Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)573–576
000–000
J. Plasma Fusion

Magnetic Field Perturbations Correlated with Large Amplitude
Lower-Hybrid Waves in a High-Voltage Linear Plasma Discharge
POPEL Sergey, TAKEDA Yasumasa1, INUZUKA Hiroshi2 and ELSÄSSER Klaus3
Institute for Dynamics of Geospheres, Leninsky pr. 38, Building. 1, 119334 Moscow, Russia
1Department of Physics, College of Science and Technology, Nihon University, Tokyo, Japan
2Department of Electrical Engineering, Faculty of Engineering, Shizuoka University, Hamamatsu, Japan
3Institut für Theoretische Physik 1, Ruhr-Universität Bochum, D-44780 Bochum, Germany(Received: 9 December 2003 / Accepted: 2 February 2004)

Abstract
Theoretical description of experiments on excitation of magnetic field perturbations correlated with large amplitude lower-hybrid (LH) wave bursts in a high-voltage linear plasma discharge is presented. It is shown that the magnetic field perturbations, which are excited in the experiments mostly in the paramagnetic sense, are associated with thedevelopment of the magneto-modulational processes. The theoretical results are compared with the experimental datausing the following parameters: the values of the magnitude of the magnetic field perturbations; the direction of thevector of the magnetic field perturbation; the correlation between the magnetic field perturbations and the electron density perturbations. It is shown that for LH waves propagating in one plane theoretical predictions are in a good agreement with the experimental results.
Keywords:
high-current linear plasma discharge, lower-hybrid wave, modulational interaction, magnetic field perturbation,
magneto-modulational process

laboratory experiments [1] differ strongly from those of tokamak plasmas. In particular, in the case of the laboratory
experiments [1] the electron plasma frequency ωpe is far
higher than the electron gyrofrequency ωBe. The equation forthe magneto-modulational processes given in Refs. [3,4]assumes that the inequality ωpe < ωBe is fulfilled. Here wepresent the main steps of the derivation of the equation forthe magneto-modulational perturbations caused by LH waves
in general case.
Although LH waves are “almost” electrostatic, we areinterested in the modulational excitation of the magnetic field
perturbations. This means that we have to take into accountlow-frequency (with the frequencies much less than those of
LH waves) transversal fields which can appear in process ofthe modulational interactions. We start from Maxwell's equations, which in Fourier representation take the form (see, e.g.,[2])1. Introduction
The experiments in a high-current linear plasma discharge carried out at the Nihon University [1] have shown
the possibility of the excitation of magnetic field perturbations related to intense lower-hybrid turbulence. The value ofthe magnitude of the perturbations is |δB| ≈ 10 G and |δB|/|B0|≈ 0.01, where B0 is the unperturbed magnetic field. The characteristic features of the observations are: 1) the magnetic
field perturbations are excited in the experiments mostly inthe paramagnetic sense; 2) there is a correlation between themagnetic field perturbations and the electron density perturbations; 3) the power spectrum of the associated electric field
fluctuations shows widely broadened profile and has multiple
peaks around the lower-hybrid (LH) frequency.
All the results show that the excitation of the magnetic
field perturbations can be associated with the development of
the magneto-modulational processes [2]. We check thisassumption.

⎡ ω22⎤⎢ 2 ε ij + kikj − δ ij k ⎥ Ek,j =⎣ c⎦

2. Equation for magneto-modulational perturbations−

The theoretical investigation of the magneto-modulational processes with the participation of LH waves has been
performed for tokamak plasmas [3,4]. The conditions of the4π iω (2) (3) ...
( jk,i + jk,i + ),c2

(1)

where j(2), j(3), ... are the quadratic, cubic, etc. (in LH electric©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: s_i_popel@mtu-net.ru
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Popel S. et al., Magnetic Field Perturbations Correlated with Large Amplitude Lower-Hybrid Waves in a High-Voltage Linear Plasma Discharge

field E) currents, respectively, εij(ω, k) is the dielectric permittivity tensor, the subscript k = {ω, k} is a four-vector
characterizing the Fourier component, ω is the frequency, kis the wave vector, δij is the Kronecker symbol, and c is thevelocity of light.
To denote the low-frequency fields we use the symbol
tilde. The direction of the unperturbed magnetic field B0 ischosen along the axis Oz. Assuming that the low-frequency~
fields obey the inequality ω~ << k zvTe, taking into accountonly the quadratic current j(2), and using the following relationship for the components of εij under the condition ω <<kzvTe [5]where the superscripts “+” and “−” denote the positive- andnegative-frequency parts of the electric field, respectively.
Using (3), (5) and (6), we obtain from (4) the equation
determining the amplitude of the quasistationary magnetic
field perturbations δB excited by intense LH waves of agiven frequency ω 0∆δ B =(7)where ∆ is the Laplace operator; b = B0 /|B0| is a unit vector
along the unperturbed magnetic field; E is the complex
amplitude of LH field, the field itself is Re(Eexp{−iω 0t});
the asterisk stands for the complex conjugate. Equation (7) isvalid for the case ω 0 << ωBe. This means that for LH waves,22/ωBe(ωpi is the ion plasma frewhen ω 0 ~ ωLH ≡ ωpi / 1 + ω pequency), Eq. (7) is valid for arbitrary relationship between
ωpe and ωBe.

2
ε xx = εyy = 1, εzz ≈ 1 + ω2 pe2 ,kz v Te

εxy = εyx = εxz = εzx = εyz = εzy = 0,(2)we find the relationship∼ ∼ ∼(2)∼ 2j ∼(2)
4π i ⎡ ωk (k · jk , ⊥ )E ∼k = ∼ ⎢ ∼2 k2, ⊥ −∼2ω ⎣ kck z εzz∼∼ 2∼
ωk ⊥(k ⊥ · j∼k(2), ⊥) ⎤+∼2 ∼2 2⎥.
kz k c⎦

3. Magnetic field perturbations(3)

Here we compare the results obtained on the basis of (7)with the data [1]. In general, Eq. (7) describes the excitationof the magnetic field perturbations which have the components both parallel and perpendicular to the unperturbed magnetic field B0. Let us find the condition of the excitation of
the magnetic field perturbations mostly in the paramagnetic
sense, i.e., when the perturbations are “almost” parallel to B0.
We denote

Here vTe is the electron thermal velocity, the subscript ⊥denotes the vector component perpendicular to the unper(2)≠ 0 then the presence ofturbed magnetic field. Thus if j~k,⊥
LH waves results in the excitation of magnetic field perturbationsc ∼δ B∼k = ∼ (k × E∼k ).
(4)
ω(2)≡ −e v⊥ f ~kdp/(2π)3 the kinetic equation forTo determine j~k,⊥

a⊥ ≡

∫

the electron distribution function f(t, r, p) is used (as it can beshown by direct calculations, the ion contribution to the magneto-modulational processes with participation of LH waves
is negligibly small in comparison with the electron one).
Here, −e is the electron charge, p(v) is the electron momentum (velocity). The distribution function f is normalized as∇δ Bz =(2)∼k ,yji≈−4π(9)(10)

Using the fact that ∂δBz /∂ z = 0 and the relationships (10) we
find from Eq. (7)∆δ B⊥ = icω 2pe(k1,z Ek+1,y Ek−2,z + k1,z Ek−2,y Ek+1,z|B0|ω 02∫−j∂ ⎡ ∂ ay
∂ ax ⎤−⎢⎥∂y ⎣ ∂x∂y ⎦∂ ⎡ ∂ ay
∂ ax ⎤−⎥,⎢∂x ⎣ ∂x∂y ⎦(11)(5)
where i and j are unit vectors in the directions of the axes Ox
and Oy, respectively. Using Fourier representation we findcω 2pe(k1,z Ek+1,x Ek−2,z + k1,z Ek−2,x Ek+1,z|B0|ω 02∫∼+ k2,z Ek+1,x Ek−2,z + k2,z Ek−2,x Ek+1,z)δ (k − k1 − k2)dk1dk2,∂ a⊥.
∂z∂δ Bz
∂ ax
∂δ Bz
∂ ay.
=,=
∂x∂z
∂y∂zsity. We assume that LH wave frequency ω0 is much lessthan ωBe.
We solve the kinetic equation using the theory of perturbations in powers of LH wave electric field and following thestandard procedure of separation of high-frequency electricfields (which are associated with LH field) and low-frequency those [2]. We find∼
+ k2,z Ek+1,y Ek−2,z + k2,z Ek−2,y Ek+1,z)δ (k − k1 − k2)dk1dk2,(8)This means that δBz does not depend on z, while∫i4π1 ω 2pe(E⊥(b · E*) + E*⊥(b · E)).
|B0| ω 02

Eq. (7) have solutions obeying the relationshipfollows ne = f(t, r, p)dp/(2π 3), where ne is the electron den-jk∼(2),x ≈21 ω pe∇ × ∇ × (E⊥(b · E*) + E*⊥(b · E)),
|B0| ω 02(6)2574Popel S. et al., Magnetic Field Perturbations Correlated with Large Amplitude Lower-Hybrid Waves in a High-Voltage Linear Plasma Discharge

∂a

∂ay−
∂x

From the other hand, we have (see, e.g., [2])x
∝ dk1dk2(Ek1E *k2ei(k1 − k2)·r∂y∫* i(k2 − k1)·r
k1+ Ek2E e)[k 2yk1x − k1yk 2x ],∫+ +−(ε 0)k E +k = 2 Σeff
k1,k2,k3 E k1E k2 Ek3(12)δ (k − k1 − k2 − k3)dk1dk2dk3,

where Σkeff1,k2,k3 is the effective nonlinear third-order responsein the fields of LH waves. The derivation of the response
for the case ω pe < ω Be is presented, e.g., in [2].
Σ keff1, k 2 , k 3
Deriving this response for the case ωpe < ωBe we use the sameprocedure as in [2]. We findwhere we take into account the longitudinal character of LH
waves (Ek,i = (ki/|k|)Ek, Ek,i* = (ki/|k|)Ek*). If for two arbitrary
LH waves with the wave vectors k1 and k2 the relationshipk1y / k1x = k2y / k2x = const(13)

is fulfilled, then the excitation of the magnetic field perturbations in the direction perpendicular to the unperturbed magnetic field is negligible, and the magnetic field perturbations
can be excited in the paramagnetic sense. This realationshipmeans that LH wave spectrum consists of the waves propagating in one plane. In this case we obtainω 2peδ Bz ∝E2

ω 02 |B0|

cos θ 0,2=−Σeff
k1,k2,k3⎡kki(k⊥ × k1⊥)z(k · k1) ⎤× ⎢ z 21z +−⎥2ωω Beω Be⎦⎣ ω(14)⎡ k 2z k 3zi(k 2⊥ × k 3⊥)z (k 2 · k 3)⎤−−⎥,⎢
22ωω Beω Be⎦⎣ ωδn|E|2=−.
n0
4π n0(Te + Ti)

The electron density perturbations δn are related to theusual development of the modulational processes with theparticipation of LH waves. They appear in the places of LH
field localization (see, e.g., [2]). Let us determine δn using
the theory of perturbations. The dielectric permittivity for LH
waves propagating under the angle θ0 with respect to theexternal magnetic filed is−n0

≡1−

ω 2pi

ω 2pe

ω

ω+22Be−

ω 2pe

ω2
δnn0

δn2 ω 20 δ Bz
sec2θ 0,∝2n0β ω pe |B0|

(15)

1(k · [δ nE]+k )n0 | k |δ nk − k1∫ n=0(k · k1) +E k1dk1| k || k1 |(20)

where β = 8πn0(Te + Ti)/|B0|2, Te(i) is the electron (ion) temperature. For the typical data of the experiments [1] (ωLH /2π= 55 MHz, ωpe /2π = 15 GHz, β ~ 0.001, δ Bz/|B0| ~ 0.01, cos
θ0 ~ me/mi) this estimate gives the value of δ n/n0 ~ 0.5. So
this is in accordance with the experimental data δ n ~ 0.5n0
measured by the microwave interferometer when strong electric field bursts give rise to enhanced electron density modulation [6]. From Eq. (19) we see that the electron density
modulation should be negative, i.e. this is a case corresponding to “density wells”.
The above results were obtained under the condition(13). Actually, it is not necessary to require the exact fulfillment of this condition in order to consider the magnetic field
perturbations in the paramagnetic sense. For this, only theinequality |δB⊥|<<|δBz| should be valid. There are two cases:
1) when | δ B ⊥ |<<| δ B z |( ω Be / ω pe ) 2 ; and 2) when|δB⊥|>>|δBz|(ωBe /ωpe)2 while |δB⊥|<<|δBz|. In the first case thewhere the electron (ωpe) and ion (ωpi) plasma frequencies correspond to the unperturbed electron and ion densities (ne0 =ni0 = n0), respectively, (ε0)k is the unperturbed dielectric function. Poisson’s equation can be written in the form

(ε 0)k Ek+ =(19)Comparing (19) and (14), we findcos2θ 0,(18)where ε (e) and ε (i) are the electron and ion parts of the unperturbed dielectric permittivity ε (the subscript 0 is omitted).
Using the explicit values of ε (e) and ε (i) in the low-fre~~ = |ω − ω | = |ω + ω | << ~k v (k= k − k1 = k2
quency limit ω123z Te
2+ k3), the fact that under the experimental conditions [1] ω pe2>> ωBe, and the condition (13), we find that the main contribution to the effective response is made by the terms which do
not contain the gyrotropic nonlinearity (proportional to vector products). Then Eq. (17) can be presented in the form
(16), where

4. Electron density perturbations

δn|k 2 + k3|2e2
2m2e |k||k1||k 2||k 3|
ε k(i2)+ k3(ε (e)
k 2 + k 3 − 1)
ε k2 + k3

where cos θ0 = kz/|k|. The estimate (14) is valid for the spectrum which is rather broad and/or consists of several peaks.
The width of LH wave spectrum in k-space determines thecharacteristic wave vector of the low-frequency magnetic
field perturbations. For the data of the experiments [1] (thecharacteristic frequency f ≈ 55 MHz, ω 0 = 2π f ≈ 3.46·108 s-1,
|E| ~ 10 kV/cm, |B0| ≈ 1.2 kG, the electron density ne ~ 1012cm-3) under the assumption that cos θ0 takes the typical (forLH waves) magnitude cos θ0 ~ me/mi, we get from Eq. (14):δBz ~ 18 G for hydrogen plasmas; δBz ~ 9 G for deuteriumplasmas. We see a good agreement of this estimate with theexperimental data δBz ~ 10 G.

ε k = (ε 0)k −

(17)

(16)

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Popel S. et al., Magnetic Field Perturbations Correlated with Large Amplitude Lower-Hybrid Waves in a High-Voltage Linear Plasma Discharge

main terms in the square brackets in the right hand side ofEq. (18) are those not containing the vector products. It is
this case that corresponds to the above calculations and theformulas (19) and (20). In the second case the dominantterms in the square brackets in the right hand side of Eq. (18)
are those containing the vector products. In this case theresult for δ n ~ (E × E*)z coincides with that found in Ref. [7].
The sign of δ n is not determined. This means that in this caseboth the density “wells” and “humps” can arise. This is in agood agreement with the data of the experiments [1] using
high-voltage linear plasma discharge.

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Plasma Phys. (Quebec City, Canada, 2000) Vol. 1, p.
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[2] S.V. Vladimirov, V.N. Tsytovich, S.I. Popel and F.Kh.
Khakimov, Modulational Interactions in Plasmas(Kluwer Academic Publishers, Dordrecht, 1995).
[3] V.N. Tsytovich and S.A. Bel’kov, Comments Plasma
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[7] V.D. Shapiro et al., Phys. Fluids B 5, 3148 (1993).

5. SummaryThus the theory developed here allows us to explain themain experimental results [1] under the assumption that LH
wave spectrum consists of the waves propagating (approximately) in one plane. The most probable reason for the excitation of the magnetic field perturbations in these experiments is the development of the magneto-modulational
processes.

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