  We examine the decay of passive scalars with small, but non zero, diffusivity
in bounded 2D domains. The velocity fields responsible for advection are smooth
(i.e., they have bounded gradients) and of a single large scale. Moreover, the
scale of the velocity field is taken to be similar to the size of the entire
domain. The importance of the initial scale of variation of the scalar field
with respect to that of the velocity field is strongly emphasized. If these
scales are comparable and the velocity field is time periodic, we see the
formation of a periodic scalar eigenmode. The eigenmode is numerically realized
by means of a deterministic 2D map on a lattice. Analytical justification for
the eigenmode is available from theorems in the dynamo literature. Weakening
the notion of an eigenmode to mean statistical stationarity, we provide
numerical evidence that the eigenmode solution also holds for aperiodic flows
(represented by random maps). Turning to the evolution of an initially small
scale scalar field, we demonstrate the transition from an evolving (i.e., {\it
non} self similar) pdf to a stationary (self similar) pdf as the scale of
variation of the scalar field progresses from being small to being comparable
to that of the velocity field (and of the domain). Furthermore, the {\it non}
self similar regime itself consists of two stages. Both the stages are examined
and the coupling between diffusion and the distribution of the Finite Time
Lyapunov Exponents is shown to be responsible for the pdf evolution.
