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The concept and use of Lagrangian Coherent Structures IFTS Intensive Course on Advaned Plasma Physics-Spring 2015 Theory and simulation of nonlinear physics.

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Presentation on theme: "The concept and use of Lagrangian Coherent Structures IFTS Intensive Course on Advaned Plasma Physics-Spring 2015 Theory and simulation of nonlinear physics."— Presentation transcript:

1 The concept and use of Lagrangian Coherent Structures IFTS Intensive Course on Advaned Plasma Physics-Spring 2015 Theory and simulation of nonlinear physics of the beam-plasma system

2 Transport processes in plasma physics The study of transport processes is of main importance in plasma physics. Different models are used to analyze the plasma behavior: Slides by Matteo Valerio Falessi

3 Transport processes in plasma physics The study of transport processes is of main importance in plasma physics. Different models are used to analyze the plasma behavior: Matteo Valerio Falessi NLED web seminar 1.two fluids and MHD; 2.Vlasov-Poisson Eulerian; 3.Vlasov-Poisson Lagrangian, i.e. PIC; 4.N-body. Slides by Matteo Valerio Falessi

4 Transport processes in plasma physics The transport process in these system is essentially the mixing of: Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

5 Transport processes in plasma physics The transport process in these system is essentially the mixing of: Matteo Valerio Falessi NLED web seminar 1.Fluid elements; 2.phase space volumes; 3.charge distribution Slides by Matteo Valerio Falessi

6 Transport processes in plasma physics The transport process in these system is essentially the mixing of: Matteo Valerio Falessi NLED web seminar 1.Fluid elements; 2.phase space volumes; 3.charge distribution Slides by Matteo Valerio Falessi

7 Analogy with transport processes in fluids Analogy with the Lagrangian advection of passive tracers in a fluid: Matteo Valerio FalessiNLED web seminar Slides by Matteo Valerio Falessi

8 Analogy with transport processes in fluids Analogy with the Lagrangian advection of passive tracers in a fluid: Matteo Valerio FalessiNLED web seminar Slides by Matteo Valerio Falessi

9 Analogy with transport processes in fluids Analogy with the Lagrangian advection of passive tracers in a fluid: Matteo Valerio FalessiNLED web seminar Advecting field obtained solving the P.D.E. Slides by Matteo Valerio Falessi

10 Analogy with transport processes in fluids Analogy with the Lagrangian advection of passive tracers in a fluid: Matteo Valerio FalessiNLED web seminar Advecting field obtained solving the P.D.E. Slides by Matteo Valerio Falessi

11 Velocity field: the double gyre Matteo Valerio Falessi 7 th IAEA Technical Meeting on Plasma Instabilities Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

12 Velocity field: the double gyre Matteo Valerio Falessi 7 th IAEA Technical Meeting on Plasma Instabilities Tracers trajectories? Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

13 Velocity field: Monterey Bay Matteo Valerio Falessi NLED web seminar Lekien Physica D 210 (2005) Slides by Matteo Valerio Falessi

14 Velocity field: Monterey Bay Matteo Valerio Falessi NLED web seminar Tracers trajectories? Lekien Physica D 210 (2005) Slides by Matteo Valerio Falessi

15 Lagrangian vs Eulerian Matteo Valerio Falessi NLED web seminar The trajectories of the particles are Lagrangian while the velocity field is Eulerian; Slides by Matteo Valerio Falessi

16 Lagrangian vs Eulerian Matteo Valerio Falessi NLED web seminar The trajectories of the particles are Lagrangian while the velocity field is Eulerian; an integration is needed to obtain the trajectories; Slides by Matteo Valerio Falessi

17 Lagrangian vs Eulerian Matteo Valerio Falessi NLED web seminar The trajectories of the particles are Lagrangian while the velocity field is Eulerian; an integration is needed to obtain the trajectories; sensitivity to initial condition problem; Slides by Matteo Valerio Falessi

18 Lagrangian vs Eulerian Matteo Valerio Falessi NLED web seminar The trajectories of the particles are Lagrangian while the velocity field is Eulerian; an integration is needed to obtain the trajectories; sensitivity to initial condition problem; complicated plots of bundles of trajectories are required to study transport processes. Slides by Matteo Valerio Falessi

19 Lagrangian vs Eulerian Matteo Valerio Falessi NLED web seminar The trajectories of the particles are Lagrangian while the velocity field is Eulerian; an integration is needed to obtain the trajectories; sensitivity to initial condition problem; complicated plots of bundles of trajectories are required to study transport processes. Let’s start with the steady state … Slides by Matteo Valerio Falessi

20 Velocity field: steady double gyre Matteo Valerio Falessi NLED web seminar Adapted from Shadden Slides by Matteo Valerio Falessi

21 Velocity field: steady double gyre Matteo Valerio Falessi NLED web seminar Streamlines are trajectories! Adapted from Shadden Slides by Matteo Valerio Falessi

22 Velocity field: steady double gyre Matteo Valerio Falessi NLED web seminar Saddle pointsSeparatrix Streamlines are trajectories! Adapted from Shadden Slides by Matteo Valerio Falessi

23 Transport processes in steady systems Matteo Valerio Falessi NLED web seminar Streamlines (Eulerian) and trajectories (Lagrangian) coincide; Slides by Matteo Valerio Falessi

24 Transport processes in steady systems Matteo Valerio Falessi NLED web seminar Streamlines (Eulerian) and trajectories (Lagrangian) coincide; transport processes, i.e. the mixing of passive tracers, can be described looking only at the velocity field; Slides by Matteo Valerio Falessi

25 Transport processes in steady systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

26 Transport processes in steady systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

27 Velocity field: steady double gyre Matteo Valerio Falessi NLED web seminar Adapted from Shadden Slides by Matteo Valerio Falessi

28 Velocity field: steady double gyre Matteo Valerio Falessi NLED web seminar Fluid elements A and B can mix while B and C diverge. Separatrix Adapted from Shadden Slides by Matteo Valerio Falessi

29 Stable and unstable manifolds Matteo Valerio Falessi NLED web seminar Adapted from Haller Slides by Matteo Valerio Falessi

30 Stable and unstable manifolds Matteo Valerio Falessi NLED web seminar Stable manifold: Points advected into the saddle point (asymptotically) Unstable manifold: Points advected into the saddle point with a backward-time evolution (asymptotically) Parcel of fluid Adapted from Haller Slides by Matteo Valerio Falessi

31 Time dependent systems Matteo Valerio Falessi NLED web seminar How to obtain transport barriers in a: 1.time dependent system? Slides by Matteo Valerio Falessi

32 Time dependent systems Matteo Valerio Falessi NLED web seminar How to obtain transport barriers in a: 1.time dependent system? 2.Numerical simulation? Slides by Matteo Valerio Falessi

33 Time dependent systems Matteo Valerio Falessi NLED web seminar How to obtain transport barriers in a: 1.time dependent system? 2.Numerical simulation? Several problems: Slides by Matteo Valerio Falessi

34 Time dependent systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

35 Time dependent systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

36 Time dependent systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

37 Time dependent systems Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

38 Lagrangian coherent structures (LCS) Matteo Valerio Falessi NLED web seminar Adapted from Haller Slides by Matteo Valerio Falessi

39 Lagrangian coherent structures (LCS) Matteo Valerio Falessi NLED web seminar Adapted from Haller Slides by Matteo Valerio Falessi

40 Lagrangian coherent structures (LCS) Matteo Valerio Falessi NLED web seminar These generalized, finite time, structures are called LCS. Repulsive LCS Adapted from Haller Slides by Matteo Valerio Falessi

41 Finite time Lyapunov exponents (FTLE) Matteo Valerio Falessi NLED web seminar Trajectories near a saddle point diverge faster; Slides by Matteo Valerio Falessi

42 Finite time Lyapunov exponents (FTLE) Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

43 Finite time Lyapunov exponents (FTLE) Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

44 Finite time Lyapunov exponents (FTLE) Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

45 FTLE: steady double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

46 FTLE: steady double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

47 FTLE: steady double gyre Matteo Valerio Falessi NLED web seminar What about the time dependent double gyre? Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

48 FTLE: time dependent double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

49 FTLE: time dependent double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

50 FTLE: time dependent double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

51 FTLE: time dependent double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

52 FTLE: time dependent double gyre Matteo Valerio Falessi NLED web seminar Shadden Physica D 212 (2005) Slides by Matteo Valerio Falessi

53 FTLE: Monterey bay Matteo Valerio Falessi NLED web seminar Lekien Physica D 210 (2005) Slides by Matteo Valerio Falessi

54 FTLE: Monterey bay Matteo Valerio Falessi NLED web seminar Recirculating water Lekien Physica D 210 (2005) Slides by Matteo Valerio Falessi

55 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

56 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

57 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

58 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; clump formation; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

59 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; clump formation; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

60 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; clump formation; spatial bunching and trapped particles; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

61 The beam plasma instability Matteo Valerio Falessi NLED web seminar Interaction of a monochromatic electron beam with a cold plasma; Langmuir resonant wave; clump formation; spatial bunching and trapped particles; transport processes in the phase space are not clear just by looking at snapshots of the simulation; O’Neil POF 14 (1971) Slides by Matteo Valerio Falessi

62 Poincaré map vs LCS Matteo Valerio FalessiNLED web seminar Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

63 Poincaré map vs LCS Matteo Valerio FalessiNLED web seminar Asymptotically periodic; Periodic behavior Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

64 Poincaré map vs LCS Matteo Valerio FalessiNLED web seminar Asymptotically periodic; single particle motion described trough Poincaré map; Periodic behavior Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

65 Poincaré map vs LCS Matteo Valerio Falessi NLED web seminar Asymptotically periodic; single particle motion described trough Poincaré map; onset of the instability? Periodic behavior Trapped particles Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

66 Poincaré map vs LCS Matteo Valerio Falessi NLED web seminar Asymptotically periodic; single particle motion described trough Poincaré map; onset of the instability? Periodic behavior Aperiodic behavior Trapped particles Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

67 Poincaré map vs LCS Matteo Valerio Falessi NLED web seminar Asymptotically periodic; single particle motion described trough Poincaré map; onset of the instability? Multi-beams? Periodic behavior Aperiodic behavior Trapped particles Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

68 Poincaré map vs LCS Matteo Valerio Falessi NLED web seminar Asymptotically periodic; single particle motion described trough Poincaré map; onset of the instability? Multi-beams? Periodic behavior Aperiodic behavior Trapped particles Tennyson Physica D 71 (1994) Slides by Matteo Valerio Falessi

69 Beam plasma instability: FTLE profiles Matteo Valerio Falessi NLED web seminar Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS; Slides by Matteo Valerio Falessi

70 Beam plasma instability: FTLE profiles Matteo Valerio Falessi NLED web seminar Backward FTLE Contour plot Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS; Forward FTLE Contour plot Recirculating particles Slides by Matteo Valerio Falessi

71 Beam plasma instability: FTLE profiles Matteo Valerio Falessi NLED web seminar Backward FTLE Contour plot Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS; phase space splitted into macro-regions with slow transport processes between them; Forward FTLE Contour plot Recirculating particles Slides by Matteo Valerio Falessi

72 Beam plasma instability: FTLE profiles Matteo Valerio Falessi NLED web seminar Backward FTLE Contour plot Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS; phase space splitted into macro-regions with slow transport processes between them; no trapped particles (asymptotic) but recirculating ones. Forward FTLE Contour plot Recirculating particles Slides by Matteo Valerio Falessi

73 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Transport phenomena during a 3-d collisionless magnetic reconnection process; Slides by Matteo Valerio Falessi

74 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

75 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

76 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

77 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

78 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Slides by Matteo Valerio Falessi

79 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Califano lec. Slides by Matteo Valerio Falessi

80 3-d Collisionless Magnetic reconnection Matteo Valerio Falessi NLED web seminar Califano lec. Slides by Matteo Valerio Falessi

81 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Borgogno POP 18 2011 Slides by Matteo Valerio Falessi

82 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Borgogno POP 18 2011 Slides by Matteo Valerio Falessi

83 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Two separated chaotic regions Borgogno POP 18 2011 Slides by Matteo Valerio Falessi

84 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Two separated chaotic regions The two regions merge Borgogno POP 18 2011 Slides by Matteo Valerio Falessi

85 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Two separated chaotic regions The two regions merge unique stochastic region Borgogno POP 18 2011 Slides by Matteo Valerio Falessi

86 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Two separated chaotic regions The two regions merge unique stochastic region Borgogno POP 18 2011 Electrons move along field lines Slides by Matteo Valerio Falessi

87 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Linear growth Two separated chaotic regions The two regions merge unique stochastic region LCS have implications on plasma transport Borgogno POP 18 2011 Electrons move along field lines Slides by Matteo Valerio Falessi

88 Matteo Valerio Falessi NLED web seminar 3-d Collisionless Magnetic reconnection Slides by Matteo Valerio Falessi

89 Matteo Valerio Falessi NLED web seminar Attractive LCS Repulsive LCS 3-d Collisionless Magnetic reconnection Slides by Matteo Valerio Falessi

90 Matteo Valerio Falessi NLED web seminar Attractive LCS Repulsive LCS 3-d Collisionless Magnetic reconnection Recirculating regions Slides by Matteo Valerio Falessi

91 Matteo Valerio Falessi NLED web seminar Summary Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields; Slides by Matteo Valerio Falessi

92 Matteo Valerio Falessi NLED web seminar Summary Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields; in a time dependent, system we cannot quantify transport looking only at the evolution of the Eulerian fields; Slides by Matteo Valerio Falessi

93 Matteo Valerio Falessi NLED web seminar Summary Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields; in a time dependent, system we cannot quantify transport looking only at the evolution of the Eulerian fields; Slides by Matteo Valerio Falessi

94 Matteo Valerio Falessi NLED web seminar Summary Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields; in a time dependent, system we cannot quantify transport looking only at the evolution of the Eulerian fields; it is not possible to split the domain into macro-regions which do not exchange tracers just by looking at the instantaneous velocity field; Slides by Matteo Valerio Falessi

95 Matteo Valerio Falessi NLED web seminar Summary Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields; in a time dependent, system we cannot quantify transport looking only at the evolution of the Eulerian fields; it is not possible to split the domain into macro-regions which do not exchange tracers just by looking at the instantaneous velocity field; Slides by Matteo Valerio Falessi

96 Matteo Valerio Falessi NLED web seminar Conclusions and future development LCS generalize these structures in a time dependent system; Slides by Matteo Valerio Falessi

97 Matteo Valerio Falessi NLED web seminar Conclusions and future development LCS generalize these structures in a time dependent system; Slides by Matteo Valerio Falessi

98 Matteo Valerio Falessi NLED web seminar Conclusions and future development Slides by Matteo Valerio Falessi

99 Matteo Valerio Falessi NLED web seminar Conclusions and future development Slides by Matteo Valerio Falessi

100 Matteo Valerio Falessi NLED web seminar Conclusions and future development Slides by Matteo Valerio Falessi

101 Matteo Valerio Falessi NLED web seminar Conclusions and future development Slides by Matteo Valerio Falessi

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