Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

TL;DR

This paper derives the evolution equation for the ensemble average of a scalar field influenced by rapidly fluctuating Gaussian noise, and explores its statistical properties under different flow conditions, revealing divergence of moments and intermittency.

Contribution

It introduces three methods to derive the ensemble average evolution equation and analyzes scalar transport in random flows, connecting statistics to geometric Brownian motion.

Findings

Ensemble average evolution equation derived via three methods.

In periodic flows, scalar correlations follow an effective diffusion equation.

In strain flows, scalar moments diverge, indicating increasing intermittency.

Abstract

We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an $n$-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the $N$-point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.