J. Plasma Fusion Res. SERIES, Vol.2 (1999) 381-384

Nonlinear Plasma Maser Driven by Electron Beamin Magnetized Plasma
NAMBU Mitsuhiro, SAIKIA Bipul J.r, GYOUBU Daisuke2 and SAKAI Jun-ichi2
Tokyo Metropolitan Institute of Technology, Hino, Tolcyo 191-0065, Japan
tCentre of Plasma Physics, Saptaswahid Path, Dispur, Guwahati 781 006, India
2l,aboratoryfor Plasma Astrophysics, Faculty of Engineeing,
Toyama University, Toyama 930-8555, Japan(Received: 9 December 1998 / Accepted: 5 August 1999)

Abstract
Simulations using a two dimensional electromagnetic and relativistic particle code show that highfrequency electromagnetic waves with right-handed polarization (R-mode) can be generated by a plasma
maser mechanism from electrostatic Langmuir waves excited by electron beam.

Keywords:
plasma maser, beam instability in magnetized plasma, Langmuir wave, right-handed polarized
electromaqnetic wave

1. Introduction

beam velocity Z6 and a homogeneous external magnetic
field are oriented in the z-direction. It is well known thatPlasma maser is a new nonlinear process in plasma
turbulence which coexists with the quasilinear process

an electron beam along a magnetic field in the z-

between electrons and the resonant mode. Since theprediction of the process [], it has attracted muchattention because multi-modes turbulence whichcontains enhanced high frequency fluctuations inaddition to low frequency turbulence are quite often
observed in laboratory and astropysical plasmas [2].

direction becomes unstable to excite Langmuir waves.
Solid lines in Figs. l(a) and l(c) show the time historyof electric field energies E2, and El for a case of ao"ld)"

Recently, the comparison between theory and simulation
was reported for the plasma maser process for Langmuir
waves [3] and electromagnetic waves [4] in the presence

ns are the electron plasma frequency, electron cyclotron= 10 with a beam of velocity va= 0.7 c, density nblno l/30, while the dotted lines show the time history for acase without the electron beam. Here, a*, d)", n6, andfrequency, electron beam density and backgroundplasma density, respectively. As seen in Fig. 1(a) andFig. l(c), both components of the electric field areof whistler turbulence.

strongly excited due to the beam instability. This meansthat the Langmuir waves are propagating obliquely tothe external magnetic field. In Fig. 1(b), the solid line
shows the time history of the electric field energy Er2 in
the presence of the beam, while the dotted line shows
the case without the beam. As seen in Fig. l(b), theelectric field E, grows after the electric field energy

2. Numerical Simulation
By using TRISTAN code [4], we present the resultsof the numerical simulations on the generation of Rmode electromagnetic waves in the presence ofLangmuir mode turbulence driven by electron beam (TlTv= 2), where 7 is the backgound temperature and Z5 isthe beam temperature. Both the electron beam with a@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author' s e-mail: mnambu@ cc.tmit.ac.jp381

Nambu M. er a/., Nonlinear Plasma Maser Driven by Electron Beam in Magnetized Plasma

associated with the beam instability becomes large.

frequency of the electromagnetic wave. We cut the low
frequency region (crr < l.63ar"), as shown in Fig. 2(a),and then perform the inverse-Fourier transformation to
obtain the time histories for the excited high-frequency
waves. As seen from Fig. 2(b), the high-frequency

To investigate the behavior of the electric field E",
we perform two-dimensional Fourier transformations intime and z which is parallel to the external magnetic
field. The results show there appears two branches ofelectromagnetic waves. The first one has broad spectrumwith maximum frequency lying around 2a* and kr= K,.
Here t. and K. are the wavenumbers parallel to theexternal magnetic field of Langmuir andelectromagnetic waves with a/ar, = 2.0 along amagnetic field can be excited only after about 35aro"/.
From the above observations we may conclude thatelectromagnetic waves propagating parallel to amagnetic field can be excited by the trigger of theelectromagnetic waves, respectively. The second branch

is observed around (D = o)pe and, ck./ar. = 1.7. The
second excited branch can be explained with the help ofCherenkov emission process by interaction with theelectron beam [5]. But the first branch can not beLangmuir waves due to the electron beam instability.
The dependence of the whole electron distributionfunction/6" on y' y) and v., are shown by Fig. 3(a), (b)and (c), respectively. In Fig. 3, for example, /s"(v,)
shows the distribution function integrated over allparticles with different vr and v.. i.e. /s"(v,) = | Ifo,O,,
vn,v,)dvrdv.. The dotted and solid lines show the whole

explained by the usual emission mechanisms because
here {21K,= l.2c and (Q - Q.)lK, = 1.1c. Here, f2 is the(a)dx(a) 3.5-Irq# "#3.#bot.5

tr** t.CIl <-l""I0%ffiIo)r.f)1- 1.5(r.fl-2.0

0.CI-2$2bo-hrj/ffi*-3.0-3.5

4.0(c)(t'l'4ll

10Nrrrl_4J-lt*w

oog -i.o-3100o*t6rot

Fig. 1 Time history of electric field energies (a) Ei, (b)Ej, and (c) E?. Solid lines show a case with a;'"/O"= 10, vr = 0.7c and nolno = 1139. Dotted lines correspond to a case without the electron beam.

Fig. 2 (a) The dispersion relation (ar vs k) for E, cutting
the low frequency region of co< 1.63roo". (b) Timehistory obtained by the inverse-Fourier transformation from (a).
382

Nambu M. et a/., Nonlinear Plasma Maser Driven by Electron Beam in Magnetized Plasma

From Fig. 4(a) we find that the growth rate increaes
with incresing beam density.
Furthermore, we study the effect of the external
magnetic field on the growth of the electromagnetic

electron distribution function at apet = 0 and (oo"/ =105.4, respectively. The electron distribution functionchanges according to the development of the Langmuir
turbulence. At the final stage of the simulation study(o)p.t = 105.4), the plateau formation in quasilinear
diffusion is observed by Fig. 3(c). This means that thewave emission. Fig. 4(b) shows the time history of Ej inthe presence of an electron beam with h6lns= l/30 and
vr = O.7c for (l) zero magnetic field, (2) weak magnetic
field ao"/{Z" = l0), and (3) strong magnetic field (aro"/
Q" = 3). From Fig. 4(b), it can be seen that there is noresonant interaction between electrons and the Langmuir
turbulence generated by the electron beam occurs quite

effectively during the simulation study.
We study the growth rate of the instability withdifferent parameters. Fig. 4(a) shows the growth ratemeasured in the simulations with different values ofelectron beam density, nrlns, (l) l/50, (2) ll40, (3) 1135,(4) ll30, (5) 1125, (6) ll2o. The growth rate is computedby making use of exponential curve fitting at a*t = 30.

enhancement of the electric field E, for unmagnetizedplasma (case (l)), compared with magnetized plasma(cases (2) and (3)).

The presence of strong magnetic field results inmore effective electromagnetic emission.
Finally, the dependence of the normalized growthrate lyltttoJ on plasma parameter [Q"/@p"] is shown in

Fig. 5. The dotted line represents a curve from theequation, 0.063 + 0.334x - 0.272 x2, where x = d).lo)p"

(a)

1(a)

6r

(6)

o.t2
T0.10
(2)

0v/v,r,(l)

(b)
35

50xlO

n/not3fir(b) 4.5-1.0-1.5-2.O

ni'i'qtv*".i""'+i"i'r,""''ffoo

0-2.5

v/v*-3.0-3.5(c)4.O

Ia0o*t\i,,Fig. 4 (a) Growth rates vs beam density for cases with E,

parallel to the magnetic field (vt = 0.7c, rtr.l{). =tron distribution function at oo"f = 0 and op"f =10). (1) nblno = 1150, l2l n,,lno = 1140, (31 nblno = 1l35, (41 nrlno = 1/30, {5) nblno= U25, and (6) n /no =1120. (bl Time history of electric field energy Ef fordifferent external magnetic field strength. (1) Thermaf noise case with electron beam (vo = 0.7c, nbl
no = 1/30) and no magnetic field. (2) Case withbeam (vo = o.7c, nalno = 1/30) and weak magnetic
field (ao"lQ" = 10). (3) Case with beam (vb = 0.7c,D6lno = 1/30) and strong magnetic field (ao.ld). =105.4, respectively.

3).

-10-505r0

v/v*

Fig. 3 {a) Dependence of the whole electron distributionfunction fo. on (a) v., (bl v, and (c) v,, respectively.
The dotted and solid lines show the whole elec-383Nambu M. et al., Nonlinear Plasma Maser Driven by Electron Beam in Magnetized plasma

0.t4

simulations, Langmuir waves are propagating obliquely
to the ambient magnetic field, whereas in the theoreticalmodel they are propagating along the magnetic field.

0.t2

3. Conclusions

0.t6

Numerical simulations show that the EM R-mode
wave is amplified by Langmuir turbulence driven byelectron beam. The results of the numerical simulations
agree well with those of the theoretical analysis [6].

0.10

0.r

0.2

0.3

0.4
o./q.

0.5

0.6

Reference

Fig. 5 Dependence of the normalized growth tatelylap"lon plasma parameter lQ"latr.l. The dotted line

tll M. Nambu, Laser Par. Beams 1, 427 (1983).

shows a curve from the equation obtained by a[2] M.Nambu, Space Sci. Rev. 44, 357 (1986).

least squares method.

t3l M. Nambu and J. Sakai, Phys. Plasmas 4,3703(1997)..

obtained by a least squares method. The growth rateenhances with the increase of the magnetic field.

[4] S. Bujarbarua, M. Nambu, B.J. Saikia, M. Eda andAccording to the detailed calculations [6], thegrowth rate obtained from theory is found tobe y/{Dr.=[5] A.F. Alexandrov, L.S. Bogdankevich and A.A.

J. Sakai, Phys. Plasmas5,2244 (1998).

Rukhadze, Principles of Plasma Electrodynamics(Springer Verlag, Tokyo, 1984).

3.83 x l0-2, while the simulations show 7/coo" = 1.15 xl0-'. The difference may be attributed to the difference

t6l M. Nambu, B.I. Saikia, D. Gyobu and J.I. Sakai,in situations between theorv and simulations. InPhys. Plasmas, to be published.

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