J. Plasma Fusion Res. SERIES, Vol.2 (1999) 73-76

Bounce-Resonant Wave-Particle Interactions in Tokamaks
with Elliptic Magnetic Surfaces
GRISHANOV Nikolay I., DE AZEVEDO Carlos A. and DE ASSIS Altair S.'
Universidade do Estado do Rio de Janeiro, RJ 20550-013, Brazil
1(.Jniversidade

Federal Fluminense, Niterdi 24024-000, Brazil(Received: 8 December 1998 / Accepted: 2O May 1999)

Abstract
The longitudinal permittivity elements of a toroidal plasma with elliptic magnetic surfaces areevaluated by solving the drift kinetic equation as a boundary-value problem for waves in the frequency
range much larger than the drift frequency. The quadratic corrections over the inverse tokamak aspectratio are taken into account in order to approximate the equilibrium magnetic field. The separate
contributions of untrapped, /-trapped and d-trapped particles to the longitudinal permittivity are writtenby the summation of bounce-resonant terms including the well-known plasma dispersion function.

Keywords:
bounce resonance, elongated tokamak, longitudinal permittivity

1. lntroduction

and the "magnetic moment" It = tin'y,[HTiETlH
instead of the pitch-angle 7(u11 = u cos% ur = D sin/).

The problems of plasma heating and current drivein large-size tokamaks by using the radio-frequency

Here, Hqs and 11e6 are the toroidal and poloidal

waves have stimulated renewed interest in the study ofthe dielectric properties of a magnetized toroidal plasma

projections of an equilibrium magnetic field, trl, for agiven magnetic surface, at the points 0 = +n12. The
modulus of .El is

taking into account the ellipticity/triangularity of themagnetic surfaces and the bounce-resonant wave-

H={Ir'ro*rI'* ^n + lco*

particle interactions there. In this paper, we evaluate theI +€cos0'

longitudinal permittivity elements of a collisionlesstoroidal plasma with elliptic magnetic surfaces. In
contrast to Refs. [1-3] related to it, we describe theH'ro + H'zro(,\untrapped and three groups of trapped particles in thegeneral case, when all these particle groups exist at theconsidered magnetic surfaces, accounting the quadratic-,"o"e'*-etl+e"or'o'*4.or'2 o"l.

JHereE=L. o =e!!.
R-hs

corrections over the inverse tokamak aspect ratio. To
solve the drift kinetic equation for the perturbed
distribution function/=/( p, 0, Ittr, ur) exp(-iat + inQ)of charged particles we introduce the new variables (r,0') instead of the quasi-toroidal coordinates (p, 0)r=9gr'l=--e' f r:-^=4(4\4' J S'+e' \o' ')Hen,
lls=---.

llO=HroH '^^+HZ
1/ Hro+Hto
rr+ Hiowhere R is the major tokamak radius; b and a are thoa2.
-;
slllb"@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e -mail: grishano @ uerj.brIJ

Grishanov N.L et al., Bounce-Resonant Wave-Particle Interactions in Tokamaks with Elliptic Magnetic Surfaces

And really, p(6) as the solution of u11 (1r, 6) = 0, in thegeneral case, has five extremums where 0 = +Tt, 0,* arccos (e/L). To have the full physical picture, we
derive the contributions of all four groups of plasma
particles to the longitudinal permittivity of a toroidal
plasma with elliptic magnetic surfaces. As a result, thephase volumes of the different groups of plasma
particles are defined by the following inequalities:

major and minor semiaxes of the elliptic cross-section of

the external magnetic surface. In this model, allmagnetic surfaces are similar each other with the sameelongation equal to b/a. Assuming that € and L ate

small, the perturbed distribution function can bepresented in the form

f<h=1f "1flexpIirq rin e(e +("o,e)],'lt

where 6 = 0'+ 0.125), sin 20' and ( = 0.25h - O.5*lQo<p<1t", -n<6<n+ €).

-for untrapped particles (zone l);

As a result, for f" we havea7

ff+iQ,t017,=Q(6),lt"3 F3 N,, -0.< 6 < e.

-for t-trapped Particles, (zone 2);(1)

It,3 tt3 Fd, -8< 0 < -0owhere-for d-trapped particles, (zone 3);o,(n)=nslt.99]'\ t+eJ

Fr3Falta, ed<e<0,-for d-trapped particles, (zone 4),sa (l +0.251)

where the reflection points Q and 0a for the trapped

4u@'particles are defined by the relationships/ "**lQQ)=#knT lr\ 4/\eE,(O\tcoS o, =exp l- inq sin 4 (s + ( cos Dl Fo ,v=L++,-\+"^of-g')" (ttu+)''2 l+e' ko=*, Fo=- ,rlcosgn=t[,*u+J

The steady-state distribution function Fs is given as
maxwellian, where Ns is the particle density, and Dt =IIu='ltTlM is the thermal velocity of particles withtemperature ? and mass M. By indexes s = +l forf, wedistinguish the perturbed distribution functions overpositive and negative values of parallel velocityD',

boundary condition for the trapped and untrapped
particles. For untrapped particles (zone l), we use theperiodicity of fr over 9. The boundary condition for therelative to II. After solving Eq. (1), the longitudinalcomponent of the current density j1 = i.H/H can bet-trapped (zone 2) and d-trapped (zones 3 and 4)
particles is the continuity of f" at the corresponding

expressed by3l

f6"*rl*nsin'JOJO

stop-points (local mirrors or reflection points) where the0 (e + (co, e)]parallel velocity is equal to zero. As usual, the equalityof denomenators of l, to zero gives us the bounce
resonance conditions of an effective wave-particle
interactions. Using Eq. (2), it is possible to evaluate theseparate contributions of all groups of plasma particles
to the longitudinal current density component.

(1-ecos0+vcos'6)-'r{l-rcosd+vcoszd)-r)r I u'|+€I +tlrt =I +0.51. (1 + e) 'l+Qe+0.51)(1+e)'
z11l +€)+4e2
Fa=210+e)+3e2-e3

The solution of Eq. (1) can be found by the specific=su@j u(0)= n,*lt -_f ,(0, p, u)dpdu Q)

Depending on the parameters p and 6 the phase volumeof plasma particles should be split in the phase volumesof untrapped and trapped particles. This separation canbe done by the condition u1 (#, 6) = 0. In contrast to thetokamak with circular magnetic surfaces, for a plasma
torus with elliptic magnetic surfaces, there is possible

2. Longitudinal Permittivity Elements
The dielectric tensor elements can be derived after
the Fourier expansions of current density and electricfield over the angle 6:+_

the existence [-3] of two additional groups of thej,,(4(l +ecos 0-vcos20)= 2 if'exp(m0),trapped particles at such magnetic surfaces where e < i,.
74

Grishanov N.l. et al., Bounce-Resonant Wave-Particle Interactions in Tokamaks with Elliptic Magnetic Surfacesn, (0) =^E *Ef') exp

elliptic functions [4,5].
The contribution of r-trapped particles to thei*' h,Thus, the harmonics 7,(i) and nfi'' are connected by thelongitudinal permittivity is

longitudinal permittivity elements:# iir'= Etr€tr'n'E\n')m =-6si/t= L letr;tr'"' +€tr;i,' +eff;!'"'" II E',i';J^'l*\where sft'f' , ef'f' , and ei:T'are the separate
contributions of untrapped, t-trapped and d-trapped

cll-u =Dr=

^LlI

k'zou'zrnt n Jo

Irfu+nq+p)2* 2"3 + 2i tftu)w(u,lf e; o;*"*p^-^

2{iot(t +0.2s1)K@4)
kopDr n f€tp' + 4vlt (l - lt)lt'o 'Bi6t=zai. s l-'(t +0.51)K(rf)+q fltu

ferp, + 4vlt (l - lt))""fr+24+zind,w(u,)fdu, @)particles, respectively, to the longitudinal permittivity(the details see in Refs. [,3]).
The contribution of untrapped particles to thelongitudinal permittivity is^ -,.

(1+0.s2)K(rf)Bi Bi" P" p2

t zK\!!ft- 2i en + nq) y, (rc, w)+2inq(, (4 w)ldp, (3)cn (r-r, w)dw

| + p dn2(K t, w)

Here, we used the following definitions:

uo=o(t +0.25L)^/2rcrc67
k olm + nq * plr, n fe2tt2 + 4vtt (l -ttiEV,=Yrcp)lo"^ 4EN^e2A)'-=:,MIK(K\Ai Qc ) =J

^fIi",",.*o['.

ert a avlt 11 a1'nftar(")- -+ 2inq\r,(r, w)l2i Qn + nq) fi, (r, w)

dnz (rc-l, w). 2a?"(r +o.il.)

eii:! =

E#t# 0,,,4

ki" u2, n3*ltl,i.' LP=t p- Jltt

.2r'ze +2itrt)wosl,

K2K(K)lr;/r-"ri,t'a'@;)". r; (r;).]dtt, (5)V t - tt - v) tt+ tl ertP + 4vtt (l - tD

l+g94.|,,2KG)'
cn$,fi)-3

wherea(r +0.25D{-zrcx1r1\tmf
sn (r, w) cn (( w)(,(K,w)=tl | + P|. r"\.)r)

k o p ur n let p' + 4vtt (lB

['.

w)- I

The contribution of d-trapped particles to theK6\=l'''
Jo -4-"t 2.+| + P

longitudinal perminivity is

4vn' -, -zv + 1f e\ +v 1t - py 1t'r,(r,w)=arctantuut.-'| + B dn2 (r-t, w)

B7dn2 (K-1,2l e'p' + 4vp (l - tt)
I - (l - v7 1t +sn (K-r, w)dn (r-', w)oi<4=t Q+F)cn2(r,w)-ll+Bcn21rc,w)[-"1.,exp_75'iffi-iQn +nq) )6a (r,vv) ++inqt{4 (r. w) I

where sn(r, w), cn(r, w), dn(r, w) are the Jacobi- p)l''o/5 cn (r, w) sn (r. w)^(r, w)d+dn2dw,

Grishanov N.l. et al., Bounce-Resonant Wave-Particle Interactions in Tokamaks with Elliptic Magnetic Surfaces

6- 2v-€-tr€\4v(l-pYp
2v+e+ e2 + 4v (I - lt)ltl
Ian',".

longirLrdinal permittivity elements, Eqs. (3)-(5), are valid

in the wide range of wave frequencies, wave numbers,
and plasma parameters in the tokamaks with an elliptic'r- d]

cross-section of the magnetic surfaces.

Xog.w)=arccosl------------]:----]-------:-'-l .

ldn'(r.
znfd dn{cU/u(K.wl='

Since the drift kinetic equation is solved as aboundary-value problem, the eff.'i.,.0 -elements are
suitable to study the wave processes with a regularw)+ alr; [dn2 tr. wl+ d-dn'(rc,r)- d]lt+(-l.
-|frequency such as the wave propagation, plasma heating
and current drive, when the wave frequency is given bydn'1r. w)+ dl

The phase coefficients Atr,Btr,and Di do not

the antenna./generator system. The expressions (3)-(4)have a natural limit to the corresponding results [3,5] forplasmas with circular magnetic surfaces if b = a and v, (+ 0. Note, the quadratic corrections over t = rlR are
important to describe more correctly the finite-e effects
in tokamaks with alR - l/3 and b > a. Tbe concrete
computer calculations of eii'''-.t.-ents evaluated inthis paper have been done in Ref. [8], where the kinetic

depend on the wave frequency that allowed us to obtainci|!'

the final expressions for eff.'i' , eff;i' andby theplasma dispersion function W(z) = exp(-221 1t + 1Zit"[i1lI explt\at]. As a result, the bounce-resonantconditions can be presented by:
up =| for the untrapped particles, where the numbers ofbounce resonances are p - 0, +1, +2, ...;
up = l, p = 1,2,3, ... for the t-trapped particles;and r, - l, p = l,2,3, ... for the d-trapped particles.

Alfv6n wave dissipation (by the trapped/untrapped

3. Gonclusions

Acknowledgments
We are grateful to F.M. Nekrasov and A.G.

electrons) is analyzed in an elongated tokamak with themain JET-parameters.

The basic feature of elongated tokamaks is apossible existence of three groups of the trapped

Elfimov for useful discussions. This work was supported
by Fundagio de Amparo d Pesquisa do Estado do Rio deJaneiro (FAPERJ) of Brazil.

particles. Together with the usual (for tokamaks withcircular magnetic surfaces) untrapped and r-trappedparticles, two additional groups of d-trapped particles
can appear at such elliptic magnetic surfaces where 2 >g. In other terms this criterion can be rewritten astll

References
F.M. Nekrasov et al., Czech J. Phys. 46, 565(1ee6).

b/a> l+€+q2/e.

t2l V.S. Tsypin et al.,Phys. Plasmas 4,3635 (1997).
[3] N.L Grishanov, C.A. de Azevedo and A.S. deThe d-particles are located near the points 0 = * arccos(e l),). The trajectories of untrapped, r-trapped and dtrapped particles are entirely different. The energetic dparticles (e.g., the fusion-born alpha particles) can excite

Assis, Phys. Plasmas 5, 705 (1998).

t4l M. Abramovitz and I.A. Stegun, Handbook ofMathematical functions (Dover Publication, NewY ork, 197 2), p. 555-587.
t5l N.I. Grishanov, C.A. de Azevedo and A.S. de

the toroidicity-induced [6] and ellipticity-induced [7]Alfv6n eigenmodes. Moreover, these new nature
particles can affect substantially the current drive

Assis, Phys. Plasmas 4, 1055 (1997).

[6] C.Z. Cheng and M.S. Chance, Phys. Fluids 29,generation and transport processes. Of course, choosing

the level and profile of an equilibrium current (or bygiven the suitable radial structure of a tokamak safety
factor, q), it is possible to realize the experimental
regimes when the d-particles are absent into the plasma.
As easily to see, the criterion I > e is not satisfied intokamaks with small elongation bla < 2 and q > 1. Thismeans that there are no d-trapped particles in the JET(as well ITER) plasmas.
The bounce resonance conditions for each of fourgroups of plasma particles (both the electrons and ions)are derived by solving the drift-kinetic equation. The3695 (1986).

t7l R. Betti and J.P. Freidberg, Phys. Fluids 83, 1865
(1991).

t8l N.I. Grishanov, C.A. de Azevedo and A.S. deAssis, Kinetic Alfv6n wave dissipation in tokamaks
with elliptic magnetic surfaces, accepted in Plasma
Phys. Control. Fusion (1999).

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