François G. Schmitt

Multifractal Cascade Dynamics and Turbulent Intermittency

Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we th...

Scaling invariance of crack surfaces

The morphology of fractured rock surfaces is studied in terms of their scaling invariance. Fresh brittle fractures of granite and gneiss were sampled with a mechanical laboratory profilometer, and (1 + 1)-dimensional parallel profiles were added to build actual maps of the surfaces. A first step in the scaling invariance description is a self-affine analysis using three independent methods. The root-mean-square and the maximum-minimum difference of the height are shown to follow a power law with the sample length. The return probability and the Fourier spectrum are also computed. All these approaches converge to a unique self-affine exponent: ζ = 0.80. Analysis over a broad statistical set provides a reproducibility error of ±0.05. No significant differences between the isotropic granite and the markedly anisotropic gneiss appear for the scaling exponents. An analysis of the profilometer shows that the two main drawbacks of the setup are not significant in these analyses. The systematic errors of the scaling analysis are estimated for the different methods. Isotropy of the scaling invariance within the mean fracture plane is shown either with the result obtained from different fracture orientations or with the two-dimensional Fourier spectrum of the surface topography itself. The analysis is brought further into the multifractal framework. The structure functions are shown to have power law behavior, and their scaling exponent varies nonlinearly with the moment order. Finally, the corresponding conserved process belongs to a universal multifractal class with α = 1.5 for the Levy index and C1 = 0.3 for the fractal codimension of the mean singularities. The three indices (ζ, α and C1) completely characterize the scale invariance. The multifractal behavior is significant for physical properties which depend on high-order moments like contact. According to this study and that of other groups, the self-affine exponent ζ is constant over a large range of scales and for different fracture modes and various materials. This opens the possibility that there exists a form of universality in the cracking process. It appears that only the prefactor of the roughness is dependent on material and mode.

Multifractal analysis of foreign exchange data

In this paper we perform multifractal analyses of five daily Foreign Exchange (FX) rates. These techniques are currently used in turbulence to characterize scaling and intermittency. We show the multifractal nature of FX returns, and estimate the three parameters in the universal multifactal framework, which characterize all small and medium intensity fluctuations, at all scales. For large fluctuations, we address the question of hyperbolic (fat) tails of the distributions which are characterized by a fourth parameter, the tail index. We studied both the prices fluctuations and the returns, finding no systematic difference in the scaling exponents in the two cases. We discuss and compare our results with several recent studies, and show how the additive models are not compatible with data: Brownian, fractional Brownian, Levy, Truncated Levy and fractional Levy models. We analyse in this framework the ARCH(1), GARCH(1,1) and HARCH (7) models, and show that their structure functions scaling exponents are undistinguishable from that of Brownian motion, which means that these models do not adequately describe the scaling properties of the statistics of the data. Our results indicate that there might exist a multiplicative ‘flux of financial information’, which conditions small-scale statistics to large-scale values, as an analogy with the energy flux in turbulence. Copyright

Multifractals and extreme rainfall events

Based on a multifractal structure hypothesis for temporal rainfall processes, a general formula relating maximum possible point rainfall accumulations is derived as a function of the duration and sample size. This formula appears to be in agreement with empirical observations. Such a result may reconcile some opposite points of view regarding extreme rainfall events, and suggests new ways of exploiting the scaling properties of rain processes.

Modeling of rainfall time series using two-state renewal processes and multifractals

The high variability of rainfall fields comes from (1) the occurrence of wet and dry events and (2) from the intermittency of precipitation intensities. To model these two aspects for spatial variability, Over and Gupta [1996] have proposed a lognormal cascade model with an atom at zero, which corresponds to combine in one model two independent cascade models, the β and the lognormal multifractal models. In the present work, we test this approach for time variability, using a high-resolution rainfall time series. We built a continuous version of the discrete β model and investigate some of its dynamical properties. We show that the β model cannot fit the probability density for the duration of the wet state. In order to model the succession of wet and dry periods we therefore use a two-state (or alternate) renewal process based on appropriate fits of the empirical densities. The synthetic series obtained this way reproduces the scaling of the original support. The intensity of the rainfall events is then modeled using the universal multifractal model, generalizing the lognormal model. We show that the fractal support of the rainfall events must be taken into account to retrieve the parameters of this model. This combination of two different models allows to closely reproduce the high variability at all scales and long-range correlations of precipitation time series, as well as the dynamical properties of the succession of wet and dry events. Simulations of the high-resolution rainfall field are then performed displaying the salient features of the original time series.

Multifractal analysis of the Greenland Ice‐Core Project climate data

Recent climatic records from the Greenland ice-core project (GRIP) ice core show that the climate proxy temperatures δ18O (18O/16O ratios) display sharp gradients and large fluctuations over all observed scales. We show that these variations are scale invariant over the range ≈400 yr to ≈40 kyr. The fluctuations corresponding to these scales are studied using multifractal analysis techniques. We estimate universal multifractal indices which characterize all the fluctuations for all the scales in this range and which are close to those obtained for turbulent temperatures at much shorter time scales. We speculate briefly on the origin of these common features intervening all over the observed range from seconds to kyr.

An amplitude-frequency study of turbulent scaling intermittency using Empirical Mode Decomposition and Hilbert Spectral Analysis

Hilbert-Huang transform is a method that has been introduced recently to decompose nonlinear, nonstationary time series into a sum of different modes, each one having a characteristic frequency. Here we show the first successful application of this approach to homogeneous turbulence time series. We associate each mode to dissipation, inertial range and integral scales. We then generalize this approach in order to characterize the scaling intermittency of turbulence in the inertial range, in an amplitude-frequency space. The new method is first validated using fractional Brownian motion simulations. We then obtain a 2D amplitude-frequency representation of the pdf of turbulent fluctuations with a scaling trend, and we show how multifractal exponents can be retrieved using this approach. We also find that the log-Poisson distribution fits the velocity amplitude pdf better than the lognormal distribution.

Stochastic equations generating continuous multiplicative cascades

Abstract:Discrete multiplicative turbulent cascades are described using a formalism involving infinitely divisible random measures. This permits to consider the continuous limit of a cascade developed on a continuum of scales, and to provide the stochastic equations defining such processes, involving infinitely divisible stochastic integrals. Causal evolution laws are also given. This gives the first general stochastic equations which generate continuous multifractal measures or processes.

Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: Comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylors microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

Turbulence intermittency, small-scale phytoplankton patchiness and encounter rates in plankton: where do we go from here?

Abstract Turbulence is widely recognized to enhance contact rates between planktonic predators and their prey. However, previous estimates of contact rates are implicitly based on homogeneous distributions of both turbulent kinetic energy dissipation rates and phytoplanktonic prey, while turbulent processes and phytoplankton cell distributions have now been demonstrated to be highly intermittent even on small scales. Turbulent kinetic energy dissipation rates and intermittent (i.e. patchy) phytoplankton distributions can be wholly parameterized in the frame of universal multifractals. Using this framework and assuming statistical independence between turbulent kinetic energy dissipation rate and phytoplankton distributions, we evaluated the effect of intermittent turbulence and the potential effects of zooplankton behavioral responses to small-scale phytoplankton patchiness on predator–prey encounter rates. Our results indicated that the effects of turbulence on predator–prey encounter rates is about 35% less important when intermittently fluctuating turbulent dissipation rates are considered instead of a mean dissipation value. Taking into account zooplankton behavioral adaptations to phytoplankton patchiness increased encounter rates up to a factor of 60.