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

JIANG Zhina,WANG Xin,and WANG Donghai

1State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences,Beijing100081

2State Key Laboratory of Tropical Oceanography,South China Sea Institute of Oceanology, Chinese Academy of Sciences,Guangzhou510301

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

JIANG Zhina∗1,WANG Xin2,and WANG Donghai1

1State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences,Beijing100081

2State Key Laboratory of Tropical Oceanography,South China Sea Institute of Oceanology, Chinese Academy of Sciences,Guangzhou510301

Negative-phase North Atlantic Oscillation(NAO)events are generally stronger than positive-phase ones,i.e.,there is a phase-strength asymmetry of the NAO.In this work,we explore this asymmetry of the NAO using the conditional nonlinear optimal perturbation(CNOP)method with a three-level global quasi-geostrophic spectral model.It is shown that,with winter climatological fl ow forcing,the CNOP method identi fi es the perturbations triggering the strongest NAO event under a given initial constraint.Meanwhile,the phase-strength asymmetry characteristics of the NAO can be revealed.By comparing with linear results,we fi nd that the process of perturbation self-interaction promotes the onset of negative NAO events,which is much stronger than during positive NAO onset.Results are obtained separately using the climatological and zonal-mean fl ows in boreal winter(December–February)1979–2006 as the initial basic state.We conclude,based on the fact that NAO onset is a nonlinear initial-value problem,that phase-strength asymmetry is an intrinsic characteristic of the NAO.

North Atlantic Oscillation,asymmetry,optimization problem

1. Introduction

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 fi eld 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;L´opez-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 veri fi ed the importance of the latitudinal position of perturbations relative to the climatological Atlantic jet in triggering NAO events. Rivi`ere and Orlanski(2007)emphasized the importance of high-frequency momentum fl ux 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 clari fi ed 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 Paci fi c 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 fl ux 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 outthatthenonlinearprocessesduringpositive-andnegativephase 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 quasigeostrophic 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.

2. Data and method

2.1.Data

The daily NAO index from the National Weather Service Climate Prediction Center(CPC)is used to de fi ne the NAO events.The observational streamfunction fi elds at 200,500, and 800 hPa are derived from the daily 0000 UTC ERAInterimreanalysisoftheEuropeanCenterforMedium-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.

2.2.Model

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:

Here,the indexi=1,2,3 refers to 200,500,and 800 hPa, respectively;Qiandψirepresent the potential vorticity and streamfunction;Jindicates the Jacobian operator of a 2D if eld;andDiare the linear dispersion operators.

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

whereandare 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.

2.3.Set-up of the short-term optimization method

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 fi eld 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 de fi ned 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.

3. CharacteristicsofthecompositeNAO events from observational evidence

The criterion to select the NAO events in this research follows that of Feldstein(2003).Brie fl y,if the NAO index fromCPC is greater(less)than 1.0(-1.0)standard deviation for fi ve or more consecutive days,then a positive(negative)NAO event is de fi nedto takeplace.The fi rst day on whichthe NAO index exceeds the threshold is noted as the onset day[lag(0)]. Accordingly,we de fi ne 32 NAO+and 16 NAO-events.

The composite evolution of the anomalous 200 hPa streamfunction fi eld 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 signi fi cant 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(2)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 strengthenedin 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).

To better compare the strength of the two NAO phases, we de fi ne 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 fi eld 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.

4. CNOPs and the triggered NAO events

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-1is 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) fl ows as the initial basic state are presented.

4.1.Winter climatological fl ow

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 CNOPNe 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 withtime(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 fi nally form the dipole NAO-anomaly(Fig.3d).

Similarly,the CNOP triggering the NAO+onset(CNOPPo)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 highlatitudeNorthPaci fi c(Fig.4a).CombinedwiththeCNOP-Po at 200 and 800 hPa,a baroclinic wave train structure can still be observed over the North Paci fi c 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 primarilyin situ.At the initial time,a wave train over the east coast of Asia to the North Paci fi c propagates downstream and ampli fi es mostly over the Paci fi c region of strong baroclinicity(Hoskins and Valdes,1990),and forms a strong meridional low-over-high dipole structure over the North Paci fi c.Another wave train over the east coast of North America to the North Atlantic ampli fi es mostly at around day 5,which propagates over the Atlantic region of strong baroclinicity(Hoskins and Valdes, 1990),and fi nally 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 disper-sion 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.

4.2.Winter zonal-mean fl ow

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

The CNOPs triggering the positive and negative NAO onset over the zonal-mean fl ow 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 Paci fi c 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 fl ow.

5. Role of nonlinear processes

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 de fi ned,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 CLOPPo 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 fl ow 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 CLOPNe 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.Thismay 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.

The NAO indices triggered by the linear and nonlinear optimal perturbations based on the climatological fl ow at the optimization time of fi ve 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 CLOPNe 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 CNOPNe is also stronger than that of CNOP-Po with the optimization time of fi ve 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 speci fi ed 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, wecalculatethenonlinearterm(perturbationself-interaction)∇-2[-J(ϕ,q)]and the linear terms(perturbation/basic state interaction)∇-2[-J(ψ,q)]+∇-2[-J(ϕ,Q)]for CNOPs,in whichqandϕrepresent the perturbation potential vorticity and streamfunction,andQandψrepresent the potential vorticity and streamfunction of the basic state.∇-2represents 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 fl ow 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 thatof 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 perturbationself-interactiondetermines thenegativephaseof NAO, whereas it only modi fi es positive NAO events.

Similarly,the index of the NAO events triggered by the optimal perturbations based on the zonal-mean fl ow 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 thatof CLOP-Po,both of which are much weaker than their respective linear evolution.For CNOPs,we fi nd that the nonlinear evolution of CNOP-Ne is stronger than that of CNOPPo with the optimization time of fi ve 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 phasestrength asymmetry of NAO is unaffected by the background fl ow.

Table 1.The NAO index induced by CLOPs and CNOPs at the optimization time(T)of fi ve and eight days using the winter climatological fl ow.

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.

6. Summary and conclusions

In this work,Medium-Range Weather Forecasts ERAInterim 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 fi rst generated with the ECMWF ERA-Interim climatological fl ow during winter 1979–2006.Under the given initial constraint condition,the CNOPs triggering the positive and negative NAO events over the winter climatological fl ow are obtained. It is found that the optimal precursors possess localized characteristics,mainly over the mid-to-high latitudes.With time, theprecursorspropagatesoutheastwardandamplify,doingso 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 sameoptimizationtime.Ifweusethewinterzonal-mean fl ow as the initial basic state forced by the climatological fl ow,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 fl ow.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 de fi nition of the objective function(Jiang et al.,2008;Mu and Jiang, 2008).In this paper,we fi nd 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 fi nd that the infl uence 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 fl uxes 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 oppositeistrue.BarnesandHartmann(2010)usedobservationsof 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.

Acknowledgements.We thank the editor and two anonymous reviewers for their insightful suggestions,which helped to greatly improve the manuscript.This study was jointly supported by the National Key Basic Research and Development(973)Project(Grant No.2012CB417200)and the National Natural Science Foundation of China(Grant No.41230420).

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(Received 12 May 2014;revised 15 September 2014;accepted 26 September 2014)

∗Corresponding author:JIANG Zhina

Email:jzn@cams.cma.gov.cn