Progress in non-linear gyrokinetic simulations ofglobal modes in tokamaks and stellarators

M. Cole, A. Mishchenko, A. KoniesA. Biancalani, M. Borchardt, R. Hatzky, R. Kleiber, P. Lauber

Max Planck Institut fur Plasmaphysik, Germany

in the framework of the:“Nonlinear Energetic-Particle Dynamics” European Enabling Research Project

14th Technical Meeting (F1-TM-49508) on Energetic Particles in MagneticConfinement Systems.

4th September 2015

This work has been carried out within the framework of the EUROfusion Consortium and has received funding fromthe European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053.The views and opinions expressed herein do not necessarily reflect those of the European Commission.

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 1 / 24

Motivation

Goal: understand plasma phenomena important for burning plasmafusion reactors

Challenges...

Finite normalised pressureElectromagneticKinetic effectsNon-linearGlobal, complex geometry - realistic tokamak, stellarator (3D)

Approach: hierarchy of numerical models (MHD, hybrid, gyrokinetic)

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 2 / 24

1 Theory and model

2 Linear validation

3 Non-linear TAE

4 Global modes in W7X

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 3 / 24

1 Theory and model

2 Linear validation

3 Non-linear TAE

4 Global modes in W7X

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 4 / 24

Hierarchy of models

more

numerically

robust

and economical

more physically

complete

bulk plasma

GK fast ion

fluid electrons

(FLU

−EU

TER

PE)

power transfer

MHD

GK fast ions

(EUTERPE)

(CKA−EUTERPE)

bulk ions and electrons, fast ions

fluid bulk plasma

electromagnetic gyrokinetics for

GK bulk ions and fast ions

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 5 / 24

EUTERPE

PIC: charge and current calculated on grid using markers

4th order Runge-Kutta scheme to solve gyrokinetic equationsof motion in phase space.

Mixed variables formulation: mitigation of cancellationproblem - Mishchenko A, Konies A, Kleiber R and Cole M 2014 Phys. Plasmas 21 092110

∂f1s∂t

+ ~R · ∂f1s∂ ~R

+ v‖∂f1s∂v‖

= − ~R(1) · ∂F0s

∂ ~R− v(1)‖

∂F0s

∂v‖

∫qiF0i

Ti(φ− 〈φ〉) δ(~R+ ρ− ~x) d6Z = n1i − n1e(

βiρ2i

+βeρ2e−∇2

⊥

)A

(h)‖ −∇

2⊥A

(s)‖ = µ0

(j‖1i + j‖1e

)Global, non-linear, collisional, δf , neglects δB‖

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 6 / 24

FLU-EUTERPE

Fluid model for electrons combined with EUTERPE’sgyrokinetic bulk and fast ion model.

∂n1e∂t

= f(u‖1e, P1e, φ,A‖)

Electron continuity equation connected to GK quantities byquasineutrality equation and Ampere’s law:

−∇⊥min0eB2

∇⊥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

µ0∇2⊥A‖, P1e = 0

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 7 / 24

1 Theory and model

2 Linear validation

3 Non-linear TAE

4 Global modes in W7X

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 8 / 24

TAE Benchmark Results

ITPA n = −6, m = 10, 11 TAE benchmark. Compare withEUTERPE, GYGLES, CKA-EUTERPE, and many others.

0 2e+05 4e+05 6e+05 8e+05

Tf [eV] (n

0f fixed)

0.36

0.38

0.4

0.42

0.44

0.46

ω,

x 1

06 r

ad

/s gap

continuum

continuum

0 2e+05 4e+05 6e+05 8e+05

Tf [eV] (n

0f fixed)

0

10000

20000

30000

γ,

ra

d/s

EUTERPEFLU-EUTERPECKA-EUTERPE

ε = 0.1, Ti = Te = 1keV, n0 = 2× 1019m−3,q(r) = 1.71 + 0.16(r/a)2

Order of magnitude speed-up compared to EUTERPE.

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 9 / 24

Model comparison

0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux

0

2e+05

4e+05

6e+05

8e+05

ω, r

ad

/s

2e+05 3e+05 4e+05 5e+05 6e+05 7e+05T

f, eV

0

10000

20000

30000

40000

50000

γ, ra

d/s

EUTERPE, mixed variablesEUTERPE, conventional schemeFLU-EUTERPE (GK bulk, fast ions)

FLU-EUTERPE (GK fast ions)

CKA-EUTERPE

global kineticstructure is formed

pure TAE

GK fast, fluid bulk

0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux

0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

m=9m=10m=11m=12m=13m=14

GK fast, hybrid bulk

0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux

0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

m=9m=10m=11m=12m=13m=14

Fully GK

0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux

0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

m=9m=10m=11m=12m=13m=14

Mishchenko A, Konies A and Hatzky R 2014 Phys. Plasmas 21 052114

Cole M et al 2015 Plasma Phys. Control. Fusion 57 054013

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 10 / 24

1 Theory and model

2 Linear validation

3 Non-linear TAE

4 Global modes in W7X

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 11 / 24

Non-linear ITPA case - saturation

Only wave-particlenon-linearity

Clearly lowersaturation amplitudein fully gyrokineticcase

Little difference inlinear growth rate

Preliminary result:how important aregyrokinetic effects forsaturation?

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 12 / 24

Non-linear ITPA case - saturation amplitude

1000 10000 1e+05

γ/s-1

0,001

0,01

0,1

1

δB

/B0 /

10

-3

CKA-EUTERPE γd=1.05.10

4s

-1

FLU-EUTERPE, resistive

VENUS-K γd= 0

HMGC γd= 1.05.10

4 s

-1

MEGA γd= 1.33 .10

4 s

-1

CKA-EUTERPE γd= 1.33.10

4 s

-1CKA-EUTERPE γd= 2.5.10

3 s

-1CKA-EUTERPE γd= 2.5.10

3 s

-1

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 13 / 24

Non-linear ITPA case - mode structure

Linear phase

0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux

0

0,2

0,4

0,6

0,8

1

|φ| (

arb

. u

nit

s)

m=12m=11m=10m=9

Saturation, γ = 4.16 × 104s−1

0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux

0

0,2

0,4

0,6

0,8

1

|φ| (

arb

. u

nit

s)

m=10m=11

Saturation, γ = 5.67 × 104s−1

0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux

0

0,2

0,4

0,6

0,8

1

|φ| (

arb

. u

nit

s)

m=10m=11

Mode structure modification can be significant in non-linear phase

Stronger at higher growth rate

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 14 / 24

Non-linear ITPA case - frequency spectra

CKA-EUTERPE: perturbative FLU-EUTERPE: non-perturbative

Frequency drop disappears in non-perturbative case

Mode structure modification could be important

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 15 / 24

Non-linear ITPA case - wave-wave interaction

Two TAE modes interacting via non-linear kinetic bulk ions (n=2,n=6)

Suppressed saturation amplitude

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1 Theory and model

2 Linear validation

3 Non-linear TAE

4 Global modes in W7X

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 17 / 24

Non-linear stellarators: perturbative approach

CKA-EUTERPE: fixed mode structure from MHD code matchedwith GK power transfer solver

‘HELIAS reactor’: scaled W7-X

A. Konies et al. - in preparation

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 18 / 24

Non-linear stellarators: perturbative approach

CKA-EUTERPE: fixed mode structure from MHD code matchedwith GK power transfer solver

‘HELIAS reactor’: scaled W7-X

A. Konies et al. - in preparation

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 19 / 24

Linear global mode - mode structure

Extend investigation to non-perturbative models

Standard ‘high mirror’ geometry, flat background plasma profiles

Te = Ti = 3keV, n0 = 1020m−3, nf/no = 0.025, Tf/Te = 300

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 20 / 24

Linear global mode - growth rate scaling

0 2,5 5 7,5 10

Fast fraction (%)

0

100

200

300

400

500

γ (k

Hz)

Mode disappears below certain level of drive

Numerical or physical problem?

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 21 / 24

Linear global mode - mode structure

Mode disappears below certain level of drive

Numerical or physical problem?

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 22 / 24

Non-linear global mode

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

|δB

/B|

t (Ω−1

)

First saturated non-linear,non-perturbative W7-X run

To do: scan temperature,density

M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 23 / 24

Conclusions

Non-linear gyrokinetic simulations now possible in tokamaks andstellarators.

Results:Successful ITPA TAE benchmark for non-linear saturation levelscalingMode-mode coupling through kinetic ion non-linearitiesdemonstratedProof of principle for non-linear stellarator simulations

Further work:Benchmark fully gyrokinetic ITPA TAE result (collaboration withNEMORB)Investigate wave-wave interactionRealistic W7X, HELIAS reactor parameters

Realistic W7X modelling within reach.M D J Cole, IAEA EP TM Progress in non-linear gyrokinetic simulations of global modes 24 / 24