1C/96/165 United Nations Educational Scientific and Cultural Organization and international Atomic Energy Agency INTERNATIONAL CENTRE EOR THEORETICAL PHYSTCS COUPLING OF in AND TRAPPED ELECTRON MODES IN PLASMAS WITH NEGATIVE MAGNETIC SHEAR J.Q. Dong Inlermi.tioTml Centre for Theoretical Physics. Triesl.e, TUi.ly and Southwestern Tnsl.itnte of Physics, P.O. "Box 432 Chengdu, 610041, People's Republic of Cliina and S.M. Maliajan TTISI.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 trapped electron (TE) modes are shown to be weakly (strongly) coupled when both the temperature gradient, and the driving mechanism of t.lie l.rapped eleet.rons are moderate or strong (weak but. (mile). Tn t.lie regime of st.rong coupling, l.here is a single hybrid mode, unsla.ble for all rji in plasmas with positive magnetic shear. For the weak coupling case, two independent unstable modes, one in the ion and the other in the electron diamagnetic direction, are found to coexist. In either situation, the negative magnetic shear exerts a strong stabilizing influence; 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 excite the two modes simultaneously in a plasma with negative shear. The results of this study are significant for tokamak experiments. M1RAMAUE- TRIESTE September "1996
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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