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    Latitudinal gradient of the PV (10−9 m−1 s−1) in a composite field of NAO events at 300 hPa that filters out synoptic-scale waves for negative and positive phases as shown in Luo et al. (2007b, their Fig. 10), in which the dashed and solid lines represent the negative (shaded) and positive values of the PV gradient, respectively: (a) negative-phase event and (b) positive-phase event.

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    (Continued)

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    The 300-hPa unfiltered geopotential height fields of 2 NAO events: (a) a negative-phase event from 25 Jan to 13 Feb 1963 and (b) a positive-phase event from 12 to 27 Jan 1975.

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    (Continued)

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    Latitudinal gradient of the total PV (10−9 m−1 s−1) for the unfiltered geopotential height fields at the 300-hPa level of the corresponding NAO events as shown in Fig. 2, in which the dashed and solid lines represent the negative (shaded) and positive values of the PV gradient, respectively: (a) negative-phase event and (b) positive-phase event.

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    (Continued)

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    Planetary-scale fields of eddy-driven negative-phase NAO life cycles for a0 = 0.17: (a) case without the effect of TPW (h0 = 0) and (b) case with the effect of TPW (h0 = 0.4). Contour interval (CI) is 0.15.

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    (Continued)

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    Total fields of eddy-driven negative-phase NAO events in the presence of the TPW fortwo types (a0) of preexisting eddy amplitudes (CI = 0.15): (a) a0 = 0.17 and (b) a0 = 0.12.

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    (Continued)

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    Meridional gradient of the planetary-scale PV of eddy-driven negative-phase NAO events corresponding to those in Fig. 4 (CI = 0.5), in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case without TPW and (b) case with TPW.

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    (Continued)

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    Meridional gradient of the TPV for an eddy-driven negative-phase NAO event in Fig. 5a, in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case with the feedback of an NAO anomaly (CI = 4) and (b) case without the feedback of an NAO anomaly (CI = 1).

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    (Continued)

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    Planetary-scale fields of eddy-driven positive-phase NAO life cycles for a0 = 0.17 (CI = 0.15): (a) case without TPW (h0 = 0) and (b) case with TPW (h0 = −0.4).

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    (Continued)

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    Total fields of the corresponding eddy-driven positive-phase NAO events as shown in Fig. 8: (a) case without TPW (h0 = 0) and (b) case with TPW (h0 = −0.4).

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    (Continued)

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    Meridional gradient of the planetary-scale PV of the corresponding eddy-driven positive-phase NAO events shown in Fig. 8 (CI = 0.5), in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case without the effect of TPW and (b) case with the effect of TPW.

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    (Continued)

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    Latitudinal gradient of the TPV for eddy-driven positive-phase NAO events in Fig. 9, in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case with the feedback of a NAO anomaly (CI = 4) and (b) case without the feedback of a NAO anomaly (CI = 1).

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    (Continued)

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    Meridional profile of the maximum mean westerly wind during the NAO life cycle, in which the dotted–dashed curve denotes the case without TPW as shown in Fig. 4a or 8a; the dotted–long-dashed curve represents the case of a positive-phase NAO with the effect of TPW; and the solid curve corresponds to the case of a negative-phase NAO with the effect of TPW.

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    Horizontal distribution of −(∂[uLυL]/∂y) for negative- and positive-phase NAO events for the same parameters as in Figs. 4b and 8b (CI = 0.05), in which the solid and dashed curves represent the positive and negative values of −(∂[uLυL]/∂y), respectively: (a) negative-phase event and (b) positive-phase event.

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    (Continued)

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    Initial fields of planetary wave (ψNAO), synoptic wave (ψ1), and eddy forcing [−J(ψ1, ∇2ψ1)P] that allow the NAO events to occur for (a)–(c) negative and (d)–(f) positive phases for the same parameters as in Figs. 4a and 8a, respectively: (a), (d) ψNAO field (CI = 0.2), in which the dashed and solid curves denote the negative and positive anomalies, respectively; (b), (e) ψ1 field (CI = 0.1), in which the dashed and solid lines correspond to the cyclone and anticyclone, respectively; and (c), (f) −J(ψ1, ∇2ψ1)P field (CI = 0.05), in which the dashed and solid lines represent the anticyclonic and cyclonic forcing, respectively.

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Dynamics of Eddy-Driven Low-Frequency Dipole Modes. Part IV: Planetary and Synoptic Wave-Breaking Processes during the NAO Life Cycle

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  • 1 Physical Oceanography Laboratory, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, China
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Abstract

Based on a highly idealized, analytical solution of the North Atlantic Oscillation (NAO) derived in Part III of this series, it is shown that wave breaking is not a necessary condition for the occurrence of NAO events. The breaking of synoptic waves can arise from the interaction between planetary and synoptic waves that gives rise to NAO events, and the type of wave breaking is dominated by the initial conditions of the two waves that determine the phase of the NAO. The planetary wave breaking (PWB) seems to be attributed to an amplification of the NAO amplitude. It is further found that both the planetary wave breaking and the cyclonic (anticyclonic) breaking of synoptic waves undergo an in-phase (out phase) evolution during the life cycles of negative (positive) phase NAO, or NAO (NAO+), events. An interesting result found is that for NAO (NAO+) events the breaking of synoptic waves is enhanced (weakened) during the growing phase, but is weakened (enhanced) during the decaying phase.

In the absence of a topographic planetary wave (TPW), PWB occurs mainly in the midlatitude regions of the Atlantic basin for NAO events, but is concentrated in subtropical and subpolar regions for NAO+ events. However, once the TPW is involved, the reversed planetary-scale potential vorticity (PV) gradient that characterizes the PWB exhibits a southwest–northeast (southeast–northwest) tilted tripole for NAO (NAO+) events, in agreement with the diagnostic results presented herein. The PWB in the subtropical Atlantic is found to occur more frequently for NAO+ events than for NAO events because the weaker subtropical mean flow is more likely to emerge during the NAO+ life cycle. In conclusion, the results of the highly idealized model used here appear to show that the PWB, synoptic wave breaking, and meridional shift of the westerly jet may be different descriptions of the NAO phenomenon.

Corresponding author address: Dr. Dehai Luo, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao 266003, China. Email: ldh@ouc.edu.cn

Abstract

Based on a highly idealized, analytical solution of the North Atlantic Oscillation (NAO) derived in Part III of this series, it is shown that wave breaking is not a necessary condition for the occurrence of NAO events. The breaking of synoptic waves can arise from the interaction between planetary and synoptic waves that gives rise to NAO events, and the type of wave breaking is dominated by the initial conditions of the two waves that determine the phase of the NAO. The planetary wave breaking (PWB) seems to be attributed to an amplification of the NAO amplitude. It is further found that both the planetary wave breaking and the cyclonic (anticyclonic) breaking of synoptic waves undergo an in-phase (out phase) evolution during the life cycles of negative (positive) phase NAO, or NAO (NAO+), events. An interesting result found is that for NAO (NAO+) events the breaking of synoptic waves is enhanced (weakened) during the growing phase, but is weakened (enhanced) during the decaying phase.

In the absence of a topographic planetary wave (TPW), PWB occurs mainly in the midlatitude regions of the Atlantic basin for NAO events, but is concentrated in subtropical and subpolar regions for NAO+ events. However, once the TPW is involved, the reversed planetary-scale potential vorticity (PV) gradient that characterizes the PWB exhibits a southwest–northeast (southeast–northwest) tilted tripole for NAO (NAO+) events, in agreement with the diagnostic results presented herein. The PWB in the subtropical Atlantic is found to occur more frequently for NAO+ events than for NAO events because the weaker subtropical mean flow is more likely to emerge during the NAO+ life cycle. In conclusion, the results of the highly idealized model used here appear to show that the PWB, synoptic wave breaking, and meridional shift of the westerly jet may be different descriptions of the NAO phenomenon.

Corresponding author address: Dr. Dehai Luo, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao 266003, China. Email: ldh@ouc.edu.cn

1. Introduction

The North Atlantic Oscillation (NAO) is a dominant low-frequency dipole mode of atmospheric circulation variability over the North Atlantic basin, which has attracted great attention in the scientific community because it may play an important role in climate (Hurrell 1995). Feldstein (2003) found that NAO events are active on a time scale as short as nearly 2 weeks and demonstrated that high-frequency eddies with periods less than 10 days are crucial for the occurrence of the NAO. In a barotropic model, Vallis et al. (2004) found that large-scale dipole modes, such as NAO, and annular modes can be produced by stochastic forcing that mimics baroclinic eddies in the Atlantic storm track.

Many previous studies have indicated that the synoptic wave breaking (SWB) tends to produce NAO events (Benedict et al. 2004, hereafter B04; Franzke et al. 2004, hereafter F04; Riviere and Orlanski 2007; Woolings et al. 2008). Recently, Abatzoglou and Magnusdottir (2006a, hereafter AM06a) noted that the planetary wave breaking (PWB) that occurs in the subtropical Atlantic may amplify the positive NAO. On the other hand, it has been demonstrated in observational studies that the NAO phenomenon is characterized by a meridional displacement of the upper-tropospheric jet where positive and negative phases correspond, respectively, to the northward and southward movements of a westerly jet (B04; F04; Riviere and Orlanski 2007), thus concluding that wave breaking is responsible for the jet shift. Furthermore, F04 and Riviere and Orlanski (2007) noted that some factors such as seeding from the Pacific and moisture in the Caribbean can influence the wave breaking, although what controls the type of wave breaking is not completely clear so far (Thorncroft et al. 1993; Riviere and Orlanski 2007). The above studies raise the following important questions:

  1. Is wave breaking a necessary condition of the NAO occurrence?
  2. Why does wave breaking occur during the NAO life cycle?
  3. What are the properties of wave breaking in both the planetary and synoptic scales?
  4. What is the relationship between wave breaking, the meridional shift of the jet, and the NAO occurrence?

In Luo et al. (2007ac, hereafter Parts IIII, respectively), we established a weakly nonlinear NAO theory to clarify how the synoptic-scale waves drive the NAO life cycle with periods of nearly two weeks, what factor determines the phase of the NAO, and how the latitudinal displacement of the westerly jet is related to the phase of the NAO. In this paper, we will use the analytical solution of the NAO obtained in Part III to address the above questions.

This paper is organized as follows: in section 2, using the composite and unfiltered fields of NAO events, a diagnostic study is performed to show the characteristics of the PWB and synoptic wave breaking in terms of the definition of the reversal of the meridional gradient of the potential vorticity (PV) as presented in McIntyre and Palmer (1983) and B04. In section 3, the latitudinal gradient of the planetary-scale and total PV can be analytically derived from the analytical solution of the NAO in Part III. The planetary and synoptic wave-breaking processes in our analytical model are presented in sections 4 and 5, respectively. In section 6, we examine the relationship between the PWB, synoptic wave breaking, and jet variability during the NAO life cycle. The precondition of the synoptic wave breaking is discussed in section 7, and the main conclusions are summarized in section 8.

2. Wave breaking seen from the composite and unfiltered fields of observed NAO events

Some observational studies have indicated that the PWB in the subtropical Atlantic can influence the NAO (AM06a), and nearly twice as many PWB events are observed during positive NAO winters than during negative NAO winters (Abatzoglou and Magnusdottir 2006b, hereafter AM06b). B04 presented evidence that the anticyclonic (cyclonic) synoptic wave breaking tends to result in the positive (negative) phase of the NAO. However, whether the wave breaking is a response to or a cause of the occurrence of the NAO is not very clear because diagnostic results cannot sufficiently show such a causal relationship. In previous diagnostic studies wave breaking is usually identified by a sign reversal of the latitudinal gradient of the PV or potential temperature (B04; AM06a). In the present study, such a definition is also used as an indicator of wave breaking to understand the characteristics of the PWB and synoptic wave breaking during the NAO life cycle. Here, the PWB is characterized by a sign reversal of the PV gradient in a composite field, but the synoptic wave breaking is defined as a sign reversal of the PV gradient in an unfiltered field. The observational datum used here is the daily mean 300-hPa geopotential height from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR).

a. Planetary wave breaking

To clearly see the characteristics of the PWB during the NAO life cycle, it is necessary to filter out the synoptic-scale eddies. For this case, the composite of 300-hPa geopotential height fields for negative- and positive-phase NAO (NAO and NAO+, respectively) events from the NCEP–NCAR reanalysis based on a daily NAO index by B04 can be considered as a planetary-scale field of NAO events in that it has removed some of the smaller-scale noise (Luo et al. 2007b). Thus, it is reasonable to calculate the meridional gradient of the composite PV (∂qC/∂y, where qC is the PV in a composite field) to see the characteristics of PWB during the NAO life cycle. Using the composite geopotential height field of NAO events presented by Luo et al. (2007b), the meridional gradient of the composite PV is shown in Fig. 1 for two phases of the NAO.

For the NAO, the composite PV gradient (CPVG) at the beginning is observed to be weak over the Atlantic basin and only exhibits a sign reversal in the western basin (−lag 8). This reversed CPVG tends to enhance when the NAO anomaly grows (from −lag 6 to lag 0), which is most prominent at the mature stage (lag 0). During the decay of the NAO there is a weakening of this overturned CPVG (from lag 2 to lag 6). This suggests that the sign reversal of the CPVG is probably due to a feedback of the NAO pattern in that both the CPVG and NAO amplitudes undergo a consistent evolution. Thus, it is concluded that during the NAO episode the PWB does not occur in the subtropical Atlantic because the sign reversal of the CPVG cannot be observed in the subtropical region, which is consistent with the finding of AM06a, who noted that the PWB is infrequent in the subtropical Atlantic for the negative phase.

It is shown in Fig. 1 that a change in the strength of the CPVG is almost the same between opposite signs of the NAO. Even so, it does not mean that the CPVG is only weakly related to the NAO. In fact, the spatial structure of the CPVG for the NAO+ is extremely different from that for the NAO, thus suggesting that the CPVG depends on the phase of NAO and that its sign reversal may be attributed to the feedback of the NAO anomaly. An interesting point found here is that during the NAO+ life cycle the reversed CPVG appears to exist mainly in the subtropical Atlantic region, implicating that during the NAO+ life cycle the PWB is more likely to occur in the subtropical Atlantic, which supports the diagnostic result of AM06a.

b. Synoptic-scale wave breaking for observed NAO events

Although the PV in an unfiltered field includes planetary-scale waves, it is reasonable to calculate the PV gradient in an unfiltered field, as done in B04, to understand why synoptic-scale waves break one way or another during the NAO life cycle. This is based on the fact that the total PV gradient (TPVG) field can reflect the characteristics of synoptic wave breaking because the relative vorticity gradient of the synoptic-scale field is dominant (B04). To find the difference of the synoptic-scale wave breaking between the two phases of the NAO, observed NAO and NAO+ events in a daily, unfiltered geopotential height field at 300 hPa from the NCEP–NCAR reanalysis are shown in Fig. 2. Figure 2a shows the life cycle of an NAO event occurring during 25 January to 13 February 1963. It is noted that on 25 January 1963 a warm ridge appears in the North Atlantic that is surrounded by two cyclones at its two sides. This Atlantic ridge is further intensified both by the persistent poleward intrusion of warm air and by the southward advection of cold air, finally forming an isolated high-over-low dipole, an NAO pattern, at the mature stage (6–8 February 1963). In this process, the NAO events are concluded to be caused by the cyclonic breaking of synoptic waves in that the trough–ridge systems are tilted along the northwest–southeast direction, similar to a cyclonic wave-breaking process noted in previous studies (B04; Riviere and Orlanski 2007). The flow characteristics bear a remarkable analogy to that of blocking flows (Berggren et al. 1949), which are also similar to the eddy-driven blocking pattern obtained theoretically by Luo (2005).

Correspondingly, Fig. 2b shows a life cycle of an NAO+ event from 12 to 27 January 1975. It is noted that on 12 January 1975 the troughs and ridges are robust in the Atlantic basin and in its upstream side, and a tongue of cold air is observed near the east coast of North America and exhibits a far southward intrusion. A large-scale low that is made of a body of cold air is formed as the intruded cold air and warm air return to the north and south, respectively (12–17 January 1975). At the same time, a large-scale high is also established in the subtropical Atlantic. The establishment of such a large-scale low-over-high dipole will strengthen the mean westerly wind through a weakening of small-scale trough–ridge systems (17–19 January 1975). In this process, the trough–ridge system undergoes a southwest–northeast tilt, which is similar to a characteristic of the anticyclonic breaking of synoptic waves (B04). Thus, NAO+ events are easily concluded to arise from the anticyclonic wave breaking (B04; Riviere and Orlanski 2007). In fact, this wave breaking can also be observed during the NAO+ life cycle even if the initial synoptic waves are not breaking (not shown). More recently, Luo et al. (2007a) found in a highly idealized model that the phase of the NAO is dominated by the spatial structures of the preexisting planetary and synoptic waves, rather than by the type of wave breaking. In the following section, we will demonstrate that the wave breaking is attributed to the interaction between planetary and synoptic waves that leads to the NAO, and that the type of wave breaking is more likely to be determined by the spatial structures of both the initial planetary and synoptic waves.

Figure 3 shows the latitudinal gradient of the total PV (∂qT/∂y, where qT is the unfiltered PV) of the daily, unfiltered geopotential height fields for the NAO and NAO+ events shown in Fig. 2. It is found that for the NAO the sign reversal of the latitudinal total PV gradient is weak at the beginning (25 January 1963), hinting that at that time the wave breaking is also weak. When the NAO anomaly further grows, the reversed TPVG is enhanced and seems to be strongest during the mature stage (Fig. 3a, 6–8 February 1963). This shows that the cyclonic wave breaking (CWB) is strongest during the mature phase of the NAO, as observed by Riviere and Orlanski (2007), but this trend reverses during the decay stage. Thus, during the NAO life cycle the wave breaking exhibits a trend consistent with the evolution of the NAO. This result is not noted in previous studies (B04; Riviere and Orlanski 2007; Woollings et al. 2008). For the positive phase, the reversed TPVG is strong at the initial stage (Fig. 3b for 12 January 1975), and then is gradually weakened when the NAO+ anomaly is amplified (Fig. 3b, 14–18 January 1975). This reversed TPVG is weakest during the mature phase because synoptic-scale waves are almost fully absorbed by the mean westerly wind (Fig. 2b, 18–19 January 1975). In particular, the reversed TPVG is no longer prominent on 19 January 1975, indicating that the wave breaking is extremely weak. During the decay phase, the reversed TPVG is enhanced, thus strengthening the anticyclonic wave breaking (AWB). Although this result is based on one or two cases, it is not difficult to conclude that the evolution of the AWB during the NAO+ life cycle seems to go through a trend opposite to the CWB observed during the NAO life cycle. Such a trend can, to a large extent, be confirmed by using the analytical solutions presented in the next section. Of course, diagnostic study of more NAO events should be presented to validate this conclusion. Furthermore, a comparison with Fig. 1 shows that during the NAO life cycle both the PWB and CWB tend to undergo a consistent evolution, but the AWB exhibits a trend opposite to that of the PWB during the NAO+ life cycle. This feature of wave breaking is not detected in previous studies (B04; Riviere and Orlanski 2007). From a highly idealized, analytical model in the following section it is proposed that the wave breaking is not a necessary condition of the NAO occurrence.

3. Analytical solutions of wave breaking in a weakly nonlinear NAO model

In this section, we will try to use the analytical solution of the NAO anomaly driven by a joint of synoptic-scale eddies and large-scale topography derived by Luo et al. (2007c) to clarify the relationship between the wave breaking, jet shift, and NAO anomalies. Thus, it is useful to review the planetary-scale solution of the NAO life cycle derived in Luo et al. (2007c). For this case, the planetary-scale solution (ψP) of the NAO evolution can be written as
i1520-0469-65-3-737-e1a
i1520-0469-65-3-737-e1b
i1520-0469-65-3-737-e1c
i1520-0469-65-3-737-e1d
i1520-0469-65-3-737-e1e
i1520-0469-65-3-737-e1f
i1520-0469-65-3-737-e1g
i1520-0469-65-3-737-e1h
where u0 is a constant westerly wind in a β-plane channel with a width of Ly, hA = −1/[β/u0 − (k2 + m2/4)], h0 is the height of the land–sea topography, xT is the relative position between the NAO center and the positive anomaly of the topographic planetary wave induced by the topography, cc denotes the complex conjugate of its preceding term, and m = ±2π/Ly. In (1g)(1h), B* is the complex conjugate of B, which is the amplitude of the NAO anomaly driven by the preexisting synoptic eddies with zonal wavenumbers equal to or greater than 9 that are of the form ψ1 = f0(x){ exp[i(1x − ω̃1t)] + α exp[i(2x − ω̃2t)]} sin[(m/2)y] + cc for α = ±1 (Luo et al. 2007c). The evolution of B is governed by a forced nonlinear Schrödinger (NLS) equation in (1h). Note that f0(x) = a0 exp[−με2(x + x0)2] denotes the eddy amplitude, in which a0 measures the intensity of preexisting eddies and both μ > 0 and x0 > 0 are chosen to allow the preexisting synoptic-scale eddies to fix the upstream side of the planetary-scale wave prior to the NAO. The other notation in (1) can be found in Luo (2005) and Luo et al. (2007a, c). In addition, the synoptic-scale solution (ψ′ = ψ1 + ψ2) used here is also identical to that derived in Luo et al. (2007c). Here, it must be pointed out that the feedback onto the synoptic-scale eddies by the NAO anomaly or the interaction between the planetary and synoptic scales is characterized by the term ψ2 as given in Luo et al. (2007c). In this paper, the NAO anomaly is defined as a wavenumber 2 disturbance in the zonal direction. As in our previous studies it might be useful to restrict our zonal average analysis to the center of the domain when relating our model to observations and other studies.

It should be pointed out that ψNAO and ψTPW in (1b) represent the NAO anomaly and topographic planetary wave (TPW), respectively. The diffluence of the planetary-scale flow is enhanced as the NAO anomaly B increases. But ψm denotes a westerly jet anomaly, which is attributed to the feedback of the NAO anomaly and is composed of two parts: the double jets driven by the NAO anomaly itself and the meridional shift of the jet induced by the interaction between the NAO anomaly and the TPW. Different from previous studies, the NAO anomaly here is not characterized by a shift in the jet (ψm). It is defined by a low-frequency planetary wave denoted by ψNAO, rather than by a rotated EOF as in B04. However, as we will note, the positive (negative) phase NAO can accompany the poleward (equatorward) shift of a westerly jet. In the present paper, the positive-phase NAO (NAO+) is characterized by ψNAO with m = 2π/Ly and α = 1, and the negative phase (NAO) corresponds to ψNAO with m = −2π/Ly and α = −1. As noted in Luo et al. (2007c), h0 > 0 is required for the NAO (m/2 = −π/Ly) because the ocean is a trough of the wavy topography used, but h0 < 0 is required for the NAO+ (m = 2π/Ly).

Using (1), the following PV gradient in a planetary-scale field can be obtained as
i1520-0469-65-3-737-e2
Calculating ∂qP/∂y in (2) can help us to better understand how the PWB takes place during the NAO life cycle. As indicated in Figs. 1 and 3, the CPVG is rather weak compared to that of an unfiltered field, thus it is reasonable to identify how the synoptic wave breaking occurs during the NAO life cycle through calculating ∂qT/ ∂y (qT = qP + ∇2ψ′, where ψ′ is a synoptic-scale field). In this paper, B(x, 0) = 0.4 is chosen as an initial value of the NAO for its two phases in order to see if an isolated dipole mode resembling an NAO pattern can be excited from a linear planetary wave prior to the NAO, and the other parameters are taken to have the same values as in Luo et al. (2007c) except for the eddy intensity a0 and xT.

4. In-phase evolution of planetary and synoptic wave breaking during the NAO life cycle

To clearly see the role of TPW in the wave breaking associated with the NAO life cycle, it should be useful to present the planetary scale and total fields of an NAO event using the analytical NAO solutions given in (1). For a0 = 0.17 the planetary-scale fields of the NAO event are plotted in Fig. 4 for h0 = 0 and h0 = 0.4, respectively. Figure 4a shows the life cycle of a NAO event with a period of nearly 2 weeks (10–20 days) observed by Feldstein (2003) and B04, whose lifetime is dependent on the initial states of planetary and synoptic waves and the setting of background westerly wind (Luo et al. 2007a). However, once the TPW is included, the NAO pattern in a planetary-scale field can exhibit a blocking high (Fig. 4b; on day 9), which is strikingly similar to a composite of the 300-hPa geopotential height fields of NAO events presented by Luo et al. (2007b; Fig. 10a). Figure 5 shows the total field of the NAO life cycle driven jointly by the synoptic-scale waves and the land–sea topography for a0 = 0.17 and a0 = 0.12. It is obvious that the theoretical solution shown in Fig. 5 looks like the temporal evolution of an observed NAO event (Fig. 2a). In this process, the CWB can be observed. For this reason, the CWB is concluded to cause NAO events in some previous studies (B04; F04; Riviere and Orlanski 2007; Woollings et al. 2008). Nevertheless, as we demonstrate here, the strong CWB is essentially due to the feedback of the NAO anomaly or the role of the term ψ2. Additionally, the total field seems more likely to resemble an observed case if the preexisting eddy intensity is slightly reduced (Fig. 5b; for a0 = 0.12).

a. Behavior of planetary wave breaking

Figure 6 shows the PV gradient of the planetary-scale field shown in Fig. 4. It is noted that in the absence of the TPW the planetary-scale PV gradient is not reversed at day 0 because the diffluent flow is rather weak. But this PV gradient will be overturned and form a meridionally oriented tripole with the further intensification of the NAO anomaly (Fig. 6a). After day 9 there is a weakening of this planetary-scale PV gradient, thus suggesting that the PWB arises from the amplification of a NAO anomaly because the enhancement of the NAO anomaly can increase the diffluence of a planetary flow (not shown). However, in the presence of the TPW the diffluence of the planetary-scale flow becomes so strong because of the role of ψm that the PV gradient at day 0 can exhibit a sign reversal even in the subtropical region (Fig. 6b, at day 0). Even so, it does not mean that the PWB results in the occurrence of the NAO event. This is because the PWB cannot be observed if the feedback of the initial NAO anomaly (initial ψm here) is removed or if the amplitude of the initial NAO anomaly is weak (Fig. 6a). As we will indicate in section 6, the interaction between the NAO anomaly and TPW will make the subtropical mean flow strong so that the sign of the PV gradient is overturned easily. Also, it is found that the reversed planetary-scale PV gradient can undergo an evolution from an intensification during the growing phase of NAO to a weakening during the decay phase and can have a southwest–northeast tilted tripole structure, similar to that of the CPVG shown in Fig. 1a. Such a pattern of the planetary-scale PV gradient cannot be observed if the TPW is absent. This implies that the topographic planetary wave in the Northern Hemisphere plays a certain role in the PWB.

b. Synoptic-scale wave breaking and the feedback of the NAO anomaly

Figure 7a shows the latitudinal gradient of the TPVG for an NAO event shown in Fig. 5a. It is found that although the TPVG at day 0 is weak, it still shows a sign reversal, indicating that synoptic-scale waves at day 0 are breaking, but such an initial wave breaking cannot be observed if the feedback of the initial NAO anomaly or the initial ψ2 is taken off (Fig. 7b, at day 0). Thus, it is natural that the occurrence of the NAO does not necessarily require the breaking of the initial synoptic waves. As pointed out by Luo et al. (2007a), NAO events can arise from the eddy forcing if the preexisting planetary wave with a high-over-low structure matches the preexisting eddy forcing having a negative-over-positive pattern. Of course, strong NAO events are more likely to be excited if the preexisting synoptic-scale eddies are of large amplitude or breaking (not shown).

It is also found that the reversed TPVG that characterizes the CWB exhibits an enhancement during the growth phase of the NAO and a weakening during the decay phase, which looks like that observed in Fig. 3a. Such a reversed TPVG pattern cannot be detected if the planetary-to-synoptic-scale interaction term (ψ2) that represents the feedback of the NAO anomaly is excluded (Fig. 7b). It is natural that the interaction between the planetary and synoptic waves can result in the CWB, as observed in Woollings et al. (2008), thus concluding that the CWB is not a necessary condition of the NAO occurrence. Also, it is noted that the CWB and PWB undergo an in-phase evolution.

5. Out-phase evolution of planetary and synoptic wave breaking during the NAO+ life cycle

The planetary-scale fields of two NAO+ events are shown in Fig. 8 for h0 = 0 and h0 = −0.4, respectively. It is noted that in the absence of the TPW (h0 = 0), the planetary-scale field exhibits a typical low-over-high dipole pattern with a period of about 2 weeks. This flow pattern is highly idealized in that other factors have been excluded. However, it shows a northwest–southeast tilted low-over-high dipole structure once the TPW is involved (Fig. 8b), which is quite similar to that observed by DeWeaver and Nigam (2000, their Fig. 11). The corresponding total fields of the two NAO+ patterns shown in Fig. 8 are displayed in Fig. 9. It is easy to see that the NAO+ life cycle in Fig. 9a looks like an observed NAO+ event in Fig. 2b. In the next subsections, our aim is focused on looking at the characteristics of wave breaking during the NAO+ life cycles as shown in Figs. 8 and 9 through calculating their respective PV gradient.

a. Characteristics of planetary wave breaking

Figure 10 shows the instantaneous PV gradient of the planetary-scale NAO+ patterns in Fig. 8. It is seen in Fig. 10a that the sign reversal of the planetary-scale PV gradient is not observed at day 0, indicating that the PWB does not occur at the beginning of the positive phase. Nevertheless, such a sign reversal of the PV gradient can exist in the subtropical and subpolar regions of the Atlantic basin as the NAO+ anomaly is intensified. In the presence of a TPW, the overturned PV gradient exhibits a southeast–northwest tilted tripole (Fig. 10b), similar to that in Fig. 2b from a composite of NAO+ events. This great similarity with Fig. 2b is attributed to the role played by the TPW in the NH. Thus, during the NAO+ life cycle the PWB is more likely to occur in the subtropical Atlantic because of a weakening of the subtropical mean flow. This offers a theoretical explanation for why the PWB is easily observed in the subtropical region of the North Atlantic basin during the NAO+ life cycle (AM06a).

b. Synoptic-scale wave breaking

Figure 11 shows the instantaneous TPVG fields of an NAO event depicted in Fig. 9a for two cases with and without the feedback of the planetary-scale NAO+ anomaly. In Fig. 11a we note that the reversed TPVG exhibits a weakening during the intensification phase of the NAO+, but an enhancement during the decay phase. In this process, the AWB is observed and has such a behavior naturally. A comparison with Fig. 10b indicates that the AWB and PWB undergo an out-phase evolution during the NAO+ life cycle. The evolution of the reversed TPVG in Fig. 11a is also quite similar to an observed case shown in Fig. 3b, but Fig. 11b does not look like this observation. This shows that the interaction between planetary and synoptic waves is crucial for the AWB. But the spatial structures of the initial planetary and synoptic waves that allow NAO+ events to occur have in fact determined the AWB. Even so, the AWB is observed to be infrequent during the NAO+ episodes, consistent with the diagnostic result of Woollings et al. (2008).

6. Relationship between the wave breaking and meridional shift of a westerly jet

B04 and F04 found in their synoptic and numerical studies that the positive (negative) phase of the NAO results from the remnants of AWB (CWB). However, AM06a noted that the strong poleward eddy momentum flux associated with the wintertime PWB may influence the NAO. Some previous studies have indicated that the wave breaking tends to push the Atlantic jet equatorward or poleward (Riviere and Orlanski 2007; AM06a). In this section, we will use the highly idealized analytical solutions in (1) to discuss the relationship between the PWB and synoptic wave breaking and the jet displacement during the NAO life cycle.

a. Planetary wave breaking and jet displacement

If one can divide the total streamfunction field into three parts—mean flow denoted by square brackets, planetary scale denoted by subscript L, and synoptic scale denoted by subscript S—it is easy to get the following mean flow equation from a barotropic vorticity equation:
i1520-0469-65-3-737-e3
where [u] denotes a zonal mean flow, uL and υL are the zonal and meridional components of the planetary-scale perturbation velocity, respectively, [uSυS] has been neglected in (3) because the scale separation assumption is used, and Ff is the residual term including other forcing and dissipation.
By using (1), it is easy to obtain the analytical expression of [uLυL] in the form of
i1520-0469-65-3-737-e4

To see that the PWB and the meridional shift of the westerly jet are concurrent events during the NAO life cycle, the latitudinal profile of the maximum westerly jet anomaly {uAm(y, t) = [uA(x, y, t)]max}for uA = −(∂ψm/∂y) in (1e) is shown in Fig. 12 for two cases with and without the effect of the TPW and for the same parameters as in Figs. 4 and 8.

It is found in Fig. 12 that in the absence of the TPW double jets are formed only in the North Atlantic, which is independent of the phase of the NAO. For this case, no PWB is observed at the beginning stage of the NAO in that the horizontal shear of the jet induced by the NAO anomaly is weaker. But in the presence of the TPW the westerly jet can exhibit a meridional shift that depends on the phase of the NAO. The meridional shift of the jet is a natural result of the interaction between the NAO anomaly and TPW. As the jet is intensified and shifts to the north or the south, the horizontal shear of the jet is increased, which results in a sign reversal of the planetary-scale PV gradient. Such a reversed PV gradient is strongest at day 9 because the horizontal shear of the jet is most prominent (Fig. 12). Thus, it is likely in our theoretical model that the PWB and jet shift can be by-products of the NAO occurrence, even though the sign reversal of the planetary-scale PV gradient and jet shift are also observed because of the role of the TPW at the beginning stage of the NAO life cycle. Of course, this result is based on a highly idealized model. Perhaps this theoretical model is unable to simulate actual wave breaking. But the result of this model at least indicates that the PWB is not a necessary condition of the NAO occurrence.

Thus, the land–sea topography in the NH can increase the likelihood of PWB through altering the zonal mean flow because of the role of the TPW. This is one reason why PWB events also occur in spring, summer, and fall, although the NAO is weak in the three seasons (AM06b). However, the PWB is strongest in winter because the eddy-driven NAO anomaly is most prominent, thus implying that the PWB, jet displacement, and NAO occurrence may be concurrent events.

In the present study, because [uSυS] has been neglected in terms of the scale separation assumption, the contribution of high-frequency transient eddies to the zonal mean flow cannot be reflected. However, the diagnostic study of AM06a appears to demonstrate that our assumption is probably acceptable because the high-frequency transient eddy momentum flux obtained by AM06a is almost independent of the NAO phase. Thus, it is inferred that the meridional shift of the jet cannot be sufficiently reflected by the high-frequency transient eddy momentum flux. As we will discuss, the low-frequency transient eddy momentum flux convergence that reflects the planetary wave propagation is able to describe the jet displacement, as shown in Fig. 12.

Figure 13 shows −(∂[uLυL]/∂y) for the parameters corresponding to those in Figs. 6b and 10b. It is found that the spatial pattern of −(∂[uLυL]/∂y) can reflect the variation in the zonal mean flow found in Fig. 12 and bears a striking resemblance to that for two phases of the NAO obtained by AM06a (their Figs. 5b,d). It is evident that during the NAO life cycle there is a poleward propagation of quasi-stationary planetary waves in that −(∂[uLυL]/∂y) is positive in the lower latitude and negative in the higher latitude (Fig. 12a). Correspondingly, the zonal mean flow is enhanced in the lower latitude (subtropical region) and weakened in the higher latitude, which corresponds to a nonlinear reflection (AM06a). During the NAO+ life cycle the equatorward propagation of quasi-stationary planetary waves is noted because of −(∂[uLυL]/∂y) being positive in the higher latitude and negative in the lower latitude (Fig. 13b). This process corresponds to a nonlinear nonreflection (AM06a). In this process the subtropical mean flow is weakened (Fig. 12), thus favoring the PWB. This can help us explain why the PWB can occur frequently in the subtropical Atlantic region during the NAO+ life cycle, but it tends to accompany the northward shift of the jet. As noted by Limpasuvan and Hartmann (1999) and AM06a, changes in the basic state associated with the NAO+ lead to a stronger equatorward wave activity flux and a weaker westerly flow over the central and eastern Atlantic near 30°N, both of which act to precondition the subtropical Atlantic to PWB.

It should be pointed out that [υLh] seems to counteract the westerly anomalies driven by −(∂[uLυL]/∂y) (not shown). As indicated by Lorenz and Hartmann (2003), the mountain torque [υLh] tends to dampen the wind anomalies. The discussions in this subsection indicate that the PWB, the associated quasi-stationary wave propagation, and jet displacement tend to accompany the occurrence of NAO. In particular, the TPW plays an important role in the north–south variability of the jet.

b. Synoptic wave breaking and jet shift

Riviere and Orlanski (2007) found that during the NAO life cycle the AWB (CWB) pushes the jet poleward (equatorward). However, the present study here appears to indicate that the wave breaking and jet displacement are likely to be attributed to the feedback of the NAO anomaly. As demonstrated in our previous studies (Luo et al. 2007a), the preexisting eddy forcing (high-frequency momentum flux) from synoptic-scale waves seems to determine the occurrence and the phase of the NAO, and concurrently the meridional shift of the jet can be observed through the interaction between the NAO anomaly and TPW (Fig. 12). The jet shift is not dependent on the type of wave breaking, but it is dominated by the phase of the NAO. Thus, the wave breaking and jet displacement seem to be different descriptions of the NAO phenomenon, which can also be observed during the NAO life cycle even if the initial synoptic waves are not breaking. This indicates that the wave breaking is not a necessary condition of either the NAO occurrence or the meridional shift of the jet. Of course, the occurrence of a strong NAO event is more likely if synoptic waves are of large amplitude or breaking (not shown).

7. Preconditions of synoptic wave breaking

As indicated by Luo et al. (2007a, c), NAO (NAO+) events can arise from the nonlinear interaction between planetary and synoptic waves as long as the initial planetary-scale wave with a high-over-low (low-over-high) anomaly matches preexisting synoptic waves having an eddy forcing with a negative-over-positive (positive-over-negative) dipole. Some previous studies have linked the type of wave breaking with the jet displacement (Thorncroft et al. 1993; Lee and Feldstein 1996; Orlanski 2003). However, Riviere and Orlanski (2007) found that the difference in wave breaking between the two phases of the NAO is not due to the meridional shear of the jet, and further noted that the relative strength between cyclones and anticyclones can be an important factor in determining the type of wave breaking. Here, we will point out that the spatial structures of the initial planetary and synoptic waves that allow the NAO to occur are crucial for the type of wave breaking.

At day 0 the preexisting planetary wave anomaly ψNAO, the preexisting synoptic wave ψ1, and associated eddy forcing −J(ψ1, ∇2ψ1)P are shown in Fig. 14 for the same parameters as in Figs. 4a and 8a. It is not difficult to see that when the preexisting planetary and synoptic waves satisfy the spatial structures as shown in Figs. 14a–c, NAO events as shown in Fig. 5 can inevitably occur through the interaction between the two scales. In this process the CWB can frequently occur. However, NAO+ events, as shown in Fig. 9, can be formed when the preexisting planetary and synoptic waves have the spatial structures shown in Figs. 14d–f. In this case, the AWB can be observed during the NAO+ life cycle. Thus, it is inferred from our highly idealized model that the type of synoptic wave breaking is more likely to be determined by the spatial structures of both the initial planetary and synoptic waves prior to the NAO. In conclusion, the wave breaking, jet displacement, and associated quasi-stationary wave propagation may be different descriptions of the NAO phenomenon.

Our analytical solution is unable to simulate actual wave breaking because it is highly idealized and based on some assumptions. However, although the analytical solution is weakly nonlinear, it can provide insight into characteristics of the NAO occurrence and wave breaking and can at least indicate that the occurrence of the NAO does not necessarily require that the initial synoptic waves are breaking.

8. Conclusions and discussion

In this study, we have used the analytical solution of the NAO life cycle obtained by Luo et al. (2007c) to examine the relationship between the breaking of planetary and synoptic waves, jet variability, and NAO occurrence. It is found that the breaking of planetary and synoptic waves and the north–south shift of the westerly jet are more likely to be different descriptions of the NAO phenomenon. The breaking of synoptic-scale waves essentially arises from the interaction between planetary and synoptic waves that gives rise to NAO events, but the PWB and meridional shift of the jet are attributed to the intensification of the planetary wave representing an NAO anomaly and its interaction with topographic planetary waves in the NH. The type of wave breaking is more likely to be determined by the spatial structures of the initial planetary and synoptic waves that control the phase of the NAO, although other factors such as the horizontal shear of the westerly jet, SST anomalies, and moisture can influence it (Thorncroft et al. 1993; Orlanski 2003; Riviere and Orlanski 2007). It appears that the cyclonic (anticyclonic) breaking of synoptic waves goes with the occurrence of the NAO (NAO+) events. It is also shown that the PWB and synoptic wave breaking (SWB) undergo an in-phase evolution for NAO events consistent with a change in the NAO amplitude, but an out-phase evolution for NAO+ events. The most interesting one of these events is that during the NAO life cycle the cyclonic wave breaking (CWB) is enhanced during the growth phase and then weakened during the decay phase. But there is an opposite trend of anticyclonic wave breaking (AWB) during the NAO+ life cycle. This hints that the CWB is more frequent during the negative NAO phase than during the positive AWB phase, in agreement with the observational finding of Woollings et al. (2008). However, they found strong evidence that it is not the NAO that is affecting the occurrence of wave breaking, but that the influence is the other way around.

Another important finding here is that in the presence of topographic planetary waves, the PWB is more likely to occur in the subtropical region of the Atlantic basin during the NAO+ life cycle than during the NAO life cycle. At the same time, the reversed planetary-scale PV gradient that characterizes the PWB looks very much like the diagnostic result presented in this paper. On the other hand, because the latitudinal shift of the jet and the NAO always accompany the wave breaking, the wave breaking may be understood to result in the occurrence of the NAO and the meridional shift of the jet (Riviere and Orlanski 2007; Woollings et al. 2008). In fact, our investigation here from a highly idealized model indicates that the wave breaking is not a necessary condition for the NAO occurrence. To a certain extent, PWB, SWB, and jet variability seem to be different descriptions of the NAO phenomenon.

It must be pointed out that all the conclusions presented here are based on the initial NAO states and preexisting synoptic eddies prespecified in our theoretical model. What drives the initial NAO mode and initial synoptic waves is not solved in this paper. But this problem is very interesting and deserves further investigation. It should be noted that in our theoretical model the synoptic eddies contribute only to the NAO anomaly because the scale separation between the mean flow, planetary waves, and synoptic waves has been assumed. In the real atmosphere, synoptic eddies can dominate the mean flow so that a scale separation may not be satisfied. This problem should be further examined by extending the present model.

Acknowledgments

The authors acknowledge the support from the National Outstanding Youth Natural Science Foundation of China under Grant Number 40325016, the National Natural Science Foundation of China (Number 4057016), and Taishan scholar funding and 111 Project (B07036). Two anonymous reviewers are highly appreciated for their useful suggestions in improving this paper.

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Fig. 1.
Fig. 1.

Latitudinal gradient of the PV (10−9 m−1 s−1) in a composite field of NAO events at 300 hPa that filters out synoptic-scale waves for negative and positive phases as shown in Luo et al. (2007b, their Fig. 10), in which the dashed and solid lines represent the negative (shaded) and positive values of the PV gradient, respectively: (a) negative-phase event and (b) positive-phase event.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 1.
Fig. 1.

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Fig. 2.
Fig. 2.

The 300-hPa unfiltered geopotential height fields of 2 NAO events: (a) a negative-phase event from 25 Jan to 13 Feb 1963 and (b) a positive-phase event from 12 to 27 Jan 1975.

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Fig. 2.
Fig. 2.

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Fig. 3.
Fig. 3.

Latitudinal gradient of the total PV (10−9 m−1 s−1) for the unfiltered geopotential height fields at the 300-hPa level of the corresponding NAO events as shown in Fig. 2, in which the dashed and solid lines represent the negative (shaded) and positive values of the PV gradient, respectively: (a) negative-phase event and (b) positive-phase event.

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Fig. 3.
Fig. 3.

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Fig. 4.
Fig. 4.

Planetary-scale fields of eddy-driven negative-phase NAO life cycles for a0 = 0.17: (a) case without the effect of TPW (h0 = 0) and (b) case with the effect of TPW (h0 = 0.4). Contour interval (CI) is 0.15.

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Fig. 4.
Fig. 4.

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Fig. 5.
Fig. 5.

Total fields of eddy-driven negative-phase NAO events in the presence of the TPW fortwo types (a0) of preexisting eddy amplitudes (CI = 0.15): (a) a0 = 0.17 and (b) a0 = 0.12.

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Fig. 5.
Fig. 5.

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Fig. 6.
Fig. 6.

Meridional gradient of the planetary-scale PV of eddy-driven negative-phase NAO events corresponding to those in Fig. 4 (CI = 0.5), in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case without TPW and (b) case with TPW.

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Fig. 6.
Fig. 6.

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Fig. 7.
Fig. 7.

Meridional gradient of the TPV for an eddy-driven negative-phase NAO event in Fig. 5a, in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case with the feedback of an NAO anomaly (CI = 4) and (b) case without the feedback of an NAO anomaly (CI = 1).

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Fig. 7.
Fig. 7.

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Fig. 8.
Fig. 8.

Planetary-scale fields of eddy-driven positive-phase NAO life cycles for a0 = 0.17 (CI = 0.15): (a) case without TPW (h0 = 0) and (b) case with TPW (h0 = −0.4).

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Fig. 8.
Fig. 8.

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Fig. 9.
Fig. 9.

Total fields of the corresponding eddy-driven positive-phase NAO events as shown in Fig. 8: (a) case without TPW (h0 = 0) and (b) case with TPW (h0 = −0.4).

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Fig. 9.
Fig. 9.

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Fig. 10.
Fig. 10.

Meridional gradient of the planetary-scale PV of the corresponding eddy-driven positive-phase NAO events shown in Fig. 8 (CI = 0.5), in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case without the effect of TPW and (b) case with the effect of TPW.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 10.
Fig. 10.

(Continued)

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 11.
Fig. 11.

Latitudinal gradient of the TPV for eddy-driven positive-phase NAO events in Fig. 9, in which the dashed and solid lines represent the negative and positive values of the PV gradient, respectively: (a) case with the feedback of a NAO anomaly (CI = 4) and (b) case without the feedback of a NAO anomaly (CI = 1).

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 11.
Fig. 11.

(Continued)

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 12.
Fig. 12.

Meridional profile of the maximum mean westerly wind during the NAO life cycle, in which the dotted–dashed curve denotes the case without TPW as shown in Fig. 4a or 8a; the dotted–long-dashed curve represents the case of a positive-phase NAO with the effect of TPW; and the solid curve corresponds to the case of a negative-phase NAO with the effect of TPW.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 13.
Fig. 13.

Horizontal distribution of −(∂[uLυL]/∂y) for negative- and positive-phase NAO events for the same parameters as in Figs. 4b and 8b (CI = 0.05), in which the solid and dashed curves represent the positive and negative values of −(∂[uLυL]/∂y), respectively: (a) negative-phase event and (b) positive-phase event.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 13.
Fig. 13.

(Continued)

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

Fig. 14.
Fig. 14.

Initial fields of planetary wave (ψNAO), synoptic wave (ψ1), and eddy forcing [−J(ψ1, ∇2ψ1)P] that allow the NAO events to occur for (a)–(c) negative and (d)–(f) positive phases for the same parameters as in Figs. 4a and 8a, respectively: (a), (d) ψNAO field (CI = 0.2), in which the dashed and solid curves denote the negative and positive anomalies, respectively; (b), (e) ψ1 field (CI = 0.1), in which the dashed and solid lines correspond to the cyclone and anticyclone, respectively; and (c), (f) −J(ψ1, ∇2ψ1)P field (CI = 0.05), in which the dashed and solid lines represent the anticyclonic and cyclonic forcing, respectively.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2440.1

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