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Scale Invariance in Liquid Water Distributions in Marine Stratocumulus. Part II: Multifractal Properties and Intermittency Issues

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  • 1 Climate and Radiation Branch, NASA/Goddard Space Flight Center, Greenbelt, Maryland
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Abstract

This is the second of two papers analyzing the internal liquid water content (LWC) structure of marine stratocumulus (Sc) based on observations taken during the First ICCP (International Commission on Cloud Physics) Regional Experiment (FIRE) 1987 and Atlantic Stratocumulus Transition Experiment (ASTEX) 1992 field programs. Part I examined wavenumber spectra and the three-decade scale range (tens of meters to tens of kilometers) over which scale invariance holds; the inability of spectral analysis to distinguish between different random processes was also underscored. This indetermination is removed in this part by applying multifractal analysis techniques to the LWC fields, leading to a characterization of the role of intermittency in marine Sc.

Two multiscaling statistics are computed and associated nonincreasing hierarchies of exponents are obtained: structure functions and H(q), singular measures and D(q). The real variable q is the order of a statistical moment (e.g., q = 1.0 yields a mean); D(q) quantifies intermittency, H(q) nonstationarity. Being derived from the slopes of lines on log(statistic) versus log(scale) plots, these exponents are only defined when those lines are reasonably straight and where this happens defines the scale-invariant range. Being nonconstant, the derived H(q) and D(q) indicate multifractality rather than monofractality of LWC fields.

Two exponents can serve as first-order measures of nonstationarity and intermittency: H1 = H(1) and C1 = 1 − D(1). For the ensemble average of all FIRE and all ASTEX data, the authors find the two corresponding points in the (H1, C1) plane to be close: (0.28, 0.10) for FIRE and (0.29, 0.08) for ASTEX. This indicates that the dynamics determining the internal structure of marine Sc depend little on the local climatology. In contrast, the scatter of spatial averages for the individual flight around the ensemble average illustrates ergodicity violation. Finally, neither multiplicative cascades (with H1 = 0) nor additive Gaussian models such as fractional Brownian motions (with C1 = 0) adequately reproduce the LWC fluctuations in marine Sc.

* Additional affiliation: Science Systems and Applications, Inc., Lanham, Maryland.

Corresponding author address: Dr. Alexander Marshak, NASA/GSFC, Code 913, Greenbelt, MD 20771.

Email: marshak@climate.gsfc.nasa.gov

Abstract

This is the second of two papers analyzing the internal liquid water content (LWC) structure of marine stratocumulus (Sc) based on observations taken during the First ICCP (International Commission on Cloud Physics) Regional Experiment (FIRE) 1987 and Atlantic Stratocumulus Transition Experiment (ASTEX) 1992 field programs. Part I examined wavenumber spectra and the three-decade scale range (tens of meters to tens of kilometers) over which scale invariance holds; the inability of spectral analysis to distinguish between different random processes was also underscored. This indetermination is removed in this part by applying multifractal analysis techniques to the LWC fields, leading to a characterization of the role of intermittency in marine Sc.

Two multiscaling statistics are computed and associated nonincreasing hierarchies of exponents are obtained: structure functions and H(q), singular measures and D(q). The real variable q is the order of a statistical moment (e.g., q = 1.0 yields a mean); D(q) quantifies intermittency, H(q) nonstationarity. Being derived from the slopes of lines on log(statistic) versus log(scale) plots, these exponents are only defined when those lines are reasonably straight and where this happens defines the scale-invariant range. Being nonconstant, the derived H(q) and D(q) indicate multifractality rather than monofractality of LWC fields.

Two exponents can serve as first-order measures of nonstationarity and intermittency: H1 = H(1) and C1 = 1 − D(1). For the ensemble average of all FIRE and all ASTEX data, the authors find the two corresponding points in the (H1, C1) plane to be close: (0.28, 0.10) for FIRE and (0.29, 0.08) for ASTEX. This indicates that the dynamics determining the internal structure of marine Sc depend little on the local climatology. In contrast, the scatter of spatial averages for the individual flight around the ensemble average illustrates ergodicity violation. Finally, neither multiplicative cascades (with H1 = 0) nor additive Gaussian models such as fractional Brownian motions (with C1 = 0) adequately reproduce the LWC fluctuations in marine Sc.

* Additional affiliation: Science Systems and Applications, Inc., Lanham, Maryland.

Corresponding author address: Dr. Alexander Marshak, NASA/GSFC, Code 913, Greenbelt, MD 20771.

Email: marshak@climate.gsfc.nasa.gov

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