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

This study is concerned with the decay behavior 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 -> 0, this rate vanishes as kappa(alpha)(0) if there exists an alpha(0)is an element of(0,1] independent of kappa such that <parallel to(-Delta)(alpha/2)theta parallel to(2)><infinity for alpha <=alpha(0). This condition is satisfied if in the limit kappa -> 0, 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 -> 0. 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 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 <theta(2)>similar to e(-gamma t), where gamma>0 is independent of kappa. (C) 2008 American Institute of Physics.