A new method of calculating the flow and magnetic fields of the sun

4 (1980) 163-173

Pergamon Press. Printed in Great Britain

Acta Astr. Sinica 20 (1979) 161-171

Chinese Astronomy

0146-6364/80/0601-0163-$07.50/0

A NEg/ M E T H O D OF T H E

OF C A L C U L A T I N G

THE

FLOW AND MAGNETIC

FIELDS

SUN

Song ~4u-tao

and

Yin Chun-lin

Purple Mountain Observatory

Received 1978 May 5.

ABSTRACT.

In this paper, an attempt is made to combine the use of the anti-

rotational operator and the non-linear operator of the b~D to find a selfconsistent solution for the flow and the magnetic fields.

The method is

illustrated by an example, which shows how the magnetic field above the solar surface changes following a motion of the feet of the field lines.

i.

INTRODUCTION

In astrophysical situations magnetohydrodynamical equations usually take the form

O__BB~ V A ( v A B ) Ot Or_ Ot

-- n v ' B =- E,

(la)

1 (vAB)AB--(v'V)v--Vp+g+~V(V'v)+v~r2v----F, 4~p v • B -- o,

(lb) (it)

P~+v-(ov)=O, Ot

(ld)

~ ( p , V, r ) ---- o. where B i s t h e m a g n e t i c f i e l d ,

v the velocity,

p the density,

(le) p pressure,

g gravity

acceleration, Z temperature, n electric resistivity, and u the viscosity coefficient. It is rather difficult to solve analytically this set of equations in the general twoand three-dimensional cases.

If we resort to numerical methods, then we shall also

encounter great difficulties because of uncertainties at the boundary. Taking cues from [1-3], we got the idea that we should perhaps try to combine methods of theory of fields and of finite differences to find solutions of (i) that satisfy some initial boundary values.

It is proved in the theory of fields that, if the rotation and divergence

of a vector field are known, then the field itself will be completely determined by its normal or tangential components at the boundary. "application of the anti-rotational operator".

We shall call this process of solution The boundary conditions and the "redundant"

equations (Ic) and (id) are implied in this process.

Following [1-3], we shall discretize

eqns. (la) and (ib) with respect to the time (but not to the spatial coordinates):

164

Solar Hydromagnetic

Field

B(.+,)_ B(.)___.I(F~ + E.+,) • (&+, -- t.), (2a)

2

~.+~,

_

v~,) _. 1

(F,

+

F~+,)

• (t,+,

--

,.).

2 0 ~

Starting

tO <

t, <

ta <

from t h e i n i t i a l

calculate

the fields

""" <

t.

<

conditions

at later

""" <

tN ~

(2b)

T~

B(°) and v (°) and t h e b o u n d a r y v a l u e s , we s u c c e s s i v e l y

instants,

B(1), v(1),..., the

precedure of calculating

the

fields at time (n + i) from those at time (n) being as follows: i)

~.+, ~ E., F . + , ~=~ F . ,

First we let

ii)

We use a sampling function to calculate the derivatives and so to find E and F. From (2a) and (2b) we calculate the iso-rotation fields

iii)

En+1 and

Fn+ 1 are only known approximately,

p~n+l) and

V~ n+l) (because

they do not satisfy the boundary conditions,

they only have the same rotation). iv)

From B~ n+l) and v~ n+l) and the corresponding boundary conditions, we apply the

"anti-rotationaloperator"

and obtain

the solution denoted by

lB~n+l),riv(n+j) which

is our

first approximation.. v)

We repeat (ii) - (iv) to get the next approximation,

and so on, until the solution

converges. At first sight, this method is not as simple and convenient as the difference method but because it does not introduce any potentials less arbitrary,

especially in astrophysical

B and v are often incompletely known.

(such as the magnetic potential),

it is

situations, where the boundary conditions for

In our case we find consistent boundary values

by trial and error: for the magnetic field we set up some normal conditions for the velocity field we set up some tangential conditions,

(or the other way round), and a few iterations

will soon reveal whether or not the boundary values are consistent, appropriate adjustments.

if not, we make some

This method appropriately reflects the interaction between the

magnetic and velocity fields.

2.

~,~THOD OF CALCULATING E AND F

The expressions for E, F (and B1) are made up of various partial derivatives, question reduces to the calculation of the latter. rather, we shall use a sampling function.

hence the

Here, we shall not use finite differences

The calculation of partial derivatives of a smooth

function of two or three variables is essentially the differentiation of a one-variable function, and in the computer, we can easily use a sampling function S(x) to approximate

f(x), and

a smooth function to

derivative of the latter. in every sub-interval

-

it is a third degree polynomial

[mj, zff+l]

S(.) " [-~j( x , + , •

use the derivative of the former as a substitute for the

$(m) is defined in [4] as follows:

-- x)'--

--Z h} (xi+,-- x)3],i+ [~3

[I 1 (x -

_

x O 2-

_

1

(x -- x,)'

1 ~xf+,__x)3]mihi (~ -

1 xj)',=i+,h~. J

-- q~(x2- - xi)3] ]i+, hi

Solar Hydromagnetic F i e l d

xo < x i < x 2 < ''' < x i < where h j = ~ . + l - Z j , rnj=S'(~),

165

... < x ,

and at the sample points, we require S(z$)=f(xj)=fj;

at the end points, S'(x 0) =f'(Xo) , S'(x n) =f'(Xn) ; and at the sub-interval end-points Xl,

x2: " ""Xn - I" S(x) is to have continuous second derivatives. It suffices to find the d e r i v a t i v e s mj a t t h e s a m p l e p o i n t s ; m0 and mn a r e found by d i f f e r e n t i a t i n g the third-order Langranges' interpolation reverse order starting

formula, while the others are calculated

Pi= 1--~,i

i = 1, 2 , - - . , ,

3.

~' :

(#~ -

1.

)

mo

# i - - ~irai+x,

=

"J-,

= --

(

' -*

1 --~

rai

(in the

w i t h m n _ 1) a c c o r d i n g t o t h e f o l l o w i n g :

hi-,+hi

~' : T '

successively

J

, =

i :

2, 3,...,,

-

,)-'

2

- - ~i-t

(3)

,.

1 , 2 , " • ", n - - 1.

THE TWO-DIMENSIONAL ANTI-ROTATIONAL OPERATOR

Formulation of the problem: in the rectangle ~"~,(xt < X < xl~ yl < y < y J) we are given the rotation, the divergence, and the normal components at boundary of the field B:

rot S(x, y) -- C(x, y)~, ¢~v B(x, y) = D(x, y); BXx, y~) = G(,:), B~(x, y,) = G,(,O, BXx,, y) = F(y), BXx,, y) = P,(y); to find the field

B(x,y).

From t h e t h e o r y o f f i e l d s , only the rotation fit

we know t h a t we c a n w r i t e ~ = B1 + B2 + B3 where B2 s a t i s f i e s

e q u a t i o n , r o t B2 = C ( x , y ) z ,

the divergence equation div Bl=D(x,y)-div

B1 i s r o t a t i o n - f r e e ,

rot

B2; B 3 i s a p o t e n t i a l

B1 = 0, c h o s e n so a s t o field

to f i t

the

boundary conditions. As m e n t i o n e d b e f o r e ,

B2 i s c a l c u l a t e d

from e q n s .

(2a) and ( 2 b ) .

As f o r 81, we u s u a l l y

take S,(x,

y)

~) ++ #( y( y _ 7"~ )7) = ~1 II • [ D ( ~ , ~)--~v B2(~, 7)] £((xx _- - ~), 2 a~a,,~ ~.

The key q u e s t i o n i n t h e t e c h n i q u e o f a n t i - r o t a t i o n grad u, t h e n t h i s equation,

that

is,

[°÷l t--'

Y~Yt

i s t o f i n d B8.

Let 6 3 =

q u e s t i o n i s reduced to the second b o u n d a r y - v a l u e problem of L a p l a c e ' s to find the harmonic function inside G(x)

[ Ou L

)' ~ l , - , J o,, I

operators

(4)

:

:

-

B,,(x,

,,)

-

=

c,(x) - B,,(~, y~) - s,,(~, r,)

F,(y)

-

B,,(x,,

y)

-

Z with the boundary values

B,Xx,, y) :

ffi=

g,(~),

l,(y).

166

Solar Magnetic Field

By the method of separation of variables, it is then not difficult to find u; taking the gradient, we then find B 3 as follows:

-2

2a

sh(m=a~

L

b

s,o

+ ]lrdSh[8~(XXl)] ) .~ t_ b

{sin --~ (:), -- ),,) 2

2b

. ~=,

[mu([x --

x,)]

b

L

+

× (-,.,,,

+ .,.,, [ , . % -

where a ~- xt -- x,, b ~ yi -- y,, ]~, ],~, gi~ g,i are the coefficients in the Fourier development of the orthogonal system (I, cos k¢). solutionS@"

We have already used the necessary condition for a

d; = 0 or,

J a. ~0 -

to. =

a Because B 3 is divergence-free,

~o -

e,o

(6~

b

[6) is certainly satisfied.

The question of finding 8 for given tangential components at the boundary,

~.(x, y,) can be t r e a t e d

=

~(,), ~.(x, y,)

=

~,(~), ~,(~,, y)

Ymy t

Ou

-

a(.) =

Y~ey]

oy!

=

~,(y)

a p a r t from a small d i f f e r e n c e i n f i n d i n g R . Here, we put 3 ~Ov - v - - ~ - ,0v in order to reduce i t to Lagranges' boundary-value

problem; we t h e n seek a harmonic f u n c t i o n s a t i s f y i n g

°!l

~(y), B,(~,, y)

similarly,

B3=rot[v(x, y)~]=~

°¢-1 oy,

=

-

~,(~) -

Ix ~ x l

=

-P(Y)

I

---

-P'(Y)

Ov

B.x,,

+

+

B . x x . , , ) - ~(,,),

y,) -

~ , . ( ~ . y,) -

B . . ( ~ . y,)

~"(x" y)

e,,(~,, y)

e,,(x,, y)

the following boundary values:

+

+

:,(~),

:

B,,(,,, y)

-?(y),

:

:

-?,(y).

S o l a r Hydromagnetic Field

As before, after finding v, we obtain B

3

167

by differentiation: ~

T

(y -

y')

Xb/

+

(~a__~_~b~ sh

(--,.sh

[.mx~y£

y,)]+

,,.sh [m.,(y£

y,)])},

sh (mb~a)

sh ( - ~ )

+

(--gmch[m~r(Y£Y')J+~''*ch[m'~(Y--Y')l)}'L

Again, we have used the n e c e s s a r y c o n d i t i o n f o r a s o l u t i o n

],0,7

]0 = 4,° -

a

-~-Z._ds~ O, or (83

f.

(6), we may note c e r t a i n

(8) i s c e r t a i n l y

satisfied.

symmetry wliich r e f l e c t s

As the a n t i - r o t a t i o n

a n a l y s i s and e x p o n e n t i a l f u n c t i o n s , 4.

(7)

b

Because B3 i s r o t a t i o n - f r e e ,

velocity fields.

a

I f we compare (7),

the i n t e r a c t i o n

(8) with (5),

between the magnetic and

o p e r a t o r i s made of m u l t i p l e i n t e g r a t i o n s ,

harmonic

i t i s e a s i l y r e a l i z e d on the computer.

THREE-DIMENSIONALANTI-ROTATIONAL OPERATOR

F o r m u l a t i o n of the problem: in the r e c t a n g l e ~ ( x l < x < xl, Yt < y < YI, zl < z -C zK) are given the r o t a t i o n ,

the d i v e r g e n c e

and

the

we

normal components a t the boundary of a

v e c t o r f i e l d B,

B,(x,, y, z) = F ( y , z ) , B~(xt, r, z) -~ F,(y, z), Bv(x , y,, z) = G(z, x), B~(x, yj, z) = G,(z, x ) , B , ( x , y, z,) = H ( x r y ) , B . ( x , y, zK) = Hi(x, y ) ; rot B(x, y, z) ~ C(x, y, z ) , div B ( x , y, z) -~ D(x, y, z ) ; to f i n d the f i e l d 8 ( x , y , z ) . given by (2a) and (2b).

As i n the two-dimensional c a s e , we w r i t e B= B1 +B 2 +B 3. P2 i s

B1 can be taken as

B , ( M ) ~ grad,. { - - ~ 1 f J l z [ D ( ~ , ~/, ~ ' ) - - d i v B2(~, ~, ~)]__.drv rMp

(9)

S o l a r Hydromagnetic Field

168

where M r e p r e s e n t s a g e n e r a l p o i n t o f t h e f i e l d

(x,y,z)

and P t h e p o i n t o f i n t e g r a t i o n

(~, q, ~); d'rp=d~dqd~ i s t h e i n t e g r a t i o n e l e m e n t . Let B3 = grad u, t h e q u e s t i o n i s t h e n r e d u c e d t o t h e second b o u n d a r y - v a l u e problem f o r the Laplaces equation in t h r e e dimensions. satisfies

That i s , we seek a harmonic f u n c t i o n Z t h a t

the f o l l o w i n g boundary c o n d i t i o n s :

Ou --~X

[

= v(y, ~) -

x~x l

B , A x , , y, ~) - - B ~ ( x , , y, ~) .=

l(y,.~),

o_~.

Ox • ~t = F , ( y . z ) - - B l . ( x t , y , z ) - - B i . ( x l , y , z ) == / , ( y . z ) ;

o.[ Oy- ,-,, = a(z, o~],.,, Ou

]

x) -

= ¢(~,

y,, .) - B,,(x. y,, .) = g(.. x),

B,,(x,

x) - B,,(x, y,, ~) - B,,(~, y,, ~) = e,(~,

= H(x, y) -- B,,(x, y, ~ , ) - B,,(x, y, z,) .= h(x, y);

g~m l

O~z . . , ~

~---H , ( x , y ) - - B t . ( x ,

y, z~) -- B,,(x,

Using t h e method o f s e p a r a t i o n o f t h e v a r i a b l e s , u(x,

x);

y. z) =

(,, - - x,)lo,o +

+ (e,o.o

we f i n d t h e s o l u t i o n as f o l l o w s :

yt)go,o + (z -- z,)ho.o + ( h o , o - lo,o)~ Z2

(y --

y2

eo.o) ~ + (h,o.o h~o)~

-

+ ~

y,. z K ) ~- h a ( x , y ) .

-

{cos [ ~ ( y - y,)] cos [ ~ ( ~ - ~,)]

.... o

Pm,.sh(ap,~,.)

( - - [m,.ch [ ( x - - x t ) p m . . ]

m+n~O

+ ]l.,.ch[(x -- xt)p,.,.l) + q,~..sh( bqm,.) × (--g.,.ch[(~

- - Y l ) q m , . ] at- g t . , . c h [ ( y . - - y t ) q . , . ] )

m~ / "

n=

+

y~)] -

!

rm,nsh( e r m , . ) X (--hra,.ch[(z -- z~)r...] where a ~ x! - - x t , +

b -~- Y l - - Y,.

. r.,,. ~

Fourier coefficients

-1- h l . . n c h [ ( z - - z , ) r m . . ] ) }

c ~ zK - - z t , +

(?y]'.

£

with base (cos) .

P,.,. ~

] ~ , . , 11. . . .

raz a +

(1o) . q,.,,,

gin,n, glm,., hm... hlm,.

are the

Here, we have used t h e n e c e s s a r y c o n d i t i o n f o r a

solution, [to,o -a

]o,o +

gto,o --

go.o + b

h,o,o --

ho.o =~ O.

(11)

¢

Because 8 3 is divergence-free, this condition is certainly satisfied.

Differentiating (i0)

Solar

with

respect

to cc, y,

computation The case writing,

is

not

of

given

very

we change

detract

from

the

we find

z,

tangential

=

expressions

the

components

region

for

Because

B3.

is

a little

the the

the

u,

values

of

following

being

set

a@>Ys

We also

write

field

A.(z,y,zl

Our problem

B,=A. in

of

2) _

have

of

the

u from

the

above

Similarly,

Taking

(12).

region

with

the

to

find

boundary

=

jcy,

the

of

values

the

>I

aZ

y, z) = Al

a2

x=3

I

x=0

values

We first

z= 0 as example:

a solution

Ky,

=

kind.

plane

,

a

X,(x,y).

first

on the SO

aY equations

, ado,

z).

problem

not

(12)

x),A,Iyeb= &(2,x);

&!a_-a14,

being

differential

Aylx*

exists

for

z).

f(O, z’)do’. J;w, zw +-J’= 0

we can find

the

as follows: 0, #{

y> z> -

h,

0, z) -

4Z,

b,

4x,

y> 0) =

{ 4x, the

z>

values

of

u on the

J:w,zw-+

Jst(o,zwz'=

‘A(& J0 =_ Ja

of

the

J

y) dx’ +

+ Aydy f

path,

J

y)dz’ +

&(x1,

can be obtained

ca’b’C’(,-f& 5 L0.0.0)

independent

x)dz’ -t

‘w, Jb

from

J

r&O,

in particular,

They are

planes.

z); (13)

x’)dx’ = g”(z, x), z’)dx’ -I- u. 3 &(z, y’)dy’ = Rx,

: &(a,

y’)dy’ + u. = ax,

other

by cyclic

Since

A. is

we can

take

z);

y),

changes.

Y>. Here u0 is

rotation-free,

this

(13)

represent

continuous

boundary

values,

the

line

integration

(14)

J:t(o,z')dx’ -I- J:jcYt, c)dy’ -I- J;x,(zt, b)dz’,

The expressions

collected

my, z),

:b(O,

the

A,,&).

other

z’)dz’ + uo = j,(y,

=&(z’, z)dz’ + I&c, JC J

-

Y> c> -

above

integration

%l -

boundary

4~9 J> z) - J: ~;(y’, z)dy’ -t J+,

9

is

now this:

does

is

u(o, y> z>-

Each of

is

= &,(x, Y), A, IPet=

boundary-value

rotation-free partial

AI,,

r>;

a, and it

To simplify

0 < x & a, 0
A,Iz=o= Rx,

Y>,

and we then

boundary

condition

the

KY> z), Allr=a- r"(Y> z); A,I,, = uy, z), AIIXS.= i,(ysz);

fCl*==Ll - a(,,

find

symmetry,

more complicated.

AI,=0 = B(z, z), 4lYCO = &xz> z); A*lu=b- &(z,

Let A=grad

of

Z toa:

rotation-free

A,I**

the

169

Field

heavy.

generality.

divergence-free,

Hydromagnetic

by separation

of

variables,

it

is

170

Solar Hydromagnetic Field

not difficult to find the continuous function u over the closed region to be ra~

u ( x , y, . ) =-

n~z

~ sh(ap.,.)

.... i

fn~Z

sia ~ +

sin

(--!.,.sh[(x

-- a)p.,.]

a

(--~.,,.sh[(y

--

p,.,., q . , . ,

b)q.,.]+~,.,.sh(yq.,,.))

n~ry

+ sir, ~sh(cr,.,.)sln---if- (--h,..,,sh[(z -i n which

+ l,.,.sh[x#.,.])

7J~X - -

sh(bq.,.) tn~x

c

c)r,.,.] + h,,.,.sh(zr,...))}.

615)

rm,. are the same as i n (10) and ~m,., I , . , . , ~m,., g,=,., h . . . .

the Fourier coefficients with the orthogonal system of sines as base.

k,.,.

are

Differentiation of

(15) then gives the expressions for ~, that is, for 83 .

6.

A PRACTICAL APPLICATION

As an application of the method set forth in this paper, we shall consider the coupling of the magnetic and velocity fields on the Sun.

According to Gold and Zirin [5,6], what makes

the magnetic field of a solar active region complicated is the movement of matter in the photosphere, with the gas dragging with it the "feet" of the magnetic field lines.

In recent

years Low and Nakagawa have examined the evolution of the force-free field caused by the motion of the "feet" of magnetic lines [8], and also the question of the dragging of the magnetic

field by the velocity field [7].

However, they have not acheived a self-

consistent solution for the magnetic and velacity fields. dimensional problem.

Here we shall consider a two-

We shall study how the magnetic field changes and evolves when the

feet of the magnetic lines are moved by the fluid.

We assume both the magnetic and velocity

fields to be two-dimensional vector fields, and the electric current to have only a zcomponent.

The fluid is taken to be homogeneous and incompressible,

resistance and viscosity are neglected, so that we have -Vp+ g = 0.

and electrical The boundary conditions

are made in accordance with the "frozen-in" principle: for the magnetic field, the normal components, and for the velocity field, the tangential components.

On the boundary y = Yl '

we use the following approximation in accordance with the in-step motion: O

and a l l

the other boundary values

consistent boundary values.

B

are set to zero.

,=,,

Fig.

"

1 shows t h e e v o l u t i o n

in the

The initial field B 0 (see Fig. 2) is a two-dimensional

potential field, V 0 ~ O, and B 0 can be calculated from the boundary values according to (S). We take the spots to have diameters of 4,7 x 109 cm, and to be 8.3x 109 cm apart, the peak value of the field to be 103 G, a density of I0/4~ x 10 -8 g cm -3, over the region -12.0 ! x < +12.0, 0 5 Y 5

11.0, (in units of 109 cm), with 24 x 20 sample points.

are given in TABLE i. The o~r~je8 illustrated in Fig. 3.

The boundary values

in the magnetic and velocity fields after 549 sec are

It can be seen that the two spots are indeed getting closer to each

Solar Hydromagnetic Field

B,.(~',y~)

boundary magnetic field

tO

" "&"~<'x

~ t ~t

l

t~f2

~t 2

l

171

/t=q

boundary flow field

/ t~tO i

etic

line

riginal

N---~ Fig. 1 Consistent

]8

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~

"

,

,

,

,

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o

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"

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'

'

*

'

°

/~<.

9 N

magnetic line

of boundary values

.

.

shift

~---S

evolution

.

.

.

. . . .

~ 3

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.

.

.

.

.

.

.

.

]0

.

.

.

]4

.

:

after

11

13

-units :

75"'~"~J, / q 9 ~

{t

2~

S

1000 G Fig. 2

The intitial magnetic field

x-step:

1.044 × 109cm;

y-step: 0.579 x 109cm

other and dragging the magnetic field with them. The distribution of the electric current is shown in Fig. 4.

(G)

(km/sec)

av.(x, yOl,=,,

(G)

B,

aB,(x, y,)l,-,,

( 1 0 9 cm)

x

i

(km/sec)

Ov.(x, Y,)I . . . .

location

(G)

(G)

( 1 0 9 cm)

i

eB,(=,y,)I ....

B,

x

location

-0.120

I-2.4XIO-*

-3.00

0.522

--3.05

--48.0

--0.60

--0.935

--40.0

2.610

1.566

--4.32

15

-t.00

--25.2

--935

3.654

16

-0.985

24.0

--940

4.698

17

-0.762

-0.490

6.86

--137

-- 455

32,8

6.786

--0.293

1.29

--34.0

--0.153

f,1XIO-*

--7.50

8.874

7.830

0.985

0.762

0.490

21

--24.0

--32.8

--6.86

20

940

19

9

10

--0.068

0.00

--9.0XlO-'

9.918

22

1.00

25.2

935

11

23

0.600

4.32

48.0

--1.566

--0.018

0.00

- - 5 . 0 X I O -a

10.962

0.935

40.0

305

--4.698 --3.654 I-2.610

455

--5.742

8

137

-6.7860

6

5.742

18

0.293

0.153

0.068

0.018

0.00

14

-- 1.29

--6.1XIO-*

0.00

0.00

0.00

13

34.0

7.50

--7.830

--8.874

9.0XlO -t

--9.918

L O x I O -s

-10.962

7,0X 10-'

-12.006

5

4

Table 1 Boundary Values

0.00

0.00

- - 7 . 0 x l O -4

12.006

24

0.120

2 . 4 X I 0 -z

3.00

--0.522

12

E

b.t. o

t,-,. o

0Q

H o

o

.%

Solar Hydromagnetic Field

4,

6

8

Change in flow field 4!

lO, - - --'r---12 14,

16

18

22

........... 3

j-p

scal(

5

/

40 G

4

j:

77.-~_x_

scale : _ ~

.42 km/s F

173

6

8

10

12

14

16

18

20

22

~

Change in 6 magnetic field ~

.

.

.

.

.

.

.

6 /

8Bl 4[

.

.

.

.

•

.

.

.

.

.

.

~4

(

2

N

~

~

S

Fig. 3 Changes in the fields after 549 seconds lO-y

8"

~I

--!

4 i

Fig. 4 Distribution of current (units: i0 -I0 c.g.s.)

ACKNOWLEDGMENT~

We thank Comrades Tong Fu and Wang Chang-bin of the Purple Mountain

Observatory for many positive and helful suggestions.

REFERENCES [ 1 ] [2 ]

[3] [4]

M. D. AItm~huler et aJ., 8olaf Phys., ~ ( 1 9 7 3 ) , 153. M . D . Altschuler et al., 8olaf Phys., 3 ( 1 9 6 8 ) , 466.

A.N. Kikhonov, "Methods of Mathematical Physics" (Chinese Translation) Volume 2, p.554. "Peking University "Methods of Calculation" (in Chinese), Peking University, Vol. l, p.,~9.

[ 5 ] T. Gold, NASA sp-50, Physics of solar flares, p. [6 ] [ 7]

389. ]~. Zirin, L ~ . U. Bympo., no. 43 (1972), 237--242. R . H . Levine, Y. Nakagaws. AP. J., 1 N ( 1 9 7 4 ) , 703.

[ 8 ] B.C. Low, AP. J., 21)-(1977), 2~4.