Global electromagnetic gyrokinetic and hybrid …Global electromagnetic gyrokinetic and hybrid...

Global electromagnetic gyrokinetic and hybridsimulations of Alfven eigenmodes

Michael Cole,

A. Mishchenko, A. Biancalani, A. Bottino, R. Kleiber and A.Konies

Max Planck Institut fur Plasmaphysik, Germany

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

Theory Seminar, Princeton Plasma Physics Laboratory

December 9, 2016

1/32

1. Introduction→ 1.1 Motivation

→ 1.2 Models

2. Nonlinear Toroidal Alfven Eigenmodes→ 2.1 ITPA benchmark

→ 2.2 Wave-wave interaction

→ 2.3 Continuum interaction

3. Realistic geometry→ 3.1 ASDEX Upgrade

→ 3.2 Wendelstein 7-X

4. Outlook and Summary

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→ 1.2 Models







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[1.1] Introduction, Motivation

Fusion devices requireexternal heating, produceenergetic Alpha particles.

These fast particles caninteract with Alfven modes byinverse Landau damping.

Interaction redistributes fastparticles→ reduction in confinement,damage to the device.

→ could be used to control

plasma profiles.

0 0,2 0,4 0,6 0,8

sqrt norm. toroidal flux

0,0

2,0×105

4,0×105

6,0×105

8,0×105

1,0×106

1,2×106

ω (r

ad s

-1)

m= 9, n= -6m= 10, n= -6m= 11, n= -6m= 12, n= -6

|φ|

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[1.2] Introduction, Models

EUTERPE code package:

Particle-in-Cell approach (δf ).

Global to last closed flux surface, with arbitrary 3D toroidalgeometry (e.g. stellarator, tokamak with RMPs, etc.).

Electromagnetic (currently δB⊥ only).

Standard model: Gyrokinetic ion species, drift kinetic electrons.

Reduced models:

gyrokinetic ion species, fluid electrons.

gyrokinetic fast ions, MHD bulk plasma (self-consistent).

gyrokinetic fast ions, prescribed background field.

Pitch-angle-scattering collision operator (not used here).

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Gyrokinetics - solve Vlasov equation for the perturbed distributionfunction, f1s, for each species,

∂f1s∂t

+ ~R · ∂f1s∂ ~R

+ v‖∂f1s∂v‖

= −~R(1) · ∂F0s

∂ ~R− v

(1)‖∂F0s

∂v‖,

with equations of motion - p‖-formulation,

~R =(v‖ −

q

m

⟨A‖⟩)~b∗ +

1

qB∗‖

~b ×[µ∇B + q∇

⟨φ− v‖A‖

⟩]v‖ = − 1

m

[µ∇B + q∇

⟨φ− v‖A‖

⟩]· ~b∗

~B∗ = ~B +m

qv‖

(∇× ~b

)quasi-neutrality equation and Ampere’s law,

−∇·n0ms

B2∇⊥φ =

∑s=i,e,f

qs n1s ,

∑s=i,e,f

βsρ2s

−∇2⊥

A‖ = µ0

∑s=i,e,f

j‖1s .

6/32


Problem: for electromagnetic simulations, numerical cancellation oftwo large terms leads to inaccuracy. ∑

s=i,e,f

βsρ2s

−∇2⊥

A‖ = µ0

∑s=i,e,f

j‖1s

Unphysical ‘skin terms’ are the result of the formulation, balancedby ‘adiabatic current’:

βeρ2e

A‖ =µ0n0e

2

meA‖ = µ0j

(ad)1e ≈ µ0qs

∫v‖

v‖qeF0e

TeA‖d

3v

Cancellation problem most severe with kinetic electrons, high βe ,

low k‖ - electromagnetic global modes.

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Unphysically large terms not present in v‖ − formulation:−∇2

⊥A‖ =∑

s=i,e,f

j‖1s .

But equations of motion are hard to solve(

∂A‖∂t

):

~R = v‖~b∗ +

1

qB∗‖

~b ×

[µ∇B + q

(∇〈φ〉+

∂⟨A‖⟩

∂t~b

)]

v‖ = − 1

m~b∗ · µ∇B − q

m

(~b∗ · ∇ 〈φ〉+

∂⟨A‖⟩

∂t

)~B∗ = ~B +

m

qv‖

(∇× ~b

)+ ~b · ∇ × A‖~b

One solution: truncate physics with Ohm’s law closure, e.g.∂A‖∂t +∇‖φ = 0.

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Fluid model: derive electron continuity equation from 0th momentof the Vlasov equation:

∂n1e

∂t= f (u‖1e ,P1e , φ,A‖)

Fluid moments connected to gyrokinetic quantities byquasineutrality equation and Ampere’s law:

−∇⊥min0

eB2∇⊥φ = n1i − n1e , j‖1i = en0u‖1e −

1

µ0∇2⊥A‖

Closures needed for pressure and E‖ (Ohm’s law):

E‖ = −∇‖φ−∂A‖∂t

= −η∇2⊥A‖

Eliminates time derivative in the v‖ equations of motion, andtruncates physics.

9/32


Fully gyrokinetic simulations still desirable. New approach: mixedvariables formulation.

Magnetic potential is split into symplectic (s, v‖-like) and

hamiltonian (h, p‖-like) components, A‖ = A(s)‖ + A

(h)‖

New perturbed guiding-centre phase-space Lagrangian:

γ = q ~A∗·d~R+m

qµdθ+qA

(s)‖~b·d~x+qA

(h)‖~b·d~x−

[mv2‖

2+ µB + qφ

]dt

Appropriate Lie transformation separates A(h)‖ and A

(s)‖ into the

Hamiltonian and symplectic structure respectively:

Γ = q ~A∗·d~R+m

qµdθ+q

⟨A

(s)‖

⟩~b·d~R−

[mv2‖

2+ µB + q

⟨φ− v‖A

(h)‖

⟩]dt

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

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Deriving equations of motion by a variational method gives rise to‘mixed’ form:

~R(1) =~b

B∗‖×∇

⟨φ− v‖A

(s)‖ − v‖A

(h)‖

⟩− q

m

⟨A

(h)‖

⟩~b∗

v‖(1) = − q

m

[~b∗ · ∇

⟨φ− v‖A

(h)‖

⟩+

∂

∂t

⟨A

(s)‖

⟩]− µ

m

~b ×∇BB∗‖

· ∇⟨A

(s)‖

⟩∂/∂t term still present, but can be eliminated with new degree of

freedom, e.g.∂A

(s)

‖∂t +∇‖φ = 0.

Ampere’s law becomes: ∑s=i,e,f

βsρ2s

−∇2⊥

A(h)‖ −∇

2⊥A

(s)‖ = µ0

∑s=i,e,f

j‖1s

No truncation of the physics, but ‘guess’ for A‖ evolution can

remove the cancellation problem from much of the solution.A. Mishchenko, M. Cole, R. Kleiber and A. Konies, Phys. Plasmas, 21 052113 (2014)

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Particles are pushed and fieldequations solved in ‘mixed

space’, where∂A‖∂t problem

not present.

After each time-step,transform to symplectic (v‖)coordinates:

f(s)

1 (Zs ,A(s)‖ ) = f

(m)1 (Zm,A

(s)‖ ,A

(h)‖ )

v(s)‖ (Zs ,A

(s)‖ ) = v

(m)‖ (Zm,A

(s)‖ ,A

(h)‖ )

In v‖ coordinates, set

A(s)‖(new) = A‖ = A

(h)‖(old)+A

(s)‖(old)

→ set problematic A(h)‖ to 0

at each time-step.

SYMPLECTIC-VARIABLE SPACE

MIXED-VARIABLE SPACE

mix

ed-v

aria

ble ∆t

mix

ed-v

aria

ble ∆t

mix

ed-v

aria

ble ∆t

pullb

ack

pullb

ack

...

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

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→ 1.2 Models







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[2.1] Nonlinear TAEs, ITPA benchmark

ITPA TAE benchmark: worldwidestandard test case for Alfven wavephysics.

B0 = 3.0 T.

Circular tokamak,R0/a0 = 10.0.

Flat bulk plasma profiles(βbulk ≈ 0.18%).

Drive from energetic particledensity profile, gradientpeaking at r/ra = 0.5(βfast ≈ 0.1%).

0 0,2 0,4 0,6 0,8

sqrt norm. toroidal flux

0,0

2,0×105

4,0×105

6,0×105

8,0×105

1,0×106

1,2×106

ω (r

ad s

-1)

m= 9, n= -6m= 10, n= -6m= 11, n= -6m= 12, n= -6

|φ|

A. Mishchenko, A. Konies and R. Hatzky, Phys. Plasmas, 16 082105 (2009) 15/32


ITPA TAE benchmark: worldwidestandard test case for Alfven wavephysics.

B0 = 3.0 T.

Circular tokamak,R0/a0 = 10.0.

Flat bulk plasma profiles(βbulk ≈ 0.18%).

Drive from energetic particledensity profile, gradientpeaking at r/ra = 0.5(βfast ≈ 0.1%).

1×104

1×105

1×106

1×107

1×108

1×109

N (markers)

380

400

420

440

460

480

ω (

10

3 r

ad

s-1

)

gyrokinetic, mixed variablesfluid-electron hybrid

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

0 200 400 600 800T/ keV

0

10

20

30

γ/1

03 s

-1

CAS3D-K (ZOW)

FLU-EUTERPE (FLR)

GYGLES (FLR)

CKA-EUTERPE (FLR)

MEGA (FLR)

NOVA-K (FLR)

LIGKA (FLR)

EUTERPE (FLR)

A. Konies et al., Proc. 24th IAEA Fusion Energy Conf., ITR/P1 (2012), M. Cole et al., PPCF, 57 054013 (2015) 16/32


1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0 50000 100000 150000 200000 250000

δ B

/B0

t (Ω-1

)

EUTERPE, mixed variables, γd=0FLU-EUTERPE, ideal, γd=0

FLU-EUTERPE, resistive, γd=3x103 rad s

-1

Linear exponential growth phase gives way to clean saturationonly with damping.

Fluid-electron, kinetic ion model requires ad-hoc (here,resistive) damping.

M. Cole et al. Phys. Plasmas, submitted 17/32


Fast particle density profiledrives the mode; non-lineareffects result in fastparticle density profileflattening (right).

Modification of modestructure (below).

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

nf/n

0r/a

Initial

EUTERPE saturated

ORB5 saturated

linear, nf (0.5)/n0 ≈ 0.0035 NL, nf (0.5)/n0 ≈ 0.0035 NL, nf (0.5)/n0 ≈ 0.007

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=12


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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

M. Cole et al. Phys. Plasmas, submitted, A. Biancalani et al. PPCF, submitted 18/32


1×104

1×105

γ/rad s-1

0.1

1

10

δB

/B0 /

10

-3

EUTERPE, ZLR, γd=1.2.10

3rad s

-1

ORB5 γd=1.7.10

3 rad s

-1

FLU-EUTERPE, FLR, γd=5.5.10

3rad s

-1

CKA-EUTERPE γd=5.6.10

3rad s

-1

δΒ/Β0 ∼ γ

2

δΒ/Β0 ∼ γ

Analytical theory predicts transition from δB/B0 ∼ c · γ2 (resonancedetuning) to δB/B0 ∼ c · γ (radial decoupling) regime.

Artificial damping, γd , no longer needed in fully gyrokinetic

treatment → accurate determination of level.M. Cole et al. Phys. Plasmas, submitted 19/32


Numerical model comparison - computational requirements forlinear ITPA TAE benchmark:

Model Markers Timestep CPU-hrs (to 105Ω−1c )

EUTERPE (p‖) 3×107 0.75 80000EUTERPE (mixed) 3×106 10.0 256FLU-EUTERPE 2×105 10.0 18CKA-EUTERPE 8×105 20.0 6

Two order of magnitude improvement for full gyrokineticsimulations - moving from p‖ formulation to mixed variablesformulation with pullback.

Fluid-electron, gyrokinetic ion hybrid model an order ofmagnitude faster, where applicable.

M. Cole et al. Phys. Plasmas, submitted20/32

[2.2] Nonlinear TAEs, Wave-wave interaction

Simplest case: two (n = −2,n = −6) TAEs interactingnonlinearly.

Through wave-particlenonlinearity: large δB/B0

(black).

Through bulk ion nonlinearity:order of magnitude lowersaturation amplitude (red).

Significant mode structuremodification via bulk ionnonlinearity.

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0 50000 100000 150000δ B

/B0

t (Ω-1

)

Wave-particle nonlinearity onlyWave-wave nonlinearity included

wave-particle only (black) with wave-wave (red)

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[2.3] Nonlinear TAEs, Continuum interaction

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.2 0.4 0.6 0.8 1

q(r

)

r/ra

ITPA caseHigh shear case

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

0

2×105

4×105

6×105

8×105

1×106

1×106

ω (

rad

s-1

)

m= 7m= 8m= 9m= 10m= 11m= 12m= 13m= 14m= 15m= 16

Steeper q profile → more complex continuum.

Higher bulk ion temperature moves mode frequency upwards→ interaction possible between the mode and the continuum.

Otherwise, parameters are the same.

A. Mishchenko et al. Phys. Plasmas 21 052114 (2014), M. Cole et al. PPCF 57, 054013 (2015)

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Apply hierarchy of models tomore realistic continuum.

Continuum interaction

creates global structure

(middle) modified by finite E‖(right); growth rate increases

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

0

2×105

4×105

6×105

8×105

1×106

1×106

ω (

rad s

-1)

m= 7m= 8m= 9m= 10m= 11m= 12m= 13m= 14m= 15m= 16

MHD bulk plasma, kinetic fast ions fluid electrons, kinetic bulk ions kinetic bulk plasma


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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


0

0,2

0,4

0,6

0,8

1

|φ| (

arb

. u

nit

s)

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


0

0,2

0,4

0,6

0,8

1

|φ| (

arb

. u

nit

s)

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

A. Mishchenko et al. Phys. Plasmas 21 052114 (2014), M. Cole et al. PPCF 57, 054013 (2015)23/32


MHD-kinetic fluid-electron


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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


0

0.2

0.4

0.6

0.8

1

|φ| (

arb

. u

nit

s)

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

1e-06

1e-05

1e-04

1e-03

1e-02

0 0.0002 0.0004 0.0006 0.0008 0.001

δB

/B0

t (s)

fluid electron, kinetic ionMHD bulk plasma

Global structure grows more quickly and saturates at a levelone order magnitude higher.

Qualitative mode structure differences persist in nonlinearphase.

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→ 1.2 Models







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[3.1] Realistic geometry, ASDEX Upgrade

Baseline case suggested by Ph. Lauber(Lauber-ITPA-14) for ‘Non-Linear En-ergetic particle Dynamics’ project -“ASDEX-like” equilibrium:

B0 = 2.248 T

R0/a0 = 3.36

κ = 1.7

No triangularity, not numericalreconstruction of experimentaldata

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Apply hierarchy of models toASDEX parameters.

Bulk plasma kinetic effectschange mode structure, increasegrowth rate.

Mode identification on-going:more investigation needed.

Complex geometry → stepping

stone to stellarator simulations.

1e-02

1e+00

1e+02

1e+04

1e+06

0 1x10-5

2x10-5

3x10-5

4x10-5

5x10-5

|φ|

t (s)

MHD-gyrokineticFluid electrons, gyrokinetic ions

MHD bulk plasma, kinetic fast ions

0 0.2 0.4 0.6 0.8 1r

|φ| (a

rb. unit

s)

m=-2m=-1m=0m=1m=2m=3m=4m=5

fluid electrons, kinetic bulk ions

0 0.2 0.4 0.6 0.8 1r

|φ| (a

rb. unit

s)

m=-1m=0m=1m=2m=3m=4m=5m=6m=7m=8

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On-going work: takeself-consistent MHD-kineticcase, add wave-particlenonlinearity.

Nonlinear saturation, withsignificant mode structuremodification.

0 0.2 0.4 0.6 0.8 1r

|φ| (a

rb.

un

its)

m=-2m=-1m=0m=1m=2m=3m=4m=5

linear phase nonlinear phase|φ| |φ|

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[3.2] Realistic geometry, Wendelstein 7-X

Global W7-Xgeometry with 3DEUTERPE code, highfast particleβf ≈ 2β0.

Cut-off in growth rateof numerical origin,related to strongcoupling of poloidaland toroidalharmonics.

Future work: reduceβf to experimentallyrelevant values.

|φ|

0 2,5 5 7,5 10

Fast fraction (%)

0

100

200

300

400

500

γ (k

Hz)

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→ 1.2 Models

2. Circular tokamak→ 2.1 Nonlinear response

→ 2.2 Kinetic mode modification

3. ASDEX-like simulations→ 3.1 Case definition

→ 3.2 Model comparison


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[4.] Outlook

Interaction of multiple effects incompletelyunderstood:

Citrin ’13: Electromagnetic effects,fast particles suppressmicroturbulence locally in tokamaks.

Xanthopoulos ’16: Full flux surfacetreatment vital to model electrostaticmicroturbulence in the Wendelstein7-X stellarator without fast particles.

Dumont ’13: Fast particle-drivenglobal EGAMs temporarily suppresselectrostatic microturbulence.

EUTERPE code package could provide a

complete model of this physics.

(Ph. Lauber et al.)

Linear EMITG in W7-X with EUTERPE

(A. Mishchenko et al.)

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[4.] Summary

Theory and numerics: novel formulations of the gyrokineticequations have led to order of magnitude improvements inperformance simulating electromagnetic global modes.

Physics results: theoretical predictions by Berk & Breizman,S. Briguglio et al. confirmed; predictions for fast particleredistribution now possible; mode structure modification andmode-mode interaction may be qualitatively crucial forpredicting fast particle transport.

On-going work: applying the tool with realistic tokamakprofiles; stellarator simulations in support of Wendelstein 7-X.

Prospects: self-consistent, global, electromagneticMHD-turbulence interaction with fast particles.

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Global electromagnetic gyrokinetic and hybrid …Global electromagnetic gyrokinetic and hybrid...

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Transcript of Global electromagnetic gyrokinetic and hybrid …Global electromagnetic gyrokinetic and hybrid...