global modelling of microinstabilitiesin stellarators and ...global modelling of...

Global modelling of microinstabilities in stellaratorsand with electromagnetic effects using XGC

M. D. J. Cole, R. Hager, A. Mishchenko, T. Moritaka, S. H. Ku, C. S. Chang

27th February 2020 PPPL Theory Seminar, M. D. J. Cole et al.

Outline


• Introduction

• Stellarator XGC

• Explicit electromagnetic XGC

• Summary and outlook

Introduction: gyrokinetics for stellarators


• Gyrokinetics is a successful model for understanding transport in fusion plasmas, e.g. microturbulence and neoclassical driven.

• Stellarator plasmas can be turbulent, and turbulence is observed to dominate transport in optimized stellarators -> one need for stellarator gyrokinetics.

• Edge physics is important for both tokamaks and stellarators, but few codes have been able to model it -> XGC was created for this purpose.

• Stellarator edge physics presents additional challenges, e.g. islands and stochastic regions.

• Gyrokinetic code XGC has been extended for stellarator physics.

Introduction: explicit electromagnetics


• The electromagnetic (ẟB⟂) gyrokinetic system requires a choice of formulation which numerically implies the use of implicit schemes or the ‘cancellation problem’ which grows with β and inversely with ms and k⟂⍴s.

• The Hamiltonian approach, affected by the cancellation problem, may be most performant.

• We implement mitigation techniques to minimize this problem and test them with delta-f simulations of circular conventional and spherical tokamaks.

• Total-f XGC is uniquely placed to treat electromagnetic edge physics, e.g. ELM onset; almost all EM GK simulations so far are with delta-f codes.

Outline


• Introduction

• Stellarator XGC



Overview: Gyrokinetic code XGC for stellarators


• XGC is a gyrokinetic Particle-in-Cell (PIC) code for high fidelity modelling:• Gyrokinetic total-f• Electromagnetic (ẟB⟂) (this talk!)• Non-linear multi-species collisions• Whole volume to first wall

• Stellarator version (this talk!) is currently:• Gyrokinetic delta-f• Electrostatic (EM ongoing…)• Collisionless• Whole volume to first wall (ongoing…)

Ultimate goal: full high fidelity XGC capability with stellarator geometry

XGC for stellarators - Numerical implementation


Potentials calc’don nodes

Nodes field aligned to calc E||

Unstructured mesh capability allows smooth treatment of edge region, with solver

M. Cole et al., Phys. Plasmas 26 032506 (2019)T. Moritaka et al., Plasma 2 (2), 179-200, (2019)M. Cole et al., Phys. Plasmas 26 082501 (2019)

PIC markers pushed in R, Z, ϕ

Verification: Linear W7-X ITG benchmark


• Defined Wendelstein 7-X high mirror VMEC equilibrium (KJM)

• R0 = 5.5 m• a0 = 0.505 m• B0 = 2.41 T• Ti(s = 0.5) = Te(s = 0.5) = 1 keV• ITG driven unstable by quasi-local

temperature gradient:

• From experience, Wendelstein 7-X is the most challenging of the stellarator geometries numerically.

T. Klinger et al., Nucl. Fusion 59 112004 (2019)



0 1 2 3 4 5 6κT

0.0

0.1

0.2

0.3

γ[v

i/a]

GENE-3D

EUTERPE

XGC

0 1 2 3 4 5κT

100

120

140

160

180

200

220

240

Mod

enu

mber

GENE-3D: m0

GENE-3D: n0

XGC: m0

XGC: n0

0 1 2 3 4 5κT

0.0

0.1

0.2

0.3

0.4

Frequ

ency

ω[v

th/a]

GENE-3D

XGC

M. Cole et al., Phys. Plasmas 26 082501 (2019)M. Maurer (GENE-3D), private communication


27th February 2020 PPPL Theory Seminar, M. D. J. Cole et al. M. Cole et al., Phys. Plasmas 26 082501 (2019)M. Maurer (GENE-3D), private communication

𝛾 “rollover”

0 1 2 3 4 5 6κT

0.0

0.1

0.2

0.3

γ[v

i/a]

GENE-3D

EUTERPE

XGC

0 1 2 3 4 5κT

100

120

140

160

180

200

220

240

Mod

enu

mber

GENE-3D: m0

GENE-3D: n0

XGC: m0

XGC: n0

0 1 2 3 4 5κT

0.0

0.1

0.2

0.3

0.4

Frequ

ency

ω[v

th/a]

GENE-3D

XGC

W7-X and QUASAR comparison


• Original EUTERPE case included Wendelstein 7-X and LHD.

• Extend this to PPPL’s QUASAR/NCSX, keeping T, n profiles the same.

Negative iota shear (positive q shear) case used – lowest turbulent transport with local

GENE simulations.

S. Lazerson et al., Phys. Plasmas 26 022509 (2019)

W7-X and QUASAR comparison


• Original EUTERPE case included Wendelstein 7-X and LHD.

• Extend this to PPPL’s QUASAR/NCSX, keeping T, n profiles the same.

• Normalised growth rates are almost the same for all three devices (see right, LHD growth rates followed W7-X).

M. Cole et al., Phys. Plasmas, under review (2020)

W7-X and QUASAR comparison – mode numbers


|φ|2

0 -50 -100-150-200-250n

0

50

100

150

200

250

m

0.2

0.4

0.6

0.8

1|φ|2

0 -30 -60 -90n

0

30

60

90

m

0.2

0.4

0.6

0.8

1

W7-X QUASARM. Cole et al., Phys. Plasmas, under review (2020)

W7-X and QUASAR: mode structure differences


|φ|

0 π/3 2π/3φ

π

0

-π

θ

0.2

0.4

0.6

0.8

1|φ|

0 π/5 2π/5φ

π

0

-π

θ

0.2

0.4

0.6

0.8

1

W7-X QUASAR


W7-X and QUASAR comparison – mode structures


|φ|

0 π/3 2π/3φ

π

0

-π

θ

0.2

0.4

0.6

0.8

1|φ|

0 π/5 2π/5φ

π

0

-π

θ

0.2

0.4

0.6

0.8

1


Poloidal shift predicted theoretically due to radial shear in iota, mode frequency; stronger at higher growth rate.

Y. Camenen et al., Nucl. Fusion 51 073039 (2011)

Global nonlinear verification: XGC1 tokamak benchmark


0 0.2 0.4 0.6time [ms]

0

5

10

15

20

25

Mea

n el

ectro

stat

ic p

oten

tial XGC1

XGC-S

0.16 0.18 0.2 0.22 0.24 0.26t (ms)

0

0.5

1

s

Ion heat flux (Jm 2 s-1 )

0

500

1000

1500

XGC1

XGC-S


Global nonlinear ITG in QUASAR: heat flux


φ

1.2 1.4 1.6 1.8r (m)

-0.2-0.1

00.10.2

z (m

)

-4

-2

0

2

4

t = 0.15ms

φ

1.2 1.4 1.6 1.8r (m)

-0.2-0.1

00.10.2

z (m

)

-50

0

50

t = 0.3ms


Global nonlinear ITG in QUASAR: heat flux


φ

1.2 1.4 1.6 1.8r (m)

-0.2-0.1

00.10.2

z (m

)

-4

-2

0

2

4

t = 0.15ms

φ

1.2 1.4 1.6 1.8r (m)

-0.2-0.1

00.10.2

z (m

)

-50

0

50

t = 0.3ms


Poloidal localisation (global feature) weakened at peak

Progress towards a whole volume capability


0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

Loss

frac

tion

(%)

Launch position (ρ)

Loss fraction at t=0.05s, 50 keV ion, W7-X from VMEC and HINT

VMECHINT

• Poisson solver test with constant charge distribution (above left – electrostatic potential; right – relative error to analytical solution).

• HINT3D equilibrium test with islands, stochastic regions (left, core).

T. Moritaka et al., Plasma 2 (2), 179-200, (2019)

Outline


• Introduction

• Stellarator XGC



Electromagnetic gyrokinetics: cancellation problem


Symplectic formulation (v||) Hamiltonian formulation (p||)

• No cancellation problem.

• Implicit methods required (for XGC: B. Sturdevant, L. Chacón, M. Adams)

• Cancellation problem scales as:

• Explicit schemes possible (RK4 etc.).

p|| = mv|| + qA||

Electromagnetic gyrokinetics: mixed formulation


A. Mishchenko, M. Cole, R. Kleiber and A. Könies, Phys. Plasmas 21 052113 (2014)A. Mishchenko, A. Könies, R. Kleiber and M. Cole, Phys. Plasmas 21 092110 (2014)

• Combine Hamiltonian and symplecticformulations in derivation of GK system:

• Introduces new degree of freedom:

• Cancellation problem occurs only in A||(h):

reset phase space at each timestep (right).

p|| = mv|| + qA||(h)

ITG-KBM transition benchmark


A||

φ

• Hamiltonian equations alone (mi/me=100) -> realistic electron mass with mitigation techniques.

M. Cole et al., in preparation

Electromagnetics in spherical tokamaks


• Predicting the KBM transition point is a key research question for spherical tokamaks.

• XGC NSTX-like case, with mitigation techniques:• R0 = 0.85 m• a0 = 0.67 m• B0 = 0.45 T• q0 = 1.21, q95 = 3.86• 𝜅 = 2.2• βs=0.25 = 2.34%

• 𝛾 = 54 kHz (159 kHz without elongation)

A||

0 1 2r (m)

-1

-0.5

0

0.5

1

z (m

)

-1

-0.5

0

0.5

1×10-13

φ

0 1 2r (m)

-1

-0.5

0

0.5

1

z (m

)

-3

-2

-1

0

1

2

3

×10-8

Outline


• Introduction

• Stellarator XGC



Summary

• Delta-f electrostatic core physics with XGC-S has been successfully benchmarked with EUTERPE, GENE-3D.

• First physics studies in NCSX/QUASAR geometry have been performed into the turbulent phase.

• State-of-the-art explicit electromagnetic techniques have been implemented in XGC.

• Proof of principle KBM simulations in geometry similar to NSTX have been performed successfully.


Outlook• KBMs may be key to optimized stellarator confinement –

combine stellarator and EM developments (total-f?)

• Whole volume stellarator version with islands and stochastic regions under development.

• Comparison to experiment planned with PPPL powder dropper on LHD.

• KBM modelling in spherical tokamaks such as NSTX-U –KBM threshold is a key physics question for STs.

• Model MTMs in NSTX-U to understand e- heat flux.

• High fidelity turbulence modelling can be used to develop reduced models for stellarators (machine learning?).


K. Aleynikova and A. Zocco, J. Plasma Phys. 84 0602 (2018)

W7-X (linear GENE flux tube)

Backup: electrostatic GK equations


global modelling of microinstabilitiesin stellarators and ...global modelling of...

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