A kinematic mechanism for the positive feedback between the North Atlantic Oscillation (NAO) and synoptic eddies are depicted based on observational data analyses. Using three-point rescaled covariance statistics of band-pass-filtered (2–8 days) synoptic eddy fields, we examined observed eddy structure changes associated with winter-mean NAO anomalous flow. It is demonstrated that the NAO flow anomalies systematically deform the structures of recurring synoptic eddies to generate seasonal-mean eddy-vorticity flux anomalies predominately directed to the left-hand side of the NAO flow anomalies. These anomalous eddy-vorticity fluxes converge into the cyclonic center and diverge from the anticyclonic center of the anomalous NAO flow, and thus enhance the NAO flow.
 Beyond monthly-mean timescale, the North Atlantic Oscillation (NAO) is known to be the dominant mode of atmospheric circulation over the northern hemisphere. Many research works have shown that the interaction between low-frequency flow and synoptic eddy is indispensable for dominant climate modes [e.g., Cai and Mak, 1990; Robinson, 1991; Lau and Nath, 1991; Branstator, 1995; Lorenz and Hartmann, 2003; Jin et al., 2006a, 2006b]. The NAO variability has been shown to be maintained or enhanced by transient eddy forcing [e.g., Lau, 1988; Branstator, 1992; Nakamura et al., 1997]. Increasing observed and numerical studies provide evidence that synoptic eddy positively feeds back onto the NAO and other low-frequency climatic modes through the two-way interaction between the synoptic eddy and low-frequency flow (SELF).
 Recently, Jin et al. [2006a, 2006b] proposed a theoretical framework with a dynamical closure for the SELF feedback. Based on this study, eddy-vorticity flux always tends to point to the left side of low-frequency flow and this simple relation is referred to as “left-hand rule” [Kug and Jin, 2009]. We here focus on the dynamical processes giving rise to the positive SELF feedback for the NAO reflected by the “left-hand rule”. We then shall show observational evidence for a simple physical picture of a kinematic mechanism on how the NAO organizes recurring synoptic eddies and harvests their eddy vorticity, which in return enhances the NAO.
2. Data and Methodology
 National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data are used for forming the monthly and daily mean 30-year (Jan 1978 to Feb 2008) datasets with 2.5° × 2.5° horizontal resolution [Kalnay et al., 1996]. The stream function and vorticity fields are calculated from zonal and meridional winds at 300 hPa pressure level. To separate synoptic-eddy component, the daily mean zonal and meridional winds are band-pass filtered in the period of 2–8 days using Lanczos filter (using 41 weights [Duchon, 1979]). The low-frequency variability is defined as seasonal mean value. We calculate anomalous seasonal-mean horizontal eddy-vorticity fluxes and the associated convergence as the eddy-vorticity forcing. We then compute eddy-induced stream function tendency by applying 2D Laplacian inversion to the eddy-vorticity forcing. The divergent component of the eddy-vorticity flux is examined here because only it affects the low-frequency flow. All analyses are done for the boreal winter time (December to following February, DJF).
 We here propose a weighted three-point covariance method to examine the synoptic eddy structure in the NAO region. This method is based on one-point correlation method introduced by Wallace and Gutzler  and later by Blackmon et al. , and is developed from the calculation of one-point covariance used by Jin et al. [2006a] to depict the structural changes of synoptic eddy flow. Thus, we first calculate one-point covariance field of eddy stream function (ψ′) at 300 hPa, where the primary base point is chosen nearby the NAO southern action center and also within the Atlantic storm track action center. From this field, we then find two points of the nearest negative centers, one upstream and another downstream.
 The rescaled one-point covariance field for each of these three points is defined as
where subscript j = 0, −1, 1, denotes the primary base point and its upstream and downstream points, respectively, τ stands for time lag, ts is year. The bar is for seasonal mean, σc expresses climatological standard deviation. The field Cj has the unit of stream function. We then define the three-point weighted average of Cj as
This definition is from the typical pattern of one-point correlation or covariance field. The field C represents synoptic eddy structure with the unit of stream function. By examining the typical wave-length and magnitude in C, we find that our method captures the synoptic wave-packet structure better than one-point covariance field. After calculating C in every winter, we can construct climatological mean and define its seasonal anomalies as the departure from the mean.
 It is well-known that the NAO flow is closely related to synoptic eddy activity and there is a positive feedback between them [e.g., Lau, 1988]. As shown in Figure 1b, it is apparent that the anomalous eddy-vorticity fluxes generally follow the left-hand rule suggested by Kug and Jin . Namely, they are systematically directed toward the left of the NAO flow anomalies. Thus, the eddy-vorticity fluxes converge into the cyclonic flow and diverge from the anticyclonic flow. The convergence (negative eddy forcing) and divergence (positive eddy forcing) of the eddy-vorticity fluxes enhance the NAO cyclonic and anticyclonic flows, respectively.
 To depict the detailed dynamical processes for the positive eddy feedback, we examine how the anomalous eddy-vorticity fluxes and related eddy forcing are conduced. We here use observational data to diagnose the eddy-structure statistics by calculating three-point covariance fields under the climatological mean and the strong NAO conditions, respectively (see Figure 2).
 The climatological mean of three-point rescaled covariance field (Figure 2a) shows a wave-packet-like structure of synoptic eddy field. The amplitude of the packet naturally decays as the distance from the primary base point increases. Overall, it exhibits slightly tilted eddy structure as evident from the depicted phase lines defined as zonal extremes of eddy structure, presumably due to the climatological jet stream. This pattern represents typical synoptic eddy structure in this region under normal condition. To obtain the statistical synoptic eddy structure under the strong NAO conditions, we first regress the anomalous three-point covariance field of each season with the seasonal NAO index. We then multiply a factor of two to this regressed anomalous eddy structure pattern and add it onto the climatological mean, where the two-time inflation is only used to make structure change more clear.
 It is evident in Figure 2b that eddy structure systematically changes from normal to NAO conditions. The NAO anomalous flow systematically deforms the synoptic eddy structure. The phase lines are anticyclonic slanted consistent with the kinematic effect of the NAO anomalous flow via its related differential eddy-vorticity advection. Therefore, the eddy-structure patterns are zonally slanted where zonal winds are strong westerly on the north side and easterly on the south side of southern NAO center. In addition, the eddy-structure patterns are meridionally stretched or squeezed where meridional winds are strong northerly on the east side and southerly on the west side of southern NAO center. In general, zonal deformation is prevailing due to much stronger zonal wind anomalies of the NAO flow. This observational evidence clearly documents that the NAO flow alters the synoptic eddy structure in a statistical sense.
 Our next focus will be on the question how the deformation of eddy characteristics leads to a positive feedback onto the NAO flow. We now use the three-point covariance field to separate synoptic eddy statistics into two components: normal and anomalous eddy-structure patterns. Because this covariance field is normalized or more precisely rescaled into the physical dimension of stream function field, we can use the anomalous eddy-structure field to calculate the associated anomalous eddy-vorticity-structure pattern by simply applying to it with a Laplacian operator. As is shown in Figure 3a, relative to the upper left (right) and lower right (left) of the anticyclone center in normal eddy-structure pattern, there are positive (negative) vorticity anomalies with a quadrupole distribution.
 Using the normal eddy-flow-structure and anomalous eddy-vorticity-structure patterns (denoted as ′c and ζ′a, respectively), we can directly assess the quantitative anomalous eddy-vorticity fluxes (viz., ′cζ′a) associated with the NAO (Figure 3b), where rotational component has been removed. These estimated anomalous eddy-vorticity fluxes tend to follow the left-hand rule with poleward fluxes on the northern flank of NAO anticyclonic center and equatorward fluxes on the other flank. Besides the meridional eddy-vorticity fluxes, the zonal component of eddy-vorticity fluxes with relatively smaller magnitude is clear over the areas on the eastern and western sides of southern centers of the NAO. In these areas, meridional winds are relatively strong and so induce zonal eddy-vorticity fluxes. Thus, all of anomalous eddy-vorticity fluxes diverge from anomalous anticyclonic circulation in the NAO pattern. As a result, this leads to an eddy-induced negative vorticity tendency over anticyclonic center of the NAO and thus reinforces the NAO.
 Although only based on the three-point covariance field, the eddy-vorticity-flux pattern in Figure 3b is clearly consistent with that shown in Figure 1b, indicating that there is an eddy-induced positive feedback onto the NAO mode under the kinematic control of the ambient NAO flow anomalies. The advantage of using three-point covariance statistics here is that the reconstructed eddy structures and corresponding calculated eddy-vorticity fluxes delineate a simple kinematic mechanism for the positive eddy feedback between synoptic eddies and the NAO, which is further illustrated schematically in Figure 4.
 Without losing the generality, we consider an anticyclonic anomalous circulation similar to southern center of positive-phase NAO. As the ubiquitous synoptic cyclones and anticyclones pass through the anticyclonic NAO flow anomalies, they are anomalously deformed during their relatively short lifetime by differential NAO-related advections. The resultant anomalous eddy structures give rise to anomalous zonal and meridional eddy-vorticity fluxes that are always preferably directed to the left of the anomalous flow. The zonal component of the anomalous flow associated with the NAO tends to slant synoptic eddies to yield meridional eddy-vorticity fluxes directed toward the left, whereas the meridional component of the anomalous flow tends to stretch or squeeze eddies to yield zonal eddy-vorticity fluxes directed also to the left. Thus the NAO flow anomalies organize synoptic eddies and get amplification by harvesting their vorticity.
 Recent studies suggest that the NAO onset accompanies strong nonlinear changes in the wave-breaking process of synoptic eddies [Benedict et al., 2004; Franzke et al., 2004; Rivière and Orlanski, 2007; Woolings et al., 2008], essentially based on the weather view for the NAO [Feldstein, 2003]. The dynamical mechanism described here should be not inconsistent with the wave-breaking mechanism proposed for the NAO onset. In particular, the anticyclonic eddy-structure change in Figure 2 is quite similar to the anticyclonic-type wave-breaking, indicating that some linkages may exist between them. Actually, what we are focusing on is not onset but the self-maintenance of the NAO by considering the statistical effect of transient synoptic eddies which leave their impacts on the time-mean flow during the limited lifetime.
4. Summary and Discussion
 This study focuses on the depiction of the dynamical processes for the NAO interacting with synoptic eddies. We used the observational data to construct the statistical structure of synoptic eddies and the changes of that structure in association with the NAO. It is demonstrated that as the leading climate mode in the northern hemisphere the anomalous flow of the NAO gains reinforcement by kinematically altering the statistical structure of transient synoptic eddies in the northern Atlantic storm track region. Eddy-vorticity fluxes tend to be systematically modulated by and feed back onto the seasonal-mean anomalies associated with the NAO.
 It was demonstrated by Lau  that the changes in storm track such as anomalies in synoptic eddy variance and eddy-vorticity fluxes are well correlated with the other leading low-frequency modes such as the NAO. The evidence documented in this work provides additional observational facts that underline this relationship between the low-frequency flow anomalies and changes in synoptic eddy statistics. Based on regression analysis, we showed the clear evidence for the kinematic mechanism of synoptic eddy-structure changes under the influence of the NAO-related anomalous flow. In the presence of the NAO-related flow anomalies, synoptic eddies in storm track region are systematically slanted and stretched to generate anomalies in time-mean eddy-vorticity fluxes that preferentially points to the left-hand side of anomalous flow associated with the NAO. Thus, the eddy-vorticity fluxes tend to converge into the center of the cyclonic flow and diverge from the center of the anticyclonic flow of the NAO. As a result, there is a positive eddy feedback or eddy-induced instability for climate modes such as the NAO, and synoptic eddy serves as a fundamental energy source for the NAO and other climate modes [e.g., Lau, 1988; Branstator, 1992; Nakamura et al., 1997].
 It can be further inferred that such the kinematic mechanism suggested here should be at work for the eddy-mean flow interactions in association with other climate modes (e.g., the PNA) as well as more general low-frequency flow variability. Also, the relationship between the kinematic mechanism and the wave-breaking mechanism will be discussed in further work.
 This work is jointly supported by National Science foundation (NSF) grants ATM 0652145 and ATM 0650552 and NSF of China (NSFC) grants 40705021 and 40805028, the Meteorological Special Project (GYHY200806005), and the National Science and Technology Support Program of China (2007BAC29B03). J.-S. Kug is supported by KORDI (PE98425, PE98401, and PP00720).