Finite orbit width effect on the neoclassical toroidal ...
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Finite orbit width effect on the neoclassical toroidal viscosity in the superbanana-plateau regime
Seikichi MATSUOKAJapan Atomic Energy Agency
Collaborators: Yasuhiro IDOMURA (JAEA), Shinsuke SATAKE (NIFS)
7th APTWG@Nagoya Univ., June 7th, 2017
This work is supported in part by the MEXT, Grant for Post-K priority issue No.6: Development of Innovative Clean Energy, and in part by NIFS Collaborative Research Program NIFS16KNST103
Acknowledgement
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Outline
1. Introduction
2. Numerical verification of NTV by global kinetic simulations
3. Finite orbit width effect on NTV
4. Another application of GT5D for 3D geometry
5. Summary
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Background & motivation - 3D effect on tokamak
3
3D effect is a key issue for plasma confinement in tokamaks.- Small resonant 3D perturbation affects on tokamak plasmas.
- stability: ELM mitigation - transport: Neoclassical Toroidal Viscosity (NTV) , Rotation
Discrepancy of NTV prediction: νb*-dependency
SBP Theory
νb*-dependency of NTV from SBP theory and FORTEC-3D.(Satake, PRL2011)
FORTEC-3DClarify the cause of the discrepancy by using two different types of global kinetic simulations.
- [Shaing ,PPCF2009] νb*-independent NTV (Superbanana-plateau theory)
- [Satake, PRL2011] νb*-dependency by FORTEC-3D code.
Purpose
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Superbanana-plateau theory for NTV
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- Local, bounce-averaged drift-kinetic equation. - Toroidal precession, 〈ωB〉bc = 0, gives the resonant condition. - Magnetic shear shifts the resonance κ2 towards the boundary [Shaing, JPP2015]. - Non-axisymmetric part of δB is only retained through the perturbed radial drift.- Independent of collisionality.
v‖
v⟂
Schematic view of SBP resonance
Trapped
PassingSBP Resonance 〈ωB〉bc = 0
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Outline
1. Introduction
2. Numerical verification of NTV by global kinetic simulations
3. Finite orbit width effect on NTV
4. Another application of GT5D for 3D geometry
5. Summary
5
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Target plasma profiles & numerical tools
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Circular tokamak with δB- Bax = 1.91 T
- a0 = 0.47 m / Rax = 2.35 m
- 1/ρ* = 150
- q = 0.854 + 2.184 (r/a0)2 (positive shear)
- Er = 0 (fixed)
- νb* ≃ 0.12 (base case) In the superbanana-plateau regime scan: νb* × 0.01, 0.1, 1, 5, 10, 50
- δB/Bax = 0.5 % with m/n = 7/5
- resonant surface with q = 1.4 at r/a0 ≃ 0.5.
Radial profile of perturbation and νb*.
q = 1.4 (7/5) at r/a0 ≃ 0.5
�Bmn(s)
Bax
= �mn exp
"�✓s� s
0
�
◆2
#
0
0.001
0.002
0.003
0.004
0.005
0.006
0.3 0.4 0.5 0.6 0.710-2
10-1
100
101B1/Bax
νb*
νSBP
B 1/B
ax
ν* b
r/a0
Global kinetic code- GT5D; Full-f Eulerian code for gyrokinetic simulations - FORTEC-3D; δf Monte Carlo (particle) code for drift-kinetic (neoclassical) simulations
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νb*-dependency of NTV arises in global sims.
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NTV of global kinetic simulations reproduce similar νb*-dependency over the wide ranges of collisionalities.
- SBP theory gives constant NTV. - GT5D/FORTEC-3D gives lower NTV. - GT5D/FORTEC-3D shows νb*-
dependency.
νb*-dependency of NTV at r/a0 ≃ 0.5
10-1
100
101
102
10-3 10-2 10-1 100 101
Superbanana-plateau
NTV
[Pa]
νb*
GT5DFORTEC-3D
NTV of SBP theory (Er = 0) is evaluated as;
he⇣ ·r · Pi = �⌘1
ni
mi
v2th
Rax
r✏
2⇡
d ln pi
dr
⌘1 = n�
✓5
2
◆4K (res)
2res
�1� 2
res
�|↵2
n + �2n|
res ' 0.83
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No resonant structure in global simulations.
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f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
f~ [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
GT5D FORTEC-3D
Velocity space structures of non-axisymmetric part of f are successfully verified.
Non-axisymmetric part of f at (r/a0, θ) ≃ (0.5, 0)
νb* = 0.12δB/Bax = 0.5%
νb* = 0.12δB/Bax = 0.5%
- No resonant structures along the boundary (barely-trapped) region. - Rather complicated structures are observed. - Especially, a clear large scale structure appear in trapped region. - Complicated structures survive for smaller δB ≃ 0.05 % —> determined by unperturbed orbit.
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Outline
1. Introduction
2. Numerical verification of NTV by global kinetic simulations
3. Finite orbit width effect on NTV
4. Another application of GT5D for 3D geometry
5. Summary
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- Banana width ⊿b/a0 ≃ 0.17 for v/vth ≃ √2; variation of q ≃ 1.2 - 1.59. - Barely-trapped particle feel the perturbation only for a fraction of the bounce period.
• Perturbation becomes less effective. - Bounce-average of dr/dt significantly decreases as m increases.
Absence of resonance results from the large finite orbit width of barely-trapped (resonant) particles.
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m=mim=me
650 700 750 800 850R [nrm.]
-100
-50
0
50
100
Z [n
rm.]
-1.8
-1.4
-1
-0.6
-0.2
0.2
0.6
1
1.4
1.8
A
BC D
Barely-trapped particle orbits with contour plot of (b×∇B1)·∇r mass (banana-width) dependency of 〈dr/dt〉bc
0
1
2
3
4
5
100 1000 10000
-<dr
/dt>
bc [a
.u.]
a0/ρti
large banana
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FOW generates finite-l mode, causes phase-mixing.
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- Non-zero (finite) mode structure appearsalong the bounce motion.
- Finite-l along the bounce motion causes the phase-mixing as;
f- [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
f- [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
f- [a.u.]
-4 -3 -2 -1 0 1 2 3 4v||/vth
0 0.5
1 1.5
2 2.5
3 3.5
4
v ⊥/v
th
νb* = 0.0012
δB/Bax = 0.5%
@t�fl + il!bc�fl = 0
Non-axisymmetric part of f at (r/a0, θ) ≃ (0.5, 0)
-0.02
-0.01
0
0.01
0.02
0 10 20 30 40 50
δn [a
.u.]
t/(R0/vth)
m=mim=me
δn sampled along the bounce motion
B
C
D
A
- Phase mixing generates fine scale structures in lower νb*.
- Makes NTV smaller in lower νb*.10-1
100
101
102
10-3 10-2 10-1 100 101
Superbanana-plateau
NTV
[Pa]
νb*
GT5DFORTEC-3D
smaller NTVby phase mixing
Fine scale structures in trapped region
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Outline
1. Introduction
2. Numerical verification of NTV by global kinetic simulations
3. Finite orbit width effect on NTV
4. Another application of GT5D for 3D geometry
5. Summary
12
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Full-f gyrokinetic simulations for 3D field
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GT5D+VMEC
0.8495
1.072
1.2945
6.270e-01
1.517e+00label_1
- Solves gyrokinetic equation.
- Global full-f model.
- VMEC equilibrium for 3D field.
- Eulerian approach. • Conservative Morinishi scheme.
- Radial electric field solver. • Ambipolar condition of neoclassical transport.
- Neoclassical benchmark has been initiated.
- LHD inward shifted configuration with Rax = 3.6 m and Bax = 3.0 T.
- a0 = 0.63 m.
- Ti,ax = 0.91 keV, and ne,ax = 3.5×1018 m-3.
Benchmark case parameters
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0
5x1017
1x1018
1.5x1018
2x1018
2.5x1018
0 200 400 600 800 1000 1200 1400step [F3D]
nx=180"ex90/out.07_r051250" u 3
(Preliminary) NC Benchmarks w/ and w/o Er
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@ρ ≃ 0.51- NC particle flux of GT5D+VMEC shows fairly good agreement with FORTEC-3D.
- 1/ν-regime; νb* ≃ 0.15 @ ρ ≃ 0.51.
- (bottom) Er is determined according to
the ambipolar condition of Γ.
GT5D
FORTEC-3D
Γ i [1
/m2 /
s]
-4000-3500-3000-2500-2000-1500-1000
-500 0
0 100 200 300 400 500 600 700 800 900 1000step [F3D]
GT5DF3D
-8x1017-6x1017-4x1017-2x1017
0 2x1017 4x1017 6x1017 8x1017 1x1018
0 100 200 300 400 500 600 700 800 900 1000step [F3D]
GT5DF3D@ρ ≃ 0.51
GT5D
FORTEC-3D
GT5D
FORTEC-3D
Geodesic Acoustic ModeEr [V/m]
Γ i [1
/m2 /
s]
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Summary
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- NTV of GT5D well reproduces the ν-dependency of NTV and velocity space structure of FORTEC-3D simulations.
- Large banana width of the unperturbed orbit plays a key role in NTV physics.
1. Radial drift caused by perturbation significantly decreases. ̶> Smaller NTV.
2. Finite-l mode along the bounce motion causes the phase mixing ̶> ν-dependency.
FOW effect on NTV in Superbanana-Plateau regime
GT5D+VMEC- Global full-f gyrokinetic simulation code for 3D geometries.
- Equilibrium from VMEC is incorporated into GT5D via the newly developed interface.
- First neoclassical benchmarks w/ and w/o Er show good agreements with FORTEC-3D.