Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Search for finite dimensional attractors in atmospheric turbulence

  • 75 Accesses

  • 11 Citations

Abstract

The question is investigated whether the dynamics of turbulent wind fields in the atmospheric boundary layer can be satisfactorily described by a low-dimensional deterministic system. Special emphasis is laid on the detection of a possibly existing, underlying strange attractor. Fast response wind measurements of an ultrasonic anemometer with a sampling rate of 21 Hz were carried out over periods of several days in the near surface boundary layer. The correlation dimension of the resulting time series, several million data points long, is estimated by means of the Grassberger-Procaccia algorithm. No sign of a low dimensional attractor can be detected. By comparison with different types of random noise, the existence of an attractor with dimension lower than six can be excluded in the present data sets.

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

References

  1. Essex, C., Lookman, T., and Nerenberg, M. A. H.: 1987, ‘The Climate Attractor over Short Timescales’,Nature 326, 64–66.

  2. Fraedrich, K.: 1986, ‘Estimating the Dimensions of Weather and Climate Attractors’,J. Atmos. Sci. 43, 419–432.

  3. Grassberger, P.: 1986, ‘Do Climatic Attractors Exist?’,Nature 323, 609–612.

  4. Grassberger P. and Procaccia, I.: 1983, ‘Characterization of Strange Attractors’,Phys. Rev. Lett. 50, 346–349 and ‘Measuring the Strangeness of Strange Attractors’,Physica D,9, 189–208.

  5. Hentschel, H. G. E. and Procaccia, I.: 1983, ‘The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors’,Physica D 9, 435–444.

  6. Keppenne, C. L. and Nicolis, C.: 1989, ‘Global Properties and Local Structure of the Weather Attractor over Western Europe’,J. Atmos. Sci. 46, 2356–2370.

  7. Lorenz, E. N.: 1963, ‘Deterministic Nonperiodic Flow’,J. Atmos. Sci. 20, 130–141.

  8. Kolmogorov, A. N.: 1941, ‘The Local Structure of Turbulence in an Incompressible Viscous Fluid for Very Large Reynolds Numbers’,Dokl. Akad. Nauk SSSR 30, 301–305.

  9. Mañé, R.: 1981, ‘On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps’, inLecture Notes in Mathematics, Vol. 898, Springer Verlag, New York, pp. 230–242.

  10. Manneville, P.: 1990,Dissipative Structures and Weak Turbulence, Academic Press, Boston.

  11. Nerenberg, M. A. H. and Essex, C.: 1990, ‘Correlation Dimension and Systematic Geometric Effects’,Phys. Rev. A 42, 7065–7074.

  12. Nicolis, C. and Nicolis, G.: 1984, ‘Is There a Climatic Attractors?’,Nature 311, 529–532.

  13. Percival, D. B. and Walden, A. T.: 1993,Spectral Analysis for Physical Applications, Cambridge University Press, Cambridge.

  14. Poveda-Jaramillo, G., and Puente, C. E.: 1993, ‘Strange Attractors in Atmospheric Boundary-Layer Turbulence’,Boundary-Layer Meteorol. 64, 175–197.

  15. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P.: 1992,Numerical Recipes in Fortran. The Art of Scientific Computing, Cambridge University Press, Cambridge.

  16. Priestley, M. B.: 1981,Spectral Analysis of Time Series. Volume 1: Univariate Time Series, Academic Press, London.

  17. Richardson, L. F.: 1992,Weather Prediction by Numerical Process, University Press, Cambridge.

  18. Ruelle, D.: 1990, ‘Deterministic Chaos: the Science and the Fiction’,Proc. R. Soc. London A 427, 241–248.

  19. Smith, L. A.: 1984, ‘Intrinsic Limitations on Dimension Calculations’,Physics Lett. A 133, 283–288.

  20. Takens, F.: 1981, ‘Detecting Strange Attractors in Turbulence’, inLecture Notes in Mathematics, Vol. 898, Springer Verlag, New York, pp. 366–381.

  21. Temam, R.: 1989,Infinite Dimensional Systems in Mechanics and Physics, Springer, Berlin.

  22. Theiler, J.: 1986, ‘Spurious Dimensions from Correlation Algorithms Applied to Limited Time-Series Data’,Phys. Rev. A 34, 2427–2432.

  23. Tsonis, A. A. and Elsner, J. B.: 1988, ‘The Weather Attractor over Very Short Time Scales’,Nature 333, 545–547.

  24. Tsonis, A. A.: 1992,Chaos from Theory to Applications, Plenum Press, New York (1992), pp. 274.

  25. Tsonis, A. A., Elsner, J. B., and Georgakakos, K. P.: 1993, ‘Estimating the Dimension of Weather and Climate Attractors: Important Issues about the Procedure and Interpretation’,J. Atmos. Sci. 50, 2549–2555.

  26. Zeng, X., Pielke, R. A., and Eykholt, R.: 1992, ‘Estimating the Fractal Dimension and the Predictability of the Atmosphere’,J. Atmos. Sci. 49, 649–659.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Weber, R.O., Talkner, P., Stefanicki, G. et al. Search for finite dimensional attractors in atmospheric turbulence. Boundary-Layer Meteorol 73, 1–14 (1995). https://doi.org/10.1007/BF00708927

Download citation

Keywords

  • Boundary Layer
  • Random Noise
  • Wind Field
  • Surface Boundary
  • Atmospheric Boundary Layer