JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 90, NO. D3, PAGES 5661-5664, JUNE 20, 1985
The Origin of the 100,000 Year Cycle in a
Simple Ice Age Model
ROELOFK. SNIEDER•
NOAA GeophysicalFluid DynamicsLaboratory,
Princeton University, Princeton, New Jersey
A one-dimensional nonlinear ice age model developed by J. Imbrie and J. Z. Imbrie was used to
investigatethe 100,000year cyclein climate records.It was already known that the model could mimic
the 100,000year cycle in the climate response,but it was not clear how this strong responsewas created.
It is shown in this paper that the interferencebetween the spectral componentsin the solar heating of
19,000 years and 23,000 years gives rise to an amplitude modulation, and how this modulation is
convertedby the nonlinearityto a 100,000year cyclein the climateresponse.
1.
INTRODUCTION
dy
The spectral properties of climate records have received
considerableinterest in geophysicalresearch.Geological data
show fairly conclusiveevidencethat the climate variations in
the past have a pronouncedspectralcomponentcorresponding to a cycle of about 100,000 K. (In this paper, time will
always be measured in units of 1000 years, which will be
denoted by K = 1000 years.) These climate variations are
thought to be paced by variations in the orbital parametersof
the earth [see Hays et al., 1976; lmbrie and lmbrie, 1980].
It turns out, however, that the fluctuations in the earth's
orbital parametersgive rise to variationsin the effectivesolar
heating which have dominant periods only of 19, 23, and 41
K, [see Berger, 1977]. (The 19 and 23 K cycle are caused
mostly by the combinedeffectsof eccentricityvariations and
precession;the 41 K cycle is causedby obliquity variations.)
The 100 K cycle in climate recordscannot be explained by
variations in the earth's orbital parametersif the climate respondslinearly to the solar heating. However, in a nonlinear
climate systemthe nonlinearitycould give rise to an energy
transferwithin the spectrum,which could create a strong 100
K cyclein the climateresponse.
Wigley [1976] showedthis explicitly for a simplenonlinear
equilibriummodel which essentiallysquaredthe forcing.Birch-
l+b
dt
Tm
dy
1- b
dt
Tm
(x-
y)
x > y
(x - y)
x < y
(1)
where y(t) is taken as a negative measure of ice volume.
II found a strong 100 K cycle in the climate responsey(t),
by usinga realisticheatingfunctionx(t) which containedmost
energy in cyclesof 19, 23, and 41 K. They found the best fit
with observedclimate variations of the last 150 K by choosing
the following parameter setting: Tm= 17 K, b = 0.6. This
means that the II systemis highly nonlinear. It was, however,
not clear why this nonlinearity gave rise to a very strong 100
K cyclein the response.
The goal of this paper is to clarify the sourceof the 100 K
cycle. In order to do this, the model was integrated using
severaldifferent heating functions x(t). The spectra of the responseswere usedto trace down the origin of the 100 K cycle.
2.
METHODS OF ANALYSIS
The heatingfunctionsthat have beenusedin this studyare
superpositionsof three oscillations:
field[1977]showe•thesamebehavior
bothfora "halfwave
x(t) = A• cos ro•t + A 2 cos ro2t + A 3 COSrO3t
(2)
rectifier" equilibrium model, as well as for Weertman's ice
sheet model. In this paper the 100 K cycle in the model of
Imbrie and Imbrie is discussed limbtie and lmbrie, 1980,
heareafterreferred to as II.]
The model of II is not an equilibrium model; a phase lag
between the radiative forcing and the climate responseis allowed. Their model is simple enough to understand what
mechanismtransfersenergy to the 100 K cycle. II introduce
the nonlinearity by assumingthat ice sheetsdecay faster than
they grow. (In II, several reasonsare given to support this
assumption.)If the solar heatingis denotedby a function x(t),
and if the climate responseis denotedby a functiony(t), then
the ice sheeteffectwasparameterizedby II as
The frequenciesro•, to2,and to3 correspondto periods of 19,
23, and 41 K. Equation (1) was integrated for 700 K; the last
655.5 K were used to determine the spectral density of the
response.The time step and the record length were chosenin
such a way that the spectral density was calculated for
ro = 2r•/655.5 K and multiples of this frequency,so that the
spectral density was evaluated for a period of 109 K. The
parameters Tmand b had the same values as in II, that is,
Tm= 17 K and b = 0.6, except in the last experiment, where
the nonlinearity parameter (b) was varied.
The mean of the response was subtracted in order to
remove undesirablelow-frequencyeffects.Next, the response
was windowed [seeBlackmanand Tukey, 1958] and then Fourier transformed.The plots of the spectraldensity were normalized with respectto the maximum of the spectralcomponents.
x Now at Departmentof TheoreticalGeophysics,
Universityof
Utrecht, The Netherlands.
3.
Copyright1985by the AmericanGeophysicalUnion.
Paper number 5D0095.
0148-0227/85/005D-0095 $05.00
RESPONSE TO HARMONIC
FORCING
FUNCTIONS
The system(1) was first integrated with heating functions
that contained only one frequency component, that is, only
5661
5662
SNIEDER' ICE AGE MODEL
reason
forthisis thatthelinearsuperposition
of twooscil-
lationswithequalamplitudegivesriseto an amplitudemoducos(o•o + 6)t + cos(o•o -- 6)t = 2 cos 6t cosO•ot
0
100
200
300
400
TIME (1000 YRS)
500
600
(3)
Resultingin a modulationwith a period givenby
-
2n
(4)
Fig. 1. Time seriesof the heating (dashedline) and the climate response(solidline) for a harmonicforcingwith a periodof 41 K.
one of the Ai in (2) was differentfrom zero. This will be called
a harmonic heating function. As an example,time seriesof the
heating and the climate responseare shown in Figure 1. In
this casethe heating occurredonly at 41 K. The spectrumof
the climate responseis shown in Figure 2.
The main feature of the spectraof the responsesto harmonic heating functions is the strong peak at the frequency at
which the systemis forced.The spectralenergyof the climate
responseis containedwithin a narrow spectralband (or equivalently, the climate responseis almost sinusoidal)despitethe
fact that the system is highly nonliner. This is becausethe
climate adjusts itself in such a way that the time-averaged
climate
is warmer
than it would
be in the case of a constant
solar heating with the same mean. This has the effect that the
ice sheetspendsabout the sameamount of time "growing"as
it is "decaying," despite the fact that the time constantsfor
growth and decayare quite different(see(1)).
The fact that the responseis not exactly sinusoidalcan be
seenin Figure 2, where the overtonesof the 41 K cyclecan be
seenat 20.5 and 13.67 K. These peaks would be absentif the
responsewere exactly sinusoidal.
The important point of the experimentswith a harmonic
forcing is that the spectrum of the responsecontains essentially only the frequencycomponent of the forcing. This indicates that the strong 100 K responsein the II model is not
causedby any sort of internal resonanceof the system.As
already noted by Le Treut and Ghil [1983], this is becausethe
model of II consistsof only one first-order differential equation.
The equilibriumsolutionof the II systemis givenby Yeq=
x, as can be seen from (1). This means that the transient
solution has a much smaller amplitude than the equilibrium
solution (Figure 1) [see Held, 1982].
4.
RESPONSE TO FORCING FUNCTIONS
CONTAINING TWO FREQUENCIES
If a heating function that containstwo frequencycomponents is used, a more interestingresponseis generated.The
FREQUENCY (2•'/1000 YRS)
0
0.1
0.2
0.3
0.4
0.5
0.6
,
,
i
,
,
I
,
This time will be referredto as the "beat period." It has to be
rememberedthat the function cos 6t changessign twice in
every time interval of length T•, so that the amplitude of the
modulation has a period T•/2. This result also holds if the
amplitudes of the two constitutive oscillationsare not equal,
althoughin that case,the modulation is weaker.
The beat period can be calculatedby taking (3) and defining
T/= 2•r/(oa
o - 6) and T•= 2•r/(Oao
+ 6),sothat
T•- 2T•Tj/IT•- TjI
(5)
A table of the beat period of three different frequencycombinationsthat were studiedis givenbelow'
Case
T•
T2
1
2
3
19
23
19
23
41
41
218.5
104.8
70.8
The time seriesof the heating and the climate responsefor
case 1 is shown in Figure 3; the spectrumof the responseis
shownin Figure 4. The amplitudemodulation is clearlyvisible
in Figure 3. Note that the climate responsedependsonly on
the amplitude of the beat and is insensitiveto the sign of the
modulation. This is causedby the fact that the "growth time"
of the ice sheetsis so much larger than the "decay time." This
has the effect that the climate
becomes warmer
whenever
the
amplitude of the beat increases,and this occurswith a period
of 109 K.
It is thereforenot surprisingthat the most important feature
of the spectrumis the strong spectral component at 109 K.
This componentcorrespondsto an oscillationwith a period of
a half beat time. As already noted, the reasonfor this is that
the climate responseis only determinedby the amplitudeof
the modulation, but not by the sign of the modulation. This
modulation effect is so strong that the 109 K cycle contains
even more energy than the 19 and 23 K cyclesat which the
systemis forced.
Note
that
the II
model
behaves
similar
to the model
of
Wigley [1976] or the "half wave rectifier" model of Birchfield
[1977]. In all thesemodelsthe strong 100 K cycleis generated
because a positive responseis in some sense favored to a
negative response.The only differenceis that the models of
Birchfield and Wigley show a strong responseat the "sum
frequency"(correspondingto a period of 10.4 K), whereasthis
,
0
100
200
300
TIME
218 109
41
23
400
(1000 YRS)
500
600
-
19
PERIOD (1000 YRSI
Fig. 2. Spectrumof the climateresponsein Figure 1.
Fig. 3. Time seriesof the heating (dashedline) and the climate
response(solid line) for a heating containing only oscillationswith
periodsof 19 and 23 K.
SNIEDER: ICE AGE MODEL
spectralcomponentis very weak in the II system.(This spectral componentcanjust be seenat the end of the frequency
scalein Figure 4.) This indicatesthat the treatmentof parametric forcing of Le Treut and Ghil [1983] does not apply
very well to the II system,sincetheir theory predictsthat the
responseat the sum frequencyand the differencefrequency
will be of equal strength.
The spectraof the responseto forcingfunctionscontaining
frequenciesat 19 and 41 K or 23 and 41 K are similar to
Figure 4, exceptthat the spectralcomponentsin the climate
responsedue to the amplitude modulationsare very weak.
5663
100
200
300
400
TIME {1000 YRS)
500
600
-
Fig. 5. Time seriesof the heating (dashedline) and the climate
response(solid line) for the sameheatingas in Figure 3, but with
b = 0.9.
This is becausethe concept of amplitude modulation is only
6.
useful if
CONCLUSION
The nonlinear dynamicalsystemof II has been integrated
«T• >>max(T/,T•)
usingheatingfunctionsthat weresuperpositions
of oscillations
with periodsof 19,23, and 41 K. The climateresponseto these
This condition is certainly not satisfiedfor cases2 and 3 (see
heating functionscontainsessentiallyone frequencyif the
the tabulation above).
heatingis harmonic.This confirmsthat the 109 K cyclein the
responseis not an internal resonanceof the system[see Le
5. VARIATIONS OF THE NONLINEARITY PARAMETER
To see the role of the nonlinearity more clearly, several
Treut and Ghil, 1983].
If the systemis forcedwith a heatingfunctioncontaining
two frequencycomponents,the climate responsehas a pronouncedspectralcomponentthat is determinedby the amplitude modulation of the heating. This effect is particularly
strongif the systemis forcedat 19 and 23 K.
case,"that is, b - 0.9, are shownin Figure 5. Note the nonlinThe 109 K peak in the spectrumof the climateresponseof
ear sawtooth-likeresponse.The shapeof the envelopeof the the II systemappearsthereforeto be causedby an interference
climateresponseis not sinusoidalat all, resultingin a climate of the 19 and 23 K cycles,which givesrise to an amplitude
responsethat containsmany overtonesof the 109 K cycle. modulation.The nonlinearityconvertsthis amplitudemodulaThis has the effect that for b- 0.9 there is less energy in the tion to a 109 K cycle.Justas in Wigley [1976] or in the half
109 K cyclethan for b- 0.6, despitethe fact that the nonlin- wave rectifiermodelof Birchfield[1977], this happensbecause
earity is stronger.
a positive responseis favored to a negativeresponse.Note
This effectcan also be seenin Figure 6, in which the energy that the 19 and the 23 K cyclesare presentbecausethe comin the 109 K cycleis shownas a functionof the nonlinearity bined effectsof eccentricityand precession
lead to a splitting
parameter.For b = 0 the 109 K cyclehas zero energy,which of the precession
cycleinto two cycleswith periodsof 19 and
is only to be expectedfor a linearsystemforcedonly at 19 and 23 K [see Hays et al., 1976]. The nonlinearitythereforeex23 K. For b = 1 the solution of (1) is y(t)= x.... so that in tracts the variations in the eccentricity and transfers the
that casethe 109 K cycledoesnot contain any energyeither.
energyfrom the 19 and 23 K cyclesto the 109 K cycle.This
There clearly has to exist a maximum for intermediateb meansthat it is ultimatelythe eccentricitycyclethat generates
values. It turns out that this maximum is attained for b = 0.72.
the 109 K cyclein the climaticresponse.
It follows that II made a fortunate choice by assuming
b = 0.6, because this value is close to the value of b that
maximizesthe energyin the 109 K cycle.(It is interestingto
1.0
note that II determinedtheir value of b by fitting model results to real data of the last 150 K without consideringany
spectralcomponentseparately.)
experiments
weredonein whichthe nonlinearityparameter(b)
was varied, usingthe heatingfunctionof case1. This casewas
chosenbecauseit gaveriseto an extremelystrong109 K cycle.
The heatingand the climateresponsein the "very nonlinear
FREQUENCY (2a-/1000 YRS)
0
0.1
0.2
!
!
0.3
0.4
[
!
0.5
[
0.6
,
0.2
218
109
41
23
19
PERIOD (1000 YRS)
Fig. 4. Spectrumof the climate responsein Figure 3.
0.4
0.6
NONLINEARITYPARAMETER (b)
0.8
1.0
-
Fig. 6. Energy in the 109 K cyclein the climate responseas a function of the nonlinearityparameter,for the heatingshownin Figure 3.
5664
S•IœDœR:
IcœAGE MODœL
Acknowledgments. The author would like to thank A. Oort and S.
Manabe for usefulcomments.The researchwas supportedby grant
ATM
8218761 A01 from the National
Science Foundation.
Held, I. M., Climate modelsand the astronomicaltheory of the ice
ages,Icarus, 50, 449-461, 1982.
Imbrie, J., and J. Z. Imbrie, Modelingthe climaticresponse
to orbital
variations, Science,207, 943-954, 1980.
Le Treut, H., and M. Ghil, Orbital forcing,climatic interactions,and
glaciationcycles,J. Geophys.Res.,88, 5167-5190, 1983.
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(ReceivedApril 30, 1984;
revisedJanuary29, 1985;
acceptedJanuary31, 1985.)