THEORY FOR STATIONARY NONLINEAR WAVE PROPAGATION IN COMPLEX MAGNETIC GEOMETRY

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THEORY FOR STATIONARY NONLINEAR WAVE

PROPAGATION IN COMPLEX MAGNETIC

GEOMETRY

T. Watanabe, H. Hojo, K. Nishikawa

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 12, Tome 38, Dicembre 1977, page C6-111

THEORY FOR STATIONARY NONLINEAR WAVE

PROPAGATION IN COMPLEX MAGNETIC GEOMETRY

T. WATANABE, H. HOJO (*) and K. NISHIKAWA (*)

Institute of Plasma Physics, Nagoya University, Nagoya, Japan

RBsum6. - Nous prksentons nos efforts recents pour former un schema systematique de calcul, pour la propagation d'onde non lineaire dans le profil de plasma auto-consistant dans une geometric

de champ magnetique complexe.

Les hypotheses et/ou approximations de base, sont : i) utilisation du modkle a deux fluides sans collision avec une equation d'etat ; ii) restriction a une propagation en regime permanent, et iii) existence de surface magnetique modifiee, modification due

a

la force de Coriolis.

Nous discutons quatre situations : i) propagation de champ faible sans ecoulement permanent, ii) intensite de champ arbitraire avec Bcoulement dans un systkme axisymetrique ; iii) limite de champ faible dans le cas ii et iv. intensite de champ arbitraire dans un tore non axialement symi- trique. Sauf dans le cas no iii, nous obtenons un principe variationnel simple, semblable a celui de Seligar et Whitham, en introduisant des coordonnkes appropriees. Dans les cas i et iii, nous obtenons des rksultats explicites donnznt la modification quasi lineaire du profil.

Abstract. - We present our recent efforts to derive a systematic calcdation scheme for nonlinear wave propagation in the self-consistent plasma profile in complex magnetic-field geometry.

Basic assumptions and/or approximations are i) use of the collisionless two-fluid model with an equation of state ; ii) restriction to a steady state propagation and iii) existence of modified magnetic surface, modification due to Coriolis' force.

We discuss four situations : i) weak-field propagation without static flow, ii) arbitrary field strength with flow in axisymmetric system, iii) weak field limit of case ii) and iv) arbitrary field strength in nonaxisymmetric torus. Except for case iii), we derive a simple variation principle, similar to that of Seligar and Whitham, by introducing appropriate coordinates. In cases i) and iii), we derive explicit results for quasilinear profile modification.

1. Introduction. - During the last decade, a consi- derable progress has been achieved in the understand- ing of the basic physical effects associated with the propagation of an intense high-frequency wave in a plasma [I]. These include particle trapping, sto- chastic orbit modification of resonant particles, modification of the average plasma profile due to the ponderomotive force and the associated effects on the wave propagation, sideband excitation by parametric instabilities, formation of a high-energy tail in the particle distribution function, etc. However, most of the investigations concerning these nonlinear effects have been restricted to the case of a plasma i n a simple geometrical configuration, such as an isotropic plasma, a plasma in a uniform magnetic field, etc. The aim of the present research is to extend some of these analyses to the situations where the plasma is confined by a magnetic field of complex geometrical structure. The principal problem here is to derive a self-consistent set of general and relatively simple formulas which are valid for a large class of magnetic field configurations and which are at the same time useful for numerical analyses.

(*) Faculty of Science, Hiroshima University, Hiroshima, Japan.

Toward this goal, we have recently been developing a formulation based on the collisionless two-fluid description of the plasma [2, 31. Particle aspects are entirely neglected here and the principal nonlinear effects are those due to the ponderomotive force of the oscillating field. In addition, we make the following assumptions and/or approximations :

1) We assume a steady state propagation of the wave having periodic dependence on time ;

2) We assume the existence of a generalized magnetic surface for each species of the particle, the generalization being made in order to include the effect of Coriolis' force due to the plasma flow ; and

3) We assume an equation of state which relates

the pressure P to the density n, the generalized flux function $ and the time t for each particle species. Assumption 1) excludes the sideband instabilities ;

they are related to the problem of the stability of the steady state obtained. Assumption 3) is introduced to close the fluid equations by equations of continuity and motion. Based on these assumptions and/or approximations, we have investigated several situations which cover a large class of problems

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C6-112 T. WATANABE, H. HOJO AND K. NISHIKAWA

relevant to the thermonuclear fusion research. In all cases that we have investigated, we arrive at a sur- prisingly simple variation principle which is of similar structure to that derived by Seligar and Whitham [4].

The variation principle will be useful for numerical analyses since the differential operations contained in it are of lower order than those of the original set of equations.

Since our work is at a stage of day-by-day progress, we here present our work according to the historical development of the theory. First, in section 2, we consider the situation where the oscillating field is weak and the dominant nonlinear mechanism is the density profile modification due to the ponderomotive force along the magnetic field [2]. The plasma flow is neglected as it is unessential and the pressure is assumed to be a function of the density alone. Three important results obtained from this analysis are i) derivation of a simple general expression for the ponderomotive force for given oscillating field profile, ii) derivation of the stationary density profile obtained by the static force balance along the magnetic field, and iii) derivation of a variation principle to self- consistently determine the oscilating field profile. Then in section 3, by restricting ourselves to an axially symmetric system, we extend the analysis to include other nonlinear effects as well, that is, the modification of the plasma flow, that of the magnetic field and the higher-order nonlinear contributions [3]. Here a major progress is made by the introduction of the modified scalor and vector potentials which take into account the effects of the centrifugal force and Coriolis' force, respectively, due to the plasma flow. Use of the Clebsch representation for the velocity components then allows us to derive a variation principle similar to reference [4]. The special case of weak oscillating field in an axisymmetric system is discussed in section 4, where we derive explicit expressions or equations for the quasilinear modification of various physical quantities. An important simpli- fication is achieved here by the introduction of a generalized displacement vector or polarizability. Finally in section 5, we consider a general toroidal system without axial symmetry and derive a similar variation principle by using Hamada-like coordinates in which the magnetic field line becomes straight. This formulation will be applicable not only to nonaxisymmetric confinement systems, such as the stellarator, the heliotron, the bumpy torus, etc., but also to the tokamak subject to a nonaxisymmetric oscillating field. Brief summary is given in the last section.

2. Weak Field Propagation in Self-Consistent Den- sity Profile. - In this section, we consider the situation where the oscillation field is weak and the dominant nonlinear effect is the density profile modification due to the ponderomotive force along the static magnetic field, as in the case of a sufficiently low$ plasma. As mentioned in section 1, we neglect the static plasma flow and assume that the pressure is a known function of the density alone, P = P(n). Within these approximations and/or assumptions, the formulation presented here is valid for arbitrary magnetic field configuration, provided that there exist magnetic surfaces determined by the average magnetic flux function

$,,,

which is defined by the relation

-

B.v$, = 0 , (1) where B is the static part of the magnetic field ; it is equal to the unperturbed magnetic field, B,, since

we are neglecting the static plasma flow.

Our starting equation is the equation of motion in the collisionless two-fluid model,

where

H(n) =

1

dP(n)/n

and the other notation is self-explanatory. The suffix designating the particle species is suppressed for brevity. Denoting the time average by the bar, we can then write the static force balance as

m X E

-

(q/c) v x B, = - v ( ~ Y

+

H )

, (3) where B, is the oscillating part of the magnetic field and Y is the electrostatic potential with

E

=

-

vY.

The left-hand side of this equation is the ponderomotive force due to the oscillating field. We calculate it in the lowest order approximation, that is, by using the linear approximation for v,

where the dot denots the time derivative and the suffix 1 depicts the oscillating component. Using the relations,

v . v v = V i v l 2 / 2 - v x (V x \ v ) . (5)

and introducing the oscillating displacement vector R by the rations,

k

= v and L = 0, we get

-

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THEORY FOR STATIONARY NONLINEAR WAVE PROPAGATION C6-113

Taking the component along B,, we find after some algebra [2] the following relation,

-

B,

.

m [ z

-

(qlc) v x B1] = Bo

.

V(mI2)

{ I

v

1 +

(qlmc)

@

x B,)

.

R

)

(8)

where the second line can be derived by using eq. (4) for

We can express the polarization vector qR and the pressure perturbation, H1 = H'(E) n,, in terms of E, by using the linear approximation. Equation (9) then implies that the ponderomotive force along B, is derivable from a potential which is a known function of oscillating field.

Using eq. (9), we integrate eq. (3) along B, to obtain

where

f

is an arbitrary function of ;jr;m. On the left-hand side, is expressible in terms of by the Poisson equation and

H

can be written, correct to second order, as

K

= H(E)

+

H"@) 212. Then knowing the equation of state, P = P(n), we can in principle solve eq. (10) for ii in terms of the oscillating variables. - In the particular case of isothermal variation, i. e. P = E T (T = constant), with the ions singly-ionized and with local charge neutrality,

Fz,

= ?ii, we obtain

- -

xexp

("

4s RS.El

+C

S ~s[n,21/n~ -Rs. V(ns,l~)]

2(Te

+

Ti)

1

where N(qm) is the density in the absence of the oscillating field and the suffix s denotes the particle species. This is a highly nonlinear equation for 2, but if the oscillating field is sufficiently weak, we can approximate ii in the exponent by ~ ( q , ) .

We now derive a variation principle for the self- consistent determination of the oscillating field. To this end, we use Arnpke's equation with the magnetic field eliminated by Faraday's induction law :

where on the right-hand side we retained only those nonlinear effects that arise from the density profile modification. To lowest order, we can replace d2/dt2 by

-

a2, where o is the wave frequency. We then take the scalor product of eq. (12) with the field variation 6E1 and integrate over the entire space. Assuming a boundary condition that the surface integral vanishes, we then obtain,

The right-hand side can be written correct to second order as 121,

from which we finally obtain the following variation principle,

The form of this variation principle has resemblance to that in the fluid theory [4]. However, here the variation is taken with respect to El only, all the other variables being determined in terms of El by the linear theory and eq. (10). Therefore, our variation principle will be extremely useful for numerical analyses.

3. General Formulation for Axisymmetric System.

-

In general, an intense oscillating field causes an average plasma flow due to the ponderomotive force across the static magnetic field. Therefore, one can- not neglect the static flow 7, particularly for a strong field case. Now, if we note the relation

(9,

we find that the effect of the plasma flow can be taken into account by introducing the modified scalor and vector potentials,

x

and a , by the relations

The modification of the scalor potential is due to the centrifugal force, while that of the vector potential is due to Coriolis' force. Note that as qs/c is nothing but the canonical momentum of the fluid. Using these modified potentials, we can write the equation of motion as

where Q, is the modified magnetic field defined by

which statisfies the source-free condition,

To proceed with our argument, we now restrict ourselves to the axisymmetric case, that is, the case

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where both the plasma and the oscillating field are axially symmetric as in the case of the r. f. end plugging of plasmas in an axially symmetric linear machine, the case of the m = 0 mode propagation in a tokamak, etc. In the cylindrical coordinate system, (r, cp, z), this implies that all quantities are indepen- dent of cp. Suppressing the suffix s, we can then write from eq. (20),

where $ is the generalized flux function defined by the relation

a.v$

=

o .

(22)

From eqs. (19) and (21), we have

$ = 2 nr a,

+

S(t)

,

(23) where S(t) is an integration constant which is an arbitrary periodic function of time.

We now rewrite the equation of motion (18). First, its cp-component yields

and Maxwell-Poisson system of equations,

For the (r, z)-components, we use the Clebsch representation [I], (u, h), defined by the relations

After some calculation [3], we then obtain from the

(r, z)-components of eq. (18) the following equations :

Here the pressure P is assumed to be a known function of n, $ ant t , P = P(n, $, t), and u,, v, and v, are determined in terms of u, $, h and A by using eqs. (17), (23) and (25). Equations (24), (26) and (27), together with the continuity equation,

..

A, = c2 rV - V(rA,)

1 +

₄_nc

C

_,

_q,n, v,,

,

..

A, = c 2

iaAr

---I\- aA

a

a p

c - -

+

4 nc

C

q, ns v,,

,

a~

,

az

ar ar at .s

are the basic equations which describe the axysimmetric system. If we use the Coulomb guage, V.A = 0, and denote

equations (30) and (3 1) are combined to make a single equation,

where the go-component of the magnetic field is given by

We now show that these set of equations can be derived from the variation principle, m u , , $,, A,, A,, W, YI = 0

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THEORY FOR STATIONARY NONLINEAR WAVE PROPAGATION C6-115

Note the similarity of this variation principle to eq. (15). We first calculate 6P = (aP/a$) 6$

+

(aP/an) 6n by

using the relation (26) which yields

The velocity variation 6v can be calculated from eqs. (17) and (25) as

4

"'

) a $ + -

- - - - 6W+-v,6A,. (39)

-

v . 6 ~ = - v.VGU

-

v.V$6h

-

i

hv.V

+

--

2 nmcr. me ( r ar r

az

mc

Using the relation,

E

= -

V Y -

c - l i ,

we can calculate 6

1

E

l 2

as

A similar calculation for 6

1

B

1

yields

Using these results, we can show that SLISu yields the

continuity equation (28), 6L/6$ yields eq. (24), 6L/6h yields eq. (27), 6L/6A, yields eq. (29), 6L/6 W

yields eq. (34) and 6L/6Y yields eq. (32). Note that

eq. (26) is used to calculate 6P. If the equation of

state is given by the relation P = nT($), then we

have from eq. (26)

which is a generalization of eq. (1 1).

We conclude this section by a remark that the variation principle derived here is valid for arbitrary strength of the oscillating field, provided that the field is periodic in time and that the generalized magnetic surface defined by eq. (22) can exist.

4. Quasilinear Approximation for Axisymmetric System. - In order to draw a more explicit result, we now restrict ourselves to the wealk-field limit and derive the expressions or equations for the quasilinear modification of various physical quantities in an axysymmetric system. As in section 2, we divide the

variables into the static part, denoted by bar, and the oscillating part, denoted by subscript-1. The unper-

turbed part of the static components will be designated by subscript 0, and we now have B # Bo and G # v,,

in contrast to the case of section 2. Instead of using

the Clebsch representation, we here use the more familiar cylindrical coordinate representation.

General two-fluid formulation of an axisymmetric system without oscillating field, that is the unperturbed system, is presented in ref. [5]. The unperturbed fluid

velocity can be written in the form

where F($,) is an arbitrary function of the unperturbed generalized flux function $, which from eq. (23) can be written as

where $, is the magnetic flux function. The unperturbed force balance along

a,,

together with the condition Po = Po(no), yields,

where f is another arbitrary function of i,bO (cf. eq. (10)).

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C6-116 T. WATANABE, H. HOJO AND K. NISHIKAWA

with

0

= 0. Indeed, a straightforward calculation yields [3]

no v1

+

nl v0 = no

Q

+

V x (no Q x v,)

,

(48)

from which we obtain

n, = -

S'

dt' V.(n, v1

+

n, v,) =

-

V.(no Q)

.

These equations are similar to those obtained in the absence of the plasma flow.

Using these relations, we can now calculate the second-order quasilinear modification of various variables, E,

F,

f and

B.

We first calculate these quantities in terms of

$

and the first-oder variables, and then derive an equation to determine

$.

First we consider A and

F.

They can be obtained from the force balance along fi and the Poisson equation. Correct to the second order, the static force balance along

scan

be calculated as follows :

= - [(v, x 2 )

+

(vo x n l ) ] . n l = - (Q x G).[v x (Q x

@)I

-

c V.[f1(F)

$,I

Q1 =

-

(112) V.[(Q x

6)

x

(6

x

fL)]

-

cf"(F)

$,

v&.@

- -- =

a.

V[(Q x a ) . b / 2

+

~f

"(a

$:I , (51) - -

where we have ignored the difference between a , ,

$,

and

a,

$, as it gives only higher order corrections, and in the last two equations we used the relations

n . v $ = o ; n 1 . T $ + G . v $ , = o , (52)

which follow from the definition of the generalized flux function (22). Equation (51) can be compared with

eq. (8) ; apart from the additional term which arised from the quasilinear modification of the flux function, the right-hand side of eq. (51) is of the same form as that of eq. (8). Integration along fithen yields

-

x

+

H/q =

f(F)

+

ff'(ll/)$:/2

+

(Q x ~ ) . @ 2 c . (53)

Combining this equation with the average Poisson equation,

v 2 F =

-

4 4 x q q , n s , (54)

S

and the assumed equation of state, P = P(n, $, t ) , we can determine ?i and

Y

in terms of

$,

F, B and the first order quantities.

To calculate f and B, we first note the relation,

V.(E) = 0 and introduce the flow flux function

I'

by the relation

Using the flux conservation,

v.vp

+

v,.V$, = 0

(see eq. (24)), we can write the component of FiV across the average generalized magnetic surface in the form,

From eqs. (21) and (55), the left-hand side of this

equation can be written as

while the right-hand side of eq. (56) can be calculated,

using eqs. (46) and (52), as

-

-- 6.vl.V$,

+

n1 v,.V$ =

= n . V [ - s n ( Q x

6 )

r2 Vv(/,

-

2 ZF'($) 2 / 2 1 (58)

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THEORY FOR STATIONARY NONLINEAR WAVE PROPAGATION C6-117

This determines the poloidal velocities, % and T,, which in turn determine the toroidal magnetic field by the Ampere law :

where I is the external current. The toroidal velocity, u,, and the poloidal magnetic field or the toroidal vector potential, A,, are determined from the time average of eqs. (23) and (29) :

rV

C

>

V(rA,) = - (4 711c)

C

qs[% Gs,

+

n s us,,]

.

S

(62) Our final task is to derive an equation which determines the flux function

$.

It can be obtained by taking the force balance across the generalized magnetic surface,

-

VT-( V(x

+

Hlq)

-

V

x - v, x

a,

)

= 0 . ₍₆₃₎ Using eqs. (21) and (55), we have

where

a,

is determined from eqs. (17) and (19) as

Using these relations, one finally obtains the following equation to determine$ :

where

(2

+

Hlq)

is given by eq. (53), E is determined from eqs. (53), (54) and P _{- -}= P(n, $, t), 5, and G, are determined in terms of n, v,, n and T from eq. (55) and

r

is given by eq. (59).

5. Variation Principale for Nonaxisymmetric Torus.

- The analyses presented in the preceding two sec- tions are restricted to the case where both the plasma and the oscillating field are axially symmetric. In the r. f. heating of the axially symmetric torus, however, the symmetry is destroyed by the r. f. field. The object of this section is to extend the general treatment of section 3 to nonaxisymmetric case by restricting ourselves to a toroidal system. We show below that

by introducing an appropriate coordinate system we can indeed derive a similar variation principle. The relevant coordinate system is the one in which either the modified magnetic line of force or the plasma flow line becomes a straight line, an analogue of the Hamada coordinate. Here we only show the derivation for the case in which the modified magnetic line of force is made straight.

We start from the usual toroidal coordinate system, ($, 8, cp), where 0 is the poloidal angle normalized t o 2 n :

f

d 8 = 2 n . (67) We assume the existence of the generalized magnetic surface defined by eq. (22), which in the present coordinate system is written as

where

g-1/2

=

c(V$ X V8) . v q

.

₍₆₉₎ From eq. (68), we find

where the generating function @ can be written as

with .Z being a periodic function of 8 and cp, because the left-hand sides of eqs. (70) and (71) are to be single-valued functions of the position. Then by introducing the transformation

we can derive the following formula (see Appendix I) :

where for simplicity we assumed that G($, t ) is non- vanishing. For G = 0, one can derive a relation of the form

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that

- g($,

t) is nothing but the safety factor of the A straightforward calculation now shows that the torus. following variation principle derives the above set of

Substituting eq. (77) into eq. (19) and using the fact that a is a single-valued function of the position,

we can derive the following expression for the modified 6UTS,

ij,,

P,, A,

y1

= 0 (85)

vector potential, with

where p is a single-valued function of the position. Namely, ~ L I J ? derives eq. (8313

~ L I ~ T

derives eq- (84), Equation of motion (18) then becomes 6L/6p derives the equation of continuity, 6L/6A derives the Ampbre equations and 6L/6Y derives the I d ( $ V p

+

{ I ? d $ v g + Vp ,

1

=

Poisson equation. Note that the above derivation 2 nc at is valid for arbitrary amplitude of the r. f. field.

We now assume an equation of state in the form

In general, n can be a function of as well. This dependence, however, disappears if the modified magnetic surface is covered quasi-ergodically by a single line of force. The quasi-ergodicity breaks down when the safety factor takes a rational value. Thus the assumption (80) is equivalent to excluding the existence of rational surfaces. With this assumption (80) is equivalent to excluding the existence of rational surfaces. With this assumption, we can write

Substituting this expression into eq. (79) and equating the coefficients of v?, V q and

VF

separately, we obtain after some calculations presented in Appendix I1 the following three equations :

where f is an arbitrary function of

F,

5

and t. These equations combined with the equation of continuity (28) and the Maxwell-Poisson system of equations determine the steady state propagation in nonaxisymmetric torus. In particular, eq. (82) with eq. (80) determines the pressure profile as before and p is determined from the equation of continuity.

6. Summary. - We have derived a simple variation principle for the following three cases :

i) Weak oscillation field without static flow, 7 = 0, whence B = B,, for an arbitrary magnetic field configuration ;

ii) Arbitrary field strength in axisymmetric system using the Clebsch representation for the poloidal

velocity components ; and

iii) Arbitrary field strength in nonaxisymmetric torus using the Hamada-like coordinate in which the modified magnetic line of force becomes straight. In case i) and in the weak-field limit of case ii), we have derived expressions or equations for the quasilinear modification of various physical quantities, _{- - -} i. e.

n

and

Y

in case i) and n, Y , v and

B

in case ii). In all cases, we have assumed the collisionless two-fluid model with an equation of state (P = P(n) in case i) and P = P(n, $, t ) in cazes ii) and iii)), steady state propagation of a periodic wave and existence of the modified magnetic surface defined by eq. (22).

Although the general variation principle obtained in cases ii) and iii) is quite complicated and is nothing but an alternate representation of the basic set of differential equations, the one derived in section 2 for case i) is extremely simple since the variation is taken only with respect to the oscillating electric field. This variation principle can be generalized to include some of the finite Larmour radius effects [2] and has also been applied to investigate the steady state in r. f. plugging of plasma at a line cusp [3].

Finally, we are extending the above formulation to the following problems :

i) Quasilinear calculations for nonaxisymmetric

torus ;

ii) Derivation of the variation principle for general nonaxisymmetric system ;

iii) Problems associated with the existence of rational surface ; and

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THEORY FOR STATIONARY NONLINEAR WAVE PROPAGATION C6-119

As applications of the above formulation, we are planning several numerical calculations, including the r. f. plugging, lower-hybrid propagation, etc. This work is supported by the Scientific Research Fund of the Ministry of Education, Science and Culture in Japan.

Appendix I. - Proof of Equation (75)

This Appendix provides a proof that the coordinate stretching given by eqs. (73) and (74) transforms the generalized magnetic line of force to a straight line.

Noting the relation

n.v$

= G ~ . v + = 0 ,

we first write f2in the form

Obviously, we have

where

-

g - = (v& X v$).v*

,

(A. 4)

which can be calculated from eqs. (73) and (74) as follows :

On the other hand, use of eqs. (70) through (72) yields

Comparing eqs. (A.2) and (A.3) with eqs. (A. 6) and (A. 7), we obtain

N N

b = KIG = K(+, t ) , a = 1 ,

which proves eq. (75).

Appendix 11. - DBrivation of Bquations (82), (83) and (84)

We fist substitute eq. (81) into eq. (79) to obtain

Equating the coefficients of v$, VKand Vq, and using the relation

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C6-120 T. WATANABE, H. HOJO AND K. NISHIKAWA

Combining eq. (A. 10) with eq. (A. 1 I), we have

from which we obtain

( A . 13)

-

where f is an arbitrary function of t,b,

t

=

cp f

kz

and t. Equating eq. (A. 12) to eq. (A. 13) immediately yields eq. (82). Equation (83) can be derived from eqs. (A. 11) and (A. 13), while eq. (84) obtains by substitution of

eq. (A. 13) into eq. (A. 9).

References [I] See, for instance SAGDEEV, R. and GALEEV, A. A., Nonlinear

Plasma Theory (Benjamin, New York), 1969 ; WEI-

LAND, J. and WILHELMSSON, H., Coherent Nonlinear

Interaction of Waves in Plasma (Pergamon Press,

Oxford, 1977) ; DAVIDSON, R. C., Methods in Non- linear Rasma Theory (Academic Press, New York and London), 1972 ; Adv. Plasma Phys. 6 (A. Simon and W. B. Thompson, ed., John Wiley & Sons, New York, London, Sydney, Toronto), 1976.

[2] WATANABE, T., LEE, Y. C. and KYOJI NISHIKAWA, Plasma Phys. Nonlinear Theory and Experiments (Proc.

of Nobel Symposium on Nonlinear Effects in Plasmas,

ed. by H. Wilhelmsson, Plenum Press Co. New York and London) 1977, p. 142.

[3] WATANABE, T., HOJO, H. and KYOJI NISHIKAWA, Lecture Notes of College on Theoretical and Computational Plasma Physics (Trieste, Italy, held in March 22-April 9,1977), to be published by IAEA.

141 SELIGAR, R. L. and WHITHAM, G. B., Proc. Roy. Soc. A

305 (1 968) 1.