1. Introduction

The boundary interaction of the surface heat flux with the sea surface temperature (SST) holds the key to understanding the nature of the coupled atmosphere–ocean system. Dominant heat flux forcing has been well established by both observations and numerical models (Kushnir et al. 2002). However, our current understanding of the surface heat flux feedback—the surface heat flux response to the SST anomalies—remains limited. Because of the causality problem, the heat flux response to the SST is difficult to observe, especially in midlatitude regions with strong atmospheric internal variability. To study this problem, ensemble model experiments that suppress the atmospheric internal variability have been widely performed. However, the results are strongly dependent on the specific model employed. Some studies have shown a negative feedback (Rahmstorf and Willebrand 1995; Kushnir and Held 1996), whereas others have reported a positive feedback (Latif and Barnett 1994; Peng et al. 1995). These controversial results require an observational benchmark for calibrating model performance.

Frankignoul et al. (1998) proposed a statistical method [hereinafter termed equilibrium feedback assessment (EFA)] for separating the ocean forcing signal from the atmospheric internal variability. The method was applied in an observational study of the surface heat flux feedback over the eastern and central North Atlantic, where the ocean current is weak and the local response assumption is reasonable (Frankignoul et al. 1998). The study revealed a local negative feedback with an amplitude of approximately . Their succeeding studies (Frankignoul and Kestenare 2002; Frankignoul et al. 2002; Park et al. 2005) further confirmed the dominant local negative response of the heat flux to the SST in both the North Pacific and North Atlantic Oceans. As the main component of the surface heat flux, the turbulent heat flux exerts a local negative feedback over most ocean basins (Tanimoto et al. 2003; Wen et al. 2005).

In addition to the dominant local feedback, a nonlocal feedback response of the heat flux to the SST may also occur. As recognized by Palmer and Sun (1985), the atmospheric response to the warm SST anomaly of the Gulf Stream Extension (GSE) region is a downstream ridge response in the central North Atlantic. The results of their model diagnostic indicated that transient eddy activity plays an important role in tilting the atmospheric response to the upstream SST anomaly. The downstream warm SST-ridge response is also identified from the observations of Ciasto and Thompson (2004). They found a significant impaction of the wintertime SST anomalies over the GSE region on the Northern Hemisphere annular mode on intraseasonal time scales. Similarly, Liu and Wu (2004) reported a downstream warm SST-ridge response of the atmosphere over the Aleutian Islands to the Kuroshio Extension SST anomaly, based on their coupled GCM experiment. In addition to these studies, many others have shown that the SST anomalies over the North Hemisphere western boundary current region play an important role in large-scale atmosphere–ocean interaction (Czaja and Frankignoul 2002; Liu et al. 2006; Frankignoul and Sennechael 2007; Minobe et al. 2008; Kwon et al. 2010). In general, a nonlocal atmospheric response would facilitate the underlying boundary air–sea interaction and result in the heat flux nonlocal feedback. In addition to the atmospheric dynamic adjustments, the ocean advection could also add to the nonlocal heat flux feedback to the underlying SST anomalies, such as in Gulf Stream region where the current is strong. The heat advection can induce the variation of the heat content and heat storage in the vicinity of the current region, and then modify the SST variability there on interannual and longer time scales. The nonlocal SST anomalies may re-form the local heat flux feedback in these areas and result in the interannual-to-decadal climate variations (Kelly and Dong 2004). Therefore, it is more plausible that a nonlocal heat flux feedback may be observable.

However, it is a challenge to separate the nonlocal from the local feedback using this technique, because of the complex interferences from the interrelated SST forcings. The conventional approach to find the SST influence in a given region is to filter out the dominant external forcing with regression and then to assess the climate impacts using the residual variability, such as removing the ENSO signal to study midlatitude air–sea interaction (Vimont et al. 2001; Alexander et al. 2002; Zhong and Liu 2008). But, this approach is effective only if there is a single dominant external forcing and the forcing is known a priori. In general, atmospheric anomalies might be influenced by multiple ocean forcings that interact with each other in a complex way (Lau et al. 2006). To address this problem, Liu et al. (2008, hereafter LWL) generalize the EFA method from the univariate ocean forcing into the multiple ocean forcings, using the generalized equilibrium feedback assessment (GEFA), to exclusively identify the impact of each SST forcing. Unlike EFA, GEFA can automatically separate each contribution of the regional SST to the atmosphere from the interrelated SST forcings, without any prefiltering. The GEFA method has been applied to distinguish the atmospheric geopotential height response to global SST variability modes (Wen et al. 2010), and to evaluate the attribution of the regional SST variability modes to the U.S. precipitation variability (Zhong et al. 2011). Their studies serve as a demonstration of the potential utility of GEFA in identifying multiple surface feedbacks to the atmosphere in observations.

In this paper, we aim to detect the nonlocal heat flux feedback in the North Atlantic from the observations. The GEFA method is employed to do the investigation. The results show a robust nonlocal positive feedback of the GSE SST to the downstream heat flux in the subpolar region. The paper is organized as follows. In section 2, the data and methods are introduced. A nonlocal response is identified for a three-region case in section 3, which is then further confirmed in an extended six-region case in section 4. The last section summarizes the main points of this study.

2. Data and methods

The data used for the study come from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996). The variables include the surface turbulent heat flux (where the downward direction is positive), SST, sea level pressure, and sea surface wind on a T62 Gaussian grid. The domain investigated is 20°–60°N, 80°W–0°, covering the North Atlantic basin. In terms of the largest SST variability in the basin (Fig. 1a), the North Atlantic is divided into three regions: the first region is located at the GSE (35°–55°N, 60°–30°W), the second lies in the subpolar region (40°–60°N, 25°–5°W), and the third is in the subtropic region (20°–35°N, 40°–15°W). Monthly data over the period 1958–2011 were anomalies from their seasonal cycle and detrended with a third-order polynomial filter before the analysis.

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(a) Standard deviation of the monthly SST (°C) from the NCEP–NCAR reanalysis data for 1958–2011. The three regions studied in section 3 are marked, where the regions 1, 2, and 3 correspond to the GSE and the subpolar and subtropic regions, respectively. (b) The pattern of the first EOF of the SST. The above three regions are further subdivided into six regions, as marked (see section 4).

Citation: Journal of Applied Meteorology and Climatology 52, 3; 10.1175/JAMC-D-11-0257.1

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To detect the nonlocal heat flux feedback in the North Atlantic, the GEFA method was used. The technique is superior at identifying the impact of each SST forcing within a unified framework. It generalizes the EFA’s univariate ocean forcing as multivariate ocean forcings, such that the quasi-equilibrium heat flux equation is expressed as

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Its vector form is

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where is the surface turbulent heat flux, is the variable of SST with as the full feedback matrix, and is the atmospheric internal variability. Matrix includes all the feedback information between the entire SST field and the heat flux field, with representing the atmospheric heat flux response of the th region to the individual SST forcing at the th region . Taking advantage of the fact that the leading ocean cannot be forced by the lagging atmosphere, the GEFA estimation of the full feedback matrix can be derived after multiplying the transposed leading SST as

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with the sampling error

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Here, is a lag time longer than the persistent time of the atmospheric internal variability ( is taken to be 1 month in this calculation), and represents a lagged covariance matrix with T as the sample size. For seasonal GEFA estimation, the data from three consecutive monthly anomalies are used to increase the realization numbers and thus reduce the sampling error (Liu et al. 2012a,b). Taking the summer feedback as an example, the matrix during June–August is calculated by using the leading SST during May–July in Eq. (2.3). The winter (December–February) feedback matrix is calculated similarly.

One thing worth mentioning is that when using GEFA, the sampling error would be rapidly amplified with the increase in spatial resolution, as discussed in LWL. This is because the covarying SST makes the covariance matrix in Eq. (2.4) singular and thus produces a large sampling error. To optimize the GEFA estimation, an empirical method—the truncated empirical orthogonal function (EOF) method—was introduced (LWL; Fan et al. 2011). Through the reconstruction of SST anomalies using the leading EOF modes, this approach reduces the forcing field noise and, in turn, the sampling error. In this paper, the truncated EOF method is used in a six-region case to check the stability of the nonlocal feedback of the GSE SST on the downstream heat flux.

In addition, the univariate EFA estimation (Frankignoul et al. 1998) of the heat flux response to the SST is also calculated in this paper. The EFA feedback matrix is denoted as

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and where indicates the covariance between and . The matrix is also called the total feedback, with indicating the total heat flux response of the th region to the SST in the th region. Its relation with the GEFA feedback matrix is determined by mutual matrix , as

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For the total feedback matrix , the total feedback not only includes feedback from its own region but also involves the influence of other regions’ SST forcings weighted by . The mutual matrix reflects the relation between the SSTs of the different regions. Generally, because of the covariance of the SST of the forcing field, the feedback matrix differs from the total matrix .

3. Nonlocal feedback identified in a three-region case

In the following, we apply GEFA to study the ocean–atmosphere thermal feedback over the North Atlantic. The assessment is first conducted for a simple three-region case, where the regional heat flux and the SST are denoted as and (the numbers 1, 2, and 3 denoting the GSE, subpolar, and subtropic regions respectively, as shown in Fig. 1a). To provide a better understanding of the GEFA estimation, the EFA result is given as well. As shown in Table 1, the annual GEFA feedback matrix , the total EFA feedback matrix (both: ), and the mutual matrix are calculated using Eqs. (2.3), (2.6), and (2.9), respectively. It reveals the general picture of the heat flux feedback in the North Atlantic. Considering the seasonality, we further calculate the winter and summer feedbacks shown in Tables 2 and 3, respectively.

Table 1.

The annual GEFA feedback matrix , EFA feedback matrix , and mutual matrix for the three-region case. The matrices , , and are calculated by Eqs. (2.3), (2.6), and (2.9), respectively, using year-round data. Note that in the calculations, each row and column uses time applied to the quantities in the columns and applied to the quantities in the header row, where the lag time is taken to be 1 month. The symbols , , and in the left column of the table indicate the heat fluxes in regions 1, 2, and 3, respectively, and , , and in the top row denote the SSTs in regions 1, 2, and 3, respectively. For the feedback matrices, the diagonal elements denote the local feedback, whereas the off-diagonal elements indicate the nonlocal feedback. Each column () indicates the response of the heat flux in different regions to the SST anomaly of a certain region (e.g., for the first column in matrix , , , and denote the responses of the heat flux in regions 1, 2, and 3 to the SST anomaly in region 1, respectively). Similarly, each row () shows the response of the heat flux in a certain region to the SST anomaly in different regions. In matrices and , negative values indicate a negative feedback and vice versa. The boldface italic values indicate the significance at the 95% confidence level, and the only boldface values are at the 90% confidence level, as tested by a Monte Carlo method in which the year of the heat flux was scrambled 1000 times. As to the mutual matrix , the element reflects the relation between the th region SST and the th region SST . It represents the weight of the contribution (GEFA feedback) of the interrelated region SST to the total EFA feedback for the region of interest. For the significance in matrix , the rule is the same as that in feedback matrices and .

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Table 2.

As in Table 1, but for the winter (December–February) GEFA feedback matrix , EFA feedback matrix , and mutual matrix .

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Table 3.

As in Table 1, but for the summer (June–August) GEFA feedback matrix , EFA feedback matrix , and mutual matrix .

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From Table 1, it is clear that the annual feedback matrix is dominated by significant negative feedback along the diagonal, consistent with our understanding that the turbulent heat flux response is dominated by local negative feedback. This dominant local feedback is consistent with the estimation in the total matrix , indicating the damping of the turbulent heat flux to the SST anomalies. It varies by season, with response magnitudes of approximately in winter (see Table 2) and in summer (see Table 3). The strong (weak) response in winter (summer) is in line with the results in Park et al. (2005). They suggested that the negative turbulent heat flux feedback mainly stems from the weak moisture–temperature adjustment of the surface air to the underlying SST anomalies. This implies that local cooling and drying responses of surface air to warm SST anomalies tend to counteract the downward surface heat flux forcing (Nonaka and Xie 2003). The physical understanding of the air–sea thermal feedback can be evidenced by the lagged regression of the surface pressure and wind on the GSE SST, especially in winter (Fig. 2b). The warm SST anomalies over the GSE region tend to induce the local offshore cold, dry surface wind and, in turn, the anomalous heat released from the ocean. Overall, the dominant local negative feedback indicates that GEFA can be used to assess the full heat flux response to the SST.

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Regression of the SLP [black contours; contour interval (CI) = 0.3 hPa K⁻¹] and surface winds (vectors; m s⁻¹ K⁻¹) against the SST variability in the GSE region (region 1 in Fig. 1a) with the SST leading by 1 month, using (a) year-round, (b) winter, and (c) summer data, respectively. A wind vector is plotted when the correlation is statistically different from zero at the 95% level. The shading indicates the instantaneous regression of the SST onto the winter SST variability in the GSE region, with white solid (black dashed) lines for positive (negative) values (CI = 0.3).

Citation: Journal of Applied Meteorology and Climatology 52, 3; 10.1175/JAMC-D-11-0257.1

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Despite the dominant diagonal elements in , there is a significant off-diagonal element in Table 1, which represents a nonlocal positive feedback response of the subpolar region’s heat flux to the GSE region’s SST. Furthermore, the nonlocal feedback presents a strong seasonality. In winter (Table 2), its response magnitude is high, up to , which is comparable to its local response amplitude, , in the subpolar region . The winter nonlocal GEFA response is also consistent with the corresponding total feedback () in both sign and magnitude. As shown for the mutual matrix (in Table 2), the GSE SST is only slightly correlated with the SST in the other two regions () (some hints are also seen from the spatial regression of the winter SST on the GSE SST, as shaded in Fig. 2b), and therefore there is little distortion in the heat flux response to the GSE SST forcing (). In comparison with winter, the summer response is weak, (in Table 3), but it is still positive and above the 90% confidence level. Physically, this remote positive feedback suggests that a warm SST anomaly over the GSE region tends to warm the downstream subpolar SST through a downward heat flux. This downstream response seems to be caused mainly by a warm SST-ridge response, as indicated by Fig. 2. The ridge response over the subpolar region, which is strong in winter (Fig. 2b) and weak in summer (Fig. 2c), reduces the westerly wind speed and, in turn, the latent and sensible heat losses in that area—more heat loss in winter and less heat loss in summer. This downstream warm SST-ridge response in the atmosphere and the associated downstream warming effect on the SST have been previously studied both in observations and climate models (e.g., Palmer and Sun 1985; Ciasto and Thompson 2004). In the North Atlantic, it is a passage of the storm track over the GSE region. The warm SST anomalies there could strengthen the SST gradient and activate the baroclinic eddies. As a result, it generates a downstream ridge response and, in turn, the nonlocal positive heat flux response in the subpolar region. Therefore, the statistic result that the robust nonlocal heat flux feedback is detected in the GSE region, rather than in the subpolar or subtropic region, is consistent with our physical understanding.

The annual total matrix in Table 1 appears to show an additional significant nonlocal negative feedback, , implying a heat flux reduction in the subtropics in response to a warming in the subpolar region. The pattern is the same for the winter and summer total feedbacks ( in Table 2 and in Table 3). This overall response, however, is not presented in the feedback matrix and should therefore be interpreted with caution. Some insight can be gained by examining the mutual matrix , which has a significant off-diagonal subtropical SST response to the subpolar SST, with (see Table 1). With the equivalence relation in Eq. (2.8), we have

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Therefore, this seemingly nonlocal total feedback does not represent a true nonlocal response of the subtropical heat flux to the subpolar SST variability . Instead, it mainly recaptures part of the local negative response of the subtropical heat flux to the subtropical SST variability that covaries with the subpolar SST variability . The high positive correlation between the SSTs in the subpolar and subtropical regions is associated with the dominant tripole SST variability over the North Atlantic (such as Fig. 1b), which is predominantly forced by the spatially coherent atmospheric variability of the North Atlantic Oscillation (NAO; Cayan 1992; Deser and Timlin 1997).

In short, in the three-region case, GEFA confirms the dominant local negative feedback, and further identifies a significant nonlocal positive feedback from the GSE SST to the heat flux downstream in the subpolar region. This nonlocal effect is likely caused by the adjustments of the downstream surface wind to the SST anomalies over the GSE region.

4. Further confirmation in a six-region case

Next, we extend our analysis from three regions to a six-region case to examine the stability of the nonlocal effect of the GSE SST on the downstream heat flux. Each of the three regions is subdivided into a pair of subregions (Fig. 1b) in the east and west. The feedback matrix is estimated using the year-round data, as indicated in Tables 4–7. Here, the seasonal feedback is not shown in the six-region case. The stability of the seasonal nonlocal feedback is the same as that of the annual feedback. And the seasonality of the nonlocal feedback in the six-region case is also consistent with that in the three-region case.

Table 4.

As in Table 1, but for the annual GEFA feedback matrix for the six-region case. The subscript of represents the number of the leading SST EOF modes retained for the reconstruction of the SST anomalies. Here, the number 6 indicates the full information of the SST forcing field is kept in assessment.

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Table 5.

As in Table 4, but for the annual EFA feedback matrix for the six-region case.

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Table 6.

As in Table 4, but for the annual mutual matrix for the six-region case.

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Table 7.

As in Table 4, but here the annual GEFA feedback matrix is assessed using the reconstructed SST anomalies from the first four leading EOF modes.

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For the annual total feedback in Table 5, there is no big difference with an increased spatial resolution. Likewise, in , the local response along the diagonal is dominated by the negative feedback, with an amplitude of approximately . Furthermore, each pair of subregions exhibits a strong negative feedback on each other (, , and ) because of their proximity. The remote positive feedback of the GSE SST to the downstream subpolar heat flux is now dominated by the response of the eastern (western) subpolar subregion to the eastern (western) GSE subregion ( and ). In addition, the remote negative total feedback of the subpolar SST to the subtropical heat flux is now dominated by that from both subpolar subregions to both subtropical subregions (, , , and ).

In contrast, the annual feedback matrix (in Table 4) exhibits an amount of noise, as seen from the less clear diagonal dominance and the large far-off-diagonal elements. This finding is expected from the simple model study in LWL, due to the enhanced resolution and, in turn, the enhanced sampling error. Each pair of subregions is neighboring each other, and therefore their SSTs tend to covary with each other, as shown in the near-diagonal elements in (in Table 6). This covariability increases the singularity of the lagged SST covariance matrix in Eq. (2.4) and, as a result, produces a sampling error in .

Following the simple model study in LWL, the sampling error in can be reduced by truncating the SST EOF mode. The total matrix seems to be insensitive to the EOF truncation (as long as the first two leading EOFs are retained; not shown), consistent with the simple model study in LWL. In contrast, the feedback matrix shows a much greater sensitivity to the EOF truncation. Since the true is unknown in the observations, we will empirically determine an optimal from their successive convergences. We found that [explained variance (EV) = 96%] or (EV = 99%) appears to be the optimal estimator. This selection is based partly on the successive pattern correlations, , which are 0.41, 0.62, 0.82, 0.83, and 0.66 for , respectively, exhibiting a maximum stability from to [a similar conclusion can be obtained from its amplitude ratio ]. This optimal estimator is partly selected based on the physical interpretation of the matrix itself. Indeed, the in Table 7 is featured by a dominant local response along the diagonal and the negative feedback between each pair of subregions around the diagonal. Additionally, has the most significant nonlocal feedback, with a positive response of the eastern subpolar region heat flux to the eastern GSE SST anomaly , a feature that is stable with the truncation of EOFs (not shown). This result is also consistent with the SST-lead regression of wind, which exhibits a maximum value in the eastern part of the subpolar region (as Fig. 2). Finally, similar to the three-region case, the nonlocal total feedback from the subpolar region to the subtropical region (large , , , and ) disappears in (small and insignificant , , , and ), because of the covariance of the subpolar and subtropical SSTs in the mutual matrix (large , , , and in Table 6).

Similar results were obtained in a complementary six-region analysis, in which each of the three regions in Fig. 1a was divided into a pair of subregions in the north and south. The optimal truncation seems to occur for EOF truncations of 4 (EV = 93%) and 5 (EV = 97%). The nonlocal feedback impact from the GSE region on the subpolar region occurs from both GSE subregions to the southern subpolar subregion. In addition, as the resolution is further increased, the GEFA results become even more noisy (not shown), as expected. Nevertheless, further analysis suggests some clues that are consistent with the major features found in the three- and six-region cases (not shown).

The above discussion further confirms that, besides the dominant local negative feedback in the North Atlantic, there is a robust nonlocal positive feedback of the GSE SST to the downstream heat flux in the subpolar region, specifically from the GSE region to the eastern subpolar region.

5. Summary

In a case study of air–sea interaction, a statistical method—the generalized equilibrium feedback assessment—was applied to the North Atlantic to assess the observed turbulent heat flux feedback. This study not only confirms the dominant local negative feedback found in the previous work, but also reports a robust nonlocal positive feedback response of the downstream heat flux in the subpolar region to the Gulf Stream Extension SST. This nonlocal feedback is strong in winter with the response amplitude up to and weak in summer with a response amplitude of . The nonlocal effect is likely initiated by the atmospheric dynamic adjustments. An anomalous downstream easterly surface wind in the subpolar region is induced by warm SST anomalies in the GSE region, which increases the heat flux toward the ocean, thus increasing the SST in that region.

The confirmation of the dominant local feedback indicates the ability of GEFA to assess the full heat flux response to the SST. This result strengthens our confidence in the nonlocal feedback response of the downstream heat flux to the GSE SST, which was first detected by GEFA in a three-region case and further confirmed in a six-region case. As a comparison, the univariate EFA also detected the robust nonlocal feedback during winter. This result arises because of the independence of the GSE SST from the other two regions—subpolar and subtropical. Furthermore, the EFA result showed an additional significant nonlocal negative feedback from the subpolar SST to the subtropic heat flux, which actually employs the negative local feedback of its covarying subtropic SST and therefore distorts the true feedback.

Our study was limited to the North Atlantic, which is less influenced by the El Niño–Southern Oscillation (ENSO). To highlight the findings in this paper, we also examined the GEFA estimation using data from which the ENSO signal was removed and obtained similar results. In addition, we did a sensitivity test on the location of the GSE region, moving the box in Fig. 1a to the east, west, north, and south by 5°. The main conclusion of the paper is unchanged. In this study, a truncated EOF method was used to optimize the GEFA estimation in the six-region case. However, when the spatial resolution is further increased, the empirical method does little to improve the accuracy. A more comprehensive GEFA analysis, including aspects such as EOF mode feedback (Wen et al. 2010) or singular value decomposition–optimal feedback (Liu and Wen 2008), making full use of the available data, is recommended for future work.