Generalized Scale Invariance in the Atmosphere and Fractal Models of Rain


  • S. Lovejoy,

  • D. Schertzer


In recent years there has been considerable interest in stochastic rain models. By developing new ideas about scale invariance and intermittency we argue that the scope of such models can be greatly extended. The notions of scale invariance, intermittency, and the associated idea of fractal dimension have lately gained considerable ground, particularly in the context of extremely variable phenomena such as those found in the mesoscale and in hydrodynamic turbulence. We review some relevant work and argue that the atmosphere respects a symmetry principle that we call generalized scale invariance in which the statistical properties of the small and large scale are related to each other by a magnification coupled with a differential stratification (due to gravity) and differential rotation (due to the Coriolis force). We further argue that the extremely erratic (intermittent) nature of the atmosphere is characterized by scale invariant (fractal) measures leading to hyperbolic (fat-tailed) probability distributions of the fluctuations. The standard statistical methods based on exponential fall offs in both correlations and probability distributions are inappropriate when the variability is of this type. We show how both the scaling and intermittency may be exploited to develop extremely variable stochastic models of rain. Although the models examined here are the simplest of a family of extremely variable processes (depending on only two radar-determined parameters), they lead to realizations possessing many realistic features of the rainfield including complexity of form, clustering and bands at all scales, as well as differential stratification and rotation. Finally, we point to weaknesses in the models (in particular, their monodimensional nature) and suggest possible improvements.