Lagrangian coherent structures in 2D turbulence

simon21

In two-dimensional turbulence coherent structures tend to emerge. While the existence of these
structures is clear from visual observations, their mathematical description is far from trivial. Several instantaneous physical quantities have been proposed and used to describe coherent structure, all yielding different answers. E.g., the picture on the left shows the spatial distribution of the norm of the potential vorticity gradient for a barotropic turbulence simulation.
At the same time, from the point of view of particle mixing, there should be a unique definition for coherent structures as regions of qualitatively similar particle behavior. We have pursued this approach, aiming to locate Lagrangian coherent structures (LCS) in two-dimensional turbulence.
  

It turns out that analytic criteria can be developed that, when applied to a turbulent  velocity field, will reveal the location of LCS. In this fashion, one can take experimentally or numerically generated velocity fields and use the software tools we have developed to find all LCS. This gives a "Lagrangian cross section" of turbulence, highlighting the boundaries of qualitatively different mixing regions. A sample result is shown here (see the picture on the right). Our Finite-Time Stability (FTS) algorithm has been applied to the same velocity field which was visualized in the picture above. The analysis rendered two types of LCS boundaries: repelling material lines (finite-time stable manifolds) and attracting material lines (finite-time unstable manifolds). This picture shows that there is an abundance of distinguished Lagrangian structures in turbulence. Our results are Galilean invariant, and hence are independent of the presence of frame-dependent features, such as instantaneous stagnation points.
  

Our current research includes (1) the extraction of LCS from pictures like the above using tools from image processing (2) the study of the effect of the above structures in the alignment of passive scalar gradients (3) the change of the above geometry for diffusive scalar transport (4) the development of accurate mixing measures based on the knowledge of LCS boundaries. This effort has been partially supported by ONR and NSF.

References:
  

Haller, G., Finding finite-time invariant manifolds in two-dimensional velocity fields CHAOS 10 (2000) 99-108.
Haller, G., and Yuan, G., Lagrangian coherent structures and mixing in two-dimensional turbulence (with. G. Yuan) Physica D (2000), in press