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

TL;DR

This paper investigates the decay behavior of passive scalars in three-dimensional flows with bounded velocity gradients, showing that diffusion anomaly is unlikely under broad spectral conditions and clarifying the decay rates in different regimes.

Contribution

It establishes bounds on scalar decay rates and conditions under which diffusion anomaly cannot occur in three-dimensional flows with bounded velocity gradients.

Findings

Decay rate bounded by physical parameters.

Diffusion anomaly ruled out for spectra steeper than $k^{-1}$.

Decay rate vanishes more slowly in the Batchelor regime.

Abstract

This study is concerned with the decay behaviour of a passive scalar $\theta$ in three-dimensional flows having bounded velocity gradients. Given an initially smooth scalar distribution, the decay rate $d<\theta^2>/dt$ of the scalar variance $<\theta^2>$ is found to be bounded in terms of controlled physical parameters. Furthermore, in the zero diffusivity limit, $\kappa\to0$, this rate vanishes as $\kappa^{\alpha_0}$ if there exists an $\alpha_0\in(0,1]$ independent of $\kappa$ such that $<|(-\Delta)^{\alpha/2}\theta|^2><\infty$ for $\alpha\le\alpha_0$. This condition is satisfied if in the limit $\kappa\to0$, the variance spectrum $\Theta(k)$ remains steeper than $k^{-1}$ for large wave numbers $k$. When no such positive $\alpha_0$ exists, the scalar field may be said to become virtually singular. A plausible scenario consistent with Batchelor's theory is that $\Theta(k)$ becomes increasingly shallower for smaller $\kappa$, approaching the Batchelor scaling $k^{-1}$ in the limit $\kappa\to0$. For this classical case, the decay rate also vanishes, albeit more slowly -- like $(\ln P_r)^{-1}$, where $P_r$ 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 $\kappa$-independent and non-vanishing 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 $<\theta^2>\sim e^{-\gamma t}$, where $\gamma>0$ is independent of $\kappa$.