Transition From Weak to Strong Energetic Ion Transport in Burning Plasmas

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Transition From Weak to Strong Energetic Ion Transport in Burning Plasmas

F. Zonca 1), S. Briguglio 1), L. Chen 2), G. Fogaccia 1), G. Vlad 1)

1) Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

2) Department of Physics and Astronomy, University of California, Irvine, CA 92717-4575.

Vilamoura, Portugal, November 1–6 2004 20th IAEA Fusion Energy Conference

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Background

2 A burning plasma is a self-organized system, where collective effects associ- ated with fast ions (MeV energies) and charged fusion products may alter their confinement properties and even prevent the achievement of ignition.

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Background

2 Simulation results indicate that, above threshold for the onset of resonant Energetic Particle Modes (EPM) (L. Chen, PoP 94), strong fast ion transport occurs in avalanches (IAEA 02).

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Avalanches and NL EPM dynamics

^{(IAEA 02)}

|φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-3

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 4 - 2 0 2 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δαH

r/a

= 60.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-2

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 4 - 2 0 2 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δαH

r/a

= 75.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

.001 .002 .003 .004 .005 .006 .007 .008 .009

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 4 - 2 0 2 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δαH

r/a

= 90.00 t/τ_A0

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Background

2 Such strong transport events occur on time scales of a few inverse linear growth rates – 100÷200 Alfv´en times – and have a ballistic character (R.B.

White, PF 83) that basically differentiates them from the diffusive and local nature of weak transport.

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Background

2 Such strong transport events occur on time scales of a few inverse linear growth rates – 100÷200 Alfv´en times – and have a ballistic character (R.B.

White, PF 83) that basically differentiates them from the diffusive and local nature of weak transport.

2 Observations on JT-60U (and other tokamaks) have confirmed macroscopic and rapid energetic particle radial redistributions in connection with the so called Abrupt Large amplitude Events (ALE) (K. Shinohara, NF 01).

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ALE on JT-60U

(K. Shinohara, et al., Nucl. Fus. 41, 603, (2001)

Courtesy of K. Shinohara and JT-60U ALE = Abrupt Large amplitude Event

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Fast ion transport: simulation and experiment

2 Numerical simulations show fast ion radial redistributions, qualitatively similar to those by ALE on JT-60U.

0 0.01 0.02

0 0.2 0.4 0.6 0.8 1

βH

r/a

Before EPM

After EPM

G. Vlad et al., EPS02, ECA 26B, P-

4.088, (2002). K. Shinohara etal PPCF 46, S31 (2004) Courtesy of M. Ishikawa and JT-60U

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Issues arising

2 Can strong fast ion transport and avalanches have any relevance to burning plasmas? Do they cause global losses or just radial redistributions?

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Issues arising

2 A realistic answer cannot be independent on the dynamic formation of the plasma scenario: nonlinear wave-particle dynamics, sources, sinks, long time scales (transport).

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Issues arising

2 A realistic answer cannot be independent on the dynamic formation of the plasma scenario: nonlinear wave-particle dynamics, sources, sinks, long time scales (transport). Unrealistic for now.

2 Applications to plasma equilibria given a priori. High B-field, low fast-ion energy density operations are favored. Applications to ITER indicate fairly benign behaviors except for the reversed shear operations. Hybrid scenario is optimal ⇒ G. Vlad et al., IT/P3-31 in Thursday Poster Session.

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Issues arising

2 Therefore, it is crucial to theoretically assess the potential impact of fusion product avalanches on burning plasma operation in the perspective of direct comparisons with experimental evidence.

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Issues arising

2 Therefore, it is crucial to theoretically assess the potential impact of fusion product avalanches on burning plasma operation in the perspective of direct comparisons with experimental evidence.

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Nonlinear Dynamics: local vs. global processes

2 Mode saturation via wave-particle trapping (H.L. Berk et al. PFB 90, PPR 97) has been successfully applied to explain pitchfork splitting of TAE spectral lines (A. Fasoli et al. PRL 98): local distortion of the fast ion distribution function because of quasi-linear wave-particle interactions.

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Nonlinear Dynamics: local vs. global processes

2 Compton scattering off the thermal ions (T.S. Hahm and L. Chen PRL 95):

locally enhance the mode damping via nonlinear wave-particle interactions 2 Mode-mode couplings generating a nonlinear frequency shift which may enhance the interaction with the Alfv´en continuous spectrum (Zonca et al.

PRL 95 and Chen et al. PPCF 98): locally enhance the mode damping via nonlinear wave-wave interactions.

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Nonlinear Dynamics: local vs. global processes

2 Compton scattering off the thermal ions (T.S. Hahm and L. Chen PRL 95):

locally enhance the mode damping via nonlinear wave-particle interactions 2 Mode-mode couplings generating a nonlinear frequency shift which may enhance the interaction with the Alfv´en continuous spectrum (Zonca et al.

PRL 95 and Chen et al. PPCF 98): locally enhance the mode damping via nonlinear wave-wave interactions.

2 EPM is a resonant mode (L. Chen PoP 94), which is localized where the drive is strongest: global readjustments in the energetic particle drive is expected to be important as well.

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Nonlinear Dynamics of a single-n coherent EPM

2 NL dynamics of a single-n coherent EPM: neglect local phenomena and consider only global NL EPM dynamics. Consistency check a posteriori . 2 Treat hot particle distribution consisting of a background plus a perturba-

tion on meso time and space scales: the background is frozen in time.

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Nonlinear Dynamics of a single-n coherent EPM

2 NL dynamics of a single-n coherent EPM: neglect local phenomena and consider only global NL EPM dynamics. Consistency check a posteriori . 2 Treat hot particle distribution consisting of a background plus a perturba-

tion on meso time and space scales: the background is frozen in time.

2 The modification in the energetic particle distribution function in the pres- ence of finite amplitude fluctuations is derived in the framework of nonlinear Gyrokinetics

2 The non-adiabatic fast ion response – δH_k – is obtained from the NL gyroki- netic equation (Frieman & Chen PF 82). For details see TH/5-1 in Friday Poster Session.

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Initial value radial envelope problem

2 Using both time scale separation, ω = ω⁰ + i∂_t as well as spacial scale separation, θ_k ⇒ (−i/nq⁰)∂_r with ∂_r acting on A(r, t) only, the initial value radial envelope problem meso time and space scales becomes:

[D_R(ω, θ_k;s, α) + iD_I(ω, θ_k; s, α)] A⁰

| {z }

= δW_KTA⁰

| {z } ,

LINEAR DISPERSION LIN. ⊕ NL EN. PART.RESP.

2 e_Hδφ/T_H = A(r, t) = A⁰(r, t) exp(−iω⁰t), with |ω0⁻¹∂_t ln A⁰(r, t)| 1

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Initial value radial envelope problem

2 Using both time scale separation, ω = ω⁰ + i∂_t as well as spacial scale separation, θ_k ⇒ (−i/nq⁰)∂_r with ∂_r acting on A(r, t) only, the initial value radial envelope problem meso time and space scales becomes:

[D_R(ω, θ_k;s, α) + iD_I(ω, θ_k; s, α)] A⁰

| {z }

= δW_KTA⁰

| {z } ,

LINEAR DISPERSION LIN. ⊕ NL EN. PART.RESP.

2 e_Hδφ/T_H = δA(r, t) = A⁰(r, t) exp(−iω⁰t), with |ω0⁻¹∂_t lnA⁰(r, t)| 1

2 Nonlinear (n = 0, m = 0) distortion to the hot particle distribution on meso time and space scales: for details see TH/5-1 in Friday Poster Session.

∂

∂tH_z = 2k_θ²ρ²_H ω_cH k_θ

T_H m_H

∂

∂r

IIm

QF⁰

ω

¯ ω_d

¯

ω_d − ω

Γ²|A|²

H

.

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2 Assume isotropic slowing-down and EPM NL dynamics dominated by pre- cession resonance.

[D_R(ω, θ_k; s, α) + iD_I(ω, θ_k; s, α)]∂_tA⁰ = 3π¹^/² 4√

2 α_H

1 + ω

¯

ω_dF ln

ω¯_dF

ω − 1

+iπ ω

¯ ω_dF

∂_tA⁰ + iπ ω

¯

ω_dF A⁰3π¹^/² 4√

2 k_θ²ρ²_H T_H

m_H ∂_r²∂_t⁻¹ α_H |A⁰|² .

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[D_R(ω, θ_k;s, α) + iD_I(ω, θ_k;s, α)]∂_tA⁰ = 3π¹^/² 4√

2 α_H

1 + ω

¯

ω_dF ln

ω¯_dF

ω − 1

+iπ ω

¯ ω_dF

∂_tA⁰ + iπ ω

¯

ω_dF A⁰3π¹^/² 4√

2 k_θ²ρ²_H T_H

m_H ∂_r²∂_t⁻¹ α_H |A⁰|²

| {z }

. DECREASES DRIVE@ MAX|A⁰|

INCREASES DRIVE NEARBY

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Avalanches and NL EPM dynamics

^{(IAEA 02)}

|φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-3

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 60.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-2

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 75.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

.001 .002 .003 .004 .005 .006 .007 .008 .009

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 90.00 t/τ_A0

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[D_R(ω, θ_k;s, α) + iD_I(ω, θ_k;s, α)]∂_tA⁰ = 3π¹^/² 4√

2 α_H

1 + ω

¯

ω_dF ln

ω¯_dF

ω − 1

+iπ ω

¯ ω_dF

∂_tA⁰ + iπ ω

¯

ω_dF A⁰3π¹^/² 4√

2 k_θ²ρ²_H T_H

m_H ∂_r²∂_t⁻¹ α_H |A⁰|²

| {z }

.

DECREASES DRIVE@ MAX|A⁰| INCREASES DRIVE NEARBY

2 Assume localized fast ion drive, α_H = −R⁰q²β_H⁰ = α_H⁰ exp(−x²/L²_p) ' α_H⁰(1 − x²/L²_p), with x = (r − r⁰).

2 In order to maximize the drive the EPM radial structure is nonlinearly displaced by

(x⁰/L_p) = γ_L⁻¹k_θρ_H (T_H/M_H)¹^/² (|A⁰|/W⁰) ,

2 x⁰ is the radial position of the max EPM amplitude and W⁰ indicating the

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Avalanches and NL EPM dynamics

^{(IAEA 02)}

|φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-3

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 60.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.05 0.1 0.15 0.2 0.25 x 10 ^-2

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 75.00

t/τ_A0 |φ_m,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

.001 .002 .003 .004 .005 .006 .007 .008 .009

r/a 8, 4

9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4

- 2 0 2 4 δαH

= 90.00 t/τ_A0

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2 During convective amplification, radial position of unstable front scales lin- early with EPM amplitude.

0 0.002 0.004 0.006 0.008 0.01

0.3 0.35 0.4 0.45 0.5 0.55 0.6

A0

(r/a)

Y = M0 + M1*X M0 -0.021507 M1 0.061857 R 0.99903

2 Real frequency chirping accompanies convective EPM amplification in order to keep ω ∝ ω¯_d

∆ω = s ω¯_dF|x (x⁰/r) (ω⁰/ ω¯_dF|x=0) .

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Avalanches and transport in NL EPM dynamics

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/a

ωτ τ/τ = 45.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9r/a 1 0

0.1 0.2

A0 A0 (r/a)(β /β ) H H0 τ/τ = 45.00_A0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ωτ_A0 τ/τ = 132.00 (r/a)(β /β ) H H0

0 0.1 0.2

A0 τ/τ = 132.00_A0

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Threshold conditions for Avalanches

2 Consistency check for neglecting wave particle trapping: based on the dis- placement ¯x⁰ of the avalanche front when trapping becomes important

2 Threshold for weak avalanche, or avalanche onset: ¯x⁰>

∼ |nq⁰|⁻¹.

1 γ_L

k_θρ_H(T_H/m_H)¹^/²R⁻0 ¹

>∼ (k_θL_p)⁻¹^/² ,

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Threshold conditions for Avalanches

2 Consistency check for neglecting wave particle trapping: based on the dis- placement ¯x⁰ of the avalanche front when trapping becomes important

2 Threshold for weak avalanche, or avalanche onset: ¯x⁰>

∼ |nq⁰|⁻¹.

1 γ_L

k_θρ_H(T_H/m_H)¹^/²R⁻0 ¹

>∼ (k_θL_p)⁻¹^/² ,

2 Strong avalanche condition: ¯x⁰>

∼ W⁰, the characteristic NL EPM width.

1 γ_L

k_θρ_H(T_H/m_H)¹^/²R0⁻¹

∼> W0²/∆² . ∆ ≈ (L_p/k_θ)¹^/²

2 W0²/∆² 1, due to the short scale of the nonlinear distortion: onset for EPM induced avalanches is close to the linear excitation threshold.

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CourtesyofS.D.PinchesandJET-EFDA

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Conclusions

2 Numerical simulations and theoretical analyses demonstrate ballistic fast ion transport above the EPM linear excitation threshold.

2 Nonlinear mode dynamics is an avalanche process: i.e. the radial propaga- tion of an unstable front.

2 Threshold conditions for the avalanche onset are given: typical values are close to the linear excitation threshold.

2 Good qualitative agreement of numerical simulations with local theoretical calculations of EPM convective amplification.

2 Time scales and qualitative phenomenology similar to experimental observations: ALE on JT-60U and fast chirping in Alfv´en Cascade experiments on JET.

2 Applications to ITER indicate fairly benign behaviors except for the reversed