Dissipation enhancing properties for a class of Hamiltonian flows with closed streamlines

TL;DR

This paper investigates how certain Hamiltonian flows with closed streamlines can enhance dissipation of passive scalars in high Péclet number regimes, showing exponential convergence to streamline averages and identifying key time-scales.

Contribution

It introduces a novel analysis of dissipation enhancement in Hamiltonian flows with closed streamlines, including a model problem and asymptotic expansion techniques.

Findings

Exponential convergence to streamline average on subdiffusive time-scale.

Time-scale T_ν depends on the behavior of the period function near elliptic points.

Analysis extends to a broader class of Hamiltonian flows.

Abstract

We study the evolution of a passive scalar subject to molecular diffusion and advected by an incompressible velocity field on a 2D bounded domain. The velocity field is $u = \nabla^\perp H$, where H is an autonomous Hamiltonian whose level sets are Jordan curves foliating the domain. We focus on the high P\'eclet number regime ($Pe := \nu^{-1} \gg 1$), where two distinct processes unfold on well separated time-scales: streamline averaging and standard diffusion. For a specific class of Hamiltonians with one non-degenerate elliptic point (including perturbed radial flows), we prove exponential convergence of the solution to its streamline average on a subdiffusive time-scale $T_\nu \ll \nu^{-1}$ ,up to a small correction related to the shape of the streamlines. The time-scale $T_\nu$ is determined by the behavior of the period function around the elliptic point. To establish this result, we introduce a model problem arising naturally from the difference between the solution and its streamline average. We use pseudospectral estimates to infer decay in the model problem, and, in fact, this analysis extends to a broader class of Hamiltonian flows. Finally, we perform an asymptotic expansion of the full solution, revealing that the leading terms consist of the streamline average and the solution of the model problem.