In this paper, it is demonstrated by using a weakly nonlinear model that low-frequency annular modes with a dipole meridional structure can be excited by the eddy fluxes from synoptic-scale eddies. If the storm track organized by synoptic- scale eddies is zonally confined in a narrow localized region as observed in the Northern Hemisphere (NH), the eddy-driven dipolar pattern will exhibit both a relatively short zonal scale and a zonal asymmetry during its life cycle, which is attributed to the strong downstream energy dispersion of Rossby waves. Such a zonal asymmetry is found to be sensitive to the relative position between the preexisting storm track and dipole mode. However, if the preexisting storm track is zonally confined in a rather wide localized region, as observed in the Southern Hemisphere (SH), the Rossby wave dispersion almost disappears. In this case, the eddy-driven dipole mode with a relatively large zonal scale exhibits a zonal symmetry, which is almost insensitive to its position relative to the preexisting storm track. This sheds light on why eddy-driven dipole modes in the SH are more likely to be annular modes.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Observations show that in winter the large-scale circulation in the Northern and Southern Hemisphere (NH and SH respectively, hereafter) troposphere exhibits a prominent month-to-month variability, which is dominated by a strong zonally symmetric or annular component [Thompson and Wallace, 1998, 2000]. In recent years, there is a debate in some literatures as to whether the NH and SH annular modes (SAM and NAM respectively, hereafter) are fundamentally zonally symmetric or asymmetric modes of variability [Wallace, 2000]. The NAM is more zonally asymmetric near the surface than the SAM because of the greater zonal asymmetry of the NH lower boundary, as noted by Cash et al. . The strong zonal symmetry of the SAM is probably due to the effect of less topography in the SH. However, model experiments with zonally symmetric boundary conditions indicate that there is still a robust zonal asymmetry of atmospheric variability [Hendon and Hartmann, 1985; Cash et al., 2002]. Thus, it is conjectured that the lower boundary condition in the NH is not the only factor for generating the zonal asymmetry of the NAM. Apart from producing stationary planetary waves that give rise to the asymmetric structure of the NAM [Körnich et al., 2006], the land-sea topography in the NH tends to zonally localize the storm tracks in the upstream sides of the ocean basins. In numerical experiments, Vallis et al.  and Cash et al.  noted a positive link between the strength of the EOF asymmetries of the NAM and the strength of the zonal localization of the storm tracks. However, it is not clear why the zonal localization of storm tracks can influence the longitudinal distribution of annular modes. The aim of this paper is to provide a theoretical explanation for why the zonal localization of the storm tracks can result in the zonal asymmetry of annular modes and why eddy-driven low-frequency dipole modes in the SH are more likely to be annular.
2. Weakly Nonlinear Eddy-Driven Annular Mode Model
2.1. Analytical Solution of Annular Modes
 It has been recognized in both observations and models that both the NAM and SAM are equivalent barotropic [Thompson and Wallace, 1998; Cash et al., 2002] and eddy-driven [Limpasuvan and Hartmann, 1999, hereinafter referred to as LH99; Feldstein, 2003]. However, a prominent difference between the storm tracks in the NH and SH is that the NH storm tracks are zonally distributed in a relatively narrow region, but the SH storm tracks seem to distribute more uniformly along the midlatitude circle even though there is a regional variability of the eddy activity [Williams et al., 2007, Figure 2]. Thus, we infer that the discrepancy between annular modes in the NH and SH are probably related to the distinctness of storm tracks in the NH and SH.
 It is possible that the weakly nonlinear North Atlantic Oscillation (NAO) model established by Luo et al. [2007a] can be used to examine how the spatial structure of annular modes depends upon the strength of the storm track localization because the NAM and the NAO are in fact two sides of one coin [Wallace, 2000].
 As was done by Luo et al. [2007a], the planetary-scale wave solution (ψP) and the evolution equation of eddy-driven dipole anomaly in a nondimensional form can be obtained as
where m = ±2π/Ly, Ly is the width of the β plane channel used, cc denotes the complex conjugate of its preceding term, B represents the amplitude of an eddy-driven dipole anomaly, ψm denotes the mean flow anomaly induced by the eddy-driven dipole mode, whose um = −∂ψm/∂y exhibits a double jet anomaly during the dipole mode life cycle [Luo et al., 2007a], u0 is the uniform mean westerly wind and the other parameters such as qn and gn are given in Luo  and Luo et al. [2007a]. Note that f0(x) = a0 exp[(x + x0)2] for = 0.24, as given in Luo et al. [2007a], denotes the spatial distribution of the storm track eddies of an intensity of a0, in which μ reflects the localization strength of the preexisting storm track and x0 measures the relative position between the preexisting storm track center and dipole anomaly.
 In this paper, a finite-difference scheme, as used by Luo  and Luo et al. [2007a], is used to solve equation (1d) for any initial condition. At the same time, the initial condition of the dipole anomaly is chosen to be a linear Rossby wave in order to see if an isolated dipolar pattern from an initial dipole anomaly (a linear Rossby wave) can be excited by the forcing of preexisting synoptic-scale eddies in the form of
where i (i = 1, 2) is the zonal wavenumber of the synoptic-scale eddies and the other notation can be found in by Luo  and Luo et al. [2007a].
2.2. Choice of the Zonal Scales of the Preexisting Eddy Forcing and Planetary Wave
 Although the storm track organized by synoptic-scale eddies and associated dipole mode represented by a developing quasi-stationary wave are dependent on each other [Held et al., 2002], this problem has been discussed by Luo et al. [2007a, 2007b] and will not be examined here because it is beyond the scope of this paper.
 Observed annular modes are dominated by zonal wavenumber one (LH99), while the preexisting dipole modes should have smaller scale on the NH than on the SH. In this case, the preexisting eddy forcing is chosen to be wavenumber 1 so as to allow an excitation of annular modes with the same wavenumber [Ting et al., 1996].
 Without the loss of generality, as an example a high phase annular mode (m = 2π/Ly and α = 1) with zonal wavenumber 1 is only considered here. In fact, a similar treatment is also feasible for a low phase (not shown). In addition, we fix the parameters k = k0, 1 = 9k0, 2 = 10k0, a0 = 0.2, x0 = 0, u0 = 1.1 and Ly = 6 (11 m/s and 6000 km in the dimensional form), but allow a varying parameter μ to perform a parameter-sweep experiment.
3. Relationship Between the Longitudinal Distribution of Annular Modes and the Zonal Localization and Position of Localized Storm Tracks
 As an example, for μ = 2.4, μ = 0.8 and μ = 0.12 the initial field of ψ′0 is shown in Figure 1. It is noted that the storm tracks organized by preexisting synoptic-scale eddies from μ = 0.8 to μ = 2.4 are close to those observed in NH, but μ = 0.12 crudely corresponds to the case in SH.
 To clearly see the role of the storm track localization, it is useful to look at the time evolution of M(x, t) = ∣B∣ under the eddy forcing by allowing the initial conditions of dipole modes to be the same for three cases considered above. For B(x, 0) = 0.4, M(x, t) is shown in Figure 2 for μ = 2.4, μ = 0.8 and μ = 0.12.
 It is found that a spatially isolated wave can be formed by the zonally localized storm tracks because storm tracks are zonally localized in two Hemispheres, implying that annular modes in two Hemispheres are spatially isolated. The zonal wave width of this isolated wave increases as μ deceases. Thus, it is likely that strong isolated dipole modes are easily formed in the NH because the NH storm track for μ = 2.4 and μ = 0.8 is more zonally localized than the SH counterpart for μ = 0.12. On the other hand, we note that the energy dispersion of the spatially isolated wave is more distinct as the zonal localization of the storm track or μ increases, as shown in Figure 2 for μ = 2.4 and μ = 0.8. This process tends to cause a strong zonal asymmetry of annular modes, as observed in the NH. This may be a possible reason for why the NAM becomes more easily zonally asymmetric.
Figure 3 shows the eddy-driven dipole anomaly ψA for μ = 2.4, μ = 0.8 and μ = 0.12. It is found that for μ = 2.4 the eddy-driven dipole mode is almost standing and has a relatively short zonal scale during the growing phase, but exhibits a downstream development during the decaying phase. Such a feature becomes less distinct as μ further decreases, which is clear in Figure 3c for μ = 0.8. Hence, eddy-driven dipole mode can become zonally asymmetric, particularly during the decaying stage, as μ varies between μ = 2.4 and μ = 0.8. However, for μ = 0.12 the eddy-driven dipole mode exhibits a rather large zonal scale and non-dispersion, and its zonal symmetry is dominant during its total process, which looks like that in the SH. The mean of these events will exhibit inevitably a zonally symmetric structure [Cash et al., 2002]. Another interesting point is that the eddy-driven dipole mode is stronger for μ = 0.12 than those for μ = 2.4 and μ = 0.8 because the preexisting eddies for this case seem to contain more total energy. However, the strength difference between the annular modes in the NH and SH is less evident in that the storm tracks are generally stronger in the NH than in the SH [Williams et al., 2007].
 If the eddy amplitude f0(x) is taken as f0(x) = b0 + a0 exp[−μɛ2(x + x0)2] for which b0 is a constant and denotes a uniform eddy activity, then ψ′0 can better represent the preexisting storm track in the SH. However, if μ is increased, the eddy-driven dipole mode can show a local dipole and is more likely to be zonally asymmetric (not shown). In other words, the regional variation of the storm track tends to produce a local eddy-mean flow interaction and then form a zonally asymmetric, localized dipole [Cash et al., 2002]. In the NH eddy-driven dipole modes become more zonally asymmetric in that the storm tracks are more zonally localized. This provides a theoretical explanation for why eddy-driven dipole modes are more likely to be annular in the SH [Thompson and Wallace, 1998, 2000; LH99].
 For μ = 2.4 and the other parameters same as in Figure 3, eddy-driven dipole anomaly ψA is shown in Figure 4 for x0 = 2.87/2 and x0 = 2.87 respectively. It is shown that the zonal asymmetry of the NAM is sensitive to the relative position between the preexisting storm track and the center of dipole modes. However, this phenomenon cannot be observed if the storm track is zonally distributed in a rather wide region as in the SH (not shown). Thus, it is inferred that a strong zonally symmetric dipolar pattern with a relatively large zonal scale is easily formed in the SH. In the atmosphere, observed annular modes are zonally isolated in both hemispheres [LH99; Cash et al., 2002], in which the NAM exhibits a strong zonal asymmetry compared to the SAM. Our study here documents that the longitudinal structure of annular modes is, to a large extent, dominated by the strength of the zonal localization of the storm track (Figure 2). The present theoretical model can not only capture the characteristics of annular modes in the NH and SH noted by Cash et al.  and Vallis et al. , but also identify the underlying mechanism of the zonal asymmetry of the NAM. It is shown that the energy dispersion of Rossby waves in a rather localized storm track is more likely to cause a strong zonal asymmetry of the NAM. This result is also correct for other choices of parameters (not shown).
4. Conclusion and Discussions
 In this paper, the impact of zonally localized storm tracks, as observed in NH and SH, on the longitudinal distribution of annular modes is investigated using a highly idealized, weakly nonlinear theoretical model proposed by Luo et al. [2007a]. It is found that if the preexisting storm track organized by synoptic-scale eddies is zonally confined in a relatively narrow region as observed in the NH, eddy-driven dipole modes can exhibit a relatively short zonal scale and a strong zonal asymmetry, which are sensitive to the relative position between the zonally localized storm track and the center of annular modes. At the same time, eddy-driven dipole modes can undergo strong downstream energy dispersion during the decay stage, thus easily causing a strong zonal asymmetry of dipole modes in a time mean sense. However, if the preexisting storm track is zonally distributed in a rather wide region as observed in the SH, eddy-driven isolated dipolar pattern can have both a strong zonal symmetry and a relatively large zonal scale, which is almost insensitive to its position relative to the preexisting storm track because the downstream energy dispersion of Rossby waves does not easily take place. This provides a theoretical explanation for why eddy-driven dipole modes in the SH are more likely to be annular.
 On the other hand, Ting et al.  noted that the linear stationary wave activity in a response to the fluctuations in zonal-mean mean flow can exhibit a zonal asymmetric structure. In this paper, a relatively symmetric dipole mode is chosen as a preexisting stationary wave in order to highlight the role of the zonal localization of the storm track. Such a consideration can help us understand what factors influence the zonal asymmetry of annular modes. On this basis, we could theoretically identify why the NAM can exhibit a zonal asymmetry found in a numerical models with relatively uniform lower boundaries [Vallis et al., 2004; Cash et al., 2005]. Nevertheless, the asymmetric variability of observed annular modes in the real atmosphere is a rather complicated process, which deserves further study.
 The authors acknowledge the support from the National Natural Science Foundation of China (4057016) and the National Outstanding Youth Natural Science Foundation of China under Grant 40325016, Taishan Scholar funding and FANEDD. The work of Wen Zhou is supported by City University of Hong Kong research grant (7002136). Two anonymous reviewers are highly appreciated for their useful suggestions in improving this paper.