Characterizing scale dependence of effective diffusion driven by fluid flows

We study the scale dependence of effective diffusion of fluid tracers, specifically, its dependence on the Péclet number, a dimensionless parameter of the ratio between advection and molecular diffusion. Here, we address the case that length and time scales on which the effective diffusion can be described are not separated from those of advection and molecular diffusion. For this, we propose an alternate method for characterizing the effective diffusivity without relying on the scale separation. For a given spatial domain inside which the effective diffusion can emerge, a time constant related to the diffusion is identified by considering the spatiotemporal evolution of a test advection-diffusion equation, where its initial condition is set at a pulse function. Then, the value of effective diffusivity is identified by minimizing the $L_{\infty }$ distance between solutions of the above test equation and the diffusion one with mean drift. With this method, for time-independent gyre and time-periodic shear flows, we numerically show the scale dependence of the effective diffusivity and its discrepancy from the classical limits that were derived on the assumption of the scale separation. The kinematic origins of the discrepancy are revealed as the development of the molecular diffusion across flow cells of the gyre and as the suppression of the drift motion due to a temporal oscillation in the shear.

Figure 1

Schematic diagram of the proposed characterization of effective diffusion.

Figure 2

Illustration of streamlines of the vector field (15).

Figure 3

Numerically computed ${\mathrm{\Lambda }}_{\mathrm{\Omega },\sigma }$ for each $L_{\mathrm{\Omega }}$. Here the variance parameter $\sigma$ is fixed at 0.04. The blue cross marks denote the calculated values, and the dashed line denotes ${\mathrm{\Lambda }}_{\mathrm{\Omega },\sigma }=0.05$.

Figure 4

Simulation results of effective diffusion for the gyre flow (15): (a) $L_{\mathrm{\Omega }}$ dependences of $E_{\mathrm{\Omega },\alpha }$ (red, thin) and ${\tau }_{\mathrm{\Omega },\alpha }$ (blue, thick); (b-1) $D$ dependences of ${\overline{D}}_{\mathrm{\Omega },\alpha }$ (red, thin) and ${\tau }_{\mathrm{\Omega },\alpha }$ (blue, thick); and (b-2) $D$ dependency of ${\sigma }_{\mathrm{\Omega },\alpha }$.

Figure 5

Simulation results of effective diffusion for the shear flow (21): (a) ${\tau }_{\mathrm{\Omega },\alpha }$ (solid) (b) $E_{\mathrm{\Omega },\alpha }$ (solid) and ${sup}_{\mathit{x}\in \mathrm{\Omega }}\rho (\mathit{x},{\tau }_{\mathrm{\Omega },\alpha })$ (dashed); (c) ${\overline{D}}_{\mathrm{\Omega },\alpha }$ (blue, solid), ${\overline{D}}_{xx}$ (black, solid) and ${\overline{D}}_{\beta }$ (dashed) for each time-period ${\tau }_0$.