Global modes of climate variability


  • O. de Viron,

    Corresponding author
    1. Géodésie et Gravimétrie, Univ Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, UMR 7154 CNRS, F‒75013 Paris, France
    • Corresponding author: O. de Viron, Géodésie et Gravimétrie, Univ Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, UMR 7154 CNRSF‒75013 Paris, France. (

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  • J. O. Dickey,

    1. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA
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  • M. Ghil

    1. Geosciences Departement and Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure, Paris, France
    2. Department of Atmospheric and Oceanic Sciences, and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, USA
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[1] The atmosphere, hydrosphere, and cryosphere form a fully coupled climate system. This system exhibits a number of large‒scale phenomena, such as the El Niño–Southern Oscillation, the Asian Monsoon, the North Atlantic Oscillation, and the Madden‒Julian Oscillation. While these modes of variability are not exactly periodic, they are oscillatory in character, and their state is monitored using so‒called climate indices. Each of these scalar indices is a combination of several climate variables. Here we use a comprehensive set of 25 climate indices for time intervals that range between 1948 and 2011 and estimate an optimal set of lags between these indices to maximize their correlation. We show that most of the index pairs drawn from this set present a significant correlation on interannual time scales. It is also shown that on average, about two thirds of the total variability in each index can be described by using only the four leading principal components of the entire set of lagged indices. Our index set's leading orthogonal modes exhibit several interannual frequencies and capture separately variability associated with the North Atlantic and the North Pacific. These modes are associated, in turn, with large‒scale variations of sea surface temperatures.

1 Introduction

[2] The Earth's atmosphere, oceans, cryosphere, and continental hydrology exchange mass, momentum, and energy on all time scales. As a consequence, global‒ or regional‒scale climate variables—such as the sea surface temperature (SST), the rainfall, the surface pressure, or the wind speed fluctuate more or less regularly. Many of these fluctuations are known as modes or oscillations, and their states are monitored by using scalar‒valued climate indices. Some of the best known oscillations extend over large areas of the globe; they include the El Niño–Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO), the Pacific Decadal Oscillation (PDO), and the Madden‒Julian Oscillation (MJO). Certain oscillations are more localized, i.e., associated with the climate of smaller regions, e.g., the Sahel Rainfall; however, many teleconnections [Wallace and Gutzler, 1981] between the latter indices are also known to exist.

[3] Numerous authors have studied such teleconnections and the, possibly lagged, correlations between up to four indices [Ambaum et al., 2001; Moonley and Munot, 1993; Tsonis et al., 2007; Wyatt et al., 2011; Wang et al., 2012]. Other studies have considered the dynamics of coupled nonlinear oscillators as a possible source for such correlations [Ghil and Mo, 1991; Kimoto and Ghil, 1993; Feliks et al., 2010] or studied the network of time series of a single climate field, such as the SSTs, at the nodes of a regular grid [Tsonis and Swanson, 2008; Donges et al., 2009].

[4] In this study, we investigate whether and to which extent a large set of climate indices which presumably capture the coupled dynamics of several climate fields may exhibit lagged correlations on a global scale and, if so, what possible causes such a phenomenon may have.

[5] In section 2, we describe the data used and how they were preprocessed. In section 3, we investigate the connections between the climate indices. Section 4 makes the link between our results and variations of the SST field. The spectral content of the retrieved modes is discussed in section 5. Concluding remarks follow in section 6.

2 Data Sets and Their Preparation

[6] For this investigation, we use a set of 25 climate indices, regional as well as global; their complete list with acronyms, time intervals of availability, and source is given in Table 1.

Table 1. Climate Indices Used in This Studya
    Unweighted LagWeighted Lag 
AbbreviationIndex NameStartsEnds(months)(months)Address
  1. a

    For each index, we list its name, time range, optimal set of lags with respect to the Atlantic Meridional Mode Index (both unweighted and weighted), and Internet address for the source of the data; ESRL stands for

AAOAntarctic Oscillation19792011−3.5−4.4ESRL/
AMMSSTAtlantic Meridional Mode194820110.00.0ESRL/
AMONAtlantic Multidecadal Oscillation19482011−0.4−0.6ESRL/
AOArctic Oscillation19802011−5.2−4.8ESRL/
ATLTRIAtlantic Tripole SST EOF19482009−1.7−5.6ESRL/
BRAZILRAINNortheast Brazil Rainfall Anomaly194820005.39.4ESRL/
CARCaribbean SST19512011−0.3−0.8ESRL/
EOFPACTropical Pacific SST EOF194820081.01.1ESRL/
EPOEast/North Pacific Oscillation195020114.03.5ESRL/
GMSSTGlobal Mean Land/Ocean Temperature194820112.33.2ESRL/
INDIAMONCentral Indian Precipitation19482000−1.30.8ESRL/
MJOMadden‒Julian Oscillation19782011−3.5−4.4
NAONorth Atlantic Oscillation19502011−4.5−4.6ESRL/
NOINorthern Oscillation194820071.01.0ESRL/
NPNorth Pacific Pattern19482011−1.8−3.6ESRL/
PACWARMPacific Warmpool194820093.94.1ESRL/
PDOPacific Decadal Oscillation194820111.81.9ESRL/
PNAPacific North American195020111.11.8ESRL/
SAHELRAINSahel Standardized Rainfall19482001−4.13.4ESRL/
SOISouthern Oscillation195120111.41.6ESRL/
SWMONSOONSW Monsoon Region Rainfall194820110.51.7ESRL/
TNATropical Northern Atlantic194820110.30.5ESRL/
TSATropical Southern Atlantic194820112.71.2ESRL/
WHWPWestern Hemisphere Warm pool194820111.71.8ESRL/
WPWestern Pacific Index19502011−2.3−0.6ESRL/

[7] This set of time series consists of the entire set of climate indices from NOAA's Earth System Research Laboratory, with four exceptions: (1) only the Southern Oscillation Index (SOI) has been kept among the different ENSO‒related time series, since all of these are highly correlated with each other, and their inclusion would have led to climatically superfluous numerical difficulties; (2) the atmospheric angular momentum has been removed, since previous studies showed that its interannual variability was linked to ENSO [Stefanick, 1982]; and (3) the solar constant variability was dropped, as it is not a climatic index per se. Finally, (iv) the MJO index was added to the set because its interannual modulation is a topic of particular interest to the authors [e.g., Marcus et al., 2001].

[8] To separate the interannual contribution, we started by removing a composite seasonal cycle, i.e., the average value over all the months of January in the time series has been subtracted from each January value, and so on. The linear trend of each time series was then fitted and removed. Finally, a tapered low‒pass filter with a cutoff frequency of 12 months has been applied by convolution of the time series with sin(2πft)/(2πft), where f is the cutoff frequency [Owen, 2007]. The resulting time series for each index has been divided by its own standard deviation, in order to normalize the different indices with respect to each other.

[9] In section 5, we used the global SST fields of Kaplan et al. [1998]. These SSTs have monthly resolution and are given on a 2.5°×2.5° grid.

3 Connections Between Climate Indices

3.1 Cross Correlations

[10] In order to study the connections between the indices, we start by cross‒correlating pairs of indices, up to a maximum lag of 1 year. The correlation coefficient used in this study is the Pearson product‒moment coefficient. Cross correlations are computed by shifting one time series by a given number of months with respect to the other.

[11] We first computed cross correlations for lags between +1 year and −1 year, in steps of 1 month, and then computed the significance for the lag with the highest cross correlation. The significance is computed using a Student's t test, after estimating the number of degrees of freedom of the time series [cf. Von Storch and Zwiers, 1999].

[12] The number of degrees of freedom was estimated for each pair of time series as the harmonic mean of the number of degrees of freedom from the two time series over the common time interval; this number was computed for each of the two series as the ratio between the common time interval and the decorrelation time of the series in question, i.e., the lag at which its autocorrelation drops to 1/e.

[13] Figure 1 summarizes the cross‒correlation results. For each index pair, a colored square indicates the maximum value of the cross correlation for those pairs of indices for which the statistical significance exceeds 95%; all other correlations appear as white squares. The figure shows that more than 60% of the pairs are significantly cross‒correlated.

Figure 1.

Cross‒correlation map between pairs of indices. The size of each square is proportional to the length of the common time interval of availability; the square showing the correlation between two time series with a common interval of 20 years will be twice larger than that showing the correlation between two time series with a common interval of 10 years. Only correlations that are significant at the 95% level are colored: red means the correlation is positive, blue indicates anticorrelation, while white squares indicate correlations that are not significant at this level. The results for each pair appear twice, for better legibility.

3.2 Principal Components (PCs) and Optimization Procedure

[14] To investigate the common content in our set of time series {xi(t)i=1,…,n}, we apply principal component (PC) analysis [Von Storch and Zwiers, 1999] and project them onto the eigenvectors of their variance‒covariance matrix. This way, the set of 25 climate indices is expressed as a linear combination of the eigenvectors {Xk(t)k=1,…,K}, Kn=25, which are orthogonal at zero lag,

display math

[15] When several time series are interrelated, much of the variance is associated with only a few PCs, Kn. As we want to allow lags between the different indices, we used a slightly different version of PC analysis and optimized a set of lags between the indices so that the variance captured by four PCs is maximized. The choice of K=4 PCs was based on Monte Carlo tests [Overland and Preisendorfer, 1982], which showed that the first four PCs are significant and that adding a few more does not modify the picture in any substantial way.

[16] Given the relatively large dimension of the set of climate indices and, a fortiori, of the set of possible lags between them, we started with a random set of lags and used a genetic algorithm [Goldberg, 1989] to converge to the optimal set. Since the intervals of availability of the 25 indices differ, we used 126 different time intervals, including in each of them only the indices available in that interval. The 126 intervals correspond to 14 starting times, from 1955 to 1981 by steps of 2 years, and nine ending times, from 1995 to 2011 by steps of 2 years; the interval lengths thus range from 15 to 55 years, with a median value of 35 years. The optimal set of lags was computed for each time interval.

[17] If a connection between two indices does exist, the difference between the lags of those indices with respect to an arbitrary origin would be consistent over the 126 runs, whereas the absence of a robust connection between the two would manifest itself as a random distribution of the difference between their lags. We tested each lag‒difference distribution against a uniform distribution by using a χ2 test and found that the distributions so tested were significantly distinct from random (at the 95% level) for about 60% of the pairs of indices. Altogether, more than 80% of the index pairs have either a cross correlation that is significant at the 95% level or a robust lag, or both.

[18] Next, an overall least square fit was performed in order to find a globally optimal set of lags, based on the best lags found between pairs of modes over all the runs. We also estimated a weighted solution, in which each run is weighted according to the variance captured by the first four modes; both solutions are provided in Table 1. The difference between the weighted and unweighted lags only exceeds one full month in three cases, all of which represent relatively local phenomena, and it is larger than one full year in only one of these three. We then used these optimized lags to compute four PCs over the subset of indices that are defined over the entire interval, 1955–2011.

[19] Each of the indices was then reconstructed as a linear combination of the four PCs over its own interval of availability, and we computed for each of them the variance so described. The contribution of each of the four PCs to each of the 25 modes and the total variance captured in each case are shown in Figure 2. It is clear from the figure that the variance described by a linear combination of the four PCs is below 50% for only six of the 25 indices. It also allows one to group the indices dominated by the same PC.

Figure 2.

Variance captured by each of the four leading PCs for each of our 25 climate indices, sorted from the least “explained” to the best “explained” index, i.e., from the one that is most independent of the other 24 to the one that is least so. The height of the color bars indicates the variance contribution of the leading PCs, from the first to the fourth: red, green, blue, and purple, in this order. The full list of the index acronyms on the abscissa is given in Table 1.

[20] The group associated with the first PC (red bars) includes mostly Atlantic‒related indices the Atlantic Multidecadal Oscillation (AMO), the Tropical North Atlantic mode, the Atlantic Meridional Mode, Caribbean SSTs, and the Atlantic Tripole along with a few other indices, such as the global mean SST and the Western Hemisphere warm pool. Some of these links, like the one between the AMO and the Tropical North Atlantic mode, have been recognized in previous studies [Trenberth and Shea, 2006]. The AMO involves the buoyancy‒driven meridional overturning of the entire Atlantic Ocean [Delworth et al., 1993; Chen and Ghil, 1996] and it does seem to have a near‒global climate impact [Knight et al., 2006; Wang et al., 2008].

[21] The group of indices that is associated with the second PC (blue bars) appears to be linked with the Pacific Ocean. This PC captures much of the PDO, ENSO, the Tropical Pacific SST pattern, the Northern Oscillation, and the North Pacific pattern, with smaller contributions to other modes, such as the MJO, the Antarctic Oscillation, and the NAO. Pohl et al. [2010] already documented a connection between ENSO, the Antarctic Oscillation, and the MJO, and so did Giannini et al. [2001] for ENSO and the NAO. The near‒global effects of ENSO have, of course, been a matter of intensive study over the last three decades [Philander, 1990; Sarachik and Cane, 2010].

[22] The third PC (green bars) is associated mostly with the Arctic Oscillation, the NAO, and the Tropical Southern Atlantic index. Links between the Arctic Oscillation and the NAO have also been studied but are known to exhibit various complexities in their details [Ambaum et al., 2001; Kravtsov et al., 2006]. The last PC is associated with smaller contributions to the variance of a few tropical and Atlantic indices.

4 Teleconnections and the SST Field

[23] In studying the dynamics of the climate system on interannual and longer time scales, it is natural to investigate the role of the oceans and of their variability [NationalResearch Council, 1995]. Consequently, we computed a lag correlation between the global SST field [Kaplan et al., 1998] and our four leading PCs. In Figure 3, we show a map of the lagged cross‒covariances between the SST field and each of the four PCs, at the lag that maximizes the sum of the absolute values of the correlations over the whole map. The cross‒covariance has been set to zero and thus appears in white on the respective map, wherever the cross correlation is not significant at the 95% level.

Figure 3.

Lagged cross‒covariances between the local SST (in °C) and each of the four leading PCs, plotted for the lag that yields the highest sum of absolute values of the cross correlations: (a–d) Modes 1–4, each at the respective optimal lag. Color bar gives the cross‒covariance values; zero values (in white) include all cross correlations that are not significant at the 95% level. Note that as the PCs are normalized to unit standard deviation, the cross‒covariance at a grid point only depends on the standard deviation of the SST there, projected onto the given PC. To better show the most significant areas for each mode, the maps are centered on the Greenwich meridian in Figures 3a and 3c, while being centered on the dateline in Figures 3b and 3d.

[24] As expected for global‒scale phenomena, the cross‒covariances are quite substantial over large areas of the World Ocean. The SST pattern correlated with the first PC is consistent with the SST signature of the AMO [Messié and Chavez, 2011], and the second one with the classical pattern of the PDO [Weare et al., 1976; Mantua et al., 1997; Chao et al., 2000]. The cross‒covariances in Figure 3a are largest in the North Atlantic's subpolar seas, off Japan, and in several marginal seas, including the Mediterranean and the Baltic.

[25] The SST field is lagging here by roughly 1 year, while it is leading by about 1 year in the other three panels. Note that the time origin of the PCs is arbitrary, as it is based on a lagged set of indices; hence, only the relative lags are meaningful.

[26] In Figure 3b, the cross‒covariances are largest in the Kuroshio extension and, with the opposite sign, along parts of the West Coast of North America, in good agreement with the dominance of the PDO for this PC. For the third PC, however, the cross‒covariances are not particularly consistent with the Arctic Oscillation. This lack of consistency may be due, at least in part, to the complex relationships between the Arctic Oscillation and the NAO [Ambaum et al., 2001; Kravtsov et al.,2006]. The cross‒covariances here are largest in the South Pacific Convergence Zone and off the West Coast of Africa; the latter region is dominated by the Benguela Current and prominent also in Figure 3a. Finally, in Figure 3d, the area of significant cross‒covariances is much smaller than in the first three panels, being essentially restricted to the Tropical Pacific, the California Current, and the Caribbean.

5 Spectral Content

[27] Finally, we inquired into the extent to which more or less regular behavior in time might be associated with the four leading modes. First, each PC was analyzed using the multi‒taper method (MTM), without pre‒filtering. We found peaks at periods close to 2.5 years and 3.5 years for all the PCs, 5.7 years for mode 3, and 4.3 years for PCs 2 and 3; see Table 2, right column. The peaks were tested using a median‒filter robustness test [Mann and Lees, 1996]. Various taper numbers n, with 5≤n≤10, and the corresponding resolution parameters p, with p=2n−1, were tested, without any noticeable change of the results.

Table 2. Main Periods (in Years) of the Four Leading PCs and Their Significance Level From Spectral‒Density Estimates Using the Multi‒Taper Method (MTM) and the Maximum‒Entropy Method (MEM)a
  1. a

    Note that singular spectrum analysis (SSA) has been used as a pre‒filter for the MEM spectrum. Only the periods found for the total reconstruction are given here, and not the ones on the individual SSA modes. All the MTM results are significant at the 99% level according to the robustness test of Mann and Lees [1996]; please see text (section 5) for the SSA + MEM tests.

1> 20 
2> 20 
3> 20 

[28] Each PC was then analyzed using singular spectrum analysis (SSA) [cf. Ghil et al., 2002 and references therein]. The SSA decomposition was performed iteratively: an embedding window M is chosen in order to optimize the frequency concentration of the first mode, i.e., of the first pair of eigenvectors associated with nearly equal variances; the associated reconstructed component is then subtracted, and the residual is analyzed the same way, until eight modes or pairs have been estimated. The embedding window M is chosen large for the first SSA run, in order to capture the long‒periodic behavior, whereas we use M≈140 months, i.e., roughly 12 years, for the next runs.

[29] The maximum‒entropy method (MEM) spectrum has been computed both on the sum of the eight SSA modes (see results in Table 2) and on each mode separately. In the first case, a long‒term component, with a period longer than 20 years, and a periodic term around 2.5 years were identified for each PC; in addition, a 4.5 year component was found in PC 2 and a 5.3 year component in PC 3. Analyzing the modes separately allowed us to confirm the results obtained on the sum of the eight modes and to detect periodic components of lesser amplitude or longer period: 3.5 years and 10 years for PC 1; 3.5 years and 8.5 years for PC 2; 3.6 years and 10 years for PC 3; and 3.2 years and 10 years for PC 4. These results are robust when changing the MEM order within a reasonable range.

[30] The shorter periods of 3.2–3.6 years were confirmed by the MTM analysis in each of the four PCs. While not present in the MTM analysis with a high significance, it is of some interest to find the 10 year peak present in all four PCs, when analyzing the modes separately.

[31] Table 2 summarizes the results from these analyses. All four PCs have a highly significant peak, obtained by both methods, in the range of 2.3–2.8 years, which is most likely associated with the global effects of ENSO's quasi‒biennial oscillation [Philander, 1990; Sarachik and Cane, 2010; Ghil et al., 2002]. In addition, modes 2 and 3 also exhibit a longer and likewise highly significant period, at 4.3–5.7 years. This peak might, in turn, be associated with the quasi‒quadrennial periodicity of ENSO, modified maybe by the effects of the NAO's 7–8 year periodicity [Ghil et al., 2002; Feliks et al., 2010]. A longer period, of 21 years or even longer, is present only in the SSA analysis of all four modes; it could be due to the 17–21 year period in the PDO [Chao et al., 2000; Ghil et al., 2002], but cannot be significant in our data set, whose length barely exceeds twice the period under consideration.

6 Conclusions

[32] We have thus shown here that 25 different climate indices associated with a great variety of climatic fields and geographic regions share a very substantial fraction of their variability. This common fraction can be captured and described by using no more than four leading modes of variability. Much of this variability, in turn, is correlated with the SST field. The preferred periodicities apparent in these modes reflect mainly the quasi‒biennial and quasi‒quadrennial periodicities of ENSO. The short records, of not much more than 50–60 years, do not allow one to determine unequivocally the longer periods of 7–8 years of the NAO or the 17–21 years of the PDO, but some trace of such periodicities is also apparent in our analysis.


[33] This study was supported by the U.S. National Science Foundation and the Office of Naval Research (MG), and by the Institut Universitaire de France and the Centre National des Etudes Spatiales through the TOSCA program (OdV). The work of JOD is a phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, sponsored by the National Aeronautics and Space Administration (NASA). All but the MJO index time series were provided through the Earth System Research Laboratory web site at The MJO index time series comes from KNMI's Climate Explorer web site It is a pleasure to thank the editor (Noah Diffenbaugh) and two anonymous reviewers for their help in improving the paper.

[34] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.