Dispersion and reaction in random flows: Single realization versus ensemble average

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 realizations of the flow, with a Gaussian core characterized by an effective diffusivity, and large-deviation tails characterized 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-realization description with that which applies to the average of the concentration over an ensemble of flow realizations. We show that the single-realization 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 realizations. 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 realization and can be deduced from the single-realization large-deviation rate function. We test our predictions against numerical simulations of the alternating sine flow.

Figure 1

Evolution of an initial diamond-shaped scalar patch (upper-right panel) through the successive steps (6)–(6) of the alternating flow with $a=\pi$ and $\kappa ={10}^{-2}$. The computation uses a ${1024}^2$ discretization of the domain ${[-4\pi ,4\pi ]}^2$.

Figure 2

Same as Fig. 1 but with an average over 1000 realizations of the flow.

Figure 3

Magnitude of the effective velocity ${\mathit{v}}_n$ and of the (relative) difference between the effective diffusivity ${\mathrm{K}}_n$ and the ensemble average $\overline{\mathrm{K}}=\left(\kappa +a^2/4\right){\mathbb{I}}_2$ as functions of $n$ for the alternating sine flow for $a=\pi$ and $\kappa ={10}^{-2}$. Here ${\mathit{v}}_n$ and ${\mathrm{K}}_n$ are defined in Eqs. (20) and (21a) and estimated by numerical sampling of the random walk (22) using ${10}^4$ realizations of the Brownian motion.

Figure 4

Comparison of the single-realization and ensemble-average large-deviation rate functions $g$ and $\overline{g}$ for the alternating sine model with $a=\pi$ and $\kappa ={10}^{-2}$. Left panels: contours of $g$ (top) and $\overline{g}$ (bottom) (using the same contour levels for both panels). Right panels: $g$ and $\overline{g}$ along the $x$ axis (top) and along a diagonal axis (bottom). The diffusive approximation is also shown (dashed).

Figure 5

Evolution of the concentration for the reacting alternating sine model in Sec. 5b with $a=\pi$, $\kappa =0.1$ and $\gamma =2$. The prediction of the front location according to (54), with the single-realization $g$ estimated as in Sec. 4b is shown as the solid blue line; its counterpart using the ensemble-averaged $\overline{g}$ is shown as the dashed black line. The computation uses a ${4096}^2$ discretization of a ${[-64\pi ,64\pi ]}^2$ domain.