Identifying Shifts in Modes of Low-Frequency Circulation Variability Using the 20CR Renalysis Ensemble

Vladimír Piskala aFaculty of Science, Charles University, Prague, Czech Republic

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Radan Huth aFaculty of Science, Charles University, Prague, Czech Republic
bInstitute of Atmospheric Physics, Czech Academy of Sciences, Prague, Czech Republic

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Abstract

Principal component analysis (PCA) is a widely used technique to identify modes of low-frequency variability of atmospheric circulation and their spatial changes. However, it turns out that PCA is highly sensitive to the period analyzed and the length of the time window used. Its results can vary considerably if the period is shifted by even 1 year. We present temporal variability of modes from the late nineteenth century using moving PCA of winter (DJF) monthly mean 500-hPa height anomalies for 20–50-yr moving periods with 1-yr step. We employ the congruence coefficient to compare spatial patterns of the modes and identify their substantial changes. Shorter moving periods are more susceptible to sudden fluctuations in mode patterns from one period to the next, while longer periods yield more stable results. We strongly recommend applying a moving PCA to detect spatial changes in modes of low-frequency variability, as it unveils any hidden sudden changes in the modes. These changes can be influenced by many aspects, such as data quality, sampling variability, and length of the analyzed period. Spatial patterns of the Atlantic–European modes are more stable across ensemble members than those over the Pacific and North America, especially before the 1920s. During this period, North Atlantic and European modes explain more variance in the ensemble mean than in ensemble members, while the reverse holds for Pacific and North American modes. In data-sparse regions, modes in ensemble members exhibit greater variability. The process of averaging then leads to weaker modes in the ensemble mean, explaining less variance compared to ensemble members.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Vladimír Piskala, vladimir.piskala@natur.cuni.cz; Radan Huth, radan.huth@natur.cuni.cz

Abstract

Principal component analysis (PCA) is a widely used technique to identify modes of low-frequency variability of atmospheric circulation and their spatial changes. However, it turns out that PCA is highly sensitive to the period analyzed and the length of the time window used. Its results can vary considerably if the period is shifted by even 1 year. We present temporal variability of modes from the late nineteenth century using moving PCA of winter (DJF) monthly mean 500-hPa height anomalies for 20–50-yr moving periods with 1-yr step. We employ the congruence coefficient to compare spatial patterns of the modes and identify their substantial changes. Shorter moving periods are more susceptible to sudden fluctuations in mode patterns from one period to the next, while longer periods yield more stable results. We strongly recommend applying a moving PCA to detect spatial changes in modes of low-frequency variability, as it unveils any hidden sudden changes in the modes. These changes can be influenced by many aspects, such as data quality, sampling variability, and length of the analyzed period. Spatial patterns of the Atlantic–European modes are more stable across ensemble members than those over the Pacific and North America, especially before the 1920s. During this period, North Atlantic and European modes explain more variance in the ensemble mean than in ensemble members, while the reverse holds for Pacific and North American modes. In data-sparse regions, modes in ensemble members exhibit greater variability. The process of averaging then leads to weaker modes in the ensemble mean, explaining less variance compared to ensemble members.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Vladimír Piskala, vladimir.piskala@natur.cuni.cz; Radan Huth, radan.huth@natur.cuni.cz

1. Introduction

Principal component analysis (PCA) is a statistical tool often used to describe atmospheric circulation in terms of modes of intraseasonal or interannual variability (referred to simply as “modes” in further text). It has been widely used for this purpose since the 1960s (Kutzbach 1967, 1970; Craddock 1973; Wallace and Gutzler 1981). Since then, results obtained by PCA have been shown to vary depending on several subjective methodological choices. The palette of possible settings has been discussed in many studies; it concerned, for example, the role of grid spacing and distribution (Craddock 1973; Karl et al. 1982; Huth 2006), the similarity matrix (Brinkmann 1999; Huth 2006), the number of principal components (PCs) (Richman and Lamb 1985; O’Lenic and Livezey 1988; Serrano et al. 1999; Wilks 2016; Hynčica and Huth 2020), and the rotation of PCs (Richman 1986; Barnston and Livezey 1987; Rogers and McHugh 2002; Hannachi et al. 2007; Huth 2007; Mezzina et al. 2020; Huth and Beranová 2021). The length of analyzed periods also varies from study to study, being 20 years (Jung et al. 2003; Wang et al. 2012; Moore et al. 2013), 30 years (Panagiotopoulos et al. 2002; Chien et al. 2019), 40 years (Thompson and Wallace 2000; Handorf and Dethloff 2012), 50 years (van den Dool et al. 2000; Huth 2006), or longer.

PCA is also used to detect changes in the spatial structure of patterns of modes (typically shifts) over time. Based on a comparison of two or more shorter periods, there is evidence for an eastward shift of both the North Atlantic Oscillation (NAO) (Jung et al. 2003; Moore et al. 2013; Wang et al. 2012), and the Pacific–North American pattern (PNA) (Lee et al. 2012; Chien et al. 2019). Studies also differ in the number of periods that are compared. Jung et al. (2003) and Lee et al. (2012) used only two nonoverlapping periods, while Moore et al. (2013) investigated changes using four nonoverlapping periods. Zhang et al. (2008), Wang et al. (2012) and Chien et al. (2019) took a different approach. They used moving PCA with the step of one year (Wang et al. 2012) and five years (Zhang et al. 2008; Chien et al. 2019). The successive periods then overlap and consist of almost identical data. For example, Wang et al. (2012) recognized that the eastward shift of the northern center of NAO has been more pronounced since the 1970s than the shift in the 1920s, and the southern center shifted eastward in the late nineteenth century. In addition, in the middle of the twentieth century, the southern center shifted to the north.

The long-term reanalyses allow us to track spatial changes of mode patterns over the entire twentieth century and even further back in time. One of the longest datasets available is the 20CRv2, which covers the period 1871–2012. This surface-input reanalysis is based on the assimilation of observations of surface pressure, monthly sea surface temperature, and sea ice concentration (Compo et al. 2011). Data from the upper atmosphere are not included because there are no such observations or there are only a few of them until the 1950s. Therefore, some processes or patterns may be poorly reproduced or even completely absent in the free atmosphere compared to full-input reanalyses. Several deficiencies of the 20CR have been noticed: Belleflamme et al. (2013) recognized an overestimation of 500-hPa geopotential height over Greenland in summer, Stryhal and Huth (2017) pointed out that the frequency of circulation types with high SLP over the European continent is overestimated, while circulation types with zonal advection are underestimated. Hynčica and Huth (2020) described differences in the spatial structure of atmospheric modes produced by different reanalyses and concluded that the 20CRv2 provides the most different results compared to other widely used reanalyses. Compo et al. (2011), Wang et al. (2012), and Krueger et al. (2013) draw attention to possible inhomogeneities in the 20CRv2 that could be due to the change in station density around the 1940s. Despite this, the 20CRv2 provides a sufficiently long dataset that can be easily divided into several independent periods. Additionally, the 20CRv2 is composed of 56 ensemble members with the intention to capture uncertainty inherent in the data and reanalysis model. As pointed out by Wang et al. (2013), ensemble members may provide considerably different results than the ensemble mean.

The aim of this paper is to describe spatial shifts of patterns of the modes in the Northern Hemisphere extratropics since the second half of the nineteenth century, to compare results obtained using the ensemble mean and ensemble members, and to provide an overview of possible effects that could occur when using moving PCA. To this end, we compare spatial patterns of modes obtained for 20-, 30-, 40-, and 50-yr-long moving periods. We use moving PCA with a 1-yr step to examine how much the shift in the mode structure can be affected by the choice of periods that are compared.

2. Data and methods

We employ winter (DJF) monthly 500-hPa geopotential heights in the 20CRv2 (Compo et al. 2011). Anomalies were calculated by subtracting the respective long-term monthly mean. The first complete winter season begins in December 1871. The winter seasons are referred to by the year of January. That is why the first winter season is marked as 1872. We use both the ensemble mean and all 56 ensemble members. We are interested in circulation in the northern extratropics, which is why the regular grid (2° × 2°) extends from 20°N northward.

The grid point values are weighted by the square root of the cosine of the latitude to compensate for decreasing areas of grid boxes toward the pole (Thompson and Wallace 2000; Jackson 2003). Orthogonally rotated PCA in S-mode (columns of the data matrix representing grid points and rows representing months) is used to detect atmospheric modes of variability (Compagnucci and Richman 2008) with the covariance used as a similarity metric (Huth 2006).

There are more than 20 rules on how to determine the number of PCs, but none of them can be applied universally (Peres-Neto et al. 2005). The scree plot (Fig. 1) shows the percentage of variance explained by 25 leading principal components (PCs) for the ensemble mean (black) and ensemble members (gray). Scree plots are a broadly used tool for determining the appropriate number of PCs to retain and rotate. O’Lenic and Livezey (1988) recommend putting the cutoff between the retained PCs and noise at the rightmost step in the graph, behind which the explained variance decreases approximately exponentially. The selection usually leads to higher numbers, such as nine modes. If fewer PCs (e.g., three or five) were rotated, the resulting modes would likely be distorted or mixed together. The effects of such an “underrotation” were already described and discussed in detail by O’Lenic and Livezey (1988). Another, more pragmatic reason for rotating nine PCs, resulting in nine modes, is the consistency with other hemispheric studies (Barnston and Livezey 1987; Panagiotopoulos et al. 2002; Huth 2006; Hynčica and Huth 2020). The retained PCs are orthogonally rotated by the varimax rotation (Richman 1986) to simplify the structure of modes and enhance their interpretability. Spatial patterns (loadings) of PCs are presented as correlations of the time series of the corresponding mode with 500-hPa height anomalies.

Fig. 1.
Fig. 1.

Scree plot of the first 25 PCs of 500-hPa height anomalies in the Northern Hemisphere extratropics for the ensemble mean (black line) and for 56 ensemble members (gray lines) for the full period (1872–2012).

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

At first, we perform PCA for the ensemble mean in the full period 1872–2012 (Fig. 2). Using the scree plot (Fig. 1), we identify the number of PCs for the full period. We opt for retaining nine PCs; the terminology (Table 1) is consistent with other hemispheric studies such as Barnston and Livezey (1987), Panagiotopoulos et al. (2002), Huth (2006), and Hynčica and Huth (2020).

Fig. 2.
Fig. 2.

Loadings of modes (identified by abbreviations from Table 1) for the ensemble mean in the full period and the percentage of variance they explain. Positive (negative) values are indicated by solid (dashed) lines, the contour interval is 0.2, and the zero contour is omitted.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

Table 1.

Abbreviations and full names of circulation modes.

Table 1.

Then we perform PCA for each ensemble member in the full period. Based on the scree plot (Fig. 1), we find it sensible to set the number of PCs in each ensemble member the same as for the ensemble mean, that is nine. The ensemble mean PC loadings are then paired with the corresponding loadings for each ensemble member (and this way attributed to the known modes) on the basis of their mutual similarity, which is quantified by the congruence coefficient (Richman and Lamb 1985; Huth 2006). It is defined as uncentered correlation (Harman 1976), that is,
rAB=j=1Najbjj=1Naj2j=1Nbj2,
where aj and bj are values of patterns to be compared at gridpoint j, j = 1, …, N. Richman and Lamb (1985) advocate the use of the congruence coefficient to quantify the similarity of PC loadings instead of the correlation coefficient because the former is sensitive to the magnitude of values, unlike the latter. Since the polarity of loadings is irrelevant, we are interested in the absolute value of the congruence coefficient. The congruence coefficient is calculated between all PC loadings for the ensemble mean and all loadings for all ensemble members. Each PC in every ensemble member is then identified with the mode with which it has the highest agreement. Table 2 (top half) shows congruence coefficients for the 26th member as an example. In this case, the one-to-one correspondence of modes between the ensemble mean and the ensemble member is clear as the maximum value in each column (corresponding to the ensemble mean) coincides with maximum values in rows (corresponding to the ensemble member): PC1 is identified as NAO, PC2 as PNA, etc.
Table 2.

Congruence coefficients between loadings for the ensemble mean in the full period (columns) and loadings for the 26th ensemble member in the full period (top half of table) and the first ensemble member in period 1872–1891 (bottom half of table). The highest congruence coefficient in each row is highlighted in boldface.

Table 2.

The same procedure is applied to all moving 20-, 30-, 40-, and 50-yr periods. Anomaly fields were calculated for each moving period separately. In total, PCA is performed for 424 moving periods (Table 3) in the ensemble mean and in all 56 ensemble members. The settings of PCA of the moving periods remain the same as for the full period, including retaining and rotating nine PCs. However, the one-to-one correspondence is not always clear-cut between modes in the full period and those in moving periods, suggesting that some modes blend one with another while others are absent. An example of such behavior, which is discussed later in section 4, is shown in Table 2 (bottom half). In this particular case, PCs 5 and 7 can both be identified with the NAO, while PCs 1 and 6 can be identified with the PNA; analogously, PCs 1 and 6 can be assigned to the PNA. None of the PCs have the highest congruence coefficient with the EP and EU2 modes. A mode in the ensemble mean for the full period that has no counterpart in the moving period will be referred to as “unassigned” in the sense that no PC is assigned to them.

Table 3.

The length and number of analyzed moving periods.

Table 3.

In the cases with unassigned modes, we applied a modified approach to avoid assigning two PCs to one mode. First, we calculate the congruence coefficient between the ensemble mean NAO loading and all nine PC loadings. The PC with the highest coefficient is assigned to NAO and then withdrawn. Next, we calculate the congruence coefficient between the PNA loading and the remaining eight PCs, and so on in the order of explained variance for the full-period ensemble mean.

To provide context to our results, we also briefly examine the newer version of the Twentieth Century Reanalysis, 20CRv3 (Slivinski et al. 2019). It starts in 1836 and ends in 2015 and is composed of 80 ensemble members. Its original spatial resolution of 1° × 1° was downsampled to 2° × 2° to maintain comparability with 20CRv2. The 20CRv3 was analyzed in a manner analogous to 20CRv2. Results related to it are provided in section 5.

3. Full period: 1872–2012

The loading patterns of modes for the ensemble mean in the full period are shown in Fig. 2. The spatial patterns are shown as correlations of corresponding loading scores with anomalies, with the solid (dashed) lines representing positive (negative) values. The number of modes, their spatial structure, and intensity are consistent with other studies (Barnston and Livezey 1987; Cheng et al. 1995; Panagiotopoulos et al. 2002; Huth et al. 2006; Hynčica and Huth 2020). NAO and PNA are the two leading modes in terms of explained variance, which is also in accord with other studies. However, the ranking of several weaker modes differs from other studies: e.g., EU1 is the third leading mode, while WPO is only the fifth, although WPO is usually reported to explain more variability than EU1 (Barnston and Livezey 1987; Huth et al. 2006; Liu et al. 2014; Hynčica and Huth 2020).

Next, we compare ensemble mean modes with modes in individual ensemble members. Figure 3a shows the degree of similarity of loadings between the ensemble mean modes and the corresponding modes in the ensemble members. The median of congruence coefficients is above 0.95 for all modes. It means that there is a good agreement between modes in the ensemble members and the ensemble mean. However, higher variability of the similarity degree, which is mainly due to a few outliers, occurs for PNA, WPO, TNH and EU2.

Fig. 3.
Fig. 3.

(a) Boxplots of congruence coefficients between the loadings of the modes in the ensemble mean and in the 56 ensemble members and (b) variance explained by the modes in the ensemble mean (black dots) and in the ensemble members (boxes), all for the full period. The whiskers point to maximum and minimum values, while the boxes represent the first and third quartiles and the central line is the median.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

For the majority of modes, the variance explained in the ensemble mean differs substantially from the median of explained variance in ensemble members (Fig. 3b). For four modes (NAO, EU1, EA, EP), the value for the ensemble mean is larger than for any ensemble member; for one mode (EU2), the explained variance for the ensemble mean is almost at the upper edge of values of ensemble members. Specifically, while NAO is the first leading mode in the ensemble mean, explaining 13.0% of the variance, the explained variance of NAOs ranges between 11.1% and 12.1% in the ensemble members. Furthermore, NAO is only the second leading mode after PNA in most ensemble members (PNA is the leading mode in 53 out of 56 members). Conversely, the explained variance for the ensemble mean is lower than in most ensemble members for PNA, WPO, and TNH. To sum up, there is an obvious tendency for the modes in the Atlantic–European sector to explain more variance in the ensemble mean than in the ensemble members, while the opposite holds for the modes in the Pacific–North American sector. There is also a high variability in the explained variance of WPO and TNH among the ensemble members. The ranking of WPO varies from the third up to the eighth leading mode, while this range is from the third to the ninth for TNH. Thus, the ranking of the modes may strongly depend on which member of the ensemble is analyzed.

The higher intermember variability shown in Fig. 3b indicates that some members contain higher variability in the Pacific–North American sector, while others do not. However, the process of averaging reduces the variability in the ensemble mean. Wang et al. (2013), and Krueger et al. (2013) pointed out that the members differ from each other due to the lack of assimilated observations before the 1940s. Compo et al. (2011) note that the variability of 500-hPa geopotential height anomalies is generally smaller over the North Pacific region before 1948.

4. Moving PCA

In total, we analyze 24 168 moving periods (i.e., 424, which is the sum of values in Table 3, times 57, which is 56 ensemble members plus one ensemble mean). Figure 4 shows the number of unassigned modes for all periods for the ensemble mean (left) and the ensemble members (right). For the ensemble mean, all nine modes are identified in 42% of the moving periods, one mode is unassigned in 54% of the moving periods, while two modes are unassigned in 4% of the moving periods. The situation is illustrated in Table 2 (bottom half) for the period 1872–91 and the first ensemble member: While two modes of this particular moving period and ensemble member bear the closest resemblance to NAO (PC5 and PC7) and two modes are most closely similar to EU1 (PC1 and PC6), there is no mode that would be most similar to EP and EU2. The unassigned modes occur most often (in 63% of cases) when two PCs are identified with EU1 and none with NAs. In 21% of cases with unassigned modes, two PCs are marked as TNH and none as NAs.

Fig. 4.
Fig. 4.

(left) The number of unassigned modes in the ensemble mean and (right) the number of ensemble members with 0, 1, or 2 unassigned modes, all for moving periods of (top to bottom) 20–50 years.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

The numbers of ensemble members with none, one, and two unassigned modes are shown in the right part of Fig. 4. Interestingly, one can see three distinct phases there, particularly for 40- and 50-yr periods. The first phase approximately covers periods starting in the nineteenth century when the majority of ensemble members have no unassigned modes. Then, for periods ending before about 1995, ensemble members with at least one unassigned mode prevail. The third phase includes the most recent periods in which the ensemble members with no unassigned mode are most frequent again.

a. Ensemble mean versus ensemble members

Figures 5 and 6 show the congruence coefficient (left) and the explained variance (right) for all moving periods of all lengths. The ensemble mean is represented by the black line, and the ensemble members are shown in gray. There are substantial differences between the ensemble mean and ensemble members, which are more pronounced in the early periods (Figs. 5 and 6) when both the congruence coefficient and the explained variance are more spread out among the members. Differences in the explained variance diminish toward the present, actually vanishing around the middle of the twentieth century when the spread among ensemble members becomes negligible, while differences in the degree of similarity remain larger. For NAO, the variance explained by the ensemble mean is generally higher than that of the ensemble members; that is, NAO tends to be stronger (relative to the total amount of variance present in the data) in the ensemble mean than in the individual members (Fig. 5). In contrast, PNA has a tendency to be weaker in the early periods in the ensemble mean than in the ensemble members (Fig. 6), which reverses approximately at the beginning of the twentieth century since when more variance tends to be explained by PNA in the ensemble mean.

Fig. 5.
Fig. 5.

NAO: (left) Congruence coefficient with the loadings for the full period and (right) variance explained for the moving periods of (top to bottom) 20–50 years for the ensemble mean (thick black line) and for the ensemble members (gray).

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for PNA.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

The PNA pattern is fully developed in the ensemble members in the early periods even over data-sparse regions of the North Pacific (Fig. 7). Note that the centers of the PNA pattern in ensemble members are typically more spatially extensive and stronger than in the ensemble mean, for which particularly the centers over northwestern and southeastern North America are rather weak. However, the ensemble members are only slightly constrained by observed data, because of which the location and the extent of the centers varies from one ensemble member to another. This is why the spread of the degree of similarity in the early periods is so wide (Fig. 6, left). The averaging of the ensemble members, which are relatively slightly similar to each other, results in reduced variability in the ensemble mean and, consequently, the variance explained by a mode in the ensemble mean is lower than that in the ensemble members (Fig. 6, right). In contrast, the ensemble mean modes in regions that are relatively data-rich even in the beginning of the study period (i.e., NAO in the late nineteenth century) do not lose variance due to averaging of ensemble members because the latter bear strong mutual similarity (Fig. 5, right). The modes of individual ensemble members can explain relatively more of the total variance simply because the total variance for the ensemble mean is lower. This explains the higher percentage of explained variance of NAO for the ensemble mean than for the ensemble members (Fig. 5).

Fig. 7.
Fig. 7.

Loadings of PNA for period 1876–1915 for the ensemble mean and five randomly chosen ensemble members. Positive contours are solid, negative contours are dashed, and zero contours are omitted. The contour interval is 0.2.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

This effect is also observed for most of the other modes (Fig. 8). In this case, the congruence coefficient (left) and the explained variance (right) are shown only for the 40-yr moving periods. There is strong evidence that Atlantic and European modes (EA, EU1, and EU2 in addition to NAO discussed above) tend to explain more variance for the ensemble mean than for individual ensemble members in the early periods. On the other hand, the Pacific and North American modes, except for EP (i.e., PNA, WPO, and TNH), explain relatively less variance in the ensemble mean than in the ensemble members. The reason why EP behaves similarly to the Euro-Atlantic modes is unclear. Starting approximately from the period 1920–59, the spread of ensemble members in explained variance becomes very low and explained variance no longer differs between the ensemble mean and members. This is likely a reflection of an increase in the amount of assimilated observations since the 1920s (Compo et al. 2011) and since the 1950s even in remote areas (Cram et al. 2015).

Fig. 8.
Fig. 8.

As in Fig. 5, but for all modes other than NAO and PNA and for moving periods of 40 years only.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

These findings are consistent with Wang et al. (2013), who found that the ensemble mean fields are not suitable for the calculation of cyclone statistics in areas with too few assimilated observations, advocating the use of ensemble members. Woollings et al. (2014) demonstrated that the long-term decrease of interseasonal variability in the period before 1940 in the Pacific region is a symptom of insufficient observational coverage. Therefore, they recommend using ensemble members that contain variability and can help to avoid misleading results. Our study provides the same recommendation regarding modes of variability: the lack of assimilated observations in remote regions such as the Pacific, parts of Asia, and the Arctic in the early periods of the 20CRv2 biases the strength and shape of the modes in the ensemble mean; hence the ensemble members should be used for their detection instead.

b. The length of moving periods

The length of the period plays a key role in determining the intensity and shape of the mode. The degree of similarity is highly fluctuating when using the 20-yr-long periods (Figs. 5 and 6, left columns). The fluctuations occur even for longer periods; the sharp drops are, nevertheless, becoming less frequent and more intermittent (occurring only for one or two consecutive moving periods) for them.

Using longer periods, there are also fewer unassigned modes (Fig. 4). A shift of one year replaces a relatively larger part of the data for short than for long periods. Therefore, we assume that this is a plausible explanation for higher fluctuations for shorter periods. Another explanation is a manifestation of a larger sampling uncertainty due to a shorter period of analysis (Hynčica and Huth 2020). Nevertheless, the sudden changes in the degree of similarity occur even for 50-yr periods, and even in the most recent times (Fig. 5).

We argue that the examination of changes in the spatial structure of modes, including their shifts, using only two relatively short periods (e.g., Jung et al. 2003; Lee et al. 2012) may be misleading. There is a risk that one of the examined modes could be affected by sampling uncertainty, demonstrated by a sharp drop in the similarity with the full-period mode. Thus, the observed shift of a mode or the change in its explained variance between two short periods may not be a reflection of real changes in the analyzed field. In such a case, different results could have been obtained had the analyzed period been shifted by a year. The shorter the periods that are used, the higher the risk of such effects. We therefore advocate the use of moving PCA to eliminate such potential misinterpretations.

c. Temporal changes in modes

Figures 5 and 6 show the congruence coefficient and explained variance for the two leading modes, NAO and PNA, respectively, for the moving periods of all lengths. Animations 1 and 2 in the online supplemental material show the development of the spatial pattern of NAO and PNA in the ensemble mean for 40-yr moving periods. As one would expect, the change in the appearance of the modes between two consecutive periods is negligible in most cases. The two consecutive periods contain almost identical data and thus the outputs of PCA are practically the same.

However, it is interesting that this is not always the case: The degree of similarity and explained variance can differ considerably even between two consecutive periods. These sudden drops and rises of the congruence coefficient and explained variance occur in both the early and most recent periods. It suggests that PCA is sensitive to the choice of the analyzed period. Even a small change of data in the form of a 1-yr shift can result in substantially different results. Changes in the degree of similarity indicate spatial shifts in the patterns of modes over time. We suggest that sudden drops and rises of the congruence coefficient may have multiple causes, such as changes in the variability between successive moving periods and sensitivity of PCA to the number of PCs rotated. A sharp change in the data (such as the introduction of new measurements, which can be viewed as an inhomogeneity) affects a number of moving periods. With the moving period sliding forward, more and more data from the period after the change are included in the analysis and may have a progressively stronger impact on its results. In fact, such a change may manifest itself by a gradual change in the spatial structure of a mode (hence its similarity with the full-period mode) and its explained variance; alternatively, a sudden shift occurs when the data after the change outweigh the data before it. However, sharp changes in data do not necessarily imply a spatial shift of modes; they can be manifested solely by a change in the variance explained by the modes.

Three types of shifts can be recognized. First, it is a slow and gradual shift, the example of which is a slight eastward shift of the northern NAO center over the century (supplemental animation 1). This change is in accordance with Jung et al. (2003). In contrast to the findings of Moore et al. (2013), we do not observe an eastward shift of the southern center. Conversely, there is no sign of an eastward shift of the PNA pattern (supplemental animation 2), reported by Lee et al. (2012) and Chien et al. (2019). The centers of action are mostly oscillating around their geographical means during the twentieth century. Shifts of this kind are likely to reflect real changes in atmospheric circulation as they are typically accompanied by changes in composites and autocorrelation maps (e.g., Jung et al. 2003, not shown here).

The second type is a rapid and sudden shift between successive running periods without changes in the explained variance, e.g., NAO between 1949–88 and 1950–89 (supplemental animation 1). Although the pattern of the mode changes, the explained variance remains the same. Such changes in patterns of modes are unlikely to describe decadal-scale changes in circulation since the changes are too extensive while the successive periods consist of almost identical data. The most likely reason for this type of shifts is the extra variance added by a strong anomaly, such as the winter of 1988/89, which was one of those with the strongest positive NAO index (Jones et al. 1997). To interpret the sudden changes in spatial patterns of modes in terms of PCA, we first note that autocovariance structures form a continuum, from which PCA picks individual structures (patterns) as PCs (modes). The sudden changes are cases when two covariance structures with a similar relevance compete for becoming PC patterns, and PCA picks one of them in the first and the other in the second of the consecutive periods.

The third type is a sudden shift accompanied by a marked change in explained variance. A part of the variance may be transferred from one mode to another; as a result, the patterns of both modes are shifted and modified and thus, the similarity with the full-period modes decreases. Alternatively, the variance pertaining to two (or more) modes may be merged into a single PC, which results in the existence of unassigned modes as discussed before. This effect may be a result of under- or overrotation, that is, retaining and rotating fewer or more modes than what is optimum, which is well documented and discussed by O’Lenic and Livezey (1988). However, it is not always so, as our experiments with different numbers of rotated components indicated (not shown): the lack of one-to-one correspondence, manifesting in unassigned modes, is in some cases not eliminated even for different numbers of rotated components. It is appropriate to note that individual determination of the number of components to rotate, which would allow the optimum number to be found, in every analysis is impossible because of the huge total number of such analyses (on the order of tens of thousands). Analogous sudden changes between consecutive time periods were detected in Euro-Atlantic circulation (weather) regimes (Dorrington and Strommen 2020; Dorrington et al. 2022). Although the regimes are defined by cluster analysis, the data dimension is reduced by PCA before their calculation and the behavior of the regimes may, therefore, be a reflection of the behavior of modes described here.

It is interesting to focus on NAO in 1960–99 (supplemental animation 1) when the congruence coefficient drops suddenly. The pattern of the mode is shifted slightly southward in comparison with the previous period (1959–98). During the following periods, the congruence coefficient is gradually rising and the pattern is slowly moving back toward the north. This sudden drop and gradual rise of the congruence coefficient are captured by all ensemble members (Fig. 5); therefore, this effect is likely to be real. Wang et al. (2012) also described the northward shift of the NAO, but they mentioned a northward shift of the southern center in the mid-twentieth century, not a simultaneous northward shift of both centers after 1959–98 as we observed. It is interesting to note that the rapid development and shift of mode centers, which is associated with a clear upward trend in the congruence coefficient, is also evident in the PNA (supplemental animation 2) during the early periods (Fig. 6).

5. Results for 20CRv3

To put our results into a somewhat broader context, we present selected analyses for the newer version of the reanalysis, namely 20CRv3. The spatial structure of the modes for the entire period of 20CRv3 (1836–2015) is similar to those in 20CRv2 with one substantial difference: a multicenter pattern is detected, which seems to be a merger of TNH and NAs patterns, which are missing in 20CRv3. This mode (not shown) is stable with respect to rotation, that is, is present regardless of the number of PCs rotated.

Time series of explained variance are shown in Fig. 9 for NAO and PNA and a 40-yr moving window. Results for 20CRv3 support most of the previous findings: The spread of ensemble members grows toward the past and is larger in the Pacific (PNA) than Atlantic (NAO) sector. The sudden drops and peaks in explained variance occur with similar frequency; most of them are present in the ensemble mean as well as in the majority of ensemble members. The drops and peaks tend to occur simultaneously in both reanalyses. NAO explains less variance in the ensemble members than in the mean before about the middle of the twentieth century. The behavior is different from 20CRv2 for PNA, for which the mean is connected with more variance than the ensemble members in the early twentieth century and is near the middle of the ensemble in most of the nineteenth century. The reason for this is unclear, but may be related to the fact that the very early period (1836–71) is included in the definition of modes, in which the spatial appearance of modes may substantially differ from later periods due to the lack of observational constraint over the data-void Pacific Ocean.

Fig. 9.
Fig. 9.

The congruence coefficients of the (left) NAO and (right) PNA with the loadings for both the ensemble members and means of 20CRv2 (green) and 20CRv3 (blue) datasets. The analysis focuses on a moving period of 40 years.

Citation: Journal of Climate 36, 22; 10.1175/JCLI-D-22-0620.1

We plan to conduct a similar analysis for multiple reanalysis products as a next step, into which 20CRv3 will also be included. Therefore, we do not delve into more details on 20CRv3 here.

6. Conclusions

In this study, we analyze modes of low-frequency variability of atmospheric circulation in the northern extratropics in winter. The analysis is based on monthly mean anomalies of 500-hPa geopotential heights in the 20CRv2. We use both the ensemble mean as well as all 56 ensemble members. We apply moving PCA with a 1-yr step to detect spatial changes in the modes over time.

The Euro-Atlantic modes tend to explain more variability in the ensemble mean than in ensemble members. In contrast, the opposite holds for the Pacific–North American modes. This applies to both the full period 1872–2012 and moving periods in the early parts of the study period approximately before 1920. This is likely due to an uneven and sparse distribution of assimilated data particularly in the North Pacific in the early periods, resulting in the reanalysis being slightly constrained by observations there. Modes detected in ensemble members differ one from another more in data-sparse areas. The process of averaging leads to the loss of variability there and thus to modes detected in the ensemble mean being weaker, that is, explaining less variance, than in the ensemble members. Therefore, we argue that using only the 20CRv2 ensemble mean to detect modes of variability is insufficient and recommend that ensemble members should be used instead to capture the uncertainty in the data.

The shape and intensity of a mode may vary even between two consecutive moving periods, although these periods consist of almost identical data. Such sudden changes are more frequent for shorter moving periods and occur in both early and recent periods. We argue that these changes have multiple causes: the inhomogeneities in data, the sensitivity of modes to the number of PCs retained for rotation, and the truncation and rotation process that is performed independently for each period. The occurrence of the sudden changes is more or less random. However, the shorter periods, the more frequent the sudden changes are. Therefore, we advocate using a moving PCA, which allows these sudden changes to be identified, instead of using just two or a few fixed periods for examination of long-term changes of modes, results of which may be biased by these random effects. Based on the presented results, we imply that any other results and applications, using PCA performed for one or two periods, may be biased. With a slight change in the period analyzed, significantly different results can be obtained.

Acknowledgments.

This research was supported by the Czech Science Foundation, project 17-07043S. VP was also supported by the Grant Agency of Charles University, student project 426216. Support for the Twentieth Century Reanalysis Project is provided by the U.S. Department of Energy, Office of Science Biological and Environmental Research, by the National Oceanic and Atmospheric Administration Climate Program Office, and by the NOAA Earth System Research Laboratory Physical Sciences Laboratory.

Data availability statement.

The data that support the findings of this study are openly available in the NOAA Physical Sciences Laboratory (PSL) at https://psl.noaa.gov/data/gridded/data.20thC_ReanV2.html (Compo et al. 2011) and https://portal.nersc.gov/project/20C_Reanalysis (Slivinski et al. 2019).

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Supplementary Materials

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  • Barnston, A. G., and R. E. Livezey, 1987: Classification, seasonality and persistence of low-frequency atmospheric circulation patterns. Mon. Wea. Rev., 115, 10831126, https://doi.org/10.1175/1520-0493(1987)115<1083:CSAPOL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Belleflamme, A., X. Fettweis, C. Lang, and M. Erpicum, 2013: Current and future atmospheric circulation at 500 hPa over Greenland simulated by the CMIP3 and CMIP5 global models. Climate Dyn., 41, 20612080, https://doi.org/10.1007/s00382-012-1538-2.

    • Search Google Scholar
    • Export Citation
  • Brinkmann, W. A. R., 1999: Application of non-hierarchically clustered circulation components to surface weather conditions: Lake Superior basin winter temperatures. Theor. Appl. Climatol., 63, 4156, https://doi.org/10.1007/s007040050090.

    • Search Google Scholar
    • Export Citation
  • Cheng, X., G. Nitsche, and J. M. Wallace, 1995: Robustness of low-frequency circulation patterns derived from EOF and rotated EOF analyses. J. Climate, 8, 17091713, https://doi.org/10.1175/1520-0442(1995)008<1709:ROLFCP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Chien, Y.-T., S.-Y. S. Wang, Y. Chikamoto, S. L. Voelker, J. D. D. Meyer, and J.-H. Yoon, 2019: North American winter dipole: Observed and simulated changes in circulations. Atmosphere, 10, 793, https://doi.org/10.3390/atmos10120793.

    • Search Google Scholar
    • Export Citation
  • Compagnucci, R. H., and M. B. Richman, 2008: Can principal component analysis provide atmospheric circulation or teleconnection patterns? Int. J. Climatol., 28, 703726, https://doi.org/10.1002/joc.1574.

    • Search Google Scholar
    • Export Citation
  • Compo, G. P., and Coauthors, 2011: The Twentieth Century Reanalysis Project. Quart. J. Roy. Meteor. Soc., 137 (654), 128, https://doi.org/10.1002/qj.776.

    • Search Google Scholar
    • Export Citation
  • Craddock, J. M., 1973: Problems and prospects for eigenvector analysis in meteorology. J. Roy. Stat. Soc., 22D, 133145, https://doi.org/10.2307/2987365.

    • Search Google Scholar
    • Export Citation
  • Cram, T. A., and Coauthors, 2015: The International Surface Pressure Databank version 2. Geosci. Data J., 2, 3146, https://doi.org/10.1002/gdj3.25.

    • Search Google Scholar
    • Export Citation
  • Dorrington, J., and K. J. Strommen, 2020: Jet speed variability obscures Euro-Atlantic regime structure. Geophys. Res. Lett., 47, e2020GL087907, https://doi.org/10.1029/2020GL087907.

    • Search Google Scholar
    • Export Citation
  • Dorrington, J., K. J. Strommen, and F. Fabiano, 2022: Quantifying climate model representation of the wintertime Euro-Atlantic circulation using geopotential-jet regimes. Wea. Climate Dyn., 3, 505533, https://doi.org/10.5194/wcd-3-505-2022.

    • Search Google Scholar
    • Export Citation
  • Handorf, D., and K. Dethloff, 2012: How well do state-of-the-art atmosphere-ocean general circulation models reproduce atmospheric teleconnection patterns? Tellus, 64A, 19777, https://doi.org/10.3402/tellusa.v64i0.19777.

    • Search Google Scholar
    • Export Citation
  • Hannachi, A., I. T. Jolliffe, and D. B. Stephenson, 2007: Empirical orthogonal functions and related techniques in atmospheric science: A review. Int. J. Climatol., 27, 11191152, https://doi.org/10.1002/joc.1499.

    • Search Google Scholar
    • Export Citation
  • Harman, H. H., 1976: Modern Factor Analysis. 3rd ed. University of Chicago Press, 487 pp.

  • Huth, R., 2006: The effect of various methodological options on the detection of leading modes of sea level pressure variability. Tellus, 58A, 121130, https://doi.org/10.1111/j.1600-0870.2006.00158.x.

    • Search Google Scholar
    • Export Citation
  • Huth, R., 2007: Arctic or North Atlantic Oscillation? Arguments based on the principal component analysis methodology. Theor. Appl. Climatol., 89 (1–2), 18, https://doi.org/10.1007/s00704-006-0257-1.

    • Search Google Scholar
    • Export Citation
  • Huth, R., and R. Beranová, 2021: How to recognize a true mode of atmospheric circulation variability. Earth Space Sci., 8, e2020EA001275, https://doi.org/10.1029/2020EA001275.

    • Search Google Scholar
    • Export Citation
  • Huth, R., L. Pokorná, J. Bochníček, and P. Hejda, 2006: Solar cycle effects on modes of low-frequency circulation variability. J. Geophys. Res., 111, D22107, https://doi.org/10.1029/2005JD006813.

    • Search Google Scholar
    • Export Citation
  • Hynčica, M., and R. Huth, 2020: Modes of atmospheric circulation variability in the northern extratropics: A comparison of five reanalyses. J. Climate, 33, 10 70710 726, https://doi.org/10.1175/JCLI-D-19-0904.1.

    • Search Google Scholar
    • Export Citation
  • Jackson, J. E., 2003: A User’s Guide to Principal Components. John Wiley and Sons, 592 pp.

  • Jones, P. D., T. Jonsson, and D. Wheeler, 1997: Extension to the North Atlantic oscillation using early instrumental pressure observations from Gibraltar and south-west Iceland. Int. J. Climatol., 17, 14331450, https://doi.org/10.1002/(SICI)1097-0088(19971115)17:13<1433::AID-JOC203>3.0.CO;2-P.

    • Search Google Scholar
    • Export Citation
  • Jung, T., M. Hilmer, E. Ruprecht, S. Kleppek, S. K. Gulev, and O. Zolina, 2003: Characteristics of the recent eastward shift of interannual NAO variability. J. Climate, 16, 33713382, https://doi.org/10.1175/1520-0442(2003)016<3371:COTRES>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Karl, T. R., A. J. Koscielny, and H. F. Diaz, 1982: Potential errors in the application of principal component (eigenvector) analysis to geophysical data. J. Appl. Meteor., 21, 11831186, https://doi.org/10.1175/1520-0450(1982)021<1183:PEITAO>2.0.CO;2.

    • Search Google Scholar
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  • Fig. 1.

    Scree plot of the first 25 PCs of 500-hPa height anomalies in the Northern Hemisphere extratropics for the ensemble mean (black line) and for 56 ensemble members (gray lines) for the full period (1872–2012).

  • Fig. 2.

    Loadings of modes (identified by abbreviations from Table 1) for the ensemble mean in the full period and the percentage of variance they explain. Positive (negative) values are indicated by solid (dashed) lines, the contour interval is 0.2, and the zero contour is omitted.

  • Fig. 3.

    (a) Boxplots of congruence coefficients between the loadings of the modes in the ensemble mean and in the 56 ensemble members and (b) variance explained by the modes in the ensemble mean (black dots) and in the ensemble members (boxes), all for the full period. The whiskers point to maximum and minimum values, while the boxes represent the first and third quartiles and the central line is the median.

  • Fig. 4.

    (left) The number of unassigned modes in the ensemble mean and (right) the number of ensemble members with 0, 1, or 2 unassigned modes, all for moving periods of (top to bottom) 20–50 years.

  • Fig. 5.

    NAO: (left) Congruence coefficient with the loadings for the full period and (right) variance explained for the moving periods of (top to bottom) 20–50 years for the ensemble mean (thick black line) and for the ensemble members (gray).

  • Fig. 6.

    As in Fig. 5, but for PNA.

  • Fig. 7.

    Loadings of PNA for period 1876–1915 for the ensemble mean and five randomly chosen ensemble members. Positive contours are solid, negative contours are dashed, and zero contours are omitted. The contour interval is 0.2.

  • Fig. 8.

    As in Fig. 5, but for all modes other than NAO and PNA and for moving periods of 40 years only.

  • Fig. 9.

    The congruence coefficients of the (left) NAO and (right) PNA with the loadings for both the ensemble members and means of 20CRv2 (green) and 20CRv3 (blue) datasets. The analysis focuses on a moving period of 40 years.

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