J. Plasma Fusion Res. SERIES, Vol.2 (1999) 347-349

Controversies in Ouasilinear Theory and Wave Structure
Formation in Current-Garrying Plasma and at Beam-Plasmalnteraction
MASLOV Vasyl I. and BAKAI Alexandr S.
NSC Kharkov Institute of Physics & Technology, Kharkov 310108' Ukraine(Received: 9 December 1998 / Accepted: 13 May 1999)

Abstract
The paper considers the development of a so called "bump-in-tail" and electric current instabilities
in plasma. The process of turbulence excitation by "hot" electron beam in plasma-filled cylinder and byelectric current in plasma is unstable concerning arising of correlations between an electric field andparticles' movement, when the wave-particle correlation length becomes longer than the propagation
distance of resonant electrons during the half-period of their oscillations in wave field. In this case aconvective movement of particles on the phase space and ordered lattice or chain of packets in space (r,Vor,) appear. Packets of the lattice are transparent for electrons but packets of the chain reflect theresonant ones.

Keywords:
ordered wave packets, nonequilibrium plasma, solitary perturbationand their interaction due to resonant particles is ofpractical interest as well as the evolution of ion-acoustic

1. Introduction
Phase Vor, and group V* velocities of plasma waves

propagation in metallic magnetized plasma-filled
cylinder [] are limited from above and Vg,3 Vpt 3 V",
unlike Vor, of Langmuire waves. V" is the maximum
phase velocity. Such dispersion relation of plasma

turbulence.

waves (and of ion-acoustic oscillations) leads to several
differences of their packets evolution at "hot"-electron-development in comparison with Langmuir waves,

Nonlinear particle dynamics in the wave field withVon- V*> Vo (Vo = 2{\,lk is the width of wave-particleresonance) is similar to dynamics in Langmuir waves.
During the period of particle oscillations 2trl{\, (Qr,=studied in[2-4].

k(erpylm")tt2; (rp*, is the amplitude) in wave fields the2. Formation of Ordered Structures ofWave Packets in Plasmas

beam-plasma instability (also at ion-acoustic instability)

particles propagate on the distance L = nt(Vpr- Vs,)lkvt.
For oscillations with Vrn- Vg, > V,. this distance islonger than wavelength L > ), and hence the packet isThis problem is considered here and is interesting
because of the start of a new stage of beam relaxation as

well as ion-acoustic instability development. This stage
is characterized by increasing of correlation and byformation of structute of coherent packets. As manytransparent for particles. If the velocity distributionfunctionf of particles, falling on packet, has positive
inclination af"lav > 0 then after packet z > L in thebeam-plasma experiments are carried out in plasma
waveguide, an investigation of the wave evolution in it@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: maslov @ rocketmail.com
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Maslov V.I. er aL, Confinvenies in Quasilinear Theory and Wave SEucture Formation in Cunent{arrying Plasma and at Beam-plasma Interaction

The growth rate of the well amplitude

resonant region I V - Vpnl < Vo the derivative changes its

sign dfldV < 0. Therefore particles damp oscillations in
the resonant region at z > l, keeping up the length ofpacket equal to I,. The latter provides that.the area inapproximately equals

phase space, occupied by one packet, does not change at

The well is formed without a threshold in plasma withT- to"(reor ^vl^)"'.

a growing of packet's amplitude

2voL= l"(vr^-vr).

(3)electron beam, with a threshold in current-carrying
plasma and with a small threshold in current-carrying(1)

dusty plasma.

Because after passing particles through packet
derivative 4foldV changes its sign in the resonant region
and in the nonresonant region derivative remains

Ifthe beam velocity V6 (or electron current velocity
V6 in current-carrying plasma) is higher than V5 > V" +Volvpr=u" then after the beam relaxation in resonant

previous, hence after packet steepenings on theregion Von''n < V < V" + V,, the electron distributionfunction is nonequilibrium in the nonresonant region V

distribution function are formed at velocities y = lVrn!
%. | . On these steepenings the growth rates of wave
excitation are maximum. Therefore new packets with
Vpn,z = lVp6 ! Vr,(Vpn) | are excited intensively afterpacket with Vpr,. This phenomenon leads to ordering ofwave fields. Hence if the propagation distance ofresonant electrons during the half-period of theiroscillations in wave fields L becomes shorter thancorrelation length /" = n(Vpn- Vdlh (11 is the linear
growth rate of wave amplitude) the linear process turns> V"+ Vr,. Therefore the wave fields become unstable
relative to modulation. Namely, the packets of the chain
with Vor, = V" and with a large distance between themtrap electrons with V = V"+ Vu with nonequilibrium
distribution function and wave fields are excited further.
Similar behavior of wave turbulence is realized also
in infinite plasma with electron beam in conditions 0 <Vblvbth- (n"ln,)tt2 < l. Here n6, tr.6 atE the densities ofplasma and beam. It is determined by the fact that thehigh-frequency waves in infinite plasma have adispersion relation similar to the waves in metallic
magnetized plasma-filled cylinder if warm electron
beam strongly influences on dispersion relation ofwaves. In latter case the frequencies of excited waves

out to be unstable concerning the arising of correlationsof a particle movement and an electric field. This
problem has been investigated in [5-1 l]. In this case thelinear relaxation process turns into the ordered turbulent

process which is characterized by a convective
movement of particles on the phase space and by theapproximately equal cr = a6 for large wavevectors ft andfrequencies are proportional to k for small wavevectors

appearance of the ordered structures of wave packets inspace (2, Vo5).

if maximum phase velocity Vpn'u* - V66(no/n6)tt2
satisfies 0 < Vu - Vor,'u^( Vu*,.

Wave packets with Vor, near maximum one % at
amplitude reaching values V,. >> Vpr, - V* reflect themain part of resonant electrons falling on the back front

The electron flow in nonequilibrium plasma canexcite not only reflecting potential wells but alsoof packet and all resonant electrons falling on the first
front of packet. It is determined by the fact that thetransparent potential humps, for example, Bunemanpotential hump in current-carrying plasma [2].

propagation distance (relative to the packet's envelope)of resonant electrons during half-period of their

3. Gonclusion

oscillations in wave field Z = tt(Vph- Vr,)lkVt becomesmuch shorter than the wavelength ,1,. Therefore after
interaction with the packet the resonant electrons remain
on its fronts, their velocities change on opposite ones

The formation of ordered structures of wave
packets in plasma with electron beam and in currentcarrying plasma has been investigated.

and separatrix is not slammed behind them. Hence wave
packets with Vpr,

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Beam-Plasma Interaction
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