A global analysis of the fractal properties of clouds revealing anisotropy of turbulence across scales

Abstract. The deterministic motions of clouds and turbulence, despite their chaotic nature, nonetheless follow simple statistical power-law scalings: a fractal dimension D relates individual cloud perimeters p to measurement resolution, and turbulent fluctuations scale with separation distance through the Hurst exponent ℌ. It remains uncertain whether atmospheric turbulence is best characterized by split isotropy that is three-dimensional with ℌ = 1/3 at small scales and two-dimensional with ℌ = 1 at large scales, or by wide-range anisotropic scaling with an intermediate value of ℌ. Here, we introduce an “ensemble fractal dimension” D_e – analogous to D – that relates the total cloud perimeter per domain area 𝒫 as seen from space to measurement resolution, and show theoretically how turbulent dimensionality and cloud edge geometry are linked through ℌ =D_e − 1. Observationally, by progressively coarsening the resolution of cloud mask arrays from various global satellite platforms and a numerical simulation of a tropical domain we find the scaling D_e ~ 5/3, or ℌ ~ 2/3, a value nearly consistent with a previously proposed “23/9D” anisotropic turbulent scaling. Remarkably, the same scaling links two “limiting case” estimates of 𝒫 evaluated at the planetary scale and the Kolmogorov microscale, as separated by 11 orders of magnitude, suggesting that cloud and turbulent behaviors are constrained by basic planetary parameters.