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    Composite of zonal mean zonal winds at 300 mb in the Atlantic sector (90°W–0°) during the positive and negative phases of the NAO life cycle, in which the solid and dashed curves denote the positive and negative phases, respectively, and the point line correspond to the difference between positive and negative phases.

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    Climatological wintertime mean barotropic streamfunction anomalies (in 107 m2 s−1 for (a) positive and (b) negative phase of the NAO, in which the solid and dashed lines represent positive and negative anomalies, respectively.

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    Horizontal distribution of land–sea contrast topography with wavenumber 2 for the parameters with m/2 = −(π/Ly) and h0 = 0.4 or m/2 = (π/Ly) and h0 = −0.4 in which the solid and dashed lines represent the topographic ridge and trough, respectively, and ψC anomalies of the topographically induced climatological stationary wave in which the solid and dashed curves represent the positive and negative anomalies, respectively: (a) topographic distribution for xT = 0, (b) topographic distribution for xT = −2.87/2, (c) ψC anomaly for xT = 0, and (d) ψC anomaly for xT = −2.87/2. The contour interval (CI) is 0.2.

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    Instantaneous fields of the interaction between planetary- and synoptic-scale waves during the negative phase of the NAO for m = −(2π/Ly), h0 = 0.4, and xT = −2.87/2: (a) planetary-scale field (CI = 0.15), (b) synoptic-scale field, in which the dashed and solid lines represent the cyclone and anticyclone, respectively (CI = 0.3), and (c) total field (CI = 0.3).

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    As Fig. 4 but for xT = 0: (a) planetary-scale field (CI = 0.15) and (b) synoptic-scale field (CI = 0.3)

  • View in gallery

    As Fig. 5 but for xT = −2.87.

  • View in gallery

    Instantaneous anomalies of anomalous stationary waves and anomalous plus climatological stationary waves for the same parameters as in Fig. 4, in which the solid and dashed lines denote the positive and negative anomalies respectively: (a) ψA anomaly (CI = 0.2) and (b) ψA + ψC anomaly (CI = 0.2).

  • View in gallery

    Instantaneous fields of the interaction between planetary- and synoptic-scale waves during the positive phase of the NAO for m = (2π/Ly), h0 = −0.4, and xT = −2.87/2: (a) planetary-scale field (CI = 0.15), (b) synoptic-scale field (CI = 0.3), and (c) total field (CI = 0.3).

  • View in gallery

    As Fig. 8 but for xT = 0: (a) planetary-scale field (CI = 0.15) and (b) synoptic-scale field (CI = 0.3).

  • View in gallery

    As Fig. 9 but for xT = −2.87.

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    Instantaneous fields of zonal mean wind anomalies for the negative phase of the NAO shown in Fig. 4 and for the positive phase shown in Fig. 8, in which the dashed and solid lines correspond to the negative and positive westerly anomalies, respectively: (a) negative phase (CI = 0.1) and (b) positive phase (CI = 0.1).

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    As Fig. 11 but for xT = 0.

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Dynamics of Eddy-Driven Low-Frequency Dipole Modes. Part III: Meridional Displacement of Westerly Jet Anomalies during Two Phases of NAO

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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

In this paper, the north–south variability of westerly jet anomalies during the two phases of the North Atlantic Oscillation (NAO) is examined in a theoretical model. It is found that the north–south variability of the zonal mean westerly anomaly results from the interaction between the eddy-driven anomalous stationary waves with a dipole meridional structure (NAO anomalies) and topographically induced climatological stationary waves with a monopole structure, which is dependent upon the phase of the NAO. The westerly jet anomaly tends to shift northward during the positive NAO phase but southward during the negative phase. Synoptic-scale eddies tend to maintain westerly jet anomalies through the excitation of NAO anomalies, but the climatological stationary wave and its position relative to the eddy-driven anomalous stationary wave appear to dominate the north–south shift of westerly jet anomalies.

On the other hand, it is shown that when the climatological stationary wave ridge is located downstream of the eddy-driven anomalous stationary wave, the storm track modulated by the NAO pattern splits into two branches for the negative phase, in which the northern branch is generally stronger than the southern one. However, the southern one can be dominant as the relative position between anomalous and climatological stationary waves is within a moderate range. The storm track for the positive phase tends to drift northeastward when there is a phase difference between the NAO anomaly and climatological stationary wave ridge downstream. Thus, it appears that the relationship between the NAO jets and storm tracks can be clearly seen from the present theoretical model.

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

In this paper, the north–south variability of westerly jet anomalies during the two phases of the North Atlantic Oscillation (NAO) is examined in a theoretical model. It is found that the north–south variability of the zonal mean westerly anomaly results from the interaction between the eddy-driven anomalous stationary waves with a dipole meridional structure (NAO anomalies) and topographically induced climatological stationary waves with a monopole structure, which is dependent upon the phase of the NAO. The westerly jet anomaly tends to shift northward during the positive NAO phase but southward during the negative phase. Synoptic-scale eddies tend to maintain westerly jet anomalies through the excitation of NAO anomalies, but the climatological stationary wave and its position relative to the eddy-driven anomalous stationary wave appear to dominate the north–south shift of westerly jet anomalies.

On the other hand, it is shown that when the climatological stationary wave ridge is located downstream of the eddy-driven anomalous stationary wave, the storm track modulated by the NAO pattern splits into two branches for the negative phase, in which the northern branch is generally stronger than the southern one. However, the southern one can be dominant as the relative position between anomalous and climatological stationary waves is within a moderate range. The storm track for the positive phase tends to drift northeastward when there is a phase difference between the NAO anomaly and climatological stationary wave ridge downstream. Thus, it appears that the relationship between the NAO jets and storm tracks can be clearly seen from the present theoretical model.

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

North Atlantic Oscillation (NAO) confined in the Atlantic sector is an important low-frequency mode in the Northern Hemisphere (NH) winters (Hurrell 1995; Visbeck et al. 2001). This mode resembles in essence the Artic Oscillation (AO) or northern annular mode (NAM; Wallace 2000; Feldstein and Franzke 2006). Some studies have indicated that the NAO anomalies arise from the forcing of synoptic-scale eddies (Feldstein 2003; Vallis et al. 2004; Cash et al. 2005; Gerber and Vallis 2005) and is a nonlinear initial-value problem (Benedict et al. 2004; Franzke et al. 2004).

In recent years, many investigators have demonstrated that synoptic-scale eddies tend to reinforce and maintain the zonal-mean wind anomalies associated with the NAO or annular modes, in particular in the Southern Hemisphere (SH; Yu and Hartmann 1993; Robinson 1996; Feldstein 1998; Feldstein and Lee 1998; Hartmann and Lo 1998; Limpasuvan and Hartmann 2000; Lorenz and Hartmann 2001). DeWeaver and Nigam (2000a, b) noted that during the NAO life cycle, stationary waves contribute more importantly to the zonal-mean jet anomalies than synoptic-scale eddies in NH winters. However, Lorenz and Hartmann (2003) found that in NH winter quasi-stationary waves can reinforce the zonal wind anomalies, but baroclinic eddies seem more important. The distinction between the subtropical jet and the eddy-driven midlatitude jet is less clear compared to the SH (Bals-Elsholz et al. 2001; Lee and Kim 2003), indicating that the stationary waves forced by large-scale mountains resembling a land–sea contrast (LSC) topography in the NH probably play an important role in the variability of the westerly jet and NAM (Lorenz and Hartmann 2003; Körnich et al. 2006). It has been recognized that for the positive phase of the NAO there is a poleward shift of the westerly jet from its climatological position, and the negative phase means an equatorward shift of the westerly jet (Limpasuvan and Hartmann 2000; Lorenz and Hartmann 2003). However, it is so far unclear what underlying mechanism dominates the north–south shift of the westerly jet although DeWeaver and Nigam (2000a, b) have emphasized in a diagnostic study that the climatological stationary waves play an important role in maintaining the zonal mean wind anomalies associated with the NAO.

In a recent theoretical study by Luo et al. (2007a), the phase of the eddy-driven NAO is found to depend upon the spatial structures of preexisting planetary- and synoptic-scale waves, implying that the NAO is indeed a nonlinear initial-value problem (Benedict et al. 2004). However, the analytical solutions in this model cannot be directly used to interpret the north–south shift of the westerly jet associated with the phase of NAO although synoptic-scale eddies are found to maintain the zonal mean wind anomalies through exciting the NAO anomaly. This is because the climatological stationary waves induced by the LSC topography are excluded in this theoretical model. In this paper, the theoretical model established by Luo et al. (2007a) is directly extended to include climatological stationary waves induced by the LSC topography in order to clarify why the meridional shift of the westerly jet anomaly takes place when the NAO is in either the positive or negative phase.

This paper is organized as follows. In section 2, the composite of zonal mean westerly wind at 300 mb and the corresponding wintertime mean barotropic streamfunction anomalies for two phases of the NAO are presented. In section 3, the theoretical NAO model by Luo et al. (2007a) is directly extended to include the climatological stationary wave induced by the LSC topography and analytical solutions are presented. The relationship between the NAO-like events and storm track is described in section 4. Section 5 depicts why the westerly jet exhibits a meridional displacement during two phases of the NAO. Finally, conclusions and discussions are summarized in section 6.

2. North–south variability of the composite zonal mean wind during two phases of NAO

Figure 1 shows the composite zonal mean winds at 300 mb during the positive and negative NAO phases based upon the daily NAO index proposed by Benedict et al. (2004). It is found that, in the upper troposphere, the westerly jet tends to shift northward and exhibits a slight intensification trend during the positive NAO phase. The negative phase accompanies a southward shift of the westerly jet. This result is also observed in the lower troposphere (not shown). More recently, Wittman et al. (2005) found that the occurrence of a meridional dipole in the first EOF of a time-dependent zonal jet is a consequence of the north–south excursion of the jet center. However, what factors dominate the north–south excursion of the jet center is not examined theoretically. As has noted in the introduction, the north–south shift of the westerly jet is more distinct in the NH than in the SH. Thus, it is speculated that the meridional displacement of the westerly jet during two phases of the NAO may be tied to the LSC topography in the NH.

Figure 2 shows the climatological wintertime mean barotropic (a vertical mean between 300 and 700 mb) streamfunction anomalies for two phases of the NAO, in which the zonal mean flow (wavenumber 0) is removed. The anomalies are actually representative of the climatological stationary waves, which are concluded to be the standing waves induced by the LSC topography in that the difference between the anomalies for two phases of the NAO is less distinct. On the other hand, we can see from Fig. 2 that for two phases of the NAO the center of the positive climatological stationary wave anomaly in the Europe–Atlantic sector is mostly located in the east side of the Atlantic basin and leaned to the Europe continent (the longitudes from 10°W to 0°). Since the action centers of the NAO anomalies have been recognized to be situated in the region from the western to central Atlantic upstream of the European continent (Wallace 2000), it is inevitable to conclude that the NAO anomalies are located upstream of the climatological stationary waves. Based upon this observational evidence, in the next section we will attempt to establish a weakly nonlinear model to provide a likely explanation as to why the westerly jet anomalies undergo a north–south shift during two phases of the NAO by extending the studies by Luo (2005) and Luo and Chen (2006).

3. Analytical solutions of planetary- and synoptic-scale fields during the NAO life cycle

In this paper, to theoretically explain the north–south shift of the zonal mean wind anomalies associated with the NAO, we will still use an equivalent barotropic vorticity equation with bottom topography as in Luo (2005) in that the NAO has generally an equivalent barotropic structure in the troposphere (Wallace 2000; Vallis et al. 2004; Gerber and Vallis 2005; Wittman et al. 2005).

In the real atmosphere, it is difficult to get an exact expression of observed zonal mean flow in that it is in general complicated (Vallis et al. 2004). In this paper, to identify the physical cause of the north–south displacement of the zonal mean wind anomalies during two phases of the NAO, the background westerly wind before the planetary- and synoptic-scale waves interact is assumed to uniform in order to allow a most simplification of the NAO dynamics. Although this assumption is not consistent with observed zonal-mean flow, it can allow us to easily understand the contributions of stationary waves and transient eddies to the zonal mean wind anomalies.

As in Luo et al. (2007a), when splitting the atmospheric streamfunction into the basic-state flow (ψ0 = −u0y) with a uniform westerly wind u0, planetary-scale wave component (ψ), and synoptic-scale wave part (ψ′), the equations of the interaction between planetary- and synoptic-scale waves can be derived under the scale separation assumption (Luo 2005). In this case, the nondimensional planetary-to-synoptic-scale interaction equations in a uniform basic flow can be expressed as (Luo 2005)
i1520-0469-64-9-3232-e1a
i1520-0469-64-9-3232-e1b
where h is the topographic variable and the other notation and boundary conditions used can found in Luo (2005) and Luo et al. (2007a).

In (1a), the term −J(ψ′, ∇2ψ′)P is a planetary-scale projection of the nonlinear self-interaction of preexisting synoptic-scale eddies that drive the NAO growth and decay (Benedict et al. 2004), which is referred to hereafter as preexisting eddy forcing. As indicated by Luo et al. (2007a), the preexisting eddy forcing plays a key role in the life cycle of the NAO anomaly with a period of about two weeks. In this paper, we will focus on identifying the physical cause of the north–south shift of the zonal wind anomaly associated with the phase of the NAO, rather than investigating the physical mechanism of the NAO onset.

Recently, in a diagnostic study Feldstein (2003) has indicated that the NAO life cycle with a period of about two weeks is driven by both high-frequency and low-frequency transient eddy fluxes. The NAO decay, however, is due to the combined influence of Ekman pumping and the low-frequency transient eddy vorticity fluxes. Unfortunately, how synoptic-scale eddies drive the NAO to grow and decay is not clear theoretically. In a recent theoretical study by Luo et al. (2007a), a weakly nonlinear NAO model is established to account for the life cycle of an NAO event. In this model, the positive and negative feedbacks of the preexisting eddy forcing (PEF) are found to determine the growth and decay of the NAO event. It is, thus, deduced that the amplification and decay of the NAO is a natural variability of synoptic-scale eddies. This result is different from the diagnostic result of the NAO by Feldstein (2003), who finds that the synoptic-scale eddies maintain the NAO anomaly, but low-frequency eddies contribute toward the decay of the NAO.

To investigate the problem noted in the above it is convenient to assume ψ = εψ̃, ψ′ = ε3/2ψ̃′, h = εh′, and ψ*S = ε5/2ψ̃*S as in Luo (2005). Because there are two continents and oceans in the mid–high latitudes in the NH, the LSC topography in the mid–high latitudes can be approximated as a wavenumber-2 topography (Charney and DeVore 1979). In this case, the bottom topography denoted by h′ is assumed to be of the form
i1520-0469-64-9-3232-e2
where h0 is the amplitude of the wavy topography, cc denotes the complex conjugate of its preceding term, xT measures the position of the dipole mode considered here relative to the LSC topography, k = 2k0 and k0 = 1/[6.37 cos(ϕ0)] is the zonal wavenumber of wave 1 around the earth at given latitude ϕ0 and m = ∓2π/Ly. It should be noted that h0 > 0 represents a trough of the LSC topography in the North Atlantic sector for m = −2π/Ly, but h0 < 0 does so for m = 2π/Ly. In this paper, we consider a weak mean westerly wind u0 that satisfies u0 = β/(k2 + m2). In this case, the planetary-scale dipole mode is quasi stationary, which is allowed to resonantly interact with the synoptic-scale eddies upstream.
Under introducing slowly varying coordinates T1 = εt, T2 = ε2t, X1 = εx, and X2 = ε2x if ψ̃ and ψ̃′ are expanded as
i1520-0469-64-9-3232-e3
then their asymptotic solutions to the planetary- and synoptic-scale streamfunctions can be derived according to Luo (2005). In this case, the following planetary-scale streamfunction solution (ψP) with a zonal mean flow used to describe the life cycle of the NAO pattern can be obtained in a fast-variable form (Luo and Chen 2006)
i1520-0469-64-9-3232-e4a
i1520-0469-64-9-3232-e4b
i1520-0469-64-9-3232-e4c
i1520-0469-64-9-3232-e4d
i1520-0469-64-9-3232-e4e
i1520-0469-64-9-3232-e4f
i1520-0469-64-9-3232-e4g
where hA = −1/[β/u0 − (k2 + m2/4)], B(t, x) = εAt, ε2t, εx, ε2x), h0 = εh0, and
i1520-0469-64-9-3232-eq1
and the other coefficients and notation can be found in Luo (2005) and Luo and Chen (2006). Note that ψA is referred to as an anomalous stationary wave, which is driven by synoptic-scale eddies and quasi-stationary in time. Actually, ψA represents an NAO anomaly. But ψC is called a climatological stationary wave because it is excited by the LSC topography and is always standing. It should be pointed out that ψm1 in (4f) represents a zonal mean flow anomaly due to the amplification of the NAO anomaly, which is actually induced by the low-frequency eddy fluxes driven by the NAO anomaly itself, and ψm2 in (4g) corresponds to a zonal mean flow anomaly driven by the interaction between the NAO anomaly (ψA) and the climatological stationary wave (ψC). As we will find latter the coupling between the anomalous stationary wave ψA and the climatological stationary wave ψC dominates the meridional shift of the westerly jet linked to the phase of the NAO.
At the same time, the synoptic-scale (ψ′) streamfunction solutions can be expressed as
i1520-0469-64-9-3232-e5a
i1520-0469-64-9-3232-e5b
i1520-0469-64-9-3232-e5c
i1520-0469-64-9-3232-e5d
i1520-0469-64-9-3232-e5e
where α = ∓1, f0(x) = a0 exp[−με2(x + x0)2], the sign of α denotes the spatial structure of preexisting synoptic-scale eddies and the other coefficients and notation can be found in Luo (2005).

In (4a), m = −2π/Ly represents the negative phase of the NAO, but m = 2π/Ly corresponds to the positive phase. The preexisting synoptic-scale eddies for α = −1 in (5b) are required to excite a negative phase NAO event, but α = 1 is required to cause a positive phase NAO event. In (5d), ψNAO is the second-order synoptic-scale eddy solution, which is induced by the interaction of an eddy-driven anomalous stationary wave (NAO anomaly) with preexisting synoptic-scale eddies (ψ1), but ψT is driven by the interaction between the climatological stationary wave (ψC) and preexisting synoptic-scale eddies. As found in Luo et al. (2007a), in our model the NAO anomaly is driven by the preexisting synoptic-scale eddies, but is not influenced by a change in the synoptic-scale transients induced by the amplified NAO anomaly. This implies that the occurrence of the NAO pattern is indeed a nonlinear initial-value problem, thus supporting the viewpoint of Benedict et al. (2004). It is interesting to note that modulated synoptic-scale eddies (ψ′) can become asymmetric due to the role of ψT when the climatological stationary wave is included. In the following sections, we will demonstrate that the meridional shift of the storm track organized by the modulated synoptic-scale eddies is dominated by the phase of the NAO and the relative position between the anomalous stationary wave (NAO anomaly) and climatological stationary wave.

It must be pointed out that because both the anomalous stationary wave and the PEF have a dipole meridional structure, the anomalous stationary wave is resonantly forced by synoptic-scale eddies under certain conditions (Luo et al. 2007a). In this case, the amplitude (B) of the anomalous stationary wave forced by both synoptic eddies and the LSC topography can be described by the following nonlinear Schrödinger (NLS) equation
i1520-0469-64-9-3232-e6
where α̃ = −{[kmhA/[4(k2 + m2 + F)]} Σn=1 n(3anbn)2〈{(nm)2 − [k2 + (m2/4)]}hA + 1〉, k − (21) = Δk, ω̃2ω̃1ω = Δω(ω ≈ 0), and other coefficients can be found in Luo and Chen (2006) and Luo et al. (2007a).

In the absence of LSC topography, (6) reduces to the NAO equation derived by Luo et al. (2007a), but reduces to the blocking equation derived by Luo and Chen (2006) if α = −1, xT = 0, and m = −2π/Ly are chosen. Moreover, it is seen that ψm that is independent of both fast variables x and t actually represents a zonal mean flow anomaly. As we will note later, since ψm1 is localized due to the forcing of local synoptic-scale transients, the eddy-driven mean flow anomaly will be inevitably localized. Ting et al. (1996) noted that the zonal westerly anomalies in the Atlantic and Pacific sectors are not well correlated with each other. Limpasuvan and Hartmann (2000) noted that in the NH the jet displacement occurs mainly over the oceanic sectors and most strongly over the North Atlantic at all levels. Recently, Vallis et al. (2004) and Gerber and Vallis (2005) indicated in a statistical–dynamical model that the localized stirring provided by baroclinic eddies tends to produce a localized jet and a dipolar circulation anomaly. Although our theoretical model is highly idealized, it can better represent the local structure of westerly jet anomalies.

According to the above expressions in (4), (5), and (6), we can account for how a westerly jet anomaly is excited and maintained during the NAO life cycle. First, it is easy to see how the anomalous stationary wave (NAO anomaly) is amplified by the synoptic-scale eddies upstream through solving Eq. (6) for given initial values. Next, the amplified anomalous stationary wave acts to maintain a mean flow anomaly and to drive the westerly jet anomaly to shift northward or southward through its interaction with the climatological stationary wave. Thus, it can be concluded that the mean flow anomalies are indirectly driven by synoptic-scale eddies through the excitation of an NAO anomaly. In principle, the westerly jet anomaly induced by the anomalous stationary wave itself and its interaction with the climatological stationary wave can also be called eddy-driven westerly jet anomaly because the anomalous stationary wave (NAO anomaly) is maintained by synoptic-scale eddies.

Following (4f) and (4g), the westerly jet anomaly uA is obtained as
i1520-0469-64-9-3232-e7a
i1520-0469-64-9-3232-e7b
i1520-0469-64-9-3232-e7c

As noted above, in (7) uNAO represents a westerly jet anomaly driven by the nonlinear self-interaction of the anomalous stationary wave for ψA (the eddy vorticity fluxes driven by the NAO anomaly itself), and uTOP corresponds to a westerly jet anomaly driven by the interaction between anomalous and climatological stationary waves. It is found that in the absence of LSC topography the eddy-driven westerly jet anomaly does not depend upon the phase of the NAO anomaly because it is independent of m in sign (the phase of the NAO anomaly) and because n(anbn) has a fixed sign. However, in the presence of LSC topography the strength and position of the westerly jet anomaly can be dependent upon the phase of the NAO anomaly. Of course, the position and intensity of the westerly jet anomaly is dominated by the amplitude of the climatological stationary wave (the height of the LSC topography) and its position relative to the anomalous stationary wave. Given xT = 0 in (2), h0 > 0 and h0 < 0 are required for the negative and positive phases, respectively, of the NAO anomaly in order to allow the LSC topography to have a trough in the Atlantic sector. In this paper, without the loss of generality B(x, 0) = 0.35 is considered as an initial value of the NAO for its two phases. Also, the parameters a0 = 0.17, ε = 0.24, μ = 1.2, and xT = 2.87/2 are fixed as in Luo et al. (2007a).

4. NAO-like events and modulated storm tracks

a. LSC topography and the climatological stationary wave

For negative [m = −(2π/Ly) and h0 = 0.4] and positive [m = (2π/Ly) and h0 = −0.4] phases of the NAO, the LSC topography (h) and the induced stationary wave anomaly (ψC) are shown in Fig. 3 for xT = 0 and xT = −2.87/2, respectively. Note that the dashed lines represent the Atlantic sector and the solid lines denote the continents. It is found that for xT = 0 the topographically generated stationary monopole wave ψC has a positive anomaly in the topographic trough centered at x = 0, which is called climatological stationary wave because it is always standing. As a result, the anomalous stationary wave (B) is in phase with the climatological stationary wave. If xT = −2.87/2 is chosen, the topographically induced positive anomaly shifts eastward so that the anomalous stationary dipole (high-over-low) wave is located upstream of the topographically induced standing ridge (climatological stationary wave). For this case, a standing trough is situated in the east coast of the North America. This topographically generated wave pattern is similar to the observed climatological stationary waves found in Fig. 2.

b. Negative-phase NAO events and the associated storm track

For the negative phase of the NAO, if xT = −2.87/2 and h0 = 0.4 are chosen for the other parameters same as in Luo et al. (2007a), then the planetary- and synoptic-scale fields and total field are shown in Fig. 4 during the interaction between the planetary-scale wave and synoptic-scale eddies.

It is found that the planetary-scale filed looks similar to an NAO− pattern, a life cycle of a blocking high, driven jointly by synoptic-scale eddies and LSC topography, which is also similar to that observed by DeWeaver and Nigam (2000a, their Fig. 11b). The storm track organized by synoptic-scale eddies modulated by this NAO pattern splits into two branches around the NAO− pattern, in which one branch drifts along the northeast direction, and the other drifts southeastward. The northern branch is stronger than the southern one. Such a feature is consistent with the climatology of storm track associated with the NAO− pattern (Jung et al. 2003). Moreover, the total field shown in Fig. 4c is also found to resemble the life cycle of an NAO− event (Benedict et al. 2004).

Figure 5 shows the planetary- and synoptic-scale fields for xT = 0. We see that the planetary-scale field bears a striking resemblance to the life process of a typical omega-type blocking, and the organized storm track is still split into two branches, in which the northern one is dominant. In this case, the storm track exhibits a poleward shift. We can also note that the omega-type blocking structure is similar to a composite of NAO events presented by Benedict et al. (2004) and Luo et al. (2007b). As demonstrated by Luo (2005), the feedback of the omega-type blocking can result in a northward shift of synoptic-scale eddies around the blocking region. Thus, the northward displacement of the storm track is attributed to the feedback of the NAO− pattern like an omega-type blocking. Jung et al. (2003) noted that during the period 1958–77 the main NAO-related deep cyclone track was aligned from the Labrador Sea, over Iceland into the Arctic, but during the period 1978–97 the main NAO-related high-latitude storm track was more zonally oriented. Obviously, this observational fact can be explained by our theoretical result here because the negative phase NAO events are dominant during the period 1958–77. If there is a large phase mismatch between the anomalous and climatological stationary waves, the southern branch of the storm track can become dominant, as shown in Fig. 6 for xT = −2.87. Even so, the planetary-scale field exhibits still an NAO− pattern. The spatial structure of the planetary-scale NAO− pattern is also dominated by the relative position between the anomalous and climatological stationary waves.

As noted above, because ψC is always standing, the spatial evolution of the NAO pattern is actually represented by the anomalous stationary wave ψA driven by synoptic eddies. To confirm that the NAO is a localized low-frequency mode, we plot ψA and ψA + ψC in Fig. 7 for the same parameters as in Fig. 4. It is noted that at initial stage (day 0) the dipole structure in the planetary-scale field is zonally uniform. The high-over-low dipole structure at x = 0 tends to amplify and to be zonally localized through the interaction with synoptic-scale eddies upstream, which also exhibits a life cycle with a period of nearly two weeks (Feldstein 2003; Benedict et al. 2004). In a highly simplified model, Vallis et al. (2004) found that the zonally localized dipole mode resembling an NAO pattern tends to be excited by the zonally localized large-scale stirring from baroclinic eddies in the storm track. Thus, this suggests that the NAO pattern should be a localized mode. At the same time, the westward movement of this amplified dipole structure is observed. Thus, the growth and decay of the anomalous stationary wave forced by synoptic eddies characterizes the life cycle of an NAO anomaly. Interestingly, once the climatological stationary monopole wave is included, the anomalous plus climatological stationary wave anomaly (ψA + ψC) exhibits an asymmetric dipole structure (Fig. 7b). However, in the real atmosphere the composite of the height anomalies of NAO events actually corresponds to the anomalous stationary wave ψA because ψC as a climatological mean field has been removed. Even so, as we will see later the climatological stationary wave ψC plays a very important role in the north–south variability of the zonal wind anomaly associated with the phase of the NAO (DeWeaver and Nigam 2000a).

c. Positive phase NAO events and the associated storm track

For the positive NAO phase (α = 1 and m = 2π/Ly), Fig. 8 shows the planetary- and synoptic-scale fields and the total field during the interaction between planetary- and synoptic-scale waves for the parameters xT = −2.87/2 and h0 = −0.4. We see that the planetary-scale field exhibits a low-over-high structure, an NAO+ pattern (DeWeaver and Nigam 2000a, their Fig. 11a). Simultaneously, the organized storm track shifts northeastward (Fig. 8b), consistent with the finding of Hurrell et al. (2003), who pointed out that the positive NAO index winters are associated with a northeastward shift in the Atlantic storm activity with enhanced activity from Newfoundland into northern Europe. The total field is also found to be similar to the life cycle of an observed NAO+ event (Luo et al. 2007a). If the climatological stationary wave ψC is in phase with the anomalous stationary wave ψA or if xT = 0, the planetary-scale field still has a low-over-high structure in the Atlantic sector (Fig. 9a), but the high pressure to the south of the low is rather strong. For this case, a zonal movement of the storm track is dominant before the NAO+ anomaly matures, but this storm track shifts northeastward accompanying the further decay of an NAO+ anomaly (Fig. 9b). When ψC is removed, ψA exhibits an antisymmetric dipole (low-over-high) anomaly, a typical NAO+ anomaly.

It is interesting to note that if the climatological stationary wave ψC has a large phase difference relative to the anomalous stationary wave ψA, the storm track exhibits a noticeable northeastward shift, as shown in Fig. 10 for xT = −2.87. In this case, the planetary-scale field seems to resemble more the NAO+ pattern observed by DeWeaver and Nigam (2000a). Thus, a comparison with Figs. 8 and 9 suggests that the northeastward shift of the storm track modulated by the NAO+ anomaly is, to large extent, dominated by the relative position between the anomalous and climatological stationary waves.

It is worthy to point out that during the onset of the NAO− pattern the northward advection of the warm air and the advection of the cold air toward the middle latitude North Atlantic are observed, displaying a feature of the cyclonic wave breaking observed by Benedict et al. (2004), who noted that the NAO− pattern may arise from the cyclonic wave breaking. In contrast, the occurrence of NAO+ pattern accompanies the advection of the warm air toward the subtropical and midlatitude North Atlantic and the northward advection of the cold air. This process is similar to the main characteristics of observed NAO+ events. However, as pointed out by Luo et al. (2007a), the north–south movement of warm and cold airs linked to the phase of the NAO is attributed to the feedback of the NAO anomaly. Such a behavior of warm and cold airs can hardly be found if the eddies induced by the feedback of the NAO anomaly (term ψ2) are neglected (not shown). This indicates that the synoptic-scale wave breaking associated with the NAO pattern is a result of the NAO occurrence rather than a cause. In fact, according to the definition of wave breaking we can conclude that the synoptic-scale wave breaking is intense during the NAO− life cycle in that the northward displacement of the warm air and the southern intrusion of the cold air toward the middle latitude North Atlantic frequently occur during the growth phase of the NAO− anomaly. But, the synoptic- scale wave breaking seems weak during the NAO+ life cycle because the air displacement is reversed to that during the NAO− life cycle. This theoretical finding is consistent with the observational result obtained by Woollings et al. (2007), who find that the wave breaking is frequent during the NAO− life cycle, but infrequent during the positive NAO phase.

It should be pointed out that when the eddy forcing from preexisting synoptic- scale eddies is stronger, that is, when the storm track is stronger, the eddy-driven dipole anomaly is inevitably stronger (not shown). Thus, it is concluded that there should be two localized dipole modes corresponding to the NAO and North Pacific Oscillation (NPO) occurring in the Atlantic and Pacific basins, respectively, in that the storm tracks in the NH are mostly located in the upstream sides of the Atlantic and Pacific basins. However, because the winter storm track is much stronger in the Atlantic sector than in the Pacific sector due to the midwinter suppression of the Pacific storm track (Nakamura 1992; Cash et al. 2005), the NAO anomaly is necessarily much stronger than the NPO anomaly (not shown). This provides an explanation as to why the NAO is a dominant teleconnection pattern in the NH. Of course, there are other possible factors such as El Niño–Southern Oscillation (ENSO), the Pacific–North American (PNA) pattern and so on to affect the intensity of the NPO, as pointed out by Vallis et al. (2004). This problem is beyond the scope of the present study, which deserves further investigation.

5. Meridional displacement of westerly jet anomalies during two phases of NAO

For the same parameters as in Figs. 4 and 8, the westerly jet anomalies (uA) are shown in Fig. 11 for two phases of the NAO. It is seen that for the negative phase there is an enhancement of the zonal mean zonal wind in the higher and lower latitudes, respectively, in which the mean westerly wind is stronger in the lower latitudes than in the higher latitudes, in agreement with the observation in Fig. 1. When combining with the climatological mean westerly jet, the organized westerly jet will shift southward. For the positive phase, two positive anomalies of the zonal mean westerly wind are observed in higher and lower latitudes, respectively, but the positive anomaly is much stronger in the higher latitudes than in the lower latitudes (Fig. 10b). This suggests that the westerly jet tends to shift poleward as the NAO is in a positive phase. These theoretical results are qualitatively in agreement with the observational fact in Fig. 1.

On the other hand, we can also see that if the climatological stationary wave is in phase with the anomalous stationary wave (xT = 0), the zonal mean wind anomalies exhibit clearly a dipole structure for two phases of the NAO. For the negative phase a negative-over-positive westerly anomaly is dominant in a region from the middle latitudes to the lower latitudes (Fig. 12a), but the zonal mean wind anomaly exhibits a strong positive anomaly in the higher latitudes for the positive phase (Fig. 12b). Thus, it is natural that the zonal mean wind anomaly exhibits a poleward drift as the NAO transits from a negative phase to a positive phase, consistent with the observational finding shown in Fig. 1.

Thus, it is inferred that whether the zonal mean wind anomalies exhibit a robust poleward shift depends crucially upon the phase of the NAO and the relative position between the anomalous and climatological stationary waves. However, as indicated by Luo et al. (2007a), in the absence of both the preexisting jet and climatological stationary wave, double jets can form due to the synoptic-scale eddy driving, but are independent of the NAO phase. In this case, no meridional shift of westerly jet anomalies is observed in the absence of the climatological stationary wave although the transient eddies help maintain the zonal mean wind anomalies through the excitation of the NAO anomaly (Luo et al. 2007a, their Fig. 9). Such a north–south displacement of the westerly wind anomaly associated with the dipole anomaly is less distinct in the SH in that the climatological stationary wave is weak (Bals-Elsholz et al. 2001; Lee and Kim 2003). This provides a likely explanation for why the north–south shift of the westerly jet associated with the NAM in the NH is more obvious than that in the SH (Limpasuvan and Hartmann 2000; Lorenz and Hartmann 2003).

6. Conclusions and discussion

In this paper, the theoretical NAO model proposed by Luo et al. (2007a) is directly extended to include the climatological stationary waves induced by the land–sea contrast (LSC) topography in order to gain insight into why the north–south shift of the westerly jet anomalies accompanies the phase of the NAO. It is found that the coupling between the anomalous stationary dipole wave resembling an NAO anomaly and topographically generated climatological stationary monopole wave seems to dominate the north–south variability of the westerly jet anomaly although the transient eddies can maintain the zonal mean wind anomalies through the excitation of the NAO anomaly.

The meridional structure of the westerly jet anomaly is found to depend strongly upon the phase of the NAO and the relative position between the eddy-driven anomalous stationary wave and topographically generated climatological stationary wave. When the NAO is in positive phase, the westerly jet shifts poleward. But for the negative phase, the westerly jet shifts southward. The breaking of synoptic-scale waves during the NAO− life cycle is attributed to the feedback of the NAO− anomaly, and accompanies the equatorward shift of the westerly69 jet anomaly. When the eddy-driven anomalous stationary wave that represents an NAO anomaly is located upstream of the topographically generated stationary wave, the storm track can show a northeastward shift for the positive NAO phase. But for the negative phase the organized storm track splits into two branches, in which the northern one is generally stronger than the southern one. The southern one can be dominant compared to the northern counterpart if there is a large phase mismatch between the eddy-driven NAO pattern and the topographically driven stationary wave.

The present theoretical study also reveals that the NAO anomaly is a zonally localized low-frequency dipole mode for two phases of the NAO. Because of the role of the land–sea contrast topography in the NH, the distinction between the subtropical jet and the eddy-driven midlatitude jet becomes less clear compared to the SH where transient eddies tend to maintain double jets (Bals-Elsholz et al. 2001; Lee and Kim 2003).

Although some of theoretical blocking models in previous studies can explain the onset of omega-type blocking (Charney and DeVore 1979), they fail to describe the localized structure of the omega-type blocking and its life cycle with a period of nearly two weeks. This problem can be solved by the block–eddy interaction model proposed by Luo (2005). More recently, in Luo et al. (2007a, b) the block–eddy interaction model has been directly extended to investigate the dynamics of dipole modes associated with the two phases of the NAO. Unfortunately, this NAO model cannot capture some characteristics of the NAO anomaly such as the meridional displacement of associated westerly jet anomalies. This problem seems likely to be solved by the present study.

In a stochastic–dynamical model, Vallis et al. (2004) found that the fluctuating stirring from baroclinic eddies tends to produce both a variation in the intensity and position of the zonal jet, and a dipolar circulation anomaly. If the stirring forcing is enhanced in a zonally localized region, the resulting variability pattern is zonally localized, resembling an NAO pattern. In the present model, a localized dipole mode having a period of nearly two weeks can be excited by the preexisting synoptic-scale eddies upstream of the Atlantic basin. That is to say, when the preexisting eddy forcing from synoptic eddies is zonally localized, the resulting dipole mode must be zonally localized. This result is consistent with the finding by Vallis et al. (2004). However, the model of Vallis et al. (2004) cannot reflect the variability of synoptic-scale eddies (Atlantic storm track) induced by the NAO anomaly. But this problem can be avoided by our present model.

In a recent stochastic model, Wittman et al. (2005) suggested that the dipolar meridional structure of the AO/NAM is likely due to a variation in the mean position of the jet. This result is significantly different from our finding here. As mentioned above, the preexisting synoptic-scale eddies that mimic the Atlantic storm track tend to excite a low-frequency dipole mode resembling the NAO pattern, and a zonal mean jet appears as a response to the occurrence of an NAO anomaly. At the same time, a variability in the synoptic-scale eddies (storm track) is induced by the NAO anomaly. Thus, the relationship between the NAO, jets, and storm tracks can be clearly seen in our present model.

On the other hand, it must be pointed out that since an equivalent barotropic model is applied, the meridional shift of the baroclinicity associated with the phase of the NAO/NAM (AO/AM), as noted in Lorenz and Hartmann (2003), cannot be represented by our theoretical model in this paper. In addition, how the variability of stratospheric polar vortex impacts on the occurrence of the NAO/NAM in the troposphere is unclear although the NAO and stratospheric polar vortex are coupled in essence. These problems deserve further investigation.

Acknowledgments

The authors acknowledge the support from the National Outstanding Youth Natural Science Foundation of China under Grant 40325016, the National Natural Science Foundation of China (4057016) and Taishan scholar funding. The authors also gratefully acknowledge discussions with Drs. Bin Wang (UH) and Lixin Wu (OUC). The authors thank two anonymous reviewers for their useful suggestions in improving this paper.

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Fig. 1.
Fig. 1.

Composite of zonal mean zonal winds at 300 mb in the Atlantic sector (90°W–0°) during the positive and negative phases of the NAO life cycle, in which the solid and dashed curves denote the positive and negative phases, respectively, and the point line correspond to the difference between positive and negative phases.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 2.
Fig. 2.

Climatological wintertime mean barotropic streamfunction anomalies (in 107 m2 s−1 for (a) positive and (b) negative phase of the NAO, in which the solid and dashed lines represent positive and negative anomalies, respectively.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 3.
Fig. 3.

Horizontal distribution of land–sea contrast topography with wavenumber 2 for the parameters with m/2 = −(π/Ly) and h0 = 0.4 or m/2 = (π/Ly) and h0 = −0.4 in which the solid and dashed lines represent the topographic ridge and trough, respectively, and ψC anomalies of the topographically induced climatological stationary wave in which the solid and dashed curves represent the positive and negative anomalies, respectively: (a) topographic distribution for xT = 0, (b) topographic distribution for xT = −2.87/2, (c) ψC anomaly for xT = 0, and (d) ψC anomaly for xT = −2.87/2. The contour interval (CI) is 0.2.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 4.
Fig. 4.

Instantaneous fields of the interaction between planetary- and synoptic-scale waves during the negative phase of the NAO for m = −(2π/Ly), h0 = 0.4, and xT = −2.87/2: (a) planetary-scale field (CI = 0.15), (b) synoptic-scale field, in which the dashed and solid lines represent the cyclone and anticyclone, respectively (CI = 0.3), and (c) total field (CI = 0.3).

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 5.
Fig. 5.

As Fig. 4 but for xT = 0: (a) planetary-scale field (CI = 0.15) and (b) synoptic-scale field (CI = 0.3)

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 6.
Fig. 6.

As Fig. 5 but for xT = −2.87.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 7.
Fig. 7.

Instantaneous anomalies of anomalous stationary waves and anomalous plus climatological stationary waves for the same parameters as in Fig. 4, in which the solid and dashed lines denote the positive and negative anomalies respectively: (a) ψA anomaly (CI = 0.2) and (b) ψA + ψC anomaly (CI = 0.2).

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 8.
Fig. 8.

Instantaneous fields of the interaction between planetary- and synoptic-scale waves during the positive phase of the NAO for m = (2π/Ly), h0 = −0.4, and xT = −2.87/2: (a) planetary-scale field (CI = 0.15), (b) synoptic-scale field (CI = 0.3), and (c) total field (CI = 0.3).

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 9.
Fig. 9.

As Fig. 8 but for xT = 0: (a) planetary-scale field (CI = 0.15) and (b) synoptic-scale field (CI = 0.3).

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 10.
Fig. 10.

As Fig. 9 but for xT = −2.87.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 11.
Fig. 11.

Instantaneous fields of zonal mean wind anomalies for the negative phase of the NAO shown in Fig. 4 and for the positive phase shown in Fig. 8, in which the dashed and solid lines correspond to the negative and positive westerly anomalies, respectively: (a) negative phase (CI = 0.1) and (b) positive phase (CI = 0.1).

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

Fig. 12.
Fig. 12.

As Fig. 11 but for xT = 0.

Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS3998.1

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