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Nonlinear Modes of North American Winter Climate Variability Derived from a General Circulation Model.. Wu, Aiming; Hsieh, William W.; Zwiers, Francis W. 2003-07-31

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15 JULY 2003 2325WU ET AL.q 2003 American Meteorological SocietyNonlinear Modes of North American Winter Climate Variability Derived from aGeneral Circulation Model SimulationAIMING WUANDWILLIAM W. H SIEHDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, CanadaFRANCIS W. Z WIERSCanadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, University of Victoria, Victoria,British Columbia, Canada(Manuscript received 14 May 2002, in final form 20 December 2002)ABSTRACTNonlinear principal component analysis (NLPCA), via a neural network (NN) approach, was applied to anensemble of six 47-yr simulations conducted by the Canadian Centre for Climate Modelling and Analysis(CCCma) second-generation atmospheric general circulation model (AGCM2). Each simulation was forced withthe observed sea surface temperature [from the Global Sea Ice and Sea Surface Temperature dataset (GISST)]from January 1948 to November 1994. The NLPCA modes reveal nonlinear structures in both the winter 500-mb geopotential height (Z500) anomalies and surface air temperature (SAT) anomalies over North America,with asymmetric spatial anomaly patterns during the opposite phases of an NLPCA mode. Only during itsnegative phase is the first NLPCA mode related to the El Nin˜o–Southern Oscillation (ENSO); the positive phaseis related to a weakened jet stream. Spatial patterns of the NLPCA mode for the Z500 anomalies generally agreewith those for the SAT anomalies.Nonlinear canonical correlation analysis (NLCCA), also via an NN approach, was then applied to the mid-latitude winter GCM data and the observed SST of the tropical Pacific. Nonlinearity was detected in both theforcing field (SST) and the response field (Z500 or SAT) at zero time lag. The leading NLCCA mode for theSST anomalies is a nonlinear ENSO mode, with a 308–408 eastward shift of the positive SST anomalies duringEl Nin˜o relative to the negative SST anomalies during La Nin˜a. The leading NLCCA mode for the Z500 anomalyfield is a nonlinear Pacific–North American (PNA) teleconnection pattern. The ray path of the Rossby wavesinduced during El Nin˜o is 108–158 east of that induced during La Nin˜a. The nonlinear atmospheric response toENSO is also found in the leading NLCCA mode for the SAT anomalies.1. IntroductionSubstantial progress has been made during the pasttwo decades toward understanding the wintertime ex-tratropical atmospheric responses to the tropical forc-ings associated with El Nin˜o sea surface temperature(SST) anomalies (Trenberth et al. 1998). The mostprominent teleconnection pattern is the Pacific–NorthAmerica (PNA) pattern (Horel and Wallace 1981),which is thought to link the changes in the extratropicalcirculation to the tropical SST through Rossby wavedynamics (Wallace and Gutzler 1981; Hoskins and Ka-roly 1981).The simplest view of the atmospheric climate signalassociated with the El Nin˜o–Southern Oscillation(ENSO) phenomenon is that the atmosphere respondsCorresponding author address: Dr. Aiming Wu, Dept. of Earth andOcean Sciences, University of British Columbia, Vancouver BC V6T1Z4, Canada.E-mail: awu@eos.ubc.calinearly, with anomalies during the El Nin˜o phase beingthe reverse of those during the La Nin˜a phase. However,recent evidence shows that the responses to warm andcold events are not exactly opposite. Sittel (1994) foundthe marginal probabilities of extreme rainfall and tem-perature over the southeastern United States to be highlynonlinear functions of the phase of the Southern Os-cillation (SO). For example, it was found that, since1946, the warm event–enhanced rainfall signal has beenmuch larger than the cold event–suppressed signal inthe southeastern United States. Richman and Montroy(1996) examined the composite January temperatureand precipitation patterns over the United States andparts of Canada associated with El Nin˜o and La Nin˜aevents. Their results suggest that El Nin˜o and La Nin˜ahave their own unique characteristics in terms of tem-perature and precipitation. Asymmetric spatial patternsof Canadian surface air temperature and precipitationassociated with the SO were also detected by Shabbarand Khandekar (1996) and Shabbar et al. (1997). Furtherevidence for a nonlinear response of the North America2326 VOLUME 16JOURNAL OF CLIMATEFIG. 1. Schematic diagram of the NN model for calculatingNLPCA. The network is a standard feed-forward NN (i.e., multilayerperceptron), consisting of several layers of neurons (i.e., variables).The network maps from the input layer on the left to the encodinglayer, then to the bottleneck layer (with a single neuron), to the de-coding layer, then finally to the output layer on the right. In theapplications here, the hidden layers h(x)and h(u)both have three neu-rons, while the input x are the five leading PCs.climate to ENSO was provided by Hoerling et al. (1997)with composite analysis and numerical experiments us-ing an idealized atmospheric general circulation model,where a shift in the equatorial positions of the maximumrain responses and a phase shift of teleconnection pat-terns in the upper troposphere were found. The inherentnonlinearity in the tropical rain response may itself beresponsible for the phase shift in the extratropical te-leconnection patterns.The robustness of the nonlinear climate response toENSO’s extreme phases has also been investigated withfour GCM simulations (Hoerling et al. 2001), whichwere all found to have a 500-mb height response toextreme warm tropical Pacific SST that was twice asstrong as the response to extreme cold SST. The lon-gitudinal phase of the GCM’s teleconnections also shift-ed eastward during warm events as compared with coldevents, though this displacement is smaller than thatobserved. A nonlinear identification of the atmosphericresponse to ENSO was also addressed by Hannachi(2001) using general circulation models.Nonlinear methods are required to investigate thenonlinear behavior of the North American climate var-iability and its relation to ENSO. Standard multivariatestatistical techniques such as principal component anal-ysis (PCA) [also known as empirical orthogonal func-tion (EOF) analysis], and canonical correlation analysis(CCA) are linear methods. Composite analysis does notassume linearity, but is restricted to the analysis of dif-ferences between specific phases of the SO or equatorialSST indices. Recently, neural networks (NN; Hsieh andTang 1998) have been used for nonlinear PCA (NLPCA;Kramer 1991) and nonlinear CCA (NLCCA; Hsieh2000). NLPCA has been applied to the Lorenz three-component chaotic system (Monahan 2000), tropical Pa-cific SST, and sea level pressure fields (Monahan 2001).NLPCA was recently used to represent the quasi-bi-ennial oscillation (QBO) in the equatorial stratosphericwind (Hamilton and Hsieh 2002), and to explore thenonlinear characteristics of Canadian surface air tem-perature (Wu et al. 2002). NLCCA has been applied tostudy the nonlinear relation between the tropical Pacificsea level pressure (SLP) and SST fields (Hsieh 2001a),as well as between the wind stress and SST fields (Wuand Hsieh 2002). Also, NLCCA has been applied toforecasting the tropical Pacific sea surface temperaturesin the Experimental Long-Lead Forecast Bulletin (moreinformation available online at www.iges.org/ellfb).In this paper, the NLPCA (Hsieh 2001b) and NLCCA(Hsieh 2001a) models will be applied to study the win-tertime climate variability over North America and itsrelation to ENSO as simulated in an ensemble of six47-yr simulations produced with the Canadian Centrefor Climate Modelling and Analysis (CCCma) second-generation atmospheric general circulation model(AGCM2). This paper is organized as follows. In section2, the data and the two methods (NLPCA and NLCCA)are briefly introduced. Nonlinear modes of the winter500-mb geopotential height and surface air temperaturethat were extracted by the NLPCA are described in sec-tion 3. Correlated nonlinear modes of atmospheric andtropical Pacific SST variability that were extracted bythe NLCCA are described in section 4. Section 5 pre-sents a summary and discussion.2. Methodology and dataa. NLPCAA variable x, which consists of l spatial stations andn observations in time, can be expressed in the formx(t) 5 [x1(t),...,xl(t)], where for each i (i 5 1,...,l), xi(t)(t 5 1,...,n) is a time series of length n. PCAis used to find a scalar variable u and an associatedvector a, withu(t) 5 a · x(t), (1)so that2^\x(t) 2 au(t)\ & is minimized, (2)where ^···& denotes a sample or time mean. Here u,called the first principal component (PC), is a time seriesresulting from a linear combination of the original var-iables xi, while a, the first eigenvector of the data co-variance matrix (the first EOF), often describes a spatialpattern. The second PC can similarly be extracted fromthe residual x 2 au, and so on for the higher modes.In practice, the common algorithms for PCA extract allmodes simultaneously by calculating the eigenvaluesand eigenvectors of the data covariance matrix.The fundamental difference between NLPCA andPCA is that NLPCA allows a nonlinear continuous map-ping from x to u whereas PCA only allows a linearmapping. NLPCA is performed with a NN, such as thatdisplayed in Fig. 1, which contains three ‘‘hidden’’ lay-ers of variables (or ‘‘neurons’’) between the input andoutput layers. These layers are called the encoding, bot-tleneck, and decoding layers, respectively. Four transfer15 JULY 2003 2327WU ET AL.functions f1, f2, f3, f4are successively used to mapfrom the input layer to the output layer (x → h(x)→ u→ h(u)→ x9), where x9 is the least square approximationof x. The mappings are(x)(x)(x)h 5 f [(W x 1 b ) ], (3)k 1 k(x)(x)(x)u 5 f (w · h 1 b ), (4)2(u)(u)(u)h 5 f [(w u 1 b ) ], (5)k 3 k(u)(u)(u)x95 f [(W h 1 b ) ], (6)i 4 iand x is the input column vector of length l. The en-coding layer h(x)is described by a column vector oflength m (m is the number of the hidden neurons in theencoding layer). The parameters that control the trans-formation to this layer are W(x), which is an m 3 l weightmatrix, and b(x), a column vector of length m containingthe bias parameters. Index k ∈ [1, m]. The bottlenecklayer contains a single neuron, which represents the non-linear principal component u. The transformation be-tween encoding and bottleneck layers is controlled byan m-element weight vector w(x)and a bias parameter(x). The decoding layer contains the same number ofbneurons m as the encoding layer, and the output layeris again a column vector of length l. The transformationsbetween these layers are controlled by m-element weightand bias vectors w(u)and b(u)(decoding layer), and anl 3 m weight matrix W(u)and an l-element bias vector(u)(output layer). Transfer functions f1and f3are gen-berally nonlinear (e.g., the hyperbolic tangent function),while f2and f4are taken to be the identity function.See Hsieh (2001b) for more details.Optimal values of the weight parameters for eachmapping layer are found by minimizing the cost func-tion J [see Eq (7)], rendering the outputs to be as closeto the inputs as possible within constraints imposed bya weight penalty term (discussed below). Data com-pression is achieved at the bottleneck, with the bottle-neck neuron representing a single degree of freedom,namely u, the nonlinear principal component (NLPC).The nonlinear optimization was carried out with aquasi-Newton method. To avoid the local minima prob-lem (Hsieh and Tang 1998, p. 1859), an ensemble of30 NNs with random initial weights and bias parameterswas run. Also, 20% of the data was randomly selectedas testing data and withheld from the training of theNNs. For the NLPCA, runs where the mean-square error(mse) for the testing dataset was 10% larger than thatfor the training dataset were rejected to avoid overfittedsolutions. Then the NN with the smallest mse was se-lected as the desired solution.As noted above, the cost function used to identify theNLPCA model in this study has an extra weight penaltyterm,2(x)2J 5^\x 2 x9\ &1p (W ) , (7)Okikiwhere p $ 0 is called the weight penalty parameter.Increasing p increases the concavity of the cost function,thereby pushing the weights W(x)to be smaller in mag-nitude and consequently yielding smoother and less non-linear solutions than when p is small or zero. With alarge enough p, the danger of overfitting is greatly re-duced (Hsieh 2001b). The NLPCA was run repeatedlywith various values of the penalty parameter (rangingfrom 0.001 to 0.02), and the solution with the smallestmse was chosen as the final solution.After the first NLPCA mode has been subtracted fromthe data, the residual is again input into the NLPCAnetwork to extract the second NLPCA mode. Becauseof the noisier conditions, the penalty parameter is in-creased to the range 0.01–0.2 for the second mode.The weight penalty parameters in NLPCA (also inthe NLCCA described below) are selected by means ofa search. Future research will hopefully provide a moreobjective way to select these parameters.b. NLCCAGiven two sets of variables x and y, CCA is used toextract the correlated modes between x and y by lookingfor linear combinationsu 5 a · x and y 5 b · y, (8)where the canonical variates u and y have maximumcorrelation; that is, the coefficient vectors a and b arechosen such that the Pearson correlation coefficient be-tween u and y is maximized.In NLCCA, we follow the same procedure as in CCA,except that the linear mappings in Eq. (8) are replacedby nonlinear mapping functions using two-layer feed-forward NNs. The mappings from x to u and y to y arerepresented by the double-barreled NN on the left-handside of Fig. 2. By minimizing the cost function J 52corr(u, y), one finds the parameters that maximize thecorrelation corr(u, y). After the forward mapping withthe double-barreled NN has been solved, inverse map-pings from the canonical variates u and y to the originalvariables, as represented by the two standard feed-for-ward NNs on the right side of Fig. 2, are to be solved,where the cost function J1is the mse of the output x9relative to x (msex), and the cost function J2, the mseof the output y9 relative to y (msey) are separately min-imized to find the optimal parameters for these two NNs(see Hsieh 2001a for details).An ensemble approach is also used for the NLCCA—runs where 2corr(u, y), the msex, or the mseyfor thetesting dataset were 10% larger than those for the train-ing dataset were rejected. The NNs with the highest cor(u, y), and smallest msexand mseywere selected as thedesired solution. A weight penalty was again used(Hsieh 2001a) as in NLPCA.In brief, the major advantage of NLPCA and NLCCAis that both are free from linear constraint. This allowsus to identify principal components or canonical variatesthat vary along empirically derived curves through the2328 VOLUME 16JOURNAL OF CLIMATEFIG. 2. Schematic diagram illustrating the three feed-forward NNs used to perform NLCCA. Inthe applications here, all the hidden layers h(x), h(y), h(u), and h(y)have three neurons each.data space instead of only straight lines. Consequently,a larger fraction of the variance can be explained (rel-ative to PCA) or higher canonical correlation can bereached (relative to CCA), thus raising the possibilityof improving the interpretation and forecasting climateusing the NLCCA model. The main difficulty in theapplication of NLPCA or NLCCA is that the cost func-tion often has multiple local minima (Hsieh and Tang1998). It takes considerable computation to obtain arobust solution that approximates the ‘‘global’’ mini-mum by running the optimization many times from ran-dom initial parameters. Another drawback is overfitting,that is, fitting to the noise in the data. Using a weightpenalty and reserving part of the data as validation datacan alleviate overfitting.c. DataThe monthly mean 500-mb geopotential height(Z500) and surface air temperature (SAT) data used inthis study were produced by the CCCma AGCM2, aspectral model with T32 resolution in the horizontal, 10levels in the vertical, semi-implicit time stepping, anda full physics package (McFarlane et al. 1992; Boer etal. 1992). An ensemble of six 47-yr runs of AGCM2were carried out, in which each integration was startedfrom different initial conditions and forced by SSTsfrom the Global Sea Ice and Sea Surface Temperature(GISST) dataset (version 2.2; Rayner et al. 1996). Sim-ulations were performed from January 1948 to Novem-ber 1994, and were initially reported by Zwiers et al.(2000). For each run, anomalies were calculated by sub-tracting the monthly climatology based on the wholeperiod. Monthly anomalies were then smoothed by tak-ing a 3-month running mean and removing linear trends.Only the smoothed data for December–February (DJF)are analyzed, thus the total number of months used fromthe six AGCM2 runs is 840 [(47 3 3 2 1) 3 6]. Thedomain of interest is 208–768N, 1508E–508W coveringthe North Pacific and North America. For the SAT, onlythe data over land grids were used.Monthly SST used in this study was from the recon-structed global historical SST datasets by Smith et al.(1996) for the period 1950–2000 with a resolution of28328. As the AGCM2 runs end in 1994, the SSTdata was actually used up to November 1994 in theNLCCA {the total number of months used in NLCCAis 804 [(45 3 3 2 1) 3 6]}. Similar data processingwas done for the SST data, as for the Z500 and SATdata. The area of interest for the SST is restricted to thetropical Pacific (218S–218N, 1238E–718W).Prior to the NLPCA and NLCCA, ordinary PCA (i.e.,EOF) analysis was conducted on the Z500, SAT, andthe SST anomalies to compress the data into manageabledimensions and to insure that the estimated variance–covariance matrices that enter into CCA calculation canbe inverted (Barnett and Preisendorfer 1987). EOFs ofthe three leading modes of Z500 and SAT anomaliesare shown in Fig. 3. Variance contributions from thesethree modes of the Z500 anomalies are 32.8%, 17.8%,and 12.7%, respectively, and for the SAT anomalies,26.8%, 16.1%, and 8.9%, respectively. The SAT anom-aly pattern for each mode generally reflects the anom-alous circulation implied by the corresponding Z50015 JULY 2003 2329WU ET AL.FIG. 3. (a), (b), (c) The first three EOFs of the winter Z500 anomalies and (d), (e), (f) SAT anomalies. Solid curves denote positivecontours; dashed curves, negative contours; and thick curves, zero contours. The contour interval is 0.02 in (a), (b), and (c), and 0.03 in(d), (e), and (f). The EOFs have been normalized to unit norm. If the sign in (b) is reversed, then over the North American continent, thepatterns in the (a), (b), (c) generally agree with those in (d), (e), (f).mode, although the signs for EOF2 are opposite (Figs.3b and 3e). The accumulated variance contribution ofthe five leading modes is 77.0% for the Z500 anomalies,61.9% for the SAT anomalies, and 83.7% for the tropicalPacific SST anomalies.The five leading PCs (i.e., the EOF time series) ofthe winter Z500 and SAT anomalies from 1948 to 1994were used as the inputs to the NLPCA. For the NLCCA,only data after 1950 were used since the SST data wereavailable from January 1950. Also for the NLCCA, SSTanomalies (DJF) were repeated 6 times to pair with theatmospheric data (Z500 and SAT) from the six GCM2330 VOLUME 16JOURNAL OF CLIMATEFIG. 4. The first NLPCA mode for the winter (a) Z500 anomalies and the (b) SAT anomalies plotted as (densely overlapping) squares(which produce a thick curve) in the PC1–PC2–PC33D space. The linear (PCA) mode is shown as a dashed line. The NLPCA mode andthe PCA mode are also projected onto the PC1–PC2,PC1–PC3, and the PC2–PC3planes, where the projected NLPCA is indicated by (denselyoverlapping) circles, and the PCA by thin solid lines, and the projected data points by the scattered dots. The minimum NLPC u occurs atthe left end, and the maximum u at the right end of the curve in both (a) and (b).ensemble runs. The five leading SST PCs, and the fiveleading Z500 (or SAT) PCs were used as the inputs tothe NLCCA.3. Nonlinear modes extracted by NLPCAa. Mode 1Figures 4a,b, which show the first NLPCA mode forthe Z500 and SAT anomalies, respectively, reveal non-linear structures in both datasets. The Z500 NLPCAmode 1 explains 35.7% of the total variance, versus32.8% explained by PCA mode 1 (straight line in Fig.4a). For SAT, the NLPCA mode 1 explains 30.6% ofthe variance, versus 26.8% by the linear mode. A mea-sure of the degree of nonlinearity is the ratio (r) betweenthe MSE of the NLPCA mode 1 and that of the cor-responding PCA mode. Smaller r means stronger non-linearity. When r 5 1, the nonlinear mode is reducedto the linear mode. Here r is 0.921 for the Z500, and0.901 for the SAT, suggesting moderate nonlinearity inboth datasets, with SAT being somewhat more nonlin-ear.Changing the value of a PC (i.e., selecting a point onthe straight line in Fig. 4a) has the effect of changingthe amplitude but not the spatial structure of the cor-responding EOF pattern (Fig. 3a). In contrast, both thestructure and amplitude of the spatial pattern of theNLPCA mode change smoothly as the NLPC u (whichtraces the curve in Fig. 4a) changes value. The NLPCAmaps the bottleneck neuron u to the output layer of theNN used to represent the data. This produces values forthe first five PCs that, in turn, can be combined withthe corresponding PCA spatial patterns (i.e., the EOFs)to yield the spatial pattern corresponding to u. When utakes on its minimum value, three large anomalies ap-pear in the Z500 anomaly field over the North Pacificand the North American continent, resembling a neg-ative PNA pattern (Fig. 5a), with negative height anom-15 JULY 2003 2331WU ET AL.FIG. 5. The spatial patterns of the NLPCA mode 1 as the NLPC u takes its min and max values. The Z500 anomaly patterns for min uand max u are shown in (a) and (b), respectively, with contour intervals of 20 m, and the SAT anomaly patterns, in (c) and (d), respectively,with contour intervals of 18C.alies over Canada and the United States except westernAlaska and the southeastern United States, where thereare positive height anomalies. The SAT anomaly patternassociated with the minimum u is roughly in agreementwith the Z500 anomaly pattern, with positive anomalies(18–28C) over the southeastern United States and neg-ative anomalies over the rest of North America (Fig.5c).When u takes on its maximum value, the Z500 anom-alies (Fig. 5b) show no closed contours north of 508N,with the United States covered by negative height anom-alies, and Canada by positive anomalies, implying aweakened jet stream because the geopotential gradientis reduced in the region that is usually occupied by thejet stream. With maximum u, the SAT anomalies (Fig.5d) show that the United States and southern Canadaare cooler, while other areas of Canada and Alaska arewarmer.The NLPCA Z500 anomaly pattern for maximum u(Fig. 5b) resembles EOF1 pattern (Fig. 3a) while thepattern for minimum u (Fig. 5a) resembles EOF2 (Fig.3b). This means that the weakened jet stream state as-sociated with maximum u does not have an equally strongnegative counterpart (i.e., an enhanced jet stream state).Instead the enhanced jet steam state is overshadowed bythe negative PNA state found during minimum u.Comparing Figs. 5a with 5c, and 5b with 5d, we cansee some spatial correspondence between the anomaliesof Z500 and SAT, where positive anomalies of Z500tend to occur roughly together with positive anomaliesof SAT, and similarly for the negative anomalies, there-by revealing some consistency in the nonlinear struc-tures found by the NLPCA approach.b. Mode 2After removing the NLPCA mode 1 from the data,the residuals were again input into the NLPCA networkto extract the second NLPCA mode, which was alsofound to contain notable nonlinearity in both the Z500and SAT fields. The second Z500 NLPCA mode ex-plains 16.5% of the total variance, which is slightly morethan the 15.2% explained by the corresponding linearmode—here the linear mode is not the same as the PCAmode 2, as the latter is extracted from the residual withthe PCA mode 1 (not the NLPCA mode 1) subtracted2332 VOLUME 16JOURNAL OF CLIMATEFIG. 6. Similar to Fig. 5 but for the NLPCA mode 2.from the original data. The mse ratio between the secondNLPCA mode and the corresponding linear mode forZ500 is 0.953. There is stronger nonlinearity in the sec-ond SAT NLPCA mode, which explains 12.5% of thetotal variance, versus the 9.3% explained by the cor-responding linear mode, with an mse ratio of 0.874.The second Z500 NLPCA mode reveals mainly east–west differences over North America and the adjacentNorth Pacific. Positive anomalies lie in the west andnegative anomalies to the east at minimum u (Fig. 6a).The anomalies reverse sign and shift southwestward formaximum u (Fig. 6b). The second SAT NLPCA modeshows even larger changes in the anomaly pattern whencomparing minimum u (Fig. 6c) and maximum u (Fig.6d).The Z500 and SAT anomaly patterns correspondingto minimum u (Figs. 6a and 6c) resemble EOF3 (Figs.3c and 3f), with positive height and SAT anomaliescentered over western Canada. Spatial patterns corre-sponding to maximum u (Figs. 6b and 6d) resembleEOF2 for both Z500 and SAT (Figs. 3b and 3e) withpositive height and SAT anomalies over much of theUnited States and Canada. We can see some similaritybetween the spatial structure of the second Z500 andSAT NLPCA modes, even though NLPCA was per-formed separately on the two datasets.To investigate the relations between the NLPCAmodes and ENSO, composites of the winter (DJF) Z500and SAT anomalies during warm (El Nin˜o) event years(1958, 1966, 1969, 1973, 1983, 1987, 1988, and 1992)and cold (La Nin˜a) event years (1950, 1951, 1955, 1956,1965, 1971, 1974, 1976, and 1989) were computed. Theyears used for the composites are the same as those usedby Hoerling et al. (1997). The Z500 composites duringEl Nin˜o years and La Nin˜a years show positive andnegative PNA patterns, respectively (Figs. 7b and 7a).When El Nin˜o takes place, positive height anomaliesare dominant over North America except over the south-eastern United States. The corresponding compositeSAT anomaly field has negative SAT anomalies overthe southeastern United States and positive SAT anom-alies over the rest of North America (Fig. 7d). The com-posite Z500 and SAT anomalies during La Nin˜a yearsare basically opposite to those during El Nin˜o years.Thus, the asymmetries between El Nin˜o and La Nin˜ain the midlatitudes simulated by the CCCma AGCM areweaker than observed (Hoerling et al. 1997). Patternsdisplayed in Figs. 7a and 7c are similar to those shown15 JULY 2003 2333WU ET AL.FIG. 7. Composites of the winter Z500 and SAT anomalies for La Nin˜a and El Nin˜o states. The Z500 anomalies are shown in (a) and (b),respectively, with contour intervals of 5 m, and the SAT anomalies, in (c) and (d), respectively, with contour intervals of 0.58C.in Figs. 5a and 5c, but with weaker amplitudes. How-ever, patterns in Figs. 7b and 7d are quite different fromthose shown in Figs. 5b and 5d. Apparently, NLPCAmode 1 is related to ENSO only when u is negative. Torelate the tropical Pacific SST to the North AmericanZ500 and SAT, we turn to the NLCCA approach.4. Nonlinear modes extracted by NLCCAa. NLCCA of SST and Z500NLCCA between winter tropical Pacific SST anom-alies and simultaneous Z500 anomalies (Fig. 8) revealsnonlinearity when compared to the linear CCA solution,which is shown as a straight line. For SST (Fig. 8a),nonlinearity in the PC1–PC2plane is manifested by acurve that links La Nin˜a states at the left end to El Nin˜ostates at the right end. For Z500 (Fig. 8b), strongernonlinearity appears in the PC1–PC3and PC2–PC3planes than in the PC1–PC2plane. The first NLCCAmode for SST explains 64.6% of SST variance, versus62.3% explained by the first CCA mode. The mse ratior is 0.856. The corresponding NLCCA mode for Z500explains 23.6% of the variance, versus 21.5% explainedby the first CCA mode, with a r value of 0.953. Thecorrelation between the canonical variates (u and y)is0.702 for the nonlinear mode and 0.675 for the linearmode.As in NLPCA, one can map values of canonical var-iate u and y onto SST and Z500 anomaly patterns, re-spectively. Here, minimum and maximum u are chosento present the La Nin˜a and El Nin˜o states, respectively,and Z500 anomaly patterns are considered for the valuesof y that correspond to minimum and maximum u. TheSST field that corresponds to minimum u presents a LaNin˜a with negative anomalies (about 22.08C) over thecentral-western equatorial Pacific. The correspondingZ500 field has a negative PNA pattern with a positiveanomaly center over the North Pacific, a negative centerover western Canada and a positive center over the east-ern United States (Fig. 9a).When u takes on its maximum value, the SST fieldpresents a fairly strong El Nin˜o with positive anomalies(about 2.58–3.08C) over the central-eastern Pacific (Fig.9b). The SST warming center shifts eastward by 308–408 longitude relative to the cooling center in Fig. 9a.This asymmetric SST variation between El Nin˜o and LaNin˜a states has also been found by Hsieh (2001a) andWu and Hsieh (2002). Note that the warming in Fig. 9b2334 VOLUME 16JOURNAL OF CLIMATEFIG. 8. The first NLCCA mode between (a) the winter tropical Pacific SST anomalies and (b) the winter Z500 anomalies, plotted as(overlapping) squares in the PC1–PC2–PC33D space. The linear (CCA) mode is shown as a dashed line. The NLCCA mode and the CCAmode are also projected onto the PC1–PC2,PC1–PC3, and PC2–PC3planes, where the projected NLCCA is indicated by (overlapping) circles,and the CCA by thin solid lines, and the projected data points by the scattered dots. There is no time lag between the SST and the correspondingZ500 data.does not display a maximum off Peru, in contrast to thefirst NLPCA mode of SST (Fig. 10d of Hsieh 2001b).This difference between the first NLPCA and NLCCAmodes suggests that warming confined to the easternequatorial Pacific does not have a strong midlatitudeatmospheric response, in agreement with Hamilton(1988). The Z500 response field contains a PNA pattern(Fig. 9b) that is roughly opposite to that shown in Fig.9a, but with a notable eastward shift. The zero contoursurrounding the North Pacific anomaly lies close to thewestern coastline of North America during El Nin˜o (Fig.9b), while it is about 108–158 farther west during LaNin˜a (Fig. 9a). The positive anomaly over eastern Can-ada and the United States in Fig. 9a becomes a negativeanomaly shifted southeastward in Fig. 9b. Evidently,the Rossby wave train linking the extratropical atmo-spheric response to the tropical source changes betweenLa Nin˜a and El Nin˜o events in the CCCma model. Asthe equatorial SST anomalies shift eastward from LaNin˜a to El Nin˜o, the Rossby wave train also shifts east-ward. However, the atmosphere’s eastward shift is notas large as that in the SST anomalies. This is probablybecause the atmospheric response is a direct result oftropical heating anomalies (the response to tropical con-vection) instead of the SST anomalies, although the for-mer is caused by the latter. The amplitude of the Z500anomaly over the North Pacific is stronger during ElNin˜o than La Nin˜a, but the anomaly over western Can-ada and the United States is weaker during El Nin˜o thanLa Nin˜a (Figs. 9a and 9b).For comparison, CCA mode 1 describes a pattern ofatmospheric response to La Nin˜a that, apart from mag-nitude, is opposite to the response pattern for El Nin˜o(Figs. 9c and 9d). Note, however, that the El Nin˜o re-sponse does have somewhat stronger amplitudes. TheSST anomaly patterns extracted with CCA are also com-15 JULY 2003 2335WU ET AL.FIG. 9. The spatial patterns for the first NLCCA mode between the winter Z500 anomalies and the tropical Pacific SST anomalies as thecanonical variate u takes its (a) min value and (b) max value. The Z500 anomalies with contour intervals of 10 m are shown north of 208N.SST anomalies with contour intervals of 0.58C are displayed south of 208N. The SST anomalies greater than 118C or less than 218C areshaded, and heavily shaded if greater than 128C or less than 228C. The linear CCA mode 1 is shown in (c) and (d) for comparison.pletely symmetric between the two extremes. The anom-aly centers are located at about the average positionsbetween those shown in Figs. 9a and 9b.b. NLCCA of SST and SATNLCCA was also applied to analyze the covariabilityof SST and SAT. The first NLCCA mode for SST(shown in Fig. 10a) is very similar to that shown in Fig.8a. For SAT, considerable nonlinearity occurs betweenthe PC1and PC2(Fig. 10b). The first NLCCA mode forSST explains 65.5% of the total variance, versus 62.3%explained by the first CCA mode, with r being 0.850.The first NLCCA mode for SAT explains 23.2% of thetotal variance, versus 22.1% by the first CCA mode,with an mse ratio r of 0.967. The canonical correlationis 0.605 for NLCCA and 0.588 for CCA.Figures 11a,b show the spatial anomaly patterns forboth SST and SAT associated with La Nin˜a and El Nin˜o,respectively. When u takes on its minimum value, pos-itive SAT anomalies (about 18C) appear over the south-eastern United States, while much of Canada and thenorthwestern United States are dominated by negativeSAT anomalies. The maximum cooling center (248C)is located over northwestern Canada and Alaska (Fig.11a). When u takes on its maximum value (Fig. 11b),the warming center (38C) is shifted to the southeast ofthe cooling center in Fig. 11a, with warming over almostall of North America except the southeastern UnitedStates. The SAT anomaly patterns here are roughly con-2336 VOLUME 16JOURNAL OF CLIMATEFIG. 10. Similar to Fig. 8 but for the NLCCA mode 1 between the SAT anomalies and the tropical SST anomalies.sistent with the composite analysis from observed databy Hoerling et al. (1997). The first CCA mode for SATand SST is shown for reference in Figs. 11c,d.We note that the nonlinear SAT response is morestrongly captured by NLCCA than by composite anal-ysis (cf. Figs. 9a,b with 7a,b and Figs. 11a,b with 7c,d).This is not surprising because the averaging in the com-posite method mixes the strong and weak events, there-by producing weaker features than NLCCA. Anotherdisadvantage with the composite approach is that onemust decide, a priori, which events to include in thecomposites.5. Summary and discussionTwo nonlinear multivariate statistical techniques,NLPCA and NLCCA, developed from neural networks,were applied to investigate the nonlinear behaviour ofthe North American winter climate as simulated by theCCCma AGCM2. NLPCA was used to find the nonlin-ear modes that could account for the maximum variancein a dataset, while NLCCA was used to detect the moststrongly correlated nonlinear modes relating the atmo-spheric (Z500 or SAT) and tropical Pacific SST vari-ability. These two methods provided valuable nonlineardiagnostics of a GCM simulation.Nonlinearity is detected by NLPCA in both the winterZ500 and SAT anomaly fields, which display asym-metric spatial patterns of variability during the oppositeextremes of the nonlinear principal component (NLPC)u. The first NLPCA mode for Z500 describes a negativePNA pattern during the minimum u, and a weakenedjet stream pattern during maximum u. Although NLPCAwas conducted on the Z500 and SAT anomalies sepa-rately, the derived NLPCA modes have similar spatialpatterns.The first NLCCA mode shows there is nonlinearityin both the SST anomaly field and the related atmo-spheric response fields (Z500 and SAT). The leadingNLCCA mode for SST is a nonlinear ENSO mode,showing asymmetry between the warm El Nin˜o statesand the cool La Nin˜a states with a 308–408 longitudeeastward shift of the SST anomalies during El Nin˜orelative to the anomalies during La Nin˜a. The leading15 JULY 2003 2337WU ET AL.FIG. 11. Similar to Fig. 9 but for the spatial patterns for the NLCCA mode 1 between the SAT anomalies and the tropical SST anomalies.The contour interval for the SAT anomalies is 18C.NLCCA mode for Z500 is a nonlinear PNA telecon-nection pattern, in which the positive PNA pattern as-sociated with El Nin˜o is situated about 108–158 east ofthe negative PNA pattern associated with La Nin˜a. Aswith NLPCA, the spatial anomaly pattern of the firstNLCCA mode for the SAT generally corroborates withthe corresponding Z500 anomaly pattern. The NLCCAwas able to extract a stronger nonlinear atmosphericresponse to ENSO than the composite method.Even with NLCCA, the nonlinearity of the atmo-spheric responses to ENSO reported in this paper isweaker than that found by Hoerling et al. (1997), wherean appreciable 308 longitude phase shift between thewarm and cold event circulation composites was de-tected from observational data. However, we shouldnote that, the atmospheric sample available here froman ensemble of six runs is much bigger than is availablefor an observational study, so the opportunities for ov-erfitting and consequently finding apparent strong non-linearity are somewhat reduced. It might therefore bemore appropriate to compare our results with other mod-els. Hoerling et al. (2001) surveyed the response toENSO in four GCMs. Nonlinearity was found in all fourmodels, existing in both the strength of the midlatituderesponse and its spatial phase. Similar to our results, thespatial phase displacements of the four GCMs’ telecon-nections during warm events and cold events are smallerthan observed. The degree of nonlinear atmospheric re-sponses to ENSO is model dependent.Hoerling et al. (2001), used one-sided linear regres-sion to analyze the nonlinearity. The procedure involvescalculating linear regressions between all occurrencesof one sign of an SST index (e.g., the leading EOF PCof tropical Pacific SST anomalies) and the response2338 VOLUME 16JOURNAL OF CLIMATEFIG. 12. One-sided regressions of the winter (DJF) 500-mb height and SAT for warm and cold phases of the leading EOF of tropicalPacific SSTs. The monthly height and SAT anomalies were regressed onto the positive and negative phases of the first EOF PC of the tropicalSST anomalies separately. Shown are height and SAT anomalies associated with 11 std dev of the EOF PC1. Contour interval is5min(a) and (b), and 0.28C in (c) and (d).fields (e.g., Z500 or SAT anomalies). For comparison,this method was also applied to CCCma AGCM2’s sim-ulation and the results are shown in Fig. 12. The re-gression patterns for Z500 and SAT anomalies are gen-erally consistent with the spatial patterns given byNLCCA (Figs. 9a,b and 11a,b), with the positive PNAteleconnection shifted 108–158 eastward of the negativePNA teleconnection (Figs. 12a,b). The phase differencein the CCCma AGCM2, as indicated by the placementof zero contours, is comparable to that in version 3 ofthe National Center for Atmospheric Research (NCAR)Community Climate Model (CCM3) and the ECHAM-3 models, but weaker than that in the Geophysical FluidDynamics Laboratory (GFDL) and Medium-RangeForecast (MRF) models [see Hoerling et al. (2001) fordetails of the four models]. The negative Z500 anom-alies over the North Pacific are much stronger duringwarm events than the positive anomalies that occur dur-ing cold events. However, the anomalies over the NorthAmerican continent have similar amplitudes duringwarm and cold events (Figs. 12a,b), which can also beseen in the NLCCA results (Figs. 9a,b).Hence the CCCma AGCM2 is basically capable ofsimulating the nonlinear responses of North Americanclimate to ENSO, although the nonlinearity is somewhatweaker than that observed. NLCCA, which successfullyextracted the nonlinear mode, offers a fully nonlineardiagnostic tool to study GCM output.Acknowledgments. This work was supported by theNatural Sciences and Engineering Research Council ofCanada via a strategic grant to Hsieh, Shabbar, andZwiers, and a research grant to Hsieh. This paper wasimproved by constructive comments on an earlier draftprovided by Greg Flato and John Fyfe.REFERENCESBarnett, T. P., and R. Preisendorfer, 1987: Origins and levels of month-ly and seasonal forecast skill for United States surface air tem-peratures determined by canonical correlation analysis. Mon.Wea. 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