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We study the influence of diffusion on the scaling properties of the firstorder structure function, S_1, of a two-dimensional chaotically advectedpassive scalar with finite lifetime, i.e., with a decaying term in itsevolution equation. We obtain an analytical expression for S_1 where thedependence on the diffusivity, the decaying coefficient and the stirring due tothe chaotic flow is explicitly stated.  We show that the presence of diffusion introduces a crossover length-scale,the diffusion scale (L_d), such that the scaling behaviour for the structurefunction is analytical for length-scales shorter than L_d, and shows a scalingexponent that depends on the decaying term and the mixing of the flow forlarger scales. Therefore, the scaling exponents for scales larger than L_d arenot modified with respect to those calculated in the zero diffusion limit.Moreover, L_d turns out to be independent of the decaying coeficient, being itsvalue the same as for the passive scalar with infinite lifetime.  Numerical results support our theoretical findings.Comment: 7 pages, 3 figure

The role of diffusion in the chaotic advection of a passive scalar with finite lifetime