Fluid electron, gyrokinetic ion simulations of global ... · WENDELSTEIN 7-X EUTERPE Code Family Hierarchy of numerical models for addressing diﬀerent problems. Max-Planck-Institut

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


WENDELSTEIN 7-X EUTERPE Code Family

Hierarchy of numerical models for addressing different problems.


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‖


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


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


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‖.


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)


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)


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.


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)


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)


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)


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.


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)


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


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.


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.


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.


Documents

Fluid electron, gyrokinetic ion simulations of global ... · WENDELSTEIN 7-X EUTERPE Code Family Hierarchy of numerical models for addressing diﬀerent problems. Max-Planck-Institut