J. Plasma Fusion Res. SERIES, Vol.4 (2001) 322-325

Electron Heating by Nonlinear Landau Damping ofElectrostatic Waves in an Electron Beam-Plasma System
MAEHARA Tsunehiro", HASHIMOTO Ryuji, KAGAYAMA Yoshiaki, UTSUNOMIYA Shoheir,
SUGAWA Masao and SUGAYA Reiji
Ehime University, Matsuyama, Ehime 790-8577, Japan
Ishinonome Junior College, Matsuyama, Ehime 790-8531, Japan(Received: 5 December 2000 / Accepted: 18 August 2001)

Abstract
The large axial inhomogeneity of electron temperature due to nonlinear Landau damping ofelectrostatic waves (space chargewave and Trivelpiece-Gould mode) in an electron beam-plasma system
is occurred. At the same time, inhomogeneity of the space potential is also observed. We evaluate andcompare the forces to the plasma electrons from the pressure gradient and the inhomogeneity of the spacepotential. This suggests that the two forces are balanced and the space potential plays an important roleon maintaining the inhomogeneity.

Keywords:
nonlinear Landau damping, space chargewave, Trivelpiece-Gould mode

1. Introduction

then plasma electrons obtain a small drift velocity.
Hence, large axial inhomogeneity of the electron
temperature and density are occurred and maintained.
However, the evaluated value of the drift velocity fromthe profiles of the electron temperature and the electron

Nonlinear Landau damping is one of the mostinteresting phenomena in weak plasma turbulence. This
process has been investigated theoretically [1-5] andexperimentally [6-121. In ref. llll, Sagaya et al.
reported detailed investigations for explosive

beam excitethe Trivelpiece-Gould (T-G) mode via
nonlinear Landau damping with beam electrons.
Because the slow mode of the space charge waves is anegative-energy wave, it excites an explosive instability
and grows rapidly together with the T-G mode. The
scattered T-G mode causes an electron heating and an
energy transport via its quasi-linear electron Landau

density was not reasonable I l].
In recent years, the relationship between 'potential
and structure' has been investigated as one of the mostimportant subjects in fusion science. From the analogy,
we suspect that the potential has also something to dowith the inhomogeneity in our small linear device.
Therefore, we start to measure the space potential aswell as electron temperature and density. In this paper,
we evaluate the drift velocity by considering the force
balance between the space potential and pressure
gradient. Thus, we discuss the role ofthe space potential

damping. Namely, energy and momentum of thefor maintenance of the axial inhomogeneity by nonlinear

electron beam are transferred into plasma electrons, andLandau damping.

instabilities by nonlinear Landau damping ofelectrostatic waves in electron beam-plasma system: The
extemally launched space charge waves of the electron*Corresponding author's e-mail: maehara@ sci.ehime-u.ac.jp322@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchMaehara T. et al., Elecfton Heating by Nonlinear Landau Damping of Electrostatic Waves in an Electron Beam-Plasma System

2. Kinetic Wave Equations and Transport[1]: The externally launched fast and slow spacecharge waves of the electron beam (46, ke) excite theTrivelpiece-Gouldmode (al, kr) via nonlinear Landau
damping when the resonance condition, @o - or - (ftrro k11t)ut = ma)ce, is satisfied, where k11o and ft11r are thecomponents of wave vectors parallel to the magnetic
field, u6 is the velocity of the electron beam, al", is theelectron cyclotron frequency, and ru is an integer. Theslow space charge wave excites an explosive instability
and grows rapidly together with the T-G mode because
it is a negative-energy wave (Ss < 0) as shown in eq. (1).
The scattered T-G mode causes an electron heating andan energy transport along a magnetic field via its quasilinear Landau damping (eq. (4)). Namely, energy andmomentum of the electron beam are transferred intoEquations
The kinetic wave equations are expressed by [a,5]Art,,) Att,,,

E,,l', 7' -drurlE^f (l)#=-so0o%ldt
dtHere, ors, o1 are the nonlinear wave-particle coupling

coefficients, E, is the wave electric field, U is theabsolute value ofthe wave energy density, 56 is the signof wave energy. The subscript 0 and I denote the spacecharge wave and the Trivelpiece-Gould (T-G) mode,respectively.
The transport equations for energy and momentum
densities of beam and plasma electrons are given as

follows [4,5]:o'ua^u^le..,l'-'u -u l"@t | 'd)0
dt =@lQ)k,,I t2.
=,"'0'o4rolE,,l_""
Kilo
dt

(3)

electron temperature.

Efl, ,(J,,+=-2T,dt

plasma electrons by nonlinear Landau damping, andthen plasma electrons have a small drift along themagnetic field. This leads to the axial inhomogeneity of

3. Experimental Apparatus and Results
In this experiment, a monoenergetic electron beamAP,=- 2Tt P|t'--dt(4)with energy Ua = 3OO eV, current Io = 2 mA, anddiameter Dt = 2 mm is continuously injected along awhere P11 = (krlo)U is a momentum density of the wave,uniform magnetic field into a beam-generated plasma atargon gas pressure of 0.4 mTorr (Fig. l). The diameterof plasma is Do = 15 mm. The electron cyclotron
frequency is adjusted to a""l2n = 190 MHz. The fastand slow beam modes are excited by applying an
external RF signal with the frequency (4 = 275 MHz

y1 is the linear damping rate and U", Px are the energy

and momentum densities for beam (s = b) and plasma
electrons (s = e).
In the present work, energy and momentum transfer
by nonlinear Landau damping are represented as follows

Diffusion

ffixx

Diffusion

Coil

ffi/x
Collector

xxffi
XXffi

Probe

Pump

Vacuum Chamber(0l00mm x 1500mm)

tAr
Electron GanPo

t=o.5w

f=2i75MHz

Data Acquisition SystemFig. 1 Schematic view of the experimental apparatus. The uniformity of the magnetic field is 2.7 percent over an axial
length of 60 cm.
J25

Maehara T. et al., Electron Heating by Nonlinear Landau Damping of Elecffostatic Waves in an Electron Beam-Plasma System.

and the power P0 = 0.5 W. This parameters conespondto one of the best conditions for occurrence of the local
heating [1]. All data are obtained by the Langmuir-.- on o otl

ne1t oecm-3)

probe.

Figure 2 shows that (a) electron density n", (b)electron temperature 2,, (c) energy density n,T" and (d)space potential V" versus axial distance z from theTeelectron gun. The electron density is almost constant in(ev)1the range of = 2-3 x lOe cm-3 and is not changed somuch by applying the external RF signal. On the other1

hand, the electron temperature rises at e = 10 and = 13
cm by applying RF signal. Figure 2 (c) shows thatenergy density also becomes large at z = l0 and = 13cm. This shows that local heating occurs. At the sametime, the space potential increases at z = 10 and = 13 cm
as shown in Fig. 2(d). This suggests that the spacepotential has something to do with maintaining theinhomogeneity of the electron temperature.

f,"T"(10eeV cmvs(v)4. Discussion
It is predicted from eq. (4) that plasma electrons are

16

drifted by momentum input from T-G modes, and thisz (cm)drift maintains the inhomogeneity. Figure 2 (d) suggeststhat contribution from the space potential to maintaining

Fig. 2 (a) Electron density, (b) electron temperature, (c)the inhomogeneity is not negligible. In order to discuss
the role of the space potential, the drift velocity ofenergy density and space potential versus axial
distance z (cm). Closed circles with lines denote
applying the external RF power. Open circles
denote no RF power.

plasma electrons is evaluated. We deduce the drift
velocity from the fluid equation of motion, dvldt =-lelr"n - Y(n"T") - menevev by neglecting thecontribution from r-components and vanishing the termof dvldt:

D7=Dr*1)2(5)u,=&il\lt(6)u,=&rtY,,a)discuss by comparing t2 with the velocity of the beamelectron u6. The energy of the beam electron (300 eV)corresponds to the velocity of 10 x 108 cm/s. If thecontribution from V" is not considered, I u7 | = | ,, t t
u6, which is unreasonable. On the other hand, inconsidering the contribution from V", the obtained valueof lual = 0-+ x 108, which is somewhat larger than| ,, | * or, may be reasonable because we have large
roughness of the fittings. Hence, we can point out thatthe force from the space potential is balanced with theforce from the pressure gradient and plays an importantrole on maintaining the inhomogeneity of the electron
temperature in our experiments.

where v" is the electron-neutral collision frequency. Thisequation shows that the drift velocity consists of twoparts. One is ur, which corresponds to the pressure

gradient. The other is u2, which represents thecontribution from the space potential. Since eq. (6) and5. Summary(7) include the derivatives, we fit the experimental databy polynomial (Fig. 3). By using this fitting and v, - 10MHz, we evaluate the velocity, u1, o2 and ua Gig. 3(c)).
Since n"T" and V" show the similar behavior (Fig. 3(a),(b)), ur and u2 are almost symmetrical to the horizontal
axis (u = 0), and I u, I b""o*.r small. Here, we simply

In an electron-beam plasma system,
inhomogeneities of the electron temperature and spacepotential come from nonlinear Landau damping ofelectrostatic waves. We evaluate the drift velocity ofplasma electrons by considering the forces to the plasma324

Maehara T. et al., Electron Heating by Nonlinear Landau Damping of Electrostatic Waves in an Electron Beam-Plasma Systemelectrons from the pressure gradient and the spaceOh"T"potential. This suggests that the two forces are balanced(a)and the space potential plays an important role on
maintaining the inhomogeneity.

(loeeVcm{)

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Fig.3 (a) energy density and (b) space potential versus
axial distance z {cm). Closed circles denote

Phys. Soc. Jpn.64,2018 (1995).

experimental data for applying RF power. Curves
denote the lines fitted by using the polynominal.
(c) The drift velocity (solid curve) versus z (cm).

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Dashed and dotted lines show v, and vr,
respectively (see eq. (5)).

325