Ham iltonian form ulation of reduced Vlasov-Maxw ell equations - PowerPoint PPT Presentation

Ham iltonian form ulation of reduced Vlasov-Maxw ell equations Cristel CHANDRE Centre de Physique Théorique – CNRS, Marseille, France Contact : chandre@cpt.univ-mrs.fr

� importance of stability vs instability in devices involving a large number of charged particles interacting with fields: plasma physics (tokamaks), free electron lasers � Here: reduced models of such systems (easier simulation, better understanding of the dynamics) Outline Outline - Hamiltonian description of charges particles and electromagnetic fields - Reduction of Vlasov-Maxwell equations using Lie transforms Alain J. BRIZARD (Saint Michael’s College, Vermont, USA) - Reduced Hamiltonian model for the Free Electron Laser Romain BACHELARD (Synchrotron Soleil, Paris) Michel VITTOT (CPT, Marseille)

Motion of a charged particle in electrom agnetic fields Motion of a charged particle in electrom agnetic fields > > in canonical form 2 ⎛ ⎞ ⎟ e ( ) ⎜ ⎟ − ⎜ p A q , t ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ∂ ∂ ∂ ∂ c f g f g ( ) ( ) { } = + = ⋅ − ⋅ p q , , q , , H t eV t with f g ∂ ∂ ∂ ∂ 2 q p p q m ⎧ ⎪ ∂ { } H ⎪ � = = − p p , ⎪ H ⎪ ∂ q ⎪ ⎨ equations of motion : ⎪ ∂ { } H ⎪ � = = q q , ⎪ H ⎪ ∂ p ⎪ ⎩ ⎛ ⎞ 1 e ⎟ ⎜ = − ( ) v p A q , ⎟ t ⎜ ⎜ ⎟ ⎝ ⎠ m c > in non-canonical form: physical variables > = x q ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ ⎟ ∂ ∂ ⎟ 1 1 B f g f g e f g ( ) ( ) { } ⎜ ⎜ ⎟ ⎟ = 2 + = ⋅ − ⋅ + ⋅ × v x , , v x , , ⎜ ⎜ H t m eV t with f g ⎟ ⎟ ⎜ ⎜ ⎜∂ ⎟ ⎜∂ ⎟ ∂ ∂ ∂ ∂ 2 x v v x 2 v v ⎝ ⎠ ⎝ ⎠ m m c ⎧ { } ⎪ = = x � x , v gyroscopic bracket ⎪ H ⎪ ⎪ ⎛ ⎞ ⎨ × v B equations of motion : ⎟ e { } ⎜ ⎪ ⎟ � = = + v v , ⎜ E ⎪ H ⎟ ⎜ ⎜ ⎟ ⎪ ⎝ ⎠ m c ⎪ ⎩

Definition: Ham iltonian system Definition: Ham iltonian system - a scalar function H , the Hamiltonian { } - a Poisson bracket F G , with the properties { } { } = − antisymmetric F G , G F , { } { } { } = + Leibnitz law F GK , F G K , G F K , { } { } { } { } { } { } + + = Jacobi identity F G , , K K F , , G G K , , F 0 - equations of motion dF { } = F H , dt - a conserved quantity { } = F H , 0

Eulerian version: case of a density of charged particles Eulerian version: case of a density of charged particles ( ) - density of particles in phase space f x v , , t 1 ( ) ( ) ( ) ( ) ( ) ∑ = δ − δ − example: f x v , , t x x t v v t Klimontovitch distribution i i N i - evolution given by the Vlasov equation ⎛ ⎞ ∂ ∂ f e v f ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ v f E B ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ t m c v ⎝ ⎠ - Eulerian, not Lagrangian: ∂ d F F ⎡ ⎤ ⎡ ⎤ = = f for any observable F , we have F H , ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∂ dt t - still a Hamiltonian system ⎛ ⎞ m ⎟ ⎜ ⎡ ⎤ = ∫∫ ⎟ 3 3 2 + Hamiltonian : f d xd v f v eV H ⎜ ⎢ ⎥ ⎟ ⎣ ⎦ ⎜ ⎟ ⎜ 2 ⎝ ⎠ ⎧ ⎫ ⎪ ⎪ δ δ F G ⎪ ⎪ ⎡ ⎤ ∫∫ = 3 3 with , d xd v f , F G ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ δ δ f f ⎪ ⎪ ⎩ ⎭

Eulerian version: case of a density of charged particles Eulerian version: case of a density of charged particles ( ) ⎡ ⎤ ∫ ρ = 3 - an example: f x e d v f ( x v , ) ⎢ ⎥ ⎣ ⎦ 0 0 ⎧ ⎫ ⎪ ⎪ ∂ρ δρ δ H ⎪ ⎪ ⎡ ⎤ ∫ = ρ = 3 3 , d xd v f , H ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ ∂ δ δ t f f ⎪ ⎪ ⎩ ⎭ - functional derivatives δ F ( ) ⎡ ⎤ ⎡ ⎤ ∫ + φ 3 3 φ + φ 2 f = f + d xd v O F F ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ δ f δρ δ 1 H ( ) = δ − = 2 + - here: e x x and m v eV 0 δ δ f f 2 ∂ρ = ∂ ⎡ ⎤ ∫ − ⋅ = 3 - therefore: J with J f e d v f v ⎢ ⎥ ⎣ ⎦ ∂ ∂ t x

Vlasov- -Maxw ell equations: self Maxw ell equations: self- -consistent dynam ics consistent dynam ics Vlasov > description of the dynamics of a collisionless plasma (low density) Variables : particle density f ( x , v , t ) , electric field E ( x , t ) , magnetic field B ( x , t ) ⎛ ⎞ ∂ ∂ f e v f ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ v f E B ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ t m c v ⎝ ⎠ ∂ B = − ∇× E c ∂ t ∂ E = ∇× B − π J c 4 ∂ t ∇ ⋅ E = πρ ∇ ⋅ B = 4 0 where and

Vlasov- -Maxw ell equations... still a Ham iltonian system Maxw ell equations... still a Ham iltonian system Vlasov 2 2 + E B m ⎡ ⎤ = ∫ ∫ ∫ 3 3 2 + 3 E , , B f d xd v f v d x H Hamiltonian : ⎢ ⎥ ⎣ ⎦ π 2 8 ⎧ ⎫ ⎪ ⎪ δ δ F G ⎪ ⎪ ⎡ ⎤ ∫∫ 3 3 = , d xd v f , F G ⎨ ⎬ with ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ δ δ f f ⎪ ⎪ ⎩ ⎭ ⎡ ⎤ π ∂ δ δ δ δ 4 e f F G F G ⎢ ⎥ ∫∫ + 3 3 ⋅ − d xd v ⎢ ⎥ ∂ δ δ δ δ m v E f f E ⎣ ⎦ ⎡ ⎤ δ δ δ δ F G F G ∫ ⎢ ⎥ + π 3 ⋅ ∇× − ∇× ⋅ 4 c d x ⎢ ⎥ δ E δ B δ B δ E ⎣ ⎦ ∂ d F F ⎡ ⎤ ⎡ ⎤ = = F E , , B f F H , Equation of motion for : ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∂ dt t ∫ 3 B E − d p f Remark: div et div are conserved quantities antisymmetry, Leibnitz, Jacobi Morrison, PLA (1980) Marsden, Weinstein, Physica D (1982)

From m icroscopic to m acroscopic Vlasov- -Maxw ell equations Maxw ell equations From m icroscopic to m acroscopic Vlasov - Elimination (or decoupling) of fast time and small spatial scales for a better understanding of complex plasma phenomena D H - reduced Maxwell equations in terms of and ∇ ⋅ = πρ D 4 R ∂ D = ∇× − π H J c 4 ∂ R t = + π = − π D E 4 P , H B 4 M where ∂ P ρ = ρ + ∇ ⋅ = − ∇× − ∂ P J J c M , R R t reduced polarization density / magnetization current / polarization current density - Can we represent the reduced Vlasov-Maxwell equations as a Hamiltonian system? Hint: use of Lie transforms - Deliverables : Expressions of the polarization P and magnetization M vectors

f , E and B of f , E and B Reduced fields fields as Lie as Lie transform s transform s of Reduced ⎡ ( ) ( ) ( ) ⎤ Given a functional E x , t , B x , t , f x v , , t , we define some new fields as S ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ 1 ⎟ ⎜ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎟ − + + E , E , , E � ⎜ S S S ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎜ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ⎛ ⎞ ⎣ ⎦ ⎟ 2 ⎜ D E ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ 1 ⎜ ⎜ ⎟ ⎟ ⎡ ⎤ ⎜ ⎡ ⎤ ⎡ ⎤ ⎟ − ⎜ L ⎜ = = − + + H ⎟ e B ⎟ B S , B S S , , B � ⎟ ⎜ S ⎜ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎟ ⎟ ⎣ ⎦ ⎣ ⎦ ⎟ ⎜ ⎣ ⎦ ⎜ ⎜ ⎟ ⎟ 2 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ F ⎜ f ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ 1 ⎟ ⎡ ⎤ ⎜ ⎡ ⎤ ⎡ ⎤ ⎟ − ⎢ + + f , f , , f � S S S ⎜ ⎟ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎝ 2 ⎠ χ Remark: If the variable is only a function of x − L χ then e is only a function of x S − L = The functionals transforms into e , F F S resulting in a new Hamiltonian and a new Poisson bracket...

Polarization, , m agnetization m agnetization, , reduced reduced density density , , etc etc… … Polarization ⎛ ⎞ δ ∂ δ ⎟ 1 e S S ( ) ⎜ ∫ − L ⎟ = − = ∇× − 3 + P e 1 E c d vf � ⎜ S ⎟ ⎜ ⎟ ⎜ π δ ∂ δ 4 B m v f ⎝ ⎠ δ 1 1 ( ) S − L = − = ∇× + M e B c � S π δ 4 E ⎧ ⎪ = + π D E 4 P ⎪ so that ⎨ ⎪ = − π H B 4 M ⎪ ⎩ ⎛ ⎞ ∂ ∂ F ⎟ ⎜ − L L ⎟ ≡ ⎜ Reduced evolution operator e e F S S ⎟ ⎜ ⎟ ⎜ ∂ ∂ t ⎝ t ⎠ ⎡ ⎤ ⎡ ⎤ − L L L = = ⎢ e e ,e , F H F H S S S ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

Reduced Vlasov Vlasov- -Maxw ell Maxw ell equations equations Reduced ∂ D = ∇× − π c H 4 J ∂ t ∂ H = − ∇× D c ∂ t ∂ ∂ D D ⎡ ⎤ = + − = ∇× − π D , H H c H 4 J ⎢ ⎥ ⎣ ⎦ ∂ ∂ R t t ∂ ∂ ∂ H H M ⎡ ⎤ = + H − = − ∇× D − π + π ∇× P , H H c 4 4 c ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂ t t t ⎛ ⎞ ∂ ∂ F e v F ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ Reduced Vlasov equation v F ⎜ D H ⎟ ⎜ ⎟ ⎜ ∂ ∂ t m c v ⎝ ⎠ ⎧ ⎫ ⎪ ⎪ δ π ∂ δ 4 e f S S ⎪ ⎪ = − − ⋅ + F f f , � ⎨ ⎬ ⎪ ⎪ δ ∂ δ f m v E ⎪ ⎪ ⎩ ⎭ guiding center theory / gyrokinetics