Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck process

Lognormality was found experimentally for coarse-grained squared turbulence velocity and velocity increment when the coarsening scale is comparable to the correlation scale of the velocity (Mouri et al. Phys. Fluids 21, 065107, 2009). We investigate this large-scale lognormality by using a simple stochastic process with correlation, the Ornstein-Uhlenbeck (OU) process. It is shown that the OU process has a similar large-scale lognormality, which is studied numerically and analytically.

💡 Research Summary

The paper revisits an intriguing experimental observation made by Mouri et al. (2009): when the coarse‑graining scale Δt of turbulent velocity u(t) or its increment Δu(t,τ) is comparable to the velocity correlation time τ_c, the squared coarse‑grained quantities ⟨u²⟩_Δt and ⟨Δu²⟩_Δt follow a log‑normal distribution. To determine whether this “large‑scale lognormality” is a specific feature of turbulent dynamics or a more generic statistical consequence of correlated fluctuations, the authors employ the Ornstein‑Uhlenbeck (OU) process, the simplest continuous‑time Gaussian process with an exponential autocorrelation.

The OU process is defined by du(t)=−(1/τ_c)u(t)dt+σ dW(t), where τ_c is the correlation time, σ the noise amplitude, and W(t) a Wiener process. Its autocorrelation decays as e^{−|Δt|/τ_c}, mimicking the large‑scale memory of turbulent flows. Two coarse‑grained observables are introduced: (i) the time‑averaged squared velocity X(Δt)= (1/Δt)∫_0^{Δt} u²(s) ds, analogous to an energy density, and (ii) the time‑averaged squared increment Y(Δt)= (1/Δt)∫_0^{Δt}