1 G. Haller Division of Applied Mathematics Lefschetz Center for Dynamical Systems Brown University Finite-Time Mixing and Finite-Time Mixing and Coherent Structures Coherent Structures Collaborators: A. Banaszuk (UTRC), C. Coulliet (Caltech), F. Lekien (Caltech), I. Mezic (Har A. Poje (CUNY), H. Salman (Brown/ G. Tadmor (Northeastern), Y. Wang G.-C. Yuan (Brown)
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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:
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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?
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
).()),((2
ξξxuξ 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
t
q
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
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 SchoolDLE analysis of surface velocity
)(logmax)(log
0
0
xx
t
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!)
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|M
Z|M
)(tH)(tP)(tE
,),(),(),( *21 ttt xuxuxS
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EPH partition of 2D turbulenceEPH partition of 2D turbulence
over a finite time interval over a finite time interval II
Fully objective
picture, i.e., invariant
under time-dependent
rotations 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 eddy
turnover time!
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Example 1: Lagrangian coherent structures in barotropic turbulence simulations
Time spent in )(tE
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),()(,0
),()(|,)),((|
tt
tttt
E
H
x
xxS),( 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
),()(|,)),((|
tt
tttt
E
H
x
xxS),( 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
xByCz
zAxBy
yCzAx
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