The increasingly nonlinear response of the climate–cryosphere
system to insolation forcing during the Pliocene and Pleistocene, as
recorded in benthic foraminiferal stable oxygen isotope ratios (
The recurrent ice ages of the Pliocene and Pleistocene, as captured in
benthic foraminiferal
Bispectral analysis was conceived in the 1960s within the research field
that studies ocean waves (Hasselmann et al., 1963). It is an
accepted method of quantifying nonlinear energy transfers among nearshore
waves that reach breaking point when the seafloor shallows (Elgar and
Guza, 1985; Doering and Bowen, 1995; Herbers et al., 2000; de Bakker et al.,
2015). Since the pioneering interdisciplinary studies by
palaeoceanographer/-climatologist Teresa (Terri) King Hagelberg and
colleagues (notably the physical oceanographer/nearshore ocean wave
researcher Steve Elgar) on Pleistocene and Holocene records (Hagelberg et
al., 1991, 1994; King Hagelberg and Cole, 1995; King,
1996), relatively few studies have applied higher-order spectral analysis
methods to the palaeoclimate archive (e.g. Muller and MacDonald, 1997a, b; von Dobeneck and Schmieder, 1999; Rial and Anaclerio, 2000; Rutherford
and D'Hondt, 2000; Huybers and Wunsch, 2004; Huybers and Curry, 2006;
Liebrand et al., 2017; Da Silva et al., 2018). However, two more recent
developments make a re-appreciation for the potential of bispectra for
palaeoclimate science timely. First, advancements have been made in the
understanding and interpretation of the bispectrum
(Herbers et al., 2000; de Bakker et al.,
2015). These constitute (i) a shift of focus from bicoherence, which
quantifies the strength of the couplings between the frequencies that are
present in both the real and the imaginary parts of the complex-valued
bispectrum, to (just) the imaginary part of the bispectrum, which can be
used to compute nonconservative relative energy exchanges, if time series
are dominated by asymmetric wave forms/cycle shapes, and (ii) integration over
the imaginary part of the bispectrum to quantify conservative net energy
transfers, and potentially absolute energy transfers, if net transfers can
be scaled to the power spectrum (de Bakker et al., 2015, 2016). Second, increasingly noise-free benthic foraminiferal
Both the 40 kyr problem of the Pliocene and Early Pleistocene, and the
To quantify nonlinear energy transfers among astronomically paced cycles of
Earth's climate–cryosphere system, we use the LR04 compilation of globally
distributed records of stable oxygen isotope ratios (
Sinusoidal properties of Pliocene and Pleistocene climate cycles
(after Lisiecki and Raymo, 2005a; Lisiecki, 2010).
Skewness and asymmetry are quantified using both the third central moment
and bispectral methods (Figs. 2, S2, and S3 in the Supplement) (see Sect. 2.3.3 for bispectral method) (Elgar, 1987). Excess kurtosis is
quantified using the fourth central moment only, because no trispectra were
calculated (Fig. 2). Using third-moment quantities, skewness is determined
by Eq. (1):
Non-sinusoidal properties of Pliocene and Pleistocene climate
cycles (after Lisiecki and Raymo, 2007).
In contrast to spectral analysis (e.g. using the fast Fourier transform),
which gives the distribution of variance of a sinusoidal signal with
frequency, bispectral analysis describes the distribution of
non-sinusoidality with frequency (King, 1996). The skewness of a
cycle is described by the real part of the bispectrum, whereas the asymmetry
of a cycle is deconvolved in the imaginary part of the bispectrum. The
bispectrum shows nonconservative relative energy exchanges among
frequencies of a single time series. Energy transfers in the bispectrum
computed on the LR04 stack are expressed in ‰
The bispectrum is defined by Eq. (4):
The
Bispectral zonation scheme. Bispectra of climate cycles can be
subdivided into 15 zones that reflect unique combinations of three frequency
bandwidths. Orange, purple, and blue lines represent the main astronomical
frequencies of the eccentricity, obliquity, and precession cycles,
respectively (this colour coding is consistent throughout the paper). The
grey shaded areas represent suborbital periodicities. Difference frequencies
To help the interpretation of the bispectrum, we depict the main
astronomical frequencies in the bispectrum with coloured lines (Fig. 3).
Vertical and horizontal lines correspond to the difference frequencies
(
We follow the convention and define a triad interaction as negative or
positive if
Bispectra of Pliocene and Pleistocene climate cycles.
We note that within this study, we almost exclusively focus on the imaginary part of the bispectrum. This approach contrasts with previous palaeoclimate investigations that used the bicoherence spectrum (see, e.g. Hagelberg et al., 1991; von Dobeneck and Schmieder, 1999; Wara et al., 2000; Huybers and Curry, 2006; Da Silva et al., 2018). In short, bicoherence is a measure of the coherency between the real and imaginary parts of the bispectrum (i.e. where both parts have strong interactions) and is marked by positive values only (see, e.g. Elgar and Sebert, 1989; de Bakker et al., 2014). Conversely, the real and imaginary parts of the bispectrum transform skewed and asymmetric cycle geometries into their composite frequency components, respectively, and both parts are characterised by positive as well as negative interactions.
Adopting the bispectral method to extract cycle geometries (Figs. 2, S2, and S3) (Elgar, 1987), skewness and asymmetry are
computed from the biphase. The biphase reflects the ratio between the real
and imaginary parts of the bispectrum. The biphase is defined by Eq. (5):
To quantify power spectral density, we apply a standard integration of the
power spectrum. To obtain conservative net energy transfers per frequency
(defined by the nonlinear source term
To obtain more insight on the energy that is exchanged among eccentricity,
obliquity, and precession cycles, we integrate the spectra and bispectra over
separate zones (Fig. 3, Table A1). For bispectra, such a zonation approach
was first applied in research concerned with nearshore ocean waves to
distinguish between infragravity wave and sea-swell wave frequencies
(de Bakker et al., 2015), and is adapted here to
investigate the most important climate cycle bandwidths of the Pliocene and
Pleistocene. The boundaries between the climate cycle zones are
(arbitrarily) defined at frequencies of 17.8, 35.6, and 80.0 Myr
The geometry computations confirm the qualitative visual inspection of the
LR04 globally averaged, benthic foraminiferal
Qualitative linkage of power spectra of insolation to those of
climate via energy exchanges among climate cycles:
Non-sinusoidal cycle shapes may indicate the potential for obtaining more
information about nonlinear interactions among frequencies using higher-order spectral analysis (Hagelberg et al., 1991; King, 1996).
We present three examples of Pliocene and Pleistocene bispectra (i.e. their
imaginary parts), based on the LR04 stack, which are characterised by clear
triad interactions (Fig. 4; see also Sect. 2.3.2.). The first example is a
bispectrum across the Middle to Late Pleistocene (i.e. the
“
The second example is a bispectrum across the mid-Pleistocene transition
(i.e. the “
The third example is a bispectrum across the Late Pliocene to Early
Pleistocene interval (i.e. the “40 kyr world”), which depicts negative
and positive interactions, where both difference frequencies
Figure 5 shows three time-evolutive spectral and bispectral analyses for the
Pliocene–Pleistocene time interval. It compares the total power spectral
density of summer-half insolation at 65
The time-evolutive power spectra of insolation are marked by high spectral
power at the precession and obliquity periodicities, and the near absence of
spectral power at the eccentricity periodicities (Figs. 5a, 6, and 7a) (Hays et al., 1976; Berger, 1977). Furthermore, it is characterised
by amplitude variability in spectral power at these periodicities, which is
governed by the long-term (eccentricity) modulations of obliquity and
precession. These amplitude modulations have durations of
Integration of the spectra and bispectra per astronomical
bandwidth (top right).
Figures 5b and S3 depict the conservative net energy transfers among
(astronomically forced) climate cycles as present in the LR04 stack. The net
energy transfers show how energy is redistributed over the power spectrum
among astronomically paced and non-astronomically paced climate cycles. They
reveal strong increases in both energy gains (in red) and losses (in blue)
toward the present, and from
Disentangling “black box” climate. Conservative net energy
transfers during the Pliocene and Pleistocene over specific zones in the
imaginary part of the bispectrum (see methods section). Computational settings are as in
Fig. 5b.
Time-evolutive power spectral analysis of the LR04 record is characterised
by high-amplitude variability at the 40 kyr obliquity-paced cycle, from
approximately 4000 ka onward (Figs. 5c, 6, and 7c). In addition,
distinct
Conservation of energy in triad interactions located in the
imaginary part of Pliocene and Pleistocene bispectra. Conservativity of
To better understand the different roles of specific (bandwidths of) climate cycles in nonlinear interactions, we determine conservative net energy transfers per bispectral zone (Figs. 8, S4, and S5). These zones reflect the “layers” that make up total energy exchanges as depicted in Fig. 5b. We assess if triad interactions are conservative of energy for each of the bispectral zones and for the entire bispectrum (Fig. 9). Lastly, we recombine the bispectral zones again for frequency bandwidths involving precession-, obliquity-, or eccentricity-paced climate cycles to highlight their influences in nonlinear interactions (Fig. 10).
In Zone 1, from 1000 ka onward, we document a modest exchange of energy
between the (eccentricity-related) periodicities in the range from
Summed zonal integrations of the imaginary part of Pliocene and
Pleistocene bispectra. Net energy transfers are computed by
Overall, the zonal integrations are marked by negative triad interactions
and thus reveal a cascade of energy from the (sub)precession bandwidth to
the obliquity and (ultimately) eccentricity bandwidths through several
successive interactions (Fig. 8). These analyses further show that many
periodicities fulfil a dual role that often remains stable through time,
namely they serve simultaneously and persistently as energy provider and
receiver. For example, the 24 kyr precession-paced cycle gains energy
through
Computations of energy conservation, hereafter also referred to as
“conservativity”, indicate that both the total as well as the zonal energy
gains in triad interactions are commensurate with losses, given our assumed
coupling coefficient (Fig. 9). The dominance of negative interactions (i.e.
positive values for
To highlight the separate roles of the precession, obliquity, and eccentricity bandwidths during nonlinear interactions within the climate–cryosphere system, we recombine (i.e. sum) their respective bispectral zones of Fig. 8. To do so, we include those zones that contain either one, two, or three frequency components of a particular bandwidth (Figs. 3 and 10). The general pattern of an energy cascade is robust, but it becomes clearer that precession cycles do not contribute much to the fuelling of eccentricity-paced climate variability directly (Fig. 10a–c). The 40 kyr periodicity during the Middle and Late Pleistocene gains energy from precession-dominated interactions but loses energy in obliquity- and eccentricity-dominated interactions (Fig. 10a–c). A comparison of conservativity of the recombined zones that include at least one precession, obliquity, or eccentricity component indicates that approximately similar amounts of energy are exchanged in (triad) interactions involving obliquity as in those involving eccentricity (Fig. 10d).
By combining more than 50 individual benthic foraminiferal
Similar to the power spectrum, the bispectrum is marked by a trade-off
between resolution in the time versus frequency domains (Fig. S3).
Bispectral results become more significant, and can yield greater degrees of
freedom, if a larger number of similarly shaped cycles, or waveforms, can be
included for analysis (i.e. a lengthening of the window). For example,
studies on natural nearshore waves or those on flume-generated waves
(periodicities of seconds to minutes) use hour-long stable time series with
high sampling resolutions and wave numbers (de Bakker et
al., 2014). Bispectral analyses of such data sets are well resolved in the
frequency domain. Furthermore, they permit the computation of robust
confidence levels on the results and the selection of statistical settings
that yield high degrees of freedom (e.g. Elgar and Guza, 1985; de
Bakker et al., 2015). However, for the Pliocene and Pleistocene record,
similar statistical objectives are unattainable, because no two climate
cycles are alike and because of the low number of
In contrast to the relatively more robust bispectral results (see previous
section), absolute geometry values are more strongly affected by the
non-ergodic nature of the Pliocene–Pleistocene record. Choices of (i) window
length, (ii) method of detrending the time series, and (iii) the application of a
windowing function or not can have profound effects on the robustness of
the results (Tim E. van Peer, personal communication, 2018). These somewhat arbitrary choices affect
both ways of quantifying geometries similarly (i.e. using the central
moments or higher-order spectra; Figs. 2, S2, and S3)
(Tim E. van Peer, personal communication, 2018). The sensitivity of geometry computations to data
processing techniques was not previously appreciated in full and may have
resulted in overoptimistic confidence levels for cycle geometry values
computed on an Oligocene and early Miocene climate record
(Liebrand et al., 2017). We note that
the uncertainty in skewness and asymmetry becomes larger for cycles with
values near the mean (
For ocean waves, the relationship between the integration of the bispectrum
and absolute nonlinear energy transfers is determined by a coupling
coefficient, which is based on the second-order Boussinesq theory
(Herbers and Burton, 1997; Herbers et al., 2000). This Boussinesq
approximation describes how energy is transferred from the primary ocean
waves to higher- and lower-frequency components by near-resonant nonlinear
triad interactions. A coupling coefficient is needed (i) to ensure the
conservation of energy within a single triad interaction (i.e. all energy
lost at a certain frequency is compensated by an energy gain at the other
two frequencies, and vice versa), and (ii) to obtain absolute energy
transfers, which are directly comparable with the changes in the power
spectrum when the waves propagate toward the coast (Herbers et al., 2000;
de Bakker et al., 2014). Here, we apply a similar approach to palaeoclimate
time series and demonstrate that energy conservation can be ensured
assuming a simple coupling coefficient that multiplies the energy transfers
at each specific sum frequency after integration, with that specific
frequency (
The common denominator of (i) the 40 kyr problem of the Pliocene and Early
Pleistocene, (ii) the
To advance the understanding of the origins of Pliocene and Pleistocene climate
cycles, we explore the potential climatic and cryospheric processes that
cause the documented nonlinear triad interactions and energy transfers.
Similar to the difficulties in linking spectral peaks of benthic
foraminiferal
In contrast to the single cause for multiple nonlinear triad interactions among ocean waves, we tentatively link two mechanisms to energy transfers among climate cycles, because we observe two key features in the total and zonal integrations over the bispectrum: (i) a persistent fuelling of obliquity-paced climate cycles by precession-paced climate cycles, which we link to astronomical forcing of atmospheric and oceanic circulation (“the climatic precession motor”) (Sect. 5.2.1), and (ii) a change of obliquity-paced climate cycles from net energy sink into net energy source, which we link to a resonance of cycles of crustal sinking and rebounds with the eccentricity modulation of precession, after Northern Hemisphere (NH) land-ice mass loading passed a critical threshold during the MPT (“the cryospheric obliquity motor”) (Sect. 5.2.2.).
First, we speculate that the fuelling of the ice ages by (mainly) the
precession periodicities is largely linked to the low- to midlatitude
monsoons (Rutherford and D'Hondt, 2000; Berger et al., 2006; Bosmans et
al., 2015a). The (sub)tropical zones, where most insolation is received,
constitute Earth's heat engine, and their monsoons function as both a source
of moisture and as a meridional teleconnection to build up large polar
(land-)ice volumes. The interactions between NH land-ice volumes and the
African and Asian monsoons are well documented for large parts of the
Pliocene and Pleistocene (deMenocal, 1995; Sun et al., 2006; Cheng et
al., 2016). The monsoonal (i.e. atmospheric) redistribution of heat and
moisture was most probably aided by orbital-scale variability in the
strength of Atlantic (i.e. oceanic) meridional overturning circulation
(AMOC) (Lisiecki et al., 2008; Ziegler et al., 2010; Elderfield et al.,
2012). Based on the bispectral results, we infer that during the Pliocene
and Early Pleistocene this predominantly monsoonally driven precession motor
fuels the 40 kyr obliquity-paced ice-age cycles, aided by more linear
climatic–cryospheric responses resulting from variability in insolation at
this periodicity, either zonally/cross-equatorially integrated (Huybers
and Tziperman, 2008; Bosmans et al., 2015b) or locally at higher latitudes.
The precession motor is especially strongly expressed in
Second, we hypothesise that the role reversal of obliquity-paced climate
cycles (from net energy receiver to net energy provider) across the MPT, in
both
If our interpretations of the bispectra are indeed supportive of an
astronomically paced monsoonal/AMOC “push” of (heat and) moisture,
followed by a resonant lithospheric–asthenospheric “pull” through a
cooling of the NH polar region after an initial phase of ice growth, for
sufficiently large glacial land-ice masses to form, then they would
highlight the importance of delayed isostatic rebound and the shape (i.e.
extent, volume, and especially height) of the NH ice sheets in determining
the pattern and geometry of climate–cryosphere variability observed for the
Middle and Late Pleistocene (Roe, 2006; Bintanja and van de Wal, 2008;
Abe-Ouchi et al., 2013). However, these mechanisms do not explain why
large-scale NH glaciations only developed during the Late Pliocene and
Pleistocene, and not during earlier time periods with comparable
astronomical configurations. The Pliocene–Pleistocene climatic–cryospheric
evolution can therefore not be seen outside a context of long-term,
multi-Myr changes in climatic and tectonic boundary conditions, not captured
by the bispectral analysis presented here. For example, the “arc” in
spectral power (from
We present considerable advances in the understanding of the origins of many
Pliocene and Pleistocene climate cycles by applying advanced bispectral
analysis to the palaeoclimate archive. However, several lines of research
remain to be explored further. Here, we list those that we think need more
attention in the future:
For nearshore ocean waves, energy transfers are (almost) fully described by
the imaginary part of the bispectrum (Norheim et al.,
1998), because (i) nearshore ocean waves are dominated by asymmetric wave
forms, and (ii) the imaginary part of bispectra in bispectral theory in
general describes the majority of energy that is transferred. The importance
of the imaginary part in describing most/all energy exchanges can easily be
observed when computing power spectra of synthetically skewed and asymmetric
time series (not shown here). Such spectra indicate that the higher harmonic
peaks of skewed cycles are about an order of magnitude smaller than those of
asymmetric cycles, especially for the higher harmonic peaks (3 Although we obtain conservative energy exchanges with our simplified,
assumed coupling coefficient, additional studies are needed to scale these
exchanges to the power spectrum (Fig. 6). Coupled climate–cryosphere models
may be used to derive or approximate coupling coefficients for
palaeoclimates that are based on “first principles” (i.e. a
physicochemical understanding of the Earth system). Alternative coupling
coefficients may also change how the bispectrum redistributes energy from
climate cycles with low to those with high periodicities. However, we note
that energy conservation should be respected for every triad interaction in
the bispectrum. The proposed relationships between nonlinear triad interactions as
documented in bispectra of climate cycles and specific processes of the
climate–cryosphere system (i.e. monsoons, isostatic rebound) remain
speculative at best. Therefore, we cannot currently separate these
mechanisms from, for example, deep ocean carbon storage (e.g. Willeit
et al., 2015; Ganopolski et al., 2016; Farmer et al., 2019), AMOC shutdown
and subsequent overshoot (Liu et al., 2009; Wu et al., 2011), and/or
changing sediment cover (Clark and Pollard, 1998; Ganopolski and Calov,
2011; Willeit et al., 2019) and the amplifying effects these processes may
have had on the strength of the 40 and There is a limited amount of information that can be extracted from studying
a single climate record. Bispectral analysis of other (Pliocene–Pleistocene)
records may shed more light on the dynamics of Earth's palaeoclimate system not only in elucidating the couplings among astronomical frequencies but
also across the spectral continuum, in relation to
In this interdisciplinary study, we present a new, higher-order spectral
analysis and interpretation of Pliocene and Pleistocene climate cycles as
present in the well-established LR04 globally averaged benthic foraminiferal
We postulate two distinct fuelling mechanisms for the ice ages to explain
the energy cascade of negative interactions from climate cycles with high
frequencies to those with low frequencies: (i) a continuous
Pliocene–Pleistocene climatic “precession motor” and (ii) a Middle-to-Late
Pleistocene cryospheric “obliquity motor”. We interpret the dominant
precession fuelling of obliquity-paced climate cycles (i.e. the precession
motor) as evidence of a low-latitude, (sub)tropically driven climate
system, in which the monsoons (and oceanic meridional overturning
circulation) transport heat and moisture to higher latitudes. We argue that
the role reversal of obliquity-paced climate cycles during the MPT is
indicative of the passing of a land-ice mass loading threshold, after which
autocycles of crustal sinking and (delayed) rebound start to resonate, i.e.
become frequency and phase coupled, with the
Despite the fact that the evidence for energy transfers agrees well with
previously published mechanisms, their unprecedentedly detailed description
here is an important step forward toward a more comprehensive solution of
the problem of the ice ages, because they largely reconcile the mismatch in
power spectral density between records that capture the combined effects of
deep-sea temperatures and land-ice volumes and those of insolation.
Furthermore, we can now state with greater certainty than before that the
asymmetry of the Pliocene and Pleistocene ice ages is in very close
agreement with Neo-Milankovitchism: i.e. the classical Milankovitch theory
of astronomical climate forcing and its Neo-Milankovitchistic
derivatives/superlatives that give greater weight to Earth's internal
nonlinear responses. If the bispectral energy transfers documented here
indeed track the entire redistribution of power over the spectrum, then the
40 kyr problem of the Pliocene and Early Pleistocene and the
The raw data that went into the LR04 stack, which was recompiled for the purposes of this study, can be downloaded at
Bispectral zones defined in this study. These zones correspond to
Fig. 3. The boundaries between the climate cycle zones are (arbitrarily)
defined at periodicities of
Bispectral triads among climate cycles with astronomical
frequencies. One, two, or three astronomical frequencies (bold font) can partake in
triad interactions, and we refer to these coordinates in the bispectrum as
single, double, or triple junctions, respectively. These triads reflect
positions in the frequency–frequency domain where nonlinear energy transfers
among astronomically paced climate cycles are likely to occur. They
correspond to the crossing points of the coloured lines in Fig. 3. In the case
of a “single junction”, an astronomically paced cycle exchanges energy
with itself or with a frequency that is only slightly offset. Single
junctions where
The supplement related to this article is available online at:
DL and ATMdB designed the study, made the figures and wrote the manuscript. ATMdB adjusted the MATLAB scripts for the purposes of this study.
The authors declare that they have no conflict of interest.
We thank Xavier Bertin for inviting Diederik Liebrand to present this work at an early stage
at the Université de La Rochelle. We are grateful to Tim E. van Peer for
making us aware of the difficulties in quantifying robust geometries of
non-ergodic records. We thank Lorraine E. Lisiecki and Maureen E. Raymo, and Seonmin Ahn and colleagues for making their benthic foraminiferal
The article processing charges for this open-access publication were covered by the University of Bremen.
This paper was edited by Qiuzhen Yin and reviewed by Mathieu Martinez and one anonymous referee.