Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder
The wave eigenmodes of a radially nonuniform plasma cylinder are examined analytically in the helicon frequency range. Conditions for the existence of almost pure helicon eigenmodes with on-axis localization are pointed out. The scaling (i.e. the relationship between the plasma density, the magnetic...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1091152016-11-21T03:02:11Z Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder Shamrai, K.P. Beloshenko, N.A. Фундаментальная физика плазмы The wave eigenmodes of a radially nonuniform plasma cylinder are examined analytically in the helicon frequency range. Conditions for the existence of almost pure helicon eigenmodes with on-axis localization are pointed out. The scaling (i.e. the relationship between the plasma density, the magnetic field, and the mode frequency and wave number) for these eigenmodes is found and shown to depend on the azimuthal mode number. Аналитически исследованы волновые собственные моды радиально-неоднородного плазменного цилиндра в геликонном диапазоне частот. Указаны условия существования почти чистых геликонных мод, локализованных вблизи оси системы. Найден скейлинг (т.е. взаимоотношение между плотностью плазмы, магнитным полем и частотой, и волновым числом моды) для таких мод, и показано, что он зависит от азимутального волнового числа. Аналітично досліджено хвильові власні моди радіально-неоднорідного плазмового циліндра в геліконному діапазоні частот. Виявлено умови існування майже чистих геліконних мод, локалізованих поблизу осі системи. Знайдено скейлінг (тобто співвідношення між густиною плазми, магнітним полем та частотою, і хвильовим числом моди) для таких мод, і показано, що він залежить від азимутального хвильового числа. 2012 Article Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder / K.P. Shamrai, N.A. Beloshenko // Вопросы атомной науки и техники. — 2012. — № 6. — С. 102-104. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.35.Hr http://dspace.nbuv.gov.ua/handle/123456789/109115 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Фундаментальная физика плазмы Фундаментальная физика плазмы |
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Фундаментальная физика плазмы Фундаментальная физика плазмы Shamrai, K.P. Beloshenko, N.A. Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder Вопросы атомной науки и техники |
| description |
The wave eigenmodes of a radially nonuniform plasma cylinder are examined analytically in the helicon frequency range. Conditions for the existence of almost pure helicon eigenmodes with on-axis localization are pointed out. The scaling (i.e. the relationship between the plasma density, the magnetic field, and the mode frequency and wave number) for these eigenmodes is found and shown to depend on the azimuthal mode number. |
| format |
Article |
| author |
Shamrai, K.P. Beloshenko, N.A. |
| author_facet |
Shamrai, K.P. Beloshenko, N.A. |
| author_sort |
Shamrai, K.P. |
| title |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| title_short |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| title_full |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| title_fullStr |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| title_full_unstemmed |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| title_sort |
scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2012 |
| topic_facet |
Фундаментальная физика плазмы |
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http://dspace.nbuv.gov.ua/handle/123456789/109115 |
| citation_txt |
Scaling laws for the helicon eigenmodes in a nonuniform plasma cylinder / K.P. Shamrai, N.A. Beloshenko // Вопросы атомной науки и техники. — 2012. — № 6. — С. 102-104. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-07T22:35:19Z |
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2025-07-07T22:35:19Z |
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| fulltext |
102 ISSN 1562-6016. ВАНТ. 2012. №6(82)
SCALING LAWS FOR THE HELICON EIGENMODES
IN A NONUNIFORM PLASMA CYLINDER
K.P. Shamrai, N.A. Beloshenko
Institute for Nuclear Research NAS of Ukraine, Kiev, Ukraine
The wave eigenmodes of a radially nonuniform plasma cylinder are examined analytically in the helicon
frequency range. Conditions for the existence of almost pure helicon eigenmodes with on-axis localization are
pointed out. The scaling (i.e. the relationship between the plasma density, the magnetic field, and the mode
frequency and wave number) for these eigenmodes is found and shown to depend on the azimuthal mode number.
PACS: 52.35.Hr
INTRODUCTION
The problem of helicon wave eigenmodes in a
radially nonuniform plasma cylinder was examined in
numerous papers in application to helicon discharges,
fusion plasmas etc. (e.g., [1,2]). However, recently this
problem was addressed again [3], for the following
reason. As is known, the helicon wave dispersion
relation for an infinite uniform plasma reads
2
2
pe
z
ce
kck
ω
ωω = , (1)
where peω is the electron plasma frequency, and ω ,
zk , ⊥k and 2/122 )( ⊥+= kkk z are the mode frequency
and wave numbers, the longitudinal, transverse and total
ones, respectively. If the frequency is fixed (in a helicon
plasma, it is normally determined by an rf generator),
the dispersion relation (1) yields the scaling for plane
waves propagating along the magnetic field in the form
2
00 / zkBn ∝ where 0n is the plasma density and 0B an
ambient magnetic field. For a radially bounded
nonuniform plasma, the dispersion is normally
evaluated from Eq. (1) by assuming k⊥ ∼ a−1 where
a = min{R, Ln} (R and Ln: the cylinder radius and the
characteristic scale of density nonuniformity). For long
waves, 1<<akz , the total wave number k ∼ a−1, and
then one obtains the scaling zkBn ∝0/ ( n : a radially
averaged density).
In some experiments with the excitation of non-
axisymmetric waves (with the aximuthal number
0≠m ; normally )1+=m [4,5], it was found that the
long-wave scaling is 2
0/ zkBn ∝ , which is similar to the
plane wave case. To explain this fact, it was assumed
[3] that excited in these experiments can be radially
localized helicon modes of special type which have off-
axis localization near the surface of strongest radial
density gradient [6]. In this paper we examine on-axis
localized eigenmodes whose scaling is likely to explain
the aforementioned experiments.
1. THE MODEL
We consider a radially nonuniform plasma cylinder
of radius R immersed in a uniform axial magnetic field
of strength B0. The plasma density is assumed to
decrease from the center to periphery. We describe the
electromagnetic fields by Maxwell equation with a cold-
plasma dielectric tensor. The eigenmode fields are
represented as )exp()(),( θω imziktirt z ++−= FrF
where zk and m = 0, ±1, ±2… are the axial and
azimuthal wave numbers. We shall examine the
eigenmodes in the helicon approximation, i.e., assuming
that the longitudinal electric field 0=zE . This
approximation is valid under two conditions. First, a
plasma should be considerably nonuniform radially,
with the density at the boundary with a confining vessel
much lesser than the center density, so that the surface
mode conversion is negligible. Second, the mode axial
wave number should lie in the range
( )( ) ( )( )cepzcep ckc ωωωωωωω ////2 00 << , (2)
where 0pω is the center plasma frequency. The right
inequality in Eq. (2) signifies that the central part of the
plasma column is transparent whereas the peripheral
part is opaque for the helicon waves. These parts are
separated by the helicon cut-off surface determined by
equation
( )( ) 222 /4/)( ckemrn zcee ωωπ= . (3)
The left inequality in Eq. (2) signifies that the surface of
the helicon wave coalescence with the quasi-
electrostatic (Trivelpiece-Gould) wave is lacking in the
plasma bulk, so that the process of bulk mode
conversion of these waves is excluded [7]. If both
aforementioned conditions are true, the helicon waves
and the Trivelpiece-Gould waves are decoupled and the
former can be described in the helicon approximation.
In this approximation one puts 0=zE , and obtains
from Maxwell equations the following equations for the
θE and zB fields
z
zv
v
z
B
N
m
rk
ik
r
E
N
m
dr
dE
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
++
1
2
2
22
1
2
2 111
εε
ε θθ ;
θεε
ε
E
N
Vik
r
B
N
m
dr
dB
z
v
z
z
z
1
2
1
2
2
−
−=
−
− , (4)
where
22
2
1 1
ce
pe
ωω
ωε
−
−= and ( )22
2
2
ce
cepe
ωωω
ωω
ε
−
=
are the dielectric tensor components, ω/ckN zz = is a
longitudinal refractive index, ckv /ω= is a vacuum
wave number, and ( ) 2
2
2
1
2 εε −−= zNV . The rest of field
components are determined from algebraic equations
ISSN 1562-6016. ВАНТ. 2012. №6(82) 103
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
= z
vz
r B
rk
mEi
N
E θε
ε 2
1
2
1 ;
θENB zr −= , rz ENB =θ . (5)
One can eliminate the zB field from Eqs. (4) to obtain a
second-order equation for the θE field
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+⎥
⎦
⎤
⎢
⎣
⎡
θ
θ
ε
ε E
qr
m
Ndr
d
dr
rEd
qrdr
d
z
2
1
2
2)(1
( ) 022
2
1
2
2
2
2
1
2
2
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
+
−
+ θε
ε
ε
E
rk
m
NqN
V
N
k
zzzz
z , (6)
where
22
2
1
2
2
1
rk
m
N
Nq
zz
z
ε−
+= . (7)
2. AXISYMMETRIC MODES
Examine first the helicon eigenmodes with the
azimuthal wave number 0=m . In this case, the
parameter 1=q , and Eq. (6) simplifies to the form
( ) ( )
( ) 01
1
22
2
1
22
22 =
−
−−
+⎥⎦
⎤
⎢⎣
⎡
θ
θ
ε
εε E
NN
Nk
dr
rEd
rdr
d
zz
z
z . (8)
As long as the left inequality in Eq. (2) implies that
10
2 4ε>zN where )0(110 == rεε , with a reasonable
accuracy one can neglect 1ε in comparison to 2
zN in
Eq. (8).
Assuming that the density profile is parabolic
( )22
0 /1)( arnrn −= , (9)
where Ra ≥ , and that ωω >>ce , one can write
( )222
20
2
2 /21 ar−≈ εε , (10)
where 224
0
2
20 / cep ωωωε = . Then Eq. (8) takes the form
( ) ( ) 01 2
20 =−+⎥⎦
⎤
⎢⎣
⎡
θ
θ Erbb
dr
rEd
rdr
d , (11)
where
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−= 14
2
202
0
z
z N
kb ε , 24
2
202
2
2
aN
kb
z
z
ε
= . (12)
Introducing a dimensionless variable
dr /=ς , 4/1
2
−= bd (13)
reduces Eq. (11) to the following
( ) ( ) 01 2 =−+⎥
⎦
⎤
⎢
⎣
⎡
θ
θ ςλ
ς
ς
ςς
E
d
Ed
d
d , (14)
where
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−== 1
2 4
2
20
20
2
0
2
z
zz
N
akNbd ε
ε
λ (15)
is an eigenvalue to be determined.
Introducing a new function g
( ) gE
ς
ς
θ
2/exp 2−
= (16)
and changing the independent variable, 2ςξ = ,
converts Eq. (14) to the following
0
42
2
=+− g
d
dg
d
gd λ
ξ
ξ
ξ
ξ . (17)
With the eigenvalue specified as
p=
4
λ , (18)
where ...,3,2,1=p is a natural number, solutions of
Eq. (17) are the orthogonal Laguerre polynomials
)(1 ξ−
pL . Taking the first polynomial, 21
1 )( ξξ =−L , one
obtains finally the θE field of first radial helicon
eigenmode in the form
( )22
0
)1( 2/exp drrCE −=θ , (19)
where 0C is a constant. The value of d , Eq. (13),
determines the width of the mode localization.
The eigenvalue equation for the first radial mode,
4=λ , can be resolved to give the following
relationship
⎟
⎠
⎞⎜
⎝
⎛ ++= 22
2
2
2
0 )16/1(1122 akak
a
c
zz
ce
p ω
ω
ω . (20)
The consequence of this equation is the scaling law for
the 0=m modes:
for long waves ( 1<<akz )
akBn z∝00 / (21)
and for short waves ( 1>>akz )
22
00 / akBn z∝ . (22)
3. NON-AXISYMMETRIC MODES
For the non-axisymmetric modes with azimuthal
wave numbers 0≠m , an analytic analysis can be
performed for the axially long waves, 1<<akz . In this
case, the parameter q , Eq. (7), can be approximated as
222 / rkmq z≈ . (23)
Substituting this into Eq. (6) and neglecting 1ε in
comparison with 2
zN , one obtains
013 2
22 =⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+′+′+′′ θθθ ε Emr
N
mErEr
z
, (24)
where the prime denotes the derivative with respect to r.
Assuming again a parabolic density profile, as in
Eq. (9), and changing the variable, ruE /=θ , one
obtains the following equation
01
2
2
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+′+′′ u
r
mu
r
u μ , (25)
where the eigenvalue to be determined
cez
p
Na
m
ωω
ω
μ 22
2
02
= . (26)
Equation (25) is the mth-order Bessel equation which
has physically reasonable solution only for the positive
azimuthal numbers ( 0>m ), )( rJu m μ= .
Finally, for the long waves 1<<akz with 0>m ,
the θE field takes the form
104 ISSN 1562-6016. ВАНТ. 2012. №6(82)
)()( rJ
r
CE m
mm μθ = , (27)
where mC is a constant. Note that the right hand side of
Eq. (27) is finite at 0→r , as long as mm rE ∝)(
θ there.
The eigenvalue (26) is specified by the boundary
condition. For instance, if the plasma confining vessel is
conducting, 0)()( == RrE m
θ , one has to put
mpR βμ = where mpβ is the pth root of the mth
Bessel function. This gives the following relation for the
pth radial mode
mp
cez
p
aN
Rm
β
ωω
ω
=22
22
02
. (28)
The relationship (28) determines the scaling for the
long ( 1<<akz ) helicon modes with the azimuthal wave
numbers 0>m in the form
22
00 / akBn z∝ . (29)
It is parabolic on zk , contrary to the linear scaling for
the long 0=m modes, Eq. (21).
CONCLUSIONS
It was found that the scaling of cylindrical helicon
eigenmodes in a radially nonuniform plasma depends on
the azimuthal mode number m. For the axisymmetric
modes ( 0=m ) the scaling is parabolic on zk for the
short modes, 22
00 / akBn z∝ ( 1>>akz ), whereas it is
linear for the long modes, akBn z∝00 / ( 1<<akz ).
On the contrary, the scaling for the non-
axisymmetric ( 0>m ) long modes is parabolic on zk .
The latter finding can explain the appropriate
experimental results [4,5].
REFERENCES
1. F.F. Chen, M.J. Hsieh, and M. Light. Helicon waves
in a non-uniform plasma // Plasma Sources Sci.
Technol. 1994, v. 3, p. 49-57.
2. G. Kamelander and Ya.I. Kolesnichenko. Localized
whistler eigenmodes in tokamaks // Phys. Plasmas.
1996, v. 3, p. 4102-4105.
3. R. Boswell. Helicon mysteries: fitting a plane wave
into a cylinder // APS/DPP Meeting (Salt Lake City,
November 14–18, 2011); Bull. Amer. Phys. Soc. 2011,
v. 56, №15, DT1.00004.
http://meetings.aps.org/link/BAPS.2011.GEC.DT1.4.
4. A.W. Degeling, C.O. Jung, R.W. Boswell, and
A.R. Ellingboe. Plasma production from helicon waves
// Phys. Plasmas. 1996, v. 3, p. 2788-2796.
5. J. Prager, T. Ziemba, R. Winglee, and B.R. Roberson.
Wave propagation downstream of a high power helicon
in a dipolelike magnetic field // Phys. Plasma. 2010, v.
17, p. 013504-1-9.
6. B.N. Breizman and A.V. Arefiev. Radially localized
helicon modes in nonuniform plasma // Phys. Rev. Lett.
2000, v. 84, p. 3863-3866.
7. K.P. Shamrai. Stable modes and abrupt density jumps
in a helicon plasma source // Plasma Sources Sci.
Technol. 1998, v. 7, p. 499-511.
Article received 19.09.12
ЗАКОНЫ ПОДОБИЯ ДЛЯ ГЕЛИКОННЫХ СОБСТВЕННЫХ МОД
В НЕОДНОРОДНОМ ПЛАЗМЕННОМ ЦИЛИНДРЕ
К.П. Шамрай, Н.А. Белошенко
Аналитически исследованы волновые собственные моды радиально-неоднородного плазменного
цилиндра в геликонном диапазоне частот. Указаны условия существования почти чистых геликонных мод,
локализованных вблизи оси системы. Найден скейлинг (т.е. взаимоотношение между плотностью плазмы,
магнитным полем и частотой, и волновым числом моды) для таких мод, и показано, что он зависит от
азимутального волнового числа.
ЗАКОНИ ПОДІБНОСТІ ДЛЯ ГЕЛІКОННИХ ВЛАСНИХ МОД
У НЕОДНОРІДНОМУ ПЛАЗМОВОМУ ЦИЛІНДРІ
К.П. Шамрай, М.А. Бєлошенко
Аналітично досліджено хвильові власні моди радіально-неоднорідного плазмового циліндра в
геліконному діапазоні частот. Виявлено умови існування майже чистих геліконних мод, локалізованих
поблизу осі системи. Знайдено скейлінг (тобто співвідношення між густиною плазми, магнітним полем та
частотою, і хвильовим числом моди) для таких мод, і показано, що він залежить від азимутального
хвильового числа.
|