Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-18T09:04:23.863Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: ITER International School 2014

Published online by Cambridge University Press:  12 October 2015

Marie Farge  and
Kai Schneider
Marie Farge
Affiliation:
LMD-CNRS, Ecole Normale Supérieure 24, Rue Lhomond, 75231 Paris CEDEX 6, France
Kai Schneider*
Affiliation:
M2P2-CNRS, Aix-Marseille Université 38, Rue Frédéric Joliot-Curie, 13451 Marseille CEDEX 13, France
*
†Email address for correspondence: kschneid@cmi.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Wavelet analysis and compression tools are reviewed and different applications for the study of MHD and plasma turbulence are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics based on the wavelet coefficients. We then show how to extract coherent structures out of fully developed turbulent flows using wavelet-based denoising. Finally some multiscale numerical simulation schemes using wavelets are described. Several examples for analysing, compressing and computing one-, two- and three-dimensional turbulent MHD or plasma flows are presented.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 435810602
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Addison, P. S. 2002 The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance. Taylor & Francis.CrossRefGoogle Scholar
Alexandrova, O., Lacombe, C. & Mangeney, A. 2008 Spectra and anisotropy of magnetic fluctuations in the Earth’s magnetosheath: cluster observations. Ann. Geophys. 26, 3585–3596.CrossRefGoogle Scholar
Azzalini, A., Farge, M. & Schneider, K. 2005 Nonlinear wavelet thresholding: a recursive method to determine the optimal threshold value. Appl. Comput. Harmon. Anal. 18 (2), 177–185.CrossRefGoogle Scholar
Batchelor, G. K. 1982 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Biskamp, D. 1997 Nonlinear Magnetohydrodynamics. Cambridge University Press.Google Scholar
Bos, W. J. T., Futatani, S., Benkadda, S., Farge, M. & Schneider, K. 2008 The role of coherent vorticity for turbulent transport in resistive drift-wave turbulence. Phys. Plasmas 15, 072305.CrossRefGoogle Scholar
Bos, W. J. T., Liechtenstein, L. & Schneider, K. 2007 Small scale intermittency in anisotropic turbulence. Phys. Rev. E 76, 046310.CrossRefGoogle ScholarPubMed
Carbone, V., Regnoli, G., Martines, E. & Antoni, V. 2000 Intermittency and self-similarity in plasma edge fluctuations. Phys. Plasmas 7 (2), 445–447.CrossRefGoogle Scholar
Daubechies, I. 1992 Ten Lectures on Wavelets. SIAM.CrossRefGoogle Scholar
Deriaz, E., Farge, M. & Schneider, K. 2010 Craya decomposition using compactly supported biorthogonal wavelets. Appl. Comput. Harmon. Anal. 28, 267–284.CrossRefGoogle Scholar
Donoho, D. 1995 Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (2), 101–127.CrossRefGoogle Scholar
Donoho, D. & Johnstone, I. 1994 Ideal spatial adaptation via wavelet shrinkage. Biometrika 81, 425–455.CrossRefGoogle Scholar
Donoho, D., Johnstone, I., Keryacharian, G. & Picard, D. 1996 Density estimation by wavelet thresholding. Ann. Stat. 24 (2), 508–539.CrossRefGoogle Scholar
Dose, V., Venus, G. & Zohm, H. 1997 Wavelet analysis of fusion plasma transients. Phys. Plasmas 4 (2), 323–328.CrossRefGoogle Scholar
Dudok De Wit, T., Alexandrova, O., Furno, I., Sorriso-Valvo, L. & Zimbardo, G. 2014 Methods for characterising microphysical processes in plasmas. In Microphysics of Cosmic Plasmas, pp. 589–617. Springer.Google Scholar
Dudok De Wit, T. & KrasnoselSkikh, V. V. 1995 Wavelet bicoherence analysis of strong plasma turbulence at the Earths quasiparallel bow shock. Phys. Plasmas 2 (11), 4307–4311.CrossRefGoogle Scholar
Farge, M. 1992 Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24, 395–457.CrossRefGoogle Scholar
Farge, M., Pellegrino, G. & Schneider, K. 2001 Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett. 87 (5), 45011–45014.CrossRefGoogle ScholarPubMed
Farge, M. & Rabreau, G. 1988 Transformée en ondelettes pour détecter et analyser les structures cohérentes dans les écoulements turbulents bidimensionnels. C. R. Acad. Sci. Paris 307, 1479–1486; série II.Google Scholar
Farge, M. & Schneider, K. 2001 Coherent vortex simulation (CVS), a semi-deterministic turbulence model using wavelets. Flow Turbul. Combust. 6, 393–426.CrossRefGoogle Scholar
Farge, M. & Schneider, K. 2006 Wavelets: applications to turbulence. In Encyclopedia of Mathematical Physics (ed. Françoise, J.-P., Naber, G. & Tsun, T. S.), pp. 408–419. Elsevier.CrossRefGoogle Scholar
Farge, M., Schneider, K. & Devynck, P. 2006 Extraction of coherent events in turbulent edge plasma using orthogonal wavelets. Phys. Plasmas 13, 042304.CrossRefGoogle Scholar
Farge, M., Schneider, K. & Kevlahan, N. 1999 Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using adaptive orthonormal wavelet basis. Phys. Fluids 11, 2187–2201.CrossRefGoogle Scholar
Farge, M., Schneider, K., Pellegrino, G., Wray, A. & Rogallo, R. 2003 Coherent vortex extraction in three-dimensional homogeneous isotropic turbulence: comparison between CVS and POD decompositions. Phys. Fluids 15 (10), 2886–2896.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Grossmann, A. & Morlet, J. 1984 Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15 (4), 723–736.CrossRefGoogle Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.CrossRefGoogle Scholar
Kurien, S. & Sreenivasan, K. R. 2000 Anisotropic scaling contributions to high-order structure functions in high-Reynolds-number turbulence. Phys. Rev. E 62 (2), 2206–2212.CrossRefGoogle ScholarPubMed
Kurien, S. & Sreenivasan, K. R. 2001 New Trends in Turbulence (ed. Lesieur, M., Yaglom, A. & David, F.), EDP Sciences.Google Scholar
Mallat, S. 1998 A Wavelet Tour of Signal Processing. Academic Press.Google Scholar
Meneveau, C. 1991 Analysis of turbulence in the orthonormal wavelet representation. J. Fluid Mech. 232, 469.CrossRefGoogle Scholar
Nguyen van yen, R., del Castilo-Negrete, D., Schneider, K., Farge, M. & Chen, G. Y. 2010 Wavelet–based density estimation for noise reduction in plasma simulation using particles. J. Comput. Phys. 229 (8), 2821–2839.CrossRefGoogle Scholar
Nguyen van yen, R., Fedorczak, N., Brochard, F., Bonhomme, G., Schneider, K., Farge, M. & Monier-Garbet, P. 2012 Tomographic reconstruction of tokamak plasma light emission from single using wavelet-vaguelette decomposition. Nucl. Fusion 52, 013005.CrossRefGoogle Scholar
Nguyen van yen, R., Sonnendrücker, E., Schneider, K. & Farge, M. 2011 Particle-in-wavelets scheme for the 1D Vlasov–Poisson equations. ESAIM: Proc. 32, 134–148.CrossRefGoogle Scholar
Okamoto, N., Yoshimatsu, K., Schneider, K. & Farge, M. 2011 Directional and scale-dependent statistics of quasi-static magnetohydrodynamic turbulence. ESAIM: Proc. 32, 95–102.CrossRefGoogle Scholar
Okamoto, N., Yoshimatsu, K., Schneider, K. & Farge, M. 2014 Small-scale anisotropic intermittency in magnetohydrodynamic turbulence at low magnetic Reynolds number. Phys. Rev. E 89, 033013.CrossRefGoogle Scholar
Pope, S. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Sandborn, V. A. 1959 Measurements of intermittency of turbulent motion in a boundary layer. J. Fluid Mech. 6, 221–240.CrossRefGoogle Scholar
Schneider, K. & Farge, M. 2006 Wavelets: mathematical theory. In Encyclopedia of Mathematical Physics (ed. Françoise, J.-P., Naber, G. & Tsun, T. S.), pp. 426–437. Elsevier.CrossRefGoogle Scholar
Schneider, K., Farge, M. & Kevlahan, N. 2004 Spatial intermittency in two-dimensional turbulence: a wavelet approach. In Woods Hole Mathematics, Perspectives in Mathematics and Physics (ed. Tongring, N. & Penner, R. C.), vol. 34, pp. 302–328. World Scientific.CrossRefGoogle Scholar
Schneider, K. & Vasilyev, O. 2010 Wavelet methods in computational fluid dynamics. Annu. Rev. Fluid Mech. 42, 473–503.CrossRefGoogle Scholar
Sonnendrücker, E., Roche, J., Bertrand, P. & Ghizzo, A. 1999 The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201–220.CrossRefGoogle Scholar
Sorriso-Valvo, L., Carbone, V., Bruno, R. & Veltri, P. 2006 Persistence of small-scale anisotropy of magnetic turbulence as observed in the solar wind. Europhys. Lett. 75 (5), 832.CrossRefGoogle Scholar
Tchamitchian, P. 1987 Biorthogonalité et théorie des opérateurs. Rev. Mat. Iberoam. 3, 163–189.CrossRefGoogle Scholar
Van Milligen, B. P., Hidalgo, C. & Sanchez, E. 1995b Nonlinear phenomena and intermittency in plasma turbulence. Phys. Rev. Lett. 74 (3), 395.CrossRefGoogle Scholar
Van Milligen, B. P., Sanchez, E., Estrada, T., Hidalgo, C., Branas, B., Carreras, B. & Garcia, L. 1995a Wavelet bicoherence: a new turbulence analysis tool. Phys. Plasmas 2 (8), 3017–3032.CrossRefGoogle Scholar
Yoshimatsu, K., Kondo, Y., Schneider, K., Okamoto, N., Hagiwara, H. & Farge, M. 2009b Wavelet based coherent vorticity sheet and current sheet extraction from three-dimensional homogeneous magnetohydrodynamic turbulence. Phys. Plasmas 16, 082306.CrossRefGoogle Scholar
Yoshimatsu, K., Okamoto, N., Kawahara, Y., Schneider, K. & Farge, M. 2013 Coherent vorticity and current density simulation of three-dimensional magnetohydrodynamic turbulence using orthogonal wavelets. Geophys. Astrophys. Fluid Dyn. 107 (1–2), 73–92.CrossRefGoogle Scholar
Yoshimatsu, K., Okamoto, N., Schneider, K., Kaneda, Y. & Farge, M. 2009a Intermittency and scale-dependent statistics in fully developed turbulence. Phys. Rev. E 79, 026303.CrossRefGoogle ScholarPubMed
Yoshimatsu, K., Schneider, K., Okamoto, N., Kawahara, Y. & Farge, M. 2011 Intermittency and geometrical statistics of three-dimensional homogeneous magnetohydrodynamic turbulence: a wavelet viewpoint. Phys. Plasmas 18, 092304.CrossRefGoogle Scholar