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G. Haller Division of Applied Mathematics

Lefschetz Center for Dynamical Systems Brown University

Finite-Time Mixing and Finite-Time Mixing and Coherent StructuresCoherent Structures

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)

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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

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A Lagrangian Approach to Coherent Structures

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

Particle mixing in 2D turbulence

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is repelling over the time interval if vectors normal to it grow in arbitrarily short times within .

uI)(tluI

Attracting material line: repelling in backward time

)( 0tl

)(tl

)()( 00 xx NF h)( 0xN

)(tx:)( 0xhF

deformation field

:)( 0xN unit normal

Stability of material linesStability of material lines

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A stretch line is a material line that is repelling for locally the longest/shortest time in the flow

Definitions of hyperbolic Lagrangian structures:

maximal locally is :flowopen uT minimallocally is :slip)-(no near wall uT

A fold line is a material line that is attracting for locally the longest/shortest time in the flow

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How do we find stretch and fold lines lines from data?How do we find stretch and fold lines lines from data?

Miller, Jones, Rogerson & Pratt [Physica D, 110, 1997]: “straddle” near instantaneous

saddle-type stagnation points

of the velocity fieldBowman [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

…

Numerical approaches:

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How do we find stretch and fold lines lines from data?How do we find stretch and fold lines lines from data?

Analytic view:Analytic view: stability of a fluid trajectory x(t) is governed by

Theorem (necessary criterion):Theorem (necessary criterion): Stretch lines at t=0 maximize the scalar fieldStretch lines at t=0 maximize the scalar field

)()()( 0*

0max0 xFxFx ttt

).()),(( 2ξξxuξ Ott

Linear part is solved by:

.)()( 00 ξxFξ tt

Simplest approach:Simplest approach: look for stretch lines as places of maximal stretching:

(DLE algorithm, Haller [Physica D, 149, 2001])

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Example 1:Example 1: velocity data 2D geophysical turbulencevelocity data 2D geophysical turbulence

,, 44 Fqq

tq

QG equations in 2D.

•pseudo-spectral code of A. Provenzale•particle tracking with VFTOOL of P. Miller by G-C. Yuan,0 u

• is the potential vorticity

• is the scaled inverse of the Rossby deformation radius

• denotes the coefficient of hyperviscosity

•

`22 q

10

74 105

]2,0[]2,0[ x

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Eulerian view on coherent structures: potential vorticity gradient

Contour plot of q Contour plot of || q

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Contour plot of q22 s

Hyperbolic regions:Elliptic regions:

22 )( xyyx uu ;)()( 222yxxYyyxx uuuus

Eulerian view on coherent structures: Okubo-Weiss partition

22 s

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Stretch lines from DLE analysis

Contour plot of q at t=50 Stretch lines at t=50

(= locally strongest finite-time stable manifolds)

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Fold lines from DLE analysis

Contour plot of q at t=50 Fold lines at t=50

(= locally strongest finite-time unstable manifolds)

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Example 2:Example 2: HF radar data from Monterey BayHF radar data from Monterey BayImage by Image by Chad CoullietChad Coulliet & & Francois LekienFrancois Lekien (MANGEN, (MANGEN, http://transport.caltech.eduhttp://transport.caltech.edu))

• Lagrangian separation point

• instantaneous stagnation point

Data by Data by Jeff PaduanJeff Paduan,,Naval Postgraduate SchoolNaval Postgraduate School

DLE analysis of surface velocity

)(logmax)(log

0

0x

xt

t

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Example 3:Example 3: Experiments by Experiments by Greg VothGreg Voth and and Jerry GollubJerry Gollub (Haverford) (Haverford)

Mixing of dye in charged fluid, forced periodically in time by magnets

Dye Dye+fold lines Dye+stretch lines

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What is missing? The Eulerian physics

Question:

What is the What is the objectiveobjective Eulerian signature of intense Lagrangian Eulerian signature of intense Lagrangian mixing or non-mixing?mixing or non-mixing?

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?

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)]

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Consider

where M is the strain acceleration tensor (Rivlin derivative of S),2 uSSM

Notation:

Z(x,t) : directions of zero strain

: restriction of M to ZZ|M

.| defpositiveZM .| semidefpositiveZM

indefiniteZ|M 0S

True instantaneous flow geometry

Definitions:

Hyperbolic region: ={ pos.def.}Parabolic region: ={ pos. semidef.}Elliptic region: ={ indef. or S=0}

Z|M

Z|MZ|M

)(tH)(tP)(tE

,),(),(),( *21 ttt xuxuxS

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EPH partition of 2D turbulenceEPH partition of 2D turbulenceover a finite time interval over a finite time interval II

Fully objectivepicture, i.e., invariant under time-dependentrotations and translations

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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.

)(tH

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

)(tP)(tE

dtJ S

M,

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.

)(tE

2

,,

dtI S

M

S

M

MAIN RESULTS MAIN RESULTS (Haller [(Haller [Phys. Fluids APhys. Fluids A., 2001,to appear])., 2001,to appear])

local eddyturnover time!

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Example 1: Lagrangian coherent structures in barotropic turbulence simulations

Time spent in )(tE

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),()(,0),()(|,)),((|

tttttt

EH

xxxS),( 0xtPlot of

t=85Local minimum curves are stretch lines (finite-time stable manifolds)

Fastest converging:

Earlier result from DLE

local flux!

t=60

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Example 2:Example 2: HF radar data from Monterey BayHF radar data from Monterey BayImage by Image by Chad CoullietChad Coulliet & & Francois LekienFrancois Lekien (MANGEN, (MANGEN, http://transport.caltech.eduhttp://transport.caltech.edu))

Data by Data by Jeff PaduanJeff Paduan,,Naval Postgraduate SchoolNaval Postgraduate School

Filtering by Filtering by Bruce LipphardtBruce Lipphardt& & Denny KirwanDenny Kirwan (U. of Delaware) (U. of Delaware)

),()(,0),()(|,)),((|

tttttt

EH

xxxS),( 0xt

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How are Lagrangian coherent structures How are Lagrangian coherent structures related to the governing equations?related to the governing equations?

Answer for 2D, incompressible Navier-Stokes flows: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

.)()''(2

*max2

12maxmax

1

2

ffS

ps

All trajectories in the above region are contained in finite-time hyperbolic material lines .

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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])

Towards understanding Lagrangian structures in 3D flowsTowards understanding Lagrangian structures in 3D flows

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An example: Lagrangian coherent structures in the ABC flow

.cossin,cossin,cossin

xByCzzAxByyCzAx

Henon [1966], Dombre et al. [1986]: Poincare map for A=1, B= , C=1200 iterations used

3/2 3/1

3D DLE analysis

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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