Gyrokinetic simulations for tokamaks and stellarators
R. Kleiber1, M. Borchardt1, M. Cole1, T. Feher2, R. Hatzky2,A. Konies1, A. Mishchenko1, J. Riemann1
1Max-Planck-Institut fur Plasmaphysik, D-17491 Greifswald, Germany
2Max-Planck-Institut fur Plasmaphysik, D-85748 Garching, Germany
February 26, 2015
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the
European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053. The views
and opinions expressed herein do not necessarily reflect those of the European Commission.
R. Kleiber 7th IAEA TM 1 / 43
Introduction
Interest in global (full-volume) gyrokinetic particle-in-cellsimulations for stellarators.
Two main areas of activity:
Microinstabilities and turbulenceMHD modes and their interaction with fast particles
Even linear electrostatic simulations for stellarators can becomedifficult due to high necessary resolution.
For electromagnetic perturbations already linear simulations in atokamak can become very difficult(especially in the MHD limit, small k⊥ρ)⇒ necessity of algorithm development
MHD modes provide an excellent testing ground for codes since avery high quality of numerics is required.
R. Kleiber 7th IAEA TM 2 / 43
Overview
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 3 / 43
Electromagnetic simulations: Hierarchy of models
Fully gyrokinetic simulations are very time-consuming and difficult.Develop simplified models: Sacrifice physics for gain in speed.⇒ EUTERPE, FLU-EUTERPE, CKA-EUTERPE
more
numerically
robust
and economical
more physically
complete
bulk plasma
GK fast ion
fluid electrons
(FLU
−EU
TER
PE)
power transfer
MHD
GK fast ions
(EUTERPE)
(CKA−EUTERPE)
bulk ions and electrons, fast ions
fluid bulk plasma
electromagnetic gyrokinetics for
GK bulk ions and fast ions
R. Kleiber 7th IAEA TM 4 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 5 / 43
Full electromagnetic gyrokinetic model
All equations are derived from a Lagrangian via a variational principlein ~R, p‖, µ, α coordinates (consistency and conserved quantities)
L =∑
species
∫ f
[(q ~A+ p‖~b)· ~R+
m
qµα−
p2‖
2m− µB + q〈
p‖
mA‖ − Φ〉
]+
f0
[− q2
2m〈A‖〉2 +
m
2B2(∇⊥Φ)2
]dV dW − 1
2µ0
∫(∇⊥A‖)2dV
δB‖ neglected
〈·〉: gyro-averaging operator
dW = B∗‖ dp‖ dµ dα, dV =√g ds dϑ dϕ, B∗‖ = B + 1
qp‖~b·∇×~b
adiabatic electron model: m2B2 (∇⊥Φ)2 =⇒ e2
2kBTe0(Φ− Φ)2
neglect q〈Φ〉 for electrons
R. Kleiber 7th IAEA TM 6 / 43
Cancellation problem
Electromagnetic simulations are hampered by the cancellation problem.
Moment of p‖ does not give a physical current.Ampere’s law in p‖ formulation (dc: collisionless skin depth)
− 1
β∇2⊥A‖ +
1
d2c
A‖ = j‖i + j‖e
1d2cA‖ 1
β∇2⊥A‖ (especially for MHD modes, since k⊥ρi ≈ 0)
j‖e contains adiabatic part j‖e,ad =∫ev‖
[eT f0v‖A‖
]d3v
Numerical cancellation of the two large terms 1d2cA‖ and j‖e,ad
Different numerical representations: matrix ⇔ particles
Cancellation problem mitigated by adjustable control variate method.[Hatzky et al. 2007]
Iterative method for determining a control variateAllows strong reduction in the number of necessary markers
Nevertheless, simulations require a very long time (small time step) orare still not feasible for certain parameters.
R. Kleiber 7th IAEA TM 7 / 43
Pullback scheme I [Mishchenko et al. 2014]
Equations of motion in v‖ formulation (slab version for simplicity)
~R = v‖~b+1
B~b×∇
[φ− v‖A‖
]v‖ = − q
m
[∇‖φ+
∂A‖
∂t
]Use splitting A‖ = As
‖ +Ah‖ , introduce u‖ = v‖ + q
mAh‖ and
simplify equations by using the resulting freedom to postulate∂As‖
∂t +∇‖φ = 0
~R = u‖~b+1
B~b×∇
[φ− v‖(As
‖ +Ah‖)]− q
mAh‖~b
u‖ =q
mu‖∇‖Ah
‖
Field equations∂As‖
∂t+∇‖φ = 0 − 1
β∇2⊥(Ah
‖ +As‖) + SAh
‖ = j
R. Kleiber 7th IAEA TM 8 / 43
Pullback scheme II
Kinetic equation:∂δf
∂t= − ~R·∇f0 −
q
mu‖∇‖Ah
‖∂f0
∂u‖
second term can be reduced by keeping Ah‖ small ⇒ idea of restarting
1 Start with Ah‖ = 0, i.e. u‖ = v‖
2 Integrate system in u‖-space: Ah‖ develops, leading to u‖ 6= v‖
3 Transform from u‖-space to v‖-space (use pullback)
δf(v‖) = δf(u‖) +q
mAh‖∂f0
∂u‖, v‖ = u‖ −
q
mAh‖
4 Keep the full solution by setting As‖ to As
‖ +Ah‖ and restart
Scheme allows for much larger time steps and enables simulations inparameter regimes not accessible before.
R. Kleiber 7th IAEA TM 9 / 43
Pullback scheme II
Kinetic equation:∂δf
∂t= − ~R·∇f0 −
q
mu‖∇‖Ah
‖∂f0
∂u‖
second term can be reduced by keeping Ah‖ small ⇒ idea of restarting
1 Start with Ah‖ = 0, i.e. u‖ = v‖
2 Integrate system in u‖-space: Ah‖ develops, leading to u‖ 6= v‖
3 Transform from u‖-space to v‖-space (use pullback)
δf(v‖) = δf(u‖) +q
mAh‖∂f0
∂u‖, v‖ = u‖ −
q
mAh‖
4 Keep the full solution by setting As‖ to As
‖ +Ah‖ and restart
Scheme allows for much larger time steps and enables simulations inparameter regimes not accessible before.
R. Kleiber 7th IAEA TM 9 / 43
Pullback scheme II
Kinetic equation:∂δf
∂t= − ~R·∇f0 −
q
mu‖∇‖Ah
‖∂f0
∂u‖
second term can be reduced by keeping Ah‖ small ⇒ idea of restarting
1 Start with Ah‖ = 0, i.e. u‖ = v‖
2 Integrate system in u‖-space: Ah‖ develops, leading to u‖ 6= v‖
3 Transform from u‖-space to v‖-space (use pullback)
δf(v‖) = δf(u‖) +q
mAh‖∂f0
∂u‖, v‖ = u‖ −
q
mAh‖
4 Keep the full solution by setting As‖ to As
‖ +Ah‖ and restart
Scheme allows for much larger time steps and enables simulations inparameter regimes not accessible before.
R. Kleiber 7th IAEA TM 9 / 43
Pullback scheme II
Kinetic equation:∂δf
∂t= − ~R·∇f0 −
q
mu‖∇‖Ah
‖∂f0
∂u‖
second term can be reduced by keeping Ah‖ small ⇒ idea of restarting
1 Start with Ah‖ = 0, i.e. u‖ = v‖
2 Integrate system in u‖-space: Ah‖ develops, leading to u‖ 6= v‖
3 Transform from u‖-space to v‖-space (use pullback)
δf(v‖) = δf(u‖) +q
mAh‖∂f0
∂u‖, v‖ = u‖ −
q
mAh‖
4 Keep the full solution by setting As‖ to As
‖ +Ah‖ and restart
Scheme allows for much larger time steps and enables simulations inparameter regimes not accessible before.
R. Kleiber 7th IAEA TM 9 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 10 / 43
MHD hybrid model I
Use simplified way of describing the interaction of Alfven modeswith fast particles.
CKA code (A. Konies) solves the linearised reduced MHDeigenvalue problem.
Resulting mode structure and frequency used as a fixed input fora particle simulation ⇒ linear
Allow trajectories to react ⇒ nonlinear
Perturbative scheme
A
particle motion
CKA EUTERPE
eigenvalue problemMHD equation
Φ,
ω
gyrokintic equation
R. Kleiber 7th IAEA TM 11 / 43
MHD hybrid model II
Time derivative of quasineutrality equation
∂
∂t∇·[Mn
B2∇⊥φ
]= −∇·
[−~b 1
µ0∇2⊥A‖ + j
(0)‖
(~b×~κB
A‖ −~b×∇A‖
B
)
+ p(1)bulk
(~b×~κB
+~b×∇BB2
)+ p
(1)‖,fast
~b×~κB
+ p(1)⊥,fast
~b×∇BB2
]CKA part, EUTERPE part
Simple closure for bulk pressure (neglect compressibility andanisotropy)
∂p(1)bulk
∂t= −
~b×∇φB·∇p(0)
bulk
MHD closure (vanishing E‖)
∂A‖
∂t= −~b·∇φ
R. Kleiber 7th IAEA TM 12 / 43
MHD hybrid model: Amplitude equations I
Alfven modes are stable eigenmodes with frequency ω0
φ(~r, t) = φ0(~r) eiω0t
A‖(~r, t) = A‖0(~r) eiω0t
p(1)(~r, t) = p0(~r) eiω0t
Allow for complex time dependent amplitude
φ(~r, t) = φ(t) φ0(~r) eiω0t
A‖(~r, t) = A‖(t) A‖0(~r) eiω0t
p(1)(~r, t) = p(t) p0(~r) eiω0t
R. Kleiber 7th IAEA TM 13 / 43
MHD hybrid model: Amplitude equations II
∂φ
∂t= iω0(A‖ − φ) + 2(γ − γd)φ
∂A‖
∂t= −iω0(A‖ − φ)
γ =T
2Wγd: ad-hoc damping
Wave-particle energy transfer:
T = −∫
dWdV f(1)fast
[1
B~b×(mv2
‖~κ+ µ∇B)·∇φ∗]
Wave energy:
W =
∫dV
Mn
B2|∇⊥φ|2
Linear model: Growth rate of the mode given by γNonlinear model: Use Re(φ),Re(A‖) for fast particle trajectories
R. Kleiber 7th IAEA TM 14 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 15 / 43
Electron fluid model [Cole et al. 2014]
Linearised electron continuity equation (v‖ formulation)
∂ne
∂t+ n0
~B ·∇ue‖
B+B~vE×B ·∇
n0
B+ (∇×A‖~b)·∇
n0ue‖0
B+
n0(2~v∗ − ~vE×B)·∇BB
+∇×BB2
·(−1
e∇pe + n0∇φ
)= 0
with ~v∗ :=~b×∇peen0B
Ampere’s law and quasineutrality
ue‖ =1
en0
(ji‖ +
1
µ0∇2⊥A‖
)−∇·
(min0
eB2∇⊥φ
)= ni−ne
MHD (E‖ = 0) and pressure closure
∂A‖
∂t= −~b·∇φ ∂pe
∂t= −~vE×B ·∇pe0
More physics can be included by improving the closures.
R. Kleiber 7th IAEA TM 16 / 43
Global gyrokinetic code EUTERPE
δf particle-in-cell code
Global simulation domain: full-volume
3D stellarator equilibria (from VMEC equilibrium code)
Multiple kinetic species (ions, electrons, fast ions/impurities)
Electrostatic/electromagnetic
Linear/nonlinear
Pitch-angle collision operator (⇒ neoclassics)
⇒ Code platform for implementation of different models(GYGLES: basically a reduced, linear, axisymmetric version of EUTERPE)
R. Kleiber 7th IAEA TM 17 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 18 / 43
W7-X Toroidal Alfven eigenmode
W7-X equilibrium withβ ≈ 3%
Flat bulk plasma profilesnbulk = 1020 m−3
TAE mode found (CKA)
Alfven spectrum
0 0.2 0.4 0.6 0.8
norm. toroidal flux
0
0.2
0.4
0.6
0.8
ω/
ωA
TAE(m=6,7) mode
m = 6m = 7
m = 4
m = 5
m = 4m = 5
TAE mode
0 0.2 0.4 0.6 0.8 1
norm. toroidal flux
0
0.5
1
Re
(φ),
no
rma
lize
d
m= 7 n=-6
m= 6 n=-6
m= 5 n=-6
m= 4 n=-1 m= 5 n=-1
fast-ion density
R. Kleiber 7th IAEA TM 19 / 43
W7-X TAE: Interaction with fast particles
Destabilise TAE mode by fastparticle interaction:Maxwellian fast particlesnfast profile fixed withnfast,0 = 1017 m−3
FLR effects important
Reaction on external radialelectric field( ~E0 = Es∇s, Es = const.)
ME =1
vth〈|~E0× ~BB2
|〉
0 5e+05 1e+06 1.5e+06 2e+06
Tf / eV
0
500
1000
1500
2000
2500
3000
γ /
(1
/s)
no FLRwith FLR
-0.012 -0.008 -0.004 0 0.004 0.008 0.012
ME
1500
2000
2500
3000
γ / (
1/s
)
Ion root
Electron root
[Mishchenko et al. 2014]
R. Kleiber 7th IAEA TM 20 / 43
W7-X TAE: Interaction with ICRH fast particles
Fast ions from ICRH
Very simple model for ICRH:anisotropic Maxwellianpower deposition localised inspace
Less-pronounced finiteorbit-width effects
Strong influence of anisotropyon growth rate (more/lessparallel temperature)
0 0.2 0.4 0.6 0.8 1
norm. toroidal flux
0
0.5
1
1.5
2
Re
φ, n
orm
alize
d
m= 7 n=-6 m= 6 n=-6 m= 5 n=-6 m= 4 n=-1 m= 5 n=-1
min
. perp
. temp.
ICRH
minority density
0 500 1000 1500 2000
Tmax
(minority) / keV
0
1000
2000
3000
4000
γ / (
1/s
)
ICRH, no FLRICRH, with FLRMaxw., no FLRMaxw., with FLR
R. Kleiber 7th IAEA TM 21 / 43
Nonlinear CKA-EUTERPE: amplitude development
ITPA benchmark case:circular A = 10 tokamak,(m,n) = (10/11,−6) TAE
Without artificial dampingmode amplitude does notsaturate.
Feature robust with respect totime step and particle number:no numerical artefact
Artificial damping necessary(γd ≈ 2.5 · 103 s−1) in order toreach saturation (damping ofsame order also used by othercodes).
no damping
WENDELSTEIN 7-X
Max-Planck-Institut für Plasmaphysik, EURATOM Association
potential does not saturate
0 200 400 600 800 1000 1200 1400−40
−35
−30
−25
−20
−15
−10
−5
0All modes
t, [103 Ω−1]
log(Φ
)
0 200 400 600 800 1000 1200 14000
0.05
0.1
0.15
0.2
0.25Selected modes
t, [103 Ω−1]
|Φ|
A. Könies, R. Kleiber, T. Fehér, M. Borchardt, R. Hatzky, A. Mishchenko · Max-Planck-Institut für Plasmaphysik
damping
WENDELSTEIN 7-X
Max-Planck-Institut für Plasmaphysik, EURATOM Association
saturated electrostatic potential
0 100 200 300 400 500 600 700−35
−30
−25
−20
−15
−10
−5
0All modes
t, [103 Ω−1]
log(Φ
)
0 100 200 300 400 500 600 7000
0.01
0.02
0.03
0.04
0.05
0.06Selected modes
t, [103 Ω−1]
|Φ|
damping value (GYGLES): γdG = 3.9 · 103s−1
damping value (LIGKA- P. Lauber): γdL = 2.3 · 103s−1
damping value set: γd = 2.5 · 103s−1
A. Könies, R. Kleiber, T. Fehér, M. Borchardt, R. Hatzky, A. Mishchenko · Max-Planck-Institut für PlasmaphysikR. Kleiber 7th IAEA TM 22 / 43
Nonlinear CKA-EUTERPE
For small growth-rates theory(Berk-Breizmann model)predicts relationship betweensaturation amplitude andgrowth rate: δB/B ∼ γ2
eff
Scaling confirmed by othercodes
Symmetric frequency chirpingobserved.
1000 10000 1e+05
γeff
/s-1
0.001
0.01
0.1
1
10
δB
rad/B
0 /
10
-3
CKA-EUTERPE γd= 1.05.10
4 s
-1
CKA-EUTERPE γd= 1.33.10
4 s
-1
CKA-EUTERPE γd= 2.5.10
3 s
-1
200 400 600 800 1000 1200 1400
−1.9
−1.8
−1.7
−1.6
−1.5
−1.4
−1.3
−1.2
−1.1
−1
x 10−3
t, [103 Ω
−1]
ω / Ω
*
R. Kleiber 7th IAEA TM 23 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 24 / 43
FLU-EUTERPE benchmarks
Internal kink in a screw pinch.Scan over position of q = 1 surface.
0.5 0.6 0.7 0.8
rc / r
a
10000
20000
30000
40000
γ,
ra
d/s
MHD, shooting methodGK PIC (T
i = 5000 eV)
FLU-EUTERPE (Ti = 5000 eV)
ITPA benchmark: EUTERPE, FLU-EUTERPE and CKA-EUTERPE
0 2e+05 4e+05 6e+05 8e+05
Tf [eV] (n
0f fixed)
0.36
0.38
0.4
0.42
0.44
0.46
ω, x
10
6 r
ad/s gap
continuum
continuum
0 2e+05 4e+05 6e+05 8e+05
Tf [eV] (n
0f fixed)
0
10000
20000
30000
γ,
rad
/s
EUTERPEFLU-EUTERPECKA-EUTERPE
R. Kleiber 7th IAEA TM 25 / 43
Internal tokamak kink mode
Finite ∇T used to destabilise themode.
In fluid limit (no gyrokinetic ions),direct comparison possible withMHD code CKA:γFLU = 1.29× 106 s−1
γCKA = 1.27× 106 s−1
Strong stabilisation withJET/ITER-like aspect ratio andelongation.
Further stabilisation with inclusionof bulk ion gyrokinetic effects.
0 0.2 0.4 0.6 0.8 1r, m
0e+00
2e-04
4e-04
6e-04
8e-04
Ele
ctro
stat
ic p
ote
nti
al
MHD m=0MHD m=1MHD m=2 GK PIC m=0GK PIC m=1GK PIC m=2
0 2.5 5 7.5 10Aspect ratio
0
2.5e+05
5e+05
7.5e+05
1e+06
1.25e+06
1.5e+06
1.75e+06
2e+06
rad/s
Bulk fluid (γ)Kinetic ions (γ)Kinetic ions (ω)
[Cole et al. 2014]
Resistive layer physics lost but tokamak simulations become practical.R. Kleiber 7th IAEA TM 26 / 43
Fast particle effects: Linear fishbone
Maxwellian fast species (D),Tfast = 300 keV
Initial stabilisation of themode (perturbative effect)overcome by unstable branch.
Frequency jump with onset offast particle destabilisation.
Non-perturbative modestructure for larger fastparticle density.
0 0.01 0.02 0.03 0.04 0.05ρ
f = n
f/n
i
0
50000
1e+05
1.5e+05
2e+05
2.5e+05
Gro
wth
rat
e (r
ad/s
)
Growth rateFrequency
ρf = 0 ρf = 0.01 ρf = 0.04
R. Kleiber 7th IAEA TM 27 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 28 / 43
Limits of fluid models
For ITPA case TAE had nocontinuum interaction.
Use steeper q-profile⇒ continuum interaction.
MHD mode structure stronglymodified by interaction withKAW. [Cole et al. 2015]
Change of mode structure notcaptured by CKA-EUTERPE.
FLU-EUTERPE:follows change in mode structurebut break-down of fluid closure ?
frequency
0 0.2 0.4 0.6 0.8 1
sqrt 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)
m =
12
m =
13m
= 9
m =
14
m =
10
growth rate
2e+05 3e+05 4e+05 5e+05 6e+05 7e+05T
f eV
0
10000
20000
30000
40000
50000
γ, ra
d/s
EUTERPE, mixed variablesEUTERPE, conventional schemeFLU-EUTERPE (GK bulk, fast ions)
FLU-EUTERPE (GK fast ions)
CKA-EUTERPE
R. Kleiber 7th IAEA TM 29 / 43
Pull-back scheme: simulations for stellarator
Electromagnetic ITG for LHD-like configuration(βeq = 1.5%, βe = 0.85%)
Standard δf scheme:
Numerical instabilityquickly develops
Wrong mode structure
New scheme:
Simulation stays stable
Clean mode structure
[Mishchenko et al. 2014]
0 0.2 0.4 0.6 0.8 1
norm. toroidal flux
-0.5
0
0.5
Re φ
−43−41
−39−37
−35−33
−31−29
−27
−83
−63
−43
−23
−3
17
37
0
0.2
0.4
0.6
0.8
1
mn
Φm
,n
0 0.2 0.4 0.6 0.8 1
norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
Re φ
−43−41
−39−37
−35−33
−31−29
−27
−83
−63
−43
−23
−3
17
37
0
0.2
0.4
0.6
0.8
1
mn
Φm
,n
R. Kleiber 7th IAEA TM 30 / 43
1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model
2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators
R. Kleiber 7th IAEA TM 31 / 43
Simplified profiles for LT -Ln-scans
For global simulations profiles are a critical issue: One gains afunctional degree of freedom but looses flexibility to produceparameter sequences.
Simplified profiles:piece-wise lineard lnTi,e
ds andd lnni,e
ds
(s = Ψtor(r)Ψtor(0)).
Further simplification:keep equilibriumconfiguration fixed andchange parameters(LT, Ln) only in thesimulations.
0 0.2 0.4 0.6 0.8 10
1
2
temperature & density (η = 2.0)
sF
j / F *
F1=T
i (T
e=T
*)
F2=n
i,e
0 0.2 0.4 0.6 0.8 10
2
4
6
8
s
p / p
*
total pressure
EUTERPEVMEC
0 0.2 0.4 0.6 0.8 1
−4
−3
−2
−1
0logarithmic derivatives (η = 2.0)
s
d( ln
Fj )
/ ds
F1=T
i
F2=n
i,e
R. Kleiber 7th IAEA TM 32 / 43
Quasi-realistic profiles for LT -Ln-scans
Quasi-realistic: simplified but derived from equilibrium(assuming finite residual pressure δp0)
p
p0= 1− 2s+ s2 +
δp0
p0
Ti,e
T0=
(1
2
p
p0
)1−χ, (Te = Ti)
ni,e
n0=
(1
2
p
p0
)χ, (ne = ni)
ηi =1− χχ
= const.
⇒ ηi-scan implies LT,n-scan
Advantage: not necessary torecalculate equilibrium for newparameter settings
0 0.2 0.4 0.6 0.8 10
1
2
temperature & density (η = 2.0)
s
Fj /
F *
F1=T
i,e
F2=n
i,e
0 0.2 0.4 0.6 0.8 10
2
4
6
8
s
p / p
*
total pressure
EUTERPEVMEC
0 0.2 0.4 0.6 0.8 1
−3
−2
−1
0logarithmic derivatives (η = 2.0)
s
d( ln
Fj )
/ ds
F1=T
i,e
F2=n
i,e
R. Kleiber 7th IAEA TM 33 / 43
ITG stability diagrams (simplified profiles)
W7-X(β = 2%, T∗ = 1 keV)
0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
W7−X ( β=2% ; T*=1keV )
ηi
γ (
vT/a
)
a/Ln = 1.41
a/Ln = 2.82
a/Ln = 4.23
a/Ln = 5.64
a/Ln = 7.05
LHD(β = 1.5%, T∗ = 1 keV, R0 = 3.75 m)
0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
LHD ( β=1.5% ; T*=1keV )
ηi
γ (
vT/a
)
a/Ln = 1.41
a/Ln = 2.82
a/Ln = 4.23
a/Ln = 5.64
a/Ln = 7.05
Clear onset of linear ITG instability for ηi ≥ 1 observed.
Similar growth rates for W7-X and LHD.
R. Kleiber 7th IAEA TM 34 / 43
W7-X: ITG mode pattern
W7-X (β = 2%, T∗ = 1 keV)
Electrostatic potential (absolute values) normalized to maximum
s = 0.5
a/LT = 1.41, a/Ln = 0.0, (m0, n0) = (210,−190)
ITG mode resides in region of unfavourable curvature.
High-shear region or ’helical edge’ avoided.
R. Kleiber 7th IAEA TM 35 / 43
LHD: ITG mode pattern
LHD (β = 1.5%, T∗ = 1 keV, R0 = 3.75 m)
electrostatic potential (absolute values) normalized to maximum at s = const.
ηi = 2, s = 0.61, γ = 0.12 vT/a ηi = 1, s = 0.75, γ = 0.02 vT/a
Helical edge less pronounced than in W7-X
R. Kleiber 7th IAEA TM 36 / 43
LT -Ln-scan for simplified profiles
LHD (β = 1.5%, T∗ = 1 keV)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8
gam
ma [v_T
/a]
a/L_T
a/L_n=0.00a/L_n=1.41a/L_n=2.82a/L_n=4.23a/L_n=5.64a/L_n=7.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8
gam
ma [v_T
/a]
a/L_n
a/L_t=0.50a/L_t=1.41a/L_t=2.82a/L_t=4.23a/L_t=5.64a/L_t=7.05
R. Kleiber 7th IAEA TM 37 / 43
LT -Ln-scan for quasi-realistic profiles
LHD (β = 1.5%, T∗ = 1 keV)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8
gam
ma [v_T
/a]
a/L_T
a/L_n = 0.00a/L_n = 1.41a/L_n = 2.82a/L_n = 4.23a/L_n = 5.64a/L_n = 7.05a/L_p = 4.71
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8
gam
ma [v_T
/a]
a/L_n
a/L_t = 0.50a/L_t = 1.41a/L_t = 2.82a/L_t = 4.23a/L_t = 5.64a/L_t = 7.05
a/L_p = 4.71
Quasi-realistic profile scan yields ’configuration characteristic’.
Consistent profiles are a unique challenge of global simulations.
Reasonable assumptions about residual pressure are crucial.
R. Kleiber 7th IAEA TM 38 / 43
Radial electric field effects
W7-X (β = 2%, T∗ = 1 keV)
growth rates
−5 0 5
0.03
0.035
0.04
0.045
0.05
<M>ExB
/ 10−3
γ (
vT/a
)
model A
ion root
model B
frequencies
−5 0 50
0.2
0.4
0.6
0.8
1
1.2
1.4
<M>ExB
/ 10−3
ωm
od
es (
vT/a
)
model A
ion root
model B
Two different radial electric field models used (variation of α):
Physical: Es = αp′iqini⇒ model A
Simple: Es = −α⇒ model B
Asymmetric damping effect
R. Kleiber 7th IAEA TM 39 / 43
Conclusions
EUTERPE: global (full-volume) gyrokinetic particle-in-cell codefor stellarators.
Hierarchy of models implemented:MHD hybrid, electron fluid, fully gyrokinetic.
New numerical schemes allow faster and more robustelectromagnetic simulations.
Examples:
Fast particle interaction with TAE and internal kink mode.Electrostatic ITG with (quasi-)realistic profiles for stellarators.
Progress, especially for electromagnetic simulations, has beenmade but further testing is necessary.
R. Kleiber 7th IAEA TM 40 / 43
R. Kleiber 7th IAEA TM 41 / 43
ITPA benchmark case
International benchmark case for Alfven wave fast-particle interaction.
Circular tokamak A = 10, R = 10 m, B = 3 T, bulk ions: D
ne = 2 · 1019 m−3,Ti = Te = 1 keV
q = 1.71 + 0.16s
Tfast = 400 keV (Maxwellian)
nfast = n0 exp(−∆nLn tanh s−s0
∆n
)with n0 = 0.75 · 1017 m−3,∆n = 0.2, Ln = 0.3, s0 = 0.5
TAE mode
R. Kleiber 7th IAEA TM 42 / 43
Electromagnetic ITG for W7-X
βe-effect on growth raterelatively weak.
In contrast to tokamak noAITG/KBM branch found inβe scan.
Pullback scheme allows muchfaster (one day) simulationsthan adaptive control variatescheme.
frequency
0 1 2 3 4 5
β [%]
0.0
0.5
1.0
1.5
2.0
ω [
106 s
-1]
adiabatic electronselectromagnetic
pullback
growth rate
0 1 2 3 4 5
β [%]
0.20
0.25
0.30
0.35
0.40
γ [1
06 s-1
]
adiabatic electronselectromagnetic
pullback
⇒ More careful tests necessary to validate schemes.
R. Kleiber 7th IAEA TM 43 / 43