Excitation of internal kink mode by barely trapped suprathermal

electrons*

Youwen Sun,

Baonian Wan, Shaojie Wang, Deng Zhou, Liqun Hu and Biao Shen

* Sun Y.W. et al., Phys. Plasmas 12, 092507(2005)

Outline

• 1. Background

• 2. Dispersion relation

• 3. Threshold condition

• 4. Application to experiment

• 5. Conclusion

1. Background• Fish-bone oscillations have been observed in the neutral-beam

injection experiments on several tokamaks and two models have been proposed to explain the these bursts.

• In recent experiments on the DIII-D Tokamak and the HL-1M Tokamak, an internal kink instability driven by barely trapped suprathermal electrons produced by high-field-side off-axis Electron Cyclotron Resonance Heating (ECRH) has been observed.

• A recent paper has already investigated the sawtooth stabilization by barely trapped energetic electrons.

2. Dispersion relation• In presence of the energetic particles, the dispersion relation for the

internal kink mode is

0c kA

D W W i

Minimized ideal variational energy (γI= -ωAδWc )

The kinetic contribution coming from the energetic trapped particle

Inertial term

5/ 2 222

7 / 22 2 0

0

1/ /

2sr c d

ks b d

dEE K E rmrm

W rdr d B FB r K

(1)

(2)

ωd=K2E/(KbmrRωc),α=μ/E, K2=(2/εα)1/2[2E(k2)-K(k2)]/π, Kb=(2/εα)1/2 K(k2)/

π, k2=sin2(θb/2)=(1/αB0-1+ε)/2ε, ε=r/R,

• Two different models for the distribution function of the

suprathermal electrons 3/ 2

0( , , ) ( ) ( )F r E n r E

/0( , , ) ( ) ( ) hE TF r E n r e

(3a)

(The slowing-down distribution)

(3b)

(The exponential distribution)

n(r)= ph(r)/( 23/2πB0mKb0Em)

n(r)=21/2ph(r)/(3π3/2B0mKb0Th5/2)

*

*

• The kinetic contributions '

20 0/ 1ln 1

2b h

ks

K KW

'

3/ 2 1/ 220 02 / 1[ ( )]

3 2b h

ks

K KW Z

(4a)

(4b)

1'

0

( )hh

d rd

d

20

8( ) ( )h hr P r

B

K20=K2(α=α0), Ω=ω/ωd ,(ωd=ωdm for the slowing-down

distribution and ωd=ωdT for the exponential distribution and ωdm

(ωdT) is the bounce averaged toroidal precessional frequency of

the electrons with energy Em (Th) at K2/Kb= K20/ Kb0 and

r=rs),εs=rs/R, and Z is the plasma dispersion function.

• Because ωc<0 for electrons and ωd>0 for the barely trapped electrons, the barely trapped condition is K20/Kb0<0 (the numerical calculation shows that it is equivalent to 0.8261≤k2≤1, or 2.2813 (130.7o) ≤θb≤π (180o)). For K20/Kb0<0, the sign and the form of δWk in Eqs. (4a) and (4b) are the same as that of the deeply trapped energetic ions.

• Substituting Eqs. (4a) and (4b) into Eq.(1) respectively, the dispersion relations are given by

'20 0/ 1

/ ln 1 02I

b hA

A s

K Ki

'

3/ 2 1/ 220 02 / 1/ [ ( )] 0

3 2I

b hA

A s

K Ki Z

(5a)

(5b)

3. Threshold condition• From the imaginary parts of Equations (5a) and (5b) ， the value of

β’h at threshold for these two models are given by

', ,

20 0

2

( / )s dm

h crit ab AK K

', , 3/ 2

20 0

3

2 ( / )

rs dT

h crit bAb r

e

K K

The equations for the real frequencies at threshold are found through setting Ωi=0 and replacing β’

h by β’h, crit in the real parts of Equations

(5a) and (5b),

1ln 1r

Ir

3/ 2 1/ 2

1/ 2

1Re

2

r

I r r r

r

eZ

(6a)

(6b)

(7a)

(7b)

• Comparing the threshold conditions for these two models, we find ωrb/ωra~3.9ωdT/ωdm~3.9Th/Em, β’

h,crit,b/β’h, crit,a~3.4ωdT/ωdm~3.4 Th/Em, and

ωib/ωia~1.6 Th/Em. Consequently, there is no essential difference

between the threshold conditions given by these two distribution

models, if Em~3Th, which is the same as the result for energetic ions

Writing β’h=β’

h, crit+∆β’h and substituting the real frequencies at threshold,

from the imaginary parts of Equations (5a) and (5b) the growth rates are given by

' '

' '/

/ln 1 1/ 1/ 1 4

h h r dm

ia h h dmr r

' ' 5 / 2

' '

3/ 2 1/ 2 5/ 2 1/ 2

/1.26 /

1 52 Re Re

2 2

rh h r dT

ib h h dT

r r r r r

e

Z Z

(8a)

(8b)

4. Application to experiment• For the barely trapped electrons with a single value pitch angle distribution, K20/Kb0

can be estimated to be -0.4.

• For typical DⅢ-D parameters [7]: major radius R=1.76m, minor radius a=0.62m, normalized singular layer radius ρs=rs/a≈0.2, B0=1.77T, ne≈3.0*1019m-3, Te≈2.5keV,

ωr/2π≈10kHz, ωi≈5*103s-1, ωA≈8.2*106s-1 (assuming s≈0.2, Zeff≈2), the ECRH

power Peff ≈ 1.1MW. On the deposition radius surface, 1.4% of the electrons are the

suprathermal electrons with 7.9% of the total electron energy and 0.27% electrons have energy above 36 keV, and they possess 3.4 %of the electron energy.

• For the slowing-down energy distribution f(E)∝E-3/2, Em is estimated to be 110keV by

solving the equation , and the slowing down time is estimated to be τs≈3ms .

• Assuming βh(r)= β0 exp[-(r-rp)2/δr

2] and choosing δr/rs ~ 0.1 and rp=rs, we obtain

β’h≈0.91β0 . Using the energy balance condition for energetic electrons,

we obtain β0≈1.4%and β’h≈1.3%. In high-field-side ECRH experiments, the beta value

of the barely trapped energetic electrons is one order less than that of the total energetic electrons. Consequently, the beta value of the barely trapped suprathermal electrons in the experiment can be estimated to be β’

h,exp≈1.3‰.

3/ 2 3/ 2

36 0

3.4%/

7.9%

m mE EE E dE E E dE

25/ 2 20

0 0 00 00

2 2 ( ) 2 2 ( ) 2 /2

a a

eff s h h p r

BP R rP r dr R r r dr R r B

• For the slowing down model, we find ωdm= -K20Em/(Kb0 rsRB0)

≈114*103s-1. Then the threshold conditions are given by ωr≈ωdm/2≈57*103s-1 (ωr/2π≈9.1kHz)and β’

h, crit≈2.7‰, which are of the

same order as the experimental data.

• The value of ∆β’h/β

’h can not be found from the experimental data.

However, the required ∆β’h/β

’h value can be estimated to be about 6%

from Eq. (8a) for the experimental growth rate.

• For the exponential energy distribution f(E) exp(∝ E/Th), Th is estimated

to be 20keV using the same method. Then, we find ωdT=-K20Th/(Kb0

rsRB0)≈21*103s-1. The threshold conditions are given by

ωr≈1.94ωdT≈40*103s-1 (ωr/2π≈6.4kHz)and β’h, crit≈1.7‰, which are also of

the same order as the experimental data. The required ∆β’h/β

’h value can

be estimated to be about 19% from Eq. (8b) for the experimental growth rate.

5. Conclusion• In summary, the barely trapped suprathermal electrons can also destabilize the

internal kink mode, when their density gradient is positive within the rational surface and the beta value of them exceeds a threshold.

• With the assumption of two different models of energy distribution function of the suprathermal electrons, the threshold beta value of the barely trapped suprathermal electrons, the real frequency and growth rate of the fishbone mode are found in this paper. The threshold condition is insensitive to the form of the energy distribution function of the suprathermal electrons similar to the result of energetic ions. The calculated threshold beta value of the barely trapped suprathermal electrons and the real frequency of the mode are in reasonable agreement with the experimental observations on DIII-D .

• Since the contribution of the deeply trapped electrons can be neglected only for high-field-side ECRH experiments, this phenomenon cannot be observed in the low-field-side ECRH experiments.

Thank you !