on the magnetic field in pores

O N T H E M A G N E T I C F I E L D I N P O R E S

G. W. SIMON

Sacramento Peak Observatory, Air Force Cambridge Research Laboratories, Sunspot, N.M., U.S.A.

and

N. O. WEISS* Dept, of Applied Mathematics and Theoretical Physics, University of Cambridge, U.K.

(Received in final form 28 January, 1970)

Abstract. The magnetic field in an axisymmetric pore is current free and can be represented by a flux tube with a magnetic potential of the form AJo(kr) e -~z. For a given magnetic flux the field in this pore model is uniquely defined if the magnetic pressure balances the gas pressure at two levels. For models with fluxes of 0.5-3.0 • 102o Mx the surface radius varies from 1100-2700 km (diameters of 3-8 arc-sec) and the Wilson depression is estimated at 200 kin. As the flux increases, the field becomes nearly horizontal at the edge of the pore and eventually a penumbra is formed. The distinction between pores and sunspots is investigated; the critical flux is about 102o Mx, corresponding to a radius of 1500 km.

1. Introduction

Observations at high resolution of photospheric magnetic fields, in active regions or at the boundaries of supergranules, indicate that they are generally concentrated into

smaller discrete areas (Simon and Leighton, 1964; Sheeley, 1966, 1967; Beckers and

Schr~Ster, 1968). These include magnetic gaps or knots, pores and sunspots. Magnetic

gaps and knots are not significantly darker than normal granules and their magnetic

fields range up to about 1500 G. A sunspot has both a dark umbra and a filamentary penumbra; the magnetic field rises from about 1500 G at the outer boundary of the

penumbra to a maximum, in the umbra, that rarely exceeds 3500 G. We define a

pore as a region which is dark compared with the normal photosphere, which pos-

sesses no penumbra, and which has a field greater than 1500 G.

Observations show that the smallest pores are comparable in size to granules. The

radius generally lies between 700 and 1750 km (diameters of 2-5 arc-sec). Occasional- ly a pore may have a radius of up to 3500 km (diameter 10 arc-sec) but a penumbra

generally develops if the radius exceeds 1750 km (Bray and Loughhead, 1964). The

brightness of a pore is less than 65% of that of the normal photosphere (Bahng, 1958)

corresponding to a temperature of less than 5600 K. The magnetic field appears to

be fairly uniform. It generally exceeds 1500 G and drops abruptly in less than 0.5 arc-sec at the edge of the pore (Steshenko, 1967; Bumba, 1967b); field gradients at

the boundary may be as great as 6000 G per arc-sec (Title, 1969). There is also an inward flow of material, opposite to the normal Evershed effect (Bumba, 1967a). Pores have typical lifetimes of several hours but they develop over periods of order

* Visitor, as a member of the High Altitude Observatory Solar Project, at Sacramento Peak Observa- tory, Sunspot, N.M., U.S.A.

Solar Physics 13 (1970) 85-103. All Rights Reserved Copyright �9 1970 by D. Reidel Publishing Company, Dordrecht-Holland

86 a . w . SIMON AND N. O. WEISS

half an hour; this time scale is intermediate between the lifetimes of observed granules and supergranules and it would correspond to convection at a depth of around 1500 km (Simon and Weiss, 1968).

Our aim in this paper is to establish a model of pores that provides criteria for distinguishing them from sunspots. This simple model is based on observations. We suppose that subphotospheric convection concentrates magnetic flux into ropes with a maximum field of about 3500 G (Weiss, 1969). These flux ropes emerge at the photosphere as pores. Within the flux tube there is a force-free field, discontinuous at its boundary and in pressure equilibrium with the external gas. This field can be derived from a potential and we adopt the simplest representation in terms of the zero-order Bessel function.

The model can be described by a single parameter, the magnetic flux, which deter- mines the structure of a pore. For a given flux the radius of the flux tube and the field within it can be calculated, and the lines of force can be plotted. These results are consistent with Steshenko's (1967) observations. As the flux increases, the central field grows and the lines of force fan out. The model breaks down when the field at the edge of the pore is almost horizontal or the central field exceeds the limit of 3500 G. Thus there is a limit to the magnetic flux that can be accommodated in a pore. For greater fluxes a penumbra is formed, giving a sunspot rather than a pore. The model predicts a critical flux of 1 TWb*, with a corresponding radius of about 1500 km at the surface. This is in fair agreement with the observations.

So far only mechanical equilibrium has been mentioned. We also consider briefly the thermal equilibrium of a pore and finally we discuss the principal differences between pores and spots. The main purpose of developing this simple model is to improve our understanding of the complicated structure of a sunspot.

2. A Model of the Magnetic Field in a Pore

Cellular convection concentrates magnetic fields into flux ropes at the boundaries of cells (Parker, 1963; Weiss, 1964a, 1966). This process is halted when the energy density in the concentrated field is comparable with the kinetic energy density of the motion and therefore able locally to suppress convection. Thus the field produced by supergranular convection should be about 3000 G (Simon and Weiss, 1968) and it is indeed unusual to observe fields greater than 3500 G. These concentrated fields protrude into the photosphere. However, photospheric convection is inhibited by fields greater than 600 G (such as those observed in magnetic knots), though it requires a field of about 1600 G to balance the gas pressure in the photosphere (Weiss, 1964b, 1969). Fields greater than 1500 G are therefore associated with

darkening, as in pores and spots.

* We endeavor to follow contemporary trends by expressing magnetic fields in gauss (G) and magnetic flux in terawebers (TWb) instead of Mx (G cm~). 1 TWb = 1020 Mx and this happens to be the appropriate unit for measuring magnetic flux in pores. We also use newtons (N) instead of dynes as units of force.

ON THE MAGNETIC FIELD IN PORES 87

The observational evidence justifies a model in which the photospheric gas pressure, Po, is much less than the magnetic pressure, Pro, inside a pore, and rises through a narrow boundary layer to its outside value. (In a sunspot this boundary layer expands to become the penumbra.) The apparent surface of the pore is below the surface of the normal photosphere, but it is difficult to estimate the magnitude of this depression. However, a magnetic field of 3500 G produces a pressure of 5 x 10 4 Nm -2, which corresponds to a depth of about 350 km. We therefore take 350 km as an upper limit to the depth for which the model can be valid. Probably the surface where the optical depth z equals unity is depressed by 150-250 km in a typical pore, and the assump- tion that P.q 4 Pm is invalid below that level.

Pores are often irregular in shape, but we shall consider only a circular pore, in which the magnetic field B is axially symmetric and has no azimuthal component. Referred to cylindrical polar coordinates (r, qS, z), with the z-axis vertical, these symmetry assumptions imply that ~B/aqb=0 and B~=0. Then, since Po~Pm, the field must be force-free and

curl B x B = 0.

But the symmetry then requires that curl B = 0, so B is a potential field and

B = - V~b where V21~ ) = 0 . (1)

The magnetic potential satisfies Laplace's equation within the flux tube. At the edge of the tube we can integrate across the boundary layer to establish pressure balancing: The normal stress (magnetic pressure) inside the tube must balance the external gas pressure. Thus we can formulate the problem mathematically as a Neumann boundary value problem for the potential ~. The external gas pressure P(z) can be obtained from a model of the convection zone (e.g., Baker and Temesvary, 1966) and the potential could be calculated by the following iterative procedure. First, choose the boundary of the flux tube R(z) for z>z l ; then solve Laplace's equation for 4~ in the region r<R(z) , z>z i , subject to the bour.dary conditions that the vertical field (-&b/c3z) is given at z = z t and tends to zero at infinity, while the normal field (Ocb/~n) is zero at r = R, since the tube is bounded by lines of force; now calculate the magnetic pressure at r = R and adjust the values of R until Pm converges to P for all z.

An alternative possibility would be to expand the potential in separable solutions of Laplace's equation. For cylindrical polar coordinates the Bessel function solution

4) = AJo(kr) e -~z (2)

can be used to form a Hankel integral, while in spherical polar coordinates the solution can be expanded as multipole fields in terms of Legendre polynomials. In fact, we shall only use two very simple models. In the first, we assume that the field is given by the expression in Equation (2) with only one value of k. Exact pressure balancing is then possible only for two discrete values of z. However, since the variation of pressure near the surface of the Sun is very nearly exponential and the magnetic pressure is dominated by the exponential term in Equation (2), this turns out to be

88 G.W. SIMON AND N. O. WEISS

a fairly satisfactory model. We shall compare it with a second model which assumes a simple dipole field.

Several authors have calculated models of solar magnetic fields. Broxon (1949) compared the observed variation of B~ in a sunspot with the fields produced by a dipole and a finite current loop; Chapman (1943, 1944) represented sunspot fields by surface distributions of monopoles and dipole moment and yon Roka (1950) derived axisymmetric potential fields. More recently, Schmidt (1964) has developed a program which has been used to calculate potential fields from the surface distribution of B~ (Rust, 1970). However, none of these treatments has included pressure balancing as a boundary condition, while the more elaborate sunspot models of Schlfiter and Temesvary (1958) and Deinzer (1965) were not confined to force-free fields.

3. A Simple Model of the Field

In this section we shall first discuss the simple Bessel function model provided by Equation (2) and use it to describe the field in a pore in terms of the total magnetic flux. Then we shall compare these results with the simple dipole model. In Section 4 we shall obtain limits to the size and depth of a pore and relate the model to observations of the magnetic field in pores.

We have used a model of the convection zone computed with the program developed by Baker, Hofmeister and Kippenhahn (Baker and Temesvary, 1966), using a mixing length equal to the density scale height, to give the pressure P(z) in the undisturbed photosphere, and combined this with the Bilderberg continuum atmosphere model (Gingerich and de Jager, 1968). The height z is measured upwards from the level where the continuum optical depth z=~. An exponential dependence of P on depth, of the form

P(z) = Po exp ( - z/210), z < 0, (3)

= Po e x p ( - z/l12), z > 0,

where P (0)= Po = 104 N m - 2 and z is in kin, provides a tolerably accurate approxima- tion over the range - 500 km < z < 2000 km. A more accurate representation over the same range is given by

P(z) = exp I f (z/1000)], where f ( x ) = 9.180 - 6.529x - 8.113x 2 - 6.446x 3, x < 0.152,

(4) = 9.414 - 9.449x, x > 0.152.

A. BESSEL F U N C T I O N MODEL

The magnetic field in the pore can be represented in terms of the potential or of the Stokes stream function ~b, which is constant along a line of force:

o , - ( 5 ) B-- (B, . ,B ,~ ,B~) - - - ~ r - r ' 0 ' - ~ = - r c ~ z r '


and we shall assume that the potential has the form

4) = A J o ( k r ) e -k~ ,

whence = A r J 1 (k r ) e - k z ,

while

(6)

(7)

(s)

8~zpP,, = A 2 k 2 e-2kz [jo 2 (kr) + j2 (kR)].

We must therefore solve the flux conservation condition

F = 27~ARoJ 1 ( kRo) e -k~~

= 2~ARIJ1 (kRa) e -k~l ,

(12)

(13)

(14)

B~ = A k J x ( k r ) e -k~ and Bz = A k J o ( k r ) e -kz ,

and the total flux through a disc of radius r is 2~z~(r, z). This description of the field holds only within the flux tube, which is bounded by lines of force. Hence ~ < T, where the total flux through the pore is

F = 2=T; (9)

moreover, for any value of z, the radius of the pore is R(z ) , where

~(R, z) = ~'. (10)

The lines of force described by Equation (7) fan out and the field strength diminishes with increasing height. At any particular level, B r vanishes on the axis (r = 0) where B z has its maximum value

B o ( z ) = A k e -k~. ( l l )

The inclination of the lines of force to the vertical increases with increasing r until k r = j o (where jo =2.405, the first zero of Jo), where B~=0 and the field is horizontal. Thus the field at the boundary of the flux tube becomes horizontal when k R =Jo and for any given flux F the tube turns over at some level z = z * , where R ( z * ) = j o / k .

Then pressure balancing cannot be applied for z > z*. However, we shall assume that the model can represent a pore provided that z* >0, i.e., k R ( O ) < j o . (In other words, we require that the field should become horizontal at or above the photospheric level, never below it; in Section 4 we discuss more stringent conditions on the inclination of the field.) Above a pore, the lines of force ultimately bend round to reenter the photosphere elsewhere. The general features of this configuration are reproduced by the model and we shall assume that the field in the pore is relatively insensitive to the detailed structure of the field higher in the atmosphere.

The field is thus defined by three parameters, A, k and R(zo) , the radius of the pore at some height Zo. These can be obtained by specifying the total flux F and imposing pressure balancing at two levels, say z = z o and zi. (Alternatively, we could balance both the pressure and its derivative at z =Zo. ) Now the magnetic pressure Pm is given by

90 O . W . SIMON AND N. O. WEISS

together with the conditions for pressure balancing

8~p~o = A2k ~ e -2~z~ [ g (kRo) + J? (kRo)], (15) and 87c1~P1 = A2k 2 e -2kz~ [yo2(kR~) + d((kR~)], (16)

where Ro=R(zo ) , R l = R ( z l ) , Po=P(zo) and P I = P ( z l ) . (17)

It is convenient to work in terms of the quantities

40=kRo , ~ = k R ~ and d = ( % - z ~ ) . (18)

Then Equation (13) and Equation (15) give

[ R0=L k -) j (19)

while Equations (13) and (14) become:

- m - - - - . ( 2 0 )

These equations can be solved on a computer by the following procedure: For a given F, Zo and z~ we know Po and P~ from Equation (4). First choose a trial value for ~o and calculate Ro from Equation (19). Then k = ~o/Ro and A is obtained from Equation (13). Now Equation (14) can be solved iteratively for R~ and ~1- We then rewrite Equation (16)as

A = J d G ) + A2k2 , (21)

and solve for A, which should of course equal zero. Next a new trial value of 4o is chosen, and a new value of A obtained. An iterative procedure is followed which reduces ]AI until it is sufficiently close to zero.

It is obviously appropriate to apply pressure balancing at the surface (~=~) and at some convenient depth d. Then Zo=0 (Po~ 104 Nm -2) and different models are defined by a single parameter, the total flux F. Results for d = - z 1 =200 km (Pt 2.7 x t04 Nm -2) ar esummarized in Table I, which contains values of Bo(z), R(z), Pm(R,z)/P(z) and the inclination a(z) of the field at the pore boundary, for a number of different fluxes. The corresponding lines of force are plotted in Figures 1 and 2. In Figure 1 we show the lines of force which contain 0.1F, 0.2F, ..., 1.0F, for fluxes F = 0.2, 0.4, 1.2 and 1.873 TWb, while Figure 2 shows the shape of the pore boundary as a function of height for 11 values of the flux between 0.004 and 1.873 TWb. It will

be seen that the error in Pro:

Ae = (Pro - P)/P, (22)

is less than 10% for z<0 , though it rises to 85% at z=200 kin. This indicates that the model provides a satisfactory representation of the structure of the pore below the level of the undisturbed photosphere.

Fig. 1.

E

Fig. 2.


. . . . . . .

W-1oo _ 2 o ) 1 1 1 1 I / / / / / . . . . . 1 0 SSO 1100 1650 2200

3n n RAD I US (Era)

100 . . . . . . .

~ - m 6 _.ool t , 1 1 1 / / / / / / i , . . . . 1 0 550 1100 1650 2200

RADIUS {Era) 300| . . . . . . . -~ 2o0~ fLUX CT~,~,~ : ~.200

~_ 1oof

_2001 , - - 0 550 1166 1650 2200

RADIUS (kin) 2t00[ . . . . . . .

-~ 200~ FLUX [mWb)= 1,873 1 iOOF

~ O~rMOl.O~Fnc.R F I . ~ ~ . . . ~ . . PHOTOSPHERE.. . P.HqTOSPHEREI

1 0 550 i i O0 1650 2200

RADIUS [kin]

Computed lines of force which conta in 0.1 F, 0.2 F, .... 1.0 F, for fluxes F = 0.2, 0.4, 1.2, and 1.873 TWb, all with d = 200 km.

1 6 0 0 ~ g = 0.004 TWb

1400 0.006

1200 0.01

I000 0.02

800 0.04

200

0.1

0.2

0,4

P H

0.7 1.2

"~0 400 000 1200 1600 2000 2400 RFIDIUS (kin)

Shape of the pore boundary for d = 200 k m and 11 values of the flux between 0.004 TWb and 1.873 TWb.

92 G. W. SIMON AND N. O. WEISS

TABLE I

Computed field models for d = 200 km and various fluxes

Height z F (TWb) 0.01 0.04 0.1 0.4 1.2 1.873 (km) z* (kin) 1 159 685 420 128 12 0

200 Bo (G) 953 1 016 1 127 - - - R (km) 185 374 603 - - - Pm/P 1.85 1.80 1.69 - - - a 13.7 ~ 27.3 ~ 43.4 ~ - - -

100 Bo ((3) 1 230 1 299 1 420 1 851 - - R(km) 163 326 519 1 119 - - Pm/P 1.25 1.24 1.21 1.06 - - a 12.0 ~ 23.6 ~ 36.5 ~ 72.4 ~ - -

0 Bo (G) 1 587 1 661 1 789 2 228 2 787 3 008 R (km) 143 286 452 918 1 663 2 143 Pm/P 1.00 1.00 1.00 1.00 1.00 1.00 a 10.5 ~ 20.5 ~ 31.3 ~ 55.5 ~ 80.2 ~ 90.0 ~

- - 100 Bo (G) 2 048 2 123 2 253 2 682 3 186 3 365 R (km) 125 251 396 795 1 380 1 713 Pm/P 0.94 0.94 0.95 0.97 1.02 1.07 c~ 9.2 ~ 17.9 ~ 27.2 ~ 46.5 ~ 61.8 ~ 65.1 ~

- - 200 Bo (G) 2 643 2 715 2 838 3 229 3 642 3 764 R (km) 110 221 349 701 1 224 1 535 Pm/ P 1.00 1.00 1.00 1. O0 1.00 1.00 a 8.1 ~ 15.7 ~ 23.7 ~ 40.1 ~ 53.0 ~ 56.4 ~

MaxAp(~o) 85 80 69 + 6 + 3 + 11 Min Ap ( ~ ) - - 6 - - 6 - - 5 - - 9 - -10 - - 0

In T a b l e II, we c o m p a r e the su r face f ea tu r e s o f m o d e l s ca l cu l a t ed w i t h d i f fe ren t

va lues o f d. T h e va lues o f Ro a n d Bo(0) va ry on ly s l ight ly as z 1 is d e c r e a s e d f r o m

- 100 to - 500 kin. Th is aga in impl ie s t h a t E q u a t i o n (8) p r o v i d e s an a d e q u a t e r ep re -

s e n t a t i o n o f t he field, excep t in t he i m m e d i a t e n e i g h b o r h o o d o f ~o =Jo -

T h e l ines o f fo rce in F igu re s 1 a n d 2 f a n o u t m o r e a n d m o r e as t h e flux is i n c r e a s e d ;

u l t ima te ly B(Ro, 0) is h o r i z o n t a l a n d t h e y b e n d over a t t he sur face . Th is sets a l imi t

t o t h e m o d e l ' s va l id i ty , w h i c h is c o n v e n i e n t l y i n v e s t i g a t e d by re l a t ing t h e flux to t h e

i n c l i n a t i o n o f B at t he edge o f the po re . L e t th is field be inc l ined at s o m e angle ~o

to t he ver t ica l . T h e n

fi = c o t ~ o = Bz(R, O)/Br(R, 0) , (23)

w h e n c e

,/1

w h i c h can be

1

Jo(r - - = f l , (24)

so lved f o r 40- T h e n , f r o m E q u a t i o n s (15), (16), a n d (20),

I J ( l)l (l+flZ) P 1 (25) Po

ON THE MAGNETIC FIELD IN PORES

TABLE II

Surface features of the model for various fluxes and depths d

F (TWb) d (kin) 100 200 300 400 500

93

0.1 z* (kin) 336 420 514 604 673 Ro (kin) 453 452 452 452 452 Bo (0) (G) 1 830 1 789 1 755 1 731 1 716 cxo 34.1 ~ 31.3 ~ 28.9 ~ 26.9 ~ 25.7 ~

0.2 z* (kin) 204 256 318 380 427 Ro (krn) 643 642 641 641 640 Bo (0) (G) 2 019 1 964 1 914 1 875 1 851 c~o 45.2 ~ 42.2 ~ 39.4 ~ 37.0 ~ 35.5 ~

0.4 z* (kin) 104 128 163 198 225 Ro (km) 920 918 915 913 912 Bo (0) (G) 2 282 2 228 2 166 2 114 2 081 c~o 58. i o 55.5 ~ 52.6 ~ 50.1 ~ 48.4 ~

1.0 z* (km) 24 23 29 37 42 Ro (kin) 1 500 1 502 1 496 1 490 1 487 Bo (0) (G) 2 682 2 690 2 656 2 613 2 588 ao 75.6~ 76.0 ~ 74.5 ~ 72.6 ~ 71.6 ~

1.5 z* (kin) 7 3 3 3 4 Ro (kin) 1 871 1 887 1 888 1 884 1 884 Bo (0) (G) 2 841 2 901 2 907 2 892 2 890 ~o 82.5 ~ 85.2 ~ 85.4 ~ 84.8 ~ 84.7"

This gives r and k can then be found from Equation (20). Finally,

2g (Sg/-tPo) 1/2 {o F = k=(1 + fl2)1/2 �9 (26)

This method provides a number of models whose properties are summarized in Table III, which gives values of F, Ro, and B o (0) corresponding to various inclinations ~o for d = 0 , 100, 200, 300, 400, and 500 km. The values for d = 0 were obtained by using P and dP/dz at z = 0 (rather than P1); we have taken the pressure scale height ]d In P/dz[- 1 as 153 km, but unfortunately the values are sensitive to this rather ill- determined quantity. To determine k from P and dP/dz at z = 0, we take the derivative of Equation (15) with respect to z and, using Equations (15) and (24), obtain

k = - 2 1 4o \ a U ] o (27)

Equations (24) and (27) can then be solved for 4o and k, and the flux is given by (26). The behavior of the Bessel function model is displayed in Figures 3, 4, and 5.

Figure 3 shows the variations of the surface field and of the inclination c~ o as functions of r the arrows showing both the magnitude and inclination of B. In Figure 4 the surface radius, Ro, and the central field, Bo(0), are given as functions of flux, assuming pressure balancing at d = 100, 300, and 500 kin. These curves are seen to be not very sensitive to the choice of d. N o w for a given value of d the value of F for which z* = 0

94 G. w , SIMON AND N, O, WEISS

TABLE II I

Fluxes and surface radii as functions of the depth d

d (km)

0

100

200

300

400

500

Fixed inclination at surface (Bessel model) Fixed central field at depth d

ao 30 ~ 45 ~ 60 ~ 75 ~ 90 ~ Co 1.00 1.43 1.81 2.13 2.40 Bo (0) (G) 1 770 2 015 2 325 2 667 3 008

Bo(-- d) = 3 500 G

Bessel Dipole

F ( T W b ) 0.06 0A7 0.43 1.32 - - Ro (kin) 351 593 952 1 718 - -

F ( T W b ) 0.07 0.20 0.44 0.97 2.56 - 0.36 Ro (kin) 389 639 970 1 472 2 507 - 815

F ( T W b ) 0.09 0.23 0.49 0.96 1.87 0.81 0.22 Ro (km) 431 695 1 024 1 467 2 143 1 334 641

F ( T W b ) 0.11 0.27 0.56 1.02 1.77 0.21 0.08 Ro (kin) 472 752 1 089 1 513 2 083 664 408

F (TWb) 0.13 0.31 0.62 1.09 1.76 - - R0 (kin) 508 802 1 147 1 561 2 080 - -

F ( T W b ) 0.14 0.34 0.66 1.12 1.74 - - Ro (kin) 533 837 1 183 1 585 2063 - -

Fig. 3.

0 0.0 0.5 LO 1.5 2.0 2.5

1 . 0 , ' . . .

i 0.8

0 .6

N 8r o .2

0~ ~ 0.0 0.5 1.0 1.5 2.0 2.5 OI5TRNCE FROH CENTER ~ r

M a ~ e t i c field and inclination in the simple Bessel f ~ c t i o n model as ~nc t ions of ~ = kr. The arrows show both the magnitude and inclination of B.

ON TILE MAGNETIC FIELD IN PORES 95

and % = 90 ~ provides an upper limit to the flux that the model can describe. Other critical values can be derived by arbitrarily specifying the value o f %; e.g., % = 75 ~ 60 ~ 45 ~ etc. Figure 5 shows the dependence o f these critical fluxes on the depth d.

For fluxes less than 1 T W b or incl ination less than 75 ~ variation o f d only affects the mode l slightly, even at shal low depths, as can be seen from Table II[ and Figures 4 and 5; however , for larger fluxes, as % approaches 90 ~ m u c h greater fluxes can be a c c o m m o d a t e d if d is small. This is a consequence o f representing the field by a single separable solution. We shall discuss the limit of the model 's validity in the next section.

3200

2800

24o0

~ 2ooo 03"%

n re00 ( I J o~w

~ w 1200

O~::I.--W 800 / = = = d = I00 km 03 ~ Z / = = = d = 300 km

rr f ": "~ "- d = 500 km �9 ,s- 400

0.0 0:7 l:q 211 2.8 FLUX (TWIo)

Fig. 4. Surface radius and central field in model as a function of flux.

Fig. 5.

5.C

q.C

3.5

.~3.C

~ 2 , ~ x

"~ 2.C

1.5 '~--Bo(-dl = 35000

]. 0 7 5 ~ " x U

O. 5 60 o_

0.% 50 100 i50 2uO 250 300 350 q~o %0 500 d (kin)

Flux in Bessel model as function of d for various surface inclinations. Also flux dependence on d if central field at d is 3500 G.

96 G . w . SIMON AND N. O. WEISS

Another limit is set by the magnitude of the field at z = - d. If flux is concentrated by subphotospheric convection, then

B~ (0, - d ) ~</t ~ 3500 G. (28)

The flux corresponding to B~ (0, - d ) = H can readily be found. For then

H = A k eke, (29)

and from Equation (16),

8zcktP 1 g (41) + j2 (4t) - H 2 (30)

Also, combining Equations (13), (14), and (15) with Equation (29) yields

#(r

and F = 2rcHk - 2 ~lJ1 (41) .

(31)

(32)

Equation (30) can be solved numerically for 41, then Equation (31) for 40; next k can be found from Equation (20) and, finally, F from Equation (32). The variation of this critical flux with d is given in Table III (column 5) and shown in Figure 5. For d> 143 km, there exists a range of values of F for which the central field can exceed 3500 G and, for d>365 km, B ( r , - d ) > 3 5 0 0 G for all r<~R1.

B. DIPOLE MODEL

An alternative model is provided by a dipole field, referred to spherical polar coordinates (s, O, (~) with origin at z = - h , for which

A cos 0 A sin 2 0

~ - s 2 ' ~ - s

2A cos 0 A sin 0 B , - s3 , Bo - s3 , (33)

and A 2

87r/~P,, = -~- (4 - 3 sin 2 0). (34) S

This is less satisfactory than the Bessel function model and we shall only discuss it briefly. The analysis is straightforward: it is necessary to find A, h, and the boundary of the flux tube. Although models can be found for a wide range of fluxes (up to more than 100 TWb), the central field is always high. Indeed, it is only for fluxes less than 0.25 TWb that the dipole field is appropriate for a pore with pressure balancing at the surface and at 200 km depth. Comparison of the last two columns of Table III indicates that, while the Bessel function model (with its exponential factor) provides


a tolerable representation, the dipole model is virtually useless. This is a consequence of the variation as a power of s in Equation (34), though quadrupole or higher order fields might be more versatile.

As an example of the use of the dipole model, we shall satisfy Equation (28). From the geometry it is possible to write:

h h - d so cos 0o sl cos 01 ' (35)

where 0o and 01 are the polar angles at the flux tube boundary at the heights z = 0 and z = - d, respectively. But Equation (28) in the limiting case for a dipole field becomes

H(h - d) 3 = 2A,

so that Equation (34) can be rewritten

i_12 COS 6 01 P1 = P ( - d ) - ( 4 - 3 sine01).

32rr/~

The flux conservation condition is given by

(36)

(37)

27rA 2rcA F = 2 ~ u = s i n e 0 o = - - s i n 2 01 , (38)

SO S 1

which, combined with the pressure balance conditions obtained by applying Equation (34) at z = 0, - d , yields

Po ( 4 - 3 s i n e0o) s~ ( 4 - 3 s i n e0o) sin 1201

P~ = ( 4 ~ 3 s i n e 01)'s~ = sin le 0o (4 - 3 sin e 01i" (39)

The solution is straightforward: First Equation (37) is solved numerically for 01, then 01 is put into Equation (39), and 00 is obtained by iterative techniques. Substi- tuting from Equation (35) into Equation (38) then gives:

d cos 00 sin 2 00 h = , (40)

cos 0o sin e 0o - cos 01 sin 2 01

and finally A is determined from Equation (36). In Figure 6 we compare the Bessel and dipole models in the case that Bo ( - d ) =

3500 G, for d--200 km. The surface field Bo(0 ) is 260 G smaller in the dipole model, the flux is approximately one-fourth as big, and the surface radius less than half as large as for the Bessel model. If one makes the Bessel and dipole fluxes equal by choosing d= 300 km for the Bessel model, then the Bessel field B o (0) is 180 G smaller than the dipole field, but the flux-tube boundary radii are almost identical in size and shape. In the left-hand chart we show the flux-tube boundary with height, while the right-hand chart illustrates the surface field characteristics.

98 G.W. SIMON AND N. O. WEISS

Fig. 6.

~ -E -50

v-,~176 l- /1

-150

- 2 0 0 ' ~ '

' / I /

i I

400 650 900 1150 1400 RADIUS (km)

c--o-o BESSEL(d=3OOkm,F~.214TWb} BESS EL(d =200km,F=.806TWb )

: \ \

1500 ~ ' ' 0 350 700 1050 1400

SURFACE RADIUS (km)

Compar i son of Bessel and dipole models if B0 ( - - d) ~ 3500 G. Lef t -hand graph: F lux tube bounda ry radius vs height . R igh t -hand : Surface magne t ic field vs radius.

4. Comparison with Observations

A. VALIDITY OF THE MODEL

In the previous section we mentioned two limits to the size of a pore that can be represented by the Bessel function model. The first was the limit H set to the central field Bo ( - d ) . The model assumes that the contribution of the internal gas pressure is negligible and therefore the magnetic field cannot balance the external pressure P if 87r#P> H 2. Now observations, supported by the supergranule model of Simon and Weiss (1968), indicate that Hm 3500 G. Thus the limiting pressure/)1 < 5 • 104 Nm -2, which corresponds to a depth of about 370 kin. This provides an absolute upper limit to d. Moreover, pressure balancing at this depth with a field of 3500 G is possible only for an infinitesimal flux. As the flux increases, the depth d must diminish, as can be seen from Figure 5. For F = 1 TWb, d~<200 km and for F > 2 TWb, d~< 150 km. This is much less than the estimates of 400-800 km in a sunspot, where curvature

forces are important. Similarly, the model cannot be reliable when ~ 1 . For z>z* the lines of force

bend over and return into the surface of the Sun. So long as z*> 0, this behavior is qualitatively correct but the detailed field structure above an actual pore is unlikely to remain axially symmetric. The flux will return through nearby sunspots, pores or knots, and the field, though force-free, will not necessarily be current-free. Neverthe- less, the model may still provide a reasonable representation of the central field in the first 100-200 km above the surface.

Another limit to the flux is imposed by the inclination of the field at the edge of the pore. In part this is owing to the inadequacy of the simple model in describing


large fluxes, which causes the variation with depth apparent in Figure 5. With a more sophisticated model it should be possible to balance pressure at all depths for any flux, but the field would still be nearly horizontal at the pore boundary for fluxes greater than 2-3 TWb, as can be seen from Table III and Figure 5.

Now an inclined field is less effective at inhibiting convection, for only the vertical component of B enters into the stability criteria (Chandrasekhar, 196l; Gough and Tayler, 1966). For a given value of the total field B, if c~ is too large, convection will occur in rolls aligned with the magnetic field, so giving rise to penumbral filaments (Danielson, 1961). Thus there should exist some critical value of ~o for simple pores

which have no penumbrae. It is hard to make a precise estimate of this angle. The smallest fields observed in

magnetic knots are around 600 G (Beckers and SchrSter, 1968). Moreover, the magnetic field might be expected to inhibit convection when the magnetic energy density becomes comparable with the kinetic energy of convection; this occurs when B ~ 600 G (Weiss, 1969). We therefore expect that B~ must be greater than 600 G at the edge of a pore. But B is determined by the external pressure and thus, for our model, we find that ~o ~< 67~ So a crude balancing of kinetic energy against the energy in the vertical field yields a critical angle of 65-70 ~

On the other hand, substitution into the stability condition for a Boussinesq fluid (Danielson, 1961 ; Weiss, 1964a) gives an effective field of only 10 G, corresponding to ~o ~ 89-6~ while the vertical field at the umbral boundary in sunspots is over 1600 G and inclined at an angle of 40 ~ .

If the critical value of eo is 65-70 ~ then the critical flux is 0.6-0.8 TWb, corresponding to a radius of 1100-1300 km. This is in fair agreement with observational data. Thus we may conclude that increasing the inclination of the field at the edge of the pore permits the occurrence of convection and thereby sets a limit to the flux. Our simple model suggests a limit of around 1 TWb. Alternatively, the observed critical radius of about 1750 km corresponds to a flux of 1.3 TWb and an inclination of 82 ~ . For such fluxes the model is fairly reliable.

Perhaps the largest source of error in the model is inaccuracy in the assumed pressure variation P (z). Estimates of the pressure or the pressure scale height Hp at the level z = 1 may vary by 30~o from one atmospheric model to another. But for a fixed value of Co, Equation (27) indicates that Ro -- ~o/kocHp and, from Equation (26),

i~,w~ iO 1 I2 it-/2 the flux ~ , o ~p. Thus radii and fluxes in our pore model may have errors of 30-60% as a consequence of this effect.

B. COMPARISON W I T H OBSERVED MAGNETIC FIELD IN A PORE

The only published measurement of the magnetic field distribution across a pore is that by Steshenko (1967). He observed a large pore of diameter 5.6 arc-sec and measured the field at six points across the diameter. The radius was 2030 km and, from Figure 4, with d= 100 km, the model predicts a flux of 1.74 TWb, with a maximum field at the surface Bo(0)=2892 G. The observed fields can be measured from Figure 5 of Steshenko's paper: They correspond to a flux of 1.96 TWb, with a maximum field

100 G . W . SIMON A N D N. O. WEISS

of 2246 G at the center. Steshenko's measured fields are compared with the model in Figure 7. The two fluxes differ by about 11~; such close agreement is perhaps fortuitous. However, his field distribution is flatter than the Bessel function and does not reach as high a central peak. He gives his error as 200 G though lack of resolution might enhance this. In fact, it is difficult to make an exact comparison, for the field is measured at an optical depth between 0.I and 0.0001, where the line is formed,

3000

2500

03 ZD

2000

N 15o0 u_

~lOOO, z

cIz

~- 5OC

Fig. 7.

' r o l l . m , "

I m

�9 m � 9

ooo STESHENKO DATA 7 \ " " ~ 140oEL " ' ' I ' ' ~ " LEAST-SQUARES FIT "'.~ OF THE DATA " .~

\ , i i i i ~ o

420 840 1260 1680 2100

RADIUS OF PORE [km)

Comparison with Steshenko's observations.

and this may be 100-500 km above v = 1. Moreover, the level varies across the pore owing to the Wilson depression. To match Steshenko's measured central field of 2246 G with the model, one must solve 2246 =2892e-kz for z, and we find that z = 222 kin. Thus the central field could be matched by the model if it is measured at this level, about 400 km above r = 1 at the center of the pore.

It is also worth mentioning that the model has 'perfect' resolution, while Steshenko's observations are limited both by instrumental resolution and atmospheric seeing. If we artificially smear our model calculation by a typical amount encountered under good seeing conditions, say 1.0 or 1.5 arc-sec, the central field is reduced to about 2400 G or 2000 G, respectively, which bracket Steshenko's observation.

Thus, either by arguing that the model refers to a different height than the observation, or by introducing artificial smearing into the model, it is possible to match the central field B~(0). However, the curve Bz(r) from the model would still have the wrong shape, being steeper than the observations indicate.

The general qualitative agreement with observation is satisfactory and lends support to the model, Thus we may conclude that the field in Equation (8) provides an approximate representation of the field in a pore, down to a depth of about 200 km below the undisturbed photosphere. The diameter of the pore would be less than 7 arc- sec, with a central field less than 3000 G. Lines of force would appear as in Figure 1 and

ON THE MAGNETIC FIELD IN PORES 10l

the flux, radius and central field should be related as in Figure 4. The flux is approximately

( R 0 ) ~ F ~ 0.45 \ l O ~ J TWb, (41)

where the radius Ro is in kin. This implies an average vertical field of 1450 G. (Steshenko found fields of 1400-1750 G.)

5. Energy Transport in Pores

So far we have only considered mechanical equilibrium in pores. In this Section we comment briefly on thermal equilibrium. The observed rate of emission of energy from a pore (Bahng, 1958) is about 3 x 107 J m - z sec - 1. In the undisturbed photosphere, the flux of energy is maintained by convection, but the strong magnetic fields in pores might be expected to inhibit the convective flux.

Various radiative processes contribute to the energy emitted from a pore. The area of the pore decreases to about 40~o of its surface value at a depth of around 400 km. Radiative transport can supply the energy emitted from this outer annulus, which amounts to more than 60~ of the total. A fraction of the energy from the central core can be conducted across the boundary of the pore below z = 1. More can be radiated laterally at levels where z ~ 1 while the optical radius ZR~ 1 (Jensen and Maltby, 1965; Zwaan, 1968). However, it seems necessary to postulate some non- thermal mechanism to provide the rest of the energy.

A similar problem arises in the umbrae of sunspots (Weiss, 1969), where umbral dots indicate the presence of small scale motion. Although fields of 2000 G would suppress convection in a uniform fluid, Savage (1969) has shown that overstability is possible in a stable layer surmounted by an unstable one, if surface gravity waves are formed at the interface. Some convection may still occur in pores and sunspot umbrae.

6. Pores and Sunspots

The model discussed in this paper breaks down when the magnetic flux is so large that the lines of force bend over near the surface. It also fails when the external pressure cannot be balanced by fields of less than 3500 G. These are criteria for the development of a penumbra. The boundary layer surrounding the flux tube can no longer be neglected, for if the equation

1 VP = 4~/~ [(B" V) B - V (1B2)] + Qg, (42)

is integrated across a boundary layer of thickness h, it can be written

B2 = h i Bz-~) (43) AP + 8~r--~ \4~I2a/'

102 G . W . SIMON AND N. O. WEISS

where a is the radius of curvature of the field lines and the weight of the gas has been neglected. The curvature term can only be ignored if h,~ a. For the pore models dis-

cussed above, a ~ 300 km and for consistency the boundary layer must be narrower

than this.

In the penumbra of a sunspot, the gas pressure drops gradually and the curvature

forces cannot be neglected (Schltiter and Temesvary, 1958; Deinzer, 1965). The

transport of energy is dominated by convective rolls and the Evershed effect. How-

ever, in the umbra the magnetic field may still be represented by the simple model.

The inclination of the field at the umbral-penumbral boundary is about 40~ thus,

if the umbral radius is a, ~ 1.2 r /a and

B, = BoJ1 (kr) e -k~, B z = BoJo (kr) e -kz (44)

where k - 1 ~ 0.83a. The variation of Bz and of the inclination with r agree satisfactorily

with the observations of Beckers and SchrSter (1969) and the vertical field gradient

OBz/Oz= - k B o ~0.4 G km -1.

In conclusion, the limitations of our simple model should be stressed. The gas

pressure within the pore has been ignored; the finite boundary layer has been neg-

lected and the form of the field is oversimplified. Nevertheless, the model agrees

satisfactorily with observational data and helps in exploring the distinction between

pores and sunspots. The structure of sunspots is a bigger problem to which this is

a necessary preliminary.

Acknowledgements

We are grateful to Dr. J. M. Beckers, Dr. S. Musman, and Dr. D. Rust for their

helpful and critical comments on this work. N.O.W. would also like to thank Dr.

J. W. Evans for inviting him to visit Sacramento Peak Observatory, where this work

was done. References

Bahng, J. : 1958, Astrophys. Y. 128, 145. Baker, N. and Temesvary, S. : 1966, Tables of Convective Stellar Envelope Models, NASA, New York. Beckers, J. M. and SchrOter, E. H.: 1968, Solar Phys. 4, 142. Beckers, J. M. and Schrtiter, E. H. : 1969, Solar Phys. 10, 384. Bray, R. J. and Loughhead, R. E. : 1964, Sunspots, Chapman and Hall, London. Broxon, J. W. : 1949, Phys. Rev. 75, 606. Bumba, V. : 1967a, Bull. Astron. Inst. Czech. 18, 238. Bumba, V. : 1967b, Solar Phys. 1, 371. Chandrasekhar, S. : 1961, Hydrodynamic and Hydromagnetie Stability, Clarendon Press, Oxford. Chapman, S. : 1943, Monthly Notices Roy. Astron. Soe. 103, 116. Chapman, S. : 1944, Terrest. Mag. Atmos. Elee. 49, 37. Danielson, R. E. : 1961, Astrophys. Y. 134, 289. Deinzer, W.: 1965, Astrophys. J. 141, 548. Gingerich, O. and de Jager, C.: 1968, Solar Phys. 3, 5. Gough, D. O. and Tayler, R. J. : 1966, Monthly Notices Roy. Astron. Soe. 133, 85. Jensen, E. and Maltby, P.: 1965, Astrophys. Norveg. 10, 19. Parker, E. N.: 1963, Astrophys. J. 138, 552. Rust, D. M.: 1970, Astrophys. J. 160, 315. Savage, B. D. : 1969, Astrophys. J. 156, 707.


Schl/Jter, A. and Temesvary, S.: 1958, Electromagnetic Phenomena in Cosmical Physics (ed. by B. Lehnert), Cambridge.

Schmidt, H. U. : 1964, AAS-NASA Syrup. on Physics of Solar Flares, 107. Sheeley, N. R., Jr. : 1966, Astrophys. J. 144, 723. Sheeley, N. R., Jr.: 1967, SolarPhys. 1, 171. Simon, G. W. and Leighton, R. B. : 1964, Astrophys. J. 140, 1120. Simon, G. W. and Weiss, N. O. : 1968, Z. Astrophys. 69, 435. Steshenko, N. V. : 1967, lzv. Krymsk. Astrofiz. Observ. 37, 21 Title, A. M. : 1969, Amer. Astron. Soc. Solar Section Meeting, Pasadena, Calif. (unpublished). Von Roka, E. G.: 1950, Z. Astrophys. 27, 15. Weiss, N. O.: 1964a, Phil. Trans. Roy. Soc. A256, 99. Weiss, N. O. : 1964b, Monthly Notices Roy. Astron. Soc. 128, 225. Weiss, N. O. : 1966, Proc. Roy. Soc. A293, 310. Weiss, N. O. : 1969, Plasma Instabilities in Astrophysics (ed. by D. G. Wentzel and D. E. Tidman),

Gordon and Breach, New York. Zwaan, C. : 1968, Ann. Rev. Astron. Astrophys. 6 (ed. by L. Goldberg), Academic Press, New York.

on the magnetic field in pores

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