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Cemracs 2015 - Daily morning seminar Cirm - Luminy - France The Geometrical Gyro-Kinetic Approximation The Geometrical Gyro-Kinetic Approximation Emmanuel Frénod Introduction Methode summarize Emmanuel Frénod 1 Hamiltonian System August 11th 2015 Polar Coordinates Darboux Lie EP Inria Tonus CfP-WP14-ER-01/IPP-03 & CfP-WP15-ER/IPP-01 CfP-WP14-ER-01/Swiss Confederation-01 Joint work with Mathieu Lutz 1 LMBA (UMR 6205), Université de Bretagne-Sud, F-56017, Vannes, France. emmanuel.frenod@univ-ubs.fr http://web.univ-ubs.fr/lmam/frenod/index.html Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Charge particles submitted to Strong Magnetic Field The In Usual Coordinates : ( x , v ) = ( x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ) Geometrical Gyro-Kinetic Approximation X ( t ; x , v , s ) , V ( t ; x , v , s ) Emmanuel Frénod Introduction Methode summarize ∂ X ∂ t = V Hamiltonian System ∂ V ∂ t = q Polar m ( E ( X ) + V × B ( X )) Coordinates Darboux Lie B : Self Induced Perturbations + Strong Applied piece � �� � � �� � Forgotten → 1 ε B ε ∼ Larmor Radius E : Self Induced piece � �� � Tokamak size Forgotten Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Helicoidal trajectories - Larmor Radius The Geometrical Gyro-Kinetic Approximation Emmanuel Frénod Introduction Methode summarize Hamiltonian System Polar Coordinates Darboux Lie Source: S. Jardin’s Lectures at Cemracs’10 In Tokamak: Electron Larmor Radius ∼ 5 · 10 − 4 m Ion Larmor Radius ∼ 10 − 2 m Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Dimensionless Dynamical System The Geometrical Gyro-Kinetic Approximation Emmanuel Frénod Introduction ∼ 10 − 2 m ε ∼ Ion Larmor Radius Methode ∼ 10 − 3 summarize Tokamak size 10 m Hamiltonian System Polar Coordinates ∂ X ∂ t = V Darboux Lie ∂ t = V × B ( X ) ∂ V ε Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Simplifications The Geometrical Gyro-Kinetic B ( x ) = ( 0 , 0 , B ( x 1 , x 2 )) Approximation Emmanuel B ( x 1 , x 2 ) = ∇ × A ( x 1 , x 2 ) = ∂ A 2 ( x 1 , x 2 ) − ∂ A 1 Frénod B > 1 , ( x 1 , x 2 ) ∂ x 1 ∂ x 2 Introduction Methode Turn to dimension 2: x = ( x 1 , x 2 ) , v = ( v 1 , v 2 ) summarize Hamiltonian ∂ X System ∂ t = V , X ( 0 ) = x 0 , Polar � V 2 � Coordinates ∂ V ∂ t = 1 ε B ( X ) ⊥ V = 1 Darboux ε B ( X ) V ( 0 ) = v 0 , − V 1 Lie     V 1     X 1 X 1 x 01   V 2         ∂ X 2 X 2 x 02 1          = , ( 0 ) = ε B ( X ) V 2        V 1 V 1 v 01 ∂ t   − 1 V 2 V 2 v 02 ε B ( X ) V 1 Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Gyrokinetic model The Geometrical Gyro-Kinetic Approximation Emmanuel Frénod Introduction Methode summarize ∂ Z ∂ t = − ε J Hamiltonian ⊥ ∇ B ( Z ) , System Z ( 0 ) = z 0 B ( Z ) Polar Coordinates Darboux   ∂ B Lie � Z 1 � ( Z ) ∂ = − ε J   ∂ x 2  , Z ( 0 ) = z 0  − ∂ B Z 2 ∂ t B ( Z ) ( Z ) ∂ x 1 for magnetic moment J Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

What is hidden The Geometrical Gyro-Kinetic Approximation Emmanuel Frénod Introduction Methode ∂ t = − ε J ∂ Z ⊥ ∇ B ( Z ) , summarize Z ( 0 ) = z 0 B ( Z ) Hamiltonian System � B ( Z ) ∇ 2 B ( Z ) − 3 ( ∇ B (( Z ))) 2 � ∂ Γ ∂ t = B ( Z ) J Polar + ε , Γ( 0 ) = γ 0 Coordinates 2 B ( Z ) 2 ε Darboux ∂ J Lie ∂ t = 0 , J ( 0 ) = j 0 Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Key result The Geometrical Gyro-Kinetic IF: In coordinate system r = ( r 1 , r 2 , r 3 , r 4 ) , a Hamiltonian Dynamical Approximation System writes: Emmanuel Frénod     ∂ H Introduction  M ( r ) 0 0 ∂ r 1 Methode   0 0 ∂ H ∂ R   summarize   ∂ r 2 ∂ t = P ( R ) ∇ r H ( R ) P ( r ) =  0 0 0   c 0 Hamiltonian System 0 0 − c 0 ∂ H Polar ∂ r 4 Coordinates Darboux with ∂ H Lie = 0 ∂ r 3 ∂ M = ∂ M AND: ∂ R 4 THEN: = 0 ∂ t = 0 ∂ r 3 ∂ r 4 (Trajectory R = ( R 1 , R 2 , R 3 , R 4 ) ) Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Key result The Geometrical Gyro-Kinetic IF: In coordinate system r = ( r 1 , r 2 , r 3 , r 4 ) , a Hamiltonian Dynamical Approximation System writes: Emmanuel Frénod     ∂ H Introduction  M ( r ) 0 0 ∂ r 1 Methode   0 0 ∂ H ∂ R   summarize   ∂ r 2 ∂ t = P ( R ) ∇ r H ( R ) P ( r ) =  0 0 0   c 0 Hamiltonian System 0 0 − c 0 ∂ H Polar ∂ r 4 Coordinates Darboux with ∂ H Lie = 0 ∂ r 3 ∂ M = ∂ M AND: ∂ R 4 THEN: = 0 ∂ t = 0 ∂ r 3 ∂ r 4 (Trajectory R = ( R 1 , R 2 , R 3 , R 4 ) ) Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Key result The Geometrical Gyro-Kinetic IF: In coordinate system r = ( r 1 , r 2 , r 3 , r 4 ) , a Hamiltonian Dynamical Approximation System writes: Emmanuel Frénod     ∂ H Introduction  M ( r ) 0 0 ∂ r 1 Methode   0 0 ∂ H ∂ R   summarize   ∂ r 2 ∂ t = P ( R ) ∇ r H ( R ) P ( r ) =  0 0 0   c 0 Hamiltonian System 0 0 − c 0 ∂ H Polar ∂ r 4 Coordinates Darboux with ∂ H Lie = 0 ∂ r 3 ∂ M = ∂ M AND: ∂ R 4 THEN: = 0 ∂ t = 0 ∂ r 3 ∂ r 4 (Trajectory R = ( R 1 , R 2 , R 3 , R 4 ) ) Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Panorama The Usual Coordinates Geometrical Gyro-Kinetic ( x , v ) Approximation Polar Coordinates Emmanuel ( x , θ, v ) Frénod ∂ X 3 ∂ t = V Introduction ∂ t = 1 ∂ V Methode ε B ( X ) ⊥ V 4: Darboux Method summarize Hamiltonian System Darboux Almost 1: Hamiltonian? 2 Polar Canonical Coordinates Coordinates ( y , θ, v ) Darboux Canonical Coordinates Lie ( q , p ) 5: Lie Method H ε = ˘ ˘ H ε ( q , p ) : Lie Coordinates ∂ Q p ˘ ∂ t = ∇ H ε ( z , γ, j ) ∂ P q ˘ ∂ t = −∇ H ε Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Panorama The Usual Coordinates Geometrical ( x , v ) Gyro-Kinetic Approximation Polar Coordinates H ε ( x , v ) , P ε ( x , v ) s.t: Emmanuel   ( x , θ, v ) Frénod ∂ X 3   H ε ( v ) , � � Introduction ∂ t P ε ( x , θ, v )    = P ε ∇ x , v H ε  ∂ V Methode 4: Darboux Method summarize ∂ t Hamiltonian System Darboux Almost 1: Hamiltonian? 2 Polar Canonical Coordinates Coordinates ( y , θ, k ) Darboux Canonical Coordinates Lie H ε ( y , θ, k ) , P ε ( y ) ( q , p ) 5: Lie Method H ε ( q , p ) , ˘ ˘ P ε ( q , p )= S Lie Coordinates s.t:   ∂ Q ( z , γ, j )   ∂ t   q , p ˘ H ε ( z , j ) , � �  = S∇ H ε P ε ( z )  ∂ P = P ε ( z ) ∂ t Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Canonical Coordinates The Usual Coordinates : ( x , v ) = ( x 1 , x 2 , v 1 , v 2 ) Geometrical Gyro-Kinetic Trajectory : ( X ( t ; x , v , s ) , V ( t ; x , v , s )) ( ( X , V ) = ( X 1 , X 2 , V 1 , V 2 ) ) Approximation Emmanuel Frénod ∂ X ∂ t = V Introduction B ( x ) = ∇ × A ( x ) Methode ∂ t = 1 ∂ V summarize ε B ( X ) ⊥ V Hamiltonian System Canonical Coordinates : ( q , p ) = ( q 1 , q 2 , p 1 , p 2 ) Polar Coordinates Trajectory : ( Q ( t ; q , p , s ) , P ( t ; q , p , s )) ( ( Q , P ) = ( Q 1 , Q 2 , P 1 , P 2 ) ) Darboux Lie p = v + A ( x ) q = x , ε   � � ∂ Q H ε ( q , p ) = 1 � p − A ( q ) 2 � � ˘ �   ∂ t 2 ε   q , p ˘ � 0 �  = S∇ H ε  I 2 ∂ P S = − I 2 0 ∂ t Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Check of Canonical nature of Canonical Coordinates The   Geometrical ∂ Q Gyro-Kinetic � � Approximation   H ε ( q , p ) = 1 � p − A ( q ) 2 ∂ t � �   q , p ˘ ˘  = S∇ H ε , � Emmanuel  2 ε Frénod ∂ P Introduction ∂ t Methode ∂ Q H ε ( Q , P ) = P − A ( Q ) summarize p ˘ ∂ t = ∇ ε Hamiltonian System � � T H ε ( Q , P ) = ( ∇ A ( Q )) P − A ( Q ) ∂ P Polar q ˘ ∂ t = −∇ Coordinates ε ε Darboux T ( p − A ) = ( ∇ A )( p − A ) + ( ∇ × A ) ⊥ ( p − A ) Lie ( ∇ A ) ∂ t = P − A ( Q ) ∂ Q ε � � ⊥ � � ∂ P ∂ t − ( ∇ A ( Q )) P − A ( Q ) = ∇ × A ( Q ) P − A ( Q ) ε ε ε ε Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation

Check of Canonical nature of Canonical Coord. - 2 The Geometrical Gyro-Kinetic X = Q Approximation V = P − A ( Q ) Emmanuel ∂ Q ∂ t = P − A ( Q ) Frénod ε Introduction ε � � ⊥ � � Methode ∂ P ∂ t − ( ∇ A ( Q )) P − A ( Q ) = ∇ × A ( Q ) P − A ( Q ) summarize ε ε ε ε Hamiltonian System ∂ X ∂ t = V Polar � � Coordinates P − A ( Q ) � ∂ Q � ⊥ � � ∂ Darboux ∂ P ∂ t − ( ∇ A ( Q )) = ∇ × A ( Q ) P − A ( Q ) ε = Lie ε ∂ t ∂ t ε ε ∂ X ∂ t = V ∂ V ∂ t = ∇ × A ( X ) ⊥ V ε Emmanuel Frénod The Geometrical Gyro-Kinetic Approximation