Beam instability caused by stochastic plasma density fluctuations
Beam instability in a plasma with the density varying stochastically with respect to space or time is investigated. In case of spatial variations, the system of equations in moments of arbitrary order is obtained and analyzed. It is shown that each moment in the sequence grows faster than the preced...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Beam instability caused by stochastic plasma density fluctuations / A.V. Buts V.A. Chatskaya, O.F. Tyrnov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 128-130. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860152744481914880 |
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| author | Buts, A.V. Chatskaya, V.A. Tyrnov, O.F. |
| author_facet | Buts, A.V. Chatskaya, V.A. Tyrnov, O.F. |
| citation_txt | Beam instability caused by stochastic plasma density fluctuations / A.V. Buts V.A. Chatskaya, O.F. Tyrnov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 128-130. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Beam instability in a plasma with the density varying stochastically with respect to space or time is investigated. In case of spatial variations, the system of equations in moments of arbitrary order is obtained and analyzed. It is shown that each moment in the sequence grows faster than the preceding one. As a result an intermittent character of development of the instability arises, and the appearance of some critical length within which the amplification of a regular signal is yet possible. In case of temporal variations, the maximum time interval of signal destitution due to fluctuation interference is estimated.
|
| first_indexed | 2025-12-07T17:51:45Z |
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128 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 128-130
UDC 533.922
BEAM INSTABILITY CAUSED BY
STOCHASTIC PLASMA DENSITY FLUCTUATIONS
A. V. Buts
NSTC "Kharkiv Institute for Physics and Technology ", Kharkiv, Ukraine
E-mail: abuts@bigfoot.com
V. A. Chatskaya, O. F. Tyrnov
National Kharkiv V. Karazin University, Kharkiv, Ukraine
E-mail: Oleg.F.Tyrnov@univer.kharkov.ua
Beam instability in a plasma with the density varying stochastically with respect to space or time is investigated.
In case of spatial variations, the system of equations in moments of arbitrary order is obtained and analyzed. It is
shown that each moment in the sequence grows faster than the preceding one. As a result an intermittent character of
development of the instability arises, and the appearance of some critical length within which the amplification of a
regular signal is yet possible. In case of temporal variations, the maximum time interval of signal destitution due to
fluctuation interference is estimated.
Introduction
The theory of interaction of a beam with plasma is
developed (see, for example, [1]). The interaction
between a beam and plasma gives rise to instabilities
that could lead to microwave amplification. At the time,
fluctuations ever present in plasma also rise. Using
plasma-beam instability for amplifying signals or
heating plasma, it is necessary to know a ratio between
the growth rate of a regular signal component and that of
fluctuations. Beguiashvily et al. [2] and Virchenko et al.
[3] studied plasma-beam interaction in a plasma with the
number density varying stochastically in the x direction.
Their analysis was restricted by the correlation theory of
instability, i.e. the dynamics of variations of the first and
second moments.
This study is dedicated to the investigation of a beam
instability in a plasma with the number density varying
stochastically with respect to space or time. In case of
spatial variations, as distinct from [3], the system of
equations in moments of arbitrary order is obtained and
analyzed. It is shown that each moment in the sequence
grows faster than the preceding one. As a result of such
a dependence of the growth rate upon the number of a
moment, an intermittent character of development of the
instability arises [4], as well as the appearance of some
critical length of interaction space within which the
amplification of a regular signal is yet possible. The
latter conclusion is due to the fact that the growth rate of
the second moments is more than two times the growth
rate of a regular component of the signal. In case of
stochastic temporal variations, the equations describing
the dynamics of first and second moments are obtained,
their growth rates are found, and the maximum time
interval of signal destitution due to fluctuation
interference is estimated.
Basic equations
Let an electron beam with a neutralizing ion
background have a number density nb and velocity υ b
in the x direction in the plasma. Perturbations in the
beam number density ~nb and velocity ~υ b satisfy the
equation of motion and the continuity equation
( )∂
∂
υ υ ∂
∂
υ
t x
e
m
E x tb b b
~ ~ ,+ = − , (1)
( ) 0~~~ =++ bbbbb nn
x
n υυ
∂
∂ (2)
where ( )E x t, is the electric field intensity.
Perturbations in the ambient plasma number density
~np and velocity ~υ p satisfy the equations
( ) 0~ =+ ppp n
x
n
t
υ
∂
∂
∂
∂
, (3)
( )∂
∂
υ
t
e
m
E x tp
~ ,= − (4)
where n p is an unperturbed plasma number density
which is a stochastic function of time or space
coordinates, and its probability properties are known.
The electric field intensity and variations in beam and
ambient plasma number densities satisfy Maxwell's
equation
( ) ( )∂
∂
π
x
E x t e n np b, ~ ~= − +4 . (5)
Spatially inhomogeneous plasma
Let all variables have the functional form
( )exp − i tω , and the plasma be spatially inhomogeneous,
n n xp p= ( ) . On rearranging and choosing boundary
conditions in such a way that the constant of integration
turns zero, from (1) to (5), we obtain the equations in the
Fourier transformed electric field:
( )i
x
x E Eb bω υ ∂
∂
ε ωω ω+
+ =
2
2 0 , (6)
2
2
1)(
ω
ω
ε px −= ,
m
xne p
p
)(4 2
2 π
ω = ,
m
ne b
b
2
2 4π
ω = .
Substituting Eω(x) into (6) from
( ) ( )D x x x i x bω ε ω ω υω, ( , ) ( ) exp≡ Ε , we obtain the
129
equation similar to (3) for a spatial oscillator with a
stochastically changing frequency ω υ εb b x2 2/ ( )
d D
d x x
Db
b
2
2
2
2 0+ =
ω
υ ε ( )
. (7)
Let us consider, as an example, a case of ε 〈 0 .
Suppose plasma number density fluctuations are
n n z xp p= +0 11( ( )) , and z1 is a stationary Gaussian
process with a zero mean; then, considering that the
amplitude of fluctuation is small, substituting
z z x p p≡ 〈〈1 0
2 2 1( ) ( )ω ω ε into (7), and changing the
variables ε ω ωp p0 0
2 21= − , ω ωp p0
2 2≡ 〈 〉 , and
τ ω υ ε≡ b b px 0 , we obtain the set of equations of
the first order
! ,D u= − ! ( ( ))u z D= − +1 τ . (8)
From Equation (8), a set of equations in moments of
any order m could be obtained. To achieve this,
multiplying both sides of the first equation in (8) by
u Dm n n− −1 , and the second equation by D un m n− −1 , adding
these equations and performing the ensemble averaging,
we obtain the set of equations of the order n in moments
〉+〈−−
−〉〈−=〉 ′〈
−−+
−+−−
11
11
)1()( nmn
nnmnnm
uDznm
DunDu τ (9)
To split the correlations 〈 〉+ − −zD un m n1 1 , we use the
method of variational derivatives [5] and the relation
derived by this method
τ
τδ
τδτ d
z
zRztztzRtz
t )(
)]([)()()]([)( 〉〈=〉〈 ∫ (10)
where R z[ ] is an arbitrary functional on z, and z is a
Gaussian process with a zero mean. Substituting (10) in
(9), and calculating respective variational derivatives,
we find
〈 − ′〉 = − 〈 − + + 〉 − − + + − − 〉 +
+ − − − 〈 − − + 〉
um nDn n um n Dn m n z Dn um n
B m n m n um n Dn
τ
1 1 1 1 1
2
1 2 2
( )( )
( )( )
(11)
where
,
)()(
22
0
2
0
2222
+−−
+−−+−−
≈
≈〉〈〉〈≡〉〈 ∫
nnm
nnm
t
nnm
Dur
DuztzdDuB
σ
ττ
σ 0
2 is the variance, r0 is the dimensionless radius of
correlation of the stochastic process z( )τ .
To analyze the stability of (11), we take out all the
moments proportional to e λτ . The determinant of the
resulting set of equations, calculated applying Rauss's
algorithm, yields the following recursion relation in the
coefficients of the characteristic equation
( ) ( )Det An n
mλ λ≡ = 0 , A A mm m
0 1
2= = −λ λ, ,
( )A m m m m Bm
2
2
2
1 1
2
= − − − − −
λ λ λ ( ) ( ) ,
A n mn
m ≠ ∀ <0, ,
A
n
A m
n
A
m
n
m n B A
n
m
n
m
n
m
n
m
= −
−
−
−
−
−
−
− +
− −
−
λ
1 2
3
1
1
2
1 1
2
( ) ,
(12)
In particular, if m =1 (first moments), the
characteristic equation takes the form
λ 2 1 0− = , (13)
and if m = 2 (second moments)
λ λ3 4 2 0− − =B . (14)
Obviously, the second moment growth rate is more
than two times the first moment growth rate. Similarly to
[3], we define the variance in the dimensionless form
( )∆ = 〈 〉 − 〈 〉 〈 〉D D D2 2 2 . (15)
In order to amplify a signal, the magnitude of this
quantity should be much less than the unity (∆〈〈1),
otherwise the signal will be destructed by fluctuation
interference. Substituting the first and second moments
from (13) and (14) into (15), and returning to
dimensional variables, from the condition ∆ 〈〈 1, we
obtain the following expression for the critical length of
interaction space within which the amplification of a
regular signal is yet possible:
( )x x rm b p b p〈〈 = 4 4 2
0
2 2
0 1
2
1ω υ ω ω ε σ (16)
where σ 1
2 is the variance, r 1 is the radius of
correlation of the stochastic process z x( ) .
If the magnitude of the amplitude of fluctuations
equals zero (B=0), using the recursion relation for the
coefficients, it can be shown that both ( )Det mm = 0 ,
and λ =m is the maximum root of Equation (12).
Hence, even if the amplitude of fluctuations can be
neglected, the difference between growth rates of two
successive moments equals the unity.
Of each two successive moments, the next grows
faster than the former, but the dimensional variance does
not. The nonzero amplitude of fluctuations leads to an
additional growth in a difference between the growth
rates and, therefore, to an increase in the variance. As
Molchanov et al. [4] have shown, such a peculiarity of
growth of the moments indicates an intermittent
character of oscillator motion. Note that such a growth
of higher harmonics is characteristic of systems
described by Langevin's equations of the first order [6],
and, therefore, in accordance with [4], intermittent
motion should also exist in such systems.
Plasma with temporal fluctuations in the
density
Let us consider a case of the plasma with the number
density varying stochastically with respect to time
( )n n z tp p= +0 1 ( ) where z t( ) is a stationary Gaussian
process with a zero mean. Because the expressions
obtained are cumbersome, we restrict our investigation
to a correlation approximation. Assuming all unknowns
to vary harmonically with the x coordinate e i k x− ,
introducing the dimensionless variables
τ υ α ω υ β ω υ≡ ≡ ≡k t k kb p b b b, ,0 , and
eliminating the variable E by using its value from (5),
from (1) to (5) we obtain the following set of equations
in spatial and temporal Fourier components of
perturbations of the number density and velocity of a
beam and plasma
130
∂
∂τ
υ
~
~ ~n i n ib
b b+ + = 0 , (17)
∂ υ
∂τ
υ β α
~
~ ~ ~b
b b pi i n i n+ + + =2 0 , (18)
∂
∂τ
υ τ
~
~ ( ( ))
n
i zp
p+ + =1 0 , (19)
∂ υ
∂τ
α β
~
~ ~p
p bi n i n+ + =2 2 0 . (20)
Performing the ensemble averaging of (17) to (20),
splitting the correlation 〈 〉z p
~υ with (10), and taking out
all the variables proportional to the e i− λτ for analysis,
we obtain the set of algebraic equations of the fourth
order in first moments of number density and velocity
perturbations, the determinant of which yields the
classical dispersion relation for a beam-plasma system
222222 )1(,, βλαλβα −−≡∆−≡∆=∆∆ bpbp (21)
which has an unstable solution at points of intersection
of plasma (∆ p = 0) and beam (∆b = 0 ) resonances of
oscillators ( λ δ α= + =1 1, ) with the maximum growth
rate of ( )γ β1
1
2
2 1 3
= . (Under the natural assumption of
ω ωp b0 〉〉 .)
Having obtained a set of equations in second
moments from (17) to (20), and having performed all the
operations mentioned above, we obtain two matrix
equations of the third order
− +
− +
− +
〈 〉
〈 〉
〈 〉
= 〈 〉
〈 〉
λ
β λ
β λ
υ
υ
α
υ
2
1 1 0
2 1
0
2
1
0
2
2
2
2
2
n
n n n
n
b
b b
b
p b
p b
(22)
〉〈
〉〈−=
〉〈
〉〈
〉〈
−
−
−
pb
bp
p
pp
p
n
nnn
nB
υ
β
υ
υ
λα
λα
λ
0
2
0
1
1
2
2
2
2
2
2 (23)
in second moments of number density and velocity
perturbations of beam and plasma oscillators
( B i k b R R≡− υ σ τ σ τ0
2
0
22/ ; , are the variance and
the time of correlation of the stochastic process z( )τ ),
and one matrix equation of the fourth order
〉〈+〉〈
〉〈
〉〈
=
=
〉〈
〉〈
〉〈
〉〈
+−
+−
+−
+−
bbpp
b
p
bp
pb
bp
bp
nn
n
n
n
n
nn
υβυα
β
α
υυ
υ
υ
λβα
λα
λβ
λ
22
22
22
22
2
2
0
10
110
101
0111
(24)
in their cross-correlation coefficients. Corresponding to
Sets (22) to (24), the characteristic equation of the order
ten is similarly to Set (12).
To its analyze, we choose the structure of field under
which the instability of the first moments occurs
( k b pυ ω= ). In this case, the maximum growth rate of
the second moments is localized within the region near
λ = 2 . Expanding the characteristic equation in terms of
small parameters δ ( λ δ= +2 ) and β , we obtain the
expression in the second moment growth rate
γ β β2
2
3
2
3
2
3
2
32 1 2 9= +
B (25)
i.e. the second moment growth rate is two times the first
moment growth rate. Using (15), it is easy to estimate a
maximum time interval over which fluctuations are
unable to destroy a regular signal
( )t t km b b R〈 ≡ −9 2 1 3
0
2( )β ω υ σ τ (26)
where σ 0
2 is the variance, τ R is the correlation time of
the stochastic process z( )τ .
Conclusion
Two kinds of the systems with fluctuations are
investigated by the moment method. The first system has
spatial and the second one – temporal instabilities that
lead to amplification of an initial regular signal. The
presence of fluctuations changes the situation
principally. The fluctuations that always exist in
unstable systems lead to destruction of the regular signal
amplification. The critical length or time determines the
possibility for regular signal amplification. The method
of the variational derivatives allows us to obtain the
solution in absolutely different physical situations and
for quite general model of the fluctuations, as it is
Gaussian random signal. It is naturally to suppose that
the result obtained here has general field of application
and have to be taken into consideration if a system with
distributed interaction is analyzed.
References
1. A. B. Mikhailovskii. Theory of Plasma Instability.
New York, 1974, vol. I, 266 pp.; vol. II, 360 pp.
2. G. A. Beguiashvily, Yu. S. Monin. On the stability of
a charged particle beam in a steady-state
inhomogeneous media. // Trans. Soviet Academy of
Sci. 1969, vol. 55, no. 3, pp. 557–560 (in Russian)
3. Yu. P. Virchenko, R. V. Polovin. On the stochastic
destruction of waves growing in a stochastically
inhomogeneous media. // Ukrainian Journal of
Physics. 1988, vol. 33, no. 12, pp. 1863–1868 (in
Russian)
4. S. A. Molchanov, A. A Ruzmaikin., D. D. Sokolov.
Kinematics of a dynamo in a stochastic stream. //
Ukrainian Journal of Physics. 1985, vol. 30, no. 4,
pp. 593–628 (in Russian)
5. V. I. Klyatskin. Statistical Description of Dynamical
Systems with Fluctuating Parameters. Moscow:
«Nauka», 1975, 239 pp. (in Russian)
6. R. V. Polovin. Applied Theory of Stochastic
Processes. Kharkov: «Vyshcha Shkola», 1982, 102
pp. (in Russian)
BEAM INSTABILITY CAUSED BY
STOCHASTIC PLASMA DENSITY FLUCTUATIONS
A. V. Buts
NSTC "Kharkiv Institute for Physics and Technology ", Kharkiv, Ukraine
E-mail: abuts@bigfoot.com
V. A. Chatskaya, O. F. Tyrnov
National Kharkiv V. Karazin University, Kharkiv, Ukraine
E-mail: Oleg.F.Tyrnov@univer.kharkov.ua
Introduction
Basic equations
Spatially inhomogeneous plasma
Plasma with temporal fluctuations in the density
Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-78546 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:51:45Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, A.V. Chatskaya, V.A. Tyrnov, O.F. 2015-03-18T18:45:28Z 2015-03-18T18:45:28Z 2000 Beam instability caused by stochastic plasma density fluctuations / A.V. Buts V.A. Chatskaya, O.F. Tyrnov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 128-130. — Бібліогр.: 6 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78546 533.922 Beam instability in a plasma with the density varying stochastically with respect to space or time is investigated. In case of spatial variations, the system of equations in moments of arbitrary order is obtained and analyzed. It is shown that each moment in the sequence grows faster than the preceding one. As a result an intermittent character of development of the instability arises, and the appearance of some critical length within which the amplification of a regular signal is yet possible. In case of temporal variations, the maximum time interval of signal destitution due to fluctuation interference is estimated. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Вeams and waves in plasma Beam instability caused by stochastic plasma density fluctuations Article published earlier |
| spellingShingle | Beam instability caused by stochastic plasma density fluctuations Buts, A.V. Chatskaya, V.A. Tyrnov, O.F. Вeams and waves in plasma |
| title | Beam instability caused by stochastic plasma density fluctuations |
| title_full | Beam instability caused by stochastic plasma density fluctuations |
| title_fullStr | Beam instability caused by stochastic plasma density fluctuations |
| title_full_unstemmed | Beam instability caused by stochastic plasma density fluctuations |
| title_short | Beam instability caused by stochastic plasma density fluctuations |
| title_sort | beam instability caused by stochastic plasma density fluctuations |
| topic | Вeams and waves in plasma |
| topic_facet | Вeams and waves in plasma |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78546 |
| work_keys_str_mv | AT butsav beaminstabilitycausedbystochasticplasmadensityfluctuations AT chatskayava beaminstabilitycausedbystochasticplasmadensityfluctuations AT tyrnovof beaminstabilitycausedbystochasticplasmadensityfluctuations |