The role of eddy feedback in the excitation of the NAO

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1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 21: ) Published online 2 August 2013 in Wiley Online Library wileyonlinelibrary.com) DOI: /met.1415 The role of eddy feedback in the excitation of the NAO Gui-Rong Tan, a,b * Fei-Fei Jin, b Hong-Li Ren b,c and Zhao-Bo Sun a a Jiangsu Key Laboratory of Meteorological Disaster, Nanjing University of Information Science and Technology, Nanjing, China b Department of Meteorology, University of Hawaii, Honolulu, HI, USA c Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing, China ABSTRACT: In this paper, the role of eddy feedback in the formation and excitation of the NAO North Atlantic Oscillation) is examined in a linear barotropic model with an empirical closure for synoptic eddy and low-frequency flow SELF) feedback. The EOF-space based model captures the linear dynamics of the barotropic model linearized with respect to climatological mean flow very accurately with relative low-order truncations. The parameterized empirical linear low-dimension SELF-feedback operator also depicts the relation between eddy-induced streamfunction tendencies and the large-scale circulation to a reasonable degree. Singular vector analysis SVD) is performed on the linear low-dimension dynamic operator with SELF-feedback to exam the leading low-frequency mode of the system. It is shown that the leading SVD mode the left vector) of the linear low-dimension dynamic operator with smallest SVD value resembles the observed NAO pattern. The role of eddy feedback in the excitation and maintenance of NAO is then further examined by relating the effective external forcing patterns to the right vector of the leading SVD mode of the linear dynamic operator. It is shown that local dipolar external forcing at middle latitudes over North Atlantic is effective in exciting the NAO-like anomaly pattern, whereas the effectiveness of the external forcing over different areas sensitively depends on the eddy feedback. KEY WORDS formation and excitation of the NAO; eddy feedback; low-dimension linear operator Received 5 September 2012; Revised 7 April 2013; Accepted 12 May Introduction The North Atlantic Oscillation NAO), as one of the most prominent modes of Northern Hemisphere NH) atmospheric variability, is well known for its great impacts on the climate variation over the middle and high latitudes of the NH e.g., Walker and Bliss, 1932; Van Loon and Rogers, 1978; Wallace and Gutzler, 1981; Thompson et al., 2002). A number of studies also demonstrated NAO variability response to the global warming Hurrell, 1995; Hurrell et al., 2003). Therefore, the low-frequency modes of atmospheric circulation, particularly the NAO and its related northern annular mode NAM) or Arctic Oscillation AO) Thompson and Wallace, 2000), have been subjects of many studies. One focus of these studies is the role of synoptic eddies in the generation of the low-frequency variability of the extratropical atmospheric circulation e.g., Lau, 1988; Cai and van den Dool, 1991; Lau and Nath, 1991; Robinson, 1991a, 1991b, 2000; Branstator, 1992, 1995; Limpasuvan and Hartmann, 1999, 2000; Vallis et al., 2004). There exists a two-way interaction between synoptic eddies and low-frequency flow e.g., Cai and Mak, 1990; Qin and Robinson, 1992; Whitaker and Barcilon, 1992a, 1992b, 1995). Most results suggest that positive feedback of synoptic eddy is important in maintaining low-frequency flow variability associated with prominent climate modes e.g., Nakamura and * Correspondence: G-. R. Tan, Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing , China. tanguirong@ nuist.edu.cn Wallace, 1990; Hoerling and Ting, 1994; Lorenz and Hartmann, 2001, 2003). The low-frequency variability has been shown not only to be maintained by transient eddy forcing, but also to organize systemically the eddy activity for generating such kind of eddy forcing e.g., Lau, 1988; Lee and Feldstein, 1996; Kimoto et al., 2001; Feldstein, 2003; Watanabe and Jin, 2004). High-frequency synoptic activity interacts with low-frequency variability through nonlinear rectification Egger and Schilling, 1983; Lau and Holopainen, 1984; Lau and Nath, 1991). The SELF-feedback has been recognized as playing an essential role in generating the dominant modes of the low-frequency variability Limpasuvan and Hartmann, 2000; Robinson, 2000). One dynamic approach to gain insight in the dynamics of synoptic eddy and low-frequency flow SELF) feedback is to develop a linear dynamic closure for the SELF-feedback Jin et al., 2006a, 2006b; Pan et al., 2006; Jin, 2010). They proposed a linear dynamic framework that includes both dynamic processes linearized with respect to climate mean flow and an approximate but dynamically deduced linear closure for the SELF-feedback. This new linear framework with the inclusion of linear dynamics for the SELF-feedback was shown to be a useful tool for exploring the dynamics of the low frequency mode. Using this framework, it was shown thorough eigen value and singular value analyses that SELF-feedback is responsible for the emergence of leading modes which resemble the known atmospheric low-frequency patterns such as AO/NAO, AAO and PNA Jin et al., 2006a, 2006b). Analytical results deduced from this linear framework demonstrate that a scale-selective eddy-induced low-frequency dynamic instability is the underlying fundamental mechanism for the origin of the extratropical low-frequency variability Jin, 2010) Royal Meteorological Society

2 The role of eddy feedback in the excitation of the NAO 769 The key hypothesis of the linear dynamic closure is that the low-frequency flow anomalies can systematically alter the statistical nature of synoptic eddies such that the so-called anomalous eddy forcing onto the low-frequency flow is to a large extent slaved by the low-frequency flow itself. Kug and Jin 2009) and Kug et al. 2010a, 2010b) investigated the scale interaction between synoptic eddies and low-frequency flow and pointed out that the synoptic eddy feedback is a key process to sustain the low-frequency flow through a positive SELFfeedback process that can be diagnostically depicted by the so-called Left-Hand Rule. Observational studies by Ren et al. 2009, 2011, 2012) following this approach depicted clearly systematical changes in statistical eddy structure associated with NAO. Moreover, they demonstrated that these changes in eddy activity are to a large extent generated by and linearly related to the NAO on one hand; they are also responsible for generating the anomalous eddy forcing that maintains the NAO, on the other hand. These observations supported the view that SELFfeedback to a certain extent can be by and large viewed as a linear dynamic process. The linear dynamic framework with SELF-feedback has also been used to examine the role eddy feedback plays in shaping the atmospheric remote response to external forcing, thus shaping the atmospheric teleconnections Peng et al., 2005; Jin et al., 2006a). In fact, numerous studies have demonstrated that both tropical and extra-tropical anomaly forcing can induce NAO-like patterns in the winter atmosphere e.g., Palmer and Anderson, 1994; Trenberth et al., 1998; Rodwell et al., 1999; Watanabe and Kimoto, 2000; Cassou and Terray, 2001; Hoerling and Kumar, 2002; Peng et al., 2002, 2003, 2005; Greatbatch et al., 2003; Gerber and Vallis, 2005; Pozo- Vasquez et al., 2001, 2005). However, atmospheric responses are often found to be sensitive to specific features of the external forcing Sutton and Hodson, 2003; Mathieu et al., 2004) because these forced responses are heavily regulated by the eddy feedback Ting and Sardeshmukh, 1993; Peng and Whitaker, 1999; Peng et al., 2002; Li et al., 2006; Pan et al., 2006). This paper builds upon the approaches taken by Peng et al. 2005) and Jin et al. 2006a,b) and uses a linear barotropic framework to examine the role of SELF feedback in the formation and external excitation of the NAO. The linear dynamic framework of Jin et al. 2006a,b) and subsequent numerical and observational studies Pan et al., 2006; Ren et al., 2009; Kug et al., 2010a, 2010b) suggested that the linear dynamical closure for SELF-feedback is largely valid. However, such a dynamic closure is somewhat complicated. Peng et al. 2005) proposed a simple empirical closure for the SELF-feedback and it worked remarkably well in the simulations of NAO. Thus, a low-dimension linear framework is proposed that includes two parts: 1) the classic linear dynamics cast in low-dimension by projecting the linear model onto EOF bases, and 2) a simple empirical SELF-feedback closure also expressed using the EOF bases. With adequate yet still relatively small numbers of EOF bases, this linear lowdimension dynamic framework can efficiently capture the linear dynamics of low-frequency variability through the approaches of Peng et al. 2005) and Jin et al. 2006a,b). The relatively low-dimension of this approach provides an efficient way of exploring optimal excitations of the NAO or singular patterns of linear dynamics with considerations of SELFfeedback. This paper is organized as follows. Section 2 describes an empirical closure for SELF-feedback, based on observed monthly streamfunction and corresponding eddy induced streamfunction tendency. Section 3 gives a low-dimension EOF-space representation of linear dynamics and a linear barotropic model with empirical SELF-closure. The leading SVD modes of the reconstructed linear dynamic model are then explored to elucidate the role of eddy feedback on the formation of a leading low frequency pattern. Section 4 presents the optimal forcing of leading modes to illustrate the role of eddy feedback on the excitation of the NAO-like pattern. The sensitivity of the forced atmospheric response with eddy feedback to the remote forcing over different areas is also examined in this section. The conclusions are given in Section An empirical closure for SELF-feedback 2.1. Description of the empirical operator Following the approaches of Peng et al., 2003; Jin et al., 2006a, 2006b; Pan et al., 2006), the linearized barotropic model with the closure of SELF-feedback can be written as: ψ a + J t ψ c, ψ a) + J ψ a, ψ c + f + γ ψ a + J ψ, ψ )a = Q a 1) Here, r is a linear damping rate, ψ c, ψ a denote the monthly climatological mean and anomaly streamfunction, Q a denotes monthly-mean anomalies in the external forcing and f is the Coriolis parameter. Monthly mean eddy forcing J ψ, ψ )a is hypothesized directly related to ψ a by a linear SELF feedback operator L * f Jin et al., 2006a) as: and a linear operator L * is defined as: L ψ a = J ψ c, ψ a) + J J ψ, ψ )a L f ψa 2) ψ a, ψ c + f ) ) + γ ψ a 3) Since here the streamfunction is used instead of the vorticity, so the linear operator L is defined as Lψ a = 1 J ψ c, ψ a) ) + 1 J ψ a, ψ c + f + γ ψ a 4) and the eddy-induced streamfunction forcing are related to ψ a by a linear SELF feedback operator L f : 1 J ψ, ψ )a 1 V ζ a) L f ψ a 5) Here V σ a denotes the monthly-mean anomalies of synoptic eddy vorticity fluxes V, σ, ψ denote the 2 8 days bandpass filtered winds, relative vorticity and streamfunction), 1 is a Lapalacian inversion operator, and J means the Jacobian b x y a b y x operator defined as J a, b) = a. To separate the synoptic-eddy component, the daily mean zonal and meridional winds are band-pass filtered in the period of 2 8 days using a Lanczos filter using 41 weights, Duchon, 1979). The lowfrequency variability is defined as the monthly mean value. Empirical orthogonal functions EOF) are used to expand ψ a 2013 Royal Meteorological Society Meteorol. Appl. 21: )

3 770 G-.R.Tanet al. a) b) c) Figure 1. The NAO patterns of low-frequency variability. a) Obtained by regressing the anomalous monthly stream function at 300 hpa ) onto the NAO indices; b) eddy-induced stream function obtained by regressing the monthly averaged anomalies of eddy vorticity forcing to the NAO indices; c) eddy forcing patterns for NAO by applying the SELF-closure on to the monthly streamfunction anomalies. The unit is 10 6 m 2 s 1, contour interval is 2. and the streamfunction tendencies 1 J ψ, ψ )a which is further denoted as χ a as follows: N ψ ψ a = E ψk A k 6) N χ k=1 χ a = E χk B 0k E ψk B k 7) k=1 where E ψ, E χ are leading spatial EOF patterns of ψ a and χ a, A k, B 0k and B k are associated time coefficients of the k th leading EOF modes, but B 0k and B k mean the time coefficients of χ a projecting on the EOF modes of the streamfunction tendencies and streamfunction respectively. N ψ, N χ are the numbers of the truncated EOF modes for ψ a and χ a. The SELF operator is here mainly for the study of the Northern Hemisphere, the domain of the data for EOF analysis is in the area over N; E). Based on the linear framework of Jin et al. 2006a), χ a can be directly related to ψ a through a dynamic relationship between the two fields as χ a L f ψ a, namely, E ψ B E ψ L f A. To derive the closure operator empirically, a simple multiple linear regression method N χ k=1 is used to establish the relation between A k and B k,which is defined as B = L s A + ε s,where L s is the linear regression operator, and ε s is the error. B Eψ T E ψl f A L s A. Because Eψ T E ψ = I where I is identity matrix), the L f closure operator can be obtained from those data as L f L s. In reality, the monthly anomalous streamfunction tendency is not completely a linear function of the monthly mean streamfunction fields. There is a number of sources that may contribute to the residual term ε s, including truncation error and neglected nonlinearities as well as sampling fluctuations. All of these may hinder the accuracy of the empirically derived linear closure operator. Nevertheless, as shown in Peng et al. 2003) and also later in this section, such an empirical model tends to capture SELF-feedback reasonably well, especially for the leading low-frequency mode Validating of the empirical operator To construct the empirical closure operator, the 300 hpa streamfunction fields of the reanalysis data from national Centers for Environmental Prediction/National Center for Atmospheric Research December 1951 to February 2008) with horizontal resolution Kalnay et al., 1996) is used. The synoptic eddy fields are derived from band-pass filtered daily data as mentioned in the above section, and the low-frequency variability is defined as the monthly mean anomaly field with respect to monthly mean climatology. To examine the accuracy of the empirical closure L f,the relation between a leading mode of atmospheric low-frequency variability such as the NAO and its associated eddy forcing is examined. The observed patterns of the NAO derived from the monthly mean anomaly fields of the 300 hpa streamfunction fields ψ a is shown in Figure 1a), whereas the corresponding observed eddy forcing fields χ a are shown in Figure 1b). These patterns are obtained by linearly regressing the observed cold-season monthly mean anomaly fields ψ a and χ a onto the monthly time series of the NAO indices from Climate Prediction Center cts/precip/cwlink). The synoptic eddy forcing Figure 1b)) associated with the NAO projects positively onto the streamfunction anomaly patterns Figure 1a)). Moreover, the eddy forcing associated with the NAO is mostly concentrated in the Atlantic sector and is much weaker over the other half of the NH. These results are well known and they indicate a positive SELF-feedback as noted by many authors in earlier studies e.g. Lau, 1988; Jin et al., 2006a). With the empirical closure of eddy feedback, the observed low-frequency flow ψ a can be used to estimate the eddy forcing by L f ψ a. The results, as shown in Figure 1c), are very similar to the observed eddy forcing in Figure 1b). The pattern correlation between the parameterized eddy forcing L f ψ a and the observed χ a is 0.92 for the NAO, which is higher than those obtained by dynamically derived closure in Jin et al. 2006a). This result is also consistent with that obtained by Peng et al. 2005) who obtained excellent empirical parameterization for the NAO related to eddy forcing. In the parameterized eddy forcing associated with the NAO, not only the positive projections of stream function tendency pattern onto the streamfunction are captured, but also the magnitudes of these eddy-induced stream function forcings are well expressed in the mid-latitude and Polar regions. Only the magnitudes of parameterized negative forcing in the Polar regions are somewhat weaker than the observed see Figure 1c)). To further illustrate the validity of the empirical closure for the SELF-feedback, the parameterized eddy forcing patterns 2013 Royal Meteorological Society Meteorol. Appl. 21: )

4 The role of eddy feedback in the excitation of the NAO 771 Figure 2. Pattern correlations of the observed and parameterized eddy forcing associated with the first 10 leading EOFs of the monthly mean anomalies of the streamfunction field during the cold seasons of for northern hemisphere. of L f ψ a associated with each EOF pattern of ψ a for the period is calculated. The observed eddy forcing patterns corresponding to these EOF modes of monthly-mean anomalies of streamfunction are obtained by regressing χ a onto the time coefficients associated with each EOF of ψ a. The spatial correlations of the parameterized and observed eddy forcing patterns for the first 10 EOFs of ψ a, which explain about 78% of the total variance of the low-frequency variability of ψ a, are shown in Figure 2. It can be seen that the correlations are all above 0.8, also higher than those in Jin et al. 2006a). However, it should be noted that these skills of the empirical closure operator are not cross validation skills whereas the skill from dynamic closure operator derived in Jin et al. 2006a) is a truly independent skill. The errors of L f include truncation error, nonlinearities as well as sampling fluctuations. The truncation error, which depends on the truncated EOF modes of both streamfunction and eddy forcing in terms of streamfunction tendencies, can be estimated before the empirical operator is finally established. The accuracy of the EOF-based empirical operator L f is shown in Figure 3. The more EOF modes are retained, the largest the variance of eddy induced forcing can be explained by EOF expansion, and the correlation coefficients between the resolved forcing L f ψ a and actual χ a are also increasing steadily. However, when the retained EOF modes are larger than 10, the cross validation errors increase as the truncation errors decrease. Therefore, only 10 EOF modes for χ a were retained. The cross validation error also depends on the number of EOF modes of the streamfunction. The EOF mode number of the streamfunction is set at 50 and the accuracy of the SELFoperator L f is not higher with more EOF modes selected. To cross validate the empirical closure of the SELF-feedback, the empirical operator was established using the data about 40 years of December, January and February DJF) from 1951/1952 to 1990/1991. The correlation distribution between the observed monthly-mean eddy-induced streamfunction forcing χ a and that estimated by applying the empirical operator onto ψ a for DJF from 1991/1992 to 2007/2008 are then calculated. The correlation between the observed seasonalmean eddy-induced streamfunction forcing and that estimated by applying the empirical operator but onto the seasonalmean anomalous streamfunction field for all 57 winters from 1951/1952 to 2007/2008 is also examined. The correlation coefficients 171 months and 57 winters) are positive with the highest values over 0.8 in both Atlantic and Pacific storm track areas and the seasonal map shows even a few broad areas of higher correlation not shown). The empirical operator shows Figure 3. Cross-validated skill of linear SELF operator as a function of the number of retained EOF modes of December to February 300 mb stream-function anomalies and related tendency. Solid line: correlation of the observed χ a and the estimated by SELF linear operator for fitting skill; dashed line: explaining variance of the truncated EOF modes; dashed line with real circle marks: correlation of the observed χ a and the estimated by SELF linear operator for cross-validated skill. reasonable ability in expressing the observed eddy-induced streamfunction forcing for each month and each season through the entire 57 winters. There are significant fluctuations in pattern correlations for monthly data, indicating that the significant portion of month-mean anomalous eddy forcing cannot be linearly parameterized by the closure. These fluctuations are less for seasonal mean data, although the empirical operator may still fail for some years. Nevertheless, the correlation score is similar for the entire period, indicating that the empirical operator stands for cross validation. Overall, the parameterized empirical SELF-feedback operator captures the linear relation between eddy forcing and the large scale circulation. 3. Leading low-frequency modes of a linear barotropic model with empirical SELF-closure 3.1. EOF-space representation of linear dynamics To study the linear dynamics of low-frequency atmospheric flow dynamics, a set of fully nonlinear primitive equations numerically linearized with respect to climatological mean are often used to derive the related linear dynamical operator cf Hoskins and Karoly, 1981; Watanabe and Jin, 2004). However, this kind of linear operator is often expressed in terms of a matrix whose rank depends on model resolution. Here it is proposed to project the linear operator onto the EOFs so that the linear operator has a low-dimension in a reduced phase-space. Following Jin et al. 2006a), the linearized barotropic model with the closure of SELF-feedback can be written as: ψ a + Lψ a + L f ψ a = Q a 8) t where Q a denotes external forcing, and L f is the SELFfeedback operator, and L is the linear operator, which are obtained in Section 2. Here, r is a linear damping rate, ψ c denotes the monthly climatological mean streamfunction. The monthly mean value is calculated for winter months DJF) from 1951/1952 to 2007/2008 and f is the Coriolis parameter. To obtain the linear operator L, the numerical method proposed by Hoskins and Karoly 1981) is used with the difference that each EOF pattern of the streamfunction, instead of each spherical harmonic function, is the perturbation streamfunction field. The model is then used to evaluate and compute the tendency corresponding to each EOF pattern cf Appendix 2013 Royal Meteorological Society Meteorol. Appl. 21: )

5 772 G-.R.Tanet al. different, in order to add them together, L f is expanded by simply adding zeros to make it to a full matrix with ranks the same as the L operator. The reduced model is then tested with a localized external forcing. Results show that the linear dynamic system can capture the remote forcing pattern in the mid-latitudes, which is similar to Jin et al. 2006a). Thus, this operator with a much reduced degree of freedom captures the essential linear dynamics for low-frequency variability, similar to the full linear barotropic model with an approximate dynamic SELF-closure Jin et al., 2006b) NAO-like SVD modes with and without SELF feedback Figure 4. Cross-validated skill of linear dynamic operator. a) Crossvalidated skill of linear dynamic operator as a function of the number of retained EOFs of December to February 300 mb streamfunction anomalies and related tendency. Solid line: explaining variance of the truncated EOF modes for external forcing; dashed line: correlation of the observed tendency and the estimated by space reduced linear dynamic operator. b) The errors as a function of the number of the retained EOFs. A). The linear operator in the EOF-based space can then be obtained. For instance, if all tendencies are denoted as F E, then F E = LE ψ,solcan be obtained by projecting F E onto the EOF base as L = Eψ T F E. It should be pointed out that this method is not limited to the barotropic model and can be easily extended to any primitive equation models. In fact, in the latter case, the advantage of this method in terms of great reduction in the size of matrix for linear dynamic operator will become even more significant. The accuracy of the EOF-based linear operator L is shown in Figure 4. When more EOF modes are retained, larger variance of the low-frequency-induced tendency F can be explained by the EOF expansion, and the correlation coefficients between the resolved forcing LE ψ and actual F also increase steadily. The first 50 EOF modes can explain 98.8% of the total variance of F, and the correlation coefficient between the observed F and the resolved portion LE ψ arrives at about when all the 50 EOF modes are included Figure 4a)). The errors are given in the Equation 9): )2 ε = Foi F i / F oi 100% 9) i where F oi is the observed external forcing, and F i is the estimated. The errors also decrease as the EOF modes increase, and the error is less than 0.25% when 50 EOFs are considered Figure 4b)). It suggests that the EOF-space-based linear operator L is a very accurate approximation of the linear dynamical system for low-frequency flow variability. With the linear dynamic operator expressed in the EOF-space and the empirical operator L f, the linear dynamic operator is L + L f. Since the dimension of the two linear operators is i It was suggested in an earlier study Jin et al., 2006b) that leading SVD modes left v vector) of the linear operator L + L f resemble the observed leading patterns of extratropical lowfrequency variability of the atmospheric circulation. Following this approach, the role of the SELF-feedback in the formation of the leading modes is examined by tracing the dependence of the leading SVD modes of the linear matrix for L + μl f on the control parameter μ see Figures 5 and 6). When μ = 1.0, the leading SVD mode resembles the observed pattern Figure 5c) and f)). As μ decreases from 1.0 toward zero Figure 5a) and d)), the NAO-like leading SVD mode not only loses some of its pattern resemblance the pattern correlation between the observed pattern and SVD pattern is 1.0 and 0.29 as μ = 1.0 and μ = 0.0, respectively), but also the corresponding singular value λ μ increases significantly with the eddy feedback Figure 6). It suggests that SELF-feedback is important not only in the formation of the NAO pattern but also in increasing the singularity of the NAO mode. The increase of growth rate of the NAO due to SELF feedback is consistent with the well known notion that SELFfeedback for the NAO is strongly positive e.g., Lorenz and Hartmann, 2001, 2003; Jin et al., 2006a, 2006b; Pan et al., 2006; Ren et al., 2009). It is the strong positive feedback that tends to bring the NAO close to criticality or singularity and this increase of singularity due to the SELF-feedback greatly increases the variance of the NAO Jin, 2010). To further demonstrate this point, the change in pattern of the ratio between the observed NAO pattern and the diagnosed NAO with the reduction of the SELF-feedback is examined. This ratio is defined as < ψ, ψ NAO >/<ψ NAO, ψ NAO >.Itisa function of μ. When μ = 0.0, this value is about 0.28 Figure 6), indicating that without SELF-feedback, the NAO will lose more than a half of its identity in terms of its pattern. 4. Excitation of NAO mode Although eddy fluxes are central to the forcing and maintenance of the NAO, external excitation is also essential in maintaining the NAO variability. Since the leading left-vector of the linear dynamic operator is related to the spatial structure of the NAO, the corresponding right-vector is examined to study the optimal forcing for the NAO-like mode. The optimal forcing patterns in terms of streamfunction tendency for the case with and without SELF-feedback are shown in Figure 5. The main loadings of the patterns are similar, apart from that the ranges of centre variation are different for different eddy feedback. The main loading of the forcing pattern for the NAO is over the Atlantic and eastern Pacific sectors. However, those forcings over the Indian Ocean and western Pacific become more apparent, and those over the tropical area shift to the higher latitude area in 2013 Royal Meteorological Society Meteorol. Appl. 21: )

6 The role of eddy feedback in the excitation of the NAO 773 µ = 0.0 µ = 0.5 µ = 1.0 Figure 5. The leading SVD vectors of the linear system with a), d) none μ = 0.0); b), e) middle μ = 0.5) and c), f) strong μ = 1.0) SELF feedback for the NH cold season basic flow. The contour interval is 1.0. a), b) and c) are the left v-vectors for NAO-like pattern, while d), e) and f) are associated right u-vectors for NAO-like pattern of the linear system. The contour interval is Figure 6. Structure variation of NAO pattern as a function of eddy feedback. Dotted line, real line and dashed line denotes the pattern correlation, < ψ, ψ NAO >/<ψ NAO, ψ NAO > and ratio of singular values λ μ / λ μ=1, respectively. the Atlantic with the SELF-feedback Figure 5d) f)), while the forcings in the east Pacific also expand to the central and western Pacific. The significant differences remain in terms of the amplitudes over the centre areas of forcing patterns. In tropical Indian Ocean to Western Pacific, and mid- and high- Atlantic regions, the dependence of the sign and amplitudes in the optimal forcing on the SELF-feedback suggest that it not only serves as a positive feedback to amplify the NAO but also alters the way the NAO might be excited. It suggests that with the SELF- feedback, anomalous forcing over the Indian Ocean to the west Pacific and subtropical to higher-latitude Atlantic are easier to excite the NAO pattern than the case without SELFfeedback. The forcing anomalies over the tropical Atlantic may become less important in exciting the NAO pattern with the participation of the SELF-feedback. To illustrate further the role of the SELF-feedback in the excitation of the NAO, a set of experiments were conducted with isolated forcing in the regions with significant loadings in the optimal forcing pattern Figure 5). The areas of forcing having absolute values greater than 2 in Figure 5 are chosen as the effect s forcing sites. When the isolated forcing is located over the Indian Ocean sector, though there is a wave train induced from the resource area with a positive anomaly centre over the North Atlantic and a negative one over the Polar area, the response is weak and the anomalous range is small in the case without eddy feedback. With the eddy feedback Figure 7c)), the atmospheric remote response to the Indian Ocean forcing becomes much stronger, with a negative variation in the Polar area and the positive one over the North Atlantic as nearly four times large as in Figure 7a). It suggests that anomaly forcing over the Indian Ocean may contribute to the formation of the NAO pattern with positive phase and it is largely owing to the eddy feedback. When the isolated forcing is located over the east Pacific sector Figure 7d) f)), there is a PNA-like wave train which appears from the resources area. Though the amplitude changes not so greatly as that in Figure 7b) c), the anomalous range in the response tends to increase and expand to the regions further away. When the isolated forcing is located over the tropical Atlantic sector Figure 7g) i)), an NAO-like pattern can be induced without eddy feedback and the amplitude increases as eddy feedback increases to moderate intensity but decreases as 2013 Royal Meteorological Society Meteorol. Appl. 21: )

7 774 G-.R.Tanet al. Figure 7. The direct response of the streamfunction fields to the stable forcing in different areas with none μ = 0.0), middle μ = 0.5) and strong μ = 1.0) SELF feedback for the NH.a), b) and c) for the forcing in the Indian Ocean; d), e) and f) for the forcing in the east Pacific; g), h) and i) for the forcing at low-latitude Atlantic; j), k) and l) for the forcing at mid- and high-latitudes of the Atlantic. Blue yellow) shaded areas denote the positive negative) anomaly divergence stable forcing. The contour interval is m 2 s 1, but for a) i) two lines with contour value m 2 s 1 and m 2 s 1 are added. eddy feedback becomes very strong. This is not surprising because the optimal excitation in the tropical Atlantic area may disappear under the strong eddy feedback see Figure 5). A dipolar anomaly forcing in the Atlantic tends to generate a significant NAO-like response more effectively with eddy feedback Figure 7j) l)). As all the amplitudes of the forcings used here are identical, the anomalies of the response are comparable. Both the extratropical anomaly forcing and tropical forcing can induce NAOlike patterns in the winter atmosphere. However, under the condition of no eddy feedback, the most effective excitation for NAO-like mode in the tropical is the eastern tropical Atlantic forcing. For moderate eddy feedback, forcings over tropical eastern Pacific and Indian Ocean become important in exciting the NAO-like mode, while with the full strength of eddy feedback the most important forcing in the tropics is the one over the Indian Ocean for the excitation of the NAO-like mode. In the extra-tropics, the local dipolar forcing can induce a significant NAO-like response. 5. Conclusion and discussion In this work, a low-dimension linear model is proposed whose classic linear dynamics is derived from a dynamic model linearized with respect to the climatological mean, then projected onto the EOF-bases and additional empirical linear closure for synoptic eddy and low frequency feedback. This simple model is useful for understanding the role of eddy feedback on the excitation and formation of low frequency mode. This approach may be extended to 3-demensional models. The observed NAO-like pattern can be qualitatively reproduced by the proposed linear system, where the SELF-feedback plays a central role in the generation of NAO-like mode. Though the NAO-like mode can be produced owing to the internal linear dynamics, the external forcing is also important in its excitation. The importance of the external forcing in the excitation of the NAO-like pattern depends on the eddy feedback. Those forcings over the tropical Indian Ocean to the 2013 Royal Meteorological Society Meteorol. Appl. 21: )

8 The role of eddy feedback in the excitation of the NAO 775 west Pacific and east Atlantic area have little influence on the excitation of the NAO-like mode if there is no eddy feedback, indicating the essential role of eddy feedback in enhancing atmospheric teleconnections. As for the NAO-like mode, the most effective excitation for the NAO-like mode is the one over the Indian Ocean and the dipolar forcing over the North Atlantic. More work is needed to further test these results in a baroclinic framework. Acknowledgements This work was supported by National Science foundationnsf) Grant ATM , the National Key Technology R&D Program 2012CB955200), the Special Foundation for Public Welfare Industry GYHY , GYHY ), and the Priority Academic Program Development of Jiangsu Higher Education Institutions PAPD) of China. Appendix A: The linear advection operator in EOF space The streamfunction field is denoted as ψj ) or in vector form. Using the linear barotropic vorticity equation, the streamfunction tendency due to linear advection and friction can be obtained, which shall be denoted as Fj ) or in vector form F. Then the linear equation can be written as = F = L A.1) t The linear operator L* is J*J matrix and is normally very large in grid space. Here we propose to truncate the L* matrix using the EOF of the streamfunction as the base functions. With a set of EOFs of, which is denoted as E n j ) or in matrix form E, then we can express = EA + R ψ A.2) here A is principle component vector and A n t) is the time coefficient of n th EOFs, and R ψ is truncation error. The tendency term can also be expressed as F = EB + R F A.3) where B is principle component vector derived from the tendency and R F is the residual part that could not be expressed in the EOF space of. 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