fluid models for burning plasmas: reactive egams · 2017. 9. 18. · reactive egam two-stream...
TRANSCRIPT
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Fluid models for burning plasmas:
reactive EGAMs
M. J. Hole1, Zhisong Qu1, M. Fitzgerald2,1, B. Layden1, G. Bowden1, R. L. Dewar1
15th IAEA Technical Meeting on Energetic Particles Sep. 5–Sep. 8, 2017
[1] Australian National University, ACT 0200, Australia [2] CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon OX14 3DB, United
Kingdom
Funding Acknowledgement: Australian Research Council, UK Engineering and Physical Sciences Research Council 1
http://intranet.culham.ukaea.org.uk/
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Outline • EGAMs: Calculation of energetic geodesic acoustic modes
(EGAMs) using fluid theory Demonstration of reactive EGAMs (2 stream instability) with
radially localised modes [Qu Z.S, Hole M. J. , Fitzgerald M, PRL 116, 095004 (2016).]
Extension to solve global mode structure [Qu Z.S, Hole M. J. , Fitzgerald M, PPCF 59 055018 (2017)]
• Anisotropy and Flow: equilibrium and stability in 2D [M. J. Hole, M. Fitzgerald, Z. Qu, B. Layden]
• Development and implementation of continuum damping in 3D [G. Bowden, A. Koenies, M. J. Hole]
• Conclusions
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Geodesic Acoustic Modes (GAMs) • 𝑛𝑛 = 0,𝑚𝑚 = 0 with 𝑚𝑚 = ±1 side bands • Mainly electrostatic • Radial electric field 𝐸𝐸𝑟𝑟
- 𝐸𝐸 × 𝐵𝐵 drift - density accumulation due to geodesic curvature, pressure perturbed - radial current, restoring 𝐸𝐸𝑟𝑟
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Zoo of GAMs
Conventional GAMs
ICRH driven (E?)GAMs
NBI driven EGAMs
Frequency ~ 74𝑇𝑇𝑖𝑖 + 𝑇𝑇𝑒𝑒 + 𝑂𝑂(
1𝑞𝑞2
) Same as conventional GAMs
Determined by fast ions (can be half of the thermal GAM)
Drive Nonlinear interaction with turbulence
ICRH trapped fast ions, positive dF/dE
NBI passing fast ions, positive dF/dE
Localization Local, edge Core Global
Observation Nearly all machines JET DIII-D, ASDEX-U, LHD, HL-2A
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Energetic-particle-driven GAMs (EGAMs)
Thermal GAM frequency
EGAM frequency
1/2 thermal GAM frequency
Neturon Loss
Chirping
Counter Beam
Global Mode
[ Nazikian, R. et al. Plasma. Phys. Rev. Lett. 101, 185001 (2008).]
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Existing kinetic theory of EGAM • Hybrid theory : fluid bulk, kinetic fast ions [Fu, G.Y. PRL.
101, 185002 (2008).] • Driven unstable by inverse Landau damping
𝝎𝝎𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 < 𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮
𝝎𝝎𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 > 𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮 𝜷𝜷𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕/𝜷𝜷𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝜷𝜷𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕/𝜷𝜷𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕
𝜷𝜷𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕/𝜷𝜷𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝜷𝜷𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕/𝜷𝜷𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕
𝑂𝑂(1) fast ion response
𝝎𝝎𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮
𝝎𝝎
𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮
𝜸𝜸𝝎𝝎
𝜸𝜸𝝎𝝎
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Why develop a fully fluid theory?
• An unstable mode emerged from Landau poles when fast ions added. Excited Landau poles?
• Unstable mode exist even when beam is cold (small thermal spread), fluid treatment can be valid for fast ions.
• Can fluid theory help to understand? → Simpler, better understanding
Zarzoso, D. et al. Nucl. Fusion 54, 103006 (2014). Girardo, J.-B. et al. Phys. Plasmas 21, 092507 (2014).
𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
Bump
Complex frequency
Growing
Damping Solution to Dispersion relationship
[Zarzoso NF 2014] [Girardo PoP 2014]
EGAM
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Fluid model of EGAM
Electrons Thermal ions Fast ions No flow
𝑛𝑛𝑒𝑒 = 𝑛𝑛𝑖𝑖 +𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑇𝑇𝑒𝑒
No flow 𝑛𝑛𝑖𝑖 𝑇𝑇𝑖𝑖
𝑽𝑽𝟎𝟎 = 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝒃𝒃 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝑃𝑃∥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ,𝑃𝑃⊥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
• Large aspect ratio, circular cross section : (𝑟𝑟,𝜃𝜃,𝜑𝜑) • Radially localised electrostatic perturbations (𝑩𝑩� = 0)
𝑉𝑉� = 𝑉𝑉𝐸𝐸�𝐵𝐵0𝐵𝐵�̂�𝜃 + � 𝑉𝑉�∥𝑚𝑚𝒃𝒃 + 𝑉𝑉�𝑑𝑑𝑚𝑚𝒃𝒃 × 𝜿𝜿 𝑒𝑒𝑖𝑖𝑚𝑚𝑖𝑖
𝑚𝑚=2
𝑚𝑚=−2
Perturbed velocity 𝐸𝐸 × 𝐵𝐵
drift
Parallel velocity
Magnetic drift velocity (mode structure)
[Qu, Z.S. et al. Phys. Rev. Lett. 116, 095004 (2016).]
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• Thermal/Fast ion response : the double-adiabatic (CGL) closure
Why CGL?
− It can give the right thermal GAM frequency
~ 74𝑇𝑇𝑖𝑖 + 𝑇𝑇𝑒𝑒 + 𝑂𝑂(
1𝑞𝑞2
) (while MHD needs to artificially
manipulate the adiabatic index 𝛾𝛾) − Heat flow is not important in our case (mode frequency far
from thermal frequency of thermal/fast ions)
• Electrons : isothermal response to the field
Fluid model of EGAM
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• The momentum equation for thermal/fast ions, electrons
• Adding up species and use quasi-neutrality: 𝛻𝛻 ⋅ 𝐽𝐽 = 0
– Drop magnetic drift velocity (zero orbit width)
Local dispersion relationship – Keep all terms (finite orbit width) Global dispersion relationship
Fluid model of EGAM
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Local dispersion relationship
Bulk response (thermal GAM) Fast ion response
𝛼𝛼 = 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛𝑓𝑓𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡
𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
Bump
𝜔𝜔𝑏𝑏 = fast ion transit frequency
• For a bump on tail distribution with negligible beam thermal spread ‒ Doppler shift of the wave in the static frame of fast ions
• Three roots are presented, depends on the relationship between 𝜔𝜔𝑏𝑏, 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, q and fast ion density. 11
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𝜔𝜔𝑏𝑏 < 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺 • 𝜔𝜔𝑏𝑏 = 0.58𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, 𝑞𝑞 = 4, 𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑇𝑇𝑖𝑖 = 0.25
𝜔𝜔 = 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺
Mode unstable
𝝎𝝎𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮
𝜸𝜸𝝎𝝎
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𝜔𝜔𝑏𝑏 < 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺 • 𝜔𝜔𝑏𝑏 = 0.58𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, 𝑞𝑞 = 4, 𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑇𝑇𝑖𝑖 = 0.25
[Fu PRL 2008]
𝜔𝜔 = 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺
Mode unstable
𝝎𝝎𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮
𝜸𝜸𝝎𝝎
But no wave-particle interaction in fluid theory
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𝜔𝜔𝑏𝑏 < 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺 • 𝜔𝜔𝑏𝑏 = 0.58𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, 𝑞𝑞 = 4, 𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑇𝑇𝑖𝑖 = 0.25
No turn on threshold :
𝜔𝜔 = 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺
No Landau damping is presented in the fluid theory : this instability must not come from wave-particle interaction
Mode unstable
14 (𝛼𝛼 = 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛𝑓𝑓𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡 )
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𝜔𝜔𝑏𝑏 > 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺
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• 𝜔𝜔𝑏𝑏 = 1.76𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, 𝑞𝑞 = 2, 𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑇𝑇𝑖𝑖 = 1
𝜔𝜔 = 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺
Bifurcation
When 𝜔𝜔𝑏𝑏 >2𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, no unstable region found
Good match between kinetics (symbol) and fluid (lines)
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Reactive EGAMs cf dissipative EGAMs
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Negative energy wave 𝜕𝜕𝜔𝜔𝐷𝐷 𝜔𝜔𝜕𝜕𝜔𝜔
< 0
Positive energy wave 𝜕𝜕𝜔𝜔𝐷𝐷 𝜔𝜔𝜕𝜕𝜔𝜔
> 0
Coupled positive/negative
energy wave
Energy transferred from negative wave to positive wave, both wave amplitude grow
Similar to two stream instabilities
Dissipative EGAMs [Fu PRL 2008]
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Reactive/dissipative EGAMs
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frequency
Growth rate
Transit frequency
Dis
tribu
tion
Func
tion
𝑇𝑇𝑖𝑖 𝑇𝑇𝑓𝑓
𝑇𝑇𝑓𝑓
EGAM frequency
To be reactive: 17
𝑇𝑇𝑓𝑓 < 𝑇𝑇𝑖𝑖 →
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Analogy to two-stream instabilities
Reactive EGAM Two-stream instabilities
Dissipative EGAM Bump-on tail instabilities Cold
beam Gentle slope 18
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Application to EGAM early onset • Instant turn on (~1ms) of the mode
after the beam is turned on, much faster than slowing down time (~a few 10ms).
• 𝐹𝐹 𝐸𝐸,Λ = 𝛿𝛿 𝐸𝐸 − 𝐸𝐸0 𝛿𝛿(Λ − Λ0) • For DIII-D conditions
– 𝐸𝐸0 = 75 keV,Λ0 = 0.5, 𝑞𝑞 = 4, 𝑇𝑇𝑒𝑒 = 1.2𝑇𝑇𝑖𝑖 = 1.2 keV
Early onset ~1ms after injection
𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛0
DIII-D observed frequency range
𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛0
No turn-on threshold
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… but EGAMs are not radially localised [ Nazikian, R. et al. PRL 101, 185001 (2008).]
• Used small expansion δ = Lorbit / Lmode,
Aim: Extend fluid model to resolve the radial mode structure of reactive EGAMs. [Qu, Hole, Fitzgerald PPCF 59 (2017) 055018]
• radial structure of the dissipative EGAMs in DIII-D reproduced using hybrid simulations: [Fu G 2008 PRL101, 185002]
fast ion drift orbit width measurement of the radial wavelength
• Found
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𝐿𝐿orbit~𝑞𝑞ρℎ
𝐿𝐿orbit = 𝐿𝐿ω~1
𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺𝑑𝑑𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺𝑑𝑑𝑟𝑟
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Generalisation to resolve mode structure • 3 small unitless small parameters:
δ = Lorbit / Lmode « 1, ε = a/R « 1, 𝑛𝑛�/𝑛𝑛 « 1
• Low beta tokamak with concentric flux surfaces → Magnetic / geometric axis at (R,Z) = (Ra,0)
• Fast ion distribution function given by
with Λ ≡ µB0/E the pitch angle
• All the fast ions now have the same energy E0 and pitch angle Λ0, consistent with the early beam injection scenario. → p||f = 0 and 𝑝𝑝⊥𝑓𝑓 = 𝑛𝑛𝑓𝑓𝐸𝐸0Λ0.
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𝐿𝐿orbit~𝑞𝑞ρ|| 𝐿𝐿mode~𝑑𝑑ln 𝐸𝐸𝑟𝑟𝑑𝑑𝑟𝑟
−1
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• Include a consistent treatment of the equilibrium fast ion density profile with FOW effects
Co-passing counter-passing
• Linear Perturbation treatment: − As before, add up species and use quasi-neutrality 𝛻𝛻 ⋅ 𝐽𝐽 = 0. − Local modes: O(δ) gives dispersion relation D(ω,r)=0 − Global modes: O(δ3) calculation
Generalisation to resolve mode structure
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Local mode dispersion relation: O(δ)
Originates from density change on a flux surface due to FOW effects : responsible for distinction between passing / co-passing
𝑐𝑐𝑓𝑓 =Λ0
2 − 2Λ0, 𝜔𝜔𝑏𝑏 𝑟𝑟 = 𝑉𝑉𝑓𝑓|| /(𝑞𝑞 𝑟𝑟 𝑅𝑅0)
• Roots of D(ω,r)=0 are dispersion solutions • In the limit Λ0 → 0 (i.e. completely tangential beam), 𝑐𝑐𝑓𝑓 → 0,
and approaches previous dispersion relation (bump-on-tail distribution).
𝛼𝛼 = 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛𝑓𝑓𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡
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Global mode dispersion relation: O(δ3) • Solve order-by-order continuity equation and the
momentum equation along with the closure condition equation: 𝑝𝑝�||𝑓𝑓 ≈ 0, 𝑝𝑝�⊥𝑓𝑓 ≈ 𝑛𝑛�𝑓𝑓𝐸𝐸0Λ0 giving
• Solve dispersion relation numerically by shooting method with BC: 𝑉𝑉�𝐸𝐸 0 = 0, and outgoing wave at other end
ρ||/L ~ O(δ) d2 /dr2 ~ O(δ3)
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Select DIII-D like conditions / profiles
• B0 = 3T, R0 = 1.7 m, ε= 0.3, E0 = 75 keV, Λ0 = 0.5, D beam. • Consider counter-passing fast ions (initially), and solve global
dispersion relation for monotonic / reverse shear cases
𝛼𝛼 = 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑛𝑛𝑓𝑓𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡; 𝑛𝑛𝑓𝑓𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡 = constant
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Monotonic Shear
• Asymptotic analysis near core:
Scanning Bo from 3T to 9T Mode has outgoing wave propagation
Lmode a function of radial position
ωEGAM, global ≥ ωEGAM, local
• WKB analysis near edge: 26
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Reverse Shear
Transition in dependence of Lmode with Lorbit at ~6T
Scanning Bo from 3T to 9T
27
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Dependency on injection angle: EGAM more unstable with counter vs.co • Change sign of ωb and keep all other parameters unchanged
monotonic
reverse
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EGAMs Summary 1) A fluid model is developed to describe EGAMs. 2) Valid when fast ion energy extent is small (Tf
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EGAMs Summary 5)
– For monotonic shear: near axis: 𝐿𝐿mode ∝ 𝐿𝐿orbit at edge: 𝐿𝐿mode ∝ 𝐿𝐿orbit
– For reverse shear: 𝐿𝐿mode ∝ 𝐿𝐿orbit (except at high B)
6) EGAM more unstable with counter beam injection
7) Some consistency with DIII-D experiments.
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� Fluid models for burning plasmas: reactive EGAMs�OutlineGeodesic Acoustic Modes (GAMs)Zoo of GAMsEnergetic-particle-driven GAMs (EGAMs)Existing kinetic theory of EGAMWhy develop a fully fluid theory?Fluid model of EGAMSlide Number 9Slide Number 10Local dispersion relationship 𝜔 𝑏 < 𝜔 𝐺𝐴𝑀 𝜔 𝑏 < 𝜔 𝐺𝐴𝑀 𝜔 𝑏 < 𝜔 𝐺𝐴𝑀 𝜔 𝑏 > 𝜔 𝐺𝐴𝑀 Reactive EGAMs cf dissipative EGAMsReactive/dissipative EGAMsAnalogy to two-stream instabilitiesApplication to EGAM early onsetSlide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28EGAMs SummaryEGAMs Summary