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WENDELSTEIN 7-X
Fluid electron, gyrokinetic ion simulations ofglobal modes in tokamaks and stellarators
Michael Cole, Alexey Mishchenko, Axel Konies, Ralf Kleiber, Matthias Borchardt
Max-Planck-Institut fur Plasmaphysik, Greifswald
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Motivation
• Global modes are important in fusion devices
– e.g. TAE, sawtooth cycle
• Simulations are computationally demanding, more so for stellarators
• Gyrokinetic PIC makes full torus simulations practical, but has drawbacks
– e.g. cancellation problem
Reducing computational requirements makes physics studies possible
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X EUTERPE Code Family
Hierarchy of numerical models for addressing different problems.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X EUTERPE
• Charge and current calculated on grid using markers.
• 4th order Runge-Kutta scheme to solve gyrokinetic equations of motion inphase space.
∂f1s
∂t+ ~
R ·∂f1s
∂ ~R+ v‖
∂f1s
∂v‖
= − ~R(1) ·
∂F0s
∂ ~R− v
(1)‖
∂F0s
∂v‖
−∇ ·
∑
s=i,f
q2sns
Ts
ρ2s
∇⊥φ
=∑
s=i,e,f
qsns (1)
∑
s=i,e,f
βs
ρ2s
− ∇2⊥
A‖ = µ0
∑
s=i,e,f
j‖s
• Global, non-linear, collisional, δf , but neglects δB‖
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X FLU-EUTERPE
• Fluid model for electrons combined with gyrokinetic model for bulk and fastions.
∂n1e
∂t= f(u‖1e, φ,A‖, P1e)
• Electron continuity equation connected to GK quantities by quasineutralityequation and Ampere’s law:
−∇⊥
min0
eB2∇⊥φ = n1i − n1e, j‖1i = en0u‖1e −
1
µ0
∇2⊥A‖
• Closures needed for E‖ (Ohm’s law) and pressure:
E‖ = −∇‖φ−∂A‖
∂t= η(j‖i+j‖e),
∂P1e
∂t= −~vE·∇P0e = −
~b × ∇φ
B·∇n0T0
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X
v‖ formulation
Equations of motion, v‖-formulation:
~R = v‖
~b∗ +
1
qsB∗‖
~b ×
[
µ∇B + qs
(
∇〈φ〉 +∂⟨
A‖
⟩
∂t~b
)]
and
v‖ = −1
ms
~b∗ · µ∇B −
qs
ms
(
~b∗ · ∇ 〈φ〉 +
∂⟨
A‖
⟩
∂t
)
where~B∗ = ~B +
ms
qsv‖s(∇ ×~b) + ∇ ×
⟨
A‖
⟩
~b.
Simplified closure eliminates∂A‖
∂tin v‖-formulation:
E‖ = −∇‖φ −∂A‖
∂t= 0
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Fluid Bulk Plasma model
• Solve the gyrokinetic equation only for fast ions.
• Deriving continuity equations for both electrons and bulk ions, calculate to-tal charge density ρ = qini + qene:
∂ρ1
∂t+ ~B · ∇
(
j‖1
B
)
+(
∇ × A‖~b)
· ∇
(
j‖0
B
)
+ ρ0~v∗ ·∇B
B−
∇ × ~B
B2· ∇P1
= 0
with
~v∗ = 2~b × ∇P1e
n0meeB.
• With appropriate quasineutrality equation and Ampere’s law:
−∇⊥
min0
B2∇⊥φ = ρ1, µ0j‖1 = ∇2
⊥A‖.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X CKA-EUTERPE
• Solve reduced ideal-MHD vorticity equations to find perturbed fields (CKA):
ω2∇ ·
(
1
v2A
∇⊥φ
)
+ ∇ ·[
~b∇2⊥(~b · ∇)φ
]
+ ∇ ·
[
~b∇ ·
(
µ0j‖
B~b × ∇φ
)]
−
−∇ ·
(
2µ0
B2
[
(~b × ∇φ) · ∇p]
(~b × κ)
)
= 0
• Solve gyrokinetic equation for fast ion species, to calculate power transferwith mode (EUTERPE)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X ITPA TAE Benchmark
• TAE in large aspect ratio tokamak, low shear, simple spectrum
• Can compare EUTERPE, FLU-EUTERPE and CKA-EUTERPE
• Permits benchmark of all models with other codes worldwide
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
0 200 400 600 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)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X TAE Benchmark Results
• Can compare EUTERPE, FLU-EUTERPE and CKA-EUTERPE
0 2e+05 4e+05 6e+05 8e+05Tf [eV] (n0f fixed)
0.36
0.38
0.4
0.42
0.44
0.46
ω,
x 10
6 rad
/s gap
continuum
continuum
0 2e+05 4e+05 6e+05 8e+05Tf [eV] (n0f fixed)
0
10000
20000
30000
γ, r
ad/s
EUTERPEFLU-EUTERPECKA-EUTERPE
• Some divergence of perturbative hybrid model, but significant speed-up.
– Roughly order of magnitude separation between models in run time.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Continuum effects
• Stronger shear
– more complex spectrum.
• Mode structure modificationby fast particles captured byfluid-electron model but notperturbative hybrid.
• Limits of closure?
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
6ra
d/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
A. Mishchenko et al., Phys. Plasmas 21, 052114 (2014)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Continuum effects
• Stronger shear
– more complex spectrum.
• Mode structure modificationby fast particles captured byfluid-electron model but notperturbative hybrid.
• Limits of closure?
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ|
2e+05 3e+05 4e+05 5e+05 6e+05 7e+05Tf, eV
0
10000
20000
30000
γ, r
ad/s
EUTERPEFLU-EUTERPECKA-EUTERPE
A. Mishchenko et al., Phys. Plasmas 21, 052114 (2014)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Internal kink mode in a screw pinch
• Full gyrokinetics possible in screwpinch, but close to limits.
– Important for resistive layerphysics.
• Benchmark for fluid-electronmodel in ‘MHD regime’.
0 0.2 0.4 0.6 0.8 1r
Ele
ctro
stat
ic p
oten
tial
0.5 0.6 0.7 0.8rc / r
a
10000
20000
30000
40000
γ, r
ad/s
MHD, shooting methodGK PIC (T
i = 5000 eV)
FLU-EUTERPE (Ti = 5000 eV)
A. Mishchenko and A. Zocco, Phys. Plasmas 19, 122104 (2012)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Kink in tokamak
• Finite ∇P now needed to desta-bilise mode.
• In fluid limit, direct comparisonpossible with MHD code CKA(A. Konies):
γFLU = 1.29 × 106s−1
γCKA = 1.27 × 106s−1
0 0.2 0.4 0.6 0.8 1r
0.8
1
1.2
1.4
1.6
1.8
2
Arb
itrar
y un
its
PressureSafety factor
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
oten
tial
MHD m=0MHD m=1MHD m=2 GK PIC m=0GK PIC m=1GK PIC m=2
Lose resistive layer physics but tokamak simulations become practical.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Tokamak shaping
• Fully global code - consider sig-nificance of geometry
– Strong stabilisation withJET/ITER-like aspect ratioand elongation
• Further stabilisation with inclu-sion of bulk ion gyrokinetic ef-fects
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)
1 1.25 1.5 1.75 2Elongation
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)
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Fast particle effects - fishbone
• Pure Energetic Particle Modes(EPMs) such as the fishboneemerge when kink is destabilisedby energy transfer from fastparticles.
• Initial stabilisation of the mode(perturbative effect) overcomeby unstable fishbone EPMbranch.
• Frequency jump with onset offast particle destabilisation.
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
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Fast particle effects - fishbone
ρf = 0.0 ρf = 0.01
ρf = 0.04
M. Cole et al., Phys. Plasmas 21, 072123 (2014)Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Ballooning mode in W7-AS
Stellarator simulations with FLU-EUTERPE and EUTERPE.
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ|
−2−101 2 3 4 5 6 7 8 9 101112
−6−5
−4−3
−2−1
01
2
0
0.2
0.4
0.6
0.8
1
m
time=5.690e+03
n
Φm
,n
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
100
10000
1e+06
0 500 1000 1500 2000 2500 3000 3500
|φ| (
arb.
uni
ts)
t (Ω-1)
EUTERPEFLU-EUTERPE
• Bulk temperature gradient drives unstable modes at high β.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Ballooning mode in W7-AS
Stellarator simulations with FLU-EUTERPE and EUTERPE.
0 0.005 0.01 0.015 0.02β/4
0
5e+05
1e+06
1.5e+06
2e+06
γ, r
ad/s
W7-AS
0 0.2 0.4 0.6 0.8 1norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ|
• Scan in β: mode disappears below critical point.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald
WENDELSTEIN 7-X Summary
• Simulating global modes still presents challenges.
– Stellarator simulations especially difficult.
• Theoretical developments required to mitigate difficulties
– pullback scheme for full gyrokinetics
– electron-fluid, gyrokinetic-ion model
– perturbative hybrid model
• New models open new avenues, e.g. stellarator phenomena, fishbones, saw-tooth cycle, rapid comparison with experiments.
Max-Planck-Institut für Plasmaphysik · Teilinstitut Greifswald