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14th IAEA TM on Energetic Particles, Vienna, Austria, Sep. 1-4, 2015

- Physics of self-heating in burning plasmas -

Alfvén Acoustic Channel for Ion Energyin High-Beta Tokamak Plasmas

A. Bierwage in collaboration with

P. Lauber, N. Aiba, K. Shinohara and M. Yagi

Fusion alphas(≦ 3.5 MeV)

Bulk ions(≦ 10 keV)

Fu

sio

n

Electrons(≦ 10 keV)

Collisionalheating

Collisionalheating

Q: Are there other self-heating channels, besides collisions with electrons?

Outline

Part 1. MHD response in high-beta JT-60U plasma with N-NB

Part 2. Local analysis with a linear gyrokinetic model (LIGKA)

A: MHD Alfvén acoustic channels with ω≈ωTAE not seen in GK case for Te ≈ Ti .

But: Te ≫ Ti regime and EP effects on sound spectra deserve attention.

?

Motivation: ▶ Burning plasmas will rely on efficient self-heating.

▶ Reliable predictions for fusion performance requires knowledge of all relevant self-heating channels.

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▶ Thanks:

A.B. thanks Yasushi Todo (NIFS, Japan) for providing the code MEGA.

▶ Computational resources:

HELIOS at IFERC-CSC in Japan is provided by the Broader Approach collaboration between Euratom and Japan implemented by Fusion for Energy and JAEA.

▶ Grants:

This work has been supported by Japan Society for Promotion of Science (JSPS) Grant-in-Aid of Scientific Research number 25820443.

This work has been partly carried out within the framework of the EUROfusion Consortium and has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

Acknowledgments

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∂ρb

∂ t=−∇⋅(ρbδ ub)

ρb

∂ub

∂ t=−ρb u b⋅∇ ub − ∇ pb + (J−Jh,eff )×B

∂B∂ t

=−∇×E

E =−ub×B + ηδ Jμ0 J = ∇ ×B

∂pb

∂ t=−∇⋅(pb ub)− (Γ−1) [pb ∇⋅ub + η(J−Jh,eff )⋅δ J ]

Bulk plasma response: MHD model

High-beta JT-60U scenario simulated with global initial-value hybrid code▶ Code: MEGA [Y. Todo, NIFS, Japan]

▶ N-NB-driven JT-60U scenario serves as test case.

▶ n=0 component of fast ion pressure treated as part of total beta (3.6% on axis).

▶ First, simulate and study only MHD response, ignoring resonant wave-particle interactions.

▶ Then, include kinetic fast ions (PIC method).

Todo & Sato, Phys. Plasmas 5 (1998) 13212;Todo et al, Phys. Plasmas 12 (2005) 012503.

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Power spectra of decaying MHD oscillations(linear color scale)

ω ∝ √Γβ

▶ Most slowly decaying signal is radially localized and has a beta-dependent frequency:

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MHD sound waves & shear Alfvén waves (SAW)(logarithmic color scale)

−i BB⋅∇ → k || = 1

R0(n− m±

q (r) ) ><

0▶ Signals are aligned with maxima of sound continua with k

|| > 0.

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Interpretation as discrete MHD modes

▶ Global slow magnetosonic eigenmode (GSME).

▶ Due to nonuniformity q(r), β(r).

▶ In high-beta plasma, GSMEs overlap with SAW continua.

→ New type of mode with mixed polarization: e.g., m

S = 2 and m

A = 5,6 components

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Beta-induced Alfvén continuum

mode (BACM)

▶ Physical mechanism:

Decaying sound waves drive Alfvén continuum.

▶ High frequency of BACM facilitates resonant interactions with fast ions.

Alfvénwaves

Soundwaves

Co

up

lin

g

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N-NB ions(≦ 400 keV)

Soundwaves

Alfvénwaves

Resonantdrive

Co

up

lin

g▶ Demonstrated that coupling works also in opposite direction, from SAW to sound.

Hybrid simulation of resonant drive from N-NB ions

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Energetic ions(≦ 3.5 MeV)

Bulk ions(≦ 10 keV)

Soundwaves

Alfvénwaves

Resonantdrive

Dissipativeheating

Fu

sio

n

Co

up

lin

g

Summary for Part 1: MHD response in high-beta JT-60U

New noncollisional self-heating channel for burning plasmas.

Alfvén wave

Sound wave

MHD wavesJT-60U plasma response spectrum

BACM

Electrons(≦ 10 keV)

Collisionalheating

Collisionalheating

Findings:

▶ Plasma nonuniformity q(r), β(r) → Global slow magnetosonic eigenmode (GSME)

▶ High beta (JT-60U: β(r=0) =3.5%) → GSMEs overlap with Alfvén continua

▶ Alfvén-acoustic coupling → Beta-induced Alfvén continuum modes (BACM)

▶ Energetic particle modes (EPM), couple to sound waves via BACMs

Bierwage et al.,Phys. Rev. Lett. 114

(2015) 015002

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−∂∂ t [∇ 1

vA2 ∇⊥ϕ ] + B⋅∇

( ∇×∇×A )||⋅B

B2 + ( b×∇ A || )⋅∇μ0 j0 ||

B

= −∑a

μ0∫d2 v ea ( v̂d⋅∇ J0 f )a + ∑a

b×∇ (βa ⊥

2Ωa)⋅∇ ∇⊥

2ϕ + ∑

a

3βa⊥

8Ωa2

∇⊥

4 ∂∂ t

ϕ + B⋅∇ 1B ∑

a

βa

4∇⊥

2 A||

Current equation / gyrokinetic moment equation (GKM)

Approximations:

▶ No trapped particle effects.

▶ Sound wave coupling up to first order (geodesic).

▶ Zero orbit width and FLR only up to order (k

⊥ρ)4.

▶ Isotropic Maxw. distribution for all species a={e,i,h}.

▶ Deuterium only.

Q: Can we confirm this MHD prediction in a GK world?

→ Part 2: Local analysis with a linear gyrokinetic model

Energy density gradients(“ballooning”)

Diamagnetic current

0 = ∑a

ea na1 = ∑a

ea∫d2 v (J0(ρ ∇⊥) f )a + ∇⊥

min i0

B2 ∇⊥ ϕ(x ) +3P i⊥

4B2Ωi

2 ∇⊥4ϕ(x ) + O ((k⊥ρa )

6 )

Quasi-neutrallity equation (QN)

Charge density gradients FLR polarization terms

Current gradient (“kink”)Inertia Field line bending (FLB)

4th-order FLR inertia High-beta FLB

▶ Code: LIGKA [Ph. Lauber, Max Planck IPP, Germany]

▶ Here: Local eigenvalue analysis → Continuous spectra ω(r),γ(r).

Lauber et al.,J. Comp. Phys. 226

(2007) 447

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▶ For parameter scans, vary the above reference profiles as

→ Te(ρ)= const. x T

e

ref(ρ)

→ Th(ρ)= const. x T

h

ref(ρ)

▶ Focus on toroidal harmonic n = 3.

Profiles for N-NB-driven high-beta JT-60U plasma

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▶ Local growth parameter Г: ▶ Purpose here:

Г = 2πγ/ω = γτ = ln |Ф(t+τ) / Ф(t) | Look for changes in Alfvén continua

e.g.: Г ≶ -2 → Ф(t+τ) / Ф(t) ≶ 10% in association with sound waves.

▶ Observations:

• Sound continua are strongly damped (ГS < -2) and structure of Alfvén continuum damping (ωA,ГA) smooth. → In GK results we find no evidence of Alfvén acoustic channels with ω≈ωA .

• Compressional stabilization of Alfvén waves increases towards ωBAE. (modified by ωж ). → Conversion to KBM, drift-Alfvén-ballooning modes.

Results for Te = T

e

ref, Ti = T

i

ref, βh = 0 → β ≈ βref / 2

zoom

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▶ Observations:

• GK continua near ω≈ωBAAE < 0.1 match MHD continua, and their damping is reduced significantly (Г > -2).

• BAE accumulation pts. now marg. stable (Г > -0.01).

• Sound continua with ω > 0.1 still strongly damped ...

• ... but structure of Alfvén continua (ωA,ГA) has changed. ▶ Expectation for Te ≫ Ti regime:

• Reduced damping of sound branches with ω > 0.1. → Look for increasing changes in Alfvén continua and relation to sound waves.

Results for Te = 2.8 x T

e

ref, Ti = T

i

ref, βh = 0 → β ≈ βref

zoom

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▶ Observation:

“Hot sound” continua ωhS

(ρ) appear → Weakly damped for ω ≦ 0.1, and

correlate with changes in ωA(ρ)≈ωTAE,u.

Results for Te = T

e

ref, Ti = T

i

ref, βh0

= 1.8% → β = βref

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Energeticions

Soundwaves

Alfvénwaves

Resonantdrive

Co

up

lin

gCurrent status:

▶ MHD predicts Alfvén acoustic channels with ωBAE

<ω≈ωTAE

.

→ Try to understand how the MHD picture of such channels translates to the GK world. How can we identify them?

▶ Local GK analysis (ω,γ): Strong damping when Te ≈ T

i.

→ Results draw attention to Te ≫ Ti regime and EP effects: ω

A(ρ) changes and “hot sound” branches ω

hS(ρ) appear.

☞ GTC results suggest moderate damping of nonperturb. global BAAE modes even for Te ≈ Ti. [Z. Lin et al, poster P-19]

Ongoing work and future steps:

▶ Explore Te ≫ T

i regime.

▶ Clarify role of “hot sound” continua ωhS

(ρ).

▶ Include trapped particles, FOW effects, realistic Fh(E,μ).

▶ Global analysis and energy flows at finite amplitudes.

Relevance of this work: (once we progress to the NL regime)

▶ Better understanding of energy flows in high-beta fusion experiments (e.g., JT-60SA) and burning plasmas (DEMO).

▶ “Anomalous bulk heating” and effect on EP confinement:

A) improvement (damping reduces mode amplitudes);

B) deterioration (damping increased probability for bursts and large-amplitude relaxation events).

Discussion and future directions

Thermalions

Dissipativeheating

MHD picture ofan Alfvén acoustic

self-heating channel: