Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap
In present communication analytical and numerical investigations of electromagnetic waves with frequency near electron cyclotron frequency propagation in axisymmetrical magnetic trap are presented taking into account variation of magnetic field direction using ray tracing formalism. Influence of mag...
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| Zitieren: | Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap / E.D. Gospodchikov, O.B. Smolyakova, E.V. Suvorov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 43-45. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-174542011-02-27T12:06:48Z Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap Gospodchikov, E.D. Smolyakova, O.B. Suvorov, E.V. Методы создания и нагрева плазмы In present communication analytical and numerical investigations of electromagnetic waves with frequency near electron cyclotron frequency propagation in axisymmetrical magnetic trap are presented taking into account variation of magnetic field direction using ray tracing formalism. Influence of magnetic field inhomogeneity on efficiency on electron cyclotron resonance heating of plasma in axisymmetrical magnetic trap is investigated. Obtained analytical results can explain numbers of experimental phenomena in electron cyclotron discharge in axisymmetrical magnetic traps experiments. Аналитически и численно исследованы особенности распространения электромагнитных волн электронного циклотронного диапазона частот в аксиально-симметричных магнитных ловушках с учетом как неоднородности модуля магнитного поля, так и соответствующей ей неоднородности направления магнитного поля. Продемонстрировано, что неоднородность направления магнитного поля может качественно изменить картину распространения волн и заметно сказаться на эффективности электронного циклотронного нагрева плазмы в прямой магнитной ловушке. Аналітично та чисельно досліджено особливості поширення електромагнітних хвиль електронного циклотронного діапазону частот в аксіально-симетричних магнітних пастках при взятих до уваги як неоднорідності модуля магнітного поля, так і відповідної до неї неоднорідності напрямку магнітного поля. Продемонстровано, що неоднорідність напрямку магнітного поля може якісно змінити картину розповсюдження хвиль та помітно позначитися на ефективності електронного циклотронного нагріву плазми в прямій магнітній пастці. 2010 Article Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap / E.D. Gospodchikov, O.B. Smolyakova, E.V. Suvorov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 43-45. — Бібліогр.: 5 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/17454 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Методы создания и нагрева плазмы Методы создания и нагрева плазмы |
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Методы создания и нагрева плазмы Методы создания и нагрева плазмы Gospodchikov, E.D. Smolyakova, O.B. Suvorov, E.V. Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| description |
In present communication analytical and numerical investigations of electromagnetic waves with frequency near electron cyclotron frequency propagation in axisymmetrical magnetic trap are presented taking into account variation of magnetic field direction using ray tracing formalism. Influence of magnetic field inhomogeneity on efficiency on electron cyclotron resonance heating of plasma in axisymmetrical magnetic trap is investigated. Obtained analytical results can explain numbers of experimental phenomena in electron cyclotron discharge in axisymmetrical magnetic traps experiments. |
| format |
Article |
| author |
Gospodchikov, E.D. Smolyakova, O.B. Suvorov, E.V. |
| author_facet |
Gospodchikov, E.D. Smolyakova, O.B. Suvorov, E.V. |
| author_sort |
Gospodchikov, E.D. |
| title |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| title_short |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| title_full |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| title_fullStr |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| title_full_unstemmed |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| title_sort |
influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2010 |
| topic_facet |
Методы создания и нагрева плазмы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/17454 |
| citation_txt |
Influence of magnetic field directions inhomogeneity on longitudinal propagation of wave beams in axisymmetrical magnetic trap / E.D. Gospodchikov, O.B. Smolyakova, E.V. Suvorov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 43-45. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT gospodchikoved influenceofmagneticfielddirectionsinhomogeneityonlongitudinalpropagationofwavebeamsinaxisymmetricalmagnetictrap AT smolyakovaob influenceofmagneticfielddirectionsinhomogeneityonlongitudinalpropagationofwavebeamsinaxisymmetricalmagnetictrap AT suvorovev influenceofmagneticfielddirectionsinhomogeneityonlongitudinalpropagationofwavebeamsinaxisymmetricalmagnetictrap |
| first_indexed |
2025-07-02T18:40:34Z |
| last_indexed |
2025-07-02T18:40:34Z |
| _version_ |
1836561606306168832 |
| fulltext |
INFLUENCE OF MAGNETIC FIELD DIRECTIONS INHOMOGENEITY
ON LONGITUDINAL PROPAGATION OF WAVE BEAMS
IN AXISYMMETRICAL MAGNETIC TRAP
E.D. Gospodchikov, O.B. Smolyakova, E.V. Suvorov
Institute of Applied Physics RAS, Nizhniy Novgorod, Russia
E-mail: eggos@mail.ru
In present communication analytical and numerical investigations of electromagnetic waves with frequency near
electron cyclotron frequency propagation in axisymmetrical magnetic trap are presented taking into account variation of
magnetic field direction using ray tracing formalism. Influence of magnetic field inhomogeneity on efficiency on
electron cyclotron resonance heating of plasma in axisymmetrical magnetic trap is investigated. Obtained analytical
results can explain numbers of experimental phenomena in electron cyclotron discharge in axisymmetrical magnetic
traps experiments.
PACS 52.35.Hr, 42.25.Bs, 52.50.Sw, 42.15.Dp
1. INTRODUCTION
In axisymmetrical magnetic traps inhomogenity of
magnetic field intensity and magnetic field direction are
closely related to each other. These two types of
inhomogeneities essentially change the structure of
electromagnetic wave beams with frequencies of the order
of electron cyclotron frequency. For example it was
numerically shown [1] that inhomogeneity of magnetic
field direction could be responsible for strong refraction
of electromagnetic waves in axisymmetrical magnetic
traps when plasma density higher than the critical one.
Such refraction decreases the heating efficiency for quasi-
longitudinal launch of rf power. This effect is related to
peculiarities of the dependence of wave’s group velocity
on the direction of magnetic field near electron cyclotron
resonance (ECR) region.
In this paper we present analytical investigation of
electromagnetic waves propagation near ECR region of
axisymmetrical magnetic trap with taking into account
inhomogeneity of both magnetic field value and of its
direction. Results are compared with well known
analytical solution of Hamiltonian equations for ray
propagation near ECR region under assumption that the
direction of magnetic field is constant [2] (Note,
nevertheless, that such magnetic field does not satisfy the
Maxwell equation ). Conditions for ray
concentration near the trap axis in the vicinity of ECR
region are discussed which assumes effective wave
absorption in the trap.
0divB =
2. ANALITICAL MODEL OF WAVE
PROPAGATION NEAR ECR REGION
AXISYMMETRICAL MAGNETIC FIELD
Magnetic field in axisymmetrical system outside the
region of its sources is determined from Maxwell
equations:
( ) ( )
( ) ( )⎪
⎪
⎩
⎪⎪
⎨
⎧
∂
∂
=
∂
∂
∂
∂
−=
∂
∂
zr
zr
B
r
B
z
B
z
rB
rr
1
. (1)
Solution of these equations in paraxial approximation
may be presented as:
( )
( )⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
zF
z
rJB
zF
z
rJB
r
z
1
0
,
where ( ) ( )zFzBz =,0 is an arbitrary function, are
Bessel functions presented as power series. Near ECR-
surface magnetic field takes the form:
1,0J
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
±+= ...1
2
2
2
1
0 L
z
L
zBzF ,
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+≈
2
2
2
1
0 2
1
L
r
L
zBBz m ,
1
0 2L
rBBr −≈ , (2)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++≈
2
1
2
1
0 1
L
r
L
zBB ξ ,
with ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−±=
8
1
2 2
2
2
1
L
L
ξ . In the expression for B we take
into account first non-zero terms over both coordinates z
and r . Correspondingly terms in power series for and
resulting in the next order corrections into the value
of are omitted.
zB
rB
B
For description of wave propagation we use ray-
tracing technique in the cold plasma approximation. In
present communication we don’t discuss applicability of
this approximation in detail (detail discussion can be find
in [2, 3]). We only note that ray trajectories near ECR
resonance surfaces are characteristic of wave
equation [2, 4].
Near ECR region it is convenient to use ray
Hamiltonian in form which do not include terms which
tend to infinity at cyclotron frequency in spite of the fact,
that some of dielectric tensor components possess such
property. For example, such ray Hamiltonian for near
longitudinal propagation of ECR waves may be written
as [1]:
( ) ( ) 02,,, 2
||
||2 =−+= −
−
⊥ ε
ε
ε
NNNNzrH zr , (3)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6. 43
Series: Plasma Physics (16), p. 43-45.
where ( )B
P
ωωω
ωευε
−
−=−= −
2
|| 1 ;1 , 2
2
ω
ωυ P= ,
αα sincos zr NNN −=⊥ , αα cossin|| zr NNN += ,
zr BBtg /=α .
It can be shown, that in the vicinity of ECR region for
quasi-longitudinal propagation direction of group velocity
is determined only by sign of . Using this fact one can
see that the ray geometry depends only on the sign of
⊥N
τ∂∂ ⊥ /N ,
B
P
B
zr
NN
ωααε
ω
ω
τ
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
−=
∂
∂ ⊥ sincos||2
2
|| , (4)
or for paraxial approximation (2) on the sign of the
expression
2
1
||
0
|| 2
1sincos
L
r
B
B
zr
⎟
⎠
⎞
⎜
⎝
⎛ −≈⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂ ξεααε . (5)
If 0
2
1
|| <⎟
⎠
⎞
⎜
⎝
⎛ −ξε , rays are “attracting” to the axis of
magnetic system. If 0
2
1
|| >⎟
⎠
⎞
⎜
⎝
⎛ −ξε , rays are pushed out of
the system in radial direction. Near ECR surface
longitudinal propagated waves are strongly absorbed [3].
As result efficiency of ECR heating can be estimated as
numbers of rays which attained ECR surface. And energy
distribution profile can be estimated using transverse
coordinates of rays. So, the condition
44
0
2
1
|| <⎟
⎠
⎞
⎜
⎝
⎛ −ξε (6)
may be considered as a condition of effective wave
absorption near the trap axis.
In case 0
2
1
|| ≈⎟
⎠
⎞
⎜
⎝
⎛ −ξε terms which were neglected in
(2), become of importance. It can be shown that if these
terms are taken into account, additional critical points are
arriving. For plasma with density lower than the critical
one they are stable nodes. For plasma with density higher
than the critical one they are «saddle» points. These
points divide rays tending to the axis of a system and rays
escaping in radial direction. It differs significantly from
what takes place in the approximation of constant
magnetic field direction, where governing parameter
instead of (6) was ξε || [2]. For the traditional magnetic
trap configuration parameter ξ increases from borders to
central part of the trap, and condition 0
2
1
|| =⎟
⎠
⎞
⎜
⎝
⎛ −ξε
separates regions where effective ECR heating can, or
can’t be realized. For undercritical plasmas where 0|| >ε
region of effective ECR heating is situated near board of
magnetic trap, this result can explain low efficiency of
ECR heating in experiments where resonance surface was
situated in central part of the magnetic trap [5]. For
overcritical plasmas where 0|| <ε over way region of
effective ECR heating is situated in central part of the
trap.
3. NUMERICAL MODELING
In Figs. 1-4 results of numerical ray tracing modeling
near ECR surface are presented, for different parameters
of magnetic field distribution (for modeling we use
expression for magnetic field components including all
terms of paraxial expansion), and different values of
plasma density, which is assumed to be homogeneous.
Ray trajectories are shown by solid lines, ECR surface is
denoted by the dashed line. Results of numerical
simulations fully confirm the resume of analytical
investigation. When 0
2
1
|| <⎟
⎠
⎞
⎜
⎝
⎛ −ξε rays tend to axis of
magnetic system, for 0
2
1
|| >⎟
⎠
⎞
⎜
⎝
⎛ −ξε rays tends to infinity
in radial direction, and in case 0
2
1
|| ≈⎟
⎠
⎞
⎜
⎝
⎛ −ξε additional
critical points are presented.
a b
Fig. 1. Ray trajectories for 1=ξ : a) 8.0=υ , b) 2=υ
a b
Fig. 2. Ray trajectories for 3.0=ξ : a) 8.0=υ ,b) 2=υ
a b
Fig. 3. Ray trajectories for 3.0−=ξ : a) 8.0=υ ,b) 2=υ
0 2
0
5
1
- - - - 0 1
-
-
0
4
8
-4 -3 -2 -1 0 1
-10
-5
0
5
10
- - - 0
1
-
-
0
5
- - 0
-
-
0
4
8
- - - - 0 2
4
0
-
-
8
45
a b
Fig. 4. Ray trajectories for 5.0=ξ : a) 8.0=υ ,b) 2=υ
a b
Fig. 5. Ray trajectories for : a)zB || 8.0=υ ,b) 2=υ
In Fig.5 results of modeling under assumption of
constant magnetic field direction, and distribution of
magnetic filed strength like in case of Fig. 2 are
presented.
For constant magnetic field direction approximation ray
trajectories are appreciably different as compared to the
case of real magnetic field distribution.
4. CONCLUSION
In present communication analytical and numerical
investigations of electromagnetic waves propagation near
electron cyclotron resonance surface are presented taking
into account variation of both magnetic field and
magnetic field direction. Differences with analytical
solution for constant magnetic field direction are
demonstrated.
REFERENCES
1. E.D. Gospodchikov, O.B. Smolyakova, E.V. Suvorov.
Ray Trajectories and Electron Cyclotron Absorption in
an Axisymmetric Magnetic Confinement System //
Plasma Physics Reports. 2007, v. 33, N 5, p. 427–434.
2. A.V. Zvonkov, A.V. Timofeev // Plasma Physics
Reports. 1988, v. 14, N 10, p. 1270–1273.
3. E.D Gospodchikov, E.V. Suvorov. On Cyclotron
Absorption of Electromagnetic Waves with
Longitudinal Propagation in Magnetized Plasma //
Radiophysics and Quantum Electronics. 2005, v. 48,
N 8, p. 641-647.
4. A.D. Piliya, V.I. Fedorov // JETP. 1971, v. 60, N 1,
p. 389.
5. R. Geller. Electron Cyclotron Resonance Sources:
Historical Review and Future Prospects // Rev. of
Scientific Instruments. 1998, v. 69, N 3, p. 1302-1310.
Article received 13.09.2010
ВЛИЯНИЕ НЕОДНОРОДНОСТИ НАПРАВЛЕНИЯ МАГНИТНОГО ПОЛЯ НА
РАСПРОСТРАНЕНИЕ ВОЛНОВЫХ ПУЧКОВ В ПРЯМЫХ МАГНИТНЫХ ЛОВУШКАХ
Е. Д. Господчиков, О.Б. Смолякова, Е.В. Суворов
Аналитически и численно исследованы особенности распространения электромагнитных волн электронного
циклотронного диапазона частот в аксиально-симметричных магнитных ловушках с учетом как
неоднородности модуля магнитного поля, так и соответствующей ей неоднородности направления магнитного
поля. Продемонстрировано, что неоднородность направления магнитного поля может качественно изменить
картину распространения волн и заметно сказаться на эффективности электронного циклотронного нагрева
плазмы в прямой магнитной ловушке.
ВПЛИВ НЕОДНОРІДНОСТІ НАПРЯМКУ МАГНІТНОГО ПОЛЯ НА ПОШИРЕННЯ ХВИЛЬОВИХ
ПУЧКIВ У ПРЯМИХ МАГНІТНИХ ПАСТКАХ
Є.Д. Господчиков, О.Б. Смолякова, Є.В. Суворов
Аналітично та чисельно досліджено особливості поширення електромагнітних хвиль електронного
циклотронного діапазону частот в аксіально-симетричних магнітних пастках при взятих до уваги як
неоднорідності модуля магнітного поля, так і відповідної до неї неоднорідності напрямку магнітного поля.
Продемонстровано, що неоднорідність напрямку магнітного поля може якісно змінити картину
розповсюдження хвиль та помітно позначитися на ефективності електронного циклотронного нагріву плазми в
прямій магнітній пастці.
-6 -4 -2 0 2
-8
-4
0
4
8
- - - 0
-
-
0
5
1
2
-6 -4 -2 0 2
-10
-5
0
5
10
8.0
- - -
4.0
0.0
0.0
2.0
-
-
Institute of Applied Physics RAS, Nizhniy Novgorod, Russia
E-mail: eggos@mail.ru
|