Download SYNCHRONIZATION OF POLAR CLIMATE VARIABILITY OVER

[American Journal of Science, Vol. 312, April, 2012, P. 417– 448, DOI 10.2475/04.2012.02]
SYNCHRONIZATION OF POLAR CLIMATE VARIABILITY OVER THE
LAST ICE AGE: IN SEARCH OF SIMPLE RULES AT THE HEART OF
CLIMATE’S COMPLEXITY
J. A. RIAL
Department of Geological Sciences, University of North Carolina, Chapel Hill, North
Carolina 27599-3315; [email protected]
ABSTRACT. Evidence is presented supporting the hypothesis of polar synchronization, which states that during the last ice age, and likely in earlier times, millennial-scale
temperature changes of the north and south Polar Regions were coupled and synchronized. The term synchronization as used here describes how two or more coupled
nonlinear oscillators adjust their (initially different) natural rhythms to a common
frequency and constant relative phase. In the case of the Polar Regions heat and mass
transfer through the intervening ocean and atmosphere provided the coupling. As a
working hypothesis, polar synchronization brings new insights into the dynamic
processes that link Greenland’s Dansgaard-Oeschger (DO) abrupt temperature fluctuations to Antarctic temperature variability. It is shown that, consistent with the presence
of polar synchronization, the time series of the most representative abrupt climate
events of the last glaciation recorded in Greenland and Antarctica can be transformed
into one another by a ␲/2 phase shift, with Antarctica temperature variations leading
Greenland’s. This, plus the fact that remarkable close simulations of the time series are
obtained with a model consisting of a few nonlinear differential equations suggest the
intriguing possibility that there are simple rules governing the complex behavior of
global paleoclimate.
Key words: Polar climate synchronization, Greenland-Antarctica teleconnection,
climatic oscillators, complex systems, coupled Polar Regions, polar phase shift, dynamic paleoclimatology
introduction
Synchronization is a widespread phenomenon in complex biological and ecological systems. Some of the most advanced areas of research in neuroscience include the
study of synchrony among far away regions of the brain and its behavioral meaning
(Varela and others, 2001; Buzsáki, 2006). Available evidence suggests that synchronization is also a common process in the earth’s climate but research on detecting its
presence and documenting its consequences is still at a very early stage (Read and
Castrejón-Pita, 2010). Though synchronization has mainly been studied in relatively
low-dimensional discrete systems, in limited size networks, and in small physical
models (for instance, Castrejón-Pita and Read, 2010), the possibility that similar
dynamics may be occurring in the extended, multidimensional earth’s climate system
is stimulating new fields of research (for instance, Donges and others, 2009).
Detection of synchronization among paleoclimate time series can likewise help
explain hitherto incompletely understood processes involving the occurrence and
timing of abrupt climatic change. An example of possible synchronization in the
paleoclimate is the oscillation in the North Pacific deep-water composition, with cycles
similar to those in the North Atlantic (Lund and Mix, 1998). In the annual to decadal
time scale, examples of climatic teleconnections that could possibly be associated to
synchronization include the El Niño–Southern Oscillation (ENSO)/Indian Monsoon
teleconnection (Maraun and Kurths, 2005; Tsonis and others, 2007), the climatic
connections between the North Atlantic, the Ethiopian Plateau, and the Mediterranean reported by Feliks and others (2010), the Atlantic-Pacific rebounding caused by
freshwater fluxes (Seidov and Haupt, 2003, 2005), and the Atlantic-Pacific seesaw
(Saenko and others, 2004).
417
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J. A. Rial—Synchronization of polar climate variability over the last
The climate system is complex, whatever our definition of system complexity may
be, but if that complexity can be reduced in some measure by detecting and
recognizing long-range symmetries caused by synchronization, our understanding of
climate dynamics would greatly benefit.
The focus of this paper is on the last ice age’s millennial to centennial (subMilankovitch) rapid temperature variations recorded by stable isotope temperature
proxies in the ice cores of Greenland and Antarctica. These records exhibit large,
sudden warming events (up to 15 °C warming in high latitudes, 4-5 °C in mid-latitudes)
occurring at least 24 times during the last 100 ky (1 ky⫽1000 years; 1 ka ⫽ 1000 years
ago). Generally known as the Dansgaard-Oeschger (DO) temperature fluctuations
(Dansgaard and others, 1993), the transition to the high temperature state in the DO
occurs in human timescales (years or decades), and hence the urgency in understanding their origin and dynamics. Many researchers have proposed mechanisms for the
DO (for relevant reviews see Alley, 2007; Clement and Paterson, 2008, and references
thereof) including deep decoupling, nonlinear thresholds in ocean circulation modes,
stochastic resonance, freshwater discharges, salt and/or sea ice oscillators, et cetera
(Broecker and others, 1990; Winton and Sarachik, 1993; Winton, 1995; Manabe and
Stouffer, 1997; Ganopolski and Rahmstorf, 2001; Alley and others, 2001; Seidov and
Maslin, 2001; Schulz and others, 2002; Knutti and others, 2004; Ditlevsen and others,
2005; Stastna and Peltier, 2007; Rial and Saha, 2011). However, the causes of the DO
and abrupt climate change remain elusive (Broecker and others, 2010; Lozier, 2010).
Of particular interest in this paper is how the northern Bølling Allerød–Younger
Dryas (BA-YD), sequence (15ka-11ka) is connected to Antarctica temperature variations (particularly to the Antarctic Cold Reversal, or ACR), a topic that has received
relatively little attention even though the BA-YD is likely the most studied abrupt
climate change sequence in the paleoclimate record (Broecker and others, 2010).
Data analyses and modeling results to be introduced below indicate that the BA-YD
temperature oscillation as recorded in Greenland is frequency entrained and in ␲/2
phase lock synchrony with Antarctica’s ACR. Results also suggest that polar synchronization is a pacemaker for the millennial scale fluctuations of the DO, but does not
appear to cause the abruptness of temperature change. According to the evidence
discussed in the following pages, the abruptness is likely due to the rapid response of
Arctic sea ice to changes in ocean temperature, which are in turn driven by synchronization with the Antarctic. Polar synchronization is consistent with the hemispheric
bipolar seesaw (Crowley, 1992; Broecker, 1998; Stocker and Johnsen, 2003) but it also
explains why proxy records from tropical and subtropical latitudes show surface ocean
temperature variations similar to Greenland’s while deep ocean temperature variations are similar to Antarctica’s (Shackleton and others, 2000; Saikku and others,
2009). The seesaw hypothesis states that abrupt warming episodes in the North Atlantic
occur at or near the beginning of gradual cooling in Antarctica, implying interaction
of the polar climates through meridional heat transport and North Atlantic deep water
(NADW) production. However, as will be discussed in the Synchronization and the Bipolar
Seesaw section below, there is currently no model of the seesaw with enough explanatory power.
It is important that the terminology used is clearly defined. Here polar synchronization refers to frequency entrainment and phase lock between records of temperature variations from the Polar Regions, while some authors refer to the same records as
being asynchronous (for instance, Blunier and others, 1998; Stenni and others, 2011).
There is no contradiction in these terms since those authors use asynchronous to mean
that the temperature variations of the Polar Regions are locked out of phase, which in
fact happens in the presence of synchronization, as will be discussed shortly. However,
to avoid confusion the term asynchronous will not be used in what follows. In addition,
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ice age: in search of simple rules at the heart of climate’s complexity
1
2
1
3 4 5
67 8
10
9
11
18
16 17
12
13
19
δ18O
20
Greenland
14
15
NGRIP
Normalized amplitudes
0
-1
1
t
∫ NGRIP(ξ)dξ
0
-1
A1
A2
A3 A4
A5 A6
A7
δD , δ18O
Antarctica
1
EDC3deut
0
-1
0
20
40
60
80
100
KYears BP
Fig. 1. High pass filtered time series (periods longer than 16 ky are attenuated) of NGRIP (Greenland)
and Antarctica’s EDC3 deuterium time series compared to the time integral function (middle panel) of
NGRIP suggests a close dynamic relationship between the climates of the Polar Regions. The vertical bars
show how prominent Dansgaard-Oeschger (DO) events lag behind the temperature peaks in the Antarctic
record (labels A1-A7), a relationship replicated by the numerically computed time integral function of
NGRIP. Numbering of DO events is conventional. The time integral is performed from the past to the
present and inverted, giving Antarctica a lead phase shift of ␲/2. Raw data provided by the National Snow
and Ice Data Center, University of Colorado at Boulder, and the WDC-A for Paleoclimatology. The EDC3
time series is tuned to the integral of NGRIP.
polar synchronization is unrelated to the technique of synchronizing age models using
methane records from both Polar Regions (Blunier and Brook, 2001).
The paper is organized as follows: In the second section of the paper, titled the
polar phase shift, time series of temperature variations from the Polar Regions are
shown to transform into one another through a ⫾␲/2 phase shift, which is shown to be
fully consistent with polar synchronization. The third section of the paper, titled a
polar synchronization model, introduces and validates a simplified climate model
that closely reproduces temperature variations in Greenland and Antarctica in the
presence of polar synchronization. The fourth section of the paper, titled synchronization as a complexity-reducing, self-organizing mechanism, explains how
polar synchronization can be viewed as a self-organizing mechanism of climate change,
and the last section, discussion and concluding remarks , discusses the connection
of Heinrich events to polar synchronization.
the polar phase shift
Previous studies have suggested that the time series of stable isotope temperature
proxies from Greenland and Antarctica somehow depend on each other (Blunier and
Brook, 2001). For instance, findings by EPICA Community Members (2006) show that
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J. A. Rial—Synchronization of polar climate variability over the last
A
North-South connection
ANTARCTICA (Dome C)
GREENLAND (NGRIP)
∆φ = 0
1
∆φ = 0
1
0
0
−1
−1
∆φ = –π/6
1
∆φ = + π/6
Normalized amplitudes
Normalized amplitudes
1
0
−1
∆φ = – π/3
1
0
−1
0
−1
∆φ = + π/3
1
0
−1
∆φ = – π/2
1
∆φ = + π/2
1
0
0
−1
−1
GREENLAND (NGRIP)
0
10
20
30
40
ANTARCTICA (DomeC)
50
60
70
80
90
100
10
0
20
30
40
Kyears BP
3
2
1..5
1
5ky – 50ky...72ky
[a(t + φ) g(t)]2
[ a2 (t) g2 (t) ]1/ 2
90
100
30ky - 65ky
2..5
Similarity Function S2
2..5
5ky - 65ky
Similarity Function S2
Similarity Function S2
80
3
S 2 (φ) =
2
1.5
1
0..5
-π/3 -π/2 -2π/3
70
a(t) : Antarctica
g(t) : Greenland
Similarity function
3
0..5
0
60
(NGRIP age model)
B
2..5
50
Kyears BP
(Dome C age model)
15ky ... 30ky – 72ky
0
-π
2
1.5
1
30ky – 50ky...72ky
0..5
-π/3 - π/2 -2π/3
-π/3 - π/2 -2π/3
0
-π
-π
phase φ
phase φ
phase φ
Hilbert transform pair
a(t) and H-1[g(t)]
1
0.8
0.6
0.4
0.2
a(t) ~ H-1[g(t)]
0
−0.2
−0.4
0
10
20
30
40
50
60
70
80
90
100
g(t) and H[a(t)]
0.6
0.4
0.2
g(t) ~ H[a(t)]
0
−0.2
−0.4
GRIP age model
0
10
20
30
40
50
60
70
80
90
100
KYears BP
Fig. 2. (A) The connection between the North and South proxy records of Antarctica (DomeC) and
Greenland (NGRIP): A phase shift of ⫺␲/2 radians applied to the Antarctica record closely reproduces
Greenland’s, and conversely, a phase shift of ⫹␲/2 radians applied to the Greenland record reproduces
Antarctica’s. As in figure 1, the records are high-pass filtered to eliminate periods longer than 16 ky, so the
astronomical insolation frequencies are filtered out, though not necessarily their nonlinear effects. The
phase shifts on each step are applied to each frequency component by using the analytic function method of
Gabor (1946). Note the convention that time increases from right to left, so that Antarctica “leads”
ice age: in search of simple rules at the heart of climate’s complexity
421
the amplitude of Antarctic warm events depends linearly on the duration of the
accompanying stadial interval of the Greenland record during the MIS3 interval 50 ka
to 30 ka. Although the MIS5 interval appears not to follow this trend (Capron and
others, 2010), they also argue that for most of the last glaciation there is a “one-to-one”
assignment of each Antarctic warming with a corresponding stadial in Greenland.
Blunier and others (1998) and Blunier and Brook (2001) show that Greenland climate
change events lag Antarctica’s by 1 to 3 ky (1 ky⫽1000 years) and suggest a dominant
role of the ocean in controlling the past climate of both Polar Regions. In addition,
there is data from the South Atlantic (Barker and others, 2009) showing rapid
temperature changes simultaneous and of the opposite sign to those in the north,
suggesting a link between the abrupt DO temperature fluctuations in the Arctic and
sub-Antarctic temperature variation. Finally, evidence supporting climatic interaction
between the poles is the ␲/2 (90°) phase shift at which the Antarctic time series leads
the Arctic records, a result obtained from increasingly refined age models and
matched methane records from both poles (Steig, 2006).
This Antarctic ␲/2 phase lead can be displayed by calculating the time integral
function (or cumulative sum) of the ␦18O temperature proxy from Greenland’s
NGRIP ice core, as shown in figure 1. Except for small differences attributable to errors
in age models the Antarctic Epica DomeC ␦18O temperature proxy (bottom panel)
closely follows the time integral of NGRIP (central panel) over the last glaciation
(other pairs of N-S proxies show similar high correlations). In the frequency domain
time integration results in a ␲/2 phase lead of all Fourier components of the
integrated record with respect to the original record as well as a low-pass filter effect
(each Fourier component is divided by its frequency). The fit in figure 1 is close
enough that allows the integral of NGRIP to be used as an approximate proxy for
Antarctic temperature variation. This is useful because the NGRIP and its time integral
share the same chronology, and thus can be added to each other (see the Heinrich
Events section), or combined to draw the trajectory in phase space that describes the
approximate N-S dynamics, albeit without using the southern polar record.
One possible interpretation of figure 1 is that the abrupt northern fluctuations are
proportional to the time rate of change of southern temperatures, a seemingly causal
relationship in which Antarctica appears as the driver. However, on close analysis it is
more likely that the phase shift occurs as consequence of polar synchronization. This is
because though time integration induces the observed ␲/2 phase lead of Antarctica,
the amplitudes obtained by time integrating the Greenland record (or timedifferentiating the Antarctic record) are too large (or too small) compared to the
observed. In contrast, the ␲/2 phase shift introduced by synchronization is consistent
with the observations as it preserves the amplitude and energy levels of both records,
and no additional assumptions are needed to explain the large amplitude differences
resulting from integration (or differentiation). Synchronization means however that
there is no discernible cause-effect relationship.
Instead of integrating, a contrasting approach shows (fig. 2A) that by manipulat-
Greenland by ⫹␲/2 radians, while Greenland’s lag is ⫺␲/2 radians. Raw data provided by the National Snow
and Ice Data Center, University of Colorado at Boulder, and the WDC-A for Paleoclimatology and Jouzel and
others (2007). (B) The similarity function S2 (Rosenblum and others, 1997) is computed to confirm that the
shifted Antarctica record is close to the Greenland record, as shown qualitatively in figure 2A. The similarity
function is clearly minimized as the phase shift approaches ⫺␲/2 for different time windows (for example, 5
ky ⫺ 50 ky . . . 72 ky indicates a start time, the extent of the main interval (dash) and longer windows
spanning the interval (dots) till the end time). For reference, the dashed curve with minimum at ⫺␲/2 is the
similarity function computed for a sine/cosine pair. The Antarctica and Greenland age models are used
without modification in calculating S2. The time window beyond 72 ka is not used in calculating S2 because
the two age models differ substantially in the interval 70 –75 ka. The Hilbert transform pair panel uses tuned
records, so the timings in that interval are consistent.
422
J. A. Rial—Synchronization of polar climate variability over the last
ing their phase, it is possible to approximately reproduce the Greenland ice core
record from that of Antarctica by a phase shift amounting to ⫺␲/2 radians, and
conversely, to obtain the Antarctica core from that of Greenland by a ⫹␲/2 shift (the
shift applies to all frequency components). In figure 2B the similarity function S2
(Rosenblum and others, 1997) provides a quantitative measure of how close the
records of a(t) (Antarctica) and g(t) (Greenland) are after the phase of Antarctica is
shifted by and angle ␾ between 0 and ⫺␲. The figure shows that the minimum of the
S2(␾) function occurs, as expected, at or near ␾ ⫽ ⫺␲/2, indicating that such shift
makes the two time series the most similar. A number of time windows are tested and
all show minima at or near ⫺␲/2.
Based upon these results, a more formal description of the phase relationship
between the north and south records can be made: As shown in the lower half of figure
2B, the close match of a(t) (Antarctica) to the inverse Hilbert transform (⫹␲/2 shift)
of g(t) (Greenland), and the close match of g(t) to the Hilbert transform (⫺␲/2 shift)
of a(t), show that the polar records form, approximately, a Hilbert transform pair (see
Appendix A for a definition). This relationship is key to the analysis that follows.
Huygens’ Synchronizing Clocks
I hypothesize that the simplest explanation for the foregoing observations is that
the temperature variations of the Polar Regions over the last ice age, and likely for
earlier times, were coupled and synchronized. Accordingly, the polar climates are
thought of as two non-identical relaxation oscillators, each subtly influencing and
influenced by the other through the intervening ocean and atmosphere. Over time,
the interaction develops into a robust global scale oscillation. The close analogy with
Christiaan Huygens’ famous observations (Huygens, 1673) describing the (nonlinear)
synchronization of pendulum clocks is relevant as well as illustrative. As Huygens noted
after lengthy contemplation of numerous experimental setups, two pendulum clocks,
coupled by the weak elastic forces they exert on each other through the wooden frame
or wall on which they hung, always become synchronized, no matter how different the
initial conditions, and so long as the natural periods of their pendulums are not too
different. Through the slight motions induced on their support, each clock gently
forces the other, speeding it up or slowing it down in a continuously evolving feedback
loop until synchrony is attained, which is the lowest energy state of the oscillating
two-clock system (Bennett and others, 2002). The clocks’ synchronized state is robust;
if disturbed by some interference it re-establishes itself in a short time. Even if the
coupling is very weak, the clocks eventually synchronize and settle to a common
frequency (entrainment) and constant phase lock, which can be either in-phase,
anti-phase, or arbitrary but constant out-of-phase (Pikovsky and others, 2001; Izhikevich, 2006). In greater detail, the two most prominent examples of abrupt climate
change in the last glaciation, namely, the 18 ka to 8 ka interval, that includes the BA-YD
sequence, and the 78 ka to 68 ka interval, that contains the two largest consecutive
warming events of the entire record, are transformed into one another by ⫾␲/2 phase
shifts (fig. 3). The transformations in figure 3 are performed using the Hilbert
transform. As before, a ⫺␲/2 phase shift corresponds to the direct Hilbert transform
and a ⫹␲/2 phase shift to the inverse Hilbert transform (time runs from right to left).
Polar Synchronization: Simple Rules at the Heart of Climate’s Complexity
A semblance of simplicity in the otherwise complex climate system recordings has
been shown in figures 1 through 3. From these figures, one can expect that if age
models were perfect and noise inexistent the polar records would be exact Hilbert
transform pairs, and consequently, would satisfy strict synchronization conditions. This
can be theoretically demonstrated as follows: First, using the concept of analytic
423
ice age: in search of simple rules at the heart of climate’s complexity
(B)
4
Antarctica
ACR
1
Vostok - π/2
Normalized amplitudes
δ18O (Normalized)
(A)
0
-1
DomeC
1
3
2
1
NGRIP
0
0
8
-1
10
12
1 DomeC – π/2
16
18
(C)
0
4
Normalized amplitudes
Normalized amplitudes
14
Kyears BP
-1
BA
1
YD
NGRIP
0
Greenland
-1
Vostok
3
2
1
0
NGRIP + π/2
5
10
15
8
20
10
12
14
16
18
Kyears BP
Kyears BP
(E)
(D)
DomeC – π/2
6
6
NGRIP + π/2
Normalized amplitudes
Normalized amplitudes
4
2
0
-2
NGRIP
72.5
77.5
Kyears BP
4
2
0
DomeC
-2
82.5
72.5
77.5
82.5
Kyears BP
Fig. 3. Details from figure 2: (A) The transformation from the Antarctica record to Greenland’s
requires a phase shift equivalent to that of a Hilbert transform. The DomeC band pass component
(periods shorter than 16 ky and longer than 1 ky) is reconstructed using the SSA method (Ghil and
others, 2002). The transformation of the Antarctica Cold Reversal (ACR) into the Bølling-AllerødYounger Dryas (BA-YD) by the application of a ⫺␲/2 phase shift, or the Hilbert transform operator
(Bracewell, 1986) is consistent with N-S synchronization. (B) With the convention that time runs from past to
present, NGRIP closely resembles the Hilbert transform of the Vostok record (⫺␲/2 shift), and conversely,
in (C) Vostok resembles the inverse Hilbert transform of NGRIP (⫹␲/2 shift). After the BA-YD, the next
most significant example of abrupt climate change occurs around 75 ka (D, E). Note the convention: a
positive phase shift means a shift to earlier times. Antarctica’s lead over Greenland is ⫹␲/2. In (D) and (E)
the records are tuned to the NGRIP age model.
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J. A. Rial—Synchronization of polar climate variability over the last
40
35
|∆φ| of surrogate pairs
30
Unwrapped Phase (rad)
(A)
|∆φ| = |Tan-1[H(a)/a] — Tan-1[H(g)/g] |
25
20
15
10
5
<-- +π/2
0
−5
−10
filter band 0.067 -- 0.437 ky-1
0
10
20
30
40
50
60
70
80
90
100
Kyears BP
(B)
(C)
Frequency
Filter bandwidth 0.022ky-1
Filter bandwidth 0.042ky-1
1200
600
1000
500
800
400
600
300
400
200
200
100
0
−2π
0
−π
−π/2
0
π/2
π
2π
−2π
−π
−π/2
0
π/2
π
2π
π/2
π
2π
∆φ
∆φ
140
150
Frequency
120
100
100
80
60
50
40
20
−2π
−π
−π/2
0
π/2
π
∆φ of surrogate pairs
2π
−2π
−π
−π/2
0
∆φ of surrogate pairs
Fig. 4. (A) Absolute values of the phase difference between the Antarctic record (Dome C) and a
Greenland record (GISP2) using methane-based age models. Mean values are depicted with solid lines and
standard deviation by dashed lines. A bank of 28 band pass filters was used on both the actual and 1000 pairs
of surrogate data with similar spectral content. The Hilbert transform is applied to the resulting narrow-band
time series. The mean value of the (absolute) phase difference of the data is close to ␲/2 for most of the
100 ky of the record although it seems to drift to higher values after 75 ka, possibly a consequence of
differing age models. The surrogate data mean phase difference is highly variable, its mean is not
constant, and far from ␲/2. (B) Histograms of the phase difference for the two data time series (top), and the
ice age: in search of simple rules at the heart of climate’s complexity
425
function (Gabor, 1946; Balanov and others, 2009) construct analytic functions A(t)
and G(t) for Antarctica and Greenland respectively in the form
A共t兲 ⫽ a共t兲 ⫹ iH 关a共t兲兴,
G共t兲 ⫽ g共t兲 ⫹ iH 关g共t兲兴
where a(t) and g(t) are the observed proxy time series of Antarctica and Greenland
respectively. Here i ⫽ 公⫺1, and H[ 䡠 ] is the Hilbert transform of its argument. By
definition the instantaneous phases of the functions a(t) and g(t) are, respectively,
⌽A ⫽ Tan ⫺ 1共H 关a兴/a兲 and ⌽G ⫽ Tan ⫺ 1共H 关g兴/g兲
where the capitalized inverse tangent stands for the unwrapped phase. The condition
for 1:1 phase synchronization is that1
兩 ⌽A ⫺ ⌽G兩 ⱕ constant
(1)
(see for instance Varela and others, 2001; Balanov and others, 2009) for all time or, at
least, several millennia. On the other hand the observed relationship (figs. 2 and 3)
between a(t) and g(t), and the analysis of their phase relationship described above
allows us to write that, approximately, H[a(t)] ⫽ g(t). Then, by the properties of the
Hilbert transform, we can write that H[H[a(t)]] ⫽ ⫺a(t) from which it easily follows
that H[g(t)] ⫽ ⫺a(t), and therefore that H ⫺1[g(t)] ⫽ a(t). The Hilbert transform of
the Antarctic record is equal to the Greenland record, and conversely, the inverse
Hilbert transform of the Greenland record reproduces the Antarctic. Introducing
these results in condition (1) gives 兩⌽A ⫺ ⌽G兩 ⫽ ␲/2, since 兩tan⫺1(1/x) ⫹ tan⫺1x兩 ⫽
␲/2. This means that the two signals a(t) and g(t) are ␲/2 phase-locked, which in this
case also implies that the instantaneous frequencies of a(t) and g(t) are the same for all
time. This is because the instantaneous frequency is given by the time rate of change of
the phase, or ␻ ⫽ d⌽/dt, so that for all time t
␻A ⫽ d⌽A/dt ⫽ ␲共关da/dt兴g ⫺ a关dg/dt兴兲/共a2 ⫹ g2兲 ⫽ d⌽G/dt ⫽ ␻G
(2)
which means 1:1 frequency entrainment. These relationships only occur if a(t) and g(t)
are 1:1 synchronized (Izhikevich, 2006; Balanov and others, 2009).
It could be argued that two components of the same oscillatory system (for
instance, the trivial case of amplitude and velocity of a simple harmonic oscillator, a
sine-cosine pair) will also satisfy synchronization conditions. However, this would
happen for a single oscillatory system, in which case the definition of synchronization
does not apply (Pikovsky and others, 2001). In the case of the polar signals (figs. 2 and
3), it is reasonable to assume that the polar climates were originally two separate
systems, each self-sustaining, each capable of generating its own different rhythms, and
that through ocean/atmosphere coupling and the influence of long term astronomi1
A more general condition than (1) is 兩m⌽A ⫺ n⌽G兩 ⱕ constant, where m and n are integers. If n ⫽ 1 or
m ⫽ 1 the phases of different harmonics are close to each other. Close inspection of the paleoclimate data
suggests that m ⫽ n ⫽ 1.
surrogate data (bottom). The 28 band pass filters have a constant width. (C) Same as (B) for a wider band set
of filters. The dashed red line in the first row of (B) and (C) indicates the maximum frequency of the
surrogate data values. The phase difference of the surrogate data is normally distributed from ⫺400␲ to
⫹400␲, so near zero and in the scale of the plot, the distribution is flat, as shown by the dashed lines. The
value of ⌬␾ is well constrained around ␲/2 in both cases (top row). Smaller peaks in the top histograms
occur at distances of ␲ and 2␲ from the ␲/2 peak, as expected.
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J. A. Rial—Synchronization of polar climate variability over the last
cal insolation, have evolved into a synchronized pair, obeying conditions (1) and (2).
There is no evidence of when or how this may have happened, but to test the idea that
the polar climates are in synchrony I introduce (see the next section titled a polar
synchronization model) a simplified, polar synchronization climate model. This
model describes the Polar Regions as two nonlinear coupled oscillators, where
in-phase, anti-phase and phase lock synchronization can occur between separate
components of the two systems. Before introducing the model however, there are a few
more observations worth describing.
Differences in age models between the time series from the Polar Regions
(Blunier and Brook, 2001) can distort phase shifts and obscure the presence of
synchronization. This is somewhat prevented by using methane-based age models, but
it is nevertheless important to substantiate the existence of a constant phase shift
between north and south records using independent statistical measures. Computing
the Hilbert transform of broad-band signals may be problematic for broad-band signals
(for example, Huang and others, 1998). Though the short signals in figure 3 are
narrow-band (most power on around ⬃4 ky), to calculate the phase difference
between the entire Dome C and GISP2 (Greenland) records 28 overlapping narrow
band-pass filters on the period band 2 to 14 ky (fig. 4A) are used. The results show that
the mean of the absolute difference in phase shift between narrow-band components is
close to ␲/2 for the entire duration of the last glaciation. The same algorithm, used on
surrogate data pairs constructed from AR(1) processes with spectral characteristics
similar to the data, violates condition (1), that is, it is inconsistent with synchrony. The
histograms (figs. 4B, 4C) show that the random surrogate pairs’ phase difference is
randomly distributed in the interval (⫺2␲, 2␲) while in the actual proxy records the
phase difference shows strong peaks around ␲/2 and smaller ones at ␲/2 ⫾ ␲ and
␲/2 ⫺ 2␲. The combined histogram of the surrogate functions exhibits normal
distribution, and is locally flat on the depicted interval, as indicated by the dashed
lines.
A wavelet coherence test (Grinsted and others, 2004) was applied to the two most
prominent examples of abrupt climate change in the Greenland records (the BA-YD
and the 78 ka-68 ka interval). The test detected strong coherence between NGRIP and
Antarctic Dome C records in the period range 2 to 6 ky (fig. 5), and a clear ⫹␲/2 phase
lead of Antarctica, indicated by the arrows pointing straight down. This occurs on both
time intervals, supporting and extending the results shown in figures 2 through 4.
Data at Intermediate Latitudes
What do data show at intermediate latitudes? Figure 6 shows no clear indication
that the ␲/2 phase shift between polar records is a (smooth) function of latitude. It in
fact appears as if the phase shift emerges suddenly, not progressively. However, the
scarcity of data does not allow drawing a firm conclusion at this time. Speleothem data
from New Zealand Hollywood cave (42°S) have been interpreted as indicating globally
synchronous climate variability during the last glacial as abrupt shifts from cold dry to
wet and cool climates occur at times that coincide with accepted ages of Heinrich
events in the interval 73 ka to 11 ka (Whittaker and others, 2008, 2011). These authors
state that their findings “argue strongly for coeval climate changes in antipodean
locations, and therefore provide compelling evidence for globally synchronous climate
variability during the last glacial period.”
Perhaps the most important data set at intermediate latitudes in the context of
polar synchronization is the record of temperature proxies in sediment core MD952042 off the coast of Portugal (38°N). These records show the by now familiar phase
relationship, which in this case shows that the ␦18O planktonic and benthic proxy time
series also form an approximate Hilbert transform pair (fig. 7). Because the response
of global ice occurs at much longer periods than the millennial oscillations, the
Fig. 5. Coherence and phase relationship between NGRIP and Dome C records in the 25 ka– 0 ka and 90 ka– 60 ka time windows, where abrupt changes in
temperature are stronger than at any other time in the high-pass filtered records. In the 25 ka– 0 ka interval the age models are likely to be correct. In the 90 ka– 60 ka
segment the NGRIP age model is used. The coherence plots are equivalent to correlation coefficients in time-frequency space (Grinsted and others, 2004). Strong
coherence is observed for the period range 2 ky– 6 ky in both windows. Arrows show relative phase: in-phase point right, anti-phase point left and ⫹90 degrees point
down, so both graphs support the previous result that Antarctica leads the Arctic by ␲/2. The thick black contour is the 5% significance level against red noise. Outside
the cone of influence (light shade) edge effects cause distortion. YD: Younger Dryas; BA: Bølling-Allerød.
ice age: in search of simple rules at the heart of climate’s complexity
427
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J. A. Rial—Synchronization of polar climate variability over the last
δ18O DATA
NGRIP - VOSTOK
INV HILBERT TRANSFORMS
Lat.
NGRIP
NGRIP 75N
NGRIP + π/2
75N
Normalized amplitudes
PHASE: +15
MD95 37.8N
MD95 + π/2
37.8N
PHASE: +30
Reflct 11N
Reflct + π/2
PHASE: +45
11N
MD03 + π/2
2.5N
MD03 2.5N
PHASE: +60
Sajama + π/2
18S
SAJAMA 18S
PHASE: +75
VOSTOK
Vostok 78S
78S
PHASE: +90
0
5
10
15
20
25
0
5
10
15
20
25
5
Time (Kyears BP)
10
15
20
Time (Ky BP) NGRIP age model
Time (Kyears BP)
NGRIP
SAJAMA - π/2
0
5
10
15
20
25
Time (Kyears BP)
Fig. 6. Data from several latitudes in the interval 25 ka– 0 ka. The left column starts with the NGRIP
record which is then phase shifted in steps of 15 degrees until ␲/2, at which it mimics the Vostok ice core
record. In the central column data at several latitudes are shown and in the right column the data phase
shifted by ⫹␲/2 to test how different they become from the Vostok signal below. Variation of phase shift with
latitude is not clear, which is consequence that most of the data are from the northern hemisphere.
However, the Sajama ice core data (18°S) when shifted by ⫺␲/2 appears somewhat like the NGRIP record.
benthic time series may be regarded as more indicative of average ocean temperature
variation (Severinghaus, personal communication, 2010) than of global ice volume
(Broecker, 1978). Shackleton and others (2000) described the clear contrast between
the “triangular” form of the benthic record and the “square-wave” form of the
planktonic record, but did not provide a physical mechanism to account for the
difference. However, they noted the clear resemblance of the benthic record to the
Antarctic time series (Vostok) and that of the planktonic to Greenland’s GISP2 or
GRIP. Confirming this find, a tropical latitude (6.3°N) sediment core MD98-2181
(Saikku and others, 2009) in the Pacific Ocean shows the surface temperatures at the
site co-varying with GISP2, while deep ocean temperatures closely follow Antarctic
surface temperatures. These relationships in fact mimic those of core MD95-2042 off
the coast of Portugal.
As will be shown in the next sections, the polar synchronization model introduced
here predicts the similarity of the Arctic and Antarctic data to the shallow and deep
ocean temperatures recorded by the Atlantic and Pacific sediment cores.
a polar synchronization model
Returning to the Huygens analogy, the oscillations of the clocks are to represent
the climate fluctuations at the two poles. The wooden frame or wall from which they
hung is the analogue of the intervening ocean and atmosphere, and the weak
429
ice age: in search of simple rules at the heart of climate’s complexity
GISP2 age model
1
Normalized δ18O
0
−1
GISP2
Planktonic
kt
3
EDC3deut
2
1
Benthic
0
−1
1
0
−1
Planktonic + π/2
0
20
40
60
80
100
KYears BP
Fig. 7. A similar relationship to that between north and south polar records exists respectively between
planktonic and benthic time series in sediment core MD95-2042 (Shackleton and others, 2000). The
Planktonic record co-varies with Greenland’s GISP2 (top), while the benthic record co-varies with Antarctica’s EDC3. The inverse Hilbert transform of the Planktonic record (Planktonic ⫹␲/2; bottom) closely
mimics the benthic record. These relationships are predicted by VSO (see text for details).
interaction between the two clocks through their common support is the analogue of
meridional coupling (for example, through N-S temperature/salinity gradients and
the thermo-haline circulation, or THC). Of course in the real climate the “clocks” are
usually strongly nonlinear, probably consist of networks of clocks, rather than of a
single oscillator, the process is multidimensional and likely chaotic, and many forms of
synchrony, as well as loss of synchrony may occur. Moreover, a precise determination of
pole-to-pole temperature phase relationships from ice core data is challenging because
many factors other than temperature can affect the isotopic composition of the ice (for
example, seasonality), while strong weather noise can prevent synchronization (see
Modeling Results in the next section). These problems notwithstanding, a simplified
model that represents the poles as climate oscillators coupled by temperature and heat
storage differences between the Polar Regions helps interpret the data discussed thus
far. Bryan (1986) has shown that multiple equilibrium solutions of the THC in general
circulation models (GCMs) include equatorially asymmetric circulations that can be
triggered by thermal and salinity anomalies. Such asymmetric flows may thus play the
role of the coupling between the southern and northern hemisphere.
A simplified model is important, not only because it allows a far more rigorous
analysis than qualitatively reasoning, but also because fully coupled GCMs or climate
models of intermediate complexity (EMICs) are less helpful. One obvious reason is
because it cannot be known whether the varying idiosyncrasies involved in collective
programming may have unintentionally inhibited or promoted polar synchronization.
Further, it would take many months to years of time-consuming modeling experiments
(typical modeling speeds of 250 model years per day of computation) to be able to
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J. A. Rial—Synchronization of polar climate variability over the last
reproduce the data using an EMIC, while the physics of the process would still lie
hidden in the complexities of the code. For instance, simulations using a coupled earth
model of intermediate complexity (ECBilt-Clio) we reported elsewhere (Rial and Saha,
2011) reproduce the square versus triangular waveforms contrast between surface and
mean ocean temperature, but the unanswered question is why is this so. Though,
admittedly, an answer may be extracted from the code after extensive, time-consuming
experimentation, there is no assurance it will be. In contrast, a low-dimensional model
such as the one to be discussed shortly uses the simplest, best-known representation of
the physics of two generic coupled nonlinear oscillators and generates time series
consistent with the results of the EMIC and with the data. The simplified model
suggests that the relations between shallow and deep temperatures (fig. 7) are a
consequence of synchronization, as will be discussed in forthcoming sections. The vast
incompleteness of the simplified model is of course its main drawback.
To build the simplified polar synchronization model the following assumptions
were used: (1) The two Polar Regions are dynamically similar, each behaving as a
nonlinear, self-sustained van der Pol relaxation oscillator, but each with different
natural frequencies and feedbacks. (2) These oscillators describe polar climate variability as the nonlinear interaction between sea ice cover and mean ocean temperature, forced externally by the astronomic insolation and by weather-like stochastic
noise. (3) The two polar oscillators influence each other through dissipative coupling
proportional to the differences in temperature, salinity and heat storage between
the northern and southern oceans. The mathematical details of the model,
heretofore called the Van der Pol Synchronizing Oscillators (VSO) model, are
given in Appendix B.
Saltzman and others (1981) justified the adoption of an oscillator of the van der
Pol/Duffing type to simulate climate fluctuations near the poles as follows: During a
glaciation insulating sea ice reaches its maximum expansion accelerated by positive
feedbacks, causing the mean ocean temperature to increase, which will eventually
induce sea ice retreat. As it becomes uncovered, the ocean releases heat to the
atmosphere and cools down, causing atmospheric and oceanic temperatures to
decrease, inducing sea ice to advance. Thus, the ocean temperature drives the advance
or retreat of sea ice, whose amplitude in turn affects the ocean temperature’s time rate
of change, giving rise to natural oscillations.
The stronger the positive feedbacks, the less sinusoidal and more abrupt the
oscillations become. In principle, simple low-dimensional but relevant climate models
like Saltzman’s can capture enough physics of the climate process to serve as a
competent guide through complicated and limited data (Stommel, 1961; Källén and
others, 1979; Paillard, 1998; Rial and others, 2004; Colin de Verdiere and others, 2006;
Rial and Saha, 2011). However, whether the modeling results to be discussed shortly
represent the actual interaction between the polar climates is of course not proven. On
the other hand, remarkably, the simplified model reproduces amplitudes, phases and
frequencies consistent with observed characteristics of the data only when synchronization occurs. This plus the functional relationships described in figures 1 through 7
suggests the intriguing possibility that at least at the millennial scale there are simple
rules that govern the behavior of polar climates.
As described in Appendix B, VSO consists of four nonlinear ordinary differential
equations in four unknowns ui (i ⫽ 1, 4). u1 and u2 are the mean ocean temperature
and sea ice extent (a proxy for surface temperature) for the Northern Hemisphere,
representing the Arctic climate. u3 and u4 are the mean ocean temperature and sea ice
extent for the southern hemisphere respectively, and represent the Antarctic climate.
The coupling (mutual feedbacks) terms in VSO (eqs B.2, B.4) are assumed to be
proportional to the difference in mean oceanic temperature/salinity (reactive cou-
431
ice age: in search of simple rules at the heart of climate’s complexity
Unsynchronized
40
50
60
Synchronized
u4
u2
u4
u2
u3
u1
u3
u1
70
40
50
60
70
Kyears BP
Kyears BP
Mean Ocean Temp S, Air Temp. N
Synchronized + noise
Greenland
2
u2
0
−2
2
u3
0
Antarctica
−2
40
50
60
Kyears BP
70
70
75
80
Kyears BP
Fig. 8. VSO simulations of the north and south ice extent (u2, u4) and north and south mean ocean
temperatures (u1, u3) respectively with noise added (left bottom panel) and without noise (top two panels).
The unsynchronized example shows the relative phases and frequencies obtained with model parameters for
two different, uncoupled oscillators (q1 ⫽ q2 ⫽ 0). The synchronized simulations result as the coupling
coefficients q1 and q2 are both positive (q2 may be small or zero). As described in Appendix B, the
homologous pairs (u2, u4) and (u1, u3) synchronize in-phase in the figure (and in anti-phase if q1 is negative).
The conjugate pairs (u2, u3) and (u1, u4) synchronize out of phase by a constant shift of ␲/2, (u1 and u3
leading) for q2 large and positive. Leading u3 is consistent with the data, as shown. When noise is added
similar results obtain (left bottom panel). Parameters are chosen to produce periods in the 6 ky–10 ky range.
Similar phase shifts have long been reported and simulated with highly simplified models (for example,
Källén and others, 1979). The data use the NGRIP age model.
pling) and mean ocean heat storage between north and south (dissipative coupling).
This form of coupling is used here since it is the most general for this type of oscillators
(Hale, 1997; Balanov and others, 2009) and the true nature of the coupling is of course
unknown. There are strong indications however that coupling between polar climates
occurs. As indicated before, an equatorially asymmetric circulation that can be
triggered by thermal or salinity anomalies (Bryan, 1986) may play the role of the
coupling between the southern and northern hemisphere. For instance, the abyssal
circulation associated to the THC carries the water formed in the Polar Regions to the
rest of the ocean. Sources of cold water exist in both Polar Regions, in the Weddell Sea
and Ross Sea in Antarctica where cold water sinks and moves north, and in the high
latitude North Atlantic where North Atlantic Deep Water (NADW) sinks and moves
south overlying the denser Antarctic bottom water. The abyssal circulation is a massive
flow of cold water that occupies the major part of the ocean in volume, so it may play a
key role in the earth’s heat balance and likely in the coupling of the polar climates.
Controlled by deep waters below the thermocline (deeper than 1 km), the average
432
J. A. Rial—Synchronization of polar climate variability over the last
Fig. 9. (A) VSO results for the BA-YD window. The timing and waveforms of the two time intervals from
NGRIP and Vostok are matched closely by a realization from an ensemble computed with the stochastic version of
VSO. In the model the phase relationship between the two polar time series (␲/2 phase shift) becomes
established after synchronization has occurred (compare with fig. 10). The variables u2 and u3 represent the
sea ice extent (surface temperature) in the North and mean ocean temperature in the South, respectively.
Results as those shown are obtained with free periods ranging 1.5 ky–2 ky for north and south oscillators, q1
large and positive, q2 zero or small. Multiple realizations are obtained by running the model repeatedly with
ice age: in search of simple rules at the heart of climate’s complexity
433
temperature of the entire world ocean is only 3.51 °C (Worthington, 1981) in spite of
the vast areas of surface waters at or over 20 °C. The abyssal circulation speed is less
than one order of magnitude slower than surface currents (Pedlosky, 1996), which
makes it a more likely conduit for the interaction between the Polar Regions at the
centennial to millennial scale than the atmosphere. The atmosphere also must play a
role, likely that of an efficient waveguide at the short time scale and a slave (onedirectional synchronization) to the ocean at the centennial to millennial scale. The
likely nature of polar coupling will be discussed further in the Synchronization and the
Bipolar Seesaw section in the context of the hemispheric polar seesaw (Crawley 1992),
and in the synchronization as a complexity-reducing, self-organizing mechanism section in the context of the mechanism of synchronization.
Modeling Results
Figure 8 presents simulations for unsynchronized and synchronized, noiseless and
noisy cases for a generic interval of 30 ky without astronomical forcing. The unsynchronized case shows the evolution of the four VSO variables when there is no coupling
between north and south, as the two oscillators evolve independent of each other.
Notice however that the pair of variables describing the climate in each hemisphere,
for instance (u1, u2) are in a phase relationship that is preserved when the two
oscillators synchronize. However pairs such as (u2, u3) or (u1, u4) from different Polar
Regions show arbitrary phase relationship before sync and phase lock after sync. The
noisy simulation (lower left panel) represents the sea ice extent in the north (u2) and
the mean ocean temperature in the south (u3). Their phase relationship and waveforms are fully consistent with the records from Greenland and Antarctica in the
period 70 ka to 80 ka shown in the lower right panel.
Figure 9A shows VSO results for the BA-YD sequence in both hemispheres
compared to the actual data. As they synchronize, the simulated signals become ␲/2
phase locked, as can be clearly seen in figure 9B, which shows the phase relationship by
overlying the Hilbert direct and inverse transforms of the modeled time series. The fits
are close, considering that the model is so crude. For reference, figure 10 shows only
VSO simulations to illustrate how an un-synchronized pair of oscillators differs from
the synchronized one, all other parameters being the same. The unsynchronized pairs
are different from each other and do not resemble the data. Synchronized simulations
(right column) do resemble the data (refer to fig. 9) and predict that there will be a
mean ocean temperature record in the Northern Hemisphere that looks like u1 and a
record of sea ice extent (surface temperature) in the Southern Hemisphere that looks
like u4. The simulated synchronized records in figure 10 show relationships fully
consistent with the ocean records off Portugal shown in figure 7, where u2 represents
the planktonic “square amplitude” and u1 the benthic “triangular amplitude” record in
the Northern Hemisphere. In the unsynchronized case, however, the simulated time
series do not reproduce the data.
VSO reproduces the data—including the waveforms and phase relationships
between the north and south temperature time series, when the coupling coefficients
q1 and q2 (Appendix B) are large and positive or for q1 large positive and q2 small or
zero (fig. 9). If q1 is negative, the phase relationships are reversed, and if q2 is negative,
the result is always unstable. Though it is clear that the synchronization of natural
systems is much more complicated, the coupling of the Polar Regions poorly under-
constant parameters. Less than ten adjustable parameters are needed in the simulation. YD: Younger Dryas,
BA: Bølling-Allerød, ACR: Antarctic Cold Reversal. (B) The direct and inverse Hilbert transforms of the
modeled time series show that the modeled signals satisfy the same phase relationships as the data (compare
with fig. 2B).
434
J. A. Rial—Synchronization of polar climate variability over the last
UNSYNCHRONIZED
SYNCHRONIZED
Normalized amplitude
u2
4
3
2
u1
1
0
−1
NH
5
10
15
Kyears BP
6
Normalized amplitude
5
3
2
u3
1
0
−1
2
u1
1
0
−1
NH
5
10
15
Kyears BP
u4
5
4
3
u3
2
1
0
SH
SH
5
10
15
Kyears BP
20
5
10
15
Kyears BP
u4
u2
3
2
2
1
1
u1
u2
0
0
−1
−1
−2
−2
0ka-25ka
−2
−1
0
1
2
−3
3
−3
2
2
1.5
1.5
1
1
u1
0.5
0
−0.5
−0.5
0ka-25ka
−1
−0.5
u3
0
0.5
1
0ka-25ka
2
3
−2
−1
0
1
−1
−0.5
0
0ka-25ka
0.5
1
1.5
0.5
0
−1
−1.5
20
u4
3
−3
−3
20
6
4
−2
3
20
u4
5
u2
4
Normalized amplitude
Normalized amplitude
5
−1
−1.5
u3
Fig. 10. VSO results for the four variables corresponding to synchronized and un-synchronized cases.
On the left column q1 ⫽ q2 ⫽ 0. On the right column q1 is positive and large, q2 is positive, small. Normalized
by the noise level q1 is essentially the same value (⬃4) than for the noiseless version shown in figure 8. If q1 is
negative (not shown) the polarities of both u3 and u4 are reversed. Parametric plots for the four components
show unsynchronized and synchronized phase space trajectories in the time interval 0 ka to 25 ka.
Correlation coefficients calculated for all possible values of q1 and q2 (not shown) indicate that VSO is robust
against changes in the coupling and that q1 is the more relevant of the two coefficients.
ice age: in search of simple rules at the heart of climate’s complexity
435
stood, and the heterogeneities of the climate likely to dominate over a few differential
equations, VSO replicates diagnostic features of the observations such as waveform,
timing and phase relationship.
Lund and Mix (1998) show isotopic evidence of millennial scale oscillations in the
North Pacific deep water composition (60 ka-10 ka) with cycles similar to those in the
North Atlantic, but opposite in sign (intervals of strong ventilation in the deep Pacific
occurred opposite the pattern in the deep North Atlantic), as if in an anti-phase
synchronized relationship. Barker and others (2009) also report North-South antiphase relationships between a south Atlantic sediment core temperature proxy and
Greenland’s abrupt temperature changes. Though thus far the clearest evidence is for
a phase lock of ␲/2, an anti-phase relationship is consistent, and adds credibility to the
idea of polar synchronization, since anti-phase is just one of the three possible forms of
synchrony, all of which can occur simultaneously on independent components of
multidimensional oscillators (Balanov and others, 2009).
The influence of noise.—It may be argued that system noise may prevent synchronization. Intuitively, the presence of noise makes synchronization seem improbable, and
usually, if uncorrelated Gaussian noise is added to each element of an assembly of
synchronized oscillators they desynchronize (Goldobin and Pikovsky, 2005). High
levels of noise of course will destroy synchronization, but there exists a large number of
observations and experiments, especially in the biological sciences, showing that
coupled nonlinear extended systems (including chaotic oscillators) though subject to
large noise fluctuations can still synchronize, exhibiting temporal correlations over a
wide range of time scales (see review by Ramaswamy and others, 2010). Further, in
ensembles of weakly coupled oscillators it has been observed that coherence of the
oscillation is maximized at intermediate noise levels (Baroni and others, 2001), similar
to the case of stochastic resonance (Benzi and others, 1982). The similarity between
the noise-free and noisy results with the data (fig. 6) suggests that noise forcing may not
be important enough to prevent polar synchronization.
Competing Models and the Periodicity of the DO
Millennial-scale variability during the last deglaciation, especially the BA warming
event, has been simulated by a few studies through perturbing meltwater flux and
changing surface temperatures (Ganopolski and Rahmstorf, 2001; Knorr and Lohmann, 2003, 2007; Weaver and others, 2003). Liu and others (2009) performed the first
transient simulation with a coupled atmosphere-ocean GCM (20-14.12 ka), which
indicated that the BA results from the combined effects of transient CO2 forcing,
AMOC recovery from Heinrich event 1 and AMOC overshoot. However, the model
does not cover YD so that the characteristic north-south phase shift of the observations
is not considered.
VSO is highly idealized, but useful for visualizing the amplitudes and relative
phase two nonlinear self-sustaining oscillators will attain under prescribed coupling.
Astronomically induced variations in insolation have been included in the forcing of
the model although the relatively low frequencies of insolation are initially filtered out
of the records to focus in the 2 to 6 ky period band of interest (compare figs. 1-3). The
filtering attenuates the long periods of the signals, but it does not eliminate the
nonlinear response of the system to the insolation (for example, frequency and/or
phase modulation) in that spectral band. Modeling experiments indicate that the
FM/AM effects do not alter the dominant effect of strong coupling over any other
forcing. VSO reproduces pairs of time series and their phase relationships probably as
well as other competing models do, but VSO suggests that the observed phase relationships are only possible if synchronization is occurring. To show this figure 10 compares
synchronized and unsynchronized simulations. It is clear than only when synchronized the
two polar time series become close to the actual data (compare fig. 9).
436
J. A. Rial—Synchronization of polar climate variability over the last
One important unresolved issue is the precise origin of the internal frequencies
␻i. The spectra of the GRIP or NGRIP time series show broad power peaks centered
around periods of 4.3 ky and 1.6 ky (Rial and Saha, 2011). Saltzman and others (1981)
used actual climatic parameters to calculate natural periods of 1000 to 3000 years for
the advance and retreat of sea ice. Bond and others (1997) described evidence for a
persistent climate signal with a period of 1500⫾500 years of possible solar origin, and
many authors have reported similar findings, but the origin of the signal is still the
subject of intense research. A number of possible mechanisms exist, including diffusionconvection instability of the thermo-haline circulation (Ganopolsky and Rahmstorf,
2001; Alley and others, 2003), and oscillating solar power output (van Geel and others,
1999) among many others. Millennial-scale free oscillations in ocean temperature and
salinity due to the interplay of convective instability and turbulent mixing have long
been known to theoretically exist (Bryan, 1986; Winton, 1995; Sakai and Peltier, 1997;
Haarsma and others, 2001; Loving and Vallis, 2005; Colin de Verdiere and others,
2006). Further, simple box-type models produce millennial oscillations in ocean
temperature, salinity and density (Colin de Verdiere and others, 2006) that require no
external forcing. In addition, internal oscillations of a modeled AMOC are found in
simulations using ocean-alone and coupled models with varied boundary conditions
(Winton and Sarachik, 1993; Winton, 1995). We performed experiments with two
EMICs: University of Victoria’s ESCM model (Meissner and others, 2008) and ECBiltClio version 3. Both produced clear DO-like events under the same glacial boundary
conditions and both are in qualitative agreement with the simulations of sea ice extent
and mean ocean temperature obtained with box and van der Pol oscillators models
(Rial and Saha, 2011). In VSO, the frequencies of the internal free oscillations ␻i are
chosen to be in the range 1500 ⫾ 500 years because that range produces the best fits to
the data.
Square and Triangular Signals
Model validation.—As briefly discussed earlier in the Data at Intermediate Latitudes
section, a well-known, yet still puzzling aspect of the records from the last ice age,
whether ice or sediment cores, is that their time series contain waveforms that are
consistently of two recognizable types: square-like and triangle-like (Shackleton and
others, 2000). The former are typical of the Greenland records, the latter of Antarctica.
More interesting yet is the fact that for tropical and subtropical latitudes proxies for
ocean temperature records show the same arrangement of square and triangular
waveforms with the surface ocean temperature similar to Greenland and the deep
ocean temperature to Antarctica (Shackleton and others, 2000; Saikku and others,
2009). Moreover, methane records (whether from north or south Polar regions) show
square waveforms like Greenland, while carbon dioxide time series appear triangular,
like Antarctica. The phase relationship between the triangular and square waveforms
in the records analyzed is ␲/2 or very close to ␲/2 in all cases (figs. 1-5, 7-10). VSO
predicts this relationship between the two components of each of the northern (u1, u2)
or the southern (u3, u4) oscillators as shown in figure 9. If the two oscillators are not
coupled (q1 ⫽ q2 ⫽ 0) the components of one oscillator bear no special phase
relationship to the other (figs. 8, 10), but when the oscillators synchronize the
conjugate pairs (u2, u3) and (u1, u4) become ␲/2 shifted (figs. 8-10). VSO reproduces
the observed waveforms in the ice cores of Antarctica and Greenland, the sediment
proxies at intermediate latitudes, and is consistent with results from a fully coupled, 3D
climate model of intermediate complexity (Rial and Saha, 2011).
Model predictions.—As shown in figure 10, VSO predicts that upon synchronization
there will be a mean ocean temperature record in the Northern Hemisphere that looks
like u1 and a record of sea ice extent (surface temperature) in the Southern Hemisphere that looks like u4. That is, there are more record types than what the
ice age: in search of simple rules at the heart of climate’s complexity
437
observations have yet produced. A hint that this might be true comes from the data
reported by Barker and others (2009), which would correspond to u4, in anti-phase to
NGRIP, according to the observations (Synchronization and the Bipolar Seesaw section).
Similarly, we should then expect to find mean ocean temperature records in the NH to
look like a typical triangular Antarctic (or deep ocean) temperature, but that, if Barker
and others (2009) is a reliable proxy, should be in anti-phase with Dome C.
Synchronization and the Bipolar Seesaw
Denton and Hendy (1994) proposed that there was an in-phase relationship
between north and south climate variability, reflecting global synchronicity, but the
records have shown that the temperature variations between the poles are out of phase.
That is, the records do not support a strict anti-phase relationship (Steig, 2006) caused
by alternating north-south deepwater formation (Broecker, 1998). The bipolar hemispheric seesaw hypothesis has been the object of vigorous research (for instance,
Crowley, 1992; Broecker, 1998; Blunier and Brook, 2001; Stocker and Johnsen, 2003;
Wunsch, 2003; Seidov and others, 2005; Barker and others, 2009; Pedro and others,
2011, and many others). Its modern version states that abrupt warming in the North
Atlantic occurs at the beginning of gradual cooling in Antarctica, implying that
reorganization of oceanic circulation may be the cause of the seesaw. The observed
phase shift between Antarctic and Arctic records has been reported to be close to ␲/2
(Steig, 2006; EPICA members, 2006). I have shown (figs. 2-5, 8 and 9) that this is
consistent with the data and modeling results presented here, with ␲/2 likely the
correct phase shift between south and north, with the south leading. The results have
shown however that the phase relationships appear to depend on which component of
the climate proxies we look at. As shown in figures 8 and 9, as synchrony sets in, some
components synchronize in-phase and some others ␲/2 out of phase. Further, the VSO
model includes the possibility that there may be anti-phase synchronization (when q1 ⬍
0) for the same components that sync in-phase (when q1 ⬎ 0). VSO is too simple to
switch the sign of q1 either as a function of geography or of time, but a more elaborate
model can easily be built that reproduces in-phase and anti-phase synchronization at
different times or different locations. This is the topic of a forthcoming paper.
Most mechanisms that endeavor to explain the bipolar seesaw are associated with
ocean circulation variability. Modeling studies show that a high rate of NADW
production increases NH temperature while SH temperature decreases synchronously
due to northward heat transport from the Southern Ocean, and conversely, a slowing
down of the NADW formation cools the northern latitudes while the south warms
(Stocker and others, 1992). A convolution integral relationship between southern and
northern temperature proxy records proposed by Stocker and Johnsen (2003) replicates the waveform of the southern records from a sequence of square pulses, whose
origin is not explained, representing the DO. The convolution integral of the GRIP
time series using an adjustable time response reproduces some features of the Byrd
Antarctic signal, with a phase delay of 70° to 80° in the DO frequency band. However,
as shown in figures 2 and 3, such operation on GRIP represents only one-half of the
process. In fact, a similar relationship to that proposed by Stocker and Johnsen (2003)
can be easily obtained by just integrating the NGRIP time series (fig. 1) without
assumptions regarding a characteristic time scale. However, as discussed earlier in the
the polar phase shift section, though integration closely reproduces the waveform
of the Antarctica time series, it amplifies long period amplitudes preferentially and so
the amplitudes of the modeled time series in Stocker and Johnsen (2003) and that of
the actual Byrd record must be very different, and thus inconsistent with the data
unless new parameters are introduced. In addition, such a model would not explain
why in ocean sediment proxies the ocean’s surface temperature follows Greenland
while the deeper temperature variation follows Antarctica (Shackleton and others,
438
J. A. Rial—Synchronization of polar climate variability over the last
2000; Saikku and others, 2009), while VSO predicts this will happen when synchronization occurs (figs. 9, 10).
Schmittner and others (2003) conclude that the main mechanism producing
inter-hemispheric temperature correlations on millennial time scales during the last
glacial period was very likely fluctuations in the rate of deepwater formation in the
North Atlantic. They interpret the data as indicating a lead of Antarctica over
Greenland by 1 to 2 ky. The issue of how the signal is transmitted from one pole to
the other is still unclear, however, and other authors prefer the idea of heat piracy
(Seidov and Maslin, 2001). Rossby-type waves have been invoked, which, in a matter
of a few decades, communicate the poles meridionally (Goodman, 2001). Numerical simulations of freshwater discharges in the North Atlantic are also seen to
contribute to the phase shift (Knutti and others, 2004), but freshwater discharge in
the Southern Ocean fails to reproduce the seesaw (Seidov and others, 2005). In the
North Atlantic, in spite of multiple and sophisticated modeling efforts, freshwater
discharges have failed to drive the AMOC anywhere near collapse (Kerr, 2005).
Recently, observed and simulated data show instead that North Atlantic cooling
might be caused by wind-driven upwelling in the South Ocean (Anderson and
others, 2009; Lee and others, 2011; Pedro and others, 2011). Lozier (2010) challenges
the conveyor belt paradigm and points to the vital role of the ocean’s eddy and wind
field in establishing the variability of the ocean’s overturning. Finally, in an unusually
scathing critique of scientific consensus, Wunsch (2010) suggests that the great ocean
conveyor, the significance of repeated freshwater hosing numerical experiments, and
many seemingly supporting results from climate models must all be called into
question.
Polar synchronization is consistent with the modern version of the polar seesaw
that abrupt warming in the North Atlantic occurs at the beginning of gradual cooling
in Antarctica. As to the underlying process of mass and heat transport that connects
the two poles, polar synchronization is consistent with most bi-directional mechanisms
proposed.
synchronization as a complexity-reducing, self-organizing mechanism
Synchronization of nonlinear oscillators is of great interest in many areas of
science and technology, from medicine to engineering (Strogatz, 2000; Pikovsky and
others, 2001; Balanov and others, 2009). Over the last twenty-five years, the theoretical
and experimental study of single pairs or large number of coupled oscillators has
produced a wealth of information that transcends discipline boundaries. The system
composed of two nonlinear oscillators, in spite of its mathematical simplicity, is
complex enough to study a wide variety of synchronization forms in non-identical,
coupled, and even chaotic oscillators. For instance, when two different chaotic
oscillators become mutually coupled, a number of types of synchronization do occur as
the coupling strength increases: amplitude envelope sync, phase sync, lag sync (in the
paleoclimate data this lag would be ␲/2) and identical (in-phase) sync (for more
details and definitions see Boccaletti and others, 2002). An example of lag synchronization can be theoretically computed using two chaotic Rossler oscillators. The resulting
oscillations are such that the phase lag is ⬃␲/2 while the amplitudes are uncorrelated
(Pikovsky and Rosenbaum, 2007). Chaotic oscillations are, by definition, unstable and
aperiodic but of bounded variation and highly sensitive to initial conditions, like many
climatic variations. Thus, this simple result could be an analogue to the polar
synchronization proposed here if we assume for generality’s sake that before they
became synchronized the original climatic variations in either pole were independent
of each other, and chaotic, just as Rossler’s nonlinear oscillators. Thus, when the two
polar ice caps were firmly emplaced, about 2.8 million years ago, high frequency
(sub-Milankovitch) temperature variations at the Polar Regions were chaotic and bore
ice age: in search of simple rules at the heart of climate’s complexity
439
no relation to each other. In time, and with coupling provided by the intervening
ocean/atmosphere system, the Polar Regions began to interact, eventually becoming
coupled and synchronized. That is, the recorded paleoclimatic variations of the Polar
Regions are simple and bear simple phase relations to each other because of synchronization. Synchronization is a form of organization that brings stability and order to
chaotic systems at low energy cost, and can create simplicity in the form of periodic,
predictable behavior. In fact, coupled chaotic phenomena have been modeled in
realistic climatic and meteorological situations with synchronization resulting in
increased predictability and decreased levels of turbulence (for instance, Brindley and
others, 1995).
At long periods, in the Milankovitch band, synchronization of the uni-directional
or master-slave type may have also occurred (Stefanski and others, 1996). If some of the
climate system’s natural frequencies of oscillation happen to be close to those of the
insolation, master-slave synchronization is possible as the nonlinear climate system
adjusts its natural rhythms to the insolation’s frequencies. When synchronization is
complete, the resulting oscillation of the climate would show spectral power peaks at
frequencies not present in the forcing. This result is experimentally equivalent to
frequency or phase modulation (Rial, 1999; Gonzalez-Miranda, 2004, fig. 3.6).
What we see today in the ice and sediment cores is likely a very incomplete history
of the climate variations of the last ice age. It is quite possible that in the future archives
that recorded other variables will be found to reveal more about the true complexity of
the system.
If the two polar climates have indeed been in synchrony at the millennial scale,
they oscillate in unison, but out of phase, and thus what we see in the proxy records
may be regarded as different “components” of that modal oscillation; one “component” in Greenland and another in Antarctica. A tentative explanation of this is that
the natural “climate recording systems” in the two poles are different. The Arctic is a
small ocean surrounded by land, and thus the northern climate is strongly sensitive to
the variation in sea ice extent, and the reflectivity and heat insulation that sea ice
provides. Antarctica, in contrast, is a large ice mass surrounded by the Antarctic
Circumpolar current, controlled more by wind stress and average ocean temperature
variation than by sea ice extent. This is consistent with the fact that methane variations
co-vary with Greenland temperature while carbon dioxide variations co-vary with
Antarctica’s. Further, because of synchronization the components of the modal
oscillations must exist everywhere in the system, so that at non-polar latitudes the
sea-ice dependent square waveforms should be predominant in the shallow ocean
signal and the average ocean-dependent triangular waveforms should be predominant in the deeper ocean signal, as observed (Shackleton and others, 2000; Saikku
and others, 2009). That is, if we assume that VSO reflects the interaction between
sea ice and mean ocean temperature, there are at least four ui (i ⫽ 1-4) components to consider. Figure 10 shows that synchronization forces pairs of components
in both the north and the south to a ␲/2 phase lock. This implies that anywhere in
any hemisphere the ocean surface temperature (given by u2 and u4) should have a
“square” waveform while the mean ocean temperature (u1, u3) should have the
triangular waveform, just like the records off Portugal (fig. 7). The observed records we
have so far discussed are represented in the panel showing the synchronized time
series in figure 10 by u2 and u3. The predicted records, and thus far not observed, are
u1 and u4 in the same panel.
In the theory of synchronized chaos, if two or more oscillators are synchronized,
an increase in coupling strength may destroy the synchronized state and lead to
“oscillation death” (Pikovsky and Rosenbaum, 2007). That is, the amplitude gradually
decreases to zero and oscillations stop. Though the study of such a regime is outside
440
J. A. Rial—Synchronization of polar climate variability over the last
the scope of this paper, it is conceivable that oscillation death in the paleoclimate is
represented by the interglacial periods, the Holocene being a clear example. The
interglacials are times at which the coupling between the poles may increase as a
consequence of the increasing Milankovitch external forcing and the rapid retreat of
the ice caps, which occur roughly every 100 ky. That this may be so is suggested by the
analytic and experimental results reported by Suárez-Vargas and others (2009) indicating that oscillation death of synchronized oscillators can occur not only as the
consequence of increased coupling, but also due to changes in the amplitudes and
frequencies of the oscillators. Both these effects could be induced by the increasing
insolation through amplitude and frequency modulation, as shown by Rial (1999), Rial
and Yang (2007), and Rial and Saha (2011).
discussion and concluding remarks
I have shown evidence that a plausible explanation for the paleoclimate data
described in this paper is that for the last ice age, and likely for earlier periods, the
millennial scale paleoclimate oscillations of the Polar Regions, by adapting their
frequencies and phases to each other, became coupled and lag-synchronized. Saltzman and others (1981) showed how climate oscillations of polar temperatures at the
millennial scale can be explained by the simple, though nonlinear, interaction
between just two variables: sea ice extent and mean ocean temperature. In spite of its
simplicity Saltzman’s is actually a powerful model that assumes that mean ocean
temperature begins to increase when sea ice reaches its maximum extent. This is
because sea ice is a very effective heat insulator, while a large portion of the ocean is
still receiving solar heat. Eventually, after several hundred years of diffusive warming,
the ocean under the sea ice reaches a threshold temperature that forces sea ice to
retreat. As sea ice rapidly retreats poleward, accelerated by the ice-albedo positive
feedback, the ocean releases its stored heat and slowly cools. When the ocean becomes
cold enough, sea ice begins to grow back advancing rapidly towards the equator, again
accelerated by the ice-albedo positive feedback. The idea that rapid sea ice response in
the Arctic is causative of the observed abruptness of climate change in Greenland’s
proxy records has received some support (for example Gildor and Tziperman, 2003;
Eisenman and others, 2009). The cyclic build up and release of stored thermal energy
is best described physically as a relaxation oscillation, and mathematically as the
solution to the canonic van der Pol differential equation. It does not follow that the
many other variables that play a part in the process are rejected. The model shows that
the two-variable interaction (Rial and Saha, 2011) and the four-variable interaction in
the synchronization model VSO appear to account for most of the observed variance.
VSO not only reproduces polar time series fairly well (fig. 9), it also predicts the square
and triangular waveforms as observed in the Atlantic and Pacific proxies (Shackleton
and others, 2000; Saikku and others, 2009). Figures 8 through 10 may be interpreted to
say that the underlying physics is simpler than most competing mechanisms would
suggest. Given the natural complexity of the climate system it is remarkable that simple
rules seem to apply. If these simplified climate models are to be trusted the message
appears to be that, surprisingly, in the paleoclimate records there are simple rules that
control the emergence of order out of chaos, and synchronization appears to be one of
them.
The idea that simple rules can govern the complexities of the paleoclimate finds
support in a number of previous studies, including the suggestions of a climatic
attractor, likely a strange attractor (Nicolis and Nicolis, 1984; Tsonis and Elsner, 1989;
Lorenz, 1991). In addition, the idea that the complete complexity of the climate system
condenses beyond the weather time scales into a small number of low-dimensional
interacting components (Tsonis and others, 2011) agrees with the simplicity of VSO.
441
ice age: in search of simple rules at the heart of climate’s complexity
H2
H3
H4
H5
A1
A2
∆T (oC)
H0 H1
H5a H6
H7?
A4
A3
EDC3deut
t
∆T (oC)
A=∫ NGRIPdt
B = NGRIP
H2
H3
H4
H5
H5a H6
H7?
∆T (oC)
H0 H1
A–B
0
20
40
60
80
100
KYears BP
Fig. 11. Approximate timing of Heinrich events (vertical bars) compared to times at which the
pole-to-pole temperature gradient is largest (time series peaks, bottom panel). The time integral of NGRIP is
used as a proxy for the EDC3 Antarctic record (in light shade, on top panel) and its amplitude corrected to
reconcile with that observed in the actual record (NGRIP age model is used). Heinrich events 7, 8, and 9 are
predicted to be at 74 ka, 77 ka, and 87 ka. Heinrich events timing from Hemming (2004).
Antipodal Teleconnections
Recent studies have shown an apparent relationship between El Niño/Southern
Oscillation (ENSO) and the Indian monsoon that suggests some degree of synchronization. These climate oscillations can have enormous impacts (some negative) on the
ecosystem and on human life and thus justify a scrutiny of synchronization as a climatic
variability mechanism. Dynamically, a key factor conducive to their long-range interaction may be that the ENSO and the Indian monsoon regions of maximum activity are
antipodal to each other (just like the two poles). The earth’s spherical shape has
interesting consequences when considering the propagation of climatic influence
especially through the atmosphere, which provides a relatively smooth spherical shell
waveguide, compared to the irregular shape of the ocean basins, to rapidly communicate information to the longest distances. Every bit of information released at a source
will eventually converge with all other bits (provided it survives dissipation) at their
geographic antipode. If the signals are sufficiently strong, this process may induce the
development of normal modes of climate oscillation, which is a form of global
synchronization for wavelengths that fit the earth’s circumference an integer number
of times. The modes would first develop in the atmosphere. Later, they would be
communicated to the ocean through mass and energy fluxes. Thus, it may be of some
consequence to climate dynamics that ENSO and the monsoon (as well as the two
Polar Regions) are antipodal to each other.
Heinrich Events
Figure 11 suggests that polar synchronization may influence the occurrence of
Heinrich events (HEs). Heinrich events (Heinrich, 1988) are massive iceberg dis-
442
J. A. Rial—Synchronization of polar climate variability over the last
charges in the North Atlantic that have had a profound impact on the climate of the
Arctic, but whose influence beyond the region is uncertain. These episodic iceberg
discharges have been studied by many researchers (for example, MacAyeal, 1993;
Marshall and Clarke, 1997; Schulz and others, 2002; Hemming, 2004; MacAyeal and
others, 2006; Alvarez-Solas and others, 2010). Recently, the idea that HEs may be
affected by events happening far from the North Atlantic has been suggested by, for
instance, Sachs and Anderson (2005), who describe a pole-to-pole link represented by
peaks in sub-Antarctic ocean productivity that occurred within 1 to 2 ky of the HEs
during the past 70 ky. The precise mechanism behind such a link is not known. Precise
methane-based timing of north-south records has shown that HEs usually occur when
Greenland is at its coldest and Antarctica at or near its warmest (Clark and others,
2007). If the S-N temperature difference is estimated from the records, a strong
correlation between peaks in the S-N temperature difference and the timing of HEs
appears (fig. 11). This is suggestive of strong coupling between the hemispheres
(notice that in fig. 11 the Antarctic record is replaced with the time integral of NGRIP
to avoid the problem of using different age models, compare fig. 1). Consistently,
peaks in pole-to-pole temperature difference correlate with all HEs, which therefore
happen when the coupling terms in equations (B.2) and (B.4) are the largest, which is
when synchronization is most robust, according to VSO. The bottom panel in figure 11
predicts what could be named Heinrich events H7, H8 and H9, at approximately 74 ka,
77 ka and 87 ka respectively. Note that the strong peak at the time of recently
discovered event H5a could have been predicted by the graphical procedure.
However the physical processes that originated the synchronization of the Polar
Regions, how coupling through the ocean and atmosphere makes polar synchrony
work, or how it has worked in the past, are enormously difficult unresolved problems
that will require much further research to unravel.
acknowledgments
This research is supported by grants from the J. S. McDonnell Foundation (21st
Century Science Initiative on Complex systems), the National Science Foundation
(Paleoclimate and P2C2 programs), and the National Geographic Research Council.
Technical help by Yue Zhang, Raj Saha and Dr. Jeseung Oh in several aspects of the
research is gratefully acknowledged. All data used in this paper was provided by the
National Snow and Ice Data Center, University of Colorado at Boulder, and the WDC-A
for Paleoclimatology, National Geophysical Data Center, Boulder, Colorado.
Appendix A
the hilbert transform
The Hilbert transform H[a(t)] of a time signal a(t) is defined as (Bracewell 1986):
H 关a 共t兲兴 ⫽
冕
1
P
␲
⬁
⫺⬁
a 共␶兲
d␶.
t⫺␶
The integral is improper and P stands for the Cauchy principal value of the integral, whenever this value
exists. If we denote by v(t) the Hilbert transform of a(t), that is, H [a(t)] ⫽ v(t), and if a(t) has Fourier
transform A(␻), and the Fourier transform of v(t) is denoted by V(␻), then by the convolution property of
the Fourier transform V(␻ ⫽ ⫺isgn(␻)A(␻), where sgn(␻) is the sign function, equal to ⫹1, 0 or ⫺1 if ␻ is
positive, zero or negative, respectively. That is, the Hilbert transform does not change the magnitude of A(␻)
it changes only the phase by ⫺␲/2 if ␻ ⬎ 0 and by ⫹␲/2 if ␻ ⬍ 0. The use of the Hilbert transform to
compute the instantaneous phase and frequency of a(t) is explained in the text.
As described in the text, for positive frequencies, the Hilbert transform of the Antarctic record, H [a(t)],
is approximately equal to g(t), the Greenland record. In the frequency domain, and for positive frequencies,
this is equivalent to shifting each frequency component of the spectrum of a(t) by ⫺␲/2 radians (see figs. 2,
ice age: in search of simple rules at the heart of climate’s complexity
443
3). Conversely, the inverse Hilbert transform of the Greenland record closely reproduces the Antarctic one,
and this is equivalent to shifting each positive spectral component by ⫹␲/2 radians. A positive phase shift
means a shift to earlier times.
Appendix B
the van der pol synchronization oscillator (vso)
To represent polar climate variability I borrow from the work of climatologist Barry Saltzman, who three
decades ago showed how climate oscillations of polar temperatures may be explained by the nonlinear
interaction between just two variables: sea ice extent and mean ocean temperature. Saltzman’s two variables
play a similar role to that of displacement and velocity in a simple linear pendulum oscillation, only that here
the oscillation is a self-sustained, nonlinear one. Previously we have used a noisy van der Pol oscillator to
closely reproduce the entire GRIP record and particularly the BA-YD sequence (Rial and Saha, 2011; figs. 6,
7).
A polar synchronization model, here referred to as VSO (Van der Pol Synchronization Oscillator)
consists of two pairs of first-order ordinary differential equations, each pair is a van der Pol oscillator
describing the interaction between sea ice extent and mean ocean temperature of the Polar Region it
represents, as follows:
u⬘1共t兲 ⫽ ⫺␻12u2共t兲 ⫹ M1共t兲 ⫹ ␰共t兲
(B.1)
u⬘2共t兲 ⫽ f1共t, ui兲 ⫹ q1关u⬘1共t兲 ⫺ u⬘3共t兲兴 ⫹ q2关u1共t兲 ⫺ u3共t兲兴
(B.2)
u⬘3共t兲 ⫽ ⫺␻22u4共t兲 ⫹ M2共t兲 ⫹ ␩共t兲
(B.3)
u⬘4共t兲 ⫽ f2共t, ui兲 ⫹ q1关共u⬘3共t兲 ⫺ u⬘1共t兲兴 ⫹ q2关u3共t兲 ⫺ u1共t兲兴
(B.4)
with
f 1共t, ui兲 ⫽ ␮1关u2共t兲 ⫺ 1 u2共t兲3兴 ⫹ u1共t兲
3
f 2共t, ui兲 ⫽ ␮2关u4共t兲 ⫺ 3 u4共t兲3兴 ⫹ u3共t兲.
1
Equations (B.1) and (B.2) describe the climatic interaction between mean ocean temperature (u1) and sea
ice extent (u2) in the Arctic region while equations (B.3) and (B.4) describe mean ocean temperature (u3)
and sea ice extent (u4) in the Antarctic region. All the variables are zero-mean, measured from an arbitrary
“equilibrium” or reference value (Saltzman and others, 1981; Saltzman, 2002; Rial and Saha, 2011). It is
appropriate to call (u2, u4) and (u1, u3) homologous pairs, and (u2, u3) and (u1, u4) conjugate ones. The
distinction is useful because equations will show that homologous pairs synchronize in-phase or in
anti-phase, while conjugate pairs synchronize at a phase lock of one-quarter period (fig. 8). The ␻i are the
natural frequencies of the (linear) oscillation whose corresponding periods, Ti ⫽ 2␲/␻i, were estimated by
Saltzman and others (1981) to be in the range 1 to 3 ky. The ␮i are positive coefficients that represent the
intensity of the ice-albedo and greenhouse gas positive feedbacks (values around unity produce nonsinusoidal waveforms with sharp increases/decreases in amplitude). The cubic nonlinearity is the negative
feedback that maintains sea ice extent within reasonable limits. The coupling mechanism is the most general
for the model since it involves both the variable and its time rate of change (Hale, 1997).
The white noise functions satisfy 具␰(t)␰(s)典 ⫽ D1␦(t ⫺ s); 具␩(t)␩(s)典 ⫽ D2␦(t ⫺ s), where the noise levels
are D1 and D2. These are both set equal to 0.45, after trial and error estimates show that this value is
consistent with the observed noise amplitude. With added noise the equations (B.1) through (B.4) are
integrated using Euler’s method to solve stochastic differential equations.
Coupling is assumed linear in both the “dissipative” (coefficient q1) and the “reactive” (coefficient q2)
terms and, as indicated above, coupling should be bidirectional, so that the Polar Regions influence and are
influenced by each other. Translated into climate terms the coupling terms in VSO (eqs B.2 and B.4) are
assumed to be proportional to the difference in mean oceanic temperature/salinity (reactive) and mean
ocean heat storage between north and south (dissipative). This form of coupling is the most widely used in
this type of oscillators (Balanov and others, 2009) and is adapted here since the true nature of the coupling is
unknown.
Results of VSO with added Gaussian noise show that with noise levels commensurate with the observed
records, synchronization between north and south is possible (for example, figs. 8, 9). To maintain
444
J. A. Rial—Synchronization of polar climate variability over the last
synchrony in the presence of increased noise requires increasing the magnitude of the coupling coefficients.
In this context insolation forcing is also considered as (long-period) noise, and modeling with VSO reveals
that the main effect of insolation forcing (phase/frequency and amplitude modulation) can also be
compensated for by increasing the coupling coefficients. As synchrony is approached, the phase relationship
between synchronizing time series becomes that expected in the absence of insolation, regardless of how
much the phase and amplitude have been distorted by modulation. The robustness of synchronization
dominates over external forcing.
The external forcing functions Mi(t) represent insolation forcing, in general different at each pole and
equal to miF (t), where mi are adjustable numerical coefficients and F (t) is the normalized sum of harmonics
with the three relevant Milankovitch periods (19 ky; 23 ky; 41 ky).
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