Diffusion-limited scalar cascades

We study advection–diffusion of a passive scalar, $T$, by an incompressible fluid in a closed vessel bounded by walls impermeable to the fluid. Variations in $T$ are produced by prescribing a steady non-uniform distribution of $T$ at the boundary. Because there is no flow through the walls, molecular diffusion, $\kappa$, is essential in ‘lifting’ $T$ off the boundary and into the interior where the velocity field acts to intensify $\bnabla T$. We prove that as $\kappa \to 0$ (with the fluid velocity fixed) this diffusive lifting is a feeble source of scalar variance. Consequently the scalar dissipation rate $\chi$ – the volume integral of $\kappa |\bnabla T|^2$ – vanishes in the limit $\kappa \to 0$. Thus, in this particular closed-flow configuration, it is not possible to maintain a constant supply of scalar variance as $\kappa\to 0$ and the fundamental premise of scaling theories for passive scalar cascades is violated. We also obtain a weaker bound on $\chi$ when the transported field is a dynamically active scalar, such as temperature. This bound applies to the Rayleigh–Bénard configuration in which $T=\pm 1$ on two parallel plates at $z=\pm h/2$. In this case we show that $\chi \leq 3.252\times (\kappa \varepsilon/\nu h^2)^{1/3}$ where $\nu$ is the viscosity and $\varepsilon$ is the mechanical energy dissipation per unit mass. Thus, provided that $\varepsilon$ and $\nu/\kappa$ are non-zero in the limit $\kappa\rightarrow 0$, $\chi$ might remain non-zero.