Introduction - IAEA NA...Summary (G.Y. Fu et al) M3D-K simulations of TAE saturation with EP...

THW/2-2a (Ph. Lauber, IPP) and THW/2-2b (G. Fu, PPPL)

• The overall saturation amplitude of Alfvén and Alfvén-acoustic fluctuations at ITER is a crucial parameter for the transport of energetic α-particles and therefore the overall energy confinement.• A prediction for ITER is very difficult because:‣a ‘sea’ of potentially unstable, global modes at different radial positions and frequencies can tap energy from different regions of the energetic particle phase space ‣non-linear saturation of a multi-mode system and their interaction with the background turbulence

• G. Fu et al, see also poster THW/2-2b : importance of the background turbulence vs collisional damping for the saturation level of toroidal Alfvén eigenmodes (TAEs)

• Ph. Lauber et al, see also poster THW/2-2a: the kinetic properties of low frequency, core localised beta-induced Alfvén eigenmodes (BAEs) and their non-linear interaction with TAEs at a different radial location

Simulations of Energetic Particle-driven Instabilities with Source and Sink

G.Y. Fu, J. Lang, Y. Chen, H. Berk, E. Fredrickson, N. Gorelenkov

Princeton Plasma Physics Laboratory, Princeton, NJ, USA Institute for Fusion Studies, Austin, TX, USA

The 2010 IAEA Fusion Energy Conference

THW/2-2Rb

23rd IAEA Fusion Energy Conference 11-16 October 2010 Daejeon, Rep. of Korea

Introduction

Simulations of Energetic Particle-driven Instabilities with Source and Sink

G.Y. Fu, J. Lang, Y. Chen, H. Berk, E. Fredrickson, N. Gorelenkov

Princeton Plasma Physics Laboratory, Princeton, NJ, USA Institute for Fusion Studies, Austin, TX, USA

The 2010 IAEA Fusion Energy Conference

THW/2-2Rb

Summary (G.Y. Fu et al)

M3D-K simulations of TAE saturation with EP collision and EP source/sink

•  Far away from marginal stability, the TAEs have steady state saturation with EP collision and source/sink. The saturation level scale as A~!0.7 in agreement with analytic theory of Berk/Breizman.

•  Near marginal stability at low collision rates, TAE nonlinear evolution exhibits pulsation and frequency chirping.

TAE saturation with EP radial diffusion

•  The effects of micro-turbulence-induced radial diffusion are similar to that of Coulomb collision for TAE saturation.

•  An analytic criterion is derived for the equivalence condition between radial diffusion and collisional pitch angle scattering. The criterion is consistent with numerical results.

•  The micro-turbulence effects could be more important than Coulomb collision.

THW/2-2Rb

M3D-K is a 3D nonlinear Particle/MHD hybrid code

Ph is the energetic particle stress tensor and is calculated by solving the gyrokinetic equation using the PIC method.

The particle radial diffusion, Coulomb collision and particle source and sink are included.

Kinetic equation for energetic particles with radial diffusion, collision, and source

radial diffusion Coulomb collision EP source

M3D-K simulations show TAE saturation level increases with collision frequency in agreement with analytic theory .

With source and sink (slowing down rate )

Without source or sink (collisionless)

J. Lang et al, Phys. Plasmas, 2010

Velocity stream function U

Near marginal stability, M3D-K simulations reveal mode pulsation and frequency chirping

Mode envelope evolution Mode frequency evolution

Criterion for radial diffusion equivalent to collisional pitch angle scattering

Where "=0 is the wave particle resonance condition. Thus, the equivalent condition is

For TAE,

Simulation results for n=3 TAE evolution with EP diffusion

•  To get same nonlinear saturation level, the ratio of the normalized diffusion rate to the pitch angle scattering rate is Rd=0.6. This is consistent with the analytic result.

Estimation of Rd for experimental cases*

Dr,i

(m2/sec)

Rd

TFTR 1.5 1.5 2.4 0.80 1.0 4.0 2.0

ITER 15 1.5 6.2 2.0 5.3 1.0 5.0

  The micro-turbulence-induced radial diffusion can be more important relative to Coulomb collision for TAE saturation in ITER!

* Used estimate for EP diffusion:

ASDEX Upgrade


!"#$%&'()'*++(+)"$),"'')-'*.)/0

/"+$)%*1'*++2($(&$*#

Damping, Drive and Non-linear Effects of Kinetic Low-frequency Modes in Tokamaks

Ph. Lauber, M. Brüdgam, D. Curran, M. Sempf, V. Igochine, S. Günter, M. Maraschek, P. McCarthy, M. García-Muñoz and the ASDEX Upgrade Team

THW/2-2Ra

Max-Planck-Institut für Plasmaphysik, EURATOM-Association, D-85748 Garching, Germany; email: [email protected] Department of Physics, University College Cork, EURATOM-Association DCU, Cork, Ireland

Acknowledgements toS.D. Pinches, F. Zonca, H. Berk

mailto:[email protected]

mailto:[email protected]

ASDEX Upgrade


BAEs at ASDEX Upgrade

•strong activity of electromagnetic perturbations in the low-frequency regime (ω~ωTAE/2) found in ICRF/NBI heated discharges•their presence strongly increases fast particle losses [M. Garcia-Munoz, PRL 2008]

•compressible MHD fails to explain mode properties

ASDEX UpgradeSimple kinetic BAE dispersion relation [Mazur,Mikhailovskii,(1977)]

Comparison with the simplest kinetic BAE dispersion relation

!2BAE =

v2thi

R20

7

4+ "

Shot 20488: MHA:B31-14

Freq

uenc

y (k

Hz)

80

90

100

110

120

130140

70

234

1.8 2.0Time (s)

1.4 1.6 2.2 2.4ICRF Power [MW]

!BAE

{

708090

100110120130140

1.3 1.4 1.5 1.6 1.7Time (s)

Zoom

8090

100

• assumed Te = Ti and ECE channel close to q = 1 surface chosen

• reasonable agreement found only before 1.5s, cannot explain frequency ’dip’ during sawtooth cycle

IPP Colloquium, Garching, January 2009 53

• although simple kinetic BAE dispersion relation explains the mode frequency partially, it is not applicable because:

1) strong side band coupling (+1/q^2 contribution with q=1!)2) background diamagnetic effects in presence of steep profiles 3) fast circulating ion approximation not valid

Shot 20488: MHA:B31-14

80

90

100

110

120

130

140

Freq

uenc

y (k

Hz)

70

2

3

4

Time (s)1.4 1.6 1.8 2.0 2.2 2.4

ICRF Power [MW]

BAE

freq

uenc

y[kH

z]time[s]

ASDEX Upgrade


• kinetic BAE dispersion

relation [Zonca NF 2009, Lauber PPCF 2009]

predicts splitting of modes with different mode numbers at q=1 surface (m=n)

• dedicated experiments with upgraded soft X-ray cameras allowed to find this splitting at ASDEX Upgrade

•evolution during sawteeth can be explained (change of background gradients)

• without trapped particles, frequencies are 20-30% too large

50

60

70

80

90

Freq

uenc

y (k

Hz)

Log

4.0

4.5

5.0

5.5

6.0

6.5

7.0

2.10 2.12 2.14 2.16 2.18 2.20 2.22Time (sec)

10

100

1000

10000

Tota

l pow

er

G Shot: 25549 : SXB : J_056 npts: 298059me: 2.086 to 2.235 frq: 50.0 to 90.0 nfft: 8192 npad: 0 nstp: 1024 nrms: 1000 ncor: 200

gdc) v3.45 - User: pwl - Wed Dec 9 10:02:36 2009

time[s] 2.162.10

m=3m=4m=5m=6

freq

uenc

y[kH

z]

SXR(central 1)

#25549

mode splitting, evolution during sawteeth

[D. Curran, EPS 2010]

!

"!

#!

$!

%!

&'()*(+,-./0123

456

!

"

#

$

%

7!

"87! "87" "87# "87$9:;(./<(,3

=>?<

@:6@A+.;5B(<

/7C73.0:+0.;5B(

<DEF55F@! !8!7 !8!" !8!G !8!#

!8"!

!87H

!87!

!8!H

I/;J73

=>?<

!8"H +J$+JH

+JG+J#

B:D;D6+(F:,.K'D+,@

Evolution of Alfven Eigenmodes during the Sawtooth Cycle at ASDEX Upgrade

D. Curran1, Ph. Lauber

2, P.J. McCarthy

1, S. da Graca

3, V. Igochine

2and ASDEX Upgrade Team

1 Department of Physics, University College Cork, EURATOM-Association DCU, Cork,Ireland2 Max-Planck-Institut fur Plasmaphysik, EURATOM-Association, Garching, Germany

3 Associacao EURATOM/IST, Instituto de Plasmas e Fusao Nuclear - Laboratorio Associado, Insituto Superior Tecnico, 1049-001 Lisboa, Portugal

✎

✍

☞

✌

Introduction

•Modes with frequencies between those of toroidicity-

induced Alfven eigenmodes (TAEs) and the kink insta-

bility observed recently using soft X-ray and fast ion loss

detectors at ASDEX Upgrade during ICRH shots.

•Modes seem to exist close to the q = 1 surface: Proposed

that these are beta-induced Alfven eigenmodes (BAEs).

•Aim is to investigate effects of diamagnetic drift frequency

on BAE frequency accumulation point.

Beta-Induced Alfven Eigenmodes

0 0.2 0.4 0.6!

0

0.1

0.2

0.3

0.4

0.5

"

n=3n=4n=5n=6

Normalized Real Frequency vs. rho_pol

0.1 0.2 0.3 0.4!

-0.04

-0.03

-0.02

-0.01

0

"

n=3n=4n=5n=6

Normalized Imaginary Frequency vs. rho_pol

Figure 1: (a) Normalized real Alfven frequency continuum. (b) Nor-

malized imaginary Alfven frequency continuum.

•BAEs are electromagnetic, n �= 0 perturbations, driven by

energetic particles, usually located in the plasma core.

•Occur in the low frequency gap caused by coupling between

an Alfven wave and a sound wave, at frequency below Alf-

ven continuum minimum.

•Geodesic curvature and compressibility are responsible for

the departure from cylindrical symmetry that creates a

low-frequency gap which increases with beta.

Experimental Observations

Mode Activity

Figure 2: (a) Mode activity during successive sawtooth cycles. (b)

BAE mode frequency splitting for n = 3− 6.

•What appear to be BAEs have been observed in certain

AUG discharges during sawtooth activity.

•Modes visible between 60kHz and 80kHz observed to dip

signifigantly in frequency during the first half of the saw-

tooth cycle before chirping upwards.

•Definite mode frequency splittings observed and are found

to correspond to toroidal mode numbers n = 3, 4, 5 and 6

Electron Temperature

2.13 2.14 2.15 2.16 2.17Time (s)

2.0

2.5

3.0

keV

ECE Channel 56 : rho_pol=0.327ECE Channel 55 : rho_pol=0.350 , q=1 surfaceECE Channel 54 : rho_pol=0.373

cview (gdc) v3.55 - User: dbc - Fri Jun 11 12:54:44 2010 dbc/cview/StdSet/dbc_ss1.cvs : 25549

2.13 2.14 2.15 2.16Time (s)

0.00

0.05

0.10

0.15

0.20

0.25

0.30ECE 57 - ECE 56ECE 56 - ECE 55ECE 55 - ECE 54ECE 54 - ECE 53

T_e

(keV

)

A B C D

cview (gdc) v3.55 - User: dbc - Wed Jun 9 15:32:16 2010 dbc/cview/StdSet/dbc_ss1.cvs : 25549Figure 3: (a) Core ECE temperature channels during sawtooth cy-

cle. (b) Temperature differences between neighbouring ECE channels

surrounding q = 1 surface.

• Strong evidence of relationship between the evolution of

the core plasma temperature during the sawtooth cycle

and mode behaviour.

• q = 1 surface inferred from ECE measurements to lie at

approximately ρpol = 0.35

•Temperature differences appear as an initial steepening of

gradient followed by flattening due to precursor oscillations

which expel particles and decrease the gradient.

Numerical Analysis

BAE/GAM/KBM Dispersion Relations

•Kinetic balooning mode dispersion relation was re-derived

for the gyrokinetic model underlying the eigenvalue code

LIGKA, based on a Fourier expansion in the poloidal angle

but keeping full resonances.

•Keeping m ± 1-sidebands, geodesic curvature and sound

wave coupling by appropriate approximation of the propa-

gator integrals leads to:

General kinetic expression in the ballooning limit [Zonca,PoP

(1996,2008)]:

ω2(1− ω∗pω )− (k2

�m+k2�m+1)±

√(k2�m−k

2�m+1)

2+4�2ρpol2k2�mk

2�m+1

2(1−�2ρpol2) =

2(�

21+κ2)

v2thi

R20([−H

m−1(xm−1)−Hm+1(xm+1)]+

τ [Nm(xm−1)Nm−1(xm−1)

Dm−1(xm−1)+ N

m(xm+1)Nm+1(xm+1)Dm+1(xm+1)

ωp∗ = ω∗n + ω∗T = Ti/(eB)kθ(∇n/n)(1 + η); η = ∇T

T/∇n

n; xm = ω

k�,mvthi

•The terms in the equation represent the following factors H

(Geodesic Curvature), N (Coupling terms) and D (Sound

Wave). TAE coupling term kept for completeness.

•Equation modified to take account of plasma elongation ef-

fects (κ = 1.67) and Zeff (Zeff = 1.6.) [Gao, PoP (2008)]

Equilibrium Reconstruction

•Equilibrium reconstructions carried out for a number of ti-

mepoints during sawtooth cycle to determine realistic tem-

perature, density and q-profiles.

0.2 0.25 0.3 0.35 0.4 0.45 0.5!

1500

2000

2500

3000

3500

T_e

(eV

)

t=2.130st=2.135st=2.143st=2.150st=2.155st=2.158s

Electron Temperature vs. Normalized Minor RadiusASDEX Upgrade Shot 25549

0 0.1 0.2 0.3 0.4 0.5 0.6!

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

q

t=2.130s, q0=0.99t=2.135s, q0=0.98t=2.143s, q0=0.97t=2.150s, q0=0.96t=2.155s, q0=0.95t=2.158s, q0=0.945

Safety Factor vs. Normalized Minor RadiusASDEX Upgrade Shot 25549

Figure 4: (a) Fitted Te profiles near q = 1 surface during sawtooth

cycle. (b) Experimental Te profile from ECE diagnostic.

•As the modes are observed at the same location after the

crash, the q = 1 surface was assumed fixed at approxima-

tely ρpol = 0.35

• q0 assumed to decrease from q0 = 1.00 at the beginning of

the cycle to q0 ≈ 0.95 at the end of the cycle.

BAEs and Interaction with Sawtooth Cycle

•Figure 5 shows the BAE accumulation points for various

toroidal mode numbers at the maximum and minimum

values of the temperature gradient.

•BAE frequency minima chirp upwards in time at the end

of the sawtooth cycle and modes with higher toroidal mode

numbers experience a greater downward shift.

0.32 0.34 0.36 0.38 0.4 0.42 0.44!

0.11

0.12

0.13

0.14

0.15

0.16

0.17

"

n=4, t=2.143sn=4, t=2.158sn=3, t=2.143sn=3, t=2.158sn=5, t=2.143sn=5, t=2.158sn=6, t=2.158sn=6, t=2.143s

Normalized Real Frequency vs. rho_polASDEX Upgrade Shot 25549

Figure 5: Evolution of BAE minima with temperature gradient.

Comparison with Experiment

•Numerical results agree qualitatively with the experimental

when Doppler shift of nfrot included. frot = 3kHz inferred

from precursor oscillation frequency.

•The BAE frequency also subject to a downshift of appro-

ximately 20% as a result of trapped particle effects

•Quantitatively, the numerical results are found to be sever-

al kHz above the experimental ones.

•This could be a result of the omission of density gradi-

ent shifts, an over-estimation of Zeff effects or an under-

estimation of trapped particle effects.

Figure 6: Comparison of numerical results with experimental ones.

Diamagnetic Effects

•Diamagnetic effects become very important close to the

BAE minimum as ω∗p ∝ ∇T .

•ω∗p varies between approximately one and two thirds of

the BAE frequency, depending on the mode number.

•When the mode and diamagnetic frequency become com-

parable, for higher toroidal mode numbers, the BAE

branch appears to couple to the ion diamagnetic branch.

• Small jump in the temperature of some of the ECE chan-

nels, accompanied by a sudden up-chirp in the mode fre-

quency observed at t = 2.138s (See Figs 2 & 3)

Conclusions and Outlook

•Correlation observed experimentally between changing

temperature gradients and mode frequencies during the

sawtooth cycle: Confirmed numerically.

•Appears that this is related to the diamagnetic frequency

which becomes an appreciable fraction of the mode fre-

quency in this situation, as well as elongation and Zeff

•Hoped to extend this analysis in the future by including

variations in the density profile during the sawtooth cycle

and to apply it to the investigation of beta-induced Alfven-

acoustic eigenmodes (BAAEs).



2, P.J. McCarthy

1, S. da Graca

3, V. Igochine




✎

✍

☞

✌

Introduction










0 0.2 0.4 0.6!

0

0.1

0.2

0.3

0.4

0.5

"

n=3n=4n=5n=6


0.1 0.2 0.3 0.4!

-0.04

-0.03

-0.02

-0.01

0

"

n=3n=4n=5n=6













Mode Activity











2.13 2.14 2.15 2.16 2.17Time (s)

2.0

2.5

3.0

keV



2.13 2.14 2.15 2.16Time (s)

0.00

0.05

0.10

0.15

0.20

0.25


T_e

(keV

)

A B C D






and mode behaviour.






Numerical Analysis










(1996,2008)]:

ω2(1− ω∗pω )− (k2

�m+k2�m+1)±

√(k2�m−k

2�m+1)


2�m+1

2(1−�2ρpol2) =

2(�

21+κ2)

v2thi

R20([−H

m−1(xm−1)−Hm+1(xm+1)]+


Dm−1(xm−1)+ N



T/∇n

n; xm = ω

k�,mvthi










0.2 0.25 0.3 0.35 0.4 0.45 0.5!

1500

2000

2500

3000

3500

T_e

(eV

)

t=2.130st=2.135st=2.143st=2.150st=2.155st=2.158s


0 0.1 0.2 0.3 0.4 0.5 0.6!

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

q







tely ρpol = 0.35










0.32 0.34 0.36 0.38 0.4 0.42 0.44!

0.11

0.12

0.13

0.14

0.15

0.16

0.17

"
















Diamagnetic Effects
























2, P.J. McCarthy

1, S. da Graca

3, V. Igochine




✎

✍

☞

✌

Introduction










0 0.2 0.4 0.6!

0

0.1

0.2

0.3

0.4

0.5

"

n=3n=4n=5n=6


0.1 0.2 0.3 0.4!

-0.04

-0.03

-0.02

-0.01

0

"

n=3n=4n=5n=6













Mode Activity











2.13 2.14 2.15 2.16 2.17Time (s)

2.0

2.5

3.0

keV



2.13 2.14 2.15 2.16Time (s)

0.00

0.05

0.10

0.15

0.20

0.25


T_e

(keV

)

A B C D






and mode behaviour.






Numerical Analysis










(1996,2008)]:

ω2(1− ω∗pω )− (k2

�m+k2�m+1)±

√(k2�m−k

2�m+1)


2�m+1

2(1−�2ρpol2) =

2(�

21+κ2)

v2thi

R20([−H

m−1(xm−1)−Hm+1(xm+1)]+


Dm−1(xm−1)+ N



T/∇n

n; xm = ω

k�,mvthi










0.2 0.25 0.3 0.35 0.4 0.45 0.5!

1500

2000

2500

3000

3500

T_e

(eV

)

t=2.130st=2.135st=2.143st=2.150st=2.155st=2.158s


0 0.1 0.2 0.3 0.4 0.5 0.6!

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

q







tely ρpol = 0.35










0.32 0.34 0.36 0.38 0.4 0.42 0.44!

0.11

0.12

0.13

0.14

0.15

0.16

0.17

"
















Diamagnetic Effects






















0.190.170.150.13

0.34 0.38 0.42

ρ pol

ASDEX Upgrade


!"#$

!"#%

!"#&

!"#'

!"$

!"$$

!"$%

!"$&

(!"% (!")* (!") (!"$* (!"$ (!"#* (!"# (!"!* ! !"!* !"# !"#*

+,-./-0123456-789!:

;86<=073 456-789!:

adding trapped particles

flat T

steep T

intermediate T

!"#$"%&!'%"()"*+

%&,-'./0000/12000/130000/14000/15000060

0

786

9.!""$

9:;&!

9-<!"%#",-&!"

4=3>26/

trapped particle effects on continuum

• using local solver of linear gyrokinetic code LIGKA• frequency-downshift by 20-30%• steep background T destabilizes accumulation point

!"#$%&'((((')*(((')+(((('),(((')-((((.(

(

/0.

,1+2*.'/

3#45&$67(8.'((9(:

.;

<2

[Lauber PPCF 2009][Zonca, Chavdarovski NF 2009]

mode growth [ωA]

ASDEX Upgrade


Global solution with LIGKA (n=4)

• mode frequency in the gap for flat and intermediate Ti profile• mode frequency above the accumulation point for steep profile (kBAE)• steep background gradients already destabilize mode

TsteepTintermediate

Tflat !"#"$

"

"#"$

"#%

"#%$

"#"& "#"&$ "#"' "#"'$ "#"$ "#"$$ "#"( "#"($ "#") "#")$

*+,-./0-12-3456+*

-789:;

,8*<=37>[email protected];

+*-78C9<.6+*-789:;

!"

#

"

$

%

&

# #'$ #'& #'( #') "

*+,*-./-01+2-

345+/6

BAE eigenfunctions

!"#$%

!"#&

!"#&%

!"#'

!"!' !"!% !"!( !"!)

*+,-.,/012345,678!9

45,67:;<2345,67;89

=!"!(

=!"!'

=!"!$

!

!"!$>75<?/623@9

45,67:;<2345,67;89

continuum

Tsteep

TintermediateTflat

grow

th[%

]

ASDEX Upgrade


4.3. Enhanced Fast Particle Losses due to Multiple Modes

BAE reveals that the mode signal consists of a mixture of eigenmodes with a dominant(n,m) = (4, 4) component.

0 0.2 0.4 0.6 0.8s = !pol

0

50

100

150

200

250

Freq

uenc

y [k

Hz]

Alfven continuumTAE(n,m)=(4,4-5)BAE(n,m)=(4,4) TAE gap

BAE gap

0 0.2 0.4 0.6 0.8 1!pol

0

0.5

1

Arb

itrar

y A

mpl

itude BAEn=4

TAEn=4, m=4TAEn=4, m=5

Figure 4.26.: Frequency spectrum showing the Alfven and sound continuum (left) and thecorresponding eigenfunctions of the BAE and the TAE, both n = 4 (right).The radial extensions of the BAE and TAE eigenfunctions are drawn at theirdiscrete frequencies.

The frequency continua for n = 4 are shown in figure 4.26. The BAE is found at theupper edge of the BAE gap. The corresponding eigenfunction only consists of one dominantpoloidal harmonic m = 4. The mode is localized at the radial position ρpol ≈ 0.45 andoverlaps radially with the n = 4-TAE. The radial position of the BAE amplitude deviatesfrom the experimental measurement ρpol,exp ≈ 0.25 (Fig. 4.2). The difference can be causedby an inexact q-profile from the equilibrium reconstruction or by the missing of kinetic ef-fects in the MHD code CASTOR. Calculations with the gyrokinetic code LIGKA [68] haveshown that the radial BAE position shifts to the plasma centre [29]. Since the mode is alsodetected clearly at ρpol ≈ 0.45 in the experiment (Fig. 4.27), this allows for the applicationof the simulated MHD eigenmode for the following simulations.

Time [s]

Freq

uenc

y [k

Hz] !pol

200

150

100

50

1.00 1.10 1.20 1.30 1.40

= 0.45

Figure 4.27.: Spectrogram of a soft X-ray measurement (line of sight at half minor radius).The n = 4-BAE and the TAEs with toroidal mode number n = 3 − 5 areclearly visible.

93

4. Simulation Results

As for the single n = 4-TAE in section 4.2, the same simulation conditions (315000 mark-

ers, fast particle beta βf = 0.3%) and the same fast particle distribution functions in real

space and energy (Fig. 4.10 and 4.11) are applied.

0.0e+00 1.0e-03 2.0e-03Time [s]

1.0e-07

1.0e-06

1.0e-05

1.0e-04

1.0e-03

1.0e-02

Mod

e am

plitu

deSingle TAE n=4Single BAE n=4TAE n=4BAE n=4

8.0e-04 9.0e-04 1.0e-03Time [s]

4.4e-04

5.0e-04

5.7e-04

6.5e-04

Mod

e am

plitu

de

Single TAE n=4TAE n=4BAE n=4

Figure 4.28.: Mode evolution of the BAE and the TAE, both n = 4, which are simulatedsimultaneously (left). Right: Enlargement of the mode evolution (black boxon the left side).

At the beginning (Fig. 4.28), the TAE amplitude grows nearly with the same growth

rate as for the single TAE case and starts to saturate. However, the BAE starts at a

higher growth rate compared to the single simulation (dashed blue line) which saturates

at an amplitude of A = 3.4 · 10−4

. The BAE amplitude grows up to the TAE amplitude,

A = 5.4 · 10−4

. From this point on, t ≈ 9.0 · 10−4

s, both mode amplitudes increase further

and do not reach a saturated level. For the time t > 2.3 · 10−3

s both modes also show an

additional nonlinear growth.

0.0e+00 1.0e-03 2.0e-03Time [s]

-60

-40

-20

0

20

40

60

Gro

wth

rate

!/"

0 [%]

TAEBAE

0.0e+00 1.0e-03 2.0e-03Time [s]

-1

0

1

2

3

4

5G

row

th ra

te !/"

0 [%]

Single BAESingle TAEBAETAE

Figure 4.29.: Temporary and mean growth rate of the BAE and the TAE, both n = 4,which are simulated simultaneously. For comparison, the growth rates forthe single eigenmodes are also displayed.

Superimposed on both amplitudes is an oscillation with a frequency of ω ≈ 100 kHz which

is phase-shifted by π between the BAE and the TAE (Fig. 4.28, right). The oscillation is

94

Non-linear evolution BAE/TAE [HAGIS]

•it is well known that radially overlapping modes with similar frequencies increase fast particle transport•what happens for modes with very different frequencies? (TAE:150kHz, BAE: 50kHz, n=4, fast hydrogen ions: isotropic,slowing down)

•BAE grows faster with TAE present•once the amplitudes of BAE and TAE are equal, further growth of both modes is observed (stochastisation)

[M Brüdgam, PhD Thesis 2010]

ρ pol

arbi

trar

y un

its

ASDEX Upgrade



0.0 0.2 0.4 0.6 0.8 1.0!pol

-4e+14

-2e+14

0

2e+14

4e+14C

hang

e in

par

ticle

den

sity

"n

[m-3

]BAE positionTAE position

Redistributiondue to TAE

Redistributiondue to BAE

0.400 0.450 0.500 0.550 0.600 0.650!pol

1e+18

2e+18

3e+18

4e+18

Parti

cle

pres

sure

p +

"p

[keV

/m3 ]

Initial distributionRedistributiondue to TAEBAETAE

Decreasedgradientwrt f0

Increasedgradientwrt initialdistribution

Figure 4.32.: Total radial redistributions of the fast particles at time t = 2.0 · 10−3 s (aver-aged over t = 1.9−2.1·10−3 s) caused by a single TAE (black; as shown in Fig.4.14) and a single BAE (blue). If both modes are present, the BAE inducedparticle ensemble between both modes is further redistributed by the TAE.Right: Due to the radial localization of the BAE, the BAE mode growth isincreased because of an increased radial gradient caused by the TAE.

TAE TAEp=-1 BAE

p=0p=0

Change in phase−space for trapped particles "f#v2

Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−4

−3

−2

−1

0

1

2

3

4x 1017

BAE

TAE

Figure 4.33.: Resonance lines for the BAE and the TAE for different bounce harmonics p(left). Right: The redistribution of trapped particles caused by the BAE andthe TAE for the point in time, t = 8.0 · 10−4 s (left). The redistribution forthe low energetic ions with E = 130 keV is increased due to the larger radialgradient and the intersection of the resonance lines (left).

This process increases the mean gradient in the fast particle pressure profile ∇p fromequation (4.8) at the radial BAE position by about 8% but does not explain the increaseof the BAE growth rate by about 14%. An additional drive mechanism must be presentsimultaneously to resolve all observations.

97


Due to the phase locking of the energetic ions, a redistribution of the double-resonant par-

ticles does not happen which is in significant contrast to the gradient-driven mode growth.

A further distinction between both mechanisms is the radial direction of the transfer. Whilst

the radially outward redistributed particles transfer their energy to the waves at the radial

mode position, the particles trapped in the exchange mechanism can transfer net energy

radially both inwards and outwards depending on the ratio of the amplitudes. The crucial

condition is only a double-resonant wave-particle-coupling. However, the process is always

superposed by the gradient-drive mode growth and cannot be treated separately. Consid-

ering a radially constant fast particle distribution, the wave-particle interaction Lagrangian

determined by the equations (3.44) and (3.45) would be zero since the change in the fast

particle distribution δf (Eqn. 3.37) is governed by the gradients.

Change in phase−space for trapped particles !f"v2

Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 1018

TAE

BAE

TAEp=−1

TAE p=0

BAE p=0

Figure 4.37.: Various mode resonances are responsible for the redistribution of fast particlescaused by the BAE and the TAE at time t = 2.0 · 10−3 s. The collectiveparticle redistribution is caused by the interplay between the p = 0-BAE andp = −1-TAE resonance.

Because of both transfer mechanisms which are of the same order in this case, the BAE

amplitude approaches the TAE amplitude at the time t = 8.0 · 10−4

s and continues to

grow. It is found that the fast particle redistribution caused by the p = 0-BAE resonance is

enlarged radially due to the increased BAE amplitude (Fig. 4.37). Again, the particles are

further shifted by the TAE via the p = −1-resonance so that an extended plateau is created

between both modes as shown in figure 4.38.

Due to the enlarged mode amplitude and the radial extension of the TAE, fast particles are

redistributed by various TAE resonances with a bounce number from p = −1, ..., 1. At time

t = 2.0 · 10−3

s, the redistribution strength due to the p = −1-resonance is as large as the

strength due to the p = 0-resonance (Fig. 4.37). This differs from the redistribution due to

p = 0 at the beginning (Fig. 4.31) and is caused by the previously explained interplay with

the p = 0-BAE resonance as shown in figure 4.33.

101


Gradient-driven Mode Growth

At the beginning, the TAE is driven unstable by the p = 0-resonance for highly energetic

trapped particles (Fig. 4.31) as discussed for the single TAE case (Fig. 4.16). The BAE

amplitude is one order of magnitude smaller than the TAE and therefore does not affect

the TAE induced redistribution significantly. On the contrary, the BAE mode is influenced

significantly by the redistribution caused by the TAE due to the radial localization as shown

in figure 4.32. At the position of the BAE mode an increased radial gradient is generated

by the TAE compared to the initial distribution function. This implies a stronger growth

compared to a single BAE since ω∗ ∼ ∇p (Eqn. 4.8).

The fast particles are redistributed by the BAE within a radial range of s = 0.5 − 0.7 only

(Fig. 4.32 left). This radial region is partially located between both modes so that the TAE

redistributes the ions further radially outwards if the fast particles are also in resonance

with the TAE. This mechanism continues until the gradient flattens over the whole radial

BAE-TAE range generating a broader radial plateau than for a single mode [15]. It has to

be emphasized that the radial overlap of the modes is necessary for the collective particle

transport. If the BAE eigenfunction is localized much more to the plasma centre, both the

collective transport and the increased mode growth of the BAE caused by the TAE induced

particle redistribution vanish.


Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1.5

−1

−0.5

0

0.5

1

1.5

x 1017

TAE

BAE

Figure 4.31.: Redistribution of trapped particles caused by the BAE and the TAE in thelinear growth phase for time t = 4.0 · 10−4 s . The TAE amplitude is oneorder of magnitude larger than that of the BAE. The horizontal lines denotethe radial positions of both modes.

Such a collective transport is to be expected for fast particles which are resonant with

both modes simultaneously. Since the main contribution to the mode drive is due to trapped

particles, the resonance lines for both modes are drawn in the same resonance plot (Fig. 4.33

left). Intersections of resonance lines indicate the phase-space regions where fast ions can

be further redistributed by the TAE. As the BAE amplitude increases up to t = 8.0 · 10−4 s,

the fast particle phase-space at the radial BAE position s = 0.47 and an energy around

E = 130keV (Fig. 4.33 right, black ellipse) is redistributed due to the enhanced BAE mode

growth caused by the increased gradient (Fig. 4.32, right). The fast particles are further

redistributed by the TAE so that the radial gradient at the BAE position remains.

96


As for the single n = 4-TAE in section 4.2, the same simulation conditions (315000 mark-

ers, fast particle beta βf = 0.3%) and the same fast particle distribution functions in real

space and energy (Fig. 4.10 and 4.11) are applied.

0.0e+00 1.0e-03 2.0e-03Time [s]

1.0e-07

1.0e-06

1.0e-05

1.0e-04

1.0e-03

1.0e-02

Mod

e am

plitu

de

Single TAE n=4Single BAE n=4TAE n=4BAE n=4

8.0e-04 9.0e-04 1.0e-03Time [s]

4.4e-04

5.0e-04

5.7e-04

6.5e-04

Mod

e am

plitu

de

Single TAE n=4TAE n=4BAE n=4

Figure 4.28.: Mode evolution of the BAE and the TAE, both n = 4, which are simulatedsimultaneously (left). Right: Enlargement of the mode evolution (black boxon the left side).

At the beginning (Fig. 4.28), the TAE amplitude grows nearly with the same growth

rate as for the single TAE case and starts to saturate. However, the BAE starts at a

higher growth rate compared to the single simulation (dashed blue line) which saturates

at an amplitude of A = 3.4 · 10−4

. The BAE amplitude grows up to the TAE amplitude,

A = 5.4 · 10−4

. From this point on, t ≈ 9.0 · 10−4

s, both mode amplitudes increase further

and do not reach a saturated level. For the time t > 2.3 · 10−3

s both modes also show an

additional nonlinear growth.

0.0e+00 1.0e-03 2.0e-03Time [s]

-60

-40

-20

0

20

40

60

Gro

wth

rate

!/"

0 [%]

TAEBAE

0.0e+00 1.0e-03 2.0e-03Time [s]

-1

0

1

2

3

4

5

Gro

wth

rate

!/"

0 [%]

Single BAESingle TAEBAETAE

Figure 4.29.: Temporary and mean growth rate of the BAE and the TAE, both n = 4,which are simulated simultaneously. For comparison, the growth rates forthe single eigenmodes are also displayed.

Superimposed on both amplitudes is an oscillation with a frequency of ω ≈ 100 kHz which

is phase-shifted by π between the BAE and the TAE (Fig. 4.28, right). The oscillation is

94

Phase space analysis

• initially, the TAE is driven mainly by 400-500 keV particles at mid radius•where resonances overlap, the TAE provides additional drive for the BAE•if amplitudes are equal, an effective drive at lower energies is found

ω - ntor ωprec - p ωb = 0 δf ∗ v2 δf ∗ v2δf ∗ v2

radi

us (

s)

Energy[MeV] Energy[keV] Energy[keV]

ASDEX Upgrade



0.0 0.2 0.4 0.6 0.8 1.0!pol

-4e+14

-2e+14

0

2e+14

4e+14C

hang

e in

par

ticle

den

sity

"n

[m-3

]BAE positionTAE position

Redistributiondue to TAE

Redistributiondue to BAE

0.400 0.450 0.500 0.550 0.600 0.650!pol

1e+18

2e+18

3e+18

4e+18

Parti

cle

pres

sure

p +

"p

[keV

/m3 ]

Initial distributionRedistributiondue to TAEBAETAE

Decreasedgradientwrt f0

Increasedgradientwrt initialdistribution

Figure 4.32.: Total radial redistributions of the fast particles at time t = 2.0 · 10−3 s (aver-aged over t = 1.9−2.1·10−3 s) caused by a single TAE (black; as shown in Fig.4.14) and a single BAE (blue). If both modes are present, the BAE inducedparticle ensemble between both modes is further redistributed by the TAE.Right: Due to the radial localization of the BAE, the BAE mode growth isincreased because of an increased radial gradient caused by the TAE.

TAE TAEp=-1 BAE

p=0p=0

Change in phase−space for trapped particles "f#v2

Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−4

−3

−2

−1

0

1

2

3

4x 1017

BAE

TAE

Figure 4.33.: Resonance lines for the BAE and the TAE for different bounce harmonics p(left). Right: The redistribution of trapped particles caused by the BAE andthe TAE for the point in time, t = 8.0 · 10−4 s (left). The redistribution forthe low energetic ions with E = 130 keV is increased due to the larger radialgradient and the intersection of the resonance lines (left).

This process increases the mean gradient in the fast particle pressure profile ∇p fromequation (4.8) at the radial BAE position by about 8% but does not explain the increaseof the BAE growth rate by about 14%. An additional drive mechanism must be presentsimultaneously to resolve all observations.

97


Due to the phase locking of the energetic ions, a redistribution of the double-resonant par-

ticles does not happen which is in significant contrast to the gradient-driven mode growth.

A further distinction between both mechanisms is the radial direction of the transfer. Whilst

the radially outward redistributed particles transfer their energy to the waves at the radial

mode position, the particles trapped in the exchange mechanism can transfer net energy

radially both inwards and outwards depending on the ratio of the amplitudes. The crucial

condition is only a double-resonant wave-particle-coupling. However, the process is always

superposed by the gradient-drive mode growth and cannot be treated separately. Consid-

ering a radially constant fast particle distribution, the wave-particle interaction Lagrangian

determined by the equations (3.44) and (3.45) would be zero since the change in the fast

particle distribution δf (Eqn. 3.37) is governed by the gradients.


Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 1018

TAE

BAE

TAEp=−1

TAE p=0

BAE p=0

Figure 4.37.: Various mode resonances are responsible for the redistribution of fast particlescaused by the BAE and the TAE at time t = 2.0 · 10−3 s. The collectiveparticle redistribution is caused by the interplay between the p = 0-BAE andp = −1-TAE resonance.

Because of both transfer mechanisms which are of the same order in this case, the BAE

amplitude approaches the TAE amplitude at the time t = 8.0 · 10−4

s and continues to

grow. It is found that the fast particle redistribution caused by the p = 0-BAE resonance is

enlarged radially due to the increased BAE amplitude (Fig. 4.37). Again, the particles are

further shifted by the TAE via the p = −1-resonance so that an extended plateau is created

between both modes as shown in figure 4.38.

Due to the enlarged mode amplitude and the radial extension of the TAE, fast particles are

redistributed by various TAE resonances with a bounce number from p = −1, ..., 1. At time

t = 2.0 · 10−3

s, the redistribution strength due to the p = −1-resonance is as large as the

strength due to the p = 0-resonance (Fig. 4.37). This differs from the redistribution due to

p = 0 at the beginning (Fig. 4.31) and is caused by the previously explained interplay with

the p = 0-BAE resonance as shown in figure 4.33.

101


Gradient-driven Mode Growth

At the beginning, the TAE is driven unstable by the p = 0-resonance for highly energetic

trapped particles (Fig. 4.31) as discussed for the single TAE case (Fig. 4.16). The BAE

amplitude is one order of magnitude smaller than the TAE and therefore does not affect

the TAE induced redistribution significantly. On the contrary, the BAE mode is influenced

significantly by the redistribution caused by the TAE due to the radial localization as shown

in figure 4.32. At the position of the BAE mode an increased radial gradient is generated

by the TAE compared to the initial distribution function. This implies a stronger growth

compared to a single BAE since ω∗ ∼ ∇p (Eqn. 4.8).

The fast particles are redistributed by the BAE within a radial range of s = 0.5 − 0.7 only

(Fig. 4.32 left). This radial region is partially located between both modes so that the TAE

redistributes the ions further radially outwards if the fast particles are also in resonance

with the TAE. This mechanism continues until the gradient flattens over the whole radial

BAE-TAE range generating a broader radial plateau than for a single mode [15]. It has to

be emphasized that the radial overlap of the modes is necessary for the collective particle

transport. If the BAE eigenfunction is localized much more to the plasma centre, both the

collective transport and the increased mode growth of the BAE caused by the TAE induced

particle redistribution vanish.


Energy [keV]

Rad

ial c

oord

inat

e s

100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1.5

−1

−0.5

0

0.5

1

1.5

x 1017

TAE

BAE

Figure 4.31.: Redistribution of trapped particles caused by the BAE and the TAE in thelinear growth phase for time t = 4.0 · 10−4 s . The TAE amplitude is oneorder of magnitude larger than that of the BAE. The horizontal lines denotethe radial positions of both modes.

Such a collective transport is to be expected for fast particles which are resonant with

both modes simultaneously. Since the main contribution to the mode drive is due to trapped

particles, the resonance lines for both modes are drawn in the same resonance plot (Fig. 4.33

left). Intersections of resonance lines indicate the phase-space regions where fast ions can

be further redistributed by the TAE. As the BAE amplitude increases up to t = 8.0 · 10−4 s,

the fast particle phase-space at the radial BAE position s = 0.47 and an energy around

E = 130keV (Fig. 4.33 right, black ellipse) is redistributed due to the enhanced BAE mode

growth caused by the increased gradient (Fig. 4.32, right). The fast particles are further

redistributed by the TAE so that the radial gradient at the BAE position remains.

96


Double-resonant Mode Coupling

Regarding the oscillation in the mode growth as seen in figure 4.28, the behaviour indicatesthat a particular particle ensemble is exchanging energy and momentum in a bidirectionalway. It is found by loading only trapped or passing particles that trapped particles causethis modulation in the amplitude. A specific ensemble of particles (Fig. 4.34) is radiallylocalized between both waves whereby their trajectories pass through both modes. Theseparticles are resonant with both modes so that equation (4.1) is satisfied for each modesimultaneously. This leads to

ωtp (nTAE − nBAE)− ωb (pTAE − pBAE) = ωTAE − ωBAE = ∆ω , (4.12)

where ∆ω is the beat frequency of the two modes.

1.6 1.8 2.0 2.2Maior axis R [m]

-0.4

-0.2

0.0

0.2

0.4

Hei

ght z

[m]

BAE n=4TAE n=4E = 120 keV, !z0 = 0.3 m

for banana orbits

Intersection

Radial dimension

surface

Figure 4.34.: Example of the specific trapped particle ensemble which is localized radially

between both mode amplitudes with a bounce frequency of ωb = 100 kHz.

The red line denotes the radial localization where the particle orbits have to

be located to generate an oscillatory behaviour.

In figure 4.35, the resonance plot for the general resonance condition (4.1) is shown for theTAE and the BAE, both with n = 4, and for comparison also with n = 3 and n = 4. The

98

Phase space analysis

• for nBAE=nTAE, double resonant particles are radially trapped between the modes•both amplitudes are modulated with difference frequency•amplitude evolution is coupled

ω - ntor ωprec - p ωb = 0

ωTAE-ωBAE =ωb

ASDEX Upgrade


Conclusions & further results

• at ITER, modes with higher (n,m) will be the most unstable ones since the orbits of the energetic particles will be smaller• therefore, the radial coupling length will be smaller• however, a larger number of modes will be unstable at different radial

locations, emphasizing the importance of multi-mode studies• although the background profiles will be less steep, diamagnetic effects

will stay important since ω* ~ m

• numerical loss detector (extended HAGIS) shows good agreement with fast-ion-loss detector at ASDEX Upgrade• identification of drive and loss regions that are not necessarily equivalent

4.5. Comparison of Fast Ion Losses: Simulation vs Experiment

to resonant TAE losses coincide very well with the experimental measurements. The lostions with E ≈ 300 keV are still included as shown in figure 4.54 (blue trajectory).The small deviation in pitch angle and the absence of even higher energetic losses of the’large’ spot can be explained by two reasons: On the one hand, the assumption that theFILD detector is parallel to the magnetic field lines might not be exactly correct. Increas-ing γ by only 1◦ (Fig. 4.56) would result in a perfect line up and is within the error barfor a deviation of ±2◦. On the other hand, the distribution function in energy (Eqn. 4.3)underestimates high energies since it is assumed that a maximum energy of E0 = 1MeV iscreated by the ICRF heating. An increase of E0 to larger values could bring the desirableeffect.

TAE & BAE (isotropic, selection R ! Rmag)

Pitch angle [deg.]

Gyr

o ra

dius

[mm

]

45 50 55 60 65 70 75

20

30

40

50

60

70

80

90

1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Corrected’small’ spot (FILD)

Corrected’large’ spot (FILD)

Figure 4.61.: Loss pattern of selected fast ion losses caused by both modes in the presence

of an isotropic fast particle distribution. Only trapped particles which have

their bounce points between R = 1.6 m −1.8 m are plotted in contrast to the

total losses shown in figure 4.60 (right).

The initialization of the ICRH-generated distribution should avoid the need for a lossselection. In figure 4.62, the losses for the two mode case are shown. The loss region co-incides partly with the experiment (blue ellipses). The ’small’ spot is however missing asfor the single TAE with an isotropic loading. The simulated loss region partly agrees withthe ’large’ spot but has a wrong distribution in pitch angle. The lost particles with smallerpitch angles and the absence of low energetic particles are due to the chosen loading of thefast ions (Fig. 3.7). If trapped particles are initialized near to the plasma edge, they havetheir turning points at the HF side. This results in smaller pitch angles when the particlesare lost (Fig. 4.59). Furthermore, the fast particle initialization near Rmag excludes trappedions with low energies located on the LF side.

Although the ’ICRH-generated’ distribution is more realistic, the fast ion losses causedby both modes in the presence of an isotropic distribution coincide much better after theparticle selection. This is due to the inclusion of all possible trapped particle orbits but

119

experiment

Pitch angle [deg]

Larm

or r

adiu

s [m

m]

loss pattern

ASDEX Upgrade


Backup slides

ASDEX Upgrade



BAE p=1

TAE p=0

BAE p=0

BAE p=2

TAE p=1

Loss regionsdue to TAE

TAE p=-1

Mode drive

BAE p=1

TAE p=0

BAE p=0

BAE p=2

TAE p=1

Mode drive

Loss regiondue to BAENot accessible!

(core-localization)

Figure 4.43.: Resonance plots of both modes but the fast ion losses caused by the TAE (left)and the BAE (right) are treated separately. A difference between fast particlesthat drive the modes and particles which are ejected is clearly observed.

Due to both mechanisms, the TAE amplitude is increased further as the BAE amplitude

reaches the TAE amplitude. Due to the significantly increased TAE amplitude, an additional

region in the low energetic fast ion loss range appears as the TAE reaches the stochastic

threshold for a single mode. A broader shape of the loss distribution indicates an increasing

influence of stochastic losses. Nevertheless, a striking difference is observed between fast

particles that drive the modes and particles that are ejected (Fig. 4.43).

4.4. Fast Particle Losses for ICRH-generated distributions

In order to compare the fast ion losses with the experimental data, an implemented ICRH-

generated fast particle distribution is used in the following. It has to be investigated how

realistic the implemented distribution is and if all experimental observations can be ex-

plained. Therefore, both single modes and multiple modes in the presence of the anisotropic

distribution function are studied. The results of the mode drive and the fast ion losses are

compared to the simulated results of the isotropic loading first.

The anisotropic ICRH-generated fast particle distribution was introduced in section 3.2.5

and deviates from its isotropic counterpart especially by the narrow initialization of the pitch

angle variable around λ = 0 as shown in figure 3.8. In addition, the particles are loaded

within small poloidal angle segments which are localized around the magnetic axis position

(Fig. 3.7). The same slowing-down distribution in energy as for the isotropic distributions

is used (Eqn. 4.3).

The volume-averaged fast particle beta �βf � (Eqn. 3.82) which is related to the distribution

function determines the particle density. Due to the anisotropy in poloidal space and in the

pitch angle in an ICRH-generated distribution, an increased particle density increases the

constant C in the equation so that an enhanced weight is represented by each marker for

106


1.6 1.7 1.8 1.9 2.0 2.1Maior axis R [m]

-0.4

-0.2

0.0

0.2

0.4

0.6

Hei

ght z

[m]

E = 130 keV, !z0 = 0.38 m (p=-1, out)E = 130 keV, !z0 = 0.15 m (p=-1, in)E = 420 keV, !z0 = 0.32 m (p=0)

TAE localizationRadial

Figure 4.19.: Examples of unperturbed particle trajectories for each redistribution regionas shown in Fig. 4.17. The magenta line shows the radial extension of then = 4-TAE.

4.2.2. Fast Particle Losses

In this section the fast particle losses caused by a TAE mode are investigated at differentpoints in time. It is expected in the simulations that fast particles are ejected by the modeas the amplitude of the eigenmode increases. Due to the fact that no particle source isavailable, the fast ion losses will stop as the mode saturates.

From figure 4.21 it can be seen that at the beginning of the linear growth phase no fastparticles are ejected due to the TAE. As an amplitude of A ≈ 5.0 · 10−5 at t = 3.2 · 10−4 sis reached, particle losses appear (red). The maximal loss amplitude occurs at time t =5.0 · 10−4 where the mode amplitude has almost reached a saturated level. Thereby, thefast particle losses show an over-shoot due to weakly-confined particles in the outer plasmaregion. After the ejection of these particles, a mean loss amplitude of δf = 1.5 · 1012 isstill present due to the continuing growth of the amplitude caused by the p = −1-resonance(section 4.2.1). Although the amplitude increases slightly further, the period t > 6.5 · 10−4 sis denoted as saturation phase (Fig. 4.20).

88

separatrix

4.5. Comparison of Fast Ion Losses: Simulation vs Experiment

to resonant TAE losses coincide very well with the experimental measurements. The lostions with E ≈ 300 keV are still included as shown in figure 4.54 (blue trajectory).The small deviation in pitch angle and the absence of even higher energetic losses of the’large’ spot can be explained by two reasons: On the one hand, the assumption that theFILD detector is parallel to the magnetic field lines might not be exactly correct. Increas-ing γ by only 1◦ (Fig. 4.56) would result in a perfect line up and is within the error barfor a deviation of ±2◦. On the other hand, the distribution function in energy (Eqn. 4.3)underestimates high energies since it is assumed that a maximum energy of E0 = 1MeV iscreated by the ICRF heating. An increase of E0 to larger values could bring the desirableeffect.

TAE & BAE (isotropic, selection R ! Rmag)

Pitch angle [deg.]

Gyr

o ra

dius

[mm

]

45 50 55 60 65 70 75

20

30

40

50

60

70

80

90

1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Corrected’small’ spot (FILD)

Corrected’large’ spot (FILD)

Figure 4.61.: Loss pattern of selected fast ion losses caused by both modes in the presence

of an isotropic fast particle distribution. Only trapped particles which have

their bounce points between R = 1.6 m −1.8 m are plotted in contrast to the

total losses shown in figure 4.60 (right).

The initialization of the ICRH-generated distribution should avoid the need for a lossselection. In figure 4.62, the losses for the two mode case are shown. The loss region co-incides partly with the experiment (blue ellipses). The ’small’ spot is however missing asfor the single TAE with an isotropic loading. The simulated loss region partly agrees withthe ’large’ spot but has a wrong distribution in pitch angle. The lost particles with smallerpitch angles and the absence of low energetic particles are due to the chosen loading of thefast ions (Fig. 3.7). If trapped particles are initialized near to the plasma edge, they havetheir turning points at the HF side. This results in smaller pitch angles when the particlesare lost (Fig. 4.59). Furthermore, the fast particle initialization near Rmag excludes trappedions with low energies located on the LF side.

Although the ’ICRH-generated’ distribution is more realistic, the fast ion losses causedby both modes in the presence of an isotropic distribution coincide much better after theparticle selection. This is due to the inclusion of all possible trapped particle orbits but

119

ASDEX Upgrade


compressible effects in a kinetic description

BAE/KBM dispersion relation

!2

v2A

1 !!"p

!! k2

# = 2v2

thi

R20v

2A

! H(xm!1) + H(xm+1) + "Nm Nm+1(xm!1)

D(xm!1)+

Nm!1(xm+1)

D(xm+1)

with xm = !k#,mvth

, " = Te/Ti = 1

D(xm) = 1 + D(xe,m) + " 1 + D(xi,m) ,

N(xm) = N(xi,m) ! N(xe,m),

D(x) = (1 ! !"! )xZ(x) ! !"

! # x2 + xZ(x)(x2 ! 12) ,

2N(x) = (1 ! !"! ) x2 + xZ(x)(x2 + 1

2) ! !"! # x2(x2 + 1

2) + xZ(x)(14 + x4) ,

H(xm) = H(xm,i) + "H(xm,e),

H(xm) = 12 (1 ! !"

! )F (xm) ! #!"! G(xm) ,

2F (x) = xZ(x)(12 + x2 + x4) + 3x2

2 + x4,

2G(x) = xZ(x)(34 + x2 + x4

2 + x6) + 2x2 + x4 + x6

IPP Colloquium, Garching, January 2009 61

ASDEX Upgrade


!"

!#$%

!#

!&$%

&

&$%

#

& &$# &$" &$' &$( &$% &$) &$*

+,-+./0.12,3.

456,07!"

!#$%

#

#$%

"

"$%

&

&$%

'

'$%

(

($%

# #$" #$& #$' #$( #$% #$) #$*

+,-+./0.12,3.

456,07

elec

tros

tatic

pot

entia

l

elec

tros

tatic

-el

ectr

omag

netic

!

"!

#!

$!

%!

&'()*(+,-./012

3456

!

"

#

$

%

7!

"87! "87" "87# "87$9:;(./<(,3

=>?<

@:6@A+.;5B(<

/7C73.0:+0.;5B(

<DEF55F@

KBM modes at low frequencies

Introduction - IAEA NA...Summary (G.Y. Fu et al) M3D-K simulations of TAE saturation with EP...

Documents

Transcript of Introduction - IAEA NA...Summary (G.Y. Fu et al) M3D-K simulations of TAE saturation with EP...