Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence.

This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows. The Lyapunov theory indicates that more attention should be given to the Lyapunov vector orientation. Moreover, the properties of Lyapunov vectors and exponents are explained in light of recent results on tracer gradients dynamics. Differences between the different Lyapunov vectors can be interpreted in terms of competition between the effects of effective rotation and strain. Also, the differences between backward and forward vectors give information on the local reversibility of the tracer gradient dynamics. A numerical simulation of two-dimensional turbulence serves to highlight these points and the spatial distribution of finite time Lyapunov exponents is also discussed in relation to stirring properties. (c) 2002 American Institute of Physics.

[1]  R. Pierrehumbert Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves , 1991 .

[2]  G. Haller Lagrangian coherent structures from approximate velocity data , 2002 .

[3]  Comment on "Finding finite-time invariant manifolds in two-dimensional velocity fields" [Chaos 10, 99 (2000)]. , 2001, Chaos.

[4]  Jean-Luc Thiffeault,et al.  Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions. , 2000, Chaos.

[5]  G. Haller Finding finite-time invariant manifolds in two-dimensional velocity fields. , 2000, Chaos.

[6]  Antonello Provenzale,et al.  Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion , 1993, Journal of Fluid Mechanics.

[7]  Stephen Wiggins,et al.  Chaotic transport in dynamical systems , 1991 .

[8]  A Closer Look at Chaotic Advection in the Stratosphere. Part II: Statistical Diagnostics , 1999 .

[9]  Raymond T. Pierrehumbert Lattice models of advection-diffusion. , 2000, Chaos.

[10]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[11]  J. Ottino The Kinematics of Mixing: Stretching, Chaos, and Transport , 1989 .

[12]  A. Vulpiani,et al.  Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient , 1997, chao-dyn/9701013.

[13]  Steven A. Orszag,et al.  Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method , 1987 .

[14]  J. Weiss The dynamics of entropy transfer in two-dimensional hydrodynamics , 1991 .

[15]  B. Hua The Conservation of Potential vorticity along Lagrangian Trajectories in Simulations of Eddy-Driven Flows , 1994 .

[16]  J. Gollub,et al.  Experimental measurements of stretching fields in fluid mixing. , 2002, Physical review letters.

[17]  H. Aref Stirring by chaotic advection , 1984, Journal of Fluid Mechanics.

[18]  C. Grebogi,et al.  Chaotic advection, diffusion, and reactions in open flows. , 2000, Chaos.

[19]  G. Lapeyre,et al.  Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence , 1999 .

[20]  B. Legras,et al.  Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex , 2002 .

[21]  Antonello Provenzale,et al.  TRANSPORT BY COHERENT BAROTROPIC VORTICES , 1999 .

[22]  Guido Boffetta,et al.  Chaotic advection in point vortex models and two-dimensional turbulence , 1994 .

[23]  A. Boozer,et al.  Finite time Lyapunov exponent and advection-diffusion equation , 1996 .

[24]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[25]  A. Ōkubo Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences , 1970 .

[26]  N. Nakamura Two-Dimensional Mixing, Edge Formation, and Permeability Diagnosed in an Area Coordinate , 1996 .

[27]  Power spectrum of passive scalars in two dimensional chaotic flows. , 2000, Chaos.

[28]  Patrice Klein,et al.  An exact criterion for the stirring properties of nearly two-dimensional turbulence , 1998 .

[29]  Alignment of tracer gradient vectors in 2D turbulence , 2000 .

[30]  Bruno Eckhardt,et al.  Local Lyapunov exponents in chaotic systems , 1993 .

[31]  Tamás Tél,et al.  Advection in chaotically time-dependent open flows , 1998 .

[32]  S. Kida Motion of an Elliptic Vortex in a Uniform Shear Flow , 1981 .

[33]  E. Ott,et al.  THE ROLE OF CHAOTIC ORBITS IN THE DETERMINATION OF POWER SPECTRA OF PASSIVE SCALARS , 1996 .

[34]  V. I. Oseledec A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[35]  Guido Boffetta,et al.  Detecting barriers to transport: a review of different techniques , 2001, nlin/0102022.

[36]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[37]  George Haller,et al.  Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence , 2001 .

[38]  A. B. Potapov,et al.  On the concept of stationary Lyapunov basis , 1998 .

[39]  Stephen Wiggins,et al.  Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow , 1998 .

[40]  Raymond T. Pierrehumbert,et al.  Global Chaotic Mixing on Isentropic Surfaces , 1993 .

[41]  Thomas M. Antonsen,et al.  The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows , 1991 .

[42]  E. Shuckburgh,et al.  Effective diffusivity as a diagnostic of atmospheric transport: 1. Stratosphere , 2000 .

[43]  Brian F. Farrell,et al.  Generalized Stability Theory. Part II: Nonautonomous Operators , 1996 .