J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)253–255
000–000

Cross-Field Electron Transport and Dynamo Effect Induced byUnstable Localized Electrostatic Waves in a Magnetized Plasma
SUGAYA Reiji, MAEHARA Tsunehiro and SUGAWA Masao
Department of Physics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan(Received: 9 December 2003 / Accepted: 11 March 2004)

Abstract
Cross-field electron transport and cross-field electric field induced by linearly or nonlinearly unstable localized
electrostatic waves in an electron beam-plasma system were investigated theoretically and numerically. It was verified
that the parallel and perpendicular electron transport generates a large and abrupt dip of the electron density and astrong peak of the electron temperature in the radial direction. At the same time the cross-field electric field is produced
partially by the cross-field transport and partially by the perpendicular pressure gradient.
Keywords:
cross-field electron transport, cross-field electric field, dynamo effect, localized electrostatic waves,pressure gradient

1. Introduction

2. Transport equations

Cross-field electron transport and cross-field electric fieldwere investigated theoretically and numerically on the basisof the transport equations and magnetohydrodynamic
equations [1,2]. They are induced by linearly or nonlinearly
unstable localized electrostatic waves in an electron beamplasma system. The transport equations are derived from theθ-dependent quasilinear velocity-space diffusion equation [14]. It was verified that the parallel and perpendicular electrontransport generates a large and abrupt dip of the electron
density and a strong peak of the electron temperature in theradial direction. At the same time the cross-field electric field
is produced partially by the cross-field electron transport andpartially by the perpendicular pressure gradient, that is, it isgiven by E⊥ = B0 × ve⊥ /c – (∇⊥ pe)/en e ( pe = ne kB T e). The
small cross-field electron transport is predicted to create thestrong cross-field electric field. On the other hand, the parallel
electric field E|| = –(∇|| pe)/ene + meνenve|| /e is created by theparallel transport and pressure gradient. Here, B0 = (0, 0, B0)
is the external uniform magnetic field in the z direction, ve⊥ =(vex , vey , 0) is the cross-field drift velocity of plasma electrons,ve|| = (0, 0, vez) is the parallel drift velocity of plasma
electrons, c is the light velocity, e is the electron charge, ne isthe electron density, Te is the electron temperature, kB is theBoltzmann constant and ven is the electron-neutral collision
frequency. The radial and axial profiles of the electron density
and temperature and the electrical potential were obtained
numerically and they can explain well the experimental
observation in an electron beam-plasma system qualitatively
and quantitatively [5,6].

We consider the electron transport arising from the θdependent quasilinear velocity-space diffusion due to spatiallylocalized unstable electrostatic waves propagating almost
perpendicularly to the magnetic field. The transport equations
and magnetohydrodynamic equations for plasma electrons are
given by∂Ue
= −2γ k( e )U k ,∂t(1)

2γ ( e ) k∂Pe= − k U k − ene E|| − ∇|| pe − me neν enve|| ,
∂t
ωk(2)

ne ve|| − µ e ne E|| − ∇|| ( De ne ) = 0 ,(3)e− ene E⊥ − ne ve⊥ × B0 − ∇ ⊥ pe = 0 ,c(4)∂ne+ ∇ ⋅ Γe = 0 ,∂t(5)Γ e = ne ve − µ e ne E|| − ∇|| ( De ne +∇⋅ E = 0 ,12ne vez),
ν en

(6)
(7)

where Uk = 81π (∂ (εk′ ω k)/∂ω k)⏐Ek⏐2 is the wave energy density,
kUk /ω k is the wave momentum density, U e = 1 m e neve2 +23 m k T and P = m n v are the energy and momentum
ee e e2 e B eCorresponding author’s e-mail: sugaya@phys.sci.ehime-u.ac.jp1253©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchSugaya R. et al., Cross-Field Electron Transport and Dynamo Effect Induced by Unstable Localized Electrostatic Waves in a Magnetized Plasma

densities of plasma electrons, γ k(e) is the linear Landau andcyclotron damping rate ascribed to plasma electrons, µe = e/meven is the electron mobility, De = kB Te /meven is the electron
diffusion coefficient, εk = εk′ + iε k″ is the dielectric constant,
Ek is the magnitude of the wave electric field, ω k is the wave
frequency and k = (k⊥, 0, k||) is the wave number. In (6), themobility and diffusion of plasma electrons across themagnetic field is neglected because of ven /ωce << 1 (ωce is theelectron cyclotron frequency). On the other hand, (3) meansthat the diffusion along the magnetic field is the ambipolar
diffusion. According to (3), the electron flux Γe given by (6)is reduced to Γe = neve⊥ = (nevex , nevey , 0) under the conditionof nev2ez /ven << De ne , and the transport equation (2) becomes∂Pe2γ ( e ) k= − k U k − 2 me neν enve|| .
∂tωkthe cross-field electric field (not shown) is generated. Thismeans that the cross-field drift of plasma electrons (notshown) induced by the spatially localized electrostatic waves
produce the abrupt inhomogeneity of plasma profile in theradial direction and the strong cross-field electric field. The
transport equations (1) and (2) govern the electron
temperature and the drift of plasma electrons (vex and vez).
The generalized Ohm’s law (4) demonstrate that E y isdetermined by vex and ∇y pe. The axial electric field Ez is givenby (3) and the electric fields in y and z directions produce theelectric field in x direction so as to satisfy the Poisson’sequation (7). Thus Ex and ∇x pe create the cross-field drift vey
via E × B drift and diamagnetic drift.
The typical parameters of the experiment reported by theauthor et al. [5,6] are Te0 = 5 eV, B0 = 7 × 10-3 Tesla and a =1.5 × 10-3 m (beam radius), and hence ωcea/vte0 ≈ 2 is given.
From the numerical analysis shown partially in Figs. 1-2, wefind ve /vte0 ≈ (0.5, 0.5, 0.05) and eEa/kB Te0 ≈ (1, 1.5, 0.01) at⏐γ k(e)⏐t = 0.9 and z/ze = 9.0. Thus we find Ea ≈ (5, 7.5, 0.05)
volt, Ea ≈ 9.0 volt, Ex ≈ 3.3 × 103 volt/m, Ey ≈ 5 × 103 volt/mand E z ≈ 33 volt/m. This value is very close to theexperimental value of Vexp ≈ 10 volt. Also the drift velocities
vex , vey ≈ 6.6 × 105 m/s (1.3 eV), vez ≈ 6.6 × 104 m/s (0.013eV) are reasonably obtained.

(8)

Equations (1) and (2) predict that the electrostatic waves
create strong electron transport or acceleration along andacross the magnetic field. The generated Ohm’s law of (4)shows that the cross-field electric field E⊥ is produced by the
dynamo effect of the cross-field electron drift and the radialpressure gradient induced by the electron transport.

3. Numerical analysis
The temporal and spatial development of the transportof plasma electrons has been investigated numericallyassuming that the electrostatic waves excited in an electron
beam-plasma system are localized radially such that Uk ∝exp[–(x2 + y2)/a2], and the background plasma is spatially
uniform initially. The electrostatic waves are assumed to be
governed by the following kinetic wave equations:∂U k= 2γ N U k ,
∂t(9)

∂U k
= 2γ sN U k ,∂z(10)

4. Conclusion
It was verified theoretically and numerically that theparallel and perpendicular electron transport induced bylocalized unstable electrostatic waves generates large andabrupt inhomogeneities of the electron density andtemperature in the radial direction. Simultaneously the strong
cross-field electric field is produced by the cross-field electrontransport and the pressure gradient. The obtained results canwell explain the experimental observation in an electron
beam-plasma system.
This research was carried out partially under thecollaborating Research Program at the Institute of Laser
Engineering, Osaka University. Numerical calculation wasmade at the Computation Center of Nagoya University.

where γ N and γ sN are the temporal and spatial growth rates
resulting from the electron beam, respectively. The numericalanalysis of Eqs. (1)-(10) was performed under the parametersof ω k /ωce = 0.3, k⊥vte0 /ωce = 1, k⊥ /k|| = 0.2, ⏐γ k(e)⏐/ωce = 0.2,γ N /⏐γ k(e)⏐ = 1, γ sNvte0 /⏐γ k(e)⏐ = 0.2, ven /⏐γ k(e)⏐ = 1, aωce /vte0 = 2,and Uk(0)/ne0 kB Te0 = 0.6×10-3. Thus the temporal evolutionof the three-dimensional profiles of the system was obtained.
We show only the transverse profiles in the x-direction for
the fixed value of y/a = –0.4. Figure 1 exhibits the axial
evolution of transverse profiles of the normalized electron
density, that is, ne /n e0 versus x/a and z/z e is shown as aparameter of ⏐γ k(e)⏐t = 0.4, 0.6, 0.8 and 0.9 for the x-z profilesof (a), (b), (c) and (d), respectively, where ze = vte0 /⏐γ k(e)⏐ andne0 = ne(0). Figure 2 is the axial evolution of the transverse
profiles of the normalized electron temperature, that is, Te /Te0
versus x/a and z/ze is shown under the same parameters of⏐γ k(e)⏐t, where Te0 = Te(0). It is found that the hollow profileof the electron density and the peaked profiles of the electron
temperature develop temporally and axially. Simultaneously

References[1] R. Sugaya et al., Proceedings of International Congress
on Plasma Physics Combined with the 25th EPS
Conference on Controlled Fusion and Plasma Physics,Praha, Czech, 1998, ECA Vol. 22C, p.141 and p. 209.
[2] R. Sugaya et al., Proceedings of International
Conference on Phenomena in Ionized Gases, Nagoya,Japan, 2001, Vol. 3, p. 117 and p. 127.
[3] R. Sugaya, J. Plasma Phys. 64, 109 (2000); 66, 143(2001).
[4] R. Sugaya, Phys. Plasmas 10, 3939 (2003).
[5] R. Sugaya et al., J. Phys. Soc. Jpn. 64, 2018 (1995).
[6] T. Maehara et al., Proceedings of International
Conference on Plasma Physics, Nagoya, Japan, 1996,
Vol. 1, p. 426.

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Sugaya R. et al., Cross-Field Electron Transport and Dynamo Effect Induced by Unstable Localized Electrostatic Waves in a Magnetized Plasma

Fig. 1

Axial evolution of the transverse profiles of the electron density. ne/ne0 is shown versus x/a and z/ze as a parameter of ⏐γk(e)⏐t =0.4 for (a), 0.6 for (b), 0.8 for (c) and 0.9 for (d).

Fig. 2

Axial evolution of the transverse profiles of the electron temperature. Te /Te0 is shown versus x/a and z/ze as a parameter of⏐γk(e)⏐t = 0.4 for (a), 0.6 for (b), 0.8 for (c) and 0.9 for (d).

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