On the probability law of turbulent kinetic energy in the atmospheric surface layer

The probability density function $p(k)$ of the turbulent kinetic energy $k$ is investigated for diabatic atmospheric surface layer (ASL) flows. When the velocity components are near-Gaussian and their squared quantities are nearly independent, the resulting $p(k)$ is shown to be gamma-distributed with exponents that vary from 0.8 to 1.8. A non-linear Langevin equation that preserves a gamma-distributed $p(k)$ but allows linear relaxation of $k$ to its mean state is proposed and tested using multiple ASL data sets. The three parameters needed to describe the drift and non-linear diffusion terms can be determined from the ground shear stress and the mean velocity at height $z$ from the ground. Using these model parameters, the Langevin equation reproduces the measured $p(k)$ with minimal Kullback-Leibler divergence. Analysis of the partial autocorrelation function is conducted to investigate the validity of the exponential decay assumption in the autocorrelation function, and numerous runs exhibit non-exponential decay. Such adjustments need not revise the linearity of the drift term and can be accommodated by assuming a relaxation time scale that is not constant or time dependent. A non-constant relaxation time is one possibility to bridge the Langevin model with super-statistics.