Climatic Variability Over Time Scales Spanning Nine Orders of Magnitude: Connecting Milankovitch Cycles with Hurst–Kolmogorov Dynamics
Yannis Markonis • Demetris Koutsoyiannis Received: 9 November 2011 / Accepted: 15 October 2012 DOI 10.1007/s10712-012-9208-9
We overview studies of the natural variability of past climate, as seen from available proxy information, and its attribution to deterministic or stochastic controls. Furthermore, we characterize this variability over the widest possible range of scales that the available information allows, and we try to connect the deterministic Milankovitch cycles with the Hurst–Kolmogorov (HK) stochastic dynamics.
To this aim, we analyse two instrumental series of global temperature and eight proxy series with varying lengths from 2 thousand to 500 million years. In our analysis, we use a simple tool, the climacogram, which is the logarithmic plot of standard deviation versus time scale, and its slope can be used to identify the presence of HK dynamics. By superimposing the climacograms of the different series, we obtain an impressive overview of the variability for time scales spanning almost nine orders of magnitude—from 1 month to 50 million years. An overall climacogram slope of -0.08 supports the presence of HK dynamics with Hurst coefficient of at least 0.92. The orbital forcing (Milankovitch cycles) is also evident in the combined climacogram at time scales between 10 and 100 thousand years. While orbital forcing favours predictability at the scales it acts, the overview of climate variability at all scales suggests a big picture of irregular change and uncertainty of Earth’s climate.
If you thought before science was certain—well, that is just an error on your part. -Richard Feynman, The Character of Physical Law (1994 p. 71).
In the first half of nineteenth century, geologic evidence indicated that at least one glacial period existed in Earth’s geological history (Agassiz 1840; from Imbrie 1982). Some decades later, it became clear that during the Pleistocene (2,588,000–12,000 years before present time—BP), there were many glacial periods, known also as ice ages, followed by shorter interglacials, such as the one we experience since the onset of human civilization. Ice age lengths ranged from 35–45 thousand years in early Pleistocene to 90–120 thousand years in the last million years. During glacial periods, continental glaciers enlarged in length and volume, reaching the 40th parallel in some regions of the Northern Hemisphere, while similar phenomena have been identified in the Southern Hemisphere, too. Milankovitch (1941) provided an explanation for the ice ages based on Earth’s orbit variations, which was confirmed after some years by the first temperature reconstructions.
It is now well known that a succession of glaciation and deglaciation periods has not occurred all the time, but only in large periods defining an ‘icehouse climate’, such as the current (Pliocene-Quaternary) icehouse period that started about 2.5 million years ago, as well as the Ordovician and the Carboniferous icehouse periods, each of which lasted 50–100 million years (Crowell and Frakes 1970). In contrast, the ‘hothouse climates’ are characterized by warmer temperatures, abundance of carbon dioxide (concentrations up to 20–25 times higher than current) and complete disappearance of polar icecaps and continental glaciers. Recently, cosmic ray flux was proposed as the controlling factor of the transition between these states (Shaviv and Veizer 2003). As underlined by Kirkby (2007), this theory was both disputed (Rahmstorf et al. 2004; Royer et al. 2004) and supported (Wallmann 2004; Gies and Helsel 2005).
Additional findings showed that the climate of the Holocene (the last 12,000 years), earlier regarded static, was characterized by many climatic events, such as ‘Little Ice Age’, ‘Medieval Warm Period’, ‘Holocene Optimum’, ‘8,200 Holocene Event’ and ‘Bond Events’, deviating from ‘normal’ conditions for hundreds or thousands of years (Bond et al. 2001). For example, during the ‘Little Ice Age’ (1,450–1,850), the temperature of the Northern Hemisphere was about 0.6 oC below 1961–1990 average (Moberg et al. 2005; Pollack and Smerdon 2004), while the ‘Medieval Warm Period’ (950–1,250) was a period of warm climate in Europe and North America and has been related to other climatic events at various regions around the world (Grove and Switsur 1994), including China (Long et al. 2011), New Zealand (Cook et al. 2002) or even Antarctica (Hass et al. 2008).
The preceding ‘Younger Dryas’ episode is an even more impressive case of abrupt climate change that has occurred in the relatively recent climatic history. At the end of Pleistocene, when the last ice age ended and the retreat of the glaciers had begun, a rapid fall of temperature led the climatic system back to glacial conditions. The ‘Younger Dryas’ episode lasted for approximately 1,300 years (starting at *12,800 BP), covered spatially both Hemispheres and ended even more suddenly than it emerged when temperatures increased regionally up to 15 oC in few decades (Alley et al. 1993). Although the cause for this short return to an ice age period is still under debate, it has become clear that it is not associated with a single catastrophic event (such as the release of freshwater from the lake Agassiz in Gulf of Mexico or the impact of a comet) but is rather regarded as an integral part of natural variability (Broecker et al. 2010; Mangerud et al. 2010).
All these relatively recent events cannot be attributed to the Milankovitch cycles, whose periods are much longer (see below). Thus, it is very difficult to attribute the climate variability at multiple time scales (from decades to many millions of years) to specific quantifiable causal mechanisms that would be applicable ubiquitously. A more modest goal, which is the purpose of this study, would be to characterize this variability over the widest possible range of scales that the available evidence allows. Such characterization unavoidably uses stochastic descriptions and tools, but without neglecting the influence of identifiable deterministic forcings, such as the variations in Earth’s orbit.
Such stochastic descriptions are related to the natural behaviour discovered by the hydrologist H. E. Hurst at the same period of Milankovitch’s discovery. Hurst (1951), motivated by the design of High Aswan Dam in Nile and after studying numerous geo- physical records, observed that ‘although in random events groups of high or low values do occur, their tendency to occur in natural events is greater. This is the main difference between natural and random events’. In other words, in a natural process (e.g., river flow) events of similar type are more likely to occur in groups (e.g., a series of consecutive low flow years) compared to a purely random process (white noise) where grouping of similar states is less frequent.
Unknowingly to Hurst, A. Kolmogorov had already proposed a stochastic process that described this behaviour a decade earlier (Kolmogorov 1940), although both the process and the natural behaviour became widely known after the works of Mandelbrot and Wallis (1968), Klemes (1974) and Leland et al. (1994, 1995). Over the years, this mathematical process (or variants thereof) has been given many names, such as fractional Gaussian noise (FGN), brown noise, fractional ARIMA process (FARIMA) or self-similar process, while the natural behaviour has been called the Hurst phenomenon, long-range dependence (or memory), long-term persistence or scaling behaviour (Koutsoyiannis and Cohn 2008). Here, when referring to the relevant natural behaviour, the stochastic process (definition of which will be given in Sect. 5.1) or the related stochastic dynamics, we prefix them with the term Hurst–Kolmogorov (HK) in order to acknowledge the contribution of the two pioneering researchers.
The HK behaviour, detected in numerous time series, as detailed in Sect. 3 below, indicates fluctuations at different time scales, which may reflect the long-term variability of several factors such as solar irradiance, volcanic activity and so forth (Koutsoyiannis and Montanari 2007). The multi-scale fluctuations cannot be described adequately by classical statistics, as the latter assumes independence (or weak dependence) and underestimates the system’s uncertainty on long time scales, sometimes by two, or even more, orders of magnitude (Koutsoyiannis and Montanari 2007). This underestimation, which some regard counterintuitive, will be further demonstrated below in Sect. 6. Moreover, traditional stochastic autoregressive (AR) models cannot describe these fluctuations in an adequate way, because the autocorrelation functions of these models decay faster than those of the processes they try to model (Beran 1994).
The study of natural variability of past climate can now be based on a lot of available proxy records, some of which are discussed in Sect. 4 and analysed in subsequent sections of this study. These proxies are free of anthropogenic influences that could allegedly contribute to the observed changes. It is our aim to demonstrate some evidence of the presence of HK dynamics at different time scales (spanning nine orders of magnitude). We also examine the coexistence of deterministic controls (due to orbital forcing) and stochastic dynamics and try to identify possible connections between this stochastic dynamics and the modern, obliquity-dominated, orbital theory.
Fig. 9 Combined climacogram of the ten temperature observation series and proxies. The dotted line with slope -0.5 represents the climacogram of a purely random process. The horizontal dashed-dotted line represents the climatic variability at 100 million years, while the vertical dashed-dotted line at 28 months represents the corresponding scale to the 100-million-year variability if climate was random (classical statistics approach). For explanation about the groups of points departing from the solid straight line (with slope -0.08), see Fig. 10 and its description in the text
Fig. 10 Theoretical climacograms of an HK process with H = 0.92 and two periodic processes with periods 100 and 41 thousand years, all having unit standard deviation at monthly scale, along with the climacogram of the synthesis (weighted sum) of these three components with weights 0.95, 0.30 and 0.15, respectively; the empirical climacogram of a time series simulated from the synthesis process with time step and length equal to those of the EPICA series is also plotted
Fig. 11 Climacogram of sunspot number from original data (shown in the embedded graph) from the Royal Greenwich Observatory & USAF/NOAA (http://solarscience.msfc.nasa.gov/greenwch/spot_num.txt)
The available instrumental data of the last 160 years allow us to see that there occurred climatic fluctuations with a prevailing warming trend in the most recent past. However, when this period is examined in the light of the evidence provided by palaeoclimate reconstructions, it appears to be a part of more systematic fluctuations; specifically, it is a warming period after the 200-year ‘Little Ice Age’ cold period, during a 12,000-year interglacial, which is located in the third major icehouse period of the Phanerozoic Eon. The variability implied by these multi-scale fluctuations, typical for Earth’s climate, can be investigated by combining the empirical climacograms of different palaeoclimatic reconstructions of temperature. By superimposing the different climacograms, we obtain an impressive overview of the variability for time scales spanning almost nine orders of magnitude—from 1 month to 50 million years.
Two prominent features of this overview are (a) an overall climacogram slope of -0.08, supporting the presence of HK dynamics with Hurst coefficient of at least 0.92 and (b) strong evidence of the presence of orbital forcing (Milankovitch cycles) at time scales between 10 and 100 thousand years. While orbital forcing favours predictability at the scales it acts, the overview of climate variability at all scales clearly suggests a big picture of enhanced change and enhanced unpredictability of Earth’s climate, which could be also the cause of our difficulties to formulate a purely deterministic, solid orbital theory (either obliquity or precession dominated). Endeavours to describe the climatic variability in deterministic terms are equally misleading as those to describe it using classical statistics. Connecting deterministic controls, such as the Milankovitch cycles, with the Hurst–Kolmogorov stochastic dynamics seems to provide a promising path for understanding and modelling climate.
The paper and SI are available here: http://itia.ntua.gr/en/docinfo/1297/