Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length
The dispersion equation, describing the instability development of vortex turbulence excitation in cylindrical plasma in crossed radial electric and axial magnetic fields with taking into account the longitudinal inhomogeneity and finite time of leaving of plasma electrons and ions from the system,...
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| Zitieren: | Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length / V.I. Maslov, I.P. Levchuk, I.N. Onishchenko, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 274-276. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1122172017-01-19T03:02:17Z Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length Maslov, V.I. Levchuk, I.P. Onishchenko, I.N. Yegorov, A.M. Yuferov, V.B. Нелинейные процессы в плазменных средах The dispersion equation, describing the instability development of vortex turbulence excitation in cylindrical plasma in crossed radial electric and axial magnetic fields with taking into account the longitudinal inhomogeneity and finite time of leaving of plasma electrons and ions from the system, has been derived. It is shown that the finite length of system time and finite time of system leaving of plasma electrons and ions leads to the appearance of the instability threshold and to decrease of growth rate of its development. Отримано дисперсійне рівняння, що описує розвиток нестійкості збудження вихрової турбулентності в циліндричній плазмі в схрещених радіальному електричному та поздовжньому магнітному полях з урахуванням поздовжньої неоднорідності і кінцевого часу залишання системи електронами та іонами плазми. Показано, що кінцева довжина системи і кінцевий час залишання системи електронами і іонами плазми призводять до появи порога нестійкості та зменшення інкремента її розвитку. Получено дисперсионное уравнение, описывающее развитие неустойчивости возбуждения вихревой турбулентности в цилиндрической плазме в скрещенных радиальном электрическом и продольном магнитном полях с учетом продольной неоднородности и конечного времени ухода электронов и ионов плазмы из системы. Показано, что конечная длина системы и конечное время покидания системы электронами и ионами плазмы приводят к появлению порога неустойчивости и уменьшению инкремента ее развития. 2015 Article Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length / V.I. Maslov, I.P. Levchuk, I.N. Onishchenko, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 274-276. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx http://dspace.nbuv.gov.ua/handle/123456789/112217 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
| spellingShingle |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Maslov, V.I. Levchuk, I.P. Onishchenko, I.N. Yegorov, A.M. Yuferov, V.B. Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length Вопросы атомной науки и техники |
| description |
The dispersion equation, describing the instability development of vortex turbulence excitation in cylindrical plasma in crossed radial electric and axial magnetic fields with taking into account the longitudinal inhomogeneity and finite time of leaving of plasma electrons and ions from the system, has been derived. It is shown that the finite length of system time and finite time of system leaving of plasma electrons and ions leads to the appearance of the instability threshold and to decrease of growth rate of its development. |
| format |
Article |
| author |
Maslov, V.I. Levchuk, I.P. Onishchenko, I.N. Yegorov, A.M. Yuferov, V.B. |
| author_facet |
Maslov, V.I. Levchuk, I.P. Onishchenko, I.N. Yegorov, A.M. Yuferov, V.B. |
| author_sort |
Maslov, V.I. |
| title |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| title_short |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| title_full |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| title_fullStr |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| title_full_unstemmed |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| title_sort |
suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Нелинейные процессы в плазменных средах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112217 |
| citation_txt |
Suppression of vortical turbulence in plasma in crossed electrical and magnetic fields due to finite lifetime of electrons and ions and due to finite system length / V.I. Maslov, I.P. Levchuk, I.N. Onishchenko, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 274-276. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T03:33:14Z |
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2025-07-08T03:33:14Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 274
SUPPRESSION OF VORTICAL TURBULENCE IN PLASMA
IN CROSSED ELECTRICAL AND MAGNETIC FIELDS DUE TO FINITE
LIFETIME OF ELECTRONS AND IONS AND DUE
TO FINITE SYSTEM LENGTH
V.I. Maslov, I.P. Levchuk, I.N. Onishchenko, A.M. Yegorov, V.B. Yuferov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
The dispersion equation, describing the instability development of vortex turbulence excitation in cylindrical
plasma in crossed radial electric and axial magnetic fields with taking into account the longitudinal inhomogeneity
and finite time of leaving of plasma electrons and ions from the system, has been derived. It is shown that the finite
length of system time and finite time of system leaving of plasma electrons and ions leads to the appearance of the
instability threshold and to decrease of growth rate of its development.
PACS: 29.17.+w; 41.75.Lx
INTRODUCTION
In plasma [1-3] a turbulence has been excited in
crossed radial electrical and longitudinal magnetic fields
by gradient of external magnetic field. This turbulence is
a distributed vorticity. In this paper the excitation and
damping of similar vortical turbulence, excited in cylin-
drical plasma in crossed radial electrical 0rE and longi-
tudinal magnetic 0H fields [4], is investigated theoreti-
cally. From the general nonlinear equation, presented in
article [3], for vorticity the dispersion relation, which
describes the instability development of vortex turbu-
lence excitation, has been derived. It is shown that the
finite length of system time and finite time of system
leaving of plasma electrons and ions leads to the appear-
ance of the instability threshold and to decrease of
growth rate of its development.
EXCITATION OF VORTICES
Let us derive the dispersion relation. We take into
account that the ions pass with velocity biV through
system of length L during time, approximately equal
i
bi
L
V
τ = . We also take into account that the electrons
pass through system and are renovated in system also
during finite time, eτ . Damping of perturbations of den-
sities and velocities of electrons and ions at recovery of
their unperturbed values we describe, using i
i
1
ν ≡
τ
,
e
e
1
ν ≡
τ
.
We use the electron hydrodynamic equations
( ) ( )e o
2
th
He e
e e
V V V V V
t
Ve , V n
m n
θ
∂
+ ν − + ∇ =
∂
= ∇φ+ ω − ∇
,
( ) ( )e oee
e
e
n nn n V 0
t
−∂
+ +∇ =
∂ τ
. (1)
Also we use the ion hydrodynamic equations
( ) ( )i i
i bi i i
i
V qV V V V
t m
∂
+ ν − + ∇ = − ∇φ ∂
,
( ) ( )i oii
i i
i
n nn n V 0
t
−∂
+ +∇ =
∂ τ
(2)
and Poisson equation for the electrical potential, φ ,
( )e i i4 en q n∆φ = π − . (3)
Here V
, en are a velocity and density of electrons; thV
is the electron thermal velocity; oVθ is the electron azi-
muth drift velocity in crossed fields; iV
, in , iq , im
are the velocity, density , charge and mass of ions.
As it will be visible from the further, the dimensions
of the vortical perturbations are much larger than the
electron Debye radius, th
de
pe
Vr ≡
ω
, then the last term in
(1) can be neglected. Here
1/22
oe
pe
e
4 n e
m
π
ω ≡
, oen is the
unperturbed electron density.
From equations (1) one can derive non-linear equa-
tions
( ) ( )He He e
t z z
e e e
d V
n n n
α−ω α−ω αν
= ∂ −
, (4)
t z e z z
e
ed V V
m
+ ν = ∂ φ
describing both transversal and longitudinal electron
dynamics. Here
( )t td V⊥ ⊥≡ ∂ + ∇
, z z
∂
∂ ≡
∂
, t t
∂
∂ ≡
∂
, (5)
V⊥
, zV are the transversal and longitudinal electron
velocities, α is the vorticity, the characteristic of elec-
tron vortical motion, ze rotVα ≡
.
Taking into account higher linear terms, from (1) one
can obtain
( )
He
z t e2
e He e
e eV e ,
m m⊥ ⊥ ⊥
≈ ∇ φ + ∂ + ν ∇ φ ω ω
(6)
From (6) we derive
ISSN 1562-6016. ВАНТ. 2015. №4(98) 275
or or
He r
e He e He e He
2eE eE 1 e
rm m m ⊥
α ≡ µω ≈ − − ∂ + ∆ ϕ+ ω ω ω
( ) ( )
He
r r t e z 2
e He e
e 1 e 1e ,
m m ⊥ ⊥
+ ∂ ϕ ∂ + ∂ + ν ∇ ∇ ϕ ω ω
,
orE .∇φ ≡ ∇ϕ−
(7)
Here orE is the radial focusing electric field, ϕ is the
electric potential of the vortical perturbation;
2
peor
He
e He He oe
2eE n
rm n
ω ∆
− = ≡ ηω ω ω
, i
oe oi
qn n n
e
∆ ≡ − .
From (7)
2
pe e
He oe
n
n
ω δ
α ≈ ω
approximately follows.
Thus the vortical motion begins, as soon as the electron
density perturbation, enδ , appears.
We use that, as it will be shown below, the character-
istic frequencies of perturbations approximately equal to
ion plasma frequency, piω .
As beam ions have large mass and propagate through
system with velocity biV , we describe their dynamics in
linear approximation. We derive ion density perturba-
tion from eq.s (2)
( )
i
i io 2
i z ib i
qn n .
m k V i
∆φ
δ = −
ω− + ν
(8)
Here k , ω are wave number and frequency of perturba-
tion, biV is the unperturbed longitudinal velocity of the
ions. Substituting (8) in Poisson equation (3), one can
obtain
en
4 e
β∆φ
= δ
π
,
( )
2
pi
2
z ib i
1
k V i
ω
β = −
ω− + ν
,
e oe en n n= + δ . (9)
Let us consider instability development in linear ap-
proximation. Then we search the dependence of the per-
turbation on z , θ in the form ( )e zn exp ik z i θδ ∝ + θ .
Then from (4) we derive
( )
2
He e He z
t
e e e oe o e
e ikd
n n m n iθ θ
ω ν ω φ
= α − ω− ω + ν
,
o
o
V
r
θ
θω ≡ . (10)
From (5), (6), (9), (10) we obtain, using the radial
gradient of the short coil magnetic field, the following
linear dispersion relation, describing the instability de-
velopment
( )
( ) ( )
( )
( )
2 2
pi pe r He
2 2
o ez bi i
2 2
pe z
2 2
o e
r 1
1
i kk V i
k 0.
ki
θ
θ θ
θ θ
ω ω ∂ ω
− − −
ω− ω + τω− + τ
ω
− =
ω− ω + τ
(11)
From (11) for quick oph VV θ≈ vortical perturba-
tions we obtain
0kz = , ( ) δω+ω=ω o , ( ) ,oδω ω<<
( )
opi
o
θθω=ω=ω ,
∆
ω
ω
=ωθ
oe
2
o n
n
2
He
pe ,
e
nqnn oii
oe −≡∆ , qiγ=δω ,
τ
+
τ
−
ω
∂
ω
ω
≈γ θ
ieHe
r
pipe
q
11
2
11
r2k
. (12)
From (11) for slow oph VV θ<< vortical perturba-
tions we obtain
τ
+
τ
−γ=γ
ie
0ss
21
3
1 , [ ] 3/1
o
2
pi3/40s 2
3
θθωω
≈γ
3
Re,1
V
1k 0s
s
He
r
2
pe
o
2 γ
=ω
ω
∂ω
−=
θ
. (13)
One can see that eτ and iτ decrease growth rates
and lead to appearance of thresholds of instability de-
velopment.
Let us consider now, how finite zk 0≠ influences on
growth rate of the instability development. From (11) we
obtain the growth rate of the excitation of slow homoge-
neous turbulence with taking into account zk .
( )
( )
1/32/3
s pi o z bi4/3
1/3
2
z
2
z o z bi r
He
3 k V
2
k1 .
12k k V
r
θ θ
θ
θ θ
γ ≈ ω ω − ×
× −
+ ω − ∂ ω
(14)
From (14) one can see that both the taking into ac-
count the longitudinal dynamics of ions and electrons
results in reduction of the growth rate. The perturbations
with least zk
L
π ≈
have maximum growth rate, that is
with the largest longitudinal dimensions, close to system
length.
CONCLUSIONS
The dispersion equation, describing the instability
development of vortex turbulence excitation in cylindri-
cal plasma in crossed radial electric and axial magnetic
fields with taking into account the longitudinal inhomo-
geneity and finite time of leaving of plasma electrons
and ions from the system, has been derived. It is shown
that the finite length of system time and finite time of
system leaving of plasma electrons and ions leads to the
appearance of the instability threshold and to decrease
of growth rate of its development.
REFERENCES
1. A. Goncharov, A. Dobrovolsky, A. Kotsarenko,
A. Morozov, I. Protsenko // Physica Plasmy. 1994,
v. 20, p. 499.
2. A. Goncharov, I. Litovko. Electron Vortexes in
High-Current Plasma Lens // IEEE Trans. Plasma
Sci. 1999, v. 27, p. 1073.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 276
3. V.I. Maslov, A.A. Goncharov, I.N. Onishchenko.
Self-Organization of Non-Linear Vortexes in Plas-
ma Lens for Ion-Beam-Focusing in Crossed Radial
Electrical and Longitudinal Magnetic Fields // Proc.
of the Intern. Workshop "Collective phenomena in
macroscopic systems". World Scientific, Singapore,
2007, p. 20-25.
4. V.B. Yuferov, A.S. Svichkar, S.V. Shariy,
T.I. Tkachova, V.О. Ilichova, V.V. Katrechko,
A.I. Shapoval, S.N. Khizhnyak. About Redistribu-
tion of Ion Streams in Imitation Experiments on
Plasma Separation // Problems of Atomic Science
and Technology. Series “Physics of Radiation
Damage and Radiation Material”. 2013, №5 (87),
p. 100-103.
Article received 25.05.2015
ПОДАВЛЕНИЕ ВИХРЕВОЙ ТУРБУЛЕНТНОСТИ В ПЛАЗМЕ В СКРЕЩЕННЫХ
ЭЛЕКТРИЧЕСКОМ И МАГНИТНОМ ПОЛЯХ ЗА СЧЕТ КОНЕЧНОГО ВРЕМЕНИ УХОДА
ЭЛЕКТРОНОВ И ИОНОВ И КОНЕЧНОЙ ДЛИНЫ СИСТЕМЫ
В.И. Маслов, И.П. Левчук, И.Н. Онищенко, А.М. Егоров, В.Б. Юферов
Получено дисперсионное уравнение, описывающее развитие неустойчивости возбуждения вихревой
турбулентности в цилиндрической плазме в скрещенных радиальном электрическом и продольном маг-
нитном полях с учетом продольной неоднородности и конечного времени ухода электронов и ионов
плазмы из системы. Показано, что конечная длина системы и конечное время покидания системы элек-
тронами и ионами плазмы приводят к появлению порога неустойчивости и уменьшению инкремента ее
развития.
ПОДАВЛЕННЯ ВИХРОВОЇ ТУРБУЛЕНТНОСТІ В ПЛАЗМІ В СХРЕЩЕНИХ ЕЛЕКТРИЧНОМУ
І МАГНІТНОМУ ПОЛЯХ ЗАВДЯКИ КІНЦЕВОМУ ЧАСУ ВИХОДУ ЕЛЕКТРОНІВ ТА ІОНІВ
І КІНЦЕВІЙ ДОВЖИНІ СИСТЕМИ
В.І. Маслов, І.П. Левчук, І.М. Онищенко, О.М. Єгоров, В.Б. Юферов
Отримано дисперсійне рівняння, що описує розвиток нестійкості збудження вихрової турбулентності
в циліндричній плазмі в схрещених радіальному електричному та поздовжньому магнітному полях з ура-
хуванням поздовжньої неоднорідності і кінцевого часу залишання системи електронами та іонами плаз-
ми. Показано, що кінцева довжина системи і кінцевий час залишання системи електронами і іонами пла-
зми призводять до появи порога нестійкості та зменшення інкремента її розвитку.
INTRODUCTION
EXCITATION OF VORTICES
|