1. Introduction
There exist competing linear and nonlinear paradigms for interpreting the structure and dynamics of atmospheric variability. The linear paradigm treats the variability as small perturbations about the mean state. It focuses on preferred patterns of variability, the time-varying indices of which tend to be normally distributed. The anomalies observed in association with a given pattern are most likely to be small, and a given pattern is equally likely to occur with either polarity. The nonlinear paradigm is concerned with preferred circulation regimes in which the anomalies may be quite large and in which one polarity of the pattern of anomalies is clearly favored over the other polarity. Observational studies of teleconnection patterns have tended to be framed in terms of the linear paradigm (e.g., Panagiotopoulos et al. 2002), and studies of recurrent circulation regimes in terms of the nonlinear paradigm. Early examples of the latter include the Grosswetterlagen (Baur 1951), blocking (Berggren et al. 1949; Rex 1950), and multiple equilibria (Charney and DeVore 1979).
Observational studies based on the linear paradigm have relied mainly on regression analysis, empirical orthogonal function (EOF) analysis, and other tools derived from linear algebra, whereas studies based on the nonlinear paradigm have usually relied on cluster analysis. Among the highly cited works in the second category was the study of Mo and Ghil (1988, hereafter MG88), who used a categorical method that yielded six fuzzy clusters of the Northern Hemisphere wintertime 500-hPa height field. The spatial patterns of the clusters identified by MG88 were hemispheric in extent with multiple centers of action reminiscent of the teleconnection patterns identified in studies based on the linear paradigm.
Simpler, more robust patterns can be obtained by identifying local peaks in the probability density function (PDF) in a reduced phase space created by expanding the field to be analyzed into its leading EOFs and retaining only the leading modes in the analysis. Among the first studies based on this methodology were those of Benzi et al. (1986) and Hansen and Sutera (1986), which were focused on the bimodality of a single index. Molteni et al. (1990), Kimoto and Ghil (1993), Smyth et al. (1999), and Corti et al. (1999) considered the EOFs of the PDF of the wintertime hemispheric 500-hPa height field and obtained a consistent set of patterns. Cheng and Wallace (1993, hereafter CW) obtained a similar set of patterns employing the hierarchical clustering method of Ward (1963). Unlike the aforementioned clustering algorithms, Ward’s method does not involve truncation in EOF phase space, but, as it turns out, similar modes are obtained regardless of whether a truncation to the two leading modes is performed. For further discussion of these studies, the reader is referred to the review of Ghil and Robertson (2002). These methods can be used to categorize all days in a dataset or a subset of the days, depending on the analysis protocol.
An alternative clustering scheme known as self-organizing maps (SOMs) has been used in several recent studies of atmospheric circulation patterns (Reusch et al. 2007; Johnson et al. 2008; Johnson and Feldstein 2010; Lee et al. 2011; Lee and Feldstein 2013; Feldstein and Lee 2014). “Maps” in SOM refers not to individual clusters but to arrays of clusters. SOM is a special case of a competitive neural network with the self-organizing feature that ensures that neighboring clusters in the array are more similar than distant clusters. In contrast to Ward’s methods, SOM categorizes all days in the dataset.
In this study, we use SOM to characterize hemispheric regimes of the Northern Hemisphere wintertime 500-hPa height field. Datasets and methodology, including the metrics that we use for assessing the distinctiveness and reproducibility of the clusters, are described in the next section. In section 3, we compare clusters obtained from SOM and Ward’s method, and we consider how the SOM clusters are affected by the following:
The prescribed dimensions of the array of clusters
The inclusion of a “null cluster” to which the ambiguous days are assigned
Whether the analysis is performed in a hemispheric or a sectoral domain
2. Data and methods
Three different reanalysis datasets are used in this study:
The four-times-daily 500-hPa geopotential height (Z500), 850-hPa temperature, and mean sea level pressure data from the 45-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40 for 1957–2002; Uppala et al. 2005)
The four-times-daily Z500, 850-hPa temperature, and mean sea level pressure data from the ECMWF interim reanalysis (ERA-Interim for 1979–2014) dataset (Dee et al. 2011)
The daily Z500 only from the NOAA Twentieth-Century Reanalysis (20CR for 1919–2012; Compo et al. 2011)
The ERA-40 dataset from September 1957 through December 1978 was combined with the ERA-Interim dataset from January 1979 through March 2014 to create a continuous 57-winter dataset. It is shown in Fig. S1 (left panel) that the daily 500-hPa height fields in ERA-Interim and ERA-40 are virtually identical, except over the Arctic, and even there the rms difference is less than 10 m, not nearly enough to materially affect the results of the clustering. Both datasets are 2.5° longitude by 2.5° latitude horizontal resolution.
The 20CR was used only for the purpose of extending the cluster chronology further backward in time. This dataset is based on surface observations of synoptic pressure only. Its temporal coverage extends from 1871 through 2012, but we only used data from 1919 onward. As shown in Fig. S1 (right panel), the rms difference between daily 20CR and ERA-40 500-hPa height fields is much larger than the difference between ERA-Interim and ERA-40. However, we show in section 5 that the cluster chronologies derived from the two datasets are, nonetheless, very similar.
The 20CR dataset is provided in a 2° longitude by 2° latitude horizontal resolution, but we interpolated it to a 2.5° by 2.5° grid for computational convenience. In performing cluster analysis, we weight the data by the square root of the cosine of latitude so that the variance at each data point is weighted in accordance with the area that it represents.
Cluster analysis is carried out over the Northern Hemisphere Z500 field poleward of 20°N during the winter season, defined as the 90-day period from 1 December through 28 February. Data from November and March are also used in the filtering. The year label applied to each winter corresponds to January: for example, the 1957/58 winter is referred to as the 1958 winter. The seasonal cycle, averaged over the 57 years, is removed from the daily height field to obtain the field of Z500 anomalies. Then a 10-day low-pass filter is applied to eliminate the high-frequency variability associated with baroclinic waves. The effective time resolution of the smoothed data is the same as in CW.
Hierarchical clustering is a way of investigating the grouping of data points within a dataset simultaneously over a range of degrees of similarity by creating a cluster tree. The tree is not a single set of clusters but, rather, a multilevel hierarchy in which the clusters at one level are merged to form the clusters at the next level. The user is called upon to decide what scale or level of clustering is most appropriate for the application in question. Further details are provided in CW.
In SOM analysis the user specifies in advance the dimensions of the array of clusters in a low- (typically two-) dimensional array. Both SOM and Ward’s method make use of the minimum Euclidean distance between maps, but, in Ward’s method, it is used as a basis for determining which clusters will be merged to form a new cluster at the next level of the hierarchy, whereas, in SOM, the quantity to be minimized is the mean squared distance between the maps in the original dataset and the clusters to which they are assigned (Lee et al. 2011). For further specifics, the reader is referred to the appendix of Johnson et al. (2008). Results reported in the paper are for a 2 × 2 array, except in section 3, in which SOM is performed on arrays of various dimensions.
Choosing how many clusters to retain (in Ward’s method) or prescribe (in SOM) involves a trade-off between distinctiveness and robustness: smaller, more numerous clusters are more distinctive but more susceptible to the vagaries of sampling variability. For each cluster obtained in this analysis, we computed metrics for both, and we used them as guidance for how many clusters should be examined in detail.
A simple and widely used way of assessing the distinctiveness of a cluster is to compare the so-called “external variance”—the squared distance between its centroid and the centroid of the entire dataset—with the total variance (i.e., the mean squared distance between the individual maps in that cluster and the centroid of the entire dataset). The variance ratio (VR), obtained by dividing the former by the latter, increases as larger clusters are subdivided into smaller tighter clusters, and it reaches its limiting value of unity when each cluster has only one member.
The countervailing robustness (or reproducibility parameter) metric (RP) is a measure of how much the clusters change in response to small perturbations in the input data used in the analysis. As a measure of how much a given cluster changes, we used the spatial correlations between the Z500 anomaly charts for its centroid. We perturbed the input data by partitioning them into consecutive, nonoverlapping pentads and assigning day 1 of each pentad to the first subset, day 2 to the second subset, and so on. Each of the five subsets consists of 1026 daily maps. We compute RP between the most closely matching clusters, comparing the patterns for day 1 with those for days 2, 3, 4, and 5; those for day 2 with those for days 3, 4, and 5; those for day 3 with those for days 4 and 5; and the patterns for days 4 with day 5—a total of ten permutations—and we average the 10 values of RP to obtain a single value for each cluster. Because of the strong autocorrelation inherent in the data, which is enhanced by the low-pass filtering, the clusters derived from the five subsets are quite similar. However, as we will presently show, the perturbations in the input data are sufficiently large to enable us to distinguish the dependence of RP upon clustering protocol and to observe its decline as the prescribed number of SOM clusters is increased.
3. Cluster analysis
We begin by repeating the analysis of CW. We performed Ward’s method on the five subsets of the data individually, retaining the four most reproducible clusters with absolute values in excess of 100 m, at least one grid point in each case. We then average the corresponding clusters derived from the five subsets to obtain a single set of clusters. The values of RP are based on spatial correlations between corresponding clusters in the five subsets of the input. As explained in the previous section, each value represents the average of 10 different comparisons. The resulting clusters are shown in Fig. 1. The first three patterns correspond closely to the regimes G′, A′, and R′ in CW (their Fig. 7).
Composite 500-hPa height anomaly maps of the cluster derived from Ward’s method. Contour interval: 25 m; red (blue) contours denote positive (negative) values; the zero contours are omitted. The value of RP is shown at the top right of each panel.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
Figure 2 shows the clusters derived from SOM using a 2 × 2 array and averaged over the five subsets of the data. Strictly speaking, they should be displayed in the format of a 2 × 2 array, but no useful information is lost by lining them up in a single row with the end members being ones that consistently appear diagonally across from one another in the array. The SOM patterns are reminiscent of the most reproducible clusters obtained using Ward’s method, shown in Fig. 1, but they exhibit much higher values of RP. It is notable that the clusters derived from Ward’s method comprise only about one-half of the days in the record, whereas the SOM clusters shown in Fig. 2 comprise all days. One might have expected the method that is more selective with respect to cluster membership to yield more distinctive clusters with higher variance ratios. However it is evident from a visual comparison of the clusters in Figs. 1 and 2 that this is not the case: the variance ratios are, in fact, quite comparable. Hence, the SOM clusters also appear to be superior to the clusters derived from Ward’s method by virtue of their much larger membership.
Composite 500-hPa height anomaly maps of the cluster derived from SOM. Contour interval: 25 m; red (blue) contours denote positive (negative) values; the zero contours are omitted. The value of the variance ratio is shown at the bottom right of each panel, and the value of RP is shown at the top right.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
We tested the sensitivity of SOM to the dimensions of the prescribed array of maps using each of the configurations listed in Tables 1 and 2. Johnson et al. (2008) also considered the sensitivity of his clusters to array size but provided no objective measure of the similarity of corresponding clusters. Table 1 shows the spatial correlations between the SOM patterns based on the 2 × 2 array shown in Fig. 2 and the most similar patterns derived individually from the other array configurations, and Table 2 shows RP for the clusters derived from each of the configurations. The sets of patterns for the 2 × 2 and 1 × 4 arrays are virtually identical, and both sets are highly reproducible. Clusters for the array sizes larger than four are not as reproducible as those obtained from the 2 × 2 and 1 × 4 arrays. A cluster very similar to cluster 1 (C1) is recovered in all the array configurations, and clusters somewhat similar to C2 and C3 are recovered in most of them. It appears that, for this particular application, increasing the SOM array size beyond four appears to be counterproductive.
Spatial correlations between the SOM patterns based on the 2 × 2 array and the most similar patterns derived from the other array configurations indicated in the first column. Boldface denotes a value higher than 0.80 and an asterisk represents a value less than 0.70.
RP for the clusters derived from each of the array configurations arranged in descending order. Boldface denotes a value higher than 0.80 and an asterisk represents a value less than 0.70.
a. Comparison of the spatial patterns derived from Ward’s method and SOM
Although the Z500 patterns in the structures derived from Ward’s method and SOM, shown in Figs. 1 and 2, appear similar at first sight, there are some important distinctions between them. CW showed that the more salient features in the former are recoverable in a truncated Z500 dataset, in which only the two leading EOFs of the Z500 field are retained. The collinearity of the clusters manifests itself in symmetries: the Pacific–North America (PNA) pattern, which dominates the second EOF, is clearly apparent with opposing polarity in the second and third patterns derived from Ward’s method (Fig. 1), and the North Atlantic Oscillation (NAO) signature, which dominates the leading EOF, is present with opposing polarity in the first and fourth clusters. The patterns identified by Molteni et al. (1990), Kimoto and Ghil (1993), Smyth et al. (1999), and Corti et al. (1999) all exhibit a similar symmetry. In contrast, the SOM patterns shown in Fig. 2 do not exhibit such obvious collinearity. For example, the pattern in the second cluster is not present in any of the other clusters, and it does not resemble either of the two leading EOFs of the Z500 field. We recover an NAO-like signature, but it is clearly apparent only in the first cluster shown in Fig. 2. Because of their greater robustness, their larger numbers of member maps, and their greater linear independence, we will consider only the SOM clusters from here onward.
b. Inclusion of a null cluster
The Z500 maps for some days are almost equidistant from the centroids of two different clusters. The ambiguity in cluster membership is greatest for those days on which maps lie relatively close to the climatology. By relegating these ambiguous days to a null cluster, it should be possible to increase the variance ratios of the clusters from which they are removed. To test this idea, we excluded from cluster membership those days on which the difference between the distance of its Z500 map from the centroids of the nearest and next nearest clusters is less than some prescribed value d0 and adjusted the cluster centroids accordingly. By setting d0 = 14 m, we relegated about two-thirds of the days in the dataset to the null cluster, which is roughly equal to the fraction of the days that were left unclassified in the analysis of CW based on Ward’s method. For each of the four SOM clusters, we created composite Z500 maps for the “reserve cluster” and for the days that formerly belonged to that cluster but were relegated to the null cluster. The samples shown in Fig. 3 are typical. The amplitudes of the reserve clusters are about 1.5 times as large as those of the original clusters shown in Fig. 2. Hence, it can be said that for clusters with comparable numbers of member days, the SOM clusters exhibit about a 50% greater variance ratio than the clusters derived from Ward’s method, shown in Fig. 1. The composite for the relegated days retains the shape and about three-fourths of the amplitude of the original clusters.
As in Fig. 2, but composite of the (top) reserve members and (bottom) members relegated to the null cluster. Contour interval: 25 m.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
c. Sectoral clusters
The clusters derived from the analysis of the full Northern Hemisphere domain tend to be centered over or close to North America. To investigate the possible existence of additional flow regimes over Eurasia, we performed cluster analysis on 180° sectors centered at various longitudes. This admittedly cursory survey yielded one interesting result, shown in Fig. 4. The clusters for the Western Hemisphere, extending from the date line, across North America, to the Greenwich meridian, resemble their hemispheric counterparts shown in Fig. 2, whereas the clusters in the opposing hemisphere are entirely different. The first of these is indicative of a weakening of the climatological-mean trough over eastern Siberia and a southward shifting of the climatological jet stream across northern China, Japan, and the North Pacific. The second and fourth involve Eurasian wave trains. We have not computed RP for the eastern clusters, but there are indications that they may be quite fragile. For example, we obtain quite different results, even for the western sector, when we use the smoothly varying weighting function 
Composite maps for the sectoral clusters of (top) the Western Hemisphere, extending from the date line, across North America, to the Greenwich meridian and (bottom) the Eastern Hemisphere from the Greenwich meridian, across Eurasia, to the date line. Contour interval: 25 m. The sectoral clustering is used to determine which dates belong to which clusters, but the composite maps are plotted for the full Northern Hemisphere domain.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
4. Synoptic interpretation of the SOM clusters
The first of the SOM clusters shown in Fig. 2 (herein C1, and similarly for C2, C3, and C4) resembles the Z500 anomaly pattern associated with the negative polarity of the NAO or Northern Hemisphere annular mode (NAM). Extended episodes in which this pattern is present are characterized by an enhanced frequency of occurrence of blocking over West Greenland (Woollings et al. 2010) and enhanced intermediate frequency (6–30-day period) variability (Rennert and Wallace 2009). The primary positive center of action of C2 is located over the Gulf of Alaska, another region of frequent blocking (Hartmann and Ghan 1980; Tyrlis and Hoskins 2008). It resembles the one-point (linear) correlation pattern for its primary center of action (55°N, 145°W; not shown). Hence, it is conceivable that Z500 anomalies at the downstream centers of action could be forced by Gulf of Alaska blocking. C2 does not project strongly upon any of the widely cited (linear) teleconnection patterns. C3 projects upon the PNA pattern and, to a lesser extent, upon the east Atlantic and Eurasian patterns of Wallace and Gutzler (1981). It resembles CW’s “Rockies Ridge” cluster. C4 also comprises PNA and North Atlantic–European wave trains.
The corresponding composite (total) Z500 fields for C1, C2, and C3 are shown in Fig. 5. C1 exhibits a distinctive blocking signature centered over the southern tip of Greenland. The northern branch of the westerlies over northern Europe is substantially weaker than in the other clusters, and the southern branch over the subtropical Atlantic, the Mediterranean, and North Africa is enhanced. C2 exhibits a pronounced ridge over the Gulf of Alaska and the Bering Strait, an enhanced downstream trough over central and eastern Canada, and a strong westerly jet extending across the North Atlantic from the northeastern United States to the British Isles. In C3, the features of the climatology—the mean trough over the Gulf of Alaska and the ridges over western North America and the eastern North Atlantic—are accentuated.
Composite total Z500 fields for the first three SOM clusters. Contour interval: 100 m. The 5600-m contour is bold. The red dots mark the primary positive centers in the Z500 anomalies for (left) C1 and (middle) C2, transcribed from Fig. 2.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
Composite 850-hPa temperature anomalies for C1, C2, and C3 are shown in Fig. 6. The strongest feature in C1 is the blob of positive temperature anomalies with amplitudes up to 7°C centered over southern Baffin Island in the Canadian Arctic, just to the west of the center of the blocking high in the Z500 field. Broad swaths of weaker negative anomalies extend across midlatitude North America and northern Eurasia. The overall C1 pattern resembles the anomalies observed in association with the negative polarity of the NAO–NAM (Hurrell 1995; Thompson and Wallace 2000). C2 is attended by anomalous cold centered over the Canadian prairies and covering most of Canada and the northern United States to the east of the Rockies, as well as anomalous warmth in the ridge in the Z500 field centered over the Bering Strait. C3 is marked by anomalous warmth centered over western Canada just to the east of the upper-level ridge in the anomalously strong downslope flow in the lee of the Rockies.
Composite 850-hPa temperature anomalies for the first three SOM clusters. Contour interval: 1°C; red (blue) contours denote positive (negative) values; the zero contours are omitted.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
SLP anomaly maps for C1, C2, and C3 are shown in Fig. 7. C1 exhibits the familiar “NAO–NAM minus” pattern marked by positive anomalies center over Iceland and negative anomalies centered over the Azores. C2 features a positive center over the south coast of Alaska near Anchorage and a negative center over the North Atlantic between the British Isles and Iceland. Patterns similar to the composite SLP patterns for C1 and C2 can be recovered by performing SOM analysis directly upon the hemispheric SLP field. C3 features a negative center over the Gulf of Alaska and a positive center over the eastern North Atlantic.
Composite sea level pressure anomalies for the first three SOM clusters. Contour interval: 2 hPa. The zero contours are omitted.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
The centers of the composite SLP anomalies in C1 and C2 are arranged such that anomalous northerly geostrophic flow overlies the more prominent cold anomalies in Fig. 6, and vice versa. The strong anomalous easterly geostrophic flow in C1 favors reduced rainfall over northern Europe and it allows cold, continental air masses over the interior of Eurasia to spread westward into western Europe. The flow around the anomalous cyclone over the North Atlantic in C2 favors wet, stormy weather over the British Isles and other parts of northwestern Europe.
5. Cluster chronologies
Using the 20CR dataset, we expanded the frame of the study backward in time to the 1920 winter. Rather than performing a separate cluster analysis based on 20CR, we simply assigned days for the 1920–57 winters to the nearest of the clusters defined on the basis of the ERA data. Contingency tables for the occurrence or nonoccurrence of each cluster within the reference period 1958–2002, shown in Table S1, confirm that the ERA and 20CR datasets yield nearly identical cluster chronologies. For the benefit of the reader who might also be interested in cluster frequencies during early and late winter, we also include the months of November and March. Day-by-day records of the occurrence of clusters 1–4 are provided in graphical form in Figs. S2–S5 and as text files in the online supplement.
A year-by-year summary of the number of daily occurrences of each cluster in the 90-day December–February (DJF) winter is shown in the form of bar graphs in Fig. 8 and as a year-by-year tabulation in Table 3. C1 is the rarest of the four clusters, occurring, on average 17.6 days per winter, and C3 is the most frequent, with an average of 27.1 occurrences per winter. The number of days in which the atmosphere resides in individual clusters varies erratically from one winter to the next, with a standard deviation of approximately 15 days for all four clusters. There were no occurrences of C1 during 3 of the past 10 winters (2008, 2012, and 2014), while the 2010 winter was marked by 70 daily occurrences out of 90 winter days, by far the most of any winter since 1920. The 2014 winter was marked by 70 occurrences of C2, also a record number.
Number of days of occurrence of C1–C4 and either C1 or C2, winter-by-winter, DJF 1920–2014. Years 1920–2012 are from the 20CR dataset, and years 2013–14 are from the ERA-Interim dataset. The year designation corresponds to JF.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
Year-by-year tabulations for occurrence days of clusters C1–C4 during DJF 1920–2012 based on the 20CR dataset; 2013–14 winters are from the ERA-Interim dataset. The year designation corresponds to JF.
The 2010 and 2014 winters were both notable for the dominance of a single cluster (C1 for 2010 and C2 for 2014), and C3 was even more dominant during the winter of 1931. DJF-mean Z500 anomaly charts for these winters are shown in Fig. 9. The 2010 winter was extraordinarily warm over the Canadian Arctic and cold over Europe and parts of the eastern United States (Seager et al. 2010; Cattiaux et al. 2010). The 2014 winter was unusually cold by recent standards over much of North America and especially over central Canada and the north-central United States (Wallace et al. 2014; Hartmann 2015). It was also marked by intense storms and heavy rainfall in northwestern Europe (Huntingford et al. 2014). The 1931 winter was notable for the warmth of the central United States (Henry 1930; Hunter 1931).
DJF-mean Z500 anomaly in (left) 2010, (middle) 2014, and (right) 1931. Contour interval: 30 m; red (blue) contours denote positive (negative) values. The zero contours are omitted.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0001.1
6. Summary and discussion
We have examined the four reproducible clusters for the Northern Hemisphere wintertime 500-hPa height field as determined by self-organizing maps (SOMs). We have shown that they bear some resemblance to clusters derived from Ward’s method, but they are superior in the following respects:
They comprise all the days in the dataset, whereas the clusters derived from Ward’s method comprise only about half the days.
They are more robust with respect to small perturbations in the data matrix.
Their spatial patterns appear to be less collinear and less dominated by the two leading EOFs.
Despite their larger membership, their external-to-total variance ratios are as large as those of the reproducible clusters derived from Ward’s method.
The reason why the SOM clusters are more distinctive and more robust than the clusters derived from Ward’s method remains to be resolved. William Hsieh (2015, personal communication; University of British Columbia) speculates that it may be because, when SOM is trained, data for all days are used to shift the positions of all the centroids such that the position of each centroid is influenced by data for all days. In contrast, hierarchical clustering algorithms, such as Ward’s method, create new clusters by merging the nearest preexisting clusters, so the outcome is influenced only by those days that belong to those clusters.
Our results indicate that the hemispheric clusters capture the important circulation regimes in the Western Hemisphere and the European sector, but, for documenting the frequency of occurrence of different circulation regimes over Asia, it will be necessary to resort to sectoral clusters. A possible strategy that we have briefly explored is to analyze the Eastern and Western Hemispheres separately.
In principle, the ability to organize maps in two-dimensional arrays based on similarity is an attractive feature of SOM, but we have not found it useful for this particular application, because the number of reproducible clusters is not large enough to take advantage of it. In our cursory exploration of the dependence of our reproducibility parameter (RP) on the prescribed dimensions of the SOM matrix, the RP was largest for the clusters derived from the 2 × 2 and 1 × 4 matrices. Whether equally reproducible clusters can be recovered from SOM analyses with larger matrices [e.g., as in the studies of Johnson et al. (2008) and Johnson and Feldstein (2010)] remains to be seen. In any case, cross validation of some kind appears to be necessary to ensure that the prescribed array size is commensurate with the number of statistical degrees of freedom inherent in the dataset. Ward’s method is subject to an analogous limitation with regard to the inclusion of clusters drawn from lower ranks in the hierarchy.
We have interpreted three of the four hemispheric clusters identified in this study with circulation regimes: C1, the negative polarity of the NAO, which is often observed in association with blocking over Greenland; C2, a pattern observed in association with blocking over the Gulf of Alaska with a cold downstream trough to the east of the Rockies and a strong westerly jet across the North Atlantic, directed toward the British Isles; and C3, an enhancement of the climatological-mean stationary waves and particularly the trough over the Gulf of Alaska and the downstream ridge over the Rockies. We consider the fourth cluster, C4, to be of less interest from the point of view of synoptic climatology. C1 and C2 are both marked by a relatively high frequency of occurrence of cold-air outbreaks over North America to the east of the Rockies, and C1 is marked with anomalously cold, dry weather over Europe, with the storm track shifted southward into the Mediterranean. The identification of C1 with the Greenland blocking regime is consistent with the interpretation in Woollings et al. (2010), and it is conceivable that C2 could bear an analogous relationship to blocking over Alaska and that the less distinctive C3 and C4 could represent a partitioning of the residual nonblocking climatology.
Trends and other statistics relating to the winter-to-winter variations of the frequency of occurrence of particular clusters are subject to large sampling variability, which reflects the small number of statistical degrees of freedom (dof). In a 90-day winter, assuming a 5–10-day e-folding decorrelation time for the NAO and other Northern Hemisphere wintertime planetary-scale circulation patterns (Feldstein 2000) and two e-folding times per dof (Leith 1973), dof ≈ 5–10, which are partitioned among four different cluster time series. In the combined frequency of occurrence of C1 and C2, shown in the bottom panel of Fig. 8, the record high values for 3 of the past 5 years is consistent with the notion that high-latitude blocking has been a recurrent feature of the circulation (Francis and Vavrus 2012), but whether statistics relating to such a short segment of the record are early indications of a change in the blocking (and cluster) climatology or whether they are merely a reflection of sampling variability remains to be seen.
We also performed 2 × 2 SOM analysis on the hemispheric SLP field to see whether the composite SLP patterns shown in Fig. 7 could be recovered directly. The results were mixed. Not unexpectedly, a pattern similar to the SLP composite for C1 emerged as the leading SLP cluster with a variance ratio (22.4%) even larger than that of C1, and a pattern similar to that of the C2 composite emerged as the fourth SLP cluster (ranked in terms of variance ratio). The primary centers of action of the other two SLP clusters (not shown) lie over the Eurasian sector. The reproducibility and synoptic interpretation of these patterns remains unexplored.
Acknowledgments
We thank William Hsieh from the University of British Columbia for his help in addressing the question of why the performance of SOM is superior to that of Ward’s method. We are grateful to Michael Ghil at UCLA and Masahide Kimoto at the University of Tokyo for sharing their recollections regarding their previous studies. We thank the anonymous reviewers for their helpful comments and suggestions. This work was supported by National Natural Science Foundation of China (Grants 41175053, 41275054, and 41330420) and a Nanjing University Overseas Study Scholarship to MB. Participation of JMW was supported by the U.S. National Science Foundation Climate Dynamics Program Office under Grant ATM 1122989.
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