Particle Distribution Modification by TAE mode and Resonant Particle Orbits

POSTECH1, NFRI1,2

M.H.Woo1, C.M.Ryu1, T.N.Rhee1,,2

Outlines

-Hybrid scheme:ideal MHD background and energetic particles

-Tokamak Equilibrium and TAE

-Distribution function modification by TAE -Resonance particle interactions with the TAE wave Bouncing, circulating and potato orbits

-Conclusions

Hybrid Code Structure

Equilibrium (EFIT data)

Re-normalization of Equilibrium(Chease)

TOKAMAK Experiment

Ideal MHD Solution (KINX)

Non-inductive heating & Alpha particle

TRANSP Data etc. Hot Particle Distribution

Mode Frequency & Growth rate

Test Particle code (ORBIT)

Monte-Carlo Distribution

Calculation of Mode Frequency Modification

0||)(2

1 )()(3220

2 nak

ak WWxd

Dispersion Relation for TAE

]Re[2

1 )()(

00

nak

akr WW

K

Calculation of Mode Growth Rate

( )

0

1[Im ]

2na

i kWK

))((ln

4

|)~(4

2)(

||

2

*

2)(

ppk

k

jj

hak

kl l

l

kk

jj

hnak

FE

E

VBW

vk

JE

E

VBW

Mode Growth-rate

( ) ( )[1 ( ) ]

( ) (damping)i d

A t A t t t

t

All the values shown here can be estimated from Simulation

Kinetic & Fluid Contributions

Why Hybrid Code ?

• Full Particle simulation is not available• Saving computation time• Individual particle trajectory• Immediate validation & examination of analytical theory• Essential issues about energetic particle• Alpha particle heating• Turbulence

Limitations

• Only perturbative treatment• Not fully self-consistent• Scale larger than Gyro-radius• Slow time scale behaviors ( Drift scale, Not gyration scale)

Ideal MHD mode

Relation between displacement and Magnetic perturbation

)(, Ingmq

nqmmn

Total Perturbation Dominant m=1 Perturbation

Particle Distribution Modification by TAE mode

• Electron distribution for ECRH• Ion distribution for NBI• Electron & Ion distribution for ICRH• Fusion Born Alpha particle distribution

b=0.1

Example: Initial Maxwellian distribution (2000 particles) in simulation

b=8

•Interpolation of test particle distribution

•Smooth radial and energy profile

•These equations are used for analytical calculation of mode frequency and growth rate

2 2 2 21

1 1( , )

( ) ( )

N

pk p pk k

F Ea E E b

2 2 2 21

2( ) 1

( ) ( )

Np pk

kp p pk k

F

a E E b

2 2 2 21

2( ) 1

( ) ( )

Np pk

k p pk k

E EF

E a E E b

Initial Distribution

First orbit loss of the particles

Final Distribution

Pitch Distribution

Energy Distribution

• Energy is conserved

• Particles with pitch around 0are dominantly lost• Large bounce orbit is the main reason for loss

Radial Distribution

Theta Distribution

• Radially flattened due tofinite bounce orbit

• Barely trapped or circulatingparticles are mainly lost

For the perturbation

Energy distribution

Resonance line

•More particles near the resonance line. •Some particles gain very high energy•High energetic particles (>150 keV) are lost

Radial distribution

•Particles near the edge are lost•Flattened compared to the no B perturbation case

3/ 10B B

Particle loss due to perturbation

•Dominant particle loss begins around •First orbit loss is about 40%

210

•In average, particle gains energy•Energy gain process shows a jump

Circulating-Passing particles: High energy

800

4transit

alpha p

t t

m m

•No perturbation•Clear precession•outward-shift of orbit

|| ||

100

/ 0.7

E keV

v v

parameters

Non-Circulating, passing particles

Same parameters with low energy 1E keV

800

4transit

alpha p

t t

m m

||

100

0.3

E keV

•This is the orbit of the particle that interact most strongly with TAE mode

||

3

0.3

100

0.7

/ 10

A

E keV

B B

Particle trajectory with B perturbation

Non-Resonant case, large perturbation

Orbit modification is not much.

Resonant case with small perturbation

||

4

0.3

100

0.454

/ 10

A

E keV

B B

• Particle has a little bit smaller velocity• Particles are lost due to resonant interaction

0 00.3 2 0.3 2wave AV R B

Wave phase velocity is about

/ 0.455wave partV V

Here is alpha particle velocitypartV

Resonant case with small perturbation -jumping orbit

•A sudden jump to other smaller orbit

||

100

4

0.43(smaller pitch)

/ 0.455

/ 10

waveV V

B B

Resonant case with small perturbation- jumping orbit

||

5

0.475 ( greater pitch )

10B

B

For a very small perturbation, resonant particles change the direction of the movement,jumping to other stable orbit with pitch reversed.

Detailed Motion of Resonant Particles

Phase trajectories of particles with different pitches

Bouncing

Potato

Circulating

Conclusions

•Modification of particle distribution by the TAE mode is under investigation. •A hybrid scheme of ideal MHD and energetic particle motion is used.•Particles tend to jump from one orbit to other stable orbit•Resonant particles interacting with TAE strongly mainly come from the potato orbit.•Some particles with low pitch gain large energy and move in the reverse direction.

Thank you for your attention!