, VOL. 6, NO. 6, PAGES 729-746, DECEMBER 1991

LINEAR AND NONLINEAR COUPLINGS BETWEEN ORBITALFORCING AND THE MARINE •5180RECORD DUR•G THE LATE NEOGENE

TeresaHagelberg and Nick Pisias

Collegeof Oceanography,Oregon State University, Corvallis

Steve Elgar

Departmentof ElectricalEngineering and Computer Science, WashingtonState University, Pullman

Abstract.Previous investigations of the responseof Plio- INTRODUCTION Pleistoceneclimatic records to long-term,orbitally induced changesin radiationhave considered a linear response of Sincethe pioneering work of Hayset al. [1976],researchers . While the second-orderstatistics of powerspectra and havesuccessfully identified a linearresponse of Plio- crossspectra provide necessary information on linear Pleistoceneclimate to orbital forcing [e.g.,Pisias and Moore, processes,insight into the nonlinear characteristics of 1981; Imbrie et al., 1984, 1989; Raymo et al., 1989; andPleistocene climate is notprovided by thesestatistical Ruddimanet al., 1989], supportingthe Milankovitch quantities.Second-order statistics do notcontain the phase hypothesisof climatechange. Empirical evidence in support informationnecessary to investigatenonlinear, phase-coupled processes.Such information is providedby higher-order of thishypothesis is strengthenedby observationsof high statisticalquantities. In particular,bispectral analysis indicates coherenceand relatively constant phase between the climatic thatnontinear couplings are present in theclimatic (radiative) forcing(orbital tilt andprecession) and the response (global ice forcingat theMilankovitch frequencies. Through a linear volume)at periodsof 41, 23 and 19 kyr. Theseresults transfer,this forcing produces similar nonlinear couptings in suggesta linearrelationship between ice sheetgrowth and deep-seasedimentary oxygen isotope records (ODP site 677 decayand orbitally induced changes in thedistribution of solar and DSDP site 607) from 1.0 to 0 Ma duringthe late radiation. Neogene.This analysis suggests that during the late A tinearresponse of the climatesystem to solarinsolation ,the dominanceof the 100,000 year cycle in the changescan account for muchof thevariance in theobtiquity climate record is consistentwith a linear, resonantresponse to (41 kyr) andprecession (23-19 kyr) bandsof thepaleoclimate eccentricityforcing. In theperiod from 2.6 to 1.0 Ma, a spectrum,but it is generallythought to be insufficientto changein thenature of theclimatic response to orbitalforcing is indicated,as phase couptings present in theisotopic time explainthe dominance of the 100kyr cycleduring the seriesare not similarto thephase couplings present in the Pleistocene[Imbrie et al., 1989 and referencestherein]. Efforts insolationforcing. Third-ordermoments (skewness and to explainthe dominance of 100kyr powerin thepaleoclimate asymmetry)are used to quantifythe shape of theclimatic recordhave produced a rangeof results,as discussed below. response.From 2.6 Ma to present,an increasein the Studieswhich contend that a linearresponse cannot explain asymmetry(sawtoothness) of the oxygen isotopic records is the 100kyr cyclein thepaleoctimatic record often argue that accompaniedby a correspondingdecrease in theskewness insolationforcing in the 100 kyr eccentricityband is too small (peakedness)of the records. This indicates an evolution in the to producethe observed response or thatit is altogether natureof the phasecoupling within the climatesystem. These nonexistent[e.g., Imbrie et al., 1989;Birchfield and resultsmay provide important constraints useful in Weertman,1978]. Althoughthe directinfluence of developmentof modelsof paleoclimate. eccentricityon insolationis small(of theorder of 0.1% [Berger,1977]), eccentricity is theonly orbital parameter that influences the total amount of radiation the receives Copyright 1991 by the AmericanG6ophysical Union. annually[Berger, 1989]. In addition,eccentricity variations are observedto correlatepositively with changesin estimated Papernumber 91PA02281. summertemperature [Berger, 1978a, Hays et al., 1976].It is 0883-8305/91/91PA-02281510.00 still unknownwhether the paleoclimatic spectral peak at 100 730 Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene kyr is producedfrom (1) a linearresponse to eccentricityor (2) In this study,bispectra of the time seriesof orbitallyinduced a nonlinearinteraction between two precessionband insolationchanges and time seriesof the oxygenisotopic oscillationswhich transfers energy to the100 kyr band. To proxyof globalice volume(from ODP site677 andDSDP investigatethis problem, the relativeproportions of variance site607) arepresented. Cross spectra and bispectra are used to in thepaleoclimatic record that are related to a linearresponse showthat both the and the climatic response to eccentricityand to a nonlinearresponse to precessional containnonlinearly coupled components, but the interaction forcingmust be determined. betweenthe insolation forcing and response is primarilylinear. Paleoclimateresearch in thisarea has proceeded along two The analysissuggests that the 100 kyr powerobserved in the similaravenues. On one hand,analyses of geologicdata paleoclimaticrecord is consistentwith a linear,resonant providefundamental observations of howclimate has varied responseto insolationforcing. overthe past several million years. Successful, extensive First, a brief introductionto bispectralanalysis, including a strategieshave been developed to examinerecords within a simpleexample, is presented.Next, thebispectra of the consistentchronologic framework and in a systematicmanner calculatedrecord of radiativeforcing as well as threePlio- [e,g.,Imbrie et al., 1989]. On the otherhand, models have Pleistocenerecords ofglobal ice volume (•j180) are presented. beendeveloped which seekto describethe physics governing The time periodfrom 1.0 to 0 Ma, where 100 kyr poweris thesechanges with varyingdegrees of complexity(see the very high, is comparedto the periodfrom 2.6 to 1.0 Ma, summaryby Saltzman[1985, and references therein]). where100 kyr poweris lower. Higher-ordermoments Deterministicmodels describing dynamics of the "slow (skewnessand asymmetry) are shownto providequantitative response"of climatesystem evolution are dividedinto two insightsinto featuresof theserecords. Finally, the groupsby Saltzman[ 1985]:(1) quasi-deductivemodels, based implicationsof theseresults for paleoclimatemodel on prognosticequations for specificprocesses [e.g., Birchfield developmentis examined. and Weertman, 1978; Birchfield and Grumbine, 1985; Le Treut and Ghil, 1983; Oerlemans, 1982; Pollard, 1983; Peltier, BACKGROUND 1982] and (2) inductivemodels, which attemptto constructa dynamicalsystem of equations,based on knownphysical Bispectraltechniques have been in usefor over25 years feedbacks,from whichreasonable paleoclimatic output can be [Hasselmanet al., 1963] (seealso the reviewby Nikias and produced[e.g., Imbrie and Imbrie, 1980;Saltzman and Sutera, Raghuveer[1987]). Becausethis is the first applicationof 1984; Saltzman et al., 1984; Maasch and Saltzman, 1990]. thesetechniques to paleoclimatictime series,a brief summary Severalmodels (of bothtypes) invoke a nonlinearresponse of andan exampleof its applicationare presented. the climatesystem to ice sheetformation to explainthe 100 The bispectrumis formallydefined as the doubleFourier kyr cycle [Le Treut and Ghil, 1983; Imbrie and Imbrie, 1980; transform of the third-order autocorrelation of a time series Wigley, 1976;Birchfield and Weertman, 1978; Sneider, 1985]. [Hasselmanet al., 1963]. While thepower spectrum describes Othermodels explain this cycle as a free oscillationof the the distributionof variance(the secondcentral moment) as a climatesystem, whose phase is setby weak eccentricity functionof frequency,the bispectrum describes the distribution forcing [Saltzmanand Sutera,1984; Saltzmanet al., 1984]. of the third momentas a functionof bi-frequency.For a Finally, a few modelsincorporate stochastic effects discretelysampled time seriesx(t) with complexFourier [Hasselman, 1976; Matteucci, 1989]. coefficientsatfrequencyj•) given by X•), thepower spectrum Thusa wide rangeof classesof modelsexist, all of which is seekto explain the samephenomenon (Pleistocene climate). A generaltest of eachmodel is its ability to reproduce P•) = (X•) X*•)) (1) characteristicsthat are presentwithin the data,as well as its ability to predictcharacteristics which have not yet been(or wherethe angled brackets indicate expected value (mean) and cannotbe) determinedfrom data. To firstorder, every model the asteriskindicates complex conjugate. Similarly, the reproducessome feature or featuresof thepaleoclimatic record, bispectmmis givenby [Haubrich,1965; Kim andPowers, mostcommonly its powerspectrum (a second-orderstatistic). 1979]: Insightsinto the higher-orderstatistics of thedata can provide = (X03)Xffk)X*q)+fO) (2) additionalconstraints or testsfor evaluatingthe models. Althoughsome observations suggest a nonlinearresponse of If thereis energy (variance) atfrequencyj•, then P•) is the paleoclimatesystem to orbitalforcing, this response has nonzero.The bispectmm,however, is zeroat thehi-frequency not beenquantified, in part becausethe power spectrumdoes J•,fk,when the modesj•,fk, andj•+k are independent of one not containphase information. Power spectral analyses are another,even if energyis presentat thesefrequencies. For incapableof detectingthe phase coupling which characterizes a independentoscillations, the phases are random relative to each nonlinearinteraction. Similarly, the crossspectrum contains other,yielding a zerobispectmm upon averaging over many phaseinformation between only two recordsat a given realizations.In theory,the bispectmmis nonzeroonly if these frequencyand is thereforecapable of resolvingonly a linear threemodes are quadratically coupled, meaning that their relationship.Both the powerspectrum and cross spectrum are phasesare nonlinearlycoupled to one another. derivedfrom covariance,a second-orderstatistical quantity. Like the crossspectrum, which provides coherence and phase The presenceof quadraticnonlinear interactions between information,the complex-valuedbispectmm is oftencast in oscillationsof differentfrequencies can be addressedmost termsof its normalizedmagnitude, the bicoherence, and its effectivelythrough the useof thebispectrum, a third-order phase,the biphase (denoted b•,f10 andB•,f10, respectively). statisticalquantity. Thesquared value of bicoherence, b2•,f10, represents the Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene 731 fractionof powerat frequencyj• +fk =J•+k owing to quadratic (nonlinear)interactions of the threemodes [Kim and Powers, 1979]. Analogousto the linear coherencespectrum, the bicoherencespectrum is normalizedsuch that 0 fib fi 1 [Kim Case 1 and Powers, 1979]. The biphaseis analogousto the phaseof the crossspectrum, and rangesbetween -•z and A simpleexample demonstrates the effectof nonlinearphase -2 couplingon both the shapeof a time seriesand the detection of phasecoupling by the bispectrum.Let the time seriesx(0 0 400 800 be composedof threecosines and a low-amplitudeGaussian noisecomponent: 2 x(t) = cos(2•zflt+•l) + cos(2nf2t+•2)+ cos(2•zf3t+•3)+ noise •' Case 2 (3) W o wherefi is the frequencyand {i is thephase of eachcosine. The frequenciesof thesecosines are chosento befl= 0.010, f2= 0.043, andf3 =fl +f2 = 0.053 for easein comparisonof thissimulated record with late Pleistocene 6180 time series, 0 400 800 whichshow power spectral peaks at thesefrequencies in cycles/thousandyears. Threedifferent cases are examined.In (o) the first case(case 1), {1, •)2, and {3 are all randomvariables 2 uniformlydistributed over (0, 2•z].This caserepresents a linear (Gaussian)system where there is no nonlinearphase coupling. 0 Case 3 The phasesof eachoscillation are randomrelative to one anotherwhen averaged over many realizations. In the second case(case 2), {1 and {2 are uniformrandom variables, but {3 -2 = •)1 + q2, and thusfl, f2, andf3 are now phasecoupled. The relationshipbetween the threemodes is no longerrandom. 0 400 800 Similarly, in the last case(case 3), {1 and {•2 are uniform randomvariables, but {3 = {1 + •)2 + •z/2 (thechoice of {3 in cases2 and 3 resultsin biphasesof 0 ø and90 ø, andwaveforms with maximumskewness and asymmetry,respectively, as Fig. 1. Portionsof eachtime seriesfor cases1-3 (seetext): (a) discussedbelow). For eachcase examined, 16 256-point case1 (fl+f2 =f3' •1, •2, •3 arerandom), (b) case2 (fl +f2 realizationswere made,resulting in a 4096-pointrecord. =f3; •1 + •2 = •3), and (c) case3 (fl +f2 =f3; •1 + •2 = •3- Portionsof eachtime seriesand the correspondingpower •z/2). The unitsare arbitrary. spectrumfor eachcase are shownin Figures1 and2, respectively.In eachcase, the power spectrashow concentrationsof varianceat frequenciesfl,f2,and f3 with comesinstead from other aspects of thephysical system under randomfluctuations at otherfrequencies resulting from the examination,including the biphase[Kim et al., 1980; Elgar noise. Even thoughthe time seriesfor eachcase look and Guza, 1985]. different,the powerspectra are identical.This is becausethe In case1, the linearcase in whichno phasecoupling is powerspectrum is phaseindependent and the threecases differ present,bicoherences near the triad (0.059, 0.051, 0.110) = only in their phaserelationships. 0.370 are significantat a 90% level (Figure3a). Althoughall Figure3 showsthe bicoherencespectra for cases1, 2, and 3. bicoherencesin thiscase should be zero,any finite lengthtime Smoothedpower spectral and bispec •tml estimates were made serieswill yield somenonzero values of bicoherence.This by windowing(using a Hanningwindow) and averaging the 16 valueof bicoherencerepresents the highvalues that can be 256-pointensembles for eachrecord. Resultingestimates have expectedto occurby randomchance for a linearprocess with 30 degreesof freedom(doO. The valuescontoured in Figure3 30 dof. On the other hand,bicoherence at (0.01, 0.043, 0.053) arethe observed bicoherences at the •ad of frequencies(fl,f2, (the (fl,f2, f3) triad)is 0.167, whichis belowthe 90% fl+f2) (denotedb(fl,f2)) whichare significantat a (l-ix) = significancelevel. 0.90level (b90 % = (4.6/do01/2[Elgar and Guza, 1988]). In cases2 and 3, wherephases are coupled,b(0.01, 0.043) = Owing to the symmetryproperties of the bispectmm,it 0.917 and b(0.01, 0.043) = 0.912, respectively,indicating necessaryonly to calculatethe bispectmmwithin a triangular highbicoherence at the (0.01, 0.043,0.053) triad(Figures 3b regiondefined by 0

moments,skewness and asymmetry. Skewness is the In case1 of the aboveexample, the shapeof the waveform normalized third moment of the time series and indicates is symmetrical,and skewness and asymmetry are zero. In asymmetrywith respectto a horizontalaxis, or top-bottom Figurela, skewness= -0.12 andasymmetry = -0.16.The asymmetry.Asymmetry is thenormalized third moment of the smallnonzero third-moment estimates are causedby statistical Hilberttransform of the time series[Elgar, 1987] and indicates fluctuations. In case2, the biphasefor the (0.01, 0.043, asymmetrywith respect to a verticalaxis, or for-aft 0.053) interactionis zero,and the waveformis positively asymmetry.For a timeseries which is purelyGaussian, skewed(in Figurelb, skewness= 0.80, asymmetry= 0). In havingonly random phase relationships, the biphase is also case3, the sametriad has biphase of •/2, andthe waveformis randomfor everytriad of frequencies. asymmetricor "sawtooth"shaped (in Figurelc, skewness= 0, asymmetry= -0.79). f f f Theasymmetric record of case3 is qualitativelysimilar in 1ø2 1 2 .3 shapeto theSPECMAP stack (Figure 4), a smoothed

c• 101 0.06 Case 1 0.05 ß•_ 10 ¸ 0.04

10-1 0.05 0.02 10-2 0.01 0.00 0.04 0.08 0.12 0.04 0.08 0.12 f 1 f2 f.3 102 0.06 Case 2 101 Case 2 0.05 0.04 10 ¸ "- 0.03 10-1 0.02 • 0.01 10-2 I I I I I I , C• , , 0.00 0.04 0.08 0.12 0.04 0.08 0.12

f • f2 f3

005 • 101 0 04 o •-• 0 03 '•- 1 0 ¸ ' ½10 10-1 2 (o) 0.06o0 0102 , , ,Case, 3, 10-2 0.04 0.08 O.2 0.00 0.04 0.08 0.12 Fig. 3. Contoursof bicoherencefor cases1-3. Smoothed Frequency bispectralestimates were obtained in the samemanner as Fig. 2. Powerspectra for cases1-3. Smoothedpower spectral powerspectral estimates and have 30 degreesof freedom.The estimateshaving 30 degreesof freedomwere obtained using a minimumbicoherence value contoured (significant at a 0.90 Hanningwindow and by averaging16 256-pointensembles for level)is b=0.39with additionalcontours every 0.05: (a) case eachrecord: (a) case1, (b) case2, and(c) case3. The unitsare 1, (b) case2, and(c) case3. The unitsof frequencyare arbitrary. arbitrary. Hagelberget al.' Linearand Nonlinear Couplings During the Late Neogene 733 compositerecord of five late Pleistocene icevolume (5180) distribution,and eccentricityvariations affect the total records[Imbrie et al., 1984]. The SPECMAP recorddisplays insolation received over time. Examination of the terms that steepterminations representing the end of glacial stages, are usedto derivelong-term Earth orbital variations [e.g., followedby a slow, gradualice buildup. With minor Berger, 1978c]indicates that eccentricity,tilt, andprecession adjustmentsto the recordof case3 (reductionof the amplitudes [e, e, and esine0]are indeedcoupled to one another. However, off2 andf3] the resemblanceto the SPECMAP stackis no quantitativeestimates of the strengthof thesecouplings has striking(Figure 4). The implicationsof this similarityare beenmade. Bispectralanalysis of the insolationrecord allows thatquadratic phase couplings are of importancein the phase-coupledoscillations to be isolated. paleoclimaticrecord and that throughthe useof higherorder A 4.096 m.y. recordof July 65øN radiationsampled at 2000 statistics,these couplings can be quantified. year intervalsis shownin Figure 5a. The power spectrumof thisrecord (Figure 5b) showsconcentrations of energyat NONLINEAR COUPLINGS AMONG ORBITAL periodsof 41, 23, and 19 kyr (0.024, 0.003, and 0.053 PARAMETERS cycles/kyr,respectively). Less energetic, but significant,peaks are alsopresent near 100, 28, and 15 kyr (0.01, 0.036, 0.067 The bispectrumof the normalizedtime seriesof radiative cycles/kyr,respectively). The bicoherencespectrum for this (insolation)forcing (as calculated by Berger[1978b] is now record(Figure 5c), showsthat manymodes in the radiative examined.(Note thatin thispaper the 1978solution of Berger forcingare stronglycoupled. Significant (1- tz =0.90) phase is used. A morerecent solution [Berger and Loutre, 1988] has couplingbetween eccentricity and precession is indicatedby beenfound to yield essentiallythe sameresults in the b(0.001, 0.012) = 0.978 (regionA in Figure 5c). The bispectrumas theolder [Berger, 1978b] solution, although the bicoherencespectrum also indicates significant coupling orbital values in the new solution differ from the older betweenorbital tilt andprecession. For example,b(0.041, solutionin the time domainprior to 1 Ma. Becausethe values 0.025) = 0.960 (regionB in Figure 5c) and b(0.055, 0.023) = of the newer solution from 1.0 to 0 Ma are less well 0.969 (regionC in Figure 5c). The third componentsin these reproducedat presentthan those of the older solution,the 1978 interactingtriads have frequencies 0.066 (15 kyr) and0.078 valuesare usedhere.) Long-term(of the orderof 100kyr) (12.8 kyr), respectively,and appearas lessenergetic peaks in variationsin theEarth's insolation are produced through the radiationpower spectrum(Figure 5b). interactingmotions of the Sun andplanets relative to the The most notableinteraction is at the triad (0.00 1, 0.012, Earth. At a given latitudeon the Earth in a given ,the 0.053), (regionA in Figure 5c), which hasa bicoherenceof solarenergy available is a functionof the solarconstant, the 0.978. This high valueof bicoherenceindicates that precession eccentricityof the Earth'sorbit, the obliquityof the Earth's componentsof 19 and 23 kyr are nonlinearlycoupled to 100 axis,and the longitudeof perihelionas measuredfrom the kyr eccentricityterms (although 0.012 cycles/kyris a 83 kyr vernalequinox [Berger, 1978c]. period,this frequency represents the 100 kyr bandafter Many studieshave examined the paleoceanographicresponse averaginghas been completed). This resultis confirmedby to changesin theseorbital parameters. Studying each orbital examiningthe termsin the seriesexpansion of eccentricityand parameterseparately has been considered reasonable as precessiongiven by Berger[1977, Tables 1 and 3]. Difference obliquityvariations affect the latitudinaldistribution of interactionsbetween the individualprecession terms listed in insolation,precessional changes affect the seasonal thisreference can produce virtually all of the listedeccentricity

-3

O I • ,'t I . •. I• " /• • , •, , I \ I I %1, l/ • I I I I ^ I•/• I ß I ; ; • SPECMAP

I ¾¾, I [ . ! t • ' f 'x / •/x/ •' ' • / ?,,'" ,,,/ 2

3

4 Case

5

0 200 400 600 8OO Time (Ka) -> Fig.4. SPECMAPcomposite 5180 stack (dashed line) [Imbrie et al., 1984] compared topart of a modifiedversion of case3 (solidline) (seeFigure lc). The amplitudeoff! is the sameas in case3, but the amplitudesoff2 andf3 havebeen reduced 60% and70%, respectively. 734 Hagelbergetal.: Linearand Nonlinear Couplings During the Late Neogene terms[Berger, 1989]. This differenceinteraction indicates that phasesin theJuly 65øNradiation time series(Figure 5a) with at frequencyf2= 0.012 cycles/kyr(100 kyr band),greater than randomphases uniformly distributed over (0, 2•) are shownin 95% of thevariance is quadraticallycoupled to oscillationsat Figure6a. The powerspectrum of thisrandom phase time fl=0.41 cycles/kyr(23 kyr) andf3=0.53 cycles/kyr(19 kyr). series(Figure 6b) is identicalto thatof theradiation record The energyat 100 kyr is small in comparisonwith thatat 41, (Figure5b) becausephase information is notretained by the 23, and 19 kyr (Figure5b). However,almost all of the powerspectrum. The bicoherencespectrum (Figure 6c) of this varianceat 100kyr is quadraticallycoupled through nonlinear randomphase record clearly indicates an absenceof phase interactionswith the precessionalterms. The biphaseof the coupling,as is expectedfor a purelylinear system. Similar to (0.041, 0.012, 0.053) triadis zero,indicating that the phaseof case1 above,the two valuesabove the 0.90 significancelevel, f2 is equalto the differenceof thephases off3 andfl. Thisis (0.039, 0.037, 0.076)= 0.509 and (0.105, 0.008, 0.113) = alsoconfirmed by Berger[1977], as differences between the 0.589), representhigh bicoherence values that can be expected phasesof theindividual precession terms listed are exactly to occurthrough random chance. The resultof therandom equalto the correspondingphases of theeccentricity terms. phasesimulation shown in Figure 6 is consistentwith the In additionto significantbicoherence values, other analyses conclusionthat the high bicoherencesobserved in the radiation suggestthat the observedphase couplings do notresult from record are not a result of random fluctuations,but result from randomfluctuations. Time seriesgenerated by replacingthe phasecouplings within the orbitalrecord. Similar low values

3

0 1000 2000 3OOO 4000 Time (kyr) 102

101 • 8070

• lO 0 ) (o)

•- 0.06 ßr- 10 -1 .x 0.05

10-2 • 004 0.03 10-3 o 0.02 O.Ol 10-4- 0.00 O.04 O.O8 0.12 0.04 0.08 O. f (cycles/kyr) f l (cycles/kyr)

417 25 12.5 8.3 417 25 12.5 8.3 kyr kyr Fig. 5. July65øN insolation, for theinterval from 4.096 to 0 Ma (fromBerger, [1978b] solution]: (a) timeseries, (b) powerspectra, and (c) significant(1-o• = 0.90) contoursof bicoherence.The minimumvalue contoured is 0.57, andcontour interval is 0.10.To obtainpower spectral and bispectral estimates having 14 degrees of freedom,data wereHanning windowed, and eight ensembles of 0.512m.y. length were averaged. Regions A, B, andC arereferred to in the text. Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene 735 of bicoherencewere observedin severalother random phase nonlinearinteractions as the forcing. The responsewill also simulations(not shown). be linearlycoherent with the forcingat the forcingfrequencies. Theseresults illustrate the useof bispectralanalyses for On the otherhand, a nonlinearresponse of climateto this detectingquadratically nonlinear interactions. Strong nonlinear nonlinearforcing will produceresponses that are not couplingis detectedbetween oscillations in theprecession and necessarilycoherent with oscillationsat the samefrequencies eccentricitybands in the estimatedrecord of radiativeforcing of as the forcing. In the next section,three records of Berger[1978b]. Using the termsgiven by Berger [1977], a paleoclimateare examinedto determinethe extentof linearand differenceinteraction of precessionterms is equivalentto nonlinearresponses to orbitalforcing. eccentricity[Berger, 1989]. This interactionwas not observablethrough examination of the time seriesand power ANALYSIS OF •5180RECORDS spectra.The bicoherencespectrum also indicates significant nonlinearinteractions between precession and obliquity. In the The relativelyshort record lengths (typically 400-800 kyr) of presentsolution of Berger[1978b], the strengthof these mostpaleoclimatic time serieshas precluded application of nonlinearinteractions does not vary throughtime, and thus bispectralanalysis in the past. Varianceof bispectral analysisof a 4 m.y. recordyields the sameresults as analysis estimatesare higherthan power spectral estimates, and short of shorter(1 m.y.) records. recordsdo not haveenough degrees of freedomto yield A purelylinear response of climateto thisnonlinear forcing statisticallystable results. Recentefforts, however, have will producea paleoclimaticrecord that contains the same produceda numberof long (>1 m.y.) recordsof paleoclimatic

3

-3- 0 1000 2000 3000 4000 Time (kyr) 102

101 • 8o% (s) • 100 (o) 0.06 ßc- 10 -1 0.05

10-2 0.04 0.03 10-3 0.02 0.Ol 10-4 ,I 0.00 0.04 0.08 0.12 0.04 0.08 0.12 f (cycles/kyr) f l (cycles/kyr)

I 417 25 12.5 8.3 417 25 12.5 kyr kyr Fig. 6. Simulatedtime seriesgenerated by replacingthe phases in theJuly 65øN insolationrecord (Figure 5a) with randomphases. Power spectral and bispectral estimates were obtained as described in Figure5: (a) time series,(b) powerspectra, and (c) significant(1-at =0.90) contoursof bicoherence. 736 Hagelberget al.' Linearand Nonlinear Couplings During the Late Neogene

change[Shackleton and Hall, 1989; Ruddimanet al., 1989; apparentnonstationarity in therecord, all recordsare analyzed Raymo et al., 1989]. The bispectmmof threerelatively long in two parts,from 1.0 to 0 Ma and from 2.6 to 1.0 Ma (2.8 to •5180records from the North Atlantic and eastern equatorial 1.0 Ma at site 607). The recordsare dividedat 1.0 Ma to Pacific are now examined. obtainthe longestlate Pleistocenerecord possible. The The recordsexamined here were taken from OceanDrilling preciselocation (within +/- 100 kyr) wherethe records are Projectsite 677 (easternequatorial Pacific, 1 ø12•1, 83ø44'W) divideddoes not changethe results significantly. and Deep SeaDrilling Projectsite 607 (North Atlantic, Sothat each •5180 time series could be processed 41 ø00'N, 32ø58'W). At site 677, time seriesof plankticand identically,the time seriesfrom site607 wasoverinterpolated benthic•5180 were examined, and at site 607, benthic •5180 to matchthe samplingresolution of the site677 data. While time serieswere examined. The site 677 recordwas generated theNyquist frequency of the 607 time seriesis at 7 kyr (as and initially studiedby Shackletonand Hall [1989] and opposedto 4 kyr in the caseof the site677 records),this Shackletonet al. [1990]. The site607 recordwas generated oversamplingallows for directcomparison of the threedata and studiedby Ruddimanet al. [1989] andRaymo et al. setsafter processing.For the interval from 1.0 to 0 Ma, all [ 1989]. For eachof thesestudies the authorshave developed a smoothedspectral estimates were obtained by averagingfour chronologyfor theserecords by makinguse of orbitaltuning 128 point(256 kyr) ensemblesfor 6 degreesof freedom.For procedures.Differences between the time scale used by the interval from 2.6 to 1.0 Ma, smoothedestimates were Shackletonet al. [1990] and Ruddiman et al. [1989] and obtainedby averagingsix 128-pointensembles for 10 degrees Raymo et al. [1989] are substantial.The issueof of freedom. In eachcase the frequencyresolution is 0.0039 chronologiesand approaches to orbitaltuning is complicated, cycles/kyr,and a Harmingwindow wasused to reduceleakage. and for consistency,only the mostrecent time scaleof (Resultsusing this methodof spectralanalysis were not Shackleton et al. [1990] is used here. This time scale has been significantlydifferent than results obtained using the appliedto boththe 677 and607 records[Shackleton et al., Blackman-Tukeytruncated lag (autocovariance)method.) 1990]. Althoughit appearsthat the site677 and site 607 For the intervalfrom 1.0 to 0 Ma, the powerspectrum of bispectrapresented here are robust to slighttime scalechanges, each•5180 record has variance concentrated at theMilankovitch the potentiallycritical effects of time scalevariability on frequenciesof 0.012, 0.023, 0.043, and0.053 cycles/kyr, bispectmmestimates will be treatedin a later paper. correspondingto the 100, 41, and23, and 19 kyr bands, The time seriesof planktic(Globeriginoides ruber) and respectively(Figure 8a). While the site677 recordsshow benthic(primarily Uvigerina senticosa) •5180 from site 677, significantvariance concentrated at theprecessional frequency andbenthic (Cibicidoides) •5180 from site 607 are shown in of 19 kyr, thispeak is not prominentin the site 607 dam. The Figure7. The recordsfrom site677 are approximately2.6 planktic•5180 time series for site 677 contains additional m.y. in length,while the site 607 recordis approximately2.8 varianceat periodsof 15 kyr and 12 kyr, whichcorrespond to m.y. long. The averagesampling interval for the 677 data is sumfrequencies of orbitalprecession and obliquity, and the 2000 years,and the averagefor the 607 datais 3500 years. site607 time seriescontains significant variance near 15 kyr It is well known that a changein the characterof the as well. Neogeneglobal ice volumerecord occurred near 1 Ma [Pisias For the interval from 2.6 to 1.0 Ma, the total variance and Moore, 1981; Ruddimanand Raymo, 1988]. Oscillations contained in each time series is lower than in the interval from at around100 kyr, whichare generallynot prevalentprevious 1.0 to 0 Ma. Each time series shows variance concentrated to 1 Ma, becomevery strongin the late Pleistocene.This is primarilyat 41 kyr (0.023 cycles/kyr)(Figure 8b). Low- clearlyseen by comparingthe power spectra of theserecords frequencypower, dominant from 1.0 to 0 Ma, is diminished before(Figure 8a) and after (Figure 8b) 1 Ma. Becauseof this relativeto 41 kyr power. In eachtime series,low-frequency

-2

o - Plonkfic 677 o - Ben'!'hic 677

_

_

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (Mo) Fig.7. Timeseries of planktic (G. ruber) and benthic (primarily Uvigerina) •5180 from site 677 [Shackleton and Hall,1989] and benthic (Cibicidoides) •5180 from site 607 [Ruddiman etal., 1989; Raymo et al., 1989]. The 607 dataare offsetfrom actualvalues to allow visualcomparison. Hagelbergetal.: Linear and Nonlinear Couplings During the Late Neogene 737 varianceis distributedbetween 400 kyr and71 kyr, ratherthan 41 kyrand 23 kyrbands with each isotopic time series, and in concentratedat 100kyr. In thisolder interval, the site 677 the 19kyr bandin thesite 677 timeseries (Figure 10 and planktic•5180 record also shows power at 23 kyr. This23 Table1). The coherenceis notas strong as in theinterval kyr peakis alsopresent in thebenthic records, although to a from 1.0 to 0 Ma, however. In each time series,there is lesserextent than in thesite 677 planktic •5180. significantcoherence with insolation at frequencieslower than Coherenceand phase spectra between July 65øN insolation 100kyr. In addition,the site 607 timeseries shows high andeach •518 0 timeseries for 1.0to 0 Ma andfrom 2.6 to 1.0 coherencewith insolationnear 30 kyr. In this time interval, Ma aredisplayed in Figures9 and10 and are summarized in eachof the insolation/•5180 phase spectra are similar. Table 1. In the interval from 1.0 to 0 Ma, significant(1-ct = Returningto the 1.0 to 0 Ma records,coherence between 0.90) coherenceoccurs at thefour primary Milankovitch insolationand •5180 near 100 cycles/kyr isestimated as0.91, frequenciesfor eachtime series. The coherence spectrum at site 0.79, and0.89 for the677 planktic,677 benthic,and 607 607 is similar to that of both site 677 records,although benthicrecords, respectively. If theseestimates represent the coherencewith insolationat 23 kyr is weaker,and coherency at truevalue of coherence(these estimates are not corrected for a 100kyr occursover a widerband. With the exception of the smallbias), then between 62% and82% of thevariance in the 41 kyr bandin theinsolation/site 607 phasespectrum, the õ!80 recordis linearly related to variations indirect phasespectrum for eachrecord is similar.As notedin the eccentricityforcing. For 677 plankticand benthic data, the Introduction,this overall high coherence between insolation phaseof theresponse changes by closeto 90ø near 100 kyr, and•5180 provides evidence for a stronglinear relationship andfor eachrecord the gain at 100kyr is relativelyhigh (Table betweeninsolation forcing and the paleoclimatic response. 1). Theseresults are consistent with a responseat 100kyr Coherence with insolation is weaker in the interval from 2.6 which is linear and resonant. This is discussedbelow. to 1.0 Ma. Significantcoherence with insolationoccurs in the Thebicoherence spectra for each •5!80 recordover the 1.0 to

(cd

lO 0

10-•

10-2 ooo 0.04 0.08 0.12 0.00 0.04 0.08 0.12 0.00 0.04 0.08 0.12 f (cycles/kyr) f (cycles/kyr) f (cycles/kyr)

417I I 25I I 12.5I I 81.3 417I I 25I I 12.5I I 8.3I 417I 2 12.5 8•.3 (kyr) (kyr) (ky)

101

]o o

10-•

10-2 i i i i I i I I I I I I I i i i I I I 0.00 0.04 0.08 O.12 0.00 0.04 0.08 O. 12 0.00 0.04 0.08 O. 12 f (cycles/kyr) f (cycles/kyr) f (cycles/kyr)

417 25 12.5 8.3 417 25 12.5 8.3 417 25 12.5 8.3 (ky) (ky) Fig.8. (a) Power spectra ofisotopic records forthe interval 1.0to 0 Ma: site 677 planktic õ180 (left),site 677 benthicõ180 (middle), and site 607 benthic õ180 (fight). Estimates having 6 degrees offreedom were obtained by usinga Hanningwindow and averaging four 256 kyr subrecords. (b)Power spectra for the interval 2.6 to 1.0 Ma: site677 planktic õ180 (left),site 677 benthic õ180 (middle),and site 607 benthic õ18 0 (fight).Estimates having 10degrees offreedom were obtained using a Hanningwindow and averaging six 256 kyr subrecords. 738 Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene

0 Ma intervalare shownin Figures1 la-1 lc. Smoothed shown(compare to Figure5c, thebicoherence spectrum for a bispectralestimates were calculated in the samemanner as the 4.096 m.y. record). The estimatesfor the 4 m.y. recordshown correspondingpower spectral and cross-spectral estimates. in Figure 5c have 14 dof, and the 0.90 significancelevel is Bispectmcalculated from shortrecords have coarser frequency 0.58. The 1 m.y. record(Figure 1ld), on the otherhand, has 6 resolution,as is clearly illustratedin Figure 1ld, wherethe dof, andthe 0.90 significancelevel is 0.88 (bicoherences bicoherencespectrum of 65øN insolationfor a 1 m.y. recordis reportedbelow are significantat a 0.80 level). Owing to the

Insolation / 677 Planktic Insolation / 677 Benfhc Insolation / 607 Benfhc 1.0 0.8 0.6 V VV 0.4 0.2 I I I ,, I I I '/ V 0.00 0.04 0.08 O. O. 2 O. 2

2OO

0 A -100 -200 IV,,-, 0.00 0.04 0.08 0.12 0.00 0.04 0.08 O. 12 O.00 0.04 0.08 O. 12 f (cycles/kyr) f (cycles/kyr) f (cycles/kyr) I I I I I I I I I I I I I 417 25 12.5 8.3 41 7 25 12.5 8.3 417 25 12.5 8.3 (kyr) (kyr) Fig.9. Coherence(top) and phase (bottom) between 65øN insolation and 8180, for the interval from 1.0 to 0 Ma: site677 planktic 8180 (left), 677 benthic 8180 (middle), and site 607 benthic 8180 (righ0. The 90% significance level for coherence(horizontal line) is 0.73. The 90% confidenceinterval for phaseestimates is indicatedby vertical bars.Positive phase indicates insolation leads 5180. The data were processed asdescribed inthe caption toFigure ga.

Insolation / 677 Planktic Insolation / 677 Benfhic Insolation / 607 Benfh 1.o 0.8 0.6 0.4 FVvv' 0.2

I I I , I 0.00 0.04 0.08 0.12 0.00 0.04 0.08 O. 0.00 0.04 0,08 O.

i-" lOO

o• o c- -100 t -200/ i i i , • , i i• 0.00 0.04 0.08 0.12 0.00 0.04 0.08 O. 0.00 0.04 0.08 0.12 f (cycles/kyr) f (cycles/kyr) f (c¾cles/kyr)

417 25 12.5 8.3 417 25 12.5 8.3 2s 2.s (kv) (kyr) Fig.10. Coherence (top) and phase (bottom) between 65øN insolation and 5 ! 80, forthe interval from 2.6 to 1.0 Ma: site677 planktic 5180 (left), 677 benthic 5180 (middle), and site 607 benthic 5180 (right). The 90% significance level for coherence(horizontal line) is 0.61. The 90% confidenceinterval for phaseestimates is indicatedby vertical bars.Positive phase indicates insolation leads 5180. The data were processed asdescribed in the caption toFigure 8b. Hagelbergetal.: Linearand Nonlinear Couplings During the Late Neogene 739 low degreesof freedomfor thesetime series, the bias of the an 80% confidence interval of one other and thus are not observedbicoherences is not negligible[Elgar and Sebert, statisticallydifferent). There are also statistically significant 1989]. Althoughthe bicoherence values reported in this couplingsin the site677 recordsbetween 23 kyr band sectionhave not been correctedfor bias, sucha correction variationsand 41 kyr (b(0.043,0.023)=0.840 in the planktic wouldnot significantly change the overall results presented recordand b(0.043, 0.023) = 0.741 in thebenthic record). here. This phasecoupling, also present in the insolationforcing, For the interval from 1.0 to 0 Ma at site 677, statistically indicatesthat oscillationsat periodsof 23 kyr and41 kyr are significant(1-• =0.80) bicoherencesindicate phase coupling coupledto oscillationsat 15 kyr. Finally, statistically amongcomponents which are alsocoupled in theinsolation significantphase coupling is presentbetween 41, 100, and 30 record,as well as amongcomponents which are not coupled in kyr oscillations(b(0.023, 0.012) = 0.723 in the plankticrecord theinsolation record. In both the planktic and benthic •5180 and b(0.023, 0.012) = 0.772 in the benthicrecord). This last records,phase coupling occurs between oscillations at 0.012 interactionis not presentin the insolationforcing and could cycles/kyr(100 kyr band),0.043 cycles/kyr (23 kyr band),and representpart of a nonlinearresponse in theclimate system. 0.053 cycles/kyr(19 kyr band). For theplanktic record, The bispectmmfor site607 is differentthan that for site677 b(0.043,0.012) = 0.794, andfor thebenthic record, b(0.043, from 1.0 to 0 Ma. Statisticallysignificant phase coupling 0.012) = 0.792. The phaserelationship between these modes occursamong two triadsof frequencies,(0.051, 0.016, 0.067) haschanged, however, from a biphaseof 0ø in theinsolation and(0.051, 0.043, 0.094), with bicoherencesof 0.520 and forcingto a biphaseof-90 ø in thebenthic data and -114 ø in the 0.766, respectively.While the lattertriad is alsopresent in the plankticdata (these biphase values of -90ø and -114 ø arewithin insolationbispectmm, this phasecoupling involves higher

TABLE1. Coherence,Gain, and Phase Shift Between Insolation and •5180 Response Near 100, 41, 23, and 19 kyr Periodsfor the Intervalsfrom 1.0 to 0 andfrom 2.6 to 1.0 Ma

Record Frequency, Period, Coherency Gain Phase, cycles/kyr kyr deg

0-1 Ma

677 planktic 0.012 --100 0.9076 9.220 154 0.023 -- 41 0.9877 0.488 79 0.043 -- 23 0.9504 0.134 105 0.053 -- 19 0.9077 0.087 136

677 benthic 0.012 --100 0.7903 7.739 170 0.023 -- 41 0.9677 0.429 95 0.043 -- 23 0.9529 0.124 96 0.053 -- 19 0.9301 0.074 167

607 benthic 0.012 --100 0.8948 9.096 - 142 0.023 -- 41 0.9755 0.448 158 0.043 -- 23 0.7437 0.070 75 0.053 -- 19 0.8320 0.032 -92

1-2.6 Ma

677 planktic 0.0101 -•100 0.2409 1.221 -40 0.0257 -- 41 0.9181 0.342 148 0.0452 -- 23 0.9141 0.096 103 0.0570 -- 19 0.7407 0.055 5

677 benthic 0.0101 -100 0.2493 1.154 -48 0.0257 N 41 0.8065 0.415 179 0.0452 N 23 0.6275 0.053 101 0.0570 - 19 0.8283 0.859 67

607 benthic 0.0101 -100 0.3197 1.334 -63 0.0257 .-- 41 0.8949 0.508 171 0.0452 - 23 0.7807 0.074 93 0.0570 N 19 0.3658 0.309 -93 740 Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene

frequenciesthat are closeto the Nyquistfrequency for thistime a 0.80 level. In the site 677 data, on the other hand, the series. With this time series,it is particularlynotable that no bicoherencefor this triad is statisticallysignificant. While the significantphase coupling occurs with oscillationsat 41 kyr causeof this discrepancy between Atlantic and Pacific •5180 or 100 kyr, as in the site 677 records. For instance,while datasets warrants further study, the low bicoherencesin the powerspectral peaks are present at 100,41, and 15 kyr in the site607 datamay be an artifactof the low degreesof freedom. site 607 data,the bicoherencefor this triad is not significantat At site677, evidencefor a nonlinearresponse of the climate systemto orbitalforcing is givenby high bicoherencein the (0.023, 0.012, 0.035) triad. However, the resultsat site 677 •- 0.06 are alsonoteworthy because of the similarityof the phase _x 0.05 couplingsin the climateresponse to thosein the orbital • 0.04 forcing.The phase coupled oscillations present in•518 0 time series from this site includes several of the same modes that -• 0.03 are coupledin the forcing.This is consistentwith a linear o 0.02 responseof the climate systemto insolationforcing. Changes in thephase relationships between each mode occur, however,

•0.01 i becausethe biphasesfor everycoupled triad in thedata are close to -90 ø, whereas in the insolation record the 0.04 0.08 0.12 correspondingbiphases are near 0 ø.

0.06 0.06- 0.05 0.05

0.04 _ 0.04 0.03 0.03 0.02 0.02 •0.01 0 <• o •0.01 i I I I 0.04 0.08 O. 0.04 0.08 0.12

0.06 0.06 0.05 0.05 -(b)

0.04 0.04 _ 0.03 0.03 o o 0.02 0.02 c• 0.01 •0.01

0.04 0.08 0.12 0.04 0.08 0.12

0.06 0.06 o 0.05 -• 0.05 0.04 • 0.04 0.03 • o.o3 0.02 o 0.02 •0.01 o40.01

0.04 0.08 0.12 O.04 0.O8 0.12 f l (cycles/kyr) f l (cycles/kyr) Fig. 11. Contoursof bicoherence,for the intervalfrom 1.0 to Fig. 12. Contoursof bicoherence,for the interval from 2.6 to 0 Ma. The datawere processed as describedin thecaption to 1.0 Ma. The datawere processed as describedin thecaption to Figure 8a. The minimumbicoherence value contoured Figure 8b. The minimumbicoherence value contoured (significantat a 80% level) is 0.73 with a contourinterval of (significantat a 80% level) is 0.57 with a contourinterval of 0.05:(a) site 677 planktic •5180, Co) site 677 benthic •5180, (c) 0.05:(a) site 677 planktic •5180, (b) site 677 benthic •5180, site607 benthic •5180, and (d) 65øN insolation. and(c) site 607 benthic •5180. Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene 741

As with the radiation time series above, simulations with In summary,bispectral estimates for site 677 •5180 data synthetictime seriescan be usedto reinforcethese results. indicatethat from 1.0 to 0 Ma, phasecouplings similar to Synthetictime serieswere generatedfor the site677 and site thosepresent in the insolationtime seriesoccur. This is 607 •5180data over the 1.0 to 0 Ma timeinterval. The consistentwith a linearresponse to insolationforcing. This amplitudespectrum of eachtime serieswas preserved, but the resultis not seenin the 1.0 to 0 Ma benthic•5180 time series observedphases were replaced with randomphases. For each from site 607, however. From 2.6 to 1.0 Ma, evidencefor a record,10 simulationswere made, and thebispectmm linearresponse ofthe •5180 record is not as strong ateither calculatedfor each. At site677, althoughsome significant location. Althoughall of the time serieshave very low bicoherencesoccurred in the plankticdata (through random numbersof degreesof freedom,and correspondingly high chance,expected at an 80% level), no simulationshowed nonzerosignificance levels, Monte Carlo simulationssuggest significantbicoherence at the samesets of triadsthat are thatthe significantbicoherences observed are not caused by coupledin the insolationforcing. For the site 677 benthic random fluctuations. data,bicoherences from only one in 10 simulationsresemble thosein the insolationdata. At site 607, significant EVOLUTION OF THIRD MOMENTS bicoherencessimilar to the data occur in two simulations, consistentwith an 80% significancelevel. Theseresults add As notedabove, the "mid-Pleistocenetransition" of global strengthto the argumentthat despite the low degreesof •5180records from a mainly41 kyr oscillation inthe early freedomof theserecords, the couplingsobserved in the dataare Pleistoceneto a 100 kyr "sawtooth"shape oscillation in the indeedsignificant and are not likely to be causedby random late Pleistoceneis well known. Subsequentmodels of fluctuations. paleoclimatehave sought to reproducethis asymmetric Bicoherencesfor thesite 677 and607 •5180time series for response[Saltzman and Maasch, 1988]. The evolutionof the intervalfrom 2.6 to 1.0 Ma are shownin Figure 12. skewnessand asymmetry(normalized third moments, as Bicoherencesfrom the site 677 planktic 5180 data show one definedabove) of thesite 677 and site 607 •5180 time series are regionof statisticallysignificant (1-a = 0.80) phasecoupling now examined. Thesethird-moment quantities are relatedto that is similarto the insolationforcing (the triad consistingof the shapeof the time series(as shownin the exampleabove) oscillationsat 19, 41, and 12 kyr hasb=0.662). This same andare givenby the integratedreal andimaginary parts of the phasecoupling ispresent inthe site 677 benthic •5180, where bispectrum,respectively [Elgar and Guza, 1985;Elgar, 1987]. b=0.603. In the benthic 677 data, two other triads have The third momentsallow the observedlong-term evolution of significantbicoherences that are not similarto coupledmodes the5180 record to be quantified. in the insolationforcing, b(0.047, 0.016) = 0.620 and The evolutionof the skewnessand asymmetryof the site b(0.078, 0.027) = 0.606. At site 607, the only significant 607and site 677 •5180 records isshown in Figure 13. phasecoupling occurs among high-frequency components Skewnessand asymmetry are calculatedfor overlapping512- (b(0.055, 0.051) = 0.599). point records,that is, from 1.0 to 0 Ma, from 1.5 to 0.5 Ma, Unlike the interval from 1.0 to 0 Ma, the results from the from 2.0 to 1.0 Ma, and from 2.5 to 1.5 Ma (becauseit is 2.8 lowerPleistocene and upper Pliocene •5180 time series do not m.y. long, for site 607 estimatesfrom 2.8 to 1.8 Ma are stronglysupport a linearresponse of the paleoclimaterecord to included).There is an increasein the asymmetry insolationforcing. Only one coupledtriad that is presentin (sawtoothness)of the time seriesfrom 2.5 Ma to presentand a the insolationrecord has high bicoherencein the site 677 simultaneousdecrease in the skewness(peakedness) of the records.Orbital frequencies that contain most of the variance record. In theoldest part of the records,there is relativelyhigh in the•5180 power spectrum for 2.6 to 1.0 Ma do not appear to skewness which decreases to near zero in the late Pleistocene. be as stronglycoupled to insolationas in the 1.0 to 0 Ma Monte Carlo simulationssuggest that thesethird-moment interval. Oscillationshaving a 62-70 kyr period,present in estimatesare significantlydifferent from zero.(Recall that for the•5180 power spectra for 2.6 to 1.0 Ma, are phase coupled to no coupling(a Gaussiantime series)the third momentsare 19kyr oscillations in the site 677 benthic •5180 data. As zero.) thereis no coherencebetween insolation and •5180 at this There are largedifferences between the skewnessand frequency(Figure 10), the62-70 kyr peakmay represent a asymmetryof the •5180 records and that of the insolation record nonlinear(difference interaction) between 15 kyr and 19 kyr. (Figure 14). The asymmetryof the insolationrecord is close Together,these observations indicate a changein the natureof to zero throughoutthe past2.5 m.y., and skewnessis only the climate responsebetween 1.0 to 0 Ma and 2.6 to 1.0 Ma. slightlygreater than zero. Thus,although the climatesystem Similar to the results for the 1.0 to 0 Ma interval, numerical is respondinglinearly to insolationforcing (at leastover the simulationswith synthetictime serieshaving random phases lastmillion years),the natureof the phasecoupling within the suggestthat the observedbicoherences (Figure 12) werenot systemis evolving.This observationis consistentwith the causedby randomfluctuations. At site677 for bothplanktic resultsof Pisias et al. [1990], who showedevidence for a andbenthic •5180 data, bicoherences of the simulated, random nonconstantphase between orbital forcing and ice volumeover phaserecords were similar to bicoherencesof thedata in only2 the past700,000 years. of 10 simulations.At site 607, only one simulationhad phasecoupling similar to thatobserved in the data. In no case DISCUSSION did the significantbicoherences resemble those of the insolation data. Thus, as in the 1.0 to 0 Ma interval, these The analysesabove demonstrate that the orbital parameters simulationsadd supportingevidence that the phasecouplings which combineto causelong-term changes in solarinsolation suggestedby bispectralanalysis are not dueto random are nonlinearlycoupled to oneanother. In particular,the fluctuations. couplingbetween precession terms of 23 kyr and 19 kyr and 742 Hagelbergetal.: Linearand Nonlinear Couplings During the Late Neogene

ß- 65øN insolation 1.0 •, ,- (577 planktic 1.0 - x .- 677 benthic -•- average

0.8 x x .- 607 benthic 0.8-

0.6 0.6-

0.4 0.4-

0.2 0.2-

0.0 0.0-

-0.2 -0.2 - I I I I o.o 1 .o 2.0 o.0 J .o 2.0 Time (Ma) Time (Ma) Fig. 13. Skewness(solid line) andasymmetry (dashed line) of Fig. 14. Skewness(solid line) andasymmetry (dashed line) of •5180records versus time. Each symbol represents a 1 m.y. theaverage ofthe three •5180 records presented inFigure 13 averageand is plottedat themidpoint of the time interval. (circles),and skewness and asymmetry of 65øNinsolation Squaresindicate 677 planktic •5180; triangles, 677 benthic (asterisks)versus time. •5180;and circles, 607 benthic •5180.

F X TM the 100 kyr eccentricityparameter is very strong.With respect m(to 2o-to2 + iYto) (4) to this coupling,it is importantto note that the strengthof theprecession-eccentricity coupling in the insolationforcing is wherex is the displacement(i.e., response),F is the force relatedto the presence of the1/(1-e 2) termin theinsolation drivingthe oscillator,m is the massof the oscillator,tOo is formulaused in this study. Only the solutionof Berger the natural(resonant) frequency of the system,tO is the [1978b] includesthis term. Previous insolationsolutions frequencyof the forcing,and ), is the dampingforce. For the havenot includedit [seeBerger, 1978a] because it doesnot caseof zero damping,the magnitudeof x is relatedto the size contributesignificantly to the overallinsolation change of theforce by the factor 1/(tOo 2- tO2),and as tO approaches (around0.1%). When this term is removedfrom the tOo,this magnitude approaches infinity. In this context,the insolationformulae of Berger[1978b], the phase coupling gainof the systemresponse is betweenprecession and eccentricity is weaker. Recallingthat the only orbitalelement that can modify the total solar energy receivedby the Earthis eccentricity(although forcing at 100 02_ 1 kyr is smallin comparisonwith precessionalor obliquity m2[(tO:z_ tOo2)2, +'Y 20) 2] (5) forcing),this term is very important,as emphasized by Berger [1978a]and Berger [1989]. The correspondingphase shift of the systemresponse is given The dilemmaof whetherthe 100 kyr peakpresent in the by •5180 spectrumduring the late Pleistocene isa linearresponse to eccentricityforcing, or a nonlinearresponse to precession 0=arctan (-tog- t0z) (6) band interactions is now addressed. The insolation data indicatethat eccentricity and precession bands are highly The gainand phase change of the systemresponse for coupledin the insolationrecord. The linearcoherence between differentvalues of tOare shownin Figure15. insolationand •5180 in theeccentricity band is greater than For any amountof damping0'), maximumgain occurs 0.90, and the samephase couplings are presentin the isotopic wheretO equals tOo. In addition,a phasechange of up to 180ø record. Theseresults suggest that a nonlinearresponse is not occursas tOchanges from tO> tOoto tO< tOo. If solar requiredto explainthe observed response. A simplelinear insolationis theforcing, and the climatesystem response is resonancemodel may help to explainthe processes leading to beingmeasured, then as a small amountof insolationis the dominanceof 100 kyr oscillationsin the late Pleistocene appliedto this systemat tO= 0.01, the climatesystem ice volume record. responsebecomes greatly amplified because of the proximity A brief review of resonance demonstrates that the results of tOto tOo,the naturalresonant frequency of the climate reportedhere are consistent with a resonantresponse to system. insolationforcing in the 100 kyr band. The simplephysical Thegain of the100 kyr band in theinsolation - •5180 cross systemof a forcedoscillator with dampingis spectrais aboutan orderof magnitudehigher than the gain in Hagelberget al.' Linearand Nonlinear Couplings During the Late Neogene 743 the 41 kyr band,and about2 ordersof magnitudehigher than smallamount of eccentricityforcing at 100 kyr (approximately the gain in the 23, or 19 kyr bands(Figure 15 and Table 1). 0.1% of the total variancein the insolationand approximately In addition,the phasespectra (Figures 9a and 16) show 3 ordersof magnitudeless than the varianceof precessional evidencefor a shiftwhich corresponds to thatpredicted by forcing)may occurat or neara resonantfrequency in the linearresonance. Note, however,that in this studyJuly 65øN climaticsystem, and the largeamplitude response observed in insolationvalues are used,and the phaseof the responsein the the518 0 recordresults from this resonance. The absence of precessionband depends on the particularmonth chosen. The sucha resonantresponse in the late Plioceneand lower (•) 104

10 3 i•'---lowdamping C)-,•- 607677 benthicbenfhic

10 2 /i• highdamping

101

100

10-1

10-2 I i 0.01 0.02 0.03 0.04 0.05 0.06 coø Frequency

180

160

140 low damping

120 _ high damping

m 100

. Q_ 80

6O

40

2O _

o 0.o0 O.•o1 0.02 Frequencyo.o3 0.04 0.05 0.06 Fig. 15. (a) Gain versusfrequency for the resonancemodel described in the text (equation(4)). Solidcurves indicate the gain of the systemresponse for differentvalues of damping,¾. The verticaldashed line at toorepresents the resonantfrequency ofthe system. Symbols plotted are the estimated gain of 5180 from Table 1. Squaresindicate 677planktic 5180; triangles, 677 benthic 5180; and circles, 607 benthic 5180. (b) Phase versus frequency forthe resonancemodel described in the text(equation (4)). 744 Hagelberget al.' Linear andNonlinear Couplings During the Late Neogene

180

160 140 120

IO0

80- 60- 40-

20-

0- 0.00 c•o 0.02 0.04 0.06 Frequency Fig. 16. Phaseshift versus frequency for theresonance model described in thetext (equation(4) andFigure 15b), comparedto•i180 phase from Table 1. The model phases have been shifted by 65 ø to allow comparison of the model tothe data. Symbols plotted are the estimated phase of •i 180 from Table 1. Squares indicate 677 planktic •i180; triangles,677 benthic •i180; and circles, 607 benthic •i180. Vertical bars indicate 90% confidence intervals for the phaseestimates.

Pleistocenemay signifya changingresonant frequency as the coupled,as are precessionand obliquity. The quadraticphase climate systemevolves. couplingspresent in the insolationforcing have been compared Linearresonance is a straightforwardmethod of accotinting tophase couplings inthe climatic time series of •5180, a for the observedclimate responseof the past 1 m.y. As noted globalice volumeproxy. Power spectra,cross spectra, and in the Introduction,paleoclimate models have explored a third moments(skewness and asymmetry)of the datasuggest varietyof mechanismsaccounting for thisclimatic response. an evolutionin the climate systemresponse from late Pliocene The observationspresented here support the results of to late Pleistocene. Quantitative estimatesof this evolution Saltzman and Sutera [1984] and Saltzman et al. [1984] that are seenin increasingasymmetry (sawtoothness) and imply an inherentlysensitive climate system at the 100 kyr decreasingskewness (peakedness) from 2.6 to 0 Ma at site677 periodduring the late Pleistocene.The observationsalso and site 607. From 2.6 to 1.0 Ma, the samephase couplings supportsBerger's [1989] statementthat the influenceof that are observed in the insolation record are not observed in eccentricityforcing alone is very important. While it is likely the oxygenisotope records. From 1 to 0 Ma, benthicand that a rangeof mechanisms,both linear andnonlinear, are planktic•5180 from ODP site 677 contain the same significant actingto producethe observedclimate response, the data bicoherences as the orbital time series. The observed presentedhere are consistentwith a resonantresponse. Models bicoherences,combined with high linearcoherence between which simulateclimatic changesmust be consistentwith these orbitalforcing and climaticresponse at orbitalfrequencies observations. duringthis time period,indicate that mostof the climatic This analysishas also demonstrated differences between the (/5180)response toinsolation forcing is linear, including the climaticresponse in the late Pleistocene,which may be responseat 100 kyr. The 100 kyr peak in the/5 !80 spectra primarilya linearresponse to insolationforcing, and the early shownhere is consistentwith a linear,resonant response of Pleistocene/Pliocene,which may includea nonlinearresponse. the climatesystem to directeccentricity forcing. The observations,including the evolutionof third moments, suggestthere is an evolutionin the climaticresponse. Acknowledgments.The cooperationof Nick Shackletonand Finally, it is necessaryto considerthese results in light of MaureenRaymo in providingthe isotopicdata and the time someimportant assumptions that havebeen made to carryout scalesused in thisstudy is greatlyappreciated. Andre Berger the aboveanalysis. It hasbeen assumed that a correcttime providednew insolationvalues and offered valuable scale has been used. While it is known that different time suggestions.The carefulcomments of JohnImbrie andBob scaleswill producedifferent results, it is unknownhow Oglesbyin reviewingthis manuscript were very usefuland are sensitivebispectral estimates are to errorsin the time scale. It appreciated.Alan Mix hasprovided helpful and thorough has also been assumed that the above results are not an artifact commentsthroughout. Walter Munk, after hearingof this of the particularorbital tuning strategy used by Shackletonet studyin its preliminarystages, remarked that a linear,resonant al. [1990]. This importantissue is left to a separatestudy. climateresponse would explainthe observations.Steve Elgar'sresearch is supportedby the Office of Naval Research CONCLUSIONS (CoastalSciences and and Geophysics) and the NationalScience Foundation (Physical Oceanography). This Bispectralanalysis was usedto detectnonlinear phase work was supportedby National ScienceFotmdation grant couplingsin the time seriesof long-termsolar insolation. OCE-90-12270 (Marine Geologyand Geophysics) to Oregon Orbital eccentricityand precession are highlynonlinearly State University. Hagelberget al.: Linearand Nonlinear Couplings During the Late Neogene 745

REFERENCES A. Berger et al., pp. 269-305, D. Riedel, Norwell, Mass., 1984. Imbrie, J., A. Mcintyre, and A. Mix, Oceanicresponse to Berger,A., Supportfor the astronomicaltheory of climatic orbitalforcing in the late Quaternary:observational change,Nature, 268, 44-45, 1977. and experimentalstrategies, in Climateand Berger,A., Long-termvariations of caloricinsolation resulting Geosciences,edited by A. Bergeret al., pp. 121-163, from the earth'sorbital elements,Quat. Res., 9, 139- Kluwer Academic, Boston, Mass., 1989. 167, 1978a. Kim, Y.C., and E.J. Powers,Digital bispectralanalysis and Berger,A., A simplealgorithm to computelong term its applicationsto nonlinearwave interactions,IEEE variationsof daily or monthlyinsolation, Contrib. Trans. Plasma Sci.,PS-7(2), 120-131, 1979. 18, Inst. d' Astron.et de Geophys.G. Lemaitre, Kim Y.C., J.M. Beall, and E.J. Powers,Bispectrum and Univ. Cath. de Louvain,Louvain-la-Neuve, Belgium, nonlinearwave coupling. Phys.Fluids, 23(2), 258- 1978b. 263, 1980. Berger,A., Long-termvariations of daily insolationand Le Treut, H., and M. Ghil, Orbital forcing,climatic Quaternaryclimatic changes,J. Atmos.Sci., 35, interactions,and glaciationcycles, J. Geophys.Res., 2362-2367, 1978c. 88, 5167-5190, 1983. Berger,A., The spectralcharacteristics of pre-Quaternary Maasch,K.A., and B. Saltzman,A low-orderdynamical model climaticrecords, and exampleof the relationship of globalclimatic variability over the full betweenthe astronomicaltheory and geosciences,in Pleistocene,J. Geophys.Res.,95, 1955-1963, 1990. Climateand Geosciences,edited by A. Bergeret al., Masuda,A., and Y-Y Kuo, A note on the imaginarypart of pp. 47-76, Kluwer Academic,Boston, Mass., 1989. bispectra,Deep Sea Res.,28A(3), 213-222, 1981. Berger,A., and M.F. Loutre,New insolationvalues for the Matteucci,G., Orbital forcingin a stochasticresonance model climate of the last 10 million years,Sci. Rep. of the Late Pleistocene climatic variations, Clim. 1988/13, Inst. d'Astron.et de Geophys.G. Lemaitre, Dyn., 3, 179-190, 1989. Univ. Cath. de Louvain,Louvain-la-Neuve, Belgium, 1988. Nikias, C.L., and M.R. Raghuveer,Bispectrum estimation: A digital signalprocessing framework, Proc. IEEE, 75, Birchfield,G.E., andR.W. Grumbine,"Slow" physics of large 869-891, 1987. continentalice sheetsand underlying bedrock and its Oerlemans,J., Glacial cyclesand ice sheetmodelling, Clim. relationto the PleistoceneIce Ages,J. Geophys. Res., 90, 11,294-11,302, 1985. Change,4, 353-374, 1982. Birchfield,G.E., andJ. Weertman,A noteon the spectral Peltier, R., Dynamicsof the earth,Adv. Geophys.,24, 1-146, 1982. responseof a modelcontinental ice sheet,J. Pisias, N.G., and T.C. Moore, Jr., The evolution of Geophys.Res., 83, 4123-4125, 1978. Elgar, S., Relationshipsinvolving third momentsand Pleistoceneclimate: a time seriesapproach, Earth bispectraof a harmonicprocess, IEEE Trans.Acoust. Planet. Sci. Lett.,52, 450-458, 1981. SpeechSignal Process., 35(12), 1725-1726, 1987. Pisias,N.G., A.C. Mix, and R. Zahn, Nonlinear responsein Elgar, S., and R.T. Guza, Observationsof bispectraof the globalclimate system: evidence from benthic shoalingsurface gravity waves,J. Fluid Mech., 161, oxygenisotopic record in coreRC13-110, 425-448, 1985. Paleoceanography,5(2), 147-160, 1990. Elgar, S., andR.T. Guza, Statisticsof bicoherence,IEEE Pollard,D., A coupledclimate-ice sheet model applied to the Trans.Acoust. Speech Signal Process., 36(10), 1667- Quaternaryice ages,J. Geophys.Res., 88, 7705- 1668, 1988. 7718, 1983. Elgar, S., and G. Sebert,Statistics of bicoherenceand biphase, Raymo, M.E., W.F. Ruddiman,J. Backman,B.M. Clement, J. Geophys.Res.,94, 10,993-10,998, 1989. and D.G. Martinson, Late Pliocene variation in Hasselman,K., Stochasticclimate models, Part 1: Theory, NorthernHemisphere ice sheetsand North Atlantic Tellus, 28, 473-485, 1976. DeepWater circulation,Paleoceanography, 4(4), 413- Hasselman,K., W. Munk, andG. MacDonald,Bispectra of 446, 1989. oceanwaves, in Time SeriesAnalysis, edited by M. Ruddiman,W.F., and M.E. Raymo, NorthernHemisphere Rosenblatt,pp. 125-139,John Wiley, New York, climateregimes during the past3 Ma: possible 1963. tectonic connections,Philos. Trans. R. Soc. London, Haubrich,R., Earth noises,5 to 500 millicyclesper second,1, Ser. B, B318, 411-430, 1988. J. Geophys.Res., 70, 1415-1427, 1965. Ruddiman,W.F., M.E. Raymo, D.G. Martinson,B.M. Hays, J.D., J. Imbrie, and N.J. Shackleton,Variations in the Clement, and J. Backman, Pleistoceneevolution: earth'sorbit: pacemaker of the Ice Ages,Science, NorthernHemisphere ice sheetsand North Atlantic 194, 1121-1132, 1976. ,Paleoceanography 4(4), 353-412, 1989. Imbrie,J., and J.Z. Imbrie, Modelingthe climaticresponse to Saltzman,B., Paleoclimaticmodeling, in Paleoclirnatic orbital variations,Science, 207, 943-953, 1980. Analysisand Modeling,edited by A.D. Hecht,pp. Imbrie, J., J.D. Hays, D.G. Martinson,A. Mcintyre, A.C. 341-396, JohnWiley, New York, 1985. Mix, J.J. Morley, N.G. Pisias,W.L. Prell, and N. Saltzman,B., and K.A. Maasch,Carbon cycle instability as a Shackleton,The orbitaltheory of Pleistoceneclimate: causeof the late PleistoceneIce Age oscillations: S,u•portfrom a revisedchronology of the marine Modelingthe asymmetricresponse, Global •5leto record, inMilankovitch andClimate, edited by Biogeochern.Cycles, 2(2), 177-185, 1988. 746 Hagelberget al.: Linear andNonlinear Couplings During the Late Neogene

Saltzman, B., and A. Sutera, A model of the internal feedback simpleice agemodel, J. Geophys.Res., 90, 5661- systeminvolved in late Quaternaryclimatic 5664, 1985. variations, J. Atmos. $ci., 41(5), 736-745, 1984. Wigley, T.M.L., Spectralanalysis and the astronomicaltheory Saltzman, B., A.R. Hansen, and K.A. Maasch, The late of climaticchange, Nature, 264, 629-631, 1976. Quaternaryglaciations as the responseof a three- componentfeedback system to earth-orbitalforcing, S. Elgar, Departmentof ElectricalEngineering and J. Atrnos.$ci., 41(23), 3380-3389, 1984. ComputerScience, Washington State University, Pullman Shackleton,N.J., and M.A. Hall, Stableisotope history of the WA, 99164-2752 Pleistoceneat ODP Site 677, editedby K. Becker,et T. Hagelbergand N. Pisias,College of Oceanography, al., Proc. Ocean Drill. Program Sci. Results,111, OregonState University, Ocean Administration Building 104, 295-316, 1989. Corvallis, OR 97331-5503. Shackleton,N.J., A. Berger,and W.R. Peltier, An alternative astronomical calibration of the Lower Pleistocene timescalebased on ODF::Site 677, Trans. R. Soc. (ReceivedApril 24, 1991; EdinburghEarth $ci., 81(4), 251-261, 1990. revised August29, 1991; Sneider,R.K., The origin of the 100,000year cyclein a acceptedAugust 30, 1991.)