Skip to main content
SpringerLink
Log in

Numerical Schemes for the Analysis of Turbulence — A Tutorial

  • Chapter
  • First Online:
Book cover Space Plasma Simulation

Part of the book series: Lecture Notes in Physics ((LNP,volume 615))

Abstract

The analysis of plasma turbulence has traditionally relied on limited repertoire of methods such as Fourier analysis, correlation analysis, etc. This text gives a short overview of what deeper insight can be obtained by using techniques that exploit nonlinear properties of the data. It covers both higher order spectra and higher order statistics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. D. I. Abarbanel, R. Brown, and J. J. Sidorovich: The analysis of observed chaotic data in physical systems, Rev. Mod. Phys. 65, 1331–1392 (1993).

    Article  ADS  Google Scholar 

  2. M. J. Aschwanden, B. R. Dennis, and A. Benz: Logistic Avalanche Processes, Elementary Time Structures, and Frequency Distributions in Solar Flares, Astrophysical Journal 497, 972–993 (1998).

    Article  ADS  Google Scholar 

  3. P. Abry, P. Gonçalvès, and P. Flandrin: Wavelets, spectrum estimation, 1/f processes, in Wavelets and Statistics, edited by A. Antoniadis and G. Oppenheim, Lecture Notes in Statistics, Vol. 103 (Springer, Berlin, 1995), pp. 15–30.

    Google Scholar 

  4. E. Bacry, J.F. Muzy, and A. Arneodo: Singularity spectrum of fractal signals from wavelet analysis: exact results, J. Stat. Phys. 70, 635–674 (1993).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. R. Badii and A. Politi: Complexity (Cambridge Univ. Press, Cambridge, 1997).

    Book  MATH  Google Scholar 

  6. S. D. Bale, D. Burgess, P. J. Kellogg, K. Goetz, R. L. Howard, and S. J. Monson: Evidence of three-wave interactions in the upstream solar wind, Geoph. Res. Lett., 23, 109–112 (1996).

    Article  ADS  Google Scholar 

  7. J. Beran: Statistics for long-memory processes, (Chapman & Hall, London, 1994).

    MATH  Google Scholar 

  8. D. Biskamp: Nonlinear magnetohydrodynamics, (Cambridge Univ. Press, Cambridge, 1993).

    Book  Google Scholar 

  9. T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani: Dynamical systems approach to turbulence, (Cambridge Univ. Press, Cambridge, 1998).

    Book  MATH  Google Scholar 

  10. J.-P. Bouchaud and A. Georges: Anomalous diffusion in disordered media: statistical mechanisms, models, and physical applications, Phys. Reports 195, 127–293 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  11. D. R. Brillinger: Time series, (Holt, New York, 1975).

    MATH  Google Scholar 

  12. L. F. Burlaga: Multifractal structure of the interplanetary magnetic field, Geoph. Res. Lett. 18, 69–72 (1991).

    Article  ADS  Google Scholar 

  13. V. Carbone: Scaling exponents of the velocity structure functions in the interplanetary medium, Ann. Geoph, 12, 585–590 (1994).

    Article  ADS  Google Scholar 

  14. V. Carbone, et al.: To what extent can dynamical models describe statistical features of turbulent flows?, Europhys. Lett, 58(3), 349–355 (2002).

    Article  ADS  Google Scholar 

  15. B. A. Carreras, B. P. van Milligen, M. A. Pedrosa, et al.: Experimental evidence of long-range correlation and self-similarity in plasma fluctuations, Phys. Plasmas 6, 1885–1892 (1999).

    Article  ADS  Google Scholar 

  16. A. Chhabra and R. V. Jensen: Direct determination of the f(?) singularity spectrum, Phys. Rev. Lett. 62, 1327–1330 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  17. D. Coca, M. A. Balikhin, S. A. Billings, H. St. C. K. Alleyne, M. W. Dunlop, and H. Lühr: Time-domain identification of nonlinear processes in space-plasma turbulence using multi-point measurements, in Proc. Cluster II workshop on Multiscale/ Multipoint Plasma Measurements, document ESA SP-449, ESA, Paris (2000).

    Google Scholar 

  18. G. Consolini, M. F. Marcucci, and M. Candidi: Multifractal structure of the auroral electrojet index data, Phys. Rev. Lett., 76, 4082–4085 (1996).

    Article  ADS  Google Scholar 

  19. G. Consolini and T. Chang: Magnetic field topology and criticality in GEOTAIL dynamics: relevance to substorm phenomena, Space Science Rev. 95, 309–321 (2001).

    Article  ADS  Google Scholar 

  20. T. Dudok de Wit and V. V. Krasnosel’skikh: Wavelet bicoherence analysis of strong plasma turbulence at the Earth’s quasiparallel bow shock, Phys. Plasmas 2, 4307–4311 (1995).

    Article  ADS  Google Scholar 

  21. T. Dudok de Wit and V. V. Krasnosel’skikh: Non-Gaussian statistics in space plasma turbulence: fractal properties and pitfalls, Nonl. Proc. Geoph. 3, 262–273 (1996).

    Google Scholar 

  22. T. Dudok de Wit: Data analysis techniques for resolving nonlinear processes in plasmas: a review, in The URSI Review on Radio Science 1993-1995, edited by R. E. Stone (Oxford University Press, Oxford, 1996).

    Google Scholar 

  23. T. Dudok de Wit, V. Krasnosel’skikh, M. Dunlop, and H. Lühr: Identifying nonlinear wave interactions using two-point measurements: a case study of SLAMS, J. Geophys. Res. 104, 17079–17090 (1999).

    Article  ADS  Google Scholar 

  24. P. Escoubet, A. Roux, F. Lefeuvre, and D. LeQuéau (eds.): START: Spatiotemporal analysis for resolving plasma turbulence, document WPP-047 (ESA, Paris, 1993).

    Google Scholar 

  25. M. P. Freeman, N. W. Watkins, and D. J. Riley: Evidence for a solar wind origin of the power law burst lifetime distribution of the AE indices, Geoph. Res. Lett. 27, 1087–1090 (2000).

    Article  ADS  Google Scholar 

  26. U. Frisch: Turbulence (Cambrisge Univ. Press, Cambridge, 1995).

    MATH  Google Scholar 

  27. G. B. Giannakis and E. Serpedin: A bibliography on nonlinear system identification, Signal Processing 81, 533–580 (2001).

    Article  MATH  Google Scholar 

  28. K.-H. Glassmeier, U. Motschmann, and R. Schmidt: CLUSTER workshops on data analysis tools, physical measurements and mission-oriented theory, document SP-371 (ESA, Paris, 1995).

    Google Scholar 

  29. M. L. Goldstein and D. A. Roberts: Magnetoshydrodynamic turbulence in the solar wind, Phys. Plasmas 6, 4154–4160 (1999).

    Article  ADS  Google Scholar 

  30. P. Grassberger and I. Procaccia: Characterization of strange attractors, Phys. Rev. Lett. 50, 1265 (1983).

    Article  MathSciNet  Google Scholar 

  31. M. J. Hinichand C. S. Clay: The application of the discrete Fourier transform in the estimation of power spectra, coherence and bispectra of geophysical data, Rev. Geoph. 6, 347–363 (1968).

    Article  ADS  Google Scholar 

  32. T. S. Horbury and A. Balogh: Structure function measurements of the intermittent MHD turbulent cascade, Nonl. Proc. Geoph. 4, 185–199 (1997).

    Google Scholar 

  33. W. Horton and A. Hasegawa: Quasi-two-dimensional dynamics of plasmas and fluids, Chaos 4, 227–251 (1994).

    Article  ADS  Google Scholar 

  34. H. J. Jensen: Self-organized criticality (Cambridge University Press, Cambridge, 1998).

    MATH  Google Scholar 

  35. B. B. Kadomtsev: Plasma turbulence (Academic Press, New York, 1965).

    Google Scholar 

  36. H. Kantz and T. Schreiber: Nonlinear time series analysis (Cambridge University Press, Cambridge, 1997).

    MATH  Google Scholar 

  37. Y. C. Kim and E. J. Powers: Digital bispectral analysis and its applications to nonlinear wave interactions, IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).

    Article  ADS  Google Scholar 

  38. V. Kravtchenko-Berejnoi, F. Lefeuvre, V. V. Krasnosel’skikh, and D. Lagoutte: On the use of tricoherent analysis to detect non-linear wave interactions, Signal Processing 42, 291–309 (1995).

    Article  MATH  Google Scholar 

  39. Y. Kuramitsu and T. Hada: Acceleration of charged particles by large amplitude MHD waves: effect of wave spatial correlation, Geophys. Res. Lett. 27, 629–632 (2000).

    Article  ADS  Google Scholar 

  40. J.-L. Lacoume, P.-O. Amblard, and P. Comon: Statistiques d’ordre supérieur pour le traitement du signal (Masson, Paris, 1997).

    Google Scholar 

  41. D. Lagoutte, D., F. Lefeuvre, and J. Hanasz: Application of bicoherence analysis in study of wave interactions in space plasmas, J. Geoph. Res. 94, 435–442 (1989).

    Article  ADS  Google Scholar 

  42. B. Lef`ebvre, V. Krasnosel’skikh, and T. Dudok de Wit: Stochastization of phases in four-wave interaction, in P. G. L. Leachet al.: Dynamical systems, plasmas and gravitation (Springer Verlag, Berlin, 1999), pp. 117–129.

    Chapter  Google Scholar 

  43. L. Ljung: System identification: theory for the user (Prentice-Hall, Englewood Cliffs, NJ, 1999).

    Google Scholar 

  44. J. L. Lumley: Stochastic tools in turbulence (Academic Press, 1970).

    Google Scholar 

  45. B. Mandelbrot: Fractals and scaling in finance (Springer Verlag, Berlin, 1997).

    MATH  Google Scholar 

  46. E. Marschand C.-Y. Tu: Non-Gaussian probability distributions of solar wind fluctuations, Ann. Geoph. 12, 1127–1138 (1994).

    Article  ADS  Google Scholar 

  47. E. Marsch: Analysis of MHD turbulence: spectra of ideal invariants, structure functions and intermittency scalings, in [28] pp. 107–118.

    Google Scholar 

  48. J. M. Mendel: Tutorial on Higher Order Statistics (Spectra) in signal processing and system theory: theoretical results and some applications, Proc. IEEE 79(3), 278–305 (1991).

    Google Scholar 

  49. A. S. Monin and A. M. Yaglom: Statistical fluid mechanics (MIT Press, Cambridge, Mass., 1967), vols. 1 and 2.

    Google Scholar 

  50. S. L. Musher, A. M. Rubenchik, and V. E. Zakharov: Weak Langmuir turbulence, Phys. Rep. 252, 177–274 (1995).

    Article  ADS  Google Scholar 

  51. J.-F. Muzy, E. Bacry, and A. Arneodo: Multifractal formalism for fractal signals: the structure-function approach versus the wavelet-transform modulus-maxima method, Phys. Rev. E 47, 875–884 (1993).

    ADS  Google Scholar 

  52. S. W. Nam and E. J. Powers: Applications of higher order spectral analysis to cubically nonlinear system identification, IEEE Trans. Acoust. SpeechSignal Proc. ASSP-42, 1746–1765 (1994).

    Google Scholar 

  53. C. L. Nikias and A. P. Petropulu: Higher-order spectra analysis (Prentice Hall, New Jersey, 1993).

    MATH  Google Scholar 

  54. C. L. Nikias and M. E. Raghuveer: Bispectrum estimation: a digital signal processing framework, Proc. IEEE 75, 869–891 (1987).

    Google Scholar 

  55. G. Paladin and A. Vulpiani: Anomalous scaling laws in multifractal objects, Phys. Reports 4, 147–225 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  56. R. K. Pearson: Discrete-time dynamic models (Oxford Univ. Press, Oxford, 1999).

    MATH  Google Scholar 

  57. H. L. Pécseli and J. Trulsen: On the interpretation of experimental methods for investigating nonlinear wave phenomena, Plasma Phys. Controlled Fusion 25, 1701–1715 (1993).

    Article  Google Scholar 

  58. D. B. Percival and A. T. Walden: Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Tehcniques (Cambridge University Press, Cambridge, 1993).

    Book  Google Scholar 

  59. P. Pommois, G. Zimbardo, and P. Veltri: Magnetic field line transport in three dimensional turbulence: Lévy random walk and spectrum models, Phys. Plasmas 5, 1288–1297 (1998).

    Article  ADS  Google Scholar 

  60. C. P. Price and D. E. Newman: Using the R/S statistic to analyze AE data, J. Atmos. Solar-Terrestr. Phys. 63, 1387–1397 (2001).

    Article  ADS  Google Scholar 

  61. T. Schreiber: Interdisciplinary application of nonlinear time series analysis, report WUB-98-13 (University of Wuppertal, Wuppertal, Germany, 1998) [available at http://www.lanl.gov/abs/chao-dyn/9807001].

  62. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter: Strange kinetics, Nature 363, 31–37 (1993).

    Article  ADS  Google Scholar 

  63. T. H. Solomon, E. R. Weeks, and H. Swinney: Chaotic advection in a twodimensional flow: Lévy flights and anomalous diffusion, Physica D 76, 70–84 (1994).

    ADS  Google Scholar 

  64. L. Sorriso-Valvo, V. Carbone, P. Giuliani, P. Veltri, R. Bruno, E. Martines, and V. Antoni: Intermittency in plasma turbulence, Planet. Space Sci., 49, 1193–1200 (2001).

    Article  ADS  Google Scholar 

  65. K. Stasiewicz, B. Holback, V. Krasnosel’skikh, M. Boehm, R. Boström, and P. M. Kintner: Parametric instabilities of Langmuir waves observed by Freja, J. Geoph. Res. 101, 21515–21525 (1996).

    Article  ADS  Google Scholar 

  66. L. Rezeau, G. Belmont, B. Guéret, and B. Lembège: Cross-bispectral analysis of the electromagnetic field in a beam-plasma interaction, J. Geophys. Res. 102, 24387–24392 (1997).

    Article  ADS  Google Scholar 

  67. C. P. Ritz and E. J. Powers: Estimation of nonlinear transfer functions for fully developed turbulence, Physica D 20, 320–334 (1986).

    ADS  Google Scholar 

  68. C. P. Ritz, E. J. Powers, T. L. Rhodes, et al.: Advanced plasma fluctuation analysis techniques and their impact on fusion research, Rev. Sci. Instrum. 59, 1739–1744 (1988).

    Article  ADS  Google Scholar 

  69. C. P. Ritz, E. J. Powers, and R. D. Bengtson: Experimental measurement of threewave coupling and energy cascading, Phys. Fluids B 1, 153–163 (1989).

    ADS  Google Scholar 

  70. B. W. Silverman: Density estimation for statistics and data analysis (Chapman & Hall, London, 1986).

    MATH  Google Scholar 

  71. D. Sornette: Critical phenomena in natural sciences: chaos, fractals, selforganisation and disorder (Springer Verlag, Berlin 2000).

    Google Scholar 

  72. J. Soucek, T. Dudok de Wit, V. Krasnoselskikh, and A. Volokitin: Statistical analysis of non-linear wave interactions in simulated Langmuir turbulence data, Ann. Geoph., in press.

    Google Scholar 

  73. J. Takalo, R. Lohikoski, and J. Timonen: Structure function as a tool in AE and Dst time series analysis, Geophys. Res. Lett. 22, 635–638 (1995).

    Article  ADS  Google Scholar 

  74. B. P. van Milligen, E. Sánchez, T. Estrada, C. Hidalgo, B. Brañas, B. Carreras, and L. García: Wavelet bicoherence: a new turbulence analysis tool, ‘Phys. Plasmas’ 2, 3017–3032 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  75. D. V. Vassiliadis, D. N. Baker, H. Lundstedt, and R. C. Davidson (eds.): Nonlinear methods in space plasma physics, Phys. Plasmas 6(11), 4137–4199 (1999).

    Google Scholar 

  76. D. Vassiliadis: System identification, modeling, and prediction for space weather environments, IEEE Trans. Plasma Science, in the press.

    Google Scholar 

  77. N. Watkins, M. P. Freeman, S. C. Chapman, and R. O. Dendy: Testing the SOC hpothesis for the magnetosphere, J. Atmos. Terr. Phys., 63, 1360–1370 (2001).

    Google Scholar 

  78. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik: Hamiltonian approach to the description of non-linear plasma phenomena, Phys. Rep. 129, 285–366 (1985).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LCPE, UMR 6115 CNRS-Université d’Orléans, 3A, av. de la Recherche Scientifique, F-45071, Orléans cedex 2, France

    Thierry Dudok de Wit

Authors
  1. Thierry Dudok de Wit
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Max-Plack-Institut für Aeronomie, Max-Planck Str.2, 37191, Katlenburg-Lindau, Germany

    Jörg Büchner

  2. Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, 85741, Garching, Germany

    Manfred Scholer  & Christian T. Dum  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

de Wit, T.D. (2003). Numerical Schemes for the Analysis of Turbulence — A Tutorial. In: Büchner, J., Scholer, M., Dum, C.T. (eds) Space Plasma Simulation. Lecture Notes in Physics, vol 615. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36530-3_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-36530-3_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00698-5

  • Online ISBN: 978-3-540-36530-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation