G. Haller Division of Applied Mathematics Lefschetz Center for Dynamical Systems

1. Finite-Time Mixing and Coherent Structures G. Haller Division of Applied Mathematics Lefschetz Center for Dynamical Systems Brown University Collaborators: A. Banaszuk (UTRC), C. Coulliet (Caltech), F. Lekien (Caltech), I. Mezic (Harvard), A. Poje (CUNY), H. Salman (Brown/UTRC), G. Tadmor (Northeastern), Y. Wang (Brown), G.-C. Yuan (Brown)

2. Fundamental observation: In 2D turbulence coherent structures emerge What is a coherent structure? • region of concentrated vorticity that retains its structure for longer times (Provenzale [1999]) • energetically dominant recurrent pattern (Holmes, Lumley, and Berkooz [1996]) • set of fluid particles with distinguished statistical properties (Elhmaidi, Provenzale, and Babiano [1993]) • larger eddy of a turbulent flow (Tritton [1987]) • dynamical systems: no conclusive answer for turbulent flows - spatio-temporal complexity - finite-time nature Absolute dispersion plot for the 2D QG equations

3. A Lagrangian Approach to Coherent Structures • stretching:fluid blob opens up along a material line Particle mixing in 2D turbulence repelling material line • folding:fluid blob spreads out along • a material line attracting material line • swirling/shearing: fluid blob encircled/enclosed by neutral material lines Approach coherent structures through material stability

4. Stability of material lines is repellingover the time interval if vectors normal to it grow in arbitrarily short times within . deformation field unit normal Attracting material line: repelling in backward time

5. Definitions of hyperbolic Lagrangian structures: A stretch line is a material line that is repelling for locally the longest/shortest time in the flow A fold line is a material line that is attracting for locally the longest/shortest time in the flow

6. How do we find stretch and fold lines lines from data? Numerical approaches: Miller, Jones, Rogerson & Pratt [Physica D, 110, 1997]: “straddle” near instantaneous saddle-type stagnation points of the velocity field Bowman [preprint, 1999], Winkler [thesis, Brown, 2000]: use relative dispersion plots Poje, Haller, & Mezic [Phys. Fluids A,11, 1999]: use Lagrangian mean velocity plots Couillette & Wiggins [Nonlin. Proc. Geophys.,8, 2001]: straddling near boundary points Joseph & Legras [J. Atm. Sci., submitted, 2000]: finite-size Lyapunov exponent plots …

7. How do we find stretch and fold lines lines from data? Analytic view:stability of a fluid trajectory x(t) is governed by Linear part is solved by: Simplest approach:look for stretch lines as places of maximal stretching: Theorem (necessary criterion):Stretch lines at t=0 maximize the scalar field (DLE algorithm, Haller [Physica D, 149, 2001])

8. Example 1:velocity data 2D geophysical turbulence QG equations in 2D. • pseudo-spectral code of A. Provenzale • particle tracking with VFTOOL of P. Miller by G-C. Yuan • is the potential vorticity • is the scaled inverse of the Rossby deformation radius • denotes the coefficient of hyperviscosity

9. Eulerian view on coherent structures: potential vorticity gradient Contour plot of Contour plot of

10. Eulerian view on coherent structures: Okubo-Weiss partition Elliptic regions: Contour plot of Hyperbolic regions:

11. Stretch lines from DLE analysis Stretch lines at t=50 (= locally strongest finite-time stable manifolds) Contour plot of q at t=50

12. Fold lines from DLE analysis Fold lines at t=50 (= locally strongest finite-time unstable manifolds) Contour plot of q at t=50

13. Example 2:HF radar data from Monterey Bay Image by Chad Coulliet & Francois Lekien (MANGEN, http://transport.caltech.edu) Data by Jeff Paduan, Naval Postgraduate School DLE analysis of surface velocity Lagrangian separation point instantaneous stagnation point

14. Example 3:Experiments by Greg Voth and Jerry Gollub (Haverford) Mixing of dye in charged fluid, forced periodically in time by magnets Dye Dye+fold lines Dye+stretch lines

15. Room for improvement: • Occasional slow convergence • Shear gradients show up as stretch lines (finite time!) • Nonhyperbolic Lagrangian structures? (jets, vortex cores,…) • What do we learn? What is missing? The Eulerian physics Question: What is the objective Eulerian signature of intense Lagrangian mixing or non-mixing? Available frame-dependent results: Haller and Poje [Physica D, 119, 1998], Haller and Yuan [Physica D, 147, 2000], Lapeyre, Hua, and Legras [J. Atm. Sci., submitted, 2000], Haller [Physica D, 149, 2001. (3D flows)]

16. Consider where M is the strain acceleration tensor (Rivlin derivative of S) Notation: Z(x,t) : directions of zero strain : restriction of M to Z Definitions: Hyperbolic region: ={ pos.def.} Parabolic region: ={ pos. semidef.} Elliptic region: ={ indef. or S=0} True instantaneous flow geometry

17. EPH partition of 2D turbulence over a finite time interval I Fully objective picture, i.e., invariant under time-dependent rotations and translations

18. MAIN RESULTS (Haller [Phys. Fluids A., 2001,to appear]) Theorem 1 (Sufficient cond. for Lagrangian hyperbolicity) Assume that x(t) remains in over the time interval I. Then x(t) is contained in a hyperbolic material line over I. Theorem 2 (Necessary cond. for Lagrangian hyperbolicity) Assume that x(t) is contained in a hyperbolic material line over I. Then x(t) can • intersect only at discreet time instances • stay in only for short enough time intervals J satisfying Theorem 3 (Sufficient cond. for Lagrangian ellipticity) Assume that x(t) remains in over I and Then x(t) is contained in an elliptic material line over I. local eddy turnover time!

19. Example 1: Lagrangian coherent structures in barotropic turbulence simulations Time spent in

20. Fastest converging: Plot of local flux! t=60 t=85 Earlier result from DLE Local minimum curves are stretch lines (finite-time stable manifolds)

21. Example 2:HF radar data from Monterey Bay Image by Chad Coulliet & Francois Lekien (MANGEN, http://transport.caltech.edu) Data by Jeff Paduan, Naval Postgraduate School Filtering by Bruce Lipphardt & Denny Kirwan (U. of Delaware)

22. How are Lagrangian coherent structures related to the governing equations? Answer for 2D, incompressible Navier-Stokes flows: ( Haller [Phys. Fluids A, 2001, to appear] ) Theorem (Sufficient dynamic condition for Lagrangian hyperbolicity) Consider the time-dependent physical region defined by All trajectories in the above region are contained in finite-timehyperbolic material lines .

23. Towards understanding Lagrangian structures in 3D flows Hyperbolic Lagrangian structures fall into 10 categories Existing analytic results in 3D: • DLE algorithm extends directly • frame-dependent approach has been extended (Haller [Physica D, 149, 2001])

24. An example: Lagrangian coherent structures in the ABC flow Henon [1966], Dombre et al. [1986]: Poincare map for A=1, B= , C= 1200 iterations used 3D DLE analysis

25. Some open problems (work in progress): • Survival of Lagrangian structures (obtained from filtered data) in the “true” velocity field • Lagrangian structures in 3D (objective approach) • Dynamic mixing criteria for other fluids equations and different constitutive laws • Relevance for mixing of diffusive/active tracers