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Universal Multifractals: Theory and Observations for Rain and Clouds

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  • 1 Department of Physics, McGill University, Montreal, Quebec, Canada
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Abstract

The standard model of atmospheric motions divides the atmosphere into distinct two- and three-dimensional isotropic turbulent regimes separated by a dimensional transition, the “mesoscale gap.” It is argued that the “gap” is fictional and that the atmosphere is scaling but anisotropic at all scales. According to this alternative unified scaling model, the dynamics are governed by anisotropic (differentially stratified and rotating) cascade processes yielding highly variable multifractal fields. Just as Gaussian random variables are associated with (linear) sums of random variables, these (nonlinear) multiplicative processes are generically associated with (special) universal multifractals in which many of the details of the dynamics are irrelevant. Although an attempt is made to outline these arguments in a widely accessible form, they are not new to this paper; they provide its context and motivation. The principal purpose of this paper is to test these ideas empirically. This is done using Landsat, NOAA-9, and Meteosat cloud radiances at visible, near-infrared, and thermal infrared wavelengths with length scales spanning the range 166 m–4000 km, radar reflectivities of rain (in the horizontal, vertical, and time), and global daily rainfall accumulations. Spectral analysis, as well as the new double trace moment data-analysis technique, is applied. In each case, rather than the sharp dimensional transition predicted by the standard model, the scaling is found to be relatively well respected right through the mesoscale. The three fundamental universal multifractal exponents are then estimated and one can go on to outline how these exponents (with the help of appropriate space–time transformations) can be used to make dynamic multifractal models.

Abstract

The standard model of atmospheric motions divides the atmosphere into distinct two- and three-dimensional isotropic turbulent regimes separated by a dimensional transition, the “mesoscale gap.” It is argued that the “gap” is fictional and that the atmosphere is scaling but anisotropic at all scales. According to this alternative unified scaling model, the dynamics are governed by anisotropic (differentially stratified and rotating) cascade processes yielding highly variable multifractal fields. Just as Gaussian random variables are associated with (linear) sums of random variables, these (nonlinear) multiplicative processes are generically associated with (special) universal multifractals in which many of the details of the dynamics are irrelevant. Although an attempt is made to outline these arguments in a widely accessible form, they are not new to this paper; they provide its context and motivation. The principal purpose of this paper is to test these ideas empirically. This is done using Landsat, NOAA-9, and Meteosat cloud radiances at visible, near-infrared, and thermal infrared wavelengths with length scales spanning the range 166 m–4000 km, radar reflectivities of rain (in the horizontal, vertical, and time), and global daily rainfall accumulations. Spectral analysis, as well as the new double trace moment data-analysis technique, is applied. In each case, rather than the sharp dimensional transition predicted by the standard model, the scaling is found to be relatively well respected right through the mesoscale. The three fundamental universal multifractal exponents are then estimated and one can go on to outline how these exponents (with the help of appropriate space–time transformations) can be used to make dynamic multifractal models.

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