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Correlation Function of a Random Scalar Field Evolving with a Rapidly Fluctuating Gaussian Process

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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 for the ensemble average, we study the passive scalar transport problem with two different types of flows, random flows possessing underlying periodic spatio-temporal structures, and a random strain flow. For the former, 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. Moreover, we identify an effective equation for the passive scalar based appealing to the Hausdorff moment problem, which shows that the periodic random flow yields a deterministic enhanced effective diffusion in the coarse-grained limit. For the strain flow, we explicitly 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, allowing for an explicit representation of the probability distribution function (PDF) for the random passive scalar. Interestingly, all normalized moments (e.g., skewness, kurtosis) of this random scalar field diverge at long times, meaning that the scalar becomes more and more intermittent during its decay, however here this intermittency is arising through an emerging singularity in the core of the PDF at long times. Further, using the moment closure equations for the random strain flow, we compute integral representations for the N point correlation function and compare those predictions with Monte-Carlo simulations.

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Acknowledgements

We acknowledge funding received from the following National Science Foundation Grant Nos.:DMS-1910824; and Office of Naval Research Grant No: ONR N00014-18-1-2490. Partial support for Lingyun Ding is gratefully acknowledged from the National Science Foundation, award NSF-DMS-1929298 from the Statistical and Applied Mathematical Sciences Institute.

We thank the anonymous referee, whose comments improved the quality of the manuscript.

The calculation of the complete Picard iteration expansion in Sect. 3.3 is based on the unpublished short manuscript titled “White noise averaging and the time ordered calculus”, written by Jared C. Bronski and Richard M. McLaughlin in the 1990s. We would like to acknowledge Jared C. Bronski for his contributions to our manuscript by providing feedback on it. We would also like to express our appreciation to Andrew J. Majda for his comments and encouragement.

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Lingyun Ding & Richard M. McLaughlin

  2. Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095, USA

    Lingyun Ding

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  1. Lingyun Ding
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Correspondence to Richard M. McLaughlin.

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Ding, L., McLaughlin, R.M. Correlation Function of a Random Scalar Field Evolving with a Rapidly Fluctuating Gaussian Process. J Stat Phys 190, 201 (2023). https://doi.org/10.1007/s10955-023-03191-7

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