1C/96/165

United Nations Educational Scientific and Cultural Organizationand

international Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSTCS

COUPLING OF in AND TRAPPED ELECTRON MODESIN PLASMAS WITH NEGATIVE MAGNETIC SHEAR

J.Q. DongInlermi.tioTml Centre for Theoretical Physics. Triesl.e, TUi.ly

andSouthwestern Tnsl.itnte of Physics,

P.O. "Box 432 Chengdu, 610041, People's Republic of Cliina

and

S.M. MaliajanTTISI.II.III.C: for Elision Studies. The TJriiv<M'Ki1.y of Texas a! Austin,

Austin. Texas 78712, ISA.

ABSTRACT

In toroidal collisionless plasmas, the ion temperature gradient flTC! or //,;) and the trappedelectron (TE) modes are shown to be weakly (strongly) coupled when both the temperaturegradient, and the driving mechanism of t.lie l.rapped eleet.rons are moderate or strong (weakbut. (mile). Tn t.lie regime of st.rong coupling, l.here is a single hybrid mode, unsla.ble for allrji in plasmas with positive magnetic shear. For the weak coupling case, two independentunstable modes, one in the ion and the other in the electron diamagnetic direction, arefound to coexist. In either situation, the negative magnetic shear exerts a strong stabilizinginfluence; the stabilizing effect, is considerably enhanced by the presence of trapped particles.It. is predicl.ed t.hat. for a given Net of plasma parrnnel.ers, it will be much harder t.o excitethe two modes simultaneously in a plasma with negative shear. The results of this study aresignificant for tokamak experiments.

M1RAMAUE- TRIESTE

September "1996

1 Introduction

Transport of energy, momentum and particles in magnetically confined plasmas is often ob-

served t.o be much higher than predicted by inl.erparl.icle collisions. Turbulence induced by

plasma instabilities is generally believed to be the cause of this anomaly. However, quanti-

tative l.heories with bot.Ti predicl.ive ability and a sound Tooting in first principles physics are

still under development. Hecent results from experiments on Tokamak Fusion Test Reactor

(TFTRJ,1 Joint European Torus (JET),2 and DTTT-D* have demonstrated that particle and

energy confinements are significantly improved in reversed magnetic shear regions of tokamak

plasmas. These experiments have sparked great, interests in the investigal.ion of insl.abilit.ies

peculiar to tokamak plasmas with negative magnetic shear.

That negative; shear could exert stabilizing influence on ideal ballooning modes was known

but not appreciated in the early work of Greene and Chance.4 Around the same time, there

were indications in Kadoml.sev and Pogul.se" t.hat. negative shear tends t.o sl.abilize non-

ideal microinstabilities. Kesselei a/.b demonstrated in a recent work that the reversed shear,

indeed, has a surprisingly st.rong stabilizing influence on these non-ideal instabilities. Tn

addition, it is found'3 that the ion temperature gradient flTC! or //,;) modes are stabilized

in a port.ion of t.Tie negative shear region for a proposed discharge; wit.h optimized plasma

density, current and temperature profiles. A physical picture showing the effect of negative

shear on curvature driven insl.abilil.ies is given by Anl.onsen, Jr. <:l al.' and the influence of

the negative shear on resistive ballooning modes has been recently studied by Drake et al.8

Mol.ival.ed by t.Tie experiments nienl.ioned above and by the common belief'1'"1 t.hat. t.he

1T(J instability is likely to be the dominant mechanism for anomalous transport in tokamak

plasmas, a systematic investigation of t.Tie TTG and PVS (parallel v<;locit.y shear) driven

modes has been performed in plasmas with negative magnetic shear.11 It was demonstrated

t.Tiat. in t.oroidal geometry, when t.Tie geodesic curval.ure drift of the ions is taken into con-

sideration, the negatively sheared plasmas show not only lower growth rates but also higher

l.hresholds (for the on Net of t.Tie TTG insl.abilil.y) as compared to plasmas with posil.ive shear.

This should certainly contribute to the processes which cause the confinement improvement

in negal.ive shear regions. However, our theoretical eHtiniat.es reveal t.hat. t.Tie differences be-

tween the strengths of the instabilities for the normal and reversed shears (for 1TC!) may

not be suflicient t.o account for so dramatic an improvement, in confinement. Additional

mechanisms have to be invoked to understand the physics of confinement and of such severe

t.ransport reduction.

The search for instabilities (or lack thereof) must also contend with an extremely im-

portant experimental finding that the spectrum of density fluctuations in tokamaks has two

distinct, wings: one of these; rotates in t.Tic: ion diamagnetic drift direction while the other

does it in the opposite direction of the electron drifts. These two spectra are detected si-

multaneously or separately in tokaniak plasmas, depending on l.he discharge; condit.ionH and

plasma parameters.12'13 Definitive theoretical explanations for this important experimental

result are quite rare.

in this paper we make an attempt to look for possible answers to the questions raised

by earlier as well as recent experiments. For this purpose, we deal wil.h a model of a col-

lisionless toroidal plasmas with negative magnetic shear which includes both the 1TC and

l.he TE (trapped electron) modes. A recently developed comprehensive gyro-kinet.ic disper-

sion equation14 used for the study of low frequency drift-like instabilities is now extended to

include; the trapped particle dynamics. The conditions for the existence of these two instabil-

ities, simultaneously or separately, are discussed, and their mutual coupling and interaction

is strongly emphasized. The results are compared with similar results in plasmas with pos-

itive magnetic shear. The theoretical results are also compared with the experiments on

confinement improvement, and on microturbnlence spectra.

The contents of this work are organized as follows, in Sec. 11. the integral dispersion

equation is displayed. Tn Sex;. TTT. the; addit.ie>nal contributions from the; trappe;d particle;

dynamics to the dispersion equation are described. The numerical results and comparisons

with experiments are; presented in Sec. TV, and Sex;. V is devoted to e;onclusie)ns.

2 Integral Dispersion Equation

We begin this section with a brief description of the gyro-kinetic integral equation14 derived

for the study of low frequency drift me)ele;s. sue;h as the; TTG mode. An impurity species is

included and the curvature and magnetic gradient effects U>D(V]_. VL 0) of both the hydrogenic

and the impurity ions are; retained. The ballooning representation is use;d NO that the; linear

mode coupling due to the two dimensional character of the tokaniak magnetic field is taken

into account. The; full ion transit k\\v\\ and the; finite Larmor radius e;fFe;cts are retained.

For simplicity, the passing electron response is assumed to be adiabatic . The response of

the trapped particles will be; discusseel in the; next se;ction. The; integral dispersion equation

derived in Hef. 1-1 is easily written (after having been extended to include the second ion

species) as

where

with

[l + nil - fV) + T/ZU] o(k) = 7=K{k, k')<p{k')'lTY

A'(M') = -* 2 e~iWT

X

X+ k12

2(1 + a )

e x p l-^

2(1 + fls)ZVs

A =

- /.)

+ 7v,; — - )/,:/!>; + —-z ( I

7 , r ^ r

A, =

(1)

(2)

a = 1 +Ti

. . *2e«

TZ

- sin 61') - x(6 cos 5 -

f 0 - * ' )'(.s + l)(smO - smO') - s(0cos0- O'cosO')

To = /"o

u, = ia

f* =

1,'i — 1.2 i 1,2 7,/2 _ 7,2 , 1/2"•_L — "•/;• ~ r "• 5 "•_!_ — " - j T "• i

L-T-*.-Tl '

T-, =

= —• Lci = —, Lc, = -

T h e qiiaTil.il.icK k.k' a n d kg ar<; Tioririalix<;(l l.o / J " 1 = Q-i/vn = ( BjV;\/2'7!;•/•»,;, :?: is Tioririalix<;(l

to p,;. and ij(j = 0.1) is the modified Bessei function of order j . The symbols with the

Hiibseript, L"r: (or wit.Tionl. any snbKcripJ.) sUnul Tor t.Tic: primary ion species (liydrogenie ioTis)

while those with ' V and ' V stand respectively for the second ion species and the electrons.

All other symbols have their usual meanings such as the Ln's are the density scale lengths,

n-o's are the unperlurbed densities, Lj'a are the t.eniperat.iire scale lenglhs, q is the safely

factor, 5" = rdq/qdr is the magnetic shear and io*t = ckgTrJeBLnr, is the electron diamagnetic

drift frequency. Z is t.Tie charge Tiiiinbcr of irnpiirily ions, m's and T'H arc t.Tie species' mass

and temperature, respectively. The derivation and details of this equation are given in Ref. 1-1

and will not. be repeated here.

it must be mentioned that not all of the above mentioned parameters are independent.

For example, t.he <.|iia.si-neiit.ralit.y condition imposes the relat.ion

_1 Jz

implying

nn<l<;r t.he aK

, . = (4)

3 Contribution from trapped particles

T h e calculation for t.he t rapped electrons IN st.raiglit.lbrwa.rd and well doc.innent.<;di>!li>~ls w}i<;n

t.he finik; gyro-radius efl'ect.s are n<;gl<;cted. Tn t.he ballooning r<;pr<;sent.ation, t}i<; pe r tu rba t ion

of the t r a p p e d electron densi ty can be wr i t t en as

TK V 7T Ju Ju u; — u;;j 4F(K) ;f^.o ./--.o

(5)

where

and

q(n, K) =- - sin" V

s i n 2

2F(K)

r /1 + Vr t ~

L V

+ 4S

-2

1 F(K) , 2

r is the minor radius associated with the rational surface. R is the major radius of the plasma,

column, and K(K.)( F(K.)) is t.Tic: complete elliptic integral of the first, (second) kind.

The response of trapped ions may be similarly obtained if the finite Larmor radius ef-

fects of t.Tic: trapped ions arc: neglected. Finally, we can write: the entire trapped particle

contribution to the dispersion equation Eq. (1) as

(a V n n 7Y Jo

(6)

where

\]f = *T; jj — LO*,

(1 + ir-a

tn(.!t1 + r + (rh + rr/i) it - -

4 Numerical Results

After adding the trapped particle contribution [Eq. (6)] to Eq. ( i ) . we numerically solve the

eigenvalue problem. Special attention must be paid to the logarithmic singularity at r = 0

when /; = k'.14 The numerical methods for solving Eredholm integral equations of the second

kind are quit.e standard and w<;ll documented.1'1

For this paper, we keep only the trapped electron dynamics although the equations given

in SCCK. 2 and 3 can be used for studying t.Tic: (.rapped ion effect.s too.

4.1 //.; threshold in the presence of trapped electrons

We first neglect the complicating effects of the temperature gradient of the trapped electrons,

rjfs = 0 in our sludy of the TTG mod<;s in t.Tic: prc:K<;ncc: of t.rapp<;d electrons. The normalized

growth rate and real frequency versus //,; are plotted in Eig. 1. The other parameters are

q = 1.5. t,j = 0.05, T = 1, /', = 0 and (. = 0.1. T'}i<; solid and t.Tic: daslied lin<;s ar<; for

s = —1.5 and s = 1.5, respectively. Tor the positive shear s = 1.5 (the dashed lines), there is

no l.Tireshold )/.; valn<; for t.Tic: TTG mode: to Tx; miKtablc;: the mode, referred to as the trapped

electron-?/,; 1( mode, is unstable even for //,; = 0. The rotation direction of the mode changes

from the ion direction to that of the electrons when )/.; decreases. This has t.Tic: implication

that when ??,; decreases, an effective transformation from an 1TC! to a predominantly I E

mode: takes place.

These results are similar l.o what, has been previously reported.17

Tor the negative shear case (for the same plasma parameters including the magnitude

of the magnetic shear), the inst.abilit.y l.hreshold is around r/- <-- 1. For lower values of )/.;,

the stabilizing effect of the negative shear overcomes the combined driving forces of the TE

and TTG. and t.hc mode is stabilized completely. In t.hc st.ill unstable region, i.e, r/i ^ 1, t.he

negative shear brings down the growth rate to less than a half of its value for the positive

shear. The stabilizing cfl'cct.s of negative shear are clearly shown in Fig. 1.

The situation changes if we switch on a constant temperature gradient i]n for the trapped

elections and l.hen sl.udy t.hc mode st.abilit.y as a I'mict.ioTi of •;/.,;. Graphs in Fig. 2 pert.ain to

the same plasma parameters as in Fig. 1 except that now, we have i]n = 2 and kepi = 0.3. As

long as one is concerned with the TTG mode alone, t.he temperature gradient, of t.he trapped

electrons has a strong stabilizing effect for ??,; < 1. The mode remains stable up to an //,;,, ~ 1

even for posit.ive inagnet.ic shear (.? = "1.5). This is to be contrasted with the results displayed

in Fig. 1( r/,:. = 0) for which there exists no //,; threshold. A finite trapped electron //,, leads to

similar consequences Tor negative magnetic shear: t.he growth rate is significantly decreased

to less than a half of that for r/,:. = 0. and the threshold value?/,;, is dramatically increased

from r/;,. *-> 1 when r/t = 0 t.o r/;,. ^ 2 when r/e = 2.

Again, the growth rate of the 1TC! mode in plasmas with negative shear is less than half

of t.hat. wit.h posit.ive shear when r/t = 2 and )/.; > r/;,.. Comparing wit.h the results obtained

previously.5 it may be concluded that the effects of the sign of magnetic shear are much more

significant when the trapped electron dynamics is taken into account than it is when it is

neglected.

An important consequence of the finite trapped electron r/,:.. in addition to enhancing the

driving mechanism of t.he TE mode, is t.o push it.s real frequency away from that of t.he TTG

mode decoupling it from the latter. This issue will be discussed in the next subsection in

detail.

4.2 Co-existence of ITG and TE modes

The possibility for the co-existence of FFCi and TF modes in collisionless toroidal plasmas

is explored in this subsecl.ion. For these sl.udies. the parameter t, determining t.he ratio

of the trapped electron, is increased to 0.3. and fully ionized carbon ions are added as an

impurity. The relevant parameters are: /- = 0.3. kop- = 0.5, q = 3, (.n = 0.1. r = 1,

rjz = rji. Lci = Lr/, = 1 and Tr = T,. The results are shown in Fig. 3 (1) with magnetic shear

7

Lei. UK fii'KJ. analyse; t.Tic: re;snlts for r/t = 0 in Figs. 3 and 4 (the; line;s vvil.li sejnares).

The results in Fig. -1 with positive magnetic shear s = 1 is similar to what is shown in

Fig. 1 for £ = 1.5 (the; clashed lines in Fig. 1). Tlie se)-calle;d trapped elee:t.ron-r/; mode is

unstable even for //,; = 0 in both these cases. However, the behaviour of the mode growth

rate ve;rsus ij- for ne;gative; shear £ = —1, displayed respectively in Figure;s 3 (OK; line wit.li

squares ) and 1 (the solid line with circles ) is quite different. Now, the mode is unstable

for the negative; shear (? = —1) even when r/;=0. unlike il.K counterpart in Fig. 1 where t.Tie

mode is stable for //,; ^ ??,> ~ 1. This dramatic change is due to the competition between

t.Tie Ktrong stabilizing effect, of the; negat.ive shear and the driving mechanism dn<; to t.lie

trapped electron dynamics represented by the parameter e. The suppression overcomes the

comparatively w<;aker (f. = 0.1) driving force; and the; mode becomes stable for r/; < 0.8 for

the Fig. 1 example. In contrast, in the Fig.3 example, the strong TE drive (e = 0.3) assures

inst.abilit.y even for no ion gradient, drive; (rji = 0). Nevert.heleKS. the; mode growth rat.<;s for

s = —1 are much lower than they are for a similar plasma with s = 1. For example, when

rji = 0. t.lie growth rate for ,s = —1 (Fig. 3) is less l.Tian a half of t.liat. for ,s = "1 (Fig. 4).

Next, let us consider the cases when r/,:. ^ 0. We shall study both the 1TC! and the TE

modes simult.aneonsly. For r/t = 2, it is clearly demonst.rat.<;d in Figs. 3 and 4 t.liat. t.lie TTG

(the curves with open circles) and the TE (the curves with closed circles) modes do not

couple, and evolve independently as r/- varies. There are; two modes ce>-e;xis(.ing in a narrenv

rji interval. 1.5 ^ ?/,; ^ 1.75 for s = —1, and 1.3 ^ ?/,; ^ 1.9 for s = 1, respectively, it is also

e:lear from t.lie dat.a (Figs. 3(b) and 4(b)) t.liat. with increaKing r/e, the; difference; be;twe;e;n t.lie

real frequency of the TE and the 1TC! mode becomes larger. This surely must contribute to

t.lie ele;ce)upling bet.vveen t.lie two inst.abilit.ie;s near the; r/; t.hre;she>ld.

We now make a small digression to remind the reader of an important experimental

re;snlt: the; observeel densit.y flue:t.uat.ie)n spee:t.rnm shift.s from the; e;le;ctre)n t.e> t.lie ion dire;ction

when t.lie plasma parame;ters e:liange;.13 De;pe;neling e>n t.lie paramet.e;rK, e;ithe;r a single; wing

unidirectional or a double winged bidirectional (with both the ion and the electron features

are; pre;sent. simultaneously) spe;ctrum is detected.12 It. is quit.e; obvious t.liat. in t.lie t}ie;e)ret.ical

results obtained in this work, there could lie a possible explanation for these experimental

observaf.ie>TiK.

The intervals where the 1TC! and TE modes co-exist (both of them are unstable si-

mult.aneonsly) pe;rt.aining t.e> the; caseK slienvn in Figs. 3 an el 4 are t.e>o narrow to war-

rant comparison with experiments. After a careful parametric search, we have uncov-

ered a much bre>ader ele)main for t.lieir simult.aneons e;xist.e;ne:e. For simplicity, we; she)\v

in Tig. 5 the results for a positive magnetic shear S = 1.5. The other parameters are

rjfs = 2. q = 1.5, T = 1, k.Qpi = 0.3, (.n = 0.1 and /', = 0. The study is done for two

values of the trapped electron parameter t: = 0.1 (the dashed lines) and t: = 0.3 (the solid

linos). Tl. is clear t.Tiat. the co-existing intervals are 1 r/i ^ 2.5 and 1 ^ )/.; S 2 for c. = 0.3

and e = 0.1. respectively. The //,; values in these intervals are reasonable and common for

lokarnak plasmas. The scaling for the driving mechanism froin l.he trapped electrons for l.he

TE mode (the lines with squares) clearly emerges from Tig. 5(a); the growth rate for e = 0.3

is t.wice as high as that for i. = 0.1. One can also infer from l.he figures that l.he trapped

electrons have a marginal effect on the ITC! mode. The ion temperature gradient (//,), on

l.he ol.her hand, has a very strong stabilizing effect, on the TE mode when )/.; <L 2.

4.3 Spectrum study

The spect.ra of t.he TTG and TE modes are given in Eig. 6 for •;/.,: = 2. £ = 1.5. j / e = 2. q =

1.5. e,,. = 0.1, r = 1. t: = 0.3, / , = 0. The lines with squares and circles are for the ITC and

TE modes, respectively. T'}i<; solid and dashed lines are respectively for l.he real frequency

and the growth rate . The maximum growth rate of the kinetic ITC.i mode in toroidal plasmas

occurs at. kofH — 0.8 when the trapped electron dynamics is neglect.edM; it. is pushed to a

higher value by the trapped electron effects. The growth rate of the l'TC mode is always

higher than the TE mode for all wavelengths and the plasma parameters considered in this

study.

It is now worthwhile t.o examine l.he relevance of t.his effort, t.o experimental observations.

We begin by noting that the experimental frequencies are measured in the laboratory frame

while l.he theoretical frequencies are calculated in the plasma frame. There; may exist., for

example, a Doppler shift between the two frequency spectra caused by a plasma rotation

in l.he poloidal direction induced by a radial electric field. The calculation of l.he Doppler

shift is rather troublesome due to the uncertainties in the measurement of the electric field.

Eorl.nnat.ely, it is st.ill possible; t.o compare; l.heory with e;xpe;riment. by ele;aling with frequency

shifts between two typical frequencies rather than the frequencies themselves.

The; first e;xpe;riment.al e)bse;rval.ion e>f interest, is l.he; appearance, in l.he; microtnrbulence

spectrum, of a new peak12 at positive frequency (in the ion diamagnetic drift direction)

besides l.he peak in l.he electron direction when l.he; plasma ele;nsil.y increases. The; frequency

shift between these two peaks may be compared with the theoretical results given in Tig. 6(b)

as follows. The; experimental result.s (sex; Eig. 1 of Ref. 12) show l.hal. the shift, between l.he

mean frequencies of the two peaks (in the density fluctuation spectra) increases with the

increaKe in t.Tic: wavemnnber. The results presented in Fig. 6(b). a.lKO show that t.Tic: frequency

shift between the 1TC! and TE modes increases with the wave number of the fluctuations.

Tli IK Kimilarit.y in Healing may suggest, that t.Tic: two-peak experimental feature in ay be viewed

as the simultaneous existence of suprathermal levels of the 1TC! and the TE modes.

Nat.nrri.lly . a. detailed or a. quantitative comparison with experiments is beyond the scope

of the present work. Nevertheless, a rough comparison may provide a stimulant for further

KJ.ndieK. Tlie typical measured fluctuation spect.rnm is in the 100 kHz range and t.Tic: frequency

difference between the ion and the electron features is of the same order, it is shown in Tig. 6

t.liat. t.lic: theoret.icri.lly calculated difference between t.Tic: frequencies asKocia.l.ed with t.Tic: two

instabilities is comparable to the electron diamagnetic drift frequency LO*^. The electron drift

frequency measured in the high density experiment is U,'*,-./2TT ^ 2 X 10" Hz.12 Tlic: agreement,

between the theoretical calculations and the experimental observations is reasonably good.

Similar spect.ra. for t.lic: negative ma.gnet.ic Kliear, £ = —1.5, arc: shown in Fig. 7. Here, t.lic:

1T(J mode (the lines with squares) is studied with //,; = 2.5 while the TE mode (the lines

with circles) is inveKtigat.ed wit.li )/.; = 0.75. Thene r/; valuer are chosen KO t.liat. bot.li the T F

and the 1TG mode are optimally unstable, it should be noted that the TE mode becomes

almoKJ. Ktable wlic:n )/.; = 2.5 and so doeK the TTG when )/.; = 0.75. In fact., for nc:gat.ive shear

plasmas, it is quite difficult to find a broad enough parameter domain where both the TE

and TTG are miKtable. Thin shows, once: again and from another point of view. that, pla.Kinas

with negative magnetic shear are more stable than those with positive shear with respect to

KUCII micro-inKtabilitieK a.K TTG and T F modes.

5 Summary and Conclusions

The gyro-kinetic integral eigenvalue equation for the study of 1TG modes in collisionless

toroidal plasmas is extended to include: the trapped electron dynarnicK. The study concen-

trates on delineating the following novel features: 1) the interactions and couplings between

t.Tic: standard T F and TTG inKtabilitieK, 2) the effect of negative magnetic shear on t.Tic: sta-

bility of these modes. It is found that these two instabilities are strongly coupled when the

temperature gradients a.K well a.K the: driving mechaniKm of the: trapped electronK are: weak

but not negligibly small, in this case, it is impossible to distinguish between them, and only

one miKtable mode:, conveniently called the: trapped elect.ron-r/; mode: iK found to exist. Tn

this regime, there is no ??,; instability threshold for plasmas with positive magnetic shear. For

negative shear, the mode: iK Ktable up to r/;,. ^ 1, and for larger valueK of)/.;, the growth rates

are considerably reduced.

10

When both the temperature gradient and the driving mechanism of the trapped electrons

are moderate or strong, t.Tic: coupling between these I.WO instabilities is weak and they may co-

exist in plasmas. In this case, the threshold value for the ITC! instability is ??,; ~ 1 (??,; ~ 2) for

.? = 1 ( —1). Tn addition, the driving mechanism of the TE mode is enhanced by t.Tic: trapped

electron temperature gradient for //,; ^ 1, and is suppressed by ITC! when //,; ^ 2. However,

t.Tic: effects of t.Tic: trapped electrons on t.Tic: TTG TVKXIC; ar<; n<;gligibly small vvTic:n )/.; > rj-c.

Tor the same set of plasma parameters (including the magnitude of the shear), the growth

™I.C:N of t.Tic: TTG mid TE TVKX]<;S in plrismaK with positive: inri.gnc:t.ic:. SIK^IT ar<; twice: a.K liigTi

as the growth rates in plasmas with negative shear. The stabilizing effects of the negative

Hlic:a.r a.re dc:finit.ivc:ly shown to Tx; Tinic:li more: significant, when t.Tic: TE dynamics IN tak<;n

into account.

We TxTievc: tliat it is the first, lime that the co-exisl.encc: of an ion (t.Tic: TTG), and an elec-

tron mode f TE mode) in realistic collisionless toroidal plasmas is theoretically demonstrated.

TTIIH highly dc:sira.ble result should lend considerable; experiment.;!! significance: t.o our model.

Einally, we make an important prediction: the possibility of a simultaneous detection of the

ion and electron features in the density fluctuations in plasmas with negative magnetic shear

is much lower t.ha.n it. is in plasmas in which t.Tic: shear is positive.

Acknowledgments

One of the authors (J.Q.D.) would like to thank the staff at the International Centre for

Theoretical Physics. Trieste, Ttaly Tor their hospitality. Helpful discussions with Dr. Y.Z.

Zhang are gratefully acknowledged.

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12

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13

Figure Captions

1. (a) Mode growth rate and (b) real frequency versus //,; for i]n = 0, q = 1.5, e,,, =

0.05. r = 1. / , = 0 and r = 0.1. The solid and da.Klit-<] lines are for 3' = —1.5 mid

5 = 1.5, respectively.

2. Tlie same as Fig. 1 except, l.lial. r/t = 2 and Ar,j ; = 0.H

3. (a.) Mo<]<; grovvj.li raLc and (b) r<;a.l fr<;(]iH:ncy V<;TKIIK r/- for A:( j; = 0.5, t/ = 3. f.n. =

0.1. r = 1. e = 0.3. / , = 0.3. r/,r = ?/,;, Lci = i-,-? = 1- and carbon fully ionized as

impurity. The magnetic nlicar ,? = —1.

1. The same as Fig. 3 except that .s = 1.

5. (a.) Mo<]<; growt.Ti rak; and (b) r<;a.l rrc<.|ii<;iicy V<;TKUK r/- showing t.Tic: co-cxiKJ.cncx; of l.lic

1T(J and TE modes. The parameters are i]n = 2, s = 1.5. q = 1.5. r = 1. kepi =

0.3. <•.„ = 0.1 and / - = 0. The squares i-nid t.Tic: circles arc for j.}i<; TE and TTG modes,

respectively.

6. The spectra of TE (the lines with circles) and 1TC! (the lines with squares) modes for

,;. = 2, .? = 1.5, rit = 2, q=\ .5, i.n = 0.1, r=1, c = 0.3, / , = 0.

7. The same as Tig. 6 except that s = —1.5. and ??,; = 0.75 (TE mode) and 2.5 (1TC.I

mode), r<;specl.ively.

14

D.35

0.30

0.25

0.15t

0.10

0.05

0.00

0.40{

0.30

0.20

0.10

0.00

0.100

Fig.l

! I < > _ L .

na

: — -e —

- - -a -

^ -Q— ^

"" I1

\ ;

- , , , , i

.0 O.i

kepr0.5kepro.3

keP|-0.5

kepro.3

' 1

X

B B—

, , i 1 ,

3 1 . 0

a-

s.

—e-—a

i | : ;

1 1 11 1 1

x

i i , ,

1.5

i i . i .

o

i i i

B,

ft. „

• i i i

2.0

t ' ! ' > _

* (a ) \

-

—

—

( b ) :-

-

-

-

— H _

I • • • . "

2.5 3

15

0.50

0.40

* e 0.30

0.20

0.10

0.00

-0.02

-0 .04

co/rar

-0 .08

-0.10

-0.12

-0 .14

• I 1 I

f*l _

— -B —

e=0

e=0e=0

.3

.3

.1

( a )

MM

0.5

Fig.2

MM

( b )

i , i i i i i i , i i

1 .0 1.5 2.0 2.5 3.0

16

0.20

- 0 .0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig.3

0.25

0.10

0.05

0.00

0.20

CO /CO ° - 1 0

r *e

0.00

-0,1 0

-0 .20

I I I ! I ! ! I I i I i I I I I I I I

(b )

L _ . i t_. t ! t I I I .1 i.. L i i I i i i i

Fig.4

0.0 0.5 1.0 1.5 2.0 2 .5 3.0

n

18

0.5

0.4

e 0.3

0.2

i s

1

rt

— - ^ —

£ =

e=e=e=

i •

.3

.3

.1

.1

0.2

0.0

- 0 . 2 ! r 1 ' I I. 1 1 r [ 1 i 3 I r i l j .. I

0.0

Fig.5

0.5 1.0 1.5 2 . 0 2 . 5 3 . 0

n

19

cok p/coeri * e

0.6

0.4 -

0.2 -

0.0 -

-0 .2 -

- 0 . 4 -

- 0 . 6

Fig.6

0.3

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6

cok p/co

-0 .40 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6

k pFig.7

20