Excitation of Alfvén cyclotron instability by charged fusion products intokamaksN. N. Gorelenkov and C. Z. Cheng Citation: Phys. Plasmas 2, 1961 (1995); doi: 10.1063/1.871281 View online: http://dx.doi.org/10.1063/1.871281 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v2/i6 Published by the American Institute of Physics. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Excitation of Alfven cyclotron instability by charged fusion products in tokamaks

N. N. Gorelenkova) and C. Z. Cheng Princeton Plasma Physics Laboratory, Princeton University, P. O. Box 451, Princeton, New Jersey 08543

(Received 1 June 1994; accepted 7 March 1995)

The spectrum of ion cyclotron emission (ICE) observed in tokamak experiments shows narrow peaks at multiples of the edge cyclotron frequency of background ions. A possible mechanism of ICE based on the fast Alfven Cyclotron Instability (ACI) resonantly excited by high energy charged products (a-particles or protons) is presented here. Two-dimensional eigenmode analysis of ACI mode structure and eigenfrequency are obtained in the large tokamak aspect ratio limit. The ACI is excited via wave-particle resonances in phase space by tapping the fast ion velocity space free energy. The instability growth rates are computed perturbatively from the perturbed fast particle distribution function, which is obtained by integrating the high frequency gyrokinetic equation along the particle orbit. Numerical examples of ACI growth rates are presented for Tokamak Fusion Test Reactor (TFTR) [World Survey of Activities in Controlled Fusion Research [Nuclear Fusion special supplement 1991] (International Atomic Energy Agency, Vienna, 1991)] supershot plasmas. The fast ion distribution function is assumed to be singular in pitch angle near the plasma edge. The results are employed to understand the ICE in Deuterium-Deuterium (DD) and Deuterium-Tritium (DT) tokamak experiments. © 1995 American Institute vf Physics.

I. INTRODUCTION

New observational results of Ion· Cyclotron Emission (ICE) in the Tokamak Fusion Test Reactor (TFTR) 1-3 and Joint European Torus (JET)4,5 (see also references contained therein) near plasma edge have been reported recently. The measured ICE frequency spectra in Deuterium-Deuterium (DD) and Deuterium-Tritium (DT) discharges show peaks at multiple harmonics of deuterium cyclotron frequency t weD

(andlor harmonics of tritium cyclotron frequency-twcT fbI' DT) evaluated at the outer periphery of the plasma. The width of each peak is very narrow with !::.w/w~a/R and I ~ 10, where a and R are the minor and major radii of the tokamak plasma, respectively. The results indicate that ICE is localized spatially near the plasma edge, and there is a strong evidence that ICE is driven by charged fusion prod­ucts. A linear correlation between the ICE power and neutron emission rate over six orders of magnitude was observed in JET.4 The ICE spectrum is more complicated in TFTR. In ohmically heated DD discharges, a different set of lines in the ICE spectra, corresponding to the odd deuteron harmon~ ics, W = (21 + 1) WeD, were observed. For higher frequencies around W = lOWeD a broadband emission was also observed. The ICE spectrum intensity was found to be proportional to the evolving neutron emission.2 In TFTR D-T supershot discharges,3 the ICE spectrum was observed at the initial stage of Neutral Beam Injection (NBI), with ICE peak fre­quencies corresponding to the a-particle (or deuterium) edge cyclotron frequency. During the steady state NBI heating the ICE spectrum shows a rather complicated picture with fre­quency peaks located at tritium andlor deuterium cyclotron frequencies. These observations also show a correlation of ICE intensity with the neutron emission rate. With theoretical

alpennanent address: Troitsk Institute for Innovative and Fusion Research, Troitsk, Moscow region, 142092 Russia.

understanding of the responsible instability mechanism due to fast particles, the ICE may be employed as a diagnostic tool to investigate the properties of the fast ion population and the plasma at least near the plasma edge.

Recent theoretical investigatioIIs6- 1O considered ICE to be the i\lfven Cyclotron· Instability CAC!) driven by fast charged fusion ions which are in cyclotron resonance with the fast compressional Alfven wave. In Refs. 6-8 the insta­bility was studied in the uniform plasma approximation. In Refs. 9 and 10 two-dimensional fast wave eigenmode struc­ture was obtained in the large aspect ratio limit ( €= a/ R ~ 1), but a local analysis of the wave-particle inter­action was performed. The possibility of exciting ACI with very low fast particle density was demonstrated. However, this theory can not explain the instabilities at low cyclotron harmonic frequencies, especially for the DT case with v aO / v A< 1 such as in TFTR, where VaO is the a-particle birth velocity, and v A is the Alfven velocity near the plasma edge. In JET a-particle velocity satisfies VaO/VA>! and spectral peaks at low harmonics of fusion product ion cyclo­tron frequencies were shown to be described by the ACI with oblique wave propagation, driven by fusion products." One overly restrictive condition for the uniform plasma theory6-8 to be correct requires that the instability growth rate yP W b ,8 where W b is the fast particle bounce frequency. This condition requires much higher fast particle density than typical experimental values near the plasma edge.

The theory of cyclotron thermonuclear instability was studied earlierl2 (see also references therein). It included the drift motion of fast particles, but the fast particle finite orbit width was assumed negligible in comparison with the radial wavelength. The analytical theory reviewed by Mikhailovskii12 can not be applied directly to the experi­ments. One reason is that the radial excursion of fast particle orbit in TFTR and JET is. comparable to the minor radius. The second reason is that the final growth rate expression

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contains high bounce harmonic resonances and requires nu­merical evaluation due to complicated non local resonance condition along the particle orbit.

In this paper we will consider the resonant excitation mechanism of ICE based on ACI driven by fast particles. The free energy is mainly due to velocity gradient and an­isotropy of fast particle distribution. Our aim is to develop a two-dimensional formalism for fast particle-compressional Alfven wave interaction and to show that ACI can be un­stable to explain the ICE observation in TFfR and JET ex­periments. The structure and the spectrum of compressional Alfven eigenmodes are obtained by neglecting ion Finite Larmor Radius (FLR) effects and plasma pressure in the large aspect ratio limit. The linear growth rates due to fast ions are computed perturbatively from the perturbed fast ion distribution function, which is obtained by integrating the high frequency gyrokinetic equation along the particle orbit. The equilibrium fast ion velocity distribution function is as­sumed to be singular in pitch angle. The ACI growth rates are proportional to the free energy associated with the fusion product population.

To excite the ACI the growth rate due to fast ions must be higher than all dampings due to thermal electrons and ions. To minimize damping and to find the most unstable eigenmodes we will mainly consider modes with small par­allel wave vector kll in comparison with perpendicular wave vector k.L' Electron Landau damping can be avoided by choosing kll~wlvTe' i.e., the wave phase velocity is higher than the electron thermal velocity.9.13 Thermal ions can con­tribute to significant cyclotron damping at wi Wei-I. How­ever. the thermal ion damping can be minimized for eigen­modes that are radially localized near the plasma edge, where the thermal ion density and temperature are small. At higher cyclotron harmonic frequencies the thermal ion damping is exponentially small. In tokamak devices such as TFTR or JET fast ions with pitch angles close to the trapped-passing boundary have radial drift orbit excursion comparable to the minor radius. These fast ions, although born near the center of the discharge, contribute to a high anisotropy in their ve­locity space distribution near the plasma mid-plane edge on the low-field side. The anisotropy of the fast fusion particle velocity distribution is a function of the fast fusion particle source gradient and is higher near the plasma edge. 14 Thus. radially localized eigenmodes located in the region of high velocity space anisotropy and large fast ion population will be most easily excited. Such eigenmodes are responsible for the observed ICE spectrum peaks. The linear growth rates of the various cyclotron harmonics of the AIC have been com­puted. To find the most unstable modes, near-perpendicular wave propagation near the plasma edge has been considered. To illustrate the possibility to describe the ICE spectrum we presented calculations for typical discharge plasma param­eters of TFTR.

One of the new and most interesting results of the paper is that odd harmonics of edge deuterium cyclotron frequency can also be excited by DD fusion protons. Odd (21 + I) har­monics of deuterium cyclotron frequency correspond to half integer harmonics (I + 112) of proton cyclotron frequencies, where I is an integer. This is because wave-particle reso-

1962 Phys. Plasmas, Vol. 2, No.6, June 1995

nance can be satisfied due to large Doppler frequency shift, which can be as large as 112 of the proton cyclotron for high toroidal mode number modes with large parallel wave vector. Since previous theories predicted that ACI can occur only at harmonics of the edge cyclotron frequency of the exciting particles,11 this result resolves a long-standing problem with interpretation of the observed ICE spectra, which are similar in form (though not in amplitude) irrespective of whether the plasma is pure deuterium or a mixture of deuterium and tri­tium.

The numerical calculations presented in this paper are still far from a complete description of all details of experi­mental data. In particular, since thermal plasma damping near plasma edge is expected to be negligibly small and will be omitted, there is no information on fast particle density threshold of ACI in the present analysis. Nonlinear treatment of the problem is also beyond the scope of this paper. How­ever, in saturated steady state phase the ICE power intensity should correlate with the instability growth rates (or the fast ion density at plasma edge), but not necessarily linearly. JET experimental data4 show a linear correlation between ICE power intensity and fusion reactivity. TFTR supershot data3

show that ICE power intensity is roughly proportional to 0.7 power of fusion reactivity. These should be taken into ac­count in developing a nonlinear theory of ACI.

The paper is organized as follows. In Sec. II we present a two-dimensional eigenmode analysis for compressional AI­fven eigenmode structure and eigenfrequency, and a pertur­bative analysis for fast particle contributions. In Sec. III the kinetic fast particle contribution is obtained in a Wentzel­Kramers-Brillouin (WKB) approximation by solving a gen­eral frequency gyrokinetic equation. Sec. IV deals with nu­merical calculations of ACI growth rates using the perturba­tive method. A summary is given in Sec. V.

II. COMPRESSIONAL ALFVEN WAVE DISPERSION

A. Eigenmode equation

We will look for electromagnetic modes in the ion cy­clotron frequency range, and the compressional Alfven wave is a possible candidate. To obtain the compressional Alfven wave eigenmode solution, we neglect the fast ion kinetic effects and consider the model of inhomogeneous, magne­tized plasma in a simple toroidal configuration with a circu­lar cross section and large aspect ratio. In the absence of kinetic contribution from wave-particle interactions, the eigenmode equation is determined by the cold plasma disper­sion relation

2 2_ €2

k -~k +€I' II-€I

where

(1)

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WcJ and wpi are the cyclotron and plasma frequencies of ion species i, k is the wave vector, and Eij are elements of the plasma permeability tensor without kinetic contribution. The perturbed electric field can be presented in the form E={E 1 ,E2 ,E3}, where

b=BIB, (2)

where B is the equilibrium magnetic field. The polarization is given by the relation

ic~ El=E2 ~k ~ .

'11_ Ct (3)

The parallel electric field of the compressional Alfven wave will be neglected.

To obtain· the compressional Alfven wave eigenmode equation we consider the perturbed parallel magnetic field in the form EII=BII(r,O)exp[ -i(wt+ncp)], where 0 and cp are the poloidal and toroidal angles respectively, r is the minor radius of the magnetic surface; Ell is related to the perturbed electric field through the equation

(4)

The eqUilibrium magnetic field is chosen as B=BoRo/CRo + rcosO), where Bo and Ro are magnetic field and major radius at the plasma magnetic axis respectively. Defining kll= - ib· VInEII we obtain the differential form of the dispersion relation as

( 2 1 a2

) - (c~ 2) -V r +?a8'1 Bn=- kff-

Cl +cI-k ll Bn, (5)

where V;= Ur(al ar)rcal ar). For W~i< w 2 and k~<kl ' Eq. (5) reduces to

(

2 1 a2) _ w 2

_

V r + ? ae2 BII"'" - ;r BII' (6)

We will use Eq. (6) to study ACI eigenmodes even for W~i-w 2 . This equation has been studied in the cylindrical plasma approximation.lO,tS In toroidal plasmas the eigen­mode equation has been solved9 in terms of an eikonal rep­resentation using the small parameters 11m and c= rl Ro, where m is the poloidal mode number.

B. Radial eigenmode structure

In the cylindrical plasma approximation the perturbed quantities can be presented in the form Bn=B(r)exp[ -iwt-incp+imO], where cp=:dR and z is the coordinate along the plasma cylindrical axis. Then the eigen­mode radial structure is described by the following equation:

(V;- V(r»B(r) =0, (7)

where the potential is given by V(r)=m 2Ir2 -(w2Ivi(0» X(n(r)lno), nCr) is the plasma density with the central den­sity no, and v~(O) is evaluated anhe magnetic axis with the

Phys. Plasmas, Vol. 2, No.6, June 1995

o

ct//v/(rl... \

\ VCr)

"-. . . . . . . . '> '" '< 2t;

o

FIG. L Typical radial dependence of the potential V(r)= w2/v~ - m2/r2 on r2/a2, which forms a well at r= ro and ro is determined in Eq. (10) The eigenrnode is localized within the 2A width region.

vacuum toroidal magnetic field. If VCr) forms a potential well, localized solutions can exist inside the well. For a plasma density profile of the form

(8)

and if m~ 1, a potential well can form at r=ro, where r6Ia2=1I(I+ui)-il2Ia2, and il2Ia2=~2u;l(I+ui) I[m(l + Uin Then Eq. (7) can be approximated as a har-' monic oscillator equation which admits . localized solutions. The lowest most localized eigenmode solution has the eigenfrequencylO given by

w2= (m2d(ro)lr6)(1 + .J2( 1 + uJlu/m) (9)

and the eigenfunction given by

B(r)=b exp[ -(r-ro)2/t;2]. (10)

Note that in the potential well region the solution (10) is consistent with the ordering

aE11 _ {;;BII iJr r

m V(r)- 2"

r (11)

For Ui:s,; 1 the eigenmode is localized near the plasma edge. Thus, to form a potential well V( r) near the plasma edge, it requires a rapid variation of the plasma density profile near the plasma periphery (see Fig. 1).

To include the toroidal effects in the eigenmode equa­tion, we note that w2lv~ = (w2Iv~(0» (n(r)lno) X (1 + c cos 0)2. For high-m modes we consider a2 BII I ae2~ r2 a2 Bill ar2, which is satisfied by the cylindrical orderings, Eq. (11). Then, Eq. (6) reduces to

2 - 2 1 a BII w nCr) 2-? ---;;er = - v 2(0) -;;-(1 + c cos 0) BII' (12)

A.. 0

To the lowest order in (clm) Eq. (12) admits a solution in the form of

EII=B(r,O)exp[ -iwt+iS], (13)

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where we introduce the eikonal9 S= - n<p+ m( e+ € sin fJ) and B(r,e) is a slowly varying function of r and e.

For €- O(llm) and considering the cylindrical ordering, Eq. (11), we have aSlar~alnBlar. Substituting Eq. (13) into Eq. (6), we obtain an equation for the envelope B( r, e) which is identical to the cylindrical eigenmode equation, Eq. (7).

For larger € we consider

(14)

and require Ill~m. From Eqs. (6), (14), and (13), we obtain the recurrence relation for the coefficients B I( r)

(15)

Assuming €-O(lIm\lV) and v>2 so that m€2> I and re­quiring the series, Eq. (14), to converge, BI must be a de­creasing function of I n. For 1- O( l) the leading terms in Eq. (15) are proportional to m 2€2_€-2(v-l), and all B/ are of the same order. For larger I with I>O(m€2) we obtain

.'I. 2 A A A •

B/=m€ B/- 2 /21. Thus, Bt<B I- 2 and the senes (14) con-verges. Therefore, the series (14) contains B/ with lo;;;mE2.

Since I e~ s the phase in B can be neglected. However. we need to calculate B/ from Eq. (15) for lo;;;mE2, which requires extensive numerical calculations. In the following we will approximate B with the cylindrical radial eigenmode given by Eq. (10).

C. Perturbative analysis

Using Eq. (4) the eigenmode equation, Eq. (5), can be rewritten in terms of the perturbed electric field and is given by

(16)

Considering resonance contributions to Eq. (16) due to all plasma species to be small, we can make use of the pertur­bation method. We expand perturbed electric field and eigen­frequency as E2=E~+ oE2 and w= wo+ ow, where OE2 and ow are small kinetic corrections to the magnetohydro­dynamic (MHD) eigenmode and eigenfrequency solutions. These kinetic effects are described by the anti-hennitian part of the penneability tensor E'f 12 for each species j, which will be presented in Sec. III. Then, from Eq. (16) the growth rate is given by

1964 Phys. Plasmas, Vol. 2, No.6, June 1995

+E'f22)E2d3r /f E~[- 2~ aW

a

2

€12 k2E2 w w 1I-€1

I aW2€JJ( ~)] 3

+~~ l+(k[-€lf E2dr. (17)

Taking into account Eq. (3) and neglecting parallel electric field, Eq. (17) can be rewritten as

III. INTERACTION OF HIGH ENERGY PARTICLES WITH COMPRESSIONAL ALFVEN WAVE

(18)

In this section we will use a WKB eikonal representation for perturbed quantities

C= {( if!. e)e-iw+iS(s,¢,IJ), V C= iCV S= i~k, (19)

where if! is the poloidal magnetic flux, the eikonal S (s, if!, e) is not specified here but will be chosen explicitly in Sec. IV, and s is the coordinate along the magnetic field so that B· V == B al as. The perturbed vector and electrostatic po­tentials are chosen as

E= - V </J+ (iwlc)A,B= VxA. (20)

Then the fast particle contribution to the anti-Hennitian part of the penneability tensor can be expressed through the reso­nance part of the perturbed distribution function:

f AA 3 4i7Tf -E*·€>Ed r=- E*.jAd 3r a W

(21)

The perturbed fast particle distribution function can be ex­pressed as

_ e al e al ( VilA II) Axb 1== m a!f </J+ Bm aIL </J- -c- + 13 . VI

(22)

where I is the eqUilibrium distribution function, J1=J1(Z) is the lth order Bessel function of the first kind, Z = k lY J..! We' IL = v 1/2B is the magnetic momentum, o=v 212 is the particle energy, L[=k.1xv·blwc-lfJ, fJ is the particle gyrophase angle between v.1 and k.1 . The nona-

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diabatic part of the perturbed distribution function is con­tained in g I, which is governed by the general frequency gyrokinetic equation 16-18

(w-wD-Iwc)gl

eJ._ T[( VIIAII) V1.BlldJ[] =-(w[-w) ¢>-- J-----T * e Z ekl dz '

(23)

where

dS. 2 WD=Tt =Vllb· \l S+ wBmjLBIT+ wkmvl11T,

wB=cTleB2 k1. ·bx\lB, wk=eTleB k1. ·bx Kc, (24)

_ T ( a lWe a) T cT . W[=-- W-+-- InJ W =-k1.·bx\llnj

In afff B aIL ' * eB '

Kc is the magnetic field curvature, and T is the temperature of given species. Equation (23) can be rewritten in the form

(:t +ilWe)gl=~iX' (25)

where X is the rhs of Eq. (23). We integrate Eq. (24) along the particle characteristics l2 with the causality condition that at t ' = -00, gz vanishes, and obtain

xexp( -iwt'+iS(s',if/,O')+i f' twedt'').

(26)

where the integral is taken along the unperturbed guiding­center trajectory. Let

wct)"'" J:CWD-if)D+tWc-ZwJdtl (27)

with ( ... ) = pdt( ... )1 Tb taken along the particle orbit, Tb=2?T/lwbl and wb=2?T[p(dslvlI)r 1

• Then

izCt)=-i f""dt'X I exp[iW(t')

- iW(t) - i( w- (oD-lwc)(t ' - t)]. (28)

Note that X' exp[iW(t')] is periodic in time with the bounce period and can be decomposed19 as:

k' exp(iW{t'» - ~ Xpexp(ipwbt'), (29) p

where Xp=X exp(iW(t)-ipwbt). For the nonadiabatic part of the perturbed current [see

Eq. (21)] we have after gyrophaseaveraging

jA=e~ J v1. dv 1. dv IIG{*gz, z

(30)

where the components of the vector G{ are v LLJICZ)lz, ivlaJzlaz and vIIJI. Substituting Eq. (30) into Eq. (21) and integrating Eq. (28) we have

Phys. Plasmas, Vol. 2, No.6, June 1995

J .E*·gA·Ed3r= 4i:e~ f d 3rv 1 dvL dv il l·p

E*.G;* exp(-iW(t)+ipWbt)Xp X----~----------------~

w-lwc-wD-pwb

To make this equation more symmetric we express

where kill k 1. ~ 1 is assumed. Then we have

A ieJ _ T Xp= wT(w/-cu*)F/p ,

(31)

(32)

(33)

where Ftp=exp(iW(t)-ipWbl)Gr E and Gt={v 1. wJt X(z)/(wcz);iv1.aJ/laz; vIIJ/}. Here the equilibrium distri­bution function is assumed to be a function of the particle's integrals of motion. Returning to Eq. (31) we note that the integration is taken over five-dimensional phase space. We will extract the fast integration along the unperturbed particle trajectory associated with the variable t in Eq. (31). We transform integration variables from (R,Z,cp,v 1. ,VII) to (P ""l{I,cp,jL,fff), where

el/l(R,Z) p ",= 2 -v",R (34)

?Tme

is the longitudinal adiabatic invariant. Then we have

v 1.RdRdZdcpdv 1.dVII

=[ ~~ :R (1 +~) r1

:c dl/ldP ~dcpdjLdg, (35)

where we assume v II <:>< v "'. The particle orbit motion is asso­ciated with the variable 1/1, so that

[al/l ft' ( V~)]-l

dt= az wcR 1 + ? dl/l.

Thus, taking into account Eqs. (33) and (35), Eq. (31) re­duces to

f 8?T2e2Bf

E*·gA·Ed3r=- wcw2T dP ",djLd 0'JTb

. x~ F;p*(WI-W~)Flp w-iw -co -pw ' l,p c D b

(36)

where F{p = exp(iW(t)-ipwbt)G; .E. A similar expression was obtained previously12 in the limit of zero banana width and with F 1p · computed for well trapped and passing par-ticles. '.. '.

.The p summation can be performed in a general form. First, the resonance term is expanded as

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Retaining only the resonance contribution Eq. (36) is inte­grated over /1-. Then we make the following approximation

2: -4 J dp. (38) p

The approximation (38) is valid because we consider high frequency modes with w= we;» Wb' Finally employing Eq. (38) we obtain the resonant contribution

where we have made use of the bounce resonance conditions and the definition

= :b f dt exp( i !:(lWe+WD-W)dt' )GrE=F/.

(40)

The expression (39) is still too complicated for analytical analysis. To further simplify F[ we note that the argument in the exponential function is much larger than 11', which means that the exponential function is fast oscillating except near the region where lwe(ll(tl» + WD((1(t\» - w=O, where (J( t 1) is the poloidal angle of the particle position on the orbit at time t l' Around this point we have

ift [lwe+ WD- w]dt' = iD.'OI + if I [ ... ]dt', (41) to 1\

where

(42)

D.'ij=J:J[lwc+wD-w]dt', and to corresponds to the radi-I

ally outermost point of the particle orbit. The same expan-sion is valid for any resonant point. With a given set of variables Pcp, c:; /1- there are two types of particle orbits which correspond to different signs of parallel velocity 0'= VII Iv. Assuming up-down symmetric particle orbits there can be two resonance points at poloidal angles (Jres and - Ores determined by the resonance condition [wc(Ores)+WD(Ores)-W=O for each 0'. We will show the calculation procedure for the case when the first derivative term in Eq. (42) is larger than the second derivative term. For trapped particles there may be four resonance points. Then we have

1966 Phys. Plasmas, Vol. 2, No.6, June 1995

(43)

where we denote the resonance points 1,3 in the upper half cross section for 0'= + 1,-1, respectively, and the points 2,4 in the lower half cross section for 0' = + 1, - 1, respectively. Also, [1.3=J~:exp[i(d(lwe+WD)/dt)1,3t2/2]dt. Then we can write

[2

+ ~ '" G/'.*·E:"G/··K 271'..Li t I I I'

bi= 3,4

(44)

where we have dropped terms with ei(aij+O:kl), i=l=k, i=l=k, j=l=l, which correspond to uncorrelated phase and give zero contribution after averaging over all groups of particles, and obtained

(45)

A similar expression can be obtained for untrapped particles, with the first term in Eq. (44) for passing particles with 0'= 1 and the second term for 0'= - 1. Summing over the parallel velocity direction 0', we arrive at the following ex­pressions:

(46)

where

2_ 811' [ - {Id(lwc+ wD)ldtI 3+ C[d2(lwc+ wD)/dt2]2} 113

(47)

with the subscripts dropped, and C = 2.6943. Also, to avoid singularity at Ores = 0 the next order term from the expansion (42) is included. The coefficients F; * F[ determined by Eq. (46) contain the information about the wave-particle Doppler shifted cyclotron resonance. Namely, the two terms in the round brackets give the electric field value at the resonance point and the factors G; and G1 appear from gyro-averaging. The term I has the dimension of time and determines how long the particle was in a resonant layer.

In summary, the ACI growth rate due to fast particles is given by Eq. (18) with the anti-hermitian tensor given by Eqs. (39) and (46). These expressions take into account the high frequency wave-particle resonance in phase space along the particle orbit.

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1.2

1

0.8 E N

0.6

0.4

0.2

0 1.5 2 2.5 3 3.5 4

R,m

J:'1G. 2. Wall position, plasma boundary (dotted line) and two orbits of barely trapped particles with different pitch angle. Also shown is the reso­nance curve (dashed' curve) between resonance points 1 and 2 at which the resonance condition (53) is fulfilled

IV. ALFVEN CYCLOTRON INSTABILITY

A. Fast particle destabilization

In this section we demonstrate the possibility that ACI in tokamak plasmas can be excited by a small group of high energy particles (a-particles or protons). From Eq. (3), E z is smaller than El for w>wc and can be ignored. Thus the perturbed electric field can be expressed in the form E={EI>O,O}, where as in Eq. (10) El is chosen as

El =Eo exp[ -(r-ro)2/A2]exp[ -iwt+iS], (48)

the eikonal is S= -ncp+m«()+ € sin B) and ro and A are given in Eqs.· (9) and (10). Then kU=mlqRo-nIR,kL =mRlrRo. For the chosen eikonal the Doppler shift frequency WD defined in Eq. (24) is given by

C!)D=(mRIRo~nq) O-nvAV cp-qV 0)

=(J'kll.J2$~l- p,B/fF~?:; Z' (~ -1) (1-:- P,B). r WeO Ro 21fj'

(49) The fast particle driven growth rate can be obtained from Eqs. (18), (39), and (46) by neglecting the spatial gradient drive term, which is negligible in the high frequency regime, and is given by

'Ya . w3

..j2eacB" J z 2P,LJt - = ---z--2.. 2 L.t dP cpdfFdp,1 E1 -::r Wca WpWca J;AroRoEo 1,00_ Z

(50)

where we assume for simplicity that !€1l!=(u;lw2, wp is the plasma frequency, and the subscript a denotes the fast par­ticle. The sign of the growth rate depends on the [ff' and p, derivatives of the distribution function. We mentioned earlier that the plasma inhomogeneity and finite banana width can lead to large velocity space anisotropy of a-particle distribu­tion. The anisotropy is higher near the plasma periphery, where the spatial gradient of a-particle source is higher.14 A limiting case of the distribution function, a delta function in p, which represents .only the contribution of the barely trapped particles, is considered. The orbits of the barely trapped particles are shown by solid curves 1 and 2 in Fig. 2.

Phys. Plasmas, Vol. 2, No.6, June 1995

Such a group of particles have the maximum radial orbit deviation from the magnetic surface to. the plasma center and their fusion source is maximum in comparison with other groups. The a-particle distribution can be expressed as

!a=na(r)!u(v)o(p,;o -Ao), (51)

where ! u ( v) represents the velocity distribution, and A 0 is the pitch angle of barely trapped particles; Ao' can be obtained from the equation (eaI2Trmac) X o/(R,Z)Ii.?=AoRo,Z=O= Pcp [Eq. (34)]. Integrating by parts with the distribution (51), Eq. (50) reduces to

'Ya w3

..j2e",cB" J { . . -= ~ r='. 2 L.t dP tpX F(BZ)- F(B1) Wca wpwca YTrAroRoEo 1,00

J g' [ a - d0'naf - -. u Bo a1fj'

lwca a] z 2 p,LJt } +---- 1 E -::r wB aIL 1 z IJL=Aoif'lBo'

(52)

where F= (1 + lWcaft'l( wBp,»n aiuPEICP,LJ;ft:2) ([ff'1 Bo), W1,2 = ~i2( P cp) ([ff'1 < 1fj'2) are the energy limits determined from resonance condition at fixed P'P' The differential op­erators in Eq. (52) operate on all quantities to their right hand side. These quantities are taken at the resonance point and therefore are functions of Pcp and 0'. The resonance condi­tion is

W -!wca(r( 0), 0) - WDa(r( (), 0) = 0, (53)

where r( 0) determines the fast particle orbit. Tn' comparison with the local theory9,10 Eq. (52) has additional terms pro­portional to derivatives of 12 and E~. One can show that the P derivative terms give small contribution, while the Ei derivative terms are always destabilizing and are comparable with the p,lflzz derivative terms from the local theory. Note that the fast particle growth rate is linear in fast particle . density.

B. Damping mechanism and ICE spectrum

The ACI eigenmode frequencies are determined by the compressional Alfven waves with high poloidal and toroidal mode numbers and are given in Eq. (9). When the compres­sional Alfven waves resonate with particles satisfying the resonance condition given by Eq. (53), ACI may gain energy from fast particles or lose energy to the thermal plasma. To excite the ACI the fast ion drive must be higher than all dampings due to therinal electrons and ions. To obtain the most unstable modes we have to find a radial location where the damping is smallest, the drive is largest, and the eigen­mode is highly localized.

To minimize the electron damping we choose eigen­modes with wave phase velocity higher than the thermal ve­locity of electrons,9,13 i.e., kr~ wi v Te' so that the electron damping is exponentially small. Thermal ion cyclotron damping can be significant if ions are in cyclotron resonance with the wave at ! = wi Wci< 5 inside the plasma core. There­fore, eigenmodes localized near the plasma edge and with

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frequency corresponding to the multiples of edge back­ground ion cyclotron frequencies can be weakly damped due to low ion density. For higher cyclotron harmonics with 1= wI wei> 5 the thermal ion damping is exponentially small. The accurate calculation of thermal plasma damping mecha­nisms is beyond the scope of this paper. To maximize the fast ion drive it requires the ACI eigenfrequency to be close (ex­cept for proton driven ACI, see below) to the multiples of the fast product cyclotron frequency at the low field side in the plasma mid-plane, where the source of instability, the anisot­ropy of fast particle distribution function, is high.

Keeping in mind that our aim is to develop the formal­ism of fast particle-wave interaction, we will present the growth rate calculation for the ACI eigenmodes with negli­gible thermal plasma damping and with eigenfrequencies corresponding to the mUltiples of deuterium (a-particle) cy­clotron frequency. Such ICE spectra were experimentally ob­served in DD experiments and in the initial stage of DT experiments in TFTR supershots when the tritium population is small. We will show that the fast particles can destabilize the ACI even when the ACI eigenfrequencies are different from the fast particle cyclotron frequency. They can be in cyclotron resonance due to the finite Doppler shift in the resonance condition (53). For example fast protons can drive ACI with frequencies w=wcD(21+1)=wcp (l+1I2). It re­quires the parallel wave vector to be large at fast particle­wave resonance points. For m=n, kll=O near q= 1 where the electron temperature is large. On the plasma periphery kll = - nl R, which is large enough for the Doppler shift to the resonance condition for protons to be equal, wDp=kllvlI= wcpl2. Such large kll is achieved only at the edge where the electron temperature and, therefore, electron Landau damping are small.

The ICE spectrum peaks can be also related to the tri­tium cyclotron frequencies. As considered here deuterium plasma eigenmodes with frequency corresponding to the edge deuterium cyclotron frequency are negligibly damped and more easily excited. A similar conclusion may be done for pure tritium plasma. In DT plasmas thermal ion cyclotron damping is not negligible and is determined by the tempera­ture and the density of deuterium and tritium at the edge. This idea may be useful in estimating the ratio of tritium to deuterium densities.

C. Numerical results

We present numerical examples of ACI growth rates for typical TFTR supershot plasma parameters:20 Ro=2.52 m, a=O.9 m, the safety factor profile is given by

q(r) = qo /[ d(~2) ( 1- o.~~r2 + O. ~~r4) r2] for r~a and is a parabola outside the plasma. qo=0.85, the plasma density is n(r)=O.5XI014(l-r2Ia 2 )0.2 cm-3, the vacuum toroidal magnetic field is Bo=5 T, the central a-particle density is n £1'0 = 2 X 1010 cm - 3• the central proton density is n pO = n ao/50, and the fast particle density profile is na.p=n£1'.po(l-r2Ia2)3.75. The chosen density of fusion products corresponds to the first few hundred milliseconds

1968 Phys. Plasmas, Vol. 2, No.6, June 1995

10.4

10.5

0 0 10.6 aU

?:-

10.7

(a)

10.4

~ ~

10.5

0 0 10.6 aU

?:-

10.1

i-

10.8 i

0 (b)

2

2

4 6 0)/0)

cDedge

4 6 O)!O)

cDedge

I

8 10

8 10

FIG. 3. The ACI growth rates driven by a-particles with the velocity distribution, given by Eq. (54) for (a) vr_=vT+=0.107v,,0 and (b) vr_=0.2vaO,vT+=O.107v"o. The plasma parameters are 8 0 =5 T, Ro=2.52 m, a=0.9 m, the plasma density is n(r) = 0.5X 1014( I - r 2ta 2)O.2 cm -3, q(r} = qo /[d/d( r2)] X (1- 0.66r2/a 2

+O.ISr4/a4)r2J. for r""'a and is a parabolic outside the plasma, qo=0.S5, and the a-particle density is n,,(r)=2X 101O( l_r2ta 2 )375 cm- 3•

after DT neutral beams are injected into deuterium plasma3

at typical neutron flux of 2· 10 18 S - I. The position of the plasma column in the vacuum vessel is shown on Fig. 2 as a dotted circle.

In order to excite the ACI instabilities it is required for the eigenmodes to be localized in a potential well very close to the plasma periphery. This can be achieved by employing a density profile with very sharp variation near the plasma edge (see Fig. 1). Choosing the parameter 0";=0.2 so that ro=O.9a. Such sharp density variation near the plasma edge can be obtained in the initial stage of typically TFTR super­shot discharges.2o During the discharge the edge behavior of the density profile does not change much. The ACI growth rates are very sensitive to the sharpness of plasma density profile at the edge and typically disappear at O"i>O.2. The frequencies of the ACI eigenmodes are assumed to be equal to mUltiples of the edge deuterium cyclotron frequency. The poloidal mode number m is calculated from the expression for eigenfrequency given in Eq. (9). The toroidal mode num­ber is varied to find the most unstable mode with the con­straints kll~kl. and m=n. Due to the finite Doppler shift

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0 0

aO

?'-

(a)

0 0

aO

?'-

(b)

10.4 .,-.-'-

10.5

10.6

Q

10.7 a , R

j q i

2

10.4

10.5

10.6

I I

4 6 ro/ro

cDedge

I I

I

8

I

I

~

10

I

10.7

1O.~l~~1 ~ o 246 8 10

ro/ro cDedge

FIG. 4. The ACT growth rates driven by protons with the velocity distribu­tion, given by Eq. (54) and (a) VT-=VT+=O.058vpo and (b) vT_=O.SSvpO,VT+=O.O~8vpo. The first dashed bar at W/WcDedge= 1 corre­sponds to the case when kH-kJ.' The proton density is chosen as np(r)=OAX I09(l-r2/a2)3.7S cm-3• The plasma parameters are the· same as in Fig. 3.

frequency in the cyclotron resonance, Eqs. (49) and (53), fast particles will resonate with the wave inside the plasma while thermal ions can be in cyclotron resonance near the pla:sma edge. The dashed curve between points 1 and 2 in Fig. 2 represents the resonance line for the barely trapped a-particles at the fundamental Doppler shifted cyclotron resonance.

The a-particle driven ACI growth rates are shown in Figs. 3 (a) and (b) and the protons driven ACI growth rates are shown in Figs. 4 (a) and (b) with a velocity distribution in the form:

(54)

where v T- is an adjustable parameter, v T+

=~2Ti/(tnl+tn2)' tnl and tn2 are the mass of reacting nu­clei, Ti is their temperature which was assumed to be 50 ke V for beam heated plasma, and 7J is the Heaviside step func­tion. The distribution function Eq. (54) is a good approxima-

Phys. Plasmas, Vol. 2, No.6, June 1995

tion for edge fusion product population where the gradient of the fusion product source is large. The distribution Eq. (54) requires the fast particle orbit to be comparable to the scale length of the fusion product source. l4 Note that the parameter v T- is critical for the ACI instability. For a-particles the instability disappears at v T- > 0.25v aO' while for protons the instability occurs at v T- > 0.8v aO' The slowing-down distri­bution is stable due to high fast particle Landau damping. The destabilization from the velocity anisotropy term is not enough for instability. To drive ACI unstable one needs ad­ditional drive due to positive energy gradient. We note that the distribution given in Eq. (54) can be created at the initial stage of discharge after the NBI starts. The velocity spread is due to the Doppler shift effect associated with the velocity spread of the thermal fusion ions. The velocity distribution (54) can also be a result of anomalous fast particle loss caused by toroidal magnetic field ripples, or plasma instabili­ties such as the Toroidicity:.induced Alfven Eigenmodes (TAB), the fishbone, or other MHD modes. As a result of such losses the fast particle confinement time can be shorter than the slowing-down time, and the velocity distribution can maintain a positive gradient. Experimental evidence has been pointed out in Ref. 4 that the Edge Localized Modes (ELM) activity and the ICE amplitude are correlated; small ampli­tude ELMs do not disturb the ICE spectrum, while high am­plitude ELMs can eliminate ICE completely. This related to the anomalous fast particle loss by ELM activities. Accord­ing to our model to drive ACI unstable one needs the veloc­ity space distribution of fast particle population in the form Eq. (54), which requires the reduction in confinement time. From this point of view, to be consistent with experimental data4 one might suggest that small ELM amplitude only re­duces the fast ion confinement time to less than the slowing­down time. High amplitude ELMs scatter almost all trapped fast particles out of the plasma. After the trapped particle population replenishes near the edge, the ICE appears again.

The a-particle birth velocity is different from the pro­ton's: vA~vaO= l.3X 109 cmls <vpo=2.4X 109 cmls. From Fig. 3(a) one can see that the instability disappears for I> 6 because of a decrease in the destabilizing contribution associated with the spatial derivative of Ei. This term is destabilizing because of two reasons. The first reason is that the region with positive derivative of Ei is closer to the center than the region with negative slope of eigenfunction, the second is that the a-particle density: profile is centrally peaked. For higher cyclotron harmonics the region where particles are in resonance with the wave is smaller and the corresponding curve of resonance points (curve 1-2 in Fig. 2) moves out towards the plasma edge. Therefore the positive spatial derivative of Ei gives a smaller destabilizing contri­bution to ACI. For protons the Doppler shift is higher and the reduction of the resonance domain does not lead to such effects. In addition, the value z=kJ.Y l./(lwc)( ~2) is higher, which makes the local contribution to ACI (~JLJJ/Z2) more destabilizing.

From Figs. 3(a) and 4(a) we see that the proton growth rates are about one order smaller than the a-particle growth rate. We expect that in TFTR experiments both kinds of par-

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ticles contribute to the ACI instability. In Figs. 4(a) and (b) we show the growth rate of the proton driven ACI at w=wcD=wcpI2 as a dashed bar. This instability can be ob­tained only at kll~k.l' which is beyond the validity of our assumptions. But. we should also note that the electron damping will be small in this case because of small wave phase velocity and Ve P V A. Only trapped electrons with ve­locity Ve[I~VA can give rise to the damping, which requires additional analysis.

The ACI growth rate provides the growth time scale of the instability. From Fig. 3(a) the growth rate is on the order of 2.5 X 102 - 2.5 X 103 sec-I for TFfR DT discharges, or equivalently the growth time is about 0.4- 4 ms, with the a-particle density being na=4X 108 cm- 3 at rla=0.8.

From Figs. 10 and 11 in Ref. 4 the ICE intensity in JET DT experiments grows exponentially after each ELM event with the growth time -20 ms, which is of the same order as our calculation for TFfR DT discharges. However, before the H-mode termination X-event as shown in Fig. 9 of Ref. 4, the growth time is much shorter, which implies that the local a-particle density may be much larger than 4 X 108 cm - 3 at rl a = 0.8, as used in our calculation. Therefore, the ACI theory can be employed to estimate the density of fast fusion products at the plasma edge.

V. CONCLUSIONS

A two-dimensional theory of compressional Alfven Cy­clotron Instability has been developed to describe the ICE spectrum observed in TFfR and JET experiments. The final expression for the ACI growth rate Eq. (39) is determined by the Doppler shifted wave-particle cyclotron resonance. The source of instability can be any free energy source associated with fast particle distribution function, namely spatially or velocity space gradients. The expressions for ACI growth rates were used in numerical calculations in a toroidal plasma with circular magnetic surfaces. Fast particle reso­nance contributions resulting from the two-dimensional eigenmode structure and finite banana width effects are re­tained. We have demonstrated that the theory can describe the narrow peaks at multiples of edge deuterium cyclotron frequency. The theory requires the ACI eigenmode to be lo­calized near the very edge of plasma and the velocity distri­bution function to be very narrow in pitch angle and suffi­ciently sharp in velocity gradient with velocity spread for a-particles less than 0.25v aO, while for protons less than 0.8v ,,0. The velocity spread can be understood by taking into account the Doppler shift effect due to thermal spread of the fusion source ions. Such a distribution can exist if fast particles are lost before they are slowed down. Such anoma­lous loss of fast particles can result from toroidal magnetic field ripple, collisional scattering, or MHD activity like fish­bone, TAE modes or Edge Localized Modes. The correlation between the ELM activities and the ICE amplitude has been shown in Ref. 4; the small amplitude ELMs do not disturb the ICE spectrum, while high amplitude ELMs can eliminate the ICE completely. This is because small amplitude ELMs reduce only the confinement time to less than the slowing down time, while high amplitude ELMs scatter almost all trapped fast particles out of the plasma. After the population

1970 Phys. Plasmas, Vol. 2, No.6, June 1995

of trapped particles replenishes near the edge the ICE ap­pears again. Note that in the eigenmode analysis we have excluded the region between the plasma current channel and the wall. In this region the eigenmode has a finite amplitude and the wave-particle interaction can contribute to stability and should be included.

Note that we have obtained the growth rates only for eigenmodes with frequencies equal exactly to multiples of the edge deuterium cyclotron frequency, Based on the quasi­linear theory one would expect the line width to be linearly proportional to the ACI growth rate. The Doppler frequency shift effects will also contribute to the spectral line width. For the same poloidal and toroidal mode numbers the Dop­pler frequency shift is larger for 3 Me V protons produced in the DD reaction than the 3.5 MeV alphas produced in the DT reaction. One also expects that the widths of higher harmonic ICE spectral peaks should be wider than those of lower cy­clotron harmonic cases because the Doppler shift for higher poloidal harmonics is larger and the background ion cyclo­tron damping is smaller for higher cyclotron harmonics, For example, JET results4 show that the ICE spectral line width increases linearly with the harmonic number. Such wider peaks can form the background ICE continuum signal which should correlate with the fusion proton source produced in the DD reaction due to higher'Doppler frequency shift for protons. This conclusion agrees well with the observed cor­relation of the background continuum ICE signal with the DD fusion neutron flux in TFTR DD and DT supershot ex­periments, but not with the DT neutron fiux.2

,3 Similar ob­servations were obtained in JET DD experiments.5 However, in JET DT experiments4 it was found that the background continuum ICE signal is proportional to the total neutron flux produced by both DD and DT reactions.

The difference of ICE results between TFfR supershot and JET DT operations may be resolved by realizing that the Alfven velocity at the location where ICE is localized is about a factor of 3-5 larger in JET than in TFTR supershots so that v aO IVA < 1 in TFfR and v aO IVA> 1 in JET. Thus, if m = n both the poloidal and toroidal mode numbers of ACI are usually smaller in JET than in TFfR supershots to satisfy the resonance condition. Near plasma center k[[=O. But, near the plasma periphery kl[= - nl R due to large q, and the Doppler shift frequency is given by wD=kIlV[I. Since w=mvAlr, the ratio of the Doppler shift frequency to the ACI frequency is roughly given by wDlw=(nrlmR)

X(vlllvA). For protons V[[=vP' and the ratio can be near unity for both TFTR and JET. For alphas ulI=v"o, and the Doppler frequency spread is much larger in JET than in TFfR. Therefore, we expect the background continuum ICE signal in JET, but not in TFfR, to be proportional to the total neutron flux produced by both DD and DT reactions.

We have shown that the ACI with eigenfrequency differ­ent from the cyclotron frequency of the fast particles can be driven unstable because the cyclotron resonance condition (53) can still be satisfied due to the finite Doppler shift. It requires the parallel wave vector to be large at fast particle­wave resonance points. For proton driven ACI the Doppler shift frequency can equal the deuterium cyclotron frequency so that wDp=kllv\I= wcpl2= WeD. Such large kif is achieved

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only at the edge where the electron temperature and, there­fore, electron Landau damping are small. For example, pro­tons can destabilize the ACI with frequencies w= wcD(21 + 1) = wcp(l+ 112) in deuterium plasmas. This conclusion may be also related to the ICE spectrum peaks at tritium frequencies in tritium and DT plasmas. We consider damping mechanisms as one of the main factors which form the ICE spectrum. In a deuterium plasma, considered here, eigenmodes with frequency corresponding to the edge deu­terium cyclotron frequency are negligibly damped and more easily excited. Similar conclusion may be drawn for pure tritium plasma. In DT plasmas thermal ion cyclotron damp­ing is not negligible and is determined by the temperature and the density of deuterium and tritium at the edge. This idea may be useful in estimating the ratio of tritium to deu­terium densities.

The contribution to the ACI from other fusion products, He3 and T, is beyond the scope of this paper. The cyclotron frequencies of these particles differ from the deuterium cy­clotron frequency and, therefore, when considering the ACI driven by He3 and T, thermal ion cyclotron damping should be included. Note that the birth velocity of He3 and T fusion products is smaller than that of fusion protons. It means that the Doppler shift to the resonance condition will also be smaller. One can expect that the location of the ICE spectral peaks due to He3 and T driven ACI will be close to the multiples of their edge cyclotron frequencies ..

ACKNOWLEDGMENTS

We would like to thank Drs. C. S. Chang, R. Majeski, S. Cauffman, S. J. Zweben and R. O. Dendy for helpful discus­sions.

This work was supported by the United States Depart­ment of Energy under Contract No. DE-AC02-CHO-3073.

Phy{>. Plasmas, Vol. 2, No.6, June 1995

One of us (N. N. G.) was supported partially under the Per­sonnel Exchange n,IO of the USIRF Collaborative Program.

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