Non-linear phenomena connected with propogation of the drift waves into plasma
Non-linear heat and particle transport and restriction of wave amplitudes are discussed. Non-linear transformation of drift wave is considered under influence of outer non-uniform static electric field.
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2006 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Цитувати: | Non-linear phenomena connected with propogation of the drift waves into plasma / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 112-114. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859800214842376192 |
|---|---|
| author | Khvesyuk, V.I. Chirkov, A.Yu. |
| author_facet | Khvesyuk, V.I. Chirkov, A.Yu. |
| citation_txt | Non-linear phenomena connected with propogation of the drift waves into plasma / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 112-114. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Non-linear heat and particle transport and restriction of wave amplitudes are discussed. Non-linear transformation of
drift wave is considered under influence of outer non-uniform static electric field.
|
| first_indexed | 2025-12-07T15:12:17Z |
| format | Article |
| fulltext |
112 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 112-114
NON-LINEAR PHENOMENA CONNECTED WITH PROPOGATION OF
THE DRIFT WAVES INTO PLASMA
V.I. Khvesyuk*, A.Yu. Chirkov
Bauman Moscow State Technical University, 2nd Baumanskaya Str., 5, 105005 Moscow, Russia,
*e-mail: khves@power.bmstu.ru
Non-linear heat and particle transport and restriction of wave amplitudes are discussed. Non-linear transformation of
drift wave is considered under influence of outer non-uniform static electric field.
PACS: 52.35.Qz; 52.55.Lf; 52.25.Fi
1. PLASMA DIFFUSIVITY FORMED BY
DRIFT WAVE
The past decades of research in high temperature
magnetized plasma has shown that transport of particles
and energy across the magnetic field 0B is largely
controlled by low-frequency drift waves [1]. In spit of
existence a lot of different formulas for plasma diffusivity
usually the reference formula is the Bohm diffusivity
0eB
T
D e
B = .
For estimations particles transport in tokamak plasma
is used often so-called the gyro-Bohm diffusivity
=
0eB
cT
L
D e
n
s
gB
ρ
.
Here factor ns L/ρ <<1 ( sρ is the ion cyclotron radius,
)/)(/1( 00
1 dxdnnLn =− is the density gradient scale.
In this work the simple model of the plasma
diffusivity is proposed. The model is based on non-linear
analysis of particles dynamics under the influence of the
drift waves. Collisionless motionless magnetized slab
plasma with ie TT = is considered. It is supposed that the
uniform magnetic field is parallel to z-axis. Outward
electric field is absent. Low ion temperature approach is
used: finite size of Larmor radius is neglected. Plasma
beta is assumed to be β<<1. The only mode is considered.
The wave amplitude is supposed to be enough large and
constant. Non-linearity of consideration presented below
follows from taking into account of particles moving
along magnetic force lines. It is the main feature of this
work. Drift waves propagate in two directions
simultaneously along magnetic force lines and across
them. Consideration longitudinal particles moving in
combination with across oscillations of drift waves allows
to explain phenomenon of the collisionless diffusivity in
the magnetized plasma and derive corresponding formula
for plasma diffusivity across magnetic field.
Originally we proceed from usual consideration of
drift waves picture. It is harmonic distribution of
electrostatic field and perturbed plasma density. We
present wave along density gradient (x-axis) as sum of
narrow layers so that in every layer one can suppose
perturbed plasma density is constant. But in general case
it is incorrectly.
Then we take into account an added non-linear
process in the plane xOz. Forming of drift waves results in
non-uniformity of the perturbed plasma density along of
the magnetic field force line (Oz) because there is plasma
of different layers with different perturbed plasma
densities. Consequently it is possible to take into account
process of mutual penetration particles between
neighboring layers along force lines. Therefore
collisionless transfer of particles from domains of high
density to domains of low density arises because of
influence of the drift waves.
So we consider the drift wave picture in a certain
moment in plane xOz. There are zero line and harmonic
lines. Harmonic lines share three narrow layers by
thickness h . It is supposed average plasma perturbed
density in this moment are 0nδ (middle layer),
]/)([ 001 dxndhnn δδδ += (upper layer) and
]/)([ 001 dxndhnn δδδ −=− (lower layer). Therefore
perturbed plasma density along magnetic field lines is
non-uniform. The particles transport is considered along
straightforward layers with thickness h parallel by
magnetic field.
The particles transport between harmonic layers number
(0), (1) and (-1) is estimated. One is produced because of
the ion collisionless motion (sound velocity
is iiti mT /2=υ ). Thickness h is chosen from non-
equality h << zλ (wave length along magnetic field
direction). Choose some part inside harmonic layer (in
considered case inside layer (0), length hl 3≈ ). This
choice plays important role because one determines time
of ions penetration of ions into chosen part harmonic
layer. Taking into account relation zk/ω >> Tiυ it is clear
that penetration into chosen part from neighboring layer is
not much because of condition that hl 3≤ . It means that
perturbation of plasma density in chosen part connected
with this penetration is no much, too. Consequently below
we restrict ourselves by simple linear relations.
Features of change between chosen part and
neighboring layer depends on location of chosen part
from layer. It is important to consider three cases. First
case is: chosen part belongs to inclined domain of
sinusoidal layer. In this case plasma density doesn’t
change because increasing of density through particles
mailto:khves@power.bmstu.ru
113
exchange (for example between layers number (0) and
(1)) is compensated by decreasing of density through
contrary particles exchange (in our case between layers
number (0) and (-1)). Second case is: chosen part is in
maximum domain. In this case neighboring layer of
chosen part is the only the layer with density more than
density inside chosen layer (for example layers number
(0) and (1)). As a result plasma density inside chosen
layer increases. Third case is: chosen part is in minimum
domain. It is the particles exchange between chosen part
and layer number (-1). As a result of such a particles
change plasma density inside chosen part decreases.
Therefore real transport across magnetic field can be
realized because of particles exchange between layers in
maximum and minimum domains. Estimation of the
particles flow presents following result
−=Γ ⊥ dx
ndD )(δ ,
were
0
22 2
eB
cT
L
hhD
nyλ
π
ξ
ω
ξ
== ∗⊥ , Tizk υωξ /= .
More convenient expression is
−=Γ ⊥ dx
dn
D 0~
with the diffusivity
B
yy
D
dx
nd
n
h
eB
cT
dx
nd
n
hD
=
=⊥
)(2)(2~
0
2
00
2 δ
λ
π
ξ
δ
λ
π
ξ
.
It is necessary to note some features of the last two
formulas. Firstly Γ is diffusive flow. Secondly ⊥D~
contains 0/ eBT like to Bohm diffusivity. Thirdly factor
before Bohm diffusivity contains both the wave
parameters ( )/)(,/, dxndk zy δωλ and the plasma
parameters ( 0, nTiυ ). Finally obviously quantitative value
of factor previous BD in formula for ⊥D~ is much less
than 1. But disadvantage of this formula for ⊥D~ is
availability of indeterminate value h .
Therefore above presented analysis of transport
processes in plasma is based on consideration of ion
dynamics both under influence wave drift moving and
free moving along magnetic field direction.
Below on base of presented model features of transport of
impurities are discussed. It is well-known so-called
“ballistic” diffusivity [2]. “Ballistic” means velocity
propagation of impurities inside plasma much more than
velocity estimated using usual diffusive factors obtained
from solution of Boltzman equation. If impurities density
0in << 0n ( 0n is plasma density), then their velocity
propagation is determinate by wave processes similar to
diffusivity considered above. Consequently diffusive
velocity turns out is very large.
Other phenomenon recovered under investigation of
propagation of impurities is so-called non-local diffusivity
[3]. It means impurities flows don’t write as usual
diffusive flows. This phenomenon can be explained by
non-linearity of spatial distribution of impurities inside
plasma. Then in general case expression of diffusive flow
is
dx
dn
D i
i
0~
⊥−=Γ .
2. INFLUENCE OF OUTER ELECTRIC FIELD
The influence of the strong non-uniform electric field
perpendicular to magnetic field on drift wave propagating
inside plasma is usually analyzed as an it’s influence on
growth rate γ of a sinusoidal wave [4]. In agreement with
[4] γ decreases because of arising of non-uniform flows of
the background plasma under influence of perpendicular
magnetic and electrical fields. But such an approach not
takes into account possibility of non-linear influence of
electric and magnetic fields on wave shape that is to say
deviation one from sinusoidal shape.
Plasma waves arise as a result appearing of ensemble
of perturbed ions and electrons. Obviously outer
stationary electromagnetic field affects both on perturbed
particles and on the background plasma. Taking into
account of this influence is content of the presented work.
Usual approximation of outer electrical field is
( )E x xα= , where x is the co-ordinate coinciding with
direction of deviation of wave particles and A x A− ≤ ≤ ,
A is the wave amplitude that supposed by constant.
Influence of electrical field ( )E x and magnetic field B on
wave particles leads to appearing of particle drift velocity
( ) / /V E x B x Bα= = .
Direction of this velocity coincides with the wave
propagation direction for 0 x A≤ ≤ and it has opposite
direction for 0A x− ≤ ≤ . Consequently the wave shape is
change in time. Like situation occurs in hydrodynamics: it
is problem of increasing wave inclination and following
for one toppling over [5].
There are two distinctions between problem [5] and
considered here problem.
1) In [5] V is linear increasing function of y in bound
0 / 2y λ≤ ≤ (y is co-ordinate along direction of wave
propagation). In this work V is non-linear function and
maximal velocity corresponds to y for x=A.
2) In [5] only upper part of wave is considered. In this
work both upper and lower parts of wave are considered.
As above was indicated in lower part of drift velocity has
opposite direction to upper part direction.
Let’s consider drift wave in the plasma confined in
uniform magnetic field B directed along z-axis and non-
uniform static electric field E(x) directed along x-axis.
Wave propagates along y-axis. In this case particle drift
displacement along x-axis ),( tyXX = is proportional to
the perturbed density n~ and wave potential ϕ~. One can
suppose
CX=~ϕ , (1)
where C some generally complex constant, which takes
into account phase shift between n~ and ϕ~ in drift wave.
114
Fig. 1. Wave profile for small growth rate
Fig. 2. Wave profile for large growth rate
In the system of coordinates moving together with the
wave, equation of guiding center drift are:
yBdt
dX
∂
∂
−=
~1 ϕ , (2)
B
XE
dt
dy )(
−= . (3)
Using (1), (2), (3) and operator
dt
dy
y
X
t
X
dt
dX
∂
∂
+
∂
∂
= ,
we obtained equation
[ ] 0)( =
∂
∂
−+
∂
∂
y
XXVC
t
X
E , (4)
where BXEXVE /)(( = .
Note, Eq. (4) practically coincides with non-linear
drift wave equation [6]. Besides such an equation
describes an perturbation in a beam of non interacting
particles. Non-linear wave profiles are illustrated in Figs.
1 and 2, where dashed curves are initial linear wave
profiles. One can see formation of the steep wave
inclination.
In this paper Sec. 1 is written by V.I. Khvesyuk, Sec.
2 is written by V.I. Khvesyuk and A.Yu. Chirkov.
REFERENCES
1. W. Horton // Rev. Mod. Phys. 1999, v. 71, p. 735.
2. B.P. van Milligen et al. // Nucl. Fus. 2002, v. 42,
p. 787.
3. K.W. Gentle et al. // Phys. Plasmas. 1997, v. 4, p. 599.
4. M.Artun, W.M.Tang // Phys. Fluids. 1992. v. B4.
p. 1102.
5. G.M. Zaslavsky, R.Z. Sagdeev. Introduction in non-
linear Physics. Moscow: Nauka, 1988 (in Russian).
6. H. Tasso // Phys. Lett. A. 1997, v. 232. p. 247.
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|
| id | nasplib_isofts_kiev_ua-123456789-81796 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:12:17Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Khvesyuk, V.I. Chirkov, A.Yu. 2015-05-20T16:56:14Z 2015-05-20T16:56:14Z 2006 Non-linear phenomena connected with propogation of the drift waves into plasma / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 112-114. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Qz; 52.55.Lf; 52.25.Fi https://nasplib.isofts.kiev.ua/handle/123456789/81796 Non-linear heat and particle transport and restriction of wave amplitudes are discussed. Non-linear transformation of drift wave is considered under influence of outer non-uniform static electric field. In this paper Sec. 1 is written by V.I. Khvesyuk, Sec. 2 is written by V.I. Khvesyuk and A.Yu. Chirkov. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Non-linear phenomena connected with propogation of the drift waves into plasma Article published earlier |
| spellingShingle | Non-linear phenomena connected with propogation of the drift waves into plasma Khvesyuk, V.I. Chirkov, A.Yu. Basic plasma physics |
| title | Non-linear phenomena connected with propogation of the drift waves into plasma |
| title_full | Non-linear phenomena connected with propogation of the drift waves into plasma |
| title_fullStr | Non-linear phenomena connected with propogation of the drift waves into plasma |
| title_full_unstemmed | Non-linear phenomena connected with propogation of the drift waves into plasma |
| title_short | Non-linear phenomena connected with propogation of the drift waves into plasma |
| title_sort | non-linear phenomena connected with propogation of the drift waves into plasma |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81796 |
| work_keys_str_mv | AT khvesyukvi nonlinearphenomenaconnectedwithpropogationofthedriftwavesintoplasma AT chirkovayu nonlinearphenomenaconnectedwithpropogationofthedriftwavesintoplasma |