The Multifractal Scaling of Cloud Radiances From 1m to 1km

Abstract

The cloud radiances and atmospheric dynamics are strongly nonlinearly coupled, the observed scaling of the former from 1 km to planetary scales is prima facae evidence for scale invariant dynamics. In contrast, the scaling properties of radiances at scales <1 km have not been well studied (contradictory claims have been made) and if a characteristic vertical cloud thickness existed, it could break the scaling of the horizontal radiances. In order to settle this issue, we use ground-based photography to study the cloud radiance field through the range scales where breaks in scaling have been reported (30 m to 500 m). Over the entire range 1 m to 1 km the two-dimensional (2D) energy spectrum (E(k)) of 38 clouds was found to accurately follow the scaling form E(k)≈ k-β where k is a wave number and β is the spectral exponent. This indirectly shows that there is no characteristic vertical cloud thickness, and that "radiative smoothing" of cloud structures occurs at all scales. We also quantitatively characterize the type of (multifractal) scaling showing that the main difference between transmitted and reflected radiance fields is the (scale-by-scale) non-conservation parameter H. These findings lend support to the unified scaling model of the atmosphere which postulates a single anisotropic scaling regime from planetary down to dissipation scales.

−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies

“…This scale invariance is indirectly linked to scale invariance of cloud field transported by atmospheric turbulence. In [31], a power law is also observed for the spectrum of cloud radiances obtained In summary, the spectra of sustainable data considered in this study, display power law behaviour with an exponent spectral close to the 5/3 Kolmogorov value. The slight difference with the exact 5/3 value is usually caused by intermittency effects [5,22].…”

Empirical evidences of a common multifractal signature in economic, biological and physical systems

“…Starting from the study of turbulent flows [1][2][3][4][5][6][7], for which the theory was first developed, multifractality has been reported in many other systems, some of them completely unrelated to turbulence. These include natural images [8,9], stock market series [10][11][12][13][14][15][16][17], human gait [18,19], the heliospheric magnetic field [20], network traffic [21], phytoplankton distribution in oceans [22], rainfall distribution [23][24][25], sea surface temperature [26,27], heartbeat dynamics [28,29], distribution of chemical fields [30,31], cloud structure [32,33], many different geophysical fields [34,35] and a long etc.…”

Multifractal fluctuations in seismic interspike series

“…Different values of the same scaling exponent could be required for different segments of the same sequence, indicating a time variation of the scaling behaviour, relying to a time variation of the underlying dynamics [14]. Furthermore, different scaling exponents can be revealed for many interwoven fractal subsets of the sequence [15]; in this case the process is not a monofractal but multifractal. A multifractal object requires many indices to characterize its scaling properties.…”