The strange eigenmode in Lagrangian coordinates

Abstract

For a distribution advected by a simple chaotic map with diffusion, the ''strange eigenmode'' is investigated from the Lagrangian ͑material͒ viewpoint and compared to its Eulerian ͑spatial͒ counterpart. The eigenmode embodies the balance between diffusion and exponential stretching by a chaotic flow. It is not strictly an eigenmode in Lagrangian coordinates, because its spectrum is rescaled exponentially rapidly. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1759431͔There are two main types of coordinates used to represent fluid flow and dynamical systems. Eulerian "or spatial… coordinates are fixed in space, while Lagrangian "or material… coordinates follow parcels of fluid. Strange eigenmodes are persistent patterns in mixing-they can decay slowly, and hence remain visible in the concentration field for a long time. So far, these have been studied from the Eulerian viewpoint. Here we describe the nature of the strange eigenmode in Lagrangian coordinates for a simple map. It is not a true eigenmode because its wavelength is continuously rescaled in time.

Open-flow mixing: Experimental evidence for strange eigenmodes

“…This contrasts with closed flows, where much research effort has been invested in the study of persistent concentration patterns and strange eigenmodes [8,14,15,16,17]. In addition, much of the earlier work on chaotic mixing in open flows has focused on flows more relevant for geophysics than industry (see [39] for a review), such as Von Kármán alleys in the wake of a cylinder [30,32,33,34].…”

“…The temporal dynamics of the concentration field of a diffusive dye mixed by chaotic advection have been addressed in various experimental and numerical studies of closed flows [3,4,5,6,7,8,9,10,11,12]. Time-persistent patterns and exponential decay rates of fluctuations observed in experiments and simulations [4,5,10,12,13] have been associated with an eigenmode of the advection-diffusion operator dubbed strange eigenmode [8,14,15,16,17]. Non-hyperbolic structures of the phase space may nevertheless affect mixing dynamics [18].…”

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

“…The discovery of persistent spatial patterns in chaotic advection experiments 26 led to the study of those patterns, known as "strange eigenmodes". 18,[27][28][29] Much of this literature focuses on the rate of decay of the scalar variance [30][31][32][33][34][35] and the behavior of that rate as the diffusivity vanishes. We will not concern ourselves with this question in this study, except to mention that the relation between the spatial scales of the fluid flow and the scalar field are important there as they will be here.…”

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.…”

“…The exponential decay is entirely due to the narrowing of the domain for eligible (i.e., nondecayed) modes. This "domain of eligibility" is also known as the cone or the cone of safety [4,14]. (In two dimensions it is more properly called a wedge.)…”