  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 $\kappa$.
