INTRODUCTION
Many
factors, above all the current crisis [1] due to the Hall effect, impede the
achievement of high velocities in plasma accelerators. In order to overcome the
negative consequence of the Hall effect it is appropriate to go to systems of
the quasi-steady plasma accelerator type proposed in [2]. The experiments
generally confirmed the underlying ideas on the basis of which such plasma
accelerators are designed. The experimental investigations [3-5] show a real
opportunity to get flows of relatively dense plasma 
with 
. Such
opportunities allow using plasma accelerators in space as the electric jets and
in various applications including thermonuclear installations as well.
The
mathematical models [6,7] of the plasma-dynamics play an important role
accelerator design. For dense plasma the processes in accelerators are
investigated, theoretically and numerically, within the framework of the MHD
equations.
The
presence of a longitudinal magnetic field 
opens up new possibilities for controlling the
dynamic processes in the accelerator channel and makes to realize transonic
flow in a channel of certain geometry.
In
[8] the effect of a longitudinal magnetic field on two-dimensional axisymmetric
two-component plasma flows is determined analytically. The investigations are
carried out within the framework of the smooth channel approximation for ideal
magnetohydrodynamic equations. The longitudinal magnetic field complicates the
flow. For example, it leads to plasma rotation about the axis of the system. In this case, an analysis of
the most impotent properties of plasma flows showed that the Hall effect and
the anode flow zone can be significantly reduced owing to the longitudinal
field.
BASIC EQUATIONS
The
numerical model of axial-symmetrical flow in plasma accelerator channels in the
presence of a longitudinal magnetic field is elaborated. Computations are based
on the MHD-equations taking into account the Hall effect (
),
electrical conductivity tensor and transport coefficients in magnetic field
depending on the 
[9].
;
; 
;
;
Here 
;
;
;
- total pressure, 
- density of heavy particles, 
- electric current; 
- heat flux.
; 
;
- conductivity of medium .
and 
are known functions of
, 
, 
In
the case of axial flow symmetry 
it is possible to
introduce the vector potential 
so that 
in the numerical simulations. The toroidal
vector potential 
defines the poloidal components of the
magnetic field 
and 
.
As a result, we have 7 equations for variables 
.
PARAMETERS, BOUNDARY CONDITIONS AND NUMERICAL METHODS
As
the input dimensional units, we will take the characteristic values of the
density and temperature 
and 
at the channel inlet, the length L of the
plasma accelerator channel and the strength 
of the azimuthal magnetic field component,
where 
is the current in the
system and 
is the characteristic
radius at the inlet. Using these quantities, we form the following units of
velocity, pressure and time scales: the characteristic Alfven velocity 
, 
and 
, respectively. In this case the dimensionless parameters in
numerical investigation have the form:
(
); 
;
The dimensionless conductivity 
(magnetic Reynolds number) contains 
which can be expressed
in terms of the initial dimensional parameters and physical constant.
The
boundary condition at the channel inlet corresponds to the subsonic plasma
inflow with 
; 
; 
; 
, where 
and 
are known functions of 
. We will assume that the total electric current flowing
through the system is supplied only through the electrodes and maintained
constant. This generates the boundary condition at the inlet for azimuthal
magnetic field 
.
The
accelerator plasma dynamics are investigated in different current transport
regimes. In the electron current transport regime the streamlines of the ion
plasma component lie on the impermeable electrode surfaces. In this case the
electrodes are not equipotential. On the other hand, in the ion current
transport regime the electrodes are equipotential surfaces. In this case they
must be transparent for the plasma entering the channel across them. Most
experiments [3-5] and models [6,8,10] are based on ion current transport.
In
the present study we will consider cold plasma flow in the ion current
transport regime. In this case we have 
on the electrode surfaces: the cathode 
and the anode 
. The shapes of the electrodes are given. In the presence of
a longitudinal magnetic field it is necessary to determine the additional
condition for magnetic field components 
and . Within the framework of simple formulation of the problem
it is natural to assume that 
on the electrode
surfaces. Besides we will suppose that 
. The anode flow enters the channel from the side of the
anode. Therefore, on the anode surface we must assign two functions 
and 
. The functions 
, , 
, 
must be chosen
depending on the particular details of the problem formulation. These functions
can be determined from analytical model in order to compare the analytical and
numerical solutions.
The
channel geometry is specified by the electrode profiles, which correspond to
the possibility of transonic flow so that at mid-channel the flow velocity
passes through the local velocity of the fast magnetosonic wave. So at the
accelerator outlet the boundary conditions correspond to a supersonic plasma
flow.
The
simulations are based on the adaptation of flux-corrected transport method for
hyperbolic part of equations. The finite conductivity is taken into account in
the parabolic equations for and 
fields by means of the implicit numerical
scheme. A steady-state supersonic flow in the nozzle-type channel calculated by
the relaxation method.
RESULTS OF COMPUTATION
The
channel has the shape of a Laval nozzle of length equal to unity. The initial
dimensional and dimensionless parameters are: 
; 
; 
; 
; 
; 
; 
. For
hydrogen plasma we have 
and 
if 
and 
. Within the
framework of the model proposed the presence of a small insignificant
longitudinal magnetic field (
) makes it possible to realize transonic flow.
In
Fig.1 we have reproduced the transonic flow in the ion current transport
regime. In Fig. 1-c the length of the vectors (in centimeters) is equal to the
dimensionless value of the velocity at a given point. At mid-channel the flow
velocity passes through the local velocity of the fast magnetosonic wave. In
Fig. 1-b the curves show the level lines of the function 
characterizing the
rotation for 
. The azimuthal velocities have maxim in the neighborhood of
the anode closer to the acceleration channel exit. Under certain conditions in
the absence of a longitudinal field, a shortage of ions develops in this region
due to the Hall effect. Current crisis and the collapse of the acceleration
process accompany this. In accordance with the results of this investigation,
in the presence of a longitudinal field the density on the anode increase due
to ration.
The
two-dimensional analytic and numerical solutions are analyzed and compared with
reference to a plasma accelerator channel. Computer experiments describing the
rotatory plasma flows in presence of a longitudinal magnetic field demonstrated
the similarity of the results obtained on the basis of analytic model [10].
Fig. 1. Plasma flow in the
presence of a longitudinal magnetic field: a) electric current 
(level lines of the
function 
) and vector magnetic field distribution; b) level lines of
the ion azimuthal velocity ; c) density distribution 
and vector velocity
field of the ion component 
. The broken curve corresponds to the analytical model.
CONCLUSION
A
computer simulation of rotatory axisymmetric steady-state plasma flows in the
presence of a longitudinal magnetic field was carried out in terms of
two-dimensional time-dependent MHD-equations, taking into account the Hall
effect, electrical conductivity tensor and transport coefficient in magnetic
field depending on the . A
longitudinal magnetic field provides additional possibilities for controlling
the processes in the channel of coaxial plasma accelerator and makes it
possible to realize transonic flow with the ion current transport.
The
calculation results correspond to the analytical model of rotatory plasma flow.
ACKNOWLEDGEMENTS
The
work is supported by RBRF (grants NN 03-01-00063, 02-07-90027)
REFERENCES
1. A.I. Morozov. Physical Principles of Space Electojets. Atomizdat, Moscow, 1978.
2. A.I. Morozov. Principles of coaxial
(quasi) steady plasma accelerators (QSPA).
// Fiz.Plasmy. 1990, v. 16, N 2, pp. 131-146.
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. Investigation of high power quasi-steady-state plasma
streams flow in magnetic field. // Problems
of atomic science and technology. Series: Plasma Physics. 1999, N 3, pp.
194-196.
4. V.I. Tereshin, V.V. Chebotarev, I.E.
Garkusha, V.A. Makhlaj, N.I. Mitina, A.I. Morozov, D.G. Solyakov, S.A.
Trubchaninov, A.V. Tsarenko, H. Wuerz. Powerful quasi-steady-state plasma
accelerator for fusion experiments. // Brazilian Journal of Physics. 2002, v. 32, N 1, pp.165-171.
5. S.I. Ananin, V.M. Astashinskii, E.A.
Kostyukevich, A.A. Man’kovskii, L.Ya. Min’ko. Interferometric studies of the
processes occurring in a quasi-steady high current plasma accelerator. // Fiz. Plasmy. 1998, v. 24, N 11, pp.
1003-1009.
6. K.V. Brushlinsky, A.M. Zaborov, A.N. Kozlov, A.I. Morozov, V.V. Savelyev. Numerical simulation of plasma flows in QSPA. // Fiz. Plasmy. 1990, v. 16, N 2, pp. 147-157.
7. A.N. Kozlov. Ionization and recombination
kinetics in a plasma accelerator channel. //
Fluid Dynamics. 2000, v. 35,
¹ 5, pp. 784-790.
8. A.N. Kozlov. Influence of a
longitudinal magnetic field on the Hall effect in the plasma accelerator
channel. // Fluid Dynamics. 2003, v.
38, ¹ 4, pp. 653-661.
9. S.I. Braginskii. Transport phenomena
in plasma. // In Reviews of Plasma Physics “Problems
of plasma pheory”. Ed. by M.A. Leontovich. Gosatomizdat, Moscow, 1963, pp.
183-272.
10. A.N. Kozlov. The numerical model of the rotatory axysymmetrical plasma flows. The
comparison with analytical model. Preprint N 48. Keldysh Institute of
Applied Mathematics, Russian Academy of Sciences, Moscow, 2004.