Cyclotron resonance conditions in current-carrying plasmas
The resonant cyclotron wave-particle interactions in the cylindrical current-carrying plasma and axisymmetric toroidal plasma models for large aspect ratio tokamaks with circular, elliptic and D-shaped cross-sections of the magnetic surfaces have been analyzed. The corresponding conditions are der...
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irk-123456789-1488562019-02-19T01:27:10Z Cyclotron resonance conditions in current-carrying plasmas Grishanov, N.I. Azarenkov, N.A. Фундаментальная физика плазмы The resonant cyclotron wave-particle interactions in the cylindrical current-carrying plasma and axisymmetric toroidal plasma models for large aspect ratio tokamaks with circular, elliptic and D-shaped cross-sections of the magnetic surfaces have been analyzed. The corresponding conditions are derived by solving the linearized Vlasov equations for perturbed distribution functions of plasma particles, accounting for the geometry of a confinement magnetic field in the zero-order over magnetization parameters. It is shown that the Doppler shift at the cyclotron resonance conditions in the current-carrying plasmas is entirely different from ones in uniform magnetic field. Проаналізовано умови резонансної взаємодії заряджених частинок з хвилями в плазмовому циліндрі зі струмом та в тороідальних аксіально-симетричних моделях плазми для токамаків з круговим, еліптичним і D-подібним перерізами магнітних поверхонь. Відповідні резонансні умови отримано шляхом розв'язку лінеаризованих рівнянь Власова для збурених функцій розподілу плазмових частинок з урахуванням геометрії утримуючого магнітного поля в нульовому наближенні за параметрами замагніченості. Доведено, що доплерівський зсув в умовах циклотронних резонансів у плазмі зі струмом істотно відрізняється від аналогічних оцінок для плазми в однорідному магнітному полі. Проанализированы условия резонансного взаимодействия заряженных частиц с волнами в плазменном цилиндре с током и в тороидальных моделях плазмы для токамаков с круговым, эллиптическим и Dобразным сечениями магнитных поверхностей. Соответствующие резонансные условия получены путем решения линеаризованных уравнений Власова для возмущенных функций распределения частиц с учетом геометрии удерживающего магнитного поля в нулевом приближении по параметрам замагниченности. Показано, что доплеровская сдвижка в условиях циклотронных резонансов в токопроводящей плазме существенно отличается от аналогичных оценок в плазме с однородным магнитным полем. 2018 Article Cyclotron resonance conditions in current-carrying plasmas / N.I. Grishanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2018. — № 6. — С. 94-97. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.55.Fa, 52.50.Qt http://dspace.nbuv.gov.ua/handle/123456789/148856 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 Grishanov, N.I. Azarenkov, N.A. Cyclotron resonance conditions in current-carrying plasmas Вопросы атомной науки и техники |
| description |
The resonant cyclotron wave-particle interactions in the cylindrical current-carrying plasma and axisymmetric
toroidal plasma models for large aspect ratio tokamaks with circular, elliptic and D-shaped cross-sections of the
magnetic surfaces have been analyzed. The corresponding conditions are derived by solving the linearized Vlasov
equations for perturbed distribution functions of plasma particles, accounting for the geometry of a confinement
magnetic field in the zero-order over magnetization parameters. It is shown that the Doppler shift at the cyclotron
resonance conditions in the current-carrying plasmas is entirely different from ones in uniform magnetic field. |
| format |
Article |
| author |
Grishanov, N.I. Azarenkov, N.A. |
| author_facet |
Grishanov, N.I. Azarenkov, N.A. |
| author_sort |
Grishanov, N.I. |
| title |
Cyclotron resonance conditions in current-carrying plasmas |
| title_short |
Cyclotron resonance conditions in current-carrying plasmas |
| title_full |
Cyclotron resonance conditions in current-carrying plasmas |
| title_fullStr |
Cyclotron resonance conditions in current-carrying plasmas |
| title_full_unstemmed |
Cyclotron resonance conditions in current-carrying plasmas |
| title_sort |
cyclotron resonance conditions in current-carrying plasmas |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2018 |
| topic_facet |
Фундаментальная физика плазмы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148856 |
| citation_txt |
Cyclotron resonance conditions in current-carrying plasmas / N.I. Grishanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2018. — № 6. — С. 94-97. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT grishanovni cyclotronresonanceconditionsincurrentcarryingplasmas AT azarenkovna cyclotronresonanceconditionsincurrentcarryingplasmas |
| first_indexed |
2025-07-12T20:28:28Z |
| last_indexed |
2025-07-12T20:28:28Z |
| _version_ |
1837474376035860480 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №6(118)
94 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2018, № 6. Series: Plasma Physics (118), p. 94-97.
CYCLOTRON RESONANCE CONDITIONS IN CURRENT-CARRYING
PLASMAS
N.I. Grishanov1,2, N.A. Azarenkov1
1V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
2Ukrainian State University of Railway Transport, Department of Physics, Kharkiv, Ukraine
The resonant cyclotron wave-particle interactions in the cylindrical current-carrying plasma and axisymmetric
toroidal plasma models for large aspect ratio tokamaks with circular, elliptic and D-shaped cross-sections of the
magnetic surfaces have been analyzed. The corresponding conditions are derived by solving the linearized Vlasov
equations for perturbed distribution functions of plasma particles, accounting for the geometry of a confinement
magnetic field in the zero-order over magnetization parameters. It is shown that the Doppler shift at the cyclotron
resonance conditions in the current-carrying plasmas is entirely different from ones in uniform magnetic field.
PACS: 52.55.Fa, 52.50.Qt
INTRODUCTION
Effective schemes of plasma heating in tokamaks can
be realized by collisionless wave dissipation in the
range of ion-cyclotron and/or electron-cyclotron
frequencies (fundamental cyclotron resonance: 1 for
ions - ICR, 1 for electrons - ECR) and their
harmonics )2|(| . As is well known [1], the
electromagnetic waves are always absorbed in the
equilibrium plasma models, e.g. with the maxwellian
distribution of charged particles. However, the presence
of non-equilibrium energetic particles can lead to wave
instabilities observed as ion-cyclotron and electron-
cyclotron emissions under the ICR and ECR plasma
heating.
To estimate the wave damping/growth rates in any
plasma model we should know the conditions of the
resonant wave-particle interactions there. The
corresponding conditions can be derived automatically
by solving the linearized Vlasov equation for perturbed
distribution functions of plasma particles, accounting
for the geometry of a confinement magnetic field H0.
In this paper we discuss the cyclotron wave-particle
interactions in the cylindrical current-carrying plasma
(i.e. with a helical magnetic field) and in the two-
dimensional (2D) axisymmetric toroidal plasma models
for tokamaks with circular, elliptic and D-shaped
magnetic surfaces. The Vlasov equations are resolved in
the zero-order over the magnetization parameters, using
an approach developed in Refs. [2-6]. It is shown that
the Doppler shifts at the cyclotron resonance conditions
in the current-carrying plasmas are entirely different
from ones for plasmas in uniform magnetic field [1]:
||||k , where ...,2,1 is the cyclotron
harmonic number; cMHq /0 is the Larmor
frequency of ions )0,( ii and electrons
)0,( ee ; c is the speed of light, zkk kh|| is
the parallel wave-number relative to confinement
magnetic field 0000 /, HH z HheH .
1. CYLINDRICAL PLASMA MODEL
The simplest 1D model of tokamaks is a magnetized
current-carrying plasma cylinder with identical ends in
the helical magnetic field, where the longitudinal ohmic
current generates the poloidal magnetic field
eH 00 H in addition to longitudinal zzz H eH 00 .
In this case the length of plasma cylinder is equal to
02 R , where R0 is the major tokamak radius. As a
result, the field z000 HHH becomes helical with
substantial rotational transformation, allowing to take
into account the so-called shear effects and the radial
profiles of ohmic current by the radial dependence of
plasma safety factor q(r)=rH0z/R0H0
To develop the kinetic theory for cyclotron waves in
such plasma model one should resolve the Vlasov
equation for perturbed distribution functions ),,( vrtf
by the Fourier-decomposition over the polar angle in
velocity space:
),iiiiexp(),,(),,( | |
zkmtrftf zvr
where we have used the usual standard notation for the
radial coordinate r numbering the magnetic surfaces,
is the poloidal angle In velocity space we use the
polar coordinates ( , ) instead of the normal and
binormal components ( bn , ) by the transformation:
cos vnn , sin vbb , vh|| .
The linearized Vlasov equation for cyclotron
harmonics
f in the zero-order over the magnetization
parameters can be reduced to algebraic equations
Qfk ||||||i , (1)
where
qR
nqm
hk
r
mh
k zz
0
||
kh ,
qRdr
dq
q
r
0
1
2
2
, qR
h
r
0
, if H0 <<H0z, (2)
m and n are the poloidal and toroidal wave-numbers,
0/ Rnkz . The expressions of
Q depend substantially
on the number of cyclotron harmonic and on the
steady-state (unperturbed) distribution function of
plasma particles F. For example, if F is maxwellian:
)(
2 bn
T
EiE
M
Fq
Q
, 1 , (3)
2
22
||
5.12
exp
)(
TT
N
F ,
M
T
T
2
,
ISSN 1562-6016. ВАНТ. 2018. №6(118) 95
En and Eb are the normal and binormal components of
the perturbed electric field relative to 0H . However,
independently on the right-hand side of the Vlasov
equation (and F. functions) the wave-particle
resonance conditions in the current-carrying plasmas are
defined by denominator of
f and can be rewritten as
|||| )(k , ...,,2,1,0 . (4)
If 0 we receive the well known Cherenkov
resonance conditions: |||| k , where -corrections are
absent. If 0 we have the cyclotron resonance
conditions on the fundamental (first, 1 ) harmonic
of cyclotron frequency and their high harmonics if
2 . As one can see the cyclotron resonance
conditions in the current-carrying plasma are different
from ones in uniform magnetic field by the -terms,
accounting for the rotation of helical magnetic field
lines z000 HHH on the considered (by r)
magnetic surface. These -terms are very important to
study the wave dissipation/excitation at the so-called
rational magnetic surfaces, where ||k
changes sign.
Of course, it is necessary to distinguish the
resonances on the positive and negative cyclotron -
harmonics. If ...,3,2,1 we have the ICR conditions
|||| )( ki
under the normal Doppler effect
for resonant ions (=i) with the parallel velocities
smaller than the wave phase velocity, |||| / k . In this
case the resonant ions can effectively interact with the
left-hand polarized waves, where rotation of the
transverse electric field component )( bn EiE
coincides with the Larmor ion gyration. The ECR
conditions (for 0 ), |||| )(|| ke , are
realized under the abnormal Doppler effect for resonant
electrons (=e) with the parallel velocities larger than
the wave phase velocity, |||| / k . In this case,
electrons cannot effectively interact with the left-hand
polarized wave since their concentration is small and
gyration is opposite to rotation of )( bn EiE .
If ...,3,2,1 we have the ECR conditions
|||| )( ke
under the normal Doppler
effect for electrons with |||| / k . In this case the
resonant electrons can effectively interact with the right-
hand polarized waves, where the rotation of transverse
electric field component )( bn EiE coincides with the
electron gyration. In contrary, the ICR conditions (for
0 ), |||| )( ki , are realized under the
abnormal Doppler effect for resonant ions with
|||| / k . In this case, ions cannot effectively interact
with the right-hand polarized wave since their gyration
is opposite to rotation of )( bn EiE .
The terms proportional to dq/dr in -corrections for
cyclotron resonance conditions allow us to study the
influence of ohmic current density profiles on the wave-
particle interactions in the current-carrying plasmas:
a) if ohmic current is uniform: dq/dr=0;
b) if ohmic current decreases to plasma edge: dq/dr > 0;
c) if ohmic current increases to plasma edge: dq/dr < 0.
It should be noted, that the signs of dq/dr are
opposite in the ICR and ECR conditions under the
normal Doppler effects for ions and electrons,
increasing the Doppler shift for ions and decreasing it
for electrons.
2. AXISYMMETRIC D-SHAPED TOKAMAK
To describe an axisymmetric D-shaped tokamak we use
the quasi-toroidal coordinates (r,,) connected with the
cylindrical ones (,,z) as [6]
2
2
2
0 sincos
a
dr
rR , , sinr
a
b
z ,
where R0 is the radius of the magnetic axis; a and b are,
respectively, the minor and major semiaxes of the cross-
section of the external magnetic surface. In this model,
all magnetic surfaces have the same elongation equal to
b/a; their triangularityis smalld/a<<1. The cylindrical
components of an equilibrium magnetic field H0 are
cos21sin
2
0
00
a
drR
HH ,
0
00
R
HH ,
cos0
00
R
a
b
HH z . (5)
Here Hand Hare, respectively, the toroidal and
poloidal magnetic field maximums at a given (by r
magnetic surface. Thus,
),(),( 2
0
2
000 rgHHrH H ,
2
22
sincos1
sincoscos1
),(
rg , (6)
where
0R
r
,
2a
dr
,
1
2
2
2
a
b
h , 24 h ,
2
0
2
0
0
HH
H
h
,
2
0
2
0
0
HH
H
h
. (7)
In tokamaks, in contrast to a cylindrical current-
carrying plasma, the particle velocities || and are
not constant. To reduce the number of derivatives in the
Vlasov equation we use the standard method of
switching to new variables associated with conservation
integrals of energy and magnetic moment, introducing
the variables and instead of || and as
22
||
2
,
),(
1
22
||
2
rg
. (8)
In this case the linearized Vlasov equation for the
perturbed distribution functions of ions and electrons,
1
, )iiiexp(),,,(),,(
s
s ntrftf
vr ,
can be reduced to the first order differential equations
with respect to the poloidal angle . For example [6],
first harmonics )1( of
sf ,
satisfy the equation:
22 sincoscos1
)(1 g
96 ISSN 1562-6016. ВАНТ. 2018. №6(118)
2
,,
sincos1
i ss nqff
(9)
,),(
sincoscos1
),(~),(1i
)(i-
2
0
22
,
,
0
0
ErgF
M
qRq
s
frrg
fg
qR
s
T
s
s
where bn EEE i , 1h , 1h , qRhr 0/ ,
0
0
H
H
q ,
cM
HHq
2
0
2
0
0
. (10)
Account of centrifugal forces in Eq. (9) is reduced to
.cos21sincos1
2
cos21sincos
cos21sincos
1
2
)11(
sincos1
cos
2
3
cos21sincos
2
3
),(~
2
2
2
2
2
2
2
2
22
2
2
2
3
3
2
2
2
2
2
a
b
dr
dq
q
r
a
b
a
ba
b
b
a
a
b
a
b
b
a
r
By 1s we distinguish the perturbed distribution
functions of particles,
sf ,
, with positive and negative
parallel velocity )(-1|| r,gs relative to H0.
Describing the wave-particle interaction in tokamaks
with one minimum of H0, i.e. when , we should [5,
6] separate all particles on two groups of untrapped (u)
and trapped (t) particles by the inequalities for and :
u 0 - untrapped particles,
tu tt - trapped particles,
analyzing the condition 0),(|| . Here
2
1
u ,
2
1
t , (12)
and the angels t are the stop points of the trapped
particles on the considered magnetic surface:
2
2
2/
1
1
)1(
2/
1
arccos
t
3)
2
(
2/
1
1
22
. (13)
To find the perturbed distribution functions of
untrapped s
uf ,
,
and trapped s
tf ,
,
particles we should
resolve Eq. (9) using the corresponding boundary
conditions: the periodicity of s
uf ,
,
on , and continuity
of s
tf ,
,
at the stop points t ; introducing the new
time-like variable instead of poloidal angle as
d
rg
0
22
),(1
sincoscos1
)( . (14)
In this case, the transit-time of u-particles and the
bounce-period of t-particles are proportional to
)(2 uT and )(4 ttT , respectively.
As a result, the cyclotron harmonics of the perturbed
distribution functions of untrapped and trapped particles
can be found in the forms:
),,(iexp ,0,
,
,
,,
,
,
s
p
p
s
up
s
u ff
,
),,(iexp ,0,
,
,
,,
,
,
s
p
p
s
tp
s
t ff
, (15)
where p is a number of the bounce resonances. After the
bounce-averaging we have the following expressions for
the bounce-resonant harmonics (if 1 ):
m
m
s
up
sm
p
uT
s
up E
Z
A
TM
Fq
f
),(
),(
i
,
,,
,,
,
2
,
,,
,
m
m
s
tp
sm
p
tT
s
tp E
Z
B
TM
Fq
f
),(
),(
i
,
,,
,,
,
2
,
,,
. (16)
Here
ut
u
s
up g
I
nqp
qTR
s
Z 0
0
,
,,
)(2
,
d
rg
A
sm
psm
p
),(1
),,(iexp
),(
,,
,,,
,
,
,)(
)(2
)()()(
)()(
)(
)(2),,(
0
0
,,
,
I
T
IIg
qR
nqm
T
nqp
u
gu
t
u
t
sm
p
t
t
s
tp g
qTR
s
pZ 0
0
,
,,
2
,
t
t
d
rg
B
sm
psm
p
),(1
),,(iexp
),(
,,
,,,
,
t
t
d
rg
sm
pp
),(1
),,(iexp
)1(
,,
,
,
)()(
)(
2),,(,,,
,
t
t
sm
p nqm
T
p
,)()()(0
0
IIg
qR
s gt (17)
0
22
,1
sincoscos1
),()( d
rg
rgI g ,
0
22 sincoscos1
),(~
)(
dr
I ,
)(
2
g
u
u I
T
g , )(
4
tg
t
t I
T
g ,
2
)(
1
qqt ,
m
immeEE
2/1
`
2
4/322
sincos1
sincoscos1
)( ,
)2sin()(
4
sin)(
.
ISSN 1562-6016. ВАНТ. 2018. №6(118) 97
The zeros of denominators in (16) determine us the
conditions of the cyclotron wave-particle interactions in
D-shaped tokamaks:
0
)(2
0
0
ut
u
g
I
lnqp
qTR
s
(18)
for the untrapped particles; and
0
2
0
0
t
t
g
qTR
s
p
(19)
for trapped particles. These wave-particle resonance
conditions in axisymmetric D-shaped tokamaks involve
two energetic characteristics of particles (by and ),
the wave frequency , the integer numbers of cyclotron
(by ) and bounce (by р) resonances. For the low , as
usual, we have the conditions of the:
- Cherenkov resonance, if 0 ;
- normal ICRs (i), if ,...3,2,1 ;
- normal ECRs (e), if ...3,2,1 ,
for both the untrapped and trapped particles.
Of course, analyzing the cyclotron wave-particle
interactions in toroidal geometry we should take into
account the coefficients ),(,,
,
sm
pA and ),(,,
,
sm
pB
for untrapped and trapped particles, respectively.
CONCLUSIONS
Regarding the plasma response to perturbations in the
current-carrying plasmas the kinetic wave analysis should
take into account the so-called shear effects connected
with the fact that the equilibrium magnetic field lines
become helical and there are additional inertial
(centrifugal) forces acting on the moving charged
particles.
Specific features of the wave-particle interactions in
D-shaped tokomaks are due to that i) the resonance
conditions for untrapped and trapped particles are
different, and ii) all m-harmonics of the perturbed electric
field contribute to the perturbed distribution functions of
untrapped and trapped particles.
If triangularity is absent, i.e. if 0d , the wave-
particle resonant conditions for untrapped and trapped
particles, Eq. (18) and Eq. (19), can be readily reduced
to the corresponding expressions for tokamaks with
elliptic magnetic surfaces. If elongation is absent (b=a),
Eqs. (18), (19) have as limits the wave-particle resonance
conditions for tokamaks with circular magnetic surfaces
[2-4]. If 00 R , the cyclotron wave-particle resonant
conditions for untrapped particles in tokamaks can be
transformed to analogous conditions, Eq. (4), in the
current-carrying plasma cylinder.
The -corrections at ICR and ECR conditions for
plasma systems in the helical magnetic field are very
important analyzing the cyclotron wave
dissipation/excitation at the rational magnetic surfaces,
where 0|| k . The terms proportional to dq/dr in –
corrections allow us to study the influence of ohmic
current density profiles on the cyclotron wave-particle
interactions in the current-carrying plasma systems.
REFERENCES
1. A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Si-
tenko, K.N. Stepanov // Plasma Electrodynamics.
Oxford: “Pergamon”, 1975, v. I.
2. N.I. Grishanov, F.M. Nekrasov. Dielectric constant in
toroidal plasmas // Sov. J. Plasma Phys. 1990, v. 16,
p. 129-133.
3. N.I. Grishanov, C.A. de Azevedo, J.P. Neto.
Dielectric characteristics of axisymmetric low aspect
ratio tokamak plasmas // Plasma Phys. Controlled
Fusion. 2001, v. 43, p. 1003-1022.
4. N.I. Grishanov, N.A. Azarenkov. Dispersion relations
for field-aligned cyclotron waves in an axisymmetric
tokamak plasma with anisotropic temperature //
Problems of Atomic Sci. and Techn. Series “Plasma
Physics”. 2011, № 1(17), p. 59-61.
5. N.I. Grishanov, G.O. Ludwig, C.A. de Azevedo,
J.P. Neto. Wave dissipation by electron Landau damping
in low aspect ratio tokamaks with elliptic magnetic
surfaces // Phys. Plasmas. 2002, v. 9, p. 4089-4092.
6. N.I. Grishanov, N.A. Azarenkov. Cyclotron wave
absorption in D-shaped tokamaks // Problems of Atomic
Sci. and Techn. Series “Plasma Physics”. 2016,
№ 6(22), p. 52-55.
Article received 25.09.2018
УСЛОВИЯ ЦИКЛОТРОННЫХ РЕЗОНАНСОВ В ПЛАЗМЕ С ТОКОМ
Н.И. Гришанов, Н.А. Азаренков
Проанализированы условия резонансного взаимодействия заряженных частиц с волнами в плазменном
цилиндре с током и в тороидальных моделях плазмы для токамаков с круговым, эллиптическим и D-
образным сечениями магнитных поверхностей. Соответствующие резонансные условия получены путем
решения линеаризованных уравнений Власова для возмущенных функций распределения частиц с учетом
геометрии удерживающего магнитного поля в нулевом приближении по параметрам замагниченности.
Показано, что доплеровская сдвижка в условиях циклотронных резонансов в токопроводящей плазме
существенно отличается от аналогичных оценок в плазме с однородным магнитным полем.
УМОВИ ЦИКЛОТРОННИХ РЕЗОНАНСІВ У ПЛАЗМІ ЗІ СТРУМОМ
М.І. Гришанов, М.О. Азарєнков
Проаналізовано умови резонансної взаємодії заряджених частинок з хвилями в плазмовому циліндрі зі
струмом та в тороідальних аксіально-симетричних моделях плазми для токамаків з круговим, еліптичним і
D-подібним перерізами магнітних поверхонь. Відповідні резонансні умови отримано шляхом розв'язку
лінеаризованих рівнянь Власова для збурених функцій розподілу плазмових частинок з урахуванням
геометрії утримуючого магнітного поля в нульовому наближенні за параметрами замагніченості. Доведено,
що доплерівський зсув в умовах циклотронних резонансів у плазмі зі струмом істотно відрізняється від
аналогічних оцінок для плазми в однорідному магнітному полі.
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