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/

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

2

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

3

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

4

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).]

5

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

6

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

7

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).]

8

• 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

9

• 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

10

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

𝜔𝜔𝑏𝑏 < 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺 • 𝜔𝜔𝑏𝑏 = 0.58𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺, 𝑞𝑞 = 4, 𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝑇𝑇𝑖𝑖 = 0.25

𝜔𝜔 = 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺

Mode unstable

𝝎𝝎𝝎𝝎𝑮𝑮𝑮𝑮𝑮𝑮

𝜸𝜸𝝎𝝎

12

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

13

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 )

𝜔𝜔𝑏𝑏 > 𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺

15

• 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)

Reactive EGAMs cf dissipative EGAMs

16

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]

Reactive/dissipative EGAMs

•

frequency

Growth rate

Transit frequency

Dis

tribu

tion

Func

tion

𝑇𝑇𝑖𝑖 𝑇𝑇𝑓𝑓

𝑇𝑇𝑓𝑓

EGAM frequency

To be reactive: 17

𝑇𝑇𝑓𝑓 < 𝑇𝑇𝑖𝑖 →

Analogy to two-stream instabilities

Reactive EGAM Two-stream instabilities

Dissipative EGAM Bump-on tail instabilities Cold

beam Gentle slope 18

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

19

… 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

20

𝐿𝐿orbit~𝑞𝑞ρℎ

𝐿𝐿orbit = 𝐿𝐿ω~1

𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺𝑑𝑑𝜔𝜔𝐺𝐺𝐺𝐺𝐺𝐺𝑑𝑑𝑟𝑟

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.

21

𝐿𝐿orbit~𝑞𝑞ρ|| 𝐿𝐿mode~𝑑𝑑ln 𝐸𝐸𝑟𝑟𝑑𝑑𝑟𝑟

−1

• 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

22

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

23

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)

24

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

25

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

Reverse Shear

Transition in dependence of Lmode with Lorbit at ~6T

Scanning Bo from 3T to 9T

27

Dependency on injection angle: EGAM more unstable with counter vs.co • Change sign of ωb and keep all other parameters unchanged

monotonic

reverse

28

EGAMs Summary 1) A fluid model is developed to describe EGAMs. 2) Valid when fast ion energy extent is small (Tf

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.

30

� 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