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.