The problem of transport of a passive scalar with finite lifetime has seen in the last years a renewed interest [19,34,35,36] because it is the prototype of a wide class of physical problems, ranging from biological processes like plankton dynamics, to chemical reactions in chaotic flows. It grasps the basic relevant features of the general problem of transport by a fluid environment of substance with an intrinsic dynamics, with the appealing feature of its simplicity, which in some specific situation allows to obtain analytical solutions.
In this section I will introduce the passive scalar problem following the works of M. Chertkov, E. Hernandez-Garcia, Z. Neufeld et al., showing how the interaction between chaotic advection and linear damping can originates intermittency and smooth-filamental transition for the scalar field. These results will be extended in Section (2.5) to the active case.