Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: From non-self-similar probability distribution functions to self-similar eigenmodes

Abstract

We examine the decay of passive scalars with small, but nonzero, diffusivity in bounded two-dimensional (2D) domains. The velocity fields responsible for advection are smooth (i.e., they have bounded gradients) and of a single large scale. Moreover, the scale of the velocity field is taken to be similar to the size of the entire domain. The importance of the initial scale of variation of the scalar field with respect to that of the velocity field is strongly emphasized. If these scales are comparable and the velocity field is time periodic, we see the formation of a periodic scalar eigenmode. The eigenmode is numerically realized by means of a deterministic 2D map on a lattice. Analytical justification for the eigenmode is available from theorems in the dynamo literature. Weakening the notion of an eigenmode to mean statistical stationarity, we provide numerical evidence that the eigenmode solution also holds for aperiodic flows (represented by random maps). Turning to the evolution of an initially small scale scalar field, we demonstrate the transition from an evolving (i.e., non-self-similar) probability distribution function (pdf) to a stationary (self-similar) pdf as the scale of variation of the scalar field progresses from being small to being comparable to that of the velocity field (and of the domain). Furthermore, the non-self-similar regime itself consists of two stages. Both stages are examined and the coupling between diffusion and the distribution of the finite time Lyapunov exponents is shown to be responsible for the pdf evolution.

Scalar Decay in Chaotic Mixing

“…The folding forces them to interact with themselves in a correlated fashion. We enter the regime of the strange eigenmode [27], which has received a lot of attention lately [14,26,[28][29][30][31][32][33][34][35]. Maybe we'll hear more about that in ten years.…”

“…If the initial concentration decays for large |x 2 | (as when we have a single blob of dye), then (30) can be simplified to…”

“…The summing (and hence averaging) over blobs is manifest in Eq. (30), which contains an integral over the initial distribution θ 0 in the x 2 direction, windowed by a Gaussian.…”

The strange eigenmode in Lagrangian coordinates

“…The concentration is normalized at each iteration such that the mode with largest concentration has unit magnitude. The iterations plotted are nϭ6, 7,8,9,10,11 ͑circles͒ and nϭ12 ͑black dots͒. Most of the circles appear as large black dots, because all these points lie on top of each other.…”

“…There are powerful theories based on the distribution of Lyapunov exponents [1][2][3] that link the mixing rate of the passive scalar with the chaotic properties of the flow. It has been recently suggested, following earlier work of Pierrehumbert,[4][5][6][7][8][9][10] that the mixing properties of the flow can often be elucidated only by solving a full eigenvalue problem for the advection-diffusion operator, in an analogous manner to what is done for the kinematic dynamo. 11 The resulting eigenfunctions have been dubbed strange eigenmodes by Pierrehumbert, and are closely related to Pollicott-Ruelle resonances in ergodic theory, [12][13][14] which describe the longtime decay of correlations in mixing hyperbolic dynamical systems.…”