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

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

2

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

Superbanana-plateau theory for NTV

4

- 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

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

Target plasma profiles & numerical tools

6

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

νb*-dependency of NTV arises in global sims.

7

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

No resonant structure in global simulations.

8

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.

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

9

- 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.

10

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

FOW generates finite-l mode, causes phase-mixing.

11

- 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

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

Full-f gyrokinetic simulations for 3D field

13

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

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

14

@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]

Summary

15

- 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.