## 1. Introduction

Extratropical tropospheric large-scale flow dynamics is one of the topics that has attracted the attention of a number of atmospheric scientists since the early 1940s (Rossby 1940) and there is a plethora of research works on the subject in the literature. Two main lines of thought have emerged for which the debate is still ongoing: first, the extratropical large-scale atmosphere exhibits discrete preferred states sporadically attracting the system trajectory in its phase space; second, the system can simply be approximated by a stochastic linear system in which nonlinearity can be parameterized by a state-dependent or multiplicative noise.

The concept of multiple equilibria (Rossby 1940; Charney and DeVore 1979; Charney et al. 1981) has been explored by various researchers in various directions. For example, some have investigated multimodality in large-scale flows (e.g., Hansen and Sutera 1995; Christiansen 2005; Corti et al. 1999; Monahan et al. 2001; Hsu and Zwiers 2001) while others have looked at clustering within the state space of the large-scale flow (Molteni et al. 1990; Molteni and Tibaldi 1990; Mo and Ghil 1988; Cheng and Wallace 1993; Michelangeli et al. 1995; Hannachi 2007, hereafter H07; Straus et al. 2007). Yet other researchers have investigated the subject using concepts of quasi-stationarity within the system state space (Legras and Ghil 1985; Vautard and Legras 1988; Marshal and Molteni 1993; Haines and Hannachi 1995; Hannachi 1997; Branstator and Berner 2005). Several authors, however, have questioned the validity of the nonlinear paradigm. For example, some have suggested that low-frequency–large-scale variability is basically linear (Newman et al. 1997, 2003) and that nonlinearity is primarily decorrelating and hence can be approximated by a multiplicative noise, which can explain the skewness observed in large-scale flow (Sura et al. 2005; Sardeshmukh and Sura 2009) while others have suggested that given the relatively small sample size of observations, particularly for monthly data, the system probability density function (PDF) can be approximated by a multinormal (Toth 1991; Nitsche et al. 1994; Wallace et al. 1991; Stephenson et al. 2004). Even for monthly or seasonal averages, as shown by Branstator and Selten (2009), prominent patterns of winter Northern Hemispheric climate variability can depart from Gaussianity. Branstator and Selten (2009) examined a large ensemble of coupled general circulation model simulations for present-day and next-century climate change scenarios. They found that for both periods the PDF of the leading pair of modes of variability of winter means 300-mb meridional wind is markedly non-Gaussian. They concluded that although the linear paradigm can explain much of the behavior observed in the experiment, there are aspects that the paradigm cannot contain.

A sharp decision regarding the acceptance or rejection of one paradigm versus the other may not be very easy to obtain, but both paradigms could be made useful. For example, on the one hand, linear models (Farrell and Ioannou 1995; Whitaker and Sardeshmukh 1998) are easy to use and can be made competitive with comprehensive prediction models, in addition to their attractive features.^{1} On the other, because of nonlinearity linear models cannot have accurate long lead time forecasts, and hence the nonlinear paradigm can be used to extend the limits of the system predictability (see, e.g., Koo et al. 2002).

Large-scale preferred flow structures that persist much longer than typical midlatitude synoptic weather systems, but less than the typical time scale of processes caused by changes in radiative fluxes and surface boundary conditions, have been identified over the North Pacific (NP) and North Atlantic (NA) sectors (e.g., Dole and Gordon 1983). The North Atlantic and North Pacific are the main storm track regions with active synoptic weather systems. The midlatitude North Atlantic–Europe and North Pacific areas are also the main regions of strong departure from normality (Hannachi et al. 2009). This nonnormality is associated with (i) local high skewness over the Aleutians and Greenland, observed in the low frequency of 500-hPa geopotential height (Z500) and sea level pressure (SLP), respectively (Nakamura and Wallace 1991; Rennert and Wallace 2009), and (ii) local large negative excess kurtosis, or platykurtosis (sub-Gaussianity) over the northern parts of both basins (Sardeshmukh and Sura 2009). These regions, in particular, are locations of frequent blocking events (Woollings et al. 2008; Woollings and Hoskins 2008; Pelly and Hoskins 2003). Nonlinearity over these regions at least seems perhaps a reasonable assumption. This view is shared by many climate scientists (e.g., Dole and Gordon 1983; Branstator and Selten 2009).

Recently Woollings et al. (2010) have put forth the suggestion of a regime paradigm of the North Atlantic Oscillation (NAO). Specifically, they suggest that the NAO can be interpreted as a measure of the transition between two flow regimes; a high-latitude (or Greenland) blocking regime and a zonal (or no blocking) regime. A similar theory may exist for the North Pacific sector. The distinction between “blocked” (or low index) and “zonal” (or high index) flow regimes over the northern oceans has often been made by synopticians since the early fifties (Rex 1950; Berggren et al. 1949). It is still not clear whether this interpretation could be extended to planetary scales. As pointed out by Cheng and Wallace (1993), the wide variety of anticyclonic blocking shapes, intensities, and locations makes extremely difficult the extension of blocking to categorize hemispheric circulation flows.

Although the hemispheric regimes paradigm remains one possibility among others (e.g., linear dynamics), it seems quite reasonable that the concept of sectorial flow regimes lends itself more easily and is more defensible. Cheng and Wallace (1993), for example, conducted a hierarchical cluster analysis of low-pass Z500-hPa heights over the Northern Hemisphere (NH) and the Pacific–North American (PNA; 120°E–60°W) and North Atlantic–European (120°W–60°E) sectors. They found that the reproducibility of sectorial clusters is higher than that of hemispheric clusters.

In H07, analysis of tropospheric planetary wave dynamics was conducted using the multivariate Gaussian mixture model (Haines and Hannachi 1995; Hannachi 1997; Smyth et al. 1999; Hannachi and O’Neill 2001; Berner and Branstator 2007; Stephenson et al. 2004). In this model the PDF of the atmosphere, within its state space, is expressed as a weighted sum of individual multivariate Gaussian PDFs with different means interpreted as the circulation regime centers. The number of components in the mixture model was derived based on arguments from order statistics. This procedure was shown to overcome the major drawback of cross validation, namely the systematic increase of the number of Gaussian components with increasing sample size (Hannachi and O’Neill 2001; H07; Christiansen 2007). Two Z500 hemispheric regimes are identified, namely a positive PNA pattern over the North Pacific sector associated with a positive NAO over the North Atlantic–European sector [(+PNA)–(+NAO)] and a negative phase of the previous pattern [(−PNA)–(−NAO)].

In the light of the above findings, the main objective in this manuscript is to investigate sectorial regimes and critically look at hemispheric regimes by seeking an answer to the following question: What is the relationship between sectorial and hemispheric regimes? The strategy used here is based on the mixture model in which the number of components is identified using two procedures. The first one is similar to that of H07 and Woollings et al. (2010) and is based on order statistics, and the other is based on the classifiability of the flow patterns. The manuscript is organized as follows: Section 2 describes the data and the methodology; section 3 investigates, with a sense of revision–extension, the hemispheric circulation regimes; and section 4 investigates the North Pacific and the North Atlantic sectorial regimes. Section 5 looks into more details for the relationship between sectorial and hemispheric regimes. A summary and discussion are given in the last section.

## 2. Data and methodology

### a. Data

The data used here consist of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) geopotential height field at a 500-hPa pressure level (Z500). The data have a horizontal resolution of 2.5° × 2.5° and span the period 1 January 1949 through 31 December 2007, yielding 5324 days. The seasonal cycle is computed using the smoothed daily geopotential heights using a 9-day moving average filter. At each grid point height anomalies are then computed as the departure of unfiltered daily values from the seasonal cycle. Winter daily anomalies, defined for the months December–February (DJF) north of 20°N, are then obtained and are used next for the analysis of planetary wave behavior. For sectorial analysis two regions are considered. These are the North Pacific region (20°–90°N, 120°–280°E) and the North Atlantic–European region (20°–90°N, 100°W–50°E). To emphasize the low-frequency part, the data have been filtered by using a nonoverlapping 5-day means yielding 1064 samples. The chosen 5-day window is a balance between the desire to obtain a relatively large sample size and the filtering of high-frequency variation. The data thus obtained are labeled filtered Z500. However, to compare the results for consistency we have also considered the unfiltered daily anomalies, but the results shown will refer, unless otherwise stated, to the filtered Z500 data. The obtained winter gridded data are then weighted by the square root of the cosine of the corresponding latitude and empirical orthogonal functions (EOFs) and associated principal components (PCs) obtained for each of the three chosen regions.

### b. Methodology

*α*

_{1}, … ,

*α*are the

_{c}*c*mixing proportions of the mixture model satisfying

*and*

**μ**_{k}**Σ**

*are, respectively, the mean and the covariance matrix of the*

_{k}*k*th,

*k*= 1, …

*c*, multivariate normal density function

*g*:

_{k}*m*is the state space dimension. The mixture model, like any other model, has its own limitations. For example, a fat tail distribution requires an infinite number of Gaussians. This is obviated, however, by the natural availability of a finite observed sample. Furthermore, we can argue that the mixture model is corroborated by the observed skewness and negative kurtosis, both of which are pertinent to the bulk of the distribution reflected, for example by a shoulder-like or inflection point in the PDF, as shown later in section 4 (see also Woollings et al. 2010).

The procedure of parameter estimation is achieved in two steps, namely the estimation of the number of components and then the estimation of the remaining parameters. For a given *c*, the model parameters in (1) are estimated from the data **x**_{1}, … , **x*** _{n}* by maximizing the log-likelihood

*c*(

*m*+1)(

*m*+2) − 2]/2 unknown parameters

*,*

**μ**_{k}**Σ**

*,*

_{k}*k*= 1, … ,

*c*, and

*α*,

_{k}*k*= 1, … ,

*c*− 1. The likelihood maximization is achieved through the very efficient expectation maximization (EM) algorithm [see, e.g., Hannachi and O’Neill (2001) for more details and further references]. The number of significant components of the model, a much harder problem, is estimated using two different approaches. The first one, detailed in H07, uses arguments from order statistics. The model starts by fitting a two-component model to the data and then repeatedly adds an extra component until the latest component does not pass the significance test. The method was shown by H07 to overcome several problems such as those encountered when using cross validation (Hannachi and O’Neill 2001; Christiansen 2007). The second approach is based on classifiability and reproducibility. Reproducibility was introduced by Cheng and Wallace (1993) as a way to test the robustness of the obtained clusters. Clusters based on the full dataset are compared, using spatial pattern correlation, to clusters obtained from many random halves of the data, and the maximum correlations between clusters are then evaluated as detailed below. The reproducibility of each cluster (from the entire dataset) is measured by the statistics of the frequency distribution of the obtained correlations. Michelangeli et al. (1995) considered also the same pattern correlation in their cluster analysis to quantify the similarity between any two partitions. They defined a classifiability index by measuring the reproducibility of partitions, from random halves of the data, with respect to a background noise. In this manuscript, a similar but not identical classifiability index is defined and is detailed next. We note here that the reproducibility index can produce a finite number of clusters from unclustered data (Christiansen 2007). A further test is applied later in section 3 to check that the data are indeed clustered.

*k*,

*k*= 2, … ,

*K*of Gaussian components in the mixture model, the best solution (i.e., the best

*k*regime centers

*μ*_{1}, … ,

*) is estimated from the entire dataset using the EM algorithm. Here*

**μ**_{k}*K*represents the maximum number of possible circulation regimes, which we set, unless otherwise stated, to

*K*= 8. Next, the original data are randomly divided into 100 halves. For each of these halves the best solution

*μ*_{1}

^{h}, … ,

*,*

**μ**_{k}^{h}*h*= 1, … , 100, is obtained from the

*k*-component mixture model. The similarity between these solutions is measured using the pattern correlation matrix 𝗖 = (

*c*), where

_{ij}*c*is the pattern correlation between

_{ij}*and,*

**μ**_{i}^{h}*, or similarly the*

**μ**_{j}*j*th column of 𝗖 gives the pattern correlations between

*and*

**μ**_{j}

*μ*_{1}

*, … ,*

^{h}*. Note that 𝗖 depends on*

**μ**_{k}^{h}*h*and

*k*. The maximum of each column gives a measure of the reproducibility of the corresponding regime or center. To get an overall measure for all the regimes (i.e., for the partition), a classifiability array is defined by the following metric:

*h*= 1, … , 100, and

*k*= 2, … ,

*K*. If the partition is perfect then

*C*(

*h*,

*k*) = 1. A classifiability index is then defined as the mean of (4) over the realizations.

*μ*_{1}

^{h,0}, … ,

*μ*_{k}

^{h,0}, obtained from the

*h*th (

*h*= 1, … , 100) Markov dataset, and

*μ*_{1}, … ,

*obtained previously using the entire Z500 data. That is,*

**μ**_{k}

*μ*_{i}

^{h,0}and

*. The noise level classifiability array is then computed as*

**μ**_{j}

*μ*_{i}

^{h,0}with

*rather than with*

**μ**_{j}

**μ**_{j}^{0}makes the test of accepting clusters tougher. The appropriate number of components is then obtained by comparing the classifiability index obtained from (4) to the noise level classifiability index obtained from (5). We note here that the (same) classifiability index used by Michelangeli et al. (1995), Kageyama et al. (1999), Solman and Menendez (2003), and Solman and Le Treut (2006) shows always a noise level starting at one for two clusters in a variety of situations [e.g., Northern Hemisphere and Southern Hemisphere (SH)], which means that their procedure cannot select the two-cluster solution. Another test, based on clustering, is also conducted to check the consistency of the obtained components. The test is used after the model is fitted and is therefore described after the application in the next section.

## 3. Hemispheric analysis

The EOFs and associated PCs of the Northern Hemisphere NH Z500 are computed. These EOFs (not shown) are very similar to those reported in H07. The percentage of explained variance of the leading five EOFs are shown in Table 1. EOF1, explaining 11.5% of the total winter variance, is similar to the PNA pattern except that the North American center is shifted over southern Greenland and has another center sitting over Europe. EOF2 (9.6%) has a PNA/NAO-like structure. The North Pacific center is sitting over the southwestern tip of Alaska. The skewness *S* and excess kurtosis^{2} *K* of the leading five PCs are also shown in Table 1. These values can be compared to the standard deviations of skewness and kurtosis—that is, *n**n**n* is the number of independent sample in the data, which can be approximated by the filtered data sample size. These standard errors are approximately 0.07 and 0.14, respectively. Most of the leading PCs are significantly skewed at the 5% level. The high skewness in PC2 suggests that the EOF2 pattern will play a significant role in the regime structure exactly as in H07. PC3 and PC5 have significant kurtosis at the 5% and 1% levels, respectively. The mixture model is next applied using different EOF state space dimensions. Figure 1 shows an example of the mixing proportions for two-component (Fig. 1a) and three-component (Fig. 1b) mixture models when the leading three EOFs are used, along with the approximate 99% confidence limits. When the leading two EOFs are used, no highly significant mixture is obtained. In fact, the two-component mixture model is only significant at the 10% level. Various reasons for this behavior can be postulated. For example, the leading two PCs have no significant kurtosis but the third one has. Perhaps a more relevant factor is that EOFs constitute a suboptimal base for cluster detection. At four and five dimensions the components are again significant at the 1% level as in Fig. 1.

The assessment of the possible number of regimes was also performed as detailed in section 2 using classifiability. Figure 2 shows the classifiability index [Eq. (4), circles] along with the upper 5%–95% bounds of the classifiability distribution obtained from the spatial first-order multivariate Markov process (AR-1) noise model of the 5-day mean (filtered) Z500 data when three EOFs are used. When the leading two EOFs are used instead no significant components are obtained. When the leading four or five EOFs are used, the two-component solution becomes again significant at 5%. The classifiability index tends to be a little more conservative than the previous test. For example, no significance is obtained at the 1% level. Figures 1 and 2 both suggest that the data cannot be described as Gaussian but that the non-Gaussianity can be described by a mixture of two Gaussian components.

The regimes given by the centers *μ*_{1} and *μ*_{2} of the individual multivariate Gaussian components of the mixture model within the 3D EOF state space are shown in Fig. 3. The solutions have strong projections onto ±PNA in the North Pacific sector and ±NAO in the North Atlantic sector. These circulation flows are very similar to those reported in H07.^{3} They are quite robust to changes in state space dimensions. Even when the unfiltered daily data are used, with a two-component model the regime structures do not change and they look much like those shown in Fig. 3. We note, however, that when the classifiability index is used with the unfiltered daily Z500-mb data, the result was not as conclusive as that obtained in Fig. 2. This example points to the importance of removing high-frequency variation when significance is required.

Within the EOF state space the location of the centers of the Gaussian components tends to be controlled by the densely populated areas. The centers of these components are not in general very far apart because sharp bimodality is unlikely in this setup. It is expected, however, that along the axis joining these regime centers nonnormality will be more pronounced.^{4} The skewness *S* and excess kurtosis *K* of Z500 projected onto the regimes axis *μ*_{2}–*μ*_{1} are shown in Table 2 for three, four, and five state space dimensions. The values have now increased, and the largest skewness and kurtosis are about −0.34 and −0.38 obtained with four and five dimensions, respectively. Based on the above standard deviations of skewness and kurtosis, using the available sample size, skewness values seem significant at the 1% level, and kurtosis values seem significant at 5% and 1% in 4D and 5D state spaces, respectively. However, since skewness and kurtosis are computed along a particular direction of the largest deviation from Gaussianity there is the possibility of these significances being overestimated. To avoid this pitfall, 100 AR-1 samples of surrogate data, as described in section 2b, are generated and a two-Gaussian component model is then fitted to each sample. Skewness and kurtosis are then computed, for each sample, along the corresponding “regime axis.” Figures 4a and 4b show the histograms of the obtained skewness and kurtosis along with the observed values shown by the arrows. The observed kurtosis can be generated in this way, but the skewness far exceeds the simulated ones.

*d*between observations, obtained by transforming the 2D state space in which observations are denoted by (

*x*,

_{t}*y*),

_{t}*t*= 1, … ,

*n*, into the unit square 0 ≤

*u*,

*υ*≤ 1, known as the probability space, where

*F*(·) and

_{x}*F*(·) are the marginal cumulative distribution functions of

_{y}*x*and

_{t}*y*,

_{t}*t*= 1, … ,

*n*, respectively. The clustering index is then defined by

*K*(

*d*) =

*N*(

*d*)/

*N*

*K*function,

*N*(·) is a function giving the mean number of points that are within a distance

*d*of a target point, and

*N*

*K*(

*d*) =

*πd*

^{2}(the area of circle of radius

*d*) and consequently

*L*(

*d*) = 1. Deviation from

*L*= 1 can rigorously indicate the presence of clusters. This is a strong test and one should mention from the outset that clustering strongly implies the existence of nonlinear circulation regimes. No clustering, however, does not necessarily imply nonexistence of circulation regimes. To increase the likelihood of clustering, we project the large-scale Z500 data onto the plane spanned by the regime’s axis (i.e.,

*μ*_{2}–

*μ*_{1}) and an orthogonal to it. Figures 4c and 4d show a scatterplot of the projected data in probability space (Fig. 4c) and the clustering index

*L*(

*d*) in bold face (Fig. 4d). The shading represents the domain between the upper and lower envelopes of the clustering indexes obtained using 100 samples, with the same sample size as the 5-day mean Z500 data, simulated from a homogeneous random Poisson point process known as complete spatial randomness (CSR; Martinez and Martinez 2002). Clustering is obtained with the 5D solutions, but Fig. 4 refers to the 10D solutions and is chosen because it is more pronounced than the 5D case (not shown). There is a clear indication of clustering, particularly in the band

*d*= 0.1–0.2. Once again, to obviate the overestimation of significance we repeated the clustering procedure by projecting the surrogate data onto their clustering planes.

^{5}The obtained curves from 100 samples are shown in dotted lines in Fig. 4d. Clearly, the clustering curve of reanalyses stands well above the curves from the surrogate data and from the homogeneous Poisson process. We note here that when the above clustering plane of the reanalyses was constructed, the direction orthogonal to the regime’s axis was not specifically selected. This means that there may be other directions that could yield stronger clustering. We also note that when the clustering was performed using pairs of EOFs, no significant clustering was obtained.

In summary, these results do suggest consistency of the obtained regimes. In section 5 below a new interpretation of these hemispheric regimes is presented.

## 4. Sectorial circulation regimes

### a. North Pacific sector

The setup in this section is exactly similar to that of the NH presented in the previous section except that the region is now restricted to the Pacific–North American sector defined by 20°–90°N, 120°–280°E. The leading three EOFs are shown in Fig. 5. The leading EOF, explaining around 20% of the total winter variability, is a PNA. The second EOF (17%) is dominated by a dipole sitting over central North Pacific and Alaska, and the third EOF (15%) is dominated by a zonal dipole that could represent the polar jet modulation. Skewness and excess kurtosis of the leading five PCs are shown in Table 1. The leading PC has a particularly high skewness and excess kurtosis, and the third PC has high skewness. The leading PC1 and PC3 are highly skewed, and the kurtosis of PC1 is significant at the 5% level. The negative excess kurtosis is known as platykurtosis^{6} and reflects the flatness of the PDF relative to the Gaussian. Thus, the observed negative excess kurtosis means that the PDF is flatter than that of the Gaussian, and this helps understand and support the argument behind expressing the PDF as a weighted sum of Gaussians.

Within the two leading EOF space the data support a mixture of two bivariate Gaussian distributions. The test based on the mixing proportions gives a 1% significance level, but the test based on the classifiability index provides only a 10% significance level. When the leading three to five EOFs are used, both tests support two components at a 1% significance level. This is congruent with the high skewness and kurtosis along EOFs 1 and 3. The results of these tests obtained with the leading three EOFs are shown in Fig. 6. Figures 6a and 6b show the mixture proportions when two (Fig. 6a) and three (Fig. 6b) components in the mixture model are used along with the 99% confidence limits, and Fig. 6c shows the classifiability index along with the 1%–99% noise levels. Figure 6 strongly suggests the presence of two regimes. The obtained circulation regimes are shown in Fig. 7. The North Pacific regimes represent ±PNA. Woollings et al. (2008) also obtained two North Pacific winter circulation regimes corresponding to the high-latitude blocking and no-blocking flows, which were found to be associated more with the negative and positive phases of the west Pacific (WP) pattern, respectively. A few points are worth mentioning here regarding the difference between the Pacific regimes presented here and those of Woollings et al. (2008). The present study uses the midtropospheric geopotential height field based on NCEP–NCAR reanalyses whereas Woollings et al. (2008) used the potential temperature field on the PV = 2 PVU isopleth surface,^{7} which corresponds approximately to the tropopause level, based on the European reanalyses. The midlatitude tropopause level is known to be a location for baroclinic eddy wave breaking activity, which contributes to forcing the zonal mean flow via eddy–mean flow interaction (Edmon et al. 1980; Hoskins et al. 1983). Moreover, the variability of the WP pattern is found to be markedly linked to the Asian–Pacific jet (Nigam 2003; Linkin and Nigam 2008). The PNA pattern, however, is known to manifest itself as anomalies in the geopotential height field and is usually depicted using the 500- and 700-mb levels (Yin 1994).

The North Pacific regimes look very similar to the North Pacific part of the hemispheric regimes shown in Fig. 3. This point will be discussed in more detail later. The ±PNAs have been identified by many authors (Cheng and Wallace 1993; Kimoto and Ghil 1993a,b). The point worth mentioning here is that the same flow structures were identified by Haines and Hannachi (1995) based on dynamical grounds using a general circulation model. They found that these ±PNAs minimize the streamfunction tendency and hence can be regarded as quasi-stationary states. These regimes also turn out to be quite robust to changes in the number of EOFs used. Similar regimes are also obtained (not shown) when the unfiltered daily data are used. The direction obtained by joining both regimes is expected to have different characteristic values of the PDF. In fact, as for the hemispheric analysis, Table 2 shows the skewness and excess kurtosis of the Pacific Z500 data projected onto the regimes axis. Clearly, a large increase in skewness, which is highly significant, is obtained compared to the PCs (Table 1). For example, the skewness along the 5D solution axes is around −0.7. This skewness has also been compared to the distribution of skewness obtained from surrogate data as in section 3. As in Fig. 4a, the observed skewness is well below the histogram of surrogate skewness (not shown).

As for the hemispheric case, the North Pacific Z500 flow has been tested for clustering within a 2D state space spanned by the regime axis and another one orthogonal to it. Figure 8, which is similar to Figs. 4c and 4d, shows the projection of the large-scale Pacific Z500 flow onto the probability space (Fig. 8a), along with the clustering index and the upper and lower bounds obtained from simulated homogeneous Poisson random processes (Fig. 8b). The same procedure as in Fig. 4d is applied and the dotted lines refer to the curves obtained from 100 samples of surrogate data. There is clear evidence of departure from the no-clustering process, which suggests that the North Pacific regimes are robust. By comparing Figs. 4 and 8 one can see that there is a similarity between them. This and other points will be discussed in detail in section 5.

### b. North Atlantic–European sector

Here the North Atlantic–European sector is defined as being 20°–90°N, 100°W–50°E. The percentages of explained variance of the leading five EOFs are shown in Table 1. The first three modes of variability are shown in Fig. 9. The leading EOF pattern (Fig. 9a) is the familiar NAO with 20.7% explained variance, although the subtropical center is stretched between the eastern North American coast and western Europe. The second mode of variability (Fig. 9b), with about 13.9% explained variance, is a north–south dipole sitting over North Atlantic and Scandinavia. The third EOF (13%; Fig. 9c) is dominated by one center situated west of the British Isles and another stretching along North Africa and the Mediterranean. Skewness and excess kurtosis of the leading five PCs are also shown in Table 1. The skewness *S* of the NAO is quite large. It has been explained by Woollings et al. (2010) in terms of high-latitude blocking–zonal regime behavior. The excess kurtosis is also quite large (Table 1), particularly for the third PC (around −0.6), keeping in mind the kurtosis standard error (0.14).

The mixture model is next applied to the North Atlantic–European Z500 data. Figure 10 shows the mixing proportions along with the 99% confidence limits within the 3D EOF state space using two- (Fig. 10a) and three- (Fig. 10b) Gaussian component mixture models, respectively. The classifiability index also shows similar results to the Pacific sector, with two significant components at the 5% significance level when three to five EOFs are used. The case for three EOFs is shown in Fig. 10c. Given the high skewness and kurtosis along EOFs 1 and 3 and the fact that the largest amplitudes of the regime centers are along these EOFs, the clustering test was applied to the (PC1–PC3) plane and revealed a significant clustering within the probability space (not shown). The clustering, however, was slightly less pronounced compared to that of the North Pacific sector.

The above results again suggest the presence of two distinctive circulation flow regimes. These are plotted in Fig. 11 and they show ±NAO for three dimensions. The structure of these flows is found to be robust to changes in the state space dimension. The same structures are obtained when the unfiltered daily Z500 data are used instead (not shown). These flow structures correspond to Greenland blocking and zonal flow regimes of Woollings et al. (2008). As for the North Pacific case, these regimes look very similar to the North Atlantic part of the hemispheric regimes (Fig. 3). The skewness and excess kurtosis of the North Atlantic large-scale Z500 projected onto the regime axes are shown in Table 2 for three, four, and five state space dimensions. The distribution becomes more skewed compared to that of the individual PCs, but the kurtosis coefficients have relatively decreased. To look at the data behavior along the regimes axis, Fig. 12 shows the histograms of the North Atlantic Z500 projected onto *μ*_{2}–*μ*_{1} axis when five (Fig. 12a) and nine (Fig. 12b) EOFs are used. A kernel estimate of the PDF is also plotted. The optimal kernel width *h*_{opt} = 1.06*σn*^{−1/5}, where *n* is the sample size and *σ* the sample standard deviation, suggested by Silverman (1994) for univariate density estimation, is used. Figure 12a shows some sort of shoulder, and Fig. 12b shows a slight hint of bimodality.

## 5. Link between hemispheric and sectorial flow regimes

**x**

*,*

_{t}*t*= 1, …

*n*, where

*n*is the data sample size, is given a probability

*p*

_{k,t}

*k*th component (or regime) of the mixture. For example,

*p*

_{1,t}and 1 −

*p*

_{1,t}represent, respectively, the probabilities of

**x**

*being in class (or regime) 1 and 2. These probabilities are now computed for each of the domains separately. Figure 13 shows an example of the probability of belonging to regime 1, as a function of time, for the North Pacific (*

_{t}Although the probability (9) may be arbitrarily small, but nonzero, one can always choose a threshold value to categorically classify the trajectory points. Below we use an arbitrarily chosen value of *p* = 0.7; that is, if *p*_{1,t} ≥ 0.7 (*p*_{1,t} ≤ 0.3), then **x*** _{t}* belongs to the first (second) regime or class. The sensitivity of the results to changes in this threshold has, however, been tested and the results turn out to be quite robust.

Given that there is an overlap between both sectors in terms of regime occurrence, the idea is to attempt to seek an answer to this question: by how much does the simultaneous occurrence of both sectorial regimes contribute to the occurrence of hemispheric regimes? This is an important question given the debate that is going on within the climate community regarding the nature of planetary circulation regimes.^{8} The answer to this question should also shed light on this debate. To move one step forward, times when both sectorial regimes co-occur are identified and labeled *T*_{both}. (Table 4 shows the number of regime events in each sector for different probability thresholds and is discussed in the next section.) The time average of the hemispheric (filtered) gridded Z500 data during times when sectorial −PNA (North Pacific blocking) co-occur with sectorial −NAO (Greenland blocking) yields precisely a pattern that is very similar to the first hemispheric regime −PNA/−NAO (Fig. 3a), and similarly for the times when sectorial +PNA co-occur with sectorial +NAO. In fact, Fig. 14 shows a scatterplot within the leading three hemispheric EOFs of those Z500 data corresponding to *T*_{both} where two clear clusters can be seen. This test, though helpful, does not entirely answer the previous question. In the next step the data corresponding to *T*_{both} have been taken out of the original hemispheric (filtered) gridded Z500 data. The remaining hemispheric Z500 gridded data of size 718 (i.e., without simultaneous sectorial regimes) are labeled Z500* _{R}*. To test the effect of removing those simultaneous sectorial regime events, the whole procedure (i.e., EOFs computation and mixture model fitting) is now repeated with Z500

*. The results from this experiment are summarized in Fig. 15. Figure 15a shows the proportions of a two-component mixture within the leading three hemispheric EOFs of Z500*

_{R}*along with the 95% confidence limits.*

_{R}^{9}Similarly, Fig. 15b shows the classifiability index along with the 10%–90% bounds of the corresponding noise model. Both figures show that no significant regimes are found in the data.

The classifiability index is also compared to similar indexes using subsamples drawn from Z500 with the same sample size as Z500* _{R}*. Figure 15c shows the classifiability index of Z500

*along with 100 similar indexes obtained from the random subsamples using the leading three PCs of the filtered Z500 [see Eq. (4)]. For the two-class model only 11 values are found below the Z500*

_{R}*index. This is only slightly less than 10% significance. This slight decrease in significance is expected because of the decreasing sample size (by about one-third). The significance of the reproducibility is also consistent with the significance based on the statistics of the mixture proportions (Fig. 15a). The test has also been applied to random 40-day blocs to keep the serial correlation of the original data, but no noticeable change in the result is observed. Because the reproducibility index tends to find clusters when there is none (Christiansen 2007), a clustering test is applied next to the leading two PCs of Z500*

_{R}*, which show the strongest non-Gaussianity. The obtained skewness values of the leading three PCs are not significant even at the 10% level, but the kurtosis of PC1 and PC2 (−0.43) are significant at 5% level. Figure 15d shows the clustering index obtained (continuous), the uncertainty range from a homogeneous Poisson point process (shading), and the clustering indexes constructed, as in Fig. 8b, using 100 subsamples drawn from the leading three PCs of the filtered Z500 data (dotted). Serial correlation is taken into account when the data are sampled, but again that did not make a noticeable difference. Figure 15d shows in particular that although many of these indexes are within the shaded region because of the relatively small sample size, nearly all the clustering indexes are above the index of Z500*

_{R}*, particularly for the interpoint distance*

_{R}*d*between 0.1 and 0.3. In addition, the generated indexes look much like the full clustering index shown in Fig. 4d with a bump in the previous interpoint distance range. The same result is also obtained when the clustering plane obtained from the leading three PCs of Z500

*is used (not shown). A comparison of Figs. 4d and 15d prompts, in particular, suggestions about the effect of sectorial regime synchronization and hemispheric regime occurrence.*

_{R}The North Atlantic and North Pacific sectors have their own physical and dynamical processes in addition to being coupled. A characteristic feature of coupled nonlinear systems, such as nonlinear oscillators, is the possibility of intermittent synchronization. Although phase synchronization took its roots from simple oscillatory systems, it soon spread to other, more complex systems, including chaotic and stochastic systems. Synchronization turns out to be a robust feature in low-order chaos (Pecora and Carroll 1990) and in simplified atmospheric models (Hannachi 1999; Duane and Tribbia 2001) and has been suggested for the large-scale atmosphere (Duane 1997; Duane et al. 1999; Tatli 2007). In most cases of synchronization between two or more systems or subsystems, the phases get entrained or locked (hence, phase synchronization) while the amplitudes might remain uncorrelated or weakly correlated (Rosenblum et al. 1996). Here we have checked for synchronous behavior between the two sectors by using the probabilities of belonging to each regime [see Eq. (9)]. For each regime in each sector, phases are calculated using Hilbert transforms (e.g., Hannachi et al. 2007) of the associated probabilities. The phase difference between the two sectors is then taken as a measure of phase synchronization. Figure 16 shows an example of the absolute value of phase difference for regime 1 (Fig. 16a) and regime 2 (Fig. 16b), respectively. The shading shows the occurrence times of the corresponding hemispheric regime. The local minima correspond to when both phases are close to each other. The synchronization is enhanced during persistent phase locking. This is reflected in Fig. 16, particularly with regime 2, when hemispheric regimes occur during periods when the sectorial phases lock for some time. Examples include periods near January 1958 and December 1960 (Fig. 16a) and again near March 1987, December 1989, and February 1992/December 1993 (Fig. 16b). It is interesting to note that the shading generally avoids sharp peaks and troughs (i.e., with no persistence and hence no phase synchronization). This suggests further that phase synchronization between the North Pacific and North Atlantic sectors can play an important role in the occurrence of regional or hemispheric regimes.

Woollings et al. (2008) investigated the relationship between upper-level Rossby wave breaking [or high-latitude blocking (HLB)] and the NAO and the west Pacific pattern over the North Atlantic and the North Pacific regions, respectively. They found a weak but significant link between the occurrence of HLB in the two sectors, with the Atlantic HLB leading Pacific HLB by a few days. Woollings and Hoskins (2008) used blocking indices, as described in Pelly and Hoskins (2003), to show that the near-simultaneous occurrence of HLB events over both sectors occurs slightly more often than would be expected by chance and hence contributes to explaining the nature of the high-frequency (daily) northern annular mode (Thompson and Wallace 2000). The results found here concur with those of Woollings et al. (2008) and Woollings and Hoskins (2008). The correlation between

These probabilities are very useful in determining the dynamical/statistical characteristics of the obtained regimes. Table 3, for example, shows the skewness of *p*_{1,t} is normally associated with more persistence in regime 1 (Figs. 3a, 7a, and 11a). For example, Table 3 shows that the flow in the North Atlantic tends to persist more in regime 2 (i.e., the less blocking or zonal regime), associated with the positive phase of NAO. This is expected since the zonal flow is the dominant regime, as blocking events tend to be intermittent. Table 3 also shows that the North Pacific blocking flow is more persistent than the North Atlantic blocking. The hemispheric persistence is normally a cumulative effect of the persistence in both sectors persistence. There is a large increase in skewness between the first and the second half of the record (Table 3). This could be due to a decrease in the number of blocking events or in the persistence time of these events or a combination of both. The association between the decrease of high-latitude blocking events in the North Atlantic and the NAO was nicely illustrated in Woollings and Hoskins (2008) using the blocking index and is discussed next.

The contribution of regime 1 to NAO variability and trend has been investigated. Cumulative probabilities of belonging to regime 1 (blocking) during the winter season (DJF) are computed for each region and compared with the NAO index (data available online at http://www.cgd.ucar.edu/cas/jhurrell/indices.html). Figure 17a shows a plot of these total winter probabilities, scaled to have zero mean and unit variance, along with the DJF NAO index. Note that the NOA index has been reversed for better agreement with blocking events. A high correlation of −0.8 is obtained between the winter probability for the North Atlantic and the NAO index. A similar probability (−0.65) is obtained between the NH winter probability and the same index. In addition, Fig. 17a clearly features the decreasing trend of blocking regime in the NH and the North Atlantic in agreement with the increasing NAO trend starting around the mid-1970s. Using the PNA index (see http://www.cpc.noaa.gov/data/teledoc/telecontents.shtml), a similar conclusion is found to hold for the North Pacific regime 1 (Fig. 17b) where correlations of −0.82 and −0.66 are found between the index and winter probabilities associated with the North Pacific sector and the NH, respectively. Note again the similar decreasing trend in regime-1 probability associated with the increasing PNA trend observed in the second half of the record (Fig. 17b). We have also included the WP index for comparison with Woollings et al. (2008). A correlation of −0.52 is found between the winter North Pacific probability and the WP index.

## 6. Summary and discussion

Preferred circulation flows in low-frequency and large-scale atmosphere are investigated using winter Northern Hemisphere as well as sectorial 500-hPa NCEP–NCAR heights for the period January 1949–December 2007. This investigation is an extension of H07 and aims at looking more closely into hemispheric and sectorial circulation regimes and their relationship and contributing to the ongoing debate on the nature of planetary flow regimes. The methodology used here is based on the multivariate mixture model whereby the probability distribution function (PDF) of the atmospheric state is expressed as a finite mixture of multinormal PDFs. The means of the individual multivariate Gaussians are then interpreted as the regime centers. The number of components in the mixture model is obtained using two different approaches. The first is based on arguments from order statistics applied to the mixing proportions (H07). The second test uses classifiability and reproducibility of the regime patterns, measured by pattern correlation (Cheng and Wallace 1993; Michelangeli et al. 1995).

Nonoverlapping 5-day means of winter Z500 over the Northern Hemisphere (20°–90°N), the North Pacific (20°–90°N, 120°–280°E) and the North Atlantic–European (20°–90°N, 100°W–50°E) sectors are used and the associated EOFs/PCs are computed. Both approaches have been applied first to the NH. Using three to five EOFs, both approaches consistently support the existence of two highly significant components. Within the two leading EOF state space no such high significance is obtained, probably because of the suboptimality of EOFs for cluster detection.

The two hemispheric circulation regimes are then identified and found to project strongly onto (+PNA)–(+NAO) and (−PNA)–(−NAO). These circulation regimes are found to be robust to changes in the number of EOFs retained and have been found to be very similar to those obtained from the unfiltered daily data. To check the consistency of these results, a further clustering test was conducted. The Z500 data are projected first onto a two-dimensional plane spanned, within the EOF state space, by the axis joining the two regime centers and an orthogonal to it. The data are then transformed into a probability space; then a clustering index is computed and compared to (i) indexes derived from a homogeneous Poisson random point process and (ii) indexes derived from AR-1 surrogate data having the same lag-0, lag-1, and sample size as the height data. These surrogate data are projected onto their regime axes and associated clustering indexes are then computed. The data are found to depart significantly from the no-clustering process and the surrogate data.

The methodology is next applied to the North Pacific and the North Atlantic sectors. For the North Pacific both procedures yield two significant components when two to five EOFs are used. For three to five EOFs the significance is 1%, but for two dimensions the classifiability index produces 10% significance only. The two regimes come out as ±PNA and are very similar to the hemispheric ones restricted to the sector. They are comparable, in some aspects, to the North Pacific blocking and no-blocking regimes identified in Woollings et al. (2008). The clustering test, applied in a similar manner to the NH, shows a significant departure from a homogeneous Poisson random process. Similar results are also obtained with the North Atlantic–European sector. Two significant components are obtained, although the significance, when the classifiability index is used, is only 10% with two dimensions and 5% with three to five dimensions. The North Atlantic data are also found to depart from a homogeneous Poisson process. This departure, however, is a little less pronounced than that for the North Pacific. The PDF along the regime axes shows, depending on the state space dimension, either a shoulder or a weak signature of bimodality. The two North Atlantic regimes come out as ±NAO, similar to the Greenland blocking and no-blocking regimes (Woollings et al. 2008); like the North Pacific, they look very much like the restriction of the hemispheric regimes to the North Atlantic.

Finally, the relationship between the hemispheric and sectorial regimes is thoroughly investigated to find out how much the simultaneous occurrence of both the sectorial regimes contributes to the occurrence of hemispheric regimes. First, in each of the two sectors the probabilities of belonging to one or the other circulation regime are evaluated, and the associated Z500 data are classified. Times when both sectorial regimes (e.g., +PNA and +NAO) occur simultaneously are then identified. It is found, in particular, that hemispheric Z500 data during those coincidence times show two well-separated clusters and correspond to the two hemispheric regimes. The Z500 data corresponding to those events are next removed from the hemispheric (filtered) gridded data. The remaining hemispheric data, Z500* _{R}*, are then analyzed anew using the order statistics test and the classifiability index with AR-1 error bars. It is found that the hemispheric regime behavior is significantly reduced. The classifiability index is also compared to similar indexes generated using random subsamples from the hemispheric Z500 data. The index is found to be significantly smaller than the generated index for the two-class model. Finally, the clustering test is applied to Z500

*and reveals no clustering. Finally, phase synchronization between both sectors has been investigated. The difference between phases obtained from a Hilbert transform of the occurrence probability of each regime, in each sector, has been evaluated. It is found, in particular, that the NH regimes tend to occur quite often when both sectors synchronize. These results suggest altogether that the co-occurrence or synchronization of the sectorial, North Pacific, and North Atlantic regimes may have a nonnegligible role in the occurrence of hemispheric regimes.*

_{R}The previous conclusion leads us to another equally important question: Does the simultaneous occurrence of sectorial regimes happen by chance or is there a mechanism by which both sectors are linked? Woollings et al. (2008) found that simultaneous high-latitude blocking appear to happen slightly more than would be expected by chance. They attributed this to distortions of the polar trough, but other mechanisms such as westward propagation of long Rossby waves (Branstator 1987; Lejanäs and Madden 1992; Lau and Nath 1999) could be considered. In this study, although the correlation between *p* = 0.5 because all the data get classified in this case. Considering only the joint probability between the two sectors (NA and NP), we find that the numbers of observed simultaneous regime events (214 and 332) are slightly larger than expected by chance (208 and 325). When we consider the NH and one of the two sectors, then the number of simultaneous events becomes significantly larger than expected by chance. To get a full picture we now consider both sectors and the NH jointly. Table 5 shows the number of simultaneous events and yields the different probabilities. For example, the probability of obtaining NH regime 1 (R1) conditioned on the sectorial regime R1, Pr(NH1|NA1, NP1), is 0.84, and similarly we find Pr(NH2|NA2, NP2) = 0.96. On the other hand we find Pr(NH1|NA1, NP2) = 0.4 and Pr(NH2|NA2, NP1) = 0.4. This suggests that hemispheric regimes are well conditioned by the co-occurrence of sectorial regimes.

The North Atlantic and North Pacific sectors constitute two subsystems of a larger hemispheric system; various relationships exist between them and they have been subject to various interpretations. A particularly interesting hypothesis was suggested recently by Rennert and Wallace (2009) based on cross-frequency coupling between low and intermediate frequencies. For example, during positive (negative) NAO phase the low and intermediate frequencies interfere destructively (constructively). Since the intermediate frequencies are dominated by retrograding long Rossby waves, this cross-frequency coupling, in addition to explaining the observed skewness, can also play a role in the link between circulation regimes in both sectors. An equally interesting possible dynamical explanation for the slightly enhanced simultaneous occurrence of sectorial regimes has been proposed by Crommelin (2004), Kondrashov et al. (2004), and Selten and Branstator (2004), based on the so-called heteroclinic connection, which represents a path in phase space joining two different equilibrium states. In this scenario preferred routes exist between relatively populated regions. These paths are thought to be related to unstable periodic orbits or the remnants of a heteroclinic orbit. This interpretation has been nicely illustrated with a quasigeostrophic model of the atmosphere (Selten and Branstator 2004; Kondrashov et al. 2004) and has been suggested as a possible paradigm for the atmosphere (Crommelin 2004). A different mechanism of the existence of sectorial regimes has been presented recently by Woollings et al. (2010, manuscript submitted to *Quart. J. Roy. Meteor. Soc.*), who suggest that these flow structures result mostly from the wobbling of the eddy-driven jet. This dynamical explanation ties in quite well with the synchronization mechanism suggested above. The North Atlantic and the North Pacific can be seen as two coupled dynamical (sub)systems, which can synchronize intermittently. The coupling between the two sectors may result, for example, from waveguides (Hoskins and Ambrizzi 1993; Ambrizzi and Hoskins 1997) or from west-propagating long Rossby waves (Branstator 1987), which could set the scene for phase synchronization. These ideas are yet to be explored further in the future.

The interannual variability of regime 1 (blocking) has been investigated. Winter cumulative probabilities of belonging to the first regime in the North Atlantic and the Northern Hemisphere have been calculated and compared to the winter (DJF) NAO index. High correlations are found between this index and regime-1 events in the North Atlantic and the NH. The increasing trend of the NAO, particularly in the second half of the record, is clearly reflected by a decrease in regime 1 as observed by Woollings et al. (2008). A similar association is also found between the winter PNA index and the cumulative probabilities for the North Pacific.

## Acknowledgments

The author would like to thank Tim Woollings for his comments on an earlier version of the paper and three anonymous reviewers for their constructive and challenging comments that helped improve the manuscript. Part of this work was performed while the author was visiting the NCAS Walker Institute at Reading University.

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The classifiability index (circle) for different numbers of multivariate Gaussian components *k*, *k* = 1, … 8, in the mixture, applied to the leading three PCs of the winter NH filtered Z500. The shading shows the 5%–95% bounds of the classifiability index obtained from a spatial Markov process with the same characteristics as those of Z500 data.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

The classifiability index (circle) for different numbers of multivariate Gaussian components *k*, *k* = 1, … 8, in the mixture, applied to the leading three PCs of the winter NH filtered Z500. The shading shows the 5%–95% bounds of the classifiability index obtained from a spatial Markov process with the same characteristics as those of Z500 data.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

The classifiability index (circle) for different numbers of multivariate Gaussian components *k*, *k* = 1, … 8, in the mixture, applied to the leading three PCs of the winter NH filtered Z500. The shading shows the 5%–95% bounds of the classifiability index obtained from a spatial Markov process with the same characteristics as those of Z500 data.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Flow structures of the regime centers obtained from a two-component mixture model using the leading three PCs of the winter NH Z500 and showing (a) (−PNA)–(−NAO) and (b) its reverse. Contour interval is 10 m; negative contours are dashed. The EOF amplitudes of the regimes are respectively (0.11, 0.56, −0.4) and (−0.08,−0.51, 0.3).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Flow structures of the regime centers obtained from a two-component mixture model using the leading three PCs of the winter NH Z500 and showing (a) (−PNA)–(−NAO) and (b) its reverse. Contour interval is 10 m; negative contours are dashed. The EOF amplitudes of the regimes are respectively (0.11, 0.56, −0.4) and (−0.08,−0.51, 0.3).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Flow structures of the regime centers obtained from a two-component mixture model using the leading three PCs of the winter NH Z500 and showing (a) (−PNA)–(−NAO) and (b) its reverse. Contour interval is 10 m; negative contours are dashed. The EOF amplitudes of the regimes are respectively (0.11, 0.56, −0.4) and (−0.08,−0.51, 0.3).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Skewness and (b) kurtosis of surrogate AR-1 data along their regime axes (see text); (c) scatterplot within the probability space of the data projected onto the two-dimensional plane spanned by the (10D) NH regime axis and an orthogonal to it; (d) the associated clustering index *L*(*d*). In (a) and (b), the arrows refer to the observed values. In (d), the shading represents the bounds between the upper and lower envelopes of clustering indexes obtained using 100 simulated homogeneous Poisson random point processes whereas the dotted lines represent clustering indexes obtained from the surrogate data projected onto their clustering planes.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Skewness and (b) kurtosis of surrogate AR-1 data along their regime axes (see text); (c) scatterplot within the probability space of the data projected onto the two-dimensional plane spanned by the (10D) NH regime axis and an orthogonal to it; (d) the associated clustering index *L*(*d*). In (a) and (b), the arrows refer to the observed values. In (d), the shading represents the bounds between the upper and lower envelopes of clustering indexes obtained using 100 simulated homogeneous Poisson random point processes whereas the dotted lines represent clustering indexes obtained from the surrogate data projected onto their clustering planes.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Skewness and (b) kurtosis of surrogate AR-1 data along their regime axes (see text); (c) scatterplot within the probability space of the data projected onto the two-dimensional plane spanned by the (10D) NH regime axis and an orthogonal to it; (d) the associated clustering index *L*(*d*). In (a) and (b), the arrows refer to the observed values. In (d), the shading represents the bounds between the upper and lower envelopes of clustering indexes obtained using 100 simulated homogeneous Poisson random point processes whereas the dotted lines represent clustering indexes obtained from the surrogate data projected onto their clustering planes.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

The leading three North Pacific EOFs of the winter 500-mb filtered geopotential height field: (a) 1, (b) 2, and (c) 3.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

The leading three North Pacific EOFs of the winter 500-mb filtered geopotential height field: (a) 1, (b) 2, and (c) 3.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

The leading three North Pacific EOFs of the winter 500-mb filtered geopotential height field: (a) 1, (b) 2, and (c) 3.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a),(b) The mixture proportions for (a) two- and (b) three-component mixture models along with the 99% confidence limits, and (c) the classifiability index (circle) along with the 1%–99% bounds from a spatial first-order autoregressive process obtained using the leading three PCs over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a),(b) The mixture proportions for (a) two- and (b) three-component mixture models along with the 99% confidence limits, and (c) the classifiability index (circle) along with the 1%–99% bounds from a spatial first-order autoregressive process obtained using the leading three PCs over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a),(b) The mixture proportions for (a) two- and (b) three-component mixture models along with the 99% confidence limits, and (c) the classifiability index (circle) along with the 1%–99% bounds from a spatial first-order autoregressive process obtained using the leading three PCs over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained from a two-component mixture model using the leading three PCs of Pacific Z500, showing (a) −PNA and (b) +PNA. The EOF amplitudes of the regimes are respectively (0.61, 0.02, −0.13) and (−0.67, −0.02, 0.14). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained from a two-component mixture model using the leading three PCs of Pacific Z500, showing (a) −PNA and (b) +PNA. The EOF amplitudes of the regimes are respectively (0.61, 0.02, −0.13) and (−0.67, −0.02, 0.14). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained from a two-component mixture model using the leading three PCs of Pacific Z500, showing (a) −PNA and (b) +PNA. The EOF amplitudes of the regimes are respectively (0.61, 0.02, −0.13) and (−0.67, −0.02, 0.14). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Figs. 4c and 4d, but for the 3D state space of Z500 flow over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Figs. 4c and 4d, but for the 3D state space of Z500 flow over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Figs. 4c and 4d, but for the 3D state space of Z500 flow over the North Pacific sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 5, but for the North Atlantic–European sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 5, but for the North Atlantic–European sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 5, but for the North Atlantic–European sector.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 6, but for the North Atlantic–European sector; in (c) the 5%–95% bounds are shown instead.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 6, but for the North Atlantic–European sector; in (c) the 5%–95% bounds are shown instead.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

As in Fig. 6, but for the North Atlantic–European sector; in (c) the 5%–95% bounds are shown instead.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained using the leading three North Atlantic Z500 PCs and showing (a) −NAO and (b) +NAO. The EOF amplitudes of the regimes are respectively (−0.46, −0.19, −0.64) and (0.35, 0.15, 0.49). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained using the leading three North Atlantic Z500 PCs and showing (a) −NAO and (b) +NAO. The EOF amplitudes of the regimes are respectively (−0.46, −0.19, −0.64) and (0.35, 0.15, 0.49). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Circulation regimes obtained using the leading three North Atlantic Z500 PCs and showing (a) −NAO and (b) +NAO. The EOF amplitudes of the regimes are respectively (−0.46, −0.19, −0.64) and (0.35, 0.15, 0.49). Contour interval is 10 m; negative contours are dashed.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Histograms and PDF estimates of the North Atlantic Z500 projected onto the (a) 5D and (b) 9D regime axes. The fitted normal density function for each case is also shown. The PDF estimation is based on the optimal smoothing parameter *h*_{opt} = 1.06*σn*^{−1/5} (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Histograms and PDF estimates of the North Atlantic Z500 projected onto the (a) 5D and (b) 9D regime axes. The fitted normal density function for each case is also shown. The PDF estimation is based on the optimal smoothing parameter *h*_{opt} = 1.06*σn*^{−1/5} (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Histograms and PDF estimates of the North Atlantic Z500 projected onto the (a) 5D and (b) 9D regime axes. The fitted normal density function for each case is also shown. The PDF estimation is based on the optimal smoothing parameter *h*_{opt} = 1.06*σn*^{−1/5} (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Probability of belonging to regime 1 *p*_{1,t} vs time for the North Pacific (dashed), the North Atlantic (continuous), and the NH (dotted).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Probability of belonging to regime 1 *p*_{1,t} vs time for the North Pacific (dashed), the North Atlantic (continuous), and the NH (dotted).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Probability of belonging to regime 1 *p*_{1,t} vs time for the North Pacific (dashed), the North Atlantic (continuous), and the NH (dotted).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Scatterplot of the three leading Northern Hemispheric PCs of Z500 corresponding to those dates when the sectorial North Atlantic regimes coincide with the sectorial North Pacific regimes (i.e., ±PNA coincide with ±NAO), respectively.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Scatterplot of the three leading Northern Hemispheric PCs of Z500 corresponding to those dates when the sectorial North Atlantic regimes coincide with the sectorial North Pacific regimes (i.e., ±PNA coincide with ±NAO), respectively.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Scatterplot of the three leading Northern Hemispheric PCs of Z500 corresponding to those dates when the sectorial North Atlantic regimes coincide with the sectorial North Pacific regimes (i.e., ±PNA coincide with ±NAO), respectively.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Proportions of a two-component mixture model using the leading three PCs of Z500* _{R}* along with the 95% confidence limits, (b) the classifiability index of the same data along with 10%–90% bounds using a spatial AR-1 noise process, (c) the same classifiability index along with 100 similar indexes (dotted) obtained from subsamples of the leading three PCs of Z500 with the same sample size as Z500

*, and (d) the clustering index (continuous) obtained from the leading two PCs of Z500*

_{R}*. In (d) the shading represents the uncertainty range obtained from a simulated Poisson point process and the dotted curves represent clustering indexes obtained from 100 random subsamples as in (c).*

_{R}Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Proportions of a two-component mixture model using the leading three PCs of Z500* _{R}* along with the 95% confidence limits, (b) the classifiability index of the same data along with 10%–90% bounds using a spatial AR-1 noise process, (c) the same classifiability index along with 100 similar indexes (dotted) obtained from subsamples of the leading three PCs of Z500 with the same sample size as Z500

*, and (d) the clustering index (continuous) obtained from the leading two PCs of Z500*

_{R}*. In (d) the shading represents the uncertainty range obtained from a simulated Poisson point process and the dotted curves represent clustering indexes obtained from 100 random subsamples as in (c).*

_{R}Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

(a) Proportions of a two-component mixture model using the leading three PCs of Z500* _{R}* along with the 95% confidence limits, (b) the classifiability index of the same data along with 10%–90% bounds using a spatial AR-1 noise process, (c) the same classifiability index along with 100 similar indexes (dotted) obtained from subsamples of the leading three PCs of Z500 with the same sample size as Z500

*, and (d) the clustering index (continuous) obtained from the leading two PCs of Z500*

_{R}*. In (d) the shading represents the uncertainty range obtained from a simulated Poisson point process and the dotted curves represent clustering indexes obtained from 100 random subsamples as in (c).*

_{R}Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Absolute value of the difference between phases of sectorial probabilities for regimes (a) 1 and (b) 2. The shading represents the times of hemispheric regimes’ occurrence.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Absolute value of the difference between phases of sectorial probabilities for regimes (a) 1 and (b) 2. The shading represents the times of hemispheric regimes’ occurrence.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Absolute value of the difference between phases of sectorial probabilities for regimes (a) 1 and (b) 2. The shading represents the times of hemispheric regimes’ occurrence.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Total winter probability of belonging to regime 1 for (a) the North Atlantic (continuous) and the NH (dotted) and (b) the North Pacific (dashed) and the NH (dotted). Superimposed in (a) is the reversed sign of the NAO index (shaded), and in (b) the reversed sign of the PNA index (shaded) and the same for the WP index (thin solid). Probabilities have been scaled to zero-mean and unit variance.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Total winter probability of belonging to regime 1 for (a) the North Atlantic (continuous) and the NH (dotted) and (b) the North Pacific (dashed) and the NH (dotted). Superimposed in (a) is the reversed sign of the NAO index (shaded), and in (b) the reversed sign of the PNA index (shaded) and the same for the WP index (thin solid). Probabilities have been scaled to zero-mean and unit variance.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Total winter probability of belonging to regime 1 for (a) the North Atlantic (continuous) and the NH (dotted) and (b) the North Pacific (dashed) and the NH (dotted). Superimposed in (a) is the reversed sign of the NAO index (shaded), and in (b) the reversed sign of the PNA index (shaded) and the same for the WP index (thin solid). Probabilities have been scaled to zero-mean and unit variance.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2009JAS3296.1

Percentage of explained variance, skewness, and kurtosis of the leading individual five PCs of Z500 for the NH, North Pacific, and Atlantic sectors.

Skewness and kurtosis of Z500 projected onto the regimes axis within the three-, four-, and five-dimensional EOF state space for the NH, North Pacific, and North Atlantic sectors.

Skewness of the probability *p*_{1,t} of belonging to regime 1 (blocking) in the NH, the North Pacific, and the North Atlantic for the entire record and for the first and second halves.

Number of regime occurrences over the Northern Hemisphere (NH), North Pacific (NP), and North Atlantic (NA) for regime 1 (R1) and regime 2 (R2) along with the NA and NP regime co-occurrences as a function of threshold probability.

Number of regime co-occurrences between the Northern Hemisphere (NH), North Pacific (NP) and North Atlantic (NA) for regime 1 (R1) and regime 2 (R2) for a probability threshold of 0.5.

^{1}

For example, closed forms of the evolution equation of the system PDF moments.

^{2}

With respect to the value 3 of a Gaussian.

^{3}

Based on unfiltered Z500 daily data for the period 1949–2003.

^{4}

In fact, to find where nonnormality is strongest, other tools, such as projection pursuit, could be more appropriate (see, e.g., Christiansen 2009).

^{5}

Determined as for the height data by selecting the regime axis and an orthogonal to it.

^{6}

Or sub-Gaussianity.

^{7}

1 potential vorticity unit (PVU) = 10^{−6} m^{2} K^{−1} kg^{−1} s^{−1}.

^{8}

The issue of differentiating between sectorial and regional regimes is, however, an enduring and challenging one and is reminiscent of the NAO/Arctic Oscillation (AO) paradigms (see, e.g., Christiansen 2002).

^{9}

The two components are, however, significant at 10% level.