Proposing Stochastic Turbulent Diffusivity from Diffusing Diffusivity with Fractional Ornstein–Uhlenbeck Process and Fractional Brownian Motion

study concentrates on proposing a stochastic turbulent diffusivity for fluid particles derived from diffusing diffusivity, a unique stochastic process defined as the square of the  Ornstein–Uhlenbeck (OU) process. The diffusing diffusivity is assigned to be the square of time-dependent velocity fluctuations in a turbulent flow and is modeled using the fractional OU process with fractional Brownian motion (FBM). Three crucial properties from various fields are tied into this model: (1) mean-reverting behavior derived from the OU process, (2) long-term memory attributed to FBM, and (3) stochastic turbulent diffusivity for fluid particles. The first four ensemble statistics—mean, variance, skewness, and kurtosis—are provided for the diffusing diffusivity to identify non-Gaussian behavior, measure the variability, and investigate the deviation from classical deterministic models.

The highlight of this study is the proposal of stochastic turbulent diffusivity for fluid particles in a turbulent flow. It is defined by multiplying the diffusing diffusivity with the Lagrangian timescale, thereby linking small-scale temporal fluctuations captured by diffusing diffusivity to the macroscopic mixing effects of turbulent diffusivity. This approach ensures dimensional consistency with deterministic turbulent diffusivity while preserving its stochastic characteristics. Additionally, higher-order structure functions and wavelet-based intermittency measures are provided to examine intermittency in turbulent flows. The former provides evidence of intermittency, and the latter captures energy bursts across scales associated with turbulent diffusing diffusivity. On the other hand, the validation is conducted against the Ergodicity Breaking parameter from theoretical stochastic analysis and turbulent velocity fluctuation data from experiments, confirming the applicability of bridging diffusing diffusivity to stochastic turbulent diffusivity.