Non-linear evolution of vortices in high-current electrostatic plasma lens
The spatial structure and nonlinear dynamics of vortices in plasma lens for high-current ion-beam focusing have been investigated theoretically. Проведено теоретичне дослідження просторової структури і нелінійної динаміки вихрів у плазмових лінзах для фокусування великих іонних пучків. Проведено т...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2004
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| Cite this: | Non-linear evolution of vortices in high-current electrostatic plasma lens / A.A. Goncharov, V.I.Maslov, I.N. Onishchenko, V.L. Stomin, V.N. Tretyakov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 30-32. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859669860943921152 |
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| author | Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. Tretyakov, V.N. |
| author_facet | Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. Tretyakov, V.N. |
| citation_txt | Non-linear evolution of vortices in high-current electrostatic plasma lens / A.A. Goncharov, V.I.Maslov, I.N. Onishchenko, V.L. Stomin, V.N. Tretyakov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 30-32. — Бібліогр.: 3 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The spatial structure and nonlinear dynamics of vortices in plasma lens for high-current ion-beam focusing have
been investigated theoretically.
Проведено теоретичне дослідження просторової структури і нелінійної динаміки вихрів у плазмових
лінзах для фокусування великих іонних пучків.
Проведено теоретическое исследование пространственной структуры и нелинейной динамики вихрей в
плазменных линзах для фокусировки больших ионных пучков.
|
| first_indexed | 2025-11-30T13:28:26Z |
| format | Article |
| fulltext |
NON-LINEAR EVOLUTION OF VORTICES IN HIGH-CURRENT
ELECTROSTATIC PLASMA LENS
A.A.Goncharov*, V.I.Maslov, I.N.Onishchenko, V.L.Stomin, V.N.Tretyakov**
*Institute of Physics NASU, 252650 Kiev;
NSC Kharkov Institute of Physics & Technology, 61108 Kharkov, Ukraine;
** Kharkov National University, Kharkov 61108, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
The spatial structure and nonlinear dynamics of vortices in plasma lens for high-current ion-beam focusing have
been investigated theoretically.
PACS: 52.40.Mj
1. INTRODUCTION
It is known from numerical simulations and
experiments that vortices are long-lived structures in
vacuum. However, the acceleration of evolution of
vortices in electron plasma was observed in laboratory
experiments. Same dynamics of vortices should take
place in near wall turbulence of nuclear fusion
installations, where the crossed configuration of
electrical and magnetic fields also is realized.
The charged plasma lens, intended for focusing of
high-current ion beams, has the same crossed
configuration of fields [1]. It is important to know the
properties of vortices at the nonlinear stage of their
evolution. It has been shown theoretically in this paper,
that after reaching the quasi-stationary state the
electrons in a field of a vortex rotate around its axis with
the higher velocity in comparison with the velocity of
azimuthal drift of electrons in the fields of the lens.
Slow and quick vortices are contacting combinations of
two vortices rotated in the opposite directions.
The instability development in the initially
homogeneous plasma causes that the vortices are born
pairs. Namely, if the vortex-bunch of electrons is
generated, the vortex-hole of electrons occurs near it. It
has been shown, that at small inhomogeneous electron
density in the real experimental lens the preference is
realized in the behaviour of vortices. Namely, the vortex
- bunch goes to the region of a higher electron density
ne , and vortex – hole goes to the region of lower ne .
2. JOINT DEVELOPMENT OF TWO
INSTABILITIES
In [2] the dispersion law of oscillations, possible in
the plasma lens is presented. The obtained dispersion
law describes the joint development of two instabilities.
Namely, in a limiting case lθωθo<<kzVbi, basically, the
instability of the ion stream relative to electrons
develops. Here Vbi is the ion beam velocity, kz is the
longitudinal wave vector, lθ is the azimuthal angular
number, ωθo is the angular velocity of the electron drift
in the crossed fields. Thus, the growth rate of the
instability development in the case lθωθo<kzVbi increases
with the growth of kz. In the limiting case lθωθo>>kzVbi
the instability of electrons, drifting relative to ions in the
crossed fields in the cylindrical system with a radial
gradient of the magnetic field develops. Slow vortices
have the highest growth rate. Joint development of two
instabilities under conditions typical for experiments lθ
ωθo>kzVbi , results to that the growth rate of the slow
vortical perturbation is more for the most
homogeneous perturbation in the longitudinal
direction, as the finite dimensions of the lens allow.
At instability development the vortices are born as
follows. The non-uniform electric field E(r, t) (≡-∇
φ(r, t)), arising as a result of the instability
development, leads to the nonuniform electron
dynamics with a velocity perturbation δV(r, t)≈(e/meω
ce)[ez ,∇φ] . As a result of a nonuniform δV(r, t) the
electron bunching is performed which results in the
nonuniform distribution of the electron density
perturbation δne≈no(kδV)/(ω-lθωθo) . Last
automatically results in the vortical movement of the
electrons with a vorticity α≡ezrotV≈(ω2
pe/ωce)δne/neo .
3. SPATIAL STRUCTURE OF VORTICES
Let us describe the structure of a quick vortex in
the rest frame, rotating with the angular velocity ωph≡
Vph/rq. Let us consider a chain on θ of vortices -
bunches and vortexes - holes of electrons. Neglecting
nonstationary and nonlineary- on φ- terms, we derive
the following equation
V⊥=-(e/meωHe)[ez,Ero]+(e/mωHe)[ez,∇φ], (1)
describing the quasi-stationary dynamics of electrons
in the fields of the lens and the vortical perturbation.
From (1) we obtain the expression for radial and
azimuthal electron velocities
Vr=-(e/meωHe)∇θφ , Vθ=Vθo+(e/meωHe)∇rφ ,
Vθo=-(e/meωHe)Ero=(ω2
pe/2ωHe)(∆n/noe)r (2)
Vθ can be presented as a sum of the phase velocity of
perturbation, Vph, and velocity of azimuth electron
oscillations, δVθ , in the field of perturbation, Vθ=Vph+
δVθ . Because Vθ=rdθ/dt , we present dθ/dt as
dθ/dt=dθ1/dt+ωph ,
where ωph=(∆n/noe)(ω2
pe/2ωHe)r=rv , rv is the radius of
the vortical perturbation location. Then from (2) we
obtain
dθ1/dt=(ω2
pe/2)(∆n/noe)[1/ωHe(r)-
-1/ωHe(rv)]+(e/rmeωHe)∂rφ , dr/dt=-(e/meωHer)∂θφ (3)
At small diversions of r from rv, decomposing ωHe(r)
on δr≡r-rv and integrating (3), we derive
(δr)2-2ωHe(rv)φ/πe∆nrv(∂rωHe)r=rv=const (4)
The vortex boundary separates the trapped
electrons, forming the vortex and moving on closed
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.30-32.30
trajectories and untrapped electrons, moving outside the
boundary of the vortex and oscillating in its field. For
vortex boundary we derive the following expression
from the condition δrφ=-φo=δrcl
δr=±[2(φ+φo)ωHe(rv)/πe∆nrv(∂rωHe)r=rv+(δrcl)2]1/2 (5)
Here δrcl is the radial width of the vortex - bunch of
electrons. From (5) the radial size of the vortex - hole of
electrons follows
δrh≈2[φoωHe(rv)/πe∆nrv(∂rωHe)r=rv]1/2 (6)
From the equation of electron motion and Poisson
equation it is possible to derive approximately the
expression for the vorticity α≡ezrotV , which is
characteristic of the vortical motion of electrons
α≈-2eEro/rmωHe+(ω2
pe/ωHe)δne/neo
From here it follows that up to certain amplitude of
vortices the structure of electron trajectories in the field
of the chain on θ of quick vortices in the system of rest,
rotated with ωph≡Vph/rq , is similar to the structure,
presented in [2].
For large amplitudes of quick vortices in the region
of electron bunches the contraflows are formed. The
vortex - hole rotates in the rest frame, rotating with a
frequency ωph≡Vph/rq , in the same direction as
unperturbed plasma. The vortex - bunch rotates in the
opposite direction of rotation of unperturbed plasma at δ
ne>∆n≡noe-noi. It is seen the size of the vortex is
inversely proportional to [(∆n/noe)(ωpe/ωeH)∂rωHe]1/2 and
is proportional to φ1/2
o . That is the size of the vortex
essentially depends on the gradient of the magnetic
field. At low ∆n/2noe and ωpe/ωHe already at small
perturbations of electron density the sizes of the vortex,
δrh , can reach δrh≈R/2, R is the plasma lens radius
(3) can be integrated without decomposition ωHe(r)
on δr≡r-rv. For this purpose we approximate ωHe(r)= ω
Ho(1+µr2/R2). Then, integrating (3), we derive
2φ+πe∆nr2[1-ωHo/2ωHe(rv)-ωHe(r)/2ωHe(rv)]=const (7)
From the condition rφ=-φo=rv+δrcl and (7) we obtain the
expression, determining the boundary of the vortex -
hole of electrons,
[r2-(rv+δrcl)2][1-ωHo/ωHe(rv)]-[r4-
-(rv+δrcl)4]ωHoµ/2R2ωHe(rv)+2(φ+φo)/πe∆n=const (8)
From (8) and rφ=φo=rv+δrh we derive the expression,
determining the radial width of the vortex - hole of
electrons,
φo4R2ωHe(rv)/πe∆n]ωHoµ=
=(δrh-δrcl)(2rv+δrh+δrcl)[rv(δrh+δrcl)+(δrh
2+δrcl
2)/2] (9)
Let us consider the vortex with the small phase
velocity Vph in comparison with the drift electron
velocity, Vph<<Vθo. The spatial structure of the electron
trajectories in its field for small amplitudes of the vortex
looks like that shown in Fig.1. It is determined by that
in all lens α has an identical sign, α>0. In other words,
the radial electric field, created by the vortex is less,
than the electric field of the lens, Erv<Ero. Then in all
lens the azimuthal electron velocities have an identical
sign and there are not contraflows of electrons. The
slow vortex of a small amplitude does not have a
separatrix. For the description of the electron
trajectories we use (2). Using in them Vθ=rdθ/dt and
excluding θ, we obtain for boundary of the vortex r(θ)
r=[r2
s+(φo-φ)2/πe∆n]1/2 (10)
In the case of small amplitudes (10) becomes
δr≡r-rs=(φo-φ)/πe∆nrs (11)
From (10) we derive the radial size of the slow
vortex
δrs≡rφ=-φo-rs=[r2
s+4φo/πe∆n]1/2-rs (12)
In the case of small amplitudes (12) becomes
δrs≈2φo/πe∆nrs (13)
For the description of the slow vortex structure one
can also use the equation
dtωHe/ne≈0, dt=∂t+(V⊥∇⊥)-Vph∇θ (14)
We obtain approximately from (14) the equation,
describing the slow vortex of the small amplitude
dr/dt≈-[noωHe/∂rωHe(r)][∂τ-Vph∇θ+Vθo∇θ](1/(no+δn)) ,
dθ/dt≈Vθo
or
δr≡r-rv≈ωHe(rv) δn/norv∂rωHe(rv)
Because on r=rv , δno(r=rv)=0, on it the electron moves
with Vθo without radial perturbations. At r>rv there is a
positive radial displacement, and at r<rv - negative
radial displacement of the electrons. The radial size of
the slow vortex is inversely proportional to the radial
gradient of the magnetic field.
In the case of large amplitudes, δne>∆n (or
Erv>Ero), in the region where the electron holes are
placed, the characteristic of the vortical motion α
accepts an opposite sign, α<0. In other words, on the
axis, connecting the vortex - hole and vortex - bunch,
the inequality Erv>Ero is fulfilled, and there is an
azimuthal contraflow of electrons. Then in some
regions the electrons rotate in the direction, opposite
to their rotation in crossed fields of the lens. The slow
vortex is a dipole perturbation of the electron density,
disjointed on radius. At δne>∆n the structure of the
slow vortex is similar to the structure of the Rossby
vortex.
Fig.1.
4. SATURATION OF EXCITED
HOMOGENEOUS SLOW VORTICAL
TURBULENCE
For quick vortices the cause of the instability is the
gradient of the velocity ∂rVθo , therefore for
development of instability the nonadiabatic dynamics
of electrons is necessary. For slow vortices the reason
of the instability is the interaction of the drifting
electron stream with ions, therefore amplitude of the
saturation of the slow vortex is determined from the
condition of the ion trapping
Vtri≈Vphs . (15)
or from the condition of the electron trapping
Vtre≈(Vθo-Vphs) (16)
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.30-32.31
and is determined by smaller of them. For the plasma
lens, close to the optimum plasma lens, the saturation is
determined by electron trapping. For the plasma lens,
far from the optimum plasma lens, the saturation is
determined by ion trapping. The slow homogeneous
turbulence is not separated into single vortices.
5. NONLINEAR DYNAMICS OF VORTICES
The development of instability in initially
homogeneous plasma lens causes that the vortices are
born pairs: if the vortex - bunch of electrons is
generated, the vortex - hole of electrons occurs near it.
Let us consider how the nonhomogeneity of electron
density effects on the behaviour of vortices. Finiteness
of time of the vortices symmetrization and also the
reflection of resonant electrons from vortices - bunches
result that the vortices are asymmetrical. Namely, on
opposite on θ parties of vortices the small bunches and
holes are formed. It results in formation of
polarization azimuth electric fields Eθ, directed along e
θ. The formation of fields Eθ causes the radial drift and
spatial separation of vortices (see fig.2). In other
words, the property of preference of motion of the
vortex - hole on the peripherals of the plasma column
and the vortex - bunch to its axis is realized. The
polarization electric fields in the vortex - hole and the
vortex - hole have opposite signs. Then the velocities
of radial drift of the vortex - hole and vortex - bunch
have opposite signs. Namely, the vortex - hole goes to
the region of a lower electron density, and the vortex -
bunch goes to the region of higher electron density).
Fig. 2. The opposite radial shift of the vortex - bunch of the electron density and the vortex - hole
The resonant electrons are reflected from the vortex-
bunch. Thus the distribution of the electron density
being asymmetrical on azimuth is formed. It results in
the radial motion of the vortex - bunch of electrons and
leads to simultaneous formation of spiral distribution of
the electron density. In the case of the azimuthally
symmetrical vortex its velocity of radial drift is equal to
Vrv≈(ω2
pe/2ωHe)(R2
v/noe)(dnoe/dr)r=rv (17)
Rv is the radius of the vortex. The width of the spiral is
equal to the radial width of the vortex in the case of its
high radial velocity. In the case of a low radial velocity
of the vortex the width of the spiral is less, than the
radial width of the vortex.
When two vortices - bunches of electrons begin to
concern each other, the electrons of each vortex, taking
place near to its boundary, are reflected from the next
vortex. Thus the asymmetry is formed on the azimuth
distribution of the electron density in the neighbourhood
of each vortex. It leads to occurrence of a relative
velocity of vortices.
Vmer≈[ω2
pe(δne)/2ωHe]Rv (18)
The similar behaviour of electrons was observed in
experiments in the only electron plasma, in the
charged plasma of the lens [1,3] and in the plasma,
placed in crossed radial electrical and longitudinal
magnetic fields.
ACKNOWLEDGEMENTS
This work was supported partly by STCU under
grant 1596.
REFERENCES
1. A.A.Goncharov et al. // Plasma Physics Rep.
1994, v.20, No.5, p.499-505.
2. A.A.Goncharov, S.N.Gubarev, V.I.Maslov,
I.N.Onishchenko // Problems of Atomic Science
and Technology. 2001, No.3, p.152-154.
3. A.Goncharov, I.Litovko // IEEE Trans. Plasma
Sci. 1999, v.27, p.1073.
НЕЛИНЕЙНАЯ ЭВОЛЮЦИЯ ВИХРЕЙ В СИЛЬНОТОЧНОЙ ЭЛЕКТРОСТАТИЧЕСКОЙ
ПЛАЗМЕННОЙ ЛИНЗЕ
А.А. Гончаров, В.И. Маслов, И.Н. Онищенко, В.Л. Стомин, В.Н. Третьяков
Проведено теоретическое исследование пространственной структуры и нелинейной динамики вихрей в
плазменных линзах для фокусировки больших ионных пучков.
НЕЛІНІЙНА ЕВОЛЮЦІЯ ВИХРІВ У ПОТУЖНОСТРУМОВІЙ ЕЛЕКТРОСТАТИЧНІЙ
ПЛАЗМОВІЙ ЛІНЗІ
А.А. Гончаров, В.І. Маслов, І.М. Онищенко, В.Л. Стомін, В.Н. Третьяков
Проведено теоретичне дослідження просторової структури і нелінійної динаміки вихрів у плазмових
лінзах для фокусування великих іонних пучків.
32
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yE
rv
yE
rv
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| id | nasplib_isofts_kiev_ua-123456789-79321 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T13:28:26Z |
| publishDate | 2004 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. Tretyakov, V.N. 2015-03-31T08:50:14Z 2015-03-31T08:50:14Z 2004 Non-linear evolution of vortices in high-current electrostatic plasma lens / A.A. Goncharov, V.I.Maslov, I.N. Onishchenko, V.L. Stomin, V.N. Tretyakov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 30-32. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.40.Mj https://nasplib.isofts.kiev.ua/handle/123456789/79321 The spatial structure and nonlinear dynamics of vortices in plasma lens for high-current ion-beam focusing have been investigated theoretically. Проведено теоретичне дослідження просторової структури і нелінійної динаміки вихрів у плазмових лінзах для фокусування великих іонних пучків. Проведено теоретическое исследование пространственной структуры и нелинейной динамики вихрей в плазменных линзах для фокусировки больших ионных пучков. This work was supported partly by STCU under grant 1596. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Новые и нестандартные ускорительные технологии Non-linear evolution of vortices in high-current electrostatic plasma lens Нелінійна еволюція вихрів у потужнострумовій електростатичній плазмовій лінзі Нелинейная эволюция вихрей в сильноточной электростатической плазменной линзе Article published earlier |
| spellingShingle | Non-linear evolution of vortices in high-current electrostatic plasma lens Goncharov, A.A. Maslov, V.I. Onishchenko, I.N. Stomin, V.L. Tretyakov, V.N. Новые и нестандартные ускорительные технологии |
| title | Non-linear evolution of vortices in high-current electrostatic plasma lens |
| title_alt | Нелінійна еволюція вихрів у потужнострумовій електростатичній плазмовій лінзі Нелинейная эволюция вихрей в сильноточной электростатической плазменной линзе |
| title_full | Non-linear evolution of vortices in high-current electrostatic plasma lens |
| title_fullStr | Non-linear evolution of vortices in high-current electrostatic plasma lens |
| title_full_unstemmed | Non-linear evolution of vortices in high-current electrostatic plasma lens |
| title_short | Non-linear evolution of vortices in high-current electrostatic plasma lens |
| title_sort | non-linear evolution of vortices in high-current electrostatic plasma lens |
| topic | Новые и нестандартные ускорительные технологии |
| topic_facet | Новые и нестандартные ускорительные технологии |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79321 |
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