Averaged dynamics of time-periodic advection diffusion equations in the limit of small diffusivity

Abstract

a b s t r a c tWe study the effect of advection and small diffusion on passive tracers. The advecting velocity field is assumed to have mean zero and to possess time-periodic stream lines. Using a canonical transform to action-angle variables followed by a Lie transform, we derive an averaged equation describing the effective motion of the tracers. An estimate for the time validity of the first-order approximation is established. For particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order. Numerical simulations indicate that the validity of the above first-order estimate extends to the second order.

Averaging and spectral properties for the 2D advection–diffusion equation in the semi-classical limit for vanishing diffusivity

“…This paper is a continuation of authors' work [26] for the mean-free case f = T 0 f (t) dt = 0, and in this section we review some of the results of that paper. The functon f (t) = f 0 (t) + f 1 is assumed to be time-periodic with period T > 0.…”

“…The main idea of the authors' paper [26] was to apply a near-identity Lie transform that eliminates the explicit time dependence of the coefficients (see [26] for details). However, in the case when f 1 = 0, we write the averaged equation in an ad-hoc fashion,…”

“…In this section, we compare numerical solutions of the original equation (25) and the averaged equation to the first order (26). We consider the evolution of the tracer field on a unit disk (0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π) with zero Dirichlet boundary conditions v(r = 1, θ) = 0.…”

“…was treated in [26]. Lie-transformation based averaging techniques were used to show that the solutions converge to solutions of a autonomous self-adjoint diffusive equation.…”

A Lagrangian perspective on nonautonomous advection-diffusion processes in the low-diffusivity limit

“…The time-periodic case has been investigated in the low-diffusivity limit by Krol [24], cf. also [38,44]. Time-periodic advection-diffusion prob-lems have also been studied by Liu & Haller [30] from the Eulerian perspective.…”