Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | chrystal-ellis |
View: | 215 times |
Download: | 1 times |
The concept and use of Lagrangian Coherent Structures
M. Falessi1,2, F. Pegoraro3, N. Carlevaro2,4, G. Montani2,4, F. Zonca2
1Dipartimento di Matematica e Fisica Università di Roma tre, 2ENEA for EUROfusion - C.R. Frascati (Roma), 3Dipartimento di Fisica Università di Pisa, 4Dipartimento di Fisica Università Sapienza.
Hands-on Session 1 – May 5th 2015
IFTS Intensive Course on Advaned Plasma Physics-Spring 2015Theory 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:
Hands-on Session 1 – May 5th 2015
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 FalessiNLED web seminar
1. two fluids and MHD;
2. Vlasov-Poisson Eulerian;
3. Vlasov-Poisson Lagrangian, i.e. PIC;
4. N-body.
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in plasma physics• The transport process in these system is
essentially the mixing of:
Matteo Valerio FalessiNLED web seminar Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in plasma physics• The transport process in these system is
essentially the mixing of:
Matteo Valerio FalessiNLED web seminar
1. Fluid elements;
2. phase space volumes;
3. charge distribution
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in plasma physics• The transport process in these system is
essentially the mixing of:
Matteo Valerio FalessiNLED web seminar
1. Fluid elements;
2. phase space volumes;
3. charge distribution
under the effect of an advecting field which can be
obtained solving the dynamics equation analytically or
numerically.Hands-on Session 1 – May 5th 2015
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 seminarHands-on Session 1 – May 5th 2015
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
𝑑 �⃗�𝑑𝑡
(𝑡)=�⃗� ( �⃗�(𝑡) , 𝑡)
Hands-on Session 1 – May 5th 2015
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
𝑑 �⃗�𝑑𝑡
(𝑡)=�⃗� ( �⃗�(𝑡) , 𝑡)
Advecting field obtained solving the P.D.E.
Hands-on Session 1 – May 5th 2015
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
𝑑 �⃗�𝑑𝑡
(𝑡)=�⃗� ( �⃗�(𝑡) , 𝑡)
Advecting field obtained solving the P.D.E.
• Is it possible to understand transport processes
looking only at ?Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: the double gyre
Matteo Valerio Falessi7th IAEA Technical Meeting on Plasma Instabilities
Shadden Physica D 212 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: the double gyre
Matteo Valerio Falessi7th IAEA Technical Meeting on Plasma Instabilities
Tracers trajectories? Shadden Physica D 212 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: Monterey Bay
Matteo Valerio FalessiNLED web seminar
Lekien Physica D 210 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: Monterey Bay
Matteo Valerio FalessiNLED web seminar
Tracers trajectories?
Lekien Physica D 210 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian vs Eulerian
Matteo Valerio FalessiNLED web seminar
• The trajectories of the particles are Lagrangian while the velocity field is Eulerian;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian vs Eulerian
Matteo Valerio FalessiNLED web seminar
• The trajectories of the particles are Lagrangian while the velocity field is Eulerian;• an integration is needed to obtain the trajectories;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian vs Eulerian
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian vs Eulerian
Matteo Valerio FalessiNLED 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.
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian vs Eulerian
Matteo Valerio FalessiNLED 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 …
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: steady double gyre
Matteo Valerio FalessiNLED web seminar
Adapted from Shadden
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: steady double gyre
Matteo Valerio FalessiNLED web seminar
Streamlines are trajectories!
Adapted from Shadden
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: steady double gyre
Matteo Valerio FalessiNLED web seminar
Saddle pointsSeparatrix
Streamlines are trajectories!
Adapted from Shadden
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in steady systems
Matteo Valerio FalessiNLED web seminar
• Streamlines (Eulerian) and trajectories (Lagrangian) coincide;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in steady systems
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in steady systems
Matteo Valerio FalessiNLED 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;
• stable and unstable manifolds relative to the fixed points of split the phase space into macro-regions;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Transport processes in steady systems
Matteo Valerio FalessiNLED 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;
• stable and unstable manifolds relative to the fixed points of split the phase space into macro-regions;
• tracers starting into different regions will have qualitatively different evolution. The boundaries of these regions are transport barriers;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: steady double gyre
Matteo Valerio FalessiNLED web seminar
Adapted from Shadden
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Velocity field: steady double gyre
Matteo Valerio FalessiNLED web seminar
Fluid elements A and B can mix while B and C diverge.
Separatrix
Adapted from Shadden
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Stable and unstable manifolds
Matteo Valerio FalessiNLED web seminar
Adapted from Haller
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Stable and unstable manifolds
Matteo Valerio FalessiNLED 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 fluidAdapted from Haller
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
• Several problems:
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
• Several problems: 1. no connection between trajectories and fixed
points of ;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
• Several problems: 1. no connection between trajectories and fixed
points of ;2. finite time of the simulation;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
• Several problems: 1. no connection between trajectories and fixed
points of ;2. finite time of the simulation;3. global mixing in a long-enough time span
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Time dependent systems
Matteo Valerio FalessiNLED web seminar
• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?
• Several problems: 1. no connection between trajectories and fixed
points of ;2. finite time of the simulation;3. global mixing in a long-enough time span
• A generalization of the stable and unstable manifolds is needed to split the domain into macro-regions not mixing over a finite time span.
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian coherent structures (LCS)
Matteo Valerio FalessiNLED web seminar
𝑡
Adapted from Haller
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian coherent structures (LCS)
Matteo Valerio FalessiNLED web seminar
𝑡
Adapted from Haller
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Lagrangian coherent structures (LCS)
Matteo Valerio FalessiNLED web seminar
These generalized, finite time, structures are called LCS.
Repulsive LCS𝑡
Adapted from Haller
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Finite time Lyapunov exponents (FTLE)
Matteo Valerio FalessiNLED web seminar
• Trajectories near a saddle point diverge faster;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Finite time Lyapunov exponents (FTLE)
Matteo Valerio FalessiNLED web seminar
• Trajectories near a saddle point diverge faster;
• the separation between two points advected by the fluid is descripted by the Lyapunov exponent field
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Finite time Lyapunov exponents (FTLE)
Matteo Valerio FalessiNLED web seminar
• Trajectories near a saddle point diverge faster;
• the separation between two points advected by the fluid is descripted by the Lyapunov exponent field
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Finite time Lyapunov exponents (FTLE)
Matteo Valerio FalessiNLED web seminar
• Trajectories near a saddle point diverge faster;
• the separation between two points advected by the fluid is descripted by the Lyapunov exponent field
• two parameters: and .
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: steady double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
FTLE field for different values
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: steady double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
FTLE field for
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: steady double gyre
Matteo Valerio FalessiNLED web seminar
What about the time dependent double gyre?
Shadden Physica D 212 (2005)
FTLE field for
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: time dependent double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: time dependent double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
FTLE field for
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: time dependent double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
FTLE field for
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: time dependent double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)Δ 𝑡=10Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: time dependent double gyre
Matteo Valerio FalessiNLED web seminar
Shadden Physica D 212 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: Monterey bay
Matteo Valerio FalessiNLED web seminar
Lekien Physica D 210 (2005)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
FTLE: Monterey bay
Matteo Valerio FalessiNLED web seminar
Recirculatingwater
Lekien Physica D 210 (2005)oursHands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED web seminar
• Interaction of a monochromatic electron beam with a cold plasma;
O’Neil POF 14 (1971)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED web seminar
• Interaction of a monochromatic electron beam with a cold plasma;
• Langmuir resonant wave;
O’Neil POF 14 (1971)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED web seminar
• Interaction of a monochromatic electron beam with a cold plasma;
• Langmuir resonant wave;
O’Neil POF 14 (1971)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED web seminar
• Interaction of a monochromatic electron beam with a cold plasma;
• Langmuir resonant wave;• clump formation;
O’Neil POF 14 (1971)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED web seminar
• Interaction of a monochromatic electron beam with a cold plasma;
• Langmuir resonant wave;• clump formation;
O’Neil POF 14 (1971)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED 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)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
The beam plasma instability
Matteo Valerio FalessiNLED 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)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
Tennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;
Periodic behaviorTennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;• single particle motion
described trough Poincaré map;
Periodic behaviorTennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;• single particle motion
described trough Poincaré map;• onset of the instability?
Periodic behaviorTrapped particles
Tennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;• single particle motion
described trough Poincaré map;• onset of the instability?
Periodic behaviorAperiodic behavior Trapped
particles
Tennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;• single particle motion
described trough Poincaré map;• onset of the instability?• Multi-beams?
Periodic behaviorAperiodic behavior Trapped
particles
Tennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Poincaré map vs LCS
Matteo Valerio FalessiNLED web seminar
• Asymptotically periodic;• single particle motion
described trough Poincaré map;• onset of the instability?• Multi-beams?
Periodic behaviorAperiodic behavior Trapped
particles
Tennyson Physica D 71 (1994)
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Beam plasma instability: FTLE profiles
Matteo Valerio FalessiNLED web seminar
• Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Beam plasma instability: FTLE profiles
Matteo Valerio FalessiNLED 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
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Beam plasma instability: FTLE profiles
Matteo Valerio FalessiNLED 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
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Beam plasma instability: FTLE profiles
Matteo Valerio FalessiNLED 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
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
𝑑𝑥𝑑𝑧
(𝑧 )=− 𝜕Ψ𝜕 𝑦
,𝑑𝑦𝑑𝑧
(𝑧 )=𝜕Ψ𝜕 𝑥
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
𝑑𝑥𝑑𝑧
(𝑧 )=− 𝜕Ψ𝜕 𝑦
,𝑑𝑦𝑑𝑧
(𝑧 )=𝜕Ψ𝜕 𝑥
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
𝑑𝑥𝑑𝑧
(𝑧 )=− 𝜕Ψ𝜕 𝑦
,𝑑𝑦𝑑𝑧
(𝑧 )=𝜕Ψ𝜕 𝑥
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
• The advecting field is obtained extracting from the numerical simulation;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
𝑑𝑥𝑑𝑧
(𝑧 )=− 𝜕Ψ𝜕 𝑦
,𝑑𝑦𝑑𝑧
(𝑧 )=𝜕Ψ𝜕 𝑥
Califano lec.
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
• The advecting field is obtained extracting from the numerical simulation;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
3-d Collisionless Magnetic reconnection
Matteo Valerio FalessiNLED web seminar
𝑑𝑥𝑑𝑧
(𝑧 )=− 𝜕Ψ𝜕 𝑦
,𝑑𝑦𝑑𝑧
(𝑧 )=𝜕Ψ𝜕 𝑥
Califano lec.
• Transport phenomena during a 3-d collisionless magnetic reconnection process;
• two fluids, low model in slab geometry ( periodicity) ;
• Magnetic field lines satisfy Hamilton equations with the helical flux function as ;
• The advecting field is obtained extracting from the numerical simulation;
• The Stochasticity of the magnetic field develops from the interaction between chains of islands;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Borgogno POP 18 2011
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Linear growth
Borgogno POP 18 2011
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Linear growth
Two separated chaotic regions
Borgogno POP 18 2011
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Linear growth
Two separated chaotic regions
The two regions merge
Borgogno POP 18 2011
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Linear growth
Two separated chaotic regions
The two regions merge
unique stochastic region
Borgogno POP 18 2011
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
3-d Collisionless Magnetic reconnection
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Attractive LCS
Repulsive LCS
3-d Collisionless Magnetic reconnection
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Attractive LCS
Repulsive LCS
3-d Collisionless Magnetic reconnection
Recirculating regions
LCS act as transport barriers for iterations of the map Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Summary
• Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED 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;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;
𝑡 𝑡+Δ𝑡
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;• they describe transport processes
happening in a well defined, characteristic time, chosen tuning the parameter ;
𝑡 𝑡+Δ𝑡
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;• they describe transport processes
happening in a well defined, characteristic time, chosen tuning the parameter ;
• LCS can be approximated studying the FTLE field ;
𝑡 𝑡+Δ𝑡
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;• they describe transport processes
happening in a well defined, characteristic time, chosen tuning the parameter ;
• LCS can be approximated studying the FTLE field ;
𝑡 𝑡+Δ𝑡
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi
Matteo Valerio FalessiNLED web seminar
Conclusions and future development• LCS generalize these structures in a
time dependent system;• they describe transport processes
happening in a well defined, characteristic time, chosen tuning the parameter ;
• LCS can be approximated studying the FTLE field ;
• in plasma physics they are relatively new and could be used as a post-processing diagnostic to study transport phenomena in simulations and experiments.
𝑡 𝑡+Δ𝑡
Hands-on Session 1 – May 5th 2015
Slides by Matteo Valerio Falessi