  An elegant model for passive scalar mixing was given by Kraichnan assuming
the velocity to be delta-correlated in time. We generalize this model to
include the effects of a finite correlation time, $\tau$, using renewing flows.
The resulting equation for the 3-D passive scalar spectrum $\hat{M}(k,t)$ or
its correlation function $M(r,t)$, gives the Kraichnan equation when $\tau \to
0$, and extends it to the next order in $\tau$. It involves third and fourth
order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For
small-$\tau$, it can be recast using the Landau-Lifshitz approach, to one with
at most second derivatives of $\hat{M}$. We present both a scaling solution to
this equation neglecting diffusion and a more exact solution including
diffusive effects. We show that the steady state 1-D passive scalar spectrum,
preserves the Batchelor form, $E_\theta(k) \propto k^{-1}$, in the
viscous-convective limit, independent of $\tau$. This result can also be
obtained in a general manner using Lagrangian methods. Interestingly, in the
absence of sources, when passive scalar fluctuations decay, we show that the
spectrum in the Batchelor regime and at late times, is of the form $E_\theta(k)
\propto k^{1/2}$ and also independent of $\tau$. More generally, finite $\tau$
does not qualitatively change the shape of the spectrum during decay, although
the decay rate is reduced. From high resolution ($1024^3$) direct numerical
simulations of passive scalar mixing and decay, We find reasonable agreement
with predictions of the Batchelor spectrum, during steady state. The scalar
spectrum during decay agrees qualitatively with analytic predictions when power
is dominantly in wavenumbers corresponding to the Batchelor regime, but is
however shallower when box scale fluctuations dominate during decay (abridged).
