Scale-by-scale evidence for an inverse energy cascade in moist atmospheric turbulence

We present a scale-by-scale analysis of atmospheric turbulence based on large-eddy simulation of the BOMEX case. The simulation employs the anelastic approximation and moist thermodynamics to represent stratification and phase changes. The study focuses on the scale-by-scale transport of kinetic energy, diagnosed using second-order velocity structure functions.

Following the approach of Valente and Vassilicos (Phys. Fluids. vol. 27, 2015), we compute all terms of the Kármán–Howarth–Monin–Hill equation and analyze their balance in a six-dimensional space of scales and positions. The budgets are averaged over time and horizontally homogeneous directions, allowing their variation with scale and height to be examined.

The results reveal an inverse average energy cascade within the lower cloud layer where there is moderate liquid water content (800 – 1300m). This inverse cascade coincides with the emergence of  buoyant forcing at small scales due to phase changes and represents the main finding of the study. At higher cloud levels, the inverse-cascade signature weakens and eventually disappears.

The results show good qualitative agreement with recent airborne measurements (Nowak et. al., QJRMS vol. 151, 2025) and highlight the role of moist processes in shaping energy transfer in atmospheric turbulence.