Gyrokinetic PIC simulations of global electromagnetic modes · Keywords: electromagnetic gyrokinetics, cancellation problem, v k -formulation vs. p k -formulation vs. mixed-variable

WENDELSTEIN 7-X

Gyrokinetic PIC simulations of global electromagnetic modes

A. Mishchenko, A. Biancalani, A. Bottino, M. Borchardt, M. Cole, T. Feher,R. Hatzky, R. Kleiber, A. Konies, L. Lewerentz, H. Leyh, J. Riemann,

C. Slaby, E. Sonnendrucker, T. M. Tran, A. Zocco

Keywords: electromagnetic gyrokinetics, cancellation problem,v‖-formulation vs. p‖-formulation vs. mixed-variable formulation,

shear Alfven waves; global GAE, MAE, TAE modes;fast-particle destabilisation, EPM; internal kink mode, fishbones

Acknowledgements:J. Nuhrenberg, P. Helander, R. Schneider (Greifswald)

S. Gunter, K. Lackner, E. Poli (Garching)K. Appert, L. Villard (CRPP Lausanne)F. Zonca (EUROfusion/ER15-ENEA-03)

Alexey Mishchenko · IPP Greifswald, Stellaratortheorie

WENDELSTEIN 7-X Motivation

• Kinetic effects on MHD instabilities

– Finite ion gyroradius (e. g. comparable to boundary layer width – kink)

– Finite electric field (electron inertia, electron pressure, collisionality)

– Non-fluid “compressibility”: trapped-particle effect (kink, ballooning)

• Kinetic destabilisation of MHD-stable modes

– Fast-ion destabilisation of Alfven eigenmodes: TAE, HAE, GAE, BAE . . .

– Lower MHD destabilisation thresholds: KBM

– Interaction between (marginal) MHD and fast ions: fishbones

• EM microturbulence: global profile evolution (drift-Alfven, magnetic flutter)

1. Global approach is needed (intrinsic for MHD; needed for profiles)

2. Kinetic approach is needed (to address relevant physics)


WENDELSTEIN 7-X Gyrokinetic theory

v

v||

v

r

rc

rg

O

ǫB = rg/LB ≪ 1

ǫ = ω/ωc ∼ k‖/k⊥ ∼ qδφ/T ∼ δB/B ≪ 1

PERTURBATIVE ELIMINATION OF THE FAST GYROMOTIONAlexey Mishchenko · IPP Greifswald, Stellaratortheorie

WENDELSTEIN 7-X Very Rough Timeline for CRPP-IPP GK PIC codes

• 1995: GYGLES is written (M. Fivaz with K. Appert) at CRPP

• 1997: GYGLES goes to IPP Greifswald (R. Hatzky)

• 1999: ORB is developed by Tran at CRPP (S. Parker involved)

• 1999: EUTERPE is written by G. Jost (with Appert, Tran) at CRPP

• 2001: EUTERPE goes to IPP Greifswald (V. Kornilov, R. Kleiber, R. Hatzky)

• 2002: ORB is rewritten as ORB5 (A. Bottino) at CRPP

• 2004: complete refurbishment of EUTERPE starts (R. Kleiber, R. Hatzky)

• 2005: ORB5 goes to IPP Garching as NEMORB (A. Bottino)

All the codes share the equations solved, physics addressed and thediscretisation principles applied. Deeper core routines are often very similar.

Normalisation in EUTERPE and ORB5 is almost identical.


WENDELSTEIN 7-X Gyrokinetic theory

γ = q ~A∗(~R)d~R +M

qµdθ −

(Mv2

‖

2+ µB

)dt+ qA‖~b(~x)d~x − qφ(~x)dt

• ~x = ~R + ~ρ(θ) ⇒ gyro-dependent correction is not small, when k⊥ρ ≥ 1

• Eliminate fast gyro-phase dependence from the Lagrangian

• U S E L I E T R A N S F O R M: Γ = eGγ + dS

p‖ − GK : Γ = q ~A∗d~R +B

Ωµdθ −

(Mv2

‖

2+ µB + q〈φ〉 − qv‖〈A‖〉

)dt

v‖ − GK : Γ = q ~A∗d~R +B

Ωµdθ + 〈A‖~b〉d~R −

(Mv2

‖

2+ µB + q〈φ〉

)dt

R. G. Hahm (1988), A.J. Brizard (1988, 1994, 2000), H. Qin(1998, 1999, 2000)


WENDELSTEIN 7-X

Gyrokinetic equations (v‖-formulation)

• Gyrokinetic Vlasov equation: method of characteristics∂f1s

∂t+ ~R · ∂f1s

∂ ~R+ v‖

∂f1s

∂v‖= − ~R(1) · ∂F0s

∂ ~R− v

(1)‖

∂F0s

∂v‖

• Gyrocenter trajectories: ∂〈A‖〉/∂t appears in v‖-GK!

~R = v‖~b∗ +

1

qsB∗‖

~b ×[µ∇B + qs

(∇〈φ〉 + ∂

⟨A‖⟩

∂t~b

)]

v‖ = − 1

ms

~b∗ · µ∇B − qs

ms

(~b∗ · ∇ 〈φ〉 + ∂

⟨A‖⟩

∂t

)

~B∗ = ~B +

ms

qsv‖s(∇ ×~b) +~b · ∇ × A‖~b = B∗

‖ +~b · ∇ × A‖~b

• Gyrokinetic field equations: δgy = δ(~R + ρ − ~x)∑

s=i,f

∫q2sF0s

Ts

(φ − 〈φ〉) δgy d6Z =∑

s=i,e,f

qsns , −∇2⊥A‖ = µ0

∑

s=i,e,f

j‖s


WENDELSTEIN 7-X

Gyrokinetic equations (p‖-formulation)

• Gyrokinetic Vlasov equation: method of characteristics∂f1s

∂t+ ~R · ∂f1s

∂ ~R+ v‖

∂f1s

∂v‖= − ~R(1) · ∂F0s

∂ ~R− v

(1)‖

∂F0s

∂v‖.

• Gyrocenter trajectories: ∂〈A‖〉/∂t does not appear in p‖-GK!

~R =

(v‖ −

q

m〈A‖〉

)~b∗ +

1

qB∗‖

~b ×[µ∇B + q

(∇〈φ〉 − v‖∇〈A‖〉

)]

v‖ = − 1

m

[µ∇B + q

(∇〈φ〉 − v‖∇〈A‖〉

)]·~b∗

• Gyrokinetic field equations:∫qiF0i

Ti

(φ − 〈φ〉) δ(~R + ρ − ~x) d6Z = ni − ne

βi

ρ2i

〈A‖〉i +βe

ρ2e

A‖ − ∇2⊥A‖ = µ0

(j‖i + j‖e

)


WENDELSTEIN 7-X Discretization

• “Klimontovich” representation for perturbed distribution function:

δfs( ~R, v‖, µ, t) =

Np∑

ν=1

wsν(t)δ(~R − ~Rν)δ(v‖ − vν‖)δ(µ − µν) ,

• Maxwellian distribution for all species:

F0s = n0

(m

2πTs

)3/2

exp

[−

msv2‖

2Ts

]exp

[− msv

2⊥

2Ts

]

• Finite-element discretization for fields:

φ(~x) =

Ns∑

l=1

φl(t)Λl(~x) , A‖(~x) =

Ns∑

l=1

al(t)Λl(~x) ,


WENDELSTEIN 7-X Cancellation problem

βi

ρ2i

A‖ +βe

ρ2e

A‖ − ∇2⊥ A‖ = µ0

(j‖i + j‖e

)

• The skin terms are “generated” by p‖-formulation: Not physics!

• The electron skin term can be very large

βe

ρ2e

A‖ =µ0n0e

2

me

A‖

• The adiabatic current are “generated” by p‖-formulation: Not physics!

H1 = qs(〈φ〉s−v‖〈A‖〉s

), F (ad)

e = F0e e− H1/Te ≈ − qeF0e

Te

(φ−v‖A‖

)

• Adiabatic current coincides with the skin terms Must cancel each other!

µ0j(ad)‖s = µ0qs

∫v‖ F

(ad)s d3v =

µ0n0e2

me

A‖ =βe

ρ2e

A‖


WENDELSTEIN 7-X Cancellation problem: markers vs. grid

• The current is computed with markers (in the phase space)

• The skin terms are known analytically (computed on a real-space grid)

• The numerically different representation makes the cancellation inexact andleads to the cancellation problem

• The large terms should cancel each other

∇2⊥A‖ ≪ βe

ρ2e

A‖ ⇒ j(nonad)‖i + j

(nonad)‖e ≪ j

(ad)‖e

• The physics is described by the small rest

−∇2⊥A‖ = µ0

(j(nonad)‖i + j

(nonad)‖e

)

• The error scales with δA‖ ∼ β/(k2⊥ρ

2e) ⇒

Simulations at the MHD limit (global modes, small k⊥) are a very challenging


WENDELSTEIN 7-X Solution of the cancellation problem

• The true-particle distribution function can be expressed through the gyroki-netic fs by the pullback transform: Also in quasineutrality!

f1s = f1s + S1, F0s +qs〈A‖〉ms

∂F0s

∂v‖, ωcs

∂S1

∂θ= qs

(φ − v‖A‖

)

• Ampere’s law in terms of the true-particle distribution function

−∇2⊥A‖ = µ0

∫v‖

[f1s + S1, F0s +

qs〈A‖〉ms

∂F0s

∂v‖

]δ(~R + ~ρ − ~x)d6Z

• The true-particle distribution function must be discretized with markers

f1s(Z) =

Np∑

ν=1

wνδ(Z − Zν(t)) , F0s(Z) =

Np∑

ν=1

F0s(Zν)ζνδ(Z − Zν(t))

• Particle discretization of the complete pullback transform (true-particle dis-tribution function) is to be used both in Ampere’s law and quasineutrality


WENDELSTEIN 7-X

Solution of the cancellation problem (iterative scheme)

• Ampere’s law is used to compute for A‖. But p‖-pullback depends on A‖!

−∇2⊥A‖ = µ0

∫v‖

[f1s + S1, F0s +

qs〈A‖〉ms

∂F0s

∂v‖

]δ(~R + ~ρ − ~x)d6Z

• Solution: introduce an easy-to-compute estimator for the pullback (βe/ρ2e)

(s + L)a = j ⇒ (s + L)a = j + (s − s)a ⇒ (s + L)a = j + (s − s)a

• Employ ‖s − s‖ = O(ǫ) and solve Ampere’s law iteratively

a = a0+ǫa1+ǫ2a2+. . . , (s+L)a0 = j , (s + L)a1 = j + (s − s)a0 , . . .

skl =

∫βe

ρ2e

Λk(~x)Λl(~x) d3x , jk =

Np∑

ν=1

v‖ν wν〈Λk〉ν

jk − sklan−1l =

Np∑

ν=1

v‖ν

wν +

qs〈A(n−1)‖ 〉

ms

∂

∂v‖F0s(Zν)ζν

〈Λk〉ν


WENDELSTEIN 7-X Alternative formulation for the gyrokinetic theory.

M I X E D - V A R I A B L E F O R M U L A T I O N

P U L L B A C K M I T I G A T I O N


WENDELSTEIN 7-X Mixed-variable gyrokinetics: derivation

• Split the magnetic potential into the ‘symplectic’ and ‘hamiltonian’ parts:

A‖ = A(s)‖ + A

(h)‖

• The perturbed guiding-center phase-space Lagrangian

γ = q ~A∗ ·d~R+m

qµ dθ+q A

(s)‖~b ·d~x+q A

(h)‖~b ·d~x−

[mv2

‖

2+ µB + qφ

]dt

• “Mixed” Lie transform: A(h)‖ → Hamiltonian, A

(s)‖ → symplectic structure

Γ = q ~A∗ ·d~R+m

qµ dθ+q

⟨A

(s)‖

⟩·d~R−

[mv2

‖

2+ µB + q

⟨φ − v‖A

(h)‖

⟩]dt


WENDELSTEIN 7-X Mixed-variable formulation: equations

• The corresponding perturbed equations of motion are

~R(1) =~b

B∗‖× ∇

⟨φ − v‖A

(s)‖ − v‖A

(h)‖

⟩− q

m〈A(h)

‖ 〉~b∗

v(1)‖ = − q

m

[~b∗ · ∇

⟨φ − v‖A

(h)‖

⟩+

∂

∂t

⟨A

(s)‖

⟩]− µ

m

~b × ∇B

B∗‖

· ∇⟨A

(s)‖

⟩

• An equation for ∂A(s)‖ /∂t is needed

∂

∂tA

(s)‖ +~b · ∇φ = 0

• Ampere’s law takes the form∑

s=i,e,f

βs

ρ2s

⟨A

(h)

‖

⟩s− ∇2

⊥A(h)‖ = µ0

∑

s=i,e,f

j‖1s + ∇2⊥A

(s)‖


WENDELSTEIN 7-X Mixed-variable formulation: the algorithm

1. At the end of each time step, redefine the magnetic potential splitting:

A(s)‖(new)(ti) = A‖(ti) = A

(s)‖(old)(ti) + A

(h)‖(old)(ti)

2. As a consequence, redefine A(h)‖(new)(ti) = 0

3. The new mixed-variable distribution function must coincide with its symplectic-

formulation counterpart (pullback 0-form): f1s(Zs, A(s)‖ ) = f1m(Zm, A

(s)‖ , A

(h)‖ )

f(m)1s(new)(ti) = f

(s)1s (ti) = f

(m)1s(old)(ti) +

qs 〈A(h)‖(old)(ti)〉ms

∂F0s

∂v‖

4. Proceed, explicitly solving the mixed-variable system of equations at thenext time step ti +∆t in a usual way, but using the symplectic coordinatesas the initial conditions.

5. Perform this transformation at the end of each time step.


WENDELSTEIN 7-X Mixed-variable formulation: nonlinear algorithm

SYMPLECTIC-VARIABLE SPACE

MIXED-VARIABLE SPACE

mixed-

varia

ble ∆t

mixed-

varia

ble ∆t

mixed-

varia

ble ∆t

pullback

pullback

...

f1s(Zs, A(s)‖ ) = f1m(Zm, A

(s)‖ , A

(h)‖ )

v(s)‖ = v

(m)‖ − e

m

⟨A

(h)‖

⟩

Additional nonlinear termsappear in equations of motion[R. Kleiber et al, PoP 2016](symplectic-hamiltonian equivalenceat the 2nd order)

1. Push coordinates and weights along the nonlinear mixed-variable trajectories

2. Transform coordinates into symplectic space keeping weights constant

3. Set A(s)‖(new)(ti) = A‖(ti) = A

(s)‖(old)(ti) + A

(h)‖(old)(ti) and A

(h)‖(new)(ti) = 0.


WENDELSTEIN 7-X Safety factor and SAW continuum

S I M U L A T I O N S

Toroidal Alfven EigenmodeEnergetic Particle Mode

Internal Kink ModeCollisionless Tearing Mode

Stellarators (EM drift modes)


WENDELSTEIN 7-X Tokamak configuration

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r / ra

1.7

1.75

1.8

1.85

q

Here, the gap is formed

0 0.2 0.4 0.6 0.8 1s

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ω,

x 10

6 rad

/s

PIC TAEMHD TAE

m = 10

m = 11

m =

10

m = 11

Large-aspect-ratio, circular cross-sectionsMajor radius R0 = 10 m, minor radius ra = 1 m

Magnetic field on the axis B0 = 3.0 T,Flat bulk-plasma temperature and density (βbulk ≈ 0.18%)

Toroidal mode number n = 6


WENDELSTEIN 7-X Radial structure

0 0.2 0.4 0.6 0.8 1s

0

5e-14

1e-13

1.5e-13

|φ|

m = 9 m = 12

m = 10

m = 11

0 0.2 0.4 0.6 0.8 1s

0

5e-21

1e-20

1.5e-20

2e-20

2.5e-20

|A|||

m = 9m = 12

m = 10

m = 11

Radial pattern resulting from the PIC simulations(in some particular point of time)

It resembles a typical TAE structure.


WENDELSTEIN 7-X ITPA benchmark

0 200 400 600 800T/ keV

0

10

20

30

γ/10

3 s-1

CAS3D-K (ZOW)GYGLES (FLR)CKA-EUTERPE (FLR)MEGA (FLR)NOVA-K (FLR)LIGKA (FLR)EUTERPE (FLR)

1×104

1×105

γ/rad s-1

1

10

δB/B

0 / 1

0-3

ORB5 γd=0EUTERPE γd=0

FLU-EUTERPE, FLR, γd=4.103s

-1

FLU-EUTERPE, ZLR

CKA-EUTERPE γd=4.103s

-1

Successful worldwide benchmark (ITPA framework)Linear with and without FLR

Nonlinear (EUTERPE vs. ORB5; GK vs. reduced models)


WENDELSTEIN 7-X TAE-EPM transition

0 2e+05 4e+05 6e+05 8e+05 1e+06 1.2e+06Tf [eV] (βf kept const)

0.34

0.36

0.38

0.4

0.42

0.44

0.46

|ω|

(MH

z)

continuum

gap

EPM (II)

unstable TAE

0 2e+05 4e+05 6e+05 8e+05 1e+06 1.2e+06Tf [eV] (βf kept const)

0

10

20

30

γ (

kHz)

cont

inuu

mga

p

FOW stabilization

resonance (vthf

= vA

/ 3)

kperp

ρE ~ 1

0 0.2 0.4 0.6 0.8 1s

0

2e-13

4e-13

6e-13

|φ|

m = 9

m = 10

m = 11

m = 12

Dependency on the fast-particle temperature (βf = 0.134% kept constant)Destabilization is most effective near the resonance vthf ≈ vA/3

At large Tf , finite-orbit-width (FOW) stabilization is seenAt smaller Tf (larger nf to keep βf constant), an EPM appears


WENDELSTEIN 7-X Safety factor and SAW continuum

0 0.2 0.4 0.6 0.8 1s (sqrt norm. tor. flux)

1.5

2

2.5

safe

ty f

acto

r

q(r) = 1.71 + 0.16 (r / ra)2 (ITPA)

q(r) = 1.5 + (r / ra)2 (new!)

0 0.2 0.4 0.6 0.8 1s

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ω,

x 1

06

rad/s

PIC TAEMHD TAE

m = 10

m = 11

m=

10

m=

11

0 0.2 0.4 0.6 0.8 1sqrt norm tor. flux

0

2e+05

4e+05

6e+05

8e+05

ω ,

rad/s

m =

10

m =

10

m = 11

m = 11

TAE (MHD)

TAE (kin, Ti=9 keV)

KAWKAW

m = 12

m = 13m

= 9

m = 14

m = 10

• Simulation Concept

– Modify the ITPA benchmark parameter safety factor (magnetic shear)

– But: use flat bulk plasma density and temperature (same vA as ITPA)

• New physics (compared to ITPA)

– The resulting continuum is much more complicated than ITPA

– The gap is deformed and involves many modes; continuum resonances


WENDELSTEIN 7-X Gyrokinetic Eigenmode

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1m= 10 n= -6 Rem= 11 n= -6 Rem= 12 n= -6 Rem= 13 n= -6 Rem= 9 n= -6 Re

EV = 6.735932e-03 + i -7.948117e-07

KAW KAW

KAW

0 0.2 0.4 0.6 0.8 1

0

0.5

1m=10 (SVD)m=11 (SVD)m = 10 (MHD)m = 11 (MHD)

KAW

TAE

2e+05 3e+05 4e+05 5e+05 6e+05 7e+05Tf, eV

0

10000

20000

30000

γ, r

ad/s

GK PIC (Ti = 9 keV)ideal hybrid (CKA)GK PIC (Ti = 1 keV)

global kinetic structureis formed

ideal TAE has less interaction with fast ions?

KAW stabilised (fast-ion FOW + E||)

• Electrostatic radial pattern at different times

• KAWs are excited at continuum resonances

• Resonantly excited KAWs mix with the global eigenmode (interference)

• Kinetic mode is wider comparing to ideal mode (KAW admixtures)


WENDELSTEIN 7-X Internal kink mode

0 0.2 0.4 0.6 0.8 1r / r

a

0

0.2

0.4

0.6

0.8

1

|φ|,

arb

itrar

y un

its

ideal MHD (shooting method)GK PIC (T

i = 200 eV)

GK PIC (Ti = 5000 eV)

0 5e-05 0.0001 0.00015 0.0002t, s

1e-14

1e-13

1e-12

|Re

φ|

Flat plasma density

Damped Alfven continuum

Kink instability

0.5 0.6 0.7 0.8rc / r

a

10000

20000

30000

40000

γ, r

ad/s

MHD, shooting methodGK PIC (T

i = 200 eV)

GK PIC (Ti = 5000 eV)

Ideal-MHD internal kink mode equation:

d

dr

([µ0min0γ

2

︸︷︷︸small

+ (~k · ~B)2

]r3 dξ

dr

)− g(r)ξ = 0 , ξ ∝ φ/r

• Intertial layer: plasma inertia can compete with magnetic tension

• Poloidal plasma rotation with vθ ∝ ∂φ/∂r resolves the MHD singularity

• The width of the intertial layer λH ∝ δWMHD

• “MHD regime”: λH exceeds all microscopic kinetic scales (ρi, δe etc)


WENDELSTEIN 7-X Collisionless m = 1, n = 1 tearing mode

0 0.2 0.4 0.6 0.8 1

r/ra

0

0.2

0.4

0.6

0.8

1

|φ|,

arbi

trar

y un

its

LTe

= 1.1L

Te = 1.5 Fine-scale structure

0.46 0.48 0.5 0.52 0.54

r/ra

0

0.5

1

|φ|,

arbi

trar

y un

its

LTe

= 1.1L

Te = 1.5

rc + ρi

rc - ρi

rc - δe r

c + δe

rA(1.1)r

A(1.5)

0.4 0.5 0.6 0.7rc

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

γ,

rad/

s

PICTheory, γ ~ 1/rc

“Collisionless m = 1 tearing mode” [Porcelli 91]

γ =sq(rc)vA(rc)

R0

(δeρ2i )

1/3

rc, vA =

B20√

µ0min0

, sq =r

q

dq

dr

• Mode structure of drift-kink mode: fine scale feature at resonant flux surface

• Sub-gyro scales are involved – Kinetic Alfven Waves (KAW)resonantly excited by the drift-kink mode

• Mode dynamics as a combination of poloidal rotation,reconnection and the KAW excitation (“continuum damping”)


WENDELSTEIN 7-X Internal kink mode in tokamak (gyrokinetic, GYGLES)

0 0.2 0.4 0.6 0.8 1sqrt poloidal flux

0

5e-07

1e-06

1.5e-06

2e-06

2.5e-06

3e-06

Re

φ

m = -3m = -2m = -1m = 0m = 1m = 2m = 3

CASE 1

0 2e-05 4e-05 6e-05 8e-05 0.0001t, s

1e-18

1e-15

1e-12

1e-09

1e-06

|Re

φ|

m = -1m = 0m = 1

CASE 1

0.2 0.4 0.6 0.8 1al_den

0

50000

1e+05

1.5e+05

2e+05

γ, r

ad/s

gyg, gk, B = 1Teut, efluid, B = 1Teut, gk, B = 1Teut, efluid, B = 3Tgyg, gk, B = 3T

0 0.002 0.004 0.006 0.008 0.01n0f / n0i, n0i = 3.3 x 10

19 m

-3

1.3e+05

1.4e+05

1.5e+05

1.6e+05

1.7e+05

1.8e+05

1.9e+05

γ, r

ad/s

Internal kink mode in tokamak geometry (R0/ra = 10); strong MHD drive(pressure+current); GK-efluid comparison; fast-ion effect: MHD drive p′

fastAlexey Mishchenko · IPP Greifswald, Stellaratortheorie

WENDELSTEIN 7-X Mixed-variable simulations in LHD-like geometry

0 2000 4000 6000 8000ωci t

0.001

1

1000

1e+06

1e+09

1e+12

Re

φ

s = 0.4s = 0.5s = 0.6

−43−41

−39−37

−35−33

−31−29

−27−83

−63−43

−23−3

1737

0

0.2

0.4

0.6

0.8

1

mn

Φm

,n

Clean instability;ballooning in Fourier

0 0.2 0.4 0.6 0.8 1norm. toroidal flux

0

0.2

0.4

0.6

0.8

1

Re

φ

m = -42m = -41m = -40m = -39m = -38m = -37m = -36m = -35m = -34m = -33m = -32

Ballooning mode(radial pattern)

A. Mishchenko et al, Phys. Plasmas 21, 092110 (2014)Alexey Mishchenko · IPP Greifswald, Stellaratortheorie

WENDELSTEIN 7-X Mixed-variable simulations in W7-X stellarator

0 1000 2000 3000 4000 5000 6000ωci t

0.0001

0.001

0.01

0.1

1

|Re

φ|

s = 0.5s = 0.6s = 0.7

0 0.2 0.4 0.6 0.8norm. tor. flux

0

0.2

0.4

0.6

0.8

1

|φ|

m = 48m = 51m = 52m = 53m = 54m = 55m = 56m = 57m = 58m = 59m = 60m = 61m = 62m = 63m = 64m = 65m = 68

Electromagnetic ITG;left: evolution;right: mode structure

left: cross-sectionright: flute-like mode


WENDELSTEIN 7-X Outlook: going mainstream

• Cancellation problem has prohibited large-scale effort on GK PIC simulations(Reynders 1992, Cummings 1995)

• Reduced models have been widely used to circumvent the problem:hybrid kinetic-MHD, fluid-electron models

• Limitations of reduced models: closure issues, no micro-tearing physics

• A lot of work has been done to mitigate the cancellation problem:control variate, mixed-variable pullback scheme

• The mitigation schemes can be used both in linear and nonlinear regime

• The mitigation schemes have been validated on many examples,including the international ITPA benchmark

Fully gyrokinetic PIC simulation schemes approach mainstream


Documents

Gyrokinetic PIC simulations of global electromagnetic modes · Keywords: electromagnetic gyrokinetics, cancellation problem, v k -formulation vs. p k -formulation vs. mixed-variable