Ion cyclotron resonance for fast magnetosonic waves in small tokamaks
The absorption of fast magnetosonic waves in a small-size tokamak has been studied under the multiple cyclotron resonance for bulk ions and under the fundamental resonance for minority ions. The small plasma radius is assumed to be less than or equal to, in the order of magnitude, the transverse wav...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Ion cyclotron resonance for fast magnetosonic waves in small tokamaks / A.I. Pyatak, K.N. Stepanov, S.V. Borisko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 79-80. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-785092015-03-19T03:02:07Z Ion cyclotron resonance for fast magnetosonic waves in small tokamaks Pyatak, A.I. Stepanov, K.N. Borisko, S.V. Magnetic confinement The absorption of fast magnetosonic waves in a small-size tokamak has been studied under the multiple cyclotron resonance for bulk ions and under the fundamental resonance for minority ions. The small plasma radius is assumed to be less than or equal to, in the order of magnitude, the transverse wavelength. In this case, the solution to the linearized Vlasov equation for the distribution function of the resonance ions in the form of a power series in their Larmor radius and the expression for the contribution from resonance ions into the wave current density have been obtained. The nonlocal coupling between the wave field and current density in a small azimuth angle has been taken into account, which is associated with the motion of resonance ions along a magnetic line of force while travelling through the cyclotron resonance zone, with consideration for the nonuniform magnetic field along a line of force. Allowances have been also made for the decorrelation effects connected with the Coulomb collisions between the resonance particles during their travelling across the successive zones of the resonance, which effects lead to the dephasing of the particle-wave phase. The expressions for the RF power absorbed in the tokamak plasma have been obtained and analyzed. 2000 Article Ion cyclotron resonance for fast magnetosonic waves in small tokamaks / A.I. Pyatak, K.N. Stepanov, S.V. Borisko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 79-80. — Бібліогр.: 6 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78509 533.9 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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DSpace DC |
| language |
English |
| topic |
Magnetic confinement Magnetic confinement |
| spellingShingle |
Magnetic confinement Magnetic confinement Pyatak, A.I. Stepanov, K.N. Borisko, S.V. Ion cyclotron resonance for fast magnetosonic waves in small tokamaks Вопросы атомной науки и техники |
| description |
The absorption of fast magnetosonic waves in a small-size tokamak has been studied under the multiple cyclotron resonance for bulk ions and under the fundamental resonance for minority ions. The small plasma radius is assumed to be less than or equal to, in the order of magnitude, the transverse wavelength. In this case, the solution to the linearized Vlasov equation for the distribution function of the resonance ions in the form of a power series in their Larmor radius and the expression for the contribution from resonance ions into the wave current density have been obtained. The nonlocal coupling between the wave field and current density in a small azimuth angle has been taken into account, which is associated with the motion of resonance ions along a magnetic line of force while travelling through the cyclotron resonance zone, with consideration for the nonuniform magnetic field along a line of force. Allowances have been also made for the decorrelation effects connected with the Coulomb collisions between the resonance particles during their travelling across the successive zones of the resonance, which effects lead to the dephasing of the particle-wave phase. The expressions for the RF power absorbed in the tokamak plasma have been obtained and analyzed. |
| format |
Article |
| author |
Pyatak, A.I. Stepanov, K.N. Borisko, S.V. |
| author_facet |
Pyatak, A.I. Stepanov, K.N. Borisko, S.V. |
| author_sort |
Pyatak, A.I. |
| title |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| title_short |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| title_full |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| title_fullStr |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| title_full_unstemmed |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| title_sort |
ion cyclotron resonance for fast magnetosonic waves in small tokamaks |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78509 |
| citation_txt |
Ion cyclotron resonance for fast magnetosonic waves in small tokamaks / A.I. Pyatak, K.N. Stepanov, S.V. Borisko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 79-80. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT pyatakai ioncyclotronresonanceforfastmagnetosonicwavesinsmalltokamaks AT stepanovkn ioncyclotronresonanceforfastmagnetosonicwavesinsmalltokamaks AT boriskosv ioncyclotronresonanceforfastmagnetosonicwavesinsmalltokamaks |
| first_indexed |
2025-07-06T02:35:03Z |
| last_indexed |
2025-07-06T02:35:03Z |
| _version_ |
1836863250116902912 |
| fulltext |
Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 79-80 79
UDC 533.9
ION CYCLOTRON RESONANCE FOR FAST MAGNETOSONIC WAVES
IN SMALL TOKAMAKS
A.I.Pyatak*, K.N.Stepanov**, S.V.Borisko*
* Kharkiv State Automobile Highway & Technical University, 25, Petrovsky Str., 61002,
Kharkiv, Ukraine
** National Science Center “Kharkiv Institute of Physics and Technology”
Institute of Plasma Physics, 1, Akademichna, 61108, Kharkiv, Ukraine
FAX: 35-26-64. E-mail: stepanov@ipp.kharkov.ua
The absorption of fast magnetosonic waves in a small-size tokamak has been studied under the multiple
cyclotron resonance for bulk ions and under the fundamental resonance for minority ions. The small plasma radius is
assumed to be less than or equal to, in the order of magnitude, the transverse wavelength. In this case, the solution to
the linearized Vlasov equation for the distribution function of the resonance ions in the form of a power series in
their Larmor radius and the expression for the contribution from resonance ions into the wave current density have
been obtained. The nonlocal coupling between the wave field and current density in a small azimuth angle has been
taken into account, which is associated with the motion of resonance ions along a magnetic line of force while
travelling through the cyclotron resonance zone, with consideration for the nonuniform magnetic field along a line
of force. Allowances have been also made for the decorrelation effects connected with the Coulomb collisions
between the resonance particles during their travelling across the successive zones of the resonance, which effects
lead to the dephasing of the particle-wave phase. The expressions for the RF power absorbed in the tokamak plasma
have been obtained and analyzed.
1. Introduction
The problem on plasma heating with fast
magnetosonic waves (FMSW) due to the ion cyclotron
resonance is conventionally considered within the
framework of a short-wavelength approximation for the
waves with the transverse wavelength ! ~ ωAv
considerably below the small plasma radius ap by
analytical and numerical techniques [1,2]. For a
tokamak the condition of the cyclotron resonance
( ) ||||, vkrn c +ϑω≈ω is met for thermal resonant particles
moving according to a Larmor spiral along magnetic
field lines only in a narrow region where the exchange
of energy between the wave field and a particle is
essential. Here ( )ϑω ,rci is the cyclotron frequency
( ) cos
R
r1,
ϑ+ω=ϑω cici r , (1)
where
cm
eB
i
ci
0=ω , 0B is the magnetic field value at the
center of the toroidal chamber, r is the distance from
this center, ϑ is the small azimuth angle, R is the large
radius of the torus, the toroidicity is assumed to be
small, 1<<=ε Rrt . As the time a particle requires to
traverse a resonant zone τ0 is small, one can neglect the
effect of binary Coulomb collisions on the wave-particle
interaction during this period if the collision frequency
is sufficiently small. Having left the resonance zone,
such a particle travels along the magnetic field line a
considerably larger distance (order of qR) before it
enters the next zone of cyclotron resonance (q is the
safety factor). During this period TivqRt ~∆
( )Tivv ~|| , the particle will obtain due to collisions a
random increment of the velocity along the magnetic
field tvv colTi ∆νδ ~|| . Therefore the wave-particle
phase will experience the change tvk ∆δΦδ ||||~ . For
thermal particles 1>>Φδ , so that one can regard the
passage through the cyclotron resonance zone as
random and neglect the effect of preceding passages
through cyclotron resonance zones, i.e., the effect of all
the particle trajectory at the section t ′ from −∞ to
0τ−t . This effect happens to be of value only for fast
particles for which due to the small value of the
collision frequency the quantity 1≤Φδ . Collisional
decorrelation of the wave-particle phase is determined
in papers [3-5]. In what follows we will consider the
case of strong decorrelation when 1>>Φδ , and take
into account nonlocal effects on a small section of the
ion trajectory near the point of resonance.
Below there will be considered the problem of the
cyclotron resonance ( )ϑω=ω ,rn ci for FMSW in small-
size tokamaks, when pa≥⊥! . Thus the report takes
into account the spatial dispersion related to the
resonant particle motion along the magnetic field line in
the nonuniform magnetic field of the tokamak leading to
the nonlocal dependence of the current density of
resonant ions on the small azimuth angle ϑ, and to the
nonlocality of current over the small radius.
2. RF current density
As the toroidicity of the tokamak is small in the case
considered, we can apply a cylindrical approximation
for taking the spatial dispersion into account, and
assume that the spatial dispersion of the equilibrium
distribution function F0i is determined by the integral of
equations of motion
mailto:stepanov@ipp.kharkov.ua
80
( )ϑφ
ωω
−++= ⊥⊥ sin22
2
2
cici
vrvrX (2)
where φ is the azimuth angle in the velocity space and
( )ϑω ,rci differs weakly from ciω in this case. Then
taking into account that the ion Larmor radius civ ω⊥
is small compared with the plasma radius, one can
develop the function F0i in powers of civ ω⊥ and keep
in the expression for the RF current density j=ei ∫ f
~
vdv
the leading terms in the small parameter
1<<⊥⊥ civk ω (see the corresponding calculations for
the nonuniform over radius plasma cylinder [6]). Take
also into account that the passage of the resonance zone
results in the nonlinear variation of the wave-particle
phase against the angle ϑ′ ( ϑ′ if the variation of the
angle ϑ when the particle crosses the resonance zone):
( )( )[ ]
( ) ( ) ,sin
2
1
,
2
|||||| ϑ′ϑζ+ϑ′
−
εω
ϑω−ωζ=
=′′ϑω−ω=∆Φ ∫
nqRkn
tdtrn
tci
ci
ci
(3)
where |||| vqrciω=ζ , q(r) is the safety factor.
Then applying the technique of integrating along the
trajectories for determining the perturbed distribution
function f~ , we obtain for the left-hand polarized
component of the RF current density in the FMSW field
( ) ϑ
+ +=ζϑ ijjrj r,, the following final expression:
( ) ( )
( )
( )
),(ˆ)(
sin
)(
1
1
!2
1
||
42
||
||
222
1
0
121
2
2
||
ϑ∆×
×
ϑζ
×
×ρω
∂
∂×
×
−=
+ϑ−−
⊥
∞
∞−
π−−
−
−
=
+ϑ−
π−−−+
∫
∑
rEeLzW
eRrqe
rvv
dv
r
rr
re
p
ne
n
nj
ippn
i
v
v
Ti
n
ipi
p
p
n
p
pipipn
n
Ti
(4)
Here
ci
Ti
i
v
ω
=ρ ,
i
ii
pi m
rne )(4 0
2
2 π=ω ,
!
))...(1(
p
pnnn
p
n −−=
.
The operators ⊥∆ and L̂ act on the function +ϑEei ,
2
2
22
2
2
2
2
2 11
ϑ∂
∂+
∂
∂+
∂
∂=
∂
∂+
∂
∂=∆⊥ rrrryx
,
ϑ∂
∂+
∂
∂−= ϑ
rr
ieL i 1ˆ
W(z) is the probability integral with a complex
argument.
π
+= ∫−
z
tz dteiezW
0
22 21)( ,
4||
||
||
sin2
π−
ϑ
ζ
ζ
−
εω
ω−ω=
i
tci
ci e
qRknz . (5)
The operator
p
rr
∂
∂1 acts on functions n0i(r) and Ti(r).
3. Conclusions
The expression (4) for +j is essentially nonlocal
over r and ϑ . The nonlocal dependence over r due to
the nonuniformity of density and temperature of ions is
important for long wavelength perturbations ( pa≥! ).
The nonuniformity of the magnetic field and the
presence of the rotational transform involve the
nonlocality of currents +j over ϑ , it being important
practically always. It may be neglected only in the
region 1>>z . In this region, using the asymptotic
expression
z
izW
π
≈)( , we obtain for (4) the
expression coinciding with the current density for
adiabatic traps with the uniform magnetic field [6].
Usually for the quantity z we have 1~ || ≤ρ
r
qRkz i
and the nonlocality over ϑ is practically always
important. For the short wavelength FMSW ( pa<<! )
one should keep in the expression (4) the largest term
with p=0, which coincides in this case with the
expression for +j , obtained by different methods in [3-
5]. The value of the RF power absorbed by the plasma
( )∫ ∫ ++ϑ=
*
Re EjdrdrP
with the account of the narrow cyclotron resonance zone
coincides practically with one obtained neglecting the
nonlocality.
The work is performed with the support of the State
fund of fundamental research of the Ministry of Science
of Ukraine, project 2.4/700.
References
1. S.V. Kasilov, A.I. Pyatak, K.N. Stepanov. Nonlocal Theory
of Cyclotron and Cerenkov Absorption in Nonuniformly
Magnetized Plasma. Reviews of Plasma Physics (Edited by
B.B. Kadomtsev) vol.20, New York – London. Consultants
Bureau.1997. p.61-300.
2. M. Brаmbilla. Numerical simulation of ion cyclotron waves
in tokamak plasmas. Plasma Physics Cont. Fusion 1999. Vol.
41, №1, p.1-34.
3. S.V. Kasilov, A.I. Pyatak, K.N. Stepanov. Intern. Conf. on
Plasma Phys. New Delhi, India, (November 22-28, 1989)
vol.2, p.385-388.
4. S.V. Kasilov, A.I. Pyatak, K.N. Stepanov. Cyclotron
resonance for fast magnetosonic waves in an inhomogeneous
magnetic field in toroidal confinement devices, Nuclear
Fusion. 1990. Vol.30,№ 12, p.2467-2495.
5. S.V.Kasilov, A.I.Pyatak, K.N.Stepanov. Nonlocal
quasilinear theory of ion cyclotron absorption of fast
magnetosonic waves in tokamak plasma. Fizika plazmy 1990.
V.15, №8, p.959-968 (In Russian).
6. A.I. Pyatak, K.N. Stepanov, S.V. Borisko. Ion cyclotron
resonance for fast magnetosonic waves in adiabatic traps.
Problems of atomic science and technology, this issue.
UDC 533.9
IN SMALL TOKAMAKS
FAX: 35-26-64. E-mail: stepanov@ipp.kharkov.ua
1. Introduction
2. RF current density
3. Conclusions
References
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