In this study, scale interaction between synoptic eddies and low-frequency flow is investigated. The synoptic eddy feedback is a key process in sustaining the low-frequency flow. We show clear evidence using NCEP reanalysis data that there is a general underlying rule—the “left-hand rule”, governing the synoptic eddy feedback onto low-frequency flow. This rule states that eddy-vorticity fluxes are directed preferentially about 90 degrees toward their left-hand side, so that they converge into cyclonic flow and diverge from anticyclonic flow. Therefore, the eddy vorticity flux plays a positive role in reinforcing low-frequency flow.
 Recently, Jin et al. [2006a, 2006b] developed a dynamical closure for the synoptic eddy feedback by separating the quadratic eddy-eddy interaction term into a climatological component and an anomalous component. Using a linear dynamical closure, they showed that the slowly varying eddy feedback could be parameterized by low-frequency flow. This dynamical closure serves as an insightful framework for understanding the eddy feedback process. On the basis of this theoretical framework, we suggest here a simple relation that the eddy vorticity flux always tends to direct to the left of the low-frequency flow so that the eddy forcing amplifies the low-frequency flow and referred as the “left-hand rule”. We use the reanalysis data to examine this left-hand rule.
 In this study, 27 years (Jun 1981 to Feb 2008) of reanalysis data from National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) are used [Kalnay et al., 1996]. The stream function and vorticity fields are calculated from zonal and meridional winds at 300 hPa. The horizontal resolution is 2.5° longitude by 2.5° latitude. To separate the synoptic-eddy component, the daily mean zonal and meridional winds are band-pass filtered for a period of 2–8 days using a LANCZOS filter (using 41 weights [Duchon, 1979]). The low frequency is defined as a monthly (or seasonal) mean value. To measure the forcing of the synoptic eddies on low-frequency flow, eddy vorticity fluxes are defined as follows:
where u′, v′ and ς′ denote the band-pass filtered zonal, meridional winds and vorticity. The overbar indicates monthly (or seasonal) mean. The superscript, a, indicates an interannual anomaly from monthly mean climatology. The convergence (divergence) of the eddy vorticity flux indicates the cyclonic (anticyclonic) vorticity tendency of the low-frequency flow. Because the rotational component of the eddy vorticity flux does not influence the low-frequency flow, only the divergent component is examined here. All analyses on eddy-vorticity flux in the northern hemisphere (NH) and southern hemisphere (SH) are done for the boreal (DJF) and austral (JJA) winter periods, respectively.
 To validate the left-hand rule from the observed data, angles between the eddy vorticity fluxes and anomalous low-frequency flows are calculated at each grid point for every month. The angle is between −180° to 180°, and the angle is positive (negative) if the vorticity flux is directed to the left (right)-hand side of the low-frequency flow.
3. Theoretical Hypothesis
 As mentioned in the introduction, Jin et al. [2006a, 2006b] developed a dynamical closure for the synoptic eddy feedback by separating the quadratic eddy-eddy interaction term into a climatological eddy and an anomalous eddy. This framework implies that the low-frequency flows tend to systematically stir and deform synoptic eddies to yield net eddy-vorticity fluxes directed to the left from low-frequency flow, as illustrated in Figure 1. The synoptic cyclones and anticyclones (ζc′) in the storm-track regions are represented schematically as the transient part of background of the stormy atmospheric circulation (solid circle in Figure 1). Note that the definitions of variables and equations are corresponding to those of Jin et al. [2006a].
 When the anomalous low-frequency flow is purely zonal (denoted as a) and passes a storm track region, it modulates synoptic eddies through differential vorticity advection (−a). When the differential advections by the anomalous flows are assumed to be balanced by the decaying tendencies of transient synoptic eddies (−ζa′/τ) over their short lifetime (τ ≈ 3 ∼ 4 days [Jin et al., 2006a]), the synoptic eddies are tilted and deformed so that the anomalous vorticity patterns of synoptic eddies (ζa′) are generated. That is, the elliptical synoptic eddies are deformed to crescent-shaped, indicating that the anomalous cyclonic (anticyclonic) vorticity of synoptic eddy are generated to the east of the normal or undisturbed cyclonic (anticyclonic) eddies under the westerly low-frequency flow (a), indicating that the synoptic eddies are systematically tilted toward the center of background westerlies. The similar argument is also described by previous studies [Lau, 1988; Qin and Robinson, 1992; Kimoto et al., 2001]. These systematic changes in synoptic eddy structure result in a net anomalous eddy vorticity flux. At the east of the cyclonic (anticyclonic) synoptic eddies, the anomalous vorticity (ζa′) is positive (negative) and climatological meridional wind (vc′) is also positive (negative). Therefore, as illustrated in Figure 1, the meridional vorticity flux due to climatological eddy flow and anomalous eddy vorticity (vc′ζa′) is always positive, indicating northward. Noted that the other vorticity flux term between anomalous eddy flow and climatological eddy vorticity (va′ζc′) is mostly opposite direction, but the magnitude is small under large-scale flow [Jin et al., 2006b]. Therefore, the net anomalous meridional vorticity flux by synoptic eddies is mostly northward under westerly low-frequency flow. If the low-frequency flow is easterly, the eddy vorticity flux is southward, following a similar argument. This clearly indicates the eddy vorticity flux is directed to the left of the low-frequency flow.
 When the low-frequency flow is purely meridional, the synoptic eddies are also deformed by differential advection of the low-frequency flow (−a). Anomalous eddy vorticity patterns are induced relative to the south and north of the normal eddy patterns, as the differential advection is balanced by the decaying term of the synoptic eddies. The anomalous synoptic eddy patterns lead a net eastward (westward) eddy vorticity flux when the low-frequency flow is southward (northward). This also indicates that the eddy vorticity flux is directed to the left of the low-frequency flow. Thus, it was shown in Figures 1a and 1b that the eddy vorticity flux always tends to direct to the left of anomalous low-frequency flows regardless of the direction of low-frequency flow, indicating the left-hand rule for synoptic eddy-feedback.
 This left-hand rule of eddy-vorticity fluxes relative to low-frequency flow indicates positive feedback of synoptic eddy forcing on the low-frequency flow. If the low-frequency flow is cyclonic (anticyclonic), the eddy vorticity fluxes will converge (diverge) to the center of the cyclonic (anticyclonic) flow based on the left-hand rule, indicating that the vorticity fields of low-frequency flow are reinforced. This relation confirms the results of the pioneer work by Lau . Note that the eddy feedback will be maximized when the eddy vorticity flux is perpendicular to the left-hand side of low-frequency flow. Therefore, this positive feedback plays an important role in the dominance of climatic modes, such NAO (North Atlantic Oscillation) and PNA (Pacific-North America) patterns.
4. Observed Evidence
 To prove that the left-hand rule is at work for low-frequency anomalies in general, the NCEP reanalysis data are used. In the previous section, it is suggested that zonal and meridional low-frequency winds can induce meridional and zonal components of the eddy vorticity flux, respectively. In order to confirm this relation, the correlation is calculated between anomalies in monthly-mean wind and vorticity flux. Figures 2a and 2b show this correlation between anomalous seasonal mean zonal wind and anomalous seasonal mean meridional vorticity flux in the NH and SH. In both hemispheres, the correlation is mostly positive in spite of the existence of a small negative region. Positive correlation indicates that the westerly (easterly) wind accompanies northward (southward) vorticity flux, indicating that the eddy vorticity flux is mostly to the left-hand side of low-frequency flow. In the NH, the distinctive high correlation (more than 0.6) appears over two storm tracks and in its downstream regions over the Pacific and Atlantic Oceans. In particular, the correlation is somewhere more than 0.8 over the Atlantic Ocean. On the other hand, the correlation is quite weak over the Eurasian continent. Presumably, this weak correlation is related to weak activity of the synoptic eddy. In the SH, the correlation is zonally uniform to a large extent, which is related to relatively uniform storm activity. Except for the South American continent, the correlation coefficients are significant as more than 0.4.
 The meridional wind is also correlated with the eddy zonal vorticity flux as shown in Figures 2b and 2d. The correlation is mostly negative, indicating that northerly (southerly) wind leads eastward (westward) vorticity flux. The correlation is also higher over the storm track region. Note that the correlation pattern for the meridional wind seems more localized than that for the zonal wind. We found that the higher correlation region is also consistent with the region where the variance of the low-frequency meridional wind is stronger due to existence of the stationary wave, which shows a somewhat localized pattern. A strong signal of low-frequency flow may lead to a clear relationship with the eddy vorticity flux.
 As shown in Figure 2, the anomalous monthly-mean eddy vorticity fluxes are significantly related to anomalous low-frequency flows and are directed to the left-hand side of the low-frequency flows, supporting the left-hand rule. In addition, we also examined the correlation between zonal (meridional) wind and zonal (meridional) eddy vorticity flux. However, their correlation is quite weak and mostly not significant. This implies that the eddy vorticity flux favors a perpendicular direction to the low-frequency flow as well as to the left-hand side, which maximizes the eddy feedback on the low-frequency flow.
 To show further evidence for the left-hand rule, the angles between vorticity flux and low-frequency are calculated using monthly mean data. From 81 sample sizes (3 months in each of 27 years), the probability of positive angles indicating left-hand rule (0–180°) is counted at each grid point. If the angle is randomly distributed, the probability for positive angles will be 50%. Figure 3 shows a probability for the left-hand side. The probabilities, more (less) than 50%, indicate the anomalous eddy vorticity fluxes tend to be directed preferentially to the left (right)-hand side of the anomalous monthly mean flows. The probability is larger than 50% over most extra-tropical and high-latitude regions in the NH, indicating that the left-hand rule is at work. In particular, the left-hand rule operates strongly over the storm track regions and their downstream regions in the north Pacific and Atlantic sectors, which is consistent with high correlation regions in Figures 2a and 2b. In those regions, the probability is more than 80%, indicating that the eddy vorticity flux was mostly to the left-hand side of the low-frequency. In the SH, the left-hand rule also operates well. There are higher probabilities (more than 50%) over the most of the whole SH. Also, the higher probability is related to the higher correlation in Figures 2c and 2d. These results all support the left-hand rule for the eddy feedback on the low-frequency flow.
 As discussed before, the angles between an anomalous low-frequency flows and eddy vorticity fluxes are related to the effectiveness of the eddy feedback. If the angle is 90 degree, the eddy feedback will be maximized to amplify low-frequency, while the damping effect of the eddy feedback will be maximized when the angle is −90°. To examine in more detail the distribution of the angles, the angles are classified into 12 groups with 30° intervals. Figure 4 shows a probability distribution for the angle over the PNA region (180–130W, 30–60N) and NAO region (60W–0, 40–70N), where the eddy feedback is strong. Note that the distance from the origin indicates probability for each angle. If the relation between the low-frequency wind and eddy vorticity flux is random, the distribution will follow the thick circle, indicating a constant one-twelfth probability. However, the distribution clearly shows maximum probability at 90° and minimum probability at −90° in both PNA and NAO regions. The probability at 90° is almost twice than of the random probability (one-twelfth). We also checked that the distributions for the whole NH and SH are also similar to those in Figure 4 though the maximum probability is reduced. These results clearly support that the low-frequency flow organizes synoptic eddies to maximize their feedback to sustain the low-frequency flow itself.
5. Summary and Discussion
 In this study, we examined the left-hand rule for synoptic eddy feedback on low-frequency flow, implied in the theoretical framework by Jin et al. [2006a, 2006b]. The left-hand rule indicates that the anomalous eddy vorticity fluxes always direct preferentially toward the left-hand side of anomalous low-frequency flows in stormy regions. The observed analysis clearly supports this left-hand rule: the eddy vorticity fluxes tend to be perpendicular to and directed to the left of the low-frequency flows, so that the eddy feedback is most effective in maintaining low-frequency flows.
 As shown in Figures 2 and 3, the left-hand rule between low-frequency flow and synoptic eddy forcing, is much stronger over the storm track regions and their downstream regions. Based on theoretical argument addressed in Section 3, the regional-dependency can be controlled by three factors: i) Activity of synoptic eddies, ii) Strength of low-frequency flow, and iii) Decaying of synoptic eddies. It is expected that the relation will be stronger where the signal of synoptic eddies and low-frequency flow is stronger. Therefore, the left-hand rule works well over the storm track region. In addition, the left-hand rule will work well where synoptic eddies are decaying. As discussed in Figure 1, the anomalous pattern of synoptic eddies is a result of balance between the differential vorticity advection term and the eddy-decaying terms. In this case, the anomalous synoptic eddy vorticity is in phase with the meridional wind of climatological synoptic eddy. However, if eddy decaying term is weak and the differential vorticity advection is balanced with the other terms, they can be out of phase so that the left-hand rule can be contaminated with other factors. That is why the left-hand rule is stronger over the downstream region of the storm track rather than over upstream region of the storm track where synoptic eddies are developing. However, further investigation will be required to prove this process.
 This left-hand rule has significant implications for climate studies. Climatic flow anomalies shall be strengthened by this positive eddy feedback under the left-hand rule, which explains why low-frequency flow can survive in a short-lived stormy environment even without external forcing. In a separate paper, we demonstrate that the left-hand rule always plays a role in maintaining the prominent climate modes such as NAO, PNA and AO in the NH and AAO in the SH. Further investigation shall follow in order to understand the role of this positive eddy feedback in the dynamics and predictability of the low-frequency variability.
 This work is supported by National Science foundation (NSF) grants ATM 0652145 and ATM 0650552, NSF of China (NSFC) grant 40528006, and the Meteorological Special Project (GYHY200806005) of China. J.-S. Kug is supported by KORDI (PE98401, PP00720). The manuscript is edited by Diane Henderson.