  This study revisits the problem of advective transfer and spectra of a
diffusive scalar field in large-scale incompressible flows in the presence of a
(large-scale) source. By ``large-scale'' it is meant that the spectral support
of the flows is confined to the wave-number region $k<k_d$, where $k_d$ is
relatively small compared with the diffusion wave number $k_\kappa$. Such flows
mediate couplings between neighbouring wave numbers within $k_d$ of each other
only. It is found that the spectral rate of transport (flux) of scalar variance
across a high wave number $k>k_d$ is bounded from above by $Uk_dk\Theta(k,t)$,
where $U$ denotes the maximum fluid velocity and $\Theta(k,t)$ is the spectrum
of the scalar variance, defined as its average over the shell $(k-k_d,k+k_d)$.
For a given flux, say $\vartheta>0$, across $k>k_d$, this bound requires
$$\Theta(k,t)\ge \frac{\vartheta}{Uk_d}k^{-1}.$$ This is consistent with recent
numerical studies and with Batchelor's theory that predicts a $k^{-1}$ spectrum
(with a slightly different proportionality constant) for the viscous-convective
range, which could be identified with $(k_d,k_\kappa)$. Thus, Batchelor's
formula for the variance spectrum is recovered by the present method in the
form of a critical lower bound. The present result applies to a broad range of
large-scale advection problems in space dimensions $\ge2$, including some
filter models of turbulence, for which the turbulent velocity field is advected
by a smoothed version of itself. For this case, $\Theta(k,t)$ and $\vartheta$
are the kinetic energy spectrum and flux, respectively.
