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 transcript:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Slides by Matteo Valerio Falessi

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 Electrons move along field lines Slides by Matteo Valerio Falessi

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 Electrons move along field lines Slides by Matteo Valerio Falessi

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

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

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

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

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

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

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

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

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

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

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

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

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

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