Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field
Results of numerical simulations of axisymmetric plasma flows in accelerator channel with longitudinal magnetic field are presented. The investigations of two-dimensional flows are carried out within the framework of onecomponent MHD-model and two-component model taking into account the Hall effec...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Cite this: | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field / A.N. Kozlov, A.M. Zaborov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 138-140. — Бібліогр.: 11 назв. — англ. |
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| author | Kozlov, A.N. Zaborov, A.M. |
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| citation_txt | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field / A.N. Kozlov, A.M. Zaborov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 138-140. — Бібліогр.: 11 назв. — англ. |
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| description | Results of numerical simulations of axisymmetric plasma flows in accelerator channel with longitudinal magnetic
field are presented. The investigations of two-dimensional flows are carried out within the framework of onecomponent
MHD-model and two-component model taking into account the Hall effect. It is found that the current
attachments are appeared in case of respective strong longitudinal magnetic field.
|
| first_indexed | 2025-12-07T15:19:58Z |
| format | Article |
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138 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 138-140
FORMATION OF THE CURRENT ATTACHMENTS IN PLASMA
ACCELERATOR CHANNEL UNDER INFLUENCE OF THE LONGITUDINAL
MAGNETIC FIELD
A.N. Kozlov, A.M. Zaborov
Keldysh Institute for Applied Mathematics, RAS, Moscow, Russia, e-mail: ankoz@keldysh.ru
Results of numerical simulations of axisymmetric plasma flows in accelerator channel with longitudinal magnetic
field are presented. The investigations of two-dimensional flows are carried out within the framework of one-
component MHD-model and two-component model taking into account the Hall effect. It is found that the current
attachments are appeared in case of respective strong longitudinal magnetic field.
PACS: 52.30.Cv, 52.59.Dk, 52.65.-y
1. INTRODUCTION
In accelerators, the plasma dynamics is investigated in
various current transport regimes [1-2]. In the electron
current transport regime, the lines of the ion plasma
current lie on the surfaces of the impermeable electrodes
(cathode and anode) and the electrode surfaces are
nonequipotential. In contrast, in the ion current transport
regime, the electrode surfaces are equipotential and
should be permeable to the plasma. Exclusively the Hall
effect governs the self-consistent plasma flow through the
electrodes. Ignoring the Hall effect ( ie VV = ), we arrive
at the regime of impermeable continuous equipotential
electrodes. Under real conditions ( ie VV ≠ ), this regime
can be regarded as degenerate. Most experiments [3-5]
and models [6-7] are based on ion current transport.
In one-fluid model, Hall effect is ignored and problem
does not depend on the electrode polarity. This model
provides a qualitative description of the above regimes.
The paper is aimed at discussing the qualitative features
of rotating plasma flows in a longitudinal magnetic field
in accelerator channels. Applying a longitudinal field
rz HH >> in addition to the traditional azimuthal
magnetic field ϕH opens up new possibilities for
controlling the dynamic processes.
The hierarchism of numerical models implies that, in
the first step, the plasma dynamics should be investigated
in a one-fluid MHD-model with allowance for the finite
conductivity of the medium. Such a model is developed in
[8]. Some aspects of the formulation of the problem and
the results of first numerical experiments within the
framework of two-component MHD-model were reported
in [9]. The corresponding transport coefficients were
obtained in explicit form in [10]. The numerical
experiments were tested against the two-dimensional
plasma flows considered earlier in analytical model [11].
2. ONE-COMPONENT MHD-MODEL
We will consider a two-dimensional axial-
symmetrical plasma flow when the two electrodes profiles
reproduced in Fig. 2 specify channel geometry. In the
presence of a longitudinal field and the arising rotation
whole three components of field and velocity participate
in model. To be specific, we will investigate a plasma
formed from atomic hydrogen when the inertia of
electrons ( mmm ie =<< ). The medium is assumed to be
quasineutral nnn ei == . Within framework of the one-
component approximation ( VVV == ie ) the
construction of the model is based on the traditional
MHD-equations taking into account of conductivity: We
have the following equations in the dimensionless form
( ) 0=•∇+ Vρ
∂
ρ∂
t
; HjV
×=∇+ P
td
d
ρ (1)
2jV νερ =•∇+ P
td
d ; ( ) ( )jHVH
ν
∂
∂
×∇−××∇=
t
Here, TPPP ei ρβ=+= is the total pressure;
( )1/ −= γβε T is the intrinsic energy per unit mass;
Hj ×∇= is the electric current. We will restrict our
attention to the case of a single-temperature mixture
TTT ei =≈ . In accordance with Ohm’s law, the
electrical field is given by the relation
HVjE ×−=ν (2)
The initial quantities are related to the dimensionless
parameters of the problem as follows: 2
00 /8 HPπβ =
is the ratio between the gas-kinetic and magnetic pressure
at entrance; σπν 0
2 4/Re/1 VLcm == is magnetic
viscosity, which is inversely proportional to Reynolds
number with Spitzer conductivity 2/3
0Re Tm σ= .
op cRJH /20 = , pJ - discharge current, 000 2 TnkP = ,
000 4/ ρπHV = - characteristic Alfven velocity.
The equations and boundary conditions define flow
dynamics. The conditions at the channel inlet ( )0=z
correspond to subsonic plasma inflow with ( ) ( )rfr 1=ρ ,
( ) ( )rfrT 2= , where 1f and 2f are known functions.
We will assume that the total electric current flowing in
the system is maintained constant. This generates the
boundary condition at the inlet constrHr == 0ϕ . The
inflow is carried out along the coordinate lines, for
example. Plasma at entrance is no rotating 0=ϕV .
We specify longitudinal field at entrance 0≠zH . For
any β value in the radial-equilibrium case, traditional
conditions 1=ρ , 1=T yield constHH zz == 0 at 0=z .
For 1<<β , the plasma can be injected in an arbitrary
manner, for instance, in accordance with the analytic
model of [13], in which ( ) 22
0 /,0 rrrz ==ρ . In this case,
it is possible to compare results with the analytic solution.
At the outlet 1=z the boundary conditions
correspond to a supersonic plasma flow in transonic case.
mailto:ankoz@keldysh.ru
139
We suppose that the electrodes with given profiles are
equipotential 0=τE and non-penetrable 0=nV ones.
It is necessary additional relation in the presence of
longitudinal magnetic field. Equality 0=nH is typical
and natural condition in plasmadynamics. This condition
leads to conservation law of magnetic flux along channel.
3. TWO-COMPONENT MHD-MODEL
The two-component model and computation of axial-
symmetrical plasma flow in the presence of a longitudinal
magnetic field are based on the MHD-equations taking
into account the Hall effect ( ie VV ≠ ), electrical
conductivity tensor and transport coefficients in magnetic
field depending on the eex τω= [10]. Ohm’s law and
the electrical field can be represented in the form
[ ] 2RRHVjE ++−= 11 ,νA (3)
1A are known functions of eex τω= .
Dimensionless parameters of model: β , ν , ee τω
and 04/ ρπξ Lemc i= is the local parameter
characterizing the Hall effect in the two-fluid model.
In [9] two-component model of two-dimensional flow
corresponds to the ion current transport regime. In this
case on anode equipotential ( 0=τE ) surface the density
( )zρ and azimuthal velocity ( )zVϕ were chosen in
accordance with analytical model to compare the
numerical and analytical solutions. However, this
formulation of problem did not give possibility to
investigate the plasma dynamics for different parameters,
including the longitudinal magnetic field.
The present investigation is aimed at discussing the
numerical model in the ion current transport regime with
the self-consistent plasma flow through the electrodes. In
the absence of longitudinal magnetic field the model with
the self consistent plasma flow was used earlier [6] to
investigate the processes with the traditional azimuthal
component of field ϕH . In the presence of the
longitudinal magnetic field ( )rz HHH <<<<ϕ in the
case of the self-consistent plasma flow it is assumed that
on the equipotential ( 0=τE , 0=ϕE ) electrodes the
jumps and breaks of thermodynamic values are absent.
4. MAIN RESULTS
In the present computations the initial dimensional
parameters correspond to the experiments within the
framework of QSPA program. For example, if
320
0 106.3 −⋅= mn , KT 4
0 103.2 ⋅= , kAJ p 300= ,
mL 6.0= , dimensionless parameters are 005.0=β ,
8.8120 =σ . In addition, as the longitudinal magnetic
field value we will take, for example, 1.00 =zH . This
sufficient small value 0
zH makes it possible to produce
the transonic flow in accordance with the analytical
model. The channel geometry (Fig. 2) and density at inlet
correspond also the analytical investigation.
The steady-state flows are calculated by the relaxation
method for the initial time-dependent equations. As result
of calculations, at a mid-channel the flow velocity passes
through the local velocity of the fast magnetosonic wave.
We can observe the peculiarities of the vector magnetic
field distribution. The value zH increases as a function
of r. It has maxim at the surface of the external electrode
in the neighborhood of the narrowest section of the
accelerator channel. The azimuthal velocity ( )rzV ,ϕ
increases along the radial and axis directions. In other
words, a small longitudinal magnetic field leads to the
increasing plasma rotation, which have maxim in the
neighborhood of the external electrode closer to the
channel exit. Nevertheless, at the outlet the kinetic energy
part %100
z
K
ε
ε ϕ
ε = associating with the rotation is less
the kinetic energy of flow along the axis. This value is
equal to %7=εK in calculation in one-component
model and %12=εK in two-component model.
The longitudinal magnetic field determines behavior
of density ( )rz,ρ in neighborhood of the external
electrode. In the presence of zH the level lines of the
function ϕHr (plasma current) change the inclination in
the neighborhood of the external electrode.
Simultaneously, at this place the density increases due to
plasma rotation. In Fig. 1 we have illustrated this effect.
Here, we can see the density distribution along the
external electrode ( )0, rrz =ρ for different values 0
zH .
Continuous curves 1 and 2 correspond to calculations
within the framework of one-component MHD-model in
cases 00 =zH and 1.00 =zH respectively. Thus, the
density enlarges near the external electrode. Due to this
circumstance there is possibility to overcome or weaken
the current crisis in plasma accelerator channel.
Moreover, we can compare the calculation of steady-
state plasma flow with the analytical solution (dotted lines
in Fig.1). These solutions may be different because the
analytical model was constructed in the smooth channel
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
4
3
2A
2
1A
1
ρ
Z
Fig. 1. Density distribution along external electrode:
lines 1 (1 ) – calculation (theory) in case 00 =zH ;
lines 2 (2 ) – calculation (theory) in case 1.00 =zH ;
line 3 – calculation at 15.00 =zH ; line 4 - 2.00 =zH
140
approximation for ideal magnetohydrodynamic equations
of cold plasma ( 0=ν , 0=β ). We have not observed the
principal qualitative distinctions of two solutions.
Further investigations based on the full MHD-model
made it possible to reveal the following regular features.
The increase of the longitudinal magnetic field leads to
the growth of the density in the neighborhood of the
external electrode. The curves 3 and 4 in Fig. 1 were
calculated for 15.00 =zH and 2.00 =zH , respectively.
Starting with the some critical value of longitudinal field,
we have observed the qualitative changing of flow. The
value of critical field is a function of initial parameters. In
case of 2.00 >zH the current layer is formed in the
moving plasma in the neighborhood of the external
electrode. In Fig. 2 we have plotted the current layer in
case of 25.00 =zH . The dotted line in this figure
corresponds to the value 0=ϕH . Such closing of plasma
current in plasma on the external electrode points to the
possible formation of current attachments in case of
sufficient strong longitudinal field 0
zH .
The numerical experiments carried out within the
framework of the two-component model confirmed the main
peculiarities of plasmadynamic processes, found before by
means of the one-component model. Dash dotted line in fig.1
corresponds to the two-component model in case 15.00 =zH .
Also we observe excellent accordance the two-component
model with analytical model. In case of two-component model
it is possible to detect that azimuthal velocity have greater
values in contrast one-component model. Normal component
of plasma velocity on anode is very small value during the
inflow through the external electrode. The increase of the
longitudinal magnetic field also leads to the formation of
current attachments as well as current layers in moving
plasma. In case of two-component model we observe the
enlargement of angle between electrode and current layer.
5. CONCLUSIONS
We observed that the weak longitudinal field can
generate transonic flows on different conditions at the
inlet. In this case at channel outlet the rotation energy part
is much less kinetic energy of plasma flow along the axis.
A longitudinal magnetic field having effect along the
channel leads to the rotational plasma motion, gradually
intensifying it. As a result the density increases near the
external electrode. This circumstance makes it possible to
weaken the current crisis phenomenon in plasma
accelerator channel. At the same time, the formation of
the current layers and attachments to the external
electrode is observed in the moving plasma in case of
respective strong longitudinal magnetic field.
ACKNOWLEDGEMENTS
This work is supported by the Russian Foundation of
Basic Research (grants Nos. 06-02-16707, 05-07-90026,
and 06-01-00312) and RAS (programs Nos. 9, and 14).
REFERENCES
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Moscow: Fizmatlit. 2006 (in Russian).
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3. V.I. Tereshin, V.V. Chebotarev, I.E. Garkusha,
V.A. Makhlaj, N.I. Mitina, D.G. Solyakov,
S.A. Trubchaninov, A.V. Tsarenko, H. Wuerz. //
Problems of atomic science and technology. Series:
Plasma Physics. 1999, 3, p. 194-196.
4. S.I. Ananin, V.M. Astashinskii, E.A. Kostyukevich,
A.A. Man’kovskii, L.Ya. Min’ko. // Fiz. Plasmy.
1998, v. 24, 11, p. 1003-1009 (in Russian).
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V.S. Mainashev, A.I. Morozov, V.L. Podkoviirov,
Iu.V. Skvortsov // Fiz. Plasmy. 1990, v.16, 2, p.96.
6. K.V. Brushlinsky, A.N. Kozlov, A.I. Morozov //
Fiz. Plazmy. 1985, v. 11, . 11, p. 1358 (in Russian).
7. K.V. Brushlinsky, A.M. Zaborov, A.N. Kozlov,
A.I. Morozov, V.V. Savelyev // Fiz. Plasmy. 1990,
v. 16, 2, p. 147-157 (in Russian).
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. , .
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.
.
. , .
.
.
.
Fig.2. Plasma current (level lines ϕrH ) at 25.00 =zH
|
| id | nasplib_isofts_kiev_ua-123456789-82152 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:19:58Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kozlov, A.N. Zaborov, A.M. 2015-05-25T15:58:22Z 2015-05-25T15:58:22Z 2006 Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field / A.N. Kozlov, A.M. Zaborov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 138-140. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.30.Cv, 52.59.Dk, 52.65.-y https://nasplib.isofts.kiev.ua/handle/123456789/82152 Results of numerical simulations of axisymmetric plasma flows in accelerator channel with longitudinal magnetic field are presented. The investigations of two-dimensional flows are carried out within the framework of onecomponent MHD-model and two-component model taking into account the Hall effect. It is found that the current attachments are appeared in case of respective strong longitudinal magnetic field. This work is supported by the Russian Foundation of Basic Research (grants Nos. 06-02-16707, 05-07-90026, and 06-01-00312) and RAS (programs Nos. 9, and 14). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma dynamics and plasma wall interaction Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field Article published earlier |
| spellingShingle | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field Kozlov, A.N. Zaborov, A.M. Plasma dynamics and plasma wall interaction |
| title | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| title_full | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| title_fullStr | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| title_full_unstemmed | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| title_short | Formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| title_sort | formation of the current attachments in plasma accelerator channel under influence of the longitudinal magnetic field |
| topic | Plasma dynamics and plasma wall interaction |
| topic_facet | Plasma dynamics and plasma wall interaction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82152 |
| work_keys_str_mv | AT kozlovan formationofthecurrentattachmentsinplasmaacceleratorchannelunderinfluenceofthelongitudinalmagneticfield AT zaborovam formationofthecurrentattachmentsinplasmaacceleratorchannelunderinfluenceofthelongitudinalmagneticfield |