by spatial resonance of Alfven wavePlasma heatingLiu Chen and Akira Hasegawa

Bell Laboratories, Murray Hill, New Jersey 07974

(Received 3 January 1974)

Heating of a collisionless plasma by utilizing the spatial resonance of shear Alfven wavesis proposed ahd application to toroidal plasmas is discussed. The resonance ~xists due tothe nonuniform Alfven speed. This heating scheme is analyzed inonedimerisidn includingthe effects of a shear magnetic field and plasma compressibility. For plasmas with smoothnonuniformities (I k J) I » 1, k.L is the wavenumber prependicular to the ambient magneticfield and the nonuniformity direction, and 1 is the scale length of the nonuniformi ty), theenergy absorbed per unit surface area per driving cycle is [bo2(JLok.L)-1]. Here, bo is theflux density of the driving magnetic field evaluated at the resonant point. With sharpnonuniformities( I k j} I « 1), absorption is large if the surface eigenmode is excited. Thecorresponding value is [bo2(JLok .L)-l(k.Ll)~l]. Otherwise, it is [bO2(JLok.L)-1(k.Ll)].

I. INTRODUCTION In Sec, II we describe the theoretical model and derive

. . . . . ' . the wave equation. In Sec, III, we show the existence ofPlasma heatmg IS one of the most Imp~rtant Issues.m the collisionless energy absorption by examining the properties

success of controlled, thermontlclea~fusion. Inrarticul.ar, of the wave equation near the singular (resonant) pointheating beyond the temperattlre achIeved by OhmIc heatmg where wave phase mixing takes place, Section IV containsin a toroidal machine is a very crucial problem. I.n such a the solution of a compkte boundary value problem after

regime one promising way is to use electromagnetI: raves, the geometry and the plasma equilibrium properties are

In this regime, heating by waves must rely on collIsIOnle~s specified. An explicit expression for the energy absorption

dissipation. Ion cycJotronr~sona.n,ceheating,t lower hybrId rate is derived and then evaluated, Conclusions as wellresonan,ce hea,ting,2 ()r pa~aIIletr~(;~x:~ita.tion3 are some. of a as discussions of the practical application to toroidal plasmafew methods proposed so far,H<l\"ever1due t()the relatIvely heating are in Sec. V.short wavelengths, these methods have intrinsic difficulty incoupling the wave energy to the plasma. Here, we propose II THEORETICAL MODEL AND THE WAVEthe use of resonance of a shear Nfven wave that has a much EQUATIONlonger wavelength. . . . .,

We adopt ideal magnetohydrodynamic equatIons, For

In a nonun,iform plasma, the Alfven speed VA is afunction simplicity of analysis, we take a one-dimensional layer model

of position in the direction of the nonuniformity x, The with straight magnetic field lines. Thus, the equilibrium

shear Al£ven wave whose dispersion relation is given by properties of the plasma (mass density pm, pressure P, and

"'= k11vA,wherekll is the wavellumber parallel to the mag. the confining magnetic field B) vary only in the x direction,

netic field, meets the resonant condition at a local point Here, B(x) has a shear component; i,e"

(plane) Xo in space where the excited frequency "'0 satisfies"'0 = kIIVA(XO) , Hence, if a surface magnetohydrodynamic B(x) = Bz(x)ez + By(x)ey. (1)wave is excited by ,an externaL coupler, the wave will bephase mixed by this resonance and its energy will be dis- (In a toroidal plasma, x, y, and z correspond to radial,sipated totheplasma,Thesurfacewavehe~em~ans.acutoff poloidal, and toroidal directions, respectively.) P(x) andelectromagnetic wave whose wa,veequatIon IS gIven by B(x) satisfy the equilibrium conditionV2<p == 0 «p is the magnetic potential), and not a surfaceeigenmodeoftp.e pla~ma. We use the su~face wave rather d/dx (P + J32/2p.o) = O.than the magnetosonlc wave- as the couplIng wave becausethe magnetosonicwave.willpropagate t~rough the pl~sma Linearizing the standard set of magnetohydrodynamicand may produceundesirable~fJ:~~~.s. rhI!;~a.y,j:)e achIeved equatiolls, from the equatiop- of motion we haveby choosing a large kJ., wave .nmnber perpendicular to theden~ity gradiynt ;as well as to theambientmagne~ic fie.ld oPm~ - (BoV)2g = -p.oVp - B{BoV)(Vog), (2)(kB m the cylIndrIcal plasma), stIch that k,J.VA > "-'0 IS satls, P.

fied for . t. he.m . inimum .. ..Alfven spee.d.The e.. X

. istenc .e.. of Alfv . en h ~ . hfl ' dd ' ..l . .. . .t. . td ."' p+bB/. .. " , ... ... dd ' w ere ~ IS t e Ul ISp acemen vec or an I:' = ° p.o

wave resonance m?OnUllIfor~ ld,eal .ma~netohy ro ynamic is the total erturbed. ..'r~sstire. b, the perturbed magnetic

plasmas as

. wel .l as ItS po.. t. e. ntial a,pplIcatIOn to .plasma heat. fild. ' l Pd . . ~ h P hthMll t '

. .. .. . . .. ' h e IS re ate to ~ t roug e axwe equa IOnsing have been noted by Grad.4 AnalytIcally, the wave p ase ,... .. I

mixing takes place bec<Lus.e the shear Alfven wave has a B ( 3)

continuous spectrum in a nonuniform. plasma. The cor, b = (BoV)g - B(Vog) - (goV) .responding energy absorption rate, however, has not been . . .". . .

calculated untjlrecently.'fp.e cal~uJa~ions ofGro~slnaI1.n ~on:bmIn,g t~e adIabatIc equatIOn of state an,d the con-

and Tataronis5 are based 6n the)ncolllpressibility assl1mp- tmUltY,equatIOn, we can express the perturbed plasma pres.

tion and do not include shear'magnetic field. Both restric. sure P m terms of g, as

tions are removed in our work,6 Therefore, our results shouldbe applicable to toroidal plasmas such asin tokamaks, p = -~zdP/dx - ')IP(Vog). (4)

1399 The Physics of Fluids, Vol. 17, No.7, July 1974 Copyright@ 1974 American Institute of Physics 1399

~~

= !J.odpjdx.

~

Here,

a(x)

and

(3(x) = IJ.oI'/B2.,

We further assume that theplasmaisofiow{3(r{3.,« 1) sothat ClJO2» 'Y{3kI12VA~aJ;ld a(:r) ,,"""',1 T'Y{3,""", 0(1) .S)lbstitut-

ing Eq~. (6) and (8)intoEq. (7), wet]:J.eJ;larrive at thewaV~ equation for ~'"

d ( eaB2 d~" ),, .-'-. ", '7 - Eh = O.

dx ak.1.2B2"";Edx

= 1 f "{{3(N2j «(N2 - "{{3kIi2VA2)

Equation (9) contains the shear Alfven,sonic, and theioI1.-ItPiJUsticwave&, . . theties, these three" .' .

a singular solutionThis singularityhence, the energysipated energytrum as in thethe process is.time. In "could destroy thewill be converted into'

III.

dW = !LlILzRedt 2

1"'2

J.~'!'.d~."'1,

Here, £11 and L. are the size of,the pla,sma, in they and it di-recti()fis, Xl and Xz (Xl < X2) are t):J.yboundaties of the plasmain the x direction, J and E are related to § and bthrough

14JJ Phys. Fluids, Vol. 17, No.7, July 1974

~~

(11)

~~

~

~

(12)

~

Near X.= Xc, E(X) """'(dEr/dx)~o(x -xc) + iEi. Here,Er = ReE andEi := ImE,Since.E(';t), , 0, the wave equation(9) can b,e approxiniated as . ... .

d2~

+ . 1 d~~' k2t - 01..2 .

+ '."-d - .L "'" - ,(~- X-Xc ~u X

~

(13)

whert~~' = Ei(XQ)/(dEr/dX)Xo. In the limit~' ~O, Eq. (13)has aJogaritlmiic singularity at X.."'" xo.This singularityhas <llso.been noted byPridmo.re-Brown for a cold plasma;7Thus, nearx=;to we have

~"""" C In(x -,. Xo + io').(9)(14)

~~

(15)

(10)

1400

~~

~

PLASMA (X>O)

space x > O. Furthermore, we take E(X) to have linear pro-files (Fig. 1),

0 ::; x ::; a,E(X)

x~ a.

Here, K = (En -EI)/a, EI = -kIl2(0)B2(0),'EII =w2J.LOPtn(a) -kIl2(a)B2(a) , and Er(Xo) = O. TheexterIla:l 'dtiying sourceis represented by a sheet currentloca.~~d in the Vacuum(x < 0) at x = -h(h > 0),

= IolJ(x + h) exp[i{kllY k.z - wt) J.J.(x, t)

Corresponding to toroidalplasll1as, We have! kyl , , (minorradius)-l »Jk~.l~(m<\-jor,radil.ls)-\ IE.I » I Byl anda(x) , , 0(1) forlow-,8 plasmas. Therefore, kJ.2(X) ~ky2

and. I E I N I kll2B21 « I akJ.2B21. Equation (9) then be-comes approximately

d2~", dE/aX d~- + ~ - - ki~ = O.

dx2 E dx

Equation (18) describes the coupling betwe~n the (cut-off)magnetosonic wave and the shear Alfven .wave in a non~

uniform, 10w-,8 plasma. Using the E(X) given here .and the

boundary condition that ~'" vanish at x = 00, we have the

following solution for x ~ a:

~z(ll)(x) = C1exp[ - I ky I (x - a)].

For 0 ~ x ~ a, we use the normalized variable

IX i= I kill (x + Ell K) == E(X) I kll IlK\",j

and Eq. (18) becomes

d2~",(l) . 1 d~",(I)dX + X dX -~.

Equation (20) is theorder modified Bessel

~z(l)(x) = C1[DIlo(X) + D2Ko(X) J.

The two constants, Dr andD2, are determined by two match-ing conditions at x == a: ~,,(I)(a) = ~:i:(II)(a)andfrom

. d~ (I) d~ (II)p(I)(a) = p(II)(a}, ax (a) =~x(a),

Dl = Xa[Kl(Xa) - Ko(Xa)]

Phys. Fluids, Vol. 17, No.7, July 19741401

and

D2 = Xa[Il(Xa) + Io(Xa) J. (23)

Here,Xa=lkIlIEU/K.

FIG. 1.. Model used inthe boundary value cal-culations. E(X) =",2p.OPm(x) - kIl2(X).B2(X).

As toCh it is related to the external driving current

through two boundary conditions at the plasma-vacuum

surfa<;eatx ="IO,~Inth~ vacuum, one can neglect the dis-

placement current and the wave magnetic field b. can be

decomposed into two parts: b. = bl + b2. Here,

J. represents the drivingsheet current.

bi = (blx, bIy, Qh)

corresponds to the driving field; i.e., V x bl = /-101. andb2 = (b2"" b2y, bh) is the induced field; i.e., V x b2 = O.With b2 =VIfi2(X, I), we have from v.b2 = 0 and that b2vanish at X = - 00 :(16)

b2 = 1[;2"\1 exp[i(kyY + k.z - ikzx - wt)], (24)

where k", = (ky2 + k.2) 1/2 ~ I ky I and i[;2 isa constant. Also,since J. is located at x = - h, for x > - h we have

bl> = i[;1'IV exp[i(kyY + kzz + ik",x - wt) J. (25)(17)

The first boundary condition at x = 0 is the continuityof the normal (x) component of the magnetic field;

bl:l?(O) + b2,,(O) = [e".v x (~ x Bi)]z=o. (26)

Here the subscript i denotes quantities inside the plasma.Using Eq. (24), we obtain

(18)

{;2 = [~kll(O)B(OH",(O) - blz>(O)J/k",. (27)

The second boundary condition at x = 0 is the pressurebalanc;:e condition;

[-'YJ.toP(v,~) + Bi'ba.=o[ ( dB 2 dB .2)]= B.. (bl + b2) + H", _ I ' - -

d ' . (28). (.X.X "'=0

(19)

Subscript 'V denotes quantities in the vacuum. Using Eq.(4) and the equilibrium condition, this boundary condi-tion can be simplili~d to

iJ.op(O) = [Bv. (bI + b2) J",=o. (28')

(20) Equation (zS') indicates that the coupling is achieved bya J )( B force which drives a magnetic compression in the

eroth- plasma. From Eqs. (6), (8), (24), and (25), Eq. (28')

becomes

[ E(X)(XB.~(d~!dX) ] . =ikll(O)B(O)[~l + ~2]. (29)akJ.2B2 - E(X) z-o

Noting E{O) =; .EI = -kIl2(O)B2(O), a(O) = 1, k.J.2 ~ ky2Ik.J.2B21 »leII,Eq. (27) and that ~l = -bb>(O)/k:., weobtain, £tofu Eq. (29),

. . . [ k~ d~~]$kll(O)B(O) . ~~(O) . - - . - (0) = 2b1z>(0). (30)

. . ky2 dx(22)

L. Chen and A. Hasegawa 1401

Substituting Eq. (21) into Eq. (30), we find with C' == C1D2andk",:::::.lku/,

iC'kll(O)B(O)r = 2blz>(O),

where

The energy absorption rate then can be obtaiJ).ed by sub-stituting the above results into Eg. (12). Let us note thatwitp. Wi ~ 0+ and,," = ilET/ d:l; > 0,

[ d~ ] 10, x > xo,1m -= ~z* = (34)

dx' 1I"IC'12K/IE(X) I, x<io.

With X2 > :1;0 and Xl = Xo -1], we obtain the fpllowingrate

for energy absorbed near the singularity at X ,= :110:

dW1 - ~ L L I C' 12, K, (3 -)- 11" y z , .:>

dt 2 #0 a(xo)k.L2(xo)

"'~

LL IC'12 K (' 3-' )- 11" y . . .:>2 - LlO a(xo)ki

As one can expect, because K = I dET/dx Izo and from Eg.

(21), I C' 12 = I C 12,Eg. (35) is identical toEg. (1S) derived

in the preceding section. The total energy absorption ratethen is obtained by taking ,~ > Xo and Xl = 0, and it is

dW =~7rLyL~ICI12. K

dt 2 ua kJ..2(O) + kI12(O)

I C/ 12 K

CUo

LL .. . ---,..'

~ - 7r y .'~

k 2

2 p.Oy

(36)

= 211"wo(LyL./1 ky D(I ky I Xo)-l[lbi,,>(O) 12/JLoJ(11'.1)'-2,

( 36')

where we have used Jj;q. (31) fo.rj C',lr~J'Hl:wit}lj fly 1»1 hi

uoloI b1,,>(O) I ~ - exp[ -I kg Ih]. (37)

2

CQmparing Eq. (35') with Eq. (36), it is inteJ;estipg to. nQtethat within Qur assumptiQns most Qf the energy !j.ps(.)!.:ptiQn

{1/[1 + 'Y{1(xo) ] fractiQn Qf the tQtal energy absorbed,

'Y{3(xo) «1} Qccurs near the singularity. Thus, the smallbut finite (3 seems Qnly to. spread Qut the abso.rptiQn regi()l1

and has little effect Qn the tQtal energy absQrptiQn rate.

NQw, let us evaluate I r I which depends Qn the value

Qf I kyll . Here, l"" DCa) is the .scale lengtbQf the 11Qn-

unifQrmity. When the plasma 11QnunifQrmity is smQQth

1402 Phys. Fluids. Vol. 17, No.7, July 1974

(I kyll» 1), then

I Xo I = I kyxo I» 1,

(31)IX-a 1= Iky(a-xo) I» 1,

and we have ft<)Ill Eqs. (32) and (33):

I r I ~ (211")1/2(1 ky I XO)-1/2 expel ky I xo].

The corresponding dW 1 dt is

dfV .{ l.I.rlo } 2 dt~CJ)o(LyL./1 ky I 2"" exp[ - 1 ky I (h + xo)] 1/1.0

= CJ)o(LyL./1 ky i) Ib1,.>(xo}'1211L0,

With a sharp nonuniformity (ikilll« 1),I Xa I « 1 and

I . .[ . (2x. o..- a) . rl~ - ~11"

I ky I xo(a - xo)

~

(38)

~~

(39)

then I Xo I ,

(40)

(41)

I r I has its minimum value I r ImiD ~ 'II" and dW /dt becomesapproximately, :

dW 2

-~-"'o(LyLz/1 ky I) I kyll-ll b1y>(O) 12/~o. (42)dt 11'

That dW I dt is large when the. nonuniformity is sharp and"'02 ~ ",,2 can be readily understood by noting that "'. cor-responds to the frequency of the weakly damped surfaceeigenmode.9 For either O<Xd«!aor a:> Xo» la, I r Iis approximately. .

Irl~[ 2xo-a I 1kllxo(a - xo) ""' I kIll! ' (43)

and

dW

Tt""' 271""'o(LyLzl) I blo?(O) 12!,uO. (44)

~

V. CONCLUSIONS AND DISCUSSIONS

We have proposed a sch,yme of heating a collisionlessplasma by utilizing the spatial resonance of the shear ;'\lfvenwave. The existence of the resonance is due to the non-uniform;'\lfven speed. Waves excited by the external source

will then be:ph,!.tseIllixed at the resonant point xo, where

"'0 = kll(xo) V,1 (xo). For a 10w-/3 plasma, we have analyzed

this scheme in One dimension. including a shear magnetic

field!.tnd plasma compressibility. A model boundary value

p).'()1;>leJJ;ljs solv!':d, and an expression of ~he energy absQpp-tion r!!,te is derived and evaluated in the various limits. Wehave found that tbe small but finite fJ only spreads out theabsorption region but bas little effect on the total energyabsOl'ption. For plasmas with smooth nommiformities

(I k.LII» 1), the energy absorbed per unit surface areaper driving cycle is [bo2(/-Iok.L)-I]. Here, k.L is the wave-

L. Chen and A. Ha~egawc;l 1402

FIG. 2. Schematic diagram of proposed setting of heating coil usingshear Alfven wave resonance. Many turns in the poloidal direction areshown to emphasize the desirability of kJ. > kll to make the magneto-sonic wave cutoff. However, in the real system even if 111 '" 1 or 2, thiscondition is satisfied.

~

number perpendicular to the ambient magnetic field andthe direction of the nonuniformity, 1 is the scale length ofthe nonuniformity, and bo is the flux density of the drivingmagnetic field evaluated at x = Xo. With sharp nonuni-

formi ties (i k III « 1), the surface eigenmodes can beexcited and the corresponding energy absorption is[bO2(,uokl)-I(kll)-I]. Otherwise, it is [bo2(,uokl)-I(kll)].

The above results oqtained in a planar geometry are ex-pected to also be applicable, at least qualitatively, tocylindrical geometries, such as toroidal devices. The onlyquantitative modification may be that in this case I bo Idecays as r-m rather than exponentially. Here, r is theradius and m is the poloidal azimuthal mode number. Note

m can be of the order of unity and, hence, I bo I can have

sufficiently large magnitude. Furthermore, since we use

the cutoff mbde with respect to the magneto sonic as well

as electromagnetic waves, there is no radiation loss. Con-sequently, the only loss is due to Ohmic loss in the coils andhigh heating efficiency is expected.

The above results can be compared with heating by transittime damping,lO which also uses Alfven waves. Aside fromnumerical factors, the above expressions give heating rateslarger than those of transit time damping by a factor of 1/{3.

~

In practical applications we must consider the following:

1. Prevention of plasma loss: Although the use of a wavewith a long wavelength provides an easier coupling andmakes possible the use of a cheaper power source, it mayprovide a large scale perturbation in the plasma, causingundesirable loss of the plasma. Use of the surface wave (nota surface eigenrrlOde) rather than the magnetosonic wavewill substantially reduce this problem.

2. Uniform heating: To prevent localized heat deposit,the frequency must be swept so that the resonaIlt condition

"'0 = kIIVA(X) is satisfied for major portions of the plasma.

3. Coupler design: Because one cannot place a metalboundary inside the plasma, the standing wave in the par-allel direction, which is essentially needed for the resonantcondition, must be provided by the coupler design. One wayis to provide a periodic coil in the toroidal direction woundparallel to the toroidal axis as shown in Fig. 2.ky must bechosen smaller than 1/ Pi to prevent finite Larmor radiuscoupling between shear and compressional waves.

4. Coupling to electrons: The present method uses one-fluid approximation of the plasma. Hence, the dissipated

1403 Phys. Fluids, Vol. 17, No.7, July 1974

energy presumably goes to the ions. However, if a chargeseparation is produced parallel to B, electrons are involvedand may be heated. This may be avoided if the wave fre-quency is chosen to be higher than the electron drift wavefrequency, kJ.VTe2 d lnpm/dx/wce.

S. Effect of finite ion Larmor radius: When the wave-number in the x (nonuniformity) direction becomes large

near the resonant point, the finite ion Larmor radius modifiesthe dispersion relation of the shear Alfven wave. In thelowest order, the modified dispersion relation for T. ~ Tiis given by

W2~ k112vA2[1 + 2(k,? + k.2)p;2J.

This dispersion relation shows that the wave is cutoff in thelower density side of the resonant point, "'0 = kllVA (Xo), but

is propagative in the higher density side. This produces amode conversion to the standing shear Alfven wave in theradial direction which is trapped within the resonant column,similar to the case of the Buchsbaum-Hasegawa resonance,uThe details of the mode conversion and the associated heat-ing rate will be published elsewhere.

ACKNOWLEDGM ENTSThe authors appreciate discussions with S. J. Buchsbaum,

D. E. Baldwin, R. W. Gould, W. Grossmann, and J. Tatar-onis.

APPENDIX: DERIVATION OF EQ. (12) USING

POYNTING VECTOR

In terms of the Poynting vector P = (1/ P,o) E* x b,dW / dl can be written as

dW = tLyLz Re[P(xl) - P(x2)}ex. (A1)

dl

Since E = iw(g x B), we have Ell = 0 and

EJ. = -iwo~xB. (A2)

Putting Eq. (A2) into Eg. (A1), we obtain

dW = ~ ~ LyL. Reli[(Bbll~x*)Xl - (Bbll~x*)X2JIdt 2 !La

1 "'0 L ( *)= - ~ lIL. 1m Bbll~x Xl'" (A3)2 iJ.o

which is identical to Eq. (12).

1 T. H. Stix and W. R. Palladino, Phys. Fluids 1, 713 (1958).2 S. Puri and M. Tutter, Nuc!. Fusbn 13, 55 (1973) and references

therein.

3 W. M. Hooke and S. Bernabei, Phys. Rev. Lett. 29,1218 (1972).4 H. Grad, Phys. Today 22, December (1969), p. 34.5 W. Grossmann and J. Tataronis, Z. Phys. 261, 203, (1973),261,217

(1973) J. Tataronis, Bull. Am. Phys. Soc. 18, 1286 (1973).

6 A. Hasegawa and L. Chen, Phys.Rev. Lett. 32, 454 (1974).7 p. C. Pridmore-Brown, Phys. Fluids 9,1290 (1966).8 B. B.Kadomtsev, in Review of Plasma Physics (Consultants Bureau,

New York, 1966), Vo!. 2, p. 153.

9 L. J. Lanzerotti, H. Fukunishi, A. Hasegawa, and L. Chen, Phys.

Rev. Lett. 31, 624 (1973). L. Chen and A. Hasegawa, J. Geophys.

Res. 79,1033 (1974).10 F. Koechlin and A. Samain, Phys. Rev. Lett. 26, 490 (1971).11 S. J. Buchsbaum and A. Hasegawa, Phys. Rev. Lett. 12, 685 (1964).

L. Chen and A. Hasegawa 1403