Lower hybrid resonance: field structure and numerical modeling
The behavior the electromagnetic fields in the vicinity of the lower hybrid resonance point is studied in case of 1D plasma non-uniformity. The first of two found solutions of Maxwell’s equations is singular and describes the wave travelling to the lower hybrid resonance layer. This wave if fully...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1152342017-03-31T03:02:25Z Lower hybrid resonance: field structure and numerical modeling Moiseenko, V.E. Wauters, T. Lyssoivan, A. Plasma heating and current drive The behavior the electromagnetic fields in the vicinity of the lower hybrid resonance point is studied in case of 1D plasma non-uniformity. The first of two found solutions of Maxwell’s equations is singular and describes the wave travelling to the lower hybrid resonance layer. This wave if fully absorbed without reflections. Another solution which is regular describes the standing wave. To extend the range of validity of the solutions found, they are matched to the WKB solutions. Three possibilities for numerical solving the wave propagation problem in presence of the lower hybrid resonance zone are discussed in the paper. Поведение электромагнитных полей в непосредственной близости от нижнего гибридного резонанса изучено в случае одномерной неоднородности плазмы. Первое из двух найденных решений уравнений Максвелла сингулярно и описывает волну, падающую на слой нижнего гибридного резонанса. Эта волна полностью поглощается без отражений. Другое решение, которое регулярно, описывает стоячую волну. Чтобы расширить область применимости найденных решений, они приведены к виду ВКБ-решений. Обсуждаются три возможности для численного решения задачи распространения волн в присутствии зоны нижнего гибридного резонанса. Поведінка електромагнітних полів в безпосередній близькості від нижнього гібридного резонансу вивчено в разі одновимірної неоднорідності плазми. Перше з двох знайдених рішень рівнянь Максвелла сингулярно і описує хвилю, що падає на шар нижнього гібридного резонансу. Ця хвиля повністю поглинається без відбиття. Інше рішення, яке регулярно, описує стоячу хвилю. Щоб розширити область застосовності знайдених рішень, вони приведені до вигляду ВКБ-рішень. Обговорюються три можливості для чисельного рішення задачі поширення хвиль у присутності зони нижнього гібридного резонансу. 2016 Article Lower hybrid resonance: field structure and numerical modeling / V.E. Moiseenko , T. Wauters , A. Lyssoivan // Вопросы атомной науки и техники. — 2016. — № 6. — С. 44-47. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Hr http://dspace.nbuv.gov.ua/handle/123456789/115234 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Plasma heating and current drive Plasma heating and current drive |
| spellingShingle |
Plasma heating and current drive Plasma heating and current drive Moiseenko, V.E. Wauters, T. Lyssoivan, A. Lower hybrid resonance: field structure and numerical modeling Вопросы атомной науки и техники |
| description |
The behavior the electromagnetic fields in the vicinity of the lower hybrid resonance point is studied in case of
1D plasma non-uniformity. The first of two found solutions of Maxwell’s equations is singular and describes the
wave travelling to the lower hybrid resonance layer. This wave if fully absorbed without reflections. Another
solution which is regular describes the standing wave. To extend the range of validity of the solutions found, they
are matched to the WKB solutions. Three possibilities for numerical solving the wave propagation problem in
presence of the lower hybrid resonance zone are discussed in the paper. |
| format |
Article |
| author |
Moiseenko, V.E. Wauters, T. Lyssoivan, A. |
| author_facet |
Moiseenko, V.E. Wauters, T. Lyssoivan, A. |
| author_sort |
Moiseenko, V.E. |
| title |
Lower hybrid resonance: field structure and numerical modeling |
| title_short |
Lower hybrid resonance: field structure and numerical modeling |
| title_full |
Lower hybrid resonance: field structure and numerical modeling |
| title_fullStr |
Lower hybrid resonance: field structure and numerical modeling |
| title_full_unstemmed |
Lower hybrid resonance: field structure and numerical modeling |
| title_sort |
lower hybrid resonance: field structure and numerical modeling |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Plasma heating and current drive |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/115234 |
| citation_txt |
Lower hybrid resonance: field structure and numerical modeling / V.E. Moiseenko
, T. Wauters
, A. Lyssoivan // Вопросы атомной науки и техники. — 2016. — № 6. — С. 44-47. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT moiseenkove lowerhybridresonancefieldstructureandnumericalmodeling AT wauterst lowerhybridresonancefieldstructureandnumericalmodeling AT lyssoivana lowerhybridresonancefieldstructureandnumericalmodeling |
| first_indexed |
2025-07-08T08:22:46Z |
| last_indexed |
2025-07-08T08:22:46Z |
| _version_ |
1837066320485548032 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2016. №6(106)
44 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 44-47.
LOWER HYBRID RESONANCE: FIELD STRUCTURE AND
NUMERICAL MODELING
V.E. Moiseenko
1
, T. Wauters
2
, A. Lyssoivan
2
1
Institute of Plasma Physics of the NSC KIPT, Kharkov, Ukraine;
2
Laboratory for Plasma Physics, ERM/KMS, Brussels, Belgium
The behavior the electromagnetic fields in the vicinity of the lower hybrid resonance point is studied in case of
1D plasma non-uniformity. The first of two found solutions of Maxwell’s equations is singular and describes the
wave travelling to the lower hybrid resonance layer. This wave if fully absorbed without reflections. Another
solution which is regular describes the standing wave. To extend the range of validity of the solutions found, they
are matched to the WKB solutions. Three possibilities for numerical solving the wave propagation problem in
presence of the lower hybrid resonance zone are discussed in the paper.
PACS: 52.35.Hr
INTRODUCTION
The upper and lower hybrid resonances (UHR and
LHR) appear in cold magnetized plasma when the
perpendicular diagonal component ee ˆ of the
dielectric tensor ̂ nullifies (here e is a unitary vector
perpendicular to the steady magnetic field). In case of
LHR the WKB solutions predict a regular behavior of
the fast magnetosonic wave (FMSW). The wave number
of the slow wave (SW) diverges on approach to the
LHR layer.
The LHR phenomenon is a base for the lower hybrid
heating and current drive. 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 LHR zone, the slow wave converts
into 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.
Presence of the singularity hampers a numerical
modeling of wave propagation in plasma when a LHR
exists in the calculation domain. A simplest way to
proceed is usage of the penalty method in which the
singularity in the LHR point is avoided by adding
locally an artificial imaginary part to . A more
rigorous option is usage of the analytical solutions in the
LHR area. The analytical continuation of the Maxwell’s
equations to the complex plane is the most rigorous
approach.
SW FIELD STRUCTURE AT LHR VICINITY
The problem is considered in slab geometry with
non-uniformity of plasma along the x coordinate. The
magnetic field is directed along z. Using smallness of
two parameters 1/,,, 2
0
2
0
22 xyz kgkkkk and
1|| , one can obtain an equation for the
slow wave [1] from Maxwell’s equations:
0
1
||
2
||
x
sw
x E
k
E
dx
d
dx
d
(1)
with yxig ee ̂ , zz ee ˆ
|| ,
)( 2
0
2
||
||2
kkksw , and in these terms, the
dispersion equation for the slow wave is 22
swkk . Other
components of the electric field vector are
dxEikE xyy and
dxE
k
kki
E x
z
z
z
20
2
. At the
lower hybrid resonance =0 and ksw=. For the
propagating wave the WKB solution of this equation is
)exp(
||
dxki
k
C
E sw
sw
x
. (2)
Here and further the constants are denoted by C. Note
here that
4/3
xE and
4/1
,
zy EE . Thus, on
approach to LHR point all components of the electric
field increase. In the vicinity of LHR the above equation
is approximated by
0
2
2
y
x
a
y
dx
d
, (3)
where xEy ,
0
2
||
/
x
z
x
k
a
. Here it is assumed
that the the LHR point is at x=0. Making substitution
xs and syv / , one can come to the Bessel
equation for v.
2
2 2
1 1
(4 ) 0.
d d
v v a v
ds s ds s
(4)
The approximate solution of this equation is
1
1
(2 ),xE Z u
u
(5)
ISSN 1562-6016. ВАНТ. 2016. №6(106) 45
where axu , 111 YCJCZ YJ for u>0. If u<0 then
111 KCICZ KI . To match the solutions, the
expansions of the solutions near the matching point,
u=0, is made. Then
(1)
1( 0) (2 )
2 1 1
( ln ),
2 2
Y
x
Y
C
E x Y u
u
C
u
u
(6)
(2)
1( 0) (2 ) .J
x J
C
E x J u C
u
(7)
Here γ = 0.57722 is Euler-Mascheroni constant.
Continuation of expression for )1(
xE to x=-0 results in
addition of imaginary unity due to logarithm. Addition
of )2(
xiE to the first solution compensates this. Finally
1 1
(1)
1
[ (2 ) (2 )], 0
2
1
(2 ), 0,
x
Y u iJ u u
u
E
K u u
u
(8)
1
(2)
1
(2 )
, 0
(2 )
, 0.
x
J u
u
u
E
I u
u
u
(9)
Other components are dxEikE xyy and
dxEikE xzz .
0 0
(1)
0
[ (2 ) (2 )], 0
2
2 (2 ), 0,
x
Y u iJ u u
E dx
a K u u
(10)
0(2)
0
(2 ), 01
(2 ), 0.
x
J u u
E dx
a I u u
(11)
The first solution has a logarithmic singularity that
indicates on residual wave damping. This is confirmed
by global behavior of the solution: At u>0 it describes a
wave traveling from infinity to the point u=0. There is
no reflected wave and, therefore, there is non-zero
valued energy flux of negative sign (see also [2, 3]). At
u<0 the solution represents a standing wave with zero
energy flux. This means that the power is absorbed
locally at the LHR point.
It is necessary to note that a more general problem is
analyzed in Ref. 4. The authors obtain the solution in
integral form which possibly may be reduced to the
explicit form given here. Also Ref. 5 should be
mentioned in this context.
VIDENING OF ZONE OF SW SOLUTION
VALIDITY
The solutions found (8)-(11) are valid in a narrow
zone |x|<<L. Here L is the characteristic scale of
variation of the dielectric tensor components. Beyond
this zone the WKB approximation comes to play.
Within it, the electromagnetic field structure is given by
formula (2) and the following formulas
)exp(
2
||
dxki
k
kC
E sw
sw
y
y
, (12)
).exp(sgn
||
||
dxki
k
kC
E sw
z
sw
z
(13)
To match the solution (8-11) with the WKB solution
(2), (12), (13) which, in contrast to the analytical
solution, accounts for not only single term in Laurent
expansion of 2
swk over x, the solution (8)-(11) is
modified so that its asymptotical behavior coincides
with the WKB solution, but the accuracy approximation
in the vicinity of the LHR point remains in the same
frame as before. In general, the modification consists in
replacing of the argument x by x+o(x) and any constant
C by C+O(x). This procedure can be justified by a
representation of transition from equation (1) to
equation (3) as application of a similar procedure
instead of omission of some small terms. Following this,
the Bessel function arguments in (8-11) is substituted
with
.0,
0,
0
2
0
2
xdxk
xdxk
x
sw
x
sw
(14)
The modified solution can be represented in the form:
)2,1(||
2/1
)2,1(
x
sw
x X
k
E
, (15)
where (1) 1 1
1
[ ( ) ( )], 0
2
( ), 0,
x
Y iJ x
X
K x
1(2)
1
( ), 0
( ), 0,
x
J x
X
I x
)2,1(
2/3
||
2/1
)2,1(
yz
sw
yy X
k
ikE
, (16)
)2,1(
2/3
||
2/1
2
0
2
)2,1( )(
yz
swz
z
z X
kk
kki
E
, (17)
where
0 0(1)
0
[ ( ) ( )], 0
2
( ), 0,
yz
Y iJ x
X
K x
0(2)
0
( ), 0
( ), 0.
yz
J x
X
I x
The obtained solutions at |x|<<1 behave similarly to
above obtained analytical solution (note, that they have
different amplitude). At WKB zone, they repeat WKB
solutions. For high enough |kz|, the vicinity of LHR and
46 ISSN 1562-6016. ВАНТ. 2016. №6(106)
WKB zones overlap, and the above solutions are valid
everywhere.
FMSW FIELD STRUCTURE
The equation for FMSW can be obtained from the
Maxwell’s equations neglecting Ez. It can be written in
the following self-conjugate form:
0
yy BEE
dx
d
A
dx
d
, (18)
where
2
0
22
2
0
2
kkk
kk
A
zy
z and
2
0
22
2
0
2
0
22
24
02
0
kkk
gk
dx
d
k
kkk
gk
kB
zy
y
zy
.
As mentioned before, the LHR point is a regular point
for FMSW. If the LHR zone is narrow, the polynomial
representation of the FMSW field is quite accurate. To
find it, the Taylor expansion for the coefficients A and B
should be made. To lowest order, the couple of the basic
solutions is:
2
0
0)1(
2
1 x
A
B
E
x
x
y
, (2) 20
0
.
2
x
y
x
A
E x x
A
(19)
The x component of the electric field could be then
found from the x component of Maxwell’s equations.
NUMERICAL TREATMENT OF LHR
Numerical modeling that uses discretization with
finite difference or finite element methods is hampered
by singularities at LHR location. The matrix of the
linear equations system which is produced by the
discretization may be ill-conditioned. Even well-
conditioned matrix may produce wrong results since the
singularity cannot be reproduced by polynomial or other
smooth functions.
The singularity disappears in natural way if the
dissipative effects, such as binary collisions, are
accounted for in the dielectric tensor. However, if the
collision frequency is much less than the radio-
frequency, too fine mesh is necessary to reproduce the
electric fields.
One of the methods to fix the singularity is the
penalty method which consists in adding an artificial
imaginary part to that covers the LHR vicinity only
(see Ref. 1). The method is quite practical, but its
apparent disadvantage is that it could result in some
inaccuracy in the solution.
Another approach could be realized using the analytical
solutions in the LHR zone. The solutions presented in
preceding sections of the paper could be used and
matched with the numerical solutions at the left and
right margin of the LHR zone. This approach is more
complicated than the penalty method both in
programming and in amount of the calculations, but the
accuracy expected should be higher. However, the
accuracy of the calculations has an upper limit
determined by the accuracy of the approximate
analytical solutions.
A prospective method for treating LHR is the
method of the analytical continuation. Following this
algorithm, the coordinate x is extended to the complex
plane. All quantities and functions in Maxwell’s
equations are assumed to be analytical. The path of
integration is chosen so that it circumvents the LHR
point. In this way the singularity is avoided. The method
has an advantage of numerical convergence that is
provided by application of discretization at the whole
domain. This method is tested in the section below.
APPLICATION OF ANALYTICAL
CONTINUATION TO MAXWELL’S
EQUATIONS
Analytical continuation is implemented at 1D
cylindrical code [6] to the radial coordinate. The
coordinate is modified to r*=r+iu. In the numerical
experiments 222
0 ]/)(1[ LrrLCu ac at the segment
L-r0<r<L+r0 and zero outside it. The plot of the
modified radial coordinate is given in Fig. 1.
Fig. 1. Modified radial coordinate
Fig. 2. The relative error in electric fields at the area
where the integration contour is at real axis compared
with the calculation without analytical continuation.
The curve ‘penalty’ corresponds to applying the penalty
method in the area
ISSN 1562-6016. ВАНТ. 2016. №6(106) 47
In the particular case under consideration the
excursion of r* to the complex plane is local and made
around the regular point (no LHR point in the domain).
The solution at the area where the excursion is made
should depend on the integration path. But outside this
area the solution should be independent on which
excursion was made, and this is the point to check.
Fig. 2 shows the relative difference in electric fields
at the area where the integration contour is at real axis
compared with the calculation without analytical
continuation. The difference quickly goes to zero when
making the mesh denser. This calculation demonstrates
applicability of the analytical continuation to the wave
propagation problems.
CONCLUSIONS
The behavior the electromagnetic fields in the
vicinity of the LHR point is studied in case of 1D
plasma non-uniformity. Analytical solutions for slow
wave are found in the LHR vicinity. The first of two
solutions is singular and describes the wave travelling to
the LHR layer. This wave if fully absorbed without
reflections. Another solution which is regular describes
the standing wave. To extend the range of validity of the
solutions found, they are matched to the WKB
solutions. The fast magnetosonic wave has no
singularities in the LHR zone and could described with
polynomials.
Presence of the singularity hampers a numerical
modeling of wave propagation in plasma when a LHR
exists in the calculation domain. A simplest way to
proceed is usage of the penalty method in which the
singularity in the LHR point is avoided by adding
locally an artificial imaginary part to . A more
rigorous option is usage of the analytical solutions in the
LHR area. For the slow and fast waves, the above
obtained solutions may be used. The analytical
continuation of the Maxwell’s equations to the complex
plane is the most rigorous approach, and its applicability
is checked using a numerical example.
ACKNOWLEDGEMENTS
The authors are thankful to D. Van Eester and
D. Grekov for useful discussions.
The work is supported in part by Science and
Technology Center in Ukraine and National Academy
of Sciences of Ukraine, grant No. 6057, and by National
Academy of Sciences of Ukraine, grant П-3-22.
REFERENCES
1. V.E Moiseenko, Yu.S. Kulyk, T. Wauters,
A.I. Lyssoivan. Optimization of self-consistent code for
modeling of RF plasma production // Problems of
Atomic Science and Technology. Series “Plasma
Physics”. 2015, v. 1, № 21, p. 56.
2. A.A. Rukhadze, V.S. Savodchenko, S.A. Triger //
Zhurn. prikl. mekhaniki i tekhn. fiziki. 1965, v. 6, p. 58
(in Russian).
3. G.M. Zaslavskiy, S.S. Moiseev, R.Z. Sagdeev // DAN
SSSR. 1964, v. 158, p. 1295 (in Russian).
4. D.L. Grekov and K.N. Stepanov // Ukrainskiy
fizicheskiy zhurnal. 1980, v. 25, p. 1281 (in Russian).
5. I. Fidone, G. Granata // Nucl. Fusion. 1971, v. 11,
p. 133.
6. V.E Moiseenko // Problems of Atomic Science and
Technology. Series “Plasma Physics”. 2003, v. 1, № 9,
p. 82.
Article received 01.10.2016
НИЖНИЙ ГИБРИДНЫЙ РЕЗОНАНС: СТРУКТУРА ПОЛЕЙ И ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ
В.Е. Моисеенко, T. Wauters, A. Lyssoivan
Поведение электромагнитных полей в непосредственной близости от нижнего гибридного резонанса
изучено в случае одномерной неоднородности плазмы. Первое из двух найденных решений уравнений
Максвелла сингулярно и описывает волну, падающую на слой нижнего гибридного резонанса. Эта волна
полностью поглощается без отражений. Другое решение, которое регулярно, описывает стоячую волну.
Чтобы расширить область применимости найденных решений, они приведены к виду ВКБ-решений.
Обсуждаются три возможности для численного решения задачи распространения волн в присутствии зоны
нижнего гибридного резонанса.
НИЖНІЙ ГІБРИДНИЙ РЕЗОНАНС: СТРУКТУРА ПОЛІВ І ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ
В.Є. Моісeєнко, T. Wauters, A. Lyssoivan
Поведінка електромагнітних полів в безпосередній близькості від нижнього гібридного резонансу
вивчено в разі одновимірної неоднорідності плазми. Перше з двох знайдених рішень рівнянь Максвелла
сингулярно і описує хвилю, що падає на шар нижнього гібридного резонансу. Ця хвиля повністю
поглинається без відбиття. Інше рішення, яке регулярно, описує стоячу хвилю. Щоб розширити область
застосовності знайдених рішень, вони приведені до вигляду ВКБ-рішень. Обговорюються три можливості
для чисельного рішення задачі поширення хвиль у присутності зони нижнього гібридного резонансу.
|