Analyzing fluid flows via the ergodicity defect ergodicity defect Sherry E. Scott FFT 2013 Norbert Wiener Center FFT 2013, Norbert Wiener Center February 22, 2013 Support: ONR-MURI Ocean 3D+1(N00014-11-1-0087)
O tli Outline B Background & Motivation kg o nd & Moti tion General Idea: Ergodicity Defect (ED) General Idea: Ergodicity Defect (ED) With Jones, Redd, Mezi ć & Kuznetsov ED & other metrics & some results With Rypina Pratt & Brown With Rypina, Pratt & Brown ED & Other fluid flow aspects ED & Other fluid flow aspects Future & preliminary work
Lagrangian Data in the Ocean L i D t i th O Float/drifter trajectory data ALACE float http://www.seabird.com/products/sp ec_sheets/41data.htm
Analyze Key Structures - Lagrangian Coherent Structures (LCS) LCS: “Organized patterns of Trajectories” Trajectories ( NOAA website)
Wh Why LCS? LCS? Understand transport of f materials/flow properties transport barriers? where/how transport happens?
M ti Motivation - Spaghetti plot ti S h tti l t Complex fluid flow Complex fluid flow & wide range of trajectory behavior www.oceancurrents.rsmas.miami.edu/ .../ analysis / Analysis .htm
Background: Th The Idea Id Understand flows/systems in terms of f f how trajectories sample/cover the space space Ergodicity? god c ty
Background: D fi iti Definition of Ergodicity f E di it Given a measure ( µ ) preserving flow T T is ergodic if the only T ‐ invariant sets A are trivial, ‐ i.e. are such that or ( A ) 0 ( A ) 1 A is T invariant if A is T ‐ invariant if 1 A T A
Theory: Boltzmann & The Ergodic Hypothesis Tx x x T 2 x x x 2 n x f ( x ), f ( Tx ), f ( T x ),..., f ( T ) f at first n po int s of trajectory at x n 1 i f ( T x ) f n i 0 Time avg = space avg? (~1860s)
Theory: ED & Characterization of Ergodicity & C a ac e a o o god c y (George Birkhoff (1931)) T is ergodic if for all f ll a e x a.e. x n 1 1 1 1 1 r f L ( ) f ( T ( x )) fd lim n ( X ) n r 1 X * time average , f ( x , T ) space average , f i.e, ergodic if for all integrable functions, “time-average = space-average”
Ergodicity Defect (General Idea) Thi k f Think of ergodicity in terms of di it i t f “time average of observables = space average of observables” average of observables Ergodicity defect evaluates difference Ergodicity defect evaluates difference between time average and space average for a collection of observables (analyzing functions) b bl ( l i f ti )
Characterization of Ergodicity (due to George Birkhoff (1931)) T is ergodic if for all n 1 1 r f L ( ) f ( T ( x )) fd lim n n r 1 X * time average , f ( x , T ) space average , f 1 , x A f ( x ) so f area ( A ) A , 0 0 else else A average time spent in A = time average x X
Ergodicity Defect (ED) on unit square Ergodicity Defect (ED) on unit square Analyzing functions are 2 dimensional Haar Analyzing functions are 2 dimensional Haar father wavelets ( s ) ( s ) ( s ) s ( ( x , , y y ) ) ( ( x ) ) ( ( y y ) ) i , , i 1 , ,... 2 i i i i i i i i 1 1 2 2 1 1 2 2 1 1 2 2 Cc Corresponds to Partition of unit square into q squares each of q 2 s 2 area 1 (where s is the spatial scale) 2 s 2
Deviation from Ergodicity with respect to Haar scaling functions Haar ergodicity defect The ergodicity defect of T with respect to the Haar g y p partition at scale s is given by s s s 2 2 1 ( ) ( s ),* 2 d ( s , T ) ( x , T ) dx j s s 2 1 2 j 1 X d(s,T) measures the degree of ergodicity - if T is ergodic, d=0 - the normalization factor is chosen such that d(s,Id) = 1 d(s,Id) 1 We call this d(s,T) the Haar ergodicity defect
ED in 2 dimensions for a trajectory– • Take mapped trajectory in unit square T k d t j t i it • Partition the unit square into squares of length and s 2 2 equal area l s 2 • Space average = s • Use number of trajectory points inside jth square N j to estimate the average time spent in each square (time average)
ED in 2 dimensions – Numerical Algorithm For a trajectory with initial conditions F j i h i i i l di i x 0 , t 0 Time average N ( s ) for jth square 2 2 ) s j 2 d d ( ( s s ; ; x x , t t ) ) ( ( s s ) 0 0 j 1 N Space average “Ergodic” (most complex) trajectory: d d 0 0 Stationary (least complex) trajectory: y ( p ) j y 2 d 1 s 1 as s 0
ED in 3 dimensions – Numerical Algorithm • • T k t j Take trajectory mapped into unit cube t d i t it b • Partition the unit cube into smaller cubes of s 3 s length and equal volume 3 • s Space average = • • Use number of trajectory points Use number of trajectory points N N j ( s ( s ) ) inside jth cube to estimate the average time spent in each cube (time average) Partition of cube for s=1/2 x 0 , t For a trajectory with initial conditions 0 N ( s ) 3 s j 3 2 d d ( ( s s ; ; x x , t t ) ) ( ( s s ) ) 0 0 0 0 j 1 N
ED & LCSs: C Conceptual illustration t l ill t ti Move into different Move into different regions Stable manifold given by black curve Complexities for trajectories along stable manifold are similar to each other (all similar to hyperbolic point) but DIFFERENT FROM Complexities of trajectories on opposite sides of stable manifold p j pp which also often differ in complexity Manifolds correspond to level sets of ED values
ED & Lagrangian Coherent Structures (LCSs) Compute the ergodicity defect of d Compute the ergodicity defect of d mean individual fluid particle trajectories Take the mean over scales of interest - Distinguish each trajectory by the manner in which it samples the i hi h i l h space (i.e., by its complexity)
ED & LCSs: D ffi Duffing Oscillator Example O ill t E l Blue curve = stable manifold from a direct evolution method c Have minimizing ridges of (left) maximizing ridges of (right) d
ED & LCSs: T Two Measures of Complexity M f C l it c Correlation dimension measures area occupied by a trajectory 1 1 For F(s) = F F( ) 2 2 ( N j s ( ) ) 2 N j Use Use to estimate to estimate c F F ( ( s s ) ) s s d d Ergodicity defect Ergodicity defect measures the manner in which the trajectory samples the space p p c d Small Large
ED & LCSs: D ffi Duffing Oscillator Example O ill t E l Blue curve = stable manifold from a direct evolution method c Have minimizing ridges of (left) maximizing ridges of (right) d
ED & LCSs: Numerically generated flow field from Regional Ocean Model System velocities c (on left) (on right) d
Oth Other Methods: M th d (1) Finite Time Lyapunov Exponent (FTLE) (1) Finite Time Lyapunov Exponent (FTLE) - separation rates between trajecs (George Haller) ( g ) (2) Correlation Dimension, c - how trajecs fill/cover the space ( (Procaccia et al) l) (3) M functions - arclengths of trajecs arclengths of trajecs ( A. Mancho) (4) Ergodic quotient (Mezic et al) (4) Ergodic quotient (Mezic et al)
Identifying LCSs: Add Addressing a Challenge i Ch ll Often data is not amenable to f traditional analysis methods such as FTLE FTLE if drifter trajectories are sparse and if drifter trajectories are sparse and non-uniformly spaced then individual trajectory methods have j y an advantage
ED & LCSs: Advantages with sparse & non-uniform data 2550 drifters 640 drifters (left) d (middle) FTLE using Lekien and Ross (2010) method (right) conventional FTL (darkest color =stable manifold)
Ergodicity Defect & Polynyas ( (3D + time dependence ) d d ) persistent open water where we would expect to find sea ice Note: 3D data primarily from floats/drifters/gliders etc i e from trajectories i.e., from trajectories
Polynyas ( (3D + time ) )
Not polynya but upwelling flow ( (3D + time ) ) • Coastal upwelling Coastal upwelling
ED & an Upwelling flow (Rivas & Samelson) ( (3D + time example) i l ) Strong Vertical Velocity in Ocean? Use Ergodicity Defect to Identify Vertical LCS? Does 3D Defect Does 3D Defect (sampling in x,y, & z ) give more/different info than just 2D? info than just 2D? Color=bathymetry Numerical model off Oregon coast in 2005
ED & an Upwelling flow (Rivas & Samelson) ( (3D + time example) i l )
3D ED & Upwelling flow at different depths
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