Mechanism of transition from subdiffusive transport to normal diffusion for passive tracers in a plane periodical flow induced by Gaussian noise

Abstract

The known analytical solution describing a two-dimensional viscous flow with vortices under a driving force is generalized. It is shown that a periodic pattern of asymmetric vortices arises when the force amplitude exceeds a critical value. The transport of an ensemble of passive particles through the resulting structure has been studied. The envelope of each vortex contains a stagnation point, and the tracer motion along it without noise is infinitely long. It is shown that the macroscopic transport of an ensemble of tracers is extremely slow in the noise-free case. The latter leads to the mixing of the tracer ensemble. We study the peculiarities of such transport using a special flow model. Such a model has the form of a series of recurrence mappings. The recurrence mappings are constructed for the ensemble transport through a unit cell using the numerical solution of the Langevin equations in the presence of random particle displacements caused by molecular diffusion. The presence of diffusion allows particles to penetrate into the vortex, which leads to an additional slowing down of transport. The effect of noise on the variance of the tracer ensemble is investigated. It is shown that the growth of the standard deviation with time can be described by a power law. On an intermediate time scale, which grows exponentially long, as the noise strength tends to zero, a subdiffusive behavior is observed. The dependence of the subdiffusion exponent on the molecular diffusivity is obtained.