FROM SCALAR CASCADES TO LIE CASCADES: JOINT MULTIFRACTAL ANALYSIS OF RAIN AND CLOUD PROCESSES

Summary

ABSTRACT There are two primary approaches to modeling rainfall; stochastic modeling and deterministic integration of nonlinear partial differential equations which model the atmospheric dynamics. The statistical advantages of the former could be combined with the physical advantages of the latter by exploiting cascade models based on scale invariant symmetries respected by the equations. Carried to its logical conclusion, this approach involves considering the atmosphere as a space-time multifractal process admitting either a vector, tensor or even only a nonlinear representation. The process is then defined by two groups which respectively specify the rule required to change from one scale to another and the corresponding transforms of fields. Both groups are characterized by their generators, hence by their Lie algebra. We show how to extend existing cascades beyond scalar processes, showing preliminary numerical simulations and data analyses, as well as indicating how to characterize and classify the scale invariant interactions of fields.

INTRODUCTION

The limitations of standard deterministic dynamical and of phenomenological stochastic modeling of rain

Geophysical fields show abundant evidence of nonlinear variability resulting from strong nonlinear interactions between different scales, different structures, and different fields. This variability is quite extreme and is associated with catastrophic events such as earthquakes, tornadoes, flash floods, extreme temperatures, volcanic eruptions. Another fundamental characteristic of this variability is the very large range of scales involved, which often extends from 10,000 km to 1 mm in space, and from geological scales to millisecond in time.