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Equilibrium and non-equilibrium of two-arch structures in the solar atmosphere
Chin.Astron.Astrophys.(1989)13/4,432-441 a translation of
Acta Astrohpys.Sin.11989/9/4,363-372
3, Pergamon Press plc. Printed in Great Britain 0275-1062/89$10.00+.00
EQUILIBRIUM ANDNON-EQUILIBRIUM OFTWO-ARCH STRUCTURES IN THESOLAR ATMOSPHERE’ SUN Kai Departrent of Geophysics, Peking University LIU Bu-lin Beijing Astronomical Observatory, Academia Sinica
Received
1988 January
9
We have derived a similarity solution in the form of a power series of the nonlinear magnetostatic equilibrium equation and identified the region of convergence in the parameter space. Within this region we have equilibrium configurations, outside it, non-equilibrium configurations. An example of nonlinear equilibrium configuration is shown graphically. Our results can be used to explain the generation of multi-ribbon flares, the two-arch structures and the structures of the magnetic fields above elongated sunspot groups. Abstract
Key words:
1.
Sun--solar --nonlinear
activity--equilibrium--non-equilibrium differential equations
INTRODUCTION
The magnetostatic equilibrium field in the solar atmosphere has complex structures and the present available solutions of the magnetostatic equilibrium equation are unable to describe them in any detail. The search for new solutions is of great concern. Some solar astrophysicists have recognized that such solutions can help us to understand not only the solar magnetic field and the observed multi-wavelength results of the surface structures but also the causes of such phenomena as the solar flares. (See e.g., Ill). The fact that quiet prominences mainly show horizontal ribbon structures allows us to set up two-dimensional models of magnetostatic equilibrium. Among the early papers on this subject are [2J and [3]. Solutions of the equilibrium equation including gravity have been obtained in [4-91. In this paper we shall use the similarity method to solve the two-dimensional equation. The equation, expressed in non-dimensional is a set of nonlinear partial differential equations (linear variables, with respect to the highest order derivative). The ordinary differential equation encountered in the solution is also nonlinear and we have found a series solution of this latter with a determinate region of convergence. We shall use this solution to explain the magnetostatic
‘Program
supported
by The National
Natural
Science
Foundation
Equilibrium
of Two-Arch Structure
433
equilibrium structures in the solar atmosphere. Section 2 gives the basic equations used here. Section 3 derives the power series similarity solution and discusses its convergence. Section 4 discusses further some aspects of the convergence of the power series relevant to the solution of the problem. Section 5 gives a brief summary. 2. BASIC EQUATION In rectangular are
coordinates
the equations
of magnetostatic
equilibrium
i$(vxB)xB-vp-ppp~-o,
(1)
v-B-0
(2)
To these we add the law of perfect
gas
$ PT
P’
Here, d is the magnetic field, p is the pressure, p is the density, g is the gravity, regarded as a constant along the negative z-direction. z is the unit vector, m is the mean particle mass, T is the temperature and k is the Boltzmann constant. We assume all the variables are independent of y and that t= const. Then B can be expressed as
(z, B,(F), -$g,
B-
where F= F(x,z). As in Ref.[Sl, (11 and (2) can be transformed V’F
f
&
4nP(F)e
-2
(3)
for an isothermal into + +
p:(p)}
-
atmosphere,
equations
0
(4)
equivalent to taking P= Pf Fbexpt-z/H’), H’ = k27mg being the pressure scale height of the isothermal atmosphere. Butting x= 1X, z= iz, IBI = &, F= &F, p= fP$tColexpf-z/e/H’), we dimensionalize 141 to give
where X = l/IT, PO = EZMJ/&J~,V2= a2/aX2t a’/a-i-‘. 1 and 00 are the characteristic length and magnetic field intensity, respectively, Co is a constant representing the underlying hydrostatic pressure and p0 can be regarded as the characteristic pressure. We shall restrict our discussion in this paper to the case where &tFl =const. and .+ #, -& IV(F) I - 4r43* be the pressure scale where a.b are two parameters. Let X= l/X = P/i height in units of the characteristic length 1. For simplicity we shall omit the overbars in (5) which becomes
434
SUN Kai
Equation
3.
(6)
will
be the
basic
I LIU Bu-lin
equation
of
this
paper.
BOWERSERIES SIMILARITY SOLUTION
As is well-known, if a partial differential invariant with respect to a one-parameter F’ -
F’(t,
equation in two variables is (e) Lie transformation group
E, F, E)
x* - I*(*,
z, F, 6)
z* - z’(x.
z,
(7)
F, 6)
then the partial differential equation can be changed into differential equation. We shall use this property to solve Expanding (7) at c= 0, we have F* -
F + E&
an ordinary equation (6 ).
I, F) + 0(6’)
r* - X + 6$(X, 3, P) +.0(6’) a* -2
(8
+ er(x, I, F) + O(s’)
here T), 5, r are simplify equation consider the Lie
infinitesimals. (61. Experience transformations
F* -
F +7(x,
I* -
x +
g(x, “)E
3, F)k
z* -
2 +
7(x, 2)s
-
We shall use the expressions shows that in the derivation of the form
F + Flr(x,
(8) to we need
Z)E,
with, that explained in Using a method similar to, but not identical [lO,lll, we can find the infinitesimals that will make equation (61 First, using the same method as [lO,lll, we have invariant.
(9)
Equilibrium of Two-Arch Structure
435
where 6 is a solution of equation (61. Equation (6) is invariant under transformation (9) if and only if any solution of (6), f=e(x,t), satisfies the following partial differential equation (11) If we proceed further along the lines of 110,111 then we shall not get good results. But if we first set &
- 0,
‘--r,
(12)
and then proceed as in [lO,ll], then we can be sure of finding a good transformation group that will leave (6) invariant. This method is new and was first used by us in our paper [121. Substituting equations (61, (9) and (10) into equation (11) and collecting the terms of the same order derivative of 6, we obtain the determining equations 2A,-L, -;:..-0 Zh.
r_ - r,,
-
t. + r. F(h.,
+ h..) -(h
-
-
0
0 + 4,nzh,F%-+’ 2r.)4noFbe
-
‘T
(13)
F’c-i
-ii _ 0
Solution of these equations gives the infinitesimalsthat preserve equation (6): FL -FFA
(14)
Here, A, E and D are three constants of integration. We shall confine our discussion to the case of D= 0. The characteristic equation of the similarity solution is then dx -Ecos
-1.
2H
da ea”H
E sin ~cztrH
2H
+
dF AH(b
-
1)
-z
(15)
First we derive the ordinary differential equation corresponding to (6) for the case b% 1. In this case, the characteristic equation is precisely the equation shown at (15). Integrating the first equality we obtain the similaritv variable
436
SUN Kai
p#4
AH(6-1)
+
x
sinm
E
t-
COI-
& LIU Bu-lin
x
(16)
2H
Integrating
the second equality,
we obtain (171
F(x, 2) - f(C)(e *RH/ws (z/ZH))* Substituting equation for 6,
equation (17) in equation f(C) in the case bf 1: +
A'H'(b E'
l>'
-
f,t
(6)
we obtain
the differential
2(b + 1) c/g* + 2@ 1)f + 16xd-W - 0 (b + 1>’
;
-
(b-l)
(161
In our previous paper [12] we have already found the similarity solution of equation (6) in the cases of bf 0 and 1. In this paper we shall describe the solution of the nonlinear equation (18) for b= 2 and 3 solution of with particular empahsis on b= 3. Consider a power series the form f - 00+ O,C_’ + I&’
+
(191
. . - + 0.p + * * *
Since the equation
involves the parameters A, E, If, a, the coefficients hence the properties of the whole solution will also depend on these parameters, in general. in the power series,
Substituting
the power series
dn + 11% + (a +
(19)
in equation
2)(n - l)o._, * $(b
i.i 4e
bf bo#.f..
[ W+.,P.. c
I>,>,>... d+,,+,,+...l. .+,+r+...-r
I>’ - no, . 2 b-l- I~+a.* b-1
.
e!f!g!***
(18) gives
1*
23
(b-l)
(20)
16noH’- 0
are ordered as indicated to avoid duplicate counts, the suffices and the cases of i= j or j= k must also be counted once only. Leaving only terms involvinq &I on the left side we have b+l + I6xoH’bot’ a. *(a + 1) -226++2 (b - 1) b-1 Here,
I
- -(*
-
I)(n - 2) $$ i.r,,r.-,
-
16n.H’
(b - l)‘lLl b br e!f!g!*
c i&i+,*...
,>,>,%’ .i+,,+rk+...-. .+,+#+...-b This
is a recurrence
formula
valid for any value of b. For dependence of the coefficients f(C) - alr
+ er
-*
e$jef.. .
for the coefficients b = 3, the explicit
(21)
of
the series
expression on the parameters is
- 4 (F
p
+ 244
A’ff - 4 n&f%; - 4 -+ lroH”17;* [ ( )(
o,C_’ - J (F
T
+ 2*.Wa:
and is
showing the
+ 2..Ip.:)
0,5-’
+**.(2’) >Io,iy’
437
Equilibrium of Two-Arch Structure
with a~,=0 and al, a2 arbitrary. Equation (21) can now be written in a form more convenient for calculation on the computer: (n - l)(n - 2)4 fg 0.
-
.-1
a.4
n'--a+2
-
16rcrP 2 i.1. i+,+ i-0 -.
/(n'-3n+ 2) (23)
For estimating the size of &I we have
(24) where a~n-l~max is the largest absolute value among at, a2, . ..a.,-~. The factor (n'tn-2)/2 in the inequality 124) is the number of terms under the summation sign in (23). Expression (24) can be simplified into (25) A= l(nt2)/(n-2)1/2.It takes its maximum value of 5/2 at n=3 tends to l/2 as n increases. If we take
and
16noH'.$1 "~.-o.LzkU..l' < + we can ensure that the second term of l&l < l/2. This is to require (26)
I < ;/I/Bo~~HJ ! =(.-w.a)mrx For example, for a=O.l, H=2,
we require
E’ > 8A’H’
(27)
and if (26) holds, then for any arbitrary al and a2 less than I, we shall have 1~1 < 1. Over a large part of the 1x, aI domain we can ensure c-' <'I.Within this part, if l&l (1, the negative power series (19) certainly converges, and the nonlinear equation (6) certainly has a solution. We now give an example of a convergent series with the parameters satisfying the above conditions. We took the following parameter values: A = I, H=
2, E = 6, a = 0.1, al = 0.2, b = 3, a;!= -0.18.
and a calculation on the computer using the recurrence relation (231 gave the following coefficients:
0,- -0.16931, U,A 7.6721 x IO-', 0, - - I.8707 x 10-2
a,- 0.1523H a, -
-0.11i01
o.- 8.1802 X IO-'
438
SUN Kai
ay -
-1.02991
011 -2.Jill
x
01, -
-2.1999
at, -
I.8373
a,, -
-
x
& LIU Bu-lin
X IO-’
B,* -
10-j
(11% - 2.8926 X IO-’
-5.1682
x
10-z
x lo-’
I?,, -
-1.3508
1W’
8‘. -
4.1584
x
0,s -
7.8217
X IO-’
1.34 14 x
lo-’
x
lo-’
IO-’ (28)
This numerical example demonstrates the convergence of the series when the parameters satisfy the stated conditions. As a negative example, if we take A = 5, H=
2, a = 1, al
1, a;:= 30.11,
q
then we shall get the following coefficients: ~1 -
-200.53
0, -
-6038.0
Q, -
-1543H
a, -
87567
0, -
1.2012 x
a, -
-2.9636
a,, -
4.5556
IO’ X 10’
x 10“
x
IO
o, -
-6.0342
x
e0 -
-1.3515
x
a,, -
6.4315
x
10~ 10‘0
10“
8,) -
-3.0970
x IO”
at+ -
-1.4858
x
911 -
-7.062’)
x 10’5
tit‘ -
2.2093
X 10”
011 -
3.2739
10”
01” -
-
x
1.3515 x
IO’S
to+”
(28')
and so on, The series will certainly not converge. This means, for some parameter vaiues, there may not exist any configurationsof magnetostatic equilibrium.
4. FURTHER DISCUSSION ON CONVERGENCE In the last Section we gave a detailed discussion of the case b=O. We shall now discuss particularly the quantities of direct physical interest, & and &, in the same case. We shall concentrate at the location e= 0 and make only auxiliary remarks concerning t>O, First, we have
&_E_L.C 02 + (_*,t-’
-lH -
co,
I (or
.A
c o,r +
2H
2&p
-
* . . . * .)
_.cL -zH
For z = 0. we have
a&-” -I- f * -)
(291 r 1
Equilibrium
of
Two-Arch
Structure
439
(30)
-~(.,5;‘+2a,S+3.,S;‘+ 2H where
Expression
(30)
can also
be written
as
l2H +o,
The ratio
of
(
3 C’ + . . . . . . I ) * 1_2All&,_X I:’ 2H
I-
two successive
terms
is (31’)
and tends Next,
to
at+l/ai
as
i tends
to
infinity.
we have
(1 -i)sinX--
1
2H(,+9n79 The ratio
of
2AH 2H
_
successive
terms
is
2AH E+
Jnm
x
) and tends
to
at+l/ai
(32)
E (
as
i tends
to
1
(33)
infinity.
Thus, as long as C-’ < 1 and an is bounded, BX and Bz must converge. .z = 0 and sin(x/2X) = -2.&f/E, Co-’ assumes the maximum value (Gil)‘.,‘ For the parameter E= 6. we have
-
I/
J
1 -=
values
(34)
E’ in the
At
above
convergent
example,
4= 1, H=
2,
SUN Kain & LIU Bu-lin
440
( C’L..
-
(L’L.
-
( 5LT’L‘ .... . .
1.34164 1.80000 2.4 1495
(35)
hence the question of convergence (35) shows that the maximum value the series. We can certainly find
has to be considered further. does not increase very greatly series ai such that
and make the
converge.
series
(31)
and (32)
However, along
The above exercise shows that the region of convergence is more restricted near s=O. If we identify the location with the photosphere then the possiblity of equilibrium turning into non-equilibrium should be somewhat greater. A further study showed that, for the parameters in the convergent example, if we restrict to ~~0.73545, then Co-’ will always be less than 1. The range OirS0.73545 is below 3% of the width of Fig. 1. When we consider that region of convergence indicated.
5.
the coefficients should be larger
may be positive than the above
and negative, discussion
the
BRIEF SUMMARY
In this paper we have given a similarity power-series solution of the equation of magnetostatic equilibrium. We have described in detail the solution in the b= 3 case. For a certain range of parameter values, the series is certainly convergent. For a specific set of parameter values, the series is explicitly given at (281. In this example, the series is certainly convergent for s>O.73545. The magnetic field configuration corresponding to this series is shown in Fig.l. The configuration in Fig1 corresponds to field lines starting from a fairly strong and concentrated positive pole at the centre and ending in weaker and more scattered negatire poles on the sides. It can be taken to be a nonlinear two-arch structure. We can regard al, a2 and the parameters of the equation as slow functions of time and as they show change, the configuration changes. When, following the changes in the parameters, the series changes from convergence to divergence, the corresponding equilibrium configuration collapses. For example, if the parameters change from the values in the convergent example above to the divergent example while Hand b remain constant, then equilibrium will certainly pass into non-equilibrium , and the structures shown in Fig. 1 will disappear and we shall see a rapid change in the field. This could be the cause of the multi-ribbon flares [131.
Equilibrium of Two-Arch Structure
441
2nH
41rH
2xH
Fig. 1 An example of nonlinear magnetostatic equilibrium. The values of the magnetic equipotentials are l-0.0258, 2-0.0774, 3-0.1290, 4-0.1806, 5-0.1978, 6-0.2150, 7-0.2322, 8-0.2838
From the choice of P(F) we have, on omitting the overbars, 1 -Fb+'+ b+l
C+,(-$.)
and our nonlinear solution can give the particular distribution of pressure in equilibrium with the magnetic field configuration. Fig. 1 is a two-dimensionaldiagram. In the y-direction we may image small changes representing arcade structures. It helps us to visualize the field over elongated sunspot groups and provides a reference when studying the arcades seen as the limb. The transition point from equilibrium to non-equilibrium is a difficult subject as it depends on so many parameters. REFERENCES
[ll
Hood, A.W. and Priest, E.R., Solar Phenomena in Stars and Stellar Systems (1981) D.Reidel Publ. [21 Dunge:r,J.W., Mon.Not.R.Astron.Soc113 (19531, 180. [31 Brown, A., Astrophys. J., 128 (19581, 646. Low, B.C., Astrophys. J., 212 (19771, 234. (41 Priest, E.R. and Milne, A.M., Solar Phys. 65 (1980). 315. (51 Birn, J., Goldstein. H. and Shindler, K., Solar Ph.vs.57 (19781, 81. [61 Jockers, K., Solar Phys. 56 (19781, 37. [71 Heyvaerts, J., Lasry, J.M. and Witomskey, G., Lecture Notes Math., [81 782 (19801, 160. Melville, J.P., Hood, A.W. & Priest, E.R., Solar Phys., 92 (1984). 191 [lOI Ovsiannikov, L.V., Group Analysis of Differential Equations, (19821, Academic Press. I.111Bluman, G.W. and Cole, J.D., Similarity Method for Differential Equations (1974) Springer-Verlag. 1121 SUN Kai, ZHANG Dong-mei and WANG Yan, Kexue Tongbao 32 (1987) 1066. 1131 TANG F., Solar Ph.vs.,102 (1985) 131.
Acta Astrohpys.Sin.11989/9/4,363-372
3, Pergamon Press plc. Printed in Great Britain 0275-1062/89$10.00+.00
EQUILIBRIUM ANDNON-EQUILIBRIUM OFTWO-ARCH STRUCTURES IN THESOLAR ATMOSPHERE’ SUN Kai Departrent of Geophysics, Peking University LIU Bu-lin Beijing Astronomical Observatory, Academia Sinica
Received
1988 January
9
We have derived a similarity solution in the form of a power series of the nonlinear magnetostatic equilibrium equation and identified the region of convergence in the parameter space. Within this region we have equilibrium configurations, outside it, non-equilibrium configurations. An example of nonlinear equilibrium configuration is shown graphically. Our results can be used to explain the generation of multi-ribbon flares, the two-arch structures and the structures of the magnetic fields above elongated sunspot groups. Abstract
Key words:
1.
Sun--solar --nonlinear
activity--equilibrium--non-equilibrium differential equations
INTRODUCTION
The magnetostatic equilibrium field in the solar atmosphere has complex structures and the present available solutions of the magnetostatic equilibrium equation are unable to describe them in any detail. The search for new solutions is of great concern. Some solar astrophysicists have recognized that such solutions can help us to understand not only the solar magnetic field and the observed multi-wavelength results of the surface structures but also the causes of such phenomena as the solar flares. (See e.g., Ill). The fact that quiet prominences mainly show horizontal ribbon structures allows us to set up two-dimensional models of magnetostatic equilibrium. Among the early papers on this subject are [2J and [3]. Solutions of the equilibrium equation including gravity have been obtained in [4-91. In this paper we shall use the similarity method to solve the two-dimensional equation. The equation, expressed in non-dimensional is a set of nonlinear partial differential equations (linear variables, with respect to the highest order derivative). The ordinary differential equation encountered in the solution is also nonlinear and we have found a series solution of this latter with a determinate region of convergence. We shall use this solution to explain the magnetostatic
‘Program
supported
by The National
Natural
Science
Foundation
Equilibrium
of Two-Arch Structure
433
equilibrium structures in the solar atmosphere. Section 2 gives the basic equations used here. Section 3 derives the power series similarity solution and discusses its convergence. Section 4 discusses further some aspects of the convergence of the power series relevant to the solution of the problem. Section 5 gives a brief summary. 2. BASIC EQUATION In rectangular are
coordinates
the equations
of magnetostatic
equilibrium
i$(vxB)xB-vp-ppp~-o,
(1)
v-B-0
(2)
To these we add the law of perfect
gas
$ PT
P’
Here, d is the magnetic field, p is the pressure, p is the density, g is the gravity, regarded as a constant along the negative z-direction. z is the unit vector, m is the mean particle mass, T is the temperature and k is the Boltzmann constant. We assume all the variables are independent of y and that t= const. Then B can be expressed as
(z, B,(F), -$g,
B-
where F= F(x,z). As in Ref.[Sl, (11 and (2) can be transformed V’F
f
&
4nP(F)e
-2
(3)
for an isothermal into + +
p:(p)}
-
atmosphere,
equations
0
(4)
equivalent to taking P= Pf Fbexpt-z/H’), H’ = k27mg being the pressure scale height of the isothermal atmosphere. Butting x= 1X, z= iz, IBI = &, F= &F, p= fP$tColexpf-z/e/H’), we dimensionalize 141 to give
where X = l/IT, PO = EZMJ/&J~,V2= a2/aX2t a’/a-i-‘. 1 and 00 are the characteristic length and magnetic field intensity, respectively, Co is a constant representing the underlying hydrostatic pressure and p0 can be regarded as the characteristic pressure. We shall restrict our discussion in this paper to the case where &tFl =const. and .+ #, -& IV(F) I - 4r43* be the pressure scale where a.b are two parameters. Let X= l/X = P/i height in units of the characteristic length 1. For simplicity we shall omit the overbars in (5) which becomes
434
SUN Kai
Equation
3.
(6)
will
be the
basic
I LIU Bu-lin
equation
of
this
paper.
BOWERSERIES SIMILARITY SOLUTION
As is well-known, if a partial differential invariant with respect to a one-parameter F’ -
F’(t,
equation in two variables is (e) Lie transformation group
E, F, E)
x* - I*(*,
z, F, 6)
z* - z’(x.
z,
(7)
F, 6)
then the partial differential equation can be changed into differential equation. We shall use this property to solve Expanding (7) at c= 0, we have F* -
F + E&
an ordinary equation (6 ).
I, F) + 0(6’)
r* - X + 6$(X, 3, P) +.0(6’) a* -2
(8
+ er(x, I, F) + O(s’)
here T), 5, r are simplify equation consider the Lie
infinitesimals. (61. Experience transformations
F* -
F +7(x,
I* -
x +
g(x, “)E
3, F)k
z* -
2 +
7(x, 2)s
-
We shall use the expressions shows that in the derivation of the form
F + Flr(x,
(8) to we need
Z)E,
with, that explained in Using a method similar to, but not identical [lO,lll, we can find the infinitesimals that will make equation (61 First, using the same method as [lO,lll, we have invariant.
(9)
Equilibrium of Two-Arch Structure
435
where 6 is a solution of equation (61. Equation (6) is invariant under transformation (9) if and only if any solution of (6), f=e(x,t), satisfies the following partial differential equation (11) If we proceed further along the lines of 110,111 then we shall not get good results. But if we first set &
- 0,
‘--r,
(12)
and then proceed as in [lO,ll], then we can be sure of finding a good transformation group that will leave (6) invariant. This method is new and was first used by us in our paper [121. Substituting equations (61, (9) and (10) into equation (11) and collecting the terms of the same order derivative of 6, we obtain the determining equations 2A,-L, -;:..-0 Zh.
r_ - r,,
-
t. + r. F(h.,
+ h..) -(h
-
-
0
0 + 4,nzh,F%-+’ 2r.)4noFbe
-
‘T
(13)
F’c-i
-ii _ 0
Solution of these equations gives the infinitesimalsthat preserve equation (6): FL -FFA
(14)
Here, A, E and D are three constants of integration. We shall confine our discussion to the case of D= 0. The characteristic equation of the similarity solution is then dx -Ecos
-1.
2H
da ea”H
E sin ~cztrH
2H
+
dF AH(b
-
1)
-z
(15)
First we derive the ordinary differential equation corresponding to (6) for the case b% 1. In this case, the characteristic equation is precisely the equation shown at (15). Integrating the first equality we obtain the similaritv variable
436
SUN Kai
p#4
AH(6-1)
+
x
sinm
E
t-
COI-
& LIU Bu-lin
x
(16)
2H
Integrating
the second equality,
we obtain (171
F(x, 2) - f(C)(e *RH/ws (z/ZH))* Substituting equation for 6,
equation (17) in equation f(C) in the case bf 1: +
A'H'(b E'
l>'
-
f,t
(6)
we obtain
the differential
2(b + 1) c/g* + 2@ 1)f + 16xd-W - 0 (b + 1>’
;
-
(b-l)
(161
In our previous paper [12] we have already found the similarity solution of equation (6) in the cases of bf 0 and 1. In this paper we shall describe the solution of the nonlinear equation (18) for b= 2 and 3 solution of with particular empahsis on b= 3. Consider a power series the form f - 00+ O,C_’ + I&’
+
(191
. . - + 0.p + * * *
Since the equation
involves the parameters A, E, If, a, the coefficients hence the properties of the whole solution will also depend on these parameters, in general. in the power series,
Substituting
the power series
dn + 11% + (a +
(19)
in equation
2)(n - l)o._, * $(b
i.i 4e
bf bo#.f..
[ W+.,P.. c
I>,>,>... d+,,+,,+...l. .+,+r+...-r
I>’ - no, . 2 b-l- I~+a.* b-1
.
e!f!g!***
(18) gives
1*
23
(b-l)
(20)
16noH’- 0
are ordered as indicated to avoid duplicate counts, the suffices and the cases of i= j or j= k must also be counted once only. Leaving only terms involvinq &I on the left side we have b+l + I6xoH’bot’ a. *(a + 1) -226++2 (b - 1) b-1 Here,
I
- -(*
-
I)(n - 2) $$ i.r,,r.-,
-
16n.H’
(b - l)‘lLl b br e!f!g!*
c i&i+,*...
,>,>,%’ .i+,,+rk+...-. .+,+#+...-b This
is a recurrence
formula
valid for any value of b. For dependence of the coefficients f(C) - alr
+ er
-*
e$jef.. .
for the coefficients b = 3, the explicit
(21)
of
the series
expression on the parameters is
- 4 (F
p
+ 244
A’ff - 4 n&f%; - 4 -+ lroH”17;* [ ( )(
o,C_’ - J (F
T
+ 2*.Wa:
and is
showing the
+ 2..Ip.:)
0,5-’
+**.(2’) >Io,iy’
437
Equilibrium of Two-Arch Structure
with a~,=0 and al, a2 arbitrary. Equation (21) can now be written in a form more convenient for calculation on the computer: (n - l)(n - 2)4 fg 0.
-
.-1
a.4
n'--a+2
-
16rcrP 2 i.1. i+,+ i-0 -.
/(n'-3n+ 2) (23)
For estimating the size of &I we have
(24) where a~n-l~max is the largest absolute value among at, a2, . ..a.,-~. The factor (n'tn-2)/2 in the inequality 124) is the number of terms under the summation sign in (23). Expression (24) can be simplified into (25) A= l(nt2)/(n-2)1/2.It takes its maximum value of 5/2 at n=3 tends to l/2 as n increases. If we take
and
16noH'.$1 "~.-o.LzkU..l' < + we can ensure that the second term of l&l < l/2. This is to require (26)
I < ;/I/Bo~~HJ ! =(.-w.a)mrx For example, for a=O.l, H=2,
we require
E’ > 8A’H’
(27)
and if (26) holds, then for any arbitrary al and a2 less than I, we shall have 1~1 < 1. Over a large part of the 1x, aI domain we can ensure c-' <'I.Within this part, if l&l (1, the negative power series (19) certainly converges, and the nonlinear equation (6) certainly has a solution. We now give an example of a convergent series with the parameters satisfying the above conditions. We took the following parameter values: A = I, H=
2, E = 6, a = 0.1, al = 0.2, b = 3, a;!= -0.18.
and a calculation on the computer using the recurrence relation (231 gave the following coefficients:
0,- -0.16931, U,A 7.6721 x IO-', 0, - - I.8707 x 10-2
a,- 0.1523H a, -
-0.11i01
o.- 8.1802 X IO-'
438
SUN Kai
ay -
-1.02991
011 -2.Jill
x
01, -
-2.1999
at, -
I.8373
a,, -
-
x
& LIU Bu-lin
X IO-’
B,* -
10-j
(11% - 2.8926 X IO-’
-5.1682
x
10-z
x lo-’
I?,, -
-1.3508
1W’
8‘. -
4.1584
x
0,s -
7.8217
X IO-’
1.34 14 x
lo-’
x
lo-’
IO-’ (28)
This numerical example demonstrates the convergence of the series when the parameters satisfy the stated conditions. As a negative example, if we take A = 5, H=
2, a = 1, al
1, a;:= 30.11,
q
then we shall get the following coefficients: ~1 -
-200.53
0, -
-6038.0
Q, -
-1543H
a, -
87567
0, -
1.2012 x
a, -
-2.9636
a,, -
4.5556
IO’ X 10’
x 10“
x
IO
o, -
-6.0342
x
e0 -
-1.3515
x
a,, -
6.4315
x
10~ 10‘0
10“
8,) -
-3.0970
x IO”
at+ -
-1.4858
x
911 -
-7.062’)
x 10’5
tit‘ -
2.2093
X 10”
011 -
3.2739
10”
01” -
-
x
1.3515 x
IO’S
to+”
(28')
and so on, The series will certainly not converge. This means, for some parameter vaiues, there may not exist any configurationsof magnetostatic equilibrium.
4. FURTHER DISCUSSION ON CONVERGENCE In the last Section we gave a detailed discussion of the case b=O. We shall now discuss particularly the quantities of direct physical interest, & and &, in the same case. We shall concentrate at the location e= 0 and make only auxiliary remarks concerning t>O, First, we have
&_E_L.C 02 + (_*,t-’
-lH -
co,
I (or
.A
c o,r +
2H
2&p
-
* . . . * .)
_.cL -zH
For z = 0. we have
a&-” -I- f * -)
(291 r 1
Equilibrium
of
Two-Arch
Structure
439
(30)
-~(.,5;‘+2a,S+3.,S;‘+ 2H where
Expression
(30)
can also
be written
as
l2H +o,
The ratio
of
(
3 C’ + . . . . . . I ) * 1_2All&,_X I:’ 2H
I-
two successive
terms
is (31’)
and tends Next,
to
at+l/ai
as
i tends
to
infinity.
we have
(1 -i)sinX--
1
2H(,+9n79 The ratio
of
2AH 2H
_
successive
terms
is
2AH E+
Jnm
x
) and tends
to
at+l/ai
(32)
E (
as
i tends
to
1
(33)
infinity.
Thus, as long as C-’ < 1 and an is bounded, BX and Bz must converge. .z = 0 and sin(x/2X) = -2.&f/E, Co-’ assumes the maximum value (Gil)‘.,‘ For the parameter E= 6. we have
-
I/
J
1 -=
values
(34)
E’ in the
At
above
convergent
example,
4= 1, H=
2,
SUN Kain & LIU Bu-lin
440
( C’L..
-
(L’L.
-
( 5LT’L‘ .... . .
1.34164 1.80000 2.4 1495
(35)
hence the question of convergence (35) shows that the maximum value the series. We can certainly find
has to be considered further. does not increase very greatly series ai such that
and make the
converge.
series
(31)
and (32)
However, along
The above exercise shows that the region of convergence is more restricted near s=O. If we identify the location with the photosphere then the possiblity of equilibrium turning into non-equilibrium should be somewhat greater. A further study showed that, for the parameters in the convergent example, if we restrict to ~~0.73545, then Co-’ will always be less than 1. The range OirS0.73545 is below 3% of the width of Fig. 1. When we consider that region of convergence indicated.
5.
the coefficients should be larger
may be positive than the above
and negative, discussion
the
BRIEF SUMMARY
In this paper we have given a similarity power-series solution of the equation of magnetostatic equilibrium. We have described in detail the solution in the b= 3 case. For a certain range of parameter values, the series is certainly convergent. For a specific set of parameter values, the series is explicitly given at (281. In this example, the series is certainly convergent for s>O.73545. The magnetic field configuration corresponding to this series is shown in Fig.l. The configuration in Fig1 corresponds to field lines starting from a fairly strong and concentrated positive pole at the centre and ending in weaker and more scattered negatire poles on the sides. It can be taken to be a nonlinear two-arch structure. We can regard al, a2 and the parameters of the equation as slow functions of time and as they show change, the configuration changes. When, following the changes in the parameters, the series changes from convergence to divergence, the corresponding equilibrium configuration collapses. For example, if the parameters change from the values in the convergent example above to the divergent example while Hand b remain constant, then equilibrium will certainly pass into non-equilibrium , and the structures shown in Fig. 1 will disappear and we shall see a rapid change in the field. This could be the cause of the multi-ribbon flares [131.
Equilibrium of Two-Arch Structure
441
2nH
41rH
2xH
Fig. 1 An example of nonlinear magnetostatic equilibrium. The values of the magnetic equipotentials are l-0.0258, 2-0.0774, 3-0.1290, 4-0.1806, 5-0.1978, 6-0.2150, 7-0.2322, 8-0.2838
From the choice of P(F) we have, on omitting the overbars, 1 -Fb+'+ b+l
C+,(-$.)
and our nonlinear solution can give the particular distribution of pressure in equilibrium with the magnetic field configuration. Fig. 1 is a two-dimensionaldiagram. In the y-direction we may image small changes representing arcade structures. It helps us to visualize the field over elongated sunspot groups and provides a reference when studying the arcades seen as the limb. The transition point from equilibrium to non-equilibrium is a difficult subject as it depends on so many parameters. REFERENCES
[ll
Hood, A.W. and Priest, E.R., Solar Phenomena in Stars and Stellar Systems (1981) D.Reidel Publ. [21 Dunge:r,J.W., Mon.Not.R.Astron.Soc113 (19531, 180. [31 Brown, A., Astrophys. J., 128 (19581, 646. Low, B.C., Astrophys. J., 212 (19771, 234. (41 Priest, E.R. and Milne, A.M., Solar Phys. 65 (1980). 315. (51 Birn, J., Goldstein. H. and Shindler, K., Solar Ph.vs.57 (19781, 81. [61 Jockers, K., Solar Phys. 56 (19781, 37. [71 Heyvaerts, J., Lasry, J.M. and Witomskey, G., Lecture Notes Math., [81 782 (19801, 160. Melville, J.P., Hood, A.W. & Priest, E.R., Solar Phys., 92 (1984). 191 [lOI Ovsiannikov, L.V., Group Analysis of Differential Equations, (19821, Academic Press. I.111Bluman, G.W. and Cole, J.D., Similarity Method for Differential Equations (1974) Springer-Verlag. 1121 SUN Kai, ZHANG Dong-mei and WANG Yan, Kexue Tongbao 32 (1987) 1066. 1131 TANG F., Solar Ph.vs.,102 (1985) 131.