1. Introduction

The North Atlantic Oscillation (NAO), which is usually defined by the difference in sea level pressure between Iceland and the Azores, is the dominant pattern of atmospheric variability over the North Atlantic region, especially in winter (Hurrell 1995; Kushnir 1999). The NAO is strongly correlated with large-scale changes in sea surface temperatures (SSTs) across the basin (Bjerknes 1964). However, the role that air–sea interactions play in the dynamics of the NAO is not fully understood. It is generally believed that atmospheric forcing dominates interactions over the North Atlantic basin and generates SST anomalies through turbulent heat fluxes or anomalous wind stress (Frankignoul 1985; Cayan 1992). Alternatively, some suggest that the ocean participates in the dynamics via the influence of SST anomalies. Because the well-mixed ocean upper layer has a large heat capacity, an oceanic thermal signal can persist for several months, which allows for a persistent thermal forcing of the overlying atmosphere (Frankignoul 1985; Kushnir et al. 2002).

Evidence for oceanic forcing of the NAO is suggested by studies of climate models (e.g., Rodwell et al. 1999) and observational data (e.g., Czaja and Frankignoul 1999, 2002, hereafter referred to as CF99 and CF02, respectively). The latter (CF99; CF02), which use maximum covariance analysis (MCA), indicate a significant correlation between the wintertime NAO (the first MCA mode of 500-hPa anomalies in their analysis) and the leading mode of anomalous SSTs from the previous summer. The link between ocean and atmosphere at such long leads seems to be a result of the long persistence of the North Atlantic SST anomalies and thus serves as evidence for the oceanic forcing of the NAO (Kushnir et al. 2002).

However, simple correlation techniques can be problematic when used to assign causal order in highly coupled systems such as the atmosphere and the ocean. The correlation between the wintertime NAO anomaly and SST anomalies from earlier seasons may be due to the persistence, or even influence, of preceding atmospheric anomalies, which also influence the underlying SSTs during the previous seasons. Thus the results described by Kushnir et al. (2002) cannot exclude the possibility that the apparent oceanic forcing signal is not uniquely provided by the ocean, but also may have existed within the atmosphere.

To expand simple correlations between the NAO and SST anomalies, we introduce the notion of Granger causality (Granger 1969). Granger causality has been used to investigate physical systems (Kaufmann and Stern 1997; Kaufmann et al. 2003) and is based on the notion of predictability. In general, in a coupled system that involves two interacting fields, Granger causality tests whether past values of one field (X) statistically help to predict the current values of the other field (Y) better than using past values of Y alone. Should past values of X contain information about current values of Y beyond that contained in the preceding Y sequence (or any other variables contained in the information set), variability in the X field is said to “Granger cause” (hereafter cause for simplicity) variability in the Y field. Similarly, we can test whether previous values of Y cause variability in the present values of X.

In this paper, we use Granger causality to examine the relationship between the NAO and SSTs. Section 2 describes the methodology that is used to test for Granger causality. Section 3 describes the datasets, and the results are described in section 4. The implications for the link between SST anomalies and the NAO anomaly are summarized in section 5.

2. Methodology

We test for the presence and direction of Granger causality between the NAO and SST anomalies following the procedure outlined by Kaufmann and Stern (1997). This procedure includes two steps. In the first step, the two-way interactions between the NAO anomaly and each gridpoint SST anomaly are described using a vector autoregression (VAR) given by Eqs. (1) and (2):

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where α, β, and γ are regression coefficients, e's are error term, and s is the lag length, which is determined with a likelihood ratio test developed by Sims (1980). Equations (1) and (2) are derived from a structural VAR (not shown), in which the current values of NAO and SST are a function of lagged values of the dependent variable and the current and lagged values of the independent variables (Enders 1995). The structural VAR specification [as well as the standard form given in Eqs. (1) and (2)] is consistent with the understanding that the current (winter, e.g.) values of the seasonal NAO and SST anomalies depend not only on the concurrent anomalies of the other variable, but also on their past values, that is, from the previous fall, summer, etc.

To determine the direction of causal order, we estimate restricted forms of Eq. (1) or Eq. (2) in which the causal variable is eliminated. For example, to test whether SST anomalies cause variability in the NAO, we estimate a restricted form of Eq. (1) in which we eliminate the SST anomalies. This is done statistically by restricting γ in Eq. (1) to zero as follows:

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Conversely, we test whether the NAO causes SST anomalies by estimating a restricted version of Eq. (2) in which we eliminate the lagged values of the NAO index.

The next step is to test whether the restricted estimates are statically significantly different from the unrestricted estimates. To do this, we calculate a test statistic as follows:

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in which RSS is the sum of the residuals squared; the subscripts r and u refer to the restricted and unrestricted versions of Eq. (1) or Eq. (2), respectively; T is the number of observations; k is the number of regressors in the unrestricted version of the equation; and s is the number of coefficients restricted to zero in Eq. (3). The test statistic can be evaluated against an F distribution with s and T − k degrees of freedom in the numerator and denominator, respectively, in order to evaluate the null hypothesis that the variable eliminated from Eq. (1) (SST in the given example) does not cause variability in the dependent variable (i.e., the NAO). For this paper, values of ω that exceed the 5% threshold reject the null hypothesis of no causal order, which indicates that the variable eliminated from the unrestricted equation causes variability in the dependent variable.

A statistically significant increase in the RSS of the restricted version of Eq. (3) relative to the unrestricted version of Eq. (1) indicates that the lagged values of SST have information about the current value of NAO beyond that in the lagged values of NAO alone. Eliminating the lagged values of SST reduces the explanatory power of the VAR, which increases the residual sum of squares. The ω statistic quantifies the information uniquely contained in the lagged SST values and represents the statistical power of Granger causality relative to simple correlation techniques.

As written, Eqs. (1) and (2) assume that air–sea interactions are constant over seasons. That is, the 1-season lag interaction between the SST and NAO is the same regardless of the seasons under consideration (i.e., spring/ summer, summer/fall, or fall/winter). To relax this restrictive assumption, we allow the coefficients to vary with lag length and seasons. Because winter is the season of interest, we modify Eqs. (1) and (2) as follows:

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In Eqs. (5) and (6), only winter values of the anomalous NAO and gridpoint SSTs appear on the left-hand side. These winter values are predicted based on values from previous seasons. Because the “current” time is winter, a lag length of 1 denotes the previous fall, a lag length of 2 denotes the previous summer, and so on. The timing of these lags implies that the statistical methodology tests whether the SST (NAO) field from previous seasons contains information about the winterime NAO (SSTs) that is not contained in the preceding values of the NAO (SSTs) itself.

We recognize that the detection of Granger causality does not necessarily imply a physical causal mechanism between the two fields. Conclusions about the presence and direction of causality depend on the validity of the statistical models. Estimates may be biased by the omission of relevant variables [e.g., the summer extent of snow cover over northern North America and northern Eurasia (Saunders et al. 2003)] that are in fact the causal variables. Despite these limits, the causality test is more rigorous than lagged correlation statistics because it removes “false positives” in which relations with lagged values are in fact the result of autocorrelation within the predicted field (see below).

3. Datasets

The NAO index time series is defined in Hurrell (1995) and Jones et al. (1997). Data for sea surface temperatures are taken from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis monthly mean SST dataset, which has global coverage with 192 (longitude) × 92 (latitude) grid points at Triangular-62 resolution (approximately 2° latitude and longitude). A description and evaluation of the reanalysis SST dataset can be found in Hurrell and Trenberth (1999). This analysis uses SSTs within the domain of 75N°–20°S and 180°–180°. The extension of the SST domain beyond the North Atlantic basin reflects the consideration that SSTs over other oceans [e.g., the tropical Indo-Pacific region (Hoerling et al. 2001)] also may influence the atmospheric circulation over the North Atlantic. Both the NAO index and the SST data are available monthly from January 1948 to December 2000. Monthly anomalies are created by subtracting the monthly climatology (1948–2000) from the monthly data. In addition, a linear trend is removed from the NAO time series and the SST time series at each grid point (conclusions are the same when a time trend is explicitly included in the statistical models as opposed to detrending the data a priori). Seasonal-average anomalies are calculated by averaging the three monthly values in the corresponding season. The seasons are defined as March–April–May (MAM) for spring, June–July–August (JJA) for summer, September–October–November (SON) for fall, and December– January–February (DJF) for winter.

4. Analysis results

First, we regress the detrended winter NAO index onto the concurrent (winter) SST field (Fig. 1a). The correlation map (Fig. 1b) shows the well-known tripole pattern over the North Atlantic basin. Near the southeast of Greenland (GL; 50°–65°N, 15°–55°W) and in the subtropics (ST; 10°–25°N, 20°–60°W), the SST is negatively correlated with the NAO. Between these two negative centers is an area of positive correlations located over the extension of the Gulf Stream (GS; 25°– 45°N, 30°–60°W). This tripole pattern is consistent with many other studies, including those mentioned above. For example, the leading MCA mode of SST in CF02, termed the North Atlantic Horseshoe (NAH), has a similar structure but is shifted to the northeast. Figure 1b also shows a similar but weaker correlation pattern over the North Pacific Ocean. It reflects the fact that the NAO is closely related to the Arctic Oscillation (AO) and indeed has an impact beyond the Euro–Atlantic half of the Northern Hemisphere (Thompson and Wallace 1998, 2001).

Next, for SST anomalies at each grid point, we test both whether the preceding NAO causes variability in the wintertime SSTs and whether the preceding SSTs cause variability in the wintertime NAO. The tests of Granger causality are run on VARs [Eqs. (5) and (6)] with lag lengths of 2, 3, and 4 seasons. Figure 2 shows the results obtained by setting maximum lag lengths to 3 seasons (the previous spring). Figure 2a indicates that preceding seasonal anomalies in the NAO Granger cause wintertime SST variations in only small and sparse areas. Conversely, Fig. 2b identifies a large, spatially consistent area of the Gulf Stream extension where SSTs Granger cause anomalies of the wintertime NAO. This region corresponds to the positive center on the correlation map (Fig. 1) with a northeastern shift. The Fig. 2b values, which represent the difference between the r² statistic for the unrestricted and the restricted equations, indicate that SSTs from previous seasons reduce the unexplained variance by up to 20%, relative to the lagged values of the NAO. In contrast, Fig. 2b indicates that there is no statistical evidence for a causal influence of SSTs near Greenland or the subtropical regions upon the NAO.

The Granger causality tests are also applied using alternative maximum lag length settings (s = 2, 4). For s = 2 seasons, the influence of the NAO on the SST field remains insignificant. However, the influence of the NAO increases when s increases to 4 seasons (not shown). This increase may be associated with the persistence of subsurface temperature anomalies that reappear the following winter (Rodwell and Folland 2002). In contrast, all tests of the causal relations from preceding SSTs to the wintertime NAO confirm the results in Fig. 2b; that is, over the North Atlantic basin, only SSTs around the Gulf Stream extension have a significant causal relationship with the winter NAO.

Figure 2b also identifies some areas over the Indian Ocean and the western/tropical Pacific where SSTs have a significant causal relationship with wintertime NAO variations. Generally, these relationships are weaker and less persistent than those found in the Gulf Stream extension. On the other hand, if realistic, they seem to support the idea that oceanic influences on the NAO may not be restricted to the North Atlantic basin. Although these causal relationships deserve more investigation, they will not be discussed further in this paper.

One limitation of using the NAO index (as defined in Jones et al. 1997) is that it may not capture the movements of the centers of the NAO pattern in seasons other than winter. To address this issue, we repeat the analysis using another NAO index compiled by the U. S. Climate Prediction Center (CPC). The CPC index is computed from the rotated empirical orthogonal function analysis of monthly mean 500-mb geopotential heights (Barnston and Livezey 1987) and therefore is expected to better reflect the seasonal spatial changes of the NAO. The results obtained using the CPC index are quantitatively the same as those in Fig. 2. This similarity indicates that the seasonal shifts of the NAO pattern do not seem to have a major influence on the results shown in Fig. 2.

As discussed previously, the results of Fig. 2 suggest that the preceding NAO does not contain significant explanatory information about wintertime SSTs over most parts of the North Atlantic basin (other than that found in the preceding SSTs themselves). In contrast, SSTs around the Gulf Stream from the previous summer and spring contain information related to the NAO during the following winter that is not contained in the previous values of the NAO. The first result indicates weak atmospheric forcing at seasonal time scales. This lack of a causal relationship may be due to the fact that the persistence time scale of the NAO is on the order of 20–30 days and that the surface turbulent heat fluxes are dominated by short time-scale weather changes (Frankignoul 1985; Deser and Timlin 1997). In fact, repeating our analysis with monthly data shows that the NAO anomalies from the previous two or three months significantly cause SST anomalies during wintertime (not shown). For seasonal anomalies, however, these high-frequency signals are filtered out, and the causal influence may be restricted. On the other hand, the large heat capacity of the ocean allows an SST signal to persist for months to seasons. Therefore, it is possible for a persistent SST signal to initiate and force a seasonal-type anomaly in the atmosphere. These results are similar to those reported by CF02; however, the apparent tripole pattern (NAH) discussed in that paper disappears in our analysis using the Granger causality technique. This discrepancy may be partially due to the fact that we test the whole SST field instead of decomposing it into several leading modes. Furthermore, it suggests that the anomalous SSTs around the Gulf Stream may be more important in initiating disturbances of the atmospheric circulation over wintertime North Atlantic, while the tripole pattern may be generated by positive feedbacks between these two fields (CF02).

To compare the results generated by Granger causality with those from simple lagged correlation analysis, we calculate the correlation coefficients between the wintertime NAO and SSTs from the previous fall and summer (Fig. 3). In both Figs. 3a and 3b, the NAO time series is correlated with the earlier SSTs around the Gulf Stream (the averaged value of the correlation coefficients r in this region is about 0.5). These results are consistent with those in Fig. 2b. We note that such agreement is not always present. For example, the winter NAO is significantly correlated with gridpoint SSTs during the following spring, that is, when the NAO leads the SSTs by a season (not shown). However, the Granger causality test of the influence of the wintertime NAO on the following springtime SSTs (not shown) indicates that much of this correlation is actually related to the state of the wintertime SST field itself and therefore cannot be uniquely attributed to the preceding NAO field. As such, the Granger causality test is stricter and more reliable than simple lagged correlations.

To validate our analysis, we compile three SST indices by spatially averaging SSTs over the previously identified GL and ST regions, which show significant concurrent correlations with the wintertime NAO (Fig. 1), along with the GS region, which shows significant causal relations with the wintertime NAO (Fig. 2). The SST indices averaged for the proceeding June–November are shown in Fig. 4. All the concurrent SST indices correlate well with the NAO in winter, and the signs of the correlations agree with the negative–positive–negative tripole pattern (Table 1). However, only the values from the Gulf Stream SST region have significant (p < 0.05) lead correlations with the winter NAO. In addition, the Granger causality test also indicates that only the GS SST causes variability in the NAO (not shown). These results are consistent with modeling studies by Wu and Gordon (2002), which indicate that Gulf Stream variability in autumn can influence the wintertime atmosphere over the NAO region on decadal/interdecadal time scales. Here we find a similar relationship within the observed and analyzed fields, indicating that this influence also may be apparent on shorter time scales.

5. Conclusions

Here, we use the notion of Granger causality to investigate the relationship between the NAO and SSTs over the North Atlantic basin. Although the concurrent correlation map of the wintertime NAO and SSTs shows the well-known tripole pattern, the influence of the preceding seasons' NAO anomalies on the wintertime SST field are very limited. Instead, the preceding state of the ocean shows a significant causal relation with the dominant wintertime atmospheric circulation pattern. This influence is mainly centered over the Gulf Stream (GS) area. These results agree with simple lagged correlation maps of the winter NAO and preceding SSTs from earlier seasons; however, we believe the Granger causality is a more rigorous test for the causal order within the coupled system. To further evaluate these results, we define three SST indices by area-averaging the SSTs near the centers of the typical tripole pattern, including the Greenland (GL), the GS, and subtropics (ST), respectively. It is found that only the GS SST index shows significant lead correlations with the winter NAO. In addition, the GS index is also the only one that “Granger causes” anomalies in the winter NAO. These results suggest that the anomalous Gulf Stream SSTs are important in initiating disturbances of the atmospheric circulation over the wintertime North Atlantic. In addition, the results suggest that the full tripole pattern (e.g., as identified in CF02) may instead represent the overall effect of positive feedbacks between the ocean and the atmosphere in this region. It is recognized that these results are statistically based and are limited by the specification of the model used in this study. In addition, the ocean–atmosphere dynamics related to these different processes need to be investigated further. However, using the Granger causality test in this context allows us to better delineate possible sources of forcing within this coupled ocean–atmosphere system.