Advection, diffusion \& dispersion: effective diffusion for transient mixing vs. stirring with steady sources \& sinks

The effective diffusion coefficient $\kappa_{eff}$ of a flow is often defined in terms of passive tracer particle dispersion. For some high P\'eclet number flows $\kappa_{eff}$ may be as large as $\kappa_{molec} \times Pe^{2}$ where $\kappa_{molec}$ is the molecular diffusion coefficient and the P\'eclet number $Pe = U\ell/\kappa_{molec}$ is defined in terms of characteristic velocity ($U$) and length ($\ell$) scales in the flow. On the other hand for stirring in the presence of steady sources and sinks an equivalent diffusion coefficient $\kappa_{eq}$ may be defined in terms of (statistical steady state) passive scalar concentration variance suppression. A theorem states that $\kappa_{eq} \le \kappa_{molec} \times Pe \times (L/\ell)$ as $Pe \rightarrow \infty$ where $L$ is a characteristic length scale of the sources-sink distribution. We discuss the origin and resolution of this discrepancy: effective diffusion coefficients proportional to $Pe^{2}$ arise in the large time asymptotic limit of particle dispersion while equivalent diffusion coefficients defined by concentration variance suppression for scalars sustained by steady sources are dominated by short-time transport characteristics of the flow. The theories may be reconciled by considering a time dependent effective diffusion coefficient that includes the transient---and not just time asymptotic---tracer particle dispersion.