## 1. Introduction

Interest in the dynamics of the stratosphere has significantly increased in recent years. This interest has been driven by the quest to seek a holistic understanding of the climate system by jointly studying the different components of the system including the stratosphere, which was somewhat neglected in the past. In addition, owing to the larger stratospheric forecast skill, compared to the troposphere, there is the possibility that this skill can be useful to extend the tropospheric forecast skill (Kuroda 2008) through the troposphere–stratosphere connection (Charney and Drazin 1961; Baldwin and Dunkerton 2001). This forecast skill improvement could be particularly prominent during sudden stratospheric warming (SSW) events (Reichler et al. 2005; Kuroda 2008). This has indeed stimulated a number of research investigations on the effect of SSW on predictability skill (Mukougawa and Hirooka 2004; Hirooka et al. 2007; Ichimaru 2010; Allen et al. 2006).

SSW events are the most dramatic meteorological phenomena in the stratosphere (O’Neill 2003), where the generally coherent structure of the cold westerly flow of the polar night vortex is disturbed and breaks down, yielding a reversal in the meridional temperature gradient and appearance of easterly winds poleward of 60° latitude. This regime change is spectacularly fast with a time scale less than about a week where the rate of polar temperature change can exceed 10 K day^{−1} (Tung and Lindzen 1979; Andrews et al. 1987, ch. 11; Limpasuvan et al. 2004). The importance of these SSW events seems to be particularly prominent during midlatitude tropospheric blocking (Tung and Lindzen 1979; Woollings et al. 2010), although this is challenged by Taguchi (2008). There is the suggestion that stratospheric predictability can be significantly enhanced during SSW events where the predictability limit can be stretched beyond 20 days (Mukougawa and Hirooka 2004). For example, investigations by Mukougawa et al. (2005) and Hirooka et al. (2007), based on the warming events of December 2001 and January 2004, reveal that the predictability limit, which depends on the SSW pattern, ranges between 9 and 20 days. These studies, although few in number, do show the importance of SSW events for predictability [see, e.g., the discussion in Charlton-Perez et al. (2008)] and can be useful for a deeper understanding of the physical processes involved, hence the importance of developing methodologies that enable a systematic identification of SSW events from observations and model simulations.

SSWs are broadly classified into two types dubbed “minor” and “major” warming events (see, e.g., Andrews et al. 1987; Charlton and Polvani 2007). Various definitions exist in the literature for minor and major warmings—for example, McInturff (1978), Labitzke (1982), Andrews et al. (1987), and notably that proposed by the World Meteorological Organization (WMO)—but they all pick up approximately the same events. One of earliest WMO definitions of minor warming involves only an increase in polar temperature by at least 20 K for more than about 5 days, whereas major warmings involve in addition a reversal of zonal-mean zonal wind north of 60°N at the 10-hPa level. For more details on major stratospheric warmings the reader is referred to Charlton and Polvani (2007), who conducted a comprehensive comparative analysis of SSW events from 40-yr European Centre for Medium Range Weather Forecast (ECMWF) Re-Analysis (ERA-40) and National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) reanalysis data, in addition to providing a list of all the dates of SSW events. Likewise, the distortion of the polar vortex during SSW events is not arbitrary but takes specific morphologies depending on the prevailing zonal wavenumber of the vortex, namely wave 1 and wave 2. In general, most wavenumber-1 and wavenumber-2 structures correspond, respectively, to so-called displaced and split vortices (O’Neill 2003). Although the frequency of major SSW events tends to be a little smaller than that of minor SSW events, the frequencies of wavenumber-1 and wavenumber-2 SSW events tend to be about equal (e.g., Charlton-Perez et al. 2008).

In view of this, and owing to the importance of SSWs for climate variability and predictability, it is of great importance to develop methods that enable a systematic identification and, if necessary, a categorization of SSW events. This will also serve as a good test of models and whether they simulate realistic warmings. If this could be achieved it would ultimately obviate the need to use various and nonsystematic categorization of SSWs, but this will undoubtedly require painstaking efforts. The topic of SSW classification is still in its infancy and to date only a few studies have looked at this using clustering approaches. Recently, Coughlin and Gray (2009, hereafter CG09) examined 43 yr of daily winter Northern Hemisphere (NH) polar stratospheric data using ERA-40 in an attempt to categorize polar stratospheric winter states. They applied *k*-means clustering to a set of 154 stratospheric variables that capture stratospheric variability and found that the NH winter stratosphere has only two well-separated states: a disturbed state representing SSW events and an undisturbed state of the cold vortex. They also reported that no further division is observed within the disturbed state of the polar vortex regarding either the amplitude of the warming or the wave-1/wave-2 type of the vortex and suggest that the observations consist of a continuum of warming events. However, the analysis of CG09 only used standard zonally averaged quantities and as such does not provide a complete picture of the vortex evolution. New insights are therefore required to describe the full structure of the vortex.

Given the coherent nature of the polar vortex, a natural way to analyze stratospheric variability is to use information from the geometry of the vortex. This information is provided precisely by the geometric moments, which reflect the stretching, deformation, and wobbling of the vortex (Melander et al. 1986; Waugh 1997; Waugh and Randel 1999; Dritschel 1993; Matthewman et al. 2009; Mitchell et al. 2011). In this manuscript we examine the most relevant geometric moments of the Arctic polar vortex at an 850-K isentropic surface from ERA-40 in an attempt to categorize SSW events regarding both the warming amplitude and the wave structure, which can be used to check for the continuum hypothesis. The data and the methodology are presented in sections 2 and 3, respectively. The results from analyzing the vortex moments are presented in section 4 and a summary and discussion are presented in section 5.

## 2. Data

The data used here to compute the geometric moments are the same as those used in Mitchell et al. (2011) and consist of the potential vorticity field at 850-K isentropic surface taken from ERA-40 data (Uppala et al. 2005). Daily data of 850-K isentropic potential vorticity (PV) field for the winter period, December–March (DJFM), from 1958 to 2002 are used. The data are departures from the seasonal cycle. In addition, to analyze the vertical structure of the vortex, we have also used data similar to those in CG09, to which the reader is referred for more details. These data also consist of daily winter (DJFM) stratospheric variables from ERA-40. A total of seven polar stratospheric variables are used. These are zonal average temperature at three latitudinal bands, zonal average zonal wind at two latitudinal bands, and finally the wave-1 and wave-2 components of geopotential height at 60°N. These wave amplitudes are then normalized by the zonal wind values at 60°N (CG09) and hence will also be referred to as relative wave amplitude in the text. These variables are taken at 11 stratospheric pressure levels from 100 to 1 hPa. These variables are all zonally averaged except the geopotential height wave 1 and wave 2. Temperature values are obtained as averages over three bands: 60°–70°N, 70°–80°N, and 80°–90°N. The zonal wind values are based on averages over the 60°–70°N and 70°–80°N latitudinal bands.

## 3. Methodology

Two main approaches are used in this paper and are presented below. We have used hierarchical clustering to cluster the vortex moments. In addition, data image visualization techniques borrowed from pattern recognition are used in the analysis. Later in the analysis we have used extended empirical orthogonal functions (EEOFs), known also as multivariate singular spectrum analysis (SSA). This is used to identify and filter out the dominant oscillating signal from the vortex area.

### a. Hierarchical clustering

Clustering is an exploratory tool in multivariate analysis used to cluster the data sample by separating it into groups or clusters that could provide a deep understanding of the underlying structure or nature of the data (e.g., Hair et al. 1995). The clustering is obtained using a similarity measure to allow the elements in each group to be as “homogeneous” as possible. The Euclidean distance, the most widely used similarity measure, is used in this study although various other measures also exist but are not considered here. Clustering methods are essentially of two types: hierarchical and nonhierarchical (see, e.g., Gordon 1999; Hastie et al. 2001). In this manuscript we focus on hierarchical clustering, which is described briefly below; for more details the reader is referred to the previous references.

In the hierarchical clustering algorithm a bunch of fully nested sets are obtained. The smallest sets are the clusters obtained as the individual elements (or observations) whereas the largest set is obtained as the whole dataset. The algorithm then proceeds by successively dividing or agglomerating the existing sets depending on whether we start from the whole dataset or from the individual observations. For example, if we start from the individual elements as clusters then the algorithm consists of different steps involving successively a merger between any existing closest clusters based on the chosen similarity measure until we are left with only one group (i.e., the whole dataset).

Given an interpoint distance matrix, different types of agglomerations known as linkages are used by the merger. The most widely used algorithms are as follows:

Single linkage: based on using the minimum distance, where the distance between clusters is defined as the distance between the closest pair of observations.

Complete linkage: the distance is as in single linkage, except that the most distant pair of observations is used.

Centroid linkage: based on the distance between the centroids of the two clusters.

Average linkage: based on the average distance between all pairs where one member of the pair comes from each cluster.

The single linkage algorithm tends to produce chainlike structures in the data where observations on the ends of the chain might be very similar. On the other hand, spheroidal linkages, such as centroid and average linkages, tend to form spheroidal shapes. The average linkage method tends to combine clusters with small variances and produce clusters with approximately similar variance. In the present context, however, we know that SSWs are relatively small in number because they do not happen very often. Also, the data turn out to be strongly clustered with nonspheroidal shapes (not shown). Here we have chosen to use the complete linkage to avoid the previous pitfalls in addition to the chaining or lineage problem.

Once an agglomeration merger is chosen, the result of the hierarchical clustering is normally presented in the form of a treelike graph or dendrogram representing the nested structure of the partition and showing the links between objects starting from the whole set down to the individual elements or vice versa. The dendrogram is composed of the root that is linked to the whole set, the nodes representing the points of splitting, and branches whose lengths reflect the distance between the objects. Each node corresponds to a numerical value representing the distance between the two clusters during the merger; the set of these numbers is normally shown on the horizontal or vertical axis of the dendrogram, depending on the display mode. Cutting the tree at various height values of the dendrogram yields different clusters.

As an illustration, we consider a very simple case of two well-separated standard bivariate Gaussian clusters placed respectively at (−4, 0) and (4, 0) and with sample size 50 each (Fig. 1a). The two clusters are (randomly) mixed so as to make two time series of sample size 100 each. The two-dimensional data points are then agglomerated using the complete linkage. The obtained dendrogram (Fig. 1b) shows the links between objects as U-shaped lines facing to the right. The individual points in this diagram are represented on the right-hand side of the panel where only points numbered 20, 40, 60, 80, and 100 are shown for purposes of clarity. The numbers on the horizontal axis represent the distance between objects. For example, the distance (see the definition of “complete linkage” above) between the two clusters is about 11.9 (see Figs. 1a,b). As the distance increases from zero, the data get increasingly agglomerated. Cutting the tree at different levels yields different numbers of clusters. For example, if the tree is cut along the dashed line (Fig. 1b) then one gets two clusters given by the two obtained branches.^{1}

^{2}One particularly robust method of finding the number of clusters is the so-called gap statistic (Tibshirani et al. 2001), which is summarized below. The gap statistic was proposed by Tibshirani et al. (2001) to estimate the number of clusters in a dataset and can be applied to any clustering technique. Unlike other approaches such as the silhouette plot, which was applied by CG09, or the upper-tail rule, which can be applied only in hierarchical clustering, the gap statistic is based on comparing the within-cluster dispersion with that from a probabilistic reference model or null distribution, normally taken to be a homogeneous Poisson point process, as used here. For a given number of clusters

*k*the sum of the pairwise distances for all points in a given cluster

*m*,

*D*, for

_{m}*m*= 1, … ,

*k*, is first calculated, then the average of these sums obtained to yield

*n*is the size of the

_{m}*m*th cluster. A comparison between the “within dispersion index” log(

*W*) of the data and that expected from the null distribution provides the gap statistic:

_{k}*W**

_{k}), appearing on the rhs of (2), is the “within dispersion” drawn from the null distribution and

*E*(·) is the expectation operator. The null hypothesis of a single cluster can then be rejected or accepted depending on the evidence provided by (2). Here, a Monte Carlo simulation is used to generate

*N*samples of log(

*W**

_{k}) based on the null distribution and to compute the corresponding mean

*W**

_{k})

*s**

_{k}. Finally, to account for the simulation error in

*E*[log(

*W**

_{k})], the inflated standard deviation,

*s*=

_{k}*s**

_{k}

*k*

_{opt}is then given by the smallest

*k*such that

### b. Data image

One efficient way to visualize multidimensional data is to use what is known in pattern recognition as a data image (Ling 1973; Minnotte and West 1999). This consists of mapping the multivariate data into an image framework where each pixel reflects the magnitude of each observation. When viewing the data image, regions of high contrast in one or more variables indicate a significant difference in that diagnostic or collection of diagnostics for different parts of the data array. This can be used as an exploratory tool in the clustering procedure. The data image can be applied to the original space–time data matrix or to a dissimilarity (or distance) matrix. Several variants of this image can be incorporated. For instance, rows and columns of the dissimilarity (or similarity) matrix can be reordered based on some clustering criterion such as hierarchical clustering, as in Minnotte and West (1999). This ordering can also be applied to the original data matrix. This representation allows clusters to emerge when the data image is constructed, particularly when it is used in conjunction with the dissimilarity matrix. In this case the clusters appear discernable along the diagonal of the image.

As an illustration, let us go back to our simple example discussed in Fig. 1a. The data image of the two time series 1 and 2 is shown in Fig. 1c and represents simply a color-coded image of the data points; that is, for each variable each color represents one value of that variable and similar colors represent similar values. Note that for purposes of clarity the shading is different between both variables. No structure is obtained because the temporal order of the time series is basically random (i.e., the time series jumps randomly between both the clusters). Similarly, Fig. 1d shows the data image of the dissimilarity matrix. Here we have a square (interpoint) distance matrix where again each color represents one value. In Fig. 1d dark and light shadings represent respectively small and large interpoint distances. The diagonal line of Fig. 1d represents the zero value. Again, no structure is obtained although one could see scattered dark islands, which can be brought together using clustering as shown next. In Fig. 1e we show the data image of the dissimilarity matrix where the data are now arranged into two blocs associated with the two clusters. Note that the order here is fixed by the clusters and so no axis label is shown. The data image now clearly shows the contrast between the dark and light shadings. The two dark shading blocs represent the two clusters.

### c. Stratospheric vortex and geometric invariant moments

*q*(

*x*,

*y*), the (

*k*+

*l*)th order moment is defined by

*m*, for different values of

_{kl}*k*and

*l*, identify in principle the function

*q*(

*x*,

*y*) and vice versa. Here, the quantity

*q*(

*x*,

*y*) is taken to represent the 850-K potential vorticity field at a given day. Normally the moments are computed with respect to the vortex centroid and yield the central moments.

*q*, whence the central (or relative) moments are then obtained as

_{b}*x*

*y*

*q*−

*q*), and the integral is taken only over the region of the vortex satisfying

_{b}*q*(

*x*,

*y*) ≥

*q*. In Waugh and Randel (1999) the vortex edge

_{b}*q*was obtained as the mean PV at the maximum meridional PV gradient, whereas in Matthewman et al. (2009) it was obtained by averaging the PV field poleward of 45°N nine days before the onset of an SSW event. This modification was aimed at removing contributions from low PV values that could have possibly originated in the tropics. In this paper

_{b}*q*is computed as in Mitchell et al. (2011) and is briefly summarized below; for details the reader is referred to that paper. The isentropic PV field is first transformed using equivalent latitudes (Nash et al. 1996), and then the maximum PV gradient is obtained. This maximum is chosen to be consistent with the peak in zonal mean zonal wind along PV isolines. The vortex edge is then obtained as the PV value associated with the equivalent latitude of the maximum meridional PV gradient. The obtained daily fields of the vortex edge are further spatially smoothed in order to reduce the level of filamentation, which occurs particularly in the upper layers of the stratosphere.

_{b}*x*

*y*

*J*

_{01}=

*J*

_{10}= 0. Four moment invariants are computed as outlined above and are considered here. These are (i) the latitude centroid

*y*

*A*=

*J*

_{00}/

*q*, which measures the vortex area and its intensity; (iii) the aspect ratio (AR),

_{b}*r*in (7) is precisely the aspect ratio given in (6). These geometric moments are easily computed after a transformation to Cartesian coordinates is first performed, then a back transformation is applied to get the centroid latitudes in the original polar coordinates (see Mitchell et al. 2011).

## 4. Analysis of the vortex moments

### a. Identification of the polar vortex groups

Mitchell et al. (2011) computed four geometric moments—the centroid latitude (Latc), the aspect ratio, the area, and the kurtosis of the vortex (see also Matthewman et al. 2009)—using the winter (DJFM) daily NH 850-K PV field from ERA-40. The aspect ratio measures the elongation of the vortex and the excess kurtosis measures its “peakedness.” A stretched vortex [large AR; see (6)] will normally have small kurtosis and so some information will be shared between the two parameters. The main point of concern here, however, is that, being a fourth-order moment, the excess kurtosis [(7)] will be dominated by a small number of large values or “outliers” in exactly the same way as the sample kurtosis from conventional time series analysis. The skewness of the daily PV kurtosis time series [(7)] is in fact 5.8, which is to be compared to the values 1.8, −0.7, and 0.6 associated with the aspect ratio, the centroid latitude, and the area of the vortex respectively. Figure 2 shows an excerpt of the (scaled) time series of the AR, Latc, area, and the excess kurtosis of the vortex. The high skewness of the excess kurtosis can be clearly seen in Fig. 2. In the remainder of the paper we therefore focus our analysis only on the remaining three moments: AR, Latc, and the area of the vortex.

Seasonality of the polar vortex is obviously exhibited by the prominent strengthening of the flow occurring in the peak winter season. A strong seasonality is indeed observed in the zonal polar wind and polar cap temperature. Since the vortex area is linearly related to the vortex and hence to the polar zonal wind and polar temperature, it inherits this strong seasonality (see also Mitchell et al. 2011). By contrast, the remaining geometric moments (i.e., the centroid latitude, aspect ratio, and excess kurtosis) show much weaker seasonality.

Because the activity of the winter polar vortex peaks in December–January (i.e., around the northern winter solstice), some sort of phase locking is obtained between the strength of the vortex and the seasonal cycle. Because of this phase locking the peak activity remains even if the data are deseasonalized by removing the mean annual cycle. Figure 3 shows the autocorrelation function of the vortex area versus lag. The period of oscillation is about 120–130 days, which coincides roughly with the DJFM winter season used in this paper, and results from the phase locking mentioned above. A similar behavior is observed even when a smooth seasonal cycle is removed from the data. This is because the seasonal cycle is not fixed but varies from year to year (see Pezzulli et al. 2005). Later we deal with this signal explicitly using SSA (see the appendix). The other two moments, however, are not affected by the periodic signal (not shown) and we start our analysis by focusing on them.

#### 1) Aspect ratio and centroid latitude

We start first by exploring the dendrogram and the data image of the aspect ratio and the centroid latitude of the polar vortex. Figure 4a shows the data image of AR and Latc following the temporal order where obviously no structure is apparent. The data image of the dissimilarity matrix is shown in Fig. 4b, where again there is no evidence of any structure. The dendrogram (Fig. 4c) shows the hierarchy of clusters in the data. Each node in the tree corresponds to the distance between the merging clusters.

In the right-hand side of the tree the number of branches and leaves represent the sample size at individual observations. Clustering, however, very often focuses on the left-hand side of the dendrogram (e.g., Fig. 4c) normally associated with a small number of clusters. Note how the data image based on the tree order (Fig. 4d) is getting more discernable compared to Fig. 4a, where “homogeneous” groups start to agglomerate and emerge. In the perfect case where the clusters are perfectly separated one gets horizontal bands associated with the clusters. Here one can already see some horizontal bands near the top (Fig. 4d).

To analyze the clustering of the vortex moments we show in Fig. 5 the gap statistic of AR and Latc. The figure shows the “within-cluster” index log(*W*) [see (1)] of the data and that expected from the reference distribution (upper panel) along with the gap statistic (lower panel). The vertical bars in the bottom panel represent one standard deviation above and below the gap statistic. Note the maximum of the gap statistic at three clusters, which also satisfies (3). Most importantly, the gap for two clusters is well below that for three clusters and (3) cannot be satisfied for *k* = 2 even if we use 2.6*s*_{k+1} (i.e., 99% confidence level) instead of *s*_{k+1}. Therefore the data clearly support three clusters in contrast to the result of CG09, who found only two. In Fig. 6 we examine the data image of the dissimilarity matrix based on the three-cluster solution and, for comparison, we also show the cases of two and four clusters. The case of two clusters (Fig. 6a) shows two clusters along the main diagonal of the data image. The first class in the upper left corner (Fig. 6a) is much smaller than the second class. This can also be seen^{3} from Fig. 4c. The optimal three-cluster case (Fig. 6b) shows two small classes and a large one. These clusters are obtained from the previous one (i.e., the two-cluster solution) by splitting the large cluster (Fig. 6a). Since the gap for four clusters is the closest to the optimal solution, we decided to examine its data image for comparison. This is displayed in Fig. 6c. Note that there are four clusters in Fig. 6c but one cluster is hard to see because it is so tiny, and hence insignificant, that it does not show up in the data image in the way Fig. 6c is plotted. Note that this also can be inspected from the dendrogram (Fig. 4c).

To check the robustness of the optimal number of clusters we have conducted a Monte Carlo resampling test. One hundred subsamples, each with a sample size that is half the total sample size, are randomly and repeatedly drawn from the data and submitted to the same analysis presented above. To keep the serial correlation of the data the resampling procedure is based on randomized two-month blocs. Figure 7 shows an example of the gap statistic curves from the surrogate data along with the curve for the whole data (Fig. 7a) and the frequency of the optimal number of clusters *k*_{opt} [see (3); Fig. 7b]. A local peak can be seen at three clusters. Although the four-cluster case seems to show large values (Fig. 7a), the uncertainty is larger compared to the three-cluster case. More precisely, when the reference (or null) distribution is used, the frequency of the optimum number of clusters (Fig. 7b) shows evidence that, besides the fact that the data are highly clustered (zero frequency at *k*_{opt} = 1), the case *k*_{opt} = 3 is the most likely. Interestingly, the case *k*_{opt} = 2 is competitive to the optimal number (Fig. 7b).

#### 2) Vortex categorization and the vortex area

Given that three neat clusters have been identified using the aspect ratio and the centroid latitude, as shown in the previous section (see Figs. 5, 6, and 7b), we would like to include next the vortex area and to see whether we can still identify the previous clusters to categorize the polar vortex. This will serve as a further consistency test of the identified groups. However, and as discussed above (see Fig. 3), the latter moment seems to be dominated by a periodic signal resulting from seasonal phase locking. To check, identify, and filter out the latter signal we use here the data-adaptive EEOF (or SSA) filtering method (see the appendix). The vortex area time series has been subjected to the EEOF analysis as outlined in the appendix, by computing first the extended data using the delay coordinates [(A1)] and then computing the spectrum and the eigenvectors of the associated covariance matrix; the result is summarized in Fig. 8. The spectrum of the (delay) covariance matrix of the vortex area (Fig. 8a) shows a clear pair of (nearly) identical eigenvalues, with a fair amount of explained variance (30%), well separated from the rest of the spectrum. A window length *M* = 400 days is used here, but we have checked that the signal is quite robust to changes in this parameter. The extended EOFs associated with the leading two eigenvalues are plotted in Fig. 8b and show clearly two sine waves in quadrature, an unambiguous indication of the existence of the roughly four-month periodic signal in the data. The leading two extended PCs (EPCs) are displayed in Fig. 8c and are showing again the *π*/2 phase shift between them. This is also supported by the phase diagram of EPC1 versus EPC2 (Fig. 8d).

To filter out the previous periodic signal, the EEOFs and EPCs are used to reconstruct the vortex area time series. Climate variations are in general nonstationary. For example, seasonality is not fixed but changes from year to year (see, e.g., Pezzulli et al. 2005). The merit of the EEOF method, compared to other standard methods such as Fourier transform, is in its being data adaptive and hence taking into account this changing seasonality. The filtering procedure is dictated by the need to remove the part of the signal associated with the leading pairs of (nearly) equal eigenvalues representing the oscillation. To better approximate the oscillating signal we have used the leading four EEOFs/EPCs. The leading two EEOFs did not do a good job in removing the periodic signal. The choice of four EEOFs/EPCs is motivated by the desire to get a better reconstruction of the periodic modulated signal and constitutes a compromise between under- and overfitting of the periodic signal. Note also that the fifth to the ninth EEOFs are degenerate and have therefore been avoided altogether. Figure 9a shows the reconstructed time series superimposed upon the vortex area time series for the first 12.5 winters of the record. The periodic signal is clearly visible in the vortex area time series and the reconstructed series tracks the signal very well. The filtered time series (Fig. 9b) is obtained by subtracting the reconstruction from the original time series, and it can be seen that the signal has been well filtered out. The autocorrelation function of the filtered series (not shown) does not show any oscillation.

After the periodic signal has been removed the three moments AR, Latc, and the (filtered) vortex area are jointly analyzed. Figure 10 shows the frequency of the optimal number of clusters using a Monte Carlo sampling method, similar to that presented in the previous section, applied to the above three variables before (Fig. 10a) and after (Fig. 10b) the periodic signal has been taken out. Note that when the periodic signal is not removed two (misleading) clusters emerge (Fig. 10a). However, when the signal is filtered out *k*_{opt} = 3 emerges again as the most probable solution. The data image of the similarity matrix of these variables is computed using the optimal number of three clusters and the result is shown in Fig. 11. The case for the AR, Latc, and the original (nonfiltered) vortex area series (Fig. 11a) is also shown for comparison. Note that when the periodic signal was not removed the cluster sizes (Fig. 11a) are not similar to those of Fig. 6b. In addition, apart from the smallest cluster (lower right corner of Fig. 11a), the remaining two clusters are not well separated from the background. Figure 11b, on the other hand, shows nicely an image (but with slightly less contrast) similar to that of Fig. 6b. The small cluster in the upper left corner of Fig. 11b is only slightly larger than the corresponding one in Fig. 6b. Based on these moments we analyze next the characteristics of these vortex groups.

### b. Analysis and characteristics of the vortex groups

In this section we set out to analyze the characteristics of each group using the vertical structure of the zonal wind, temperature, and wave amplitude in addition to the horizontal structure of the associated PV fields. Table 1 shows, in the form of a contingency table, the percentage of each group when the two moments AR and Latc or the three moments AR, Latc, and the filtered vortex area are used. The good agreement between the two is evident (see also Figs. 6b and 11b). The first group, which we label D, has 10%–13%; the second and largest group, which we label U, has 82%–86%; and the last and smallest group, which we label S, has about 5% of the winter daily events. Figure 12 shows the average (or centroid) of the 850-K PV distribution for each group when AR and Latc are used. The group D (Fig. 12, left) shows that the vortex has shifted over Scandinavia, reflecting very much a displaced vortex (e.g., Charlton and Polvani 2007). The group U (Fig. 12, middle) shows a typically undisturbed state of the polar vortex. Finally, the last cluster S (Fig. 12, right) shows a stretched structure reminiscent of the split vortex (if we keep in mind that this is an average). In fact, this state looks very much like the state of the vortex two days before splitting (not shown).

To have an idea of the state of the vortex during splitting, we show in Fig. 13 an example of the evolution of the PV field on the isentropic 850-K surface when the polar vortex was in group S around mid-January 1971, based on the latitude, the aspect ratio, and the filtered area of the vortex. The vortex started to split around 15 January, and by 17 January the daughter vortices departed from each other but were still linked through filamentation. Three days later the two daughter vortices became well separated. Note also the positions of both the vortices on longitudes 60°E and 120°W, respectively. The first vortex remained close to Scandinavia and by the end of January it has moved back over the pole. The second vortex moved equatorward over western Canada and then weakened toward the end of January.

To complement the picture obtained from the two-dimensional 850-K PV field (Fig. 12), we examine next the vertical structure of the obtained groups. Figure 14a shows the vertical structure of the zonally averaged polar temperature anomalies, with respect to the winter means, over the 80°–90°N latitude bin for each group. Group U has the coldest temperature throughout most of the stratospheric depth compared to the remaining two groups in accordance with the cold undisturbed state of the vortex. The vertical polar temperature profile for group U is between −5 and 0 K, with the coldest part located in the middle of the stratosphere between 30 and 10 hPa. Group D (displaced) shows by far the largest temperature in most of the stratosphere (Fig. 14a). The temperature anomaly increases from around 7 K at 100 hPa to around 17 K in the middle of the stratosphere, between 30 and 10 hPa, and then decreases upward thereafter with a nearly constant lapse rate of about 2 K hPa^{−1}. The warming of the last group, S, is less than that of group D by about 6 K between 100 and 10 hPa, above which the temperature profiles become similar.

The zonally averaged polar zonal wind profile, averaged over the latitudinal band 70°–80°N, is shown in Fig. 14b and also reflects the significant difference between the three groups. The zonal wind profile for group U increases with height from around 10 m s^{−1} at 100 hPa to about 26 m s^{−1} at 5 hPa. This is exactly the opposite of group D, for which the wind profile decreases with height from 7 m s^{−1} at 100 hPa to about −7 m s^{−1} at 1 hPa, with the largest decreasing rate again between 30 and 10 hPa. Group S, on the other hand, shows an increase of zonal wind from 8 to 13 m s^{−1} between 100 and 25 hPa and then a slow decrease upward to reach 6 m s^{−1} at 1 hPa. The temperature profiles averaged over the 60°–70°N and 70°–80°N latitudinal bands have also been computed. For group U the profiles (not shown) are nearly identical to the 80°–90°N profile (Fig. 14a). For group D the profiles are similar but the maximum temperature decreases to about 13 and 8 K for the 70°–80°N and 60°–70°N bands, respectively, and for group S the associated maximum temperature values are respectively 10 and 7 K. The zonal wind profiles for the 60°–70°N band (not shown) are slightly larger than those for the 70°–80°N band (14b) by about 3 m s^{−1} for group D and S, and up to 10 m s^{−1} for group U, particularly above 5-hPa level. Note that the temperature profiles for vortex split and vortex displacement (Fig. 14) are not significantly different from the results of Charlton and Polvani (2007), which, as they put it, look counterintuitive. We recall, however, that we have chosen latitude bands for temperature (80°–90°N) and zonal wind (70°–80°N) that are quite close to the pole. The winds, in particular, will not be in general easterly here since they are just easterly around 60°N. Also, with a splitting event we should expect slightly less polar warming since the jet (and the ageostrophic descent and the warming associated with it) is not displaced as close to the pole as in a displacement event [for details, see, e.g., O’Neill (2003).].

The wave structure of the different groups is examined next using the mean wave-1 and wave-2 profiles shown in Fig. 15. Wave 1 (Fig. 15a) is largest for groups U and D compared to group S, with the largest difference located near 10 hPa. For group U, the maximum amplitude is located higher up in the stratosphere near 5 hPa, whereas it is located between 20 and 10 hPa for group D and decreases quite rapidly upward thereafter. This is entirely in agreement with the vertical profile of zonal wind for these two groups (Fig. 14b), where the upward decreasing zonal wind of group D becomes easterly between 20 and 10 hPa, inhibiting hence vertical wave propagation. On the other hand, the wind remains westerly for group U but becomes stronger above 5 hPa, and this also limits the upward wave propagation since vertically propagating waves are exhibited, particularly in low-speed westerlies (Charney and Drazin 1961). Wave 1 for group S also propagates upward but remains significantly weaker than for groups U and D. As expected, the structure of wave 2 (Fig. 15b) is spectacularly different between group S and groups U and D. The wave-2 amplitude for the latter two groups is weak throughout the depth of the stratosphere with no sign of vertical propagation. On the other hand, the wave-2 amplitude for group S is much larger than that for the other two groups, by a factor of more than 2 in the lower stratosphere. Wave 2 for group S propagates vertically to reach its maximum amplitude at around 20-hPa level and then decreases upward thereafter. This is also in agreement with the vertical profile of zonal wind for group S, with a moderate maximum at around the same level (Fig. 14b).

The same analysis has been conducted using the three variables, namely the aspect ratio, the centroid latitude, and the filtered vortex area. The results are very similar to those discussed above. The 850-K PV fields of the corresponding clusters (Fig. 16) are quite similar to those shown in Fig. 12, although the splitting case (i.e., group S) is clearer than that shown in Fig. 12. In addition, the filamentation or vortex stretching is also more pronounced in Fig. 16 for the displaced case, with a much clearer comma shape of the vortex compared to the same displaced case of Fig. 12. So clearly the inclusion of the vortex area results in a better structure of the vortex organization. For comparison, Fig. 17 shows the vertical profiles of the polar (80°–90°N) temperature (Fig. 17a), polar (70°–80°N) zonally averaged zonal wind (Fig. 17b), and wave-1 and wave-2 relative amplitude (Figs. 17c,d). The temperature profiles for groups D and S are smaller in amplitude, by about 4–5 K, compared to those of Fig. 14a. The polar zonal winds (Fig. 17b) are also similar to those of Fig. 14b except that the wind for group D decreases to zero, on average, only at 1 hPa and presumably becomes easterly, on average, above this level. Note that many of the vertical profiles of the individual events for group D (not shown) become easterly below 1 hPa. The wave-1 and -2 profiles (Figs. 17c,d) are very similar, with a slightly stronger wave 2 for group S compared to those shown in Fig. 15.

## 5. Summary and discussion

An investigation has been conducted to analyze winter sudden stratospheric warming (SSW) events for possible identification and determination of different groupings that could characterize the dynamics of the stratospheric polar vortex. The analysis is based on stratospheric Northern Hemispheric winter ERA-40 data and uses hierarchical clustering combined with data image techniques applied to geometric moments of the 850-K potential vorticity field. Four moments are considered; the area, the aspect ratio, the centroid latitude, and the excess kurtosis. The kurtosis, a fourth-order moment measuring the peakedness of the vortex, is very skewed and is dominated by a small number of extreme (or outlier) values and is therefore discarded. The vortex area, on the other hand, is found to be dominated by a periodic signal resulting from the phase locking between the peak activity of the polar vortex and the seasonal cycle. The analysis has focused primarily on the aspect ratio and the centroid latitude.

Three strongly significant clusters are found based on the gap statistic and data image when the aspect ratio and the centroid latitude are used. The clusters have sizes of 10%, 86%, and 4%; we label them respectively D, U, and S. The 850-K potential vorticity map of the cluster centroids reveals that cluster D shows a displaced vortex over Scandinavia whereas group U represents the basic undisturbed state of the cold polar vortex. The last and smallest group S shows a stretched vortex very reminiscent of the split vortex. Since the vortex area contains information on the broadness of the vortex, it was included next in the analysis after the periodic signal has been filtered out. Very similar results are obtained when the AR, Latc, and filtered vortex area are used together. The 850-K potential vorticity map of the clusters is quite similar to that obtained when only AR and Latc were used but has more fine structures.

The picture of the obtained groups has been complemented by examining the profiles of the remaining variables that characterize SSW (i.e., polar temperature, zonal wind, and wave-1 and -2 amplitudes). We discuss first the case when AR and Latc are used. The zonally averaged polar temperature anomaly averaged over the 80°–90°N latitude band reveals that group U has the coldest temperature between 0 and −5 K, on average, throughout the depth of the stratosphere in accordance with the state of an undisturbed cold vortex. The largest temperatures are observed with group D, in accordance with its warming state, where the temperature increases, on average, from 7 K at 100 hPa to around 17 K in the middle of the stratosphere between 30 and 10 hPa. The maximum temperature anomaly for the smallest group S is about 10 K between 100 and 10 hPa. Similarly, the strongest polar (70°–80°N) zonal winds are obtained for group U in which the wind increases, on average, from 10 m s^{−1} at 100 hPa to about 26 m s^{−1} at 5 hPa. The smallest wind is obtained with group D for which the wind decreases, on average, from around 7 m s^{−1} at 100 hPa to about −7 m s^{−1} at 1 hPa with a maximum decreasing rate between 30 and 10 hPa. A moderate wind profile is obtained with the smallest group S where an increase from 8 to 13 m s^{−1} is obtained between 100 and 25 hPa.

It is clear from this analysis that groups U, D, and S respectively correlate well with the cold, displaced, and split warming states of the winter polar vortex. The relative wave-1 and wave-2 amplitudes of the geopotential height at 60°N are used to discriminate between these groups regarding vortex splitting or displacement. As expected, wave-1 amplitude is found to be largest for groups U and D, with the maximum for group U located (5 hPa) above that of group D (20–10 hPa) and a rapid decrease with height thereafter in agreement with the zonal wind profile. The wave-1 amplitude of group S is, however, smaller than that of the previous two groups. The wave-2 amplitude of group S, on the other hand, is spectacularly different from the associated amplitudes of groups U and D, which are similar and much weaker than that of group S. Wave 2 for the latter group reaches a maximum at around 20 hPa, also in agreement with the associated zonal wind profile.

The polar temperature and zonal winds as well as wave-1 and wave-2 profiles of the three groups based on all three of the parameters (aspect ratio, centroid latitude, and the filtered vortex area) are found to be quite similar to those discussed above. The temperature profiles are a little smaller than those discussed above. The zonal wind profile for group D decreases with altitude to reach zero at around 1 hPa and presumably becomes easterly, on average, above that level, but many individual wind profiles within this group show easterly flow even below that level. The wave-1 and wave-2 relative amplitudes are very similar to those discussed above. These findings show that the waves associated with split and displaced vortex are distinct and hence do not support the claim for a continuum of SSW as suggested by CG09.

The use of the vortex geometric moments proves to be prominent in identifying the different states of the winter polar vortex and their morphology. The vertical profiles of temperature, zonal wind, and wave amplitudes do overlap as might be expected since the clustering procedure is applied to the moments derived from a single vertical level. However, since the polar vortex is a coherent three-dimensional structure, a natural extension of this work is to examine the same geometric moments at different vertical levels. The use of moments at various levels will ensure a holistic and coherent picture of the structure of the vortex and its groups. The procedure can also be applied to climate model simulations as a test of how well they represent the natural variability of the stratospheric vortex with and without anthropogenic climate change to test the effect of the latter on the SSW events.

## Acknowledgments

The ERA-40 reanalysis data are provided by ECMWF. We thank three anonymous reviewers and the editor for their constructive comments. DM is funded through a Natural Environment Research Council (NERC) scholarship. AH and LG are funded by NERC Centre for Atmospheric Science (NCAS).

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## APPENDIX

### Extended EOFs

*w*,

_{t}*t*= 1, 2, …

*n*, the method consists first in constructing an

*M*-dimensional time series 𝘄

*, for*

_{t}*t*= 1, 2, … ,

*n*−

*M*+ 1, using the delay coordinate as

*M*in (A1) is known as the window length or delay parameter, and sometimes it is also referred to as the embedding dimension, a concept that is rooted in the theory of dynamical systems (see, e.g., Takens 1981) and has to be chosen beforehand. The covariance matrix

**x**

*= (*

_{t}*x*

_{t1}, …

*x*), for

_{tp}*t*= 1, … ,

*n*to yield the new data in the delay coordinate:

*t*= 1, … ,

*n*−

*M*+ 1.

EOFs of the extended data (A1) or (A3) are then computed along with the associated eigenvalues. The obtained EOFs and associated principal component time series are normally referred to as EEOFs and extended PCs (EPCs), respectively. Oscillations are then identified by the existence, in the spectrum of the covariance matrix of the extended data, of separated pairs of identical eigenvalues associated with pairs of (extended) EOFs in quadrature. The singular value decomposition of the delay data matrix (A1) or (A3) yields a decomposition of the data into its extended EOFs and extended PCs and can be used to filter the data by selecting a subset of EEOFs and associated EPCs to yield the filtered or reconstructed fields or signals (see Hannachi et al. 2007).

Time series of the (scaled) AR, Latc, area, and excess kurtosis of the daily winter polar vortex for the first 1000 winter (DJFM) days (i.e., 8.26 winters), starting on 1 Dec 1958.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Time series of the (scaled) AR, Latc, area, and excess kurtosis of the daily winter polar vortex for the first 1000 winter (DJFM) days (i.e., 8.26 winters), starting on 1 Dec 1958.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Time series of the (scaled) AR, Latc, area, and excess kurtosis of the daily winter polar vortex for the first 1000 winter (DJFM) days (i.e., 8.26 winters), starting on 1 Dec 1958.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Autocorrelation function of the vortex area at 850 K. The dashed lines show the boundaries of the 5% significant autocorrelations.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Autocorrelation function of the vortex area at 850 K. The dashed lines show the boundaries of the 5% significant autocorrelations.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Autocorrelation function of the vortex area at 850 K. The dashed lines show the boundaries of the 5% significant autocorrelations.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR and Latc time series and the (b) dissimilarity matrix, (c) dendrogram, and (d) data image based on the tree order. The axes in (a) and (b) represent time (No. of winter days); in (d) the data samples have been reordered following the rightmost part of the dendrogram.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR and Latc time series and the (b) dissimilarity matrix, (c) dendrogram, and (d) data image based on the tree order. The axes in (a) and (b) represent time (No. of winter days); in (d) the data samples have been reordered following the rightmost part of the dendrogram.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR and Latc time series and the (b) dissimilarity matrix, (c) dendrogram, and (d) data image based on the tree order. The axes in (a) and (b) represent time (No. of winter days); in (d) the data samples have been reordered following the rightmost part of the dendrogram.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(top) Log(*W*) of AR and Latc (stars) and that expected from a uniform distribution (circles). (bottom) The corresponding gap statistic. The vertical bars in the bottom panel refer to 1 standard deviation above and below the gap statistic.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(top) Log(*W*) of AR and Latc (stars) and that expected from a uniform distribution (circles). (bottom) The corresponding gap statistic. The vertical bars in the bottom panel refer to 1 standard deviation above and below the gap statistic.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(top) Log(*W*) of AR and Latc (stars) and that expected from a uniform distribution (circles). (bottom) The corresponding gap statistic. The vertical bars in the bottom panel refer to 1 standard deviation above and below the gap statistic.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data images of the dissimilarity matrix of AR and Latc when (a) two, (b) three, and (c) four clusters are used. The axes are similar to those of Fig. 4b, except that now the data samples are grouped into clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data images of the dissimilarity matrix of AR and Latc when (a) two, (b) three, and (c) four clusters are used. The axes are similar to those of Fig. 4b, except that now the data samples are grouped into clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data images of the dissimilarity matrix of AR and Latc when (a) two, (b) three, and (c) four clusters are used. The axes are similar to those of Fig. 4b, except that now the data samples are grouped into clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Gap statistic curves from 100 randomized samples from AR and Latc time series and (b) frequency *k*_{opt}. The subsamples have a sample size equal to half that of the data and are drawn by randomizing 2-month blocs.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Gap statistic curves from 100 randomized samples from AR and Latc time series and (b) frequency *k*_{opt}. The subsamples have a sample size equal to half that of the data and are drawn by randomizing 2-month blocs.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Gap statistic curves from 100 randomized samples from AR and Latc time series and (b) frequency *k*_{opt}. The subsamples have a sample size equal to half that of the data and are drawn by randomizing 2-month blocs.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Spectrum of the (delay) covariance matrix of (a) the vortex area, the leading two (b) EEOFs and (c) EPCs, and (d) the phase diagram of EPC1 vs EPC2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Spectrum of the (delay) covariance matrix of (a) the vortex area, the leading two (b) EEOFs and (c) EPCs, and (d) the phase diagram of EPC1 vs EPC2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Spectrum of the (delay) covariance matrix of (a) the vortex area, the leading two (b) EEOFs and (c) EPCs, and (d) the phase diagram of EPC1 vs EPC2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Original vortex area time series (thin) and the reconstructed periodic signal (thick) based on the leading four EEOFs/EPCs and (b) the filtered time series obtained by subtraction of the reconstructed signal.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Original vortex area time series (thin) and the reconstructed periodic signal (thick) based on the leading four EEOFs/EPCs and (b) the filtered time series obtained by subtraction of the reconstructed signal.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

(a) Original vortex area time series (thin) and the reconstructed periodic signal (thick) based on the leading four EEOFs/EPCs and (b) the filtered time series obtained by subtraction of the reconstructed signal.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Frequency of optimal number of clusters obtained from 100 subsamples from the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area. The subsamples are drawn as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Frequency of optimal number of clusters obtained from 100 subsamples from the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area. The subsamples are drawn as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Frequency of optimal number of clusters obtained from 100 subsamples from the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area. The subsamples are drawn as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area, using three clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area, using three clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Data image of the (a) AR, Latc, and original vortex area and (b) AR, Latc, and filtered vortex area, using three clusters.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

850-K PV field for groups D, U, and S based on the Latc and AR of the vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

850-K PV field for groups D, U, and S based on the Latc and AR of the vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

850-K PV field for groups D, U, and S based on the Latc and AR of the vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Example of daily PV field evolution during a split vortex event taken from group S.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Example of daily PV field evolution during a split vortex event taken from group S.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Example of daily PV field evolution during a split vortex event taken from group S.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the winter (DJFM) (a) zonally averaged polar temperature anomalies averaged over the 80°–90°N latitude band and (b) the 70°–80°N latitude band for the zonally averaged zonal wind for the obtained three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR and Latc of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the winter (DJFM) (a) zonally averaged polar temperature anomalies averaged over the 80°–90°N latitude band and (b) the 70°–80°N latitude band for the zonally averaged zonal wind for the obtained three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR and Latc of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the winter (DJFM) (a) zonally averaged polar temperature anomalies averaged over the 80°–90°N latitude band and (b) the 70°–80°N latitude band for the zonally averaged zonal wind for the obtained three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR and Latc of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the mean (a) wave-1 and (b) wave-2 relative amplitude geopotential height at 60°N for the same groups as in Fig. 14.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the mean (a) wave-1 and (b) wave-2 relative amplitude geopotential height at 60°N for the same groups as in Fig. 14.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of the mean (a) wave-1 and (b) wave-2 relative amplitude geopotential height at 60°N for the same groups as in Fig. 14.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

As in Fig. 12, but for the Latc, AR, and filtered vortex area.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

As in Fig. 12, but for the Latc, AR, and filtered vortex area.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

As in Fig. 12, but for the Latc, AR, and filtered vortex area.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of (a) polar (80°–90°N) temperature, (b) polar (70°–80°N) zonal wind, and (c) wave-1 and (d) wave-2 relative geopotential height amplitude from the three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR, Latc, and (filtered) area of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of (a) polar (80°–90°N) temperature, (b) polar (70°–80°N) zonal wind, and (c) wave-1 and (d) wave-2 relative geopotential height amplitude from the three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR, Latc, and (filtered) area of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Vertical profile of (a) polar (80°–90°N) temperature, (b) polar (70°–80°N) zonal wind, and (c) wave-1 and (d) wave-2 relative geopotential height amplitude from the three groups D (continuous), U (dashed), and S (dashed–dotted) obtained using the AR, Latc, and (filtered) area of the polar vortex.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3585.1

Contingency table showing the proportion (%) of the three groups D, U, and S when AR and Latc or AR, Latc, and filtered vortex area are used.

^{1}

For this simple example the tree clearly shows two distinct branches or clusters.

^{2}

The silhouette method (Rousseeuw 1987) measures the difference between the similarity of any point to other points in its own cluster and similarities that it has with points in other clusters. The upper-tail rule uses the distribution of a clustering criterion associated with each hierarchical level to select the best number of clusters. The reader is referred to the previous references for more details.

^{3}

If one cuts the tree with a vertical line to obtain only two clusters (near the left-hand side of the dendrogram), then the first cluster, composed of data that fall in the upper branch, is much smaller than the other.