero contouris omitted. Eliassen-Palm flux vectors divided by thebackground density) are shown; the longest vector isabout 1.4×108 m3s−2. Figure courtesy of V. Limpasu-van, adapted from Limpasuvan and Hartmann [2000].


Plate 8. Climatological zonal wind (contours) and re-gression of AM in zonal wind (shading) for two versionsof a simplified GCM with the same reference tempera-ture profile in the troposphere and stratosphere but with(a) no topography and (b) wave-2 topography of ampli-tude 3000 m. The topography affects the position andstrength of the jets and the AM structure; the AM ex-hibits stratosphere-troposphere coupling representativeof the active season only in the second case. From Gerberand Polvani [2009].


Plate 9. (a) AM index composited on strong negativeNAM events in the stratosphere, which correspond tostratospheric warmings, for the simplified GCM config-ured as in Plate 8b. (b) Composite of a measure of thevertical component of the wave activity (bars) and itsmean for the previous forty days (solid line) for the sameNAM index. In the configuration where the simplifiedGCM has a realistic AM structure, it is able to representdownward propagating NAM events in a realistic manner,although the persistence of the NAM in the troposphereand stratosphere is generally longer than observed. FromGerber and Polvani [2009]


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(a) ∆U(GHG, HIGHTOP)

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(b) ∆U(GHG, LOWTOP)

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(c) ∆U(O3, HIGHTOP)

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(d) ∆U(O3, LOWTOP)

Plate 10. (a) Northern Hemisphere December-January-February zonal mean zonal wind response to CO2 dou-bling in the high-top version of the Canadian Middle At-mosphere Model (CMAM) with a lid at 10−4 hPa. Con-tour interval is 1 ms−1 and dashed contours are negative.(b) As in (a), but for the low-top version of CMAM witha lid at 10 hPa but otherwise configured to be as consis-tent as possible with the high-top version. (c) SouthernHemisphere December-January zonal mean zonal windresponse to Antarctic ozone depletion in the high-topCMAM. (d) As in (c), for the low-top CMAM. It is ap-parent that the model lid has little impact on the tro-pospheric AM response in these models. Adapted fromSigmond et al. [2008]; Shaw et al. [2009].

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ANNULAR MODES OF THE TROPOSPHERE AND ...pauluis/Olivier_Pauluis_Homepage...garding terminology, the term \Annular Modes" was rst used in Thompson and Wallace [2000] (which was received