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

We study the influence of diffusion on the scaling properties of the first order structure function, $S_1$, of a two-dimensional chaotically advected passive scalar with finite lifetime, i.e., with a decaying term in its evolution equation. We obtain an analytical expression for $S_1$ where the dependence on the diffusivity, the decaying coefficient and the stirring due to the 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 structure function is analytical for length-scales shorter than $L_d$, and shows a scaling exponent that depends on the decaying term and the mixing of the flow for larger scales. Therefore, the scaling exponents for scales larger than $L_d$ are not modified with respect to those calculated in the zero diffusion limit. Moreover, $L_d$ turns out to be independent of the decaying coeficient, being its value the same as for the passive scalar with infinite lifetime. Numerical results support our theoretical findings.
Our analytical and numerical calculations rest upon the Feynmann-Kac representation of the advection-reaction-diffusion partial differential equation.