Wavelet transforms and their applications to MHD and plasma turbulence: a review

Farge, Marie; Schneider, Kai

doi:10.1017/s0022377815001075

Cited by 43 publications

(44 citation statements)

References 51 publications

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“…Wavelets are waveforms that locally quantify the amplitude of a field in a given range of scales (see for instance Cohen & Ryan (1995); Van Den Berg (2004); Farge et al (2010); Farge & Schneider (2015) for a more detailed introduction to wavelets and their application in physics and turbulence). They are constructed by dilating and rotating an initial wavelet, generally called the mother wavelet.…”

Section: Appendix A: Morlet Wavelets and Windowed Fourier Transformsmentioning

confidence: 99%

“…A simpler way to describe such processes involves a hierarchical multiscale approach: the small scale interactions lead to the formation of local structures at intermediate scales, that in turn interact to form structures at larger scales, etc. This requires to properly separate the variability of the process under study at different scales, which is precisely the purpose of the wavelet transform (Cohen & Ryan 1995;Van Den Berg 2004;Farge et al 2010;Farge & Schneider 2015).…”

Section: Introductionmentioning

confidence: 99%

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The RWST, a comprehensive statistical description of the non-Gaussian structures in the ISM

Allys

Levrier

Zhang

et al. 2019

A&A

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The interstellar medium (ISM) is a complex non-linear system governed by the interplay between gravity and magnetohydrodynamics, as well as radiative, thermodynamical, and chemical processes. Our understanding of it mostly progresses through observations and numerical simulations, and a quantitative comparison between these two approaches requires a generic and comprehensive statistical description of the emerging structures. The goal of this paper is to build such a description, with the purpose to permit an efficient comparison independent of any specific prior or model. We start from the Wavelet Scattering Transform (WST), a low-variance statistical description of non-Gaussian processes, developed in data science, that encodes long-range interactions through a hierarchical multiscale approach based on the Wavelet transform. We perform a reduction of the WST through a fit of its angular dependencies, allowing to gather most of the information it contains into a few components whose physical meanings are identified, and that describe, e.g., isotropic and anisotropic behaviours. The result of this paper is the Reduced Wavelet Scattering Transform (RWST), a statistical description with a small number of coefficients that characterizes complex structures arising from non-linear phenomena, in particular interstellar magnetohydrodynamical (MHD) turbulence, free from any specific prior. The RWST coefficients encode moments of order up to four, have reduced variances, and quantify the couplings between scales. To show the efficiency and generality of this description, we apply it successfully to three kinds of processes that are a priori very different: fractional Brownian motions, MHD simulations, and Herschel observations of the dust thermal continuum in a molecular cloud. With fewer than 100 coefficients when probing 6 scales and 8 angles on 256×256 maps, we were able with the RWST to perform quantitative comparisons, to infer relevant physical properties, and to produce realistic synthetic fields.

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Section: Appendix A: Morlet Wavelets and Windowed Fourier Transformsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The RWST, a comprehensive statistical description of the non-Gaussian structures in the ISM

Allys

Levrier

Zhang

et al. 2019

A&A

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“…A review on wavelet transforms and their applications to MHD and plasma turbulence can be found in Ref. [15].…”

Section: Introductionmentioning

confidence: 99%

Wavelet methods for studying the onset of strong plasma turbulence

et al. 2018

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Recent simulations have demonstrated that coherent current sheets dominate the kinetic-scale energy dissipation in strong turbulence of magnetized plasma. Wavelet basis functions are a natural tool for analyzing turbulent flows containing localized coherent structures of different spatial scales.Here, wavelets are used to study the onset and subsequent transition to fully developed turbulence from a laminar state. Originally applied to neutral fluid turbulence, an iterative wavelet technique decomposes the field into coherent and incoherent contributions. In contrast to Fourier power spectra, finite time Lyapunov exponents (FTLE), and simple measures of intermittency such as non-Gaussian statistics of field increments, the wavelet technique is found to provide a quantitative measure for the onset of turbulence and to track the transition to fully developed turbulence. The wavelet method makes no assumptions about the structure of the coherent current sheets or the underlying plasma model. Temporal evolution of the coherent and incoherent wavelet fluctuations is found to be highly correlated (Pearson correlation coefficient of > 0.9) with the magnetic field energy and plasma thermal energy, respectively. The onset of turbulence is identified with the rapid growth of a background of incoherent fluctuations spreading across a range of scales and a corresponding drop in the coherent components. This is suggestive of the interpretation of the coherent and incoherent wavelet fluctuations as measures of coherent structures (e.g., current sheets) and dissipation, respectively. The ratio of the incoherent to coherent fluctuations R ic is found to be fairly uniform across different plasma models and provides an empirical threshold for turbulence onset. The utility of this technique is illustrated through examples. First, it is applied to the Kelvin-Helmholtz instability from different simulation models including fully kinetic, hybrid (kinetic ion/fluid electron), and Hall MHD simulations. Second, the wavelet diagnostic is applied to the development of turbulence downstream of the bowshock in a global magnetosphere simulation. Finally, the wavelet technique is also shown to be useful as a de-noising method for particle simulations.

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“…For plasmas with important small-scale and transient turbulent structures, the biorthogonal wavelet decomposition and reduced-order methods based on wavelets [20], such as multiresolution DMD [44] and multi-resolution biorthogonal decomposition [48], show significant promise. These would be ideal for the analysis of coherent structures with small spatial and temporal correlation lengths; for instance, quasi-coherent modes of this sort appear to be universal in the edge of Alcator C-Mod plasmas [26].…”

Section: Resultsmentioning

confidence: 99%

Characterizing magnetized plasmas with dynamic mode decomposition

et al. 2020

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Accurate and efficient plasma models are essential to understand and control experimental devices. Existing magnetohydrodynamic or kinetic models are nonlinear, computationally intensive, and can be difficult to interpret, while often only approximating the true dynamics. In this work, data-driven techniques recently developed in the field of fluid dynamics are leveraged to develop interpretable reduced-order models of plasmas that strike a balance between accuracy and efficiency. In particular, dynamic mode decomposition (DMD) is used to extract spatio-temporal magnetic coherent structures from the experimental and simulation datasets of the HIT-SI experiment. Three-dimensional magnetic surface probes from the HIT-SI experiment are analyzed, along with companion simulations with synthetic internal magnetic probes. A number of leading variants of the DMD algorithm are compared, including the sparsity-promoting and optimized DMD. Optimized DMD results in the highest overall prediction accuracy, while sparsity-promoting DMD yields physically interpretable models that avoid overfitting. These DMD algorithms uncover several coherent magnetic modes that provide new physical insights into the inner plasma structure. These modes were subsequently used to discover a previously unobserved three-dimensional structure in the simulation, rotating at the second injector harmonic. Finally, using data from probes at experimentally accessible locations, DMD identifies a resistive kink mode, a ubiquitous instability seen in magnetized plasmas.

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Wavelet transforms and their applications to MHD and plasma turbulence: a review

Cited by 43 publications

References 51 publications

The RWST, a comprehensive statistical description of the non-Gaussian structures in the ISM

The RWST, a comprehensive statistical description of the non-Gaussian structures in the ISM

Wavelet methods for studying the onset of strong plasma turbulence

Characterizing magnetized plasmas with dynamic mode decomposition

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