Coupling Modes Among Action Centers of Wave–Mean Flow Interaction and Their Association with the AO/NAM

15 JANUARY 2012 Z H A O E T A L . 447

Coupling Modes among Action Centers of Wave–Mean Flow Interaction and Their Association with the AO/NAM

NAN ZHAO AND SUJIE LIANG State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

YIHUI DING National Climate Center, Beijing, China

(Manuscript received 2 December 2010, in ﬁnal form 29 June 2011)

ABSTRACT

The Arctic Oscillation/Northern Hemisphere annular mode (AO/NAM) is attributed to wave–mean flow interaction over the extratropical region of the Northern Hemisphere. This wave–mean flow interaction is closely related to three atmospheric centers of action, corresponding to three regional oscillations: the NAO, the PNA, and the stratosphere polar vortex (SPV), respectively. It is then natural to infer that local wave– mean flow interactions at these three centers of action are dynamically coupled to each other and can thus explain the main aspects of the three-dimensional coherent structure of the annular mode, which also provides a possible way to understand how the local NAO–PNA–SPV perspective and the hemispheric AO/NAM perspective are interrelated. By using a linear stochastic model of coupled oscillators, this study suggests that two coupling modes among the PNA, NAO, and SPV are related to the two-dimensional pattern in sea level pressure of the AO. Although both of them may contribute to the AO/NAM, only one is related to the three- dimensional equivalent barotropic structure of the NAM, while the other one is mainly restricted to the troposphere. So the equivalent barotropic structure of the NAM, as usually revealed by the regression of the zonal wind against the AO index, is the manifestation of just one coupling mode. Another coupled mode is a baroclinic mode that resembles the NAM only in the troposphere. However, this similarity in spatial structures does not imply that the total variability of the AO/NAM index can be explained by those of the NAO–PNA–SPV or their coupling modes, because of the existence of the variability that may contribute to the AO/NAM, produced outside of these three regions. It is estimated that the coupling modes can jointly explain 44% of the variance of the AO/NAM index.

1. Introduction and cause the fluctuation of 10–20 and 30–60 days to the AO/NAM, respectively. Particularly, this wave–mean The Arctic Oscillation (AO) (see Thompson and flow interaction is concentrated on three regional centers: Wallace 1998, 2000; Wallace 2000), also known as the 1) the interaction between a synoptic eddy and the North Northern Hemisphere annular mode (NAM), is usu- Atlantic jet stream in the troposphere, which in recent ally regarded as a result of wave–mean flow interaction years is known to be characterized by the anticyclonic/ (Limpasuvan and Hartmann 1999, 2000; DeWeaver and cyclonic synoptic wave breaking; 2) that between syn- Nigam 2000; Eichelberger and Holton 2002; Lorenz and optic eddy and the Northern Pacific jet stream; and 3) the Hartmann 2003; Vallis et al. 2004; Riviere and Orlanski interaction between upward propagating quasi-stationary 2007; Benedict et al. 2004; Franzke et al. 2004; Feldstein waves and the polar vortex in the stratosphere. On these 2003; Feldstein and Franzke 2006). Researchers suggest issues, one can refer to a review by Thompson et al. that synoptic eddy and quasi-stationary waves can inter- (2002). In general, the AO represented by the leading act with zonal flow in the troposphere and stratosphere empirical orthogonal function (EOF) of monthly mean sea level pressure (SLP) has also three centers of action over both oceanic basins and the Arctic region, coinciding Corresponding author address: Dr. Nan Zhao, Chinese Academy of Meteorological Sciences, No. 46, South Zhongguancun Ave., with those of wave–mean flow interaction. However, due Beijing 100081, China. to the following facts of 1) the weak correlation between E-mail: [email protected] the SLP in the two centers of the Atlantic and the Pacific

DOI: 10.1175/2011JCLI4240.1

Ó 2012 American Meteorological Society Unauthenticated | Downloaded 09/26/21 12:34 PM UTC 448 JOURNAL OF CLIMATE VOLUME 25 and 2) that the AO pattern cannot be identified in principal component analysis (PCA) methods or other a physically consistent way in EOF analysis applied to purely statistical tools, some dynamical model is neces- various fields of the whole Northern Hemisphere, the sary in the data analysis and mode recognition. For this dynamical relevance of the AO/NAM has been ques- purpose, in this paper we will develop a coupled linear tioned by many researchers (Deser 2000; Ambaum et al. oscillator model with random forcing and thus seek for 2001; Wallace and Thompson 2002;Dommenget and coupling modes related to the AO/NAM from this model. Latif 2002; Christiansen 2002a,b). In fact, these doubts Overall, the objective of this research is to examine can be attributed to one question of how wave–mean how local wave–mean flow interaction at different centers flow interactions localized to three regional centers can of action can dynamically form the three-dimensional be coordinated to form one three-dimensional coherent coherent structure of the annular mode so as to better structure of the AO/NAM, even if the mechanisms, the understand how the local NAO–PNA–SPV perspective phases, and time scales of the interactions in these cen- and the hemispheric AO/NAM perspective are inter- ters are rather different. related to each other. In section 2, we establish such On the other hand, the local eddy–jet interaction in a model and discuss its theoretical aspects. In section 3, the two basins can be reflected well by the North At- coupling modes of this model and their physical nature lantic Oscillation (NAO) and the Pacific–North Ameri- are derived and discussed. The last section gives a sum- can pattern (PNA) (Walker and Bliss 1932; Wallace and mary and further discussion on related issues. Gutzler 1981), two regional modes confined to the Euro- Atlantic and the Pacific sectors, respectively, and found 2. Stochastic model of coupled oscillators much earlier than the AO. Both phases of the NAO and PNA are associated with basinwide changes in the The description of the coordinated behavior of the strength and location of the North Atlantic/Pacific jet above three regional oscillations resulting from local stream and storm track. So, unlike the AO, the NAO wave–mean flow interactions needs a well-designed cou- and PNA patterns are physical modes rather than sta- pled dynamical model. Such a model can only be obtained tistical artifacts. In addition, the fluctuation of the strato- statistically from the observed data. Traditional multi- sphere polar vortex (SPV), induced also by wave–mean variate statistical tools like PCA or EOF yields only flow interaction, is an observable physical entity and is statistical modes in context of the optimal variance con- surely another physical mode that must be incorporated tribution and need not be explained as physical modes. into the understanding of this issues as well. In a point Consequently, it is of key importance to construct em- of view of synoptic wave breaking over the Atlantic, pirically from observed datasets a dynamical model that is Feldstein and Franzke (2006) suggests that the NAO and able to depict the coupling dynamics of these oscillations. AO/NAM are indistinguishable in a statistical sense and a. Regional oscillators and the coupling among them offers a somewhat different perspective on the role of the NAO in the formation of the AO/NAM. However, As mentioned above, the centers of action of wave– this may be true only at the time scale of 10–20 days, mean flow interaction in the troposphere are situated at or the life cycle of synoptic wave breaking over the At- two jet streams over the two basins of the Atlantic and lantic. Contributions from the other two regional oscil- the Pacific and represented by the NAO and the PNA. lations to the AO/NAM, such as the fluctuation of the The positive and negative phases of the NAO are found SPV at 30–60 days, cannot be completely ignored. All of to correspond to a meridional shift of the jet stream these naturally raise another question as to how the local caused by the anticyclonic and cyclonic synoptic wave NAO–PNA–SPV perspective and the hemispheric AO/ breaking over the Atlantic, respectively (Riviere and NAM perspectives are interrelated to each other. Orlanski 2007; Benedict et al. 2004; Franzke et al. 2004, As the regional wave–mean flow interactions can be Feldstein 2003). The positive phase of the PNA is asso- reflected by the NAO, PNA, and SPV, it can be inferred ciated with an enhanced Pacific jet stream and an east- that an appropriate combination of the NAO, PNA, and ward shift in the jet exit region, while the negative phase SPV may be well enough to depict the three-dimensional is associated with a westward retraction of that jet stream. quasi-zonally symmetric variability like the AO/NAM, Another center is on the stratospheric polar vortex. In which is regarded as the result of hemispheric scale wave– midlatitudes, upward propagating Rossby waves are re- mean flow interaction, as long as this quasi-zonal sym- fracted by the SPV more or less strongly toward the metric variability does exist. The question is what kind of tropics, depending on the strength of the SPV. If the polar combination of these three regional oscillations can rep- vortex is weak, more waves are refracted into it. When resent the physical ways of the coupling among them and these waves break they decelerate the vortex signifi- give rise to physically relevant modes. Rather than using cantly. In the high phase the strong vortex refracts more

Unauthenticated | Downloaded 09/26/21 12:34 PM UTC 15 JANUARY 2012 Z H A O E T A L . 449 wave activity into the tropics, and the absence of wave the two jet streams with the stratospheric polar vortex. breaking into the vortex makes it stronger because of the In addition, change of the stratospheric polar low may relaxation toward radiative equilibrium. In any case, the also alter tropospheric circulation, which provides an- basic activities of the local flows near these centers of ac- other way of the troposphere–stratospheric coupling tion, which are indicated by the NAO, PNA, and SPV, can (Baldwin and Dunkerton 1999, 2001). be viewed as single oscillators coupled, more or less, one b. Linear stochastic model of coupled oscillators to another. Generally, a single linear oscillator like a single pendulum with frequency v, damping coefficient l,and Let xNAO, xPNA, and xSPV be the daily indices of the subjected to a stochastic external forcing F is described by NAO, PNA, and SPV. Then vector 0 1 d2y dy x 1 l 1 v2y 5 F. (1) B NAO C 2 @ A dt dt y 5 xPNA (3) x We want to illustrate roughly the local dynamics of wave– SPV mean flow interaction by this model, because its left-hand denotes the state of the coupled oscillator system. In- side can depict the local flow or jet stream that is domi- stead of using the vector autoregressive process of order nated by some quasi-stationary wave, while the forcing F 2, that is, vector AR(2) model, in this study we choose represents the random eddy forcing or nonlinear in- differential equations: teraction among different waves. Equation (1) is equivalent to a so-called autoregressive process, AR(2), for the d2y dy discrete variable y , 1 A 1 By 5 f, (4) n dt2 dt

yn 5 a1yn21 1 a2yn22 1 «n, (2) to describe the coupled system because this form has more evident physical meaning. Matrix A and B are constant if «n is a white noise process with Gaussian distribution. and represent the properties such as frequency, damping The order p of autoregressive process AR(p) can be rate, and coupling strength of the coupled system. Vector f estimated by partial autocorrelation function (see, e.g., can be viewed as some stochastic external forcing, which is von Storch and Zwiers 1999). A useful property of an caused by nonlinear terms or eddy forcing of short waves AR(p) process is that the partial autocorrelation function or other variability. Usually, f is assumed to satisfy the so- uk,k 5 0whenk . p,butup,p 6¼ 0, where k is the lag. So, called white noise approximation: for a single linear oscillator like (1) or (2) without cou- * 5 s d 2 5 pling to another oscillator, one can expect that u2,2 6¼ hfm(s)fn (r)i mn (s r); m, n 1, 2, 3, (5) 0anduk,k 50whenk . 2. However, once an oscillator is coupled to others, one can expect p . 2. Table 1 shows where angle brackets denote the ensemble mean, d(s 2 r)is the partial autocorrelation functions calculated from the Dirac delta function, and smn the covariance (vari- daily indices of the AO, NAO, PNA, and SPV. The upper ance when m 5 n). and lower bound of the 95% confidence interval is 0.0217 For discrete variable yn, we define and 20.0217, respectively. It is found that the orders of a linear stochastic process governing each of the three dy y 2 y d2y y 1 y 2 2y [ n n21; [ n11 n21 n. (6) oscillators are greater than 2, but do not exceed 6. It is dt Dt dt2 Dt2 also obvious that u2,2 is far greater than uk,k for k 5 3, 4, and 5, indicating that they are mainly described by os- Thus, (4) can be constructed empirically from observed cillators like (1) or (2) and the extra degrees of freedom series by using multivariate linear regressions. When we may come from the coupling to other oscillators. So, if the discretize (4) by (6), it reverts to the AR model, but coupled system of these three oscillators is approximately according to the definition (6) of derivatives, this AR a closure system, the order of this coupled system may be model can be written in the form of an analogy with 6. Therefore, we can expect to establish the dynamical ordinary differential equations in (4), so the physical model by coupling three oscillators like (1). meaning of every term remains unchanged. The dataset The physical mechanism of these coupling seems evi- for the analysis are the daily indices of the NAO and the dent and sound: planetary Rossby wave plays a key role in PNA taken from the Climate Prediction Center (CPC). the coupling of the three oscillators. Its eastward propa- Other data utilized includes the National Centers for En- gation connects the two jet streams over the Atlantic and vironmental Prediction–National Center for Atmospheric the Pacific, while its upward propagation in winter links Research (NCEP–NCAR) reanalysis data of daily mean

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TABLE 1. The partial autocorrelation functions (boldface numbers) calculated from daily indices of the AO, NAO, PNA, and SPV. The upper and lower bounds of the 95% confidence are 0.0217 and 20.0217, respectively. ukk k 5 1 k 5 2 k 5 3 k 5 4 k 5 5 k 5 6 k 5 7 k 5 8 k 5 9 k 5 10 p AO 0.919 20.524 0.255 20.059 0.026 0.022 0.002 0.004 0.005 0.005 6 NAO 0.923 20.495 0.164 20.058 20.011 20.003 20.001 0.005 0.008 20.004 4 PNA 0.937 20.508 0.145 20.046 20.047 0.000 0.000 20.018 20.013 20.018 5 SPV 0.964 20.366 20.078 20.005 0.005 0.025 0.017 0.011 0.021 0.005 6 height. Only winter (November/March) data for 58 multiple determination R2 for each single regression years from 1950 to 2007 are utilized for the following equations give a quick answer (von Storch and Zwiers study. The SPV index is calculated from the averaged 1999). For the first, second, and third regression equation, daily 50-hPa geopotential height anomalies over the we have R2 5 0.3203, 0.2927, and 0.3169, respectively, polar cap poleward of 658N and is normalized by stan- indicating a quite acceptable ability for the model (4) to dard deviation. The seasonal cycle has been removed depict the linear coupling dynamics. from all data before the analysis. In the following study, c. Basic theory of coupling modes the SPV index is weighted by the ratio of the masses of the layers of 100;0 hPa and 800;200 hPa, wherein The coupled oscillator system of (4) can be decoupled SPV and NAO–PNA are defined and can be represen- by introducing a transform tative for the whole layer, respectively, y 5 Mq (7) 100 hPa weight 52 , SPV 800 hPa 2 200 hPa in which q is a new vector variable and M the matrix of transformation. Equation (4) then becomes which also reflects the relative importance of the stratosphere and the troposphere. d2q dq 1 M21AM 1 M21BMq 5 M21f. (8) Matrices A and B are composited of regressive co- dt2 dt efficients, which are obtained with a significance level of 0.05 as If such a matrix M exists that A and B are diagonalized 0 1 simultaneously, then (4) can be decoupled into three 0:090 0:004 0:083 independent equations of oscillators as B C A 5 @ 0:001 0:072 0:013 A; 2 d qn dqn 2 0:000 0:001 0:027 1 l 1 vnq 5 f ; n 5 1, 2, 3 (9) 0 1 dt2 n dt n n 0:467 20:038 0:208 B C B 5 @ 0:017 0:442 0:157 A. (see, e.g., Thomson 1981). This can be achieved if matrices A and B have the same set of eigenvectors, 20:003 0:006 0:591 say, vn with n 5 1, 2, and 3. If so, a coupling mode can be written as qnvn,wherevn is the spatial configura- Roughly speaking, the diagonal elements of both A and B tion and qn the time coefficients, which is the solution are much greater than most of the other elements, which is of (9): further evidence for the applicability of the coupled os- ð cillator model. The residual parts of the multivariate lin- 1 t 2[ln(t2s)]/2 ear regressions can then serve as an estimation of the qn 5 fn(s)e sinvdn(t 2 s) ds. (10) vdn 0 stochastic external forcing, that is, f.Thejustificationof this kind of model can be performed by checking the 2 2 1/2 Here vdn 5 (vn 2 ln/4) . With the white noise ap- white noise approximation for stochastic external forcing proximation (5) employed in the derivation, it can be f. Figure 1 gives the cross correlation functions (including proven that autocorrelation function of qn is given by auto correlation functions) of the three components of f.It displays approximately a white noise feature for f as de- ! 2l t/2 2lnvdn scribed by (5). So, principally the regressive model (4) can r t 5 n v t 2 v t ( ) e cos dn 2 sin dn , (11) describe the coupling dynamics of these three oscillators. 2vn 2 ln When we think about how well the linear stochastic model (4) fits the coupling dynamics, the coefficients of and theoretical covariance between different qn is

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FIG. 1. Cross-correlation functions (autocorrelation functions for m 5 n). They display quite well white noise feature for f as described by Eq. (5).

(l 1 l )s where I and 0 are unitary and zero matrix, respectively. hq q i 5 m n mn . m n 2 2 2 2 2 2 2 2 2 By expanding x, y, and f in these eigenvectors, we have (vm 2 vn) 1 vmln 1 vnlm 2 lmln/4

(12) 6 6 y yn 0 yn Alternatively, if such a matrix M does not exist, coupling 5 å pn ; 5 å fn , (16) x 5 x f 5 x modes in more general meaning can be obtained in the n 1 n n 1 n following way. By denoting where pn satisﬁes dy 5 x (13) dp dt n 5 l p 1 f . (17) dt n n n one can rewrite (4) as One often obtains real eigenvectors for the coupling d y 0 I y 0 modes in the former case, while in the latter case com- 5 1 . (14) dt x 2B 2A x f plex eigenvectors are very common. Unlike real eigenvectors, which represent just standing wave modes, each The coupling modes of which are given by the following conjugate pairs of complex eigenvectors describe one eigenvalue problem: propagating wave mode via two alternating patterns represented by the real and imaginary parts of the ei- 0 I y y n 5 l n , (15) genvector. This can be demonstrated as below. Let p 5 2B 2A x n x n n pr 6 ipi be a conjugate pair of complex eigenvectors and

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TABLE 2. Parameters of decoupled oscillators of Eq. (9).

n 5 1 n 5 2 n 5 3 ln 0.0260 0.0910 0.0727 2 vn 0.5918 0.4655 0.4423 T 5 2p/vn (days) 8.2 9.2 9.5

their time coefﬁcients be a 5 ar 6 iai, then the physical mode represented by them is ap 1 a*p* 5 2arpr 2 2aipi.

3. Coupling modes associated with the NAM/AO a. Results of calculation If the columns of M are selected as the eigenvectors of A, that is, vn with Avn 5 lnvn, then A can be diagonalized. However, B needs not to be diagonalized simultaneously. This is also the case if the columns of M are selected as the eigenvectors of B. Therefore, model (4) cannot always be decoupled into (9). In this study, we find that, if columns of M are selected as the eigenvectors of A, matrix B can approximately be diagonalized with FIG. 2. The configurations of coupling modes in a three- great precision, which is demonstrated by matrix M21 AM dimensional Cartesian coordinate system spanned by the NAO, and M21 BM as below: PNA, and SPV. The vectors in red, green, and blue thick lines represent mode v1, v2, and v3, respectively. In this study, the vector 0 1 reg. AO (black thick line) is employed to represent the AO/NAM. 0:026 0 0 The components of this vector are defined as the regressive co- B C M21AM 5 @ 00:091 0 A; efficients of the respective daily indices of the NAO, PNA, and SPV on that of the AO/NAM. The vectors in red, green, and blue 000:072 thin dashed lines represent EOF3, EOF2, and EOF1 calculated 0 1 0:592 20:006 0:004 from the three indices, respectively. B C M21BM 5 @ 0:056 0:466 0:018 A somethingincommonwiththeNAO,whilemodev3 0:060 20:020 0:442 0 1 represents an opposite phase between the jet streams over 0:592 0 0 the two basins, indicated by a weak NAO and a strong B C ’ @ 00:466 0 A PNA and vice versa. On the contrary, mode v1 is indeed a coupling mode between the troposphere and strato- 000:442 sphere. It has all three components in an appropriate configuration resembling that of the AO/NAM, including As a result, parameters of decoupled oscillators of (9) an in-phase relationship between the jet streams over the are obtained and given in Table 2. The periods of free two basins. These can also be demonstrated in detail as oscillation are around 8–10 days, reflecting the time below. scales of dominant long Rossby waves associated with Figure 2 shows the configurations of these coupling jet streams and the polar vortex. modes in a three-dimensional Cartesian coordinate sys- Three dynamical modes reflecting the coupling be- tem spanned by the NAO, PNA, and SPV. The vectors in havior of the PNA, NAO, and SPV are obtained as the red, green, and blue thick lines represent mode v1, v2, and eigenvectors of A as given below: v3, respectively. To compare them with the AO/NAM, 0 1 0 1 0 1 we need a standard vector that can represent the AO/ 0:779 0:998 0:355 NAM in this space. In this study, the components of this v 5 @ 20:152A; v 5 @ 20:058A; v 5 @ 0:934A. 1 2 3 vector are defined as the regressions of the respective 0:609 20:006 0:026 daily indices of the NAO, PNA, and SPV on that of the AO/NAM, which is demonstrated by the vector in the As indicated in the vectors given above, v2andv3are black thick lines and denoted by reg. AO. The vectors modes basically confined to the troposphere. Mode v2has in red, green, and blue thin dash lines represent EOF3,

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FIG. 3. Regression maps of the 1000-hPa geopotential height versus time series (a) X1, (b) X2, (c) X3, and (d) the daily AO index.

EOF2, and EOF1 calculated from the three indices, re- the same as Fig. 3, but is for 50-hPa geopotential spectively. It is found that EOF1 and EOF2 are almost height. Again it is seen that although both v1andv2 consistent with mode v2andv3, whereas the EOF3 is are very similar to the AO/NAM in the troposphere, largely separate from the v1. This means that the coupling their structures in the stratosphere are quite different. mode v1, v2, and v3 obtained from a physical model are Mode v1 is barotropic like the AO/NAM, it has an quite different from the EOFs obtained by optimizing the annular structure in both troposphere and strato- variance contribution of orthogonal sets of vectors. An- sphere, while mode v2 in the stratosphere is of wave- gles included by reg. AO and v1, v2, and v3are308,148, number 1, rather than an annular structure. These and 968, respectively. So mode v1andv2 may spatially results are in agreement with recent work of Zhao et al. resemble that of the AO/NAM, while v3 may be totally (2010), in which two normal modes similar to v1andv2 different from the AO/NAM. are found to be related to the AO/NAM, corresponding By projecting observed y on v1, v2, and v3, we can get to the pulsation and wandering of the stratosphere polar the time series X1, X2, and X3 of these modes. Figure 3 vortex, respectively. Mode v3ismainlyassociated displays the regressions of 1000-hPa geopotential height with the PNA pattern and shows no annular feature against these time series and daily AO index. Figure 4 is at all.

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FIG. 4. As in Fig. 3 but for the 50-hPa geopotential height.

b. Validation and discussions shown the importance of multiplicative noise in climate (e.g., Majda et al. 2003, 2009). In addition, the low values Validation of the results above is necessary because of of R2 for each of the single regression equations that were the approximations that are made both in the construc- mentioned before indicate that residual f is quite large in tion of the model (4) and in the way of its decoupling. As magnitude. All of these may result in inaccuracy of the is indicated in Fig. 1, residual f can just approximately be linear stochastic model that we are using. These defects of viewed as a white noise process. Moreover, it can be the model are reﬂected by the statistical property of the found from Fig. 5, the cumulative distribution functions time coefﬁcients or qn, and one may expect to see some (cdf) of the projections of this residual on v1, v2, and v3, discrepancy between the statistical property of qn and or f1, f2,andf3, deviate greatly from Gaussian (normal) that of Xn. distribution, demonstrating a nonlinear feature of the As can be indicated in Table 3, hXmXni represents original process against the linear model (Fan and Yao the observed covariance of each Xn and can be com- 2005). The deviation from Gaussianity can also possibly pared with their theoretical counterparts hqmqni given by be explained by multiplicative noise. Recent studies have (12). It is clearly seen that the observed and theoretical

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FIG. 5. Cumulative distribution functions (cdf) of the projections of the residual f on v1 (blue), v2 (green), and v3 (red) (or, f1, f2, and f3) on a stretched Gaussian coordinate. Each corresponding long-short dashed line shows the cdf of the normally distributed variable with the same variance. The large deviation from Gaussian distributions can be seen clearly.

0 1 0 1 0 1 0 1 covariances are roughly consistent. Also, smaller magni- 0:82 0:00 0:92 0:00 tudes when m 6¼ n indicate that these coupling modes are B C B C B C B C y1 5 @20:37A6 i@20:00A; y2 5@0:26 A6 i@0:29A; nearly uncorrelated. Nevertheless, Fig. 6 shows great discrepancy between the observed (red stem plots) and 0:44 0:07 0:03 0:00 0 1 0 1 theoretical [blue line plots, given by (11)] autocorrela- 0:61 0:35 tion functions, which is mainly due to the deviation of B C B C y3 5 @ 0:71A6 i@ 0:00A. random forcing f from an exact white noise process and the deviation from Gaussianity. 0:04 0:00 However, this need not deny the utility of the model. The imaginary parts are small as compared with their Since the configurations of the linear coupling modes v n corresponding real parts, which can be demonstrated by are merely decided by the homogenous parts of the the ratios kIm(y )k/kRe(y )k (50.067, 0.306, and 0.376 model and are independent of the random forcing f, n n for n 5 1, 2, and 3, respectively). Figure 7 displays v1, v2, linear coupling modes obtained above are not affected and v3 as well as the real and imaginary parts of y1, y2, by the inaccuracy of the model and can be regarded as and y3. It is evident that the structures of v1, v2, and v3 some physical modes. Although the forcing f cannot reflect very well those of the real parts of y1, y2, and y3, affect the configurations of the coupling modes, the while the imaginary parts of y1, y2, and y3 are almost approximations in the way of decoupling of the equations do affect them. Since M21BM is just approximately TABLE 3. Comparison between theoretical and observed diagonal and coupling mode v1, v2, and v3 obtained by covariance among each Xn. the decoupling procedure of (8) are merely approximations, it is necessary to make a comparison between hqmqni/hXmXni n 5 1 n 5 2 n 5 3 these approximate coupling modes and those obtained m 5 1 0.079/0.026 0.000/0.000 0.003/0.000 by the exact way as indicated in (14)–(16). The latter are m 5 2 0.000/0.000 0.623/0.590 0.163/0.000 complex modes: m 5 3 0.003/0.000 0.163/0.000 0.631/0.625

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FIG. 6. This figure shows great discrepancy between the observed (red stem plots) and theoretical (blue line) autocorrelation functions. uncoupled modes and of less importance. So, to a large interrelated. By using a linear stochastic model of three extent, v1, v2, and v3 can approximate y1, y2, and y3, or coupled oscillators, this study suggests that two coupling the exact coupling modes. Since the AO/NAM is origi- modes among the PNA, NAO, and SPV have a plausible nally defined by the leading EOF that can be viewed as link to the two-dimensional pattern in sea level pressure some real mode or standing wave, in this study we prefer of the AO. It may then be concluded that both of them to choose v1, v2, and v3 rather than y1, y2, and y3 so that contribute to the AO or NAM. However, only one of them we can make a comparison between the AO/NAM and is related to the three-dimensional equivalent barotropic the coupling modes. structure of the NAM, while the other one is a baroclinic mode that resembles the NAM only in the troposphere. The equivalent barotropic structure of the NAM, as usu- 4. Summary and discussion ally revealed by the regression of the zonal wind against The Arctic Oscillation/Northern Hemisphere annular the AO index, is the manifestation of just one coupling mode is usually attributed to the wave–mean flow inter- mode. These, to some extent, can explain the question action over the extropical region of the Northern Hemi- as to how local wave–mean flow interaction at the dif- sphere. This wave–mean flow interaction has three centers ferent centers of action can dynamically form the three- of action, corresponding to three local oscillations: the dimensional coherent structure of the annular mode. North Atlantic Oscillation, Pacific–North American pat- On the other hand, the results above give the follow- tern, and the stratospheric polar vortex (SPV), respec- ing way to understand the relationship between the local tively. It is then inferred that local wave–mean flow NAO–PNA–SPV perspective and the hemispheric AO/ interactions at these three centers of action are dynami- NAM perspective. From two coupling modes associated cally coupled to each other and can explain the main as- with the AO/NAM, that is, v1 and v2, it can be found pects of the three-dimensional coherent structure of the that 1) the PNA plays a nearly trivial role in both modes, annular mode, which also provides a possible way to un- 2) the SPV plays an essential role only in one mode, and derstand how the NAO–PNA and AO perspectives are 3) the NAO plays significant roles in both modes. These

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FIG. 7. (top) Modes v1, v2, and v3 and (middle) the real and (bottom) the imaginary part of complex modes y1, y2, and y3. may be attributed to the fact that both the NAO and the of the AO/NAM. It is then necessary to estimate how SPV correspond to a meridional shift of the jet streams much the coupling modes can explain the variance of the and thus are related to quasi-zonal symmetric variabil- AO/NAM index. This can be achieved by a multiple ity, while the PNA just corresponds to a strengthening/ linear regression model with the AO/NAM index as the weakening of the jet stream and provides little contri- response variable and X1 and X2 as the factors. The bution. Obviously, the NAO is associated with the AO/ coefﬁcient of multiple determination, R2, is equal to

NAM in two categories of ways—a barotropic way that 0.44, indicating that X1 and X2 jointly represent 44% of coordinates the NAO with the SPV to form the three- the variability in the AO/NAM index. Even so, they can dimensional structure of the AO/NAM and a baroclinic represent well life cycles of the AO/NAM because the way in which the structure of the AO/NAM is restricted dynamical processes of many life cycles of the AO/NAM to the troposphere. are confined to the three centers of action. In terms of their similarity in spatial structures, we have Finally, there is also an intuitive way for the justifica- tied the AO/NAM to the coupling modes of three regional tion of the model employed in this study. In the tropo- oscillations: the NAO, PNA, and SPV. However, this does sphere, jets over two oceanic basins are just local sectors not mean that the total variability of the AO/NAM index of the planetary waves in the middle and upper tropo- can be explained by those of the NAO–PNA–SPV or sphere, with their dominant portion being wavenumbers coupling modes of them. This is simply due to the exis- 0, 1, 2, and 3. As a result, the order of two coupled os- tence of the variability that may also contribute to the AO/ cillators in the troposphere, which is 4, is sufficient for the NAM outside of these three patterns. For example, in description of the main aspects of these planetary waves. the Pacific sector, the west Pacific teleconnection pattern In the stratosphere, the circulation is very much annular corresponds to a meridional shift of the jet stream (e.g., and wavenumbers 0 and 1 are dominant because plane- Franzke and Feldstein 2005), which might be a better tary waves with wavenumber .2 are trapped in the tro- pattern than the PNA to be used to explain the emergence posphere. So, one oscillator of order 2, which is also

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