Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients

Abstract

This study is concerned with the decay behavior of a passive scalar θ in three-dimensional flows having bounded velocity gradients. Given an initially smooth scalar distribution, the decay rate d⟨θ2⟩∕dt of the scalar variance ⟨θ2⟩ is found to be bounded in terms of controlled physical parameters. Furthermore, in the zero diffusivity limit, κ→0, this rate vanishes as κα0 if there exists an α0∊(0,1] independent of κ such that ⟨∣(−Δ)α∕2θ∣2⟩<∞ for α⩽α0. This condition is satisfied if in the limit κ→0, the variance spectrum Θ(k) remains steeper than k−1 for large wave numbers k. When no such positive α0 exists, the scalar field may be said to become virtually singular. A plausible scenario consistent with Batchelor’s theory is that Θ(k) becomes increasingly shallower for smaller κ, approaching the Batchelor scaling k−1 in the limit κ→0. For this classical case, the decay rate also vanishes, albeit more slowly—like (lnPr)−1, where Pr is the Prandtl or Schmidt number. Hence, diffusion anomaly is ruled out for a broad range of scalar distribution, including power-law spectra no shallower than k−1. The implication is that in order to have a κ-independent and nonvanishing decay rate, the variance at small scales must necessarily be greater than that allowed by the Batchelor spectrum. These results are discussed in the light of existing literature on the asymptotic exponential decay ⟨θ2⟩∼e−γt, where γ>0 is independent of κ.