Exploring the Phase-Strength Asymmetry of the North Atlantic Oscillation Using Conditional Nonlinear Optimal Perturbation

The North Atlantic Oscillation (NAO) is a very important low-frequency variability mode of the Northern Hemisphere characterized by a meridional dipole pattern in the sea level pressure field and a meridional shift of the westerly jet in the North Atlantic region (Walker and Bliss, 1932; Feldstein, 2003; Luo et al., 2007a, 2007b). It affects regional weather and climate (Hurrell, 1995; Hurrell and van Loon, 1997; Pablo and Soriano, 2007; López-Moreno and Vicente-Serrano, 2008; Song et al., 2011), and even global circulation (Wallace, 2000; Jeong and Ho, 2005; Hong et al., 2008; Wang et al., 2013). The NAO has a wide range of variability from several days to decades (Feldstein, 2003; Ostermeier and Wallace, 2003; Moore et al., 2013).

In recent years, a series of studies have been focused on the characteristics and mechanisms of NAO events on the intraseasonal time scale. (Feldstein, 2003) suggested that NAO events complete their life cycle in about two weeks. (Benedict et al., 2004) investigated the synoptic characteristics of individual NAO events and found that the formation of NAO events originate from synoptic-scale waves. Franzke et al. (2004), using numerical experiments, further verified the importance of the latitudinal position of perturbations relative to the climatological Atlantic jet in triggering NAO events. (Rivière and Orlanski, 2007) emphasized the importance of high-frequency momentum flux on the NAO pattern by analyzing reanalysis data and numerical sensitivity experiments.

Considering that the NAO is a nonlinear initial-value problem (Benedict et al., 2004; Franzke et al., 2004), (Luo et al., 2007a) established a weakly nonlinear nondimensional barotropic model with scale-separation and uniform westerly wind assumptions and clarified the dynamical mechanism of synoptic-scale waves driving the life cycle of the NAO with a period of nearly two weeks. Furthermore, (Luo and Cha, 2012) extended the above model by removing the uniform westerly wind assumption, and explored the effect of the meridional shift of the North Atlantic jet from its mean position on the formation of different NAO phases. Subsequently, (Drouard et al., 2013) performed short-term simulations using a three-level quasi-geostrophic global model and analyzed the propagation of synoptic waves in the eastern Pacific in the presence of a large-scale ridge or trough anomaly and their downstream impact on the NAO.

Many studies have illustrated the feedback of eddies on the persistence of low-frequency patterns (Robinson, 2000; Feldstein, 2003; Gerber and Vallis, 2007). (Luo et al., 2007a) theorized that the eddy forcing arising from pre-existing synoptic-scale waves is crucial for the growth and decay of the NAO. Besides, they found that negative-phase NAO (NAO-) events occurred repeatedly within the NAO region after the initial NAO- event had decayed. However, for positive-phase NAO (NAO+) events, isolated dipole blocking downstream of the North Atlantic region could occur after the initial NAO+ event had decayed. These results imply that longer-persisting NAO- events are more easily maintained than NAO+ events. Furthermore, (Barnes and Hartmann, 2010) explored the persistence of the NAO using observations of the 3D vorticity budget in the Atlantic sector. They pointed out that the eddy vorticity flux plays a positive eddy feedback role in the midlatitude region and is strongest during the negative-phase NAO, which induces the greater persistence of this phase of the NAO. (Jiang et al., 2013) explored the dynamics of the onset of NAO with the conditional nonlinear optimal perturbation (CNOP) method. By comparing the linear and nonlinear evolutions of CNOP, they pointed out that the nonlinear processes during positive- and negative-phase NAO onset may play different roles. In fact, by comparing Figs. 6b and 7b in the work of (Feldstein, 2003), it is apparent that the nonlinear interaction between eddies is stronger during negative NAO events than positive NAO events.

In the present work, we extend the study of (Jiang et al., 2013) and explore the role of nonlinear processes in NAO phase-strength asymmetry (i.e. the fact that negative NAO events are usually stronger than positive ones) by comparing NAO events triggered by linear optimal perturbation and CNOP. Short-term simulations with a three-level quasi-geostrophic global model are carried out.

The remainder of the paper is organized as follows: In section 2, the data and method are presented. An analysis of the observational composite NAO events is shown in section 3, followed in section 4 by a presentation of the CNOPs triggering NAO onset and their nonlinear evolution. The role of nonlinear processes is revealed in section 5 by comparison with linear optimal perturbation. Finally, conclusions are given in section 6.

The daily NAO index from the National Weather Service Climate Prediction Center (CPC) is used to define the NAO events. The observational streamfunction fields at 200, 500, and 800 hPa are derived from the daily 0000 UTC ERA-Interim reanalysis of the European Center for Medium-Range Weather Forecasts (ECMWF) (Dee et al., 2011). Our focus is on boreal winter in the months of December-February (DJF) for the period 1979-2006.

The model used is a three-level quasi-geostrophic global spectral model proposed by (Marshall and Molteni, 1993), which has already been used in previous studies of the NAO (Jiang et al., 2013; Drouard et al., 2013). The model equations are as follows:

\begin{equation} \label{eq1} \dfrac{\partial Q_i}{\partial t}=-\bm{J}(\psi_i,Q_i)-\bm{D}_i(\psi_1,\psi_2,\psi_3)+\bm{S}_i . \end{equation}

Here, the index i=1,2,3 refers to 200, 500, and 800 hPa, respectively; Q_i and ψ_i represent the potential vorticity and streamfunction; J indicates the Jacobian operator of a 2D field; and D_i are the linear dispersion operators.

S_i are the potential vorticity forcing terms, which are estimated using the following expression (Drouard et al., 2013):

\begin{equation} \label{eq2} \bm{S}_i=\bm{J}(\overline{\psi_i},\overline{Q_i})+\bm{D}_i(\overline{\psi_1},\overline{\psi_2},\overline{\psi_3}) , \end{equation}

where \(\overline\psi_i\) and \(\overlineQ_i\) are the climatological states generated from the ERA-Interim reanalysis data in boreal winter (DJF) during 1979-2006 with a T21 truncation at three levels, to make the simulation consistent with the observational analysis.

The CNOP is an initial perturbation, which makes the objective function in the nonlinear regime acquire a maximum at the optimization time with some given constraint (Mu and Duan, 2003). Many studies have revealed that CNOP is a useful tool for exploring the effect of nonlinear processes on weather and climate predictability (Sun and Mu, 2011; Wang et al., 2012; Qin and Mu, 2012; Qin et al., 2013). (Duan and Mu, 2006) and (Duan et al., 2008) explained ENSO amplitude asymmetry using this method. Inspired by their work, and considering that the formation of the NAO is a nonlinear initial-value problem (Franzke et al., 2004; Luo et al., 2007a, 2007b), CNOP is applied to explore the phase-strength asymmetry of the NAO.

The particular set-up of the nonlinear optimization method used here has been described previously in (Jiang et al., 2013); however, for convenience, we provide a simple introduction as follows: First, an empirical orthogonal function (EOF) is applied to a long-term streamfunction anomaly field to obtain the typical NAO pattern. Then, an NAO index, which is similar to the standardized principal component time series of the computed EOF, is defined to quantify the intensity of an NAO event. The CNOPs triggering the NAO- (NAO+) onset are the initial perturbations that make the difference of the NAO indices between the perturbed basic state and the reference state at the optimization time acquire a minimum (maximum) under some initial constraint condition. That is, for the initial anomalies satisfying the given constraint condition, the NAO event caused by the CNOPs is the strongest during all those induced by other initial perturbations. In this study, a total energy norm (Franzke and Majda, 2006; Jiang et al., 2008) is used as the initial constraint.

The criterion to select the NAO events in this research follows that of (Feldstein, 2003). Briefly, if the NAO index from CPC is greater (less) than 1.0 (-1.0) standard deviation for five or more consecutive days, then a positive (negative) NAO event is defined to take place. The first day on which the NAO index exceeds the threshold is noted as the onset day [lag(0)]. Accordingly, we define 32 NAO+ and 16 NAO- events.

The composite evolution of the anomalous 200 hPa streamfunction field for NAO events from ERA-Interim reanalysis from lag(-2) days to lag(10) days is shown in Fig. 1. As seen in Fig. 1a, at lag(-2) days, a statistically significant NAO- pattern with a high-over-low dipole anomaly is shown over the east of Greenland and the midlatitude North Atlantic Ocean. This dipole pattern then propagates westward and reaches maturity at lag(3) days over Greenland and the midlatitude North Atlantic Ocean. At lag(10) days, the NAO- has weakened. From Fig. 1b we can see that the NAO+ with a low-over-high dipole anomaly over Greenland and the midlatitude North Atlantic Ocean forms at lag(-2) days and, by lag(4) days, it is largely strengthened in situ. At lag(8) days, only weak remnants of the NAO+ are found. Comparatively, the life cycle of the composite NAO- events is longer than that of the NAO+ events. Another distinct difference between the two phases of NAO events is that the amplitude of the composite NAO- events is remarkably stronger than that of the NAO+ events in the mature stage. This characteristic can also be found in Luo et al. (2012, Fig. 1), though their criterion for selecting NAO events was based on different persistence periods (i.e., 3 days).

Figure 1.  Composite evolution of the anomalous 200 hPa streamfunction field from lag(-2) days to lag(10) days (contour interval: 2.5×10⁶ m² s^-1) for the (a) NAO- events and (b) NAO+ events. Shaded areas are statistically significant above the 90% confidence level according to a two-sided Student's t-test.

To better compare the strength of the two NAO phases, we define an NAO strength index, which is the absolute value of the difference of the two streamfunction anomalous centers of the NAO pattern between higher and lower latitudes. The temporal evolution of the NAO strength index in the 200 hPa streamfunction field from lag(0) to lag(8) days is shown in Fig. 2. The data clearly show that the composite NAO- events are stronger than the corresponding NAO+ events during their life cycles, which illustrates the characteristics of NAO phase-strength asymmetry from the observational point of view.

Figure 2.  Geopotential fields of the CNOPs triggering NAO- onset (gpm, shaded) with an optimization time of eight days and its nonlinear evolution based on the winter climatological streamfunction field (contour interval: 10⁷ m² s^-1) at 500 hPa: (a) at day 0; (b) at day 2; (c) at day 5; and (d) at day 8.

In this section, we seek to identify the optimal perturbations triggering NAO onset with an optimization time of eight days. An upper-bound of the initial constraint magnitude 4 J kg-1 is chosen in order to make the amplitude of the initial perturbation approximately 30 gpm. The results for the winter climatological and zonal mean (an average over all longitudes) flows as the initial basic state are presented.

Figure 3 presents the CNOP triggering the NAO- onset (CNOP-Ne) with an optimization time of eight days and its nonlinear evolution at 500 hPa. It is found that CNOP-Ne at 500 hPa is mainly concentrated in and around the mid-to-high latitudes of North America (Fig. 3a). The wave trains at lower-to-middle levels over North America present a baroclinic structure, which is westward with height (not shown). Initially, the CNOP-Ne propagates southeast with time (Fig. 3b). When the perturbations reach the east coast of North America, they begin to propagate along the classical regions of strong baroclinicity over the Atlantic (Hoskins and Valdes, 1990) to the northeast (Fig. 3c), and finally form the dipole NAO- anomaly (Fig. 3d).

Figure 3.  Temporal evolution of NAO strength index for positive and negative NAO phases at 200 hPa from lag(0) to lag(8) days.

Similarly, the CNOP triggering the NAO+ onset (CNOP-Po) with an optimization time of eight days and its nonlinear evolution at 500 hPa are shown in Fig. 4. It can be seen that the CNOP-Po at 500 hPa is mainly located over the high-latitude North Pacific (Fig. 4a). Combined with the CNOP-Po at 200 and 800 hPa, a baroclinic wave train structure can still be observed over the North Pacific at lower-to-middle levels (not shown), more upstream than that of the CNOP-Ne with the same optimization time (Fig. 3a). This is consistent with the observational evidence revealed by (Feldstein, 2003) that the formation of NAO- appears to be primarily in situ. At the initial time, a wave train over the east coast of Asia to the North Pacific propagates downstream and amplifies mostly over the Pacific region of strong baroclinicity (Hoskins and Valdes, 1990), and forms a strong meridional low-over-high dipole structure over the North Pacific. Another wave train over the east coast of North America to the North Atlantic amplifies mostly at around day 5, which propagates over the Atlantic region of strong baroclinicity (Hoskins and Valdes, 1990), and finally forms the dipole NAO+ anomaly (Fig. 4d). Comparatively, strong energy dispersion of CNOP-Po exists during its evolution; whereas, the perturbation energy of CNOP-Ne mainly concentrates over the North Atlantic region. This may partly explain why the NAO+ is weaker than the NAO- at the optimization time. In the theoretical model of (Luo et al., 2007a), they also attributed the energy dispersion of NAO+ events as causing the frequent occurrence of European blocking events and, accordingly, NAO persistence asymmetry in the two phases. The difference here is that we focus on the perturbation evolution during an NAO onset stage, whereas (Luo et al., 2007a) paid attention to what will happen after an NAO event decays.

Figure 4.  As in Fig. 3, but for CNOP-Po.

To further explore the effect of the background flow on the asymmetric characteristics of the NAO, the CNOPs and their nonlinear evolution based on the winter zonal-mean flow as the initial basic state are examined.

The CNOPs triggering the positive and negative NAO onset over the zonal-mean flow with an optimization time of eight days and their nonlinear evolution at 500 hPa at day eight are presented in Fig. 5. It is found that the CNOP-Ne is distributed over the midlatitude Western Hemisphere. Comparatively, the CNOP-Po is composed of two wave trains, one upstream over the North Pacific Ocean and the other over the east coast of North America and the North Atlantic. At the optimization time, an NAO-like pattern can be observed, in which one anomaly forms over southern Greenland, accompanied by another anomaly with an opposite sign to its south. Comparatively, the NAO- is still stronger than the NAO+, meaning that the phase-strength asymmetry is an intrinsic characteristic of the NAO, unaffected by the initial background flow.

Figure 5.  Geopotential fields of the CNOPs (gpm, shaded) triggering positive and negative NAO events with an optimization time of 8 days and its nonlinear evolution at day 8 based on the winter zonal-mean streamfunction field as the basic state (contour interval: 10⁷ m² s^-1) at 500 hPa: (a) CNOP-Ne; (b) nonlinear evolution of (a) at day 8; (c) CNOP-Po; (d) nonlinear evolution of (c) at day 8.

To better illustrate the role of nonlinear processes, in this section we calculate the conditional linear optimal perturbation (CLOP) triggering the NAO onset. To obtain the CLOP, a new objective function is defined, which is similar to that of CNOP but the nonlinear evolution of the initial perturbation is replaced by its tangent linear integration. In addition, a very small value is chosen as the initial constraint. After the CLOP is obtained, because of its linear characteristics, the energy norm of CLOP is then scaled to the same as that of CNOP. The CLOP triggering the NAO+ (NAO-) onset is called CLOP-Po (CLOP-Ne). The spatial pattern of CLOP-Po is similar to that of CLOP-Ne, but with an opposite sign.

The CLOP-Ne with an optimization time of eight days and its linear and nonlinear evolution at day eight based on climatological flow is presented in Fig. 6. Comparing with Fig. 3a, it is apparent that, at the initial time, CLOP-Ne has another strong negative anomaly over the east coast of Asia, whereas CNOP-Ne has another strong positive anomaly over North Canada. The linear and nonlinear evolution of CLOP-Ne can both develop into a dipole NAO- structure (Figs. 6b and c). Comparatively, the NAO- induced by CNOP-Ne (Fig. 3d) is stronger than that triggered by CLOP-Ne. This may be due to the strong positive anomaly over North Canada for CNOP-Ne making a positive contribution to the northern center of the NAO- dipole pattern. Because of the linear characteristics, CLOP-Po and its linear evolution are similar to CLOP-Ne and its linear evolution, but with an opposite sign. The nonlinear evolution of CLOP-Po is shown in Fig. 6d. It seems that the southern center of the NAO+ dipole pattern is somewhat displaced northwestward. The two anomalous centers of the NAO+ dipole pattern are weaker than that in Fig. 4d.

Figure 6.  Geopotential fields of the CLOPs triggering NAO- onset with an optimization time of 8 days and its linear and nonlinear evolution at 500 hPa: (a) CLOP-Ne (contour interval: 5 gpm); (b) linear evolution of (a) at day 8 (contour interval: 50 gpm); (c) nonlinear evolution of (a) at day 8 (contour interval: 50 gpm); (d) nonlinear evolution of CLOP-Po at day 8 (contour interval: 50 gpm).

The NAO indices triggered by the linear and nonlinear optimal perturbations based on the climatological flow at the optimization time of five and eight days are illustrated in Table 1. It is clear that the linear evolution of CLOP-Ne and CLOP-Po are symmetric. The nonlinear evolution of CLOP-Ne is stronger than that of CLOP-Po with the same optimization time, both of which are weaker than their respective linear evolution. In addition, the nonlinear evolution of CNOP-Ne is also stronger than that of CNOP-Po with the optimization time of five and eight days, respectively. Though the phase-strength asymmetry of NAO events can be revealed by both the nonlinear evolution of CLOPs and CNOPs, we also notice that the nonlinear evolution of CNOP is stronger than that of CLOP with the same specified constraints, which implies CNOP is the most optimal perturbation triggering the NAO onset in the nonlinear regime.

To better illustrate the physics of nonlinear processes, we calculate the nonlinear term (perturbation self-interaction) ∇^-2[-J(φ,q)] and the linear terms (perturbation/basic state interaction) ∇^-2[-J(ψ,q)]+∇^-2[-J(φ,Q)] for CNOPs, in which q and φ represent the perturbation potential vorticity and streamfunction, and Q and ψ represent the potential vorticity and streamfunction of the basic state. ∇^-2 represents the inverse Laplace operator. The projection (Feldstein, 2003) of the above terms on the typical NAO pattern at 200 hPa for CNOP-Ne and CNOP-Po with an optimization time of eight days based on the climatological flow is shown in Fig. 7. It can be seen that the CNOP-Po self-interaction contributes to a positive or negative effect during the NAO+ onset, far less than the CNOP-Po/basic state interaction. However, the CNOP-Ne self-interaction contributes more than the CNOP-Ne/basic state interaction before day 5, which both promote the evolution of NAO- events. After day 5, the CNOP-Ne self-interaction contributes slightly less than that of the CNOP-Ne/basic state interaction. It is evident that the perturbation/basic state interaction during NAO- onset is much stronger than that during NAO+ onset. The perturbation self-interaction determines the negative phase of NAO, whereas it only modifies positive NAO events.

Figure 7.  Projection of the nonlinear term (perturbation self-interaction) ∇^-2[-J(φ,q)] and the linear terms (perturbation/basic state interaction) ∇^-2[-J(ψ,q)]+∇^-2[-J(φ,Q)] for CNOP-Po and CNOP-Ne on the typical NAO pattern at 200 hPa for CNOP-Ne and CNOP-Po with an optimization time of eight days based on the climatological flow, in which "CNOP-Po N" represents CNOP-Po self-interaction, "CNOP-Po L" represents CNOP-Po/basic state interaction, "CNOP-Ne N" represents "CNOP-Ne" self-interaction, and "CNOP-Ne L" represents CNOP-Ne/basic state interaction.

Similarly, the index of the NAO events triggered by the optimal perturbations based on the zonal-mean flow at different optimization times is also shown in Table 2. Again, the linear evolution of CLOP-Ne and CLOP-Po are symmetric. The nonlinear evolution of CLOP-Ne is stronger than that of CLOP-Po, both of which are much weaker than their respective linear evolution. For CNOPs, we find that the nonlinear evolution of CNOP-Ne is stronger than that of CNOP-Po with the optimization time of five and eight days, respectively. However, the nonlinear evolution of CNOPs is much stronger than that of their respective CLOPs. In spite of whether the nonlinear evolution of CNOPs or the nonlinear evolution of CLOPs can reveal the phase-strength asymmetry of NAO, linear approximation greatly underestimates the NAO strength. In addition, it also illustrates that the phase-strength asymmetry of NAO is unaffected by the background flow.

The results of the above two sets of experiments with different basic states suggest that the phase-strength asymmetry is an intrinsic characteristic of the NAO induced by nonlinear processes during NAO onset. Perturbation energy is inclined to accumulate over the North Atlantic sector during NAO- onset, but dissipates easily during the formation of NAO+ events.

In this work, Medium-Range Weather Forecasts ERA-Interim reanalysis data are used to reveal a phase-strength asymmetry of the NAO, i.e., the composite NAO- events are stronger than NAO+ events during boreal winter (DJF) 1979-2006. This asymmetry is explored using the CNOP method with a quasi-geostrophic T21 three-level spectral model.

For consistency, the forcing term of the model is first generated with the ECMWF ERA-Interim climatological flow during winter 1979-2006. Under the given initial constraint condition, the CNOPs triggering the positive and negative NAO events over the winter climatological flow are obtained. It is found that the optimal precursors possess localized characteristics, mainly over the mid-to-high latitudes. With time, the precursors propagate southeastward and amplify, doing so most rapidly when they reach the classical regions of strong baroclinicity. At the optimization time, an NAO-like pattern can be observed over the North Atlantic. Comparatively, it seems that the NAO- is stronger than the NAO+ with the same optimization time. If we use the winter zonal-mean flow as the initial basic state forced by the climatological flow, the NAO events with the correct spatial scale and phase-strength asymmetry can still be triggered. The propagation path and evolution of the optimal perturbations can be clearly revealed with this global spectral model.

In contrast, according to our sensitivity experiments, no NAO events can be triggered by CNOPs if the forcing term is generated with the winter zonal-mean flow. This suggests that zonal asymmetric forcing is crucial for the growth of the NAO. (Franzke et al., 2004) used sensitivity experiments to verify that the zonally symmetric basic state cannot show realistic NAO characteristics of the correct spatial and temporal scale, which is consistent with our results. In addition, as we know, CNOP is dependent on the norm used in the definition of the objective function (Jiang et al., 2008; Mu and Jiang, 2008). In this paper, we find that it is not sensitive to the initial norm used with the streamfunction squared norm or the total energy norm.

Furthermore, by projecting the linear term and nonlinear term on the NAO pattern and comparing with the NAO indices induced by CNOPs and CLOPs, we find that the influence of nonlinear processes on NAO- onset is greater than that on NAO+ onset. The perturbation self-interaction greatly promotes the negative NAO onset, whereas it sometimes promotes, and sometimes prohibits, the positive NAO onset. In addition, it is the different role played by the perturbation self-interaction that induces the phase-strength asymmetry of NAO events. We can attempt to understand this in terms of the strength asymmetry of the mean westerly wind in the mid-to-high latitudes associated with the phase of NAO events. During the onset of NAO+ events, the westerly jet core is shifted poleward (DeWeaver and Nigam, 2000a, 2000b; Luo et al., 2007b; Jiang et al., 2013), which leads to eddy fluxes producing a poleward transport of heat and, accordingly, the baroclinicity in the region of the shifted jet is reduced (Robinson, 2000). Subsequently, less eddies are produced. In this case, the nonlinear processes induced by eddies become less important. For NAO- events, the opposite is true. (Barnes and Hartmann, 2010) used observations of the 3D vorticity budget in the Atlantic sector to attribute the stronger positive feedback during NAO- events to an association with anomalous northward eddy propagation away from the jet. Therefore, in future work, a complex global circulation model should be used to further examine the dynamical mechanisms that dominate the phase-strength asymmetry of the NAO.