Gyrokinetic Maxwell- Vlasov model: concepts, code & theory...Vlasov model: concepts, code & theory...

Gyrokinetic Maxwell-Vlasov model: concepts,

code & theory

Natalia Tronko, NMPP

In collaboration with E.Sonnedrücker, A.Bottino

NEMORB team A.J. Brizard, St. Michael’s college

European Enabling Research project on gyrokinetic codes verification

•  Removing fast scale of motion from particles dynamics: Modern approach: near-identity (invertible) phase space transformations

GK Maximal ordering

GK dynamical reduction: concept

(x,p) !�X, pk, µ, ✓

�

6D 4D+1

•  Ordering parameters: two step transformation •  B0 curvature effects: guiding-center:

•  Eletromagnetic fluctuations: gyrocenter:

Ecole des Houches 2015

µ =mv2?2B

{µ, ✓} = 1

NEMORB [Hahm 1988]

•  Gyrokinetic theory [Frank’s lecture]: important tool for low-frequency plasma turbulence investigation

[Brizard 2007]

Magnetic momentum

Canonically conjugated Action-angle variables

✏B ⇠ ✏� ✏B = ✏2�

GK dynamical reduction: main challenge


•  The impact on field’s dynamics?

•  Polarization effects: fields&particles and not evaluated at the same position

X

r

⇢✏X = T�1

✏�

⇥T�1✏B x

⇤= x� ⇢✏

Systematic coupling between fields and reduced particles is necessary for derivation of self-consistent GK M-Vl model !!!

��1(X)

��1(r)= �3 (X+ ⇢✏ � r)

Gyrokinetic Field theory formalism

Demystification of the GK theory

Self-consistent GK equations for simulations: why?

•  Systematically derived analytical model:

Reduced (GK) particles dynamics systematically coupled with fields dynamics: §  Consistent orderings and good conservation properties

•  Energy, Momentum & phase space volume conservation

Field theory formalism: §  Systematically introduced approximations: self-consistency of the model §  Noether’s method for consistently conserved quantities derivation

•  PIC codes: NEMORB (Garching, TOKAMAK) & EUTERPE (Greifswald, STELLERATOR)

§  Eliminating all sources of possible inconsistencies before discretisation §  Discretisation of Lagrangian action via finite element method

[Bottino,Sonnendrücker, JPP, 2014; in press]


Euler or Lagrange?

• Lagrangian: evolution of fluid element

• Eulerian: fixed labels evolution of density

[Low 1958 Sugama 2000]

[Cendra et al 1998] (6D): Analogy with fluids description [Brizard 2000] (8D:extended phase space)

Lagrangian • Independent fields variations • Reconstruction of Vlasov equation from characteristics

Eulerian •  Constrained variations

•  Vlasov: the dynamical field

•  Direct derivation of conservation laws


F0 (x(t),v(t)) = Ft (x,v)

Variational principles: How to couple fields and particles?

Lp =⇣ecA+ pkbb

⌘· X+

mc

eµ✓ �Hgy

Defines model: how much physics we want to include: electrostatic, electromagnetic linearized polarization etc…

All the approximations should be done at this point in order to keep energetic consistency of the model!!!


[Sugama2000] •  Particle’s Lagrangian

•  Eulerian variational principle: extended 8D phase space: d8Z ⌘ d6Z dt dw

•  Lagrangian variational principle: reduced phase space measure d6Z ⌘ B⇤

k dX dµ dpk d✓

Hgy ⌘ ✏�H1 + ✏2�H2

AL[�1,A1,Z] =

Zd

4x

8⇡

⇣✏

2� |r?�1|2 � ✏

2� |B1|2

⌘�

X

sp

Zd

6Z F Lp(Z, Z)

©◊

©◊ AE [�1,A1,F1] =

Zd

4x

8⇡

⇣✏

2� |r?�1|2 � |B0 + ✏�B1|2

⌘�X

sp

Zd

8Z

(F0 + ✏�F1) H1 � ✏

2�

2F0 H2

�

Full 2nd order derivation vs NEMORB model: Reduced particles dynamics


•  Second order FLR decomposition is necessary: to catch up all terms from physical

model

•  Common 1st order dynamics (1st order FLR decomposition)

H1 = e h 1gci ⌘ eD�1gc �

pk

mcA1kgc

E

Hnemo

2 ⌘ e2

2mc2⌦A1kgc

↵2 � mc2

2B2|r?�1|2 ,

H full2 ⌘ e2

2mc2

DA2

1kgc

E� mc2

2B2

��r?�1 �pk

mcr?A1k

��2

H full2 �!

DA2

1kgc

E=

*✓A2

1k + ⇢0 ·rA1k +1

2⇢0⇢0 : rrA1k

◆2+

= A21k +m

⇣ ce

⌘2 µ

B

�r?A1k

�2+m

⇣ ce

⌘2 µ

BA1k r2

?A1k

Hnemo

2 �!⌦A1kgc

↵2=

✓A2

1k +1

2⇢0⇢0 : rrA1k

◆2

= A21k+m

⇣ ce

⌘2 µ

BA1k r2

?A1k

©◊

©◊

Magnetic fields


Differences in magnetic field definitions:

B1 = r⇥⇣bbA1k

⌘= bb⇥rA1k +A1kr⇥ bb

Maxwell’s constraint is violated within physical model r ·B1 = 0

B = B0 +B1

|B|2 =��r?A1k

��2•  NEMORB&EUTERPE model

•  Full 2nd order GK

What is the impact? Conservation of the phase space volume is violated!!!! Errors in numerical scheme!

⇠ ✏�✏B

First variation: concept

�AL =

Z t2

t1

�L dt

0 = �Z�LL

�Z+ ��1

�LL

��1+ �A1k

�LL

�A1k

•  Vlasov equation is reconstructed from characteristics •  Reconstruction of conserved quantities via fluid moments calculation

or via direct derivation from the Noether theorem

Particle’s trajectories

Poisson equation

Ampere equation


First variation: Eulerian Equations of motion & Noether’s terms

�L = ��1

✓1

4⇡r2

?�1 +�Hgy(X)

��1(r)

◆+ �A1k

✓bb ·r⇥B+

�Hgy(X)

�A1k(r)

◆

�Z

dW �S {F ,Hgy}+ @⇤

@t+r · �

•  Sources of polarization and magnetization in Ampère and Poisson equations: coupling fields and reduced particle’s dynamics

•  Energy conservation derivation from Noether’s terms

�Hgy(X)

��(r)=

� h�gc(X)i��(r)

= e⌦�3 (X+ ⇢gc � r)

↵


Exact derivatives: Noether’s field terms

0 = �AE = �

Z t2

t1

dt �LE

GK Quasineutrality equation


�AL

��1= 0

�X

sp

ZdW

1

B⇤kr?

B⇤

kF0mc2

B2r?�1

�=

X

sp

e

ZdWF1

⌦�3gc

↵

•  Full 2nd order GK theory

•  NEMORB&EUTERPE model No coupling with Ampere’s law via linear polarization

�AE

��1= 0&

•  Averaging procedure suitable only in uniform field approximation ⌦�3gc

↵:Z

d6Z (F0 + ✏�F1) h�(X+ ⇢0 � r)i =Z

dWd3X (F0 + ✏�F1)⌦e⇢0·r�(X� r)

↵

=

ZdW (F0 + ✏�F1)�

1

2

ZdW d3X h⇢0⇢0i| {z }

12k

2?⇢2

0

: rr (F0 + ✏�F1) �(X�r) ⌘ J0 (F0 + ✏�F1)

�✏�X

sp

ZdW

1

B⇤kr?

B⇤

kF0mc2

B2r?

⇣�1 �

pkmc

A1k

⌘�=

X

sp

e

ZdW (F0 + ✏�F1)

⌦�3gc

↵

GK Ampere’s equation


�AL

�A1k= 0

1

4⇡r2

?A1k = �Z

dV dWepkmc

F1

⌦�3gc

↵+

ZdW

e2

2mc2�A1kF0

�+

1

2

ZdW

1

B⇤kr2

?

⇣B⇤

kµ

BF0

⌘A1k

�Z

dW1

B⇤kr?

⇣B⇤

kµ

BF0

⌘r?A1k �

ZdW

⇣F0

c pkB2

⌘r?

h pkmc

r?A1k �r?�1

i

�Z

dW1

B⇤kr?

⇣F0B

⇤kc pkB2

⌘r?

h pkmc

r?A1k �r?�1

i

1

4⇡r2

?A1k = �Z

dV dWepkmc

F1

⌦�3gc

↵+

ZdW

e2

2mc2�A1kF0

�+

1

2

ZdW

1

B⇤kr2

?

⇣B⇤

kµ

BF0

⌘A1k

+

ZdW

µ

BF0 r2

?A1k

•  First 3 terms: common •  Terms in red: neglected •  Terms in blue: main differences

⇠ r2?F0⇠ r?F0

•  NEMORB model

•  Full 2nd order GK theory

Common terms of NEMORB&EUTERPE

�AE

�A1k= 0& ©◊

GK Vlasov equation


✓dgcF1

dt= �dgyF0

dt

◆⌘ @F0

@pk{pk, H0 + ✏H1}gc +rF0 · {X, H0 + ✏H1}gc

@F1

@t= �{F1, H0}gc � {F0, H1}gc � ✏{F1, H1}gc

To get perturbed distribution function evolution use gyrocenter charactersitics & gradients of background distribution

The way Vlasov equation is organised in NEMORB

�AE

�F = 0 �! {H1,F} = 0�AL

�Z= 0 �! dgyZ

dt= {H0 +H1,Z}

Z = (pk,X)

•  NEMORB Vlasov equation

•  Eulerian 2nd order variational principle: neglecting nonlinear drive of second order

©◊ ©◊ Eulerian variational principle Characteristics reconstruction

dgypkdt

dgyX

dt

GK Energy conservation


•  Noether’s method:

@E@t

+r · S = 0

Action symmetries

Conservation laws

t ! t+ �t

Efull =Z

d6Z (F0 + ✏F1)⇣H0 � ✏ e

pk

m

⌦A1kgc

↵⌘

+✏2

2

Zd6Z F0

✓e2

c21

mA2

1k +µ

B

�r?A1k

�2+

µ

BA1kr2

?A1k

◆

+✏2

2

Zd6Z F0

mc2

B2

✓|r?�1|2 �

⇣ pk

mc

⌘2 ��r?A1k��2◆

+1

8⇡

Zd3X

�✏2|r?�1|2 + |B0 + ✏B1|2

�.

•  Containing second order FLR effects and contribution from energy of magnetic field

Code diagnostics


•  Particles dynamics: balance between kinetic energy & those of first order perturbative fields

H =p2k

2m+ µB + J0(�1gc �

p2k

2mA1kgc)

dH

dt= 0 !

dgypk

dt

pk

m+

dgyX

dt· µrB =

� erJ0(�1gc �p2k

2mA1kgc) ·

dgcX

dt+

e

cJ0A1k

dgcpk

dt= � d

dt

"J0(�1gc �

p2k

2mA1kgc)

#

•  Electrostatic Power Balance diagnostics in NEMORB code, CYCLONE case

Conclusions


Comparison between full second order gyrokinetic model and NEMORB:

•  Second order Polarization effects differences, coupling of Ampere and Poisson equations

•  Magnetic field perturbation: conservation of phase space volume

•  Necessity to identify test cases, in which differences between models lead to different physics

•  Derivation of exact order second energy density containing part of field energy

•  Perspectives: test new Poisson, Ampere equations & equations and energy diagnostics in NEMORB

Gyrokinetic Maxwell- Vlasov model: concepts, code & theory...Vlasov model: concepts, code & theory...

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