Transport and mixing in fluids from geometric, probabilistic, and topological perspectives*

Transcription

1 Transport and mixing in fluids from geometric, probabilistic, and topological perspectives* Shane Ross Associate Professor, Engineering Science and Mechanics, Virginia Tech Visiting Faculty, ICMAT Joint work with M. Stremler, D. Schmale, P. Vlachos, F. Lekien, P. Tallapragada, A. BozorgMagham, S. Naik, P. Rao, S. Raben, P. Grover, P. Kumar ICMAT PDE and Fluid Mechanics Seminar (23 Oct 2013) (*sorry, no movies linked in this version)

2 Motivation: complex fluid motion, mixing, and control Oceans 1 Atmosphere 2 1 Special material surfaces in an ocean model (Harrison, Siegel, Mitarai [2013]) 2 Special material surfaces over North America: orange = repelling, blue = attracting 3 Many other approaches, see work of Wiggins, Mancho, Froyland, Bollt, Padberg-Gehle, etc. ii

3 Lagrangian transport in fluid experiments A particle image velocimetry (PIV) fluid experiment (Hubble [2011]); Vlachos lab (Virginia Tech/Purdue) iii

4 Lagrangian transport in fluid experiments Eulerian analysis Lagrangian analysis Data processing by S. Raben (Georgia Tech) iv

5 Motivation: application to real fluid data Fixed points, periodic orbits, or other invariant sets and their stable and unstable manifolds organize phase space Many systems defined from data or large-scale simulations experimental measurements, observations e.g., from geophysical fluids, ecology, fluid experiments Data-based, aperiodic, finite-time, finite resolution generally no fixed points, periodic orbits, etc. to organize phase space Let s first look at lobe dynamics for analytically defined systems v

6 Phase space transport via lobe dynamics Suppose our dynamical system is a discrete map 1 f : M M, e.g., f = φ t+t t, flow map of time-periodic vector field and M is a differentiable, orientable, two-dimensional manifold e.g., R 2, S 2 To understand the transport of points under the f, consider invariant manifolds of unstable fixed points Let p i, i = 1,..., N p, denote saddle-type hyperbolic fixed points of f. 1 Following Rom-Kedar and Wiggins [1990] vi

7 Partition phase space into regions Natural way to partition phase space Pieces of W u (p i ) and W s (p i ) partition M. p 1 p 2 p 3 Unstable and stable manifolds in red and green, resp. vii

8 Partition phase space into regions Intersection of unstable and stable manifolds define boundaries. p 1 q 1 q 4 q 2 q 3 q 5 p 2 p 3 q 6 viii

9 Partition phase space into regions These boundaries divide the phase space into regions p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 R 4 q 6 R 5 ix

10 Label mobile subregions: atoms of transport Can label mobile subregions based on their past and future whereabouts under one iterate of the map, e.g., (..., R 4, R 4, R 1, [R 1 ], R 2,...) p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 R 4 q 6 R 5 x

11 Lobe dynamics: transport across a boundary W u [f 1 (q), q] W s [f 1 (q), q] forms boundary of two lobes; one in R 1, labeled L 1,2 (1), or equivalently ([R 1 ], R 2 ), where f(([r 1 ], R 2 )) = (R 1, [R 2 ]), etc. for L 2,1 (1) L 2,1 (1) R 2 q f -1 (q) p i L 1,2 (1) p j R 1 xi

12 Lobe dynamics: transport across a boundary Under one iteration of f, only points in L 1,2 (1) can move from R 1 into R 2 by crossing their boundary, etc. The two lobes L 1,2 (1) and L 2,1 (1) are called a turnstile. L 2,1 (1) R 2 q f (L 1,2 (1)) f -1 (q) p i L 1,2 (1) f (L 2,1 (1)) p j R 1 xii

13 Lobe dynamics: transport across a boundary Essence of lobe dynamics: dynamics associated with crossing a boundary is reduced to the dynamics of turnstile lobes associated with the boundary. L 2,1 (1) R 2 q f (L 1,2 (1)) f -1 (q) p i L 1,2 (1) f (L 2,1 (1)) p j R 1 xiii

14 Identifying atoms of transport by itinerary In a complicated system, can still identify manifolds... Unstable and stable manifolds in red and green, resp. xiv

15 Identifying atoms of transport by itinerary... and lobes R 3 R 2 R 1 Significant amount of fine, filamentary structure. xv

16 Identifying atoms of transport by itinerary e.g., with three regions {R 1, R 2, R 3 }, label lobe intersections accordingly. Denote the intersection (R 3, [R 2 ]) ([R 2 ], R 1 ) by (R 3, [R 2 ], R 1 ) xvi

17 Lobe dynamics intimately related to transport n = 0 n = 1 n = 2 n = 3 n = 5 n = 7 xvii

18 Lobe dynamics: example Restricted 3-body problem: chaotic sea has unstable fixed points. xviii

19 Compute a boundary pril 8, : M. Dellnitz et al. 0.2 R 2 f -1 (q) 0.2 (a) L 2,1 (1) (b) L 2,1 (1) (c) L 2,1 (1) ẋ 0 p R 1. q f -1 (q) (c) L (b) 1,2 (1) L 1,2 (1) (a) L 1,2 (1) R R B = U+[p,q]US+[p,q] x Fig. 6. Transport using lobe dynamics for the same Poincaré surface of section shown in Fig. 2(b). (a) The boundary B between two regions is shown as the thick black line, formed by pieces of one branch of the stable and unstable manifolds of the unstable fixed point p. We can call the region inside of the boundary R 1 (in cyan) and the outside R 2 (in white). The pips q and f 1 (q) are shown as black dots along the boundary and the turnstile lobes that will determine the transport between R 1 and R 2 are shown as colored regions. In (b), we see more details of the turnstile lobes. This is a case of a multilobe, self-intersecting turnstile discussed in Sec A schematic of this situation is shown in Fig. 4. In this case we define the xix

20 Transport between two regions The evolution of a lobe of species S 1 into R 2 Dellnitz, Junge, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Physical Review Letters xx

21 Transport between two regions Species Distribution: Species S 1 in Region R 2 Phase Space Volume F 1,2 = flux of species S 1 into region R 2 on the nth iterate T 1,2 = total amount of S 1 contained in R 2 immediately after the nth iterate 1 n = Iterate of Poincare Map 10 xxi

22 Lobe dynamics: fluid example A microfluidic mixer A channel with spatially periodic flow structure, due, e.g., to grooves or wall motion xxii

23 Lobe dynamics: fluid example A microfluidic mixer modeled as time-periodic Stokes flow streamlines for τ f = 1 tracer blob (τ f > 1) piecewise constant vector field (piecewise steady flow) t [nτ f, (n + 1)τ f /2), top streamline pattern t [(n + 1)τ f /2, (n + 1)τ f ), bottom streamline pattern System has parameter τ f, which we treat as a bifurcation parameter critical point τ f = 1 xxiii

24 Lobe dynamics: fluid example Poincaré map 2 for τ f > 1 period-3 points bifurcate into groups of elliptic and saddle points, each of period 3 2 Computations of Mohsen Gheisarieha and Mark Stremler (Virginia Tech) xxiv

25 Lobe dynamics: fluid example Structure associated with saddles some invariant manifolds of saddles xxv

26 Lobe dynamics: fluid example Can consider transport via lobe dynamics pips, regions and lobes labeled xxvi

27 Stable/unstable manifolds and lobes in fluids material blob at t = 0 xxvii

28 Stable/unstable manifolds and lobes in fluids material blob at t = 5 xxviii

29 Stable/unstable manifolds and lobes in fluids some invariant manifolds of saddles xxix

30 Stable/unstable manifolds and lobes in fluids material blob at t = 10 xxx

31 Stable/unstable manifolds and lobes in fluids material blob at t = 15 xxxi

32 Stable/unstable manifolds and lobes in fluids material blob and manifolds xxxii

33 Stable/unstable manifolds and lobes in fluids material blob at t = 20 xxxiii

34 Stable/unstable manifolds and lobes in fluids material blob at t = 25 xxxiv

35 Stable/unstable manifolds and lobes in fluids Saddle manifolds and lobe dynamics provide template for motion xxxv

36 Stable/unstable manifolds and lobes in fluids Concentration variance; a measure of homogenization Log(CV) t Homogenization has two exponential rates: slower one related to lobes Fast rate due to braiding of ghost rods (discussed later) xxxvi

37 Transport in aperiodic, finite-time setting Data-driven, finite-time, aperiodic setting e.g., non-autonomous ODEs for fluid flow Recall the flow map, x φ t+t t (x), where φ : R n R n x.. t 0 +T φ t0 (x) xxxvii

38 Identify regions of high sensitivity of initial conditions Small initial perturbations (line segments) δx(t) grow like δx(t + T ) = φ t+t t = dφt+t t (x + δx(t)) φ t+t (x) (x) δx(t) + O( δx(t) 2 ) dx t. φ t0 t 0 +T (x + δx) x + δx. δx(t 0 ) x.. φ t0 δx(t 0 +T) t 0 +T (x) xxxviii

39 Identify regions of high sensitivity of initial conditions Small initial perturbations (line segments) δx(t) grow like δx(t + T ) = φ t+t t = dφt+t t (x + δx(t)) φ t+t (x) (x) δx(t) + O( δx(t) 2 ) dx t xxxix

40 Invariant manifold analogs: FTLE-LCS approach The finite-time Lyapunov exponent (FTLE) for Euclidean manifolds, σt T (x) = 1 T log dφ t+t t (x) dx measures maximum stretching rate over the interval T of trajectories starting near the point x at time t Ridges of σt T are candidate hyperbolic codim-1 surfaces; analogs of 280 S.C. Shadden et al. / Physica D 212 (2005) stable/unstable manifolds; Lagrangian coherent structures (LCS) 2 p i j+1 p i 1 j p i j+1 p ij p i+1 j p i 1 j p ij p i+1 j p i j 1 p i j 1 (a) σ = 3x4 4x 3 12x (1+4y 2. (b) Side view. ) 2 cf. Bowman, 1999; Haller & Yuan, 2000; Haller, 2001; Shadden, Lekien, Marsden, 2005 xl

41 Invariant manifold analogs: FTLE-LCS approach Autonomous double-gyre flow xli

42 Invariant manifold analogs: FTLE-LCS approach xlii

43 Invariant manifold analogs: FTLE-LCS approach 2 y x Invariant manifolds LCS Time-periodic oscillating vortex pair flow xliii

44 Invariant manifold analogs: FTLE-LCS approach We can define the FTLE for Riemannian manifolds 3 σt T (x) = 1 T log Dφ t+t t =. 1 T log Dφ t+t t (y) max y 0 y with y a small perturbation in the tangent space at x. p 2 v 1 p 1 p 2 p 1 p i p 1 p 2 v 2 p 3 p i p i M p j v j T p i M p 3 p 1 p 2 p j p j p N p j p N T pi M v 1 p i v j v 2 3 Lekien & Ross [2010] Chaos xliv

45 Transport barriers on Riemannian manifolds repelling surfaces for T > 0, attracting for T < 0 3 cylinder Moebius strip Each frame has a different initial time t 3 Lekien & Ross [2010] Chaos xlv

46 Atmospheric flows: Antarctic polar vortex ozone data xlvi

47 Atmospheric flows: Antarctic polar vortex ozone data + LCSs (red = repelling, blue = attracting) xlvii

48 Atmospheric flows: Antarctic polar vortex air masses on either side of a repelling LCS xlviii

49 Atmospheric flows: continental U.S. LCSs: orange = repelling, blue = attracting xlix

50 Atmospheric flows and lobe dynamics orange = repelling LCSs, blue = attracting LCSs satellite Andrea, first storm of 2007 hurricane season cf. Sapsis & Haller [2009], Du Toit & Marsden [2010], Lekien & Ross [2010], Ross & Tallapragada [2011] l

51 Atmospheric flows and lobe dynamics Andrea at one snapshot; LCS shown (orange = repelling, blue = attracting) li

52 Atmospheric flows and lobe dynamics orange = repelling (stable manifold), blue = attracting (unstable manifold) lii

53 Atmospheric flows and lobe dynamics orange = repelling (stable manifold), blue = attracting (unstable manifold) liii

54 Atmospheric flows and lobe dynamics Portions of lobes colored; magenta = outgoing, green = incoming, purple = stays out liv

55 Atmospheric flows and lobe dynamics Portions of lobes colored; magenta = outgoing, green = incoming, purple = stays out lv

56 Atmospheric flows and lobe dynamics Sets behave as lobe dynamics dictates lvi

57 Ecology and transport barriers lvii

58 2D curtain-like structures bounding 3D air masses lviii

59 Aerial sampling of airborne diseases on either side of LCS lix

60 Filament with high pathogen values sandwiched by LCS 12:00 UTC 1 May :00 UTC 1 May :00 UTC 1 May km 100 km 100 km (a) (b) (c) (d) Spore concentration (spores/m 3 ) :00 12:00 00:00 12:00 00:00 12:00 30 Apr May May 2007 Time Tallapragada et al [2011] Chaos; Schmale et al [2012] Aerobiologia; BozorgMagham et al [2013] Physica D lx

61 Filament with high pathogen values sandwiched by LCS 12:00 UTC 1 May :00 UTC 1 May :00 UTC 1 May km 100 km 100 km (a) (b) (c) Sampling location (d) (e) (f) Tallapragada et al [2011] Chaos; Schmale et al [2012] Aerobiologia; BozorgMagham et al [2013] Physica D lxi

62 Stirring fluids, e.g., with solid rods lxii

63 Topological chaos through braiding of stirrers Topological chaos is built in the flow due to the topology of boundary motions lxiii

64 Thurston-Nielsen classification theorem (TNCT) Thurston (1988) Bull. Am. Math. Soc. A stirrer motion f is isotopic to a stirrer motion g of one of three types (i) finite order (f.o.): the nth iterate of g is the identity (ii) pseudo-anosov (pa): g has Markov partition with transition matrix A, topological entropy h TN (g) = log(λ P F (A)), where λ PF (A) > 1 (iii) reducible: g contains both f.o. and pa regions h TN computed from braid word, e.g., σ 1 1 σ 2 log(λ P F (A)) provides a lower bound on the true topological entropy lxiv

65 Topological chaos in a viscous fluid experiment lxv

66 Topological chaos in a viscous fluid experiment lxvi

67 Modeling the atmosphere Stirring fluids with coherent structures (?) Hurricane Andrew lxvii

68 Stirring with periodic orbits, i.e., ghost rods lxviii

69 Identifying periodic points in cavity flow example tracer blob for τ f > 1 At τ f = 1, parabolic period 3 points of map τ f > 1, elliptic / saddle points of period 3 streamlines around groups resemble fluid motion around a solid rod τ f < 1, periodic points vanish lxix

70 Identifying periodic points in cavity flow example period-τ f Poincaré map for τ f > 1 At τ f = 1, parabolic period 3 points of map τ f > 1, elliptic / saddle points of period 3 streamlines around groups resemble fluid motion around a solid rod τ f < 1, periodic points vanish lxx

71 Stirring protocol braid topological entropy period-τ f Poincaré map for τ f > 1 y 3τ f τ b Periodic points of period 3 act as ghost rods Their braid has h TN = from TNCT (b) Actual for flow h flow = h TN is an excellent lower bound (a) (c) x 2τ f τ f t x (d) lxxi

72 Topological entropy continuity across critical point topological entropy as a function of τ f lxxii

73 Topological entropy continuity across critical point topological entropy as a function of τ f also showing Thurston-Nielson lower bound h T N due to the braid on 3 strands lxxiii

74 Topological entropy continuity across critical point topological entropy as a function of τ f h T N continues to be a lower bound until about τ f = lxxiv

75 Identifying ghost rods? Poincaré section for τ f < 1 no obvious structure! Note the absence of any elliptical islands No periodic orbits of low period were found Is the phase space featureless? lxxv

76 Probabilistic approach Take probabilistic point of view Partition phase space into loosely coupled regions Almost-invariant sets regions with a long residence time 3 3-body problem phase space is divided into several invariant and almost-invariant sets. 3 Dellnitz, Junge, Koon, Lekien, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Int. J. Bif. Chaos lxxvi

77 Almost-invariant sets / almost-cyclic sets Identify almost-invariant sets (AISs) Relatedly, almost-cyclic sets (ACSs) 1 Create box partition of phase space B = {B 1,... B q }, with q large Consider a q-by-q transition (Ulam) matrix, P, where P ij = m(b i f 1 (B j )), m(b i ) the transition probability from B i to B j using, e.g., f = φ t+t t, often computed numerically P approximates P, Perron-Frobenius operator which evolves densities, ν, over one iterate of f, as Pν Typically, we use a reversibilized operator R, obtained from P 1 Dellnitz & Junge [1999], Froyland & Dellnitz [2003] lxxvii

78 Almost-invariant sets / almost-cyclic sets A set B is called almost invariant over the interval [t, t + T ] if ρ(b) = m(b f 1 (B)) m(b) 1. Can maximize value of ρ over all possible combinations of sets B B. In practice, AIS identified from spectrum of P or graph-partitioning Dellnitz, Froyland, Sertl [2000] Nonlinearity example spectrum of P lxxviii

79 Identifying AISs / ACSs by spectrum-partitioning Invariant densities are those fixed under P, P ν = ν, i.e., eigenvalue 1 Essential spectrum lies within a disk of radius r < 1 which depends on the weakest expansion rate of the underlying system. The other real eigenvalues identify almost-invariant sets Dellnitz, Froyland, Sertl [2000] Nonlinearity lxxix

80 Boxes are vertices Identifying AISs Coarse / ACSs dynamics by graph-partitioning represented by edges P has graph representation where nodes correspond to boxes B i and transitions between them are edges of a directed graph Use graph theoretic algorithms in combination with the multilevel structure use graph partitioning methods to divide the nodes into an optimal number of parts such that each part is highly coupled within itself and only loosely coupled with other parts by doing so, we can obtain AISs and transport between them lxxx

81 Identifying ghost rods : almost-cyclic sets For τ f > 1 case, where periodic points and manifolds exist... Agreement between ACS boundaries and manifolds of periodic points Known previously 1 and applies to more general objects than periodic points, i.e. normally hyperbolic invariant manifolds (NHIMs) 1 Dellnitz, Junge, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Phys. Rev. Lett.; Dellnitz, Junge, Koon, Lekien, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Int. J. Bif. Chaos lxxxi

82 Identifying ghost rods : almost-cyclic sets For τ f > 1 case, where periodic points and manifolds exist... Agreement between ACS boundaries and manifolds of periodic points Known previously 1 and applies to more general objects than periodic points, i.e. normally hyperbolic invariant manifolds (NHIMs) 1 Dellnitz, Junge, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Phys. Rev. Lett.; Dellnitz, Junge, Koon, Lekien, Lo, Marsden, Padberg, Preis, Ross, Thiere [2005] Int. J. Bif. Chaos lxxxii

83 Identifying ghost rods : almost-cyclic sets Poincaré section for τ f < 1 no obvious structure! Return to τ f < 1 case, where no periodic orbits of low period known What are the AISs and ACSs here? Consider P t+τ f t induced by family of period-τ f maps φ t+τ f t, t [0, τ f ) lxxxiii

84 Identifying ghost rods : almost-cyclic sets Top eigenvectors of R from one period of the flow map for τ f = 0.99 reveal structure ν 2 ν 3 ν 4 ν 5 ν 6 lxxxiv

85 Identifying ghost rods : almost-cyclic sets The zero contour (black) is the boundary between the two almost-invariant sets. Three-component AIS made of 3 ACSs of period 3 ACSs, in effect, replace periodic orbits for TNCT lxxxv

86 Identifying ghost rods : almost-cyclic sets ghost manifolds The zero contour (black) is the boundary between the two almost-invariant sets. Three-component AIS made of 3 ACSs of period 3 ACSs, in effect, replace periodic orbits for TNCT Also: we see a remnant of the stable and unstable manifolds of the saddle points, despite no saddle points ghost manifolds? lxxxvi

87 Identifying ghost rods : almost-cyclic sets Almost-cyclic sets stirring the surrounding fluid like ghost rods Movie shown is second eigenvector for P t+τ f t for t [0, τ f ) lxxxvii

88 Identifying ghost rods : almost-cyclic sets y 3τ f τ b (a) x 2τ f (b) τ f (c) t x (d) Braid of ACSs gives lower bound of entropy via Thurston-Nielsen One only needs approximately cyclic blobs of fluid But, theorems apply only to periodic points! Stremler, Ross, Grover, Kumar [2011] Phys. Rev. Lett. lxxxviii

89 Almost-cyclic sets are leaky Particles starting in an ACS will eventually escape ACS braid points started in an ACS 1 1 Stremler, Rao, Ross [2013] APS DFD lxxxix

90 Topological entropy vs. bifurcation parameter topological entropy as a function of τ f h TN shown for ACS braid on 3 strands xc

91 Eigenvalues/eigenvectors vs. bifurcation parameter Eigenspectrum of P changes with the parameter τ f xci

92 Eigenvalues/eigenvectors vs. bifurcation parameter Top eigenvalues of R as parameter τ f changes xcii

93 Eigenvalues/eigenvectors vs. bifurcation parameter Genuine eigenvalue crossings? Eigenvalues generically avoid crossings if there is no symmetry present (Dellnitz, Melbourne, 1994) But eigenvectors can cross (Bäcker, 1998) xciii

94 Eigenvalues/eigenvectors vs. bifurcation parameter Movie shows change in eigenvector along thick red branch (a to f), as τ f decreases. Grover, Ross, Stremler, Kumar [2012] Chaos xciv

95 Bifurcation of ACSs For example, braid on 13 strands for τ f = 0.93 Movie shown is second eigenvector for R t+τ f t for t [0, τ f ) Thurson-Nielsen for this braid provides lower bound on topological entropy xcv

96 Bifurcation of ACSs A 1 A 2 A 3 A 4 B1 B2 B3 B4 B5 C1 C2 C3 C4 A 1 A 2 A 3 A 4 B1 B2 B3 B4 B5 C1 C2 C3 C4 (a) Initial state (b) First half-period A 1 A 2 A 3 A 4 B5 C4 C3 C2 C1 B4 B3 B2 B1 C2 C3 C4 B5 C1 A4 A3 A2 A1 B4 B3 B2 B1 (c) Second half-period (d) State after 1 period xcvi

97 Bifurcation of ACSs Braid word (action of this braid) representation of braid on 13 strands σ 10 σ 11 σ 10 σ 12 σ 11 σ 10 σ 4 σ 5 σ 3 σ 6 σ 4 σ 2 σ 7 σ 5 σ 3 σ 1 σ 6 σ 4 σ 2 σ 5 σ 3 σ 4 σ 3 σ 2 σ 3 σ 1 σ 2 σ 3 σ 9 σ 8 σ 10 σ 7 σ 9 σ 11 σ 6 σ 8 σ 10 σ 12 σ 7 σ 9 σ 11 σ 8 σ 10 σ 9 From word, can determine if f.o. or pa (and topological entropy) Use braidlab software of Jean-Luc Thiffeault (U. Wisconsin-Madison) xcvii

98 Sequence of ACS braids bounds entropy strands 16 strands 3 strands strands strands For various braids of ACSs, the calculated entropy is given, bounding from below the true topological entropy over the range where the braid exists Grover, Ross, Stremler, Kumar [2012] Chaos xcviii

99 Coherent sets in the atmosphere that braid t Sets form braid on three strands xcix

100 Southern California coast: highly mixed marine ecosystem Fish larva transport, Cheryl Harrison, OSU; Harrison, Siegel, Mitarai [2013], Mitarai et al [2009] c

101 Southern California coast: highly mixed marine ecosystem Sea surface height (streamlines if ocean surface velocity) ci

102 Speculation: trends in eigenvalues/modes for prediction cii

103 Speculation: trends in eigenvalues/modes for prediction Duffing system with small noise: six largest eigenvalues of the reversibilized discretized transfer operator in dependence of the bifurcation parameter (Junge, Marsden, Mezic 2004) ciii

104 Predict critical transitions in geophysical transport? Different eigenmodes can correspond to dramatically different behavior. Some eigenmodes increase in importance while others decrease Can we predict dramatic changes in system behavior? e.g., predicting major changes in geophysical transport patterns?? civ

105 Optimal navigation in an aperiodic setting? Selectively jumping between coherent air masses using control Moving between mobile subregions of different finite-time itineraries p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 R 4 q 6 R 5 cv

106 Optimal navigation in an aperiodic setting? Selectively jumping between coherent air masses using control Moving between mobile subregions of different finite-time itineraries p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 R 4 q 6 R 5 cvi

107 Optimal navigation in an aperiodic setting? Selectively jumping between coherent air masses using control Moving between mobile subregions of different finite-time itineraries 2 p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 q 6 R 4 1 R 5 cvii

108 Optimal navigation in an aperiodic setting? Selectively jumping between coherent air masses using control Moving between mobile subregions of different finite-time itineraries FTLE shown in grayscale; bright lines are LCS separating coherent sets; green=passive; red=control cviii

109 Some parting words on transport & mixing in fluids What are robust descriptions of transport which work in data-driven aperiodic, finite-time settings? Geometric methods finite-time lobe dynamics / symbolic dynamics Probabilistic methods almost-invariant sets, almost-cyclic sets Topological methods periodic braids, Thurston-Nielsen classification Interesting links between some of these notions cix

110 The End Thank You Main Papers: Grover, Ross, Stremler, Kumar [2012] Topological chaos, braiding and bifurcation of almost-cyclic sets. Chaos 22, Tallapragada & Ross [2013] A set oriented definition of the finite-time Lyapunov exponent and coherent sets. Communications in Nonlinear Science and Numerical Simulation 18(5), Raben, Ross, Vlachos [2013] Computation of finite time Lyapunov exponents from time resolved particle image velocimetry data, Experiments in Fluids, to appear. Stremler, Ross, Grover, Kumar [2011] Topological chaos and periodic braiding of almost-cyclic sets. Physical Review Letters 106, Tallapragada, Ross, Schmale [2011] Lagrangian coherent structures are associated with fluctuations in airborne microbial populations. Chaos 21, Lekien & Ross [2010] The computation of finite-time Lyapunov exponents on unstructured meshes and for non-euclidean manifolds. Chaos 20, cx