STATISTICAL DESCRIPTION OF STOCHASTIC ORBITS IN A TOKAMAK

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STATISTICAL DESCRIPTION OF STOCHASTIC ORBITS IN A TOKAMAK

A. Rechester, N. Rosenbluth, R. White

To cite this version:

A. Rechester, N. Rosenbluth, R. White. STATISTICAL DESCRIPTION OF STOCHASTIC ORBITS IN A TOKAMAK. Journal de Physique Colloques, 1980, 41 (C3), pp.C3-351-C3-358.

�10.1051/jphyscol:1980390�. �jpa-00219909�

JOURNAL DE PHYSIQUE Colloque C3, supplement au n ^O 4, Tome 41, avril 1980, page C3-351

STATISTICAL DESCRIPTION OF STOCHASTIC ORBITS I N .4 TOKAMAK

A.B. Rechester, N.M. Rosenbluth

*

and R.B. White

""

B e l l L a b o r a t o r i e s , M u r r a y H i l l , N . J . 0 7 9 7 4 , U . S . A .

* ~ n s t i t u t e f o r A d v a n c e d S t u d y , P r i n c e t o n , N . J . 0 8 5 4 0 , U . S . A .

* * ~ l a s r n a P h y s i c s L a b o r a t o r y , P r i n c e t o n U n i u e r s i t y , P r i n c e t o n , N . J . 0 8 5 4 0 , U . S . A .

ABSTRACT

We have summarized the results of ana- lytical and numerical studies concerning particle orbits in the presence of low frequency fluctuations of magnetic and electric fields.

Introduction

Magnetic confinement in a Tokamak is achieved with the strong toroidal mag- netic field Bo and a poloidal field B [l].

P A single particle (electron or ion) moves primarily along the magnetic field and also rotates rapidly around its guiding center which is a particle position averaged over the gyromotion. A statis- tical description of a plasma is given by the kinetic theory. The kinetic theory which is applied specifically for a toroi- dal geometry of a Tokamak is called neo- classical theory [l]. The validity of such a description is mainly due to the stochastic 'nature of particle collisions.

But at the high temperatures of a Tokamak plasma (T % 10'-10*~~) the Coulomb cross section is relatively small. That is probably why the neoclassical theory has failed to explain energy transport in

Tokamaks. It has been suggested recently [2-6,101 that low frequency magnetic fluctuations (6B/B0 % could account for the experimentally observed electron energy transport. When such fluctuations are present the particle orbits become stochastic. The nature of this stochasti- city is very different from that intro- duced by the particle collisions. But it also permits description of the particle orbits statistically. There are two important quantitative characteristics of this statistical description [3]. First, particles are diffusing radially as they move along the field lines with a diffu- sion coefficient D (in cm) and, second, the neighboring orbits diverge from each other exponentially with a characteristic correlation length LC. We have found earlier [ 3 ] that radial electron heat transport can be expressed through these two quantities as:

Here vll is an average electron velocity

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980390

C3-352 ^{JOURNAL DE PHYSIQUE}

a l o n g t h e magnetic f i e l d and A i s an e l e c t r o n mean f r e e p a t h . I n p r e s e n t d a y Tokamaks, A v a r i e s from a b o u t

l o 5

cm i n t h e c e n t e r o f d i s c h a r g e t o 1 0 cm n e a r t h e w a l l s ( A ^% T 2 ) . The p a r t i c l e o r b i t s i n

t h e p r e s e n c e o f low f r e q u e n c y f l u c t u a t i o n s o f e l e c t r i c f i e l d i s v e r y s i m i l a r t o t h a t due t o magnetic f l u c t u a t i o n s w i t h t h e c o r r e s p o n d i n g v a l u e s o f D and L ( s e e

C

l a t e r ) . I n h o t h c a s e s t h e motion a l o n g t h e magnetic f i e l d i s c o u p l e d w i t h t h e r a d i a l d i f f u s i o n . Due t o t h i s c o u p l i n g , A/Lc i s an i m p o r t a n t p a r a m e t e r . The pur- p o s e of t h i s p a p e r i s t o summarize t h e r e s u l t s of a n a l y t i c a l and n u m e r i c a l

The e q u a t i o n s o f motion f o r t h e p a r - t i c l e g u i d i n g c e n t e r a r e

We always assume i n t h i s p a p e r t h a t t h e s h e a r , L-I = dB /Bodx i s c o n s t a n t . The

P

argument o f exponent i n E q . ( 3 ) i s e q u a l t o

T y p i c a l l y f o r t h e low f r e q u e n c y d r i f t t y p e modes wR/nv a 1 0 - ~ - 1 0 - ~ . T h i s i m p l i e s

I I

t h a t t h e e l e c t r o n dynamics i s much f a s t e r t h a n t h e t i m e e v o l u t i o n o f f l u c t u a t i o n s s t u d i e s o f D and LC. which can b e c o n s i d e r e d a s s t a t i c f o r o u r

A s i m p l i f i e d magnetic f i e l d o f i n t e r e s t problem, b a 6 ( w ) .

can be w r i t t e n i n t h e s l a b geometry: I n t h e c a s e o f low f r e q u e n c y f l u c t u a - t i o n s o f e l e c t r i c f i e l d t h e o n l y d i f f e r - ( 2 )

e n c e i s t h a t t h e r a d i a l ( x ) motion i s

Magnetic f l u c t u a t i o n s 6B c o n s i d e r e d t o b e + 2

g i v e n by t h e v e l o c i t y , c

( ~ X S ) / B .

v e r y s m a l l

< < B~ << B~ = c o n s t

.

~ n o r d e r t o model t o r o i d a l and p o l o i d a l ^{A l l} l a t e r formulae c a n be w r i t t e n f o r . p e r i o d i c i t y we assume t h a t t h e system i s t h i s c a s e by a s i m p l e s u b s t i t u t i o n

p e r i o d i c i n t h e z d i r e c t i o n w i t h t h e p e r i o d 2rR and i n t h e y d i r e c t i o n w i t h t h e

I

p e r i o d 2 s a . ) The c o o r d i n a t e x r e p r e s e n t s A Simple Example o f S t o c h a s t i c O r b i t s t h e r a d i a l d i r e c t i o n . Then 6 6 can b e We c a n i l l u s t r a t e t h e d i f f e r e n c e b e t - w r i t t e ~ i n t h e form ' ween s t o c h a s t i c and n o n s t o c h a s t i c o r b i t s

3

I -

ⁱ (my/a-nz/R-wt) by mapping t h e t r a j e c t o r y o f a p a r t i c l e on 6B = Bo 1 b m n ( x f w ) e

mtn a s i n g l e c r o s s s e c t i o n . We s o l v e numeri-

dw

+

c . ~ . c a l l y .Eqs. ( 4 ) and f i n d t h e p a r t i c l e co- ( 3 ) o r d i n a t e s xi, yi a t t h e s u c c e s s i v e d i s t a n - I£ t h e s p e c t r u m of f l u c t u a t i o n s i s d i s c r e & c e s zi = 2aRi, i = 0 , 1 , 2 ,

...

a l o n g t h e

-f -f

t h e n bmn(xtw) = b m l n ( x ) 6 ( w - w ~ , ~ ) . r t r a j e c t o r y . C o n s i d e r a s i m p l e example

when only 2 static harmonics are present in Eq. (3): ml = 1, nl = 0, m2 = 1 , n2 = 1, b10 = bll = E - 6 ( w ) /2i. Figures 1-5 are the results of these calculations when we gradually increase the parameter

E (we actually used another parameter s, see later, and plot x and y in dimension- less units). Mapping points lying on smooth lines correspond to nonstochastic or integrable orbits while scattered points correspond to stochastic orbits.

The width of a separatrix on Fig. 1 can be calculated from the equations [ 7 1 :

-06 -04 -02 0 -02 0 4 0 6

FIG. 1, S = 0 . 5

Here x is called a rational surface.

mn

Figures 1-5 also illustrate the so- called resonance (islands) overlapping criterion. Introduce the dimensionless parameter

FIG. 2 , S =0.7

Here x mn and x m l n 1 are any two neighboring rational surfaces. Then the transition from integrable orbits (Fig. 1, s = 0.5) to stochastic orbits (Fig. 5, s = 2) takes place roughly at s = 1 181. The transi- tion region is very complicated as one can see in Figs. 2-4. Obviously, the

stochastic transition is meaningful only for the case of a discrete ,spectrum of fluctuations. In the case of a continuous spectrum we are always in the stochastic region. Figure 6 illustrate that the com- puter can be quite creative.

Analytical Results

The quantitative measure of stochas- ticity is the Kolmogorov entropy. It characterizes the exponential divergence

0 0

-0.6 -0.4 -0.2 0

0.2 0.4 0.6

FIG. 3, S = 0.7

c3-354 JOURNAL DE PHYSIQUE

FIG. 6

of close orbits and is defined by the formula

1 d ( z ) , h = lim lim

EZ - -

z+- d o +

- ^do

Here d is a distance between two close orbits: d = J(xl-x)

+

^{(y'-y) and} z is length of an orbit, which plays the role of a dynamical variable. The Kolmogorov entropy can be calculated on a computer using the equations of motion (4)

.

^We

will describe these calculations later.

Let us now just mention the main results of these computations. 1. The well de- fined limit exists in Eq. ( 9 )

.

^{2. This}

limit does r~ot depend on a value of do if it is small enough. 3 . h always

approaches zero for integrable orbits.

4. h converges to positive values for stochastic orbits. 5. In the case of well developed stochasticity (s >> 11, h

is almost independent of initial condi- tions. The property 5 is an indication that h can be considered as a statistical quantity. It is in that case when one would expect to calculate h using statis- tical theory, namely, replace dynamical averaging ( z + -) in Eq. (9) by phase- space averaging with a proper distribu- tion function. We have developed such a statistical theory. Our main result can be written in the form [ 9 1

Here < > is an averaging over x. An

X

approximate expression for LC has been previously given by Xronunes, Xleva and

Overman [ 5 1 . We have g e n e r a l i z e d Eq. ( 1 3 ) o f p a p e r [93 f o r t h e c a s e o f c o n t i n u o u s spectrum

.

The main a p p r o x i m a t i o n o f o u r t h e o r y i s Ls/Lc < < 1 which means a s t r o n g s h e a r . The c a s e o f z e r o s h e a r h a s been r e c e n t l y c o n s i d e r e d by Kadomtsev snd Pogutse [ l o ] . The d e r i v a t i o n o f Eqs.

( 1 0 , l l ) r e q u i r e s a knowledge o f t h e 2 p o i n t d i s t r i b u t i o n f u n c t i o n [91 w h i l e t h e e v o l u - t i o n o f p a r t i c l e d i s t r i b u t i o n a s t h e y move a l o n g t h e i r t r a j e c t o r i e s i s d e s c r i b e d by a one p o i n t d i s t r i b u t i o n f u n c t i o n f ( x , y ; z ) . f averaged o v e r t h e p o l o i d a l a n g l e e v o l v e s a c c o r d i n g t o d i f f u s i o n e q u a t i o n

w i t h t h e d i f f u s i o n c o e f f i c i e n t g i v e n b y t h e q u a s i l i n e a r formula .[I11

Numercial R e s u l t s

L e t u s now t u r n t o t h e r e s u l t s of o u r computer c a l c u l a t i o n s . We used t h e f o l l o w i n g d i m e n s i o n a l u n i t s : 2 ~ a = 1, 2nR = 1, Ls = 1, and c o n s i d e r e d a s t a t i c model w i t h

exp li2nm14Jmn) bmnx = E

2 i 6 ( w )

.

^{( 1 4 )}

Here m = 1,2,

...,

^{M; n = 0,1,2,}

...

^{N, and}

p h a s e s 14Jmn a r e randomly choosen numbers between 0 and 1. We have i n t e g r a t e d Eqs.

( 4 ) w i t h t h e c o n s t a n t s t e p s i z e Az. We

have used t h e d i r e c t i n t e g r a t i o n scheme f o r t h e f i r s t e q u a t i o n and t h e i n d i r e c t scheme f o r t h e second e q u a t i o n . T h i s method a l l o w s u s t o make an a r e a p r e s e r v - i n g s t e p p i n g i n t e g r a t i o n . Thus we have

Here

We have e l i m i n a t e d t h e t e r m B (0)Az/Bo i n P

t h e r i g h t hand s i d e o f Eq. (15) by a simple s u b s t i t u t i o n . E q u a t i o n s (15,16) have been d e r i v e d e x a c t l y f o r a s l i g h t l y d i f f e r e n t model [ 9 ] . Xolmogorov e n t r o p y h a s been c a l c u l a t e d d i r e c t l y u s i n g t h e n u m e r i c a l method d e s c r i b e d by C a s a r t e l l i e t a l . [12]

and Eqs. (15,161. To o b t a i n r e a s o n a b l e a c c u r a c y a p p r o x i m a t e l y

l o 4

s t e p s were n e c e s s a r y . We have found h i s a p p r o a c h i n g

z e r o below t h e s t o c h a s t i c t r a n s i t i o n , which t a k e s p l a c e a t ^E~ r _{2 /} i n good agreement _{~ ~}

w i t h t h e r e s o n a n c e o v e r l a p p i n g c r i t e r i o n . Above t h e s t o c h a s t i c i t y t h r e s h o l d h i s converging t o a p o s i t i v e v a l u e which i s a l m o s t i n d e p e n d e n t o f i n i t i a l c o n d i t i o n s . The r e s u l t s o f t h e s e c a l c u l a t i o n s a r e p r e s e n t e d i n F i g . 7. E q u a t i o n s ( 1 0 , l l ) a p p l i e d t o t h i s s p e c i f i c model g i v e

C3-356 JOURNAL DE PHYSIQUE

Here N is the number of test particles.

P

Initially all particles were taken at x (0) = 0. For integrable orbits D was

P

always approaching zero regardless of number of test particles. Above the stochastic transition (E > cM)D did not approach any limit and was widly fluctu- ating as z increased in the case of just few test particles. As the number

FIG 7

of test particles was increased the level of fluctuation decreased as 1 / K

.

^The

P Equation (17) is plotted as solid curves

-

relative fluctuations D/D are plotted in in Fig. 7. One can see a remarkable Fig. 9 and are defined by the expression.

agreement with the theoretical formula

N 1/2

(17). These results do not depend on a

D _{p p=l}

.

⁽²⁰⁾

step size Az in the region hAz < 1. But \

in the region hAz > 1, h is given by a different theoretical formula 191

Fig. 8

10 I 1 8 1 1 1 1 1 1

The corresponding numerical results are

$15

- -

-

D

.I0 -

-

.05 -

Oo ~Ibo

2h

^{3b0 4A0 5b0 6bo 7A0} sbo goo N~

Fig. 9

Thus, when N + m,D approaches a well de- P

fined limit, which does not depend on presented in Fig. 8 for the case Az = 1. y (0) even when all test particles were

9

We have also found some dependence of h very close initially. We have presented on the choice of phases @mn. For example, the results of these computations in Fig.

i f we take @mn = 0, than h is given by the 10. Equation (13) applied to our model expression [131: h = R ~ T M E . gives

Let us now summarize our numerical results for D. We used for calcdlations

~ q s . (15,161 and expression

FIG 10

The agreement with the quasilinear formula (21) is excellent. No dependence on phases or step size has been found.

Conclusions

We have found that the theoretical formula for the correlation length LC, Eqs. (10,ll) and for the stochastic diffusion coefficient D, Eq. (13) agrees very well with the direct numerical compu- tations using the model equations (15,16)

.

Such a good agreement can be used to justify the statistical arguments under- lying the derivation of Eqs. (10,ll and 13). Statistical description is valid only above the stochasticity threshold, which takes place for finite values of bmn and agrees with the overlapping criterion in the case of a discrete spectrum. A large step size of integra- tion or correlation of mode phases produces a logarithmic dependence of the Kolmogorov entropy (l/Lc) which is

different from that given by Eqs. (10,ll).

D is insensitive to these factors and well described by the quasilinear formula (13)

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for all cases we have studied.

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