Recent advances in the variational formulation of reduced Vlasov-Maxwell equations Alain J. Brizard Saint Michael’s College Plasma Theory Seminar Princeton Plasma Physics Laboratory Thursday, February 2, 2017 Alain Brizard (SMC) Plasma Theory Seminar - PPPL
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Recent advances in the variational formulation ofreduced Vlasov-Maxwell equations
Alain J. BrizardSaint Michael’s College
Plasma Theory SeminarPrinceton Plasma Physics Laboratory
Thursday, February 2, 2017
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
I. Vlasov-Maxwell Variational Principles
• Variational formulations: Lagrange, Euler, or Euler-Poincare
δA =
∫δL d4x = 0 → δL = δLV +
1
4π
(E · δE − B · δB
) Lagrange variational principle (Low, 1958)
δALV =
∑∫δL f0 d
6z0
Euler variational principle (Brizard, 2000)
δAEV = −
∑∫ (δF H + F δH
)d8Z
Euler-Poincare variational principle (Cendra et al., 1998)
δAEPV =
∑∫ (δf LEP + f δLEP
)d6z
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Constrained variations on electromagnetic fields
δE = −∇δΦ− c−1∂tδA and δB = ∇× δA
• Reduced polarization & magnetization
LredV(· · · ; Φ,A; E,B) →
Pred ≡ δLredV/δE
Mred ≡ δLredV/δB
Reduced polarization charge density
− ∇δΦ · δLredVδE
→ δΦ (∇ ·Pred)
Reduced polarization & magnetization current densities
−1
c
∂δA
∂t· δLredV
δE−∇× δA · δLredV
δB→ δA ·
(1
c
∂Pred
∂t+∇×Mred
)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
OUTLINE
• Part I. Guiding-center Vlasov-Maxwell Equations
Pre-gyrokinetic theory: No background-fluctuation separation Vlasov-Maxwell fields (f ,E,B) satisfy standard guiding-center
plays a crucial role in momentum & angular-momentumconservation (Brizard & Tronci, 2016)
• Part II. Parallel-symplectic Gyrokinetic Equations
Parallel-symplectic representation: gyrocenter Poisson bracketcontains terms due to perturbed magnetic field: 〈A1‖gc〉
The Parallel-symplectic representation is equivalent to theHamiltonian representation since the gyrocenter magneticmoment µ is the same in all representations (Brizard, 2017)
Guiding-center toroidal angular momentum conservation law
Toroidal covariant component Pgcϕ ≡ Pgc · ∂x/∂ϕ
∂Pgcϕ
∂t+ ∇ ·
(Tgc ·
∂x
∂ϕ
)= ∇
(∂x
∂ϕ
): T>gc ≡ 0
• Symmetric guiding-center stress tensor
T>gc ≡ Tgc
Guiding-center stress tensor Tgc was previously only assumedto be symmetric (e.g., Similon 1985).
Guiding-center polarization is crucial in establishing symmetry
TgcV ≡ PCGL + Σµ∫ (
X⊥ p‖ b + p‖ b X⊥)
Fµ dp‖
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Summary of Part I
Variational formulations (Lagrange, Euler, & Euler-Poincare) ofguiding-center Vlasov-Maxwell equations have been derived.
Guiding-center Vlasov-Maxwell theory is a pre-gyrokinetictheory that does not separate background and perturbedVlasov-Maxwell fields.
Exact energy-momentum & angular-momentum conservationlaws rely on the symmetry of the guiding-center Vlasov-Maxwellstress tensor.
The symmetry of the guiding-center Vlasov-Maxwell stresstensor depends on the complete representation of theguiding-center magnetization as the sum of the intrinsicmagnetic-dipole and the moving electric-dipole contributions.
Extended gyrokinetic Vlasov equation: 0 = F , Hgygy
0 =
∫Fµ, Hgygy dw =
∂(B∗ε‖ Fµ)
∂t+
∂
∂za
(za B∗ε‖ Fµ
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Gyrokinetic variational principle
δAgy = −∫
d8Z[Hgy
∂
∂Zα(Fgy δZα
)+ Fgy
(ε⟨
T−1gy
(e δψ1gc
)⟩+ ε
ep‖mc〈δA1‖gc〉
)]−∫
d4x
4π
(ε2 δΦ1 ∇2Φ1 + ε δA1 ·∇×B
)= −
∫d8ZB∗ε‖
[δS
Fµ, Hgy
gy
− ε ec〈δA1‖gc〉 Fµ
(∂Hgy
∂p‖−
p‖m
)]−[∫
d4x
4π
(ε2 δΦ1 ∇2Φ1 + ε δA1 ·∇×B
)+
∫d8Z Fgy
(ε⟨
T−1gy
(e δψ1gc
)⟩)]= O(ε3)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Summary of Part II
Equivalent representations of guiding-center and gyrokineticVlasov-Maxwell equations are available.
Equivalent gyrokinetic Vlasov-Maxwell equations can be derivedby variational principle.
Future work will look at truncated parallel-symplecticgyrokinetic Vlasov-Maxwell equations and derive itsenergy-momentum conservation laws by Noether method.
Lectures Notes on Gyrokinetic Theory(graduate-level textbook)