Climatic Variability Over Time Scales Spanning Nine Orders of Magnitude: Connecting Milankovitch Cycles with Hurst–Kolmogorov Dynamics

Abstract

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.

Appendices

Appendix 1: Proof of Equation (2)

We assume a fully deterministic, strictly periodic process composed of a single harmonic with period T, described by,

$$ x\left( t \right) \, = \sqrt 2 { \cos }( 2\pi t/T + b) $$

(6)

It can be seen that during the time interval [t ₁, t ₂], where $2 { }\pi t_{ 1, 2} /T + b = \pm { \arccos }(x/\sqrt 2 )$, the process $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} \left( t \right)$ takes on values greater than or equal to x, provided that $- \sqrt 2 \le x \le \sqrt 2$ (see Fig. 12). The length of the interval [t ₁, t ₂] is (T/π) arccos$\left( {x/\sqrt 2 } \right)$. Consequently, if we treat the process x(t) stochastically, it follows that its marginal distribution function is

$$ {F}\left( {x} \right) \, = \, 1 - \arccos ({\text{x}}/\sqrt 2 )/\pi , \quad- \sqrt 2 \le {\text{x}} \le \sqrt 2 $$

(7)

Fig. 12

Sketch to illustrate the proof of (7). For the ease of the illustration, it was assumed b = π, but this does not affect the result nor the length of the time interval, which does not depend on b

Taking its derivative with respect to x, we find its marginal density function as

$$ {f}\left( x \right) \, = \, 1/(\pi \sqrt {2 - x^{2} } ),\; \quad- \sqrt 2 \le x \le \sqrt 2 $$

(8)

By application of definitions of mean and variance, we readily obtain that the mean of the process is 0, and its variance is 1. Likewise, the process autocovariance is:

$$ R\left( \tau \right) = \text{cov} [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} \left( t \right), \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} \left( {t + \tau } \right)\left] { = E} \right[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} \left( t \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} \left( {t + \tau } \right)] $$

(9)

where τ is lag time. If x(t) = x then $t = \left( {T/ 2\pi } \right) \, \left[ { - b + { \arccos }\left( {x/\sqrt 2 } \right)} \right]$ (one of the infinitely many possibilities), so that $t + \tau = \left( {T/ 2\pi } \right) \, \left[ { - b + { \arccos }\left( {{x \mathord{\left/ {\vphantom {x {\sqrt 2 }}} \right. \kern-0pt} {\sqrt 2 }}} \right)} \right] + \tau$ and $x\left( {t + \tau } \right) = \sqrt 2 { \cos }\left[ { 2 { }\pi \tau /T + { \arccos }(x/\sqrt {2)} } \right]$. Consequently,

$$ R\left( \tau \right) \, = \sqrt 2 \int\limits_{ - \sqrt 2 }^{\sqrt 2 } {x{ \cos }[ 2\pi \tau /T + { \arccos }(x/\sqrt 2 )]f\left( x \right){\text{ d}}x} $$

(10)

which after algebraic manipulations becomes

$$ R\left( \tau \right) \, = { \cos }\left( { 2\pi \tau /T} \right) $$

(11)

Interestingly, this does not depend on t, thus behaving like a stationary process.

The climacogram value at scale k can be calculated from the variance ${\text{Var }}\left[ {\underline{x}_{i}^{(k)} } \right]$, that is,

$$ {\text{Var}}\left[ {\underline{x}_{i}^{(k)} } \right] = E\left[ {\left\{ {\underline{x}_{i}^{(k)} } \right\}^{2} } \right] = \frac{1}{{k^{2} }}\int\limits_{0}^{k} {\int\limits_{0}^{k} {E\left[ {x\left( t \right)x\left( s \right)} \right]} } dtds $$

(12)

or

$$ {\text{Var}}\left[ {\underline{x}_{i}^{(k)} } \right] = \frac{1}{{k^{2} }}\int\limits_{0}^{k} {\int\limits_{0}^{k} {R\left( {t-s} \right){\text{ d}}t{\text{d}}s} = \frac{1}{{k^{2} }}\int\limits_{0}^{k} {\int\limits_{0}^{k} {\cos [2\pi \left( {t-s} \right) / {\text{T] d}}t{\text{d}}s} } } $$

(13)

which after algebraic manipulations becomes

$$ {\text{Var}}\left[ {\underline{x}_{i}^{(k)} } \right] = [T/(\pi k)]^{ 2} { \sin }^{ 2} (\pi k/T) $$

(14)

By taking the square root of ${\text {Var}}\left[ {\underline{x}_{i}^{(k)} } \right]$, which by definition is the standard deviation σ ^(k), this gives Eq. (2).

From (2), we readily infer, that for increasing k, there appears a series of maxima at values k = αT/2, with α any odd integer, so that |sin(π k/T)| = 1. This series is described by σ ^(k) = T/(π k), which is an upper envelope curve of the climacogram. Obviously, across this envelope, θ = d(ln σ ^(k))/d(ln k) = −1. However, the local slope of the climacogram is not constant but varies. We can easily determine it from $\theta : = {\text{d}}({ \ln }\sigma^{\left( k \right)} )/{\text{d}}({ \ln }k) = (k/ 2){\text{d}}\{ { \ln }[(\sigma^{\left( k \right)} )^{ 2} ]\} /{\text{d}}k = (k/ 2){\text{d}}({ \ln }\{ [T/(\pi k)]^{ 2} { \sin }^{ 2} (\pi k/T)\} /{\text{d}}k$ which after algebraic manipulations becomes

$$\theta = \, (\pi k/T){ \cot }(\pi k/T)- 1 $$

(15)

It can be seen that θ tends to ±∞ whenever k/T is integer.

Appendix 2: Assessment of the interpolation effect on the climacogram

In order to examine possible effects of the linear interpolation to the data, we compared it with the closest-point estimation. In the latter, we estimate the unknown value at a specific time using the value of the given time series at the point closest to this specific time (without doing any calculation). As shown in Fig. 13, where the climacograms estimated by both interpolation methods for the GRIP time series are depicted, the results of the two methods are virtually indistinguishable, which means that in the case examined the climacogram is practically indifferent to the type of interpolation.

Fig. 13

Climacogram for the GRIP data set calculated from the time series constructed by linear interpolation in comparison with that of the time series constructed by closest-point estimation