• Barnes, E. A., , and D. L. Hartmann, 2010: Dynamical feedbacks and the persistence of the NAO. J. Atmos. Sci., 67, 851865.

  • Barnston, A. G., , and R. E. Livezey, 1987: Classification, seasonality, and persistence of low-frequency atmospheric circulation patterns. Mon. Wea. Rev., 115, 10831126.

    • Search Google Scholar
    • Export Citation
  • Benedict, J. J., , S. Lee, , and S. B. Feldstein, 2004: Synoptic view of the North Atlantic Oscillation. J. Atmos. Sci., 61, 121144.

  • Blackmon, M. L., , Y.-H. Lee, , and J. M. Wallace, 1984a: Horizontal structure of 500-mb height fluctuations with long, intermediate, and short time scales. J. Atmos. Sci., 41, 961979.

    • Search Google Scholar
    • Export Citation
  • Blackmon, M. L., , Y.-H. Lee, , J. M. Wallace, , and H.-H. Hsu, 1984b: Time variation of 500-mb height fluctuations with long, intermediate, and short time scales as deduced from lag-correlation statistics. J. Atmos. Sci., 41, 981991.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1992: The maintenance of low-frequency atmospheric anomalies. J. Atmos. Sci., 49, 19241946.

  • Branstator, G., 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, 207226.

    • Search Google Scholar
    • Export Citation
  • Cai, M., , and M. Mak, 1990: Symbiotic relation between planetary and synoptic-scale waves. J. Atmos. Sci., 47, 29532968.

  • Cai, M., , and H. M. van den Dool, 1991: Low-frequency waves and traveling storm tracks. Part I: Barotropic component. J. Atmos. Sci., 48, 14201436.

    • Search Google Scholar
    • Export Citation
  • Chang, E. K. M., 2009: Are band-pass variance statistics useful measures of storm track activity? Re-examining storm track variability associated with the NAO using multiple storm track measures. Climate Dyn., 33, 277296, doi:10.1007/s00382-009-0532-9.

    • Search Google Scholar
    • Export Citation
  • Chang, E. K. M., , S. Lee, , and K. L. Swanson, 2002: Storm track dynamics. J. Climate, 15, 21632183.

  • DeWeaver, E., , and S. Nigam, 2000: Zonal-eddy dynamics of the North Atlantic Oscillation. J. Climate, 13, 38933914.

  • Duchon, C., 1979: Lanczos filtering in one and two dimensions. J. Appl. Meteor., 18, 10161022.

  • Feldstein, S. B., 2003: The dynamics of NAO teleconnection pattern growth and decay. Quart. J. Roy. Meteor. Soc., 129, 901924.

  • Franzke, C., , S. Lee, , and S. B. Feldstein, 2004: Is the North Atlantic Oscillation a breaking wave? J. Atmos. Sci., 61, 145160.

  • Gerber, E. P., , and G. K. Vallis, 2009: On the zonal structure of the North Atlantic Oscillation and annular modes. J. Atmos. Sci., 66, 332352.

    • Search Google Scholar
    • Export Citation
  • Hartmann, D. L., 1995: A PV view of zonal flow vacillation. J. Atmos. Sci., 52, 25612576.

  • Hartmann, D. L., , and F. Lo, 1998: Wave-driven zonal flow vacillation in the Southern Hemisphere. J. Atmos. Sci., 55, 13031315.

  • Hartmann, D. L., , and P. Zuercher, 1998: Response of baroclinic life cycles to barotropic shear. J. Atmos. Sci., 55, 297313.

  • Holopainen, E., , and C. Fortelius, 1987: High-frequency transient eddies and blocking. J. Atmos. Sci., 44, 16321645.

  • Hoskins, B. J., , I. N. James, , and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci., 40, 15951612.

    • Search Google Scholar
    • Export Citation
  • Hurrell, J. W., , and C. Deser, 2009: North Atlantic climate variability: The role of the North Atlantic Oscillation. J. Mar. Syst., 78, 2841, doi:10.1016/j.jmarsys.2008.11.026.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 2010: Eddy induced instability for low-frequency variability. J. Atmos. Sci., 67, 19471964.

  • Jin, F.-F., , L.-L. Pan, , and M. Watanabe, 2006a: Dynamics of synoptic eddy and low-frequency flow interaction. Part I: A linear closure. J. Atmos. Sci., 63, 16771694.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., , L.-L. Pan, , and M. Watanabe, 2006b: Dynamics of synoptic eddy and low-frequency flow interaction. Part II: A theory for low-frequency modes. J. Atmos. Sci., 63, 16951708.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437471.

  • Kok, C. J., , J. D. Opsteegh, , and H. M. van den Dool, 1987: Linear models: Useful tools to analyze GCM results. Mon. Wea. Rev., 115, 19962008.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , and F.-F. Jin, 2009: Left-hand rule for synoptic eddy feedback on low-frequency flow. Geophys. Res. Lett., 36, L05709, doi:10.1029/2008GL036435.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , F.-F. Jin, , J.-H. Park, , H.-L. Ren, , and I.-S. Kang, 2010a: A general rule for synoptic-eddy feedback onto low-frequency flow. Climate Dyn., 35, 10111026, doi:10.1007/s00382-009-0606-8.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , F.-F. Jin, , and H.-L. Ren, 2010b: Role of synoptic eddies on low-frequency precipitation variation. J. Geophys. Res., 115, D19115, doi:10.1029/2009JD013675.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., 1988: Variability of the observed midlatitude storm tracks in relation to low-frequency changes in the circulation pattern. J. Atmos. Sci., 45, 27182743.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., , and E. O. Holopainen, 1984: Transient eddy forcing of the time-mean flow as identified by geopotential tendencies. J. Atmos. Sci., 41, 313328.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., , and M. J. Nath, 1991: Variability of the baroclinic and barotropic transient eddy forcing associated with monthly changes in the midlatitude storm tracks. J. Atmos. Sci., 48, 25892613.

    • Search Google Scholar
    • Export Citation
  • Lim, G.-H., , and J. M. Wallace, 1991: Structure and evolution of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 48, 17181732.

    • Search Google Scholar
    • Export Citation
  • Limpasuvan, V., , and D. L. Hartmann, 1999: Eddies and the annular modes of climate variability. Geophys. Res. Lett., 26, 31333136.

  • Limpasuvan, V., , and D. L. Hartmann, 2000: Wave-maintained annular modes of climate variability. J. Climate, 13, 44144429.

  • Löptien, U., , and E. Ruprecht, 2005: Effect of synoptic systems on the variability of the North Atlantic Oscillation. Mon. Wea. Rev., 133, 28942904.

    • Search Google Scholar
    • Export Citation
  • Lorenz, D. J., , and D. L. Hartmann, 2001: Eddy–zonal flow feedback in the Southern Hemisphere. J. Atmos. Sci., 58, 33123327.

  • Lorenz, D. J., , and D. L. Hartmann, 2003: Eddy–zonal flow feedback in the Northern Hemisphere winter. J. Climate, 16, 12121227.

  • Luo, D., 2005: A barotropic envelope Rossby soliton model for block–eddy interaction. Part I: Effect of topography. J. Atmos. Sci., 62, 521.

    • Search Google Scholar
    • Export Citation
  • Luo, D., , A. R. Lupo, , and H. Wan, 2007: Dynamics of eddy driven low-frequency dipole modes. Part I: A simple model of North Atlantic Oscillations. J. Atmos. Sci., 64, 328.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., and Coauthors, 2001: North Atlantic climate variability: Phenomena, impacts and mechanisms. Int. J. Climatol., 21, 18631898.

    • Search Google Scholar
    • Export Citation
  • Mullen, S. L., 1987: Transient eddy forcing of blocking flows. J. Atmos. Sci., 44, 322.

  • Nakamura, H., , and J. M. Wallace, 1990: Observed changes in baroclinic wave activity during the life cycles of low-frequency circulation anomalies. J. Atmos. Sci., 47, 11001116.

    • Search Google Scholar
    • Export Citation
  • Nakamura, H., , M. Nakamura, , and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in blocking formation. Mon. Wea. Rev., 125, 20742093.

    • Search Google Scholar
    • Export Citation
  • Pan, L.-L., , F.-F. Jin, , and M. Watanabe, 2006: Dynamics of synoptic eddy and low-frequency flow interaction. Part III: Baroclinic model results. J. Atmos. Sci., 63, 17091725.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1990: A nonacceleration theorem for transient quasi-geostrophic eddies on a three-dimensional time-mean flow. J. Atmos. Sci., 47, 18251836.

    • Search Google Scholar
    • Export Citation
  • Qin, J. C., , and W. A. Robinson, 1992: Barotropic dynamics of interactions between synoptic and low-frequency eddies. J. Atmos. Sci., 49, 7179.

    • Search Google Scholar
    • Export Citation
  • Ren, H.-L., , F.-F. Jin, , J.-S. Kug, , J.-X. Zhao, , and J. Park, 2009: A kinematic mechanism for positive feedback between synoptic eddies and NAO. Geophys. Res. Lett., 36, L11709, doi:10.1029/2009GL037294.

    • Search Google Scholar
    • Export Citation
  • Ren, H.-L., , F.-F. Jin, , J.-S. Kug, , and L. Gao, 2011: Transformed eddy PV flux and positive synoptic eddy feedback onto low-frequency flow. Climate Dyn., 36, 23572370, doi:10.1007/s00382-010-0913-0.

    • Search Google Scholar
    • Export Citation
  • Rivière, G., , and I. Orlanski, 2007: Characteristics of the Atlantic storm-track eddy activity and its relation with the North Atlantic Oscillation. J. Atmos. Sci., 64, 241266.

    • Search Google Scholar
    • Export Citation
  • Robinson, W. A., 1991: The dynamics of low-frequency variability in a simple model of the global atmosphere. J. Atmos. Sci., 48, 429441.

    • Search Google Scholar
    • Export Citation
  • Robinson, W. A., 2000: A baroclinic mechanism for the eddy feedback on the zonal index. J. Atmos. Sci., 57, 415422.

  • Rogers, J. C., 1997: North Atlantic storm track variability and its association to the North Atlantic Oscillation and climate variability of northern Europe. J. Climate, 10, 16351647.

    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1983: The propagation of eddies in diffluent jetstreams: Eddy vorticity forcing of ‘blocking’ flow fields. Quart. J. Roy. Meteor. Soc., 109, 737761.

    • Search Google Scholar
    • Export Citation
  • Strong, C., , and G. Magnusdottir, 2008a: How Rossby wave breaking over the Pacific forces the North Atlantic Oscillation. Geophys. Res. Lett., 35, L10716, doi:10.1029/2008GL033578.

    • Search Google Scholar
    • Export Citation
  • Strong, C., , and G. Magnusdottir, 2008b: Tropospheric Rossby wave breaking and the NAO/NAM. J. Atmos. Sci., 65, 28612876.

  • Vautard, R., , B. Legras, , and M. Déqué, 1988: On the source of midlatitude low-frequency variability. Part I: A statistical approach to persistence. J. Atmos. Sci., 45, 28112843.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., , and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon. Wea. Rev., 109, 784812.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., , G.-H. Lim, , and M. L. Blackmon, 1988: Relationship between cyclone tracks, anticyclone tracks and baroclinic waveguides. J. Atmos. Sci., 45, 439462.

    • Search Google Scholar
    • Export Citation
  • Watanabe, M., 2009: Self-limiting feedback between baroclinic waves and a NAO-like sheared zonal flow. Geophys. Res. Lett., 36, L08803, doi:10.1029/2009GL037176.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and A. Barcilon, 1992a: Type B cyclogenesis in a zonally varying flow. J. Atmos. Sci., 49, 18771892.

  • Whitaker, J. S., , and A. Barcilon, 1992b: Genesis of mobile troughs in the upper westerlies. J. Atmos. Sci., 49, 20972107.

  • Woollings, T., , B. J. Hoskins, , M. Blackburn, , and P. Berrisford, 2008: A new Rossby wave-breaking interpretation of the North Atlantic Oscillation. J. Atmos. Sci., 65, 609626.

    • Search Google Scholar
    • Export Citation
  • Woollings, T., , A. Hannachi, , B. Hoskins, , and A. Turner, 2010: A regime view of the North Atlantic Oscillation and its response to anthropogenic forcing. J. Climate, 23, 12911307.

    • Search Google Scholar
    • Export Citation
  • Yu, J.-Y., , and D. L. Hartmann, 1993: Zonal flow vacillation and eddy forcing in a simple GCM of the atmosphere. J. Atmos. Sci., 50, 32443259.

    • Search Google Scholar
    • Export Citation
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    (a) Schematic diagram for the new theoretical framework of interactions among climatological mean flow, low frequency flow, and basic and anomalous synoptic eddy flow. (b) Detrended winter NAO index anomalies from 1978/79 to 2007/08, where the gray dashed lines indicate ±80% standard deviation. (c) NAO-index-regressed patterns of 300-hPa streamfunction (contours; 106 m2 s−1) and eddy forcing in terms of streamfunction tendency (shading; m2 s−2). (d) Climatological synoptic eddy streamfunction structure (contour interval 1 × 106 m2 s−1, zero line omitted) and anomalous eddy vorticity structure (shading; 0.5 × 10−6 s−1) by regressing the three-point zero-lag covariance fields at lag-0 day onto the NAO index. Panels (c) and (d) are redrawn from Figs. 1b and 3a of Ren et al. (2009). (e) Conceptual diagrams for the synoptic eddy–NAO interaction in terms of the (left) F3 and (right) F4.

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    Snapshots of the synoptic eddy structure patterns in terms of the SES fields from lead day 2 to lag day 2 under (left) positive and (right) negative NAO conditions. Gray contours denote the NAO patterns, as in Fig. 1c. Black contours (color shades) represent the climatological (NAO-associated) eddy structure. Thick curves are phase lines of the eddy structures in the climatology (black) and under the NAO conditions (red). Anticyclonic (cyclonic) eddy structure patterns are denoted by red shading and solid contours (blue shading and dashed contours); interval in the black contours is 1 × 106 m2 s−1 , all zero lines omitted; and cross signs denote selected base points at 45°N, 295°E; 45°N, 325°E; and 45°N, 352.5°E for the positive NAO phase (NAO+), and 45°N, 300°E; 45°N, 332.5°E; and 45°N, 355°E for the negative phase (NAO−).

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    Center composites for (a),(b) the eddy structure patterns in Fig. 2 and (c),(d) the anomalous eddy flow structure (blue contours), generated from lead day 2 to lag day 2 by choosing basic reference points with centers of the central anticyclone at lag 0 in Fig. 2 for the climatological and NAO conditions, respectively. Black dots (yellow squares) are the centers of the contoured (shaded) eddies; unit throughout is 1 × 105 m2 s−1. Here the NAO patterns (gray contours) are referred to as a background flow to examine eddy structure change conveniently.

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    Eddy vorticity fluxes (vectors; 10−6 m s−2) and induced eddy forcing (shading; m2 s−2) under the NAO conditions. (a),(c),(e) Summation of two integrated anomalous fluxes, , and (b),(d),(f) observed anomalous fluxes for (top) the composites in NAO+, (middle) the composites in NAO−, and (bottom) the regression by the NAO index. Black contours denote the NAO pattern. PCC[(a),(b)] = 0.671, PCC[(c),(d)] = 0.728 and PCC[(e),(f)] = 0.725.

  • View in gallery

    Maps for (a),(b) the basic eddy flow structure (contours; 106 m2 s−1) and the anomalous eddy vorticity structure patterns (shading; 10−6 s−1), and (c),(d) the anomalous eddy flow structure (contours; 106 m2 s−1) and the basic eddy vorticity structure patterns (shading; 10−6 s−1). All patterns are obtained and derived from Fig. 3. Black arrows illustrate the generated eddy vorticity fluxes.

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    Snapshots of the fluxes of the basic eddy velocity multiplied by anomalous eddy vorticity (vectors; 10−6 m s−2) and the induced eddy forcing (shading; m2 s−2) from lead day 2 to lag day 2 in the (left) positive and (right) negative NAO phases. Gray contours for the NAO flow are as in Fig. 2.

  • View in gallery

    (a),(b) Center composites of the BA fluxes (vectors) and the induced eddy forcing (shading), where the NAO patterns (gray contours) are only referred to as a background flow. The maps are generated using the fields in Fig. 6 with the basic reference points taken at the centers of the central climatological anticyclone on lag-0 day in Fig. 2. (c),(d) Lead–lag composites of the BA fluxes and the induced eddy forcing by using the maps in Fig. 6 from lead day 2 to lag day 2, where the black contours denote the NAO. Units are 10−6 m s−2 for vectors and m2 s−2 for shading.

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    As in Fig. 7, but for the AB fluxes .

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    (a),(c) Meridional and (b),(d) zonal component of the observed total eddy fluxes (vectors; 10−6 m s−2) and the induced eddy forcing patterns (shading; m2 s−2): black contours are for the NAO patterns.

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    As in Fig. 9, but for the integrated eddy fluxes, .

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    Schematic diagrams of the mechanisms for the (a)–(d) zonal eddy-slanting feedback and (e)–(h) meridional eddy-shifting feedback of synoptic eddies onto low-frequency flow. Thick green solid ellipses with counterclockwise (clockwise) arrows represent cyclonic (anticyclonic) low frequency flow like the negative (positive) phase NAO flow. (a),(b),(e),(f) Solid thin circles indicate the idealized synoptic eddy patterns with cyclonic (red) and anticyclonic (blue) eddies. Dashed circles denote the changed eddy structure as a result of the anomalous climatic flow. Black stars are the centers of basic eddy structures. (c),(g) Dashed circles stand for the anomalous eddy flow structures with cyclonic (red) and anticyclonic (blue) circulations. Smaller (bigger) shaded ovals stand for the anomalous (climatological) eddy vorticity, where yellow is positive and blue is negative. (b),(c),(f),(g) Gray dashed and black solid arrows represent the eddy vorticity fluxes before and after the eddy feedback accumulation, respectively. (d),(h) Dark gray solid arrows denote the net eddy vorticity fluxes and their convergence (divergence) is represented by the yellow (blue) grid.

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    (a) Observed 300-hPa climatological eddy vorticity fluxes and (b) integrated BB fluxes . Shading is the 300-hPa climatology of zonal wind (m s−1). (c),(d) Integrated AA fluxes in the (c) positive and (d) negative NAO phases. Vectors need to be multiplied by 10−6 m s−2. The black contours are for the NAO patterns.

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Anatomy of Synoptic Eddy–NAO Interaction through Eddy Structure Decomposition

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  • 1 Department of Meteorology, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, Honolulu, Hawaii, and Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing, China
  • | 2 Department of Meteorology, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, Honolulu, Hawaii
  • | 3 Department of Meteorology, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, Honolulu, Hawaii, and Numerical Prediction Center, National Meteorological Center, China Meteorological Administration, Beijing, China
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Abstract

A method of eddy structure decomposition is proposed to detect how low-frequency flow associated with the North Atlantic Oscillation (NAO) organizes systematically synoptic eddy (SE) activity to generate in-phase and upstream feedbacks. In this method, a statistical eddy streamfunction (SES) field, defined by the three-point covariance of synoptic-scale streamfunction, is introduced to characterize spatiotemporal SE flow structures. The SES field is decomposed into basic and anomalous parts to represent the climatological SE flow structure and its departure. These two parts are used to calculate the basic and anomalous eddy velocity, eddy vorticity, and thus eddy vorticity flux fields, in order to elucidate those two SE feedbacks onto the NAO. This method is validated by the fact that the observed anomalous eddy vorticity flux field can be reproduced well by two linear terms: the basic eddy velocity field multiplied by anomalous eddy vorticity field and the anomalous eddy velocity field multiplied by basic eddy vorticity field. With this method, it is found that, in the positive and negative phases, the NAO flow tends to induce two different types of anomalous SE flow structure, which are largely responsible for generating the net meridional and zonal eddy vorticity fluxes that, in return, feed back onto the NAO. The two processes that are related to these two different types dominate in the in-phase and upstream feedbacks, which are delineated conceptually into two kinematic mechanisms associated with zonal-slanting and meridional-shifting changes in the SE structure. The present observational evidence supports the theory of eddy-induced instability for low-frequency variability and also provides insights into the reason for the asymmetry between the SE feedbacks onto the two NAO phases.

Corresponding author address: Dr. Hong-Li Ren, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, Honolulu, HI 96822. E-mail: honglir@hawaii.edu

Abstract

A method of eddy structure decomposition is proposed to detect how low-frequency flow associated with the North Atlantic Oscillation (NAO) organizes systematically synoptic eddy (SE) activity to generate in-phase and upstream feedbacks. In this method, a statistical eddy streamfunction (SES) field, defined by the three-point covariance of synoptic-scale streamfunction, is introduced to characterize spatiotemporal SE flow structures. The SES field is decomposed into basic and anomalous parts to represent the climatological SE flow structure and its departure. These two parts are used to calculate the basic and anomalous eddy velocity, eddy vorticity, and thus eddy vorticity flux fields, in order to elucidate those two SE feedbacks onto the NAO. This method is validated by the fact that the observed anomalous eddy vorticity flux field can be reproduced well by two linear terms: the basic eddy velocity field multiplied by anomalous eddy vorticity field and the anomalous eddy velocity field multiplied by basic eddy vorticity field. With this method, it is found that, in the positive and negative phases, the NAO flow tends to induce two different types of anomalous SE flow structure, which are largely responsible for generating the net meridional and zonal eddy vorticity fluxes that, in return, feed back onto the NAO. The two processes that are related to these two different types dominate in the in-phase and upstream feedbacks, which are delineated conceptually into two kinematic mechanisms associated with zonal-slanting and meridional-shifting changes in the SE structure. The present observational evidence supports the theory of eddy-induced instability for low-frequency variability and also provides insights into the reason for the asymmetry between the SE feedbacks onto the two NAO phases.

Corresponding author address: Dr. Hong-Li Ren, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, Honolulu, HI 96822. E-mail: honglir@hawaii.edu

1. Introduction

A remarkable phenomenon in the stormy extratropical atmosphere is the coexistence of high-frequency synoptic eddies (SEs) with a lifetime of 2–8 days and low-frequency (LF) climate modes with a lifetime of typically more than 2 weeks (e.g., Wallace and Gutzler 1981; Blackmon et al. 1984a,b). The North Atlantic Oscillation (NAO) is a dominant mode of atmospheric circulation in the North Atlantic (NA) and the entire Northern Hemisphere, residing downstream of the major storm track region in the NA. In recent decades, many studies have examined this predominant climate mode because it accounts for such a large fraction of atmospheric variability in the NA and has a significant impact on regional and hemispheric weather and climate (Marshall et al. 2001; Hurrell and Deser 2009). Thus, the maintenance mechanism of the NAO has been a subject of continued interests in the field of atmospheric large-scale dynamics (e.g., Rogers 1997; Löptien and Ruprecht 2005).

The internal dynamical feedback between atmospheric SE activity and LF flow has long been recognized as playing an essential role in maintaining extratropical atmospheric LF variability (Kok et al. 1987; Lau 1988; Cai and Mak 1990). The interaction between synoptic eddies and low-frequency flow (referred to as SELF) is indispensable for generating and maintaining zonal indices and dominant climatic modes (e.g., Cai and Mak 1990; Cai and van den Dool 1991; Lau and Nath 1991; Robinson 1991, 2000; Branstator 1992, 1995; Limpasuvan and Hartmann 1999, 2000; Lorenz and Hartmann 2001, 2003; Jin et al. 2006a,b; Pan et al. 2006; Jin 2010). The concept that SEs positively feed back onto the NAO and other dominant climatic modes through the two-way interaction of SELF has drawn strong supports from an increasing number of observational and numerical studies (e.g., Lorenz and Hartmann 2001, 2003; Jin et al. 2006a,b; Pan et al. 2006; Ren et al. 2009).

The SELF interaction involves two interdependent aspects. First, anomalous SE activity plays a key role in maintaining month-to-month extratropical climatic variability (Kok et al. 1987; Lau 1988; Vautard et al. 1988). For example, eddy forcing by synoptic-scale transients is well known as a leading mechanism for sustaining blocking flow (Hoskins et al. 1983; Shutts 1983; Mullen 1987; Nakamura et al. 1997; Luo 2005). The monthly or seasonal mean variance fields of SE, referred to as storm track activity, show systematic variability and a linkage with LF variability (e.g., Lau 1988; Chang et al. 2002). Previous studies have also investigated storm track variability in the NA region and its association with the NAO (Rogers 1997; Gerber and Vallis 2009). Many works have shown that SE activity has a significant impact on the NAO variability (DeWeaver and Nigam 2000; Luo et al. 2007; Watanabe 2009). Eddy forcing by synoptic systems make a positive contribution to sustainment of the NAO (Löptien and Ruprecht 2005; Barnes and Hartmann 2010). The NAO is also affected by SE activity in another way, as the onset of the daily NAO (Feldstein 2003) is closely associated with a wave breaking process of SEs (Benedict et al. 2004; Franzke et al. 2004; Rivière and Orlanski 2007; Strong and Magnusdottir 2008a,b; Woollings et al. 2008), which is different from the subject of the present study—SELF feedback.

The second aspect of the two-way SELF interaction is that anomalous SE activity can be generated partly by LF flow, as high-frequency SEs are systematically organized by ambient mean flow anomalies. For example, Branstator (1995) showed that the structure of storm track activity is modified by the dominant large-scale LF circulation, and again suggested that SE activity could be organized by, and feed back onto, the recurrent LF climatic modes. Of note, SEs may be elongated or split by the ambient blocking circulation, thereby generating eddy vorticity feedback onto the blocking flow (Shutts 1983; Nakamura and Wallace 1990). This two-way interaction between high- and low-frequency waves has been demonstrated in many theoretical, observational, and numerical works (e.g., Lau 1988; Cai and Mak 1990; Plumb 1990; Cai and van den Dool 1991). Robinson (1991) proposed a mechanism for such a scale interaction based on an analysis of two kinds of eddy feedback in a simple atmospheric model, namely in-phase and quadrature feedbacks, which contribute to a positive in-phase and upstream development of LF flow. Qin and Robinson (1992) proposed a barotropic mechanism for the interaction between SEs and LF eddies, where SE distortion caused by the shearing/stretching effect of LF flow plays a key role in generating the in-phase and quadrature feedbacks. Whitaker and Barcilon (1992a,b) examined a mechanism involving the structural modification of baroclinic wave packets in the type-B cyclogenesis due to a zonally varying basic flow. Other studies, based on baroclinic life cycle experiments and observational analyses, reported structural changes of baroclinic waves under different barotropic shears or zonal indices (Yu and Hartmann 1993; Hartmann 1995; Hartmann and Lo 1998; Hartmann and Zuercher 1998). These previous works indicated that systematic changes in SE structure are crucial in the SELF interaction, thereby providing the motivation for the present study to further examine the SE structure change associated with the NAO and its relation to NAO maintenance.

Although the SELF interaction was outlined in the aforementioned studies, the physical processes that SEs are organized by and, in turn, generate in-phase/quadrature feedbacks onto LF flow remain unclear. The classical framework of atmospheric climate dynamics (denoted as F3) usually involves interactions among three components of flow: climatological mean flow, LF flow, and SE flow (Fig. 1a), which has been followed by many previous studies on eddy–mean flow feedback. For example, Hoskins et al. (1983) proposed the concept of the E vector, using the variance and covariance of eddy velocity to represent the SE feedback onto the mean flow and to determine the eddy shape and sense of relative group velocity. Further, Lau (1988) suggested a positive SELF feedback by examining the in-phase relationship between the eddy forcing field induced by eddy vorticity fluxes and the anomalous mean circulation. Indeed, one can easily derive a transform from the E vector to eddy vorticity flux. However, based on the conventional F3, neither the E vector nor eddy vorticity flux can be used to depict the processes of how the SE structure is changed and, thus, how the eddy feedback is induced under the modulation of LF flow. This limitation arises because the change in the eddy structure is necessarily relative to a “climatological” eddy structure, which is not included in the F3.

Fig. 1.
Fig. 1.

(a) Schematic diagram for the new theoretical framework of interactions among climatological mean flow, low frequency flow, and basic and anomalous synoptic eddy flow. (b) Detrended winter NAO index anomalies from 1978/79 to 2007/08, where the gray dashed lines indicate ±80% standard deviation. (c) NAO-index-regressed patterns of 300-hPa streamfunction (contours; 106 m2 s−1) and eddy forcing in terms of streamfunction tendency (shading; m2 s−2). (d) Climatological synoptic eddy streamfunction structure (contour interval 1 × 106 m2 s−1, zero line omitted) and anomalous eddy vorticity structure (shading; 0.5 × 10−6 s−1) by regressing the three-point zero-lag covariance fields at lag-0 day onto the NAO index. Panels (c) and (d) are redrawn from Figs. 1b and 3a of Ren et al. (2009). (e) Conceptual diagrams for the synoptic eddy–NAO interaction in terms of the (left) F3 and (right) F4.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

To better understand the general SELF feedback, Jin et al. (2006a) proposed to separate the SE flow into “basic” and “anomalous” parts in a statistical sense, where the basic part represents a climatological measure of the SE flow and the anomalous part an eddy activity anomaly induced by LF flow. They developed a new framework (denoted as F4, which involves four parts; see Fig. 1a) by combining the basic eddy flow with the climatological mean flow, and realized a linear dynamical closure for the SELF feedback in which the slowly varying eddy feedback is directly expressed by LF flow through a linear operator (Jin et al. 2006a,b; Pan et al. 2006). Based on the F4, further diagnostic works revealed a left-hand preference in the direction of various SE fluxes relative to LF flow (Kug and Jin 2009; Kug et al. 2010a,b; Ren et al. 2011). That is, all of the eddy vorticity, eddy temperature, eddy moisture, and transformed eddy potential vorticity fluxes can be organized systematically by LF flow and tend to be directed to the left-hand side of the LF flow. Essentially, this left-hand preference in the eddy vorticity flux direction indicates the positive eddy feedback. To depict the physical processes involved in this phenomenon, Ren et al. (2009) proposed a kinematic mechanism based on the idea of separating the SE flow in the F4. This can be conveniently applied to examine the SE structure change that is induced by LF flow (e.g., the NAO). Indeed, the regressed eddy vorticity forcing pattern by the NAO index in Fig. 1b is not only in phase with, but also upstream of, the NAO, as shown in Fig. 1c. Jin (2010) proposed a theory of eddy-induced instability for LF variability and derived theoretically the in-phase and quadrature SE feedbacks in a barotropic framework. These recent studies provide guidance in seeking an effective approach to examine the physical mechanisms of the two types of SE feedback onto the NAO flow because, to our knowledge, these mechanisms are still unclear. In this respect, we will follow up the study of Ren et al. (2009) on representing physically the basic and anomalous SE structures, as redrawn in Fig. 1d, and propose a method for decomposing the SE structure to anatomize the SELF interaction. Here this method will be applied to the NAO-associated in-phase and upstream SE feedbacks in order to examine the NAO-induced SE structure changes and anomalous eddy structure patterns and to demonstrate their roles in generating the two SE feedbacks onto the NAO.

Figure 1e shows a conceptual diagram of this main idea, which illustrates how the physical mechanisms in the SE–NAO interaction can be detected by introducing the decomposition of SE flow structure, in contrast with the conventional means. The remainder of this paper is organized as follows. The data are introduced in section 2, and a statistical eddy streamfunction (SES) field is defined in section 3. Based on this field, we first examine the spatiotemporal features of SE structure in section 4 and propose a method of eddy structure decomposition in section 5. SE feedback statistics for the NAO are diagnosed in section 6, and the mechanisms that underlie the SE–NAO interaction are identified in section 7. A summary and discussion are given in section 8.

2. Data

National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data are used to construct monthly and daily mean 30-yr datasets for the period from January 1978 to February 2008 (Kalnay et al. 1996). The streamfunction and vorticity fields are calculated with zonal and meridional winds at the 300-hPa pressure level. To separate the SE component, the daily mean winds are bandpass filtered in the period of 2–8 days using a Lanczos filter with 41 weights (Duchon 1979). The LF spectrum is defined by the seasonal mean. All analyses are performed in boreal winter [December–February (DJF)].

To measure the feedback of the SEs onto the LF flow, we define the seasonal-mean eddy vorticity flux and the induced eddy forcing in terms of streamfunction tendency as
e1
e2
respectively, where , , and denote the 2–8-day bandpass-filtered zonal wind, meridional wind, and vorticity, respectively; the overbar indicates seasonal mean, and its climatology and anomaly are denoted by and . Also, is the horizontal divergence operator, the horizontal inverse Laplacian operator, and the eddy-induced tendency. Convergence (divergence) of eddy vorticity fluxes corresponds to a negative (positive) streamfunction tendency, indicating a cyclonic (anticyclonic) vorticity tendency of LF circulation. Note that only the irrotational component Fi in eddy vorticity flux can influence LF flow. Eddy vorticity flux is taken as by following the approach of Holopainen and Fortelius (1987), where is the gradient operator and the rotational component Fr of eddy flux is removed accordingly.

The NAO index, constructed by a rotated principal component analysis (Barnston and Livezey 1987), is from NCEP (http://www.cpc.ncep.noaa.gov/data/teledoc/telecontents.shtml). In this study, the wintertime index is obtained by averaging the monthly mean index in DJF. Figure 1b gives the NAO index curve without a trend for 30 winters from 1978/79 to 2007/08. To ensure that summation of the number of positive and negative anomaly years is nearly half of the total years, we define seven positive anomaly years with index values greater than 0.80 standard deviations and six negative anomaly years with index values less than −0.80 standard deviations.

3. Statistical eddy streamfunction field

First, we construct diagnostics of the synoptic eddy structure and its change. Jin et al. (2006a) used a one-point covariance field of the SE flow in their framework, which captures the eddy structure associated with the anomalous LF-mode flow. Their notions of basic and anomalous eddy flow were developed by Ren et al. (2009), who used a three-point normalized covariance field to examine the SE structure and change associated with the NAO. This field is a natural extension of the one-point correlation statistics first introduced by Wallace and Gutzler (1981) and subsequently applied broadly in synoptic-scale transient diagnoses (e.g., Blackmon et al. 1984a,b; Wallace et al. 1988). Lim and Wallace (1991) used a similar one-point regression map to examine the structure and evolution of baroclinic waves.

To make the eddy structure more clear and robust over the entire center of the NAO, Ren et al. (2009) extended the calculation of the one-point statistics to three-point weighted statistics because the former is more local on a synoptic scale. In their scheme, the normalized one-point covariance field of the eddy streamfunction is first calculated at the 300-hPa level where the primary base point is chosen near the southern center of the NAO and still within the Atlantic storm track action center. From this field, the two base points at the nearest negative centers, one upstream and the other downstream, can be identified.

The normalized one-point covariance field for each of the three base points is defined as
e3
where x, y, and t signify the spatiotemporal coordinates; the subscript j = 0, −1, 1, denotes the primary base point and its upstream and downstream points, respectively; τ is the time lag (day). The overbar denotes the DJF average for year ts, and σc is the climatological standard deviation calculated from the entire record at a particular base point. Then, the three-base-point weighted average of the fields in Eq. (3) is defined as
e4

Here the negative sign comes from the fact that the one-point covariance field always starts with a positive center at the base point and is flanked by wavelike patterns with alternating negative and positive centers located upstream and downstream. The field , defined as the statistical eddy streamfunction field with the unit of streamfunction, represents the statistical SE structure captured by the normalized covariance pattern and changes from year to year. Based on Eq. (3), the SES field can be constructed to characterize the spatiotemporal evolution of the SE structure for each season. The main advantage of this three-point averaged SES field is that it has a better performance than the one-point SES field in capturing the synoptic wave packet structure by factoring the typical wavelength into the spacing of the three-point SES field. Although one-point covariance maps can also capture well the wave-packet-like eddy structures, the three-point SE structure pattern has markedly different amplitude from the one-point pattern (not shown). In general, the amplitude of the wave packet shows a gradual decay with the increasing of distance from the primary base point. For example, Jin et al. (2006a) used complex empirical orthogonal functions to identify typical SE packet patterns, as shown in their Fig. A2. Referred to their patterns, the three-point pattern features a slower decay in the amplitude of the wave packet from the central positive center to the other centers than the one-point pattern, which helps to produce more realistic SE structure. Indeed, there are no apparent differences in extracting subsequent conclusions based on the one-point and three-point statistics.

Using the 30-yr wintertime SES dataset in which the 90 days after 1 December are selected for each year, the average of the SES field in all 30 winters is taken as the climatological (or basic) SE structure and its deviations as the anomalous eddy structure in each winter. Indeed, the climatological (or basic) eddy in this paper is referred to as the climatological-mean measure of synoptic-scale statistics (e.g., the SES) rather than the direct climatology of individual SEs or the climatological departure from zonal symmetry. To identify the changed eddy structure associated with the NAO, we take the composite of the SES field in terms of the NAO index as the eddy structure under the NAO condition. The difference between the NAO-associated and climatological eddy structures is defined as the anomalous eddy structure under the NAO. More precisely, the eddy structures derived from the SES field are called an eddy flow/streamfunction structure. If using synoptic-scale vorticity and wind velocity, the eddy vorticity and velocity structures can be obtained in the same way. Instead, for simplicity, they are directly derived from the eddy streamfunction structure field.

For constructing the SES fields, a key question is how to select the base points properly. Two groups of base points are chosen (see Fig. 2), located in the major eddy activity region where the synoptic-scale teleconnectivity (which reflects a robustness of the eddy structure represented as the one-point correlation) reaches a maximum in the NA, by referring to the teleconnectivity map of Blackmon et al. (1984a) in their Fig. 16. More importantly, to better examine the spatial relationships between SE structure changes and the barotropic effects of the NAO, we prefer to locate the primary base points as close as possible to the centers of the southern lobes of the two NAO phases. As seen in Fig. 2, the two NAO phases have almost same central latitude but different central longitudes, which explains why the chosen base points are the same latitudinally but different longitudinally in the two phases.

Fig. 2.
Fig. 2.

Snapshots of the synoptic eddy structure patterns in terms of the SES fields from lead day 2 to lag day 2 under (left) positive and (right) negative NAO conditions. Gray contours denote the NAO patterns, as in Fig. 1c. Black contours (color shades) represent the climatological (NAO-associated) eddy structure. Thick curves are phase lines of the eddy structures in the climatology (black) and under the NAO conditions (red). Anticyclonic (cyclonic) eddy structure patterns are denoted by red shading and solid contours (blue shading and dashed contours); interval in the black contours is 1 × 106 m2 s−1 , all zero lines omitted; and cross signs denote selected base points at 45°N, 295°E; 45°N, 325°E; and 45°N, 352.5°E for the positive NAO phase (NAO+), and 45°N, 300°E; 45°N, 332.5°E; and 45°N, 355°E for the negative phase (NAO−).

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

4. Synoptic eddy structure and its spatiotemporal changes

Many studies have shown that SEs can be deformed or changed during their short lifetime by LF flow (e.g., Qin and Robinson 1992). Particularly, the NAO-induced kinematic eddy structure change has been observed to be responsible for the positive eddy feedback onto the NAO (Ren et al. 2009). Here we further examine the spatiotemporal characteristics of this kind of kinematic eddy structure change. Figure 2 shows the evolutions of the composites of the SES fields in the two NAO phases. We simply (and qualitatively) define the zonal extreme line of an eddy as its phase line. As seen in Fig. 2, the climate-mean SE flow field features a wave-packet-like structure whose amplitude naturally decays with increasing distance from the primary base point and is bigger in the SE-developing region than in the decaying region. This climatic pattern looks slightly top-heavy and tilted in the northeast–southwest orientation, presumably due to actions of the jet stream and baroclinicity. In fact, Fig. 2 also shows the life cycle of the SE structure: the circular SE originates in the western boundary area, develops gradually, becomes elongated meridionally, tilts zonally with an eastward propagation, and is finally damped in the downstream area of the NAO. The slight difference between the two groups of base points has a negligible impact on the characteristics of the climatological SE structures.

The SE structures under the NAO show quite small changes relative to the climatology, which is essentially because the NAO flow is small compared with the climatic mean flow. To make the SE structure change clearer, the differences between the composite and climatological structure patterns are multiplied by a factor of 2—this factor is only used in Fig. 2 and subsequently in Figs. 3a and 3b. By adding the enhanced differences onto the climatological patterns, the new composite patterns in the two NAO phases (shading in Fig. 2) are obtained. These patterns appear stronger in western regions than in eastern regions because the observed synoptic-scale variances are centered in the upstream region of the NAO (Chang 2009). Figure 2 shows that the SE structure in the NAO phases is systematically deformed relative to that in the climatology. The deformations are mainly characterized by two types of kinematic SE structure change associated with two barotropic effects caused by wind anomalies of the NAO; that is, the shearing effect of the zonal wind tends to produce a zonally slanted eddy structure change, while the zonal variation in the zonal wind forms a stretching effect that results in the eddy structure being zonally stretched in the north (south) and contracted in the south (north) of the southern lobe of the positive (negative)-phase NAO. These two effects have been revealed in previous theoretical and numerical studies (Qin and Robinson 1992; Jin 2010). Also, in Fig. 2, when the SEs propagate eastward, the climatological phase lines gradually slant clockwise, and the phase lines of the changed eddy structure are tilted clockwise by the positive-phase NAO flow and tilted counterclockwise by the negative-phase flow. In particular, the angles between the NAO-associated and climatological phase lines are enlarged by degrees with SEs passing, indicating an accumulated effect of the SE structure change modulated by the NAO flow.

Fig. 3.
Fig. 3.

Center composites for (a),(b) the eddy structure patterns in Fig. 2 and (c),(d) the anomalous eddy flow structure (blue contours), generated from lead day 2 to lag day 2 by choosing basic reference points with centers of the central anticyclone at lag 0 in Fig. 2 for the climatological and NAO conditions, respectively. Black dots (yellow squares) are the centers of the contoured (shaded) eddies; unit throughout is 1 × 105 m2 s−1. Here the NAO patterns (gray contours) are referred to as a background flow to examine eddy structure change conveniently.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

To clearly show features of eddy structure change, we introduce a new center composite scheme, similar to the average over the life cycle of an eddy (see appendix A for details). Figures 3a and 3b show the standard patterns of climatological and changed SE structures in the two NAO phases through this center composite, using the maps in Fig. 2 from lead day 2 to lag day 2. One of the most prominent features in the patterns is that the phase lines show a clockwise tilt in the positive NAO phase and a counterclockwise tilt in the negative phase, relative to those in the climatology. This indicates that the SE structure is subject to an anticyclonic-type slanted change in the positive phase and to a cyclonic-type change in the negative phase under the shearing effect of the NAO flow. In addition, the NAO-related eddy structure patterns, in contrast to the climatology, feature a zonal extension in the north (south) and a zonal contraction in the south (north) of the southern lobe in the positive (negative) NAO, which tends to yield a changed eddy structure with a top (bottom)-heavy bell shape.

We further identify the centers of the unchanged and changed SEs, shown in Figs. 3a and 3b. It is clear that all statistical eddy centers, relative to the climatological positions, shift northward in the positive NAO phase and southward in the negative phase, and roughly reach a maximum displacement east of the NAO centers when they move eastward continuously along the midline at ~45°N. Here the barotropic stretching effect of the NAO flow contributes to a bell-shaped distortion of the SE structure and thus the meridionally shifted eddy center, where the eddy structure change is marked by eddy deformation and an induced meridional shift of the eddy centroid. Another possibility is that the NAO-related meridional variations of the NA jet stream and local baroclinicity affect the meridional positions of the eddy structure centers, where the eddy structure change is manifested by a meridional shift of the eddy center. These two types of SE structure change, reflecting the barotropic and baroclinic effects of the NAO flow, are difficult to separate in an observational study, and hence their individual contributions to the eddy center shift are difficult to distinguish.

Qin and Robinson (1992) suggested that the formation of a bell-shaped SE is due to the barotropic deformation effect caused by the LF eddy. Following their method, we also diagnose the deformation rate by different wind components of the NAO (not shown). It is found that the deformation rate patterns reflect essentially the distribution of divergence of the NAO wind. For example, in the southwest (northwest) of the southern lobe of the positive-phase NAO, the convergence (divergence) of the zonal wind tends to contract (dilate) the SE structure in the east–west direction, and this zonal distortion is balanced by the divergence (convergence) of the meridional NAO wind. Also, an opposite pattern of eddy structure change can be formed in the east of the same NAO lobe because the distribution of deformation rate in the east is opposite to that in the west. However, the results in Figs. 2 and 3 support a uniform bell-shaped eddy structure change and eddy center shift in both the east and west of the NAO regions. In a theoretical study, Jin (2010) proved that such a triangular/bell-shaped SE structure change in LF flow is essentially generated under the combined influences of eddy propagation and dispersion.

Overall, the SE structure has been observed to be changed from its climatological pattern to zonally slanted and meridionally shifted patterns under the modulation of the NAO.

5. Eddy structure decomposition

a. Decomposing synoptic eddy flow structure

In a statistical sense, the changed SE flow structure can be regarded as the summation of a basic part and an anomalous part. Thus, the SE flow structure, represented by the SES field, is separated as follows (hereafter, the statistics of SE structures are expressed with a hat to distinguish them from observational variables):
e5
where denotes the general SE flow structure, is the climatological (basic) SE structure and is the anomalous SE structure caused by LF flow. In this study, and correspond to the SE flow structure under the NAO condition and its departure from , respectively. Thus, our method, based on the SES field, can be used to depict the SE flow separation first noted by Jin et al. (2006a). In terms of Eq. (5), the eddy structure decomposition is proposed to represent the separation of the anomalous SE structure from the climatological. Clearly, this eddy decomposition method is not applied to an individual eddy, but to the statistical eddy structure.

The anomalous SE structure patterns in Figs. 3c and 3d are taken as differences of the NAO-associated eddy structure patterns from the climatological ones in Figs. 3a and 3b. Here the multiplication factor of 2 has been removed from the anomalous structure. As an illustration of the eddy structure decomposition, Figs. 3c and 3d show a clear contrast between two different types of anomalous eddy structures in the two NAO phases—that is, the positive (negative)-polarity NAO flow prefers a meridional dipole of anomalous eddy structure that is anchored in phase (in quadrature) with the anticyclones or cyclones in the basic eddy structure. Thus, the anomalous eddy structure associated with the NAO involves two primary components: the meridional in-phase dipole (IPD) and the meridional in-quadrature dipole (IQD), where “in phase” and “in quadrature” are relative to the climatological SE structure, not to the NAO. It is clear that the two types of dipole appear in both of the NAO phases but with different preferences, where the IQD-type pattern was presented previously by Ren et al. (2009).

Overall, the IQD- and IPD-type anomalous eddy structures are clearer over the western/central region of the NAO in the two phases, especially around 35°W. The IQD-type pattern is essentially due to the zonal-slanting eddy structure change under the shearing effect of the NAO, whereas the IPD-type pattern is caused by the bell-shaped distortion of eddy structure under the stretching effect of the NAO and is also probably from the contribution of the meridional shift of local baroclinicity by the NAO flow. Therefore, the different phases of NAO flow preferentially generate the different patterns of anomalous eddy structure. The contrast between the patterns is of primary importance to different eddy feedback processes in the two NAO phases.

b. Decomposition of eddy vorticity flux structure

The method of eddy structure decomposition is further used to diagnose the dynamical SE feedback onto the LF flow. Based on the SE flow structure, we can obtain the synoptic eddy velocity and eddy vorticity structures (denoted as and , respectively, where and ). Similar to Eq. (5), these structures are separated into
e6
e7
The subscripts c and a denote climatological (basic) and anomalous SE structures, respectively.
Based on Eqs. (6) and (7), the decomposition of eddy vorticity flux structure is derived as
e8
Here the eddy flux structure is separated into four parts: the basic eddy velocity multiplied by basic eddy vorticity (BB), the basic eddy velocity multiplied by anomalous eddy vorticity (BA), the anomalous eddy velocity multiplied by basic eddy vorticity (AB), and the anomalous eddy velocity multiplied by anomalous eddy vorticity (AA).
Considering the phase propagation of SE structure, we integrate Eq. (8) over the life cycle of SE by using the eddy flux structure fields on different lead–lag days as follows:
e9
where the brace pair is referred to as the eddy phase integration operator (see appendix A for details). We take composites of the left-hand term in Eq. (9), in terms of the NAO index, as the eddy fluxes for the two NAO phases. Here is the nonlinear term. The climatology of eddy flux by ignoring the small term in , where is taken as the average of the similar results from the two groups of base points.
In the anomalous eddy flux, , where is generally much smaller than the two other dominant linear terms and may be neglected. The integrated anomalous eddy vorticity flux structure is rewritten as follows:
e10
In the following, the two linear components of will be examined. For reference, the pattern of and the NAO-associated patterns are shown in appendix B (see Fig. B1).

6. Anatomy of dynamical eddy feedback onto the NAO

Ren et al. (2009) showed that the NAO-related kinematic changes of SE structures are responsible for the eddy vorticity feedback onto the NAO. However, the authors only focused on the eddy forcing induced by the basic eddy velocity multiplied by anomalous eddy vorticity (i.e., the BA part in the present study). In fact, in terms of Eq. (10), the net anomalous eddy flux is dominated by the two linear parts: BA and AB. Thus, their different roles in generating the two dynamical SE feedbacks onto the NAO are revealed in the following.

We first examine . As shown by the composites in Figs. 4a–d, the patterns of the integrated eddy vorticity flux anomalies show remarkable similarity to the observed, where pattern correlation coefficients (PCCs) between the two corresponding eddy forcing fields reach 0.67 in the positive NAO phase and 0.73 in the negative phase. As a contrast, the NAO-index-regressed patterns are also presented in Figs. 4e and 4f by referring to Ren et al. (2009), which show the same degree of similarity. It is clear that the positive (negative) eddy forcing, accompanied by divergence (convergence) of the anomalous eddy fluxes, covers in the southern center of the positive (negative)-polarity NAO. This indicates an in-phase SE feedback onto the NAO. The upstream feedback responsible for the retrograde propagation of planetary waves is also clear in Fig. 4 for the observed and integrated cases; that is, the eddy forcing centers tend to be upstream of the southern NAO lobe, which is more apparent in the positive NAO phase than in the negative. These results confirm the suitability and feasibility of delineating the SE feedbacks onto the NAO by using the method of eddy structure decomposition. Moreover, a strong asymmetry in the eddy forcing patterns between the two NAO phases is found, which is likely responsible for the two eddy forcing centers (see Fig. 4f) associated with the upstream and in-phase feedbacks. Further, we speculate that this asymmetry in eddy forcing patterns is linked to the apparent asymmetry between the anomalous eddy structure patterns in Figs. 3c and 3d.

Fig. 4.
Fig. 4.

Eddy vorticity fluxes (vectors; 10−6 m s−2) and induced eddy forcing (shading; m2 s−2) under the NAO conditions. (a),(c),(e) Summation of two integrated anomalous fluxes, , and (b),(d),(f) observed anomalous fluxes for (top) the composites in NAO+, (middle) the composites in NAO−, and (bottom) the regression by the NAO index. Black contours denote the NAO pattern. PCC[(a),(b)] = 0.671, PCC[(c),(d)] = 0.728 and PCC[(e),(f)] = 0.725.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

To verify this speculation, we next examine the physical processes of the different types of SE feedback associated with the BA and AB parts of flux. We present the center composites of and in Figs. 5a and 5b, and those of and in Figs. 5c and 5d, obtained from Figs. 3c and 3d, to illustrate how and are calculated from the decomposed eddy structure components in detail. The patterns of appears similar to those of , although the IPD around the central anticyclone in is more robust than that in in the positive NAO phase, whereas the IQD in is weaker than that in in the negative phase.

Fig. 5.
Fig. 5.

Maps for (a),(b) the basic eddy flow structure (contours; 106 m2 s−1) and the anomalous eddy vorticity structure patterns (shading; 10−6 s−1), and (c),(d) the anomalous eddy flow structure (contours; 106 m2 s−1) and the basic eddy vorticity structure patterns (shading; 10−6 s−1). All patterns are obtained and derived from Fig. 3. Black arrows illustrate the generated eddy vorticity fluxes.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

a. In-phase and downstream feedback induced by the BA flux

Using and , the BA part () of the anomalous eddy vorticity flux is estimated. Figure 6 shows that the eddy flux and induced eddy forcing patterns of this part, when propagating eastward, are region-dependent and are more intense over the eddy damping than over the eddy-developing regions in both the NAO phases. It reflects the modulation effects of LF flow on the SE feedback. The eddy fluxes tend to diverge over the southern center region in the positive NAO phase and converge over the same region in the negative phase on each lag day. It is interesting that the eddy forcing patterns all have a positive projection onto the positive-polarity NAO and a negative projection onto the negative NAO when eddies with different phases travel through this region, implying a net accumulation of positive eddy feedback. The eddy forcing patterns also tend to be downstream of the positive-phase NAO.

Fig. 6.
Fig. 6.

Snapshots of the fluxes of the basic eddy velocity multiplied by anomalous eddy vorticity (vectors; 10−6 m s−2) and the induced eddy forcing (shading; m2 s−2) from lead day 2 to lag day 2 in the (left) positive and (right) negative NAO phases. Gray contours for the NAO flow are as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

To obtain the average patterns of eddy flux and forcing structure, two kinds of composite analysis are conducted. First, Figs. 7a and 7b present the center composites as produced by averaging the patterns in Fig. 6 from lead day 2 to lag day 2 in terms of the basic reference points that move with eddy group propagation. The eddy forcing patterns project onto the NAO with the same signs, indicating a positive feedback. Actually, the patterns in Figs. 7a and 7b are a combination of two types, that is, the IPD and IQD types, of eddy flux structure patterns, which can be derived from the IPD- and IQD-type anomalous eddy structures, respectively. To describe the formation of the IPD-type flux pattern, the middle anticyclone in Fig. 5a is taken as an example in the positive NAO phase. In the north (south) of the anticyclone, we find negative (positive) and eastward (westward) , and thus obtain westward . Also, westward flux is obtained in the adjacent cyclones, but, in the negative NAO phase, the weak IPD-type structure makes it difficult to form a clear zonal flux over the southern NAO center, as seen in Fig. 7b. To depict the formation of the IQD-type flux pattern, the middle anticyclone in Fig. 5b is taken as an example in the negative NAO phase. At the upper left (right) corner of the central anticyclone, we find negative (positive) and northward (southward) and thus obtain southward . Then, northward can be obtained at the lower left (right) corner of this anticyclone. Thus, all of around the middle anticyclone and adjacent cyclones converge into the NAO, forming a negative eddy forcing pattern and enhancing the negative-phase NAO flow. In the positive NAO phase, an almost reverse but weaker pattern also contributes to the positive feedback.

Fig. 7.
Fig. 7.

(a),(b) Center composites of the BA fluxes (vectors) and the induced eddy forcing (shading), where the NAO patterns (gray contours) are only referred to as a background flow. The maps are generated using the fields in Fig. 6 with the basic reference points taken at the centers of the central climatological anticyclone on lag-0 day in Fig. 2. (c),(d) Lead–lag composites of the BA fluxes and the induced eddy forcing by using the maps in Fig. 6 from lead day 2 to lag day 2, where the black contours denote the NAO. Units are 10−6 m s−2 for vectors and m2 s−2 for shading.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

Figures 7c and 7d show the second kind of composite that we term “lead–lag composite,” which is obtained through a local point-to-point average of the fields on different lead–lag days in Fig. 6. This composite reflects approximately the accumulation effect of the eddy feedback of a number of individual eddy group structures with different propagating phases onto the NAO. It is clear in Figs. 7c and 7d that the wavelike irregularities in eddy flux patterns in Figs. 6, 7a, and 7b are completely filtered out. The resultant averaged BA fluxes diverge in the positive-phase NAO region and converge in the negative-phase region, indicating a positive eddy feedback onto the NAO. The eddy forcing patterns are thus in phase with the NAO, especially in the negative phase. In contrast, the positive NAO phase has a much clearer zonal component of eddy flux over the southern lobe than does the negative phase, where the westward fluxes of BA form a divergence in the east of the southern lobe and a convergence in the west; consequently, the eddy feedback pattern tends to be downstream of the NAO pattern.

b. Out-of-phase and upstream feedback induced by the AB flux

Now, we move to the AB part () of the eddy flux, which is estimated from and . The time evolutions of this AB flux and induced forcing patterns on lead–lag days are also examined (not shown). Their characteristics are almost opposite to those in Fig. 6 but the magnitudes are different. Figures 8a and 8b shows the center composites of . The formation of the AB flux over the southern NAO center in Fig. 8a is largely associated with the IPD-type anomalous eddy structure that dominates in the positive NAO phase and tends to generate zonal fluxes. As seen in Fig. 5c, in the middle of the primary cyclone (anticyclone), is positive (negative) and eastward (westward); thus, is eastward. In the negative NAO phase, the AB fluxes in Fig. 8b are generated mainly from the IQD-type anomalous eddy structure, which tends to form meridional fluxes. In Fig. 5d, we obtain northward (southward) in the north (south) of the middle anticyclone where is negative and southward (northward), and of the two adjacent cyclones where is positive and is northward (southward).

Fig. 8.
Fig. 8.

As in Fig. 7, but for the AB fluxes .

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

The lead–lag composites in Figs. 8c and 8d show an out-of-phase relationship between the patterns of eddy forcing and the NAO, reflecting a negative feedback. The eddy flux and eddy forcing patterns are opposite to those in Figs. 7c and 7d, which act to cancel partly the feedback of the BA flux onto the NAO flow. In the positive NAO phase, the eastward eddy fluxes diverge in the west of the southern lobe of the NAO and converge in the east; thus, the induced eddy forcing centered upstream of the NAO pattern. A comparison of Figs. 7c and 7d against Figs. 8c and 8d indicates that the eddy forcing induced by is less than that by in the positive NAO phase, but greater in the negative phase. As a result, the net eddy feedback shown in Fig. 4 (i.e., the summation of the two parts of eddy flux) appears upstream of and in phase with the NAO. In addition, the accumulation effect of multieddy feedback acts to filter out the wavelike irregularities in the eddy flux patterns, validating the concept of eddy phase integration.

7. Identification of mechanisms for eddy vorticity feedback onto NAO

a. Eddy feedback accumulation

As seen in Fig. 5, the IPD-type anomalous eddy structure tends to induce a larger zonal eddy flux rather than a meridional one, whereas the IQD-type tends to induce a meridional flux rather than a zonal one. When all of the eddy forcing patterns with different phases are projected onto the NAO, as shown by the integrated maps in Fig. 4 or the lead–lag composite maps in Figs. 7 and 8, the meridional (zonal)-component fluxes induced by the IPD (IQD) tend to counteract each other so that only the zonal (meridional) fluxes by the IPD (IQD) emerge, thereby resulting in an accumulative eddy forcing. We term this process “eddy feedback accumulation.”

Considering the eddy feedback accumulation, we infer that the physical mechanism operating through the IPD (IQD)-type anomalous eddy structure is largely responsible for the observed zonal (meridional) eddy vorticity fluxes shown in Fig. 9. The meridional eddy fluxes make a primary contribution to the in-phase feedback onto the NAO flow in the two phases and to an additional upstream effect in the eddy forcing fields over the southern center, especially in the positive NAO phase. The eddy forcing by the zonal fluxes features an east–west distribution, where the positive forcing lies upstream of the positive-phase NAO center and the negative forcing lies downstream. An almost opposite pattern of eddy forcing is observed in the negative NAO phase, but this upstream feedback appears to be relatively weak.

Fig. 9.
Fig. 9.

(a),(c) Meridional and (b),(d) zonal component of the observed total eddy fluxes (vectors; 10−6 m s−2) and the induced eddy forcing patterns (shading; m2 s−2): black contours are for the NAO patterns.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

In Fig. 10, the integrated meridional and zonal eddy fluxes derived from the decomposed eddy structures reveal a high degree of similarity to Fig. 9. The in-phase eddy feedback is robust in both positive and negative NAO phases (Figs. 10a,c), especially in the negative phase. An upstream eddy forcing pattern is also clear in Fig. 10b, consistent with that in Fig. 9b. Indeed, there is no clear upstream feedback pattern in Fig. 10d, compared to that in Fig. 9d. These indicate that the positive-phase NAO is closely related to the physical processes associated with both the IPD- and IQD-type anomalous eddy structures, whereas the negative-phase NAO is fed back dominantly by the physical process associated with the IQD type only.

Fig. 10.
Fig. 10.

As in Fig. 9, but for the integrated eddy fluxes, .

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

b. Kinematic eddy feedback mechanisms due to eddy structure change

In nature, the physical processes for the two SE feedbacks are intermixed although, with the eddy decomposition method, the anomalous eddy vorticity flux structure is separated into the two dominant linear parts with opposite signs. Note that the anomalous eddy structures caused by the NAO flow are dominated by the two primary patterns (meridional IPD and IQD), generated from the two types of eddy structure change. Thus, the two feedback processes in the SE–NAO interaction can be further depicted conceptually as kinematic zonal eddy-slanting (ZES) and meridional eddy-shifting (MES) mechanisms, which correspond to the IQD- and IPD-type anomalous eddy structure patterns, respectively. Although the two typical patterns are difficult to obtain from pure observational diagnoses, it is fortunately found that, based on the anterior analyses, the two NAO phases provide such particular cases in reality. The IQD- and IPD-type patterns were observed to be dominant in the negative and positive NAO phases, respectively. Therefore, we choose the two NAO cases as typical examples to delineate the ZES and MES mechanisms and show schematic diagrams of these mechanisms in Fig. 11.

Fig. 11.
Fig. 11.

Schematic diagrams of the mechanisms for the (a)–(d) zonal eddy-slanting feedback and (e)–(h) meridional eddy-shifting feedback of synoptic eddies onto low-frequency flow. Thick green solid ellipses with counterclockwise (clockwise) arrows represent cyclonic (anticyclonic) low frequency flow like the negative (positive) phase NAO flow. (a),(b),(e),(f) Solid thin circles indicate the idealized synoptic eddy patterns with cyclonic (red) and anticyclonic (blue) eddies. Dashed circles denote the changed eddy structure as a result of the anomalous climatic flow. Black stars are the centers of basic eddy structures. (c),(g) Dashed circles stand for the anomalous eddy flow structures with cyclonic (red) and anticyclonic (blue) circulations. Smaller (bigger) shaded ovals stand for the anomalous (climatological) eddy vorticity, where yellow is positive and blue is negative. (b),(c),(f),(g) Gray dashed and black solid arrows represent the eddy vorticity fluxes before and after the eddy feedback accumulation, respectively. (d),(h) Dark gray solid arrows denote the net eddy vorticity fluxes and their convergence (divergence) is represented by the yellow (blue) grid.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

For the ZES mechanism, without losing generality, we consider a cyclonic LF circulation as the southern lobe of the negative-phase NAO. As seen in Fig. 11a, SEs (cyclone/anticyclone) that pass by this region are deformed kinematically during their relatively short lifetime by the background flow. That is, the zonal LF flow acts to slant the SEs counterclockwise relative to the climatological SE structure, which yields a meridional IQD-type anomalous eddy flow structure. This IQD-type structure can generate the two almost opposite parts of anomalous eddy vorticity flux: BA and AB (gray dashed arrows in Figs. 11b and 11c), where the BA fluxes converge but the AB fluxes diverge over the region. The effect of eddy feedback accumulation makes zonal components of these fluxes tend to cancel each other when numerous SEs pass. Thus, only the meridional fluxes (black solid arrows) are preserved. Owing to the observational fact that the BA part is greater than the AB in the meridional fluxes, the net meridional fluxes act to converge and enhance the LF flow, indicating an in-phase SE feedback, as shown in Fig. 11d.

For the MES mechanism, an anticyclonic circulation is considered as the southern lobe of the positive-phase NAO. Under the influence of the background flow, the SE center is subject to a northward shift. This occurs, on one hand, because the eddy structure is barotropically distorted by the anticyclonic LF flow so as to form a top-heavy bell-shaped eddy structure (Fig. 11e) and, on the other hand, because the anticyclonic flow may also make the eddy centers shift northward due to the northward shift of local baroclinicity. In terms of the eddy decomposition method, the IPD-type anomalous eddy structure can be identified from the climatological structure and then the two eddy flux parts with almost opposite signs (BA and AB; see gray dashed arrows in Figs. 11f and 11g) are derived, where the BA (AB) fluxes are basically westward (eastward). Due to the eddy feedback accumulation, the meridional components of the eddy fluxes tend to counteract each other; thus, only the zonal eddy fluxes (black solid arrows) are preserved. Because the AB part is observed to be greater than the BA in the zonal fluxes, the resultant net eddy fluxes are zonally directed to the east, where their maxima lie over the central latitudes and longitudes of the LF circulation. Such a zonal-flux pattern induces a positive vorticity tendency downstream of the LF flow center and a negative tendency upstream. This indicates an upstream reinforcement of the LF flow, that is, an upstream SE feedback, as shown in Fig. 11h.

Likewise, we can apply the ZES mechanism to the positive NAO phase, and the MES mechanism to the negative NAO phase, although it is difficult to observe directly these patterns in reality. Indeed, the two kinematic mechanisms are expected to work for the LF flow in general and for other climate modes in particular, even though the NAO may possess the most typical manifestations of these two mechanisms for the SELF feedback in the upper troposphere.

8. Summary and discussion

Many studies have shown that the SE structural change and induced feedback play a key role in the two-way interaction between SE and LF flow. Based on the framework that SE flow can be separated into basic and anomalous parts in a statistical sense (Jin et al. 2006a), we sought to anatomize the physical process of the SE–NAO interaction by proposing the method of eddy structure decomposition, as an extension of the previous study on a kinematic mechanism for positive SE feedback (Ren et al. 2009). In this method, to characterize the spatiotemporal SE structure we used the three-point covariance statistics of eddy streamfunction to construct the seasonal SES field in a space and time-lag domain. This SES field can be easily decomposed into its climatological and anomalous parts, which are regarded as the physical representations of the theoretical basic and anomalous SE flow. These two parts are used to estimate the anomalous eddy vorticity flux field and examine the physical processes of the SELF feedback.

With this method, we showed that the NAO flow can form the two primary barotropic SE structure changes: zonal eddy slanting and meridional eddy shifting. The former, associated with the shearing effect of the NAO flow, yields the IQD-type anomalous eddy structure. The latter generates the IPD-type anomalous structure from the bell-shaped eddy structure deformation, and from a meridional eddy structure shift due to local baroclinicity shift under the modulation of the NAO. We found the asymmetry that the positive-phase NAO prefers the IPD-type eddy structure and the negative phase prefers the IQD type, which provides two typical observed cases for illustrating the two distinct processes in the SE–NAO interaction. Thus, we decomposed the statistics of the dynamical SE feedback induced by the changed eddy structures under the NAO conditions and showed that the net anomalous eddy vorticity flux and forcing patterns, integrated by the SES-based fields, agree well with the observed patterns. This validates our method proposed to elucidate in-phase and upstream feedback for the NAO.

It is found that the IQD (IPD)-type anomalous eddy structure is largely responsible for generating the net meridional (zonal) anomalous eddy vorticity flux through the effect of eddy feedback accumulation. In this study, the two processes that form the in-phase and upstream SE feedbacks onto the NAO have been delineated conceptually into the kinematic ZES and MES mechanisms. Through separating the anomalous eddy vorticity flux structure into the two linear parts (BA and AB), we showed that the net eddy forcing induced by the meridional fluxes from the ZES mechanism makes a positive contribution to the enhancement of the NAO flow because the BA part is greater than the AB part in the meridional fluxes. In contrast, the net forcing induced by the zonal fluxes from the MES mechanism contributes to an upstream reinforcement of the positive-phase NAO flow because the AB part is larger than the BA part in the zonal fluxes, but the negative NAO phase has no clear upstream feedback by the MES mechanism. Still, the observed relationships between the BA and AB parts of the eddy vorticity flux need to be further verified. The kinematic ZES and MES mechanisms, indeed, help us understand how the NAO gains the in-phase and upstream reinforcement by organizing SEs and harvesting upscale eddy vorticity.

This study also gives a further observational validation of the linear dynamic framework of SELF feedback proposed by Jin et al. (2006a). The observed anomalous eddy vorticity fluxes are reproduced well with the basic and anomalous SE structures through the eddy decomposition method. Of note, the observed climatological-mean eddy vorticity flux field is also reproduced well by the basic eddy velocity field multiplied by basic eddy vorticity field (Fig. B1), indicating that our method is also adequate for examining climate issues related to SE feedback. Moreover, Jin et al. did not take the nonlinear term into account, which can be represented by the anomalous eddy velocity field multiplied by anomalous eddy vorticity field. As seen in appendix B (Fig. B1), this nonlinear part of eddy flux has a negligible magnitude compared to the two dominant linear parts. This demonstrates that the SE feedback onto the NAO is largely linear with respect to the climatic basic state, defined as the combination of the climatological mean flow and the climatological SE flow structure. The nonlinear question of the SE feedback in the conventional framework can be transformed into a largely linear question in the new framework. Indeed, it appears in Fig. B1 that this nonlinear part can impact the NAO as a downstream feedback.

The present results have proved that the method of eddy structure decomposition can be used as an effective tool to anatomize the physical processes and mechanisms involved in the two-way SELF interaction. It is also expected that this method could help us understand some of observed features of LF flow and climate modes. For example, the negative NAO phase might be easier to maintain due to a clearer in-phase SE feedback than for the positive phase. Our speculation that the asymmetry of eddy forcing is attributed to the asymmetry of anomalous eddy structure has been preliminarily verified here. This may give a new explanation of why the negative NAO phase tends to be more persistent than the positive phase (Woollings et al. 2010; Barnes and Hartmann 2010). Indeed, the contrast of the two NAO phases in this study is only for validating this method in anatomizing the SELF interaction. Further studies that focus on the asymmetries of the SE feedback between the two NAO phases may provide deeper insights into the asymmetries of the LF flow anomalies between the two phases.

It is worthy of note that the choice of the base points in generating the SES fields has an impact on the anomalous SE structures, although this choice has been expected to reflect a more barotropic than baroclinic effect of the NAO on SEs. By changing the base points slightly to the north, we rechecked the patterns in Figs. 3c and 3d and found that the anomalous eddy structures become correspondingly stronger to the north (not shown). This shows that the meridional shifts of SE activity associated with the different NAO phases may affect the SE structure change and hence the eddy forcing, which was likely underestimated here. Thus, this sensitivity to the choice of base points should be kept in mind when applying the eddy decomposition method.

Acknowledgments

This work is jointly supported by National Science foundation (NSF) Grant ATM 1034439, 973 Program of China (2010CB950404), NSF of China Grants 40805028, China Meteorological Special Project (GYHY201206033, GYHY200906015), and the National Science and Technology Support Program of China (2007BAC29B03). The authors especially appreciate the two anonymous reviewers for their invaluable suggestions, and also give thanks to M. Stuecker and A. Levine for their careful proofreading and revising.

APPENDIX A

Definitions of Center Composite and Eddy Phase Integration

a. Center composite based on eddy structure evolution

It is found in Fig. 2 that, for some particular eddies, their corresponding structure or other statistics tend to follow themselves eastward. Thus, to examine typical patterns of the SE structure change and related eddy activity statistics that move with an individual eddy, we design a center composite scheme in a Lagrangian point of view. The idea is that the statistics of eddy activity are averaged in reference to the centers of the particular eddy in the climatological SE structure on the different lead–lag days as this eddy continuously propagates eastward during these days. Definition of the center composite field is written as
ea1
where A is the variable/statistic for the composite analysis, M is the number of the determined lag days, τ denotes any lag day, τ0 means τ = 0 day, and x and y are the longitude and latitude coordinates at any spatial point in C map, respectively; Xr, Yr is the basic reference point, which is generally taken as the center of some particular eddy in the climatological SE structure pattern at τ0. For example, the basic reference point in Fig. 3 is taken as the center coordinates of the middle anticyclone in the climatological eddy structure pattern at τ0; thus this anticyclone is regarded as the reference eddy. Also X(τ), Y(τ) are the coordinates of that basic reference point at lag τ ≠ 0. According to Fig. 2, we can exactly position the coordinates of these points X(τ), Y(τ) by tracking the center of the same reference eddy at each τ and eventually obtain the center composites for the different NAO phases.

b. Eddy phase integration for reconstructing time-mean eddy statistics

The SE structure, as a function of space and lag time, features a continuous eastward propagation during the short life time of an eddy. The eddy statistics at any lag, estimated by using the decomposed SES fields, has an instantaneous phase θ. Thus, to reconstruct the observed time-mean eddy statistics (e.g., variance and covariance), we need to integrate these estimated eddy statistics with the different propagation phases from 0 to 2π, that is, over an entire life cycle of the SE. If one follows an individual eastward propagating eddy, shown in Fig. 2, it roughly spends an entire period of θ from 0 to 2π evolving from lag −2 to 2 days, taking lag 0 as the phase θ = π. Therefore, we design an approach of eddy phase integration to transform the accumulation of the all-phase eddy activity statistic field B, that is, , into a simple empirical formula
ea2
Subscripts denote the lag day. The trapezoid formula is used in this eddy phase integration with an increment of π/4 corresponding to a 1-day lag, and the brace pair is called the eddy phase integration operator.

APPENDIX B

Patterns of and

Figures B1a and B1b compare the patterns of the observed and integrated climatological eddy vorticity fluxes, defined as and , respectively. The observed climatological eddy flux in Fig. B1a is strongly poleward with a convergence over the northeastern coastal regions of North America and a weak divergence over the central NA. Such a convergence of mean eddy fluxes contributes to a SE forcing that maintains the climatological time-mean flow (Lau and Holopainen 1984). In contrast, the integrated climatological eddy flux structure in Fig. B1b is remarkably similar. The magnitudes of the integrated fields are almost 80% of the observed values. These results indicate that the observed climatological eddy vorticity flux field can be reproduced well by the basic eddy velocity field multiplied by the basic eddy vorticity field. Moreover, the AA flux is shown in Figs. B1c and B1d. It appears that these fluxes can generate a downstream feedback onto the NAO, but with a small magnitude of eddy forcing, compared to that of the two linear fluxes.

Fig. B1.
Fig. B1.

(a) Observed 300-hPa climatological eddy vorticity fluxes and (b) integrated BB fluxes . Shading is the 300-hPa climatology of zonal wind (m s−1). (c),(d) Integrated AA fluxes in the (c) positive and (d) negative NAO phases. Vectors need to be multiplied by 10−6 m s−2. The black contours are for the NAO patterns.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-11-069.1

REFERENCES

  • Barnes, E. A., , and D. L. Hartmann, 2010: Dynamical feedbacks and the persistence of the NAO. J. Atmos. Sci., 67, 851865.

  • Barnston, A. G., , and R. E. Livezey, 1987: Classification, seasonality, and persistence of low-frequency atmospheric circulation patterns. Mon. Wea. Rev., 115, 10831126.

    • Search Google Scholar
    • Export Citation
  • Benedict, J. J., , S. Lee, , and S. B. Feldstein, 2004: Synoptic view of the North Atlantic Oscillation. J. Atmos. Sci., 61, 121144.

  • Blackmon, M. L., , Y.-H. Lee, , and J. M. Wallace, 1984a: Horizontal structure of 500-mb height fluctuations with long, intermediate, and short time scales. J. Atmos. Sci., 41, 961979.

    • Search Google Scholar
    • Export Citation
  • Blackmon, M. L., , Y.-H. Lee, , J. M. Wallace, , and H.-H. Hsu, 1984b: Time variation of 500-mb height fluctuations with long, intermediate, and short time scales as deduced from lag-correlation statistics. J. Atmos. Sci., 41, 981991.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1992: The maintenance of low-frequency atmospheric anomalies. J. Atmos. Sci., 49, 19241946.

  • Branstator, G., 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, 207226.

    • Search Google Scholar
    • Export Citation
  • Cai, M., , and M. Mak, 1990: Symbiotic relation between planetary and synoptic-scale waves. J. Atmos. Sci., 47, 29532968.

  • Cai, M., , and H. M. van den Dool, 1991: Low-frequency waves and traveling storm tracks. Part I: Barotropic component. J. Atmos. Sci., 48, 14201436.

    • Search Google Scholar
    • Export Citation
  • Chang, E. K. M., 2009: Are band-pass variance statistics useful measures of storm track activity? Re-examining storm track variability associated with the NAO using multiple storm track measures. Climate Dyn., 33, 277296, doi:10.1007/s00382-009-0532-9.

    • Search Google Scholar
    • Export Citation
  • Chang, E. K. M., , S. Lee, , and K. L. Swanson, 2002: Storm track dynamics. J. Climate, 15, 21632183.

  • DeWeaver, E., , and S. Nigam, 2000: Zonal-eddy dynamics of the North Atlantic Oscillation. J. Climate, 13, 38933914.

  • Duchon, C., 1979: Lanczos filtering in one and two dimensions. J. Appl. Meteor., 18, 10161022.

  • Feldstein, S. B., 2003: The dynamics of NAO teleconnection pattern growth and decay. Quart. J. Roy. Meteor. Soc., 129, 901924.

  • Franzke, C., , S. Lee, , and S. B. Feldstein, 2004: Is the North Atlantic Oscillation a breaking wave? J. Atmos. Sci., 61, 145160.

  • Gerber, E. P., , and G. K. Vallis, 2009: On the zonal structure of the North Atlantic Oscillation and annular modes. J. Atmos. Sci., 66, 332352.

    • Search Google Scholar
    • Export Citation
  • Hartmann, D. L., 1995: A PV view of zonal flow vacillation. J. Atmos. Sci., 52, 25612576.

  • Hartmann, D. L., , and F. Lo, 1998: Wave-driven zonal flow vacillation in the Southern Hemisphere. J. Atmos. Sci., 55, 13031315.

  • Hartmann, D. L., , and P. Zuercher, 1998: Response of baroclinic life cycles to barotropic shear. J. Atmos. Sci., 55, 297313.

  • Holopainen, E., , and C. Fortelius, 1987: High-frequency transient eddies and blocking. J. Atmos. Sci., 44, 16321645.

  • Hoskins, B. J., , I. N. James, , and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci., 40, 15951612.

    • Search Google Scholar
    • Export Citation
  • Hurrell, J. W., , and C. Deser, 2009: North Atlantic climate variability: The role of the North Atlantic Oscillation. J. Mar. Syst., 78, 2841, doi:10.1016/j.jmarsys.2008.11.026.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 2010: Eddy induced instability for low-frequency variability. J. Atmos. Sci., 67, 19471964.

  • Jin, F.-F., , L.-L. Pan, , and M. Watanabe, 2006a: Dynamics of synoptic eddy and low-frequency flow interaction. Part I: A linear closure. J. Atmos. Sci., 63, 16771694.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., , L.-L. Pan, , and M. Watanabe, 2006b: Dynamics of synoptic eddy and low-frequency flow interaction. Part II: A theory for low-frequency modes. J. Atmos. Sci., 63, 16951708.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437471.

  • Kok, C. J., , J. D. Opsteegh, , and H. M. van den Dool, 1987: Linear models: Useful tools to analyze GCM results. Mon. Wea. Rev., 115, 19962008.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , and F.-F. Jin, 2009: Left-hand rule for synoptic eddy feedback on low-frequency flow. Geophys. Res. Lett., 36, L05709, doi:10.1029/2008GL036435.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , F.-F. Jin, , J.-H. Park, , H.-L. Ren, , and I.-S. Kang, 2010a: A general rule for synoptic-eddy feedback onto low-frequency flow. Climate Dyn., 35, 10111026, doi:10.1007/s00382-009-0606-8.

    • Search Google Scholar
    • Export Citation
  • Kug, J.-S., , F.-F. Jin, , and H.-L. Ren, 2010b: Role of synoptic eddies on low-frequency precipitation variation. J. Geophys. Res., 115, D19115, doi:10.1029/2009JD013675.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., 1988: Variability of the observed midlatitude storm tracks in relation to low-frequency changes in the circulation pattern. J. Atmos. Sci., 45, 27182743.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., , and E. O. Holopainen, 1984: Transient eddy forcing of the time-mean flow as identified by geopotential tendencies. J. Atmos. Sci., 41, 313328.

    • Search Google Scholar
    • Export Citation
  • Lau, N.-C., , and M. J. Nath, 1991: Variability of the baroclinic and barotropic transient eddy forcing associated with monthly changes in the midlatitude storm tracks. J. Atmos. Sci., 48, 25892613.

    • Search Google Scholar
    • Export Citation
  • Lim, G.-H., , and J. M. Wallace, 1991: Structure and evolution of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 48, 17181732.

    • Search Google Scholar
    • Export Citation
  • Limpasuvan, V., , and D. L. Hartmann, 1999: Eddies and the annular modes of climate variability. Geophys. Res. Lett., 26, 31333136.

  • Limpasuvan, V., , and D. L. Hartmann, 2000: Wave-maintained annular modes of climate variability. J. Climate, 13, 44144429.

  • Löptien, U., , and E. Ruprecht, 2005: Effect of synoptic systems on the variability of the North Atlantic Oscillation. Mon. Wea. Rev., 133, 28942904.

    • Search Google Scholar
    • Export Citation
  • Lorenz, D. J., , and D. L. Hartmann, 2001: Eddy–zonal flow feedback in the Southern Hemisphere. J. Atmos. Sci., 58, 33123327.

  • Lorenz, D. J., , and D. L. Hartmann, 2003: Eddy–zonal flow feedback in the Northern Hemisphere winter. J. Climate, 16, 12121227.

  • Luo, D., 2005: A barotropic envelope Rossby soliton model for block–eddy interaction. Part I: Effect of topography. J. Atmos. Sci., 62, 521.

    • Search Google Scholar
    • Export Citation
  • Luo, D., , A. R. Lupo, , and H. Wan, 2007: Dynamics of eddy driven low-frequency dipole modes. Part I: A simple model of North Atlantic Oscillations. J. Atmos. Sci., 64, 328.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., and Coauthors, 2001: North Atlantic climate variability: Phenomena, impacts and mechanisms. Int. J. Climatol., 21, 18631898.

    • Search Google Scholar
    • Export Citation
  • Mullen, S. L., 1987: Transient eddy forcing of blocking flows. J. Atmos. Sci., 44, 322.

  • Nakamura, H., , and J. M. Wallace, 1990: Observed changes in baroclinic wave activity during the life cycles of low-frequency circulation anomalies. J. Atmos. Sci., 47, 11001116.

    • Search Google Scholar
    • Export Citation
  • Nakamura, H., , M. Nakamura, , and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in blocking formation. Mon. Wea. Rev., 125, 20742093.

    • Search Google Scholar
    • Export Citation
  • Pan, L.-L., , F.-F. Jin, , and M. Watanabe, 2006: Dynamics of synoptic eddy and low-frequency flow interaction. Part III: Baroclinic model results. J. Atmos. Sci., 63, 17091725.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1990: A nonacceleration theorem for transient quasi-geostrophic eddies on a three-dimensional time-mean flow. J. Atmos. Sci., 47, 18251836.

    • Search Google Scholar
    • Export Citation
  • Qin, J. C., , and W. A. Robinson, 1992: Barotropic dynamics of interactions between synoptic and low-frequency eddies. J. Atmos. Sci., 49, 7179.

    • Search Google Scholar
    • Export Citation
  • Ren, H.-L., , F.-F. Jin, , J.-S. Kug, , J.-X. Zhao, , and J. Park, 2009: A kinematic mechanism for positive feedback between synoptic eddies and NAO. Geophys. Res. Lett., 36, L11709, doi:10.1029/2009GL037294.

    • Search Google Scholar
    • Export Citation
  • Ren, H.-L., , F.-F. Jin, , J.-S. Kug, , and L. Gao, 2011: Transformed eddy PV flux and positive synoptic eddy feedback onto low-frequency flow. Climate Dyn., 36, 23572370, doi:10.1007/s00382-010-0913-0.

    • Search Google Scholar
    • Export Citation
  • Rivière, G., , and I. Orlanski, 2007: Characteristics of the Atlantic storm-track eddy activity and its relation with the North Atlantic Oscillation. J. Atmos. Sci., 64, 241266.

    • Search Google Scholar
    • Export Citation
  • Robinson, W. A., 1991: The dynamics of low-frequency variability in a simple model of the global atmosphere. J. Atmos. Sci., 48, 429441.

    • Search Google Scholar
    • Export Citation
  • Robinson, W. A., 2000: A baroclinic mechanism for the eddy feedback on the zonal index. J. Atmos. Sci., 57, 415422.

  • Rogers, J. C., 1997: North Atlantic storm track variability and its association to the North Atlantic Oscillation and climate variability of northern Europe. J. Climate, 10, 16351647.

    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1983: The propagation of eddies in diffluent jetstreams: Eddy vorticity forcing of ‘blocking’ flow fields. Quart. J. Roy. Meteor. Soc., 109, 737761.

    • Search Google Scholar
    • Export Citation
  • Strong, C., , and G. Magnusdottir, 2008a: How Rossby wave breaking over the Pacific forces the North Atlantic Oscillation. Geophys. Res. Lett., 35, L10716, doi:10.1029/2008GL033578.

    • Search Google Scholar
    • Export Citation
  • Strong, C., , and G. Magnusdottir, 2008b: Tropospheric Rossby wave breaking and the NAO/NAM. J. Atmos. Sci., 65, 28612876.

  • Vautard, R., , B. Legras, , and M. Déqué, 1988: On the source of midlatitude low-frequency variability. Part I: A statistical approach to persistence. J. Atmos. Sci., 45, 28112843.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., , and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon. Wea. Rev., 109, 784812.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., , G.-H. Lim, , and M. L. Blackmon, 1988: Relationship between cyclone tracks, anticyclone tracks and baroclinic waveguides. J. Atmos. Sci., 45, 439462.

    • Search Google Scholar
    • Export Citation
  • Watanabe, M., 2009: Self-limiting feedback between baroclinic waves and a NAO-like sheared zonal flow. Geophys. Res. Lett., 36, L08803, doi:10.1029/2009GL037176.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and A. Barcilon, 1992a: Type B cyclogenesis in a zonally varying flow. J. Atmos. Sci., 49, 18771892.

  • Whitaker, J. S., , and A. Barcilon, 1992b: Genesis of mobile troughs in the upper westerlies. J. Atmos. Sci., 49, 20972107.

  • Woollings, T., , B. J. Hoskins, , M. Blackburn, , and P. Berrisford, 2008: A new Rossby wave-breaking interpretation of the North Atlantic Oscillation. J. Atmos. Sci., 65, 609626.

    • Search Google Scholar
    • Export Citation
  • Woollings, T., , A. Hannachi, , B. Hoskins, , and A. Turner, 2010: A regime view of the North Atlantic Oscillation and its response to anthropogenic forcing. J. Climate, 23, 12911307.

    • Search Google Scholar
    • Export Citation
  • Yu, J.-Y., , and D. L. Hartmann, 1993: Zonal flow vacillation and eddy forcing in a simple GCM of the atmosphere. J. Atmos. Sci., 50, 32443259.

    • Search Google Scholar
    • Export Citation
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