2 - Multifractals and rain

Summary

ABSTRACT Scaling models and analyses of rain have now been around for over ten years, a period in which the corresponding scale invariant notions have seen rapid development. We review these developments concentrating on multifractals that are believed to provide the appropriate theoretical framework for scaling nonlinear dynamical systems. Although early scaling notions were geometric rather than dynamic, they contributed towards establishing and testing scaling ideas in rain and in determining the limits of scaling in both time and space. The problematic of passive scalar clouds and (continuous) turbulent cascades, provided them with a sound physical basis. Building on these advances, later analysis methods (particularly Double Trace Moment technique) made it possible to obtain robust estimates of the basic multifractal parameters. Continuous (and universal) cascades allow us to exploit these parameters to make dynamical models. We also discuss various applications of multifractals to rain including multifractal objective analysis, statistics of extreme values, multifractal modelling, space-time transformations, the multifractal radar observer's problem, stratification, and texture of rain.

INTRODUCTION

Stochastic models of rain, atmospheric scaling and multifractals

The atmosphere is probably the most familiar highly nonlinear dynamical system; the nonlinear terms are roughly ≈10¹² (the Reynolds number) times larger than the linear (dissipation) terms, and structures vary over 9–10 orders of magnitude in space (≈ 1 mm to 10⁴ km) and at least as much in time (≈10⁻³ s on up).