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We examine the decay of passive scalars with small, but non zero, diffusivityin bounded 2D domains. The velocity fields responsible for advection are smooth(i.e., they have bounded gradients) and of a single large scale. Moreover, thescale of the velocity field is taken to be similar to the size of the entiredomain. The importance of the initial scale of variation of the scalar fieldwith respect to that of the velocity field is strongly emphasized. If thesescales are comparable and the velocity field is time periodic, we see theformation of a periodic scalar eigenmode. The eigenmode is numerically realizedby means of a deterministic 2D map on a lattice. Analytical justification forthe eigenmode is available from theorems in the dynamo literature. Weakeningthe notion of an eigenmode to mean statistical stationarity, we providenumerical evidence that the eigenmode solution also holds for aperiodic flows(represented by random maps). Turning to the evolution of an initially smallscale scalar field, we demonstrate the transition from an evolving (i.e., {\itnon} self similar) pdf to a stationary (self similar) pdf as the scale ofvariation of the scalar field progresses from being small to being comparableto 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 examinedand the coupling between diffusion and the distribution of the Finite TimeLyapunov Exponents is shown to be responsible for the pdf evolution.Comment: 21 pages (2 col. format), 11 figures. Accepted, to appear in PR

physical review. e, statistical physics, plasmas, fluids, and related interdisciplinary topics

Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: From non-self-similar probability distribution functions to self-similar eigenmodes