Recent advances in the variational formulation of reduced ... · 2/2/2017  · I. Vlasov-Maxwell...

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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

Transcript of Recent advances in the variational formulation of reduced ... · 2/2/2017  · I. Vlasov-Maxwell...

Page 1: Recent advances in the variational formulation of reduced ... · 2/2/2017  · I. Vlasov-Maxwell Variational Principles Variational formulations: Lagrange, Euler, or Euler-Poincar

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

Page 2: Recent advances in the variational formulation of reduced ... · 2/2/2017  · I. Vlasov-Maxwell Variational Principles Variational formulations: Lagrange, Euler, or Euler-Poincar

I. Vlasov-Maxwell Variational Principles

• Variational formulations: Lagrange, Euler, or Euler-Poincare

δA =

∫δL d4x = 0 → δL = δLV +

1

(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

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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

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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

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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

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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

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• 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=

c(Jgc + c ∇×Mgc)

≡ 4πΣµ∫ [

∂Lgc∂A

Fµ +∇×(∂Lgc∂B

)]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

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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

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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

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III. Guiding-center Euler Variational Principle

• Guiding-center Euler action (Brizard & Tronci, 2016)

Agc = −Σµ∫FµHgc d

6Z +

∫d4x

(|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

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• 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

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Guiding-center Lagrangian density

Eulerian variation

δLgc =1

(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

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Eulerian variation (H ≡ B− 4πMgc)

δLgc ≡ −Σµ∫

B∗‖ δS Fµ/B∗‖ , Hgc dp‖ dw

+δΦ

(∇ ·E − 4π %gc

)+δA

4π·(

1

c

∂E

∂t−∇×H +

cJgc

)+∂δJ∂t

+ ∇ · δΓ

Noether components

δJ ≡ Σµ∫δS Fµ dp‖ dw −

E · δA4π c

δΓ ≡ Σµ∫δS Fµ

dgcX

dtdp‖ dw −

1

(δΦ E + δA×H

)

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Guiding-center Vlasov-Maxwell variational principle

0 = δAEgc = −Σµ

∫B∗‖ δS Fµ/B

∗‖ , Hgc d

6Z

+

∫δΦ

(∇ ·E − 4π %gc

)d3x dt

+

∫δA

4π·(

1

c

∂E

∂t−∇×H +

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

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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

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Guiding-center energy conservation law

∂Egc∂t

+ ∇ ·Sgc = 0

Guiding-center energy density (Kgc = µB + p2‖/2m)

Egc ≡ Σµ∫

Fµ Kgc dp‖ +1

(|E|2 + |B|2

) Guiding-center energy-density flux

Sgc ≡ Σµ∫

Fµ Kgc X dp‖ +c

4πE×H

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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

(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

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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

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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.

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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

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• 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

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• 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

)

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• 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

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Parallel-symplectic Gyrokinetic Euler Variational Principle

• Gyrokinetic Euler action

Agy = −∫FgyHgy d8Z +

∫d4x

(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

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• 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

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• Gyrokinetic variational principle

δAgy = −∫

d8Z[Hgy

∂Zα(Fgy δZα

)+ Fgy

(ε⟨

T−1gy

(e δψ1gc

)⟩+ ε

ep‖mc〈δA1‖gc〉

)]−∫

d4x

(ε2 δΦ1 ∇2Φ1 + ε δA1 ·∇×B

)= −

∫d8ZB∗ε‖

[δS

Fµ, Hgy

gy

− ε ec〈δA1‖gc〉 Fµ

(∂Hgy

∂p‖−

p‖m

)]−[∫

d4x

(ε2 δΦ1 ∇2Φ1 + ε δA1 ·∇×B

)+

∫d8Z Fgy

(ε⟨

T−1gy

(e δψ1gc

)⟩)]= O(ε3)

Alain Brizard (SMC) Plasma Theory Seminar - PPPL

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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