RESTORATION OF TWO-DIMENSIONAL RADIATION FOR AN OPTICALLY THICK PLASMA

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RESTORATION OF TWO-DIMENSIONAL

RADIATION FOR AN OPTICALLY THICK PLASMA

V. Pickalov, N. Preobrazhensky

To cite this version:

V. Pickalov, N. Preobrazhensky. RESTORATION OF TWO-DIMENSIONAL RADIATION FOR AN OPTICALLY THICK PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-855-C7-856.

�10.1051/jphyscol:19797413�. �jpa-00219414�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, vaqe C7- 855

RESTORATION (X TWO-DIMENSIONAL RADIATIW FOR AN OPTICALLY THICK PLASMA

V.V. PickaIov and N.G. Preobrazhensky.

I n s t i t u t e f o r Pure & Applied Mechanics, S i b e r i a n Division, Academy o f Science, U.S.S.R. Novosibirsk, 630090.

Ab8tract.A concept of local diagnostics for an optically thick plasma without axial symmetry is proposed.

It

is possi- ble to reduce the problem in two-dimen- sional case to the system of one-dhen- sional problems by decomposition in po- lar and cartesian coordinates.

Recently [ I

]

the authors proposed

an

algorithm of data conversion to the lo- cal emissivities for an optically thick plasma of arbitrary configuration. Simu- lation procedure proved the reasonable

lution as B priori assumption. If some additional information about solution is available the mathematical formulation of the problem can be efficiently simp- lified, E.g, the possibility of factori- zation in local coefficients:

k(=,rl=

k,(=)k2(%

, E ( z ,

I ) = E

(=) E 2 ( 1 )

1

permits the retrieval of E(=,y)by means of decomposition in Cartesian coordina- tes. In this case only two ortogonal di- rections are eufficient for observation and the problem comes to the system of accuracy for optical densities from 0 two integral eqsions : Y

I,(=,- ^E, (*)-I e,

~*)exp[-k7(~~%$<:c.r)dr]dq till 10 and the level of experimental er- -

- _-

^,-,

+ - x ^{( 2 )}

rors 5-1076

p] .

By decomposition tech-

I

( I ) = F ~ (7).

!

^{- &,}

-

( r ) e z p [ - kp(y)! $ ( a r ) d r ]dl

-PO

nique in polar coordinates one can reduce Another kind of 21 priori information the two-dimensional integral equation for

local emissivities

&(r,8)

to the system of linear integral equations

8,

s

~ , B ~ C 0 , 2 1 7 ,

P E C - R , R I ~ ,

1 = 1 , 2

...,

kL (@) is the absorption coefficient for i-ri%

, f (

q,t)is the intensity of esca- ped radiation for i-ring arid sngle

3

, The system (1) can be solved successful- ly with due regard for smooth;lese of so-

related to the form of isolines for basic plasma parameters can be prominent,

In

the special and the most popular axisym- metrical case (isolines are concentric

circles) the problem of local diagnostics is described by one-dimensional Abel or Freeman-Katz

3

equations.

In

the more

[ 1

general case of isolines which are con- vex closed curves without self-intersec-

tions

W

( x , g , b ) = O one can find the spatial distribution of emiseivities solving the Volterra I-st kind integral

equation

,

^{I t}

I(?,= e=p(-S T b t ) ~

^{E [}

^3:ap( P ⁹

^{+k 1}

I $ 3 t,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797413

The "fan-shaped1* measurement scheme (Pig.la) is supposed here and the diffe- rential d S along the line of sight .DL for two-place function -;c = 3C

<

^{b) is given}

t = t ,

is easily obtained from

(3);

as to the special case

T-=-T

⁺ taking place for the system of shifted ellipsea one comes to the gene- ralized Freemn-Katz equation

~~ometrical arrangement of obsemra- tion for a plasma with the known isolines is sketched

in

Fig.la, The result of re- storation for the model function & ( x , Y ) 5

=€(f&l-tis shown in Pig. 1 2 b ( x %

*

)

and

is related to the system of shifted ellipses

Also in Fig.lb are plotted the IIexpeari- mental1* function

I

(%>with the noise le- vel of

5%

(Ooo0) and the coefficient of random error amg1i.f ication ₌

%

I d I

(---- )

in

the method of regularization when the equation ( 3 ) is solved (k=O).

Fig.2 corresponds to the results of simulation in the problem of asymmetrical plasma field diagnostics for the lack

of

information about isolines. Modified "oni- on peelingff strategy [ 4 ] was used for eq.

(1 1, maximum optical density being

--I

and the initial noise level 1%.

Our results nnd theory show reaso-

nable accuracy for retrieval of rather complicated plasma fields

as

in laser

ex-

plosion of targets, high temperature, tocamac or stellarator plasma etc.

REFERENCES

1. Pickalov V,V.

,

Preobrazhensky N, G.

In: Proc. XI11 ICPIG, Berlin, v.1, 1977, 201.

2. Picblov V.V.

In:

Abel inversion and its generaliaations. Novosibirsk, 1978, 25 (in Russian),

3.

M e m a n N., Katz S, JOSA, 1960,

,

826.

4.

Chen

F.

P.

,

Goulard R. JQSRT, 1976,

16 ,

819.

Fig.1 Geometrical arrangement of obser- vation and the example of

restoration: optically thin case, Fig, 2 Modified

*'Onion pe- eling" me- thod (layer decompositi- on) :model function (-

-1 and re

-

sult of res- toration (---

--I;

optical density Z

=I