abrupt climate change: chaos and order at orbital and millennial scales

7/25/2019 Abrupt climate change: chaos and order at orbital and millennial scales

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Abrupt climate change: chaos and order at orbital

and millennial scales

J.A. Rial*

Wave Propagation Laboratory, Geological Sciences Department, University of North Carolina, Chapel Hill, NC 27599-3315, USA

Accepted 1 October 2003

Abstract

Successful prediction of future global climate is critically dependent on understanding its complex history, some of which is

displayed in paleoclimate time series extracted from deep-sea sediment and ice cores. These recordings exhibit frequent

episodes of abrupt climate change believed to be the result of nonlinear response of the climate system to internal or external

forcing, yet, neither the physical mechanisms nor the nature of the nonlinearities involved are well understood. At the orbital

(104 105 years) and millennial scales, abrupt climate change appears as sudden, rapid warming events, each followed by

periods of slow cooling. The sequence often forms a distinctive saw-tooth shaped time series, epitomized by the deep-sea

records of the last million years and the DansgaardOeschger (D/O) oscillations of the last glacial. Here I introduce a simplified

mathematical model consisting of a novel arrangement of coupled nonlinear differential equations that appears to capture some

important physics of climate change at Milankovitch and millennial scales, closely reproducing the saw-tooth shape of the deep-sea sediment and ice core time series, the relatively abrupt mid-Pleistocene climate switch, and the intriguing D/O oscillations.

Named LODE for its use of the logistic-delayed differential equation, the model combines simplicity in the formulation (two

equations, small number of adjustable parameters) and sufficient complexity in the dynamics (infinite-dimensional nonlinear

delay differential equation) to accurately simulate details of climate change other simplified models cannot. Close agreement

with available data suggests that the D/O oscillations are frequency modulated by the third harmonic of the precession forcing,

and by the precession itself, but the entrained response is intermittent, mixed with intervals of noise, which corresponds well

with the idea that the climate operates at the edge between chaos and order. LODE also predicts a persistentf 1.5 ky oscillation

that results from the frequency modulated regional climate oscillation.

D 2004 Elsevier B.V. All rights reserved.

Keywords: climate change; paleoclimatology; DansgaardOeschger oscillation; complexity; emergent behavior

1. Introduction

Paleoclimate records over many time scales exhibit

episodes of rapid, abrupt climate change, which may

be defined as sudden climate transitions occurring at

rates faster than their known or suspected cause

(Rahmstorf, 2001; National Academy of Sciences,

2002). Abrupt climate change is believed to be the

result of instabilities, threshold crossings, and other

types of nonlinear behavior of the global climate

system (Jouzel et al., 1994; Clark et al., 1999; Alley

et al., 1999; Rahmstorf, 2000), but neither the phys-

ical mechanisms involved nor the nature of the non-

0921-8181/$ - see front matterD 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.gloplacha.2003.10.004

* Tel.: +1-919-966-4553; fax: +1-919-966-4519.

E-mail address: [email protected] (J.A. Rial).

www.elsevier.com/locate/gloplacha

Global and Planetary Change 41 (2004) 95109


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linearities are well understood. Fig. 1 shows selected

examples of abrupt climate change in the form of

rapid warming episodes followed by much slower

cooling episodes. Each warming/cooling sequenceusually repeats at nearly equal time intervals, giving

the time series a characteristic quasi-periodic saw-

tooth appearance that, remarkably, appears indepen-

dent of time scale (as shown in the enlargement) and

displays an unclear relation to astronomical forcing.

Throughout most of the paleoclimate proxy data

there is a frequent repetition of this same theme:

sudden and fast warming followed by much slower

cooling. Since many of these abrupt warming events

have happened often in the recent past, it is reasonable

to expect them in the near future, hence the urgent

need to improve our understanding of the physical

processes involved. By itself,Fig. 1already provokesa number of obvious and stimulating questions, such

as, why are warming episodes generally so much

faster than cooling ones (saw-tooth)? How can rapid

climate change be triggered by slow change in orbital

parameters? Does self-similarity of response mean

similarity of processes regardless of timescale? What

nonlinear processes are at work? Do the ice core

records reflect a climate system operating between

order and chaos?

Fig. 1. Samples of climate change across different time scales and proxy records. Note that the saw-tooth shape, which is created by the fast

warming/slow cooling sequence, appears to be independent of time scale, showing an intriguing self-similarity. Main warming periods are

indicated by colored vertical stripes. This same theme of fast warming followed byslower cooling is repeated throughout the paleoclimate

records. Data fromRaymo (1997),GRIP Project Members (1993),Petit et al. (1999),Sachs and Lehman (1999).(Figure fromRial et al. (2004),

with permission from Kluwer Science Publishers).

J.A. Rial / Global and Planetary Change 41 (2004) 9510996


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This paper describes a simplified dynamic model

of the climatesystemthat successfully reproduces the

time series inFig. 1and suggests answers to some of

the questions posed above. From orbital to millennialtime scales the model appears to capture some of the

essential physics of global climate change, closely

replicating the familiar saw-tooth shape of the time

series, the mid-Pleistocene climate switch and the

intriguing Dansgaard Oeschger oscillations, all of

which exemplify abrupt climate change.

Many simplified climate models consist of two or

more coupled ordinary differential equations con-

trolled by a few carefully selected parameters. A

typical model may consist of an energy balance

equation (Budyko et al., 1987; Schneider, 1992)

coupled to an equation describing the dynamics of

ice sheets (e.g., Ghil and Childress, 1987). I t i s

generally acknowledged that the best models will

be those that contain aminimum of adjustable param-

eters(Saltzman, 2002)and are robust with respect to

changes in those parameters. For over two decades,

and in spite of their obvious limitations, simplified

climate models have helped understand the driving

forces of long-term climate variability, and explain

many features of the paleoclimate record (e.g., Ghil

and Tavantzis, 1983; Kallen et al., 1980; Oerlemans,

1982; Saltzman and Sutera, 1987; Saltzman andMaasch, 1991; Saltzman and Verbitsky, 1994; Ghil,

1994; Li et al., 1998; Berger et al., 1999; and many

others).

2. LODE: a model based on the logistic delay

differential equation

Here I introduce a conceptual, deterministic (and

chaotic) climate model that simulates ice sheet dy-

namics using a nonlinear, logistic delay-differentialequation (DDE) borrowed from ecology (e.g. Hutch-

inson, 1948; May, 1973; Driver, 1977). DDEs have

been used recently to model the irregular oscillations

of ENSO (Battisti and Hirst, 1989; Tziperman et al.,

1994), and in paleoclimatology a Verhulst (logistic)

DDE was introduced byRial and Anaclerio (2000)to

simulate the ice ages. The model introduce here is a

generalization of the latter approach, with the logistic

equation tightly coupled to a standard energy balance

equation that accounts for global surface temperature

change. The model structure and dynamics are sug-

gested by the close analogy between the growth and

decay of the ice caps under a changing environment

and the evolution of a single-species population oforganisms subject to variations in the availability of

resources. Governed internally by competing positive

and negative delayed feedbacks and externally by

(usually periodic) environmental influences, both pro-

cesses develop quasiperiodic oscillations showing

only partial resemblance to the forcing (e.g., Gurney

and Nisbet, 1998; Rial, 1999). Heretofore called

LODE for its use of the logistic-delay differential

equation, the model is written as:

dLtdt

lLts 1LtsKt

1

CdTt

dt Q1aL ABTt 2

where L(t) represents a dimension of the ice cap

comparable to the d18O proxy, such as ice volume.

l is the ice sheets time equilibration constant in

ky 1, and K(t) = 1 + e(t)T(t) is the temperature-depen-

dent carrying capacity of the system (1 ky = 1000years). Internal or external forcing is introduced

through the function e(t), which amplitude-modulates

T(t), as described by Eq. (3) below. As shall be

discussed in detail later, the most important charac-

teristic of LODE is that it transforms amplitude

modulation of the global temperature T(t) into fre-

quency modulation of ice volume L(t), which is

consistent with evidence of frequency modulation in

many paleoclimate proxies and explains the spectral

presence of frequencies not in the astronomical forc-

ing and the absence of others that should be present(Rial, 1999). The term inside square brackets in Eq.

(1) represents the negative feedback (decrease in

atmospheric moisture asL(t) grows large and temper-

ature decreases), while the outside product is the

positive feedback (exponential ice growth when L(t)

is small). Both feedbacks are delayed by time s, which

is a parameterized measure of the systems thermal

and mechanical inertia.

Global heating or cooling is described by the LHS

of Eq. (2), and depends on L(t) through the planetary

J.A. Rial / Global and Planetary Change 41 (2004) 95109 97


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albedo a(L). Cis the heat capacity of the system, 4Q

the solar constant and A, B are satellite-determined

constants that include the greenhouse effect of the

atmosphere (Budyko et al., 1987). The equation iscalibrated so that forL(t) = 1 the albedoa(L) = 0.3 and

T(t) = 1 5 jC.

To simulate forcing, I use the general representation:

et e0 1XNi1

aicosxit/i

( )ut 3

where e0 is a small quantity with respect to unity

( < 0.1) and the sum is over the Fourier components

of any known or assumed external (astronomical)

forcing (Berger and Loutre, 1991). The cosine func-

tions amplitude-modulate the global temperature T(t)

and therefore the carrying capacity K(t). The function

u(t) represents any other internal or external forcing

not accounted for by the first term, and will be of

use in the simulation of the mid-Pleistocene transi-

tion, to be discussed later. Solutions of nonlinear

systems of the forms (1) (2) with simpler external

forcing are known to include ultra-harmonic and

sub-harmonic responses (Gurney and Nisbet, 1998)

and synchronization (phase and frequency locking),

for all of which there appears to be abundantevidence in the paleoclimate record (Clark et al.,

1999; Imbrie et al., 1993; Rial, 1999; Ghil and

Childress, 1987; Hagelberg et al., 1994, etc.) The

system of Eqs. (1) and (2) combines two desirable

characteristics: simplicity in the formulation (small

number of parameters, few equations) and enough

complexity of the dynamics (infinite-dimensional

delay differential equation).

2.1. The physics of LODE

In its simplest form, the carrying capacity K(t) is

unity and the delay s is zero. LODE then reducesto a description of the competition between what

can be construed as the ice-albedo positive feed-

back, represented by the linear term on the RHS of

Eq. (1), and the precipitation-temperature negative

feedback, represented by the quantity within square

brackets. The familiar elementary solutions are

sigmoidal growth or exponential decay functions,

depending on whether the initial value L(0) is less

or greater than unity (e.g., Boyce and DiPrima,

1997).

At the next level of complexity, a positive time

delay s transforms Eq. (1) into a delay-differential

equation, one that contains much richer and varied

solutions, including damped oscillatory and steady-

state periodic (Gopalsamy, 1992). The delayed

equation yields damped oscillations of L(t) about

the carrying capacity for small s. If s becomes long

compared to the natural response time of the

system, the oscillations will become strong, and

will grow in amplitude, period and duration (Gur-

ney and Nisbet, 1998). As in the logistic equation

for growth (May, 1973), here the product sl is a

bifurcation parameter, which when crossing thethreshold value p/2 makes the solutions to Eqs.

(1) and (2) undergo a Hopf bifurcation (Marsden

and McCracken, 1976) and settle to a stable limit

cycle with fundamental period Pf 4s. Thus, given

an astronomical periodicity say, 95 ky, a model

climate system can be easily constructed that oscil-

lates at a similar period by selecting the delay s to

be f 24 ky; which makes the climates period

Fig. 2. LODE closely reproduces the saw-tooth shape of the data. Numerical experiments with LODE provide an explanation: letting thefeedback delay time s increase while the ice sheets time response constantl is kept fixed increases the saw-tooth asymmetry while the

abruptness (the slope of the warming episode) increases. For instance, with 1/l= 11 ky and s = 23 ky, the duration of the warming episode is

about 1/5 of the total sequence (i.e., 20 ky out of 100 ky), very close to that observed in typical deepsea records(Fig. 1).The saw-tooth in the

model is thus the direct consequence of the feedback delay being much longer than the response time. Physically this means that the feedback

delay includes not only the thermal but also the mechanical inertia of the ice sheet. (b) The model also reproduces the so-called mid-Pleistocene

transition (MPT), which in the scale of the last three million years can be considered as an abrupt climate change. Around 950 ky ago, the then

predominant 41 ky glaciation period switched to a f 100 ky period, of greater amplitude, apparently without a corresponding change in the

forcing. LODE transforms amplitude modulation into frequency modulation, so that to simulate the frequency switch at the MPT it suffices to

modulate the temperature T(t) with the smoothed step function shown. The amplitude change gets transformed into a proportional frequency

change. (c) and (d) show an example of a complete simulation ofthe time series and its power spectrum compared to deep-sea sediment data

(Site 849 arbitrarily chosen for comparison. Compare with Fig. 4a). Note the near absence of power at 413 ky in both theoretical and observed

spectra. Both the theoretical waveform and spectrum are qualitatively and quantitatively close to their observed counterparts.



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f 96 ky and the product sl slightly larger than the

bifurcation threshold p/2, since lf 1/14 ky (Imbrie

et al., 1993). This is a model climate system that

oscillates close enough to the astronomical forcing tosynchronize and resonate with it. Further, the ampli-

tude ratio of maximum to minimumL(t) is as large as

310 for values ofsl slightly above the bifurcation

threshold (May, 1973), which explains the large ice

volume changes from glacial to interglacial.

Further, LODE will synchronize with an external

forcing frequency that may be just close enough to1/P(Pikovsky et al., 2001). In fact, numerical experi-

ments show that LODE synchronizes and resonates

with forcing that is within 15% of its natural

frequency 1/P. The implications for modeling are



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clear: one does not need to know the exact resonant

frequency of the actual climate system, little external

forcing can produce a strong climate response at a

given period, and resonance of the climate systemwith the astronomical forcing is robust to variations

in the models parameters.

Perhaps the most important nonlinear feature of

LODE is its property of transforming amplitude

modulation into frequency modulation. This effect

develops as T(t) computed in Eq. (2) feeds back into

Eq. (1), whose solution in turn provides the updated

input to Eq. (2) through the albedo. Since there is

evidence in the paleoclimate record that the climate

system somehow induces frequency modulation of the

astronomical forcing(Rial, 1999; Rial and Anaclerio,

2000) it is important to understand at least how the

model does it. To illustrate,Fig. 3shows two different

(normalized) outcomes from LODE whose only dif-

ference is in the amplitude-modulating coefficiente(t)

of T(t). To keep matters simple, e(t) is set to be a

numerical constant, first equal to 0.15, say. This

produces a climate model that oscillates with a period

of 90 units. With all other factors the same, an

increase ofe(t) to 0.85 shortens the period to 40 units,

as shown. This happens because the increase in e

raises the mean value of K(t), which is the threshold

across which dL(t)/dtchanges sign. The higher valueof the threshold makes the growth rate higher because

the nonlinear quadratic term in Eq. (1) decreases. A

faster growth rate will go over the threshold at an

earlier time and thus the change in sign of dL/dtwill

occur sooner than before, also pulling L(t) down in a

shorter time (Fig. 3). Physically, the equations pro-

duce short glacial periods as global temperature

increases and longer glacial periods as temperature

drops, which is the expected behavior. Therefore, in

general, when the carrying capacityK(t) changes with

time, the frequencies will increase or decrease in

proportion to its changing amplitude, resulting infrequency modulation ofL(t).

3. Comparison to data and validation of LODE

In what follows proxy paleoclimate data in the

form ofd18O time series from deep-sea sediments and

from ice cores are compared to the synthetic time

series L(t) representing normalized ice volume. It is

important to state that these comparisons are legiti-

mate, since variations in the isotopic compositions

(dD and d18O) of the ice roughly coincide with

variations ofd18O in sea water, and both are linked

to the waxing and waning of continental ice sheets, a

fact that has permitted researchers to place records

from ice cores and deep-sea sediments in a common

temporal framework over the last 420,000 years(Petit

et al., 1999).

3.1. The saw-tooth, a manifestation of abrupt climate

change

The importance of the delay s in Eq. (1) cannot beoverstated. If s< 1/l the solution to Eq. (1) is a

quickly damped single sinusoid. Increasing ls

increases the duration of the sinusoid, which

approaches steady state near the bifurcation threshold.

At the bifurcation, ls (which physically is the ratio of

the delay response of the ice sheet to its thermal

equilibrium time) is equal to p/2. Beyond this thresh-

old the time series becomes saw-toothed (Fig. 2a,b)

Fig. 3. Example of amplitude modulation transformed into frequency modulation by LODE. Amplitude increase (decrease) ofK(t) results in

frequency increase (decrease) of the signal (see text for details). Pis the fundamental period of the output response (in arbitrary units).



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reflecting the thermal asymmetry of the equation.

That is, the cooling-to-warming duration ratio in each

period is p/2 at the bifurcation and grows as the

bifurcation parameter increases. For instance, it canbe numerically shown that for fixed feedback delay s,

an increase in l makes the warming time grow

increasingly shorter (the ice responds faster but feed-

back is comparatively slow), which makes the saw-

tooth become increasingly pronounced. The reverse is

also true; that is, an increasing delay with fixed time

response makes the warming duration as short as one-

tenth of the total warming/cooling period, as is

typically observed in deep-sea sediment records (see

Fig. 4).

Mathematically, it is the excitation of the higher

harmonics P/n (n = 2,3,4,. . .

) of the fundamental pe-

riod P= 4s what creates the characteristic saw-tooth

shape of the time series beyond the p/2 threshold.

Physically, it is the difference between response time

1/l and the feedback delay swhat results in the saw-

tooth asymmetry of the time series. It is reasonable

that the feedback delay is longer than the equilibration

time response since it is assumed that the former

includes the mechanical as well as the thermal inertia

of the ice cap-controlled climate system. The mechan-

ical inertia contribution to the delay represents the

bulk motion dynamics of the ice sheet, which likelydepends on the geology and topography of the under-

lying substratum as well as on the internal deforma-

tion and internal friction of the ice mass (e.g., Clark et

al., 1999).

As shown in Fig. 4 LODE closely reproduces

the saw-tooth waveform in several distinct time-

scales and varied data sets. Therefore, if LODE

correctly describes this aspect of the paleoclimate,

the saw-tooth is a consequence of the characteristic

thermomechanical constants of the system, and does

not require external (orbital) forcing to occur,though external forcing may intensify the asymme-

try of the saw-tooth, as suggested by numerical

experiments.

3.2. Modeling the Mid-Pleistocene climate transition

(MPT)

Around 950 ky ago, the then predominant 41 ky

glaciation period switched to a f 100 ky period, of

greater amplitude, apparently without a corresponding

change in the orbital forcing(Clark et al., 1999). This

still unexplained switch in periodicity is usually called

the mid-Pleistocene transition (MPT), which in the

scale of the last three million years can be consideredas an abrupt climate change (see reviews by Hinnov,

2000; Elkibbi and Rial, 2001). At the MPT it is

estimated that the ice mass increased by about

1.05F0.201019 kg, equivalent to an ice sheet areaexpansion of 3.1F0.71012 m2 (Mudelsee andSchultz, 1997). This unprecedented expansion re-

sulted in the great ice ages of the late Pleistocene

and Holocene, with their characteristic glacialinter-

glacial intervals of nearly 100 ky and typical saw-

tooth waveform.

LODEs nonlinear transformation of amplitude

modulation into frequency modulation is the key to

the simulation of the MTP frequency switch. The

simulation uses the hyperbolic tangent function:

ut e11e2tanhgtt0

utHt0 e11e2

utbt0 e11e2 4

that acts a s a smooth step (seeFig. 2b). Heret0is the

age of the MPT (f 920 ka), ande1and e2are positive

constants selected to produce the appropriate frequen-

cies before and after t0, as illustrated in Fig. 4a. The

factor g controls the sharpness of the transition; its

value is estimated fromMudelsee and Schultz (1997)

and provides a good fit to the data.

The behavior of the model is very similar to that

illustrated in Fig 3. If u(t)f 0 the model climate

system oscillates at its longest period, which forsf 24 ky is f 95 ky, but as the step is passed, a

higher value is given tou(t) causing the free period to

shorten to f 45 ky when u(t)f 1. This means that

by judiciously selecting e1 and e2 in Eq. (4) it is

possible to switch the natural frequency of the system

and construct a model that oscillates with frequencies

close to those present in the external forcing. Small

discrepancies between the climate frequencies and the

astronomical forcing frequencies are of no conse-

quence since the former will synchronize to the



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Milankovitch forcing periods of 95 and 41 ky (this is

analogous with the synchronization of the circadian

biological clock to the24-h day cycle; see for instance

Pikovsky et al., 2001). Then, if the climate systemworks like LODE, the paleoclimate time series data

show not the actual climate systems internal free

periods (which one can only guess) but the periods

of the orbital forcing, after synchronization with the

climate has taken place. Although this is a mechanis-

tic and simplistic view, in which the climate is

represented by a nonlinear oscillator responding to

external forcing, it appears to be validated by the

paleoclimate data, as shown inFig. 4 and as will be

discussed in what follows.

The way LODE causes the climate system to

switch frequency at the MPT makes physical sense

since, by design, the simulated MPT change to longer

period also causes the coupling between Eqs. (1) and

(2) to weaken (the coupling parameter is smaller) and

consequently, the ice sheet growth rate dL(t)/d t

becomes less strongly dependent on global tempera-

ture. This resembles the actual climate if one assumes

that large enough ice sheets created their own climate,

driving the temperature T(t) to a lower equilibrium

level after the MPT. In the model of course the switch

is accomplished by the ad-hoc function (4) which

could be thought of as an external forcing. There wasin fact a drop in mean global insolation atf 1000 ka

(Berger and Loutre, 1991), such that the insolation

since the MPT has been at a lower mean level than it

was before the MPT. The step-like drop is noticeable

in low-pass filtered insolation time series. It is tempt-

ing to speculate that this subtle astronomically in-

duced cooling was amplified by feedbacks, and drove

the system into a much colder regime that trigger long

period glaciations of the last million years. Long

period changes in orbital forcing may indeed cause

substantial changes in ice response. For instance, arare orbital congruence of minima in obliquity and

eccentricity that occurred f 23 Ma may have favored

ice-sheet expansion on Antarctica (Zachos et al.,

2001).

In Fig. 2c the synthetic time series for L(t) com-

puted with LODE is very similar to the data, bothqualitatively and quantitatively after the relevant as-

tronomical (Milankovitch) frequencies 413, 120 and

41 ky (in their known proportions) are included in the

forcing.

Comparison with diverse paleoclimate data is

shown in Fig. 4a,b. Besides the switch in frequency

at the MPT this figure shows that LODE closely

fits the saw-tooth shape and the variable duration of

the f 100 ky glaciation cycles. The fact that the

duration of these cycles is variable (Raymo, 1997)

and closely fitted by LODE further supports the

idea that frequency modulation is a better choice

than amplitude modulation as the cause of the

multi-peak spectral characteristics of the data (Rial,

1999, 2004).

3.3. Modeling the DansgaardOeschger oscillations

Perhaps the most puzzling feature of recent pale-

oclimate records, highly relevant to understanding

future global climate change, is the fast-warming/

slow-cooling sequence found in the d18O time series

of Greenlands ice cores known as the DansgaardOeschger (D/O) oscillations (Jouzel et al., 1994;

Alley et al., 1999). The D/O typically show very

sudden, 6 10 jC warming episodes lasting a few

centuries or perhaps even a few decades, followed

by millennia of relatively slow cooling. Remarkably,

reconstructed sea surface temperatures (SST) in the

tropical Atlantic(Fig. 1)mimic the D/O record in the

3060 ka interval, and similar recordings are found

in the subtropical Pacific and tropical Indian oceans

(Rahmstorf, 2001; Sachs and Lehman, 1999). Initial-

ly the D/O appeared to represent regional climaticfluctuations dominated by the response of the Green-

lands ice cap and the north Atlantic, but their

Fig. 4. (a) Comparison between the LODE results (L(t), red) and selected deep-sea sedimentd18O time series. The synthetic time series is the

same in all cases, constructed as in Fig. 2c. Note the close fit to the saw-tooth shaped amplitudes. The shift in frequency around 950 ka is

difficult to fit in its detail, but periods and waveforms before and after the switch are closely matched. (b) Fits to Site607 d18O time series and to

the temperature record (Deuterium) from Vostok ice core. The latter is compared with both LODEs calculated temperature T(t) and ice volume

L(t) (sameL(t) as that compared to Site607). T(t) fits the Vostok temperature record slightly better than L(t), which gives some credence to the

model. All parameters are the same.



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ultimate cause is still a mystery (Rahmstorf, 2003).

Recently, the global reach of the D/O has been

documented (Leuschner and Sirocko, 2000), while

Hinnov et al. (2002) have demonstrated statistical

coherence of D/O cycles between the North and

South hemispheres. Interestingly, the D/O quasi-pe-

riodic oscillations from 45 to 29 ka are consistent

with frequency modulation of af

2.73 ky carrierby a f 7.5 ky modulator (see Fig. 5).

Hence, to simulate the D/O, the prominent 2.75 ky

oscillation(Fig. 5)is taken to be the natural period of

the regional climate system dominated by the Green-

land ice cap. This is assumed to be the case because it

is consistent with the fact that the resonantf 95 ky

period of the global ice sheets(Imbrie et al., 1993) is

longer by a factor of 30, nearly the proportion of

global ice volume to Greenlands ice volume duringthe last glacial, which was f 1/30 times the volume

Fig. 5. The power spectrum (FFT of autocorrelation) of the Dansgaard Oeschger oscillations between 29 and 45 ka (GRIP record) is consistentwith that of a frequency modulated signal (red bars) with a 7.5 ky period signal, (the main forcing of the D/O) modulating a carrier of 2.75 ky.

The separation between peaks in the spectrum is nearly constant and equal to f 1/7.5 ky 1.

Fig. 6. (a) Fitting the details of the D/O oscillations in the GRIP ice core interval 29 45 ka is accomplished by assuming forcing by the third

harmonic (7.5 8 ky) of the 23 ky precession signal. By assuming the ice system to have a 3 ky natural period, amplitude modulation ofT(t) by

the 7.5 ky forcing is transformed into frequency modulation, which also generates strong oscillations at 12 ky. (b) Oscillations similar to those

in (a) but seemingly stretched in time and centered at 80 ka are easily fit by a simple expansion of the time scale, consistent with forcing by the

precessions 23 ky component. Note how the generalaspect of theGRIP signal is that of a mixture of ordered forcing and chaotic behavior,

typical of what is understood to be a complex system (Rind, 1999),where order can emerge spontaneously out of chaos. c,d) Comparison of

LODEs results with the Vostok (Deuterium) and Greenland GISP2 records. Note that the same synthetic time series that fit the Greenland

records consistently fit Antarcticas, and although the latter is much noisier, the amplitude and relative pha ses of the computed timeseries are

consistent with those of Greenlands. Figures on selected peaks correspond to the numbered interstadials (Dansgaard et al., 1993).



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of the entire ice sheets (Saltzman, 2002). Therefore,

an equally linear estimate for Greenlands time re-

sponse constant would be 14 ky/30 or about 0.47 ky,

which happens to work well in the model. Thus, all

the physical parameters used to simulate the D/O

oscillations are scaled-down by a factor of 30. This



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means that in Eq. (1) one takes s =P/4 = 0.7 ky, and to

cross the bifurcation threshold the productsl must be

greater thanp/2, so 1/lcan be set to f 0.48 ky, which

is, not coincidentally, about 1/30 of that of the globalice volumes 14 ky. Although the assumption that the

thermal constants are linearly related to the ice caps

volume is probably violated, it is possible that any

nonlinearity is small, since the computed periods and

phases of the component oscillations are very close to

the actual ones(Fig. 5).

The segment of the D/O between 29 and 45 ka, that

consists of at least two distinct groups of regular

oscillations with periods ranging from f 3 k y t of 1.5 ky, is closely reproduced by LODE (Fig. 6).

These periods are determined by spectral analyses of

the two time series GISP and GRIP, which have

slightly different timescales, hence the uncertainty.

The estimated Fourier periods are 7.5, 3 and 1.67

ky, which are probably correct within 10%. Fig. 5

shows clearly that the spectral peak spacing is ap-

proximately 7.5 8 ky, suggesting frequency modula-

tion by that period, which being numerically close to

one-third of the 23 ky suggests that the ice cap is

forced by the third harmonic of the precession-in-

duced insolation. This behavior is not unusual; in

nonlinear self-excited mechanical, electrical and bio-

logical systems, frequency entrainment to subhar-monics and high-harmonics of the forcing is well

known (e.g.,Minorsky, 1962;Stoker, 1950;Pikovsky

et al., 2001). The theoretical possibility exists of

course that the D/O might be entrained by the fifth

harmonic of the obliquity (41 ky/5f8 ky), but a 1:3

frequency entrainment ratio is generally more likely to

occur than a 1:5 one (Pikovsky et al., 2001). In

practice, climate variability in the sub-Milankovitch

band found in deep-sea sediment records has been

identified and interpreted (Hagelberg et al., 1994) as

higher harmonics of the precession band.For ages older than 45 ka the record of the D/O

becomes less structured; but beginning at 65 ka an

almost identical sequence of oscillations as that

between 36 and 45 ka can be seen, though it lasts

almost exactly three times longer (f 23 ky). The

waveform is, significantly, a stretched version of

those in the 3645 ka interval(Fig. 6b). Why would

the climate system behave in this way is not a simple

matter to understand. The modeling results can be

interpreted as indicating that the ice cap and its

regional climate respond to the forcing in a sort of

capture and escape mode (Simiu, 2002), sometimes

being captured (entrained) by the precession and

sometimes by its higher harmonics. Which responseactually occurs may be controlled (triggered) by am-

bient noise. From the point of view of nonlinear

dynamics it appears as if the GRIP record of the lastf 100 ky shows a climate system operating at the

edge between order and chaos, that is, between a

recognizable (albeit nonlinear) response to the astro-

nomical forcing, and chaotic noise.

Recently, a different explanation of the D/O oscil-

lations has been proposed based on the process known

as stochastic resonance, or SR (Alley et al., 2001;

Ganopolski and Rahmstorf, 2001, 2002). To explain

the D/O, SR requires the presence of a weakf 1.5 ky

signal (of unknown origin) that becomes perceptible

due to the presence of an appropriate level of noise.

Based on the close match with the data(Fig. 6ad)I

suggest however that the D/O signal may rather be

linked to orbital forcing (precession-induced insola-

tion), and that the f 1.5 ky oscillation (Bond et al.,

1997)is caused by frequency modulation of the 2.73

ky natural period of the ice cap by the 7.5 ky

harmonic of the precession (1/1.58 = 1/2.75 + 2/7.5).

Precession forcing would also explain why, though

the response is strongest in the Arctic, the noisierAntarctic record of temperature shows similar oscil-

lations, which, though weaker and phase shifted, seem

closely reproduced with the same time series that fit

the D/O in the Artic (Fig. 6c,d).

LODEs success at reproducing abrupt climate

change at many scales suggests that climate variability

at orbital and millennial scales results from an astro-

nomically forced complex climate system with emer-

gent behavior (Rind, 1999) typified by the sudden

warming events that characterize abrupt climate

change. LODEs results do not rule out SR, but pointto the possibility that rather than a single harmonic

signal at 1.5 ky, there may be a more complex signal

in the record, perhaps enhanced by SR, but of external

origin.

4. Concluding remarks

Summarizing, LODE provides useful insights into

the nature of abrupt climate change. The mysterious



13/15

saw-tooth appears to be just the consequence of the

difference between the thermal inertia of the ice and

the thermo-mechanical feedback response of the ice

cap, so it is of internal origin (although it may beenhanced by external forcing). If LODE correctly

mimics it, the climate system transforms amplitude

modulation of global temperature into frequency

modulation of global ice extent. This is fully consis-

tent with the observations(Rial, 1999, 2003)and with

the well-known absence of spectral power at 413 ky

(Imbrie et al., 1993)that is replicated in the synthetic

spectrum ofFig. 2d.The key assumption in the model

is that the carrying capacity K(t) is modulated by the

global temperature, which turns out to produce the

correct frequency behavior of the system, as shown by

the close simulation ofthe mid-Pleistocene frequency

shift (Figs. 2b,c and 4).

The modeling results also suggest that although the

D/O oscillations are entrained by the third harmonic

of the precession forcing, and by the precession itself,

the response is intermittent, occurring in relatively

short intervals, as shown in Fig. 6a,b. Only during

these intervals LODE can closely replicate the data,

which corresponds well with the idea that the climate

is a complex system (Rind, 1999), where order can

emerge spontaneously from chaos. It is apparent that

through frequency modulation of the Greenland icecaps natural oscillation by the third harmonic of the

precession forcing, LODE generates apersistent spec-

tral sideband with period off 1.5 ky(Fig. 5).Could

such be the origin of the D/O mystery 1470 year

cycle? In fact, it has been proposed that the high

stability of this 1470year precise clock oscillation

points to an origin outside of the climate system

(Rahmstorf, 2003). The LODE results are consistent

with this idea, inasmuch as the mystery oscillation

results from frequency modulation by an external

driver. But the complexity of the climate systemstrongly contributes too, through its highly nonlinear

response, which causes synchronization, entrainment

of high harmonics and frequency modulation.

4.1. Internal vs. external forcing

What has been described in the foregoing indicates

that, if it behaves at all like LODE, the earths climate,

while weakly driven by external astronomical forcing,

generates most of the intriguing features (including

abrupt climate change) of the time series through

internal nonlinear processing of the forcing. The

built-in nonlinear drivers of the models nonlinearity

are the delayed feedbacks and the coupling betweentemperature and ice extent. These are apparently

enough to simulate the important features in the time

series, including the saw-tooth shape, the MPT, the D/

O oscillations and the frequency modulation of the

external forcing.

In spite of its obvious limitations, it is possible

that LODE captures some essential physics of the

climate system. The D/O records are fascinatingly

mysterious, and one is driven to think that these

time series reflect the typical behavior of a complex

climate system, whose emergent behavior may in

fact be the frequent abrupt and sharp warming

episodes.

Acknowledgements

The work reported here was supported by grant

ATM #0241274 from the National Science Founda-

tions Paleoclimate program. It is a pleasure to

acknowledge useful comments by Bruce Bills and

the thoughtful, thorough revision of the manuscript by

Linda Hinnov.

References

Alley, R.B., Clark, P.U., Keiwin, L.D., Webb, R.S., 1999. Making

sense of millennial-scale climate change. In: Clark, R.B., Webb,

P.U., Keiwin, L.D. (Eds.), Mechanisms of Global Climate

Change at Millennial Time Scales. AGU Geophysical Mono-

graph, vol. 112, pp. 385394.

Alley, R.B., Anandakrishnan, S., Jung, P., 2001. Stochastic reso-

nance in the North Atlantic. Paleoceanography 16, 190198.

Battisti, D.S., Hirst, A., 1989. Interannual variability in the tropicalatmosphere/ocean system: influence of the basic state and ocean

geometry. J. Atmos. Sci. 46, 16871712.

Berger, A., Loutre, M.F., 1991. Insolation values for the climate of

the last 10 million of years. Quat. Sci. Rev. 10 (4), 297317.

Berger, A., Li, X.S., Loutre, M.F., 1999. Modeling northern hemi-

sphere ice volume over the last 3 Ma. Quat. Sci. Rev. 18, 111.

Bond, G., Showers, W., Cheseby, M., Lotti, R., Almasi, P., de

Menocal, P., Priore, P., Cullen, H., Hajdas, I., Bonani, G.,

1997. A pervasive millennial-scale cycle in North Atlantic Ho-

locene and glacial times. Science 278, 12571266.

Boyce, W.E., DiPrima, R.C., 1997. Elementary Differential Equa-

tions, sixth ed. Wiley, New York, NY.



14/15

Budyko, M., Ronov, A.B., Yanshin, A.L., 1987. History of the

Earths Atmosphere. Springer-Verlag, New York, NY.

Clark, P.U., Alley, R.B., Pollard, D., 1999. Northern Hemisphere

ice sheet influences on global climate change. Science 286,

1104 1111.Dansgaard, W., Johnsen, S.J., Clausen, H.B., Dahl-Jensen, D., Gun-

destrup, N.S., Hammer, C.V., Hvidberg, C.S., Stefensen, J.P.,

Sveinbjornsdottir, A.E., Jouzel, J., Bond, G., 1993. Evidence

for general instability of past climate from a 250 kyr ice-core

record. Nature 364, 218220.

Driver, R.D., 1977. Ordinary and Delay Differential Equations.

Springer-Verlag, New York, NY.

Elkibbi, M., Rial, J.A., 2001. An outsiders review of the astro-

nomical theory of the climate: is the eccentricity-driven in-

solation the main driver of the ice ages? Earth-Sci. Rev. 56,

161177.

Ganopolski, A., Rahmstorf, S., 2001. Rapid changes of glacial

climate simulated in a coupled climate model. Nature 409,

153158.

Ganopolski, A., Rahmstorf, S., 2002. Abrupt glacial climate changes

due to stochastic resonance. Phys. Rev. Lett. 88 (3), 1 4.

Ghil, M., 1994. Cryothermodyamics: the chaotic dynamics of pale-

oclimate. Physica, D 77, 130159.

Ghil, M., Tavantzis, J., 1983. Global Hopf bifurcation in a simple

climate model. SIAM J. Appl. Math. 43, 10191041.

Ghil, M., Childress, S., 1987. Topics in Geophysical Fluid Dynam-

ics: Atmospheric Dynamics, Dynamo Theory and Climate Dy-

namics. Springer-Verlag, New York, NY. 485 pp.

Gopalsamy, K., 1992. Stability and Oscillations in Delay Differen-

tial Equations of Population Dynamics. Kluwer Academic Pub-

lishing, Boston.

GRIP Project Members, 1993. Climate instability during the lastinterglacial period recorded in the GRIP ice core. Nature 364,

203207.

Gurney, W.S.C., Nisbet, R.M., 1998. Ecological Dynamics. Oxford

Univ. Press, New York, NY.

Hagelberg, T.K., Bond, G., deMenocal, P., 1994. Milankovitch

band forcing of sub-Milankovitch climate variability during

the Pleistocene. Paleoceanography 9, 545 558.

Hinnov, L.A., 2000. New perspectives on orbitally forced stratig-

raphy. Annu. Rev. Earth Planet. Sci. 28, 419 475.

Hinnov, L.A., Schulz, M., Yiou, P., 2002. Interhemispheric space-

time attributes of the DansgaardOeschger oscillations between

0 100 ka. Quat. Sci. Rev. 21, 1213 1228.

Hutchinson, G.E., 1948. Circular causal systems in ecology. Ann.

N.Y. Acad. Sci. 50, 221 246.Imbrie, J., Berger, A., Boyle, E.A., Clemens, S.C., Duffy, A.,

Howard, W.R., Kukla, G., Kutzbach, J., Martinson, D.G., McIn-

tyre, A., Mix, A.C., Molfino, B., Morley, J.J., Peterson, L.C.,

Pisias, N.G., Prell, W.L., Raymo, M.E., Shackleton, N.J., Togg-

weiler, J.R., 1993. On the structure and origin of major glacia-

tion cycles 2. The 100,000-year cycle. Paleoceanography 8 (6),

699735.

Jouzel, J., Lorius, C., Johnsen, S., Grootes, P., 1994. Climate insta-

bilities: Greenland and antarctic records. C. R. Acad. Sci. Paris t.

319 (serie II), 6577.

Kallen, E., Crafoord, C., Ghil, M., 1980. Free oscillations in a

climate model with ice-sheet dynamics. J. Atmos. Dynamics

36, 2292 2303.

Leuschner, D.C., Sirocko, F., 2000. The low-latitude monsoon cli-

mate during Dansgaard Oeschger cycles and Heinrich events.

Quat. Sci. Rev. 19, 243254.Li, X.S., Berger, A., Loutre, M.F., 1998. CO2 and Northern Hemi-

sphere ice volume variations over the middle and late Quater-

nary. Clim. Dyn. 14, 537544.

Marsden, J.E., McCracken, M., 1976. The Hopf Bifurcation and its

Applications. Springer-Verlag, NY.

May, R.M., 1973. Stability and Complexity in Model Ecosystems.

Princeton Univ. Press, Princeton, NJ.

Minorsky, N., 1962. Nonlinear Oscillations. Van Nostrand-Rein-

hold, Princeton, NJ.

Mudelsee, M., Schultz, M., 1997. The mid-Pleistocene climate tran-

sition: onset of the 100 ka cycle lags ice volume build-up by 280

ka. Earth Planet. Sci. Lett. 151, 117123.

National Academy of Sciences, 2002. Abrupt Climate Change:

Inevitable Surprises. Web site:http://www.nap.edu/books/

0309074347/html/.

Oerlemans, J., 1982. Glacial cycles and ice-sheet modeling. Clim.

Change 4, 353 374.

Petit, J.R., Jouzel, J., Raynaud, D., Barkov, N.I., et al., 1999. Cli-

mate and atmospheric history of the past 420,000 years from the

Vostok ice core, Antarctica. Nature 399, 429436.

Pikovsky, A., Rosenblum, M., Kurths, J., 2001. Synchronization.

Cambridge Univ. Press, Cambridge, UK.

Rahmstorf, S., 2000. The thermohaline ocean circulation: a system

with dangerous thresholds? Clim. Change 46, 247256.

Rahmstorf, S., 2001. Abrupt climate change. In: Steele, J., Thorpe,

S., Turekian, K. (Eds.), Encyclopedia of Ocean Sciences. Aca-

demic Press, London, pp. 16.Rahmstorf, S., 2003. Timing of abrupt climate change: a precise

clock. Geophys. Res. Lett. 30, 15101514.

Raymo, M.E., 1997. The timing of major climate terminations.

Paleoceanography 12, 577 585.

Rial, J.A., 1999. Pacemaking the ice ages by frequency modulation

of Earths orbital eccentricity. Science 285, 564 568.

Rial, J.A., 2003. Earths Orbital Eccentricity and the Rhythm of

the Pleistocene Ice Ages: the Concealed Pacemaker (This

volume).

Rial, J.A., Anaclerio, C.A., 2000. Understanding nonlinear

responses of the climate system to orbital forcing. Quat. Sci.

Rev. 19, 17091722.

Rial, J.A., Pielke, R.A., Beniston, M., Claussen, M., Canadell,

J., Cox, P., Held, H., de Noblet, N., Prinn, R., Reynolds,J.F., Salas, J.D., 2004. Nonlinearities, feedbacks and critical

thresholds within the earths climate system. Clim. Change

(in press).

Rind, D., 1999. Complexity and climate. Science 284, 105 107.

Sachs, J.P., Lehman, S., 1999. Subtropical North Atlantic temper-

atures 60,000 to 30000 years ago. Science 286, 756759.

Saltzman, B., 2002. Dynamical Paleoclimatology. Academic Press,

San Diego, CA.

Saltzman, B., Maasch, K., 1991. A first-order global model of late

Cenozoic climate change. Clim. Change 5, 201210.

Saltzman, B., Sutera, A., 1987. The mid-Quaternary climate transi-

http://%20http//www.nap.edu/books/0309074347/html/http://%20http//www.nap.edu/books/0309074347/html/http://%20http//www.nap.edu/books/0309074347/html/


15/15

tion as the free response of a three-variable dynamical model.

J. Atmos. Sci. 44, 236241.

Saltzman, B., Verbitsky, M., 1994. Late Pleistocene climate trajec-

tory in the phase space of global ice, ocean state, and CO 2:

observations and theory. Paleocenography 9 (6), 767 779.Schneider, S.H., 1992. Introduction to climate modeling. In: Tren-

berth, K.E. (Ed.), Climate System Modeling. Cambridge Univ.

Press, Cambridge-New York, pp. 3 23.

Simiu, E., 2002. Chaotic Transitions in Deterministic and Stochas-

tic Dynamical Systems. Princeton Univ. Press, Princeton, NJ.

Stoker, J.J., 1950. Nonlinear Vibrations in Mechanical and Electri-

cal Systems. Interscience Publishers, New York. 273 pp.

Tziperman, E., Stone, L., Cane, M.A., Jarosh, H., 1994. El Nino

chaos: overlapping of resonances betwen the seasonal cycle

and the Pacific ocean atmosphere oscillator. Science 264,7274.

Zachos, J.C., Shackleton, N., Revenaugh, J.S., Palike, H.,

Flower, B.P., 2001. Climate response to orbital forcing

across the Oligocene Miocene boundary. Science 292,

274278.


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