Two-dimensional steady-state pressure structure in a coronal loop

T W O - D I M E N S I O N A L S T E A D Y - S T A T E P R E S S U R E S T R U C T U R E

IN A C O R O N A L L O O P

I. Superposition of Two Chandrasekhar-Kendall Functions ( n = 0 , m = 0 ; n = 1, m = 0 )

S U R E S H C H A N D R A and L A L A N P R A S A D

Department of Physics, University of Gorakhpur, U.P., India

(Received 8 June, 1988; in revised form 7 January, 1991)

Abstract. By expressing the magnetic field and fluid velocity in terms of two Chandrasekhar-Kendal l functions (n = 0, m = 0; n = 1, m = 0) we investigated the steady-state pressure profile inside a solar coronal loop. For constant density loops, we found a two-dimensional (radial and axial) structure of pressure. This work is the modified version of the work of Krishan (1985). At the base of the loop, the pressure is found to increase steeply outwards along the radius, whereas at the apex it decreases slowly. The radial variation of pressure is found to be minimum around L/5, where L is the length of the loop measured from one foot to another one. But Krishan (1985) found that the rate of increase of pressure at the base was nearly equal to the rate of decrease of pressure at the apex, and the pressure was found nearly constant at L/4. For axial variation, we found that along the loop axis the pressure increases from the base up to z = 3L/8 and then decreases up to the apex, whereas at the surface, the pressure decreases from the base up to the apex. Krishan (1985), however, found the axial variation to be linear.

1. Introduction

Observations of solar active regions in optical, UV, EUV, and X-ray wavelengths generally show the loop structures associated with magnetic flux tubes in the corona. These observations show that the corona is highly structured. Most of the parts in the active regions are occupied by these arc-like structures, hence, they play an important role for the energy and pressure balance in the region. Temperature distribution in the coronal loops has been discussed by a number of authors (e.g., Chandra, 1984; Chandra and Narain, 1982; Craig, McClymont, and Underwood, 1978; Galeev etal., 1981; Hood and Priest, 1980; Levine and Withbroe, 1977; Roberts and Frankenthal, 1980; Sario etal., 1981; Takakura, 1984; Wragg and Priest, 1981, and references cited therein). Krishan (1983) investigated temperature structure in the loop by expressing the magnetic field and fluid velocity in terms of a single Chandrasekhar-Kendall function (C-K function) in the state n = 0, m = 0. The single C - K function corresponds to a force-free state whereas a superposition of two or more C - K functions does not do so. She found that the temperature increased radially outwards. That model advocated that hotter lines were produced near the surface of the loop whereas the cooler ones were generated in the cool core of the loop. This situation remained the same throughout the loop, as the temperature did not vary along the loop.

In the subsequent work, Krishan (1985) used a superposition of two C - K functions (n = 0, m = 0; n = 1, m = 0) to express the magnetic field and fluid velocity, and found

Solar Physics 134: 99-107, 1991. �9 1991 Kluwer Academic Publishers. Printed in Belgium.

100 SURESH CHANDRA AND LALAN PRASAD

a two-dimensional (radial and axial) pressure structure. For radial variation she (Krishan, 1985) showed that the rate of increase of pressure radially outwards at the

base was nearly equal to the rate of decrease of pressure at the apex, and the pressure was found nearly constant at z = L/4. The axial variation was found to be linear.

The work of Krishan (1985) is quite appreciable. However, her results are not free

from mistakes. Therefore, in order to improve her results, we reinvestigated the problem and found some changes in the two-dimensional pressure profile in a coronal loop in comparison to that of Krishan (1985). In other words, the results presented in this paper are the modified version of the work of Krishan (1985).

2. Pressure in a Coronal Loop

For plasma in a coronal loop of radius R and length L with rigid, perfectly conducting walls at r = R, the magnetohydrodynamical equations are

~V - - + ( V . V ) V = (V x B) x B _ __VP + g , ( la ) Ot p

~B - V x (V x B ) , ( lb)

&

V . B = 0 = V . V , ( l c )

where p is the mass density assumed to be constant in the region (Krishan, 1985; Montgomery et aI., 1978), p the pressure inside the loop, B the magnetic field defined

in Alfv6n speed units, V the fluid velocity, and g is the acceleration due to gravity. The boundary conditions on V and B are

V,(r = R) = O, B,(r = R) = 0. (2)

For a loop of length less than the scale height, the effect of gravity (g) can be neglected and the steady-state pressure gradient is given by

Vp/p = (V • B) x B - (V. V)V. (3)

Since we intend to reinvestigate the work of Krishan (1985) we also expressed the magnetic field and fluid velocity in terms of two C - K functions (n = 0, m = 0; n = 1, m = 0) as (Krishan, 1985)

B = Co 2o ~oa(0, 0, q) + C121 r a(1, 0, q ) , (4)

V = Co2o t/oa(0, 0, q) + C1~ 1 tha(1 , 0, q), (5)

where the suffices 0 and 1 stand for (n = 0, m = 0) and (n = 1, m = 0), respectively. The ~s and t/s are the expansion coefficients and Cs are the normalization constants. The general expressions for the normalization constants and the C - K functions (in cylindri-

TWO-DIMENSIONAL STEADY-STATE PRESSURE STRUCTURE IN A CORONAL LOOP~ 1 101

cal coordinates) are

"~nmq

. 2 R2j2 -~ ,}2nm q m + l ( ~ n m q R ) - ~ )nmqeJm(~nmqR)Jm+l(~)nmqR) X

• k2 }1-1/2 , (6)

where

(7 a ( n , m , q ) = d~ + - - - -

+ do(

ikn ~ ) 2nmq ~r t~(n, m, q) +

0 mk, ] O(n, m, q) + d~ 7"2mq O(n, m, q), ar r2nmq/ 2nm q

(7)

tp(n, m, q) = Jm (7~mq r) exp (im 0 + ik, z) ,

&2q 2 = 7nmq + kn ,

k n = 2 ~ n / L , n = 0 , _ + l , +_2, + 3 , . . . , m = 0 , +1, +2, +3 . . . . .

7nmq > 0 ;

2~,,q is the eigenvalue corresponding to the eigenfunction a(n, m, q) and Jm (Tnmq r) is the Bessel function.

One of the possibilities to satisfy the relation V x (V • B) = 0 is V to be parallel to B, i.e., ~; oc q;. For equipartition of energy between the velocity and magnetic field, as assumed by Krishan (1985) at the scales 2 o and 21, r = q;. This ansatz ~o = ~/o and r = ~h is very helpful in further treatment of the problem. We also used this assumption and by using Equations (4) and (5) in Equation (3), the steady state pressure gradient may be expressed in terms of three partial differential equations:

+ 212(7o - 21)Jo(71r)Jl(?o r) + ?,(7~ - 21 ?o)Jo(yor)Jl(71r)} e ~k`~ +

+ (CI rh r 7~)2 J2(ylr) e2"<~] ' (8a)

@ - o , (Sb) dO

@ &

- p[iklCoC12~?oth {21YlJl(yor)Jl(?l r) + 721Jo(?or)Jo(71r) } e ik'~ +

+ / k l ( G ~1 v[)2 {j12(7,r) + &~(710} e2;k'z] �9 (8c)


For the same situation, the corresponding equations obtained by Krishan (1985) may, however, be written as

{22~ ~1 8p _ (Co)lo r/o 7o) 2 j 2 ( 7 o r ) + CoG12o2 ~,]o ~,] 1 ao(7or)J1(71 r) + 8r r

] + 72(70 - 21)Jo(7 , r )J l (7or ) + 71(72 - 21 7o)Jo(7or)Jl (71r)~ e ik~z ,

]

(9a)

8p = 0 , (9b) 8O

@ - - ik1CoC122~O?~l {21 7aJ , (7or)J l (~ar) + 7 2 J o ( 7 o r ) J o ( h r ) } e ik,z .

8z

(9c)

Obviously, some terms in the expressions of Krishan (1985) are missing. Besides that, the density term is not available in her equations, probably she took it equal to unity.

The solution of Equation (8) can be obtained as

P P = Po + ~ [(Co2o t/o 7o)2{ 1 - J~(~o r) - J2(7or)} + 2COC1 tlo th22{72 -

- [2171Jl(7or)Ja(Ta r) + ~2Jo(7ot')Jo(71r)] cos(klZ) } +

-t- ( C 1/'~1 71) 2 {1 - [J02(71 F) ~- J12(~lr ) ] c o s (2klZ)}], (10)

w he rep o is the pressure at r = 0, z = 0. This expression differs from the corresponding expression of Krishan (1985):

P = Po + �89 [(Co 20 t/o 7o) 2 { 1 - Jo2(?o r) - J2(7 o r)} + 2C o C a ~/o qa 22 { 72 -

- [2 a 7aJ1(7or)Ja(71r) + 72Jo(7or)Jo(y l r ) ] cos (klZ)} ] . (11)

3. Model Calculations and Discussion

With the help of the Equations (2) and (4) (or (5)) one finds that 71 is given by the zeroes of J l ( h R ) , i.e., h R _~ 3.8 and, therefore, 21R --- 4. 7o is determined from the ratio of the toroidal and poloidal fluxes:

~t _ R J I ( 7 o R )

LJo(7om (12)

Taking L / R = 5 and O_,/Op = 0.1 we get 7o R = 2oR _~ 1. As an illustration we assume R = 1 0 9 cm; the values of the normalization constants are

C o = 8.482 x 10- 6 c m - 1/2,

TWO-DIMENSIONAL STEADY-STATE PRESSURE STRUCTURE IN A CORONAL LOOP, I 103

C 1 = 3.670 x 10 -6cm -1/2 .

The total energy E is given by

E = f ( V 2 + B 2) d3x. (13)

Using Equations (4) and (5) in Equation (13) one gets

E = 2(22 t/2 + 2~ t/~). (14)

Using E = 1028 ergs and choosing rh/r/o = 0.2, we obtained

r/o = 5.522 x 10 22 erg u2 cm, (15)

t h = 1.104 x 1022 erg 1/2 cm.

Using the above values of parameters, we calculated (p - Po)/P from Equation (10) for various values of 7o r and z. In Figure l(a) the values of (p - Po)/P are plotted as a

0.6 ~ (o) " L/2

0.4

t

0.2

0 0 0.2 0.4 0.6 0.8 1.0

(bl

0 0

Z = L/2

. . - - ~ / I I 1 I I I I I I' 0.2 0.,4 0.6 0.8 1.0

~ or

Fig. 1. Radial variation of pressure in a steady state loop for yo R = 1, and z = 0, L/4, L/2. (a) Shows the present values of (p -Po)/P as a function of 7o r (Equation (10)). (b) Exhibits the values of (p - P o ) as a

function of 7o r (Equation (11)).

function of 7o r for three values of z = 0, L/4, L/2, whereas Figure 2(a) shows the variation of (p - Po)/P as a function of z for different values of 7o r. In order to exhibit the difference explicitly, the values of (p - Po) obtained from Equation (11) are plotted in Figures l(b) and 2(b). (The results are symmetrical about the apex, z = L/2, of the loop.) Besides the major difference between the values obtained from Equation (10) and those from Equation (11), Figures l(b) and 2(b) show that the calculations of Krishan (1985) are not correct as she reported different values o f (p - Po) than the present ones.

I (bl

I~~ 2

IO'2//

O ~ / O 1 l I [ l 1


0 L/8 L I 4 3L/8 L/2 0 LiB L/4 3L/8 L/2

Z

Fig. 2. Axial variation of pressure in a steady state loop for 7oR = 1, and 7o r = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. (a) Shows the present values of (p - Po)/P as a function of z (Equation (10)). (b) Exhibits the values of

(P - Po) as a function of z (Equation (11)).

A major test may be that she showed that (p - Po) varied linearly with z for a definite value of 70 r (Figure i (b) in her paper). Equation (11), for convenience, may be expressed as

P -Po = A(r) + B(r)cos(klz) , (16)

where A(r) and B(r) are functions of r only. This equation obviously shows that for a

given value of r, (p - Po) cannot vary linearly with z.

In the present investigation, at the base of the loop, the pressure is found to increase steeply outwards along the radius whereas at the apex it decreases slowly. The radial

variation of pressure is found to be minimum around L/5. The surface pressure at the

base of the loop is nearly equal to the surface pressure at the apex of the loop. It shows that for constant electron density in the loop, the hotter lines are formed throughout the

loop. However, at the base, these are formed near the surface. On the other side, the cooler lines can be formed near the base in the cool core of the loop. Figures l(b) and

2(b) can be interpreted that the hotter lines are formed near the axis at the apex and not throughout the loop, whereas the cooler lines can be formed near the base in the cool core. For axial variation, we found that along the loop axis the pressure increases

from the base up to z = 3L/8 and then decreases up to the apex, whereas at the surface the pressure decreases from the base up to the apex.

For each curve in Figure 1 two different values for z (one of them being zero) are used, and for each curve in Figure 2 two different values for r (one of them being zero) are used. Therefore, in order to get explicit variations we applied the condition p = p ' at


r = r' a n d z = z ' . Then the pressure p is given b y

1 2 2 2 ( p (p - p ' ) / p = ~(Co2o rio) [{J~ 7o r ) + J ] ( 7 o r ' ) } - {JZ(7or ) + J2(7or)}] +

+ C o C 1 ) ~ r l o r h 7 2 [ { J o ( y o r ' ) J o ( T a r ' ) + ' ~ 1 J l ( 7 o r ' ) J l ( 7 , r ' ) } • 71

X COS

71

X COS + J,~(71,)} cos ( ~ ) 1 (17)

F o r radia l var ia t ion z = z ' , a n d the var ia t ion is shown in Figure 3. F o r axial var ia t ion

r = r ' , and the var ia t ion is shown in Figure 4.

Fig. 3.

0 . 5

0 . 4

0 .3

0 .2

a. 0.1

I i"1 O

- O . I

- -0 .2

Z s O "

m

" ' - - 4 2

I I I I I 1 1 I I 1 0 0 2 0 . 4 0 . 6 0 .8 1.0

,4"o1.- Variation of (p - p' )/p as a function of 7o r (Equation (17)) in a steady-state loop for yo R = 1, and

z = z' = O, L/4, L/2.


0.6

0.5

"(or--O

0.2

0.4

0.3

0 .4

el,

-a-

I n

0.2

O.I

0 .6

Fig. 4.

~ r -O.I [

-O. 2

o

0 . 8

I I I 1 1 I ...... I I

L / 8 L / 4 3 L / 8 ~ 2

Z

Variation of (p -p')/p as a function of z (Equation (17)) in a steady-state loop for 7o R = 1, and ?o r = 7o r ' = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0.

4. Conc lus ions

The model of Krishan (1985) for two-dimensional steady-state pressure structure in a coronal loop is quite appreciable, but is not free from mistakes. The results presented in this paper are the modified version of the model of Krishan (1985).

Priest (1982) mentioned that the cool cores of the loops were found by Jordan (1975) and Foukal (1975). Jordan (1975) found that the temperature and pressure in the core of a loop can either be larger or smaller than the ambient coronal values, but were always lower than in a surrounding sheath. Foukal (1975) found that the prominant active region loops connecting the sunspots had cool cores with a temperature lower than 2 x 105 K, while the ambient coronal temperature was probably 2 x 106 K. In the present model we found that the pressure in the core near the foot of a loop (z < 3L/16) is lower than in the surrounding sheath, but for 3L/16 <,G z ~ L/2 the situation was reversed. A reason for getting larger pressure in cores than in the surrounding sheath


for 3L/16 < z < L / 2 may be assigned to consideration of the expressions for magnetic

field and fluid velocity considered in this paper. However, these features are similar to

those found by Krishan (1985). Since we intended to present a modified version of the

model of Krishan (1985), therefore, we have also considered the same expressions for

magnetic field and fluid velocity. In a further work we intend to consider other

expressions for magnetic field and fluid velocity.

Acknowledgements

We are thankful to the learned referee for constructive suggestions. Financial supports

from the Council of Science and Technology, U.P., Lucknow and the University Grants

Commission, New Delhi are thankfully acknowledged.

References

Chandra, S.: 1984, Bull. Astron. Soc. lndia 12, 152. Chandra, S. and Narain, U.: 1982, Bull. Astron. Soc. India 10, 18. Chandrasekhar, S. and Kendall, P. C.: 1957, Astrophys. J. 126, 457. Craig, I. J. D., McClymont, A. N., and Underwood, J. H.: 1978, Astron. Astrophys. 70, 1. Foukal, P. V.: 1975, Solar Phys. 43, 327. Galeev, A. A., Rosner, R., Serio, S., and Vaiana, G. S.: 1981, Astrophys. J. 243, 301. Hood, A. W. and Priest, E. R.: 1980, Astron. Astrophys. 87, 126. Jordan, C.: 1975, in S. R. Kane (ed.), 'Solar Gamma-, X- and EUV Radiation', IAU Symp. 68, 109. Krishan, V.: 1983, Solar Phys. 88, 155. Krishan, V.: 1985, Solar Phys. 97, 183. Lavine, R. H. and Withbroe, G. L.: 1977, Solar Phys. 51, 83. Montgomery, D., Turner, L., and Vahala, G.: 1978, Phys. Fluids 21, 757. Priest, E.R.: 1982, Solar Magnetohydrodynamics, D. Reidel Publ. Co., Dordrecht, Holland. Roberts, B. and Frankenthal, S.: 1980, Solar Phys. 68, 103. Serio, S., Peres, G., Vaiana, G. S., Golub, L., and Rosner, R.: 1981, Astrophys. J. 243, 288. Takakura, T.: 1984, Solar Phys. 91, 311. Wragg, M. A. and Priest, E. R.: 1981, Solar Phys. 70, 293.

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Two-dimensional steady-state pressure structure in a coronal loop