Climatic variations in the Pacific and Atlantic Oceans are critical for the modelling and the prediction of global climate. Atmospheric teleconnections point out to the communication between the two Oceans. This is referred to as the atmospheric bridge, called teleconnections (Wallace and Gutzler, 1981; Klein et al., 1999; Rogers and Coleman, 2003). A number of publications presented evidence that these two regions are teleconnected to each other (Enfield and Mayer, 1997; Vecchi and Soden, 2007). The climate conditions of these two regions are described with a number of climate indices. Among these indices, the Pacific Decadal Oscillation (PDO) index (Mantua et al., 1997) and the North Atlantic Oscillation (NAO) index (Visbeck et al., 2001) are commonly accepted to reflect the major climate conditions of the Pacific and Atlantic Oceans.

In this study, we investigated the interconnection between the Pacific and Atlantic Oceans using these two representative climate indices using a spectral analysis. Until now, most of studies on spectral analysis of NAO and PDO focused on either finding abrupt changes (Schwing et al., 2003) or identifying a range of significant frequencies during a certain time period using frequency analyses and assuming an a priori basis (e.g. a mother wavelet in wavelet analysis) (Torrence and Compo, 1998; Labat et al., 2004). However, in the current study an empirical frequency technique, called empirical mode decomposition (EMD) (Huang and Wu, 2008; Lee and Ouarda, 2010) was used to analyse the NAO and PDO indices.

The empirical signals and connections hidden in the NAO and PDO indices are shown through the EMD components for different frequency levels. The variations and the correlation structures of high-frequency components according to the phase of the low-frequency components are investigated. Furthermore, global spatial patterns of climate variables are also analysed during the long-term warm and cold periods identified from the low-frequency EMD components of two indices.

2. Methods and data

2.1. Empirical mode decomposition

To investigate the association of long-term oscillations as well as short-term modes between NAO and PDO indices, EMD analysis was applied (Huang et al., 1998; Huang and Wu, 2008). EMD is an algorithm that extracts nonstationary oscillation components into the intrinsic mode functions (IMFs) from a targeted time series in order to yield physically meaningful explanations of the time series. To find significant components, a significance test (Wu and Huang, 2004) was developed. This test was used to compare the relationship of the mean period and the mean normalized energy of the IMFs of the target time series with the IMFs from normal random independent sequences. Unlike other time frequency analyses (e.g. wavelet analysis), the EMD procedure directly decomposes the original signal into a finite number of components (Lee and Ouarda, 2010, 2011, 2012).

The procedure to obtain IMFs from a time series, X, is as follows: (1) identify all of the local extrema and connect all local maxima (minima) with a cubic spline as the upper (lower) envelope; (2) obtain the first component (h) by finding the difference between the data and the local mean of the upper and lower envelopes (m) as h = X − m; (3) substitute X by h and repeat steps (1) and (2) until a certain criterion in which the component (h) is guaranteed to retain enough physical sense of both amplitude and frequency modulations (Huang et al., 1998) is met; (4) designate the final h as the jth IMF, c_{j}, and the residue is r_{j} (i.e. r_{j} = r_{j−1} − c_{j} where r_{0} = X); and (5) repeat steps (1)–(4) by treating the residue r_{j} as the original data until the final residue becomes a monotonic function. The final residue (r_{n}) becomes c_{n+1}. Then the original time series, X, is presented as the sum of the estimated IMFs as

(1)

Note that lower order components have a higher frequency level and vice versa. For example, the highest frequency level component is c_{1}.

2.2. Data description

NAO is a large seesaw in atmospheric mass between the Icelandic low and the Azores high (Visbeck et al., 2001). A zonal-average version was introduced by Li and Wang (2003a, 2003b) and used in the current study instead of the instrumental record-based indices (Cullen et al., 2002). The NAO index (NAO_{LW}) is denoted as

(2)

where is the standardized monthly sea-level pressure (SLP) anomalies over the longitude of 80°W–30°E at the latitude of θ. The yearly NAO index was formed by averaging the monthly NAO index.

The PDO index represents the leading principal component of sea-surface temperature (SST) anomalies in the North Pacific Ocean, polewards of 20°N. Among a number of PDO indices, the most commonly used one developed by Mantua et al. (1997) was used.

In the current study, SLP (Pa), precipitable water (PW; kg m^{2}), and SST (Kelvin) are selected to determine the global climate effects along with the long-term oscillations of the two indices. The global SLP and PW data were acquired from the dataset of the National Centers for Environmental Prediction and the National Center for Atmospheric Research (NCEP/NCAR) reanalysis data (Kalnay et al., 1996), for the period 1948–2008. National Oceanic and Atmospheric Administration (NOAA) extended, reconstructed data were used for SST for the period 1900–2008 (Smith et al., 2007). Yearly anomalies were estimated from the monthly dataset by subtracting the monthly mean relative to years 1961–1990 and averaging the monthly anomalies for each year.

3. Results

3.1. EMD analysis on the NAO and PDO indices

An EMD analysis indicated the presence of a considerable interconnection between the two indices in both long-term and short-term frequencies, as shown in Figure 1. The average of the two summed critical long-term oscillations (4th and 5th components) from the EMD analysis and its significance test (Wu and Huang, 2004, 2005) as well as the time series of the NAO and PDO indices are presented in Figure 1(a). Moreover, the most significant long-term oscillation component (the 5th) from the significance test (Wu and Huang, 2004), with around a 70 to 80 year cycle, shows that this long-term oscillation pattern exists in both signals (Figure 1(b)). The magnitudes and cycles for the long-term oscillation component of both signals match well. The slight deviation between the two 5th components (i.e. the PDO's 5th component is slightly shorter than the NAO's, Figure 1(b)) is inevitable because random noise processes and other components might slightly perturb the 5th oscillation process during the EMD analysis (Huang et al., 1998; Huang and Wu, 2008). There could be an internal (Haines, 1994) or external (McGuffie and Henderson-Seller, 2005) driving force or a combination of both (Latif et al., 2009) that induces this long-term oscillation simultaneously in the Pacific and Atlantic Oceans. In short, Figures 1(a) and (b) reveal that the two indices share the same long-term oscillation (a 70 to 80 year cycle).

Furthermore, the magnitude of the cross-correlation between the two indices and the serial correlation of each index vary according to the phase of the long-term oscillation, especially over the last 60 years (see Figures 1(c) and (d)). As shown in Figure 1(c), the cross-correlations between the residuals of the NAO and PDO indices with a 21 year window (10 years before to 10 years after the centre) are significantly negative after 1970 to the present, while around 1950–1970 no significant cross-correlation is observed. The lag-1 serial-correlation of the residuals (the original data minus the 4th and 5th components) shown in Figure 1(d) is negative for the NAO index and positive for the PDO index during 1955–1965, while outside this period, not much significant correlation is observed.

In order to draw the spatial implications of the observed long-term variations shown in Figure 1, the spatial patterns of the global climate variables as well as their correlation structures are analysed in the following three sections.

3.2. Global climate variation and long-term oscillation of climate indices

The long-term oscillation presents a negative phase centered on the year 1960 (Neg60, years 1955–1965) and a positive phase centered on the year 1990 (Pos90, years 1985–1995). The spatial patterns of the selected climate variables (i.e. SLP, PW, and SST) over these two periods (Neg60 and Pos90) are displayed in order to reveal the existence and the global effect of the long-term oscillation (refer to Figure 2).

The mean yearly anomalies of the SLP for Neg60 and Pos90 are displayed in Figure 2(a-1) and (a-2). In these figures, blue regions indicate values lower than the mean of 1961–1990, whereas red regions indicate higher values. Notably low SLP is then observed over North Africa and the middle of Asia, which represent arid climate zones in Figure 2(a-1). The lower-than-normal pressure over the arid zone is related to the precipitation, because rainy days are more likely in a lower-pressure condition than at higher pressure. A considerably high positive quantity of PW over North Africa is observed in Figure 2(b-1) as expected.

On the contrary, Figure 2(a-2) shows that the mean yearly anomalies of the SLP during the Pos90 are the opposite of the anomalies of the Neg60, such that a lower SLP over both poles and a slightly higher SLP over North Africa and the middle of Asia were present during the Pos90. The higher SLP over North Africa might be expected to lower the precipitation of this region, as shown in Figure 2(b-2), in opposition to the years 1955–1965. In Figures 2(c-1) and (c-2), the mean yearly anomalies of the SST over the Neg60 and the Pos90 show opposite spatial patterns over various regions. The positive-anomaly regions during the Neg60, such as the North Atlantic near Greenland and the middle Pacific, present negative values during the Pos90. Notice that, in the region between 35°N and 65°N, the Pacific and Atlantic Oceans show opposite tendencies during the Pos90 period (35°N, cold Pacific and warm Atlantic; 65°N, warm Pacific, cold Atlantic) shown in Figure 2(c-2), while this spatial tendency is not obvious during the Neg60 (Figure 2(c-1)). The negative-anomaly regions, especially the Southern Hemisphere during the Neg60, show a slightly positive anomaly during the Pos90. The overall spatial tendency of SST is summarized in Figure 2(c-3).

The difference maps of the mean yearly anomalies between the Pos90 and the Neg60 (the anomalies of the Pos90 minus the ones of the Neg60) for SLP, PW, and SST are illustrated in Figures 2(a-3), (b-3) and (c-3), respectively. The red regions indicate an increase of a variable during the Pos90 period compared to the Neg60, while the blue regions indicate a decrease in the variable. As shown in Figure 2(a-3), the Pos90 SLP is higher than the Neg60 in Africa and Eurasia. In particular, the desertic regions (North Africa and the middle of Asia) and southern Europe show a relatively significant increase. For the NAO index (the difference of the SLP 35°N and the SLP 65°N, Equation (2)), it is obvious that the decrease of the SLP over the Azores High (blue) region and the increase of the Icelandic Low (red) region led to the variation of the NAO index from the negative phase over the Neg60 period to the positive phase over the Pos90 period.

The spatial pattern of precipitable water might be expected to follow the conditions of the SLP shown in Figure 2(b-3). The most significant difference is observed around the North African (Sahara desert) region. The high negative values (dark-blue region) indicate the significant reduction in precipitation related to the desertification of this region. The region where the significant precipitation reduction occurred includes the watershed of the Niger River. The time series of the Niger River at Koulikoro, which covers the western part of the Niger River basin, is shown in Figure 3. It reveals that a much higher volume of stream flow was observed during the Neg60 than during the Pos90 period. This coincides with the results shown in Figures 2(b-1), (b-2) and (b-3). In contrast, the Amazon region shows an increase in precipitable water along with South Australia. The magnitude is not very significant compared to the Niger River basin.

3.3. Cross-correlation study between climate indices and global gridded variables

Further investigation of the long-term variation of the cross-correlations between the residuals of the NAO and PDO indices, as shown in Figure 1(c), was carried out in this section. The cross-correlations between the PDO index and the global SLP over the extended Neg60 (1950–1970) and the extended Pos90 (1980–2000) periods were estimated, and the results are presented in Figures 4(a) and (b), respectively. During the extended Neg60 period, the PDO index was negatively correlated with the whole Pacific region and the middle and north Atlantic, while it was positively correlated with the eastern side of Greenland and the south Atlantic (Figure 4(a)). Notice that the negative cross-correlation of the North Atlantic Ocean near 35°N was stronger during the extended Pos90 (Figure 4(b)). At the same time, a positive cross-correlation with the PDO index is observed above 65°N of this region during this period. Therefore, these spatial patterns lead to the conclusion that the negative correlation between the PDO and NAO indices for the extended Pos90 (see Figure 1(c)) was influenced by the negative correlation of PDO with the 35°N region and the positive correlation of PDO with the 65°N region. The proof of this cross-correlation structure is provided in Equation (A4) of Appendix A. for the case of the NAO index (NAO_{LW} in Equation (2)).

3.4. Serial correlation study of global gridded variables related to climate indices

The long-term variation of the lag-1 correlation estimated from the residuals of the PDO and NAO indices shown in Figure 1(d) was further analysed for the global climate variables (i.e. SLP and SST). The lag-1 correlation of the global SLP and SST was estimated for the extended Neg60 and the extended Pos90 periods and is presented in Figure 5. During the extended Neg60 (Figure 5(a-1)), the SLP had a stronger temporal self-dependent structure globally than during the extended Pos90 (Figure 5(a-2)). During the extended Neg60, a strong positive serial correlation is observed in Africa, South America, and middle Asia, whereas a significant negative serial correlation is seen in Greenland and nearby northern Europe. The strong negative SLP lag-1 serial correlation in the mid-Atlantic (35°N) and Greenland (65°N) described in Figure 5(a-1) for the extended Neg60 (blue line of Figure 1(d)) might drive the negative serial correlation in the NAO index during the same period (refer to Equation (A7) of Appendix A). No significant persistence over the SST can be observed (Figure 5(b-1) and (b-2).

4. Remarks and conclusions

The results of the current study reveal that (1) the representative climate indices of the Pacific and Atlantic Oceans, PDO and NAO, share the same long-term oscillation pattern over the Neg60 and Pos90 periods; and (2) according to the phase condition of the long-term oscillation, the cross-correlation between the two indices and the autocorrelation structure of each index varies in strength for the extended Neg60 and the extended Pos90. The global climate condition over the negative and positive periods shows a particular spatial tendency in SLP, PW, and SST. The burgeoning negative period starting from around year 2005 is also analysed. Somewhat similar spatial tendency to Neg60 is observed in SLP, PW, and SST variables even though it is not clear because of the short available record (data not shown).

Overall, a long-term oscillation with a 70 to 80 year cycle may exist in the Pacific and Atlantic regions, and global climate variables follow the conditions of the long-term cycle. This long-term cycle is also discussed by Keenlyside et al. (2008) for the North Atlantic sector. If the long-term oscillation continues to vary smoothly and cyclically similarly to the last several decades presented in the current study, the spatial evolution of climate variables might be predictable with a time-series simulation technique (Smith et al., 2007; Lee and Ouarda, 2010). However, the number of long-term cycles is still too limited to lead the firm conclusion. Further physical study and the analysis of long proxy records of climate indices (Shen et al., 2006) might be useful to confirm the long-term oscillation because the current time series of the two climate indices contain only one full 70 to 80 year cycle.

5. Appendix A. Lag-1 serial correlation of NAO and cross-correlation between NAO and PDO

Assume that Z_{t} represents the NAO index introduced in the original manuscript with time index t and that X_{t} and Y_{t} are the scaled SLP over the Atlantic region for 35°N and 65°N, respectively (Li and Wang, 2003). Therefore, the NAO (Z_{t}) is represented as

(A1)

Assume that A_{t} is another variable indicating the PDO index,

(A2)

The expectation is

(A3)

With zero mean for all the variables, the covariance is

(A4)

If cov(AX) < 0 and cov(AY) > 0, then it follows that cov(AZ) < 0.

As the variance is generally always greater than zero, this implies that, if A and X are negatively correlated and A and Y are positively correlated, then A and Z are negatively correlated.

For lag-1 serial correlation,

(A5)

Without loss of generality, E(X_{t}Y_{t−1}) = E(Y_{t}X_{t−1}) = 0; then

(A6)

With zero mean for all the variables,

(A7)

If cov(X_{t}X_{t−1})< 0 and cov(Y_{t}Y_{t−1})< 0, then surely cov(Z_{t}Z_{t−1})< 0.

As the variance is generally always greater than zero, this implies that, if X_{t} and X_{t−1} are negatively correlated and Y_{t} and Y_{t−1} are negatively correlated, then Z_{t} and Z_{t−1} are negatively correlated as well.