Gyrokinetic particle simulations of reversed shearAlfven eigenmodes (RSAEs) in DIII-D tokamak

W. DengUniversity of California, Irvine, USA

Z. Lin1,2, I. Holod1, X. Wang3,1, Z. Wang1, Y. Xiao1, H. Zhang2,1

W. Zhang4,1, E. Bass5, D. Spong6, M. Van Zeeland5

1 University of California, Irvine, USA2 FSC, Peking University, China

3 IFTS, Zhejiang University, China4 University of Science and Technology of China

5 General Atomics, USA6 Oak Ridge National Laboratory, USA

Supported by US DOE SciDAC GSEP Center

1/28

Introduction

RSAE: shear Alfven wave in the toroidalgeometry, localized near qmin, drivenunstable by energetic particles (fast ions),frequency up-chirping (up-sweeping).

Motivations to study RSAE: fast ion loss;grand cascade for measuring when andwhere qmin reaches an integer.

RSAE in local linear ideal MHD limit insimple geometries is quitewell-understood [Berk et al., PRL 2001;Breizman et al., PoP 2003, 2005; etc.]

Global effects, kinetic effects, nonlineareffects, etc. are still poorly understood.

RSAE simulations by global gyrokinetictoroidal code (GTC)

ωRSAE ≈vAR

∣∣∣∣ mqmin− n

∣∣∣∣

Experimental spectrogramshowing frequency up-chirping ofRSAEs driven by energeticparticles [Tobias et al., PRL 2011]

2/28

Outline

1 GTC gyrokinetic model for Alfven eigenmodes

2 Verification of gyrokinetic model for RSAEZero-β limit and benchmark with XHMGCIon finite-β and FLR effectsEquilibrium current effect

3 Gyrokinetic simulations of RSAE in DIII-D discharge #142111Damping and driving mechanismsMode frequencies and structuresRSAE frequency up-sweeping and RSAE to TAE transitionComparisons among GTC, GYRO and TAEFL

3/28

Outline

1 GTC gyrokinetic model for Alfven eigenmodes

2 Verification of gyrokinetic model for RSAEZero-β limit and benchmark with XHMGCIon finite-β and FLR effectsEquilibrium current effect

3 Gyrokinetic simulations of RSAE in DIII-D discharge #142111Damping and driving mechanismsMode frequencies and structuresRSAE frequency up-sweeping and RSAE to TAE transitionComparisons among GTC, GYRO and TAEFL

4/28

GTC gyrokinetic model for Alfven eigenmodesThermal ions and fast ions are simulated using gyrokinetic PIC approach:

(∂t + X · ∇+ v‖∂v‖ )[f0(X, µ, v‖) + δf(X, µ, v‖, t)] = 0

[Brizard and Hahm, RMP 2007]

Electrons are simulated using the electromagnetic fluid-kinetic hybrid model [Lin and

Chen, PoP 2001; Holod et al., PoP 2009]:I Fluid part:

0 = ∂t δne−δB · ∇(

c

4πeB0b0 · ∇ ×B0

)+ B0 · ∇

(n0e δu‖e

B0

)+B0vE · ∇

(n0e

B0

)− n0e(δv∗e + vE) ·

∇B0

B0

+c∇×B0

B20

·[−∇ δP‖ee

+ n0e∇ δφ]

+ nonlinear terms

δP‖,⊥,e =

(Teδne + n0e δψ

∂Te

∂ψ0

)+ δP

(1)‖,⊥,e + δP

(2)‖,⊥,e + · · ·

I Parallel electric field is also solved through electron δE‖ = −b0 · ∇ δφeff :

δφeff =Te

en0e

(δne − δψ

∂n0e

∂ψ0

)+ δφ

(1)eff + δφ

(2)eff + · · ·

I δP(n)‖,⊥,e and δφ

(n)eff are obtained by solving electron drift-kinetic equation order

by order: Lδhe = −δL f0e − L0 δf(0)e − 〈δL〉 δf (0)

e .5/28

Reduction to ideal MHDGyrokinetic Poisson’s equation [Lee, JCP 1987] & Ampere’s law:

Z2fnf

Tf(δφ− δφf ) +

Z2i niTi

(δφ− δφi) =∑

α=e,i,f

Zα δnα

c

4π{∇ × [∇× (δA‖b0)] · b0}b0 =

∑α=e,i,f

δJα‖

In the linear long wave-length limit, the gyrokinetic model reduces to the idealMHD equation:

ω(ω − ω∗P )

v2A

∇2⊥ δφ− iB0 · ∇

{b0 · ∇ × [∇× (k‖ δφb0)]

B0

}− iωcδB⊥ · ∇

(b0 · ∇ ×B0

B0

)−iω 4π

c∇ ·(

b0

B0×∇ · δP

)= 0

[W. Deng et al., submitted to Nuclear Fusion, 2011]

The gyrokinetic model in GTC has been verified by the simulations of three

kinds of Alfven eigenmodes:

I TAE [Nishimura et al., PoP 2007 & 2009]I RSAE [Deng et al., PoP 2010]I BAE [H. Zhang et al., PoP 2010]

6/28

Outline

1 GTC gyrokinetic model for Alfven eigenmodes

2 Verification of gyrokinetic model for RSAEZero-β limit and benchmark with XHMGCIon finite-β and FLR effectsEquilibrium current effect

3 Gyrokinetic simulations of RSAE in DIII-D discharge #142111Damping and driving mechanismsMode frequencies and structuresRSAE frequency up-sweeping and RSAE to TAE transitionComparisons among GTC, GYRO and TAEFL

7/28

Antenna excitation of RSAEEquilibrium current drive is turned off to ensure existence of eigenmode.

q-profile:

1.61.71.81.9

22.12.22.32.42.52.62.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r/a

q

qmin = 1.69

n = 4 Alfven continua andpower spectrum:

r/a

ω/(

v A/R

0)

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8XHMGC

ωGTC = 0.135vA/R0

ωXHMGC = 0.160vA/R0

δφ poloidal contour:

GTC

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

XHMGC

m-decomposed δφ:

0 0.2 0.4 0.6 0.8 1

|δφm|(

a.

u.)

r/a

m = 6

m = 7

m = 8

GTC

0 0.2 0.4 0.6 0.8 1r/a

|δφ(

r)| [

a.u.

]

(1,4)(2,4)(3,4)(4,4)(5,4)(6,4)(7,4)(8,4)(9,4)

XHMGC

8/28

Antenna excitation of higher radial-harmonic RSAEδφ poloidal contour:

GTC

m-decomposed δφ:

0

0 0.2 0.4 0.6 0.8 1

<eδφ

(a.

u.)

r/a

m = 6

m = 7

m = 8

GTC

ωRSAE−2 = 0.131vA/R0 (l = 1)ωRSAE−1 = 0.135vA/R0 (l = 0)

500 550 600 650 700

Time (ms)

60

80

100

120

140

f (k

Hz)

(a)

RSAE-1

RSAE-2

0

5

10

(b)

qRSAE-1

RSAE-2

–200

0

200

(c)

0.0 0.2 0.4 0.6 0.8 1.0

–5

0

10

(d)

ρ

|δT

e| (

eV)

φ(º

)δT

eco

s(φ)

(eV

)

Higher radial-harmonic observed inDIII-D [Van Zeeland, NF 2009]

9/28

Drift-kinetic fast ion excitation of n = 4 RSAER0/Lnf0 = 36.6, vf/vA = 0.3, ρf/a = 0.03, k⊥ρf = 0.4, nf0/ne0 = 0.01

Fast ion density profile:

0

0.002

0.004

0.006

0.008

0.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r/a

nf0/ne0

GTC:(ωr, γ) = (0.107, 0.0159) vA

R0

XHMGC:(ωr, γ) = (0.145, 0.013) vA

R0

δφ poloidal contour:

GTC

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

XHMGC

m-decomposed δφ:

0 0.2 0.4 0.6 0.8 1

|δφm|(

a.

u.)

r/a

m = 6

m = 7

m = 8

GTC

0 0.2 0.4 0.6 0.8 1r/a

|δφ(

r)| [

a.u.

]

(1,4)(2,4)(3,4)(4,4)(5,4)(6,4)(7,4)(8,4)(9,4)

XHMGC

Similar mode structure modification by fast ions is also seen in DIII-D experiment

and TAEFL simulation [Tobias et al., PRL 2011] 10/28

Drift-kinetic thermal ion, damping rate measurementvi/vA = 0.08, ρi/a = 0.008, kθρi = 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

ωA/(v A/R

0)

r/a

w/ thermal ionGTC w/ ther. ion

w/o thermal ionGTC w/o ther. ion

GTC antenna: (ωr, γ) = (0.199, 0.0106)vA/R0

GTC initial perturbation:(ωr, γ) = (0.198, 0.011)vA/R0

XHMGC initial perturbation:(ωr, γ) = (0.218, 0.017)vA/R0

0

0 100 200 300 400 500 600 700

δφ(a

.u.)

t/(R0/vA)

δφsat ∝ [(ω20 − ω2

ant)2 + 4γ2ω2

ant]−1/2

00.1 0.15 0.2 0.25 0.3

δφsa

t(a

.u

.)

ωant/(vA/R0)

Ant. data

Best fit

Similar damping rate measurement by antenna is also used in JET [Fasoli et al.,

PRL 1995 & PPCF 2010]11/28

Kinetic thermal ion and FLR effects on the mode structure

MHD + drift-kinetic fast ions, noFLR, ωr = 0.107vA/R0,

γ = 0.0159vA/R0

Drift-kinetic thermal &fast ions, no FLR,ωr = 0.168vA/R0,γ = 0.0174vA/R0

MHD + gyrokinetic fastions, ωr = 0.108vA/R0,γ = 0.0090vA/R0

[W. Deng et al., Phys. Plasmas, 17, 112504 (2010)]

12/28

RSAE may not exist with equilibrium current drive

In a tokamak with concentric circular flux surfaces and in single n,m limit, the vorticity equation reduces to the RSAE equation:

1

r

d

dr

(rΛ

d

drδφ

)− m2

r2Λ δφ− D

rδφ = 0

Λ represents the Alfven continuum [Zonca et al., PPCF 1996]:

Λ =ω2

v2A

− k2‖−

(7

4+TeTi

)2v2i

v2AR

20

D determines whether an eigenmode (RSAE) exists near the qmin

continuum extremum [Breizman et al., PoP 2005]:

D = k‖dk‖

dr+ rk‖

d2k‖

dr2−3k‖

dk‖

dr− rk‖

d2k‖

dr2+Df +Dp +Dt

13/28

No RSAE with equilibrium current in a simple case√〈δφ2〉f radial-time contour plots for (n,m) = (4, 6) RSAE:

Without equilibrium currentEigenmode exists, oscillation at differentlocation has the same frequency.Amplitude damps slowly.

With equilibrium currentEigenmode doesn’t exist, oscillation’sfrequency at different location is the localcontinuum frequency.Amplitude damps quickly.

14/28

Outline

1 GTC gyrokinetic model for Alfven eigenmodes

2 Verification of gyrokinetic model for RSAEZero-β limit and benchmark with XHMGCIon finite-β and FLR effectsEquilibrium current effect

3 Gyrokinetic simulations of RSAE in DIII-D discharge #142111Damping and driving mechanismsMode frequencies and structuresRSAE frequency up-sweeping and RSAE to TAE transitionComparisons among GTC, GYRO and TAEFL

15/28

DIII-D #142111 spectrogram and ∼ 750ms profiles from EFIT/ONETWO

[Van Zeeland et al., PoP 2011; Tobias et al., PRL 2011]

GTC GYRO-0.8

-0.4

0

0.4

0.8

1.2 1.4 1.6 1.8 2 2.2

Z(m)

R(m)

q=7.18

q=5

q=4

q=3.5

q=3.25

q=3.33

TAEFL

q-profile:

33.5

44.5

55.5

66.5

77.5

0 0.2 0.4 0.6 0.8 1ρ

q

qmin = 3.18

Temperature:

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

ρ

T/keV

Ion

e−

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

ρ

Tf/keV

Density:

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

ρ

n/(1012cm−3)

Ion

e−

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

ρ

nf/(1012cm−3)

GTC, GYRO and TAEFL use the same geometry from EFIT and the same plasmaprofiles from ONETWO for rigorous benchmark.

16/28

Damping and driving mechanisms identified & measured

Case Description (ωr, γ)/(2π)/kHz γ/ωr and dampingor driving mechanism

(I) Zero temperature ∼ 0ideal MHD (73.8,∼ 0) continuum damping

Finite δE‖, adiabatic e− −0.0357

(II) with Te ← Te + 7Ti/4, (87.3,−3.12) radiative dampingkinetic ion with Ti ← 0.01Ti added on top of case (I)

Same as case (II) except −0.00586(III) for real Te & Ti (85.1,−0.499) ion Landau damping & gradient

profiles recovered driving added on top of case (II)

Drift-kinetic e− added −0.0190

(IV) on top of case (III) (85.1,−1.61) e− kinetic damping addedon top of case (III)

Same as case (III) except 0.0733(V) that fast ions are added in (92.6, 6.79) fast ion gradient driving

added on top of case (III)

Drift-kinetic e− added 0.0671

(VI) on top of case (V) (92.0, 6.17) e− kinetic damping addedon top of case (V)

Damping rate calculation requires non-perturbative, fully self-consistent modestructure.

17/28

Alfven continua, mode frequencies and widthsZero temperautre:

Finite-β in slow sound approx. [Chu1992]:

Accurate finite-β zoomed in to see BAAE[Gorelenkov et al., PoP 2009] gap:

Accurate finite-β [Cheng1986]:

18/28

Mode structures of cases (I) and (II)(I) Zero T ideal MHD

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

(II) Finite-β artificially on e−

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

19/28

Mode structures of cases (III) and (IV)(III) GK bg ion, no fast ion, ad. e−

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

(IV) GK bg ion, no fast ion, DK e−

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

20/28

Mode structures of cases (V) and (VI)(V) GK bg & fast ion, adiabatic e−

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

(VI) GK thermal & fast ion, DK e−

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

21/28

qmin scan reveals RSAE freq. up-sweeping and RSAE to TAE transition

50

60

70

80

90

100

110

3.053.13.153.23.253.33.35

ω/(2π

)/kH

z

qmin

Experiment

GTC

n = 3

50

60

70

80

90

100

110

3.053.13.153.23.253.33.35

ω/(2π

)/kH

z

qmin

Experiment

GTC

n = 4

0

2

4

6

8

10

12

14

3.13.153.23.253.33.350

0.05

0.1

0.15

0.2

γ/(2π

)/kH

z

Mod

ew

idth

(FW

HM

)

qmin

γ

Mode width

n = 3

[W. Deng et al., submitted to NuclearFusion, 2011]

[Van Zeeland et al., PoP 2011; Tobias etal., PRL 2011]

22/28

Closer look at transition from RSAE to TAE (1)

qmin = 3.22

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

qmin = 3.18

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

23/28

Closer look at transition from RSAE to TAE (2)

qmin = 3.16

0.1 0.2 0.3 0.4 0.5 0.6

|δφm|(

a.u

.)

ρ

m = 8

m = 9

m = 10

m = 11

qmin = 3.14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

|δφm|(

a.u

.)

ρ

TAE1TAE2

m = 8

m = 9

m = 10

m = 11

m = 12

24/28

Btor and J direction effect (qmin = 3.22)B�,J�

B�,J⊗

B⊗,J�

B⊗,J⊗

25/28

Comparisons among GTC, GYRO and TAEFL

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

qmin=3.16 qmin=3.22 qmin=3.30

GTC

GYRO

TAEFL

GTC

GYRO

TAEFL

26/28

Summary (1)

Electromagnetic gyrokinetic simulation model used in GTC ispresented and can be shown to reduce to ideal MHD theory in thelong wave-length limit. GTC simulations of TAE, RSAE and BAEhave been verified.

In tokamak with circular flux surfaces, some basic RSAE featuresare verified:

I Frequency raising by kinetic thermal ion pressure.I Fast ions non-perturbatively modify the RSAE mode structure due

to radial symmetry breaking.I With plasma current taken into account, RSAE may not exist.

[W. Deng et al., Phys. Plasmas, 17, 112504 (2010)]

27/28

Summary (2)

Simulations of DIII-D discharge #142111 near 750msI Damping and driving mechanisms of RSAE identified and measured.I Simulation frequency close to experiment in both up-sweeping

RSAE region and RSAE to TAE transition region.I The mode structure becomes upside down when Btor is reversed.I Good agreements in frequency, growth rate and mode structure in

comparisons among GTC, GYRO and TAEFL.

[W. Deng et al., submitted to Nuclear Fusion, 2011]

Next step: nonlinear RSAE study, which is feasible based onprevious GTC nonlinear simulations:

I Nonlinear ITG & CTEM turbulence simulations in reversed shearplasmas [W. Deng & Z. Lin, Phys. Plasmas, 16, 102503 (2009)]

I Nonlinear BAE simulations [H. Zhang et al., to be submitted toPRL]

28/28