PALEOCEANOGRAPHY,
LINEAR
AND NONLINEAR
COUPLINGS
VOL. 6, NO. 6, PAGES 729-746, DECEMBER 1991
BETWEEN
ORBITALFORCINGAND THE MARINE •5180RECORD
DUR•G
THE LATE NEOGENE
TeresaHagelbergandNick Pisias
Collegeof Oceanography,
OregonStateUniversity,Corvallis
SteveElgar
Department
of ElectricalEngineering
andComputer
Science,
WashingtonStateUniversity,Pullman
Abstract.Previousinvestigations
of theresponseof PlioPleistocene
climaticrecordsto long-term,orbitallyinduced
INTRODUCTION
changes
in radiation
haveconsidered
a linearresponse
of
Sincethepioneeringworkof Hayset al. [1976],researchers
havesuccessfully
identifieda linearresponse
of PlioPleistoceneclimateto orbitalforcing[e.g.,Pisiasand Moore,
climate. While the second-order
statisticsof powerspectraand
crossspectra
providenecessary
information
onlinear
processes,
insightintothenonlinear
characteristics
of Pliocene
andPleistocene
climateis notprovidedby thesestatistical
quantities.Second-order
statistics
donotcontainthephase
information
necessary
to investigate
nonlinear,
phase-coupled
processes.
Suchinformation
is provided
by higher-order
statistical
quantities.In particular,
bispectral
analysis
indicates
thatnontinearcouplings
arepresentin theclimatic(radiative)
forcingat theMilankovitchfrequencies.
Througha linear
transfer,thisforcingproduces
similarnonlinear
couptings
in
1981; Imbrie et al., 1984, 1989; Raymo et al., 1989;
Ruddimanet al., 1989], supportingtheMilankovitch
hypothesis
of climatechange.Empiricalevidencein support
of thishypothesis
is strengthened
by observations
of high
coherence
andrelativelyconstant
phasebetweentheclimatic
forcing(orbitaltilt andprecession)
andtheresponse
(globalice
volume)at periodsof 41, 23 and 19 kyr. Theseresults
suggest
a linearrelationship
betweenice sheetgrowthand
deep-sea
sedimentary
oxygen
isotope
records
(ODPsite677
decayandorbitallyinduced
changes
in thedistribution
of solar
and DSDP site 607) from 1.0 to 0 Ma duringthe late
radiation.
Neogene.Thisanalysis
suggests
thatduringthelate
A tinearresponse
of theclimatesystemto solarinsolation
changes
canaccountfor muchof thevariancein theobtiquity
(41 kyr) andprecession
(23-19kyr) bandsof thepaleoclimate
spectrum,
but it is generallythoughtto be insufficient
to
explainthedominance
of the 100kyr cycleduringthe
Pleistocene,the dominanceof the 100,000year cyclein the
climate record is consistentwith a linear, resonantresponseto
eccentricity
forcing.In theperiodfrom2.6 to 1.0Ma, a
change
in thenatureof theclimaticresponse
to orbitalforcing
is indicated,
asphasecouptings
presentin theisotopictime
seriesarenot similarto thephasecouplingspresentin the
insolationforcing.Third-ordermoments(skewness
and
asymmetry)
areusedto quantifytheshapeof theclimatic
response.
From2.6 Ma to present,anincrease
in the
asymmetry
(sawtoothness)
of theoxygenisotopic
records
is
accompanied
by a corresponding
decrease
in theskewness
(peakedness)
of therecords.Thisindicates
anevolution
in the
natureof thephasecouplingwithinthe climatesystem.These
resultsmayprovideimportantconstraints
usefulin
development
of modelsof paleoclimate.
Copyright 1991
by the AmericanG6ophysicalUnion.
Papernumber91PA02281.
0883-8305/91/91PA-02281510.00
Pleistocene[Imbrie et al., 1989 andreferencestherein]. Efforts
to explainthedominance
of 100kyr powerin thepaleoclimate
recordhaveproduced
a rangeof results,asdiscussed
below.
Studieswhichcontendthata linearresponse
cannotexplain
the 100kyr cyclein thepaleoctimatic
recordoftenarguethat
insolationforcingin the 100kyr eccentricity
bandis toosmall
to producetheobserved
response
or thatit is altogether
nonexistent
[e.g.,Imbrieet al., 1989;Birchfieldand
Weertman,1978]. Althoughthedirectinfluenceof
eccentricity
on insolation
is small(of theorderof 0.1%
[Berger,1977]),eccentricity
is theonlyorbitalparameter
that
influences the total amount of radiation the Earth receives
annually[Berger,1989]. In addition,
eccentricity
variations
are
observedto correlatepositivelywith changesin estimated
summertemperature
[Berger,1978a,Hayset al., 1976].It is
still unknownwhetherthepaleoclimatic
spectralpeakat 100
730
Hagelberget al.: LinearandNonlinearCouplingsDuringtheLateNeogene
kyr is produced
from(1) a linearresponse
to eccentricity
or (2)
a nonlinearinteractionbetweentwo precession
band
oscillations
whichtransfers
energyto the100kyrband.To
investigatethisproblem,therelativeproportions
of variance
in thepaleoclimaticrecordthatarerelatedto a linearresponse
to eccentricity
andto a nonlinearresponse
to precessional
forcingmustbe determined.
Paleoclimateresearchin thisareahasproceeded
alongtwo
similaravenues.On onehand,analysesof geologicdata
providefundamental
observations
of howclimatehasvaried
overthepastseveralmillionyears.Successful,
extensive
strategies
havebeendevelopedto examinerecordswithina
consistentchronologicframeworkandin a systematicmanner
[e,g.,Imbrieet al., 1989]. On theotherhand,modelshave
beendevelopedwhichseekto describethephysicsgoverning
thesechangeswith varyingdegreesof complexity(seethe
summaryby Saltzman[1985,andreferences
therein]).
Deterministicmodelsdescribingdynamicsof the "slow
response"
of climatesystemevolutionaredividedinto two
groupsby Saltzman[ 1985]:(1) quasi-deductive
models,based
on prognostic
equations
for specificprocesses
[e.g.,Birchfield
and Weertman, 1978; Birchfield and Grumbine, 1985; Le Treut
and Ghil, 1983; Oerlemans, 1982; Pollard, 1983; Peltier,
1982] and (2) inductivemodels,whichattemptto constructa
dynamicalsystemof equations,basedon knownphysical
feedbacks,from whichreasonable
paleoclimaticoutputcanbe
produced[e.g.,ImbrieandImbrie, 1980;SaltzmanandSutera,
1984; Saltzman et al., 1984; Maasch and Saltzman, 1990].
In thisstudy,bispectraof the time seriesof orbitallyinduced
insolationchangesandtime seriesof theoxygenisotopic
proxyof globalice volume(fromODP site677 andDSDP
site607) arepresented.Crossspectraandbispectraareusedto
showthatboththeradiativeforcingandtheclimaticresponse
containnonlinearlycoupledcomponents,
but theinteraction
betweentheinsolationforcingandresponse
is primarilylinear.
The analysissuggests
thatthe 100 kyr powerobservedin the
paleoclimaticrecordis consistent
with a linear,resonant
response
to insolationforcing.
First,a brief introductionto bispectralanalysis,includinga
simpleexample,is presented.Next, thebispectraof the
calculatedrecordof radiativeforcingaswell asthreePlio-
Pleistocene
records
ofglobal
icevolume
(•j180)
arepresented.
The time periodfrom 1.0 to 0 Ma, where 100 kyr poweris
very high,is comparedto the periodfrom 2.6 to 1.0 Ma,
where100 kyr poweris lower. Higher-ordermoments
(skewness
andasymmetry)are shownto providequantitative
insightsinto featuresof theserecords.Finally, the
implicationsof theseresultsfor paleoclimatemodel
developmentis examined.
BACKGROUND
Bispectraltechniques
havebeenin usefor over25 years
[Hasselmanet al., 1963] (seealsothereviewby Nikias and
Raghuveer[1987]). Becausethisis thefirst applicationof
thesetechniques
to paleoclimatictime series,a brief summary
andan exampleof its applicationarepresented.
The bispectrumis formallydefinedas thedoubleFourier
Severalmodels(of bothtypes)invokea nonlinear
response
of
theclimatesystemto ice sheetformationto explainthe 100
kyr cycle [Le Treut andGhil, 1983;Imbrie andImbrie, 1980;
Wigley, 1976;BirchfieldandWeertman,1978;Sneider,1985].
Othermodelsexplainthiscycleasa freeoscillationof the
climatesystem,whosephaseis setby weak eccentricity
forcing [Saltzmanand Sutera,1984; Saltzmanet al., 1984].
Finally, a few modelsincorporate
stochastic
effects
functionof frequency,thebispectrum
describes
thedistribution
of thethird momentasa functionof bi-frequency.For a
discretelysampledtime seriesx(t) with complexFourier
[Hasselman, 1976; Matteucci, 1989].
coefficients
atfrequencyj•)
givenbyX•), thepower
spectrum
Thusa wide rangeof classesof modelsexist,all of which
seekto explainthe samephenomenon(Pleistoceneclimate).
A generaltestof eachmodelis its ability to reproduce
characteristics
thatarepresentwithin the data,aswell asits
ability to predictcharacteristics
whichhavenot yet been(or
cannotbe) determined
fromdata. To firstorder,everymodel
reproduces
somefeatureor featuresof thepaleoclimatic
record,
mostcommonlyits powerspectrum(a second-order
statistic).
Insightsinto thehigher-order
statistics
of thedatacanprovide
additionalconstraints
or testsfor evaluatingthemodels.
Althoughsomeobservations
suggesta nonlinearresponse
of
thepaleoclimatesystemto orbitalforcing,thisresponsehas
not beenquantified,in part becausethe power spectrumdoes
not containphaseinformation.Powerspectralanalysesare
incapableof detectingthephasecouplingwhichcharacterizes
a
nonlinearinteraction. Similarly, the crossspectrumcontains
phaseinformationbetweenonly two recordsat a given
frequencyandis thereforecapableof resolvingonlya linear
relationship.Both thepowerspectrumandcrossspectrumare
derivedfrom covariance,
a second-order
statistical
quantity.
The presence
of quadraticnonlinearinteractions
between
oscillations
of differentfrequencies
canbeaddressed
most
effectivelythroughtheuseof thebispectrum,
a third-order
statisticalquantity.
is
transform of the third-order autocorrelation of a time series
[Hasselman
et al., 1963]. While thepowerspectrum
describes
the distributionof variance(the secondcentralmoment)as a
P•) = (X•) X*•))
(1)
wheretheangledbracketsindicateexpectedvalue(mean)and
theasteriskindicatescomplexconjugate.Similarly,the
bispectmmis givenby [Haubrich,1965;Kim andPowers,
1979]:
= (X03)Xffk)
X*q)+fO)
(2)
If thereisenergy
(variance)
atfrequencyj•,
thenP•) is
nonzero.The bispectmm,however,is zeroat thehi-frequency
J•,fk,whenthemodesj•,fk,
andj•+kareindependent
ofone
another,evenif energyis presentat thesefrequencies.For
independent
oscillations,
thephasesarerandomrelativeto each
other,yieldinga zerobispectmmuponaveragingovermany
realizations.In theory,thebispectmmis nonzeroonly if these
threemodesarequadratically
coupled,meaningthattheir
phasesare nonlinearlycoupledto oneanother.
Like thecrossspectrum,whichprovidescoherence
andphase
information,the complex-valuedbispectmmis oftencastin
termsof its normalizedmagnitude,thebicoherence,andits
phase,
thebiphase
(denoted
b•,f10 andB•,f10,respectively).
Thesquared
value
ofbicoherence,
b2•,f10,
represents
the
Hagelberg
et al.: LinearandNonlinear
Couplings
DuringtheLateNeogene
731
fraction
ofpoweratfrequencyj•
+fk =J•+kowingtoquadratic
(nonlinear)interactionsof the threemodes[Kim andPowers,
1979]. Analogousto the linear coherencespectrum,the
bicoherencespectrumis normalizedsuchthat0 fib fi 1 [Kim
andPowers,1979]. The biphaseis analogousto the phaseof
the crossspectrum,and rangesbetween-•zand
A simpleexampledemonstrates
the effectof nonlinearphase
couplingon both the shapeof a time seriesand the detection
of phasecouplingby the bispectrum.Let the time seriesx(0
be composedof threecosinesanda low-amplitudeGaussian
noisecomponent:
Case
1
-2
0
400
800
2
x(t) = cos(2•zflt+•l) + cos(2nf2t+•2)+ cos(2•zf3t+•3)+ noise
(3)
•'
Case 2
W o
wherefi is thefrequencyand{i is thephaseof eachcosine.
The frequencies
of thesecosinesarechosento befl= 0.010,
f2= 0.043, andf3 =fl +f2 = 0.053 for easein comparison
of
thissimulated
record
withlatePleistocene
6180timeseries,
whichshowpowerspectralpeaksat thesefrequencies
in
cycles/thousand
years. Threedifferentcasesareexamined.In
thefirst case(case1), {1, •)2,and {3 are all randomvariables
uniformlydistributedover (0, 2•z].This caserepresents
a linear
(Gaussian)systemwherethereis no nonlinearphasecoupling.
The phasesof eachoscillationarerandomrelativeto one
anotherwhenaveragedovermanyrealizations.In thesecond
case(case2), {1 and{2 are uniformrandomvariables,but {3
= •)1 + q2, and thusfl, f2, andf3 arenowphasecoupled.The
relationshipbetweenthe threemodesis no longerrandom.
Similarly,in the lastcase(case3), {1 and{•2are uniform
randomvariables,but {3 = {1 + •)2 + •z/2(thechoiceof {3 in
cases2 and 3 resultsin biphasesof 0øand90ø,andwaveforms
with maximumskewnessandasymmetry,respectively,as
discussed
below). For eachcaseexamined,16 256-point
realizationswere made,resultingin a 4096-pointrecord.
Portionsof eachtime seriesandthe corresponding
power
spectrumfor eachcaseare shownin Figures1 and2,
respectively.In eachcase,thepowerspectrashow
concentrations
of varianceat frequenciesfl,f2,andf3 with
randomfluctuationsat otherfrequencies
resultingfrom the
noise. Even thoughthe time seriesfor eachcaselook
different,thepowerspectraareidentical.This is becausethe
powerspectrumis phaseindependent
andthethreecasesdiffer
only in their phaserelationships.
Figure3 showsthebicoherence
spectrafor cases1, 2, and3.
Smoothed
power
spectral
andbispec
•tmlestimates
weremade
by windowing(usinga Hanningwindow)andaveragingthe 16
256-pointensemblesfor eachrecord. Resultingestimateshave
30 degreesof freedom(doO.The valuescontoured
in Figure3
aretheobserved
bicoherences
at the•ad of frequencies
(fl,f2,
fl+f2) (denotedb(fl,f2)) whichare significantat a (l-ix) =
0.90level(b90
%= (4.6/do01/2
[Elgar
andGuza,
1988]).
Owing to the symmetrypropertiesof the bispectmm,it
necessary
only to calculatethebispectmmwithin a triangular
regiondefinedby 0 <fl <fN,f2 <fl, andfl+f2 <fN, where
fN is theNyquistfrequency[Kim andPowers,1979]. It is
importantto notethat the directionof the nonlinearinteraction
is not given by the bispectrum,that is, suminteractions
(interactionsatfl andf2 thatproducef3 TM
fl+f2) are not
distinguished
from differenceinteractions
(interactions
atf3 and
f2 thatproducef! =f3-f2, for example). This information
0
400
800
(o)
2
Case
0
3
-2
0
400
800
Fig. 1. Portionsof eachtime seriesfor cases1-3 (seetext): (a)
case1 (fl+f2 =f3' •1, •2, •3 arerandom),
(b) case2 (fl +f2
=f3; •1 + •2 = •3), and (c) case3 (fl +f2 =f3; •1 + •2 = •3•z/2). The unitsare arbitrary.
comes
instead
fromotheraspects
of thephysical
system
under
examination,includingthe biphase[Kim et al., 1980;Elgar
and Guza, 1985].
In case1, the linearcasein whichno phasecouplingis
present,bicoherences
nearthetriad(0.059,0.051,0.110)=
0.370 are significantat a 90% level (Figure3a). Althoughall
bicoherences
in thiscaseshouldbe zero,any finite lengthtime
serieswill yield somenonzerovaluesof bicoherence.This
valueof bicoherence
represents
thehighvaluesthatcanbe
expectedto occurby randomchancefor a linearprocess
with
30 dof. On the otherhand,bicoherenceat (0.01, 0.043, 0.053)
(the(fl,f2, f3) triad)is 0.167, whichis belowthe 90%
significancelevel.
In cases2 and 3, wherephasesarecoupled,b(0.01, 0.043) =
0.917 andb(0.01, 0.043) = 0.912, respectively,indicating
highbicoherence
at the(0.01, 0.043,0.053) triad(Figures3b
and3c). In case2, thevalueof bicoherence
indicatesthatat
frequency
0.053,approximately
84% of thevarianceis
quadratically
coupledto theothertwo components
of thetriad.
The valueis lessthan 1.0 owingto the additionof noise.
Biphaseestimatesarerelatedto theshapeOfthewaveform
(time series)[MasudaandKuo, 1981;Elgar, 1987]. The
waveformshapeis oftendescribed
by two normalized
third
732
Hagelberg
et al.: LinearandNonlinearCouplings
DuringtheLateNeogene
moments,skewness
andasymmetry.Skewness
is the
normalized third moment of the time seriesand indicates
asymmetry
withrespectto a horizontal
axis,or top-bottom
asymmetry.
Asymmetry
is thenormalized
thirdmomentof the
Hilberttransformof thetimeseries[Elgar,1987]andindicates
asymmetry
withrespectto a verticalaxis,or for-aft
asymmetry.For a timeserieswhichis purelyGaussian,
havingonlyrandomphaserelationships,
thebiphase
is also
randomforeverytriadof frequencies.
1ø2
f
f
1
f
2 .3
c• 101
In case1 of the aboveexample,the shapeof the waveform
is symmetrical,
andskewness
andasymmetry
arezero.In
Figurela, skewness
= -0.12 andasymmetry
= -0.16.The
smallnonzerothird-momentestimatesarecausedby statistical
fluctuations. In case2, the biphasefor the (0.01, 0.043,
0.053) interactionis zero,andthe waveformis positively
skewed(in Figurelb, skewness
= 0.80, asymmetry
= 0). In
case3, thesametriadhasbiphaseof •/2, andthewaveformis
asymmetric
or "sawtooth"
shaped
(in Figurelc, skewness
= 0,
asymmetry= -0.79).
Theasymmetric
recordof case3 is qualitatively
similarin
shapeto theSPECMAPstack(Figure4), a smoothed
0.06
Case
1
0.05
ß•_
10 ¸
0.04
0.05
10-1
0.02
10-2
0.01
0.00
0.04
0.08
0.12
0.04
102
f1
0.08
0.12
f2 f.3
0.06
101
Case 2
Case
0.05
2
0.04
10 ¸
"- 0.03
0.02
10-1
10-2
0.00
f•
I
I
0.04
•
I
I
0.08
I
0.01
I
0.12
,
C• ,
0.04
0.08
,
0.12
f2 f3
005
•
o
'•-
10
2
0.06
,Ca
3
101
0 04
•-• 0 03
10 ¸
½10-1
10
-2
0.00
0.04
(o) o
02 ,
0 01
0.08
0.12
Frequency
Fig. 2. Powerspectrafor cases1-3. Smoothed
powerspectral
estimates
having30 degreesof freedomwereobtainedusinga
Hanningwindowandby averaging16256-pointensembles
for
eachrecord:(a) case1, (b) case2, and(c) case3. The unitsare
arbitrary.
'
,
,
,
0.04 0.08 O.2
Fig. 3. Contours
of bicoherence
for cases1-3. Smoothed
bispectral
estimates
wereobtainedin thesamemanneras
powerspectral
estimates
andhave30 degrees
of freedom.The
minimumbicoherence
valuecontoured
(significantat a 0.90
level)is b=0.39withadditionalcontours
every0.05:(a) case
1, (b) case2, and(c) case3. The unitsof frequencyare
arbitrary.
Hagelberg
et al.' LinearandNonlinear
Couplings
DuringtheLateNeogene
733
composite
record
offivelatePleistocene
icevolume
(5180)
distribution,andeccentricityvariationsaffectthe total
records[Imbrie et al., 1984]. The SPECMAP recorddisplays
steepterminationsrepresentingthe endof glacialstages,
followedby a slow, gradualice buildup. With minor
adjustments
to therecordof case3 (reductionof the amplitudes
off2 andf3] the resemblanceto the SPECMAP stackis
striking(Figure4). The implicationsof this similarityare
thatquadraticphasecouplingsareof importancein the
paleoclimaticrecordandthatthroughthe useof higherorder
statistics,thesecouplingscanbe quantified.
insolation received over time. Examination of the terms that
NONLINEAR
COUPLINGS
AMONG
are usedto derivelong-termEarthorbitalvariations[e.g.,
Berger,1978c]indicatesthateccentricity,tilt, andprecession
[e, e, andesine0]are indeedcoupledto oneanother.However,
no quantitativeestimatesof the strengthof thesecouplingshas
beenmade. Bispectralanalysisof the insolationrecordallows
phase-coupled
oscillationsto be isolated.
A 4.096 m.y. recordof July 65øN radiationsampledat 2000
year intervalsis shownin Figure 5a. The power spectrumof
thisrecord(Figure5b) showsconcentrations
of energyat
periodsof 41, 23, and 19 kyr (0.024, 0.003, and0.053
cycles/kyr,respectively).Lessenergetic,but significant,peaks
are alsopresentnear 100, 28, and 15 kyr (0.01, 0.036, 0.067
cycles/kyr,respectively).The bicoherencespectrumfor this
record(Figure5c), showsthatmanymodesin theradiative
forcingare stronglycoupled.Significant(1- tz =0.90) phase
couplingbetweeneccentricityandprecession
is indicatedby
b(0.001, 0.012) = 0.978 (regionA in Figure 5c). The
bicoherence
spectrumalsoindicatessignificantcoupling
betweenorbitaltilt andprecession.For example,b(0.041,
0.025) = 0.960 (regionB in Figure 5c) and b(0.055, 0.023) =
0.969 (regionC in Figure 5c). The thirdcomponents
in these
interactingtriadshavefrequencies
0.066 (15 kyr) and0.078
(12.8 kyr), respectively,andappearaslessenergeticpeaksin
the radiationpowerspectrum(Figure5b).
The mostnotableinteractionis at the triad (0.00 1, 0.012,
0.053), (regionA in Figure 5c), which hasa bicoherenceof
0.978. This high valueof bicoherence
indicatesthatprecession
components
of 19 and 23 kyr are nonlinearlycoupledto 100
kyr eccentricityterms(although0.012 cycles/kyris a 83 kyr
period,thisfrequencyrepresents
the 100kyr bandafter
averaginghasbeencompleted).This resultis confirmedby
examiningthe termsin the seriesexpansionof eccentricityand
precession
givenby Berger[1977, Tables1 and 3]. Difference
interactions
betweenthe individualprecession
termslistedin
thisreferencecanproducevirtuallyall of thelistedeccentricity
ORBITAL
PARAMETERS
The bispectrumof the normalizedtime seriesof radiative
(insolation)forcing(ascalculatedby Berger[1978b]is now
examined.(Note thatin thispaperthe 1978solutionof Berger
is used. A morerecentsolution[BergerandLoutre,1988]has
beenfoundto yield essentiallythe sameresultsin the
bispectrum
astheolder[Berger,1978b]solution,althoughthe
orbital values in the new solution differ from the older
solutionin the time domainprior to 1 Ma. Becausethe values
of the newer solution from 1.0 to 0 Ma are less well
reproducedat presentthanthoseof the older solution,the 1978
valuesare usedhere.) Long-term(of theorderof 100kyr)
variationsin theEarth'sinsolationareproduced
through
interactingmotionsof the Sun andplanetsrelativeto the
Earth. At a given latitudeon the Earth in a given season,the
solarenergyavailableis a functionof the solarconstant,the
eccentricityof theEarth'sorbit, the obliquityof theEarth's
axis,andthe longitudeof perihelionas measuredfrom the
vernalequinox[Berger,1978c].
Many studieshaveexaminedthepaleoceanographic
response
to changesin theseorbitalparameters.Studyingeachorbital
parameterseparately
hasbeenconsidered
reasonable
as
obliquityvariationsaffectthelatitudinaldistributionof
insolation,precessional
changesaffectthe seasonal
-3
O
I
I
I
2
/
•
,'t
I
\ I
I
¾¾, I
.
•.
%1,
I•
l/ •
[ . ! t•
?,,'"
"
I I
'
f
I
'x /
,,,/
/•
•
I
^ I•/•
•/x/
I
,
ß
•,
,
I ;
; •
SPECMAP
•' ' •
3
Case
4
5
0
200
400
Time
600
8OO
(Ka) ->
Fig.4. SPECMAP
composite
5180stack
(dashed
line)[Imbrie
etal.,1984]compared
topartofa modified
version
of case3 (solidline) (seeFigure lc). The amplitudeoff! is the sameasin case3, but theamplitudesoff2 andf3
havebeenreduced60% and70%, respectively.
734
Hagelberg
etal.: LinearandNonlinear
Couplings
DuringtheLateNeogene
terms[Berger,1989]. This differenceinteractionindicatesthat
at frequency
f2= 0.012 cycles/kyr(100 kyr band),greaterthan
95% of thevarianceis quadratically
coupledto oscillations
at
fl=0.41 cycles/kyr(23 kyr) andf3=0.53 cycles/kyr(19 kyr).
The energyat 100 kyr is smallin comparison
with thatat 41,
23, and 19 kyr (Figure5b). However,almostall of the
varianceat 100kyr is quadratically
coupledthroughnonlinear
interactions
with theprecessional
terms.The biphaseof the
(0.041, 0.012, 0.053) triadis zero,indicatingthatthephaseof
f2 is equalto thedifferenceof thephasesoff3 andfl. Thisis
alsoconfirmedby Berger[1977],asdifferences
betweenthe
phases
of theindividualprecession
termslistedareexactly
equalto thecorresponding
phasesof theeccentricity
terms.
In additionto significant
bicoherence
values,otheranalyses
suggestthatthe observedphasecouplingsdo notresultfrom
randomfluctuations.
Time seriesgenerated
by replacing
the
phasesin theJuly65øNradiationtime series(Figure5a) with
randomphasesuniformlydistributedover (0, 2•) are shownin
Figure6a. The powerspectrumof thisrandomphasetime
series(Figure6b) is identicalto thatof theradiationrecord
(Figure5b) becausephaseinformationis notretainedby the
powerspectrum.The bicoherence
spectrum
(Figure6c) of this
randomphaserecordclearlyindicates
anabsence
of phase
coupling,asis expectedfor a purelylinearsystem.Similarto
case1 above,thetwo valuesabovethe0.90 significance
level,
(0.039, 0.037, 0.076)= 0.509 and (0.105, 0.008, 0.113) =
0.589), representhighbicoherence
valuesthatcanbe expected
to occurthroughrandomchance.The resultof therandom
phasesimulationshownin Figure 6 is consistentwith the
conclusionthatthehighbicoherences
observedin theradiation
record are not a result of random fluctuations,but result from
phasecouplingswithin the orbitalrecord. Similarlow values
3
0
1000
2000
4000
3OOO
Time (kyr)
102
101
•
•
8070
)
lO 0
(o)
•-
0.06
.x
0.05
10-2
•
004
10-3
o 0.02
ßr- 10 -1
0.03
O.Ol
10-40.00
O.04
O.O8
0.04
0.12
25
12.5
kyr
O.
f l (cycles/kyr)
f (cycles/kyr)
417
0.08
8.3
417
25
12.5
8.3
kyr
Fig. 5. July65øNinsolation,
for theintervalfrom4.096to 0 Ma (fromBerger,[1978b]solution]:(a) timeseries,
(b) powerspectra,and(c) significant(1-o•= 0.90) contoursof bicoherence.The minimumvaluecontoured
is 0.57,
andcontour
intervalis 0.10.To obtainpowerspectral
andbispectral
estimates
having14degrees
of freedom,data
wereHanningwindowed,
andeightensembles
of 0.512m.y.lengthwereaveraged.
RegionsA, B, andC arereferred
to in the text.
Hagelberget al.: LinearandNonlinearCouplings
DuringtheLateNeogene
of bicoherence
wereobservedin severalotherrandomphase
simulations(not shown).
Theseresultsillustratethe useof bispectralanalysesfor
detectingquadraticallynonlinearinteractions.Strongnonlinear
couplingis detectedbetweenoscillations
in theprecession
and
eccentricity
bandsin theestimatedrecordof radiativeforcingof
Berger[1978b]. Using the termsgivenby Berger[1977], a
differenceinteractionof precession
termsis equivalentto
eccentricity[Berger,1989]. This interactionwasnot
observablethroughexaminationof thetime seriesandpower
spectra.The bicoherencespectrumalsoindicatessignificant
nonlinearinteractionsbetweenprecession
andobliquity. In the
presentsolutionof Berger[1978b],thestrengthof these
nonlinearinteractionsdoesnot vary throughtime, andthus
analysisof a 4 m.y. recordyieldsthesameresultsasanalysis
of shorter(1 m.y.) records.
A purelylinearresponseof climateto thisnonlinearforcing
will producea paleoclimatic
recordthatcontainsthesame
735
nonlinearinteractionsas the forcing. The responsewill also
be linearlycoherentwith theforcingat theforcingfrequencies.
On the otherhand,a nonlinearresponseof climateto this
nonlinearforcingwill produceresponses
thatare not
necessarily
coherentwith oscillationsat the samefrequencies
as the forcing. In the next section,threerecordsof
paleoclimateareexaminedto determinetheextentof linearand
nonlinearresponses
to orbitalforcing.
ANALYSIS OF •5180RECORDS
The relativelyshortrecordlengths(typically400-800 kyr) of
mostpaleoclimatictime serieshasprecludedapplicationof
bispectralanalysisin thepast. Varianceof bispectral
estimatesarehigherthanpowerspectralestimates,andshort
recordsdo nothaveenoughdegreesof freedomto yield
statisticallystableresults.Recentefforts,however,have
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
10-3
0.02
0.03
0.Ol
,I
10-4
0.00
0.04
0.08
0.04
0.12
0.08
0.12
f l (cycles/kyr)
f (cycles/kyr)
I
417
25
12.5
kyr
8.3
417
25
12.5
kyr
Fig. 6. Simulatedtime seriesgeneratedby replacingthephasesin theJuly65øN insolationrecord(Figure5a) with
randomphases.Powerspectralandbispectral
estimates
wereobtainedasdescribed
in Figure5: (a) timeseries,(b)
powerspectra,and (c) significant(1-at=0.90) contoursof bicoherence.
736
Hagelberget al.' LinearandNonlinearCouplings
DuringtheLateNeogene
apparentnonstationarity
in therecord,all recordsareanalyzed
in two parts,from 1.0 to 0 Ma and from 2.6 to 1.0 Ma (2.8 to
1.0 Ma at site 607). The recordsare dividedat 1.0 Ma to
obtainthelongestlatePleistocenerecordpossible.The
preciselocation(within+/- 100kyr) wheretherecordsare
divideddoesnot changetheresultssignificantly.
change[Shackletonand Hall, 1989;Ruddimanet al., 1989;
Raymo et al., 1989]. The bispectmmof threerelativelylong
•5180
records
fromtheNorthAtlantic
andeastern
equatorial
Pacific are now examined.
The recordsexaminedhereweretakenfromOceanDrilling
Projectsite677 (easternequatorialPacific,1ø12•1,83ø44'W)
andDeep SeaDrilling Projectsite607 (NorthAtlantic,
41ø00'N,32ø58'W). At site677, time seriesof plankticand
Sothateach•5180
timeseries
could
beprocessed
identically,the time seriesfrom site607 wasoverinterpolated
to matchthe samplingresolutionof the site677 data. While
theNyquistfrequencyof the 607 time seriesis at 7 kyr (as
opposedto 4 kyr in the caseof the site677 records),this
oversamplingallowsfor directcomparisonof the threedata
setsafter processing.For the intervalfrom 1.0 to 0 Ma, all
smoothedspectralestimateswereobtainedby averagingfour
128point(256 kyr) ensembles
for 6 degreesof freedom.For
benthic
•5180wereexamined,
andatsite607,benthic
•5180
time serieswere examined. The site677 recordwasgenerated
andinitially studiedby ShackletonandHall [1989] and
Shackletonet al. [1990]. The site607 recordwasgenerated
andstudiedby Ruddimanet al. [1989] andRaymoet al.
[ 1989]. For eachof thesestudiestheauthorshavedevelopeda
chronologyfor theserecordsby makinguseof orbitaltuning
procedures.
Differencesbetweenthetimescaleusedby
the interval from 2.6 to 1.0 Ma, smoothedestimateswere
Shackletonet al. [1990] and Ruddiman et al. [1989] and
Raymo et al. [1989] are substantial.The issueof
obtainedby averagingsix 128-pointensembles
for 10 degrees
of freedom.In eachcasethe frequencyresolutionis 0.0039
cycles/kyr,anda Harmingwindowwasusedto reduceleakage.
(Resultsusingthismethodof spectralanalysiswerenot
significantlydifferentthanresultsobtainedusingthe
Blackman-Tukeytruncatedlag (autocovariance)
method.)
For the intervalfrom 1.0 to 0 Ma, thepowerspectrumof
each•5180recordhasvarianceconcentrated
at theMilankovitch
frequencies
of 0.012, 0.023, 0.043, and0.053 cycles/kyr,
corresponding
to the 100,41, and23, and19 kyr bands,
respectively(Figure8a). While thesite677 recordsshow
significantvarianceconcentrated
at theprecessional
frequency
of 19 kyr, thispeakis not prominentin the site607 dam. The
chronologies
andapproaches
to orbitaltuningis complicated,
andfor consistency,
only themostrecenttime scaleof
Shackletonet al. [1990] is used here. This time scale has been
appliedto boththe677 and607 records[Shackleton
et al.,
1990]. Althoughit appearsthatthe site677 andsite607
bispectrapresentedherearerobustto slighttime scalechanges,
thepotentiallycriticaleffectsof time scalevariabilityon
bispectmmestimateswill be treatedin a later paper.
The time seriesof planktic(Globeriginoides
ruber)and
benthic
(primarily
Uvigerina
senticosa)
•5180
fromsite677,
andbenthic
(Cibicidoides)
•5180
fromsite607areshown
in
planktic
•5180
timeseries
forsite677contains
additional
Figure7. The recordsfrom site677 are approximately2.6
m.y. in length,while the site607 recordis approximately2.8
m.y. long. The averagesamplingintervalfor the 677 datais
2000 years,andtheaveragefor the607 datais 3500 years.
It is well known that a changein the characterof the
Neogeneglobalice volumerecordoccurrednear 1 Ma [Pisias
andMoore, 1981;RuddimanandRaymo, 1988]. Oscillations
at around100 kyr, whicharegenerallynotprevalentprevious
to 1 Ma, becomevery strongin the late Pleistocene.This is
clearlyseenby comparingthepowerspectraof theserecords
before(Figure8a) andafter (Figure8b) 1 Ma. Becauseof this
varianceat periodsof 15 kyr and 12kyr, whichcorrespond
to
sumfrequencies
of orbitalprecession
andobliquity,andthe
site607 time seriescontainssignificantvariancenear15 kyr
as well.
For the interval from 2.6 to 1.0 Ma, the total variance
contained in each time series is lower than in the interval from
1.0 to 0 Ma. Each time series shows variance concentrated
primarilyat 41 kyr (0.023 cycles/kyr)(Figure8b). Lowfrequencypower,dominantfrom 1.0 to 0 Ma, is diminished
relativeto 41 kyr power. In eachtime series,low-frequency
-2
o
o
-
Plonkfic
677
-
Ben'!'hic
677
_
_
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time (Mo)
Fig.7.Timeseries
ofplanktic
(G.ruber)
andbenthic
(primarily
Uvigerina)
•5180
fromsite677[Shackleton
and
Hall,1989]andbenthic
(Cibicidoides)
•5180
fromsite607[Ruddiman
etal.,1989;Raymo
etal.,1989].The607
dataare offsetfrom actualvaluesto allow visualcomparison.
Hagelberg
etal.:Linear
and
Nonlinear
Couplings
During
theLate
Neogene
737
varianceis distributed
between400 kyr and71 kyr, ratherthan
concentrated
at 100kyr. In thisolderinterval,thesite677
41 kyrand23 kyrbands
witheachisotopic
timeseries,
andin
the19kyrbandin thesite677 timeseries(Figure10and
planktic
•5180
record
alsoshows
power
at23kyr. This23
Table1). Thecoherence
is notasstrongasin theinterval
kyr peakis alsopresentin thebenthicrecords,
although
to a
from 1.0 to 0 Ma, however. In eachtime series,thereis
lesser
extent
thaninthesite677planktic
•5180.
significant
coherence
withinsolation
at frequencies
lowerthan
100kyr. In addition,
thesite607timeseriesshowshigh
Coherence
andphasespectra
between
July65øNinsolation
andeach•518
0 timeseriesfor 1.0to 0 Ma andfrom2.6 to 1.0
Ma aredisplayed
in Figures9 and10andaresummarized
in
coherence
with insolationnear30 kyr. In thistimeinterval,
eachoftheinsolation/•5180
phase
spectra
aresimilar.
Returningto the1.0to 0 Ma records,
coherence
between
Table 1. In the interval from 1.0 to 0 Ma, significant(1-ct =
0.90) coherence
occursat thefourprimaryMilankovitch
insolation
and•5180near100cycles/kyr
isestimated
as0.91,
frequencies
foreachtimeseries.
Thecoherence
spectrum
at site
0.79,and0.89 for the677 planktic,677 benthic,and607
benthicrecords,
respectively.
If theseestimates
represent
the
truevalueof coherence
(theseestimates
arenotcorrected
for a
smallbias),thenbetween62% and82% of thevariancein the
607 is similar to that of both site 677 records,although
coherence
with insolationat 23 kyr is weaker,andcoherency
at
100kyroccursovera widerband.Withtheexception
of the
41 kyr bandin theinsolation/site
607 phasespectrum,
the
phasespectrum
for eachrecordis similar.As notedin the
Introduction,
thisoverallhighcoherence
betweeninsolation
õ!80 record
islinearly
related
tovariations
indirect
eccentricity
forcing.For677plankticandbenthicdata,the
phase
of theresponse
changes
by closeto90ønear100kyr,
andforeachrecordthegainat 100kyr isrelativelyhigh(Table
1). Theseresultsareconsistent
witha response
at 100kyr
and•5180
provides
evidence
fora strong
linear
relationship
betweeninsolationforcingandthepaleoclimatic
response.
which is linear and resonant. This is discussedbelow.
Coherencewith insolation is weaker in the interval from 2.6
Thebicoherence
spectra
foreach
•5!80record
overthe1.0to
to 1.0 Ma. Significant
coherence
withinsolation
occursin the
(cd
lO 0
10-•
10-2
0.04
ooo
0.08
0.12
0.00
f (cycles/kyr)
I
417
I
I
I
25
I
12.5
0.04
0.08
0.12
0.00
0.04
f (cycles/kyr)
f (cycles/kyr)
I 81
.3
I
417
I
I
I
25
(kyr)
I
12.5
0.12
0.08
I
I
8.3
I
2
417
8•.3
12.5
(ky)
(kyr)
101
]o o
10-•
10-2
0.00
i
i
i
0.04
i
0.08
I
i
O.12
I
0.00
f (cycles/kyr)
417
25
(ky)
12.5
I
I
0.04
I
I
0.08
I
I
O. 12
i
0.00
417
25
12.5
i
I
0.08
I
I
O. 12
f (cycles/kyr)
f (cycles/kyr)
8.3
i
0.04
8.3
417
25
12.5
8.3
(ky)
Fig.8.(a)Power
spectra
ofisotopic
records
fortheinterval
1.0to0Ma:site
677planktic
õ18
0 (left),
site
677
benthic
õ180(middle),
andsite607benthic
õ180(fight).Estimates
having
6 degrees
offreedom
wereobtained
by
using
aHanning
window
andaveraging
four256kyrsubrecords.
(b)Power
spectra
fortheinterval
2.6to1.0Ma:
site677planktic
õ18
0 (left),
site
677benthic
õ18
0 (middle),
andsite
607benthic
õ18
0 (fight).
Estimates
having
10degrees
offreedom
wereobtained
using
a Hanning
window
andaveraging
six256kyrsubrecords.
738
Hagelberget al.: LinearandNonlinearCouplingsDuringtheLateNeogene
shown(compareto Figure5c, thebicoherence
spectrum
for a
4.096 m.y. record). The estimatesfor the 4 m.y. recordshown
in Figure5c have 14 dof, and the 0.90 significancelevel is
0.58. The 1 m.y. record(Figure 1ld), on theotherhand,has6
dof, andthe0.90 significance
level is 0.88 (bicoherences
reportedbeloware significantat a 0.80 level). Owing to the
0 Ma intervalare shownin Figures1la-1 lc. Smoothed
bispectralestimates
werecalculatedin thesamemannerasthe
corresponding
powerspectralandcross-spectral
estimates.
Bispectmcalculatedfromshortrecordshavecoarserfrequency
resolution,as is clearlyillustratedin Figure 1ld, wherethe
bicoherence
spectrumof 65øN insolationfor a 1 m.y. recordis
Insolation /
677 Planktic
Insolation /
677 Benfhc
Insolation /
607 Benfhc
1.0
0.8
V
0.6
0.4
0.2
I
VV
I
I
0.04
0.00
0.08
O.
O.
0.12
0.00
,,
I
I
'/ V
I
2
O.
2
2OO
0
-100
-200
A
IV,,-,
0.00
0.04
0.08
0.04
I
25
O. 12
O.00
0.04
f (cycles/kyr)
f (cycles/kyr)
417
0.08
12.5
8.3
41 7
I
I
I
I
25
0.08
O. 12
f (cycles/kyr)
I
12.5
I
8.3
I
417
I
I
I
I
25
(kyr)
I
12.5
8.3
(kyr)
Fig.9.Coherence
(top)andphase
(bottom)
between
65øNinsolation
and8180,fortheinterval
from1.0to0 Ma:
site677planktic
8180(left),677benthic
8180(middle),
andsite607benthic
8180(righ0.The90%significance
level for coherence(horizontalline) is 0.73. The 90% confidenceintervalfor phaseestimatesis indicatedby vertical
bars.Positive
phase
indicates
insolation
leads
5180.Thedatawereprocessed
asdescribed
inthecaption
toFigure
ga.
Insolation /
677 Planktic
Insolation /
677 Benfhic
Insolation /
607 Benfh
1.o
0.8
0.6
FVvv'
0.4
0.2
I
I
0.00
i-"
0.04
0.08
0.12
0.00
I
,
I
0.04
0.08
O.
0.00
0.04
0,08
O.
0.04
0.08
O.
0.00
0.04
0.08
0.12
lOO
o•
o
c- -100
-200/
t
0.00
i
i
0.04
i ,
•
, i
0.08
i•
0.12
0.00
f (cycles/kyr)
f (cycles/kyr)
417
25
12.5
8.3
417
25
12.5
(kv)
f (c¾cles/kyr)
8.3
2s
2.s
(kyr)
Fig.10.Coherence
(top)andphase
(bottom)
between
65øNinsolation
and5!80,fortheinterval
from2.6to1.0Ma:
site677planktic
5180(left),677benthic
5180(middle),
andsite607benthic
5180(right).The90%significance
level for coherence(horizontalline) is 0.61. The 90% confidenceintervalfor phaseestimatesis indicatedby vertical
bars.Positive
phase
indicates
insolation
leads
5180.Thedatawereprocessed
asdescribed
inthecaption
toFigure
8b.
Hagelberg
etal.: LinearandNonlinear
Couplings
During
theLateNeogene
low degrees
of freedomfor thesetimeseries,thebiasof the
observed
bicoherences
is notnegligible[ElgarandSebert,
1989]. Althoughthebicoherence
valuesreportedin this
sectionhave not been correctedfor bias, sucha correction
wouldnotsignificantly
changetheoverallresultspresented
here.
For the interval from 1.0 to 0 Ma at site 677, statistically
significant
(1-• =0.80) bicoherences
indicatephasecoupling
amongcomponents
whicharealsocoupledin theinsolation
record,aswell asamongcomponents
whicharenotcoupledin
theinsolation
record.
Inboththeplanktic
andbenthic
•5180
records,
phasecouplingoccurs
betweenoscillations
at 0.012
cycles/kyr
(100kyr band),0.043cycles/kyr
(23 kyr band),and
0.053cycles/kyr(19 kyr band). For theplankticrecord,
b(0.043,0.012) = 0.794,andfor thebenthicrecord,b(0.043,
0.012) = 0.792. The phaserelationship
betweenthesemodes
haschanged,
however,froma biphase
of 0øin theinsolation
forcingto a biphaseof-90 øin thebenthicdataand-114øin the
plankticdata(thesebiphase
valuesof -90øand-114øarewithin
739
an 80% confidence interval of one other and thus are not
statistically
different).Therearealsostatistically
significant
couplings
in the site677 recordsbetween23 kyr band
variationsand41 kyr (b(0.043,0.023)=0.840in theplanktic
recordandb(0.043,0.023) = 0.741 in thebenthicrecord).
This phasecoupling,alsopresentin theinsolationforcing,
indicatesthatoscillationsat periodsof 23 kyr and41 kyr are
coupledto oscillationsat 15 kyr. Finally, statistically
significantphasecouplingis presentbetween41, 100, and30
kyr oscillations(b(0.023,0.012) = 0.723 in theplankticrecord
andb(0.023, 0.012) = 0.772 in thebenthicrecord). This last
interactionis not presentin the insolationforcingandcould
representpartof a nonlinearresponse
in theclimatesystem.
The bispectmmfor site607 is differentthanthatfor site677
from 1.0 to 0 Ma. Statisticallysignificantphasecoupling
occursamongtwo triadsof frequencies,
(0.051,0.016,0.067)
and(0.051, 0.043, 0.094), with bicoherences
of 0.520 and
0.766,respectively.
While the lattertriadis alsopresentin the
insolationbispectmm,thisphasecouplinginvolveshigher
TABLE1. Coherence,
Gain,andPhase
ShiftBetween
Insolation
and•5180
Response
Near100,41,
23, and 19 kyr Periodsfor theIntervalsfrom 1.0 to 0 andfrom 2.6 to 1.0 Ma
Record
Frequency,
cycles/kyr
Period,
kyr
Coherency
Gain
Phase,
deg
0-1 Ma
677 planktic
0.012
0.023
0.043
0.053
--100
-- 41
-- 23
-- 19
0.9076
0.9877
0.9504
0.9077
9.220
0.488
0.134
0.087
154
79
105
136
677 benthic
0.012
0.023
0.043
0.053
--100
-- 41
-- 23
-- 19
0.7903
0.9677
0.9529
0.9301
7.739
0.429
0.124
0.074
170
95
96
167
607 benthic
0.012
0.023
0.043
0.053
--100
-- 41
-- 23
-- 19
0.8948
0.9755
0.7437
0.8320
9.096
0.448
0.070
0.032
- 142
158
75
-92
1-2.6 Ma
677 planktic
0.0101
0.0257
0.0452
0.0570
-•100
-- 41
-- 23
-- 19
0.2409
0.9181
0.9141
0.7407
1.221
0.342
0.096
0.055
-40
148
103
5
677 benthic
0.0101
0.0257
0.0452
0.0570
-100
N 41
N 23
- 19
0.2493
0.8065
0.6275
0.8283
1.154
0.415
0.053
0.859
-48
179
101
67
607 benthic
0.0101
0.0257
0.0452
0.0570
-100
.--41
- 23
N 19
0.3197
0.8949
0.7807
0.3658
1.334
0.508
0.074
0.309
-63
171
93
-93
740
Hagelberget al.: LinearandNonlinearCouplingsDuringtheLateNeogene
frequenciesthatare closeto theNyquistfrequencyfor thistime
series. With this time series,it is particularlynotablethat no
significantphasecouplingoccurswith oscillationsat 41 kyr
or 100 kyr, as in the site677 records.For instance,while
powerspectralpeaksarepresentat 100,41, and 15kyr in the
site607 data,the bicoherencefor this triad is not significantat
a 0.80 level. In the site 677 data, on the other hand, the
bicoherence
for thistriad is statisticallysignificant.While the
cause
ofthisdiscrepancy
between
Atlantic
andPacific
•5180
•-
0.06
_x
0.05
datasetswarrantsfurtherstudy,the low bicoherences
in the
site607 datamay be an artifactof the low degreesof freedom.
At site677, evidencefor a nonlinearresponse
of theclimate
systemto orbitalforcingis givenby high bicoherence
in the
(0.023, 0.012, 0.035) triad. However, the resultsat site 677
are alsonoteworthybecauseof thesimilarityof thephase
couplingsin the climateresponseto thosein the orbital
•
0.04
forcing.
Thephase
coupled
oscillations
present
in•518
0 time
-•
0.03
o
0.02
series from this site includes several of the same modes that
•0.01
are coupledin the forcing.This is consistentwith a linear
responseof the climate systemto insolationforcing. Changes
in thephaserelationships
betweeneachmodeoccur,however,
becausethebiphasesfor everycoupledtriadin thedataare
i
0.04
0.08
0.12
close to -90 ø, whereas in the insolation record the
corresponding
biphases
arenear0ø.
0.06
0.06-
0.05
0.04
0.05
0.04
_
0.03
0.03
0.02
•0.01
0.02
0
<•
0.04
o
•0.01
0.08
i
O.
I
I
0.04
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
c• 0.01
I
0.08
0.12
-(b)
_
o
o
•0.01
0.04
0.08
0.12
0.04
0.08
0.12
0.06
0.06
0.05
-•
0.05
0.04
•
0.04
•
0.03
0.02
o.o3
o
•0.01
o
0.02
o40.01
0.04
0.08
0.12
f l (cycles/kyr)
O.04
0.O8
0.12
f l (cycles/kyr)
Fig. 11. Contoursof bicoherence,for the intervalfrom 1.0 to
0 Ma. The datawereprocessed
asdescribed
in thecaptionto
Figure 8a. The minimumbicoherencevaluecontoured
(significantat a 80% level) is 0.73 with a contourintervalof
Fig. 12. Contoursof bicoherence,for the interval from 2.6 to
1.0 Ma. The datawereprocessed
asdescribed
in thecaptionto
Figure 8b. The minimumbicoherencevaluecontoured
(significantat a 80% level) is 0.57 with a contourintervalof
0.05:(a)site677planktic
•5180,
Co)site677benthic
•5180,
(c)
site607benthic
•5180,
and(d)65øNinsolation.
0.05:(a)site677planktic
•5180,
(b)site677benthic
•5180,
and(c)site607benthic•5180.
Hagelberget al.: LinearandNonlinearCouplingsDuringtheLate Neogene
741
As with the radiation time series above, simulations with
In summary,
bispectral
estimates
forsite677•5180
data
synthetictime seriescanbe usedto reinforcetheseresults.
Synthetictime serieswere generatedfor the site677 and site
indicatethat from 1.0 to 0 Ma, phasecouplingssimilarto
thosepresentin the insolationtime seriesoccur. This is
consistentwith a linearresponse
to insolationforcing. This
resultis notseenin the 1.0to 0 Ma benthic•5180time series
607 •5180dataoverthe 1.0 to 0 Ma timeinterval.The
amplitudespectrumof eachtime serieswaspreserved,but the
observedphaseswerereplacedwith randomphases.For each
record,10 simulationsweremade,andthebispectmm
calculatedfor each. At site677, althoughsomesignificant
bicoherences
occurredin the plankticdata(throughrandom
chance,expectedat an 80% level), no simulationshowed
significantbicoherenceat the samesetsof triadsthatare
coupledin the insolationforcing. For the site 677 benthic
data,bicoherences
from only onein 10 simulationsresemble
thosein the insolationdata. At site 607, significant
from site 607, however. From 2.6 to 1.0 Ma, evidencefor a
linear
response
ofthe•5180
record
isnotasstrong
ateither
location. Althoughall of the time serieshavevery low
numbersof degrees
of freedom,andcorrespondingly
high
nonzerosignificance
levels,MonteCarlosimulations
suggest
thatthesignificantbicoherences
observedarenotcausedby
random fluctuations.
EVOLUTION
OF THIRD
MOMENTS
bicoherencessimilar to the data occur in two simulations,
consistent
with an 80% significancelevel. Theseresultsadd
strengthto theargumentthatdespitethelow degreesof
freedomof theserecords,thecouplingsobservedin thedataare
indeedsignificant
andarenotlikelyto be caused
by random
fluctuations.
Bicoherences
for thesite677 and607 •5180timeseriesfor
the intervalfrom 2.6 to 1.0 Ma are shownin Figure 12.
As notedabove,the "mid-Pleistocene
transition"
of global
•5180
records
fromamainly
41kyroscillation
intheearly
Pleistoceneto a 100 kyr "sawtooth"shapeoscillationin the
late Pleistocene
is well known. Subsequent
modelsof
paleoclimatehavesoughtto reproducethisasymmetric
response[Saltzmanand Maasch, 1988]. The evolutionof
skewness
andasymmetry(normalizedthirdmoments,as
Bicoherences
fromthesite677planktic
5180datashow
one
defined
above)
ofthesite677andsite607•5180
timeseries
are
regionof statisticallysignificant(1-a = 0.80) phasecoupling
thatis similarto the insolationforcing(the triad consistingof
oscillationsat 19, 41, and 12 kyr hasb=0.662). This same
now examined.Thesethird-momentquantitiesarerelatedto
the shapeof the time series(asshownin theexampleabove)
andaregivenby theintegratedreal andimaginarypartsof the
bispectrum,respectively[Elgarand Guza, 1985;Elgar, 1987].
The third momentsallow the observedlong-termevolutionof
phase
coupling
ispresent
inthesite677benthic
•5180,
where
b=0.603. In the benthic 677 data, two other triads have
significantbicoherences
thatarenot similarto coupledmodes
in the insolationforcing,b(0.047, 0.016) = 0.620 and
b(0.078, 0.027) = 0.606. At site607, the only significant
phasecouplingoccursamonghigh-frequency
components
(b(0.055, 0.051) = 0.599).
the5180record
tobequantified.
The evolutionof the skewnessand asymmetryof the site
607andsite677•5180
records
isshown
inFigure
13.
stronglysupporta linearresponseof the paleoclimaterecordto
insolationforcing. Only onecoupledtriad that is presentin
theinsolationrecordhashighbicoherencein the site677
records.Orbitalfrequencies
thatcontainmostof thevariance
Skewnessandasymmetryarecalculatedfor overlapping512point records,that is, from 1.0 to 0 Ma, from 1.5 to 0.5 Ma,
from 2.0 to 1.0 Ma, and from 2.5 to 1.5 Ma (becauseit is 2.8
m.y. long, for site 607 estimatesfrom 2.8 to 1.8 Ma are
included).Thereis an increasein theasymmetry
(sawtoothness)
of the timeseriesfrom 2.5 Ma to presentanda
simultaneous
decrease
in theskewness
(peakedness)
of the
record. In theoldestpartof therecords,thereis relativelyhigh
inthe•5180
power
spectrum
for2.6to1.0Madonotappear
to
skewness which decreases to near zero in the late Pleistocene.
be as stronglycoupledto insolationas in the 1.0 to 0 Ma
interval. Oscillationshavinga 62-70 kyr period,presentin
MonteCarlo simulationssuggestthatthesethird-moment
estimates
are significantlydifferentfrom zero.(Recallthatfor
no coupling(a Gaussiantime series)the thirdmomentsare
Unlike the interval from 1.0 to 0 Ma, the results from the
lower
Pleistocene
andupper
Pliocene
•5180
timeseries
donot
the•5180
power
spectra
for2.6to1.0Ma,arephase
coupled
to
19kyroscillations
inthesite677benthic
•5180
data.As
thereis no coherence
betweeninsolationand•5180at this
frequency(Figure10), the62-70 kyr peakmayrepresent
a
nonlinear(differenceinteraction)
between15 kyr and 19 kyr.
Together,theseobservations
indicatea changein thenatureof
the climateresponsebetween1.0 to 0 Ma and 2.6 to 1.0 Ma.
Similar to the results for the 1.0 to 0 Ma interval, numerical
simulationswith synthetictime serieshavingrandomphases
suggest
thattheobserved
bicoherences
(Figure12) werenot
causedby randomfluctuations.At site677 for bothplanktic
andbenthic
•5180data,bicoherences
ofthesimulated,
random
phaserecordsweresimilarto bicoherences
of thedatain only2
of 10 simulations.At site 607, only one simulationhad
phasecouplingsimilarto thatobservedin the data. In no case
did thesignificantbicoherences
resemblethoseof the
insolation data. Thus, as in the 1.0 to 0 Ma interval, these
simulationsaddsupportingevidencethatthephasecouplings
suggested
by bispectralanalysisarenot dueto random
fluctuations.
zero.)
Thereare largedifferencesbetweenthe skewness
and
asymmetry
ofthe•5180
records
andthatoftheinsolation
record
(Figure 14). The asymmetryof the insolationrecordis close
to zerothroughout
the past2.5 m.y., andskewness
is only
slightlygreaterthanzero. Thus,althoughtheclimatesystem
is respondinglinearlyto insolationforcing(at leastover the
lastmillion years),the natureof the phasecouplingwithinthe
systemis evolving.This observationis consistentwith the
resultsof Pisias et al. [1990], who showedevidencefor a
nonconstant
phasebetweenorbitalforcingandice volumeover
the past700,000 years.
DISCUSSION
The analysesabovedemonstrate
thattheorbitalparameters
whichcombineto causelong-termchangesin solarinsolation
arenonlinearlycoupledto oneanother.In particular,the
couplingbetweenprecession
termsof 23 kyr and 19 kyr and
742
Hagelberg
etal.: LinearandNonlinear
Couplings
During
theLateNeogene
•,
1.0
x
x
0.8
x
,-
(577 planktic
.-
677
benthic
.-
607
benthic
1.0
-•-
0.6-
0.4
0.4-
0.2
0.2-
0.0
0.0-0.2
o.o
1 .o
65øN
insolation
average
0.8-
0.6
-0.2
ß-
-
-
I
I
o.0
2.0
I
J .o
I
2.0
Time (Ma)
Time (Ma)
Fig. 13. Skewness(solidline) andasymmetry(dashedline) of
Fig. 14. Skewness(solidline) andasymmetry(dashedline) of
•5180
records
versus
time.Eachsymbol
represents
a 1m.y.
theaverage
ofthethree
•5180
records
presented
inFigure
13
averageandis plottedat themidpointof the timeinterval.
(circles),andskewness
andasymmetryof 65øNinsolation
(asterisks)versustime.
Squares
indicate
677planktic
•5180;
triangles,
677benthic
•5180;
andcircles,
607benthic
•5180.
F
the 100kyr eccentricity
parameteris very strong.With respect
to this coupling,it is importantto notethatthe strengthof
theprecession-eccentricity
couplingin theinsolationforcingis
related
tothepresence
of the1/(1-e
2) termin theinsolation
formulausedin this study. Only the solutionof Berger
[1978b] includesthis term. Previousinsolationsolutions
havenot includedit [seeBerger,1978a]becauseit doesnot
contributesignificantlyto the overallinsolationchange
(around0.1%). When thisterm is removedfrom the
insolationformulaeof Berger[1978b],thephasecoupling
betweenprecession
andeccentricity
is weaker.Recallingthat
theonlyorbitalelementthatcanmodifythetotalsolarenergy
receivedby theEarthis eccentricity(althoughforcingat 100
kyr is smallin comparison
with precessional
or obliquity
forcing),thistermis veryimportant,asemphasized
by Berger
[1978a]andBerger[1989].
The dilemmaof whetherthe 100 kyr peakpresentin the
X TM
m(to2o-to2
+ iYto)
(4)
wherex is the displacement
(i.e., response),
F is the force
drivingthe oscillator,m is the massof the oscillator,tOois
thenatural(resonant)frequencyof the system,tOis the
frequencyof theforcing,and), is thedampingforce. For the
caseof zerodamping,the magnitudeof x is relatedto thesize
oftheforcebythefactor1/(tOo
2-tO2),
andastOapproaches
tOo,thismagnitudeapproaches
infinity. In thiscontext,the
gainof the systemresponseis
02_
1
m2[(
tO:z_
tO
o2)2,
+'Y20
)2]
(5)
Thecorresponding
phaseshiftof thesystemresponse
is given
by
•518
0 spectrum
during
thelatePleistocene
isa linear
response
0=arctan
(-togt0z)
to eccentricityforcing,or a nonlinearresponseto precession
(6)
band interactions is now addressed. The insolation data
indicatethateccentricity
andprecession
bandsarehighly
coupledin theinsolationrecord.The linearcoherence
between
insolation
and•5180
intheeccentricity
bandisgreater
than
0.90, andthe samephasecouplingsarepresentin theisotopic
record. Theseresultssuggestthata nonlinearresponseis not
requiredto explaintheobserved
response.A simplelinear
resonance
modelmay help to explaintheprocesses
leadingto
the dominanceof 100 kyr oscillations
in the latePleistocene
ice volume record.
A brief review of resonance demonstrates that the results
reportedhereareconsistent
with a resonantresponse
to
insolationforcingin the 100 kyr band. The simplephysical
systemof a forcedoscillatorwith dampingis
The gainandphasechangeof thesystemresponse
for
differentvaluesof tOare shownin Figure15.
For any amountof damping0'), maximumgain occurs
wheretOequalstOo.In addition,a phasechangeof up to 180ø
occursas tOchangesfrom tO> tOoto tO< tOo. If solar
insolationis theforcing,andtheclimatesystemresponse
is
beingmeasured,thenas a smallamountof insolationis
appliedto this systemat tO= 0.01, theclimatesystem
response
becomesgreatlyamplifiedbecauseof theproximity
of tOto tOo,thenaturalresonantfrequencyof theclimate
system.
Thegainof the100kyrbandin theinsolation
- •5180cross
spectra
is aboutan orderof magnitude
higherthanthegainin
Hagelberget al.' LinearandNonlinearCouplings
DuringtheLateNeogene
the41 kyr band,andabout2 ordersof magnitudehigherthan
the gain in the 23, or 19 kyr bands(Figure 15 and Table 1).
In addition,thephasespectra(Figures9a and 16) show
evidencefor a shiftwhichcorresponds
to thatpredictedby
linearresonance.Note, however,thatin this studyJuly 65øN
insolationvaluesareused,andthephaseof theresponsein the
precession
banddependson the particularmonthchosen.The
(•)
743
smallamountof eccentricityforcingat 100kyr (approximately
0.1% of the total variancein theinsolationand approximately
3 ordersof magnitudelessthanthevarianceof precessional
forcing)mayoccurat or neara resonantfrequencyin the
climaticsystem,andthe largeamplituderesponse
observedin
the5180 recordresultsfromthisresonance.
Theabsence
of
sucha resonantresponsein the latePlioceneandlower
104
i•'---low
damping ,•- 677benthic
10 3
C)-
607
benfhic
/i• high
damping
10 2
101
100
10-1
I
10-2
0.01
i
0.02
coø
0.03
0.04
0.05
0.06
0.05
0.06
Frequency
180
160
low damping
140
120
_
high damping
m 100
.
Q_ 80
6O
40
2O
_
o
1
0.o0 O.•o
0.02
o.o3
0.04
Frequency
Fig. 15. (a) Gain versusfrequencyfor theresonance
modeldescribed
in the text (equation(4)). Solidcurvesindicate
the gain of the systemresponsefor differentvaluesof damping,¾. The verticaldashedline at toorepresentsthe
resonant
frequency
ofthesystem.
Symbols
plotted
aretheestimated
gainof5180fromTable1. Squares
indicate
677planktic
5180;triangles,
677benthic
5180;andcircles,
607benthic
5180.(b)Phase
versus
frequency
forthe
resonance
modeldescribed
in thetext(equation(4)).
744
Hagelberget al.' Linear andNonlinearCouplingsDuringtheLate Neogene
180
160
140
120
IO0
8060402000.00
c•o
0.02
0.04
0.06
Frequency
Fig. 16. Phaseshiftversusfrequencyfor theresonance
modeldescribed
in thetext (equation(4) andFigure15b),
compared
to•i180phase
fromTable1.Themodel
phases
havebeenshifted
by65øtoallowcomparison
ofthemodel
tothedata.Symbols
plotted
aretheestimated
phase
of•i180fromTable1.Squares
indicate
677planktic
•i180;
triangles,
677benthic
•i180;andcircles,
607benthic
•i180.Vertical
barsindicate
90%confidence
intervals
forthe
phaseestimates.
Pleistocenemay signifya changingresonantfrequencyasthe
climate systemevolves.
Linearresonanceis a straightforward
methodof accotinting
for the observedclimate responseof the past 1 m.y. As noted
in the Introduction,paleoclimatemodelshaveexploreda
varietyof mechanisms
accountingfor thisclimaticresponse.
The observations
presented
heresupporttheresultsof
Saltzman and Sutera [1984] and Saltzman et al. [1984] that
imply an inherentlysensitiveclimatesystemat the 100 kyr
periodduringthe late Pleistocene.The observations
also
supportsBerger's[1989] statementthat the influenceof
eccentricityforcing alone is very important. While it is likely
that a rangeof mechanisms,
bothlinear andnonlinear,are
actingto producethe observedclimateresponse,the data
presentedhereare consistentwith a resonantresponse.Models
which simulateclimatic changesmustbe consistentwith these
observations.
This analysishasalsodemonstrated
differencesbetweenthe
climaticresponsein the late Pleistocene,whichmay be
primarilya linearresponse
to insolationforcing,andtheearly
Pleistocene/Pliocene,
whichmay includea nonlinearresponse.
The observations,includingthe evolutionof third moments,
suggestthereis an evolutionin the climaticresponse.
Finally, it is necessaryto considertheseresultsin light of
someimportantassumptions
thathavebeenmadeto carryout
theaboveanalysis.It hasbeenassumedthata correcttime
scale has been used. While it is known that different time
scaleswill producedifferentresults,it is unknownhow
sensitivebispectralestimatesare to errorsin the time scale. It
has also been assumed that the above results are not an artifact
of theparticularorbitaltuningstrategyusedby Shackleton
et
al. [1990]. This importantissueis left to a separatestudy.
CONCLUSIONS
Bispectralanalysiswasusedto detectnonlinearphase
couplingsin the time seriesof long-termsolarinsolation.
Orbital eccentricityandprecession
are highlynonlinearly
coupled,asareprecession
andobliquity.The quadraticphase
couplingspresentin the insolationforcinghavebeencompared
tophase
couplings
intheclimatic
timeseries
of•5180,
a
globalice volumeproxy. Power spectra,crossspectra,and
third moments(skewnessandasymmetry)of the datasuggest
an evolutionin the climate systemresponsefrom late Pliocene
to late Pleistocene. Quantitativeestimatesof this evolution
areseenin increasing
asymmetry(sawtoothness)
and
decreasing
skewness
(peakedness)
from 2.6 to 0 Ma at site677
and site 607. From 2.6 to 1.0 Ma, the samephasecouplings
that are observed in the insolation record are not observed in
the oxygenisotoperecords.From 1 to 0 Ma, benthicand
planktic
•5180fromODPsite677contain
thesame
significant
bicoherences as the orbital time series. The observed
bicoherences,
combinedwith highlinearcoherence
between
orbitalforcingandclimaticresponseat orbitalfrequencies
duringthis time period,indicatethat mostof the climatic
(/5180)
response
toinsolation
forcing
islinear,
including
the
response
at 100kyr.The100kyrpeakinthe/5
!80 spectra
shownhereis consistentwith a linear,resonantresponseof
the climatesystemto directeccentricityforcing.
Acknowledgments.
The cooperation
of Nick Shackletonand
MaureenRaymo in providingthe isotopicdataandthetime
scalesusedin thisstudyis greatlyappreciated.AndreBerger
providednew insolationvaluesandofferedvaluable
suggestions.The carefulcommentsof JohnImbrie andBob
Oglesbyin reviewingthismanuscriptwerevery usefulandare
appreciated.Alan Mix hasprovidedhelpfuland thorough
commentsthroughout.Walter Munk, after hearingof this
studyin its preliminarystages,remarkedthat a linear,resonant
climateresponsewouldexplainthe observations.Steve
Elgar'sresearchis supported
by theOffice of Naval Research
(CoastalSciencesandGeologyandGeophysics)
andthe
NationalScienceFoundation(PhysicalOceanography).This
work was supportedby NationalScienceFotmdationgrant
OCE-90-12270(MarineGeologyandGeophysics)
to Oregon
StateUniversity.
Hagelberg
etal.: LinearandNonlinear
Couplings
DuringtheLateNeogene
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