transition from weak to strong energetic ion transport in

ENEA-UCI F. Zonca, L. Chen, S. Briguglio, G. Fogaccia and G. Vlad 1

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

20th IAEA FEC


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 altertheir confinement properties and even prevent the achievement of ignition.

20th IAEA FEC


Background


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

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Avalanches and NL EPM dynamics (IAEA 02)

|φm,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, 411, 412, 413, 414, 415, 416, 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

= 60.00t/τA0

|φm,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, 411, 412, 413, 414, 415, 416, 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

= 75.00t/τA0

|φm,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, 410, 411, 412, 413, 414, 415, 416, 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

= 90.00t/τA0

20th IAEA FEC


Background



2 Such strong transport events occur on time scales of a few inverse lineargrowth rates – 100÷200 Alfven times – and have a ballistic character (R.B.White, PF 83) that basically differentiates them from the diffusive and localnature of weak transport.

20th IAEA FEC


Background



2 Such strong transport events occur on time scales of a few inverse lineargrowth rates – 100÷200 Alfven times – and have a ballistic character (R.B.White, PF 83) that basically differentiates them from the diffusive and localnature of weak transport.

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

20th IAEA FEC


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 experiment2 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 burningplasmas? 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 theplasma scenario: nonlinear wave-particle dynamics, sources, sinks, long timescales (transport).

20th IAEA FEC


Issues arising


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

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

20th IAEA FEC


Issues arising




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

20th IAEA FEC


Issues arising




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

2 Simulation, Theory and Experiment are equally important.

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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 TAEspectral lines (A. Fasoli et al. PRL 98): local distortion of the fast iondistribution function because of quasi-linear wave-particle interactions.

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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 mayenhance the interaction with the Alfven continuous spectrum (Zonca et al.

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

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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 mayenhance the interaction with the Alfven continuous spectrum (Zonca et al.

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

2 EPM is a resonant mode (L. Chen PoP 94), which is localized where thedrive is strongest: global readjustments in the energetic particle drive isexpected 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 andconsider 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.

20th IAEA FEC


Nonlinear Dynamics of a single-n coherent EPM

2 NL dynamics of a single-n coherent EPM: neglect local phenomena andconsider 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 nonlinearGyrokinetics

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

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Initial value radial envelope problem

2 Using both time scale separation, ω = ω0 + i∂t as well as spacial scaleseparation, θk ⇒ (−i/nq′)∂r with ∂r acting on A(r, t) only, the initial valueradial envelope problem meso time and space scales becomes:

[DR(ω, θk; s, α) + iDI(ω, θk; s, α)]A0︸︷︷︸

= δWKT A0︸︷︷︸

,

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

2 eHδφ/TH = A(r, t) = A0(r, t) exp(−iω0t), with |ω−1

0 ∂t lnA0(r, t)| � 1

20th IAEA FEC


Initial value radial envelope problem

2 Using both time scale separation, ω = ω0 + i∂t as well as spacial scaleseparation, θk ⇒ (−i/nq′)∂r with ∂r acting on A(r, t) only, the initial valueradial envelope problem meso time and space scales becomes:

[DR(ω, θk; s, α) + iDI(ω, θk; s, α)]A0︸︷︷︸

= δWKT A0︸︷︷︸

,

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

2 eHδφ/TH = δA(r, t) = A0(r, t) exp(−iω0t), with |ω−1

0 ∂t lnA0(r, t)| � 1

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

∂

∂tHz = 2k2

θρ2

H

ωcH

kθ

TH

mH

∂

∂r

[

IIm(

QF0

ω

ωd

ωd − ω

)

Γ2|A|2]

H

.

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

[DR(ω, θk; s, α) + iDI(ω, θk; s, α)] ∂tA0 =3πε1/2

4√

2αH

[

1 +ω

ωdF

ln(

ωdF

ω− 1

)

+iπω

ωdF

]

∂tA0 + iπω

ωdF

A0

3πε1/2

4√

2k2

θρ2

H

TH

mH

∂2

r∂−1

t

(

αH |A0|2)

.

20th IAEA FEC




4√

2αH

[

1 +ω

ωdF

ln(

ωdF

ω− 1

)

+iπω

ωdF

]

∂tA0 + iπω

ωdF

A0

3πε1/2

4√

2k2

θρ2

H

TH

mH

∂2

r∂−1

t

(

αH |A0|2)

︸︷︷︸

.

DECREASES DRIVE@ MAX |A0|INCREASES DRIVE NEARBY

20th IAEA FEC



|φm,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, 411, 412, 413, 414, 415, 416, 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

= 60.00t/τA0

|φm,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, 411, 412, 413, 414, 415, 416, 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

= 75.00t/τA0

|φm,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, 410, 411, 412, 413, 414, 415, 416, 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

= 90.00t/τA0

20th IAEA FEC




4√

2αH

[

1 +ω

ωdF

ln(

ωdF

ω− 1

)

+iπω

ωdF

]

∂tA0 + iπω

ωdF

A0

3πε1/2

4√

2k2

θρ2

H

TH

mH

∂2

r∂−1

t

(

αH |A0|2)

︸︷︷︸

.

DECREASES DRIVE@ MAX |A0|INCREASES DRIVE NEARBY

2 Assume localized fast ion drive, αH = −R0q2β′

H = αH0 exp(−x2/L2

p) 'αH0(1 − x2/L2

p), with x = (r − r0).

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

(x0/Lp) = γ−1

L kθρH (TH/MH)1/2 (|A0|/W0) ,

2 x0 is the radial position of the max EPM amplitude and W0 indicating thetypical EPM radial width in the NL regime

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|φm,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, 411, 412, 413, 414, 415, 416, 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

= 60.00t/τA0

|φm,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, 411, 412, 413, 414, 415, 416, 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

= 75.00t/τA0

|φm,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, 410, 411, 412, 413, 414, 415, 416, 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

= 90.00t/τA0

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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*XM0 -0.021507M1 0.061857R 0.99903

2 Real frequency chirping accompanies convective EPM amplification in orderto keep ω ∝ ωd

∆ω = s ωdF |x0(x0/r) (ω0/ ωdF |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 10

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

r/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2A0 A0

(r/a)(β /β ) H H0

τ/τ = 45.00A0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/a

ωτA0

τ/τ = 132.00 (r/a)(β /β ) H H0

r/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2A0

τ/τ = 132.00A0

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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>∼ |nq′|−1.

1 � γL

kθρH(TH/mH)1/2R−1

0

>∼ (kθLp)−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>∼ |nq′|−1.

1 � γL


0

>∼ (kθLp)−1/2 ,

2 Strong avalanche condition: x0>∼ W0, the characteristic NL EPM width.

1 � γL


0

>∼ W 2

0/∆2 . ∆ ≈ (Lp/kθ)

1/2

2 W 2

0/∆2 � 1, due to the short scale of the nonlinear distortion: onset for

EPM induced avalanches is close to the linear excitation threshold.

2 If neither of these conditions is satisfied, EPM will saturate either via wave-particle trapping or other local phenomena.

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Possible EPMs during Alfven Cascades in JETC

ourt

esy

ofS.D

.P

inch

esan

dJE

T-E

FD

A

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Conclusions

2 Numerical simulations and theoretical analyses demonstrate ballistic fastion 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 areclose to the linear excitation threshold.

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

2 Time scales and qualitative phenomenology similar to experimental obser-vations: ALE on JT-60U and fast chirping in Alfven Cascade experimentson JET.

2 Applications to ITER indicate fairly benign behaviors except for the reversedshear operations. Hybrid scenario is optimal ⇒ G. Vlad et al., IT/P3-31 inThursday Poster Session.

20th IAEA FEC

transition from weak to strong energetic ion transport in

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