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. Particle mixing in 2D turbulence
A Lagrangian Approach to Coherent Structures
Particle mixing in 2D turbulence
stretching: fluid blob opens up along
a material line
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 repelling over 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. saddle-type stagnation points
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
Contour plot of
Elliptic regions:
Hyperbolic regions:

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

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

13. DLE analysis of surface velocity
Example 2: HF radar data from Monterey Bay
Image by Chad Coulliet & Francois Lekien (MANGEN,
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. Occasional slow convergence
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, (3D flows)]

16. where M is the strain acceleration tensor (Rivlin derivative of S)
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. Earlier result from DLE Local minimum curves are stretch
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,
Data by Jeff Paduan,
Naval Postgraduate School
Filtering by Bruce Lipphardt
& Denny Kirwan (U. of Delaware)

22. Answer for 2D, incompressible Navier-Stokes flows:
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-time hyperbolic 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