Non-Linear Variability in Geophysics: Scaling and Fractals

consequences of broken symmetry -here parity-is studied. In this model, turbulence is dominated by a hierarchy of helical (corkscrew) structures. The authors stress the unique features of such pseudo-scalar cascades as well as the extreme nature of the resulting (intermittent) fluctuations. Intermittent turbulent cascades was also the theme of a paper by us in which we show that universality classes exist for continuous cascades (in which an infinite number of cascade steps occur over a finite range of scales). This result is the multiplicative analogue of the familiar central limit theorem for the addition of random variables. Finally, an interesting paper by Pasmanter investigates the scaling associated with anomolous diffusion in a chaotic tidal basin model involving a small number of degrees of freedom. Although the statistical literature is replete with techniques for dealing with those random processes characterized by both exponentially decaying (non-scaling) autocorrelations and exponentially decaying probability distributions, there is a real paucity of literature appropriate for geophysical fields exhibiting either scaling over wide ranges (e. g. algebraic autocorrelations) or extreme fluctuations (e. g. algebraic probabilities, divergence of high order statistical moments). In fact, about the only relevant technique that is regularly used -fourier analysis (energy spectra) -permits only an estimate of a single (power law) exponent. If the fields were mono-fractal (characterized by a single fractal dimension) this would be sufficient, however their generally multifractal character calls for the development of new techniques.

Table of Contents:
1 — Turbulence and Geophysical Fluid Dynamics.- Ch. 1 — Energy spectrum and intermittency of semi-geostrophic flow.- Ch. 2 — Helicity fluctuations and coherence in developed turbulence.- Ch. 3 — Deterministic diffusion, effective shear and patchiness in tidal flows.- Ch. 4 — Scaling nonlinear variability in geodynamics: Multiple singularities, observables, universality classes.- Ch. 5 — Scaling and structure functions in shear flows.- 2 — Data Analysis Techniques.- Ch. 6 — Correlation dimension and data sample size.- Ch. 7 — On the determination of the codimension function.- Ch. 8 — Multifractal analysis techniques and the rain and cloud fields from 10?3 to 106m.- Ch. 9 — Eigen-structure of eddies; continuous weighting versus conditional sampling.- 3 — Modeling and Analysis of Clouds, Rain, and Other Atmospheric Fields.- Ch. 10 — Atmospheric boundary layer variability and its implications for CO2 flux measurements.- Ch. 11 — A fractal study of dielectric breakdown in the atmosphere.- Ch. 12 — On lognormality and scaling in spatial rainfall averages?.- Ch. 13 — Physically based rain and cloud modeling.- Ch. 14 — Fractal characterization of intertropical precipitation variability and anisotropy.- 4 — Modeling and Analysis of the Climate, Oceans and in Solid Earth Geophysics.- Ch. 15 — Fractal nature of surface geometry in a developed sea.- Ch. 16 — Fractal linear models of geophysical processes.- Ch. 17 — Extreme variability of climatological data: Scaling and intermittency.- Ch. 18 — On the existence of low dimensional climatic attractors.- Ch. 19 — The third generation WAM models for wind-generated ocean waves.- Ch. 20 — 1/f geology and seismic deconvolution.- Ch. 21 — The stochastic coherence and the dynamics ofglobal climate models and data.- 5 — Remote Sensing.- Ch. 22 — Landsat observations of fractal cloud structure.- Ch. 23 — Fractal dimension analysis of horizontal cloud pattern in the intertropical convergence zone.- Ch. 24 — Radiative transfer in multifractal clouds.