Transport and diffusion in the embedding map

TL;DR

This study investigates how inertial particles move and spread in a 2D incompressible flow modeled by a dissipative embedding map, revealing diverse transport behaviors and their dependence on system parameters.

Contribution

It introduces a detailed analysis of transport properties in a 4D dissipative map modeling impurity dynamics, highlighting the connection between dynamical regimes and transport characteristics.

Findings

Recurrence time distributions vary with phase space regions.

Diffusion exhibits normal, subdiffusive, and superdiffusive behaviors.

Phase diagrams link transport regimes with system parameters.

Abstract

We study the transport properties of passive inertial particles in a $2-d$ incompressible flows. Here the particle dynamics is represented by the $4-d$ dissipative embedding map of $2-d$ area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter $\alpha$, and the dissipation parameter $\gamma$. We obtain the statistical characterisers of transport for this system in these dynamical regimes. These are, the recurrence time statistics, the diffusion constant, and the distribution of jump lengths. The recurrence time distribution shows a power law tail in the dynamical regimes where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion constant shows behaviour of three types - normal, subdiffusive and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behaviour, as well as the behaviour of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes, and in the behaviour of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterisers of transport. We discuss the implications of our results for realistic systems.