# Dispersion and reaction in random flows: single realisation vs ensemble average

## TL;DR

This paper investigates the dispersion of passive scalars in random, spatially periodic flows, contrasting single realisation and ensemble-averaged descriptions, and explores implications for reactive scalars modeled by FKPP equations.

## Contribution

It demonstrates that while effective diffusivities are the same for single realisations and ensemble averages, their large-deviation rate functions differ, affecting tail predictions and reactive front behavior.

## Key findings

- Single-realisation and ensemble-average diffusivities are identical.
- Rate functions differ, with ensemble overestimating tail concentrations.
- Reactive scalar fronts are well-described by single-realisation rate functions.

## Abstract

We examine the dispersion of a passive scalar released in an incompressible fluid flow in an unbounded domain. The flow is assumed to be spatially periodic, with zero spatial average, and random in time, in the manner of the random-phase alternating sine flow which we use as an exemplar. In the long-time limit, the scalar concentration takes the same, predictable form for almost all realisations of the flow, with a Gaussian core characterised by an effective diffusivity, and large-deviation tails characterised by a rate function (which can be evaluated by computing the largest Lyapunov exponent of a family of random-in-time partial differential equations). We contrast this single-realisation description with that which applies to the average of the concentration over an ensemble of flow realisations. We show that the single-realisation and ensemble-average effective diffusivities are identical but that the corresponding rate functions are not, and that the ensemble-averaged description overestimates the concentration in the tails compared with that obtained for single-flow realisations. This difference has a marked impact for scalars reacting according to the Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) model. Such scalars form an expanding front whose shape is approximately independent of the flow realisation and can be deduced from the single-realisation large-deviation rate function. We test our predictions against numerical simulations of the alternating sine flow.