Ivan L’Heureux

Are Hurst exponents estimated from short or irregular time series meaningful

We show that several time series analysis methods that are often used for detecting self-affine fractal scaling and determining Hurst exponents in data sets may lead to spurious results when applied to short discretized data series. We show that irregularities in the series, such as jumps or spikes (as are often found in geophysical data) may lead to spurious scaling and consequently to an incorrect determination of the Hurst exponent. We also illustrate the statistical error in measuring Hurst exponent in series where self-affine fractal scaling does exist. Users should be aware of these caveats when interpreting the results of short time series analysis.

Late Quaternary Climate Response at 100 kyr: A Noise-Induced Cycle Suppression Mechanism

Late quaternary climate proxies suggest the presence of a strong cycle at a period of about 100 kyr. It is thought that this cycle could be due to variations in the eccentricity of the Earth’s orbit, as part of the Milankovitch forcing. However, based on simple energy balance arguments, the eccentricity variations are too small to explain the strength of the climatic response. Some amplification mechanisms based on ice sheet dynamics or ocean circulation models have been suggested to explain this paradox. But recently (Wallmann 2014), a different explanation was proposed. There, a non-linear biogeochemical model coupling seawater alkalinity, dissolved phosphate, dissolved inorganic carbon, and atmospheric carbon dioxide without any orbital forcing was developed. As the parameters vary, the system may undergo a Hopf bifurcation and exhibits self-organized oscillations with a period that has the appropriate order of magnitude but remains larger than 100 kyr. In this contribution, I revisit Wallmann’s model by adding a weak stochastic periodic Milankovitch forcing at 100 kyr in the spirit of stochastic resonance phenomena. It is seen that for sufficiently high noise intensity, a noise-induced cycle suppression occurs, whereby the self-sustained oscillation of biogeochemical origin is destroyed and a strong signal persists at 100 kyr. This mechanism could thus provide an amplification mechanism for the presence of a strong response under the influence of a weak Milankovitch forcing.

Diagenetic Self-Organization and Stochastic Resonance in a Model of Limestone-Marl Sequences

Banded patterns in limestone-marl sequences (“rhythmites”) form widespread sediments typical of shallow marine environments. They are characterized by alternations of limestone-rich layers and softer calcareous-clayey material (marl) extending over hundreds of meters with a thickness of a few tens of meters. The banded sequences are usually thought to result from systematic variations in the external environment, but the pattern may be distorted by diagenetic nonlinear processes. Here, we present a reactive-transport model for the formation of banded patterns in such a system. The model exhibits interesting features typical of nonlinear dynamical systems: (i) the existence of self-organized oscillating patterns between a calcite-rich mode (“limestone”) and a calcite-poor one (“marl”) for fixed environmental conditions and (ii) bistability between these two modes. We then illustrate the phenomena of stochastic resonance, whereby the multistable system is driven by a small external periodic signal (the 100,000 years’ Milankovitch cycle comes to mind) that is too weak to generate oscillations between the states on its own. In the presence of random fluctuations, however, the system generates transitions between the calcite-rich and calcite-poor states in statistical synchrony with the external forcing. The signal-to-noise ratio exhibits many maxima as the noise strength is varied. Hence, this amplification effect is maximized for specific values of the noise strength.

Transition rates for stochastic delay differential equations

An approach allowing the analysis of stochastic delay differential equations using Fokker-Planck equations has been recently proposed. It is used here as a basis for studying delayed systems with noise induced rate processes. Simulation results agreeing with numerical computations of transition rates are presented.