on the nature of the pliocene/pleistocene glacial cycle lengthening

Global Perspectives on Geography (GPG) Volume 1 Issue 1, February 2013 www.as-se.org/gpg

9

On the Nature of the Pliocene/Pleistocene Glacial Cycle Lengthening Nadezda N. Ivashchenko1, Vladimir M. Kotlyakov2, Dmitry M. Sonechkin3, and Nadezda V. Vakulenko4 1Hydrometeorological Research Centre of Russia, Moscow, Russia; 2 Institute of Geography RAS, Moscow, Russia; 3, 4P.P. Shirshov Oceanology Institute RAS, Moscow, Russia 1 [email protected]; 2 [email protected]; 3 [email protected]; 4 [email protected]

Abstract

Although numerous endeavours have been undertaken to understand the nature of the 40-to-100 ka transition in the glacial cycle lengths that took place about 1.0-1.5 ma BP, this phenomenon remains a mystery up till now. A specially designed wavelet-based technique was used to treat paleoclimatic records related to those times in concepts of the mathematical dynamical system theory. The well known fact that the global climate system underwent a rhythmic behaviour of the ~41 ka period (the main period of the Earth’s axis obliquity) was interpreted during the Pliocene as an evidence of the limit cycle attractor in the climate system dynamics. Under the stress of gradual climate system cooling, the magnitude of this limit cycle attractor increased from the Early to Late Pliocene. But this increase was not monotonous because of the ~1.2 ma-long beat of obliquity. When this magnitude exceeded a certain level, the climatic limit cycle attractor lost its stability, and a new, more complex, attractor arose via the period doubling bifurcation that is well-known in the dynamical system theory. During the Pleistocene, the magnitude of the new-arisen attractor was essentially enhanced because of resonances with insolation variations that were induced by combinational tones of the Earth’s orbit eccentricity, and the attractor period trebled, doubled and trebled by turns depending on phase-locking to either of those tones.

Keywords

Pliocene/Pleistocene Glacial Cycles; Orbital Insolation Variations; Climate Cooling Trend; Period Doubling/Trebling Bifurcation; Wavelet Analysis

Introduction

Since pioneering researches of L. Agassiz, J. Adhemar, J. Croll, and M. Milankovitch paleoclimatologists accept a theory (usually called the Milankovitch theory) to explain the glacial/ interglacial alternation in the Earth climate history during the Pliocene/Pleistocene epoch (from about 5-6 million years BP up to present). According to this theory the glacial/ interglacial alternation are driven by latitudinal and seasonal re-distributions of the ingoing solar radiation (insolation)

in one’s turn stipulated by quasi-periodic wobbles in the Earth’s orbital motion. In its classical form, this theory claimed that variations of the precession-induced northern near-polar summer insolation were the most important for the continental ice sheets to grow or melt. But there is an essential problem with the Milankovitch theory connected with the well-registered lengthening of the Pliocene/Pleistocene glacial/interglacial cycles from ~40 ka to ~100 ka near the beginning of the Pleistocene. In spite of this, the magnitude of the main ~41-ka obliquity periodicity did not decrease; rather it increased slightly at the moment.

This research was stimulated by some interesting considerations of the problem with a stress on the role of obliquity variations that have been published by some Western colleagues. Taking these into account, the problem was looked into an unprejudiced manner relying on the experience of the researchers in studies of the low- and super low-frequency weather variability in which the ideas and methods of the contemporary mathematical nonlinear dynamical system theory were successfully used. Namely, the concept of qualitative changes in the dynamical system behaviour under stress of quantitative changes in external forces called bifurcations.

Firstly, following the other reports and other researchers, it was taken into consideration that the Pliocene/Pleistocene climate was incessantly affected by quasi-periodic insolation variations that were excited by amplitude and frequency modulated wobbles of Earth’s eccentricity and obliquity during the whole time period under study. For example, one of the most powerful eccentricity wobbles of the 94,945 year period was modulated by the million-year long periodicities like 3466,974; 2035,441 (this eccentricity period was indicated as 2305,441 in the famous paper perhaps because of a typographic error); 1306,618; and 1282,495 years generating some combinational tones at very close frequencies 1/94,945+1/3466,974=1/92,414;

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www.as-se.org/gpg Global Perspectives on Geography (GPG) Volume 1 Issue 1, February 2013

10

1/94,945-1/2035,441=1/99,590;

1/94,945-1/1306,618=1/102,384;

and 1/94,945-1/1282,495=1/102,535 year-1 etc. In turn the million-year long modulations of these tones generated second order tones at the frequencies from ~1/110,000 to ~1/160,000 year-1 etc. At the same time, a spectral gap existed over a range of eccentricity frequencies ~1/200 - 1/270 year-1.

Thus, a multiscale hierarchy of combinational tones was maintained during the Pliocene/Pleistocene epoch. A similar hierarchy was also inherent to the obliquity wobbles. Moreover, the most powerful million-year long tones were common for both eccentricity and obliquity, and so insolation variations turned out to be very intricate in their character everywhere on the Earth, and their instantaneous frequencies varied in an apparently erratic (but very ordered in fact because the spectrum of these insolation tones was discrete) manner. On this ground, these insolation variations were described as “Climatic Polyphony”. This Polyphony made the understanding of the cause/effect interrelations between insolation and the Pliocene/Pleistocene climate difficult.

Some of the earlier researchers attributed variations observed in paleoclimatic records to either of such orbital parameter tones. But it should be stressed that responses of the climate system to insolation variations could produce peaks at the frequencies of the above combinational tones under the obligatory of nonlinearity condition of the climate system. Besides, the magnitudes of the nonlinear climatic responses to the tones will be noted to be more or less dependent on closeness (resonance) between current external frequencies and the frequencies inherent to the climate system itself (the eigen frequencies). Indeed, such a resonance existing at a time the magnitude of the respective climatic response was needed to be large even where the forcing magnitude was weak. The period of this response could be a multiple of the forcing period according to the resonance ratio. But if the forcing frequency was essentially lower than any of the eigen frequencies, then the respective climatic response needs to passively reproduce the forcing period, i.e. it was the case when the nonlinear climate system should behave itself as almost linear one.

Secondly, the consideration of possible external climate forces was not restricted by the quasi-periodic insolation variations. Paleoclimatic time series covering the Pliocene/Pleistocene epoch clearly

demonstrate a general cooling trend, and an increase of the higher-frequency climate variability in parallel to this trend. This trend can be attributed to changes in the Earth’s system ability to accumulate ingoing solar radiation, to emit infra-red radiation into space, and to exchange its warmness with the deeper ocean. The Earth albedo might not have been large and almost invariable during the very warm Early Pliocene. Perhaps, the emission of infra-red radiation also was poor variable because of a more or less constant level of greenhouse gases concentrations. However, according to and many others, the latest re-organizations of the general oceanic circulation, in particular those connected with the rise of the Northern Hemisphere ice sheets and the Panama seaway closure about 3-4 ma BP, could excite climatic instabilities at more frequencies as before. The researchers agree with these assumptions. But, it is assumed that like the opinions of, other numerous and still unrecognized, physical processes could be responsible for the increase of the Late Pliocene climate instability. Therefore, the emphasis of this study was on some consequences of independent climatic instability increase from its concrete substance; and the assumption that the climate evolution during the Pliocene/Pleistocene epoch consisted in a wandering of different resonances between external and eigen frequencies that results in sudden breaks in the typical climate behaviour (climatic attractor), called bifurcations (in the mathematical dynamical system theory) was inevitable.

Method of Study

A specially designed wavelet-transform (WT) technique of the paleoclimatic time series was developed and used for this analysis. The earlier researcher was the first to apply WT to some paleoclimatic records in order to portray a time- and scale-dependent character of the Pliocene/Pleistocene glacial-interglacial cycles. Some WT pictures in papers were also referred to. But when WT was applied properly it can give more than simple display of nonstationary paleoclimate variations.

The continuous WT of a real record )(tX was introduced by French mathematicians about two-three decades ago. The mathematical construction of WT is defined as a convolution

( ) dttXabtGaabW )(~5.0),( ∫

∞

∞−

−−= (1)


11

where )(tG is an analyzing wavelet that is a complex-

valued function in general ( G~ denotes the complex conjugate of G ), and so the result of WT consists of two (real and imaginary) parts:

Im),(Re),(),( abWiabWabW +≡ .

This result can be represented on the two-dimensional ),( ab -plane as a pattern either of ),(Re abWT and

),(Im abWT taken separately or as the WT-amplitude

( )),(2Im),(2

Re),( abWTabWTSQRTabWTAM +=

and phase

( )),(Re/),(Imtan),( abWTabWTaabWTPH =

also taken separately.

Because of finiteness of every paleoclimatic record, some finite limits of the integration must be used in (1), which implies the distortions of the WT-pattern within some boundary areas that are wider in larger WT scales. Areas of such distortions are outlined in all figures of the WT-patterns shown in this paper. Besides, every paleoclimatic record is represented by its finite data sample for a number of points of time, and so a numerical summation must be used instead of the integration in (1). Such a summation is easy to compute if the sample points are equidistant. Unfortunately, it is not the case in paleoclimatology. Therefore, a preliminary interpolation of the data sampled on a sequence of equidistant points usually is computed. This interpolation can distort WT-pattern. To partly reduce this source of distortion, a special FORTRAN routine capable of working directly with irregularly sampled data points was developed and used in this study,

The Morlet wavelet function

{ } { } 5,exp2/2exp25.0)( >−−= CiCtttG π

used for wavelet functions especially suitable to investigate amplitude and frequency modulated oscillations in time series was adaptive to this study. The chosen value C=6.2035 ensured strict equality of the Fourier and wavelet scales, i.e. under this choice temporal variations in ),( abW are most sensitive to the Fourier harmonic of )(tX with period equal to a ,

and the WT-amplitude is maximal at the wavelet scale

a if a maximum exists at the frequency aπ2 in the

power spectrum of the time series under WT. Temporal variations of ),( abWTAM corresponding to the same value a evidenced the existence of an amplitude modulation of the respective Fourier

harmonic. Displacements of the ),( abWTAM -local maxima up and down from the wavelet scale a evidenced the existence of a frequency modulation of this harmonic.

The )(tX value can be reconstructed or any of its band-pass filtered ingredients by means of an inversion of WT. One of the most often used of the several different inversion formulas consists of simple added up values of the real part ),(Re abWT

corresponding to the same value b over either the full range or a sub-range max)min( aa − of wavelet scales a

daata

aWTtX ),(

max

minRe)( ∫= (2a)

Using imaginary part ),(Im abWT instead of real

one,

daata

aWTtY ),(

max

minIm)( ∫= (2b)

a complex completion can be formed from

)()()(~ tYitXtZ +=

the record under study to create an analog of its Hilbert transform

[ ] ∫∞

∞−−−= dsstsXPtXH )/()(1)( π

where P is the Cauchy principal value of the integral. Certainly, the Hilbert transform is a meaningful tool of time series study only if the variability of time series is concentrated on a rather narrow range of time scales. It is not the case for paleoclimatic time series but the use of (2) band-passes paleoclimatic series. This band-passing makes the Hilbert transform to be meaningful.

Indeed, the two-dimensional plane ( )YX , can be

treated as a state-subspace of the dynamical system of


12

82123

41

comp

lex re

con.5

8 - 29

ka

Coole

r

Warm

er

Co

oling

trend

Smaller LargerObliquity beats

complex recon. 58 - 29 ka

obliquity

FIG. 1 WAVELET-TRANSFORMED AMPLITUDE PATTERN OF THE LR04 BENTIC O18

δ TIME SERIES [16]. THE ORIGINAL SERIES IS SHOWN BY THE BACK LINE WITH ITS VERTICAL AXIS BEING INVERTED TO SHOW GLACIATIONS BY MINIMA. A TWO-

PARAMETRIC DIAGRAM IS FRAMED INTO THE WT PATTERN THAT SHOWS A DEPENDENCE OF MAGNITUDE OF CLIMATIC VARIATIONS FROM THE MILLION YEAR LONG BEAT OF OBLIQUITY (GREEN LINE LOWER THE LR04 SERIES) AND THE GENERAL

CLIMATE COOLING (BLACK SMOOTH LINE IMPOSED ON THE LR04 SERIES). THE CLIMATIC VARIATIONS ARE GIVEN BY A COMPLEX LR04 SERIES RECONSTRUCTION BAND-PASSED OVER THE 58-29 KA RANGE OF WAVELET SCALES FOR THE PLIOCENE (RED AND BLUE LINES LOWER THE OBLIQUITY SERIES). TWO TIME MOMENTS WHEN THE ~41 KA LONG CLIMATIC LIMIT CYCLE

COULD LOSE ITS STABILITY ARE MARKED BY TWO PINK STARS, AND THE MOMENT OF THE REALIZED PERIOD-DOUBLING BIFURCATION OF THIS CYCLE IS MARKED BY THE RED STAR ON THE DIAGRAM

interest, and, even if this dynamics is many-dimensional, such a subspace can admit to study some of its very long-living peculiarities, e.g. its steady and periodic (limit cycle-like) states.

Results

Searching Qualitative Changes in the Pliocene / Pleistocene Climate Evolution

FIG. 1 illustrates how WT technique works on an example of a ~5.3 ma long time series stacked by 57

bentic foraminifera O18δ records and called LR04. The same analysis was prepared for another long time series with similar results. The WT-amplitude pattern of the LR04 series convincingly corroborates the well known fact that the Pliocene climate variations were very powerful near the time scale of the main obliquity period (~41 ka), and much less powerful over longer and shorter time scales of precession and eccentricity.

To investigate this fact quantitatively a complex 58-29 ka reconstruction of LR04 was computed (shown by red and blue lines slightly lower the original LR04 series in FIG. 1) using (2). The ordinate scales of both original and reconstructed LR04 series are inverted to show maximal glaciations as minima of these series graphs in all figures of this paper.

Comparing this reconstruction with the obliquity time series (shown by a green line between the original LR04 series and the reconstructed series in FIG. 1) it could be visually observed that the Pliocene climatic cycles were in phase with the main ~41 ka long obliquity oscillations. Besides, the ~1.2 ma-long beat of these climatic cycles was in phase with the equally long-term obliquity beat. Perhaps it was due to the fact that all eigen frequencies of the Pliocene climate were essentially higher than the frequencies of the simultaneous powerful obliquity oscillations. Therefore, the climate system slave responding to the


13

~41 ka-long insolation force. As a result, a limit cycle-like attractor was excited and maintained in the state-space of the climate system.

Numerous powerless variations exist in the original LR04 time series with shorter than 29 ka, and longer than 58 ka periods. Even where some of these variations are artifacts of observational noises and biases, their reality is beyond all reasonably doubt. Therefore, it is more precise to suggest a limit cycle-like “skeleton” of a seemingly chaotic Pliocene climate attractor. But this “skeleton” appears to be a initial good indicators to represent the most important features of the whole chaotic attractor as the inspected afore-mentioned powerless climatic variations revealed no in-phase behaviour with the simultaneous precession (eccentricity) higher (lower)- frequency variations.

After thorough comparison of the obliquity time series with the time series of the complex LR04 58-29 ka reconstruction it could be observed that the magnitude of the climatic limit-cycle as well as the contribution of this cycle to the general Pliocene climate variability depended on two factors: the magnitude of the obliquity ~1.2 ma-long beat, and the general Pliocene climate cooling. A two-parametric diagram quantifying combined influence of these two external forces on the magnitude of the climatic limit cycle-like attractor is framed into the WT-amplitude pattern in the right lower side of FIG. 1. It was observed that the limit cycle magnitude at a time moment ~2.5 ma BP (marked by a pink star on the right border of the diagram) was 2-3 times more than the magnitude at the beginning of the Pliocene (marked by three brown-filled circles near the upper border of the diagram). Observation on the WT-amplitude pattern, indicates that a small domain of a very high WT-amplitude (colored by deep red) shows the position of this event within the main WT-amplitude pattern in FIG. 1. It was the first time when the climatic limit cycle-like attractor could lose its stability. But the limit cycle magnitude later decreased because of the fading of the obliquity beat. This explained how the climatic limit cycle-like attractor preserved its stability in this event.

The next attempt to destabilize the limit cycle-like attractor took place at a more recent time (~1.5 ma BP). This event is indicated by the second pink star in the framed diagram, and a larger domain colored by purple within the main WT-amplitude pattern in Fig. 1. However, this attempt was also failed to succeed. The real change of the climatic limit cycle-like attractor

shape (a bifurcation in terms of the dynamical system theory) took place after a short while (~1.250 ma BP). This bifurcation event is marked by red star in the diagram. Both external forces (the progressive climate cooling and the quasi-periodic insolation variations induced by ~1.2 ma-long obliquity beat) were responsible for this bifurcation. But the progressive climate cooling was decisive because the obliquity beat magnitude was not very large at this time moment.

Portraying the Pliocene/Pleistocene Climatic Attractors

There are two questions now. The first one is how the new shape of the climatic attractor arose as a result of this bifurcation? The second is what kind of external forces drove the newly-arisen climatic attractor during the followed times of the Pleistocene?

FIG. 2a helps to answer the first question showing a zoom of the LR04 WT-amplitude pattern over the time interval from 1.5 ma BP up to present as well as the time series of the complex LR04 164-29 ka reconstruction. Thin red vertical lines drawn over the main minima of the real part of the reconstruction divide into 20 pieces the time interval under consideration. These vertical lines also pass through (or very close to) main minima in the original LR04 time series. According to its temporal length, each of the pieces is marked by a fat horizontal strip at the respective wavelet scale within the WT-amplitude pattern. The strips of the pieces 20 and 19 (corresponding to the number of the second unsuccessful attempt of the climatic limit cycle- like attractor to lose its stability) are colored by pink (namely these cycles marked by second pink star at the diagram framed into FIG. 1). The strips of the pieces 18-15 (when the ~41-ka limit cycle-like attractor was stable) are colored by black, and those of the pieces 14 and 13 (the bifurcating cycles) are colored by red (and marked by red star in the diagram framed into FIG. 1). The strips of the pieces 12-1, representing the new-arisen climatic attractor, are colored by green. Note that almost all the strips are situated at or very close to the wavelet scales corresponding to the main local maximum of the WT-amplitude. The most essential deviations of these strips from the maximum are seen for the pieces 6, 5, and 2. The piece 6 seems to be too short, and the pieces 5 and 2 seem to be too long. It is assumed that these deviations are caused by some defects in construction of LR04 series because the respective climatic cycles appears to be very variable in paleoclimatic records used to construct LR04, and in


14

FIG. 2 (A) - A ZOOM OF THE WAVELET-TRANSFORM

AMPLITUDE PATTERN SHOWN IN FIG. 1, AND ITS REAL AND IMAGINARY PARTS OF A RECONSTRUCTION OF LR04 OVER

THE 164-29 KA RANGE OF WAVELET SCALES. THIN RED VERTICAL LINES DRAWN OVER THE MAIN MINIMA OF THE REAL PART DIVIDE THE SERIES ONTO 20 PIECES (CLIMATIC

CYCLES), AND FAT HORIZONTAL STRIPS BETWEEN CONSEQUENT RED VERTICAL LINES MARK POSITIONS OF

EACH CLIMATIC CYCLE ON THE TIME-SCALE PLANE OF THE WT PATTERN. (B) – A RESPECTIVE WT-PATTERN OF THE

ANNUAL EQUATORIAL INSOLATION TIME SERIES [27]), AND ITS COMPLEX RECONSTRUCTION BAND-PASSED OVER THE 164-29 KA RANGE OF WAVELET SCALES (VERTICAL AXIS IS INVERTED). THIN RED VERTICAL LINES ARE DRAWN OVER

THE MAIN MINIMA OF THE REAL PART OF THE INSOLATION RECONSTRUCTION. ALL INTERVALS BETWEEN CONSEQUENT

LINES RELATED TO THE CLIMATIC CYCLES AFTER THE BIFURCATION ARE DENOTED BY A LETTER (A, B, OR C)

SIGNALING THE CYCLE DOUBLING OR TREBLING.

many other paleoclimatic series. For example, computing WT amplitude patterns for the Dome C – EPICA deuterium ( Dδ ) and sodium ( ssNA ) time series, and doing the same partitions of these patterns (FIG. 3 and 4), it could be observed that these patterns varied slightly compared to LR04 WT-pattern in FIG. 2(a). As a result, the strips of the pieces 6, 5, and 2 coincided better with the respective local WT-amplitude maxima in the Dδ and ssNA series, this coincidence however remained incomplete: the piece 6 appears to be even longer than it should be, especially for the ssNA series.

The complex 164-29 ka reconstruction for the LR04 series (FIG. 2a) was used to portray the climatic attractor on a two-dimensional state-subspace

( )YX , of the climate system during each of the

divided time pieces (FIG. 5). FIG. 5 shows such shapes for the pieces 20 - 15 to be perfectly circular, i.e. representative by sine and cosine of time as the periods varied from 37 up to 45 ka, i.e. around the ~41 ka obliquity period. The climatic limit cycle magnitude was larger for the pieces 20 and 19, and smaller for four next pieces according to simultaneous increases and decreases of the ~1.2 ma-long obliquity beat (see FIG. 1). These facts showed that the climate response to the obliquity force remained almost linear from the beginning of the Pliocene up to ~1.25 ma BP. After this, a domain of moderately large WT-amplitudes shifted in direction of the wavelet scale 82 ka within the WT-amplitude patterns as shown in FIG. 1 and 2a signaling that the period of the climate system response to the obliquity 41-ka forcing began to lengthen. The duration of the pieces 14 and 13 turned out to be equal to ~52 ka and ~76 ka respectively (FIG. 5), and so one-to-one correspondence between the 41 ka obliquity forcing and the climatic limit cycle-like attractor was lost. The shapes of these pieces still remained to be circular, and the new climatic attractor did not form. It was just the time of a qualitative change (bifurcation) of the climatic attractor.

Durations and shapes of the climatic cycles corresponding to the pieces 12 – 1 were quite different from the previous ones (FIG. 5). It can also be seen in FIG. 6 where these cycles are represented on the same two-dimensional state-subspace for the Dome C-EPICA Dδ and ssNA time series. All these shapes reveal one or two loops before the closure of the climatic state-trajectory. This fact indicates nonlinearities in some responses of the climate system to the obliquity force. Periods of the climatic cycles were around the doubled (~82 ka) obliquity in the first instance (the cycles 12-7). Then these periods were lengthened up to the trebled (~123 ka) obliquity period (the cycle 6, and maybe also 5), returned back to the doubled period (the cycles 4 and 3), and again were lengthened up to the trebled period (the cycles 2 and 1). These changes in the cycle length were noticed by earlier Western researchers. This study note in addition that an inner symmetry was inherent to the afore-enumerated length changes for the 6 most recent cycles: the more distant longer cycles were from the interglacial period that arose from ~400 ka BP, moreover, it was also indicated on a symmetric reversal of the saw-tooth shapes of these 6 cycles. This symmetry was the result of a frequency modulation of climatic variations, and this possibility has already been mentioned in Introductory Section.


15

41

82

123

Ins

olatio

n 0

(58-29

inve

rted)

ka BP

Scale

(ka)

comple

x reco

nstruc

tion 1

64 - 29

1 2 3 4 5 6

FIG. 3 WAVELET-TRANSFORM AMPLITUDE PATTERN OF THE DOME C – EPICA DEUTERIUM ( Dδ ) TIME SERIES [28, 29]. THE ORIGINAL SERIES (VERTICAL AXIS IS INVERTED TO SHOW GLACIATIONS BY MINIMA, BLACK LINE), AND ITS COMPLEX RECONSTRUCTION OVER THE 164-29 KA RANGE OF WAVELET SCALES (RED AND BLUE LINES). THE RECONSTRUCTED

INSOLATION VARIATIONS [27] ARE RESPECTIVELY SHOWN BY BROWN LINE. THIN GREEN VERTICAL LINES ARE DRAWN OVER THE MAIN MINIMA OF THE REAL PART OF THE Dδ RECONSTRUCTION, AND GREEN FAT HORIZONTAL STRIPS BETWEEN CONSEQUENT VERTICAL LINES MARK POSITIONS OF EACH CLIMATIC CYCLE ON THE TIME-SCALE PLANE OF THE WT-

PATTERN INSOLATION RECONSTRUCTION WITHIN THE SCALE RANGE 58-29 KA IS SHOWN BY BROWN LINE

Drivers of the Pleistocene Climate

Rial was noted as the first researcher considering the existence of a frequency modulation in the Pleistocene climate variations. He postulated the existence of a parametric excitation (as in the well-known Mathieu equation) of these variations at a carrier frequency 1/95 ka-1 modulated by the essentially lower frequencies 1/413, and 1/826 ka-1. However, it is difficult to imagine that there is such low frequency as 1/95 ka-1 among the eigen frequencies of the climate system. For this reason, it is inappropriate to interpret this frequency modulation as a kind of the parametric excitation of the climate system. Rather, it could be interpreted as an evidence of a wandering of the climate system response amongst different resonances between the doubled and trebled frequencies inherent to the new-arisen climatic attractor and external frequencies. Namely, this attractor could resonate with either of eccentricity combinational tones such as almost doubled tones 1/92,414+1/412,885=1/75,512; 1/94,945+1/412,885=1/77,194;

1/99,590+1/412,885=1/80,237; 1/102,535+1/412,885=1/82,138 year-1, and almost trebled ones

1/92,414-1/412,885=1/119,063;

1/94,945-1/412,885=1/123,298 year-1 etc. even where insolation oscillations induced by these tones were very weak.

The outcome of the comparison between the WT-pattern for a part (0 – 1500 ka BP) of LR04 series shown in FIG. 2a and the WT-pattern for an annual mean equatorial insolation series shown in FIG. 2b helps to answer the second question posed in the beginning of Subsection B. Did what kind of external forces drive the new-arisen climatic attractor during the followed Pleistocene times? This immediately explains how the equatorial insolation was chosen for such comparison, although daily maximum of the summer insolation at 65o n.l. driven by precession is considered to be the most important quantity in the frame of the classical Milankovitch theory. Firstly, according to Huybers,


16

ka

1 2 3 4 5 6

123

comp

lex re

c.58 -

14 ka

comp

lex re

c.164

- 29 k

a

BP

164 - 29 ka

23

41

82

123

(ka)

58 - 14 ka

ssNA

flux

FIG. 4 THE SAME AS IN FIG. 3 BUT FOR THE DOME C – EPICA SEA-SALT SODIUM ( ssNa ) TIME SERIES

excess over a threshold of the summer insolation integrated over the whole warm half year is a better candidate to be “driver”, and obliquity is the prime factor determining this excess at any latitude. Because there is one-to-one correspondence between near-polar and equatorial insolation variations driven by obliquity, insolation variations at either of latitudes could be considered with the same success (see references for more details). As the equatorial insolation variations are much more sensitive to eccentricity than near-polar insolation variations, they are therefore preferable where both obliquity and eccentricity oscillations are of interest. Besides, several recent papers had indicated that some roots of the glacial/interglacial cycles were near the equator during the Pliocene/Pleistocene epoch.

It could be observed that the vertical axis of the insolation real and imaginary parts of the 164-29 ka reconstruction is inverted in order to have insolation minima to be coincided with minima in the climatic reconstruction shown in FIG. 2(a). Thin red vertical lines are drawn over all main minima of the real part of the insolation reconstruction to divide the time

FIG. 5 SHAPES OF ALL LATE PLIOCENE/PLEISTOCENE CLIMATIC CYCLES AS THESE ARE SEEN ON A TWO-

DIMENSIONAL STATE-SUBSPACE OF A COMPLEX RECONSTRUCTION BAND-PASSED OVER THE 164-29 KA

RANGE OF WAVELET SCALES FOR THE LR04. THE LENGTHS OF ALL CYCLES (IN KA) ARE INDICATED

interval under consideration into 20 pieces similar to such drawing over the minima of the LR04 reconstruction shown in FIG. 2(a). It could then be assumed that the insolation minima led the climatic minima in majority of all 20 division pieces with the exception of pieces 7, 4, 3, and 1. In these latter pieces, a discrepancy between pairs of dividing lines is noted


17

to be within a limit of accuracy of the original LR04 series timing. One can also observe that the position of the WT-amplitude local maximum within the eccentricity range of the wavelet scales (shown by a fat dashed black sine-like curve on the WT-amplitude pattern in FIG. 2b) varied from ~123 to ~82 ka and back with the main modulating period of ~413 ka. The time moments when the lengths of climatic pieces were close to 82 ka (the numbers 3, 4, 8, 9, 13, and 14) coincided well with the upper positions of this sine-like curve. The elongated climatic pieces (the numbers 1, 2, 5, 6, and 7) also coincided with the lower positions of the curve. The time of the period doubling bifurcation (the pieces 13 and 14) particularly coincided well with the upper position of the curve.

FIG. 6 THE SAME AS IN FIG. 5 BUT FOR THE DOME C – EPICA

Dδ AND ssNa TIME SERIES

These lead/lag interrelations of the insolation and climatic cycles doubted as the time scale of the LR04 series were tuned to 650 n.l. insolation variations. In order to avoid such doubts, the same WT-analyses

were done for two O18δ (benthic and planktonic foraminifera) time series created by Huybers with no tuning to insolation variations. FIG. 7 shows how real parts of these series 58-29 ka reconstructions (black and green lines respectively) interrelate with two respective equatorial insolation real parts reconstructed over 58-29 ka (brown line) and 164-58 ka (bold blue line shown with a vertical shift to avoid overlapping) wavelet scales. Note that time shifts between benthic and planktonic minima do not exceed 3-4 ka, and that there is no interrelation of the main benthic minima which restrict the pieces under consideration with the respective insolation minima. At the same time, the time shifts between seven insolation minima were seen in the 58-29 ka reconstruction before the period doubling bifurcation and seven main benthic minima (these shifts are indicated by whole numbers of ka bottom of the figure) varied from 7 to 13 ka. On the average, these shift values correspond to the phase lags of benthic

reconstruction cyclic variations which are equal to one quarter of the 41 ka obliquity cycle.

Two benthic O18δ cycles corresponding to the pieces 14 and 13 have already affected by the period doubling bifurcation. Their phases did not therefore interrelate with the insolation phases. Two shift numbers corresponding to these cycles are marked by red; one of them is negative, another one is positive and almost equal to one half of the 41 ka obliquity cycle. The benthic cycles 12-6 were of doubled periods in LR04, and time shifts of their minima in respect of the 58-29 ka insolation minima varied from 1 to 13 ka. Considering their shifts in respect of the 164-58 ka insolation minima, the pieces 12 – 10 revealed no interrelations with these later minima like the previous pieces. But such shifts were positive, i.e. eccentricity induced insolation tones were leaders, for the pieces 9 – 6, and so resonant excitation of these latter benthic

and planktonic O18δ cycles could occur.

FIG. 7 COMPLEX RECONSTRUCTION OF THE BENTHIC O18

δ

(BLACK LINE), PLANKTONIC O18

δ (GREEN LINE), ANNUAL EQUATORIAL INSOLATION TIME SERIES BAND-PASSED OVER THE 58-29 KA RANGE OF WAVELET SCALES (VERTICAL AXIS ARE INVERTED TO SHOW GLACIATIONS BY MINIMA), AND

ANNUAL EQUATORIAL INSOLATION TIME SERIES (FAT BLUE LINE SHIFTED DOWNWARDS TO PREVENT OVERLAPPING

WITH PREVIOUS GRAPHS) RECONSTRUCTED OVER THE 164-58 KA RANGE OF WAVELET SCALES (ALL SERIES FROM [27]).

BLACK WHOLE NUMBERS IN THE BOTTOM OF THE FIGURE INDICATE TIME SHIFTS BETWEEN THE MAIN GLACIATIONS OF THE PLEISTOCENE AS THESE ARE SEEN IN THE 164-29 KA

REAL PART OF THE BENTIC O18

δ TIME SERIES AND RESPECTIVE MINIMA IN THE 58-29 KA REAL PART OF THE

ANNUAL EQUATORIAL INSOLATION. POSITIVE AND NEGATIVE NUMBERS INDICATE WHICH INSOLATION CYCLE

LED (LAG) THE SIMULTANEOUS BENTIC CYCLE. THE LEAD/LAG NUMBERS FOR TWO BIFURCATING CLIMATIC

CYCLE ARE SHOWN BY RED.


18

Starting from about 550 ka BP up to present time shifts

between benthic O18δ minima and minima of the 58-29 ka insolation minima were similar (from 4 to 14) to those of the afore-mentioned pieces. In particular, two shifts (for the beginning of the piece 1 and for the transition to the Holocene) were equal to 11 and 5 ka, i.e. rather similar to the estimations of these shifts given in a recent paper. So, one can suppose that the obliquity induced insolation variations were indeed climate drivers in conformity with the opinion of Western paleoclimatologists mentioned in our Introductory Section. But the shift was negative (-17 ka) for the piece 2 (~250 ka BP), i.e. the respective obliquity induced insolation variations and climatic ones were out-of-phase. This study believes that another driver came into the scene during this event. This driver could be connected with a resonant enhancement of climatic cycles by either of eccentricity induced insolation tones from those indicated at the beginning of this paper. Comparing five most recent 164-58 ka benthic minima with the respective 164-58 ka insolation minima (bold blue line) one can be convinced that it was the case indeed: eccentricity induced insolation variations led in all five pieces for about 25-10 ka.

Last but not the least, the role of precession in the Late Pleistocene climate variations should be in consideration. The WT-pattern shown in FIG. 4 for the ssNA time series, characterizing climate variations within near-polar area of the Southern Hemisphere, helps to explain this question. This pattern reveals a shift of a domain of increased WT-amplitudes that is seen in the WT-patterns of the LR04 and Dδ time series near the 41 ka wavelet scale after ~450 ka BP (see Figs 1, 2a, 3) to the 23 ka wavelet scale. It shows that variations in the scale of precession became essential during the Late Pleistocene for the near-polar climate. However, considering graphs of the 58-14 ka and 164-29 ka reconstructions shown in FIG. 4, one could be convinced that those were lagged in comparison with variations of the obliquity and eccentricity scales. Thus, precession could not be one of the Late Pleistocene climate drivers although its influence may not be negligible. It could be assumed that the energy injected into the climate system within the scales of obliquity and eccentricity began to transfer to shorter scales due to the essential increase of the climate dynamics instability. Such a transfer could imply the origin of the saw-tooth shape of the Pleistocene glacial cycles.

It is well-known in mathematics that the ideal saw-tooth shape can be revealed as the following sequence of harmonics

,2,1,)sin(1)( =∑ −= mm

tmmtT ω

where the period of the first harmonic ωπ /2=l is equal to the “saw” period (approximately 100 ka in our case), and the power spectrum of such ideal saw-tooth follows the “-2” power law. In such a spectrum for the Vostok Dδ time series was computed, and it was found that the Vostok power spectrum followed the “-2” power law in general. However, periods of some intermediate spectrum peaks slightly deviate from integers ωπ mml /2= to coincide better with

precession and halved precession periods (~22, ~18, and ~12 ka). Besides, the amplitudes of these peaks are more powerful in comparison with the amplitude

governed by 2/1 m . It could therefore be assumed that the shorter-term climatic variations excited by a nonlinear energy transfer from the eccentricity and obliquity time scales to shorter scales were captured and enhanced by resonances with near-polar insolation variations induced by precession.

Conclusions

In recent years there have been some indications that obliquity induced insolation variations were the most essential for the Pliocene/Pleistocene climate, and that these variations in particular affected the character of the Late Pliocene and Early Pleistocene glacial/interglacial cycles.

The aim of this study was to establish that a combined action of these insolation variations together with a progressive global climate cooling predetermined the Pliocene/Pleistocene climate dynamics. To prove this assumption, the study formulated for the first time the problem of the Pliocene/Pleistocene climate dynamics as the bifurcation problem well-known in the contemporary mathematical dynamical system theory, and developed a specially designed wavelet-based technique of paleoclimatic time series analysis to portray the orbital scale climatic variations and trace their changes across the Pliocene/Pleistocene epoch. To sum up, two-parametric bifurcation diagram was constructed. This diagram showed how simultaneous quantitative changes in the magnitude of the ~1.2 ma long obliquity beat and the climate cooling trend


19

influenced on the character of climatic variations – the climatic “attractor”.

The detailed results of this study include the following:

Because the internally inherent frequencies of the Pliocene climate were essentially higher than the main 1/41 ka-1 obliquity frequency, the climate response to the insolation forcing induced by obliquity had to be almost linear, i.e. concentrated at the forcing frequency itself with a delay of about one quarter of the forcing period. Roughly speaking, “skeleton” (essentially low-passed variations of instantaneous climate states) of this response was periodic (like a limit cycle attractor well-known in the dynamical system theory) during the Early and Mid Pliocene.

When the general climate cooling began to be excessive at the Late Pliocene, the climatic limit cycle-like attractor lost its stability, and a new attractor arose via the well-known (to mathematicians) period-doubling bifurcation, i.e. the “skeleton” of the climatic attractor remained periodic but with approximately doubled period.

During the Pleistocene, insolation variations induced by combinational tones of the Earth’s orbit eccentricity came into the scene. Although the magnitudes of these tones were very subtle, the magnitude of the new-arisen climatic attractor turned out to be enhanced in comparison with the magnitude of the earlier existed attractor because of resonances with these tones, and the attractor period doubled and trebled by turns depending on phase-locking to either of these tones.

During the Late Pleistocene, the progressive cooling trend began to destabilize even the new-arisen climatic attractor. As a result of this destabilization, a nonlinear energy transfer arose from the scales of eccentricity and obliquity, where this energy was injected into the climate dynamics, to shorter scales. The shorter-scale climatic variations resulting from this transfer were captured and enhanced by resonances with precession induced insolation variations.

ACKNOWLEDGEMENTS

This study was fulfilled under financial supports of the Russian Foundation for Basic Research (Grants 09-

05-00155 and 12-05-00136) and the Program of Fundamental Researches No. 17 of the Presidium of the Russian Academy of Sciences.

REFERENCES

A. Berger, J.L. Melice, M.F. Loutre, “On the origin of the 100-

kyr cycles in the astronomical forcing” Paleoceanology,

vol.20, 2005.

A.Berger, “Long-term variations of daily insolation and

Quaternary climatic changes” Journal of the Atmospheric

Sciences, vol. 35, pp. 2362-2367, 1978.

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

the last 10 million years” Quaternary Science Reviews, vol.

10, pp. 297-317, 1991.

A.C. Ravelo, D.H. Andreasen, M. Lyle, A.O Lyle, M,W, Wara,

“Regional climate shifts caused by gradual global cooling

in the Pliocene epoch” Nature, vol. 429, pp. 263-267, 2004.

D.J. Lunt, P.J. Valdes, A.Haywood, I.C. Rutt, “Closure of the

Panama seaway during the Pliocene: implications for

climate and Northern Hemisphere glaciation” Climate

Dynamics, vol. 30, doi: 0.1007/s00382-007-0265-6, 2007

D.M. Sonechkin, “Stochasticity in the atmospheric general

circulation models” Hydrometeoizdat: Leningrad - in

Russian, 1984.

EPICA community members, “Eight glacial cycles from an

Antarctic ice core” Nature, vol. 429, pp. 623-628, 2004.

EPICA community members, “One-to-one coupling of

glacial climate variability in Greenland and Antarctica”

Nature vol. 444, pp. 195-198, 2006.

E.W. Bolton, K.A. Maasch, J.M. Lilly, “A wavelet analysis of

Plio-Pleistocene climate indicators: A new view of

periodicity evolution” Geophysical Research Lett , vol. 22,

№ 20, pp. 2753-2756, 1995.

E. Tziperman, M.E. Raymo, P. Huybers, C. Wunsch,

“Consequences of pacing the Pleistocene 100 kyr ice ages

by nonlinear phase locking to Milankovitch forcing”

Paleoceanology,vol. 21, PA4206, doi;10.1029/2005PA001241,

2006.

Han-Shou Liu, B.F. Chao, Wavelet spectral analysis of the

Earth’s orbital variations and paleoclimatic cycles”

Journal of the Atmospheric Sciences , vol. 55, pp. 227-236,

1998.

Han-Shou Liu, “Glacial-interglacial changes induced by

pulse modulation of the incoming solar radiation” Journal


20

of the Atmospheric Sciences, vol. 103, D20, pp. 26147-26164,

1998.

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

of Earth’s orbital eccentricity” Science, vol. 285, pp. 564-

571, 1999.

J.L. Melice, A. Coron, A. Berger, “Amplitude and frequency

modulations of the Earth’s obliquity for the last million

years” Journal of Climate, vol. 14, pp. 1043-1054, 2001.

J. Nie, J. King, and X. Fang, “Late Pliocene – Early

Pleistocene 100-ka problem” Geophysical Research Lett, vol.

35, 2008.

J. Zachos, M. Pagani, L. Sloan, E. Thomas, K. Billups,

“Trends, rhythms, and aberrations in global climate 65

ma to present” Science, vol. 292, pp. 686-693, 2001.

K.T. Lawrence, Liu Zhonghui, T.D. Herbert, “Evolution of

the Eastern Tropical Pacific through Plio-Pleistocene

glaciation” Science, vol. 312, pp. 79-83, 2006.

L.E. Lisiecki, M.E. Raymo, “A Pliocene-Pleistocene stack of

57 globally distributed bentic O18δ records”

Paleoceanology, vol. 20, PA1003, doi: 0.1029/2004PA001071,

2005.

M.E. Raymo, K. Nisancioglu, “The 41 kyr world:

Milankovitch’s other unsolved mystery” Paleoceanology,

vol. 18, doi: 0.1029/2002PA000791, 2003.

M.E. Raymo, P. Huybers, “Unlocking the mysteries of the ice

ages” Nature, vol. 451, pp. 284-285, 2008.

N.V. Vakulenko, A.S. Monin, D.M. Sonechkin, “The key role

of precession cycle modulation in the alternation of Late

Pleistocene glacial and interglacial epochs” Doklady Earth

Science, vol, 391A, pp. 917-920, 2003.

N.J. Shackleton, M.A. Hall, D. Pate, “Pliocene stable isotope

stratigraphy of ODP site 846” Proc. Ocean Drill, Program

Sci. Results, vol.138, pp. 337-356, 1995.

N.J.Shackleton, S. Crowhurst, T. Hagelberg, N.G. Pisias, D.A.

Schneider, “A new late Neogene time scale: Application

to Leg 138 sites” Proc. Ocean Drill. Program Sci. Results,

vol. 138, pp. 73-101, 1995.

N.V. Vakulenko, V.M. Kotlyakov, A.S. Monin, D.M.

Sonechkin, “Significant features of the calendar of the

Late Pleistocene glacial cycles” Izvestiya, Atmospheric and

Oceanic Physics, vol. 43, pp. 713-721, 2007.

N.V. Vakulenko, V.M. Kotlyakov, A.S. Monin, D.M.

Sonechkin, “Symmetry of Late Pleistocene glacial cycles

in records of the Antarctic Vostok and Dome C stations”

Doklady Earth Sciences, vol. 407(2), pp. 326-329, 2006.

N,V, Vakulenko, V.M. Kotlyakov, A.S. Monin, D.M.

Sonechkin, “Synchronism of Late Pleistocene glacial

cycles with insolation variations on the equator” Doklady

Earth Science, vol. 413, №. 2, pp. 292-296, 2007.

P. Huybers, C. Wunsch, “Obliquity pacing of the late

Pleistocene glacial terminations” Nature, vol 434, pp. 491-

494, 2005.

P. Huybers, “Early Pleistocene glacial cycles and the

integrated summer insolation forcing” Science, vol. 313,

pp. 508-511, 2006.

R.N. Drysdale, J.C. Hellstrom, G. Zanchetta, A.E. Fallick, M.F.

Sanchez Goni, I. Couchoud, J. McDonald, R. Maas, G.

Lohmann, I. Isola, “Evidence for obliquity forcing of

glacial termination II” Science, vol. 325, pp. 1527-1531,

2009.

S.G. Filander, A.V. Fedorov, “Role of tropics in changing the

response to Milankovitch forcing some three million

years ago” Paleoceanology, vol. 18, 2003.

S.Sosdian, Y.Rosenthal, “Deep-sea temperature and ice

volume changes across the Pliocene - Pleistocene climate

transitions” Science, vol. 325, pp. 306-310, 2009.

T.J. Crowley, W.T. Hyde, “Transient nature of late

Pleistocene climate variability” Nature,vol. 456, pp. 226-

230, 2009.

T.K. Lowenstein, R.V. Demicco, “Elevated Eocene

atmospheric CO2 and its subsequent decline” Science, vol.

313, p. 1928, 2006.

Y. Ashkenazy, E. Tziperman, “Are the 41 kyr glacial

oscillations a linear response to Milankovitch forcing?”

Quaternary Science Reviews, vol. 23, 2004.

Y. Ashkenazy, E. Tziperman, “Scenarios regarding the lead

of equatorial sea surface temperature over global ice

volume” Paleoceanology, vol.21, 2006.

Y. Ashkenazy, “The role of phase locking in a simple model

for glacial dynamics” Climate Dynamics, vol. 27, 2006.

on the nature of the pliocene/pleistocene glacial cycle lengthening

Documents

attractor period

ka period

milankovitch theory

global climate system

climatic limit cycle

climate system dynamics

glacial cycle lengths

main period