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.Comment: 6 pages with 8 figure

Hilbert Statistics of Vorticity Scaling in Two-Dimensional Turbulence

In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed $\omega(x,y)$ in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang Transform, to a vorticity field obtained from a $8192^2$ grid-points direct numerical simulation of the 2D turbulence with a forcing scale $k_f=100$ and an Ekman friction. The measured joint probability density function $p(C,k)$ of mode $C_i(x)$ of the vorticity $\omega$ and instantaneous wavenumber $k(x)$ is separated by the forcing scale $k_f$ into two parts, which corresponding to the inverse energy cascade and the forward enstrophy cascade. It is found that all conditional pdf $p(C\vert k)$ at given wavenumber $k$ has an exponential tail. In the inverse energy cascade, the shape of $p(C\vert k)$ does collapse with each other, indicating a nonintermittent cascade. The measured scaling exponent $\zeta_{\omega}^I(q)$ is linear with the statistical order $q$, i.e., $\zeta_{\omega}^I(q)=-q/3$, confirming the nonintermittent cascade process. In the forward enstrophy cascade, the core part of $p(C\vert k)$ is changing with wavenumber $k$, indicating an intermittent forward cascade. The measured scaling exponent $\zeta_{\omega}^F(q)$ is nonlinear with $q$ and can be described very well by a log-Poisson fitting: $\zeta_{\omega}^F(q)=\frac{1}{3}q+0.45\left(1-0.43^{q}\right)$. However, the extracted vorticity scaling exponents $\zeta_{\omega}(q)$ for both inverse energy cascade and forward enstrophy cascade are not consistent with Kraichnan\rq{}s theory prediction. New theory for the vorticity field in 2D turbulence is required to interpret the observed scaling behavior.Comment: 13 pages with 10 figure

Equivalent Effect Function and Fast Intrinsic Mode Decomposition

The Equivalent Effect Function (EEF) is defined as having the identical integral values on the control points of the original time series data; the EEF can be obtained from the derivative of the spline function passing through the integral values on the control points. By choosing control points with different criteria, the EEF can be used to find the intrinsic mode function(IMF, fluctuation) and the residue (trend); to fit the curve of the original data function; and to take samples on original data with equivalent effect. As examples of application, results of trend and fluctuation on real stock historical data are calculated on different time scales. A new approach to extend the EEF to 2D intrinsic mode decomposition is introduced to resolve the inter slice non continuity problem, some photo image decomposition examples are presented

Intermittency measurement in two dimensional bacterial turbulence

In this paper, an experimental velocity database of a bacterial collective motion , e.g., \textit{B. subtilis}, in turbulent phase with volume filling fraction $84\%$ provided by Professor Goldstein at the Cambridge University UK, was analyzed to emphasize the scaling behavior of this active turbulence system. This was accomplished by performing a Hilbert-based methodology analysis to retrieve the scaling property without the $\beta-$limitation. A dual-power-law behavior separated by the viscosity scale $\ell_{\nu}$ was observed for the $q$th-order Hilbert moment $\mathcal{L}_q(k)$. This dual-power-law belongs to an inverse-cascade since the scaling range is above the injection scale $R$, e.g., the bacterial body length. The measured scaling exponents $\zeta(q)$ of both the small-scale \red{(resp. $k>k_{\nu}$) and large-scale (resp. $k<k_{\nu}$)} motions are convex, showing the multifractality. A lognormal formula was put forward to characterize the multifractal intensity. The measured intermittency parameters are $\mu_S=0.26$ and $\mu_L=0.17$ respectively for the small- and large-scale motions. It implies that the former cascade is more intermittent than the latter one, which is also confirmed by the corresponding singularity spectrum $f(\alpha)$ vs $\alpha$. Comparison with the conventional two-dimensional Ekman-Navier-Stokes equation, a continuum model indicates that the origin of the multifractality could be a result of some additional nonlinear interaction terms, which deservers a more careful investigation.Comment: 23 pages, 7 figures. This paper is published on Physical Review E, 93, 062226, 201

Multi-level segment analysis: definition and application in turbulent systems

For many complex systems the interaction of different scales is among the most interesting and challenging features. It seems not very successful to extract the physical properties in different scale regimes by the existing approaches, such as structure-function and Fourier spectrum method. Fundamentally these methods have their respective limitations, for instance scale mixing, i.e. the so-called infrared and ultraviolet effects. To make improvement in this regard, a new method, multi-level segment analysis (MSA) based on the local extrema statistics, has been developed. Benchmark (fractional Brownian motion) verifications and the important case tests (Lagrangian and two-dimensional turbulence) show that MSA can successfully reveal different scaling regimes, which has been remaining quite controversial in turbulence research. In general the MSA method proposed here can be applied to different dynamic systems in which the concepts of multiscaling and multifractal are relevant.Comment: 18 pages with 9 figures, accepted by Journal of Statistical Mechanic

Multidimensional Analytic Signals and the Bedrosian Identity

The analytic signal method via the Hilbert transform is a key tool in signal analysis and processing, especially in the time-frquency analysis. Imaging and other applications to multidimensional signals call for extension of the method to higher dimensions. We justify the usage of partial Hilbert transforms to define multidimensional analytic signals from both engineering and mathematical perspectives. The important associated Bedrosian identity $T(fg)=fTg$ for partial Hilbert transforms $T$ are then studied. Characterizations and several necessity theorems are established. We also make use of the identity to construct basis functions for the time-frequency analysis

Compressed Sensing for Time-Frequency Gravitational Wave Data Analysis

The potential of compressed sensing for obtaining sparse time-frequency representations for gravitational wave data analysis is illustrated by comparison with existing methods, as regards i) shedding light on the fine structure of noise transients (glitches) in preparation of their classification, and ii) boosting the performance of waveform consistency tests in the detection of unmodeled transient gravitational wave signals using a network of detectors affected by unmodeled noise transientComment: 16 pages + 17 figure

Moment-based cosh-Hilbert Inversion and Its Applications in Single-photon Emission Computed Tomography

The inversion of cosh-Hilbert transform (CHT) is one of the most crucial steps for single-photon emission computed tomography with uniform attenuation from truncated projection data. Although the uniqueness of the CHT inversion had been proved \cite{Noo2007}, there is no exact and analytic inverse formula so far. Several approximated inversion algorithms of the CHT had been developed \cite{Noo2007}\cite{You2007}. In this paper, we proposed a new numerical moment-based inversion algorithm

Lagrangian Statistics and Intermittency in Gulf of Mexico

Due to the nonlinear interaction between different flow patterns, for instance, ocean current, meso-scale eddies, waves, etc, the movement of ocean is extremely complex, where a multiscale statistics is then relevant. In this work, a high time-resolution velocity with a time step 15 minutes obtained by the Lagrangian drifter deployed in the Gulf of Mexico (GoM) from July 2012 to October 2012 is considered. The measured Lagrangian velocity correlation function shows a strong daily cycle due to the diurnal tidal cycle. The estimated Fourier power spectrum $E(f)$ implies a dual-power-law behavior which is separated by the daily cycle. The corresponding scaling exponents are close to $-1.75$ and $-2.75$ respectively for the time scale larger (resp. $0.1\le f\le 0.4\,{day^{-1}}$) and smaller (resp. $2\le f\le 8\,{day^{-1}}$) than 1 day. A Hilbert-based approach is then applied to this data set to identify the possible multifractal property of the cascade process. The results show an intermittent dynamics for the time scale larger than 1 day, while a less intermittent dynamics for the time scale smaller than 1 day. It is speculated that the energy is partially injected via the diurnal tidal movement and then transferred to larger and small scales through a complex cascade process, which needs more studies in the near future.Comment: 8 pages with 5 figure

Synchrosqueezed Curvelet Transform for 2D Mode Decomposition

This paper introduces the synchrosqueezed curvelet transform as an optimal tool for 2D mode decomposition of wavefronts or banded wave-like components. The synchrosqueezed curvelet transform consists of a generalized curvelet transform with application dependent geometric scaling parameters, and a synchrosqueezing technique for a sharpened phase space representation. In the case of a superposition of banded wave-like components with well-separated wave-vectors, it is proved that the synchrosqueezed curvelet transform is capable of recognizing each component and precisely estimating local wave-vectors. A discrete analogue of the continuous transform and several clustering models for decomposition are proposed in detail. Some numerical examples with synthetic and real data are provided to demonstrate the above properties of the proposed transform.Comment: 32 page