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Are the 41 kyr glacial oscillations a linear response to Milankovitch forcing?

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Quaternary Science Reviews 23 (2004) 1879–1890
Are the 41kyr glacial oscillations a linear response toMilankovitch forcing?
Yosef Ashkenazy
a,
, Eli Tziperman
b
a
Environmental Sciences, Weizmann Institute, Rehovot 76100, Israel
b
Department of Earth and Planetary Sciences and Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Received 7 January 2004; accepted 20 April 2004
Abstract
The characteristics of glacial oscillations changed drastically
0
:
8Ma ago, at the ‘‘mid-Pleistocene transition’’. During the past0.8Ma the
100kyr glacial–interglacial oscillations were strongly asymmetric (i.e., long glacial intervals of growth followed byrapid intervals of deglaciation). The 40kyr oscillations prior to the mid-Pleistocene transition were of a smaller amplitude and less-asymmetrical looking. The smaller amplitude, apparently symmetric form and period that matches that of obliquity, suggests thatthese oscillations were a linear response to Milankovitch forcing, while the 100kyr oscillations are attributed either to somenonlinear self-sustained variability due to a mechanism internal to the climate system itself or to nonlinear ampliﬁcation of theinsolation forcing. The signiﬁcant strengthening of the 100kyr eccentricity power in the past
800kyr is one of the intriguingquestions of climate history.Here we show that glacial–interglacial oscillations pre-mid-Pleistocene transition are, in fact, signiﬁcantly asymmetric. Thisasymmetry may contradict a straight forward linear Milankovitch explanation, and we therefore suggest that the glacial oscillationsbefore and after the transition may both be explained as self-sustained variability (although the possibility of nonlinear response toinsolation forcing still exists). The role of Milankovitch forcing is in setting the phase of the oscillations (e.g. time of terminations)and their period, rather in being the main driving force of the oscillations. This is demonstrated using a simple model based on thesea ice switch mechanism of Gildor and Tziperman (Paleoceanography 15 (2000) 605).
r
2004 Elsevier Ltd. All rights reserved.
1. Introduction
Earth climate of the last 2.7 million years (2.7Ma)was dominated by repetitive and drastic glacial–inter-glacial oscillations, the ice-ages. These cycles werecharacterized by a 100kyr period during the past0.8Ma (time of the mid-Pleistocene transition), and bya 40kyr period before that (Fig. 1). It is generallybelieved that orbital forcing plays a signiﬁcant role inthis variability (Milankovitch hypothesis), althoughwhat that role is still somewhat unclear. Orbital changesoccur due to changes in eccentricity (100kyr time scale),obliquity (40kyr), and precession (20kyr) (Milanko-vitch, 1941; Paillard, 2001). The extreme scenario thateccentricity variations on 100kyr are responsible for theexistence of 100kyr signal during the past 0.8Ma is notlikely to be valid due to the weak power of eccentricitychanges. It is more likely that the 100kyr oscillationsresulted from an internal variability of the climatesystem (although see Hagelberg et al., 1991) whileobliquity and precession variations phase lock (andmodulate) the 100kyr cycles. By phase locking it ismeant that the timing of the terminations is set by theMilankovitch forcing. More precisely, one may view theMilankovitch forcing as an external clock felt by theclimate system. The 100kyr oscillations are then weaklyinﬂuenced by this external clock, and adjust their timing
ARTICLE IN PRESS
0277-3791/$-see front matter
r
2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.quascirev.2004.04.008
Corresponding author. Tel.: +972-8-934-4924; fax: +972-8-934-4124.
E-mail addresses:
yossi.ashkenazy@weizmann.ac.il(Y. Ashkenazy), eli@eps.harvard.edu (E. Tziperman).
to be in phase with the changes in solar radiation. Inaddition to the pronounced 1/100kyr frequency thatexists in the power spectrum of the past 0.8Ma, thereare secondary frequencies of 1/40 and 1/20kyr. Theselast two frequencies are believed to result due to linearresponse of the climate system to insolation (e.g.,Saltzman, 1990; Imbrie et al., 1992, 1993).The phase locking to Milankovitch is more clearlydemonstrated in a model scenario: Suppose we initializesome model of the glacial cycles with different initialconditions for the land ice volume and other variables,and run the model forward in time, simulating glacialcycles. Then all the different initial conditions areinﬂuenced by the same external Milankovitch clock,and therefore all of them will adjust their internaloscillation according to the pacing of the externalinsolation forcing; thus they will converge after sometime into a single time series in which the cycles aresomehow paced (Hays et al., 1976) by the Milankovitchforcing. This phase locking gives a new meaning to the‘‘pacemaker’’ idea: Milankovitch forcing is the pace-maker in the sense that glacial cycles are phase-locked toit. Such phase locking is likely the mechanism behind thesuccessful ﬁtting of model results to glacial records, forexample by Saltzman (1987) and by Paillard (1998), as
these models and the actual ice ages are likely both phaselocked to the Milankovitch variations in solar forcing.Now, the above discussion of phase locking mostlyreﬂects work done on the 100kyr cycles. The phaselocking requires the oscillations to be nonlinear, and the100kyr oscillations are indeed both large-amplitude andasymmetric (long glaciation followed by rapid deglacia-tions), both signs of nonlinearity. It seems that the40kyr oscillations are more often thought of as beinglinearly forced by Milankovitch forcing, especially byobliquity (Raymo and Nisancioglu, 2003; Tzipermanand Gildor, 2003).Clark and Pollard (1998) suggest that the areal extentof the Laurentide ice-sheet was approximately the samebefore and after the mid-Pleistocene transition, althoughthe maximal ice-volume was larger by a factor of 3/2during the 100kyr cycles (assuming the isotopic signal istaken as a mostly ice volume proxy). They thereforepropose that the ice-sheet of the 40kyr glacial cycles waslying on a thin (
30m) deformable regolith layer thatcould only support a small glacier bottom stress andthus a lower elevation of the ice-sheet (
2km).Successive ice-ages during the 40kyr glacial oscillationsremoved the soft layer and uncovered a hard bed-rockthat could support a larger bottom stress and a higherelevation ice-sheet (
3km). Based on above Clark andPollard (1998) developed a ice-ﬂow model to explain themid-Pleistocene transition. The 40kyr oscillations in thismodel are most likely due to Milankovitch variationsrather than being self-sustained. That is, the oscillationswould not exist without the Milankovitch variations.Raymo and Nisancioglu (2003) suggested that thenorthern hemisphere summer insolation gradient (domi-nated by the obliquity signal) somehow drives the ice-volume variations of the 40kyr oscillations. However, aswe will show below, the insolation gradient is symmetric(increases and decreases in time at the same rate) unlikethe ice-ages prior to the mid-Pleistocene transition,which are shown below to be asymmetric. Tzipermanand Gildor (2003) tried to attribute the mid-Pleistocenetransition to deep ocean cooling, and the 40kyroscillations as a linear response to Milankovitch forcing,although their results for the 40kyr oscillations did notmatch observations in some critical aspects. Addition-ally, recent studies (Ruddiman, 2003, 2004) related themid-Pleistocene transition to a global cooling trend, tolinear response to changes in Milankovitch forcing, andto feedback mechanisms related to greenhouse gases.We note that other nonlinear mechanisms (not necessa-rily self-sustained internal variability) such as (i)different sedimentation rates during glaciation/deglacia-tion periods or (ii) variable response time that results ina time varying phase of the climate system toMilankovitch forcing, may also produce the asymmetrywe observe for the 41kyr glacial cycles of the mid-Pleistocene.Nonlinearity does play an important role in somemodels for the 40kyr oscillations, even if the oscillationscannot exist without Milankovitch forcing. Examplesare the multiple-state model of Paillard (1998), wherethe mid-Pleistocene transition was simulated by assum-ing a gradual increase in the maximal ice-volumethreshold, and where a good agreement with the proxyrecords was found. Berger et al. (1999) used the morecomplex LLN-2D model (Galle´e et al., 1991, 1992) withdecreasing atmospheric CO
2
in the last 2.7Ma tosimulate the mid-Pleistocene transition. They showeda reasonable agreement with the data. Again, the
ARTICLE IN PRESS
2500 2000 1500 1000 500 0
Time
[kyr BP]
2345
δ
1 8
O
I c e V o l u m e
DSDP607
40kyr100kyr
Fig. 1. A typical record of benthic Foraminifera
d
18
O (Deep SeaDrilling Project (DSDP) Site 607, Raymo et al., 1989; Ruddiman et al.,1989) which is (at least partly) a proxy for global ice-volume; therecord is not orbitally tuned and the chronology is based on theassumption of constant sedimentation rates between several magneticreversals (see Raymo and Nisancioglu, 2003). While the time periodbetween 2.5 and 0.8Ma ago is characterized by frequent, lessdominant, and more symmetric ice-ages of
40kyr, the last 0.8Maare characterized by less frequent, more pronounced, and moreasymmetric ice-ages.
Y. Ashkenazy, E. Tziperman / Quaternary Science Reviews 23 (2004) 1879–1890
1880
oscillations in this model may only exist if Milankovitchforcing is present. Still, nonlinear interactions may, inprinciple, make the Milankovitch-driven oscillationseven before the mid-Pleistocene transition somewhatasymmetric. We note that the obliquity power of glacialcycles pre mid-Pleistocene is similar to obliquitypower post mid-Pleistocene while the 100kyr power ismuch more pronounced in the past 800kyr or so.This situation is plausible in both self-sustained mechan-ism and nonlinear response to Milankovitch forcingscenario.The objectives of the present study are to show thatthe 40kyr glacial oscillations are, in fact, asymmetricand hence to propose that these oscillations are self-sustained nonlinear oscillations rather than a linearresponse to Milankovitch forcing. We also discuss thepossible role of the Milankovitch forcing in affecting thephase and period of the glacial cycles. The highlyidealized model we use is derived from the sea-ice-switchmodel of Gildor and Tziperman (2000) and Tziperman
and Gildor (2003). To simulate the mid-Pleistocenetransition we use the mechanism of Clark and Pollard(1998) mentioned above.The paper is organized as followed. In Section 2, weanalyze the asymmetry of proxy records of the 40kyrglacial cycles. In Section 3, we present our ice-agesmodel and in Section 4, we use the model to discuss theabove objectives. We conclude in Section 5.
2. The asymmetry of the 40kyr glacial oscillations
In this section we analyze the asymmetry of the 40kyrice-ages (between 2.5 and 1Ma ago). We use the timeseries shown in Fig. 2a, which is a segment of a deep seaForaminifera
d
18
O proxy from site 607 of the Deep SeaDrilling Project (DSDP) (Raymo et al., 1989; Ruddimanet al., 1989), and which is presumably mostly a measureof global ice-volume. The time series was ﬁrst linearlyinterpolated to have equal time intervals of 1kyrbetween data points. The age model uses no orbitaltuning and is constructed by assuming constantsedimentation rates between several known magneticreversals (see Raymo and Nisancioglu, 2003). While theglacial–interglacial oscillations during 2.5–1Ma ago areapparently symmetric, a closer look shows that most of the ice-ages are asymmetric, with a slow buildup of theice-sheet that followed by a more rapid melting (somesuch asymmetric cycles are marked by arrows in Fig. 2).The asymmetry of a time series is stored in the Fourierphases of the series, which contain the informationregarding the time direction of the series. It is possible to‘‘destroy’’ the asymmetry of a time series by randomiz-ing its Fourier phases, leaving the power spectrum andthe probability distribution of the time series unchanged(Schreiber and Schmitz, 2000) (see Appendix A fordetails of the phase randomization procedure). Fig. 2bshows a so-called ‘‘surrogate’’ time series, that isobtained by randomizing the phases of the recordshown in Fig. 2a. The probability distribution and thepower spectrum of the srcinal data (Fig. 2a) and of thesurrogate, phased randomized, time series (Fig. 2b) areidentical as expected (Appendix A); see Figs. 2c and d.The surrogate time series are symmetric by construction.Many surrogate time series may be produced from agiven time series, and this allows us below to test thestatistical signiﬁcance of our results for the symmetry of the 40kyr glacial cycles.The asymmetry of an evenly sampled time series
x
i
can be measured as follows. Deﬁne an
l
-size incrementof the time series to be
ð
x
i
þ
l
x
i
Þ
. Count now thenumber of positive increments versus the number of negative increments of the time series for different sizes
l
. If the positive to negative ratio,
p
=
n
, is larger than onethen the series generally increases gradually anddecreases rapidly. Fig. 3a shows the positive to negativeratio of the 40kyr data shown in Fig. 2a as well as the
p
=
n
ratio of ten phase randomized, symmetric, surrogatetime series. The
p
=
n
ratio of the data is signiﬁcantlylarger than one while the
p
=
n
ratio of the surrogate(symmetric) time series is scattered around one. Theasymmetry ratio
p
=
n
of the data is clearly abovethe 90% conﬁdence level since it is located well outsidethe range of the
p
=
n
ratio of the ten surrogate data.
ARTICLE IN PRESS
2 2.5 3 3.5 4
δ
18
O
0300
C o u n t
Histogram
DataRand. data
234
δ
1 8
O
40kyr (DSDP 607)
0 0.02 0.04 0.06
Freq.,
f [1/kyr]
00.2
S(f)
Power Spec.
2500 2000 1500 1000
Time
[kyr BP]
23
δ
1 8
O
(a)
Data
(b)
Phase randomized data
I c e V o l u m e
41kyr
(c) (d)
Fig. 2. (a) A segment of the 40kyr oscillations from the record shownin Fig. 1. The arrows indicate some of the many asymmetric
40kyrglacial cycles with gradual glaciation followed by more rapiddeglaciation. (b) A surrogate series of the segment shown in (a). Wedestroy the asymmetry of the data by shufﬂing the Fourier phases,leaving the histogram (c) and the power spectrum (d) of the srcinaldata unchanged after the surrogate data test; see text and AppendixAppendix A for more details.
Y. Ashkenazy, E. Tziperman / Quaternary Science Reviews 23 (2004) 1879–1890
1881
Thus, in the 40kyr oscillations the ice-sheet buildupgradually and melts more rapidly, similar to the ice-agesof the past 0.8Ma. The
p
=
n
ratio also reﬂects theperiodicity of the data since for time lags larger than half a period (for the 40kyr ice ages, half a period is about20kyr) the ratio becomes smaller than one. We repeatedthe analysis for an additional record [Ocean DrillingProject (ODP) Site 677 (Shackleton et al., 1990)] andobtained similar results (Fig. 3b); the chronology thatwe use is based on a constant sedimentation rateassumption (4.24cm/kyr) for the past 2.5Ma.A previous study (Raymo and Nisancioglu, 2003)suggested that the 41kyr oscillations are driven by thevariations in the northern hemisphere summer insola-tion gradient due to obliquity changes. We calculate theasymmetry ratio
p
=
n
for the insolation gradient pre-sented in Raymo and Nisancioglu (2003) and ﬁnd that itis symmetric (Fig. 3c). Tziperman and Gildor (2003)
also suggested that the 41kyr oscillations may belinearly driven by the standard 65
N July insolation,yet we ﬁnd this insolation time series to be symmetric aswell.Fig. 3d shows the positive to negative ratio of anidealized sawtooth time series (increasing linearly for35kyr and decreasing linearly for 5kyr). This ﬁgureshows that indeed the
p
=
n
ratio reﬂects the asymmetry of the series, since for lag 1=1
p
=
n
¼
7 while the ratiobetween the increasing times and decreasing times is 35/5=7. The
p
=
n
ratio is periodic in the lag, as thesawtooth time series itself, and it becomes smaller than 1for lags higher than 20kyr (half of the period).It should be re-emphasized that given a symmetricMilankovitch forcing, a linear climate dynamics mustresult in symmetric climate signal. Given our ﬁndingabove that the 41kyr oscillations are signiﬁcantlyasymmetric, we thus conclude that ice-ages pre-mid-Pleistocene transition cannot be the result of a directlinear response to the insolation forcing. Our resultssuggest that ice-ages dynamics pre- and post-mid-Pleistocene transition may both be due to self-sustainednonlinear variability which is indeed typically asym-metric. If this is the case, then the 40kyr oscillationswould have existed regardless of the Milankovitchforcing. Admittedly, having asymmetric oscillationsdoes not rule out the scenario in which the 40kyr iceages are not self sustained, but rather are forced byMilankovitch variations in the insolation, and madeasymmetric by some nonlinear ice sheet or climatefeedbacks.As will be shown below using a simple model,Milankovitch forcing may still play an important rolein our proposed self-sustained glacial oscillations, insetting the phase and affecting the period of theoscillations during both the 40 and 100kyr ice ages.We ﬁnally note that (i) Hagelberg et al. (1991) and (ii)King (1996) demonstrated that the asymmetry of the ice-ages increased drastically after the mid-Pleistocenetransition. (iii) Raymo (1992) inspected for DSDP 607
d
18
O that ‘‘after 2.2Ma, rates of ice sheet decay almostalways exceed rates of ice growth for a given glacial-interglacial cycle.’’ (iv) Zhonghui and Herbert (2004)
ARTICLE IN PRESS
0.811.21.40.811.21.40.811.21.40 10 20 30 40 50 60
Lag,
l
1357
A s y m m e t r y ,
p / n
(d)
Artificial data
(b)
ODP 677
(a)
DSDP 607
(c)
Summer insolation 7025N
Fig. 3. (a) The asymmetry of the 40kyr oscillations proxy record(DSDP 607 shown in Fig. 2a), calculated as the ratio between thepositive increments and the negative increments for different time lags(solid line, see text for details). The same analysis for ten symmetric,surrogate phase randomized series is shown by dashed lines. Theasymmetry in the proxy time series is clearly statistically signiﬁcant asshown by the comparison with the surrogate data. The asymmetry of the data is well outside the shown vertical bars (marking mean
1standard deviation of the surrogate data) and outside the range of the10 surrogate data, indicating 90% conﬁdence level. Thus, the 40kyrice-ages are clearly asymmetric. (b) Same as (c) for ODP 677. (c) Thesame analysis applied to the average meridional gradient summerinsolation shows that this obliquity-dominated signal is symmetric.Thus, the 40kyr ice ages cannot be the result of a linear response toMilankovitch forcing. (Insolation (Laskar, 1990; Paillard et al., 1996)is averaged between the vernal and autumnal equinoxes, and thegradient is calculated as the difference between 70
N and 25
N. Ananalysis of the July 65
N insolation of Berger and Loutre (1991) showssimilar results.) (d) The asymmetry of an idealized sawtooth time seriesthat is linearly increasing for 35kyr and decreasing for 5kyr. The
p
=
n
ratio for lag 1 is 7, representing the ration between increasing time of 35kyr and the decreasing time of 5kyr. See text for details.
Y. Ashkenazy, E. Tziperman / Quaternary Science Reviews 23 (2004) 1879–1890
1882
reconstructed the SST of eastern equatorial Paciﬁc andpointed out the asymmetry of the temperature signalduring the glacial cycles even before the mid-Pleistocenetransition.
3. The model
We now use a simple model in order to examine thepossibility that self-sustained glacial oscillations areresponsible for the asymmetric 40kyr oscillations, andto study the possible role of Milankovitch forcing. Forthis purpose, we simplify the sea ice switch model of Gildor and Tziperman (2000) and Tziperman andGildor (2003), although the point we wish to make hereis independent of the sea ice switch mechanism. We referthe reader to the above papers for a detailed justiﬁcationof the model assumptions used below.Assume changes to northern hemisphere ice-volume
V
to be due to the difference between net snow precipita-tion over land
P
and total ablation
S
(melting, ice-sheetsurges, wind erosion etc.); ablation is assumed to dependonly on Milankovitch summer radiation at 65
N, so thatit equals
S
þ
S
M
I
ð
t
Þ
, where
I
ð
t
Þ
is the July insolation at65
N (Berger and Loutre, 1991) normalized to zero
mean and unit variance and
S
M
is a constant.We also assume that precipitation rate
P
is small forlarge ice-volume and large for small ice-volume. This isdue to the temperature precipitation feedback: as icesheets grow, their albedo effect cools the atmospherictemperature, and therefore reduces accumulation rate.A simple formulation of accumulation reducing with icevolume is of the form
P
ð
no sea ice
Þ¼
p
0
kV
, where
p
0
and
k
are constants:
p
0
is the precipitation rate when theice-sheets are completely melted and
k
is the growth rateconstant of the ice-sheet.Following Gildor and Tziperman (2000) we assumethat when the ice-volume reaches a certain speciﬁedmaximal ice-volume
V
max
, the atmospheric temperaturebecomes sufﬁciently low such that a signiﬁcant sea-icecover rapidly forms. Precipitation over land ice thenreduces very sharply because (i) drop in temperature dueto the increased albedo of the sea-ice, (ii) reducedevaporation due to insulating sea-ice cover, and (iii) thestorm track is shifted by the sea ice away from the ice-sheet. Thus accumulation in the presence of sea ice maybe written as
P
ð
with sea ice
Þ¼ð
p
0
kV
Þð
1
a
si
on
Þ
where
a
si
on
is the relative area of the sea ice when‘‘on’’. This reduced accumulation in the presence of seaice results in land ice withdrawing. When land ice-volume drops below a certain minimal ice-volume
V
min
(resulting in warming due to ice albedo feedback) thesea-ice melts rapidly and precipitation returns to itssrcinal rate without sea ice.Finally, combining the above expressions for theablation and accumulation, the ice-volume mass balancemay be written asd
V
d
t
¼ð
p
0
kV
Þð
1
a
si
Þ
S
S
M
I
ð
t
Þ
;
ð
1
Þ
where
a
si
is the relative area of the sea-ice(
a
si
¼
a
si
on
4
0 when sea-ice is ‘‘on’’ and
a
si
¼
0 whensea-ice is ‘‘off’’).A technical point regarding the model nonlinearitymay be due now. Eq. (1) seems linear in the ice volume
V
. However, because of the dependence of the sea icearea
a
si
on the ice volume at the two threshold points
V
max
and
V
min
, the model is, in fact, nonlinear. Thenonlinearity is represented in this ‘‘piecewise linear’’formulation, where the nonlinearity only occurs at twopoints. In any case, the model solution below shows self-sustained oscillations which are only possible in non-linear models (Fig. 4). Appendix B brieﬂy explains thedifference between self-sustained, nonlinear, and linearoscillations in the present context.We note that the model ignores the red spectrum of the climate records (Kominz and Pisias, 1979; Pelletier,1997; Ashkenazy et al., 2003; Wunsch, 2003) whichimplies that noise must play an important role. Suchnoise may be added to this model, yet is not needed formaking our main point here.There are seven parameters for the model (1):
V
min
,
V
max
,
a
si
on
,
p
0
,
k
,
S
, and
S
M
(Table 1). However, it ispossible to switch to normalized ice-volume unitswithout changing the model’s dynamics (e.g. by dividing
V
by
V
min
). This leaves six independent modelparameters.The minimal ice-volume threshold is approximated by
V
min
¼
3
10
6
km
3
which is approximately the presentday northern hemisphere ice-volume (Hartmann, 1994).The maximal ice-volume is set to
V
max
¼
45
10
6
km
3
;the total ice-volume increase during the last glacialmaximum is
50
10
6
km
3
(Mix et al., 2001) and weassume that most of it (42
10
6
km
3
) accumulated inthe northern hemisphere ice-sheets and the rest(8
10
6
km
3
) in the southern hemisphere ice-sheet and
ARTICLE IN PRESS
400 300 200 100 0
Time
[kyr BP]
01020304050
V ( t ) [ 1 0
6
k m
3
]
V
min
V
max
g l a c i a t i o n
d e g l a c i a t i o n
Fig. 4. A time series of the model’s ice volume without Milankovitchinsolation forcing. The ice volume,
V
ð
t
Þ
, grows when sea-ice cover isabsent, till it reaches a maximal threshold
V
max
. Then the sea-ice coveris assumed to be extensive, and the land ice retreats due to ablationuntil it reaches the minimal ice volume threshold
V
min
.
Y. Ashkenazy, E. Tziperman / Quaternary Science Reviews 23 (2004) 1879–1890
1883

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