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Linear and Nonlinear Couplings Between Orbital Forcing and The PALEOCEANOGRAPHY, 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 climate. 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 Pliocene 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 Pleistocene,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 Earth 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 radiative forcing 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 similar avenues.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,
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