Comparison of collision operators for the geodesic acoustic mode

Yang Li1),2)*, Zhe Gao 1) 1)Department of Engineering Physics, Tsinghua University, Beijing 100084, CHINA

2)Southwestern Institute of Physics, Chengdu 610041, CHINA

*Email: liyang@sunist.org

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This work is supported by NSFC, under Grant Nos. 10990214, 11075092,11261140327 and 11325524, MOST of China, under Contract No. 2013GB112001, and Tsinghua University Initiative Scientific Research Program

Outline

•Background and Motivation•Theoretical Model•Analytical and Numerical Results•SummarySafety factor effectCollision frequency effectInfluence of different collision operators

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Background • Collisional model of GAM: The frequency and damping rate are basic properties of

GAM. Since GAM is usually observed in edge high collisional frequency region. It may be important to investigate collisional GAM model.

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Motivation

• To employ more kinds of collision operator

• To obtain analytical dispersion relation of collisional GAM and numerical results

• To analyze the influence of safety factor, collision frequency and different collision operators

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Theoretical Model: Drift Kinetic Model

• Since the finite Lamor radius effect can be neglected in this problem, the drift kinetic equation can be employed

• Only consider a radial electric field

• Qusineutral condition

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1, 1, ˆcos sinc sr rE e

,

0r pi e

j j j

5

0

1ˆ ˆ

1 cosB

q

B e e

Theoretical Model: Collision Operators

• Five collision operators are used, including three Lorentz-types and two Krook-types• Krook operator with number conservation term (Rewoldt et al, 1986)

• Krook operator with number and energy conservation term (Rewoldt et al, 1986)

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2 22

2 211 1

21

22 2 31

3 2x

iixii ii x

C f x x f e dxf f x e dx z dd z

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Theoretical Model: Collision Operators

• Lorentz operator independent of energy

• Lorentz operator with an energy-dependent collision frequency(Rewoldt et al, 1986)

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21(1 )iC z f

z z

21(1 )iC x z f

z z

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Theoretical Model: Collision Operators

• Full Hirshman-Sigmar-Clarke form collision operator(Hirshman et al, 1976)

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2 11 11i

fC f x z zS f

z z

Momentum conservation term

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Method

• Decomposition Fourier series(poloidal angle and toroidal angle)Legendre polynomial (perturbation method)Hermite polynomial (Krook type operators)

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Solving process10

( ) ( ) 011gc gc

ff Bf e C

t t Umé ù¶¶ ¶ê ú+ + ×Ñ + + + × =ê ú¶ ¶ ¶ë û

v v v v EP P

( )( )

( )

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,, , 1 , 1

, , 1 , 1 ,

1

2 1 2 3

1

2 1 2 3

4 2

3n n

s n c n c n

c n s

s n

cn nn s

xx C f

x

n ni f f f

n n

n ni f f f

n nCf

wd

w

d- +

- +

æ ö+ ÷ç- - + ÷ç ÷çè ø- +æ ö+ ÷ç- + + ÷

+- =

=ç ÷çè ø- +

( )

( )

( )

( )

,0

,2 ,1 ,3

,1 ,0 ,2

,3 ,

2

,1 ,0

2

,2

,1

,32

4

2 3

3 7

2

5

3

5

3

3

2

3

s

s c c

c s s

c s

c s

s

c

c

x xf C f

xx

i f

i f f f

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x

C f

x C f

C fi f f

w

w

w

w

- =

- =

- -

æ ö÷ç- - + ÷ç ÷çè ø

æ ö÷ç- + + =÷ç ÷çè

=

ø

- +

2 21s c s

c s c

i f xzf x z C f

i f xzf C f

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Dispersion relation for Krook operator with number conservation term

2

2

3

814 8

ii

ii

q

ng

nt

=+ +

τ=1

11

Analytical result is corresponding to numerical result of infinite q.

For Krook operator with number and energy conservation term

( )

22 5 1

03 12 ii

qi q

i i i

tw

w w n w

æ ö÷ç ÷- + ç- + - =÷ç ÷÷ç - +è ø

1 2 1 22

2 2 2

7 1 5

4 3 7 4 4 3ii

rii

q qq q

nw t t

t n

æ ö æ ö÷ç ÷ç÷= + - ® +ç ÷ç÷ ÷ç ç÷ç è ø+ +è ø

2

2

1

8314 8

ii

ii

q

ng

nt

=+ +

12

For Lorentz operator independent of energy

2

2

1447 4

ii

ii

q

ng

nt

=+ +

1 2 1 22

2 2 2

7 12 5

4 7 4 144 3ii

rii

q qq q

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For Lorentz operator with an energy-dependent collision frequency

Since it is difficult to obtain an analytical result, we only present numerical result here.

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For full Hirshman-Sigmar-Clarke form collision operator15

(a) Lorentz operator independent of energy(b) Lorentz operator with an energy-dependent collision frequency(c) Full Hirshman-Sigmar-Clarke form collision operator(d) Krook operator with number and energy conservation term(e) Krook operator with number conservation term

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Summary17

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Summary

• Number conservation was vital for all modes. All of the operators conserves number.

• Energy conservation is important for determiating the eigenfrequency of the GAM, but momentum conservation is not. In physics, the density quasi-neutrality governs the GAM dynamics and the collisional damping means energy transferring from the GAM to random thermal motion.

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Thanks !

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