Propagation of the fast magnetosonic wave through the generalized Budden barrier
Propagation of the fast magnetosonic wave through the generalized Budden barrier, which is formed by the ion-ion hybrid resonance and the accompanying L-cutoff, is studied. Analytical expressions for the transmission, reflection and conversion coefficients are derived. It is shown that the nonzero r...
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| Zitieren: | Propagation of the fast magnetosonic wave through the generalized budden barrier / Ye.O. Kazakov, I.V. Pavlenko, I.O. Girka // Вопросы атомной науки и техники. — 2010. — № 4. — С. 90-93. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860134540087918592 |
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| author | Kazakov, Ye.O. Pavlenko, I.V. Girka, I.O. |
| author_facet | Kazakov, Ye.O. Pavlenko, I.V. Girka, I.O. |
| citation_txt | Propagation of the fast magnetosonic wave through the generalized budden barrier / Ye.O. Kazakov, I.V. Pavlenko, I.O. Girka // Вопросы атомной науки и техники. — 2010. — № 4. — С. 90-93. — Бібліогр.: 8 назв. — англ. |
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| description | Propagation of the fast magnetosonic wave through the generalized Budden barrier, which is formed by the ion-ion hybrid resonance and the accompanying L-cutoff, is studied. Analytical expressions for the transmission, reflection and conversion coefficients are derived. It is shown that the nonzero reflection from the barrier arises in case of the wave incidence from the resonance side, and the conversion coefficient can reach the value 48.6% for the cutoff incidence case. The obtained results generalize the formulas of the Budden theory in case of the different fast wave wavelength at the opposite sides of the ion-ion hybrid resonance.
Решена задача распространения быстрой магнитозвуковой волны через обобщенный барьер Баддена, который образован ион-ионным гибридным резонансом и связанной с ним L-отсечкой. Получены аналитические выражения для коэффициентов прохождения, отражения и конверсии. Показано, что имеет место ненулевое отражение от барьера в случае падения волны со стороны резонанса, а коэффициент конверсии может достигать величины 48.6% при падении волны со стороны отсечки. Полученные результаты обобщают формулы теории Баддена на случай различной длины волны по разные стороны от ион-ионного гибридного резонанса.
Розв’язано задачу про поширення швидкої магнітозвукової хвилі крізь узагальнений бар’єр Баддена, який утворений іон-іонним гібридним резонансом та L-відсічкою, що пов’язана з ним. Здобуто аналітичні вирази для коефіцієнтів проходження, відбиття та конверсії. Показано, що має місце ненульове відбиття від бар’єру для випадку падіння хвилі зі сторони резонансу, а коефіцієнт конверсії може сягати величини 48.6% за умови падіння хвилі зі сторони відсічки. Здобуті результати узагальнюють формули теорії Баддена на випадок різної довжини хвилі по різні боки від іон-іонного гібридного резонансу.
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ТЕРМОЯДЕРНЫЙ СИНТЕЗ (КОЛЛЕКТИВНЫЕ ПРОЦЕССЫ)
PROPAGATION OF THE FAST MAGNETOSONIC WAVE
THROUGH THE GENERALIZED BUDDEN BARRIER
Ye.O. Kazakov, I.V. Pavlenko, I.O. Girka
V.N. Karazin Kharkov National University, Kharkov, Ukraine
E-mail: kazakov_evgenii@mail.ru
Propagation of the fast magnetosonic wave through the generalized Budden barrier, which is formed by the ion-
ion hybrid resonance and the accompanying L-cutoff, is studied. Analytical expressions for the transmission, reflec-
tion and conversion coefficients are derived. It is shown that the non-zero reflection from the barrier arises in case of
the wave incidence from the resonance side, and the conversion coefficient can reach the value 48.6% for the cutoff
incidence case. The obtained results generalize the formulas of the Budden theory in case of the different fast wave
wavelength at the opposite sides of the ion-ion hybrid resonance.
PACS: 52.50.Qt
1. INTRODUCTION. BUDDEN THEORY
The Ion Cyclotron Resonance Heating (ICRH) is
widely used in modern tokamaks [1]. The ICRH an-
tenna, which is located either at the high field side
(HFS) or at the low field side (LFS) of the trap,
launches the fast magnetosonic wave (FW) into the
plasma. The wave propagates to the plasma center, and
is either absorbed at the fundamental and harmonic cy-
clotron resonance layers by ions, or is converted to the
small-scale plasma mode at the ion-ion hybrid (IIH)
resonance layer. The latter arises only in multicompo-
nent plasmas with two or more ion species with the dif-
ferent charge-to-mass ratio. In this regime, which is
known as the mode conversion, the localized electron
heating is observed [2]. The effective electron Landau
damping of the converted mode occurs due to the up-
shift of the parallel wavenumber under the presence of
the toroidal current in tokamaks [3]. Mode conversion
regime is extensively studied within the last years since
it has a number of important applications beyond heat-
ing itself [4]. To name just a few: it is used to study
electron transport, generate plasma rotation and current
drive, measure the plasma composition, as a mechanism
of impurity pump-out, etc. The successful performance
of such a heating scenario relies on the achievement of
the effective conversion conditions. Thus, the numerous
efforts have been made to understand the physics of the
mode conversion.
The propagation of the FW through the inhomoge-
neous in the direction plasma is usually described by
the wave equation
(1)
where is one of the electric field components of the
wave, and is the potential function which depends
on the dispersion relation for the FW,
. The latter is given by
(2)
Here, , and are the components of the cold plas-
ma dielectric tensor in the notation of Stix [5], is the
parallel (with respect to the magnetic field) refractive
index. In the ion cyclotron frequency range the reso-
nance denominator condition defines the ion-
ion hybrid resonance. Its frequency lies between the ion
cyclotron frequencies of the ion species, and . It is
located near the cyclotron resonance of the minority
ions, and shifts towards majority resonance with the
minority concentration increase. The IIH resonance is
accompanied by the left-hand polarized L-cutoff, which
is defined by the condition , towards the LFS.
Together they form the evanescence layer, where
(Fig.1). The hot plasma theory resolves the
IIH resonance. The more sophisticated full-wave mod-
els show that at the IIH resonance layer the FW couples
to the small-scale mode.
Fig.1. The typical spatial dependence of the FW-
refractive index for the two-ion component plasma.
The evanescence layer is formed by the ion-ion hybrid
resonance and the accompanying L-cutoff
The classical theory which describes the propagation
of the FW through the isolated IIH cutoff-resonance pair
is the Budden theory [6]. In this case the potential func-
tion is modeled by the following expression:
(3,a)
where is the wavenumber of the FW far from the
resonance, is the width of the evanescence layer,
is the location of the IIH resonance. If we normalize all
_______________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2010. № 4.
Серия: Плазменная электроника и новые методы ускорения (7), с.90-93.
90
mailto:kazakov_evgenii@mail.ru
the spatial variables on the FW wavelength, i.e. intro-
duce new variable , then the potential
(3,a) is simplified and given by
(3,b)
The dimensionless parameter is called as
the tunneling factor. Within the Budden theory it en-
tirely defines the scattering coefficients. The important
feature of the considered barrier is the asymmetry of the
scattering coefficients with respect to the side of the
wave incidence. While the transmission coefficient
equals to regardless of the incidence side,
the dependence of the reflection coefficient is essen-
tially different for the HFS and LFS cases. For the HFS
incidence (resonance side) the wave is transmitted
through the layer without any reflection. The rest of the
energy is converted to the small-scale mode. In such a
way providing the evanescence layer enough thick (by
increasing the concentration of the minority ions) the
effective mode conversion is obtained. This is not ap-
propriate for the case of the antenna location at the LFS
(as for most of present-say tokamaks). In this case the
reflection coefficient is equal to . For
the thick evanescence layers it is the dominant process.
The mode conversion coefficient,
reaches its maximal value 25%, when the evanescence
layer is semi-transparent one, .
The Budden theory implies the inhomogeneity of the
magnetic field. In a real situation due to the decrease of
the plasma density to the edge, the dispersion of the FW
is more complicated than that described by (3a). The
FW wavenumber at the HFS decreases (Fig.1), and even
the additional R-cutoff at the HFS can appear. In the
present paper the generalized Budden barrier is consid-
ered, for which the FW is assumed to have different
wavelength at the cutoff and resonance sides. The ana-
lytical formulas for the scattering coefficients are de-
rived. Comparison with the Budden results is presented.
2. GENERALIZED BUDDEN POTENTIAL
This section describes the scattering properties of the
generalized Budden barrier. It is convenient to normal-
ize all the spatial variables to the FW wavelength at the
LFS (cutoff) side. Then, the potential is written simi-
larly to (3b):
(4)
Its spatial dependence is shown in Fig.2. The pa-
rameter is the ratio of the FW wave-
length at the cutoff and resonance sides.
For both sides of the problem the analytical solution
of the wave equation in terms of the confluent hy-
pergeometric (Whitakker) functions can be written. For
region the solution is written as
(5)
where the functions and are given by
(6)
and is the Kummer’s function,
is the second independent solution of the
Kummer’s equation. The definition and properties of
these functions can be found in [7].
In region , the solution of (4) is represented as
follows:
(7)
where , , , and
(8)
Fig.2. Plot of the generalized Budden potential, which
describes the propagation of the FW through
the isolated cutoff-resonance pair
In order to find the global solution of (4) one needs
to match the coefficients and of the solu-
tions (5) and (7). Therefore, three conditions, which
connect the coefficients, should be formulated. The first
two are obtained from the solution matching at the point
. It implies the continuity of the solution function
and its first derivative . Using the expansion
of the Kummer’s functions for small arguments one
obtains:
(9)
where is the Digamma function [7].
The last matching equation depends on the considered
side of the wave incidence. For the HFS incidence, con-
dition
(10,a)
ensures that at the opposite side only the transmitted
wave exists. Similarly, for the case of the LFS inci-
dence, condition
(10,b)
suppresses the non-physical right-travelling term (pro-
portional to ) at the HFS. Condition (10,a) or (10,b)
is called as the radiating boundary condition. Its explicit
form is derived using the asymptotic expansion of (6)
and (8) for large arguments of the independent variable.
Using the matching conditions (9) and (10), the scat-
tering coefficients can be easily calculated. For conven-
ience, we introduce the following parameters:
(11)
(12)
In case of the Budden barrier with the parameter
is infinitely large.
91
The interesting feature of the generalized Budden
barrier is the fact that the transmission coefficient does
not depend on the incidence side like for the classical
Budden case. This feature represents the fundamental
reciprocity principle [8]. The transmission coefficient is
equal to
(13)
For , the formula (13) reduces to the famous Bud-
den result, .
In contrast to the Budden theory the non-zero reflec-
tion occurs for the HFS incidence case. The reflection
coefficient is equal to
(14)
Fig.3 shows the reflection coefficient as a
function of for different values of the tunneling factor.
In the vicinity of the parabolic dependence of
is clearly seen. This part of the curve is described
by the following approximate formula:
(15)
where the small parameter is introduced.
Thus, the Budden case with zero reflection is the excep-
tional one. For any the non-zero reflection from
the barrier occurs.
Fig.3. Dependence of the reflection coefficient
versus for different values of the tunneling factor
The reflection coefficient for the LFS incidence case
is given by
(16)
where stands for the complex conjugate of .
Another distinctive feature of the Budden barrier is
the upper limitation of the conversion coefficient for the
LFS incidence at the level . The conver-
sion coefficient is calculated from the energy conserva-
tion law, , using (13) and (16). Fig.4
shows the dependence of the conversion coefficient
as a function of the tunneling factor for different
values of the parameter . It is clearly seen that for
the conversion coefficient is less than the Budden
result. Vice versa, for the mode conversion coef-
ficient exceeds the Budden level. After some algebraic
manipulations, the approximate analytical formula for
the conversion coefficient is derived. It can be
presented as a sum of two terms
(17)
where the correction function is defined as
(18)
Fig.4. Mode conversion coefficient as a function
of the tunneling factor for different values of
The first term in is the Budden result. The sign
of the second term is determined by the sign of . As
shown in Fig.5, the correction function is positively
defined. Thus, for the correction term in is
positive, and the mode conversion coefficient exceeds
the result of Budden.
Fig.5. Plot of the correction function defined
by (18)
Next, we are interested in the question, what is the
highest level of the conversion coefficient that can be
achieved for the arbitrary value. We have calculated
numerically the value of the maximal conversion coeffi-
cient for the given value of . The results are
shown in Fig.6.
Fig.6. Dependence of the maximal conversion
coefficient as a function of (LFS incidence).
For exceeds the classical Budden result
92
The highest value is reached for
at . It is nearly twice greater than the
classical Budden result.
93
The dispersion of the FW, which is shown in Fig.1,
is calculated for (3He)H plasma with the concentration
of 3He ions X[3He]=4%. The parameters chosen are
typical for the JET tokamak 3He heating experiments:
f=37 MHz, BB0=3.6 T, kz=3.5 m , n-1
e0=2.5·10 cm ,
R
13 -3
0=2.96 m, a=0.9 m. For the conditions considered the
parameter is equal to . The scattering coeffi-
cients calculated using the formulas for the generalized
Budden barrier differ from the results of the classical
theory just by a few percent. Thus, the presented ap-
proximate formulas (17) and (18) give the value of the
conversion coefficient with a high accuracy for
the wide range of experimental parameters.
CONCLUSIONS
The paper describes the propagation of the FW
through the generalized Budden barrier. The assumption
that the wavelength of the FW is equal to both sides of
the barrier is neglected. The analytical solution of the
wave equation in terms of the confluent hypergeometric
functions is derived. The scattering coefficients are
found for both cases of the wave incidence. The detailed
analysis of the scattering coefficients is performed. It is
shown that the obtained results generalize the formulas
of the classical Budden theory. Particularly, it is shown
that the non-zero reflection from the barrier occurs for
the HFS incidence. For the LFS incidence the conver-
sion coefficient can reach the value 48.6% that is nearly
twice greater than the maximum within the Budden the-
ory.
REFERENCES
1. A.V. Longinov and K.N. Stepanov. High-Frequency
Plasma Heating / Ed. A.G. Litvak. New York:
“American Institute of Physics”, 1992,
p.93-238.
2. M.J. Mantsinen, et al. Localized bulk electron heat-
ing with ICRF mode conversion in the JET tokamak
// Nuclear Fusion. 2004, v.44, №1, p.33-46.
3. A.K. Ram and A. Bers. Propagation and damping of
mode converted ion-Bernstein waves in toroidal
plasmas // Phys. Fluids B. 1991, v.3, №4, p.1059-
1069.
4. M.J. Mantsinen, et al. Application of ICRF waves in
tokamaks beyond heating // Plasma Physics and
Controlled Fusion. 2003, v.45, №12A, p.A445-
A456.
5. T.M. Stix. The Theory of Plasma Waves. New York:
“McGraw-Hill”, 1962, 283 p.
6. K.G. Budden. The Propagation of Radio Waves.
Cambridge: “Cambridge University Press”, 1985,
684 p.
7. M. Abramowitz and I.A. Stegun. Handbook of Ma-
thematical Functions. New York: “Dover”, 1970,
1046 p.
8. V.L. Ginzburg. The Propagation of Electromagnetic
Waves in Plasmas. Oxford: “Pergamon Press”, 1970,
615 p.
Статья поступила в редакцию 31.05.2010 г.
РАСПРОСТРАНЕНИЕ БЫСТРОЙ МАГНИТОЗВУКОВОЙ ВОЛНЫ
ЧЕРЕЗ ОБОБЩЕННЫЙ БАРЬЕР БАДДЕНА
Е.A. Казаков, И.В. Павленко, И.А. Гирка
Решена задача распространения быстрой магнитозвуковой волны через обобщенный барьер Баддена, ко-
торый образован ион-ионным гибридным резонансом и связанной с ним L-отсечкой. Получены аналитиче-
ские выражения для коэффициентов прохождения, отражения и конверсии. Показано, что имеет место нену-
левое отражение от барьера в случае падения волны со стороны резонанса, а коэффициент конверсии может
достигать величины 48.6% при падении волны со стороны отсечки. Полученные результаты обобщают фор-
мулы теории Баддена на случай различной длины волны по разные стороны от ион-ионного гибридного ре-
зонанса.
ПОШИРЕННЯ ШВИДКОЇ МАГНІТОЗВУКОВОЇ ХВИЛІ
КРІЗЬ УЗАГАЛЬНЕНИЙ БАР’ЄР БАДДЕНА
Є.О. Казаков, І.В. Павленко, І.О. Гірка
Розв’язано задачу про поширення швидкої магнітозвукової хвилі крізь узагальнений бар’єр Баддена,
який утворений іон-іонним гібридним резонансом та L-відсічкою, що пов’язана з ним. Здобуто аналітичні
вирази для коефіцієнтів проходження, відбиття та конверсії. Показано, що має місце ненульове відбиття від
бар’єру для випадку падіння хвилі зі сторони резонансу, а коефіцієнт конверсії може сягати величини 48.6%
за умови падіння хвилі зі сторони відсічки. Здобуті результати узагальнюють формули теорії Баддена на
випадок різної довжини хвилі по різні боки від іон-іонного гібридного резонансу.
|
| id | nasplib_isofts_kiev_ua-123456789-17308 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:46:21Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kazakov, Ye.O. Pavlenko, I.V. Girka, I.O. 2011-02-25T12:02:27Z 2011-02-25T12:02:27Z 2010 Propagation of the fast magnetosonic wave through the generalized budden barrier / Ye.O. Kazakov, I.V. Pavlenko, I.O. Girka // Вопросы атомной науки и техники. — 2010. — № 4. — С. 90-93. — Бібліогр.: 8 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17308 Propagation of the fast magnetosonic wave through the generalized Budden barrier, which is formed by the ion-ion hybrid resonance and the accompanying L-cutoff, is studied. Analytical expressions for the transmission, reflection and conversion coefficients are derived. It is shown that the nonzero reflection from the barrier arises in case of the wave incidence from the resonance side, and the conversion coefficient can reach the value 48.6% for the cutoff incidence case. The obtained results generalize the formulas of the Budden theory in case of the different fast wave wavelength at the opposite sides of the ion-ion hybrid resonance. Решена задача распространения быстрой магнитозвуковой волны через обобщенный барьер Баддена, который образован ион-ионным гибридным резонансом и связанной с ним L-отсечкой. Получены аналитические выражения для коэффициентов прохождения, отражения и конверсии. Показано, что имеет место ненулевое отражение от барьера в случае падения волны со стороны резонанса, а коэффициент конверсии может достигать величины 48.6% при падении волны со стороны отсечки. Полученные результаты обобщают формулы теории Баддена на случай различной длины волны по разные стороны от ион-ионного гибридного резонанса. Розв’язано задачу про поширення швидкої магнітозвукової хвилі крізь узагальнений бар’єр Баддена, який утворений іон-іонним гібридним резонансом та L-відсічкою, що пов’язана з ним. Здобуто аналітичні вирази для коефіцієнтів проходження, відбиття та конверсії. Показано, що має місце ненульове відбиття від бар’єру для випадку падіння хвилі зі сторони резонансу, а коефіцієнт конверсії може сягати величини 48.6% за умови падіння хвилі зі сторони відсічки. Здобуті результати узагальнюють формули теорії Баддена на випадок різної довжини хвилі по різні боки від іон-іонного гібридного резонансу. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Термоядерный синтез (коллективные процессы) Propagation of the fast magnetosonic wave through the generalized Budden barrier Распространение быстрой магнитозвуковой волны через обобщенный барьер Баддена Поширення швидкої магнітозвукової хвилі крізь узагальнений бар’єр Баддена Article published earlier |
| spellingShingle | Propagation of the fast magnetosonic wave through the generalized Budden barrier Kazakov, Ye.O. Pavlenko, I.V. Girka, I.O. Термоядерный синтез (коллективные процессы) |
| title | Propagation of the fast magnetosonic wave through the generalized Budden barrier |
| title_alt | Распространение быстрой магнитозвуковой волны через обобщенный барьер Баддена Поширення швидкої магнітозвукової хвилі крізь узагальнений бар’єр Баддена |
| title_full | Propagation of the fast magnetosonic wave through the generalized Budden barrier |
| title_fullStr | Propagation of the fast magnetosonic wave through the generalized Budden barrier |
| title_full_unstemmed | Propagation of the fast magnetosonic wave through the generalized Budden barrier |
| title_short | Propagation of the fast magnetosonic wave through the generalized Budden barrier |
| title_sort | propagation of the fast magnetosonic wave through the generalized budden barrier |
| topic | Термоядерный синтез (коллективные процессы) |
| topic_facet | Термоядерный синтез (коллективные процессы) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17308 |
| work_keys_str_mv | AT kazakovyeo propagationofthefastmagnetosonicwavethroughthegeneralizedbuddenbarrier AT pavlenkoiv propagationofthefastmagnetosonicwavethroughthegeneralizedbuddenbarrier AT girkaio propagationofthefastmagnetosonicwavethroughthegeneralizedbuddenbarrier AT kazakovyeo rasprostraneniebystroimagnitozvukovoivolnyčerezobobŝennyibarʹerbaddena AT pavlenkoiv rasprostraneniebystroimagnitozvukovoivolnyčerezobobŝennyibarʹerbaddena AT girkaio rasprostraneniebystroimagnitozvukovoivolnyčerezobobŝennyibarʹerbaddena AT kazakovyeo poširennâšvidkoímagnítozvukovoíhvilíkrízʹuzagalʹneniibarêrbaddena AT pavlenkoiv poširennâšvidkoímagnítozvukovoíhvilíkrízʹuzagalʹneniibarêrbaddena AT girkaio poširennâšvidkoímagnítozvukovoíhvilíkrízʹuzagalʹneniibarêrbaddena |