Response to "Comment on 'Finding finite-time invariant manifolds in two-dimensional velocity fields' " [Chaos 11, 427 (2001)].
Lapeyre, Hua, and Legras have recently suggested that the detection of finite-time invariant manifolds in two-dimensional fluid flows, as described by Haller and Haller and Yuan, can be substantially improved. In particular, they suggested (a) a change of coordinates to strain basis before the application of Theorem 1 of Haller and (b) the use of a nondimensionalized time computed from Theorem 1. Here we discuss why these proposed steps will not result in a significant overall improvement. We verify our arguments in a more detailed computation of the example analyzed in Lapeyre, Hau, and Legras (the Kida ellipse), as well as in a two-dimensional barotropic turbulence simulation. While in both of these examples the techniques suggested by Lapeyre, Hau, and Legras reveal additional thin regions of hyperbolicity near vortex cores, they also lead to an overall loss of detail in the global computation of finite-time invariant manifolds. (c) 2001 American Institute of Physics.
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Division of Applied Mathematics, Lefschetz Center for Dynamical Systems, Brown University, Providence, Rhode Island 02912.