J. Plasma Fusion Res. SERIES, Vol.2 (1999) 16l-164

Effect of a Transverse D.C. Electric Fields on Electrostatic
lon-Cyclotron Wave Instability
SHARMA Suresh C., SUGAWA Masao and JAIN V.K.r
Department of Physics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan
tPhysics Group, School of EnvironmentalSciences, Jawaharlal Nehru (Jniversity (JNU), New Delhi-| 10067, India(Received: 7 December 1998 / Accepted: 15 April 1999)

Abstract
The temporal evolution of the current-driven electrostatic ion-cyclotron (CDEIC) instability isinvestigated in presence of a transverse d.c. electric fields in a collisional magnetized plasma. The growthrate of the instability has the largest value at the mode frequency for the appropriate magnetic field(where the effect of a transverse d.c. electric fields is larger) when the electron drift velocity is less thanthe critical value for the CDEIC instability. The growth rate is also a sensitive function of electron
collision frequency.

Keywords:
ion cyclotron waves, d.c. electric fields, electron collision frequency, growth rate

1. Introduction

in a single-ended Q-machine. Hatakeyama et al. 16)
have presented a new measurements and mechanism of

The first experimental observations of electrostatic
ion cyclotron oscillations in a laboratory plasma were

the ion cyclotron oscillations driven by applying a
positive potential to the small electrode immersed in amagnetized plasma. Ganguli and Palmadesso [7] havedriven by applying a positive potential to a small
electrode immersed in a uniform magnetic field in asingle-ended Q-machine []. Several measurements offound that the inclusion of a transverse localized electricthese oscillations and related phenomena were madeunder almost the same configuration and the results
were discussed on the basis of current-driven instability[2]. Hatakeyama et al. [3] made measurements on thefield in addition to the magnetic field aligned current

electrostatic ion cyclotron oscillations in nonuniform
configuration as in ref. [] except that the axial profileof the magnetic field. The oscillation frequency wasobserved around the ion cyclotron frequency. Later
Ganguli et al. [4] have presented a new kinetic ion
cyclotron instability driven by a nonuniform electricfield perpendicular to the external magnetic field, which
could be of importance to the situations referred to ref.

driven electrostatic ion-cyclotron instability has been
investigated in a Q-machine by Koepke et al. [8]. In thiscase, the ion cyclotron fluctuations were observed inpresence of a transverse, d.c. electric fields when theelectron drift velocity was less than the critical value for
the CDEIC instability. Recently Amatucci et al. f9)have
reported laboratory observations of reactively driven
plasma waves in the ion cyclotron frequency range
associated with a localized, transverse electric field.

[3]. Schrittwieser et al. [5] have observed electrostatic

More recently, Koepke et al. [10] have observed

lowers the threshold for excitation and changes thedispersion characteristics ofthe waves. The influence oftransverse. localized. d.c. electric fields on the current-magnetic field under the same experimental

ion-cyclotron instability driven by a slow electron drift@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchl6l

Sharma S.C. et al., Effect of a Transverse D.C. Electric Fields on Electrostatic lon-Cyclotron Wave Instability

multiple eigenmodes of the inhomogeneous energydensity driven instability with large and comparable

nl"n,"='-a lfamplitude. In this paper we develop a local theory of
current driven electrostatic ion-cyclotron instability inpresence of a transverse d.c. electric fields in acollisional magnetized plasma. Koepke et al. I8l havenot taken into account transverse d.c. electric fields intheir theoretical analysis. We have studied this effectek2,-Q 1ro +iv.-mf(a +iv.72 - a!1

k2r1a +iv.1
n,^T.
+_['^ ^I(o +i:4)2 - a|l nZp*!: +iV,) r.
- '!:o+iV") m ,=01!,",m(Am1A

(s)

no""'

using fluid theory.
The response of plasma ions can be obtained from Eq.
(5) by replacing -e, m, c, T" and va"by e, mi, (D"i, Ti andzero, respectively. Also we put vi = 0 in Eq. (5) for ions,
we obtain

2. lnstability Analysis
Consider a homogeneous collisional plasma ofequilibrium density ,?3p, temperature T(= f" for electrons
and = 4 for ions), collisional frequency v(= v" forelectrons and = 0 for ions) immersed in a static
magnetic field i, ll 2. We introduce a transverse, d.c.
electric fields E, in the x-direction. This initiates a driftof magnitude ve = cEol8, in the y-direction. Moreover,electrons have a drift (va") along the magnetic field
direction (i.e., z-axis) and we look for the excitation ofion cyclotron waves in the y-z plane. The equilibrium is^'115,**etlQ-*L ll_ rrJ^ Jr;tk,.

-f^tb, ^,77_(6)

I',kiL . T, k:,*=il-,nZ,(l)

t"Y-!z).

T, k21n,,_ ar,)*

where ri = a - kuvp.. For electron response Eq. (5) is
modified as

perturbed by an electrostatic perturbation
Q = Qog-i(t'-,k'ry

nrt= nirl^r*.rvL1*

The response of the plasma electrons to the perturbationis governed by the equation of motion which on
linearlization yields the perturbed velocity

nr"
nZp=ftt",^ t"*.a:*],eQa)(8)T"U - -t!L)
x1t.tm.

Equation (6) can be rewritten as,E rt
V '-=--acl(o +iv")2 - otll+"Tk..a\flti+,

im t(ar+iv)'-a?l nipvlv --'tmeE,.tl.

el1(ar+iv)'z-az;"k"-Qm,(b'- az"r)Tr[lT'k1r-l

mt gb|(e)- t';l,y'

we have ignored axial wave vector ft. in Eq. (9) because
Et>>k? in the present analysis. Using quasi-neutrality
approximation nplnf;o = nrlnSp, we obtain e,(o, k) +(a +iv")[(a +iv")2 - afi]kr(a +iv")
T+"mnZp

iet(a, k) = 0, wherenr"n"*€r(a.k)=(3)

C\'je +T.\kz,' ri"\r
mi-(b'-a?),€i(o, k)=- v"amk"

*,k1.

(10)
(1 1)

writing ot = (4 + iTand assuming that the wave is either
weakly damped or growing (i.e, lyl << a4), the real partof the frequency a4 is given by

where cl = a) - krvp, - krva.; -e, m and @"(= eB"lmc) are
the electron charge, mass and cyclotron frequency, respectively, and subscript I refers to perturbed quantities.
Using Eqs. (2), (3) and (4) in the continuity equation,
we obtain the perturbed electron densitye,(o-a,,k)-0,r62(r2)

Sharma S.C. et al., Effect of a Transverse D.C. Electric Fields on Electrostatic Ion-Cyclotron Wave Instability]^-a,=krv"+{ a?,+kic!,(13)where c] = (f" + T)lmi. The imaginary part of rhe
frequency is given byti(a, k)de ,(at,, k) lda,v"U(l +R)l mk1
I2-+0-+)

m,k?'Io(14)>where R = krvBlkrvt.. lf E" - 0 (in absence of atransverse, d.c. electric fields), i.e., R -+ 0 we recover
expression for the growth rate of Amatucci [11].
xl.0s

3. Results and Discussions

12In the calculations we have used plasma parameters

Fig. 1 Growth rate of the ion cyclotron wave instability(in sec'') as a function of mode frequency ar,(inrad./sec). The parameters are given in the text.

for the experiment of Koepke et aI. [8]. Using Eq. (14),
we have plotted in Fig. I the growth rate (in sec-r; of theion cyclotron wave instability as a function of mode
frequency a4 (in rad./sec) for the following parameters:
finite gyroradius parameter s = 1.0, temperatures T" = Ti
= 0.2 eY,ion sound speed c, = 9.55 x l}a cn/sec, mlmi
= 1.3 x l0-5 (potassium), guide magnetic field B. = Q. l
kG-4.0 kG, plasma density nlp = | x 10e cm-3, electron
collision frequency v"=2.24 x 105 sec-l, electron drift
velocity rd" = 6 x 107 cm,/sec, k.v4"/r = 0.75 (i.e., theelectron drift velocity v6" is less than the critical value
at lk,for the CDEIC instability), drift due to transverse
d.c. electric fields vs = EolBu where E, = 0.35 statevolt
cm-r and mode frequency (4 = 5 x 105 rad./sec to 12 x

fields. In the absence of transverse, d.c. electric fields
the growth rate has the same value as obtained by

Amatucci Il].

The growth rate of the instability

increases with the mode frequency and has the largest

value at the mode frequency for the appropriate
magnetic field (where the effect of a transverse d.c.
electric fields is larger) when the electron drift velocity

is less than the critical value for the CDEIC instability

in compliance with the experimental observations ofKoepke et al. [8]. The frequency of the ion cyclotron
waves increases with the magnetic field [cf. Eq. (13)].

105 rad./sec. From these parameters the value ofR turns
out to be << l. For R << l, the case of large v6" and small

Acknowledgement This work was partially supported by
the Ministry of Education, Science and Culture, Govt. ofvB, the electrostatic ion cyclotron waves are CDEIC

instability. The growth rate of the instability increases
with the mode frequency and has the largest value at themode frequency for the appropriate magnetic field(where the effect of a transverse d.c. electric fields islarger) when the electron drift velocity is less than thecritical value for the CDEIC instability. This result issimilar to the experimental observations of Koepke et al.
[8], where the ion cyclotron waves were observed inpresence of a transverse, d.c. electric fields when theelectron drift velocity was less than the critical value for
the CDEIC instability. The growth rate of the instability

Japan.

is also a sensitive function of electron collision

t4l G. Ganguli, Y.C. Lee and P. Palmadesso, Phys.

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Sharma S.C. et al., Effectof a Transverse D.C. Electric Fields on Electrostatic lon-Cyclotron Wave Instability

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