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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
Seventh IAEA TM on Plasma Instabilities 2
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
Seventh IAEA TM on Plasma Instabilities 3
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
Seventh IAEA TM on Plasma Instabilities 4
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].
Seventh IAEA TM on Plasma Instabilities 5
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
Seventh IAEA TM on Plasma Instabilities 6
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′′ ;
Seventh IAEA TM on Plasma Instabilities 7
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]
Seventh IAEA TM on Plasma Instabilities 8
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]
Seventh IAEA TM on Plasma Instabilities 9
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
)
Seventh IAEA TM on Plasma Instabilities 10
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
Seventh IAEA TM on Plasma Instabilities 11
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
Seventh IAEA TM on Plasma Instabilities 12
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)
Seventh IAEA TM on Plasma Instabilities 13
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
Seventh IAEA TM on Plasma Instabilities 14
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)
Seventh IAEA TM on Plasma Instabilities 15
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.
Seventh IAEA TM on Plasma Instabilities 16
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
Seventh IAEA TM on Plasma Instabilities 17
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))
Seventh IAEA TM on Plasma Instabilities 18
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)
Seventh IAEA TM on Plasma Instabilities 20
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
Seventh IAEA TM on Plasma Instabilities 21
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
Seventh IAEA TM on Plasma Instabilities 22
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”
Seventh IAEA TM on Plasma Instabilities 23
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
Seventh IAEA TM on Plasma Instabilities 24
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.