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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
orderings (Ω−1∂t 1, E‖ |E⊥|) Guiding-center magnetization (magnetic + moving-electric)
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)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
II. Guiding-center Vlasov-Maxwell Equations
• Guiding-center Lagrangian Zα = (X, p‖,w , t) & µ = label
Lgc =e
cA∗ · X−w (t−1)−
(p2‖2m
+ e Φ∗
)→
Φ∗ = Φ + µB/e
A∗ = A + p‖ c b/e
Modified electromagnetic fields (E∗,B∗):
E∗ = −∇Φ∗ − c−1∂tA∗
B∗ = ∇×A∗
→∇×E∗ = − c−1∂tB∗
∇ ·B∗ = 0
Reduced guiding-center Euler-Lagrange equations (B∗‖ ≡ b ·B∗)
X =p‖m
B∗
B∗‖+ E∗× c b
B∗‖, p‖ = e E∗ · B∗
B∗‖→
∂B∗‖∂t
= − ∂
∂za
(za B∗‖
)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center Poisson bracket
F , Ggc =B∗
B∗‖·(∇∗F ∂G
∂p‖− ∂F
∂p‖∇∗G
)− c b
eB∗‖·∇∗F ×∇∗G
+∂F
∂w
∂G
∂t− ∂F
∂t
∂G
∂w
Notation: ∇∗ ≡ ∇− (e/c)∂tA∗ ∂w
Guiding-center Hamilton equations
Hgc =p2‖2m
+ e Φ∗ − w → Zα =Zα, Hgc
gc
Liouville property
F , Ggc =1
B∗‖
∂
∂Zα
(B∗‖ F
Zα, G
gc
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Guiding-center Vlasov equation za = (X, p‖)
Fµ ≡ (2πmB∗‖ ) fµ →∂Fµ∂t
+∂
∂za
(za Fµ
)= 0
• Guiding-center Maxwell equations (Σµ ≡∑
species
∫dµ)
∇ ·E = 4π %gc ≡ − 4πΣµ∫∂Lgc∂Φ
Fµ dp‖
∇×B− 1
c
∂E
∂t=
4π
c(Jgc + c ∇×Mgc)
≡ 4πΣµ∫ [
∂Lgc∂A
Fµ +∇×(∂Lgc∂B
Fµ
)]dp‖
Guiding-center magnetization: intrinsic & moving-electric dipole
∂Lgc∂B
= − µ ∂B
∂B+ p‖
∂b
∂B· X = µgc + πgc×
p‖b
mc
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Higher-order guiding-center theory
• Higher-order guiding-center theory (Tronko & Brizard, 2015)
A∗ = A +c
e
[p‖ b − εB
µB
Ω
(R +
1
2∇× b
)]Hgc =
p2‖2m
+ µB + ε2B Ψ2 =m
2
⟨∣∣∣X + ρgc
∣∣∣2⟩ Gyrogauge vector (b ≡ e1× e2): R ≡ ∇e1 · e2
Guiding-center polarization correction: (b ·∇× b) b→ ∇× b
• Guiding-center polarization (Pfirsch, 1984; Kaufman, 1986)
πgc = e 〈ρgc〉 − ∇ ·⟨e
2ρgcρgc
⟩+ · · · =
eb
Ω× X
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Geometry of Reduced Polarization
• Guiding-center Polarization driven by ∇B-drift
Ion polarization driven by U∇ = (c b/eB)×µ∇B
π∇ =e b
Ω×U∇ = − e µ
mΩ2∇⊥B → |π∇|
e ρ⊥= ρ⊥ |∇⊥ lnB| 1
!
!
!
!
!
Average
Position
!
!
Average
Position
!
VB Drift Orbit
!
!
!
!
!
!
Polarization
Displacement
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
III. Guiding-center Euler Variational Principle
• Guiding-center Euler action (Brizard & Tronci, 2016)
Agc = −Σµ∫FµHgc d
6Z +
∫d4x
8π
(|E|2 − |B|2
) Extended Vlasov phase-space density (Brizard, 2000)
Fµ ≡ Fµ δ(w − Hgc) and Hgc ≡ Hgc − w = 0
Eulerian Hamiltonian variation → intrinsic magnetization
δHgc ≡ e δΦ∗ = e δΦ + µ b · δB
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Guiding-center Eulerian Vlasov variation
δFµ ≡ B∗‖
δS, Fµ/B∗‖
gc
+ δB∗‖ Fµ/B∗‖
+e
cδA∗ ·
(B∗‖
X, Fµ/B∗‖
gc
) Eulerian magnetic variations → moving electric dipole
e
cδA∗ =
e
cδA + p‖ δB · ∂b
∂Band δB∗‖ = δB∗ · b +
(δB · ∂b
∂B
)·B∗
Guiding-center Vlasov constraint∫δFµ d6Z = 0 → δFµ ≡
∂
∂Zα
(Fµ δZα
) Guiding-center phase-space virtual displacement
δZα ≡ δS,Zαgc + (e/c) δA∗ · X,Zαgc
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center Lagrangian density
Eulerian variation
δLgc =1
4π
(δE ·E − δB ·B
)−Σµ
∫ (δFµ Hgc + Fµ e δΦ∗
)dp‖ dw
Usefull expression
δFµHgc = − Fµ(e
cδA∗ · dgcX
dt
)+ B∗‖ δS
Fµ/B∗‖ , Hgc
gc
+∂
∂Zα
[Fµ (Noether terms)
] Identity
e δΦ∗ − e
cδA∗ · dgcX
dt= e δΦ− e
cδA · dgcX
dt
− δB ·(µgc + πgc×
p‖ b
mc
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Eulerian variation (H ≡ B− 4πMgc)
δLgc ≡ −Σµ∫
B∗‖ δS Fµ/B∗‖ , Hgc dp‖ dw
+δΦ
4π
(∇ ·E − 4π %gc
)+δA
4π·(
1
c
∂E
∂t−∇×H +
4π
cJgc
)+∂δJ∂t
+ ∇ · δΓ
Noether components
δJ ≡ Σµ∫δS Fµ dp‖ dw −
E · δA4π c
δΓ ≡ Σµ∫δS Fµ
dgcX
dtdp‖ dw −
1
4π
(δΦ E + δA×H
)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center Vlasov-Maxwell variational principle
0 = δAEgc = −Σµ
∫B∗‖ δS Fµ/B
∗‖ , Hgc d
6Z
+
∫δΦ
4π
(∇ ·E − 4π %gc
)d3x dt
+
∫δA
4π·(
1
c
∂E
∂t−∇×H +
4π
cJgc
)d3x dt
Stationarity with respect to the variations (δΦ, δA) yields theguiding-center Maxwell equations.
Stationarity with respect to the variation δS yields the extendedguiding-center Vlasov equation B∗‖ Fµ/B
∗‖ , Hgc = 0
0 =
∫dw B∗‖ Fµ/B
∗‖ , Hgc =
∂Fµ∂t
+∂
∂za
(za Fµ
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center Noether Equation
• Hamiltonian constraint: Hgc = Hgc − w = 0 ⇒ Lgc → LM
δLgc ≡ −(δt
∂
∂t+ δx ·∇
)LM =
∂δJ∂t
+ ∇ · δΓ
Energy-momentum conservation law
Space-time translations generated by δS:
δS =e
cA∗ · δx − w δt ≡ P · δx − w δt
Eulerian potential variations (gauge: δχ ≡ A · δx− Φ c δt)
δΦ ≡ δx ·E + c−1∂tδχ
δA ≡ c δt E + δx×B − ∇δχ
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center energy conservation law
∂Egc∂t
+ ∇ ·Sgc = 0
Guiding-center energy density (Kgc = µB + p2‖/2m)
Egc ≡ Σµ∫
Fµ Kgc dp‖ +1
8π
(|E|2 + |B|2
) Guiding-center energy-density flux
Sgc ≡ Σµ∫
Fµ Kgc X dp‖ +c
4πE×H
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Guiding-center momentum conservation law
∂Pgc
∂t+ ∇ ·Tgc = 0
Guiding-center momentum density
Pgc ≡ Σµ∫
p‖ b Fµ dp‖ +E×B
4π c
Symmetric guiding-center stress tensor Tgc ≡ TM + TgcV
TM ≡(|E|2 + |B|2
) I
8π− 1
4π
(EE + BB
)TgcV ≡ PCGL + Σµ
∫ (X⊥ p‖ b + p‖ b X⊥
)Fµ dp‖
CGL pressure tensor
PCGL ≡Σµ∫ [p2
‖
mb b + µB
(I− bb
)]Fµ dp‖
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
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.
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
IV. Parallel-symplectic Gyrokinetic Equations
• Gyrocenter symplectic one-form (R∗0 ≡ R + 12 ∇× b0)
Γgy =[ec
(A0 + ε 〈A1‖gc〉 b0
)+ p‖ b0
]· dX
+ µB
Ω
(dζ − R∗0 · dX
)− w dt
Gyrocenter Poisson bracket ( , 0 ≡ , 0gc)
F , Ggy =Ω
B
(∂F
∂ζ
∂G
∂µ− ∂F
∂µ
∂G
∂ζ
)+
(∂F
∂w
∂G
∂t− ∂F
∂t
∂G
∂w
)+
B∗εB∗ε‖
·(∇∗εF
∂G
∂p‖− ∂F
∂p‖∇∗εG
)− c b0
eB∗ε‖·∇∗εF ×∇∗εG
where B∗ε ≡ B∗0 + ε∇× (〈A1‖gc〉 b0), B∗ε‖ ≡ b0 ·B∗ε , and
∇∗εF ≡(∇F + R∗0
∂F
∂ζ
)−εe
cb0
(∂〈A1‖gc〉∂t
∂F
∂w+
Ω
B
∂〈A1‖gc〉∂µ
∂F
∂ζ
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Gyrocenter Hamiltonian: (Φ1gc,A1gc)→ (E1gc,B1gc)
Hgy =(µB + p2‖/2m
)+ ε e
(〈Φ1gc〉 −
⟨A1⊥gc ·
Ω
c
∂ρ0
∂ζ
⟩)− ε2
2
[⟨e ρ1gc ·
(E1gc +
1
cX + ρ0, H0gc0×B1gc
)⟩+⟨ec
A1gc ·(X + ρ0, e 〈ψ1gc〉0 +
e
mc〈A1‖gc〉 b0
)⟩] First-order gyrocenter displacement: Polarization/Magnetization
ρ1gc ≡(
d
dεT−1gy (X + ρ0)
)ε=0
= X + ρ0, S10
First-order effective potential & gyrocenter gauge function:
ψ1gc ≡ Φ1gc − A1gc ·1
cX + ρ0, H0gc0
∂S1∂t
+ S1, H0gc0 = e(ψ1gc − 〈ψ1gc〉
)Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Gyrokinetic Vlasov equation
Gyrocenter Vlasov distribution Fµ(X, p‖, t)
∂Fµ∂t
+ X ·∇Fµ + p‖∂Fµ∂p‖
= 0
Gyrocenter Hamilton equations
X =∂Hgy
∂p‖
B∗εB∗ε‖
+c b0
B∗ε‖×∇Hgy, p‖ = − B∗ε
B∗ε‖·∇Hgy − ε
e
c
∂〈A1‖gc〉∂t
Gyrocenter Liouville theorem (Jgy ≡ 2πm B∗ε‖)
∂B∗ε‖∂t
= − ∂
∂za
(za B∗ε‖
)→
∂(B∗ε‖ Fµ)
∂t= − ∂
∂za
(za B∗ε‖ Fµ
)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Gyrokinetic Maxwell equations
Gyrokinetic Poisson equation [δ3gc ≡ δ3(X + ρgc − r)]
ε∇2Φ1(r) = − 4π
∫Jgy Fµ
⟨T−1gy e δ
3gc
⟩d6Z
= − 4π(%gy − ∇ ·Pgy
) Gyrokinetic Ampere equation (B = B0 + εB1)
∇×B(r) =4π
c
∫Jgy Fµ
⟨T−1gy
(e δ3gc X + ρgc, Hgc0
)⟩d6Z
=4π
c
(Jgy +
∂Pgy
∂t+ c ∇×Mgy
) Note: Because of variational derivation, T−1gy f = f − εL1gyf
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
Parallel-symplectic Gyrokinetic Euler Variational Principle
• Gyrokinetic Euler action
Agy = −∫FgyHgy d8Z +
∫d4x
8π
(ε2 |∇Φ1|2 − |B0 + ε∇×A1|2
) Extended gyrocenter Vlasov density
Fgy ≡ B∗ε‖ Fµ δ(w − Hgy) ≡ B∗ε‖ Fµ
Eulerian gyrocenter Hamiltonian variation
δHgy = ε⟨
T−1gy
(e δψ1gc
)⟩+ ε
ep‖mc〈δA1‖gc〉
Note: ∂Hgy/∂p‖ = p‖/m +O(ε2)
Alain Brizard (SMC) Plasma Theory Seminar - PPPL
• Gyrocenter Vlasov Eulerian variation
δFgy = δB∗ε‖ Fµ + B∗ε‖ δFµ
=(ε 〈δA1‖gc〉 b0 ·∇× b0
)Fµ
+ B∗ε‖
(δS, Fµgy + ε
e
c〈δA1‖gc〉
∂Fµ∂p‖
)=
∂
∂Zα(Fgy δZα
) Gyrocenter phase-space virtual displacement
δZα ≡ δS,Zαgy + εe
c〈δA1‖gc〉 b0 · X, Zαgy
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)
to be completed by Fall of 2017
Alain Brizard (SMC) Plasma Theory Seminar - PPPL