excitation of kinetic geodesic acoustic modes by … iaea tm on plasma instabilities 1 excitation of...

Seventh IAEA TM on Plasma Instabilities 1

Excitation of Kinetic Geodesic Acoustic Modes by Drift Waves

in Nonuniform Plasmas

Zhiyong Qiu1, Liu Chen1,2 and Fulvio Zonca3,1

1 Inst. Fusion Theory & Simulation and Dept. Phys., Zhejiang Univ., Hangzhou, P.R.C.2 Dept. Phys. & Astronomy, Univ. California, Irvine, CA 92717-4575, U.S.A.

3 ENEA C. R. Frascati, C. P. 65-00044 Frascati, Italy.

March 6, 2015Seventh IAEA TM on Theory of Plasma Instabilities,

March 4 - 6, Frascati


Outline

• Motivation

• Theoretical model

1. fully nonlinear two-field DW-GAM equations2. unified theoretical framework of GAM/KGAM

• Excitation of GAM/KGAM in nonuniform plasmas

1. finite DW and KGAM linear group velocities: convective amplification2. nonuniform diamagnetic frequency: absolute instability3. nonuniform GAM frequency: additional asymmetry

• Nonlinear saturation of DW due to GAM excitation

• Summary


Motivation

2 Drift wave (DW) type turbulence induced by expansion free energy due to plasmanonuniformity is a main candidate for causing “anomalous transport”

2 DW regulation by spontaneously excited Zonal Flow (ZF)/Zonal Structures (includ-ing kinetic geodesic acoustic mode (KGAM), finite frequency component of ZF), byscattering it into stable short radial wavelength regime. Envelope modulation

2 Resonant coherent parametric decay processes:

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

kr

ω

(ω0−ω

G,k

0−k

G)

(ω0,k

0)

(ωG

,kG

)

DW

KGAM

• DW→ DW + KGAM

• Frequency/wavenumber matching condi-tions


2 Existing theories on GAM/KGAM excitation by DW turbulence: uniform plasma,fluid limit [Zonca08, Chakrabarti07]

• growth rate proportional to krΓ (Γ: normalized pump DW amplitude)

• kr determined by (ω, k) matching conditions (kr ≡ nq′θk: radial envelope)

2 Plasma nonuniformity and/or kinetic effects: qualitatively change the features of theparametric decay instability

• linear group velocities of GAM and DW due to kinetic dispersiveness

• system nonuniformities

2 Nonlinear theory must be applied to understand the experimental observation oflinearly stable GAM [Qiu15].


Theoretical Model

2 Resonant parametric decay process marginally unstable: GAM only modify the radialenvelope of DWs, parallel mode structures remain unchanged

2 Three wave parametric decay instability: δϕ = δϕd + δϕG

δϕd = Ade−inξ−iωP t

∑eimθΦ0(nq −m) + c.c.,

δϕG = AGe−iωGt + c.c.;

2 ...with Ad and AG being the radial envelopes:

Ad = ei∫kddr, AG = ei

∫kGdr + c.c..

2 ...and Φ0(nq−m) being the short radial scale structure associated with k∥ and mag-netic shear

Φ0(nq −m) ≡ 1√2π

∫ ∞

−∞e−i(nq−m)ηψ0(η)dη


2 The nonlinear equations for the DW-GAM system derived from quasi-neutrality con-dition (δH = δHL + δHNL):

n0e2

Ti

(1 +

TiTe

)δϕk − ⟨eJkδHL

i ⟩k + ⟨eδHLe ⟩k

= − i

ωk

⟨ec

B

∑b · (k′′ × k′)δϕk′δHe,k′′

⟩k− ⟨eδHNL

e ⟩k

− i

ωk

⟨ec

B

∑b · (k′′

⊥ × k′⊥) (JkJk′ − Jk′′) δϕk′δHi,k′′

⟩k.

2 ...and the nonadiabatic particle response derived from nonlinear gyrokinetic equation[Frieman82]:(

∂t + v∥∂l + iωd

)kδHk = − qs

mJk

(∂tδϕ∂E + (∇Xδϕ× b/Ωc) · ∇X

)F0

+qsm

∑Jk′(∇Xδϕk′ × b/Ωc) · ∇XδHk′′ ;


Fully nonlinear two-field equations

2 Coupled nonlinear DW-GAM equations

ωdDdAd =c

B0

TiTekθAd∂rAG, (1)

ωGEGAG =α

2

c

B0ρ2i kθ∂r

(Ad∂

2rA

∗d − c.c.

). (2)

with

Dd ≡ 1 +Ti

Te−∫ ∞

−∞ψ0(η)

⟨eJ0(γ)δH

Li

⟩dη/

(n0e2

TiAd

)

EG ≡[n0e2

Ti

(1 +

Ti

Te

)δϕG −

⟨eJGδH

Li,G

⟩+⟨eδHL

e,G

⟩]/(n0e2

TiδϕG

).

2 Fully nonlinear two-field equations without separating DW into a constant amplitudepump and a lower sideband with much smaller amplitude [Guo09]: applicable toinvestigate nonlinear DW saturation, turbulence spreading due to excitation of GAM

2 Recover the usual three-field equations by separating δϕd = δϕP +δϕS , and averagingover fast time scales [Zonca08]


Unified theoretical framework of GAM

2 GAM resonantly driven unstable by velocity space anisotropic energetic particles(EPs) [Nazikian08,Fu08]: EGAM

2 EP contribution readily included in the nonlinear equations, by replacing EG withEEGAM [Qiu10] (only l = ±1 transit resonances kept):

EEGAM = EG + 2πBδϕGeiωGt Ω2

i

nck2r

∑σ=±1

∫EdEdΛ

|v∥|∂F0h

∂E

ω2dh

ω2 − ω2tr

. (3)

2 EP contribution in EEGAM is formally linear, nonlinearity enters via evolution ofequilibrium EP distribution due to emission and reabsorption of EGAM [Zonca15]

− iωF0h = ie2ωd

16|δϕG(τ)|2

∂

∂E

[ωd(ω − iγ)

(ω − iγ)2 − (ω20r − ωtr)2

]∂

∂EF0h(ω − 2iγ) + F0h(0). (4)

2 Neglecting DWs, the coupled nonlinear equations describe nonlinear saturation ofEGAM [QiuEPS14]: wave-particle trapping in the weak drive limit [Qiu11], andpitch angle scattering [in progress]

2 This presentation: ignore EPs, and investigate the effect of kinetic particle responsesand system nonuniformities on GAM excitation [Qiu14]


Effects of nonuniformities on GAM excitation

Three field equations

2 Three field equations obtained by taking δϕd = δϕP + δϕS , with

δϕP = AP e−inξ−iωP t

∑eimθΦ0(nq −m) + c.c.,

δϕS = ASeinξ−i(ωG−ωP )t

∑e−imθΦ∗

0(nq −m) + c.c.,

2 kGρi ≪ 1, |ωG/ω0| ≪ 1,

DS(ωS ,kS , r) = D0r(ωP∗ , kP∗ , r0) +∂D0r

∂ωP∗(i∂t + ωG) +

∂D0r

∂kS

∣∣∣∣0

kS +1

2

∂2D0r

∂k2S

∣∣∣∣∣0

k2G

+∂D0r

∂r0(r − r0) +

1

2

∂2D0r

∂r20(r − r0)

2 + iDI + · · · .

2 Assuming quadratic dispersiveness for DW, and a Guassian profile for ω∗ ⇒

DS = i

(∂t + γS + iωP − iω∗

(1− (r − r0)

2

L2∗

)− iCdω∗ρ

2i

∂2

∂r2

)


2 ... we then obtain the coupled nonlinear equations describing GAM excitation:(∂t + γS + iωP − iω∗

(1− (r − r0)

2

L2d

)− iCdω∗ρ

2i

∂2

∂r2

)AS = Γ∗

0E , (5)(∂t(∂t + 2γG) + ω2

G(r)− CGω2G(r0)ρ

2i

∂2

∂r2

)E = −Γ0∂t∂

2rAS , (6)

with E the electric field of GAM, Γ0 ≡ (αiTi/ωPTe)1/2ckθδϕP/B normalized

pump amplitude2 Describe the nonlinear excitation of GAM by DW, accounting for kinetic dispersive-

ness, system nonuniformities

2 Feedbacks of GAM and DW sideband on DW pump ignored: linear growth stage ofthe parametric instability


Effects of nonuniformities on GAM excitation

2 There are three length scales due to nonuniformities in this system

• GAM continuum: LG ∼ a

• Nonuniform diamagnetic profile (ω∗(r)): Ld ∼ a

• Nonuniform pump DW (Γ(r)) consistent with ω∗(r): LP ∼√ρiLd ≪ LG, Ld

2 Problem can be simplified, taking advantage of the scale separation LP ≪ LG, Ld

• local limit: uniform plasma [Zonca08,Chakrabarti07]

• shortest time scale: ignore the effect of Ld and LG, and study the effect of LPand finite group velocities due to kinetic dispersiveness on GAM excitation

• longer time scale: ignore LG to illustrate the effect of nonuniform ω∗(r)

• systematically account for the effects of LP , Ld and LG


Uniform plasma: convective amplification

2 Here, we focus on the effect of kinetic dispersiveness, while neglecting nonuniformities

2 Two spatial- and temporal-scale expansion: ∂t = −iω + ∂τ and ∂r = ∂ζ + ikr ⇒

(∂τ + γS + VS∂ζ)AS = Γ∗0(ζ)E ,

(∂τ + γG + VG∂ζ)E =1

2Γ0(ζ)

(k2r − 2ikr∂ζ

)AS .

with VS = 2Cdω∗ρ2i kr and VG = CGω

2G(0)ρ

2i kr/ω the linear group velocities of DW sideband

and GAM. kr and ω derived from matching conditions

2 Ignoring VS and VG associated with kinetic dispersiveness [Zonca08] ⇒

(γ + γS)(γ + γG) = k2rΓ20.

• threshold on pump amplitude: γSγG < k2rΓ20

• well above threshold: γ = krΓ0 ⇒ excitation favors short wavelength KGAM⇒ motivation to investigate effects of finite group velocities (∝ kr)


2 Considering the effects of linear group velocities: convective amplification v.s.absolute instability depending on sign of VGVS(⇒ CdCG) [Rosenbluth72]

2 CdCG > 0 for typical tokamak parameters: DW parametric decay is a convectiveinstability, less interest for confinement

• Cd > 0: finite radial envelope variation due to coupling between neigh-boring poloidal harmonics

• CG > 0 for typical tokamak parameters, exceptions [Zonca08]

0 500 1000−0.02

0

0.02

t

φ G(r

0)

−200 −100 0 100 200−1

0

1

x

ampl

itude

s

0 500 1000−1500

0

1500

t

φ G(r

0)

−200 −100 0 100 200−4000

0

4000

x

ampl

itude

s

GAMDWSB

GAMDWSB

(a)

(d)

(b)

(c)

• fix Cd = 1

• upper panel: CG = 1, convective

• lower panel: CG = −1, absolute

• need to go to longer scales


Spatial - temporal evolution of the parametrically excited GAM

2 Moving into wave-frame by taking ξ = ζ − Vcτ , with Vc = (VS + VG)/2, thecoupled nonlinear equations then reduce to the nonlinear GAM equation(

∂2τ − V 20 ∂

2ξ

)E =

1

2k2rΓ

20E − ikrΓ

20∂ξE . (7)

2 Letting E = exp(iβξ)A(ξ, τ), with β = krΓ20/(2V

20 ), equation (7) reduces to

(∂2τ − V 2

0 ∂2ξ

)A =

(1

2k2rΓ

20 + βkrΓ

20 − β2V 2

0

)A ≡ η2A. (8)

2 Equation (8) has a general solution: A = A exp[ikIξ +

√η2 − k2

IV20 τ

]with kI

being the wavenumber conjugate to ξ at τ = 0. When convective damping dueto FLR effects are higher order corrections to the temporal growing (|V0∂ξ| ≪|∂τ |), equation (7) has the following time asymptotic solution

E = E0 exp

(ητ + iβ(ζ − Vcτ)−

η

2V 20 τ

(ζ − Vcτ)2

). (9)


2 Parametrically excited GAM propagates at a nonlinearly coupled group velocity

V NLG = Vc = (VS + VG)/2 ≫ VG (10)

• much larger than that predicted by linear theory

• independent of Γ0: can be obtained from kr ⇐matching condition

2 Wavevector of the parametrically excited GAM increased with pump amplitude

kNL = kr − i∂ξ lnE = kr(1 + Γ2

0/(2V20 )

)(11)

2 Frequency of the parametrically excited GAM

ωNL = ω0 + i∂τ lnE = ω0 + krΓ20Vc/(2V

20 )

= ωG +krΓ

20Vc

2V 20

+CGωGρ

2i k

2NL

2(1 + Γ20/(2V

20 ))

2. (12)

• nonlinear frequency higher than ω0 solved from matching conditions

• the frequency increment krΓ20Vc/(2V

20 ) is independent of kr

• effective “CNLG ” smaller than that of linear D. R.


2 This explains the HT-7 experiments on GAM [Kong13]:

• high frequency branch of the observed “dual GAM”: |∆ω/ω0| ∼|eδϕ/T |2(Ln/ρi)2, order of unity frequency increment for typical tokamakparameters

• overestimation of “CNLG ” if one analyzes experimental data/numerical re-

sults with linear theory [Kong13, Hager12]

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

kr

ω

Linear D. R.Fitting exp. data with Linear D. R. Nonlinear D. R.

Experimentaldata


Nonuniform ω∗(r): from convective to absolute instability

2 Uniform ω∗ case: generated DW sideband and GAM couple together, and prop-agate out of the unstable region due to VSVG > 0

2 Nonuniform ω∗(r) case: outward propagating DW sideband reflected at theturning points induced by ω∗(r) nonuniformity, and propagate through r0 again⇒ quasi-exponentially growing absolute instability

0 1000 2000 2500−10

0

10

20

30

t

log

(φG

(r0))


Nonuniform ω∗(r): nonlinear DW eigenmode

2 Taking ∂t = −iω, the coupled nonlinear GAM-DW sideband equations reducedto nonlinear DW sideband equation in kr−space [White74](

ω∗

L2d

∂2

∂k2r+ ω − ωP + ω∗ − Cdω∗ρ

2i k

2r +

ωk2rΓ20

ω2 − ω2G−CGω2

Gρ2i k

2r

)AS = 0. (13)

2 Strong drive (γ/ωG ≫ k2rρ

2i ) ⇒ Weber’s equation ⇒ nonlinear DW eigenmode

AS ∝ exp(−k2r/β

2), with the dispersion relation given as

L2d

ω∗(ω − ωP + ω∗)β

2 = 2l + 1, l = 0, 1, 2, 3

and β given by

β4L2d

ω∗

(Cdω∗ρ

2i +

ωΓ20

ω2G − ω2

)= 1.

∗Similar equation derived in [Guzdar09], and solved numerically to show the localization ofGAM due to ω∗(r)


Nonuniform GAM and DW sideband

2 Take all nonuniformities self-consistently into account

• ω∗(r) plays dominant role, and renders the convective instability intoquasi-exponentially growing absolute instability

• ωG(r) plays relatively minor role, and induces additional asymmetry, i.e.,wave packet initially propagating in smaller ωG region (radially outward)has larger kr and thus, larger growth rate and group velocity

0 1000 2000 2500−8

−4

0

4

8

10

t

log

(φG

(r0))

uniform ωG

nonuniform ωG


2 Snapshots of mode structure at different times

−200 −100 0 100 200−1000

−500

0

500

1000t=1000

−200 −100 0 100 200−4000

−2000

0

2000

4000t=1200

−200 −100 0 100 200−2

−1

0

1

2x 10

4 t=1500

−200 −100 0 100 200−2

−1

0

1

2x 10

5 t=1800

−200 −100 0 100 200−4

−2

0

2

4x 10

6 t=2150

−200 −100 0 100 200−2

−1

0

1

2x 10

7 t=2350

GAM

DW sidebandGAM

DW sideband

GAM

DW sidebandGAM

DW sideband

GAMDW sideband

GAM

DW sideband


Nonlinear DW saturation due to GAM excitation

2 Preliminary results from numerical solution of the coupled fully nonlinear two-field equations

2 Ignoring drive and damping, and study the spatial-temporal evolution of a givenDW envelope solved self-consistently from DW eigenmode equation.

2 GAM excitation peaks at maximum Ad gradient: maximum drive (cross-section,∝ |∂2

r |)2 GAM excitation: competition between nonlinear drive (parameterized by Ad(0))

and dispersiveness (parameterized by CG and |∂r|). Fix the nonlinear driveAd(0), and vary CG:

• increase CG: parametric instability becomes less unstable

• decrease CG: local growth becomes more important ⇒ generation ofshorter scale structures ⇒ faster growing and regularization by disper-siveness ⇒ parametric instability can be “bursty”


2 DW propagate to linearly stable region due to GAM excitation: turbulencespreading

−100 −50 0 50 100−1

−0.5

0

0.5

1

r

ampl

itude

s

A

d(t=0)

Re(Ad(t=500))

Im(Ad(t=500))

AG

(t=500)

mode structures

0 100 200 300 400 500 600 700 800 900 1000−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t

decrease of |Ad|2

increase of |AG

|2

energy exchange between DW andGAM

2 Energy exchange between DW and GAM

2 More investigation needed for a better understanding of DW saturation


Summary

2 Derived the general equations describing parametric excitation of GAM by DWturbulences

2 Kinetic effects and plasma nonuniformities must be taken into account to cor-rectly understand the excitation of GAM by DW in experiments

• finite group velocities of DW/GAM: convective/absolute instability

• nonuniform diamagnetic frequency ω∗(r): nonlinear DW eigenmode

• GAM continuum: additional asymmetry

2 Nonlinear theory must be applied to understand the experimental observationsof linearly stable GAM

2 Nonlinear saturation of DW due to GAM excitation: important roles played bykinetic dispersiveness. More in-depth investigation needed

∗ Acknowledgements: work supported by US DoE, ITER-CN, NSFC and EUROfusion projects.

excitation of kinetic geodesic acoustic modes by … iaea tm on plasma instabilities 1 excitation of...

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