## 1. Introduction

In this study, we pursue the ongoing question of the zonal structure of the annular modes. The annular modes (AMs) are usually defined as the first EOF of the hemisphere-wide streamflow in the Northern Hemisphere (NH) or Southern Hemisphere (SH) extratropics (Thompson and Wallace 1998, 2000). The hemisphere-wide EOF analysis produces hemispheric-scale streamflow patterns with a dipolar meridional structure and relatively little zonal structure. On the other hand, teleconnection analysis or EOF analysis on zonally limited extratropical sectors produces regional-scale streamflow patterns, like the North Atlantic Oscillation (NAO), with the hemispheric AMs’ dipole meridional structure but with a localized zonal structure (Deser 2000; Ambaum et al. 2001; Cohen and Saito 2002; Cash et al. 2002, 2005; Vallis et al. 2004). Thus, the AMs’ meridional structure appears to be robust but their zonal structure depends on the analysis technique and so is not as robust. This leads to the question of whether the hemispheric-scale pattern is physically meaningful or a mere statistical artifact (e.g., Robinson 2004; Cash et al. 2004; Watanabe 2005; Cash et al 2007). A clear answer to this question will allow us to better understand the AMs’ role in climate variability and climate change (e.g., Thompson and Wallace 1998; Thompson and Solomon 2002).

Here we systematically examine how the EOFs of the extratropical tropospheric streamflow depend on the zonal width *δλ* of the analysis region (section 2), and find results that at least partially reconcile the hemispheric- and regional-scale views of the AMs (sections 3–5). The standard hemispheric (Thompson–Wallace) definition of the AMs corresponds to *δλ* = 360°, whereas regional sector analyses that give rise to the NAO and the sector EOF patterns of Cohen and Saito (2002) and Cash et al. (2002, 2005) correspond to sector widths *δλ* = 90°–120°. We show that regional AM signatures emerge for smaller analysis sector widths, and even for an analysis domain that includes only a single longitude. For narrow sectors the AM signatures are no longer the dominant pattern of variability, but the narrow sector analysis produces a more sharply resolved picture of the AM regional structure than the wider sector analysis. The regional signatures take the form of long baroclinic wave structures (section 3c) and so provide a novel view of how the AM might be generated. Although results in the Southern Hemisphere (section 3) are simpler than in the Northern Hemisphere (section 4), the regional signatures are similar in both hemispheres. The analysis opens several research questions that we discuss in the conclusions (section 5).

## 2. Method

We are looking for a robust regional signature of the AMs in the extratropical troposphere. Baldwin (2001) shows that the tropospheric signatures of the hemispheric AMs are robust: we see essentially the same spatial structures whether NH or SH data are used, whether zonally averaged or zonally unaveraged data are used, and whether winter season monthly data or year-round daily data are used. In the same vein, we seek regional tropospheric AM signatures that occur in both hemispheres, in all seasons, and, in addition, in all zonal sectors.

Following Baldwin (2001) we use the daily extratropical surface pressure field *p _{s}*(

*λ*,

*ϕ*,

*t*) from the 1979–2004 National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis and retain all days except for leap days. (When we instead use the 1993–2004 period we find no substantial change in the results we report.) Here,

*λ*denotes longitude,

*ϕ*denotes latitude, and

*t*denotes time. We multiply

*p*(

_{s}*λ*,

*ϕ*,

*t*) by

*ϕ*

*λ*and

*t*and then remove a climatology that has been smoothed with a 31-day running mean filter from the result. We denote the resulting surface pressure anomaly field by

*p̃*

_{s}(

*λ*,

*ϕ*,

*t*).

*p̃*

_{s}(

*λ*,

*ϕ*,

*t*) in extratropical sectors with zonal boundaries

*λ*±

_{c}*δλ*/2 and meridional boundaries (as in Baldwin 2001) 90° and 22.5° latitude. We perform separate calculations for the SH (see section 3) and the NH (see section 4). We denote the sector zonal mean of

*p̃*

_{s}(

*λ*,

*ϕ*,

*t*) by

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*,

*λ*

_{c}); the notation

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*,

*λ*

_{c}) distinguishes the local dependence on the coordinates

*ϕ*and

*t*from the nonlocal dependence on the sector parameters

*δλ*and

*λ*. Figure 1 illustrates two SH sectors, one with (

_{c}*δλ*= 180°,

*λ*= 90°W = 270°) and another with (

_{c}*δλ*= 20°,

*λ*= 150°). We calculate

_{c}p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*,

*λ*

_{c}) forIn our notation

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*= 360°,

*λ*

_{c}) =

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*= 360°) corresponds to the standard zonal mean of

*p̃*

_{s}(which is independent of

*λ*), and

_{c}p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*= 0°,

*λ*

_{c}) =

*p̃*

_{s}(

*λ*

_{c},

*ϕ*,

*t*) corresponds to taking

*p̃*

_{s}at the single longitude

*λ*. The single longitude

_{c}*δλ*= 0° notation is understood to imply that we are sampling the data at the smallest applicable zonal grid scale, which is 2.5° for the NCEP–NCAR reanalysis.

In both the NH and SH, we carry out a separate principal component analysis for each *δλ* and *λ _{c}*. From this we obtain a set of unit amplitude EOFs

*υ*(

_{i}*ϕ*;

*δλ*,

*λ*), associated unit amplitude principal component (PC) time series

_{c}*u*(

_{i}*t*;

*δλ*,

*λ*), and eigenvalues Λ

_{c}*(*

_{i}*δλ*,

*λ*). Here the index

_{c}*i*indicates the rank of the EOF. The calculations are carried out using MATLAB’s singular value decomposition routine on the

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*,

*λ*

_{c}). We stress that each EOF

*υ*(

_{i}*ϕ*;

*δλ*,

*λ*) is a unit amplitude quantity that depends explictly on latitude alone. These are distinct from the regression maps that we will describe below and that can depend on longitude and height in addition to latitude.

_{c}Figure 1 plots the SH first EOFs *υ*_{1}(*ϕ*; *δλ* = 180°, *λ _{c}* = 270°) and

*υ*

_{1}(

*ϕ*;

*δλ*= 20°,

*λ*= 150°) for the analysis sectors mentioned above (see caption for sign convention and other details). Notice that the first EOF for the narrower sector is monopolar and for the wider sector is dipolar; we will see in section 3a that this behavior carries over more generally.

_{c}We define the “*λ _{c}* mean” to be the average over center longitudes

*λ*; this represents a kind of ensemble average over all sectors. We denote the

_{c}*λ*mean of any function

_{c}*f*(

*λ*) by [

_{c}*f*]

_{λc}. Thus, for example, the

*λ*mean of the

_{c}*i*th EOFs

*υ*(

_{i}*ϕ*;

*δλ*,

*λ*) is [

_{c}*υ*

_{i}(

*ϕ*;

*δ*

*λ*)]

_{λc}and can be regarded as a function of

*ϕ*and

*δλ*(see, e.g., Figs. 3a–c).

We also calculate “*λ _{c}*-mean regression maps” that depict how various fields vary coherently with a given PC time series

*u*(

_{i}*t*;

*δλ*,

*λ*). For example, to calculate the

_{c}*λ*-mean regression map for

_{c}*p*with the first EOF PC time series

_{s}*u*

_{1}(

*t*;

*δλ*= 100°,

*λ*) (see Fig. 4a), we first calculate the regression (i.e., the temporal covariance) between

_{c}*p*(

_{s}*λ*,

*ϕ*,

*t*) and each of the 36

*u*

_{1}(

*t*;

*δλ*= 100°,

*λ*) [i.e., one for each of the 36

_{c}*λ*values in Eq. (1)]. The

_{c}*u*are unit amplitude, so these maps each represent the variation in surface pressure per unit standard deviation of the PC time series. These maps are then shifted in longitude

_{i}*λ*so that they share a common sector center

*λ*, which we choose for plotting purposes to be the date line

_{c}*λ*= 180°; finally, we take the

_{c}*λ*mean of these shifted maps to produce the plot. Because of the shifting to the date line, the center longitude of the plot is arbitrary; geographic boundaries are included for reference only.

_{c}## 3. Results: Southern Hemisphere

### a. Sensivity of EOF structure to *δ**λ*

Our key results concern how the meridional structure of the EOFs of the extratropical streamflow depends on the analysis sector width *δλ*. By construction we know that as *δ**λ* → 360° the EOFs *υ _{i}*(

*ϕ*;

*t*;

*δλ*,

*λ*) become independent of

_{c}*λ*because the sector zonal mean field

_{c}p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*,

*λ*

_{c}) becomes independent of

*λ*. We also know that for

_{c}*δλ*= 360° the first SH EOF is exactly the Baldwin (2001) Southern Annular Mode (SAM) definition based on the daily (and all year) zonal-mean surface pressure

p

_{s}(

*ϕ*,

*t*;

*δ*

*λ*, = 360°).

^{1}But, as we will now show, the meridional AM structure emerges at considerably narrower sector widths

*δλ*.

Figure 2 plots the first EOFs *υ*_{1}(*ϕ*; *δλ*, *λ _{c}*) for

*δλ*= 0°, 70°, 100°. As explained in the caption, the longitude dependence in the plot reflects the dependence of the meridional structure of the

*υ*

_{1}on the sector center

*λ*. For the single-longitude calculation with

_{c}*δλ*= 0° (Fig. 2a) the

*υ*

_{1}(

*ϕ*;

*δλ*= 0°,

*λ*) are monopolar, similar to the narrow sector EOF (solid curve) in Fig. 1. For the wide sector calculation with

_{c}*δλ*= 100° (Fig. 2c), the

*υ*

_{1}(

*ϕ*;

*δλ*= 100°,

*λ*) are dipolar in most sectors, similar to the wider sector EOF (dashed curve) in Fig. 1. For the intermediate-width sectors with

_{c}*δλ*= 70° (Fig. 2b), the

*υ*

_{1}(

*ϕ*;

*δλ*= 70°,

*λ*) are not robust from sector to sector: they are monopolar for some

_{c}*λ*and dipolar for other

_{c}*λ*.

_{c}We now show more systematically how the first EOFs depend on sector width *δλ*. Figure 3a plots as a function of *ϕ* and *δλ* the *λ _{c}*-mean first EOF [

*υ*

_{1}(

*ϕ*;

*δ*

*λ*)]

_{λc}. (See section 2 for the definition of the

*λ*mean.) As Figs. 1 –2 imply, for narrower sectors (0° <

_{c}*δλ*< 50°), the

*λ*-mean pattern [

_{c}*υ*

_{1}(

*ϕ*;

*δ*

*λ*)]

_{λc}is monopolar; for wider sectors (90° <

*δλ*< 360°), the

*λ*-mean pattern [

_{c}*υ*

_{1}(

*ϕ*;

*δ*

*λ*)]

_{λc}is dipolar. The range 50° <

*δλ*< 90° involves a transition between the two patterns in which the structures are not robust from one sector to another.

Our first key result follows from Fig. 3b, which plots [*υ*_{2}(*ϕ*; *δ**λ*)]_{λc}, that is, the *λ _{c}*-mean second EOF. Even though

*δλ*= 90° represents the lower bound at which the dipolar structure dominates the variability (Fig. 3a), the dipolar structure does not disappear for narrower sectors. Instead, the dipolar structure emerges as a secondary pattern of variability for

*δλ*< 50° in Fig. 3b. At the same time as the dipolar structure switches from the first EOF for

*δλ*> 90° to the second EOF for

*δλ*< 50°, the monopolar structure switches from the second EOF to the first EOF. By contrast, the

*λ*-mean third EOF [

_{c}*υ*

_{3}(

*ϕ*;

*δ*

*λ*)]

_{λc}, which we plot in Fig. 3c, remains stable and retains a tripole structure for all

*δλ*.

### b. The regional signature of the SAM

We now show that the dipolar second EOF pattern that emerges for small sector widths in Fig. 3b is the regional signature of the SAM: it remains a significant contributor to the variability, it is robust from sector to sector, it has a regional NAO-like structure, and it is directly related to the hemispheric SAM.

We first demonstrate that the dipolar second EOF pattern [*υ*_{2}(*ϕ*; *δ**λ*)]_{λc} is a significant contributor to the variability for *δλ* < 50°. Figure 3d plots the *λ _{c}*-mean variance fraction accounted for by EOFs 1–3 as a function of

*δλ*(i.e., [Λ

_{j}/Σ

_{i}Λ

_{i}]

_{λc}for

*j*= 1, 2, 3 in our notation). The variance fraction for the

*υ*

_{2}(

*ϕ*;

*δλ*= 0°,

*λ*) is about 28%, which represents a significant portion of the variability compared to the variance fraction for the first EOF

_{c}*υ*

_{1}(

*ϕ*;

*δλ*= 0°,

*λ*), which is approximately 46%. For the third EOF

_{c}*υ*

_{3}(

*ϕ*;

*δλ*= 0°,

*λ*), the variance fraction is about 10%.

_{c}Next, we show that the narrow sector second EOF dipolar patterns *υ*_{2}(*ϕ*; *δλ*, *λ _{c}*) are robust in the sense that they are similar in all the sectors. Figure 3e plots a relative measure of this robustness, namely the squared amplitude of the

*λ*-mean EOFs as a function of

_{c}*δλ*; call this quantity “

*R*.” Because the

*υ*are unit amplitude, when the EOFs depend weakly on

_{i}*λ*[as for

_{c}*υ*

_{1}(

*ϕ*;

*δλ*= 0°,

*λ*) in Fig. 2a],

_{c}*R*will be less than but close to unity; when the EOFs depend strongly on

*λ*[as for

_{c}*υ*

_{1}(

*ϕ*;

*δλ*= 70,

*λ*) in Fig. 2],

_{c}*R*will be positive but relatively small.

^{2}By construction, as we see in Fig. 3e,

*R*→ 1 for all EOFs as

*δλ*→ 360°. But Fig. 3e also shows that for the three leading EOFs, and in particular for the dipolar EOF,

*R*> 0.8 for

*δλ*< 50° and that

*R*is almost unity for

*δλ*= 0°. The lack of robustness seen in the relatively small

*R*values in the range 50° <

*δλ*< 90° is expected from Figs. 2 and 3a,b. For the third EOF

*υ*

_{3}(

*ϕ*;

*δλ*,

*λ*),

_{c}*R*remains near unity for all

*δλ*, which is not surprising given that this EOF appears to be independent of the sector width (Fig. 3c).

Figure 3 shows that a robust and significant meridional dipolar pattern remains as the second EOF as *δλ* is decreased below 90°. We now ask how the zonal structure associated with the dipole EOFs depends on *δλ*. To answer this question, we regress the surface pressure field *p _{s}*(

*λ*,

*ϕ*,

*t*) on the PC time series corresponding to the dipolar EOFs. Figure 4 plots the

*λ*-mean regression (section 2) of

_{c}*p*with

_{s}*u*

_{1}(

*t*;

*δλ*= 100°) (see Fig. 4a) and with

*u*

_{2}(

*t*;

*δλ*= 0°,

*λ*) (see Fig. 4b). For

_{c}*δλ*> 90°, the circulation pattern more or less fills the sector, as in Fig. 4a and as seen in the sector EOFs in the simulations of Cash et al. (2002). But as

*δλ*falls below 90°, a more localized circulation pattern appears, as in Fig. 4b. The zonal details have become better resolved, and the pattern more independent of the sector width.

The localized dipolar pattern in Fig. 4a, which we interpret as the regional circulation signature of the SAM, can be obtained independently of a particular EOF analysis by using an index based on the difference in surface pressure between two locations (e.g., Wallace 2000; Gong and Wang 1999; Gerber 2006). In this case, we define a local dipole index Δ*p _{s}*(

*λ*,

_{c}*t*) =

*p*(

_{s}*λ*,

_{c}*ϕ*= 40°S,

*t*) −

*p*(

_{s}*λ*,

_{c}*ϕ*= 65°S,

*t*), that represents the difference in surface pressure between 40° and 65°S at each longitude

*λ*. The latitudes 65° and 40°S correspond to the typical centers of action of the dipolar

_{c}*δλ*= 0° EOF and are the latitudes used by Gong and Wang (1999) in their definition of the SAM. The correlation between the time series and the corresponding PC time series

*u*

_{1}(

*t*,

*δλ*= 0°,

*λ*) is very high (greater than 0.95) everywhere except in the vicinity of South America, and the

_{c}*λ*mean of this correlation is 0.93. Thus the pressure difference index Δ

_{c}*p*(

_{s}*λ*,

_{c}*t*) and the regional dipole PC time series

*u*

_{1}(

*t*,

*δλ*= 0°,

*λ*) are essentially equivalent.

_{c}In Figs. 3a,b, we see that the meridional dipolar structure slowly shifts poleward as *δλ* is increased. Figure 4 helps explain this poleward shift: notice in Fig. 4b that the meridional location of the maxima at each longitude curve poleward away from the center longitude (the date line in the plot). In the sector zonal mean, as *δλ* is increased and wider zonal sectors are sampled, these higher-latitude contributions become more pronounced, resulting in the poleward shift of the dipole.

We have tried to make clear the connection in spatial structure between the narrow sector, wide sector, and hemispheric dipole patterns. We can also connect the temporal variability of these patterns. For example Fig. 5 plots the temporal correlation between the hemispheric SAM index *u*_{1}(*t*; *δλ* = 360°) and the *δλ* = 0° dipolar EOF time series *u*_{2}(*t*; *δλ* = 0°, *λ _{c}*), as a function of center longitude

*λ*. The correlation remains very significant (between 0.37 and 0.57, with a mean of 0.48) for all longitudes.

_{c}^{3}Thus, for example, positive SAM days are typically related to positive phases of the regional dipole pattern [

*υ*

_{2}(

*ϕ*;

*δ*

*λ*= 0°)]

_{λc}.

We can explain the approximately 0.5 correlation in Fig. 5 in terms of the working hypothesis (Wallace 2000; Cash et al. 2002, 2005) that the hemispheric AM represents a superposition of regional dipole events. As a simple model, suppose (see, e.g., Cohen and Saito 2002; Gerber and Vallis 2005) that we have a set of *N* independent and identically distributed regional dipole PC time series, each corresponding to a dipole pattern of zonal width Δ*λ* = 360°/*N*. The time series are independent and so are mutually uncorrelated. We generate a hemispheric AM index by taking the arithmetic mean of the regional dipole PC time series (this would correspond to a zonal mean). The correlation between the hemispheric AM and each of the regional dipole PC time series is 1/*N**λ*/360°*λ* ≈ 90°, which is in qualitative agreement with the zonal scale of the pattern in Fig. 4b.

The AMs (the dipole patterns for *δλ* = 360°) have strong signatures in the extratropical eddy fluxes of heat and momentum, as summarized by their signature in Eliassen–Palm fluxes (e.g., Limpasuvan and Hartmann 2000; Lorenz and Hartmann 2001). These signatures provide the foundation for theories of AM variability that involve feedbacks between the transient baroclinic eddy forcing and the zonal mean flow (e.g., Robinson 2000; Lorenz and Hartmann 2001; Rashid and Simmonds 2004). We find, somewhat unexpectedly, that these signatures in the eddy flux fields are also associated with the regional dipole events, and not just hemispheric AM events. Figure 6 demonstrates an example of this; to describe it requires some additional notation. Let 〈*A*〉 represent the regression of some field *A* with a particular PC time series, and let * B* and

*B** =

*B*−

*denote the zonal mean and zonally asymmetric (“eddy”) components of some field*B

*B*. In this notation, 〈

*u*

_{1}(

*t*;

*δλ*= 360°). This is the flux anomaly associated with a “1

*σ*” SAM event and is typically dominated by contributions from synoptic-scale transient eddies (Lorenz and Hartmann 2001). On positive SAM days, the poleward transport of heat by synoptic eddies is enhanced in the extratropical troposphere, and on negative SAM days it is attenuated.

Figure 6b plots the *λ _{c}* mean of 〈

*u*

_{2}(

*t*;

*δλ*= 0°,

*λ*) time series. The plot thus represents the signature in the meridional eddy heat flux of a 1

_{c}*σ*regional dipole event. The heat flux signatures in Figs. 6a,b are nearly identical, with the hemispheric SAM regression pattern having an amplitude of about half that of the regional pattern regression. This suggests that the dynamics that couple the synoptic eddy fluxes and the dipole circulation anomalies might occur on a regional scale. A more complete discussion of the eddy mean–flow interaction aspects is beyond the scope of this study. In section 3c, we will return to the question of why the regional dipole pattern’s heat flux signature is half that of the hemispheric AM.

To summarize, we have found that the hemispheric SAM has an NAO-like regional signature with a zonal scale of about 90° longitude. The regional dipolar patterns do not dominate the variability at these scales. Instead, the variability is dominated by the *δλ* = 0° monopolar EOFs, whose *λ _{c}*-mean circulation signature in

*p*is shown in Fig. 7. But the dipole patterns are robust and connected via their spatial structure, temporal correlation, and eddy flux signatures to the SAM pattern, which is the dominant pattern at larger zonal scales. So it appears to be worthwhile to study these “building blocks” of the AMs in more detail.

_{s}### c. Some dynamical features of the regional dipole patterns

We find it instructive to compare the dynamical features of the regional dipole patterns (Fig. 4b) to the regional monopole patterns (Fig. 7). Figures 8a,b plot the *λ _{c}*-mean lag correlation for the

*u*

_{1}(

*t*;

*δλ*= 0°,

*λ*) and

_{c}*u*

_{2}(

*t*;

*δλ*= 0°,

*λ*) time series, as a function of zonal displacement from

_{c}*λ*and time lag. To obtain Fig. 8a, for example, we choose a base center longitude

_{c}*λ*, lag correlate

_{c}*u*

_{1}(

*t*;

*δλ*= 0°,

*λ*) with all the

_{c}*u*

_{1}(

*t*;

*δλ*= 0°,

*λ*

_{c}_{′}), translate these in

*λ*to obtain a dependence on

*λ*−

_{c}*λ*

_{c}_{′}, and take the

*λ*mean.

_{c}^{4}

Figure 8a shows that the monopolar pattern propagates as a zonal wavenumber-3–4 wave packet with phase velocity of about 5° longitude day^{−1} and group velocity of about 30° longitude day^{−1}. The dipole pattern in Fig. 8b, on the other hand, propagates as a weakly dispersive coherent pulse with a phase speed of about 8°–10° day^{−1}. Figure 8 also shows that the monopole patterns are less persistent for a given *λ _{c}* −

*λ*: the decorrelation time scale for the monopolar pattern is about 2 days and for the dipolar pattern is about 4 days, and significant (>0.1) correlations in the dipole pattern extend past ±10 days. The overall behavior is qualitatively consistent with the expectation—from barotropic dynamics—that zonally elongated structures like the dipole pattern are of lower frequency and have a slower zonal group speed than more isotropic structures like the monopole pattern (e.g., Hoskins et al. 1983).

_{c′}The decorrelation times of the regional patterns is short compared to that of the hemispheric AMs: we find that the decorrelation time of the SAM index *u*_{1}(*t*; *δλ* = 360°) is about 8 days, which is about twice that of regional dipole index *u*_{2}(*t*; *δλ* = 0°, *λ _{c}*) and about 4 times that of the regional monopole index

*u*

_{1}(

*t*;

*δλ*= 0°,

*λ*).

_{c}^{5}The enhanced persistence of the SAM relative to the regional dipole pattern might explain the factor of 2 difference in amplitude between the transient eddy heat flux signatures in Figs. 6a,b: since a SAM event persists longer than an individual regional dipole event, the associated organization of the transient eddy field might be stronger when related to the SAM fluctuations. Consistently, the monopole patterns, which have an even shorter decorrelation time of about 2 days, have a very weak signature in the transient eddy heat flux (not shown) compared to Fig. 6.

One of the unexpected results of our analysis is that both the monopolar and dipolar circulation patterns have a significant baroclinic signature. To show this, we plot in Fig. 9 the *λ _{c}*-mean regression with the monopole and dipole PC time series of geopotential height at selected latitudes, as a function of pressure and longitude. For example, Fig. 9a plots the regression with the monopolar EOF PC time series at 52.5°S, which represents the meridional center of action of the

*δλ*= 0° EOF in Fig. 7. This plot shows the familiar westward phase tilt with height associated with baroclinic wave packets. Lagged regression plots (not shown) show this entire structure moving as a wave group like classical baroclinic wave packets (e.g., Chang 1993), as suggested by Fig. 8a, although these wave structures are larger scale than typical synoptic wave packets.

Figure 9b plots the regression with the dipolar EOF PC time series at 65°S, which represents the poleward center of the dipolar *δλ* = 0° EOF in Fig. 4b. A section through the equatorward center at 40°S (not shown) has a similar structure with opposite sign. As was the case for the monopolar pattern, the dipolar pattern has a baroclinic structure, exhibiting a westward tilt with height. Lag regression plots (not shown) also show that this structure moves eastward in the way suggested by Fig. 8b.

The baroclinic structure seen in Fig. 9b leads to our second key result; namely that the regional signature of the AM is a propagating baroclinic wave. To quantify the baroclinic character of the monopolar and dipolar patterns, in Fig. 10 we calculate the meridional heat flux associated with the circulation patterns in Fig. 9. This figure plots the *λ _{c}* mean of

*υ**〉〈

*T**〉

^{−1}and for the dipolar pattern is 0.4 mK s

^{−1}; this is comparable to the zonal-mean heat flux signature for North Pacific synoptic baroclinic wave packets (see Fig. 8 of Chang 1993).

We emphasize the distinct physical interpretations that apply to the plot of 〈

We also point out that Fig. 10 depends strongly on the analysis sector width *δλ*: since the zonal structure of the regression patterns becomes smoother as *δλ* increases, the heat flux signature in Fig. 10 will decrease at the same time. By construction, as *δλ* → 360°, this heat flux signature will disappear. Thus, the hemispheric EOF analysis masks the baroclinic character of the regional dynamics.

## 4. Results: Northern Hemisphere

Although the NH circulation is more zonally asymmetric and has a stronger seasonal cycle than the Southern Hemisphere, Baldwin (2001) has shown that the tropospheric signature of the NAM and SAM, based on daily surface pressure for the entire year, are very similar. We therefore have applied the SH analysis of section 3 to the NH with the idea that the regional dipole patterns might be similar in both hemispheres. The results are summarized as follows:

- The
*υ*_{1}(*ϕ*;*δλ*,*λ*) converges to the Baldwin (2001) NAM [_{c}*υ*_{1}(*ϕ*;*δλ*= 360°)] for sector widths*δλ*> 180° (Fig. 11a). This is relatively large compared to the analogous convergence to the SAM in the SH, which occurs for*δλ*> 90°. - The switch in the SH case between the monopolar and dipolar first EOF patterns (Figs. 3a,b) does not occur in the NH case (Figs. 11a,b). For example, the two leading EOFs in the NH as
*δλ*> 180° are both dipolar. - Nevertheless, for
*δλ*< 30°, a monopolar first EOF and dipolar second EOF emerge with a similar structure to the SH EOFs. These NH regional EOFs are somewhat less robust (more*λ*dependent) than the SH regional EOFs, but explain a comparable portion of the variability (Figs. 11a,b,d,e). Thus, the dipolar NH second EOF, [_{c}*υ*_{2}(*ϕ*;*δ**λ*= 0°)]_{λc}, remains a significant pattern, which we interpret as the regional signature of the NAM. - The NAO comes out of this analysis as a dominant pattern for the North Atlantic sector, as long as we use a sufficiently wide analysis sector. Figure 12 shows the leading EOF
*υ*_{1}(*ϕ*;*δλ*,*λ*) for_{c}*δλ*= 0°, 30°, 70°, 100°. The first EOF in the limit*δλ*→ 0° is monopolar for all*λ*, and the second pattern in this limit is dipolar for all_{c}*λ*(not shown), as in the SH. As we expect from the smaller value of the robustness_{c}*R*in Fig. l1e, the details of its structure are more sector dependent than for the SH case in Fig. 2a. As*δλ*increases to 30°, 70°, and 100° dipole structures emerge as the dominant patterns first in the North Atlantic sector and then in the North Pacific sector. Figure 12d shows that the dominant EOF remains monopolar over the continents, even for*δλ*= 100°; this is reminiscent of the regression map of the NAM, as seen in Thompson and Wallace (2000). - The zonal structure and evolution of the regional signature of the NAM is similar to that of the regional signature of the SAM (e.g., Fig. 13) and exhibits a comparable baroclinic structure (not shown). The correlation between the hemispheric NAM and the regional pattern is somewhat weaker than for the SAM but still quite significant at all longitudes (not shown).

The relatively strong zonal asymmetry and seasonal cycle of the NH extratropical circulation appears to complicate the dependence on *δλ* of the EOF structure to a certain extent. But the key results of the SH case carry over to the NH case: the NAM, too, has a regional signature with a baroclinic structure.

## 5. Conclusions

We repeat our main findings.

- The regional signature of the annular modes emerges as the second EOF of the extratropical surface pressure
*p*at each longitude—[_{s}*υ*_{2}(*ϕ*;*δ**λ*= 0°)]_{λc}in our notation, with time series index*u*_{2}(*t*;*δλ*= 0°,*λ*) at each longitude_{c}*λ*. Although they are not the dominant pattern, these patterns still account for a significant fraction of the variance of the extratropical circulation at each longitude. They are robust in the sense that they occur in both hemispheres, that their meridional structure is similar at all longitudes and that they are found in an analysis that includes all days of the year. They are related through spatial structure, temporal correlations, similarities in frequency distribution, and eddy-forcing quantities to the hemispheric annular modes—_{c}*υ*_{1}(*ϕ*;*δλ*= 360°) in our notation, with time series*u*_{1}(*t*;*δλ*= 360°). - The regional AM signatures consist of eastward-propagating baroclinic waves of zonal scale of approximately 90° that transport heat poleward in the same manner as classical synoptic-scale baroclinic wave packets. As far as we know, this result has not been previously anticipated: typically, extratropical teleconnection patterns such as the AMs and the NAO are pictured as equivalent barotropic.

Although we have devoted some effort toward examining the detailed dependence of the meridional structure of the EOFs on *δλ*, the zonal-mean limit *δλ* → 360° and the single-longitude limit *δλ* → 0° lead to the most robust and, in our view, the most dynamically interesting results. Nevertheless, the question of why the dipole is the dominant pattern for large *δλ* but not for small *δλ*, and of what determines the scale of this transition, are still interesting. We might explain this behavior as a consequence of the tendency for wider sectors to smooth the motion and so bring out lower frequencies. Since lower-frequency eddies tend to be more zonally elongated than higher-frequency eddies (Hoskins et al. 1983), wider sector averages might favor the more persistent zonally elongated patterns. Alternatively, this behavior might relate to the view of Gerber and Vallis (2005) that the dipole hemispheric SAM structures reflect conservation of angular momentum. From this point of view, the *δλ* = 90° scale reflects the lower bound for which this angular constraint is important in the SH. In the NH, with its more zonally asymmetric circulation, the dipole NAM structure does not emerge as the dominant mode until *δλ* > 180°; this may be the relevant zonal scale for the angular momentum constraint in the NH.

One implication of point 2 above might be that the regional dipole patterns, like other baroclinic waves, are self-maintained structures that owe their existence to the background baroclinicity. We might expect that the dipolar baroclinic EOFs correspond to a secondary mode of baroclinic instability with a dipolar meridional structure (seen, e.g., in Simmons and Hoskins 1976; Hartmann 1979; Tanaka and Tokinaga 2002). Hartmann (1979) in particular has shown that the Southern Hemisphere circulation is baroclinically unstable to slowly traveling long waves with a dipolar meridional structure. A baroclinic instability–based theory might provide an explanation for the scale and propagation speed of the regional dipole patterns. Because the growth rates of baroclinic instability typically decrease with decreasing meridional scale, this theory might also account for the relatively weak contribution to the variance of the dipolar baroclinic patterns compared to the monopolar baroclinic patterns. A baroclinic instability–based theory would not explain all the signatures of the AM patterns, for example, the heat flux signature seen in Fig. 6. So it is clear that this theory would complement theories based on regional and zonal eddy mean–flow interactions (e.g., Franzke et al. 2004; Lorenz and Hartmann 2001; Vallis et al. 2004).

We have proposed in section 3b a simple description along the lines of the Gerber and Vallis (2005) model of the AMs in terms of a superposition of independent regional events. To augment this description, we need to build in the propagation characteristics of the regional dipole patterns seen in Fig. 8b, and the possible influence of the zonal mean flow via baroclinic generation of the patterns. This augmented description might explain the fact that significant correlations exist at zero time lag all the way around a latitude circle (Fig. 8b). It might also explain why the annular modes are roughly twice as persistent as the regional dipole patterns (section 3c).

To conclude, we point out that the regional dipole patterns might provide insight into the role of the AMs in stratosphere–troposphere dynamical coupling (e.g., Baldwin and Dunkerton 1999, 2001; Thompson and Wallace 2000; Thompson and Solomon 2002). This coupling is strongly seasonal and so is not handled well within our analysis, which includes all seasons. But Hartmann’s (1979) results and the fact that the regional dipole patterns that form the “building blocks” of the annular modes are long baroclinic waves suggest that an investigation of the stratospheric signatures of the regional patterns is well warranted.

We thank the Natural Sciences and Engineering Research Council of Canada and the Canadian Foundation for Climate and Atmospheric Sciences for support. Much of this work was completed at the Aspen Center for Physics 2005 Workshop on Novel Approaches to Climate. We also thank Maarten Ambaum (for correcting a mistake in the analysis related to Fig. 5), Edwin Gerber (for the idea of the using the Δ*p _{s}* index), as well as Adam Monahan, John Fyfe, and the manuscript’s reviewers for helpful comments.

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^{1}

We have verified that the first EOF *υ*_{1}(*ϕ*; *δλ* = 360°) qualitatively reproduces the results of Baldwin for his analysis period (not shown). The values for variance explained by each EOF are somewhat larger in our analysis.

^{2}

Here *R* ≥ 0 if we choose the polarity of the EOFs so that the dot product of any pair of EOFs is positive semidefinite. We can calculate *R* in the following simple example: suppose the EOFs are in pattern *a* for half of the sectors and in pattern *b* for the other half, and that *a* and *b* are orthogonal with *ab* = 0, where the dot product is the sum over latitude points of the products of the vector components. In this case *R* = [(*a* + *b*)(*a* + *b*)]/4 = 1/2.

^{3}

For 26 yr of daily data, there are over 900 separate 10-day periods. Taking 900 as an estimate for the number of degrees of freedom, any correlation above 0.1 would be highly significant.

^{4}

Again (see footnote 3), it may be safely assumed that the 0.1 correlation level is highly significant since the plot involves thirty-six 26-yr daily time series. Even though each time series is not independent, we estimate that there are at least 1800 degrees of freedom, assuming independence between 10-day periods and points separated by a zonal distance of 180°.

^{5}

This hemispheric AM time scale is somewhat short compared to that of Lorenz and Hartmann (2001), who find a decorrelation time of about 2 weeks, using a tropospheric zonal wind–based definition of the AM. The differences between the zonal wind– and the streamflow-based definitions of the annular mode are not trivial and might account for the discrepancy in time scale (A. Monahan 2006, personal communication).