The stationary modes of a heavy-current, self-constricting radiating discharge in a plasma

THE STATIONARY MODES OF A HEAVY-CURRENT, SELF-CONSTRUCTING RADIATINGDISCHARGE IN A PLASMA* P. A. DORODNITSYN

and YU, P, POPOV

Moscow 24 April 1972)

(Received

A STATIONARY heavy-current context of the one-dimensional consists of two coaxial zones, heat conduction, assumed

and in the other, the approximation

of spatial

afterglow,

is

to be applicable.

1. The dynamic plasma

discharge in a plasma is considered in the magnetohydrodynamic model; the discharge in one of which the approximation of non-linear

are usually

properties discussed

of a heavy-current theoretically

within

radiating

discharge

the framework

in a

of the system

of equations of magnetohydrodynamics (MHD). The topic is extremely complicated, and even in the simple case of axial symmetry, a solution can only be For examples of such solutions obtained by numerical methods using a computer. see e.g. [l-41. In addition to general solutions, approximate and in particular solutions are often obtained in discharge problems. Such solutions individual

stages

Examples

are to be found in (S-71, where the approximation

conduction

in the actual

or of spatial

discharge

afterglow

and may be used for estimates.

is used throughout

A stationary state of a self-constricting tic mode, which is capable of establishing the total current fairly slow.

reaches

a sufficiently

*Zh.

vGhis1.

the stability

by realizing

of non-linear

plasma is treated as an asymptoitself in the actual discharge when

high value and its variation

a solution

Mat. mat. Fiz., 13, 1, 247-253, 1973.

328

heat

the discharge.

It must be mentioned that the stability of such stationary questionable but is not usually open to analytic investigation. of checking

stationary describe

in numerical

in time is

solutions is Hence the value computations

of

Radiatipg discharge

the corresponding

non-stationary

In [3, 41 non-stationary ted numerically,

using

in a plasma

329

problem.

heavy-current

the radiation

discharges

transport

in a plasma

equation.

computations in [4, 81 showed that, under certain is realized with two zones, differing substantially

are investiga-

Analysis

of these

conditions, a stationary in their thermodynamic

mode

parameters. The present stationary

paper constructs

equations.

a two-zone

Some simplifying

model of non-linear

heat conduction

in the other the model of spatial assumed constant. It is shown that, discharge

plasma,

single-parameter

family

The analytic departure

is applicable

afterglow

given fixed problem

properties

the plasma

transport,

and

conductivity

(total current,

such a solution

is

mass of the

is not unique;

a

exists.

are realized

non-stationary

for the in one zone the

for the radiation

parameters

solutions

constructed

in the corresponding

analytically are made:

is used;

of the material),

of stationary

solutions

solution

assumptions

numerically

by a modal

problem.

2. Consider the stationary modes of a discharge represented by an infinitely long cylindrical plasma filament in uucuo, along which a total current I, = const flows. Assume that the position of the plasma-vacuum boundary is fixed, that the plasma

conductivity

the equation

of state

zones:

0,~ r < R,

is constant: of an ideal

is the linear

Below we quote the System case of a stationary

discharge,

gas. zone,

The discharge

and that the gas satisfies consists

of two coaxial

and R, < r < R, the outer zone.

I of one-dimensional

MHD equations

the inner zone of which radiates

in the outer zone the approximation case when the zones

c = u0 = const,

of non-linear

have the reverse

positions

heat conduction is considered

for the

spatially, holds

while (the

similarly).

Notice that !I’, f const in the spatial afterglow zone; in the outer zone we shall put T, = const for simplicity, since this is justified when the coefficient of radiant

heat conduction

System I:

is high.

330

V. A. Dorodnitsyn

and Yu. P. Popov

Rl

Ii = 2x

s

&

jir dr,

lo -Ii

0

ii pi

=

RI

m!L

i2

=pdoTs,

ii&i

fiE =

=

p2RoTzt i2liz P

dEIz = dr

431jir,

dT2

A pi2 fT,,

0 =G r <

DOE,

dPZ -=-_ dr

r =

= Pz

dP* -=----, dr $

= 23~ jzr dr, s

dr-

Ri,

3

43tjZr,

-

R* -=I r <

0, Rz,

where r is the radius vector drawn from the cylinder axis, p, p and 2’ are respectively the plasma pressure, density and temperature, j is the current density,

M, is the total mass of the discharge plasma, R, is the gas constant, afterglow constant, E and H, are the electric and magnetic field strengths, and H I r-H?. The subscript 1 refers to the inner, and 2 to the outer zone.

A is the spatial

The boundary

conditions

at the point r = 0 are the symmetry

conditions

H,(O) = 0, W,(O) = 0 (W (r) is the radiant

energy

flux).

At the point r = R, (the boundary

with the vacuum),

f’, @,I = 0, W, (R,) = vCT;R,, where uC is the gtefan-Boltzmann constant. opaque outer zone radiates as a black body. Matching

conditions

are imposed

It is assumed

that the optically

at the point r = R,:

E,(R,)=E,(R,),Q,+W,-=W,+, P,(R,) = Pz(R,), Q, + W;’ =

W,++ W;,

H,(R,) = H,(R,) = 211, where Q is the total Joule

heat of the relevant

zone, W” is the radiation

flux in

Radiating

discharge

331

in a plasma

FIG. 1. the negative

direction

of the r axis,

Below we quote the solution tion along the radius solutions

of certain

and Wt the flux in the positive

of System quantities

I, and in Fig. in the discharge

direction.

1 show the distribu(the family

S (7)): Hi,2 = -

210

R22

pz =

9,

T,=Ai

(f--&)“,

Rz = ARTZ+,

of

V. A. Dominitsyn

332

and Yu. P. Popov

cl

FIG.2.

a-fora>l;

b-fora
A typical feature of our solution is that it contains a free parameter, connected with the position of the zone boundary R,. Hence, whatever the fixed problem parameters system of solutions,

M,, I,, oo, cr,, R,, and A, there exists a single-parameter differing from one another in the value of q = I,/.!, = R12/

R,‘, which it is convenient that,

to take as the free parameter.

given fixed q and problem

constructed

family

includes

parameters,

all the solutions

the solution of System

It can also be shown is unique.

Hence the

I.

In Fig. 2 we show the behaviour of the parameters T,, T,, M, and R, as functions of the free parameter 7, 0 < q 4 1. It can be seen that the distributions split into two series (Fig. 2, a and Fig. 2, b), according to the relationships between the problem parameters, or more specifically, according to the value of the dimensionless combination a=-

4nM0 /Re

4~ _ (

34

~3 )

’

The fact that our single-parameter family of solutions includes all the solutions of System I enables us to point to some important features of the stationary discharge mode. For instance, when a < 1 (see Fig. 2, b), the outer

Radiating

discharge

l?IG. 3. -for for t =l,O; ----fort

in a plasma

t=O;

333

-,=2,0.

radius of the discharge cannot exceed a certain quantity Ramax. 3>, The constructed analytic solutions were realized by means of numerical computations of the system of ~~~ partial differential equations. The computations were based on a homogeneous completely conservative difference scheme by the method of successive pivotal condensations with iterations [9, 101. For details of the method and the computational program see [31. To describe the process of radiation transport we used the kinetic equation with an absorption factor averaged over the spectrum (grey body approximation). The method of computing the radiation transport equation is described in ill]. The radiation absorption factor and its temperature-dependence were chosen in such a way that the approximation of spatial afterglow is in fact realized in the inner zone,

334

V.A.Dorodnitsyn and Yu. P. Popov

FIG, 4. 1 - for m=M,; 2 - for m = 0.693M,; 3 - for m = M,; 4 - for m=0.087 M,; 5-form = 0.023 M,,.

and in the outer zone, the approximation of non-linear heat conduction. the stationary the analytic

Hence

modes obtained in the numerical computation should be close to stationary solutions

S (77).

As initial data we took the analytic solutions

for certain values of 7. The

numerical solution for the case 7 = 0.5 is illustrated

in Fig. 3.

The numerical computation showed that the initial profiles, taken from the analytic solution, remain virtually undistorted for a time exceeding by one order the characteristic gas-dynamic time, i.e. the time taken by a sound wave to travel the distance from the boundary between the plasma cylinder and the vacuum to its axis. In Fig. 4 we show the behaviour in time of certain quantities. It can be seen that a balance establishes itself in time between the total Joule heat & and the energy W radiated from the system (Fig. 4, a), which affords an indication of the degree to which the process is stationary. The trajectories of certain mass intervals, shown in Fig. 4, b, allow us to see small oscillations, damped in time.

The period of these oscillations

is equal to the characteristic

Radiating

discharge

r'0P_

in a plasma

335

I

I \ \ \ I \

zs\ ‘,

FIG. 5. ~ - fort=O;-.for t = 0.5; ----for t=l.O. gas-dynamic

time of the problem.

We can conclude asymptotically 0.5,

from the computational

results

that the solution

mode close to the analytic

into a stationary

moves

solution with q=

this solution being stable. A computation with the same problem parameter values,

but a different

value of the free parameter 11in the initial data, showed the existence another stable stationary

of

mode. In short, the numerical computations confirmed

the main conclusion obtained by analyzing the exact stationary solutions, namely, the existence of the free parameter T)in the solution. This fact is also confirmed in the more complicated model of radiation transport. 4. solutions

In addition

to the computations

S(q) were used as initial

some initial

data not included

solution

which establishes

solution

S(q) with 77= 0.66.

data,

in S(q).

itself

described

above,

a computation The results

with time proves

where the exact was performed with

are shown in Fig. close

5.

to the stationary

The

and Yu. P. Popov

V. A. ~oro~njts~n

336

From the physical of the discharge the discharge

the lack of uniqueness

can be explained process

in the non-stationary established,

viewpoint

by the fact that the asymptotic

stage

solution

for

is considered. process,

the concrete

family of solutions.

in the stationary

When account is taken of the initial data and of the way in which the stationary state is

form of the solution

For instance,

may be discovered

in [3, 41 computations

from the general

are described

of a non-

stationary observed

heavy-current discharge, during which a quasi-stationary mode is in the region of the current maximum, The form of the solution at this (e.g. the distribution of the parameters along the discharge radius) here

stage proves

to be dependent

Notice constructed conduction

finally

data.

that multi-parameter

systems

of solutions

can be similarly

for a discharge, consisting of alternating zones of non-linear heat and spatial afterglow. The number of free parameters here is N - 1,

where N is the number The authors discussions,

on the initial

of zones

in the discharge.

thank A. A. Samarskii

and G. V. DaniIov

and S. P. Kurdyumov

and L. S. Tsarev

for valuable

for performing

the numerical

computations.

REFERENCES 1.

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theory plasma, in: Topics in Plasma Atomizdat, Moscow, 1967.

of

2.

BRAGINSKII, S. I., GEL’FAND, I. M. and FEDORENKO, R. P. Theory of comprassion and ripple of the plasma column in a high-power pulse discharge, in: Plasma Physics and the Problem of Controlled Themtonuclear Investigations (Fizika plazmy i problema upravlyaemykh termoyadernykh issledovanii), Vol. 4, 201-221, Akad. Nauk SSSR, Moscow, 1958.

3.

GOL’DIN, V. YA., DANILOVA, G. V., ~LITKIN, N. N., ~Z’~A, V. A., KURDYUMOV, S. P., NI~FOROV, A. F., POPOV, YU. P., ROGOV, V. S., ROZANOV, V. B., SA~RSKII, A. A., UVAROV, V. B., TSAREVA, L. S. and CHETVERUSHKIN, B. N. Study of problems in radiation magnetohydrodynamics by numerical methods using a computer, Preprint, IPM Akad. Nauk SSSR, 36, 1971.

4.

GOL’DIN, V. YA., KALITKIN, N. N., KURDYUMOV, S. P., POPOV, YU. P., ROZANOV, V. B. and SAMARSKII, A. A. Preprint, IPM Akad. Nauk SSSR, 40, 1971.

5.

SCHLUTER, A. Dynamik des Plasmas,

6.

ROZANOV, V. B. and RUKHADZE, A. A. The radiation, dynamics, and stability of the dense plasma of heavy-current pulse discharges, FIAN Preprint, 132, 1969.

I, 2. Naturforsch., Sa, 2. 72-78,

1950.

Radiating discharge

in a plasma

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7.

ROZANOV, V. B. Possible characteristics of a source of braked radiation for laser pumps, Dolzl. Akad. Nauk SSSR, 182, 2, 326321,1988.

8.

GOL’DIN, V. YA. and CHET~RUS~~, B. N. A stationary solution of the problem of a heavy-current discharge with radiation transport, in: Collected Proceedings of the Third All-Union Conference on Low-Temperature Plasma Physics, MGU, 182, 185, Moscow, 1971.

9.

POPOV, YU. P. and SAMARSKII, A. A. Completely conservative difference schemes for the equations of ~~etohydrod~amics, Zh. u$hisf. Mat. mat. Fiz., 10, 4, 990-998, 1970.

10.

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11.

GOL’DIN, V. YA. and CHETVKRUSHKIN, 8. N. Methods for computing radiation transport in one-dimensional problems of low-temperature plasmas, IPM Preprint, 12, 1970.

USSR

CMMP

I31

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