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Nonlinear modes of decadal and interannual variability of the subsurface thermal structure in the Pacific.. Tang, Youmin; Hsieh, William W. 2003-03-18

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Nonlinear modes of decadal and interannual variability of thesubsurface thermal structure in the Pacific OceanYoumin TangCourant Institute of Mathematical Sciences, New York University, New York, New York, USAWilliam W. HsiehDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, CanadaReceived 27 November 2001; revised 19 November 2002; accepted 17 December 2002; published 18 March 2003.[1] The nonlinear principal component analysis, a neural network technique, is appliedto the observed upper ocean heat content anomalies (HCA) in the Pacific basin from1961 to 2000. By applying the analysis to high-passed and low-passed data,nonlinear interannual and decadal modes are extracted separately. The first nonlinearinterannual mode is mainly characterized by the El Nin˜o-Southern Oscillation (ENSO)structure in the tropical Pacific, with considerable asymmetry between warm El Nin˜o andcool La Nin˜a episodes; for example, during strong El Nin˜o, the negative HCA in thewestern tropical Pacific is much stronger than the corresponding positive HCA duringstrong La Nin˜a. The first nonlinear decadal mode goes through several notable phases. Twoof the phases are related to decadal changes in the La Nin˜a and El Nin˜o characteristics,revealing that the decadal changes for La Nin˜a episodes are much weaker than the changesfor El Nin˜o episodes. Other phases of the decadal mode show a possible anomaly linkfrom the middle latitudes to the western tropical Pacific via the subtropical gyre. Thedecadal changes in the HCA around 1980 and around 1990 were compared andcontrasted. INDEX TERMS: 3339 Meteorology and Atmospheric Dynamics: Ocean/atmosphereinteractions (0312, 4504); 4215 Oceanography: General: Climate and interannual variability (3309); 4522Oceanography: Physical: El Nin˜o; 1620 Global Change: Climate dynamics (3309); KEYWORDS: nonlinearprincipal component analysis, heat content, El Nin˜oCitation: Tang, Y., and W. W. Hsieh, Nonlinear modes of decadal and interannual variability of the subsurface thermal structure inthe Pacific Ocean, J. Geophys. Res., 108(C3), 3084, doi:10.1029/2001JC001236, 2003.1. Introduction[2] Among the low-frequency variability of the thermalfields in the Pacific Ocean, interannual variability anddecadal variability are the two most interesting [e.g., Wal-lace et al., 1998; Trenberth and Hurrell, 1994]. While thesetwo well-defined variabilities reside in the whole Pacificbasin within at least the upper 400-m ocean, they also showstrong regional features. The interannual variability, domi-nated by the El Nin˜o-Southern Oscillation (ENSO) phe-nomenon, is centered in the equatorial Pacific, whereas thedecadal variability is most strongly manifested in the mid-latitude North Pacific, as characterized by an ellipticalanomaly located in the subtropic gyre [Zhang et al.,1999]. Understanding and interpreting the interannual anddecadal variabilities have long been of interest [e.g., Klee-man et al., 1996, 1999], not only for their major impacts onthe regional and global climates and ecologies, but also forassessing possibly forced climate variability, such as anthro-pogenic global warming [Latif et al., 1997].[3] An important aspect of studying the low-frequencyvariability in the Pacific Ocean is to characterize the majorspatial and temporal characteristics in a low-dimensionalspace. Until very recently, this has been implemented byprincipal component analysis (PCA, also called EOF anal-ysis), and by related techniques, for example, singularspectrum analysis (SSA, also called extended EOF analy-sis), and principal oscillation pattern (POP) analysis, witheither observed data [Zhang et al., 1999] or modeled data[Miller et al., 1998]. The interannual and decadal modes aredescribed by the first few leading eigenvectors, giving thespatial patterns, and by the corresponding time series. Tofocus on a specific timescale, the data are usually filteredprior to applying PCA. For instance, for detecting decadalvariability, we used a filter which removes signals withperiods under 5 years, while for studying interannualvariability, we filtered out periods above 5 years. Theleading interannual and decadal PCA modes (Figures 1aand 1b) characterize the spatial anomaly patterns at differentfrequency oscillations (Figure 2a and 2b).[4] In this paper, a nonlinear algorithm to extract low-dimensional structure from multivariate data sets, i.e.,nonlinear principal component analysis (NLPCA), isapplied to the oceanic heat content anomalies in the upperJOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C3, 3084, doi:10.1029/2001JC001236, 2003Copyright 2003 by the American Geophysical Union.0148-0227/03/2001JC001236$09.0029 - 1400 m (HCA) over the Pacific basin to detect nonlinearmodes of decadal-scale and interannual variability. There isno a priori reason to believe that the thermal structures inPacific Ocean are linear. As the data contain nonlinearlower-dimensional structure, the PCA will miss the non-linearity. Compared with the sea surface temperature, theupper ocean heat content is better for describing andunderstanding interannual and decadal variability [Zhanget al., 1999], as it reflects the thermocline displacement andcontains the ocean’s ‘‘memory.’’ NLPCA was developedoriginally by Kramer [1991] in the chemical engineeringliterature, was applied to the Lorenz three-componentchaos system by Monahan [2000], and to several meteoro-logical and oceanographic data sets [Monahan, 2001;Monahan et al., 2001; Hsieh, 2001; Hamilton and Hsieh,2002].[5] This paper is structured as follows: Section 2 brieflydescribes the methodology and the data. Section 3 presentsthe nonlinear interannual mode, section 4 presents thenonlinear decadal mode, section 5 presents the decadalchanges in the 1980s and the 1990s, and Section 6 is thesummary and conclusion.2. Method and Data2.1. NLCPA[6] If the data are in the form x(t)=[x1,..., xl], whereeach variable xi,(i =1,...,l), is a time series containing nobservations, the PCA method looks for u, a linear combi-nation of the xi, and an associated vector a, withutðÞ¼a C1 x tðÞ; ð1Þso thathkx tðÞC0 autðÞk2i is minimized; ð2Þwhere h...i denotes a sample or time mean. Here u, calledthe first principal component (PC), is a time series, while a,the first eigenvector of the data covariance matrix (alsocalled an empirical orthogonal function, EOF), oftendescribes a spatial pattern.[7] The fundamental difference between NLPCA andPCA is that NLPCA allows a nonlinear mapping from xto u whereas PCA only allows a linear mapping. To performNLPCA, a nonlinear mapping is made; that is,utðÞ¼f x tðÞ; w ; ð3Þwhere f denotes the nonlinear mapping function from thedata space to the u (the nonlinear PC) space, and w denotesthe parameters determining the f structure inherent to thedata set. Denoting g as the inverse mapping function from uto the data space, we havex0tðÞ¼ gu;~wðÞ; ð4Þwhere g is the f-adjoint operator. For linear PCA, g is simplythe transpose of f. Here x0(t) is the approximation to data setx(t), when the 1-D PC space is used to describe the data set.Figure 1. EOF1 of the HCA data for (a) the high-passed data (i.e., with the 61-month running meansubtracted from the original data) and (b) the low-passed data (i.e., the 61-month running mean). Thevalue in percentage is the explained variance by each mode. Contour interval is 0.2C176C, with dashedcontours for negative anomalies.Figure 2. First mode PC associated with the EOF spatialpatterns in Figure 1. For better legibility, the PCs fordifferent data sets have been shifted vertically by 0.25. Thetick marks along the abscissa indicate the start of the year.29 - 2 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEANAs in linear PCA, the cost function defined by the errorbetween x(t) and x0(t) is used to determine the parameters wand~w; that is,hkx tðÞC0x0tðÞk2i is minimized: ð5Þ[8] An important issue in NLPCA is how to derive thenonlinear operators f and g from the inherent structure of thedata set. This has been implemented by neural networks(NN) [Kramer, 1991], since NN can simulate any nonlinearcontinuous functions [Cybenko, 1989]. Figure 3a shows thearchitecture of the NLPCA, which is capable of extracting a1-D open curve approximation to the data. However, thisalgorithm cannot be used to extract closed curve solutions, asthe bottleneck neuron u is not an angular variable. Kirby andMiranda [1996] introduced a circular node or neuron, andshowed that the NLPCA with a circular node (henceforthabbreviated as NLPCA.cir) at the bottleneck is capable ofextracting closed curve solutions. The algorithm of theNLPCA.cir is identical to the architecture of the NLPCA ofKramer, except at the bottleneck layer, where instead of asingle neuron u, there are now two neurons p and q, con-strained to lie on a unit circle in the p-q plane (Figure 3b), sothere is only 1 angular degree of freedom (q) to present thenonlinear PC (NLPC). In this paper, both NLPCA andNLPCA.cir algorithms are used. When we discuss thedecadal mode, we use NLPCA.cir, since the analyzed data,obtained by smoothing the original data set with a low-passfilter, are well characterized by closed curve solutions.[9] In contrast to PCA, as the mapping function g from thePC space to the data space is nonlinear, there is not a singlespatialpatternassociatedwithanNLPCAmode.Theapprox-imationx0(t),however,correspondstoasequenceofdifferentpatterns that can be visualized cinematographically. Forlinear PCA, the approximation au (equation (2)) produces astandingwavepatternasthePCvaries,whereaswithNLPCAthe spatial pattern generally changes as the NLPC varies. Wewill use the x0(t) corresponding to a few u (q) values toexplorethechangingspatialstructuresoftheNLPCAmodes.[10] An important aspect of the NLPCA is the size of thenetwork, i.e., the number of hidden neurons m in theencoding (and also in the decoding layer) for representingthe nonlinear functions f and g. A larger m increases thenonlinear modeling capability of the network, but could alsolead to overfitted solutions (i.e., wiggly solutions which fitto the noise in the data). Based on a general principle ofparsimony, the m values were varied from 2 to 4 and theweight penalty parameters [Hsieh, 2001] were varied from0.01 to 0.05 for smoothing. For a given m, an ensemble of30 NNs with random initial weights and bias parameterswas run. Also, 20% of the data was randomly selected astest data and withheld from the training of the NNs. Runswhere the mean square error (MSE) was larger for the testdata set than for the training data set were rejected to avoidoverfitted solutions. The NN with the smallest MSE wasselected as the solution for the given m. The solutions fromdifferent m were further compared with respect to their MSEto get the optimal NN structure.2.2. Data[11] The data used are the monthly 400-m depth-averagedheat content anomalies (HCA) during 1961–2000, from thedata set of subsurface temperature and heat content pro-vided by the Joint Environmental Data Analysis Center atthe Scripps Institution of Oceanography. This data setconsists of all available XBT, CTD, MBT and hydrographicobservations, optimally interpolated by White [1995] to athree-dimensional grid of 2C176 latitude by 5C176 longitude, and 11standard depth levels between the surface and 400 m. Thisdata set has recently been successfully assimilated into aFigure 3. (a) A schematic diagram of the NN model forcalculating nonlinear PCA (NLPCA). There are three‘‘hidden’’ layers of variables or ‘‘neurons’’ (denoted bycircles) sandwiched between the input layer x on the left andthe output layer x0on the right. Next to the input layer is theencoding layer, followed by the ‘‘bottleneck’’ layer (withone neuron u), which is then followed by the decodinglayer. A nonlinear function maps from the higher dimensioninput space to the lower dimension bottleneck space,followed by an inverse transform mapping from thebottleneck space back to the original space represented bythe outputs, which are to be as close to the inputs as possibleby minimizing the cost function J = hkxC0x0k2i. Datacompression is achieved by the bottleneck, with thebottleneck neuron giving u, the nonlinear principalcomponent (NLPC). (b) A schematic diagram of the NNmodel for calculating the NLPCAwith a circular node at thebottleneck (NLPCA.cir). Instead of having one bottleneckneuron u, there are now two neurons p and q constrained tolie on a unit circle in the p-q plane, so there is only one freeangular variable q, the NLPC. This network is suited forextracting a closed curve solution.TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN 29 - 3hybrid coupled model for ENSO prediction [Tang andHsieh, 2003], and used in the study of decadal oscillations[e.g., Miller et al., 1997, 1998; Schneider et al., 1999].[12] To study the decadal mode, the data were firstsmoothed by a 61-month running mean (referred to as thelow-passed data hereinafter). The residual field between theoriginal data and the low-passed data (referred to as the high-passed data) will be used to extract the interannual mode. Toreduce the large number of spatial variables, the HCA datawere preprocessed by retaining only the first six EOF modes,which account for 41% and 93% of the variance for the high-passed data and low-passed data, respectively.3. Interannual Mode[13] The six leading PCs from the high-passed HCA areinput to the NLPCA network to extract the NLPCA mode 1(NLPCA1). Figure 4 shows the projection of the NLPCA1solution in the PC1-PC2-PC3 space. The NLPCA1 accountsfor 26% of the total variance versus 22% by the PCA mode1. The trajectory of the NLPCA1 describes a curve in thePC space, indicating nonlinearity as compared to the PCA(straight line). The NLPC, u, time series is shown in Figure5a, well characterized by irregular oscillations at 2- to5-year timescale, while Figure 5b is the frequency distribu-tion curve (FDC) for u.Wenextexaminethespatialanomaly patterns associated with some specific u values,namely those marked in Figure 5b. The neural networkmaps from u to the output PCs (x0), which when individu-ally multiplied to the associated EOF spatial pattern, andsummed over the six modes, yield the spatial anomalypattern of the NLPCA1 for the given u. As shown in Figure6, the spatial structures of this nonlinear interannual modeare mainly characterized by ENSO features in the tropicalPacific, i.e., a seesaw oscillation along the equator. Themost probable spatial pattern, corresponding to C in Figure5b, describes a neutral state, i.e., negligible anomalies in thetropical Pacific (not shown). Patterns A and B depictextreme and typical La Nin˜a episodes, respectively, whileD and E represent typical and extreme El Nin˜o, respectively(Figure 6). In the middle latitude, the interannual variabilityFigure 4. The first NLPCA mode for the high-pass filtered HCA plotted as (overlapping) squares in thePC1-PC2-PC33-D space. The linear (PCA) mode is shown as a dashed line. The NLPCA mode and thePCA mode are also projected onto the PC1-PC2plane, the PC1-PC3plane, and the PC2-PC3plane, wherethe projected NLPCA is indicated by (overlapping) circles, the PCA is indicated by thin solid lines, andthe projected data points are indicated by dots. One end of the NLPCA curve with maximum PC1value isassociated with the minimum value of the NLPC u and an extreme La Nin˜a situation, while the oppositeend of the curve corresponds to maximum u and extreme El Nin˜o. The plotted PCs have been scaled upby a factor of 10.Figure 5. (a) NLPC1, u, and (b) the frequency distributioncurve (FDC) for the NLPC1. The data have been high-passed prior to the NLPCA.29 - 4 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEANis weak, particularly during the cool episodes of ENSO, incontrast to the interannual variability in the sea surfacetemperature (SST), where there are significant anomalies inthe midlatitudes [Giese and Carton, 1999].[14] Asymmetries between El Nin˜o and La Nin˜a spatialanomaly patterns, which are absent in the linear mode, arereadily manifested in NLPCA1 (Figure 6). One notices muchstronger anomalies occurring in the western tropical Pacificduring extreme El Nin˜o (pattern E) than during extreme LaNin˜a (pattern A), even though in the eastern tropical Pacific,the anomalies are of similar magnitude. Furthermore, northof 30C176N, the anomalies are considerably stronger during ElNin˜o than during La Nin˜a (from comparing the amount ofshaded area in pattern D with that in B, and between E andA). A useful way to characterize the asymmetry between ElNin˜o and La Nin˜a is by the spatial correlation coefficient.Between pattern A and E, the correlation is C00.75, departingconsiderably from the correlation of C01 for the linear PCAmode. Another interesting nonlinear behavior is seenbetween typical El Nin˜o (pattern D) and extreme El Nin˜o(pattern E); as one proceeds from D to E, the cool anomaliesin the western equatorial Pacific intensifies as expected, butthe warm anomalies in the eastern equatorial Pacific weak-ens; that is, E is obtained from D by adding cool HCA inboth the western and eastern equatorial Pacific.[15] We can compare our NLPCA results with the con-ventional composite method. Composites of HCA for 5typical La Nin˜a years (1971/1972, 1975/1976, 1984/1985,1988/1989, 1995/1996) and 5 typical El Nin˜o years (1972/1973, 1982/1983, 1986/1987, 1991/1992, 1997/1998) areshown in Figure 7, where the warm episodes have strongerheat content anomalies in the equatorial Pacific, especiallyin the western equatorial Pacific, than the cool episodes, inagreement with our NLPCA results. Of course, the averag-ing process in the composite method does not allow adistinction between typical and extreme El Nin˜o conditionsas in the NLPCA results. Also with the composite approach,one has to somewhat subjectively decide which ENSOepisodes to include in the composite.[16] One reviewer cautioned that the data had unrealisti-cally small amplitudes in the southwestern tropical Pacificbefore the early 1980s [Lysne and Deser, 2002], comparedto other data sources, and could affect our NLPCA calcu-lations. Fortunately, the extreme u values were attained afterthe earlier defective period, as seen in Figure 5a. We alsorecomputed the NLPCA excluding the earlier defectiveperiod, and the new extreme patterns A and E (not shown)are not very different from those in Figure 6.[17] Figure 8 is the Hovmo¨ller diagrams showing the timeevolution of the HCA along the equator from the NLPCA1,the linear PCA mode 1 and the leading six linear PCAmodes. As in Figure 8a, the NLPCA1 rather well reflectsobserved features such as the eastward propagation of HCA,the oscillatory periods of 2–5 years, and the asymmetry ofanomalies between El Nin˜o and La Nin˜a episodes. Thesefeatures are absent or not obvious in the PCA mode 1(Figure 8b), indicating that the NLPCA1 approximates thedata set better than the PCA mode 1.4. Decadal Mode[18] The first nonlinear decadal mode for the low-passedHCA data extracted from the NLPCA.cir network (Figure3b) [Hsieh, 2001] is shown in the PC space (Figure 9). ThisFigure 6. Spatial anomaly patterns associated with the NLPC at A, B, D and E in Figure 5b. Thecontour interval is 0.4C176C, and areas with absolute values over 0.2C176C are shaded.TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN 29 - 5mode explains 72% of the HCA variance, versus only 38%by the first PCA mode. The NLPC q in Figure 10a showsthat the decadal variations are characterized by two jumpsin q. The first jump, occurring in the early 1980s as detectedalso by linear PCA [Zhang et al., 1999], is closely asso-ciated with the large-scale climate regime shift in thePacific Ocean around 1976. While the value at the time-point t in the low-passed data is actually averaged from theFigure 7. Composite of the HCA for several La Nin˜a and El Nin˜o years (see text), averaged over theextreme month of each episode. The contour interval is 0.4C176C, and areas with absolute values over 0.2C176Care shaded.Figure 8. Time-longitude plot of the reconstructed heat content anomalies along the equator. Thereconstructed HCA is from (a) the first NLPCA mode, (b) the first PCA mode, and (c) the first six PCAmodes. The contour interval is 0.6C176C, and areas with absolute values over 0.2C176C are shaded.29 - 6 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEANoriginal data over 61 months, thereby precluding finetemporal resolution, it nevertheless seems that the HCA(which involves subsurface temperature changes to 400 mdepth) lags the sea surface condition changes around 1976,suggesting that it may take a few years for the surfaceregime shift to penetrate into the subsurface waters. Thesecond jump in the early 1990s (Figure 10a) is mainlycaused by q jumping from C0p to p, rather than by aphysical regime shift like the first one. However, a clearcontrast between the 1980s and the 1990s has been found inmany observations such as sea level pressure, SST, low-level zonal wind, and subsurface ocean heat content anoma-lies in the Pacific [Kleeman et al., 1996; Latif et al., 1997;Ji et al., 1996].[19] Decadal dependence of ENSO predictability hasbeen found in many ENSO forecast models. While allmodels tended to have very good forecast skills in the1980s, they suffered low skills in the 1990s, even with animproved initialization strategy [Chen et al., 1997]. It hasbeen suggested that the decadal dependence of predictabil-ity may be due to the decadal changes in the mean stateleading to the decadal variability of ENSO [e.g., Wang,1995; Zhang et al., 1997]. Several possible mechanisms forchanging the mean state have been suggested by somerecent work, including the remote response in the tropicalatmosphere to the midlatitude decadal oscillations, anthro-pogenic global warming, and the interaction between trop-ical and extratropical oceans by subduction processes[Kleeman and Power, 1999].[20] The frequency distribution of the decadal mode(Figure 10b) presents a completely different shape than thatof the interannual mode shown in Figure 5b. The FDC ofthe interannual mode is roughly Gaussian, whereas that ofthe decadal mode shows several spikes distributed over thefull range of phase angles. As we lack sufficient samples tocompute the FDC of the decadal mode, the relative shortdata record leads to the spiky frequency distribution. Assuch, the spatial patterns associated with these spikes maynot be particularly meaningful. Instead, we examine thespatial patterns associated with four phases of the decadalmode, namely those corresponding to maximum p, max-imum q, minimum p, and minimum q (Figure 3b), with theirlocations in the PC space shown in Figure 9.[21] The spatial anomalies of the NLPCA1 mode corre-sponding to these four phases are shown in Figure 11, whereFigures 11b and 11d are roughly the negative version of eachother. Their basic pattern, similar to the linear PCA mode 1(Figure 1b), is characterized by an anomaly in the midlati-tudes about 40C176N and one of the same sign in the westerntropical Pacific, and by a weak anomaly of the opposite signin the eastern Pacific. The anomaly in the midlatitudesappears to connect to the anomaly in the western tropicalPacific by a clockwise circulation. Hence this ‘‘subtropicalgyre’’ pattern depicts a possible link of the decadal oscil-lation from the middle latitudes to the tropical Pacific. Such apathway of decadal signals from midlatitudes to the tropicshas also been proposed by other researchers through dataFigure 9. The first NLPCA.cir mode for low-passed HCAdata plotted as (overlapping) asterisks in the PC1-PC2-PC33-D space. The linear (PCA) mode is shown as a dashedline, and the data points are shown as dots. The circledenotes the point corresponding to min(q), the diamondcorresponds to max(p), the pentagram corresponds tomax(q), and the hexagram corresponds to min(p). Theplotted PCs have been scaled up by a factor of 10.Figure 10. (a) NLPC1, q, and (b) NLPC1 FDC. The data have been smoothed by a 61-month runningmean prior to performing NLPCA.cir. Note that q is periodically bounded within (C0p, p).TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN 29 - 7analysis and modeling [e.g., Kleeman et al., 1999; Deser etal., 1999].[22] In contrast to Figures 11b and 11d, the other pair ofpatterns in Figures 11a and 11c do not resemble each otherstrongly. The pattern in Figure 11c is characterized by an ElNin˜o-like dipole structure along the equator, with positiveanomalies in the east and negative anomalies in the west,suggesting that the pattern depicts the decadal variability ofthe ENSO mode. Our interpretation is that when the patternin Figure 11c is on, warm phases of ENSO are reinforced,while cold phases are weakened. The prevalence of warmENSO conditions in the period from 1991 to 1995 offersone example for this type of interaction between interannualand interdecadal variations. There are also notable midlati-tude anomalies in this decadal phase (Figure 11c).[23] The phase in Figure 11a reveals rather weak anoma-lies, though in the tropics, the anomalies are La Nin˜a-like.The phase would enhance cool episodes and weaken warmepisodes. But the fact that the phase in Figure 11a is muchweaker than that in Figure 11c implies that the decadalvariability for La Nin˜a episodes is much less dramatic thanfor El Nin˜o episodes.[24] This finding is consistent with the study by A. Wuand W. W. Hsieh (Nonlinear interdecadal changes of the ElNino-Southern Oscillation, submitted to Climate Dynamics,2002) using nonlinear canonical correlation analysis(NLCCA) of wind stress and SST to examine the mid-1970s climate regime shift. During 1981–1999, the locationof the equatorial easterly anomalies during cool phases ofENSO was found to be unchanged from that observed in the1961–1975 period, but during warm phases of ENSO, thewesterly anomalies were shifted eastward by up to 25C176.From the position of the wind anomalies, the delayedoscillator theory would lengthen the duration of the warmepisodes, but leave the cool episodes unchanged. Hence theNLCCA study also found much larger decadal changes inEl Nin˜o episodes than in La Nin˜a episodes.[25] To further explore the spatial structure of theNLPCA1 in the time domain, we plot the Hovmo¨llerdiagrams for the reconstructed anomalies from the NLPCA1along 40C176N and along 10C176S, the regions of the strongestdecadal variability (Figures 12 and 13). For comparison, thereconstructed anomalies from the linear PCA mode 1 arealso given.[26] As shown in Figures 12a and 13a, decadal changescan be clearly seen in the NLPCA1. Along 40C176N (Figures12a and 12c), the Pacific basin exhibited a positive anomalyduring the middle 1960s to 1981 with a magnitude of+0.6C176C–+1.0C176C around 1973–1974 centered in the Kur-oshio-extension region. The whole Pacific basin shifted to alarge negative anomaly by 1981, which persisted about 10years until 1990, when a new positive anomaly with amagnitude of +0.4C176C–+0.6C176C emerged (Figure 12c). Thispositive anomaly, which is not as wide as the earlier one inthe 1960s to 1970s, has its center shifted 10–15C176 toward theeast compared with the earlier one. Clearly the NLPCA1(Figure 12a) models the regime shifts in Figure 12c muchbetter than the PCA1 (Figure 12b), which missed the regimeshift of the 1990s completely.[27] Along 10C176S (Figures 13a and 13c), from the mid-1960s to the late 1970s, a strong positive anomaly in thewestern Pacific coincided with a weak negative anomaly inthe eastern Pacific. Around early 1981, almost the wholePacific along 10C176S shifted to a negative anomaly. ThisFigure 11. Spatial patterns corresponding to the four phases labeled in Figure 9 for the NLPCA mode 1.The contour interval is 0.1C176C, and areas with absolute values over 0.1C176C are shaded.29 - 8 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEANnegative anomaly persisted around 10 years in the easternPacific until about 1990, when the eastern Pacific shifted toa positive anomaly. In the western Pacific, the negativeanomaly persisted until the late 1990s.5. The 1980s and 1990s Decadal-Scale Changes[28] Over the last two decades, the upper ocean heatcontent experienced two prominent changes, resulting ingenerally warm conditions in the 1970s, cool conditions inthe 1980s and mixed conditions in the 1990s, as seen in lastsection and in other works [e.g., Lysne and Deser, 2002].The large-scale changes in the upper ocean thermal fieldaround 1980 and 1990 can be seen as phase transitions ofthe decadal mode. Figures 14a and 14b show the differencesin the average HCA between the 1970s and 1980s, andbetween the 1990s and 1980s, respectively. The spatialpattern in Figure 14a strongly resembles one of the phasesof the decadal mode (Figure 11d), with a spatial correlationof 0.96, while the pattern in Figure 14b moderately resem-bles Figure 11c, with a correlation of 0.72.[29] There are several hypotheses to explain the mecha-nism of the decadal changes in the upper thermal field in thePacific Ocean. The most popular one is the decadal changesin the wind stress curl affecting the gyre-scale patterns ofthe ocean circulation via the Sverdrup balance [Deser et al.,1999; Lysne and Deser, 2002]. The decadal signals in thewind stress curl are first forced into the surface ocean byEkman pumping, and then transported to the thermocline byRossby wave adjustment with the time of about 2–5 years[Deser et al., 1999].[30] The occurrence of the decadal changes in SST(Figure 15) could be almost simultaneous to the changesin the wind around 1976 and 1988. That the decadal changein the HCA occurred 2–5 years after the wind change isprobably due to the adjustment time scale of the subsurfaceocean to surface changes.[31] As the 1980s decadal changes in the HCA lagged thesurface changes by a longer time compared to the 1990sdecadal change in the HCA, this suggests that the adjust-ment timescale of the subsurface to surface changes isconsiderably longer in the 1980 change than in the 1990change. Possibly the physical processes involved in the twodecadal changes were not completely the same. For exam-ple, for the 1990 decadal change, the main anomalies in thesubsurface (Figure 14b) and the surface (Figure 15b)Figure 12. Time-longitude plot of the reconstructed heat content anomalies along 40C176N. Thereconstructed HCA is based on (a) the first NLPCA mode, (b) the first PCA mode, and (c) the first sixPCA modes (with 93% of the variance of the HCA). The contour interval is 0.2C176C, and areas withabsolute values over 0.1C176C are shaded.TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN 29 - 9occurred roughly in the same or neighboring regions,suggesting that the adjustment processes of the subsurfaceinvolved considerable vertical mixing and advection. Butfor the 1980 decadal change, the main anomalous change inthe western equatorial subsurface ocean (Figure 14a) is verydifferent from changes in the surface (Figure 15a or 15c),suggesting that the subsurface adjustment involved consid-erable horizontal transmission of the surface signal. Adjust-ment in the horizontal direction could involve the Rossbywave adjustment timescale, resulting in the longer responseFigure 13. As for Figure 12, but along 10C176S.Figure 14. Differences in mean upper ocean heat content by (a) subtracting the mean of the 1970s fromthe mean of the 1980s and (b) subtracting the mean of the 1980s from the mean of 1990s. The contourinterval is 0.2C176C. Shading denotes the regions where the two-tailed t-test for difference in means exceedsthe 95% confidence level.29 - 10 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEANtime of the subsurface to surface in the 1980 change than inthe 1990 change. In addition, a much slower process ofsubduction along the subtropical oceanic gyre may also beinvolved in the 1980s subsurface decadal change as sug-gested by Figure 14a.6. Summary and Conclusion[32] We applied the nonlinear principal component anal-ysis technique to the observed upper ocean heat contentanomalies in the Pacific basin from 1961 to 2000, andextracted the leading interannual and decadal modes. Forthe leading nonlinear interannual mode, the spatial anoma-lies are strongest in the equatorial Pacific, with an ENSOeast-west seesaw pattern. As the nonlinear mode is notlimited to a standing wave spatial anomaly pattern, it revealsconsiderable asymmetry between strong La Nin˜a and strongEl Nin˜o. During strong El Nin˜o, the negative anomaly in theequatorial western Pacific is much stronger than the positiveanomaly found in this region during strong La Nin˜a. Thisnonlinear interannual mode also manifests eastward phasepropagation along the equator (Figure 8), in contrast to thestanding wave found in the linear mode 1.[33] Four phases of the nonlinear decadal mode wereexamined. Two of them are roughly mirror images of eachother, both showing a subtropical gyre pattern with the largeanomaly in the midlatitudes circulating clockwise aroundthe subtropical gyre towards the western tropical Pacific, apossible link from the middle latitudes to the tropical Pacificin the decadal mode. Two other phases of the decadal modeare related to decadal changes in the La Nin˜a and El Nin˜ocharacteristics. Since the one associated with La Nin˜a hasmuch weaker anomalies than the one associated with ElNin˜o, it follows that the decadal changes in the character-istics of La Nin˜a episodes are much weaker than thechanges for El Nin˜o episodes.[34] Over the last 2 decades, the nonlinear decadal modeexperienced two phase shifts in 1981 and 1990, respectively,leading to the remarkable decadal changes in the upperocean heat content in the 1980s and 1990s. From theequatorial to midlatitude Pacific, positive HCA during themid-1960s to the late 1970s reversed to negative HCAaround 1981. The regime shift around 1990 was also wellrepresented by the nonlinear decadal mode; the negativeanomalies in the midlatitudes and in the equatorial region inthe 1980s reversed to positive anomalies around 1990 in thecentral midlatitude region and in the eastern equatorialPacific. Prior to the two decadal changes in HCA, windstress (curl) also changed in 1976 and 1988. While the SSTchanges were almost simultaneous with the wind changes,the HCA changes were delayed 2–5 years, corresponding tothe Rossby wave adjustment timescale of the subsurfacewaters to surface changes.[35] The HCA change around 1980 was quite differentfrom the one around 1990 in that the former occurred afterthe wind change with a much longer time delay than theFigure 15. Differences in mean SST by (a) subtracting the mean of 1967–1976 from the mean of1977–1986, (b) subtracting the mean of 1979–1988 from the mean of 1989–1997, (c) subtracting the1970s from the 1980s, and (d) subtracting the 1980s from the 1990s. The years of surface wind changeswere around 1976 and 1988, while the HCA changes were around 1980 and 1990; hence Figures 15c and15d are provided to temporally match Figure 14. The contour interval is 0.2C176C. Shading denotes theregions where the two-tailed t-test for difference in means exceeds 95% confidence.TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN 29 - 11latter. The former (Figure 14a) showed the anomaly linkfrom the midlatitudes to the western tropical Pacific via thesubtropical gyre, while the latter (Figure 14b) did not. Theformer was also more different from the corresponding SSTanomalies (Figure 15a) than the latter was from SST (Figure15b), suggesting that the signals involved more horizontaltransmission in the former than in the latter, where thesurface signals appeared to be transmitted more vertically tothe subsurface. The leading linear PCA mode was able todetect the former change but not the latter, which wasclearly detected by the leading nonlinear PCA mode.[36] Acknowledgments. The heat content data were kindly suppliedby the Joint Environmental Data Analysis Center, headed by Warren White,at the Scripps Institution of Oceanography. This work was supported byresearch and strategic grants to W. Hsieh from the Natural Sciences andEngineering Research Council of Canada.ReferencesChen, D., S. E. Zebiak, and M. A. Cane, Initialization and predictability ofa coupled ENSO forecast model, Mon. Weather Rev., 125, 773–788,1997.Cybenko, G., Approximation by superpositions of a sigmoidal function,Math. Control Signals Syst., 2, 303–314, 1989.Deser, C., M. A. Alexander, and M. S. Timlin, Evidence for a wind-drivenintensification of the Kuroshio Current Extension from the 1970s to the1980s, J. Clim., 12, 1697–1706, 1999.Giese, B., and J. 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Hurrell, Decadal atmosphere-ocean variations inthe Pacific, Clim. Dyn., 9, 303–319, 1994.Wallace, J., E. Rasmusson, T. Mitchell, V. Kousky, E. Sarachik, and H. vonStorch, On the structure and evolution of ENSO-related climate variabil-ity in the tropical Pacific: Lessons from TOGA, J. Geophys. Res., 103,14,241–14,259, 1998.Wang, B., Interdecadal changes in El Nin˜o onset in the last four decades,J. Clim., 8, 267–285, 1995.White, W. B., Design of a global observing system for gyre-scale upperocean temperature variability, Prog. Oceanogr., 36, 169–217, 1995.Zhang, R.-H., L. M. Rothstein, and A. J. Busalacchi, Interannual and dec-adal variability of the subsurface thermal structure in the Pacific Ocean:1961–90, Clim. Dyn., 15, 703–717, 1999.Zhang, Y., J. M. Wallace, and D. S. Battisti, ENSO-like interdecadal varia-bility, J. Clim., 10, 1004–1020, 1997.C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0W. W. Hsieh, Department of Earth and Ocean Sciences, University ofBritish Columbia, Vancouver, B.C., V6T 1Z4, Canada. (whsieh@eos.ubc.ca)Y. Tang, Courant Institute of Mathematical Sciences, New YorkUniversity, 251 Mercer Street, New York, NY 10012, USA. (ytang@cims.nyu.edu)29 - 12 TANG AND HSIEH: NONLINEAR MODES IN SUBSURFACE PACIFIC OCEAN


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