Continuous Multiplicative Cascade Models of Rain and Clouds

Abstract

Early scaling stochastic models of cloud and rain fields were designed to respect an unrealistically simple scale invariant symmetry called “simple scaling” or “scaling of the increments”, involving a single fractal dimension. These monofractal processes were produced by summing a large number of random structures: pulses and/or wave packets (depending on whether the model was based in real or Fourier space) and in spite of this extreme simplification yielded simulations of cloud fields with some realistic features including texture, clustering, bands and intermittency. The linear (additive) nature of these processes is however intrinsically related to the fact that they have a single fractal dimension and is in sharp contrast with the non-linear nature of the true dynamical processes and with the observed multiple fractal dimensions of the fields. From this perspective, even the relatively simplified case of stochastic modeling of passive clouds — i.e. passively advected by a turbulent velocity field — is already beyond the scope of linear stochastic approaches, because of the statistical implications of the highly non-linear distortion of the concentration field by the turbulent velocity field. Focusing on this problem, we show how to build up models corresponding to coupled cascade processes, non linearly conserving the fluxes of energy and concentration variance. These cascades, have generally been based on a series of highly artificial discrete steps, are here taken to their continuous limit. Continuous cascades are not only far more realistic than their discrete counterparts, they also admit universality classes characterized by 2 parameter generators. This means that a simple (dynamical) generator describes the entire multifractal spectrum rather than an infinite number of (geometric) dimension parameters. We also show how to numerically simulate such processes with both Gaussian and Levy generators as well as how to perform rescaling iteration on the results which is scale invariant “zooming” procedure.