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important differences that require further exploration, one of
them being the fact that the ice core records (e.g., NGRIP,
GRIP, and GISP2) show self-similarity [Rial and Yang,
2007], which suggests the presence of internal periodic os-
cillations. For instance, Figure 4 shows a reconstruction of
the three-dimensional phase-space trajectory of NGRIP us-
ing Taken
shall see shortly, these particular periods may result from the
distortion, for example, through frequency modulation, of
the original period of free oscillation if it exists. It has been
speculated that a possible source of oscillations in the D-O is
the enigmatic and pervasive 1500 ± 500 year climate cycle
discovered by Bond et al. [1997], though its origin is not
known [Ganopolsky and Rahmstorf, 2002; Alley et al.,
2003]. There is a short segment of the ice core record (32
-
19 ka) (ka = kiloyear ago), where insolation is nearest its
mean value and thus may have its least effect on the climate
s delay method [Williams, 1997]. Comparing its
phase-space trajectory with the limit cycle (Figure 4b) of a
periodic van der Pol oscillator (which is basically SIO, as
explained in Appendix A) brings about similarities between
the climate processes and the model that we will fully exploit
in what follows. As to which periodicities exist in the data,
there is some evidence in the singular-spectrum analysis
(SSA) of the high-pass-filtered series of several ice core
records (Figure 5 and Table 1). The presence of statistically
signi
'
s
intrinsic temperature variability, which means that the under-
lying ~1.5 kyr oscillation may be free from distortion. Figure
5b shows that the multitaper power spectrum of the NGRIP
record in that time interval exhibits peaks at periods of ~1629
and ~1358 years, just within Bond
'
s range.
The van der Pol equation-based SIO becomes helpful here,
since it is one step higher in complexity from the Langevin
model of Figure 3. SIO is a set of two
'
cant oscillatory frequencies corroborates the idea that
periodicities are important part of the D-O climate fluctua-
tions. SSA is a nonparametric method [Ghil et al., 2002] that
uses data-adaptive basis functions. Data-adaptive eigen-
modes can capture the basic periodicity of an irregular wave-
form without the many overtones needed in a regular Fourier
analysis. Accordingly, in the millennial band, two pairs of
oscillatory eigenvalues (each pair has the same frequency)
with periods ~4.6 and ~1.67 kyr lay outside the 95% percen-
tile, so that the null hypothesis that these eigenvalues are red
noise can be rejected with this confidence. However, as we
first-order differential
equations describing self-sustaining nonlinear oscillations of
sea ice extent and mean ocean temperature. From actual
climatic parameters, Saltzman and Moritz [1980] estimated
that the oscillator
s natural period would be in the range
1000
-
3000. Thus, it appears reasonable to prescribe natural
periods for the mean ocean temperature in the range 1500 ±
500 years in SIO. When this is done, synthetic time series are
obtained that resemble the NGRIP record, as shown in Fig-
ures 6 and 7. Here we used noise level D = 0.45; insolation
level A = 0.5; a = b = 1; and c = 0 (see Appendix A). The best
fits with the data are obtained for periods T =2
π
/
Ω
= 1400
-
1600 years, and insolation amplitude and noise levels com-
mensurate with that used in the Langevin model.
That SIO closely mimics the data leads to the idea that few
rules may exist governing the long-term evolution of climate
and that those could be understood from
'
Table 1.
Singular Spectrum Analyses of the Ice Core Data Time
Series
a
SSA Window
(0.05 kyr
sample
1
)
95% Significant
EV Pairs
(cycle kyr
1
)
Record
Name
Period
(kyr)
NGRIP 150 0.24/0.60 4.6/1.67
200 0.23/0.61 4.34/1.64
250 0.225/0.625 4.44/1.60
GRIPss09 150 0.22
-
0.24/0.66 4.54
-
4.16/1.51
200 0.24/0.68 4.16/1.47
250 0.24/0.66
-
0.68 4.16/1.51
-
1.47
GISP2 150 0.2
-
0.22/0.68 5
-
4.54/1.47
200 0.2/0.68 5.0/1.47
250 0.2/0.68 5.0/1.47
a
Monte Carlo singular spectral analysis (SSA) [Ghil et al., 2002]
of ice core time series reveals that around ~4.5 and ~1.5 ka, there are
signi
first principles. If
this is true, the D-O just reflects the (nonlinear) interactions
among sea ice, ocean temperature, and insolation during the
last ice age. It is therefore not unreasonable to state that when
it comes to understanding the essential physics, large and
complex codes like ECBILT-CLIO may be less useful than
much simpler ones.
5. RELATION BETWEEN SURFACE AND DEEP
OCEAN TEMPERATURE HISTORIES
cant eigenvalue pairs that indicate nonlinear anharmonic sig-
nals for which the null hypothesis (red noise) can be rejected at the
95% level of con
SIO produces time series for mean ocean temperature
and sea ice extent consistent with well-known properties of
proxy data, for instance, from sediment core MD95-2042
collected off Portugal [Cayre et al., 1999]. Figure 8 shows
a planktonic
dence. The results are robust to changes in
window length (should be less than N/5, where N is the total number
of points). Sampling rate is 50 years and N = 2000. Timing errors
and different age models account for the main differences in the
detected periods among ice cores. Figure 5a shows an example of
SSA for NGRIP.
18
O proxy for SST with the characteristic
“
square wave
”
or saw-toothed shape of the Greenland prox-
ies and the benthic
δ
18
O record with the also characteristic
δ
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