# On Time Scales of Intrinsic Oscillations in the Climate System

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

^{18}O, and the percent melt. The percent melt is the percentage of each year’s accumulation encompassed in melt layers by mass in an ice core. Three different ice cores from Canada and Greenland (specifically from Devon Island ice cap, Agassiz ice cap, and Camp Century) were used (see [4] and references therein). The remaining three records were: an instrumental global temperature data set (annual averages 1880–1987; see [5]), a global temperature proxy data (2000-year averages for the past two million years; see [6]), and a proxy temperature data derived from δ

^{18}O core RC11-120 (with the sampling interval of 3000 years; see [7]). For each of these records the power spectrum was estimated, and significant, at the 5% level, spectral peaks were identified by using random surrogates having the same (fitted) frequency distribution as the actual data. If a peak was shown to be significant, its period T and power P were registered. Finally, all of the significant peaks so determined were plotted in a P vs. log(T) graph (see Figure S1 of the Supplementary Material). This figure shows significant oscillations with periods up to 750 years, a break from 750 to about 20,000 years, and then further significant periodicities with timescales in the range between 20,000 to 100,000 years.

## 2. Slow Feature Analysis (SFA), Wavelets and Red-Noise Surrogates

**x**(t), to find a set of real-valued input-output functions g

_{j}(

**x**) such that the output signals:

_{j}(t): = g

_{j}(

**x**(t))

_{j}>

_{t}= 0 (zero mean),

_{j}>

_{t}= 0 (zero mean),

_{i}y

_{j}>

_{t}= 0, ∀i < j (decorrelation and order)

_{t}and $\dot{y}$ indicating temporal averaging and the derivative of y, respectively.

_{j}are distinct and hence extract different information from the input signal. For a tutorial on this method the reader could consult [29] or a more recent presentation [34]. In that tutorial, a simple example of a two-dimensional input signal x

_{1}(t) = sin(t) + cos(11t)

^{2}and x

_{2}(t) = cos(11t) is considered. Both components are quickly varying, but hidden in the signal is the slowly varying ‘feature’ y(t) = x

_{1}(t) − x

_{2}(t)

^{2}= sin(t), which can be extracted with a polynomial of degree two, namely h(

**x**) = x

_{1}−x

_{2}

^{2}.

**Z**is column orthogonal:

## 3. Results

## 4. Summary and Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mitchell, J.M. An Overview of Climatic Variability and its Causal Mechanisms. Quat. Res.
**1976**, 6, 481–493. [Google Scholar] [CrossRef] - Von der Heydt, A.S.; Ashwin, P.; Camp, C.D.; Crucifix, M.; Dijkstra, H.A.; Peter Ditlevsen, P.; Timothy, M.; Lenton, T.M. Quantification and interpretation of the climate variability record. Glob. Planet. Chang.
**2021**, 197, 103399. [Google Scholar] [CrossRef] - Zhuang, J. A Study of the Variability of the Global Climate System. Master’s Thesis, Department of Geosciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA, 1991. [Google Scholar]
- Fisher, D.A.; Koerner, R.M.; Reeh, N. Holocene climate records from Agassiz ice cap, Ellesmere Island, NWT, Canada. Holocene
**1995**, 5, 19–24. [Google Scholar] [CrossRef] - Hansen, J.; Lebedeff, S. Global surface air temperatures: Update through 1987. Geophys. Res. Lett.
**1988**, 15, 323–326. [Google Scholar] [CrossRef][Green Version] - Ruddiman, W.F.; Raymo, M.; McIntyre, A. Matuyama 41,000-year cycles: North Atlantic Ocean and northern hemisphere ice sheets. Earth Planet. Sci. Lett.
**1986**, 80, 117–129. [Google Scholar] [CrossRef] - Hays, J.D.; Imbrie, J.; Shackleton, N.J. Variations in the Earth’s orbit: Pacemaker of the Ice Ages. Science
**1976**, 194, 1121–1132. [Google Scholar] [CrossRef] - Tsonis, A.A.; Madsen, M.D. On the range of frequencies of intrinsic climate Oscillations. In Advances in Nonlinear Geosciences; Springer: Cham, Switzerland, 2018; pp. 651–660. [Google Scholar]
- Koscielny-Bunde, E.; Bunde, A.; Havlin, S.; Roman, H.E.; Goldreich, Y.; Schellnhuber, H.J. Indication of a universal persistence law governing atmospheric variability. Phys. Rev. Lett.
**1998**, 31, 729–732. [Google Scholar] [CrossRef] - Loso, M.G. Summer temperatures during the Medieval Warm Period and Little Ice Age inferred from varved proglacial lake sediments in southern Alaska. J. Paleolimnol.
**2008**, 41, 117–128. [Google Scholar] [CrossRef] - Tan, M.; Liu, T.S.; Hou, J.; Qin, X.; Zhang, H.; Li, T. 2650-Year Beijing stalagmite layer thickness and temperature reconstruction. In IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series # 2003-050; NOAA/NGDC Paleoclimatology Program: Boulder, CO, USA, 2003. [Google Scholar]
- Grudd, H. Tornetrask tree-ring width and density AD 500-2004: A test of climatic sensitivity and a new 1500-year reconstruction of north Fennoscandian summers. Clim. Dyn.
**2008**, 31, 843–857. [Google Scholar] [CrossRef][Green Version] - Mangini, A.; Spötl, C.; Verdes, P. Reconstruction of temperature in the Central Alps during the past 2000 yr from a d18O stalagmite record. Earth Planetary Sci. Lett.
**2005**, 235, 741–751. [Google Scholar] [CrossRef] - Lüdecke, H.J.; Weiss, C.O. Harmonic Analysis of Worldwide Temperature Proxies for 2000 Years. Open Atmos. Sci. J.
**2017**, 11, 44–53. [Google Scholar] [CrossRef][Green Version] - Asten, M.W.; Lin, K.E.; Weiss, C.O. On the Coherence of Natural Climate Cycles of the Past 1 ka in Multiple Proxies from Central Europe, the Arctic and East Asia; Paper PP031-0010; American Geophysical Union Fall Meeting: San Francisco, CA, USA, 2020. [Google Scholar]
- Asten, M.W. Holocene 6000-yr Climate Cycles in Temperate and Sub-Tropical SST Record—A Cosmic Ray Connection? Paper EGU2020-7285; European Geophysical Union: Vienna, Austria, 2020. [Google Scholar]
- Ge, Q.; Liu, H.; Ma, X.; Zheng, J.; Hao, Z. Characteristics of Temperature Change in China over the Last 2000 years and Spatial Patterns of Dryness/Wetness during Cold and Warm Periods. Adv. Atmos. Sci.
**2017**, 34, 941–951. [Google Scholar] [CrossRef] - Holzhauser, H. Die bewegte Vergangenheit des Grossen Aletschgletschers. In Blatter aus der Walliser Geschichte, Band XLI; Geschichtsforschender Verein Oberwallis: Brig, Switzerland, 2009; pp. 47–102. [Google Scholar]
- Nussbaumer, S.U.; Steinhilber, F.; Trachsel, M.; Breitenmoser, P.; Beer, J.; Blass, A.; Grosjean, M.; Hafner, A.; Holzhauser, H.; Wanner, H.; et al. Alpine climate during the Holocene: A comparison between records of glaciers, lake sediments and solar activity. J. Quaternary Sci.
**2011**, 26, 703–713. [Google Scholar] [CrossRef] - Asten, M.W. Sub-Milankovich millennial cycles in proxy (UK37) sea surface temperatures for the Okinawa Trough, W Mediterranean Sea, NW Atlantic Ocean and Southern Ocean. In Geophysical Research Abstracts; Paper EGU2018-12541; European Geophysical Union: Vienna, Austria, 2018. [Google Scholar]
- Lopes dos Santos, R.A.; Spooner, M.I.; Barrows, T.T.; De Deckker, P.; Damsté, J.S.S.; Schouten, S. Comparison of organic (UK’ 37, TEXH 86, LDI) and faunal proxies (foraminiferal assemblages) for reconstruction of late Quaternary sea surface temperature variability from offshore southeastern Australia. Paleoceanography
**2013**, 28, 377–387. [Google Scholar] [CrossRef][Green Version] - Bintanja, R.; van de Wal, R.S.W.; Oerlemans, J. Modeled atmospheric temperatures and global sea levels over the past million years. Nature
**2005**, 437, 125–128. [Google Scholar] [CrossRef] [PubMed] - Jouzel, J.; Masson-Delmotte, V.; Cattani, O.; Dreyfus, G.; Falourd, S.; Hoffmann, G.; Minster, B.; Nouet, J.; Barnola, J.M.; Chappellaz, J.A.; et al. EPICA dome C ice core 800KYr deuterium data and temperature estimates. In IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series # 2007-091; NOAA/NCDC Paleoclimatology Program: Boulder, CO, USA, 2007. [Google Scholar]
- Alley, R.B. GISP2 Ice Core Temperature and Accumulation Data; IGBPPAGES/World Data Center for Paleoclimatology Data Contribution Series #2004-013; NOAA/NGDC Paleoclimatology Program: Boulder, CO, USA, 2004.
- Ruan, J.; Xu, Y.; Ding, S.; Wang, Y.; Zhang, X. A biomarker record of temperature and phytoplankton community structure in the Okinawa Trough since the last glacial maximum. Quat. Res.
**2017**, 88, 89–97. [Google Scholar] [CrossRef] - Blaschke, T.; Berkes, P.; Wiskott, L. What is the relationship between slow feature analysis and independent component analysis? Neural Comput.
**2006**, 18, 2495–2508. [Google Scholar] [CrossRef] [PubMed] - Franzius, M.; Wilbert, N.; Wiskott, L. Invariant object recognition and pose estimation with Slow Feature Analysis. Neural Comput.
**2011**, 23, 2289–2323. [Google Scholar] [CrossRef] - Berkes, P.; Wiskott, L. Slow feature analysis yields a rich repertoire of complex cells. J. Vis.
**2005**, 5, 579–602. [Google Scholar] [CrossRef][Green Version] - Wiskott, L.; Sejnowski, T.J. Slow Feature Analysis: Unsupervised learning of invariance. Neural Comput.
**2002**, 14, 715–770. [Google Scholar] [CrossRef] - Wiskott, L. Estimating Driving Forces of Nonstationary Time Series with Slow Feature Analysis. 2003. Available online: http://arxiv.org/abs/cond-mat/0312317/ (accessed on 21 April 2020).
- Yang, P.; Wang, G.; Zhang, F.; Zhou, X. Causality of global warming seen from observationsa scale analysis of driving force of the surface air temperature time series in the Northern Hemisphere. Clim. Dyn.
**2015**. [Google Scholar] [CrossRef] - Tsonis, A.A.; Pan, X.; Wang, G.; Nicolis, C. On the min-max estimation of mean daily temperatures. Clim. Dyn.
**2019**, 53, 1981–1989. [Google Scholar] [CrossRef] - Grinsted, A.; Moore, J.C.; Jevrejeva, S. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Process. Geophys.
**2004**, 11, 561–566. [Google Scholar] [CrossRef] - Wiskott, L.; Berkes, P.; Franzius, M.; Sprekeler, H.; Wilbert, N. Slow feature analysis. Scholarpedia
**2011**, 6, 5282. [Google Scholar] [CrossRef] - Pan, X.; Wang, G.; Yang, P. Extracting the driving force signal from hierarchy system based on slow feature analysis. Acta Phys. Sin.
**2017**, 66, 080501. [Google Scholar] [CrossRef] - Torrence, C.; Compo, G.P. A practical guide to wavelet analysis. Bull. Am. Meteor. Soc.
**1998**, 79, 61–78. [Google Scholar] [CrossRef][Green Version] - Pan, X.; Wang, G.; Yang, P.; Wang, J.; Tsonis, A.A. On the intercomparison among major climate modes and their common driving forces. Earth Syst. Dynam.
**2020**, 11, 525–535. [Google Scholar] [CrossRef] - Dansgaard, W.; Johnsen, S.J.; Clausen, H.B.; Dahl-Jensen, D.; Gundestrup, N.S.; Hammer, C.U.; Hvidberg, C.S.; Steffensen, J.P.; Sveinbjörnsdottir, A.E.; Jouzel, J.; et al. Evidence for general instability of past climate from a 250-kyr ice-core record. Nature
**1993**, 364, 218–220. [Google Scholar] [CrossRef] - Sarachik, E.S.; Winton, M.; Yin, F.L. Mechanisms for decadal-to-centennial climate variability. In Decadal Climate Variability. NATO ASI Series (Series I: Global Environmental Change); Anderson, D.L.T., Willebrand, J., Eds.; Springer: Berlin/Heidelberg, Germany, 1996; Volume 44. [Google Scholar]
- Crucifix, M. Oscillators and relaxation phenomena in Pleistocene climate theory. Philos. Trans. Ser. A Math. Phys. Eng. Sci.
**2012**, 370, 1140–1165. [Google Scholar] [CrossRef][Green Version] - Ditlevsen, P.D.; Kristensen, M.S.; Andersen, K.K. The recurrence time of Dansgaard-Oeschger events and limits on the possible periodic component. J. Clim.
**2005**, 18, 2594–2603. [Google Scholar] [CrossRef][Green Version] - Ditlevsen, P.D.; Andersen, K.K.; Svensson, A. The DO-climate events are probably noise induced: Statistical investigation of the claimed 1470 years cycle. Clim. Past
**2007**, 3, 129–134. [Google Scholar] [CrossRef][Green Version] - Ghimire, G.R.; Jadidoleslam, N.; Krajewski, W.F.; Tsonis, A.A. Insights on streamflow predictability across scales using horizontal visibility graph-based networks. Front. Water
**2020**, 2, 17. [Google Scholar] [CrossRef] - Marshall, J.; Johnson, H.; Goodman, J. A study of the interaction of the North Atlantic Oscillation with ocean circulation. J. Clim.
**2001**, 14, 1399–1421. [Google Scholar] [CrossRef] - Kravtsov, S. Dynamics and predictability of hemispheric-scale multidecadal climate variability in an observationally constrained mechanistic model. J. Clim.
**2020**, 33, 4599–4620. [Google Scholar] [CrossRef][Green Version] - Buckley, M.W.; Marshall, J. Observations, inferences, and mechanisms of the Atlantic meridional overturning circulation: A review. Rev. Geophys.
**2016**, 54, 5–63. [Google Scholar] [CrossRef][Green Version] - Delworth, T.L.; Mann, M.E. Observed and simulated multidecadal variability in the Northern Hemisphere. Clim. Dyn.
**2000**, 16, 661–676. [Google Scholar] [CrossRef][Green Version] - Knight, J.R.; Allan, R.J.; Folland, R.J.; Vellinga, C.K.; Mann, M.E.A. Signature of persistent natural thermohaline circulation cycles in observed climate. Geophys. Res. Lett.
**2005**, 32, L20708. [Google Scholar] [CrossRef][Green Version] - Zhang, R.; Sutton, R.; Danabasoglu, G.; Kwon, Y.O.; Marsh, R.; Yeager, S.G.; Amrhein, D.E.; Little, C.M. A review of the role of the Atlantic Meridional Overturning Circulation in Atlantic Multidecadal Variability and associated climate impacts. Rev. Geophys.
**2019**, 57, 316–375. [Google Scholar] [CrossRef][Green Version] - Wyatt, M.G.; Kravtsov, S.; Tsonis, A.A. Atlantic multidecadal oscillation and Northern Hemisphere’s climate variability. Clim. Dyn.
**2012**, 38, 929–949. [Google Scholar] [CrossRef] - Kravtsov, S.; Grimm, C.; Gu, S. Global-scale multidecadal variability missing in the state-of-the-art climate models. NPJ Clim. Atmos. Sci.
**2018**, 1, 34. [Google Scholar] [CrossRef][Green Version] - Delworth, T.L.; Zeng, F. Multicentennial variability of the Atlantic meridional overturning circulation and its climatic influence in a 4000-year simulation of the GFDL CM2.1 climate model. Geophys. Res. Lett.
**2012**, 39, L13702. [Google Scholar] [CrossRef] - Latif, M.; Martin, T.; Park, W. Southern Ocean sector centennial climate variability and recent decadal trends. J. Clim.
**2013**, 26, 7767–7782. [Google Scholar] [CrossRef][Green Version] - Zhang, L.; Delworth, T.L.; Zeng, F. The impact of multidecadal Atlantic meridional overturning circulation variations on the Southern Ocean. Clim. Dyn.
**2016**, 48, 2065–2085. [Google Scholar] [CrossRef] - Newman, M.; Michael, A.A.; Toby, R.A.; Kim, M.C.; Clara, D.; Emanuele, D.L.; Nathan, J.M.; Arthur, J.M.; Shoshiro, M.; Hisashi, N.; et al. The Pacific decadal oscillation, revisited. J. Clim.
**2016**, 29, 4399–4427. [Google Scholar] [CrossRef][Green Version] - Ghil, M.; Childress, S. Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics; Springer: New York, NY, USA; Berlin/Heidelberg, Germany, 1987. [Google Scholar]
- Mukhin, D.; Gavrilov, A.; Loskutov, E.; Kurths, J.; Feigin, A. Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Sci. Rep.
**2019**, 9, 7328. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**An example of SFA analysis using record 5 of Table 1. The top panel shows the original data. The middle panel shows the SFA extracted driving forces for various embedding dimensions (m = 1, 13, 23). The bottom three show the time-averaged wavelet power spectrum of each SFA-extracted slow feature signal and the significant peaks (open circles) with power exceeding the red-noise based 5% significance level (black dashed lines). These confidence levels are obtained from 100,000 surrogate signals [36].

**Figure 2.**SFA results from all records. The x-axis is the peak periodicity and the y-axis is the SFA power. Blue points are produced from records 1–7 and red points are produced from records 8–12.

Location | Type | Period Covered | Resolution | Reference | |
---|---|---|---|---|---|

1 | Iceberg Lake, Alaska | Annual-mean varve thickness | 442–1998 | annual | [10] |

2 | Beijing, China | Summer-mean stalagmite thickness | −665–1985 | annual | [11] |

3 | Tornetrask, Sweden | annual tree-ring data | 500–2004 | annual | [12] |

4 | Spannagel Cave, Europe | stalagmite thickness | −90–1935 | interpolated to annual | [13] |

5 | Global mean | Average of a large number (tens) of temperature proxies | 1–2015 AD | annual | [14,15,16] |

6 | China | An average of 28 temperature proxies | 6–1996 AD | Uneven, interpolated to annual | [15,16,17] |

7 | Great Aletsch Glacier, European Alps | Temperature proxy | −53–2084 AD | Uneven, interpolated to annual | [15,16,18,19] |

8 | Murray Canyon, Southeastern Australia | Based on several faunal temperature proxies ^{1} | 1.1–134.8 Ky BP | Uneven, interpolated to 100-year | [20,21] |

9 | Global 1Ma Temperature | marine benthic oxygen isotopes | −1,067,900–2000 | 100-year | [22] |

10 | EPICA Dome C, Antarctica | Ice Core | −800,000–1900 | interpolated to 500-year | [23] |

11 | GISP2, central Greenland | Ice core | −48,000–1850 | interpolated to 50-year | [24] |

12 | IODP, 1202B, Okinawa Trough | SST based on U^{K’}_{37} index | 8.8–20,089 year BP | Uneven, interpolated to 10-year | [20,25] |

^{1}The proxies are constructed from algae biochemistry from offshore Southeastern Australia. This data set comes in three forms. An SST proxy estimated from U

^{K’}

_{37}index, an SST proxy estimated from the TEX

^{H}

_{86}index, and an SST proxy estimated from the LDI (long-chain diol) index. All three records give identical results. Here we only record the LDI results.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsonis, A.A.; Wang, G.; Lu, W.; Kravtsov, S.; Essex, C.; Asten, M.W. On Time Scales of Intrinsic Oscillations in the Climate System. *Entropy* **2021**, *23*, 459.
https://doi.org/10.3390/e23040459

**AMA Style**

Tsonis AA, Wang G, Lu W, Kravtsov S, Essex C, Asten MW. On Time Scales of Intrinsic Oscillations in the Climate System. *Entropy*. 2021; 23(4):459.
https://doi.org/10.3390/e23040459

**Chicago/Turabian Style**

Tsonis, Anastasios A., Geli Wang, Wenxu Lu, Sergey Kravtsov, Christopher Essex, and Michael W. Asten. 2021. "On Time Scales of Intrinsic Oscillations in the Climate System" *Entropy* 23, no. 4: 459.
https://doi.org/10.3390/e23040459