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

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:


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.



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





1. Fluid elements;

2. phase space volumes;

3. charge distribution






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


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



tracers in a fluid:


𝑑 �⃗�𝑑𝑡

(𝑡)=�⃗� ( �⃗�(𝑡) , 𝑡)




tracers in a fluid:



(𝑡)=�⃗� ( �⃗�(𝑡) , 𝑡)

Advecting field obtained solving the P.D.E.




tracers in a fluid:



(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


Velocity field: the double gyre

Matteo Valerio Falessi7th IAEA Technical Meeting on Plasma Instabilities

Shadden Physica D 212 (2005)



Velocity field: the double gyre

Matteo Valerio Falessi7th IAEA Technical Meeting on Plasma Instabilities

Tracers trajectories? Shadden Physica D 212 (2005)



Velocity field: Monterey Bay


Lekien Physica D 210 (2005)



Velocity field: Monterey Bay


Tracers trajectories?




Lagrangian vs Eulerian


• The trajectories of the particles are Lagrangian while the velocity field is Eulerian;





• The trajectories of the particles are Lagrangian while the velocity field is Eulerian;• an integration is needed to obtain the trajectories;





• 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;





• 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.





• 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 …



Velocity field: steady double gyre


Adapted from Shadden





Streamlines are trajectories!






Saddle pointsSeparatrix

Streamlines are trajectories!




Transport processes in steady systems


• 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;







• 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;










Fluid elements A and B can mix while B and C diverge.

Separatrix




Stable and unstable manifolds


Adapted from Haller



Stable and unstable manifolds


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



Time dependent systems


• How to obtain transport barriers in a: 1. time dependent system?





• How to obtain transport barriers in a: 1. time dependent system?2. Numerical simulation?






• Several problems:






• Several problems: 1. no connection between trajectories and fixed

points of ;







points of ;2. finite time of the simulation;







points of ;2. finite time of the simulation;3. global mixing in a long-enough time span







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.



Lagrangian coherent structures (LCS)


𝑡

Adapted from Haller





𝑡

Adapted from Haller





These generalized, finite time, structures are called LCS.

Repulsive LCS𝑡

Adapted from Haller



Finite time Lyapunov exponents (FTLE)


• 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 .



FTLE: steady double gyre



FTLE field for different values






FTLE field for





What about the time dependent double gyre?


FTLE field for



FTLE: time dependent double gyre








FTLE field for






FTLE field for





Shadden Physica D 212 (2005)Δ 𝑡=10Hands-on Session 1 – May 5th 2015







FTLE: Monterey bay





FTLE: Monterey bay


Recirculatingwater

Lekien Physica D 210 (2005)oursHands-on Session 1 – May 5th 2015


The beam plasma instability


• Interaction of a monochromatic electron beam with a cold plasma;

O’Neil POF 14 (1971)






• Langmuir resonant wave;







• Langmuir resonant wave;







• Langmuir resonant wave;• clump formation;







• Langmuir resonant wave;• clump formation;







• Langmuir resonant wave;• clump formation;• spatial bunching and trapped

particles;







• 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)



Poincaré map vs LCS


Tennyson Physica D 71 (1994)





• Asymptotically periodic;

Periodic behaviorTennyson Physica D 71 (1994)





• Asymptotically periodic;• single particle motion

described trough Poincaré map;

Periodic behaviorTennyson Physica D 71 (1994)






described trough Poincaré map;• onset of the instability?

Periodic behaviorTrapped particles







described trough Poincaré map;• onset of the instability?

Periodic behaviorAperiodic behavior Trapped

particles







described trough Poincaré map;• onset of the instability?• Multi-beams?


particles







described trough Poincaré map;• onset of the instability?• Multi-beams?


particles




Beam plasma instability: FTLE profiles


• Superimposition of the FTLE calculated with forward and backward time integrations: repulsive and attractive LCS;





Backward FTLE Contour plot


Forward FTLE Contour plot

Recirculating particles







• phase space splitted into macro-regions with slow transport processes between them;









• phase space splitted into macro-regions with slow transport processes between them;

• no trapped particles (asymptotic) but recirculating ones.





3-d Collisionless Magnetic reconnection


• 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 ;





𝑑𝑥𝑑𝑧

(𝑧 )=− 𝜕Ψ𝜕 𝑦

,𝑑𝑦𝑑𝑧

(𝑧 )=𝜕Ψ𝜕 𝑥








𝑑𝑥𝑑𝑧

(𝑧 )=− 𝜕Ψ𝜕 𝑦

,𝑑𝑦𝑑𝑧

(𝑧 )=𝜕Ψ𝜕 𝑥








𝑑𝑥𝑑𝑧

(𝑧 )=− 𝜕Ψ𝜕 𝑦

,𝑑𝑦𝑑𝑧

(𝑧 )=𝜕Ψ𝜕 𝑥




• The advecting field is obtained extracting from the numerical simulation;





𝑑𝑥𝑑𝑧

(𝑧 )=− 𝜕Ψ𝜕 𝑦

,𝑑𝑦𝑑𝑧

(𝑧 )=𝜕Ψ𝜕 𝑥

Califano lec.









𝑑𝑥𝑑𝑧

(𝑧 )=− 𝜕Ψ𝜕 𝑦

,𝑑𝑦𝑑𝑧

(𝑧 )=𝜕Ψ𝜕 𝑥

Califano lec.





• The Stochasticity of the magnetic field develops from the interaction between chains of islands;





Borgogno POP 18 2011





Linear growth






Linear growth

Two separated chaotic regions






Linear growth


The two regions merge






Linear growth



unique stochastic region






Linear growth





Electrons move along field lines





Linear growth




LCS have implications on plasma transport


Electrons move along field lines








Attractive LCS

Repulsive LCS





Attractive LCS

Repulsive LCS


Recirculating regions

LCS act as transport barriers for iterations of the map Hands-on Session 1 – May 5th 2015



Summary

• Studying transport processes in a plasma requires to deal with Lagrangian quantitities such as bundles of trajectories advected by the fields;




Summary


• in a time dependent, system we cannot quantify transport looking only at the evolution of the Eulerian fields;




Summary






Summary



• it is not possible to split the domain into macro-regions which do not exchange tracers just by looking at the instantaneous velocity field;




Summary



• it is not possible to split the domain into macro-regions which do not exchange tracers just by looking at the instantaneous velocity field;




Conclusions and future development• LCS generalize these structures in a

time dependent system;





time dependent system;

𝑡 𝑡+Δ𝑡





time dependent system;• they describe transport processes

happening in a well defined, characteristic time, chosen tuning the parameter ;

𝑡 𝑡+Δ𝑡







• LCS can be approximated studying the FTLE field ;

𝑡 𝑡+Δ𝑡








𝑡 𝑡+Δ𝑡








• 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



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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