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Abstract. The deterministic motions of clouds and turbulence, despite their chaotic nature, have nonetheless been shown to follow simple statistical power-law scalings: a fractal dimension D relates individual cloud perimeters p to a measurement resolution, and turbulent fluctuations scale with the air parcel separation distance through the Hurst exponent, ℋ. However, it remains uncertain whether atmospheric turbulence is best characterized by a split isotropy that is three-dimensional (3D) with H=1/3 at small scales and two-dimensional (2D) with ℋ=1 at large scales or by a wide-range anisotropic scaling with an intermediate value of ℋ. Here, we introduce an “ensemble fractal dimension” De – analogous to D – that relates the total cloud perimeter per domain area 𝒫 as seen from space to the measurement resolution, and we show theoretically how turbulent dimensionality and cloud edge geometry can be linked through H=De-1. Observationally and numerically, we find the scaling De∼5/3 or H∼2/3, spanning 5 orders of magnitude of scale. Remarkably, the same scaling relationship links two “limiting case” estimates of 𝒫 evaluated at resolutions corresponding to the planetary scale and the Kolmogorov microscale, which span 10 orders of magnitude. Our results are nearly consistent with a previously proposed “23/9D” anisotropic turbulent scaling and suggest that the geometric characteristics of clouds and turbulence in the atmosphere can be easily tied to well-known planetary physical parameters. 

A significant uncertainty in assessments of the role of clouds in climate is the characterization of the full distribution of their sizes. Order-of-magnitude disagreements exist among observations of key distribution parameters, particularly power law exponents and the range over which they apply. A study by Savre and Craig (2023) suggested that the discrepancies are due in large part to inaccurate fitting methods: they recommended the use of a maximum likelihood estimation technique rather than a linear regression to a logarithmically transformed histogram of cloud sizes. Here, we counter that linear regression is both simpler and equally accurate, provided the simple precaution is followed that bins containing fewer than ∼ 24 counts are omitted from the regression. A much more significant and underappreciated source of error is how to treat clouds that are truncated by the edges of unavoidably finite measurement domains. We offer a simple computational procedure to identify and correct for domain size effects, with potential application to any geometric size distribution of objects, whether physical, ecological, social or mathematical. 

Abstract. Cloud area distributions are a defining feature of Earth's radiative exchanges with outer space. Cloud perimeter distributions n(p) are also interesting because the shared interface between clouds and clear sky determines exchanges of buoyant energy and air. Here, we test using detailed model output and a wide range of satellite datasets a first-principles prediction that perimeter distributions follow a scale-invariant power law n(p) ∝ p-(1+β), where the exponent β = 1 is evaluated for perimeters within moist isentropic atmospheric layers. In model analyses, the value of β is closely reproduced. In satellite data, β is remarkably robust to latitude, season, and land–ocean contrasts, which suggests that, at least statistically speaking, cloud perimeter distributions are determined more by atmospheric stability than Coriolis forces, surface temperature, or contrasts in aerosol loading between continental and marine environments. However, the satellite-measured value of β is found to be 1.26 ± 0.06 rather than β = 1. The reason for the discrepancy is unclear, but comparison with a model reproduction of the satellite perspective suggests that it may owe to cloud overlap. Satellite observations also show that scale invariance governs cloud areas for a range at least as large as ∼ 3 to ∼ 3 × 105 km2, and notably with a corresponding power law exponent close to unity. Many prior studies observed a much smaller range for power law behavior, and we argue this difference is due to inappropriate treatments of the statistics of clouds that are truncated by the edge of the measurement domain.