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1 G. Haller Division of Applied Mathematics Lefschetz Center for Dynamical Systems Brown University Finite-Time Mixing and Coherent 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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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

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3 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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4 is repelling over the time interval if vectors normal to it grow in arbitrarily short times within. Attracting material line: repelling in backward time deformation field unit normal Stability of material lines

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

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6 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 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 … Numerical approaches:

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7 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): Stretch lines at t=0 maximize the scalar field Linear part is solved by: Simplest approach: Simplest approach: look for stretch lines as places of maximal stretching: (DLE algorithm, Haller [ Physica D, 149, 2001 ])

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8 Example 1:velocity data 2D geophysical turbulence 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

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

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

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11 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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12 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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13 Example 2:HF radar data from Monterey Bay Example 2: HF radar data from Monterey Bay Image by Chad Coulliet & Francois Lekien (MANGEN, Lagrangian separation point instantaneous stagnation point Data by Jeff Paduan, Naval Postgraduate School DLE analysis of surface velocity

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

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15 What is missing? The Eulerian physics Question: What is the objective Eulerian signature of intense Lagrangian 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, (3D flows) ]

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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 True instantaneous flow geometry Definitions: Hyperbolic region: ={ pos.def.} Parabolic region: ={ pos. semidef.} Elliptic region: ={ indef. or S=0}

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

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18 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. MAIN RESULTS (Haller [Phys. Fluids A., 2001,to appear]) local eddy turnover time!

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

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

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21 Example 2:HF radar data from Monterey Bay 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)

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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- time hyperbolic material lines.

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23 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 flows

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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

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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

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