Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field
Slow wave propagation in 1D non-uniform plasma with tilted magnetic field with respect of direction of non-uniformity is considered. The second order differential equation describing the slow wave is derived from the Maxwell’s equations. The analysis of this equation reveals a singular point for the...
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irk-123456789-1945932023-11-27T18:11:48Z Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field Moiseenko, V.E. Basic plasma physics Slow wave propagation in 1D non-uniform plasma with tilted magnetic field with respect of direction of non-uniformity is considered. The second order differential equation describing the slow wave is derived from the Maxwell’s equations. The analysis of this equation reveals a singular point for the solutions, which could be associated with the Lower Hybrid Resonance. The condition of the resonance is found to be dependent on the tilting angle. Among two WKB solutions only one is singular. The wave vector behaves as 1/x in LHR point for the singular solution. The amplitude diverges only for x-component of the electric field. The solution describes propagating wave both to the left and to the right of the LHR point. The analytical solution obtained in the vicinity of the LHR has a special feature of having drop of its amplitude in the LHR point because of residual damping of the wave inside the LHR location. The energy flux also makes drop down there. Розглянуто поширення повільних хвиль в одновимірній неоднорідній плазмі з похилим магнітним полем щодо направлення неоднорідності. Диференціальне рівняння другого порядку, що описує повільну хвилю, виводиться з рівнянь Максвелла. Аналіз цього рівняння виявляє особливу точку для рішень, яка може бути пов'язана з нижнім гібридним резонансом (LHR). Виявлено, що умова резонансу залежить від кута нахилу. Серед двох рішень ВКБ тільки одне є сингулярним. Хвильовий вектор поводиться як 1/x в точці LHR для сингулярного рішення. Амплітуда розходиться тільки для x-компоненти електричного поля. Рішення описує хвилю, що біжить як зліва, так і праворуч від точки LHR. Аналітичне рішення, отримане в околиці LHR, має особливість, яка полягає в падінні його амплітуди в точці LHR за рахунок залишкового загасання хвилі усередині розташування LHR. Потік енергії також падає в цій зоні. Рассмотрено распространение медленных волн в одномерной неоднородной плазме с наклонным магнитным полем относительно направления неоднородности. Дифференциальное уравнение второго порядка, описывающее медленную волну, выводится из уравнений Максвелла. Анализ этого уравнения выявляет особую точку для решений, которая может быть связана с нижним гибридным резонансом (LHR). Обнаружено, что условие резонанса зависит от угла наклона. Среди двух решений ВКБ только одно является сингулярным. Волновой вектор ведет себя как 1/x в точке LHR для сингулярного решения. Амплитуда расходится только для x-составляющей электрического поля. Решение описывает бегущую волну как слева, так и справа от точки LHR. Аналитическое решение, полученное в окрестности LHR, имеет особенность, заключающуюся в падении его амплитуды в точке LHR из-за остаточного затухания волны внутри местоположения LHR. Поток энергии также падает в этой зоне. 2019 Article Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field / V.E. Moiseenko // Problems of atomic science and technology. — 2019. — № 1. — С. 67-69. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.35.Hr http://dspace.nbuv.gov.ua/handle/123456789/194593 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics Moiseenko, V.E. Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field Вопросы атомной науки и техники |
| description |
Slow wave propagation in 1D non-uniform plasma with tilted magnetic field with respect of direction of non-uniformity is considered. The second order differential equation describing the slow wave is derived from the Maxwell’s equations. The analysis of this equation reveals a singular point for the solutions, which could be associated with the Lower Hybrid Resonance. The condition of the resonance is found to be dependent on the tilting angle. Among two WKB solutions only one is singular. The wave vector behaves as 1/x in LHR point for the singular solution. The amplitude diverges only for x-component of the electric field. The solution describes propagating wave both to the left and to the right of the LHR point. The analytical solution obtained in the vicinity of the LHR has a special feature of having drop of its amplitude in the LHR point because of residual damping of the wave inside the LHR location. The energy flux also makes drop down there. |
| format |
Article |
| author |
Moiseenko, V.E. |
| author_facet |
Moiseenko, V.E. |
| author_sort |
Moiseenko, V.E. |
| title |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| title_short |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| title_full |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| title_fullStr |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| title_full_unstemmed |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| title_sort |
slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2019 |
| topic_facet |
Basic plasma physics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/194593 |
| citation_txt |
Slow wave propagation in plasma with non-uniformity not perpendicular to the magnetic field / V.E. Moiseenko // Problems of atomic science and technology. — 2019. — № 1. — С. 67-69. — Бібліогр.: 3 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-16T21:58:34Z |
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2025-07-16T21:58:34Z |
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1837842423329325056 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2019. №1(119)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2019, № 1. Series: Plasma Physics (25), p. 67-69. 67
SLOW WAVE PROPAGATION IN PLASMA WITH NON-UNIFORMITY
NOT PERPENDICULAR TO THE MAGNETIC FIELD
V.E. Moiseenko
National Science Center “Kharkov Institute of Physics and Technology”,
Institute of Plasma Physics, Kharkiv, Ukraine
Slow wave propagation in 1D non-uniform plasma with tilted magnetic field with respect of direction of non-
uniformity is considered. The second order differential equation describing the slow wave is derived from the
Maxwell’s equations. The analysis of this equation reveals a singular point for the solutions, which could be
associated with the Lower Hybrid Resonance. The condition of the resonance is found to be dependent on the tilting
angle. Among two WKB solutions only one is singular. The wave vector behaves as 1/x in LHR point for the
singular solution. The amplitude diverges only for x-component of the electric field. The solution describes
propagating wave both to the left and to the right of the LHR point. The analytical solution obtained in the vicinity
of the LHR has a special feature of having drop of its amplitude in the LHR point because of residual damping of
the wave inside the LHR location. The energy flux also makes drop down there.
PACS: 52.35.Hr
INTRODUCTION
The slow wave (SW) plays an important role in
certain scenarios of plasma heating and current drive,
and also in wall conditioning discharge sustaining. Its
field structure is studied well within one-dimensional
model including the zone of lower hybrid resonance
(LHR) [1-3]. The LHR phenomenon is a base for the
lower hybrid heating and current drive concepts. The
mode conversion scenario of the minority heating also
includes the LHR mechanism for the wave absorption.
In a standard minority heating scenario the LHR appears
at the plasma periphery, and its role in wave
propagation and power balance is not yet studied
sufficiently.
In hot plasma in a LHR zone, the slow wave
converts into the ion Bernstein wave. In cases of radio-
frequency discharge start-up or a wall conditioning
discharge the ions are cold and the wavelength of ion
Bernstein wave becomes extremely short. Under such
conditions, it is expedient to treat LHR without account
of wave conversion.
The previous theoretical considerations it was
assumed that plasma gradients are oriented
perpendicular to the steady magnetic field. This is
almost true for fusion machines because the plasma
density is approximately constant at the magnetic
surface. However, the magnetic field module has some
variations, and the plasma dielectric tensor follows
them. For such reason it is of interest to consider a case
when the magnetic field is not perpendicular to plasma
gradients.
In this paper, a 1D non-uniform plasma with a tilted
magnetic field is considered. The second order
differential equation describing the slow wave is
derived from the Maxwell’s equations. The analysis of
this equation reveals a singular point for the solutions.
However, the point located aside of the lower hybrid
resonance found using earlier theoretical results. The
solutions obtained are also different. These solutions
and location of the singular point are discussed in this
paper.
SLOW WAVE EQUATION AT LHR
VICINITY
The problem is considered in slab geometry with
non-uniformity of plasma along the x coordinate. Time-
harmonic Maxwells equations read:
02
0 DE k . (1)
The electric displacement field in cold plasma is
EhhEhEEεD )(ˆ
|| ig . (2)
Where h is the unitary vector along the magnetic field:
)cos,0,(sin h . (3)
Uniformity of plasma and magnetic field in y and z
directions allows one to represent the electric field
through Fourier harmonics
)exp()(),,( zikyikxzyx zy EE . (4)
Closeness to the LHR means that
zy
kk
dx
d
, . Using
this and formulas (1-4), one can obtain the following
equation for the slow wave:
0 zzz cEE
dx
d
bE
dx
d
a
dx
d
. (5)
Here
0
*2
0 / dka ,
0||
2
0 /)(cossin2 dkikb z , (6)
2 2 4 2
0 || 0
2 2 2
0 ||
2
0 ||
0
sin cos ( ) /
( sin cos )
[ sin cos ( )],z
c k d
k
ik d
k
d dx
and
*2
0
2
0 kkd z ,
2
||
2* sincos ,
k0=/c.
68 ISSN 1562-6016. ВАНТ. 2019. №1(119)
WKB ANALYSIS
In the WKB analysis
x
ik
dx
d
, and one can obtain
the dispersion equation for the equation (5). As it is
expected, it could be written in the form standard for the
slow wave:
)( 2
0
2
||
||2
kkk . (7)
In the particular case under consideration
222222 sincos zyx kkkk , (8)
22222
|| cossin zx kkk . (9)
In our problem only kx is allowed to vary, and if the
LHR resonance point is reached, then 2
x
k . In this
case both k and k|| diverge. The dispersion equation
permits this if
// ||
2
||
2 kk . (10)
When 2
x
k , from formulas (8), (9) one can
obtain
22
||
2 tan/
kk (11)
and with account of this, the condition (10) could be
written as
0* , (12)
if * is real. If it is complex, the condition is
0Re * . (13)
It is interesting to note that
xx eεe ˆ* . (14)
This indicates that the LHR point occurs when the
diagonal component of the dielectric tensor in the
direction of plasma non-uniformity nullifies.
The singular solution has the wave vector component
aibkx / (15)
and the regular one has
bickx / . (16)
Note here that the quantity a nullifies if the LHR
condition (12) is met.
In WKB approximation the singular solution is
)exp(
'
'
exp dxikdx
ab
cb
E xz . (17)
Here the derivative over x is denoted by prime. The
phase of the solution logarithmically increases on
approach to the singular point. The amplitude remains
regular. The Ex component of the field
0||
2
0 /])(cossin[ dEkE
dx
d
ikE zzzx
(18)
has the singularity both in the phase and amplitude.
ANALYTICAL SOLUTIONS OF SLOW
WAVE EQUATION AT LHR VICINITY
Near the LHR point two solutions of equation (5)
could be found analytically using smallness of the
coefficient before the leading derivative, a<<bL, where
L is the characteristic spatial scale of the non-uniformity
of plasma. The approximate solution of the differential
equation (5) can be obtained neglecting its last term.
The solution reads:
a
dx
a
dx
bEz )exp( . (19)
Keeping lowest terms in Taylor series, xaa
1
and
0
bb the integrals above can be taken using analytical
continuation around point x=0.
.0)],ln(/exp[
0),ln//exp(
10
1010
0
xforxab
xforxabaib
EEz
(20)
The solutions (20) fit well to the WKB solution (17)
even in the vicinity of the LHR point since the first
exponent in (17) does not vary rapidly there.
There is a drop in the amplitude of solutions (20).
The tunneling factor is
)/Imexp( 10 abS . (21)
The drop in amplitude indicates the residual damping of
the wave in the LHR point. The x component of the
Pointing vector is
2
00
2
Im
16
3
Eb
c
x
(22)
for negative x, and for positive x the energy flux density
is smaller by the square of the tunneling factor
xx S 2
. (23)
This is so if Imb0>0. In the opposite case, the picture
reverses.
Note here that the tunneling factor decreases with
|kz|, sin2 and the non-uniformity space scale.
CONCLUSIONS
Slow wave propagation in 1D non-uniform plasma
with tilted magnetic field with respect of direction of
non-uniformity is considered. The second order
differential equation describing the slow wave is
derived from the Maxwell’s equations. The analysis of
this equation reveals a singular point for the solutions,
which could be associated with the Lower Hybrid
Resonance. The condition of the resonance can be
written as 0)ˆRe(
xx
eεe (ex here is the direction of
plasma non-uniformity). This condition gives the
conventional LHR condition when the non-uniformity
direction is perpendicular to the magnetic field. When
the magnetic field is tilted, the condition reveals the
dependence on tilting angle.
Among two WKB solutions only one is singular. The
wave vector behaves as 1/x in LHR point for the
singular solution. The amplitude diverges only for x-
component of the electric field. The solution describes
propagating wave both to the left and to the right of the
LHR point.
The analytical solution obtained in the vicinity of the
LHR is written in terms of the exponential functions and
fits well to the WKB solution. The special feature of it
ISSN 1562-6016. ВАНТ. 2019. №1(119) 69
is dropping of its amplitude in the LHR point because of
residual damping of the wave inside the LHR location.
The energy flux also makes droping there.
ACKNOWLEDGEMENTS
The work is supported in part by the National
Academy of Sciences of Ukraine, grants П-3-22, X-4-3
and ЦВ-5-20.
This work has been carried out within the framework
of the EUROfusion Consortium and has received
funding from the Euratom research and training
programme 2014-2018 under grant agreement No
633053. The views and opinions expressed herein do
not necessarily reflect those of the European
Commission.
REFERENCES
1. D.L. Grekov and K.N. Stepanov // Ukrainskiy
fizicheskiy zhurnal. 1980, v. 25, p. 1281 (in Russian).
2. I. Fidone, G. Granata // Nucl. Fusion. 1971, v. 11,
p. 133.
3. V.E. Moiseenko, T. Wauters, A. Lyssoivan //
Problems of Atomic Science and Technology. Series
“Plasma Physics” (22). 2016, № 6, p. 44.
Article received 15.01.2018
РАСПРОСТРАНЕНИЕ МЕДЛЕННЫХ ВОЛН В ПЛАЗМЕ С НЕОДНОРОДНОСТЬЮ,
НЕ ПЕРПЕНДИКУЛЯРНОЙ МАГНИТНОМУ ПОЛЮ
В.Е. Моисеенко
Рассмотрено распространение медленных волн в одномерной неоднородной плазме с наклонным
магнитным полем относительно направления неоднородности. Дифференциальное уравнение второго
порядка, описывающее медленную волну, выводится из уравнений Максвелла. Анализ этого уравнения
выявляет особую точку для решений, которая может быть связана с нижним гибридным резонансом (LHR).
Обнаружено, что условие резонанса зависит от угла наклона. Среди двух решений ВКБ только одно
является сингулярным. Волновой вектор ведет себя как 1/x в точке LHR для сингулярного решения.
Амплитуда расходится только для x-составляющей электрического поля. Решение описывает бегущую
волну как слева, так и справа от точки LHR. Аналитическое решение, полученное в окрестности LHR, имеет
особенность, заключающуюся в падении его амплитуды в точке LHR из-за остаточного затухания волны
внутри местоположения LHR. Поток энергии также падает в этой зоне.
ПОШИРЕННЯ ПОВІЛЬНИХ ХВИЛЬ У ПЛАЗМІ З НЕОДНОРІДНІСТЮ,
НЕ ПЕРПЕНДИКУЛЯРНОЮ ДО МАГНІТНОГО ПОЛЯ
В.Є. Моісeєнко
Розглянуто поширення повільних хвиль в одновимірній неоднорідній плазмі з похилим магнітним полем
щодо направлення неоднорідності. Диференціальне рівняння другого порядку, що описує повільну хвилю,
виводиться з рівнянь Максвелла. Аналіз цього рівняння виявляє особливу точку для рішень, яка може бути
пов'язана з нижнім гібридним резонансом (LHR). Виявлено, що умова резонансу залежить від кута нахилу.
Серед двох рішень ВКБ тільки одне є сингулярним. Хвильовий вектор поводиться як 1/x в точці LHR для
сингулярного рішення. Амплітуда розходиться тільки для x-компоненти електричного поля. Рішення описує
хвилю, що біжить як зліва, так і праворуч від точки LHR. Аналітичне рішення, отримане в околиці LHR, має
особливість, яка полягає в падінні його амплітуди в точці LHR за рахунок залишкового загасання хвилі
усередині розташування LHR. Потік енергії також падає в цій зоні.
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