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The Dimension and Intermittency of Atmospheric Dynamics

  • Conference paper
Turbulent Shear Flows 4

Abstract

We question the existence of a dimensional transition separating quasi-two dimensional and quasi-three dimensional atmospheric motions, i.e. large- and small-scale dynamics. We insist upon the fact that no matter how this transition should occur, it would have drastic consequences for atmospheric dynamics, consequences which have not been observed in spite of many recent experiments.

An alternative simpler hypothesis is proposed: that small scale structures are continuously deformed — flattened — at larger and larger scales by a scale invariant process. This continuous deformation may be characterised by defining an intermediate fractal dimension D el that we call an elliptical dimension. We show both theoretically and empirically that D el = 23/9 ~ 2.56. Atmospheric structures are therefore never “flat” (D el = 2), nor isotropic (D el = 3), but always display aspects of both. Larger structures are on the average, more stratified as a result of a well-defined stochastic process.

In this scheme, the intermittency must be quite strong in order to produce the well-known meteorological “animals” such as storms, fronts, etc. We propose that intermittency is characterised by hyperbolic probability distributions with exponents α. This possibility was first suggested by Mandelbrot (1974 a) for the rate of turbulent energy transfer (ε). We investigate intermittency for the wind (υ) the potential temperature (θ), and ε in terms of this hyperbolic intermittency. In particular, we find α υ = 5, α ε = 5/3, α 1n θ = 10/3, which show that the fifth moment of υ, the second moment of ε, and the fourth moment of 1n θ diverge.

We re-examine Mandelbrot’s model of intermittency and generalize it for anisotropic turbulence. We stress that it cannot be characterised by a single parameter, the dimension of the support of turbulence: we show that, except in a trivial case, several fractal dimensions intervene. We exhibit, for instance, a two-parameter model depending on the fractal dimension of the very active regions.

Finally, we sketch a direction for future work to assess this 23/9 dimensional scheme of atmospheric dynamics with hyperbolic intermittency.

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Authors and Affiliations

  1. EERM/CRMD, Météorologie Nationale, 2, Avenue Rapp, 75007, Paris, France

    D. Schertzer & S. Lovejoy

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  1. D. Schertzer
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  2. S. Lovejoy
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Editors and Affiliations

  1. Department of Mechanical Engineering, University of Surrey, GU2 5XH, Guildford, Surrey, England

    Leslie J. S. Bradbury

  2. Technische Fakultät, Friedrich-Alexander-Universität, Egerlandstraße 13, D-8520, Erlangen, Fed. Rep. of Germany

    Franz Durst (Lehrstuhl für Strömungsmechanik) (Lehrstuhl für Strömungsmechanik)

  3. Department of Mechanical Engineering, University of Manchester, Institute of Science and Technology, PO Box 88, M60 1QD, Manchester, England

    Brian E. Launder

  4. Mechanical Engineering Department, The Pennsylvania State University, 16802, University Park, PA, USA

    Frank W. Schmidt

  5. Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, SW7 2BX, London, England

    James H. Whitelaw

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Schertzer, D., Lovejoy, S. (1985). The Dimension and Intermittency of Atmospheric Dynamics. In: Bradbury, L.J.S., Durst, F., Launder, B.E., Schmidt, F.W., Whitelaw, J.H. (eds) Turbulent Shear Flows 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69996-2_2

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  • DOI: https://doi.org/10.1007/978-3-642-69996-2_2

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