The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer
The temporal evolution of the linear diocotron instability of the cylindrical annular plasma column, which is driven by the shear of the equilibrium velocity of pure electron non-neutral plasma in crossed external magnetic and own electric fields, is investigated by using the extension of shearing m...
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irk-123456789-1119032017-01-16T03:03:27Z The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer Mykhaylenko, V.V. Hae June Lee Mykhaylenko, V.S. Azarenkov, N.A. Нерелятивистская электроника The temporal evolution of the linear diocotron instability of the cylindrical annular plasma column, which is driven by the shear of the equilibrium velocity of pure electron non-neutral plasma in crossed external magnetic and own electric fields, is investigated by using the extension of shearing modes methodology onto the cylindrical geometry. That approach does not use any spectral transforms in time and gives the solution of the initial value problems for any desired time. The evolution process leads toward the convergence to the phase-locking configuration of the mutually growing eigen and forced modes. Часова лінійна еволюція діокотронної нестійкості циліндричного шару електронів, яка збуджується широм рівноважної швидкості електронів у схрещених зовнішньому магнітному та власному електричному полях, досліджується використовуючи узагальнення методології зсувних мод на циліндричну геометрію. Цей підхід не використовує спектральне перетворення по часовій змінній і дає розв’язок задачі на початкові дані для любого часу. Еволюційний процес веде до утворення конфігурації з фазовою синхронізацією взаємно зростаючих власних та вимушених мод. Временная линейная эволюция диокотронной неустойчивости цилиндрического слоя электронов, которая возбуждается широм равновесной скорости электронов в скрещенных внешнем магнитном и собственном электрическом полях, исследуется используя обобщение методологии сдвиговых мод на цилиндрическую геометрию. Этот подход не использует спектральное преобразование по времени и дает решение начальной задачи для любого времени. Эволюционный процесс приводит к образованию конфигурации с фазовой синхронизацией взаимно растущих собственных и вынужденных мод. 2013 Article The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer / V.V. Mykhaylenko, Hae June Lee, V.S. Mykhaylenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 25-29. — Бібліогр.: 4 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/111903 PACS: 52.27.Gr en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нерелятивистская электроника Нерелятивистская электроника |
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Нерелятивистская электроника Нерелятивистская электроника Mykhaylenko, V.V. Hae June Lee Mykhaylenko, V.S. Azarenkov, N.A. The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer Вопросы атомной науки и техники |
| description |
The temporal evolution of the linear diocotron instability of the cylindrical annular plasma column, which is driven by the shear of the equilibrium velocity of pure electron non-neutral plasma in crossed external magnetic and own electric fields, is investigated by using the extension of shearing modes methodology onto the cylindrical geometry. That approach does not use any spectral transforms in time and gives the solution of the initial value problems for any desired time. The evolution process leads toward the convergence to the phase-locking configuration of the mutually growing eigen and forced modes. |
| format |
Article |
| author |
Mykhaylenko, V.V. Hae June Lee Mykhaylenko, V.S. Azarenkov, N.A. |
| author_facet |
Mykhaylenko, V.V. Hae June Lee Mykhaylenko, V.S. Azarenkov, N.A. |
| author_sort |
Mykhaylenko, V.V. |
| title |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| title_short |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| title_full |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| title_fullStr |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| title_full_unstemmed |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| title_sort |
shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Нерелятивистская электроника |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111903 |
| citation_txt |
The shearing modes approach to the theory of the diocotron instability of the cylindrical electron layer / V.V. Mykhaylenko, Hae June Lee, V.S. Mykhaylenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 25-29. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 25
THE SHEARING MODES APPROACH TO THE THEORY
OF THE DIOCOTRON INSTABILITY OF THE CYLINDRICAL
ELECTRON LAYER
V.V. Mykhaylenko1, Hae June Lee1, V.S. Mykhaylenko2,3, N.A. Azarenkov2
1Pusan National University, Busan, S. Korea;
2V.N. Karazin Kharkov National University, Kharkov, Ukraine;
3Kharkov National Automobile and Highway University, Kharkov, Ukraine
E-mail: vladimir@pusan.ac.kr
The temporal evolution of the linear diocotron instability of the cylindrical annular plasma column, which is
driven by the shear of the equilibrium velocity of pure electron non-neutral plasma in crossed external magnetic and
own electric fields, is investigated by using the extension of shearing modes methodology onto the cylindrical ge-
ometry. That approach does not use any spectral transforms in time and gives the solution of the initial value prob-
lems for any desired time. The evolution process leads toward the convergence to the phase-locking configuration
of the mutually growing eigen and forced modes.
PACS: 52.27.Gr
1. BASIC EQUATIONS OF
THE NON-MODAL APPROACH
The diocotron instability [1], is the electrostatic in-
stability of the low-density non-neutral plasmas in mag-
netic field. It is driven by the shear of the equilibrium
velocity of non-neutral plasma in crossed external mag-
netic and own electric fields. In recent years, the inves-
tigations of this instability are going far beyond tradi-
tional studies of plasma stability in Malmberg-Penning
traps. The understanding the physics of this instability is
important for the development of a new type of beam
collimator system in high-energy colliders, which util-
izes pulsed hollow electron beam to kick halo particles
transversely while leaving the beam core unperturbed
[2]. The diocotron instability is considered [3] as a
promising mechanism leading to highly unstable flows
in the pulsar inner magnetosphere.
In this paper we develop the theory of the diocotron
instability of the cylindrical annular plasma column by
extending the shearing modes methodology [4] onto
cylindrical geometry. We consider the most simple
model of the confined electron plasma as an infinitely
long along the magnetic field hollow annulus with step-
function electron density profile, which, nevertheless,
requires the development of the shearing mode ap-
proach [4] to the rotating cylindrical plasma with a
radially inhomogeneous angular velocity. The basic
equation in that model is the drift-Poisson equation for
the perturbed electrostatic potential φ
( ) ( )
( ) ( )( )
2
2
, ,
= ,pe
ce
r r t
t
r b r d
φ θ
θ
ω φ δ δ
ω θ
∂ ∂⎛ ⎞+Ω ∇⎜ ⎟∂ ∂⎝ ⎠
∂
− − −
∂
(1)
where the angular velocity ( )rΩ is equal to
( )
2 2
2= 1 .
2
pe
ce
br
r
ω
ω
⎛ ⎞
Ω −⎜ ⎟
⎝ ⎠
(2)
The boundary conditions for potential φ are the
continuity of the potential across the edges =r b and
=r d , i.e. ( ) ( )= , , = = , ,r b t r b tφ ε θ φ ε θ− + with
0ε → and the same condition at =r d , the zero mag-
nitude of the potential on the conducted boundary
r R= and the conditions on the jump of the /d drφ at
=r b and =r d ,
( )
( )
( )
2
= =
= =
2
, ,1=
, ,1= .
pe
r b r b ce
r d r d
pe
ce
b t
t r r b
d
t r r
d t
d
ε ε
ε ε
ω φ θφ φ
ω θ
φ φ
θ
ω φ θ
ω θ
+ −
+ −
∂⎡ ⎤∂ ∂ ∂
−⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦
⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞+Ω −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦
∂
−
∂
(3)
We describe two areas: the electron layer, b r d„ „ ,
and vacuum in the rest of space. Eq. (1) in the vacuum
has a form
2 = 0.
t
φ∂
∇
∂
(4)
The solutions to Eq.(4) for the separate Fourier harmon-
ics ( ), ,r l tφ , determined as
( ) ( ) ( )
=
, , = , , exp ,
l
r t r l t ilφ θ φ θ
∞
−∞
∑ (5)
are
( ) ( )
( ) ( )
1
2
2 2
, , = , for 0 < < ,
, , = , 1 for < .
l
l
l
l
r l t C l t r r b
rr l t C l t r d r R
R
φ
φ − ⎛ ⎞
−⎜ ⎟
⎝ ⎠
„
(6)
In electron layer, the right hand side of Eq. (1) is equal
to zero, except the edges at =r b , and =r d i.e.
( ) ( )2 , , = 0.r r t
t
φ θ
θ
∂ ∂⎛ ⎞+Ω ∇⎜ ⎟∂ ∂⎝ ⎠
(7)
Instead of application of the commonly used spectral
transform in time, here we use other approach, which
gives easy and transparent treating of the problem con-
sidered. That approach is grounded on the transforma-
tion of Eq. (7) to the sheared coordinates = ,t t
ˆ= ,r r ( ) ˆ= ,t rθ θΩ + (8)
ISSN 1562-6016. ВАНТ. 2013. №4(86) 26
where the sheared coordinate ( )ˆ = t rθ θ − Ω is the char-
acteristic for Eq.(7). In these coordinates, we have
( )/ / = /t r tθ∂ ∂ +Ω ∂ ∂ ∂ ∂ and Eq.(7) is integrated
easily over time. That gives for the Fourier harmonic
( )ˆ, ,r l tφ of the potential, determined as
( ) ( ) ( )
=
ˆ ˆˆ ˆ, , = , , exp ,
l
r t r l t ilφ θ φ θ
∞
−∞
∑ the equation
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
2
2
22
2 2
2
2 2
1 0
ˆ ˆ ˆ, , , ,1 2
ˆ ˆ ˆˆ
ˆ ˆ1
ˆ ˆˆ ˆ
ˆ
ˆ ˆ, , = 4 , , =
ˆ
r l t d r r l t
ilt
r dr rr
d r d rl ilt ilt
r drr dr
d r
l t r l t en r l t t
dr
φ φ
φ π
∂ Ω ∂⎛ ⎞
+ −⎜ ⎟
∂∂ ⎝ ⎠
⎡ Ω Ω
− + +⎢
⎢⎣
⎤Ω⎛ ⎞
⎥+ ⎜ ⎟
⎥⎝ ⎠ ⎦
(9)
and brings into the further consideration the initial per-
turbation ( )1 0ˆ, ,n r l t of the electron density in electron
layer. The general solution to Eq.(9) is obtained
straightforwardly and is equal to
( ) ( )( ) ( )(
( ) ( )
( )( ( ) ( )1
ˆ ˆ
3
=1
ˆ ˆ1 1
1 1 1 1 0
ˆ1
4 1 1 1 1 0ˆ
ˆ, , = [ ,
2 ˆ ˆ ˆ ˆ, ,
2 ˆ ˆ ˆ ˆ, , , .
il t r
l
r ilt rl l
b
d ilt rl l
r
r t e C l t
e dr r n r l t e r
l
eC l t dr r n r l t e r
l
θφ θ
π
π
∞
+ Ω
− Ω−
− Ω+ −
⎞+ +⎟
⎠
⎤⎞+ ⎟ ⎥⎠ ⎦
∑
∫
∫
(10)
That solution is valid for any time. It does not con-
tain any singularities, which are inherent for the solu-
tions obtained with spectral transforms in time and
compose serious obstacles for the determining the ex-
plicit time dependence for the potential for the finite
time.
2. MODAL DIOCOTRON INSTABILITY
If we suppose that any initial perturbation in layer is
absent, i.e. ( )1 0ˆ, , = 0n r l t , the solution (10) in layer
b r d≤ ≤ reduces to a form
( ) ( )( ) ( ) ( )( )ˆ ˆ
3 4
=1
ˆ ˆ ˆ, , = , ,il t r l l
l
r t e C l t r C l t rθφ θ
∞
+ Ω −+∑ (11)
which describes only the surface waves, which form the
discrete spectrum of perturbations. The condition of the
perturbed potential continuity on the boundaries =r b
and =r d couples the coefficients ( )1 ,C l t , ( )2 ,C l t of
Eq.(6) with ( )3 ,C l t , ( )4 ,C l t , and gives the following
presentation for the potential in the vacuum regions:
( )
( ) ( )
=1
2
3 4
, , =
( , , ), 0 < < ,
il l
l
l
r t e r
C l t C l t b r b
θφ θ
∞
−× +
∑ (12)
( ) ( )( )
( ) ( )( )
1
2 2 2 2
=1
2
3 4
, , =
, , , < .
il l l l l l
l
l
r t e r R r R d
C l t d C l t d r R
θφ θ
−∞
− − −
× +
∑
„
(13)
We apply the boundary conditions (3) to (12) - (13),
and obtain the system of equations for ( )3 ,C l t and
( )4 ,C l t , i.e.
( ) ( ) ( ) ( )3 4
3 2
, ,
, l
C l t C l t
il d C l t
t R
∂ ⎛ ⎞
+ Ω +⎜ ⎟
∂ ⎝ ⎠
( ) ( )
( ) ( )( )
2 2 2
4
3 2 2 2
2
24
3 4
,
= , 1 1 ,
2
= , , .
2
l l
pe
l l l
ce
pe l
ce
C l td bi C l t
R d d
C i C l t b C l t
t
ω
ω
ω
ω
⎡ ⎛ ⎞ ⎤⎛ ⎞
− − +⎢ ⎜ ⎟⎜ ⎟ ⎥⎜ ⎟⎢ ⎝ ⎠ ⎦⎝ ⎠⎣
∂
− +
∂
(14)
The solution to Eqs.(14) has a modal form,
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
3 1 2
4 1 1 2 2
, = ,
, = ,
i l t l t i l t l t
i l t l t i l t l t
C l t c l e c l e
C l t c l a e c l a e
ω γ ω γ
ω γ ω γ
− + − −
− + − −
+
+
(15)
where
( ) ( )( )
1
2
1,2 2
2
= 1l ce
pe
a b l i l
ω
ω γ
ω
−
⎛ ⎞
± −⎜ ⎟⎜ ⎟
⎝ ⎠
,
and
( )
2 2 2 2
2 2 2= 1 1 ,
4
l l
pe
l l
ce
b d bl l
d R d
ω
ω
ω
⎡ ⎤⎛ ⎞ ⎛ ⎞
− + −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
(16)
( )
2 2 2 2
2 2 2
1/ 222 2 2
2 2 2
= 4 1 1
4
2 1 1 ,
l l
pe
l l
ce
l l
l l
b b dl l
d d R
b d bl
d R d
ω
γ
ω
⎧ ⎡ ⎤⎛ ⎞⎪ − −⎨ ⎢ ⎥⎜ ⎟
⎝ ⎠⎪ ⎣ ⎦⎩
⎫⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎪− − − − − ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎪⎭
(17)
which define the known frequency and growth rate for
diocotron instability in cylindrical annular plasma
column [1] with conducted boundary. It follows from
(17), that instability is absent for = 0l and = 1l . and
exists when
2 2 2
2 2 2
22 2 2
2 2 2
4 1 1
> 2 1 1 .
l l
l l
l l
l l
b b dl
d d R
b d bl
d R d
⎡ ⎤⎛ ⎞
− −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞ ⎛ ⎞
− − − −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
(18)
3. MODAL DIOCOTRON INSTABILITY
INTERPRETED IN TERMS OF EDGE
WAVES INTERACTION
The application of the transformation to shearing
coordinates (1) opens the way to effective analysis of
the diocotron instability in terms of edge waves
interaction [5], applied for the diocotron instability in
plane geometry in Ref. [4]. Writing the functions
( )3 ,C l t and ( )4 ,C l t in the complex form [4],
( ) ( ) ( ) ( ) ( ) ( ), ,3 4
3 3 4 4, = , , , = , ,i l t i l tC l t Q l t e C l t Q l t eε ε (19)
the edge perturbation of the potential can be regarded as
two edge waves with amplitudes ( )3 ,Q l t and ( )4 ,Q l t
and phases ( )3 ,l tε and ( )4 ,l tε . By substituting Eqs.
(19) into Eqs. (14) and separating the real and
imaginary parts at =r b and =r d , we obtain, that
amplitudes ( )3 ,Q l t and ( )4 ,Q l t , and the relative phase
ISSN 1562-6016. ВАНТ. 2013. №4(86) 27
3 4=ε ε ε− of the edge diocotron waves evolve
according to equations
2 2 2
23
4 2 2
2
24
3
= sin 1 1 ,
2
= sin ,
2
l
pe l
l
ce
pe l
ce
dQ b dd Q l
dt d R
dQ
b Q
dt
ω
ε
ω
ω
ε
ω
− ⎡ ⎤⎛ ⎞
− −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦ (20)
and
( )( )= cosd t
dt
ε ε βΓ + (21)
where
2 2 2 2
3 4
2 2 2
4 3
= 1 1 ,
2
l l
pe
l l
ce
Q Q b b dl
Q Q d d R
ω
ω
⎡ ⎤⎡ ⎤⎛ ⎞
Γ + − −⎢ ⎥⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦
(22)
and
( )
2 2 2
2 2 2
1
2 2 2
3 4
2 2 2
4 3
= 2 1 1
1 1 ,
l l
l l
l l
l l
b d bt l
d R d
Q Q b b dl
Q Q d d R
β
−
⎛ ⎞⎛ ⎞ ⎛ ⎞
− − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪× + − −⎨ ⎬⎢ ⎥⎜ ⎟
⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
(23)
From Eqs. (20) one can obtain the integral,
2 2 2
2 2
3 4 2 2 2= 1 1 .
l l
l l
b b dQ Q l C
d d R
⎡ ⎤⎛ ⎞
− − +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
(24)
Due to the exponential growth of amplitudes 3Q ,
4Q with time from infinitesimal beginnings, the
amplitudes become
2 2 2
2 2
3 4 2 2 21 1 ;
l l
l l
b b dQ Q l C
d d R
⎡ ⎤⎛ ⎞
≈ − −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
? (25)
then ( )tβ and Γ approaches the values
2 2 2
0 2 2 2
1/ 22 2
2 2
1/22 2 2
0 2 2
= 1 1 1
2 2
1 1 ,
= 1 1 .
l l
l l
l l
l
l l
pe
l
ce
l b d b
d R d
b b dl
d d R
b b dl
d d R
β
ω
ω
−−
⎛ ⎞⎛ ⎞ ⎛ ⎞
− − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞× − −⎢ ⎥⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞⎛ ⎞Γ − −⎢ ⎥⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
(26)
At condition (18), under which the diocotron
instability develops, 0β is less than unity and therefore,
the stationary (or fixed) points of the equation (21),
where / = 0d dtε , exist and are determined by the
equation 0cos = 0ε β+ . The solutions of this equation
are two sets of stationary points: stable (or attractors) at
( )1
0= 2 ,cosk kε π β π−− + (27)
and unstable at
( )1
0= 2 .cosk kε π β π−− − + (28)
The solution of the equation
( )0 0/ = cosd dtε ε βΓ + with initial condition 0=ε ε at
0= = 0t t , has a simple form
2
00
0
2
0 00
11 1tan = ,
2 1 11
tAe
tAe
ββε
β β
Γ
Γ
⎛ ⎞−
+ ⎜ ⎟+
− ⎜ ⎟− −⎜ ⎟−⎝ ⎠
(29)
where
( )
( )
20
0 0
20
0 0
1 tan 1
2=
1 tan 1
2
A
εβ β
εβ β
− + −
− − −
. (30)
As it follows from Eq.(29), the initial perturbations
with an arbitrary value of the initial phase of each wave,
will evolve with time to the ultimate value *ε of relative
phase,
* 0cos = ,ε β (31)
which does not depend on the initial data.
This solution of the initial value problem reveals the
linear stage of the instability development as a process
of the formation of the phase locked configuration.
Figure illustrates such configurations for separate
modes = 5l with relative phase *ε , determined by
Eq.(31). On Figure, we use the values of the parameter
/b d , for which the growth rate ( )lγ attains the
maximal values. These values are / = 0.83b d with
* = 2.17ε rad for = 5l . The time of the developing of
such configuration is comparable with the inverse
growth rate time of the diocotron instability.
Phase-locked configuration for azimuthal wave number
= 5l (a) with / = 0.838b d , / = 0.8d R ,
* = 2.136ε rad, and (b) with / = 0.8316b d ,
/ = 0.5d R , * = 2.352ε rad
4. NON-MODAL ANALYSIS
OF THE DIOCOTRON INSTABILITY
Now we obtain the complete solution of the boun-
dary and initial value problems, determined by the
condition of the continuity of the potential φ and by
Eqs. (3) with accounting for the initial perturbation of
the electron density. The condition of the potential
continuity at the inner and outer surfaces of the electron
cylinder gives the connection formulae for the functions
( )1 ,C l t , ( )2 ,C l t and ( )3 ,C l t , ( )3 ,C l t , and as a result,
presentation of solutions (6) through the functions
( )3 ,C l t and ( )4 ,C l t of the solution (11). We obtain for
the vacuum region, 0 < <r b ,
( ) ( )( ) ( ) ( )(
( ) ( )1
ˆ ˆ 2
3 4
=1
ˆ2 1
1 1 1 1 0
ˆˆ ˆ, , = , ,
2 ˆ ˆ ˆ , , ,
il t r l l
l
d ilt rl l
b
r t e r C l t C l t b
e b dr r n r l t e
l
θφ θ
π
∞
+ Ω −
− Ω− +
+
⎞+ ⎟
⎠
∑
∫
and for region >r d
a b
ISSN 1562-6016. ВАНТ. 2013. №4(86) 28
( ) ( )( ) ( ) ( )(
( ) ( )1
ˆ ˆ 2
3 4
=1
ˆ2 1
1 1 1 1 0
ˆˆ ˆ, , = , ,
2 ˆ ˆ ˆ , , .
il t r l l
l
d ilt rl l
b
r t e r C l t d C l t
e d dr r n r l t e
l
θφ θ
π
∞
+ Ω −
− Ω−
+
⎞+ ⎟
⎠
∑
∫
The application of the conditions (3) to the above
solutions gives the inhomogeneous equations for
( )3 ,C l t and ( )4 ,C l t ,
( ) ( ) ( ) ( )
( ) ( ) ( )
3 4
3 2
2 2 2
4
3 12 2 2
, ,
,
,
= , 1 1 , ,
2
l
l l
pe
l l l
ce
C l t C l t
il d C l t
t R
C l td bi C l t f l t
R d d
ω
ω
∂ ⎛ ⎞
+ Ω +⎜ ⎟
∂ ⎝ ⎠
⎡ ⎛ ⎞ ⎤⎛ ⎞
− − + +⎢ ⎜ ⎟⎜ ⎟ ⎥⎜ ⎟⎢ ⎝ ⎠ ⎦⎝ ⎠⎣
( ) ( )( ) ( )
2
24
3 4 2= , , , ,
2
pe l
ce
C
i C l t b C l t f l t
t
ω
ω
∂
− + +
∂
(32)
where functions 1,2f , determines the effect of the initial
perturbations of the electron density introduced by
solution (10) into the boundary conditions (3), and are
equal to
( ) ( ) ( )
2
1̂
1 1 1 1 0
2 2 2
1
1 2 2 2
1
2
1
1 2
1
2 ˆ ˆ, = , ,
2
ˆ 1
ˆ
ˆ 1 1 ,
ˆ
d ilt rpe
b
ce
l
l
l
l
ef l t i dr n r l t e
l
d b br l
R r d
br l
r
ω π
ω
− Ω
−
+
⎧ ⎡ ⎤⎛ ⎞⎪× − − −⎢ ⎥⎨ ⎜ ⎟
⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩
⎫⎡ ⎤⎛ ⎞ ⎪+ + −⎢ ⎥⎬⎜ ⎟
⎢ ⎥⎝ ⎠ ⎪⎣ ⎦⎭
∫
(33)
( ) ( )
( )
2
1
2 1 1 1 1 0
2
1̂
2
1
2 ˆ ˆ ˆ, = , ,
2
1 1 .
ˆ
dpe l
b
ce
ilt r
ef l t i dr r n r l t
l
be l
r
ω π
ω
+
− Ω ⎛ ⎞⎛ ⎞
× + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∫
(34)
The system (32) - (34) compose second initial value
problem in the investigation of the stability cylindrical
annular plasma column, the solution of which gives
complete linear description of the temporal evolution of
the diocotron instability. The solutions to system (32)
for ( )3 ,C l t and ( )4 ,C l t with 2l ≥ are obtained
straightforwardly and are given by
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 1
2 3
, =
ˆ , ,
i l t l t
i l t l t
C l t c l e
c l e C l t
ω γ
ω γ
− +
− −+ +
(35)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
4 1 1
2 2 4
, =
ˆ , ,
i l t l t
i l t l t
C l t c l a e
c l a e C l t
ω γ
ω γ
− +
− −+ +
(36)
where
( )
12 2
2
1,2 2 2 2
2
= 1 1 1 .
l
l ce
l
pe
d ba d i l l
R d
ω
γ
ω
−⎛ ⎞⎛ ⎞⎛ ⎞
± − −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠
The first two terms in Eqs. (35), (36) describe the
modal temporal evolution with growth rate ( )lγ (16),
of the initial perturbations of the electrostatic potential
on the boundary surfaces at =r b and =r d , which are
determined by constants 1c and 2c . The functions
( )3
ˆ ,C l t and ( )4
ˆ ,C l t are
( ) ( )
( )( ) ( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
2 2 2 2
3 2 2
1 1
1 2 1 1 1 1
0
1
2 1 1 2 1
ˆ , = 1 1
4
, ,
, , ,
l l
pe
l
ce
i l t t l t t
t
l t t
i d d bC l t l
l R d
tdt e f l t a f l t e
a f l t f l t e
ω γ
γ
ω
ω γ
−
− − − −
−
⎛ ⎞⎛ ⎞
− − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎧ ⎡× −⎨ ⎢⎣⎩
⎤+ − ⎥⎦
∫ (37)
( ) ( )
( )( ) ( ) ( )( ) ( )( )
2 2 2 2
4 2 2
1 1
1 2 2 1 1 1 1
0
ˆ , = 1 1
4
, ,
l l
pe
l
ce
i l t t l t t
t
i d d bC l t l
l R d
tdt e a f l t a f l t eω γ
ω
ω γ
−
− − − −
⎛ ⎞⎛ ⎞
− − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎧ ⎡× −⎨ ⎢⎣⎩∫
( ) ( )( ) ( )( ) }1
1 2 1 1 2 1, , .l t ta a f l t f l t eγ − ⎤+ − ⎥⎦
(38)
The functions ( )3
ˆ ,C l t , ( )4
ˆ ,C l t for any values of l
introduce the non-modal modification of the modal
evolution, that arises from the initial perturbations of
the electron density, which are sheared due to the
rotation of electron column with inhomogeneous
angular velocity ( )r̂Ω . Now, the solution for the
electrostatic potential in region < <b r d may be
presented in a simple form,
( ) ( ) ( ) ( )(0) (1) (2)
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , = , , , , , , .r t r t r t r tφ θ φ θ φ θ φ θ+ + (39)
Here ( )(0)
ˆˆ, ,r tφ θ is determined by a general
solution of the homogeneous system (32),
( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
ˆ ˆ
(0)
=2
1 2
ˆˆ, , =
ˆ
i l t il t r
l
l t l t l
r t e
c l e c l e r
ω θ
γ γ
φ θ
∞
− + + Ω
−⎡× +⎣
∑
( ) ( ) ( ) ( )( )1 1 2 2 ˆ ,l t l t la c l e a c l e rγ γ− − ⎤+ + ⎦ (40)
where the Fourier harmonic with =1l , which is stable
in the geometry considered, is omitted,
( ) ( )( ) ( ) ( )( )ˆ ˆ
(1) 3 4
=1
ˆ ˆ ˆˆ ˆ ˆ, , = , , ,il t r l l
l
r t e C l t r C l t rθφ θ
∞
+ Ω −+∑ (41)
( ) ( )( )
( ) ( )
( ) ( )
ˆ ˆ
(2)
=1
ˆ
1̂
1 1 1 1 0
1
ˆ1 1
1 1 1 1 0ˆ
2ˆˆ, , =
ˆˆ ˆ ˆ , ,
ˆ
ˆ
ˆ ˆ ˆ , , ,
ˆ
il t r
l
l
r ilt r
b
l
d ilt r
r
er t e
l
rdr r n r l t e
r
r
dr r n r l t e
r
θ πφ θ
∞
+ Ω
− Ω
− Ω
⎛ ⎛ ⎞
⎜× ⎜ ⎟⎜ ⎝ ⎠⎝
⎞⎛ ⎞+ ⎟⎜ ⎟ ⎟⎝ ⎠ ⎠
∑
∫
∫
(42)
is formed by the initial perturbations in Eq. (9). The
exact solution (41), (42) are valid for any time, at which
the linear theory of the diocotron instability is valid.
The integration of (2)φ on time by parts displays the
decay of ( )(2) , ,r tφ θ as 1t− for ( )( ) 1
t l d
−
Ω? . The
obtained asymptotics reveals that the origin of this non-
modal time dependence, which is attributed usually to
the continuous spectrum, is the non-modal effect of the
continuous shearing of the initial disturbance of the
electron density determined by ( )1̂ilt re− Ω function in
Eq. (42). For the better understanding the contents of
ISSN 1562-6016. ВАНТ. 2013. №4(86) 29
Eqs. (37), (38) it is instructive to obtain the large time,
( )( ) 1
t l d
−
Ω? , asymptotics for coefficients ( )3
ˆ ,C l t
and ( )4
ˆ ,C l t . The integration of (37) on time, in which
only the exponentially growing terms are retained,
yields ( ) ( ) ( )3 31 32
ˆ ˆ ˆ, = , , ,C l t C l t C l t+ where
( ) ( )
( ) ( )( )( )
( ) ( )
( ) ( ) ( )
1
4 2
0
31 2
ˆ0
1 1 0
1
1
2 2 2
1
2 1 2 2 2
1
2
1 2
1 2 2
1
ˆ , =
4
ˆ , ,
ˆ
ˆ
ˆ 1
ˆ
ˆ 1 1 1
ˆ
l
i l l t tpe
ce
ilt r
d
b
l
l
l
l
l
i e d
C l t e
l l
n r l t e
dr
l l r i l
d b ba r l
R r d
abr l
r R
ω γπ ω
ω γ
ω γ
−
− − −
− Ω
−
+
×
− Ω +
⎡ ⎛ ⎞⎛ ⎞
× − − −⎢ ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎝ ⎠⎝ ⎠⎣
⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − + ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎥⎝ ⎠⎝ ⎠ ⎦
∫
(43)
( )
( )
( ) ( ) ( )
( ) ( ) ( )
22 1 2
32 1 02 2
2 2
2
2 2 2 2
ˆ , = , ,
4
1 1 1 1
l
pe ilt d
ce
l
l l
l l
ed dC l t n d l t e
ll t b
ad ba d d l
R d R
l l d i l
ωπ
ω γ
ω γ
− −
− Ω
−
⎧
⎨
⎩
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + + − +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦×
− Ω +
( )
( ) ( )
( )
2 2 2
1 0
22 2 2
22
2
, ,
1 1
1 .
l
l
l
l
l
n b l t b d ba b l
l i l d R d
a
b O t
R
ω γ
−
−
⎡ ⎤⎛ ⎞⎛ ⎞
− − − −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎫⎛ ⎞− + +⎬⎜ ⎟
⎝ ⎠⎭
(44)
The same asymptotic is for
( ) ( )4 1 3
ˆ ˆ, = ,C l t a C l t (45)
experience the power-law decay with time (as 1t− in the
non-modal parts of the perturbed potential (39)
( )(2) , ,r tφ θ and ( )32
ˆ ,C l t ). Only the exponentially
growing with time function ( )31
ˆ ,C l t , presented by
Eq. (43), survives. Changing ( )ˆ ˆ =t rθ θ+ Ω in Eq. (43)
with ( )31
ˆ ,C l t determined by Eq. (43), we obtain the
observed in the laboratory frame the modal presentation
for ( )(1) , ,r tφ θ as for the normal unstable diocotron
wave. It follows from the last expression, that the
relative phase difference ε of the edge surface waves,
determines as 1 1| | ia a e ε= becomes constant and is
equal to that corresponds to the formation of the phase
locked state for the edge surface waves. This result
reveals, that the accounting for the initial perturbations
of the electron density does not destroy of the phase
locked configuration. The performed analysis displays,
that in spite of the similar spatial and temporal
dependencies with normal unstable diocotron mode,
solution actually is not a normal mode. As a solution of
the inhomogeneous system it is a forced wave, which is
resulted from the interaction of separate spatial mode of
the electrostatic potential, formed by the rotating initial
perturbation of the electron density, with the unstable
modal diocotron wave.
This work was funded by National R&D Program
through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science
and Technology (Grant No. 2012-M1A7A1A02-
034918).
REFERENCES
1. R.C. Davidson, Hei-Wai Chan, Chiping Chen, and
S. Lund // Rev. Modern Physics. (63). 1991, p. 341.
2. G. Stancari, A. Valishev, G. Annala, G. Kuznetsov,
V. Shiltsev, D.A. Still, and L.G. Vorobiev // Phys.
Rev. Lett. (107). 2011, p. 084802.
3. J. Petry, Journal Astr. and Astrophys, (503). 2009, 1.
V.V. Mikhailenko, Hae June Lee, V.S. Mikhailenko
// Problems of Atomic Science and Technology.
Series «Phys. Plasmas» (19). 2012, p. 082112.
4. N.A. Bakas, P.J. Ioannou // Physics of Fluids (21).
2009, p. 024102.
Article received 04.04.2013.
ПОДХОД СДВИГОВЫХ МОД К ТЕОРИИ ДИОКОТРОННОЙ НЕУСТОЙЧИВОСТИ
ЦИЛИНДРИЧЕСКОГО СЛОЯ ЭЛЕКТРОНОВ
В.В. Михайленко, Хай Джун Ли, В.С. Михайленко, Н.А. Азаренков
Временная линейная эволюция диокотронной неустойчивости цилиндрического слоя электронов, кото-
рая возбуждается широм равновесной скорости электронов в скрещенных внешнем магнитном и собствен-
ном электрическом полях, исследуется используя обобщение методологии сдвиговых мод на цилиндриче-
скую геометрию. Этот подход не использует спектральное преобразование по времени и дает решение на-
чальной задачи для любого времени. Эволюционный процесс приводит к образованию конфигурации с
фазовой синхронизацией взаимно растущих собственных и вынужденных мод.
ПІДХІД ЗСУВНИХ МОД ДО ТЕОРІЇ ДІОКОТРОННОЇ НЕСТІЙКОСТІ
ЦІЛІНДРИЧНОГО ШАРУ ЕЛЕКТРОНІВ
В.В. Михайленко, Хай Джун Лi, В.С. Михайленко, М.О. Азарєнков
Часова лінійна еволюція діокотронної нестійкості циліндричного шару електронів, яка збуджується ши-
ром рівноважної швидкості електронів у схрещених зовнішньому магнітному та власному електричному
полях, досліджується використовуючи узагальнення методології зсувних мод на циліндричну геометрію.
Цей підхід не використовує спектральне перетворення по часовій змінній і дає розв’язок задачі на початкові
дані для любого часу. Еволюційний процес веде до утворення конфігурації з фазовою синхронізацією взає-
мно зростаючих власних та вимушених мод.
|