Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities
The conditions are established when multiple Alfvén eigenmodes are able to withdraw a significant part of the energy of fast ions for possible transfer to another spatial region (spatial channelling). This can happen when the resonance islands of the instabilities overlap to form an extensive stocha...
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| Date: | 2015 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Cite this: | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities / M.H. Tyshchenko, Yu.V. Yakovenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 49-52. — Бібліогр.: 7 назв. — англ. |
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| author | Tyshchenko, M.H. Yakovenko, Yu.V. |
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| citation_txt | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities / M.H. Tyshchenko, Yu.V. Yakovenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 49-52. — Бібліогр.: 7 назв. — англ. |
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| description | The conditions are established when multiple Alfvén eigenmodes are able to withdraw a significant part of the energy of fast ions for possible transfer to another spatial region (spatial channelling). This can happen when the resonance islands of the instabilities overlap to form an extensive stochastic zone in the fast ion phase space. An analytical expression for the width of a resonance island induced by an Alfvén eigenmode in the phase space is derived. The number and amplitude of modes are estimated, which are required to form a stochastic zone in a given energy range. Two codes intended for numerical verification of these estimates are briefly described. First results of these codes are presented.
Устанавливаются условия, при которых множественные альфвеновские собственные моды способны отнять значительную долю энергии у быстрых ионов для возможной передачи в другую область пространства (пространственного каналирования). Это может происходить, если резонансные острова неустойчивостей перекрываются, образуя обширную стохастическую зону в фазовом пространстве быстрых ионов. Выводится аналитическое выражение для ширины резонансного острова, образованного альфвеновской собственной модой в фазовом пространстве. Оцениваются количество и амплитуда мод, необходимые для возникновения стохастической зоны в заданном диапазоне энергий. Кратко описываются два кода, предназначенные для численной проверки этих оценок. Представлены первые результаты этих кодов.
Встановлюються умови, за яких множинні альфвенові власні моди є здатними забрати значну частку енергії в швидких іонів для можливої передачі в іншу область простору (просторового каналювання). Це може траплятись, якщо резонансні острови нестійкостей перекриваються і утворюють широку стохастичну зону в фазовому просторі швидких іонів. Виводиться аналітичний вираз для ширини резонансного острова, створеного альфвеновою власною модою в фазовому просторі. Оцінюються кількість та амплітуда мод, що потрібні для утворення стохастичної зони в певному діапазоні енергій. Стисло описуються два коди, призначені для числової перевірки цих оцінок. Представлено перші результати цих кодів.
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| first_indexed | 2025-11-27T12:35:36Z |
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 49-52. 49
SPATIAL ENERGY CHANNELLING AND STOCHASTIZATION OF FAST ION
MOTION BY HIGH-FREQUENCY PLASMA INSTABILITIES
M.H. Tyshchenko
1
, Yu.V. Yakovenko
1,2
1
Institute for Nuclear Research, Kyiv, Ukraine;
2
National University of Kyiv Mohyla Academy, Kyiv, Ukraine
The conditions are established when multiple Alfvén eigenmodes are able to withdraw a significant part of the
energy of fast ions for possible transfer to another spatial region (spatial channelling). This can happen when the
resonance islands of the instabilities overlap to form an extensive stochastic zone in the fast ion phase space. An
analytical expression for the width of a resonance island induced by an Alfvén eigenmode in the phase space is
derived. The number and amplitude of modes are estimated, which are required to form a stochastic zone in a given
energy range. Two codes intended for numerical verification of these estimates are briefly described. First results of
these codes are presented.
PACS: 52.55.Hc, 52.55.Pi, 55.25.Fi, 52.35.Bj, 05.45.-a
INTRODUCTION
In many experiments on the spherical torus
(spherical tokamak) NSTX, multiple high-frequency
instabilities (the frequency / 2π ~0.5...1f =ω ( ) MHz)
are observed when the neutral beam injection power is
sufficiently high [1, 2]. These instabilities were
identified as GAEs (global Alfvén eigenmodes) and
CAEs (compressional Alfvén eigenmodes) driven by
fast ions. It was pointed out [3] that the instabilities can
channel energy and momentum of fast ions outside the
region where these ions are located (Fig. 1) which
explains the observed deterioration of the energy
confinement. This channelling can lead, in particular, to
cooling the plasma core. A necessary condition for the
spatial energy channelling to occur is the ability of the
instabilities to take away a significant fraction of the
fast ion energy.
Fig. 1. GAE frequency (horizontal line), Alfvén
continuum branches with the mode numbers m , n
and 1m , n (curves 1, 2), and the radial profile of the
beam ions (curve 3). This sketch demonstrates the
energy and momentum channelling by a GAE mode: the
mode receives energy and momentum of the beam ions
mainly inside the region 1rr but gives the energy
and momentum to electrons due to continuum damping
at err
The aim of this work is to estimate the number and
the amplitudes of the modes necessary to produce a
wide stochastic region in the fast-ion phase space. In
this case, the waves are able to receive a major part of
the power of injected ions for transfer to another spatial
region. Within this work, we obtain this estimate by
analytical means and describe the numerical tools
intended for the verification of our analytical results.
Our analysis here is restricted to GAE modes. In
addition, we do not consider cyclotron resonances
(playing probably a significant role in the excitation of
high-frequency GAE instabilities in NSTX), which
enables us to study the particle motion in the guiding-
centre approximation. We believe that this is justified at
the initial stage of our investigation; these limitations
are to be removed in future works.
The structure of this paper is as follows. In Sec. 2,
we obtain analytical estimates of the resonance island
width and apply the Chirikov criterion [4] to obtaining
the stochasticity condition. Our numerical tools and
their first results are described in Sec. 3. Finally, in Sec.
4 we summarize and discuss our results.
1. RESONANCE WIDTH
The general resonance condition can be written as
),( JJns , (1)
where Rv /|| ; )/(|| Rqv are frequencies of
toroidal motion and poloidal motion, respectively; R is
the major radius of the torus; the quantities s and n are
integers, n being always equal to the toroidal wave
number in axisymmetric geometry. We are interested in
resonances of passing particles.
One can show that in linear approximation in wave
amplitude, the guiding centre Lagrangian can be written
as follows
,)]exp(Re[
,ns
sn inistiV
WdtdJdJ=L
(2)
where ,,, JJ are the action-angle variables of the
particle unperturbed motion, , being the toroidal
and poloidal angles, respectively,
)exp()
~
(
2 2
inistiEvdd
ei
Vsn
(3)
is the so-called matrix element of the wave-particle
resonance. For interactions of fast ions with Alfvénic
50 ISSN 1562-6016. ВАНТ. 2015. №1(95)
perturbations we can neglect the contribution of ||
~
E and
A
~
, taking
~~
DvEv
, where
Dv is the drift
velocity,
~
is the scalar potential of the
electromagnetic field. Expanding the unperturbed part
of the Hamiltonian at the resonance and keeping only
the resonant term of the perturbation, we obtain the
resonance width as follows:
2/1
'
4)( sn
island
nV
J . (4)
Using the relationship
s
J
n
JW
(5)
and evaluating
snV for passing particles, we can write
the following analytical expression for the width of the
resonance islands of passing fast ions
21
21
2
~
1
)1(4
)(
B
B
RkW
W r
s
island , (6)
where
qr
sRs
22
22 €
1 , (7)
W is the particle energy; r is the radial coordinate;
is the Larmor radius;
B
is the cyclotron frequency;
rB
~
and B are the wave and equilibrium magnetic
fields; respectively; q(r) is the safety factor; s€ is the
magnetic shear; ks=(s/q-n); is the plasma cross-
section ellipticity; = / is the pitch angle cosine with
v the velocity.
Let us use Eq. (6) to evaluate the number of modes
needed to stochastize the ion motion in a certain energy
range. We take the plasma parameters of the
experiments described in Ref. [2] (B=0.45 T; the energy
of injected ions Winj = 90 keV; the electron density
ne (5…6)∙10
19
m
-3
). Then the instabilities with
1~f MHz correspond to 6~||k . Taking, in addition,
=0.8, = 1.5, a=0.85 m, R=1 m , we find that the width
of a resonance island is ( W)island/W 0.1 for the mode
amplitude 3/ ~ 5 10rB B and ( W)island/W 0.03 for
4/ ~ 5 10rB B . This means that about 5 modes with the
amplitude of 5∙10
-3
or 15 modes with the amplitude of
5∙10
-4
are sufficient to create a stochastic motion zone
reaching from Winj to Winj/2. These numbers seem
plausible for NSTX experiments.
The slowing-down process caused by the waves is
accompanied by the radial motion of the ion. The
estimates above are based on the assumption that the
radial position of the beam ions does not change during
slowing down. The ion displacement is, indeed, small at
high , as follows from Eq. (5) (which, in fact, describes
the characteristics of the quasilinear equation [5]):
2
||
2 v
m
r
B
. (8)
We take the inequalities 2/1/ 2
0||
2
|| vv (the subscript
‘0’ corresponds to the birth energy) and r
2
<a
2
/4 as
conditions that the particles are not lost while losing
over a half of their energies. Then Eq. (8) yields
130m kHz, which is satisfied for most high-
frequency GAE modes observed in NSTX.
a
b
Fig. 2. Poincare plots for the modes )5,3(),( nm (a)
and (m,n) = (-4; -6) (b). Several chains of resonance
islands are seen, the number of the islands in the chain
being equal to the poloidal resonance number s
2. NUMERICAL SIMULATIONS
The analytical results presented above are based on
the assumption that an interval of energies is stochastic
when the total width of all resonance islands is
sufficient to cover it. Although this statement seems
credible, being similar to the well-known Chirikov
criterion [4], it needs to be verified by numerical
experiment. Having this in mind, we developed two
guiding centre codes.
The first code constructs Poincaré maps to study the
width of resonance islands induced by a single Alfvén
mode. The second one is developed to study the overlap
of resonances of multiple waves and reveal the resulting
stochastic domains. The idea is that we seed a set of test
particles with different energies at a certain radius and
adiabatically (i.e., sufficiently slowly) turn on the
perturbation, then we adiabatically turn off the wave
and find the phase space regions where the energies of
ISSN 1562-6016. ВАНТ. 2015. №1(95) 51
the particles have changed [6]. Energy intervals in
which the particles become randomly displaced after the
process indicate the stochasticity and/or resonance
islands. Regions with negligible energy displacement of
the particles correspond to domains where KAM-
surfaces between islands exist. This gives us a
possibility to find the energy spaces in which the
particles are able to lose energy rapidly.
For our simulations we took the plasma parameters
described above and the safety factor profile of the form
2
00 )()()( arqqqrq a , where 1.3aq , 1.00q ,
85.0a m is the minor radius of the torus. In the
experiments [2], modes with frequencies in the range of
0.5 to 1.1 MHz, and toroidal wave numbers n from -2
to -7 were observed. For our numerical simulation we
took modes with these parameters, choosing m so that
Avk|| , where Rnqmk /)/(|| and
Av is the
Alfvén velocity, and, at the same time, the waves were
resonant with particles of interest.
In Fig. 2, Poincaré plots for different modes in
NSTX are shown to illustrate the islands produced by
GAE modes with BBr /
~
of order of 310 and the
frequency 97.0f MHz. The resonance width
evaluated from Eq. (6) is in agreement with the code
results (with discrepancy of 20...30%).
In Fig. 3, the energy intervals calculated by the
second (adiabatic) code for the same modes are shown.
The results of the two codes for a single mode are in
good agreement. We observe that energy intervals in
which the particles wander due to the mode are at the
same places as the resonance islands in Fig. 2.
a
b
Fig. 3. Consequences of adiabatic switching on
and off a mode for particles at a certain radius.
Horizontal bars show energy intervals in which the
particles become randomly displaced after the process.
Dots show particles with negligible displacements.
The calculations were carried out for the same modes
and the same mode amplitudes as in Fig. 2
Theory predicts that the widths of these intervals
should be 2 times less than the island widths (in
order to provide the same phase space volume). We
observe that this is really the case (an exception is the
resonance at injWW 8.0 in Fig. 3,b, which is
somewhat wider). Thus, the results of the two codes for
single modes are in reasonable agreement, which proves
that the approach implemented in the adiabatic code is
viable.
A typical example of calculations for several (two)
modes is shown in Fig. 4. Comparing this figure with
Figs. 2 and 3, we observe that a most part of the energy
interval in Fig. 4 is covered by horizontal bars. These
bars are extensive energy intervals where particles
wander, which appear due to the overlap of the islands
shown in Figs. 2, 3. Detailed verification of our
conclusions on the channelling threshold amplitudes is
yet to be done with this code.
Fig. 4. Domains of stochastic motion produced by
simultaneous adiabatic switching on and off the modes
(m,n)=(-3,-5) and (m,n) = (-4,-6). The mode amplitudes
are the same as in Figs. 2 and 3
DISCUSSION AND CONCLUSIONS
Our analytical estimates show that about 5 GAE
modes with the amplitudes of 3/ ~5 10rB B are
sufficient to make stochastic the motion of injected ions
in NSTX in the energy interval from 90 to 50 keV. This
means that these modes are capable to extract a half of
the fast ion energy. The radial displacements of the
particles in the course of the energy extraction are
moderate and should not result in premature losses of
the particles. The normalized mode amplitudes of order
of 35 10 are usually considered as realistic in NSTX,
and spectrograms of instabilities in NSTX (see, e. g., [1,
2]) show that 5…10 unstable modes are often observed
simultaneously. Thus, our estimates confirm the
conclusions of Ref. [3].
The numerical simulations have shown that our two
codes agree with each other and with analytical
estimates. It is planned to use them for detailed
verification of our analytical results.
It should be mentioned that our analysis does not
take account of cyclotron resonances, which are known
to be an important mechanism of the excitation of high-
frequency instabilities in NSTX [7]. This means that the
stochasticity threshold found above is, most probably,
overestimated – even lower mode amplitudes are
sufficient to take away the energy from the particles. To
include the cyclotron resonances into our consideration
(we are planning to do this in the future), we will have
to abandon the guiding-centre approximation. This will
require more time-consuming numerical simulations. In
52 ISSN 1562-6016. ВАНТ. 2015. №1(95)
addition, it will be difficult to use the Poincaré map
approach in this case. However, we expect that the
approach based on the adiabatic variation of the
perturbation amplitude will remain efficient.
ACKNOWLEDGEMENTS
This work was supported in part by the Project
#0114U000678 of the National Academy of Sciences of
Ukraine.
REFERENCES
1. N.A. Crocker et al. Internal amplitude, structure and
identification of compressional and global Alfvén
eigenmodes in NSTX // Nucl. Fusion. 2013, v. 53,
p. 043017.
2. D. Stutman et al. Correlation between electron
transport and shear Alfvén activity in the National
Spherical Torus Experiment // Phys. Rev. Lett. 2009,
v. 102, p. 115002.
3. Ya.I. Kolesnichenko, Yu.V. Yakovenko,
V.V. Lutsenko. Channeling of energy and momentum
during energetic-ion-driven instabilities in fusion
plasmas // Phys. Rev. Lett. 2010, v. 104, p. 075001.
4. B.V. Chirikov. A universal instability of many-
dimensional oscillator systems // Phys. Rep. 1979, v. 52,
p. 263-379.
5. Ya.I. Kolesnichenko, Yu.V. Yakovenko,
V.V. Lutsenko. Effects of energy-ion-driven instabilities
on plasma heating, transport and rotation in toroidal
system // Nucl. Fusion. 2010, v. 50, p. 084011.
6. R.B. White. The theory of toroidally confined
plasmas. London: Imperial College Press, 2014.
7. N.N. Gorelenkov et al. Theory and observations of
high frequency Alfvén eigenmodes in low aspect ratio
plasmas // Nucl. Fusion. 2003, v. 43, p. 228-233.
Article received 15.12.2014
ПРОСТРАНСТВЕННОЕ КАНАЛИРОВАНИЕ ЭНЕРГИИ И СТОХАСТИЗАЦИЯ ДВИЖЕНИЯ
БЫСТРЫХ ИОНОВ ВЫСОКОЧАСТОТНЫМИ НЕУСТОЙЧИВОСТЯМИ ПЛАЗМЫ
М.Г. Тищенко, Ю.В. Яковенко
Устанавливаются условия, при которых множественные альфвеновские собственные моды способны
отнять значительную долю энергии у быстрых ионов для возможной передачи в другую область
пространства (пространственного каналирования). Это может происходить, если резонансные острова
неустойчивостей перекрываются, образуя обширную стохастическую зону в фазовом пространстве быстрых
ионов. Выводится аналитическое выражение для ширины резонансного острова, образованного
альфвеновской собственной модой в фазовом пространстве. Оцениваются количество и амплитуда мод,
необходимые для возникновения стохастической зоны в заданном диапазоне энергий. Кратко описываются
два кода, предназначенные для численной проверки этих оценок. Представлены первые результаты этих
кодов.
ПРОСТОРОВЕ КАНАЛЮВАННЯ ЕНЕРГІЇ ТА СТОХАСТИЗАЦІЯ РУХУ ШВИДКИХ IОНІВ
ВИСОКОЧАСТОТНИМИ НЕСТІЙКОСТЯМИ ПЛАЗМИ
М.Г. Тищенко, Ю.В. Яковенко
Встановлюються умови, за яких множинні альфвенові власні моди є здатними забрати значну частку
енергії в швидких іонів для можливої передачі в іншу область простору (просторового каналювання). Це
може траплятись, якщо резонансні острови нестійкостей перекриваються і утворюють широку стохастичну
зону в фазовому просторі швидких іонів. Виводиться аналітичний вираз для ширини резонансного острова,
створеного альфвеновою власною модою в фазовому просторі. Оцінюються кількість та амплітуда мод, що
потрібні для утворення стохастичної зони в певному діапазоні енергій. Стисло описуються два коди,
призначені для числової перевірки цих оцінок. Представлено перші результати цих кодів.
|
| id | nasplib_isofts_kiev_ua-123456789-82065 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-27T12:35:36Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Tyshchenko, M.H. Yakovenko, Yu.V. 2015-05-24T13:42:03Z 2015-05-24T13:42:03Z 2015 Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities / M.H. Tyshchenko, Yu.V. Yakovenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 49-52. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.55.Hc, 52.55.Pi, 55.25.Fi, 52.35.Bj, 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/82065 The conditions are established when multiple Alfvén eigenmodes are able to withdraw a significant part of the energy of fast ions for possible transfer to another spatial region (spatial channelling). This can happen when the resonance islands of the instabilities overlap to form an extensive stochastic zone in the fast ion phase space. An analytical expression for the width of a resonance island induced by an Alfvén eigenmode in the phase space is derived. The number and amplitude of modes are estimated, which are required to form a stochastic zone in a given energy range. Two codes intended for numerical verification of these estimates are briefly described. First results of these codes are presented. Устанавливаются условия, при которых множественные альфвеновские собственные моды способны отнять значительную долю энергии у быстрых ионов для возможной передачи в другую область пространства (пространственного каналирования). Это может происходить, если резонансные острова неустойчивостей перекрываются, образуя обширную стохастическую зону в фазовом пространстве быстрых ионов. Выводится аналитическое выражение для ширины резонансного острова, образованного альфвеновской собственной модой в фазовом пространстве. Оцениваются количество и амплитуда мод, необходимые для возникновения стохастической зоны в заданном диапазоне энергий. Кратко описываются два кода, предназначенные для численной проверки этих оценок. Представлены первые результаты этих кодов. Встановлюються умови, за яких множинні альфвенові власні моди є здатними забрати значну частку енергії в швидких іонів для можливої передачі в іншу область простору (просторового каналювання). Це може траплятись, якщо резонансні острови нестійкостей перекриваються і утворюють широку стохастичну зону в фазовому просторі швидких іонів. Виводиться аналітичний вираз для ширини резонансного острова, створеного альфвеновою власною модою в фазовому просторі. Оцінюються кількість та амплітуда мод, що потрібні для утворення стохастичної зони в певному діапазоні енергій. Стисло описуються два коди, призначені для числової перевірки цих оцінок. Представлено перші результати цих кодів. This work was supported in part by the Project #0114U000678 of the National Academy of Sciences of Ukraine en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Магнитное удержание Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities Пространственное каналирование энергии и стохастизация движения быстрых ионов высокочастотными неустойчивостями плазмы Просторове каналювання енергії та стохастизація руху швидких iонів високочастотними нестійкостями плазми Article published earlier |
| spellingShingle | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities Tyshchenko, M.H. Yakovenko, Yu.V. Магнитное удержание |
| title | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| title_alt | Пространственное каналирование энергии и стохастизация движения быстрых ионов высокочастотными неустойчивостями плазмы Просторове каналювання енергії та стохастизація руху швидких iонів високочастотними нестійкостями плазми |
| title_full | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| title_fullStr | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| title_full_unstemmed | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| title_short | Spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| title_sort | spatial energy channelling and stochastization of fast ion motion by high-frequency plasma instabilities |
| topic | Магнитное удержание |
| topic_facet | Магнитное удержание |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82065 |
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