Looking for the 400 Kyr cycle

Franco Zavatti and Luigi Mariani


Climate of a place or a wider area (a vineyard, a valley, a continent, the whole planet) derives froma variety of astronomical or geophysical causes (table 1) often interacting each other in a, also complex, way giving place, e.g., to well characterized positive and negative feedbacks.
In table 1 we denoted with an asterisk factors subjected to cycles over a very wide range of periods (hours to millions years). As anexample, Sun presents 11-yr activity cycles and longer cycles(Suess, Eddy, Hallstatt, etc.) e Ocean circulation shows cycles like AMOC (Atlantic Meridional Overturning Circulation) that then plays a role on the surface temperature of the Atlantic Ocean by forcing typical cycles known as AMO (Atlantic Multidecadal Oscillation).
The consciousness of the exixtence of climte cycles has a long hystory.
We can quote, e.g., the Saserna, roman treaters (writers) whose work has been lost but are quoted by Columella, argued the climate of their epoch had become milder than in the old times, so that olive trees and vineyards can live well where it would been impossible before.
In the same way the idea of cyclical cycles, that incuded also a deluge as in the biblic narrative, is widely spread in many cultures (precolombians peoples, australian blackfellows, ecc.). They mantained in trouble our progenitors along many millenia.
In the first half of the XIX century Joseph Fourier, during his studies on heat propagation, realized that analyses waer much more simple if a function was represented as the sum of simple trigonometric functions with adequate parameters. Also, the researches by Fourier are near to climatology because the terrestrial atmosphere is an heat machine whose motions depend on the need to equilibrate the differences due the unequal subdivision of heat on the Planet's surface 1.

1 Fourier in 1824 uses, and he was the first one, the concept of greenhouse effect, which along with atmospheric and oceanic circulation is the basis of our planet's climate.

Again, from the XIX century, due to geomorphological work lead between 1800 and 1900 and summarized by Louis Agassiz the idea of the presence of glacial cycles takes place, what gives rise to modern climate studies about climate cycles converging to the theory by Milutin Milankovich about the astronomical causes of glacial eras proven in the following by Cesare Emiliani (1955) with his studies on ocean floor cores.
Briefly said, what Fourier, Agassiz, Milankovich and Emiliani give us has all neede to understand the spectral analisys and its usefulness.

Actual cycle study depends on instrumental data (temperature, precipitation, global solar radiation etc.). We outline that the presence of periodicity in independent series (like speleothemes, tree rings growth, grape harvest date) is an important reinforce to their reality. In applications, it will be important to link period analisys with (also hystorical) documents or narrative (e.g. legends); it follows may be of importance to link the astronomical cycles at periods of 2000 years (Bray or Hallstatt cycles) and 1000 years (Eddy's cycle) (Scafetta et al., 2016) with key periods in the Holocene like:

Table 1 - Astronomical and geophysical elements that define the climate at different scales (from a single site to the whole planet)
Astronomical factors weighting on the amount of energy at the planet's surface.
  • earth motion around its axis (rotation)*
  • eccentricity of earth's orbit*
  • solar activity*
  • galactic cosmic rays (GCRs)*
  • orbital effects by the other planets of solar system*
  • inclination of terrestrial axis (with consequences on inclination of solar rays)*
  • quasi spherical shape of the earth (with effects on inclination of solar rays)
geophysical factors that modulate the effects of astronomical factors
  • ocean and land distribution*
  • distance from sea*
  • ocean streams*
  • atmospheric circolation*
  • shape and position of mountain ridges*
  • soil characteristic*
  • living beings activity (flora, fauna, mankind)*

(*) submitted to cycles over a very large range of time scales (days to millions years).

the large variety of periods force someone to select a group of spectral scycles belonging to, ofen unknown, precise causes.
Her we used time series covering about 5 million years and selected cycles of some hundreds of thousands years.

Data Analysis
At about the end of August 2019, we have had the availability of a paper by Kent et al., 2018, where the empirical evidence for the stability of the 405-kiloyear Jupiter-Venus eccentricity cycle over at hundreds of milions of year (at least 215 Myr) was presented. We never had notice of such a spectral maximum before of this paper, so used the De Boer et al.(2014) dataset

    "Global 5 Million Year Sea Level, Temperature, and d18Osw Reconstructions"
who declared that
    Persistent 400,000-year variability of Antarctic ice volume and the carbon cycle is revealed throughout the Plio-Pleistocene
published in Nature Communications (2014). The abstract of the De Boer et al.,2014) reads:
    Marine sediment records from the Oligocene and Miocene reveal clear 400,000-year climate cycles related to variations in orbital eccentricity. These cycles are also observed in the Plio-Pleistocene records of the global carbon cycle. However, they are absent from the Late Pleistocene ice-age record over the past 1.5 million years. Here we present a simulation of global ice volume over the past 5 million years with a coupled system of four three-dimensional ice-sheet models. Our simulation shows that the 400,000-year long eccentricity cycles of Antarctica vary coherently with d13C data during the Pleistocene, suggesting that they drove the long-term carbon cycle changes throughout the past 35 million years. The 400,000-year response of Antarctica was eventually suppressed by the dominant 100,000-year glacial cycles of the large ice sheets in the Northern Hemisphere.

The DeBoer (2014) dataset (hereafter deboer2014.txt) include 5 Myr (millions of years) time span at 100 yr step values of some climatic parameters (i.e. 53000 data points) and is shown in figure 1 at 10-points step, i.e. with 5300 points represented, for the benthic δ18O (an inverse proxy for temperature).

Fig.1: The 5 Myr dataset of d18O from benthonic foraminifera, plotted at 10-point step. The bottom plot is the enlargement of the first million years of the series. To be noted the inverse y-scale, so the plot mimics the temperature, and the presence of the 25 MIS (Marine Isotope Stages) which denote the interglacials in 1-million years (with of course the relative glacial periods). The red lines are, respectively for top and bottom plots, low pass filter at 1000 and 300 data points window (100 and 30 Kyr).

We decided to separate deboer2014.txt into 8 datasets of 7000 data each (the last one of 4000 data). In such a way we had something resembling a wavelets ensamble which allowed to derive spectra (MEM spectra: they are at constant step) of different and adjacent time sections, covering 700 kyr (kilo years) each one.

At the same time we computed the Lomb-Scargle periodogram (hereafter Lomb) of the whole dataset, deriving both ~400 Kyr and ~1 Myr spectral maxima as shown in figure 2

Fig.2: Lomb periodogram, from the CRAN R suite, of the full deboer2014.txt series. The green lines define here and in figure 3 the ensemble of secondary maxima between 0.4 and 1 Myr. Dashed line is the 99% confidence level.

We also continued with the MEM spectrum, computing the two half-dataset (i.e. 26000 data points each) spectra, as shown in figure 3.

Fig.3: MEM spectra of the two half-dataset sections of deboer2014.txt. The green lines define here and in figure 2 the ensemble of secondary maxima between 0.4 and 1 Myr.

The comparison between the above spectra shows that the main ~0.4 and ~1 Myr maxima remain at about the same period with a skyrocket variation of the power during the second half section (730/103 or 7X and 230/11 or 21X); also the peak at 0.24 Myr (the leftmost one in figure 2) becomes 7 times higher (50/7) during the more recent two millions years than during the first section. The other visible peaks changed their frequency (period).

Green lines in both figure 2 and 3 define a group of secondary maxima: to be noted the 0.74 Kyr maximum (0.68 in the upper frame of figure 3) clearly visible in the Lomb spectrum of figure 2 and that confirm that the analysis of the whole dataset (figure 2) is strongly dominated by the 2.nd (more recent) section of the dataset because the 0.74 maximum clearly emerges above the little "forest" of maxima.

In order to verify within a better accuracy the variation of frequency along the time sequence, we used the 8 sections defined above to compute the MEM spectrum. The time sequence is listed in Table1.

Table 1. Deboer2014.txt. Start-End Kyr BP of the 8 sections. 100 yr step
Sec Start Kyr End Kyr Comments
1 5300.0 4600.1 7000 values
2 4600.0 3900.1
3 3900.0 3200.1 min power
4 3200.0 2500.1
5 2500.0 1800.1
6 1800.0 1100.1 max power
7 1100.0 400.1
8 400.0 0.1 4000 values

  • Spectra computed from 3500 values from line 1500 through 4500.
  • Time spread: 700 thousand years per file (8 excluded).

    A summary of the results is in figure 4

    Fig.4: The MEM spectrum of the 8 sequential and adjacent sections defined by color and, in one case, by line shape. Here only 400 Kyr and nearby maxima have been selected. The bottom plot is an enlargement in power (y-axis) of the top one.

    Analysis of the ~0.41 Kyr spectral maximum
    We can derive from figure 4 the suggestion that the power of spectral maxima evolve along the sections (i.e. with time) and a more precise list of peak's power confirm, as in figure 5, this hypothesis:

    Fig.5: Time evolution of the ~0.4 Myr spectral maximum power from the 8-sections series. The blue line is the fitting parabola of the first 6 data.

    The power variation of the 0.4 Kyr peak has nothing to do with casuality but it seems to follow a rising law (the blue line is the fitting parabola) through section 6 and then a drop not too much different from the corresponding rise. So, we can suppose that, at the end of the period 1.8-1.1 Myr (section 6), something happened, so that the power of the most important cycle in the 200-700 Kyr time lapse, begun to drop.
    We cannot know what happened before section 1 (i.e. before 5.3 Myr) but perhaps it could show a cyclic behavior with a 5.7 Myr period (4600-1100 from table 1).

    Fig.6: Time evolution of the ~0.4 Myr spectral maximum power from the 7-sections, 2-subsections each one, series. red line-and-dot accounts for the subsection, blue line-and-dot accounts for the subsection of each section. dot-dashed lines are the respective parabolic fits.

    The characteristic shape of figure 5, relative to the 8 sections, holds again for the 14 subsections, with the data of subsections 1 appearing shifted backward by one section. We cannot explain such a behaviour, only note, as in figure 7, that a positive shift of 1 section changes notably the comparisons in figure 6

    Fig.6: Time evolution of the ~0.4 Myr spectral maximum power from the 7-sections, 2-subsections each one, series when the section of subsections 1 becomes "section+1".

    Nothing of what has been found in the analysis of the power of the main peak of the actual series appears to be casual. It seems the result of an (unknown) evolution of external or internal forcings, spanning over millions of years.

    Milankovic Cycles
    The δ18O benthic by De Boer et al. (2014), while seems very good for 0.4-1 Myr (and more) spectral peaks, poses the problem that the 100, 41, 26 Kyr Milankovic cycles (the orbital cycles of eccentricity, obliquity and precession) cannot be derived from this series as e.g. it appears in figure 3. We can suppose the actual spectral maxima are too weak to be identified in the above plots, so try their "emersion" by a x1000 amplification but, as it can be seen in figure 7, a daunting result in obtained: no orbital maximum in the spectra, at all.

    Fig.7: Trying to identify the Milankovic cycles by a x1000 amplification: in the range 97-104 Kyr 6 peaks (out of 8 series) can b be identified, but nothing at all at 40 and, mainly, at 26 Kyr.

    Fig.8: MEM Spectrum of Page800 δ18O benthic 0 to 800 Kyr BP (Ka is used in place of Kyr BP). The bottom plot outlines the periods of Milankovic cycles. This figure has been already published elsewhere; here it has been slightly revised. To be noted, as a mirror of figure 7, the absence of the 400 Kyr peak.

    The data De Boer et al., 2014 used is the stacked dataset LR04 Benthic by Lisiecki and Raymo (2005) at variable step, to which a model has been applied in order to derive a dataset at 100 yr step. So we dowloaded the Lisiecki & Raymo's series and computed the Lomb and wavelet spectrum. The following figures 9 and 10 show that the spectra are the same and exclude some kind of procedural error.

    Fig.9: LOMB spectrum of the Lisiecki & Raymo (2005) dataset LR04 Stack. Bottom frame is a 0-200 Kyr enlargement of the above total. Dot-dash green lines are the 95%, white noise, confidence level

    Fig.10: Wavelet spectrum of the LR04 series computed by PAST. Due to the log2 vertical scale, the figure has been labelled with the corresponding periods in Kyr. The x-axis has been also labelled in Kyr BP.

    The last dataset also confirms that the 400 Kyr peak has low power and this is confirmed in both LOMB an wavelets

    initial and derived data, and plots, are available at the support site