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, κ , is essential in ‘lifting’ T off the boundary and into the interior where the velocity field acts to intensify ∇T . We prove that as κ → 0 (with the fluid velocity fixed) this diffusive lifting is a feeble source of scalar variance. Consequently the scalar dissipation rate χ – the volume integral of κ |∇T | – vanishes in the limit κ → 0. Thus, in this particular closed-flow configuration, it is not possible to maintain a constant supply of scalar variance as κ → 0 and the fundamental premise of scaling theories for passive scalar cascades is violated. We also obtain a weaker bound on χ when the transported field is a dynamically active scalar, such as temperature. This bound applies to the Rayleigh–Bénard configuration in which T = ±1 on two parallel plates at z = ±h/2. In this case we show that χ 3.252× (κε/νh) where ν is the viscosity and ε is the mechanical energy dissipation per unit mass. Thus, provided that ε and ν/κ are non-zero in the limit κ → 0, χ might remain non-zero.