The shell model[1,2] is extended to describe plasma turbulence. Using this model, the electromagnetic plasma turbulence is investigated. The model consists of !-dimensional three feild equations:

§30. Study on Plasma Turbulence Based on Shell Model

Yagi, M., Itoh, S.-1. (Kyushu Univ.), Fukuyama, A.

(Kyoto Univ.), ltoh, K.

The shell model[1,2] is extended to describe plasma turbulence. Using this model, the electromagnetic plasma turbulence is investigated. The model consists of !-dimensional three feild equations:

( d 2) *2 *2 . . ..

dt + v kn un = A kn [un _,- bn_, - h(unun+i - bn bn+i )]

+ B kn [u:u:_ , -b):, - h(u::,- b::, )]+ iknbn +8n (1) ( _- d _{+ 1J k b} 2) _{= A k h} . . ..

(u ,b - u bn+, )

dt n n n n + n n

( _- d _{+ X k 8} 2) _{=A k (u} . _{,8 , -} . . _{hu 8} . . _+I)

d!

where un represents the fluctuating velocity field, bn' the fluctuating magnetic field and (}n, the fluctuating temperature field. The system is normalized by using the system size L and the Alfven time L I v

In the convective nonlinearity in Eqs.(l)-(3), only the nearest neighbor interaction is kept. S is defined by S = Ra I (Q~)where Ra = aftL

I (KV) is the Rayleigh number, Q = b:L

I (J1

Po 1JV), the Chandrasekhar number, ~ = 1J IK, the magnetic Prandtl number, respectively. For the typical parameters of high temperature plasma, the viscosity due to the Coulomb collision gives the estimate:

R

10

Q

10

P

10 S

10

a ' ' m '

Figure 1 shows the time evolution of flow energy 1 ~

(red), Eu =; ^{LJI un I} and internal energy (blue), 1 ~

E ₈ = 2 ^{LJ I 8}

^{I •} Parameters are chosen as A = B = i , h = 2, v = 1J =X= 10--{;. In these parameters, the relation luJ ^=Ibn I holds. It is found that flow energy stays at some energy level for a moment, then it starts to increase and reaches at the higher energy level.

The bursting behavior of internal energy is observed in the phase of increase of flow energy.

Figure 2 shows the power spectra of

eie<;tromagnetic energy (red) and internal energy

362

(blue) at t = 300 (quasi-steady state), which are given by Eb(kJ = lbJ

^{I (2kJ and E}

(kJ = leJ ^{I (2k). It}

seems that k~

law does not hold for Eb(k) or EJ k) and E

kJ . The behavior of this model is diffemt from the results obtained by the model without Alfvenic effect [2] and the model without thermal convection but with Alfvenic effect [3].

I== %1

0.010° Wi .... llllll ... ...

0.0 200.0 400.0 600.0 800.0 1000.0

Fig.1 the time evolution of fluctuating energies.

101 10-1 10-3 10-5 10-7 8 ^10·9

5 ^10-ll 3 ^10-13

0..

10 .1 5

Vi

t ^10-17

~

10 .19

&

10

10-23 10-25

w-21

10-29 10 -31

100 101

Fig.2 the power spectrum of energy at t = 300 . The model in which the drift wave effec is incorporated should be examined as a future work.

Reference

1) C. Gloaguen, et.al, Physica 17D (1985) 154.

2) A. Brandenburg, Phys. Rev. Lett. 69 (1992) 605.

3) D. Biskamp, Chaos, Soliton & Fractals 5 (1995)

1779.