Skip to main content
SpringerLink
Log in

Scale, Scaling and Multifractals in Geophysics: Twenty Years on

  • Conference paper
Nonlinear Dynamics in Geosciences

Abstract

We consider three developments in high number of degrees of freedom approaches to nonlinear geophysics: a) the transition from fractal geometry to multifractal processes, b) the self-organized critical (SOC) generation of extremes via multifractal phase transitions c) the generalization from isotropic scale invariance (self-similar fractals, multifractals) to (anisotropic) generalized scale invariance. We argue that these innovations are generally necessary for geophysical applications. We illustrate these ideas with data analyses from both the atmosphere and the earth’s surface, as well as with multifractal simulations.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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.

Similar content being viewed by others

References

  • Bak, P., et al. (1987), Self-Organized Criticality: An explanation of 1/f noise, Physical Review Letter, 59, 381-384.

    Article  Google Scholar 

  • Chigirinskaya, Y., D. Schertzer, S. Lovejoy, A. Lazarev and A. Ordanovich (1994). “Unified multifractal atmospheric dynamics tested in the tropics, part I: horizontal scaling and self organized criticality.” Nonlinear Processes in Geophysics 1(2/3): 105-114.

    Google Scholar 

  • Dewan, E. (1997), Saturated-cascade similtude theory of gravity wave sepctra, J. Geophys. Res., 102, 29799-29817.

    Article  Google Scholar 

  • Feigenbaum, M. J. (1978), Quantitative universality for a class of nonlinear transformations, J. Stat. Phys.,’ 19, 25.

    Article  Google Scholar 

  • Frisch, U., et al. (1978), A simple dynamical model of intermittency in fully develop turbulence, Journal of Fluid Mechanics, 87, 719-724.

    Article  Google Scholar 

  • Gage, K. S. (1979). “Evidence for k-5/3 law inertial range in meso-scale two dimensional turbulence.” Journal of Atmospheric Sciences, 36: 1979.

    Article  Google Scholar 

  • Gagnon, J., S., et al. (2006), Multifractal earth topography, Nonlin. Proc. Geophys., 13, 541-570.

    Article  Google Scholar 

  • Gardner, C. (1994), Diffusive filtering theory of gravity wave spectra in the atmosphere, J. Geophys. Res., 99, 20601.

    Article  Google Scholar 

  • Klinkenberg, B., and M. F. Goodchild (1992), The fractal properties of topography: A comparison of methods, Earth surface Proc. and Landforms, 17, 217-234.

    Article  Google Scholar 

  • Kolesnikova, V. N., and A. S. Monin (1965), Spectra of meteorological field fluctuations, Izvestiya, Atmospheric and Oceanic Physics, 1, 653-669.

    Google Scholar 

  • Kolmogorov, A. N. (1962), A refinement of previous hypotheses concerning the local structure of turbulence in viscous incompressible fluid at high Raynolds number, Journal of Fluid Mechanics, 83, 349.

    Google Scholar 

  • Lavallée, D., et al. (1993), Nonlinear variability and landscape topography: analysis and simulation, in Fractals in geography, edited by L. De Cola and N. Lam, pp. 171-205, Prentice-Hall, Englewood, N.J.

    Google Scholar 

  • Lilly, D., and E. L., Paterson (1983). “Aircraft measurements of atmospheric kinetic energy spectra.” Tellus, 35A, 379-382.

    Google Scholar 

  • Lilley, M., et al. (2004), 23/9 dimensional anisotropic scaling of passive admixtures using lidar aerosol data, Phys. Rev. E, 70, 036307-036301-036307.

    Article  Google Scholar 

  • Lindborg, E., and J. Cho (2001), Horizontal velocity structure functions in the upper troposphere and lower stratosphere i-ii. Observations, theoretical considerations, J. Geophys. Res., 106, 10223.

    Article  Google Scholar 

  • Lorenz, E. N. (1963), Deterministic Nonperiodic flow, J. Atmos. Sci., 20, 130-141.

    Article  Google Scholar 

  • Lovejoy, S., and D. Schertzer (1986), Scale invariance in climatological temperatures and the spectral plateau, Annales Geophysicae, 4B, 401-410.

    Google Scholar 

  • Lovejoy, S., and D. Schertzer (1990), Our multifractal atmosphere: A unique laboratory for non-linear dynamics, Physics in Canada, 46, 62.

    Google Scholar 

  • Lovejoy, S., and D. Schertzer (1995), How bright is the coast of Brittany? in Fractals in Geoscience and Remote Sensing, edited by G. Wilkinson, pp. 102-151, Office for Official Publications of the European Communities, Luxembourg.

    Google Scholar 

  • Lovejoy, S., and D. Schertzer (1998a), Stochastic chaos and multifractal geophysics, in Chaos, Fractals and models 96, edited by F. M. Guindani and G. Salvadori, Italian University Press.

    Google Scholar 

  • Lovejoy, S., and D. Schertzer (1998b), Stochastic chaos, scale invariance, multifractals and our turbulent atmosphere, in ECO-TEC: Architecture of the In-between, edited by e. b. A. Marras, p. in press, Storefront Book series, copublished with Princeton Architectural Press. 336 Lovejoy and Schertzer

    Google Scholar 

  • Lovejoy, S., and D. Schertzer (2006), Multifractals, cloud radiances and rain, J. Hydrol., 322, 59-88.

    Article  Google Scholar 

  • Lovejoy, S., et al. (1987), Functional Box-Counting and Multiple Dimensions in rain, Science, 235, 1036-1038.

    Article  Google Scholar 

  • Lovejoy, S., et al. (2004), Fractal aircraft trajectories and nonclassical turbulent exponents, Physical Review E, 70, 036306-036301-036305.

    Article  Google Scholar 

  • Lovejoy, S., et al. (2007), Is Kolmogorov turbulence relevant in the atmosphere? Geophys. Resear. Lett. (in press).

    Google Scholar 

  • Mandelbrot, B. B. (1967), How long is the coastline of Britain? Statistical self-similarity and fractional dimension, Science, 155, 636-638.

    Article  Google Scholar 

  • Mandelbrot, B. B. (1974), Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, Journal of Fluid Mechanics, 62, 331-350.

    Article  Google Scholar 

  • Mandelbrot, B. B. (1977), Fractals, form, chance and dimension, Freeman, San Francisco.

    Google Scholar 

  • Marsan, D., D. Schertzer, S. Lovejoy, 1996: Causal Space-Time Multifractal modelling of rain. J. Geophy. Res. 31D, 26,333-26,346.

    Google Scholar 

  • Matheron, G. (1970), Random functions and their applications in geology, in Geostatistics, edited by D. F. Merriam, pp. 79-87, Plenum Press, New York.

    Google Scholar 

  • Meneveau, C., and K. R. Sreenivasan (1987), Simple multifractal cascade model for fully develop turbulence, Physical Review Letter, 59, 1424-1427.

    Article  Google Scholar 

  • Monin, A. S. (1972). Weather forecasting as a problem in physics. Boston Ma, MIT press.

    Google Scholar 

  • Nastrom, G. D., and K. S. Gage (1983), A first look at wave number spectra from GASP data, Tellus, 35, 383.

    Google Scholar 

  • Novikov, E. A., and R. Stewart (1964), Intermittency of turbulence and spectrum of fluctuations in energy-disspation, Izv. Akad. Nauk. SSSR. Ser. Geofiz., 3, 408-412.

    Google Scholar 

  • Orlanski, I. A rational subdivision of scales for atmospheric processes, Bull. Amer. Met. Soc., 56, 527-530.

    Google Scholar 

  • Parisi, G., and U. Frisch (1985), A multifractal model of intermittency, in Turbulence and predictability in geophysical fluid dynamics and climate dynamics, edited by M. Ghil, et al., pp. 84-88, North Holland, Amsterdam.

    Google Scholar 

  • Perrin, J. (1913), Les Atomes, NRF-Gallimard, Paris.

    Google Scholar 

  • Richardson, L. F. (1961), The problem of contiguity: an appendix of statistics of deadly quarrels, General Systems Yearbook, 6, 139-187.

    Google Scholar 

  • Rodriguez-Iturbe, I., and A. Rinaldo (1997), Fractal River Basins, 547 pp., Cambridge University Press, Cambridge.

    Google Scholar 

  • Ruelle, D. and F. Takens (1971). “On the nature of turbulence.” Communications on Mathematical Physics 20 & 23: 167-192 & 343-344.

    Article  Google Scholar 

  • Sachs, D., et al. (2002), The multifractal scaling of cloud radiances from 1m to 1km, Fractals, 10, 253-265.

    Article  Google Scholar 

  • Schertzer, D., and S. Lovejoy (1985a), The dimension and intermittency of atmospheric dynamics, in Turbulent Shear Flow 4, edited by B. Launder, pp. 7-33, Springer-Verlag.

    Google Scholar 

  • Schertzer, D., and S. Lovejoy (1985b), Generalised scale invariance in turbulent phenomena, Physico-Chemical Hydrodynamics Journal, 6, 623-635.

    Google Scholar 

  • Schertzer, D. and S. Lovejoy (1986). Generalised scale invariance and anisotropic inhomogeneous fractals in turbulence. Fractals in Physics. L. Pietronero and E. Tosatti. Amsterdam, North-Holland: 457-460.

    Google Scholar 

  • Schertzer, D., and S. Lovejoy (1987), Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling of Multiplicative Processes, Journal of Geophysical Research, 92, 9693-9714.

    Google Scholar 

  • Schertzer, D. and S. Lovejoy (1992). “Hard and Soft Multifractal processes.” Physica A 185: 187-194.

    Google Scholar 

  • Schertzer, D., and S. Lovejoy (1994), Multifractal Generation of Self-Organized Criticality, in Fractals In the natural and applied sciences, edited by M. M. Novak, pp. 325-339, Elsevier, North-Holland.

    Google Scholar 

  • Schertzer, D., and S. Lovejoy (1997), Universal Multifractals do Exist! J. Appl. Meteor., 36, 1296-1303.

    Article  Google Scholar 

  • Schertzer, D., S. Lovejoy, 2004: Uncertainty and Predictability in Geophysics: Chaos and Multifractal Insights, In: State of the Planet, Frontiers and Challenges in Geophysics, edited by R.S.J. Sparks, and C.J. Hawkesworth, pp. 317-334, AGU, Washington.

    Google Scholar 

  • Schertzer, D. and S. Lovejoy (2006). Multifractals en Turbulence et Gèophysique. Irruption des Gèomètries Fractales dans la Sciences. G. Belaubre and F. Begon. Paris, Acadèmie Europèeene Interdisciplinaire des Sciences: p.189-209.

    Google Scholar 

  • Schertzer, D., et al. (1993), Generic Multifractal phase transitions and self-organized criticality, in Cellular Automata: prospects in astronomy and astrophysics, edited by J. M. Perdang and A. Lejeune, pp. 216-227, World Scientific.

    Google Scholar 

  • Schertzer, D., S. Lovejoy, F. Schmitt, I. Tchiguirinskaia and D. Marsan (1997). “Multifractal cascade dynamics and turbulent intermittency.” Fractals 5(3): 427-471.

    Google Scholar 

  • Schertzer, D., S. Lovejoy and P. Hubert (2002). An Introduction to Stochastic Multifractal Fields. ISFMA Symposium on Environmental Science and Engineering with related Mathematical Problems. A. Ern and W. Liu. Beijing, High Education Press. 4: 106-179.

    Google Scholar 

  • Schertzer, D., et al. (2002), Which chaos in the rainfall-runoff process? Hydrol. Sci., 47, 139-148.

    Article  Google Scholar 

  • Schmitt, F., et al. (1995), Multifractal analysis of the Greenland Ice-core project climate data., Geophys. Res. Lett, 22, 1689-1692.

    Article  Google Scholar 

  • Schmitt, F., et al. (1994), Empirical study of multifractal phase transitions in atmospheric turbulence, Nonlinear Processes in Geophysics, 1, 95-104.

    Google Scholar 

  • She, Z., S., and E. Levesque (1994), Phys. Rev. Lett., 72, 336.

    Article  Google Scholar 

  • Steinhaus, H. (1954), Length, Shape and Area, Colloquium Mathematicum, III, 1-13.

    Google Scholar 

  • Tchiguirinskaia, I., et al. (2006), Wind extremes and scales: multifractal insights and empirical evidence, in EUROMECH Colloquium on Wind Energy, edited by P. S. Eds. J. Peinke, Springer-Verlag.

    Google Scholar 

  • Tuck, A. F., et al. (2004), Scale Invariance in jet streams: ER-2 data around the lowerstratospheric polar night vortex, Q. J. R. Meteorol. Soc., 130, 2423-2444.

    Article  Google Scholar 

  • Van der Hoven, I. (1957), Power spectrum of horizontal wind speed in the frequency range from.0007 to 900 cycles per hour, Journal of Meteorology, 14, 160-164.

    Google Scholar 

  • Wilson, K. G. (1971), “Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture.” Phys. Rev. B 4(9): 3174-3183.

    Article  Google Scholar 

  • Yaglom, A. M. (1966), The influence on the fluctuation in energy dissipation on the shape of turbulent characteristics in the inertial interval, Sov. Phys. Dokl., 2, 26-30.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Physics, McGill University, 3600 University st, Montreal, Que. H3A 2T8, Canada

    Shaun Lovejoy

  2. CEREVE, Ecole Nationale des Ponts et Chaussé, 6-8, avenue Blaise Pascal, Cité Descartes, 77455 MARNE-LA-VALLEE Cedex, France

    Daniel Schertzer

  3. CNRM, Météo-France, 1 Quai Branly, 75007 Paris, France

    Daniel Schertzer

Authors
  1. Shaun Lovejoy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Schertzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Science+Business Media, LLC

About this paper

Cite this paper

Lovejoy, S., Schertzer, D. (2007). Scale, Scaling and Multifractals in Geophysics: Twenty Years on. In: Nonlinear Dynamics in Geosciences. Springer, New York, NY. https://doi.org/10.1007/978-0-387-34918-3_18

Download citation

Publish with us

Policies and ethics

Search

Navigation