Estafette of resonances, stochasticity and control of particle motion
A new method of particle motion control in toroidal magnetic traps with rotational transform using the estafette of drift resonances and stochasticity of particle trajectories is proposed. The using the word "estafette" (relay race) means here that the particle passes through a set of reso...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Estafette of resonances, stochasticity and control of particle motion / A.A. Shishkin // Вопросы атомной науки и техники. — 2000. — № 6. — С. 32-34. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-784852015-03-19T03:02:24Z Estafette of resonances, stochasticity and control of particle motion Shishkin, A.A. Magnetic confinement A new method of particle motion control in toroidal magnetic traps with rotational transform using the estafette of drift resonances and stochasticity of particle trajectories is proposed. The using the word "estafette" (relay race) means here that the particle passes through a set of resonances consistently from one to another during its motion. The overlapping of adjacent resonances can be moved radially from the center to the edge of the plasma by switching on the corresponding perturbations in accordance with a particular rule in time. In this way particles (e.g. cold alpha-particle) can be removed from the center of the confinement volume to the plasma periphery. 2000 Article Estafette of resonances, stochasticity and control of particle motion / A.A. Shishkin // Вопросы атомной науки и техники. — 2000. — № 6. — С. 32-34. — Бібліогр.: 7 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78485 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Shishkin, A.A. Estafette of resonances, stochasticity and control of particle motion Вопросы атомной науки и техники |
| description |
A new method of particle motion control in toroidal magnetic traps with rotational transform using the estafette of drift resonances and stochasticity of particle trajectories is proposed. The using the word "estafette" (relay race) means here that the particle passes through a set of resonances consistently from one to another during its motion. The overlapping of adjacent resonances can be moved radially from the center to the edge of the plasma by switching on the corresponding perturbations in accordance with a particular rule in time. In this way particles (e.g. cold alpha-particle) can be removed from the center of the confinement volume to the plasma periphery. |
| format |
Article |
| author |
Shishkin, A.A. |
| author_facet |
Shishkin, A.A. |
| author_sort |
Shishkin, A.A. |
| title |
Estafette of resonances, stochasticity and control of particle motion |
| title_short |
Estafette of resonances, stochasticity and control of particle motion |
| title_full |
Estafette of resonances, stochasticity and control of particle motion |
| title_fullStr |
Estafette of resonances, stochasticity and control of particle motion |
| title_full_unstemmed |
Estafette of resonances, stochasticity and control of particle motion |
| title_sort |
estafette of resonances, stochasticity and control of particle motion |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78485 |
| citation_txt |
Estafette of resonances, stochasticity and control of particle motion / A.A. Shishkin // Вопросы атомной науки и техники. — 2000. — № 6. — С. 32-34. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT shishkinaa estafetteofresonancesstochasticityandcontrolofparticlemotion |
| first_indexed |
2025-07-06T02:34:06Z |
| last_indexed |
2025-07-06T02:34:06Z |
| _version_ |
1836863190235873280 |
| fulltext |
UDC 533.9
32 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 32-34
ESTAFETTE OF RESONANCES, STOCHASTICITY AND CONTROL OF
PARTICLE MOTION
Alexander A.SHISHKIN
Institute of Plasma Physics, National Science Center "Kharkov Institute of Physics and
Technology", Kharkov - 108, Ukraine and
Plasma Physics Chair, Department of Physics and Technology,
Kharkov “V.N. Karazin” National University, Kharkov-77, Ukraine
A new method of particle motion control in toroidal magnetic traps with rotational transform using the estafette
of drift resonances and stochasticity of particle trajectories is proposed. The using the word "estafette" (relay race)
means here that the particle passes through a set of resonances consistently from one to another during its motion.
The overlapping of adjacent resonances can be moved radially from the center to the edge of the plasma by
switching on the corresponding perturbations in accordance with a particular rule in time. In this way particles (e.g.
cold alpha-particle) can be removed from the center of the confinement volume to the plasma periphery.
1. Introduction
There exist different approaches to control the
particle motion and one of them is to apply magnetic
field perturbations to produce resonance trajectories -
drift islands, overlapping of the islands and stochastic
trajectories that lead, for example, to the removal of test
particles from the center of the magnetic configuration
to the periphery [1-3]. A new method of particle motion
control in toroidal magnetic traps with rotational
transform using the estafette of drift resonances and
stochasticity of particle trajectories can be described in
the following. If there are some adjacent rational drift
surfaces with the drift rotational transform
mn /* =ι , mn ′′= /*ι , mn ′′′′= /*ι , then the magnetic
perturbations with the wave numbers ( nm, ), ( nm ′′, ),
( ),nm ′′′′ can lead to some families of drift islands.
Overlapping of the adjacent resonance structure is the
reason for the stochasticity [2] of particle trajectories. If
a particle trajectory passes through the set of
perturbations this test particle can escape from the
center of the confinement volume to the periphery.
Overlapping of the adjacent resonances in the local
space within the plasma can be transferred outward in
the process of particle motion by switching on the
corresponding perturbations in accordance with a
special rule (consequently) in time
The mathematical basis of the description is rather
similar to the description of the nonlinear oscillator
(pendulum) with a quasi-periodic perturbation [4-6]. In
this paper, estafette of resonances is studied with both
analytical methods and numerical integration of guiding
center equations. Stochastic diffusion coefficients
ϑϑϑ ,,, ,, DDD rrr
are introduced. These coefficients are
useful to evaluate the deviation of the trajectories in the
radial ( r ) and the poloidal angular (ϑ ) directions after a
large number of rotations (rounds) along the torus, i.e.
in the toroidal angular (ϕ ) direction.
2. Analytical Treatment
2.1. Equation for the Drift Flux Surface Function
As is known, the particle motion can be described
with the integrals of the guiding center equations system
and one of them is the function of the drift surface
),,,,,( ||
* tVVr ⊥Ψ ϕϑ .
We assume that the flux of particle guiding centers is
conserved during the particle motion in analogy with the
magnetic flux. This leads to
consttVVr =Ψ ⊥ ),,,,,( ||
* ϕϑ (1)
and the total derivative of the drift surface function is
equal to zero
0
*
=Ψ
dt
d (2)
The equation (3.2) in variables ϕϑ ,,r and
||V ,
⊥V takes
the form
0
*
||
*
||
****
=
∂
Ψ∂+
∂
Ψ∂+
∂
Ψ∂+
∂
Ψ∂+
∂
Ψ∂+
∂
Ψ∂
⊥
⊥
Vdt
dV
Vdt
dV
Rdt
Rd
rdt
rd
rdt
dr
t ϕ
ϕ
ϑ
ϑ
(3)
After substituting the equations for
dt
Rd
dt
rd
dt
dr ϕϑ ,, and
dt
dV
dt
dV ⊥,|| from guiding center equations system into
(3.3) we obtain
( ) ( )( )
( ) ( ) 0
4
][2
2
2
*
2
||
||
*
2
2
*22
||3
*||
*
=
∂
Ψ∂
∇+
∂
Ψ∂
∇−
Ψ∇∇×++Ψ∇+
∂
Ψ∂
⊥
⊥
⊥
V
B
B
V
V
B
B
V
BVV
eB
Mc
B
V
t
BB
BB (4)
We assume that 0
||
*
=
∂
Ψ∂
V
and 02
*
=
∂
Ψ∂
⊥V
, because only
passing particles in the narrow range of
||V and
⊥V are
considered. Under the magnetic field that consists of
the basic field, perturbing field with “wave” numbers
(m, n) and additional perturbing field with “wave“
numbers ),( nm ′′ , namely
nmnm ′′++= ,,
~BBBB 0
the equation (3.4) reduces to the following one
+
∂
Ψ∂
∆
∂
∂+∆
∂
∂++
∂
Ψ∂
r
r
r
Vr
r
VV
t rr
*
nm,0,
*
nm,0, ~~~
ϑϑ
33
ϑ
ϑϑ
ϑ
ϑ ϑϑ ∂
Ψ∂
∆−∆
∂
∂+∆
∂
∂++
rr
Vr
r
Vr
r
VV rr
*
nm,0,~)(1~)(~~
+ 022
*
,,0
2
,
*
,,0
2
,2
*
,,0
2
, =
∂
Ψ∂
+
∂∂
Ψ∂
+
∂
Ψ∂
ϑϑ ϑϑϑ r
D
rr
D
r
D nmnm
r
nm
rr
(5)
Here such designations are introduced
( )[ ]BVV
eB
Mc
B
V
∇×++≡ ⊥ BBV ~2
2
~~ 22
||3
|| (6)
and
,~
, rVD rrr ∆≡ rVrVD rr ∆+∆≡ ϑϑ ϑ ~~
,
, ϑϑϑϑ ∆≡ rVD ~
,
(7)
*
,,0 nmΨ describes the drift surfaces corresponding to
magnetic field
nm ,BB0 + .
2.2. Stochastic Diffusion Coefficients
Stochastic diffusion coefficients can be evaluated using
V
RBBB
B
V
rVD rrrnmrrr
π2)~~(~ 2
,,
2
||
, +
≅∆= ,
,2)~)(~(2
~~
,,,,
2
||
,
V
RBBBB
B
V
rVrVD
nmrrnm
rr
π
ϑ
ϑϑ
ϑϑ
++
≅∆+∆=
(8)
V
RBBB
B
V
rVD nm
πϑ ϑϑϑϑϑϑ
2)~~(~ 2
,,
2
||
, +
≅∆= .
It should be noted that the sense of the diffusion
coefficients is the following
rr
rr
rD
τ
δ 2
,
)(= ,
ϑ
ϑ τ
δϑδ
r
r
rrD =,
,
ϑϑ
ϑϑ τ
δϑ 22
,
)(rD = (9)
From the expressions (9) (after substituting from (8)) we
can evaluate, for example, the time rrτ ( ϑϑτ )
necessary for the deviation of particle trajectories from
the initial surface in the radial (angular) direction, the
time ϑτ r necessary for the deviation of particle
trajectories from the initial surface in a certain angular
space under a given deviation in the radial direction.
Under the perturbations with the “wave” numbers
( nm ′′, ) the separatrix of the ( nm, ) resonance is broken
and the particle trajectories become stochastic. Thus the
particle can wander from one resonance (initial) to
another resonance – adjacent to the initial one and then
this phenomenon repeats, the particle passes step by
step through the set of resonances and can escape from
the center of the confinement volume to the periphery
and vice versa from the periphery into the center of the
confinement volume.
3. Estafette of Resonance and Helium Ion
Removal
3.1. Test particle
Estafette of drift resonances of the test particle is
demonstrated by the numerical integration of guiding
center equations. Helium ions ( 4
2 He ) with the
energy
W
=100 keV and starting pitch VVII / = 0.9 are
taken as the test particles. Starting point coordinates are
0r =3, 7, 11, 15, 19 cm, under
0ϑ =0 and
0ϕ =0. For
analysis here the configuration for the Large Helical
Device (LHD) – the most advanced steady-state helical
device, which is under successful operation at the
National Institute for Fusion Science (Toki, Japan) – is
chosen.
3.2 Drift Surfaces without Perturbations
The so-called Inward Shift configuration is taken
where the magnetic axis is shifted toward the torus
center and the rotational transform varies from
6.0)0( ≈ι till 0.1)( ≈paι . Among the rational values of
the drift rotational transform the following set is chosen
)/( 22 arι ≈ 0.75, 0.8, 0.9, 1.0.
3.3. Drift Surfaces with Perturbations
3.3.1. Splitting of Resonant Surfaces
All perturbations act (switched on) simultaneously
Under the set of perturbations:
pm =4,
pn =3; p,3,4ε =
0.003, 2/,3,4 πδ =p
; pm = 5, pn =4 p,4,5ε =0.007,
2/,4,5 πδ =p
; pm =9, pn =10; p,10,9ε =0.03, 0,10,9 =pδ , -
some families of magnetic islands in the vertical cross-
section appear (Fig.1)
330 390 450
R(cm)
-60
0
60
Z(
cm
)
Fig.1. Drift surfaces of the helium ions under the set of
perturbations indicated in Section 3.3.1
3.3.2. Stochasticity of Drift Trajectories
Perturbations switched on in the strict sequence
The perturbations are switched on in a strict sequence as
is shown in Fig. 2. The amplitudes of the perturbations
0 50000 100000 150000
0.00
0.02
0.04
Fig.2. Amplitudes of perturbations in dependence on
time
34
with pm = 5, pn =4 and pm =9, pn =10 are 1.4 times
larger than in Fig. 1. Another very important fact is that
the perturbation with 4/5/ =pp nm is switched on at the
moment of time when the radial deviation of test
particle is the largest. Therefore the particle starting at
the point with 70 =r cm, 0.00 =ϑ , 0.00 =ϕ moves from
the initial magnetic surface to the following outward
magnetic surfaces (Fig. 3).
330 390 450
R(cm)
-60
0
60
Z(
cm
)
Fig 3.Diffusion of the drift surface of the test helium ion
with the starting point 0r =7 cm
The radial variable of the test particle trajectory
increases in time and achieves a value (Fig. 4) where it
may be removed by mechanical means. It is possible to
mark out the intervals of time when the particle is in
resonance with the
0 50000 100000 150000
0
40
80
r
(c
m
)
Fig.4,Radial variable of the test particle trajectory in
dependence on time
3/4/ =pp nm and 4/5/ =pp nm perturbations and
finally the time when it leaves the plasma. Stochasticity
of the trajectories with different but rather close starting
points is shown on Fig.5.
If a deuterium ion with the energy W =7 keV (the
possible thermal energy of the bulk ion plasma in device
considered) starts at the same point its drift rotational
transform =*ι 4 / 5 this particle forms 5 islands in the
cross-section instead of 4 islands as in the case of the
helium ion with W =100 keV. The trajectory of this
particle is not stochastic. Electrons with the same
energy and the same starting point have =*ι 0.836. Their
drift surfaces have some corrugation but are not
stochastic.
330 390 450
R(cm)
-60
0
60
Z(
cm
)
Fig. 5. Diffusion of two drift surfaces for the test helium
ions with the starting points 0r =7 cm (small crosses)
and 0r =11 cm (circles)
4. Conclusions
1. Estafette of drift resonances can be used for the
removal of the test particle, particularly helium ash,
when the perturbing magnetic fields are externally
applied. For this purpose in the case of particle
trajectories with the drift rotational transform
pp mn /* =ι
the perturbing magnetic field with wave numbers
pp nm ,
leads to drift island formation. If the adjacent drift
islands overlap the particle trajectories become
stochastic. A particle passes through a set of adjacent
resonances and can leave the confinement volume.
2. The bulk plasma ions do not experience the
stochastisation of their trajectories and do not escape the
plasma.
3. Perturbing magnetic fields can be produced by a
system of coils with specific currents, for example,
similar to the coils of the local island divertor [7].
4. The method of the estafette of resonances can be
applied to other practical physics tasks and may be
considered as a new approach in the transition to chaotic
states.
Acknowledgments
The author expresses his deep gratitude to Prof. Tomas
Dolan (IAEA) for his continuous support.
References
[ 1 ]. Mynick, H.E. Phys.Fluids B 5 (1993) 1471 and
2461.
[ 2 ]. Chirikov, B.V. Phys. Rep. 52 (1979) 263.
[ 3 ]. Motojima, O., Shishkin, A.A. Plasma Physics and
Controlled Fusion 41 (1999) 227.
[ 4 ]. Mitropol’skiy, Yu.A. Averaged Method in
Nonlinear Mechanics, Naukova Dumka, Kiev 1971, 440
pages (in Russian).
[ 5 ]. Rosenbluth, M.N., Sagdeev, R.Z., Taylor, J.B.,
Zaslavski, G.M. Nuclear Fusion 6 (1966) 297.
[ 6 ]. Grishchenko, A.D., Vavriv, D. M. Tech. Phys. 42
(1997) 1115.
[7]. Shishkin, A.A., Motojima, O., Journal of Plasma
and Fusion Research SERIES, v.3 (2000) (in press).
ESTAFETTE OF RESONANCES, STOCHASTICITY AND CONTROL OF PARTICLE MOTION
Alexander A.SHISHKIN
Kharkov “V.N. Karazin” National University, Kharkov-77, Ukraine
1. Introduction
2. Analytical Treatment
3. Estafette of Resonance and Helium Ion Removal
3.2 Drift Surfaces without Perturbations
3.3.1. Splitting of Resonant Surfaces
All perturbations act (switched on) simultaneously
Fig.1. Drift surfaces of the helium ions under the set of perturbations indicated in Section 3.3.1
3.3.2. Stochasticity of Drift Trajectories
Perturbations switched on in the strict sequence
4. Conclusions
Acknowledgments
References
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