Kinetic MHD Simulation in Tokamaks

H. Naitou, J.-N. Leboeuf†,

H. Nagahara, T. Kobayashi, M. Yagi‡,T. Matsumoto*, S. Tokuda*

Joint Meeting of US-Japan JIFT Workshop on Theory-Based Modeling and Integrated Simulation of Burning Plasmas and 21COE Workshop on Plasma Theory ------Kyodai-Kaikan, Kyoto, 2003/12/15-17 ------

Yamaguchi University†University of California at Los Angeles

‡Kyushu University

*Japan Atomic Energy Research Institute

Key Words

• Sawtooth Crash• m=1/n=1 Internal Kink Mode • Kinetic MHD Model• Collisionless Magnetic Reconnection• Diamagnetic Effects• Sheared Poloidal Flow of m=1• Kelvin-Helmholtz (K-H) Instability• Vortex Generation

Outline

1. Motivations2. Basic Equations3. Results of Cylindrical Code

(a) Linear Calculations (b) Nonlinear Calculations

4. Toroidal Code (Kinetic-FAR)5. Summary

1. Motivation

• There is no complete theory to explain the sawteeth phenomena in tokamaks without inconsistency.

• Resistive MHD model is not appropriate.• Kinetic MHD model can elucidate (a) fast sawtooth crash. (b) nonlinear acceleration of the growth rate. (c) diamagnetic stabilization.

• Gyrokinetic particle simulation and gyro-reduced-MHD (GRM) simulation have revealed the fast full reconnection followed by the second phase of axis q-value less than unity.

• Linear and nonlinear studies by GRM code. ……… Summarized in this presentation.

• The vortex generation by K-H instability can be a critical issue for the complete understandings of the sawtooth crash.

2. Basic Equations

ezee

zes

ze

z

z

nDAbnbt

n

Anba

Adt

d

a

dbA

t

DAbbt

22*

22*2

22

*

222*22

)(

)( )(

)( )( )(

⊥⊥

⊥⊥⊥

⊥⊥⊥⊥⊥

∇+∇∇⋅−∇⋅φ∇×−=∂∂

∇∇μ−∇⋅⎟⎠

⎞⎜⎝

⎛ρ+∇⎟

⎠

⎞⎜⎝

⎛+φ∇⋅−=∂∂

φ∇∇+∇∇⋅−φ∇∇⋅φ∇×−=φ∇∂∂

∇⋅φ∇×+∂∂

=

×∇+=

) (

*

btdt

d

bAbb z

Safety factor profile :

Equilibrium density profile:

Key Parameters:

Assumption : Single Helicity

12

0 0 ) 1( 4 1 )(

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−= arqqrq

,85.00=q 125.2)( =aq

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −ε−=n

nlrrnrn 0

0 tanh 1 )(

,0.1)5.0( =aq

de / a, ρs / a, εn

m / n = 1

r0/a = 0.5, ln/a = 0.16

(a) Linear Calculations

3. Results of Cylindrical Model

de/a = 0.0005315, ρs/a = 0.002891, 1/0 = 417μsec

Electron Diamagnetic Stabilization of Kinetic Internal Kink Mode

Mode Structure in r-

*e/0 = 1.48 (theoretically unstable)

Mode Structure in r-

*e/0 = 1.98 (theoretically close to marginal point)

Electron and Ion Diamagnetic Effects

de/a = 0.0005315, ρs/a = 0.002891, Ti/Te = 1.0, 1/0 = 340μsec

Mode Structure in r-

A B

(b) Nonlinear Calculations

de = 0.01, ρs = 0.03

0.0

0.5

1.0

0.00 0.50 1.00 1.50 2.00

É÷*e/(2É¡0)

É¡/É¡

0

Linear Growth Rate

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Linear Mode Pattern

Movie of Vortex Generation

Magnetic Field Structure

4. Toroidal Code (Kinetic-FAR)

• Kinetic terms are included.• Can treat realistic equilibrium with shaping, finite

beta, and curvature.• Can directly compare resistive MHD with kinetic

MHD.• Two approaches based on resistive FAR (R-FAR)

code and turbulent FAR (K-FAR) code.

Made cylindrical by keeping only m=0 and n=0 component in Grad-Shafranov toroidal equilibrium and switching off toroidal terms ( e.g. curvature ).

Comparison between GRM and K-FAR

cylindrical model de = 0.01, ρs = 0.03

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.2 0.4 0.6 0.8 1

growth rate

GRMK-FAR

εn

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1

real frequency

GRMK-FAR

r

εn

Comparison Between GRM and T-FAR

GRM

r

T-FAR

r

z)

Comparison Between R-FAR and T-FAR

K-FAR

r

T-FAR

r

z)

RSTEQ Toroidal Equilibrium

0 25 50 75

75

50

25

0

R

Z

=9.8x10-3

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1

Safety factor q(r) profile

Pressure P(r) profile

q(r) P(r)

r/a

=a/R=1/3

Comparison between Toroidal and Cylindrical Cases

de = 0.01, ρs = 0.03

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.2 0.4 0.6 0.8 1

growth rate

toroidal

cylindrical

εn

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1

real frequency

toroidal

cylindrical

r

εn

Preliminary

5. Summary

• We believe that vortex generation due to K-H instability has critical effects on the nonlinear developments of kinetic internal kink modes.

• Comparison with K-H theory is underway. ------------ growth rate, threshold, etc.

• Effects of vortex generation may be important for the complete understandings of sawtooth crash phenomena.

• Kinetic modifications of FAR code are underway to tackle these issues.