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  • View in gallery

    Scatter diagrams between the vertically integrated (from 1000 to 100 hPa) variances of low-frequency flow and the vertically integrated projections of the eddy forcing derived from the (a),(c) eddy vorticity flux forcing and (b),(d) PV flux forcing for MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH. Latitudes of the NH and SH are both taken as 30°–70°.

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    Correlations between the variances of low-frequency flow and the projections of the eddy forcing derived from the (a),(c) eddy vorticity flux forcing and (b),(d) eddy PV flux forcing for all seasons (black), MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH.

  • View in gallery

    Scatter diagrams with each dot indicating the vertically integrated (from 1000 to 100 hPa) EIG rate (day−1, the same hereafter) derived from the eddy PV flux forcing (x axis) and eddy vorticity flux forcing (y axis) during the period from January 1978 to February 2008 in the (a) NH and (b) SH.

  • View in gallery

    Vertical profiles of the climatological EIG rates derived from the (a),(c) eddy vorticity flux forcing and (b),(d) eddy PV flux forcing for MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH.

  • View in gallery

    Maps of the LGR derived from the eddy vorticity flux forcing (LGR2) and eddy PV flux forcing (LGR3) in the (left) NH and (right) SH for (top) 300, (middle) 500, and (bottom) 850 hPa using all months of data.

  • View in gallery

    As in Fig. 5, but for (top to bottom) MAM, JJA, SON, and DJF in the (left) NH and (right) SH, where LGR2 is shown at the 300-hPa level and LGR3 at the 850-hPa level.

  • View in gallery

    The zonal-averaged LGR derived from the (left) eddy vorticity flux forcing and (right) eddy PV flux forcing for (top to bottom) MAM, JJA, SON, and DJF.

  • View in gallery

    Seasonal variations of the (a),(b) climatological vertically averaged zonal-mean LGR and (c),(d) extratropical NH (30°–80°N)-mean LGR.

  • View in gallery

    Seasonal variations of the climatological LGR of LFV calculated over the (a),(b) NH and (c),(d) SH regions.

  • View in gallery

    Seasonal variations of the climatological vertically integrated LGR2 (blue) and LGR3 (red) averaged over the (a) NH (30°–80°N, 0°–360°E) and SH (30°–80°S, 0°–360°E), (b) Eurasia (30°–80°N, 30°–120°E), (c) North Pacific (30°–80°N, 120°E–100°W), and (d) North Atlantic (30°–80°N, 100°W–30°E). The dashed–dotted lines in (a) are for the SH.

  • View in gallery

    Seasonal variations of the (a) climatological 300-hPa zonal-mean and (b) extratropical NH (30°–80°N) mean storm-track statistics (m2 s−2).

  • View in gallery

    Seasonal variations of the climatological 300-hPa EKE (contours; m2 s−2) and zonal wind (shaded; m s−1), zonally averaged over regions defined in Fig. 10.

  • View in gallery

    Seasonal variations of the climatological 300-hPa EKE (black) and zonal wind (gray) averaged over the regions defined in Fig. 10. The dashed–dotted lines in (a) are for the SH.

  • View in gallery

    Seasonal variations of the climatological vertically integrated denominator (black lines) of Eq. (6) and the corresponding numerators of λ2 (open-circle lines) and λ3 (filled-circle lines).

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Eddy-Induced Growth Rate of Low-Frequency Variability and Its Mid- to Late Winter Suppression in the Northern Hemisphere

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  • 1 Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing, China
  • | 2 Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing, China, and Department of Meteorology, School of Ocean and Earth Sciences and Technology, University of Hawai‘i at Mānoa, Honolulu, Hawaii
  • | 3 School of Environmental Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea
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Abstract

Synoptic eddy and low-frequency flow (SELF) feedback plays an important role in reinforcing low-frequency variability (LFV). Recent studies showed that an eddy-induced growth (EIG) or instability makes a fundamental contribution to the maintenance of LFV. To quantify the efficiency of the SELF feedback, this study examines the spatiotemporal features of the empirical diagnostics of EIG and its associations with LFV. The results show that, in terms of eddy vorticity forcing, the EIG rate of LFV is generally larger (smaller) in the upper (lower) troposphere, whereas, in terms of eddy potential vorticity forcing, it is larger in the lower troposphere to partly balance the damping effect of surface friction. The local EIG rate shows a horizontal spatial distribution that corresponds to storm-track activity, which tends to be responsible for maintaining LFV amplitudes and patterns as well as sustaining eddy-driven jets. In fact, the EIG rate has a well-defined seasonality, being generally larger in cold seasons and smaller in the warmest season, and this seasonality is stronger in the Northern Hemisphere than in the Southern Hemisphere. This study also reveals a mid- to late winter (January–March) suppression of the EIG rate in the Northern Hemisphere, which indicates a reduced eddy feedback efficiency and may be largely attributed to the eddy kinetic energy suppression and the midlatitude zonal wind maximum in the midwinter of the Northern Hemisphere.

Corresponding author address: Dr. Hong-Li Ren, Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, 46 Zhongguancun Nandajie St., Haidian District, Beijing 100081, China. E-mail: renhl@cma.gov.cn

Abstract

Synoptic eddy and low-frequency flow (SELF) feedback plays an important role in reinforcing low-frequency variability (LFV). Recent studies showed that an eddy-induced growth (EIG) or instability makes a fundamental contribution to the maintenance of LFV. To quantify the efficiency of the SELF feedback, this study examines the spatiotemporal features of the empirical diagnostics of EIG and its associations with LFV. The results show that, in terms of eddy vorticity forcing, the EIG rate of LFV is generally larger (smaller) in the upper (lower) troposphere, whereas, in terms of eddy potential vorticity forcing, it is larger in the lower troposphere to partly balance the damping effect of surface friction. The local EIG rate shows a horizontal spatial distribution that corresponds to storm-track activity, which tends to be responsible for maintaining LFV amplitudes and patterns as well as sustaining eddy-driven jets. In fact, the EIG rate has a well-defined seasonality, being generally larger in cold seasons and smaller in the warmest season, and this seasonality is stronger in the Northern Hemisphere than in the Southern Hemisphere. This study also reveals a mid- to late winter (January–March) suppression of the EIG rate in the Northern Hemisphere, which indicates a reduced eddy feedback efficiency and may be largely attributed to the eddy kinetic energy suppression and the midlatitude zonal wind maximum in the midwinter of the Northern Hemisphere.

Corresponding author address: Dr. Hong-Li Ren, Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, 46 Zhongguancun Nandajie St., Haidian District, Beijing 100081, China. E-mail: renhl@cma.gov.cn

1. Introduction

In the extratropical stormy atmosphere, the slow-varying low-frequency flow with planetary-scale patterns is well developed in the monthly- and seasonal-mean fields. Over recent decades, much attention has been paid to such low-frequency variability (LFV) as it accounts for a large proportion of atmospheric variability and significantly impacts on global weather and climate. However, a fundamental question remains unanswered: how is the LFV maintained amid the transient atmospheric circulation? Previous studies have concluded that the LFV is internal to atmospheric dynamics. The internal dynamics in atmospheric models without external forcing can display significant LFV and the pure nonlinear eddy–mean flow interactions can generate a substantial amount of LFV (e.g., Held 1983; Egger and Schilling 1983; Hendon and Hartmann 1985; Cai and Mak 1990; Cai and van den Dool 1991; Lau and Nath 1991; Robinson 1991; Branstator 1995; Whitaker and Barcilon 1995; Limpasuvan and Hartmann 1999, 2000). The dynamical feedback between short-lived atmospheric synoptic eddies and low-frequency planetary-scale flow has long been recognized as essential to the maintenance of the extratropical atmospheric LFV (Kok et al. 1987; Lau 1988; Cai and Mak 1990).

Lau (1988) first found high covariations between monthly-mean flow anomalies and anomalies of variance fields of synoptic eddy flow and thus suggested a positive feedback between synoptic eddy and low-frequency flow (SELF). Subsequently, a number of studies have shown that the SELF feedback is vital for the generation and maintenance of zonal indices and dominant climate modes and that synoptic eddies are major supplier of energy to the LFV (e.g., Cai and van den Dool 1991; Lau and Nath 1991; Robinson 1991, 2000; Branstator 1992, 1995; Lee and Feldstein 1996; Limpasuvan and Hartmann 1999, 2000; Lorenz and Hartmann 2001, 2003).

To better understand the dynamics of the general SELF feedback, Jin et al. (2006a,b) and Pan et al. (2006) proposed a linear dynamical closure scheme in which the slowly varying part of eddy feedback can be expressed directly by low-frequency flow through a formulated linear operator. Following their studies, many diagnostic analyses revealed a left-hand preference in the direction of eddy fluxes relative to that of low-frequency flow (Kug and Jin 2009; Ren et al. 2009, 2011, 2012; Kug et al. 2010a,b). In particular, both eddy-vorticity and transformed eddy potential vorticity (TEPV) fluxes are preferentially directed to the left-hand side of the low-frequency flow, which shows that the left-hand preference is a convenient diagnostic indicator of the positive SELF feedback as it relates anomalies in the mean synoptic eddy forcing to mean-flow anomalies.

Recently, Jin (2010) proposed a theory of eddy-induced instability for LFV and predicted theoretically the eddy-induced growth (EIG) of LFV through a simple formula. Based on this theory, Ren et al. (2011) defined an empirical EIG rate to conveniently measure the efficiency of the positive SELF feedback. This EIG rate, originated from the dynamical closure and the new instability theory of the LFV, provides a good indicator for the nature of the two-way SELF interaction. It reflects the fact that only a portion of total synoptic eddy forcing is available as the energy of maintaining the LFV because the portion can be explained by the LFV itself. In this study, to further reveal implications of the EIG rate in quantifying the degree that maintenance or persistence of the general LFV (not only climate modes) can be extended by contributions of the positive synoptic eddy feedback, we will apply the EIG rate to quantitatively examine spatiotemporal features of the eddy-induced instability of the LFV and its associations with LFV.

In addition, the synoptic eddy feedback is intimately associated with the intensity of eddy activity. As revealed in previous studies, storm-track intensity in the North Pacific was observed as being suppressed in midwinter—namely, the phenomenon of the midwinter suppression of North Pacific storminess, which was first pointed out by Nakamura (1992) and has since attracted much research interest (e.g., Zhang and Held 1999; Chang 2001; Nakamura and Sampe 2002; Deng and Mak 2005; Park et al. 2010; Penny et al. 2010; Lee et al. 2013). Thus, we will also examine the relationship of the EIG with the storm-track activity as well as mean zonal flow in the climatological sense. The remainder of the paper is organized as follows. Section 2 is a brief introduction to the data and method used. Basic features of the empirical EIG rate are outlined in section 3 and horizontal spatial features of the local EIG rate are shown section 4. Main features of seasonal variations of the EIG are examined in section 5 and its possible associations with storm-track activity are in section 6. Finally, a summary and discussion are given in section 7.

2. Data and method

National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR v1) reanalysis data are used for monthly and daily-mean 30-yr datasets from January 1978 to February 2008 (Kalnay et al. 1996). The variables used are geopotential height, zonal wind, meridional wind, and potential temperature derived from air temperature. The fields on 12 normal pressure levels (1000–100 hPa) are used. Streamfunction and vorticity fields are calculated from zonal and meridional wind. The low frequency is defined as monthly mean. The boreal spring, summer, autumn, and winter seasons are taken as averages of March–May (MAM), June–August (JJA), September–November (SON), and December–February (DJF), respectively. The LFV is further defined in this paper as the monthly- or seasonal-mean anomalies of a variable. To separate the synoptic eddy component, the daily-mean fields are bandpass filtered using a 2–8-day Lanczos filter with 41 weights (Duchon 1979). Here, the so-called low-frequency eddy is excluded though it may indirectly contribute to the LFV (Zhang et al. 2012).

The focus of this study is on the projection of the eddy forcing onto the LFV to measure the efficiency of the synoptic eddy feedback. The eddy forcing can be expressed conventionally in terms of the geopotential tendencies induced by the anomalous eddy fluxes. In this paper, the geopotential tendencies, which are induced by anomalous eddy vorticity fluxes and eddy potential vorticity (PV) fluxes, can be defined by conducting the 2D and 3D Laplacian inversions, respectively, as follows:
e1
where is horizontal divergence operator, (or ) is Laplacian operator, and are the 2D and 3D inverse Laplacian operators, and f is the Coriolis parameter. The methods for these Laplacian inversions and calculations of the eddy fluxes can be referred to the paper of Ren et al. (2011). In Eq. (1), the anomalous eddy vorticity and eddy PV fluxes are defined as
e2
where , , , and denote the bandpass-filtered zonal wind, meridional wind, quasigeostrophic PV, and relative vorticity; denotes monthly-mean anomaly, defined as the deviation of a variable from its climatological mean at each month. In addition, the storm track here is expressed as monthly mean of .

3. Empirical EIG rate and its basic features

To quantitatively examine the efficiency of the synoptic eddy feedback onto the low-frequency flow, we use the empirical diagnostics developed by Ren et al. (2011) to measure the EIG of LFV for the eddy vorticity feedback and eddy PV feedback cases (refer to appendix for the details of the EIG rate definition), as follows:
e3
e4
Here, and are the geopotential tendency induced by the eddy vorticity forcing in terms of the 2D inversion and the eddy PV forcing in terms of the 3D inversion, respectively, as defined in Eq. (1), and the domain of horizontal integration S here is specified over the annular mid- and high-latitude areas (30°–70°N, 0°–360°E). In addition, we may define more kinds of EIG rate if other kinds of eddy-induced tendencies can be provided, such as those produced by linear or nonlinear models.

We applied Eqs. (3) and (4) at each vertical level to estimate the EIG rates and (also denoted as and ) for the eddy vorticity feedback and eddy PV feedback, respectively. It is noteworthy that the definitions here are exactly the same as those by Ren et al. (2011) even though they emphasized more the conception of TEPV flux, which is linearly transformed from the eddy PV flux. Therefore, we will uniformly use the expression of the EIG rate derived from the eddy PV flux forcing. Unit of the EIG rate is per day in this paper.

Figure 1 shows the relationship between the variance of LFV and the covariance of eddy forcing and LFV, which correspond to the denominator and numerator of the two definitions of the empirical EIG rate in Eqs. (3) and (4). It is clear in Fig. 1 that high correlations are between the denominator and numerator, which exist in both the NH and SH extratropics, although the correlations in the NH are higher than those in the SH. These high correlations indicate that the total variance of LFV may be well explained by the eddy-induced part of the LFV in the sense of its vertical average. The slope of the lines fitted to the data points in Fig. 1 actually represents the EIG rate according to its definition in the climate-mean sense.

Fig. 1.
Fig. 1.

Scatter diagrams between the vertically integrated (from 1000 to 100 hPa) variances of low-frequency flow and the vertically integrated projections of the eddy forcing derived from the (a),(c) eddy vorticity flux forcing and (b),(d) PV flux forcing for MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH. Latitudes of the NH and SH are both taken as 30°–70°.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

We further examine the relationships between the denominator and numerator at all the pressure levels and show vertical profiles of the correlations in Fig. 2; in general, high correlations are evident at most levels. The eddy-induced LFV that only involves the effect of eddy vorticity forcing has larger values at the upper levels, whereas the eddy-induced LFV that combines the effects of eddy vorticity and eddy heat forcing has larger values at the mid- to lower levels. Noting the seasonality in the relationships between the numerator and denominator in Fig. 1, we also show the vertical profiles of correlations for the different seasons in Fig. 2. Almost all seasons have positive correlations throughout all levels, indicating that the eddy-induced forcing that projects onto the low-frequency flow may play an important role in maintaining the variance of LFV. In addition, the correlations in warm seasons are lower, compared with cold seasons, probably reflecting a stronger SELF interaction during the latter than the former. However, it is interesting that the highest correlations do not appear in the boreal winter.

Fig. 2.
Fig. 2.

Correlations between the variances of low-frequency flow and the projections of the eddy forcing derived from the (a),(c) eddy vorticity flux forcing and (b),(d) eddy PV flux forcing for all seasons (black), MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

Figure 3 uses mass weighting to compare the vertically integrated and for all months to examine the coherence between them. The EIG rate in each month is almost always positive, reflecting the positive feedback of eddies onto LFV. A good relationship exists between the two growth rates in terms of the eddy vorticity and eddy PV forcing, respectively, and the correlation is stronger in the NH than in the SH. Such a good coherence between the two EIG rates indicates the predominance of the dynamical eddy vorticity feedback in the positive eddy feedback, particularly at upper level, and the eddy heat flux tends to favor the formation of a barotropic structure of the extratropical LFV (e.g., Lau and Nath 1991; Kug et al. 2010a). Moreover, as first noted by Ren et al. (2011), is typically smaller than , which is much clearer in the SH than in the NH. This implies that the forcing induced by the eddy heat flux may positively contribute to the eddy feedback although its effect tends to be largely canceled out by the vertical integration.

Fig. 3.
Fig. 3.

Scatter diagrams with each dot indicating the vertically integrated (from 1000 to 100 hPa) EIG rate (day−1, the same hereafter) derived from the eddy PV flux forcing (x axis) and eddy vorticity flux forcing (y axis) during the period from January 1978 to February 2008 in the (a) NH and (b) SH.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

To examine the seasonal dependency of the empirical EIG rates at different levels, Fig. 4 shows the vertical structures of the climatological-mean growth rates. All of the profiles are positive at all levels, and in all seasons, except at the 100-hPa level in Fig. 4b, indicating that the LFV catches a positive growth induced by synoptic eddies throughout the entire troposphere. The warmest season typically shows the minimum growth rate, indicating that it not only has a weak variance of LFV but also a weak EIG. This seasonal variation in the EIG rate appears to be much larger in the NH than in the SH, presumably because the SH has more oceanic areas to modulate a moderate seasonality. In addition, it is more interesting that the EIG rate in the NH cold season (DJF) is not at its maximum, whereas the SH has its maximum in the cold season (JJA).

Fig. 4.
Fig. 4.

Vertical profiles of the climatological EIG rates derived from the (a),(c) eddy vorticity flux forcing and (b),(d) eddy PV flux forcing for MAM (green), JJA (red), SON (yellow), and DJF (blue) in the (a),(b) NH and (c),(d) SH.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

The positive growth of LFV due to eddy feedback actually accords the left-hand preference that has been found in the directing of the eddy vorticity flux and TEPV flux (Kug et al. 2010a; Ren et al. 2011). As seen in Figs. 4a and 4c, the eddy vorticity–induced growth rate peaks at upper levels where the dynamical feedback dominates. In contrast, the eddy PV–induced growth rate (Figs. 4b and 4d) decreases with height, which can be essentially attributed to the offset effect of the eddy heat feedback. That is to say, the divergence (convergence) of eddy heat fluxes induces local cooling (warming) and downward (upward) secondary circulation, which tends to weaken the eddy vorticity feedback at upper levels but strengthen it at lower levels, and may contribute to inducing the barotropic structure of the low-frequency flow.

To further understand the role of the EIG in maintaining the LFV, it is necessary to clarify what the value of the EIG rate dynamically means. As seen in Figs. 3 and 4, the vertically integrated EIG rate is about 0.04–0.05, averaged at all seasons, corresponding to an e-folding time scale on the order of 20–25 days, which is consistent with the results of Lau (1988). This EIG is usually insufficient to overcome the linear damping, where the damping rate at the surface corresponds to the damping time scale of about 3.5–5 days (Jin 2010; Barnes and Hartmann 2010), but it can significantly reduce the damping from surface friction. For example, in cold seasons, the observed EIG rate, λ2 at the 300-hPa level or λ3 near the surface, reaches about 0.08–0.10, representing the time scale on the order of about 10–12 days. In contrast, the e-folding time scale of the dominant low-frequency variability; for example, the first four low-frequency modes as revealed by Simmons et al. (1983), has been estimated to be on the order of 7–15 days without external forcing effect. The intercomparison between these three time scales clearly indicates the significant contribution of the EIG to reducing the linear damping and thus surviving the LFV. Thus, it can be seen that the value of the EIG rate actually reflects the time scale of the LFV induced by eddies, which has obtained a quantitative support by the above-mentioned results.

As referred to in Eq. (A6), the contribution of the linear advection terms will be zero if we take integration over a global area or a regional domain without lateral boundary flow. In this case, the growth induced by the eddy and external forcing would together act to overcome the linear damping and enhance the LFV. However, the way to define the empirical EIG rate may not be suitable to an estimation of the growth by the external forcing only based on an observational analysis but without numerical solution. Jin et al. (2006a) and Pan et al. (2006) quantitatively demonstrated the fundamental importance of the eddy forcing, compared with the external forcing, to the Pacific–North America (PNA)-like pattern by numerically solving their linearized barotropic and baroclinic models with the eddy feedback operators. Once the feedback of eddy is closed, the amplitude of the PNA-like pattern in extratropics as a response to the tropical external forcing will be more than 50% weakened. This clearly suggests the important role of the EIG to the LFV maintenance through the way that the synoptic eddy forcing not only directly sustains the LFV but also forms a singularity condition favoring the effective response of extratropical atmosphere to the tropical external forcing (Jin 2010).

4. Spatial features of the local growth rate

The differences between the results from the two hemispheres shown in Figs. 14 imply a geographical dependency of the EIG of LFV. To examine the horizontal spatial features of the EIG rate, we devise a localized approach based on Eqs. (3) and (4):
e5
where j = 2 or 3, Sf denotes a spatial definition operator with a sliding window. Here, to estimate the local growth rate (LGR) of LFV, the size of the sliding window is taken as 40° longitude × 20° latitude, by which the relatively stable results can be obtained.

Figure 5 shows distributions of the climate-mean LGR. Similar patterns of the LGR appear on the upper, middle, and lower levels, and the amplitudes of the patterns change with height—consistent with what is shown in Fig. 4. The EIG rate is almost always positive over all extratropical regions. In the mid- to high latitudes, the LGR patterns reach a meridional maximum and are mainly centered over oceanic regions, which are similar to the distribution of the midlatitude storm tracks (e.g., Chang et al. 2002). It is noted that the LGR has an essentially zonally annular structure, which may be linked with the NH and SH annular modes because their maintenance is intimately associated with the synoptic eddy feedback (e.g., Lorenz and Hartmann 2001, 2003). The annular patterns of LGR, which are more symmetric in the SH than in the NH, further confirm that the EIG plays a key role in forming the two dominant hemispheric annular modes. Moreover, the growth rate reaches the regional maximum over the North Pacific and North Atlantic, where synoptic eddy activity is the strongest and several climate modes dominate. Overall, the LGR has climatological spatial patterns and regional centers that are fairly similar to those of the variances of LFV, referred to in Fig. 9b of Jin (2010), indicating that the EIG plays an important role in maintaining the patterns and amplitudes of LFV. Also, it is noteworthy that the major centers of and at the upper level appear be located downstream of the corresponding centers at the lower level.

Fig. 5.
Fig. 5.

Maps of the LGR derived from the eddy vorticity flux forcing (LGR2) and eddy PV flux forcing (LGR3) in the (left) NH and (right) SH for (top) 300, (middle) 500, and (bottom) 850 hPa using all months of data.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

Figure 6 shows how the LGR varies with seasons where the 300- and 850-hPa levels are taken as representing and , respectively. In the NH, the LGR has similar patterns and amplitudes in spring, autumn, and winter, whereas summer is distinct, with considerably poleward-contracted patterns and relatively small amplitudes. In the SH, the austral summer is also characterized by poleward-contracted patterns and relatively small amplitudes when compared with the other seasons. Overall, the warmest season has a much weaker EIG of LFV than the cold seasons. Nevertheless, the climatic LGR is positive in all four seasons, indicating a positive eddy feedback onto the local LFV. Moreover, in Fig. 6 the major LGR centers at the upper level tend to be downstream of those at the lower level. Based on the theory of Jin (2010), this is presumably due to the observed fact that the midlatitude major storm-track centers at the upper level are usually located downstream of the centers at the lower level (not shown).

Fig. 6.
Fig. 6.

As in Fig. 5, but for (top to bottom) MAM, JJA, SON, and DJF in the (left) NH and (right) SH, where LGR2 is shown at the 300-hPa level and LGR3 at the 850-hPa level.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

Here, we take the North Atlantic region as an example to illustrate the time scale reflected by the EIG rate that has the largest values in this region. As seen in Figs. 5 and 6, the EIG rate over the region can reach about 0.14–0.16, corresponding to the time scale on the order of 6–7 days, which means the strongest synoptic eddy feedback in the globe. That is the major reason that the North Atlantic Oscillation (NAO), the dominant climate model in the NH, is gestated in this region. Barnes and Hartmann (2010), based on a vorticity budget analysis using observations, showed that the upper-tropospheric eddy vorticity forcing act to counteract the effect of the large-scale advection terms at the upper level and to indirectly maintain the low-level vorticity anomalies against frictional drag through generating a large-scale vertical circulation cell. They further demonstrated that the eddy forcing contributes to a significant extension of the persistence of the NAO vorticity anomaly owing to the fact that the decay time scale of the NAO can be extended from 3.5 to 13.2 days as a result of positive synoptic eddy feedback. In the similar way, our results show that the decay time scale of the North Atlantic LFV can be extended from 3.5–5 to 8–20 days to the biggest degree because of the eddy feedback.

To present comprehensively the vertical structure and seasonality of the LGR, we here take zonal means of the patterns in Fig. 6. As seen in Fig. 7, the EIG rate of LFV is mostly positive in the extratropics but weakly negative in the tropics. The dynamical eddy vorticity feedback contributes to a strongly positive LGR at upper levels and in the subpolar regions, serving to maintain the so-called eddy-driven jet (e.g., Athanasiadis et al. 2010). In contrast, the eddy PV feedback generates a downward positive LGR by involving the eddy heat forcing effect and induced vertical secondary circulation, which acts to partly balance surface friction damping and contributes to generating a barotropic structure in the LFV (e.g., Ren et al. 2011). In fact, and become similar when they are taken as vertical integrations (not shown). It is also evident that the NH shows a stronger seasonality in the LGR amplitude and meridional location than the SH; that is, the LGR centers become weaker and more poleward during the warm season.

Fig. 7.
Fig. 7.

The zonal-averaged LGR derived from the (left) eddy vorticity flux forcing and (right) eddy PV flux forcing for (top to bottom) MAM, JJA, SON, and DJF.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

5. Seasonal variations of the EIG rate and its mid- to late winter suppression in the NH

It is well known that the LFV usually has a larger amplitude in colder seasons, as indicated in Fig. 1 and, hence, a higher EIG rate would be expected. However, a comparison of Figs. 4 and 7 shows that the highest EIG rate in the NH does not occur in winter, which differs from the situation in the SH. For example, as seen in Fig. 7, the amplitude of the EIG rate mostly maximizes in spring or autumn for either of the EIG rates. That is to say, the EIG rate of LFV in the NH might be subject to suppression in winter, in the sense of seasonal variation, even though, at the same time, the LFV reaches its maximum amplitude and projection of the eddy forcing onto it, as revealed in Fig. 1.

To confirm that the boreal EIG rate tends to be suppressed in the winter, Fig. 8 shows the seasonal variations of the zonal-mean monthly climatological LGR, which is vertically averaged throughout the troposphere. It is most interesting to see that the NH growth rate peaks in autumn to early winter and spring, while, in contrast, the SH growth rate essentially peaks during austral winter, which is the case for both of the EIG rate definitions. Furthermore, seasonal variations of the mid- to high-latitude-averaged growth rate in the NH experience a general mid- to late winter (January–March) suppression (MLWS), which is well defined in the North Pacific, Eurasian, and North Atlantic regions. In addition, the and patterns are similar, but the latter is about one-third larger than the former. It is also noted that the SH EIG rate has a much smaller seasonal variation than in the NH, particularly in .

Fig. 8.
Fig. 8.

Seasonal variations of the (a),(b) climatological vertically averaged zonal-mean LGR and (c),(d) extratropical NH (30°–80°N)-mean LGR.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

To confirm the observed MLWS phenomenon, Fig. 9 presents the extratropics-averaged LGR from the vertical levels for the NH and SH, and at all levels the EIG rate in the SH reaches a maximum during the austral winter despite some other weaker peaks being present, whereas the situation in the NH is very different. That is, the NH EIG rate at all levels tends to peak in autumn to early winter and springtime and lessen in the mid- to late winter when the rate may have been expected to peak as it does in the SH. We also further examine the contributions from different sectors in the NH to the MLWS in the EIG of LFV, such as the LGR averaged over the North Pacific, Eurasian, and North Atlantic regions (not shown). The North Pacific sector shows the most similarity to the EIG patterns of the NH, indicating its dominant contribution to the NH MLWS in the EIG of LFV. The North Atlantic sector makes a positive, but relatively weak contribution to the MLWS, and the Eurasian sector, in spite of smaller values, still shows the MLWS, interestingly, with the minimum EIG rate in mid- to late winter rather than in summer.

Fig. 9.
Fig. 9.

Seasonal variations of the climatological LGR of LFV calculated over the (a),(b) NH and (c),(d) SH regions.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

The major features of the seasonal variations in the EIG are highlighted by the vertically integrated LGR curves shown in Fig. 10. All of the three main sectors in the NH extratropics (i.e., North Pacific, Eurasia, and North Atlantic) contribute positively to the observed NH MLWS phenomenon. The weakest EIG of LFV in the extratropics occurs during the Eurasian winter, which is induced by the upper-level eddy vorticity forcing, as also seen in Fig. 8. In addition, the suppression of EIG in the North Pacific tends to occur in midwinter, while that in the North Atlantic occurs in late winter. Another prominent feature of the NH EIG rate curves is that the largest values are always anchored in April, compared with the broad range of September–December peaks.

Fig. 10.
Fig. 10.

Seasonal variations of the climatological vertically integrated LGR2 (blue) and LGR3 (red) averaged over the (a) NH (30°–80°N, 0°–360°E) and SH (30°–80°S, 0°–360°E), (b) Eurasia (30°–80°N, 30°–120°E), (c) North Pacific (30°–80°N, 120°E–100°W), and (d) North Atlantic (30°–80°N, 100°W–30°E). The dashed–dotted lines in (a) are for the SH.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

6. Possible causes of the MLWS of EIG

The MLWS phenomenon in the EIG of the NH LFV presumably develops because the coldest season usually has the largest amplitude LFV, which may reduce the EIG rate in terms of its definition, even though the relevant eddy forcing and its projection onto the LFV are also the largest, as indicated by Fig. 1 (and further, more clearly shown in Fig. A1). This probably indicates that the efficiency of the positive eddy feedback is dependent on the amplitude of the LFV. Watanabe (2009) found a nonlinear self-limitation of the positive eddy feedback to the North Atlantic Oscillation–like zonal flow anomalies in eddy life cycle experiments, where the dipole-type zonal-flow anomalies with extremely large amplitude cannot be efficiently intensified by the baroclinic wave feedback. This means that the eddy feedback may be suppressed nonlinearly when the amplitude of the LFV increases to some degree. Our current observations strongly reflect such a self-limitation in the NH but not in the SH. Consequently, we must consider what the possible causes of the MLWS of the EIG may be.

To address this question, it is naturally thought that the synoptic eddy feedback is closely related to the mid- to high-latitude eddy activity intensity, which is usually referred to as the storm track. In other words, the seasonal variations in storm activity can influence those of the synoptic eddy feedback. Similar to Fig. 8a, in Fig. 11 the storm-track statistics () show that the zonal-mean storm-track intensity peaks in autumn to early winter and spring are similar to the seasonal variations of the zonal-mean EIG rate in the NH. The longitudinal distributions of seasonal storm-track variations in Fig. 11b further reflect the phenomenon of midwinter suppression of North Pacific storm-track activity. However, our results show that such a midwinter suppression of storminess not only occurs in the North Pacific but also in the Eurasian region, to which Ren et al. (2010) noted a similar situation in the East Asia. It is even possible to identify a weak signal of the late winter suppression in the North Atlantic. Therefore, the MLWS of the EIG may be attributed to the midwinter suppression of storm-track activity in the NH.

Fig. 11.
Fig. 11.

Seasonal variations of the (a) climatological 300-hPa zonal-mean and (b) extratropical NH (30°–80°N) mean storm-track statistics (m2 s−2).

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

To further understand the possible dynamical causes of the observed MLWS of the NH EIG, the theory proposed by Jin (2010) regarding the eddy-induced instability of LFV is conceptually employed here. His study proved theoretically that EIG is intimate to the climatological eddy kinetic energy (EKE) and zonal-mean zonal flow under certain conditions, as described by his Eq. (8):
e6
where is the climatological mean of the synoptic EKE, is the summation of the typical synoptic eddy autodecay rate, is a constant on the order of 10 m s−1, and is a group velocity for synoptic Rossby waves, which depends on the zonal-mean zonal flow and is on the order of 10–20 m s−1. It is shown in Eq. (6) that the EIG rate tends to be directly proportional to the synoptic EKE and inversely proportional to the zonal-mean zonal flow squared. Note that as this equation was derived by Jin (2010) based on a linear dynamically closed barotropic model for the low-frequency flow, we consequently use either the 300-hPa or vertically averaged EIG rate and eddy statistics from Fig. 8 to the following figures. Our purpose is to make that the observed empirical EIG rate features can be well captured by Eq. (6) in a barotropic sense.

For attribution of the MLWS of the EIG in terms of Eq. (6), Fig. 12 presents the zonal-mean extratropical EKE and zonal flow over the different regions. It shows that the climatological EKE at mid- to high latitudes in the different NH regions all tends to be at a secondary minimum (to different extents) in mid- to late winter, around February. This feature is particularly clear in North Pacific, which is coincident with the occurrence of the midwinter suppression of storminess. It indicates that the midwinter suppression of the synoptic EKE in the NH may be the dominant factor controlling the MLWS of the EIG of LFV in the NH. In contrast, the climatological zonal flow in Fig. 12 normally peaks in the coldest months and can also contribute to the MLWS of the EIG since the growth rate in Eq. (6) is inversely proportional to the mean flow squared via . In addition, as Eq. (6) implies, the EIG rate may be dependent upon the eddy autodecay rate as well; that is, a smaller growth rate will be predicted as long as eddies in mid- to late winter have a longer lifespan. This remains an open question and requires further study.

Fig. 12.
Fig. 12.

Seasonal variations of the climatological 300-hPa EKE (contours; m2 s−2) and zonal wind (shaded; m s−1), zonally averaged over regions defined in Fig. 10.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

Figure 13 further extracts the mid- to high-latitude means of the EKE and zonal flow over the different regions to highlight the fact that the EKE reaches its minimum and the zonal wind reaches its maximum around February, both of which favor generation of a mid- to late winter suppression of the EIG rate. That is, both the weakening of eddy activity and the strengthening of group velocity of synoptic Rossby waves can yield the suppression of the efficiency of SELF interaction. However, the LFV intensity reaches the maximum in mid- to late winter, although, meanwhile, the covariance between eddy forcing and LFV reaches the maximum in the NH. Evidently, the former has a larger speed to grow up than the latter, which may be due to the fast-increasing contributions of external forcing in low latitudes. In contrast, the storm tracks or EKE are somehow weakened and then the eddy forcing becomes weaker, suggesting that the efficiency of the SELF feedback on the LFV (viz., the EIG rate) will be smaller.

Fig. 13.
Fig. 13.

Seasonal variations of the climatological 300-hPa EKE (black) and zonal wind (gray) averaged over the regions defined in Fig. 10. The dashed–dotted lines in (a) are for the SH.

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

Indeed, many of mechanisms have been raised in previous studies for explaining the occurrence of the midwinter suppression of storminess in the North Pacific based on diagnosing data from observation and climate models. These mechanisms have reflected different aspects of dynamics, mainly including advection of the storminess suppression by strong westerly jets (Nakamura 1992; Harnik and Chang 2004), diabatic heating effect (Chang 2001), trapping effect of higher-altitude baroclinic waves by stronger subtropical jet (Nakamura and Sampe 2002), weakened eddy seeding–feeding processes associated with downstream development (Chang 2001; Zurita-Gotor and Chang 2005), the barotropic wind shear effect (Deng and Mak 2005), and impacts of the central Asian mountains and Tibetan Plateau (Park et al. 2010; Lee et al. 2013).

It is interesting that these mechanisms for explaining the midwinter suppression of storm-track intensity in North Pacific may provide more clues to interpret the MLWS in the EIG. Particularly, these mechanisms can be used to explain the suppression of the EKE, which is the controlling factor for the MLWS of the EIG. However, Chang (2001) also suggested the enhanced group velocity of eddy propagation as a result of the strong jet may contribute to a slower downstream growth of eddy energy over the North Pacific, which further yields an interannual variability in the reduction of the storm-track intensity. In contrast, we may find a similar story in Eq. (6): the climatologically strongest westerly jet in midwinter corresponds to the fastest group velocity of synoptic Rossby waves in the meanwhile, which tends to generate a relatively slow downstream growth of eddy energy over North Pacific and hence the midwinter suppression of storm-track intensity. Also, the zonal-mean EKE and zonal wind curves in the SH have been put into Fig. 13a, which, similar to those of the EIG rate, does not exhibit a suppression of the EKE in the austral midwinter. According to the effects of big mountains on the midwinter suppression, the presumable suspects for the differences between the hemispheres are more land–sea contrast and pronounced topography of the NH than the SH. Further demonstrations for the mechanism implied in Eq. (6) are still needed in future studies.

7. Summary and discussion

A number of past studies have also concluded that the SELF feedback plays an indispensable role in reinforcing LFV, as outlined in the introduction to this paper. The feedback that is induced by synoptic eddies can contribute to the growth or instability of the LFV (Jin et al. 2006a,b; Pan et al. 2006; Jin 2010). Therefore, how to quantitatively measure the EIG or instability of the LFV, caused by the SELF feedback, is an important issue. In this paper, we have completed a fundamental study with the aim of examining the climatological spatial features and seasonal variations of the empirical EIG rate, which is able to quantify the efficiency of the SELF feedback by projecting the eddy-induced forcing onto the LFV.

Our results show that the eddy forcing projected onto the LFV, and the variability of the LFV, are closely related in the extratropics, and their ratio has been defined as the empirical EIG rate of LFV. Generally speaking, the EIG rate, which is derived from the eddy vorticity flux forcing, is dominant in the upper troposphere, whereas the EIG rate, which is derived from the eddy PV flux forcing, is dominant in the lower troposphere. This downward redistribution of the EIG rate is mainly due to the eddy heat flux effect, which is important to balance the surface frictional damping to some degree, and may also contribute to the barotropic structure of LFV. Further results show that the local growth rate has climatological spatial patterns and regional centers that are similar to those of the LFV variances, indicating that the EIG plays an important role in maintaining the patterns and amplitudes of LFV. The vertical structures of the LGR reflect the contributions that the eddy-induced growth makes to both sustaining the eddy-driven jets and overcoming the surface friction. The SH tends to have a more uniform distribution of the local EIG rate than that in the NH, which may favor the more zonally symmetric annular mode in the SH than in the NH.

The EIG rate of LFV shows a general seasonality, with a maximum in cold seasons and minimum in the warmest season, which may explain why the warmest season has a lower-amplitude LFV when compared with the colder seasons. This seasonality appears stronger in the NH than in the SH and may account for the seasonality in the variability of LFV to some degree. Further results show that the seasonal variations of the EIG rate in the NH are more complicated than those in the SH. The EIG rate of LFV is subject to suppression in the NH mid- to late winter, although this is also the time when the LFV experiences its maximum amplitude and projection of eddy forcing onto it. This mid- to late winter suppression of the EIG in the NH essentially represents suppression of eddy feedback efficiency and may be attributed largely to the midwinter suppression of the storm-track activity (EKE) in the NH, probably implying a nonlinear self-limitation of the eddy feedback for large-amplitude LFV.

However, the attribution of the observed MLWS of the EIG remains an open question. As discussed in the context of Eq. (6), the eddy autodecay rate may play a role in suppressing the mid- to late winter EIG. In addition, the dynamics associated with the baroclinic eddy equilibration during the boreal winter (Zhang and Stone 2010) may offer an alternative interpretation of the MLWS of the eddy feedback efficiency. Furthermore, the good agreement between the seasonal variations of the empirical EIG rate and the EKE indicate that the former may be expected to agree well with the theoretical EIG rate as predicted by the theory of eddy-induced instability of LFV proposed by Jin (2010), which would validate this theory to a great degree. All of these points will be considered further in future studies. In addition, note the fact that the eddy vorticity and heat fluxes can make different contributions to the EIG of the LFV; we may distinguish the different roles of the two kinds of eddy fluxes by examining the growth rate λ3 generated from the eddy vorticity part and from the eddy heat part in the eddy PV flux, respectively. Also, we may further examine the roles of the eddy-induced vertical secondary circulation in contributing eddy feedback by comparing the difference between λ2 and λ3 derived from the eddy vorticity flux forcing. These will be further studied in the future.

This study directly quantified the features of the positive synoptic eddy feedback onto the LFV in extratropics. On one hand, the positive EIG provided by eddies, serving as an important energy source for LFV, may substantially offset the natural frictional and thermal damping. Thus, an improved understanding of the SELF feedback, as indicated in this study, may enable us to better utilize the systematic low-frequency information. By diagnosing the EIG rate, such low-frequency information can be gleaned from transient synoptic eddies and eventually increase our ability to improve monitoring and prediction of climate variability in the mid- to high latitudes. On the other hand, the EIG rate, as the diagnostic to measure the efficiency of the SELF interaction, also represents the degree that the persistence of the LFV can be extended by the positive eddy feedback. Indeed, the relative importance of the EIG in reinforcing and maintaining LFV, compared to other factors, such as linear advection terms, external forcing, low-frequency eddy, and so on, remains an open question notwithstanding some previous studies.

Acknowledgments

This work is jointly supported by National Science foundation (NSF) Grant of China (41375062), China Meteorological Special Projects (GYHY201406022, GYHY201206016), U.S. NSF Grant (ATM 1034439), and 973 Program Grant of China (2010CB950404). JSK was supported by the National Research Foundation of Korea (Grant NRF-2009-C1AAA001-2009-0093042) funded by the South Korean government (MEST).

APPENDIX

Eddy-Induced Growth Rate of Low-Frequency Variability

As usual, the geopotential tendency induced by the eddy vorticity flux forcing and the eddy PV flux forcing, respectively, can be expressed as follows:
ea1
ea2
Here, the anomalous eddy fluxes and their induced anomalous geopotential tendencies are the same as the definitions in Eqs. (1) and (2). Equation (A1) indicates that the tendency is estimated from the eddy vorticity forcing based on the vorticity equation framework, and Eq. (A2) indicates that the tendency is obtained from the eddy PV forcing under the quasigeostrophic PV equation framework, referring to the paper of Lau and Holopainen (1984). Using either the 2D or 3D inversion operator is just dependent on the purpose to do diagnosis, as many previous studies did (e.g., Lau 1988; Lau and Nath 1991; Kug et al. 2010a). In this paper, we, still along this line, aim to define the two kinds of EIG rate using the 2D and 3D Laplacian inversions, respectively.
Lau (1988) demonstrated the close association between anomalies in the monthly-mean-flow field and systematic anomalies in the monthly-mean eddy statistics such as the eddy forcing. Jin et al. (2006a) further hypothesized that part of the eddy forcing can be directly related to the low-frequency flow via a dynamical closure between the two fields, which, in a barotropic framework, can be expressed as
ea3
where Lf is the linear operator representing synoptic eddy feedback. Therefore, a linear barotropic geopotential equation, similar to the version as derived by Jin et al. (2006a), can be expressed as
ea4
Here, L is the linear advection operator, r the damping coefficient (<0), and is the external forcing. Based on the dynamic closure, Jin (2010) proposed a theory on the eddy-induced instability of low-frequency variability and formulated a theoretical eddy-induced growth rate by which the contribution of Lf can be simply expressed as a constant.
Motivated by abundant observed evidence for the positive eddy feedback, Ren et al. (2011) assumed the linear relationship between the eddy vorticity–induced forcing and low-frequency flow as
ea5
where is a linear coefficient for the bulk growth rate (>0) or damping rate (<0), representing the positive or negative eddy feedback, and R2 is the residual term, representing the remainder of eddy forcing that is not explained by low-frequency flow. If we approximately take R2 as zero, then Eq. (A4) can be changed into
ea6
It is indicated in Eq. (A6) that the total derivative of low-frequency flow is determined by the contrast between r and , given that .
Then, it can be similarly assumed that
ea7
where represents the growth rate (>0) or damping rate (<0) induced by the eddy PV forcing, and R3 is the residual term.
To quantify the eddy-induced growth for the low-frequency flow, as written in Eqs. (A5) and (A7), Ren et al. (2011) adopted an empirical expression as follows:
ea8
where j = 2 and 3 denotes the eddy vorticity and eddy PV feedback cases, respectively. It is clearly seen in the definition equation that the EIG rate will be positive (negative) when the eddy forcing pattern is in phase (out of phase) with the LFV pattern, which is actually used as a diagnostic to measure the SELF feedback efficiency instead of its intensity. Moreover, the maintenance of the LFV generated from the eddy forcing also expresses to what degree the persistence of the LFV can be extended by the positive eddy feedback. In addition, we here show the seasonal variations of the climatological vertically integrated denominator of Eq. (A8) and the corresponding numerators of λ2 and λ3, as shown in Fig. A1. It is quite clear that both the denominator and numerators are the biggest during February in the NH and during July in the SH.
Fig. A1.
Fig. A1.

Seasonal variations of the climatological vertically integrated denominator (black lines) of Eq. (6) and the corresponding numerators of λ2 (open-circle lines) and λ3 (filled-circle lines).

Citation: Journal of the Atmospheric Sciences 71, 7; 10.1175/JAS-D-13-0221.1

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