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Notes on Joseph and Legras (2002, JAS)


  • “Hyperbolic trajectories in 2D time-dependent flows quite often pass close to ‘persistent’ instantaneous saddle points of the streamfunction map, but are different from them.”
  • FTLE corresponds to a “fuzzy” view of hyperbolic manifolds. FSLE provides a less fuzzy distribution and “allows the manifold structures to emerge, more naturally, as the threshold rate of separation is increased” (The threshold rate of separation might refer to the final distance).
  • r = delta_f/delta_i is the ratio of the final distance delta_f to the initial distance delta_i. When r is a few units, the FSLE represents the diffusion properties at delta_i. When r is large, the stable and unstable manifolds are obtained by plotting the extrema of FSLE.
  • Particles near a hyperbolic point separate exponentially while particles far from a hyperbolic point separate linearly.
  • “Maximum separation lines may sometimes generate artifacts”. With too short a time and low spatial resolution, the method cannot distinguish between exponential and linear growth.
  • They describe in the appendix three examples where the manifolds are computed from the FLSE and from an independent technique and they show that the two estimates resemble well. They did not use, however, a more typical geophysical flow with his wide range of spatial and temporal scales. They expect, however, that “hyperbolic structures at various scales will be revealed by varying the distance over which the FSLE is calculated” [italics are mine].
  • They suggest to encompass the study of hyperbolic structures with other methods, such as escape time plots (Rom-Kedar et al. 1990), patchiness (Malhotra et al. 1998), or other techniques described in Malhotra and Wiggins (1999), Haller (2000), Haller and Yuan (2000) and Haller (2001).
  • They use PV as a test for the identification of unstable manifold. In two cases, the hyperbolic point calculated from FSLE is not associated with a PV filament along the unstable manifold and they, thus, do not conclude that these points are real. They might be btu they prefer not conclude so with the failure of the PV test.


  • Why could we not ‘simply’ find the locations where the streamfunction of a geostrophic flow converges in one direction and diverges in another one as the locations of stable/unstable manifolds?
  • Case of the parallel shear flow: an unstable manifold is obtained with the method. Is this not an unstable manifold in reality? Why? Why the rate of separation would be different than at other locations?
  • When the flow is nondivergent, what is the consequence on the interpretation of FSLE?

Interesting references

  • Malhotra et al. 1998: Patchiness: A new diagnostic for Lagrangian trajectory analysis in time-dependent fluid flows. Int. J. Bifurcation Chaos, 8, 1053–1093.
  • Malhotra, N., and S. Wiggins, 1999: Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow. J. Nonlinear Sci., 8, 401–456.