Advances in stellarator gyrokinetics Per Helander and T. Bird, F. - - PowerPoint PPT Presentation

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Advances in stellarator gyrokinetics Per Helander and T. Bird, F. - - PowerPoint PPT Presentation

Oct 26, 2022 110 likes •449 views

Max-Planck-Institut fr Plasmaphysik Advances in stellarator gyrokinetics Per Helander and T. Bird, F. Jenko, R. Kleiber, G.G. Plunk, J.H.E. Proll, J. Riemann, P. Xanthopoulos Per Helander IAEA 2014 1 Max-Planck-Institut fr Background

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SLIDE 1

Max-Planck-Institut für Plasmaphysik

Per Helander 1 IAEA 2014

Advances in stellarator gyrokinetics

Per Helander

and

  • T. Bird, F. Jenko, R. Kleiber, G.G. Plunk, J.H.E. Proll, J. Riemann, P. Xanthopoulos
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SLIDE 2

Max-Planck-Institut für Plasmaphysik

Per Helander 2 IAEA 2014

  • Wendelstein 7-X will start experiments in 2015

– optimised for low neoclassical transport

  • Turbulence?
  • Electrostatic instabilities:

– ion-temperature-gradient (ITG) driven modes – trapped-electron modes

Background

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Max-Planck-Institut für Plasmaphysik

Per Helander 3 IAEA 2014

W7-X from above

Bad curvature q‘(r) < 0 everywhere

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Max-Planck-Institut für Plasmaphysik

Per Helander 4 IAEA 2014

  • EUTERPE

– global, particle-in-cell, linear in 3D – see poster TH/P4-49 by A.Mishchenko

  • GENE

– radially local (flux-tube or full-surface), continuum, nonlinear

  • Both codes: electromagnetic, collisions etc.

– here: collisionless, electrostatic instabilities

Gyrokinetic stellarator codes

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Max-Planck-Institut für Plasmaphysik

Per Helander 5 IAEA 2014

  • Linear ITG growth rate with Boltzmann electrons vs ion

temperature gradient in W7-X: Benchmark

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Max-Planck-Institut für Plasmaphysik

Per Helander 6 IAEA 2014

  • Global, linear ITG simulations in W7-X (EUTERPE)

W7-X vs LHD W7-X

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SLIDE 7

Max-Planck-Institut für Plasmaphysik

Per Helander 7 IAEA 2014

  • Global, linear ITG simulations in LHD (EUTERPE)

W7-X vs LHD LHD

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SLIDE 8

Max-Planck-Institut für Plasmaphysik

Per Helander 8 IAEA 2014

  • ITG turbulence with Boltzmann electrons (GENE): rms

potential fluctuations Nonlinear simulations

DIII-D W7-X

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Max-Planck-Institut für Plasmaphysik

Per Helander 9 IAEA 2014

  • Nonlinear simulations with Boltzmann electrons (grad Te=0,

r*=1/150):

– heat flux

ITGs with Boltzmann electrons

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SLIDE 10

Max-Planck-Institut für Plasmaphysik

Per Helander 10 IAEA 2014

  • So far, in W7-X comparable to that in a typical tokamak, but

“softer“:

– depends on r*

Turbulent transport (ITG ae)

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SLIDE 11

Max-Planck-Institut für Plasmaphysik

Per Helander 11 IAEA 2014

Trapped-electron modes

Bad curvature trapped particles

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SLIDE 12

Max-Planck-Institut für Plasmaphysik

Per Helander 12 IAEA 2014

  • Instability requires

where

  • In an orbit-confining (omnigenous) stellarator

Trapped-electron modes

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Max-Planck-Institut für Plasmaphysik

Per Helander 13 IAEA 2014

  • But the precession frequency can be written

so

  • Stability is thus promoted by “the maximum-J“ condition

Maximum-J configurations

Rosenbluth, Phys. Fluids 1968

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SLIDE 14

Max-Planck-Institut für Plasmaphysik

Per Helander 14 IAEA 2014

  • The quantity,

is an adiabatic invariant. E = energy.

  • Hence, if a low-frequency instability moves a particle radially,

then implying that it costs energy to move a particle radially

  • utward

Physical picture

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SLIDE 15

Max-Planck-Institut für Plasmaphysik

Per Helander 15 IAEA 2014

  • Theorem: collisionless trapped-electron and trapped-ion

modes are stable if for all species a.

  • Favourable bounce-averaged curvature.
  • In a maximum-J device, the precession drift is reversed

compared with a tokamak

– no resonance with drift waves.

Trapped-particle modes

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Max-Planck-Institut für Plasmaphysik

Per Helander 16 IAEA 2014

  • Simulations with and without kinetic electrons

(grad Te=grad Ti):

– growth rate for the most unstable wave number

  • Kinetic electrons are stabilising.

ITGs and TEMs with kinetic electrons

Boltzmann electrons Kinetic electrons

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SLIDE 17

Max-Planck-Institut für Plasmaphysik

Per Helander 17 IAEA 2014

  • Simulations with and without kinetic electrons (grad Te=0):

– kinetic electrons in a flux tube

ITGs with kinetic electrons

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SLIDE 18

Max-Planck-Institut für Plasmaphysik

Per Helander 18 IAEA 2014

  • Another case:

W7-X, HSX and DIII-D

HSX simulations by Benjamin Faber, Madison

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SLIDE 19

Max-Planck-Institut für Plasmaphysik

Per Helander 19 IAEA 2014

  • ITG and TEM modes exist in stellarators, but display

qualitative differences.

– turbulent fluctuations much less evenly distributed.

  • Wendelstein 7-X is, to some approximation, a maximum-J

device.

– most orbits have favourable bounce-averaged curvature

  • Strongly stabilising for trapped-particle instabilities.
  • ITG modes also benefit from stabilising action of the (kinetic)

electrons.

  • Less turbulent transport than in tokamaks?

– too early to say

Conclusions

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SLIDE 20

Max-Planck-Institut für Plasmaphysik

Per Helander 20 IAEA 2014

Extra Material

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SLIDE 21

Max-Planck-Institut für Plasmaphysik

Per Helander 21 IAEA 2014

  • Linear, flux-tube, electrostatic GENE simulations in DIII-D

and W7-X

– no ion temperature gradient

Gyrokinetic calculation of TEMs

Proll, Xanthopoulos and Helander, submitted to PoP

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SLIDE 22

Max-Planck-Institut für Plasmaphysik

Per Helander 22 IAEA 2014

  • Simulations with and without kinetic electrons (grad Te=0):

– growth rate for the most unstable wave number

  • Kinetic electrons are stabilising.

ITGs with kinetic electrons

Boltzmann electrons Kinetic electrons

Proll, Xanthopoulos and Helander, submitted to PoP

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SLIDE 23

Max-Planck-Institut für Plasmaphysik

Per Helander 23 IAEA 2014

  • In a maximum-J device, the precession drift is reversed

compared with a tokamak, since

– no resonance between precessing electrons and drift waves

Another argument for stable TEMs

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SLIDE 24

Max-Planck-Institut für Plasmaphysik

Per Helander 24 IAEA 2014

  • Linear, collisionless, electrostatic gyrokinetics.

– energy balance:

  • Substitute the solution of the gyrokinetic equation for fast-

moving partices

– at marginal stability

Energy balance

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SLIDE 25

Max-Planck-Institut für Plasmaphysik

Per Helander 25 IAEA 2014

  • Conventinal drift-wave ordering
  • Expanding in the inverse aspect ratio

– few trapped particles,

gives electron drift-wave frequency

  • In next order, instability from wave-particle resonance only

if

– impossible unless

Outline of calculation

Helander et al, PPCF 2012

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SLIDE 26

Max-Planck-Institut für Plasmaphysik

Per Helander 26 IAEA 2014

  • Linear, collisionless, electrostatic gyrokinetics in ballooning

space:

  • Multiply by J0f* and integrate over phase space.
  • Energy balance:

Energy balance

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SLIDE 27

Max-Planck-Institut für Plasmaphysik

Per Helander 27 IAEA 2014

  • For fast-moving particles
  • the energy transfer at marginal stability becomes
  • Stabilising action if bounce-averaged curvature is favourable:

Energy balance, cont‘d

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SLIDE 28

Max-Planck-Institut für Plasmaphysik

Per Helander 28 IAEA 2014

  • Conventinal drift-wave ordering
  • Expanding in the inverse aspect ratio

– few trapped particles,

gives electron drift-wave frequency

  • In next order, instability from resonant denominator only

if

– impossible unless

Algebra

Helander et al, PPCF 2012

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SLIDE 29

Max-Planck-Institut für Plasmaphysik

Per Helander 29 IAEA 2014

  • TEMs result from overlap between

– bad curvature and – trapping regions

Trapped-electron modes

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Max-Planck-Institut für Plasmaphysik

Per Helander 30 IAEA 2014

Trapped-electron modes

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Max-Planck-Institut für Plasmaphysik

Per Helander 31 IAEA 2014

Trapped-electron modes

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SLIDE 32

Max-Planck-Institut für Plasmaphysik

Per Helander 32 IAEA 2014

Trapped-electron modes

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SLIDE 33

Max-Planck-Institut für Plasmaphysik

Per Helander 33 IAEA 2014

Trapped-electron modes