Magnetic-field accumulation in supergranules

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

ALFRED CLARK, JR. and H. KEVIN JOHNSON Dept. of Mechanical and Aerospace Sciences, University of Rochester, Rochester, N.Y., U.S.A.

(Received 27 June, 1967; revision received 25 July, 1967)

Abstract. Some theoretical models illustrating the growth of magnetic fields around the boundaries of supergranulation cells are given. The evolution of a magnetic field in a prescribed velocity field is calculated for the idealized case of hexagonal cells. The calculations are carried out both for a perfect conductor and a fluid of finite conductivity. The theory is developed in such a way as to be virtually free of specific assumptions about the depth dependence of the supergranulation flow. The results demonstrate the tendency of the field to accumulate at those boundary points where the flow is most strongly converging.

1. Introduction

From recent observations (LEIGHTON, NOYES, and SIMON, 1962; SIMON and LEIGHTON, 1964), it is known that the solar supergranulation accumulates vertical magnetic

fields in the boundary regions between supergranules, where the flow is strongly converging. This 'channeled' magnetic field is believed to be the cause of the chromospheric

emission netwolk (PIKEL'NER, 1963; SIMON and LEIGIqTON, 1963), since the field is responsible for enhanced heating of the chromosphere (OSTERBROCK, 1961). Within the framework of this theory, the problem of the formation of the chromospheric network conveniently falls into two parts: the production of the channeled field by the supergranulation, and the formation of the emission network in the regions of

enhanced field. It is the purpose of this paper to present some calculations illustrative of the first of these - the accumulation of field by the supergranulation. We first review briefly the previous work on this problem.

The first such calculation was made by PARKER (1963). He calculated the develop-

ment of an initially uniform magnetic field in a prescribed two-dimensional flow which was periodic in the horizontal coordinate. Parker 's model clearly illustrated the field growth in the boundary regions between supergranules. The converging flow in these regions produces large field gradients as well. For this reason, Parker 's work was extended to include the effects of finite electrical conductivity (CLARK, 1966), and it was found that the time scale for Ohmic diffusion is comparable with a supergranule lifetime (about 20 hr).

Both of the models described above are two-dimensional. In the present work, calculations are presented for three-dimensional models; the calculations illustrate the variable field growth around the boundary of a cell, and the dependence of this growth on the flow convergence. The case considered in detail is the somewhat artificial one of a network of hexagons. Even though the actual supergranules are

somewhat irregular in shape, one may make some comparisons between the present theory and observations by means of qualitative correspondences. For example,

Solar Physics 2 (1967) 433-440; �9 D. Reidel Publishing Company, Dordrecht-Holland

434 ALFRED CLARK~ JR. AND H. KEVIN JOHNSON

the hexagon vertices (where the model flow is most strongly converging) may be associated with the actual small regions bordering on several supergranules where the real flow is presumably strongly converging.

The calculations presented here are based on kinematical hydromagnetic theory - that is, the evolution of the magnetic field in a prescribed velocity field is calculated. While the early stages of field growth are adequately described by the calculations, the later stages of flow-field interaction are beyond the scope of the kinematical theory. The calculations in the remainder of the paper are summarized briefly below.

In Section 2 the equations for a perfectly conducting fluid are developed, and some results are given for hexagonal cells. In Section 3, the theory is extended to include

finite conductivity. I t is shown that under certain general assumptions the models for the surface field become virtually independent of the (unknown) depth dependence of the supergranulation flow. The results for the hexagonal cells illustrate the tendency of the field to accumulate rapidly in the regions of strongest flow convergence.

2. Perfectly Conducting Fluid

In kinematical hydromagnetic theory, the basic equation is the induction equation which (for a perfectly conducting fluid) has the form

~B - curl(V x B). (1)

~t

The computational procedure is to calculate the time development of the magnetic field B from a given velocity field V. We consider first some properties that a suitable

velocity field should have. At the visual ' top ' of the supergranulation, the motions appear to be entirely

horizontal.* Thus if z is the vertical coordinate and W the corresponding velocity

component, then there is a plane z = 0 on which W--0 - a kind of 'leveling-out' surface for the flow. The horizontal velocity components U and V (corresponding to the x- and y-directions respectively) will presumably become small very rapidly with height above the plane z=0 . It is also reasonable to suppose that U and V decrease somewhat with depth below the surface z = 0 . Then U and V are maximum for z = 0,

so that ~ U/Oz and ~ V/~z should both vanish for z = 0. Our basic assumptions about the

flow are then OU ~V

- - W = 0 f o r z = 0 . (2) ~z ~z

For simplicity the flow is also assumed incompressible so that div V = 0. It will be shown that these flow properties allow the calculation of the magnetic field at the

top of a supergranule without further knowledge of the depth-dependence of the flow.

* The absence of observable vertical motions has been a matter of some concern in the literature (MIcHARD, 1965; SIMON and LEIOnTON, 1964; ZIRIN, 1966). From the continuity equation it follows that the vertical velocity at a depth h is W = - - ~ (aU/Ox + OV/Oy)dz ~ Uo(h/L), where /20 ~ �89 km/sec, L ~ 15000 km. Since h(v = 1) ~ 300 km, the vertical velocity sought is only ,~ 10 m/sec.

MAGNETIC-FIELD ACCUMULATION IN SUPERGRANULES 435

We let Bx, By, and B= be the components of the magnetic field, with bx, by, and b~ denoting the same quantities evaluated at z = 0 - thus b=(x, y, t)=B~(x, y, 0, t), etc. Similarly we let u, v, and w stand for U, V, and W evaluated at z=0. Then with the aid of Equation (2) and the conditions div B = 0, div V = 0, we may evaluate Equation (1) at z = 0 to get

3b~ 3b~ 3b~ 3u 3u + u 8x + v 3y = bx3xx + bYgv' (3) 3~

3by 3by v3by 3v 3v 3~- + Uffxx + ~yy -- b ~ x + by3y' (4)

and 3

3b=o~t + 3x3 (ub=) + ~y(Vb=) = 0. (5)

The important point is that Equations (3) - (5) form a closed set for the surface field b(x, y, t): from a knowledge of b(x, y, 0), u(x, y, t), and v(x, y, t), b may be calculated for all times. Since Equation (5) is independent of b= and by, the study of

Y

(0, L)

//J, C (-ffL, --ffL)

; - D (--ffL, O) ---,~..- x

Fig. 1. S c h e m a t i c o f t h e s t r e a m l i n e s o f t h e h o r i z o n t a l m o t i o n s a t t h e t o p o f a h e x a g o n a l supergranule.

436 ALFRED CLARK, JR. AND H. KEVIN JOHNSON

the distribution of the vertical field is not complicated by the possible presence of horizontal field components,

The idealized model chosen to represent the supergranulation network is an array of uniform hexagonal cells, shown in Figure 1 with a schematic of the streamline pattern for the horizontal motions on the plane z=0. The choice of the velocity field is somewhat arbitrary, since neither detailed observations nor a dynamical theory are available. It is convenient, then, to use the simple flow field discovered by CHRISTOPHERSON (1940) for a hexagonal ceil. The properties of this flow were discussed in detail (in connection with the B4nard problem) by CHANORASEKHAR (1961). The horizontal velocity components on z = 0 are

QL~/3 &b QL~3 dq~ u - v . . . . (6)

4re c3x' 4rc c3y' where

c o s (x, y) = - 2 cos \ L x / j j \ 3L / ' (7)

and where Q is a constant of the order of the maximum horizontal velocity. Since the vertical field component is the one of primary interest, we will consider only Equation (5) for b~. The equation is solved by the method of characteristics, which requires the solution of

dx dy db~ - - = u , - v , - 0 b ~ , (8) dt dt dt

where ~u 8v Q 4Ir ~b

0 = Ox + ~?y 3L~/3 (9)

The last of Equations (8) shows clearly the quantitative relation between flow convergence (measured by 0) and field amplification.

In general the integration of (8) is complicated; for points or lines which are invariant under the flow, however, it is much simpler. Thus the field at the invariant points A, B, C, and D (Figure 1), as well as the cell center, is readily calculated: cell center

bz = b~ ~ e -6t/~, (10)

edge midpoint (B or D) (0) ,2 t / r b~=b~ ~ , (11)

vertex (A or C) (0) ~3t/* b z = b z ~ , (12)

where 3Lx/3

- ( 1 3 ) 2nQ

In each case bz (~ is the initial value of b_, at the point in question. The field at the center is depleted very rapidly. The growth at the vertex is more rapid than the growth at


the edge midpoint. The time history of the field at other boundary points is also of interest. The integration of (8) along the boundary segment CD yields

( s )2 {1 - s2"~ {1 - 4 s2"~ 2 bz(�89 y, t) = b (lL,/3, Y0, 0) 2oo \1 - \1 ' (14)

where

, ) = , So = s in~,3L] , (15)

and where Yo is determined by

et/~=(s){1-s2"~+{1-4s2"] ~ s o \ 1 - s U ~ 1 ~ 4 S 2 J "

(16)

Tables of values of bz, constructed from Equations (14)-(16), have shown that the field growth at points between C and D is intermediate between the rates given by Equations (11) and (12). We consider now the applications of these results to supergranulation.

S~MON and LEIGm:ON (1964) have observed horizontal velocities in the range 0.3-0.5 km/sec; these are presumably average values so the choice Q=0.5 km/sec seems reasonable for the maximum velocity. The average cell diameter, according to Simon and Leighton, is about 32000 km, so we take L = 16000 km. Then the time scale (13) is v,-~7.36 hr. Thus after a time ~, the field at the edge midpoint has been amplified by a factor of about 7.4 ~e2), whereas the vertex field has increased by a factor of about 20 (e3). After a time 2~ - just under 15 hr - the midpoint field has increased by a factor of 55, whereas the vertex field has been amplified by a factor of over 400. Since the initial fie!d strengths are probably in the neighborhood of one or two gauss, it is clear that flow convergence is capable of producing rather large fields (subject of course to dynamical effects which could limit the field strengths to an equipartition value of 20 or 25 gauss). It is also clear that the vertex field soon becomes dominant.

The actual supergranulation cells are not hexagons, but there are relatively small interstitial regions, bordering on three or more supergranules, where the flow must be strongly converging. In comparing the theoretical predictions with observation, then, we may identify these interstices with hexagon vertices in the idealized model. A quantitative map of the field strength around the boundary of a supergranule is precluded by insufficient resolution; however, there is considerable indirect evidence of the 'vertex' accumulation. BUMBA and HOWARD (1965a) found that the initial brightening in calcium emission, indicating the birth of an active region, takes place in the space between several supergranules. A later study of flares by the same authors (BUMBA and HOWARD, 1965b) showed that in every case observed, the first brightening of the flare occurred in the border region between several network cells. Other observational results which may be relevant are the very small clumps of re?atively strong magnetic fields discovered by SHEELEY (1966) in quiet regions of the


sun. SHEELEY (1967) has estimated that fields of several hundred gauss regions as small as 500 km in extent.

occur in

3. Fluid of Finite Conductivity

Since the conductivity of the solar material is so high, the assumption of a perfect conductor is reasonable in most situations. An exceptional case occurs whenever a strongly converging flow crowds field lines together and enhances Ohmic diffusion. This is the situation in the boundary region between adjacent supergranules, so the effects of finite conductivity in the present problem are not without interest.

For a fluid of finite conductivity, the induction equation takes the form

8B 8t curl(V • B) + t/V2B, (17)

where t/is the magnetic diffusivity. Equation (17) is based on the usual assumptions of hydromagnetic theory - Ohm's law with a constant scalar conductivity, and negligible displacement currents. Normally, the diffusive term would prevent one from obtaining a closed set of equations for the surface field b. In the present case, however, we may still do this by a reasonable approximation. The only regions of the flow where diffusion is likely to be of importance are the cell boundaries. In these regions, the large components of the field gradient responsible for diffusion are the horizontal ones built up by the converging flow. Thus we may make the approximation V 2 ~V~ = ~2/0X2. . t_ ~2/(~y2. Then the equations for the field components on the plane z= 0 are obtained by adding the components of t/VZb to the right-hand sides of Equations (3)-(5). In particular, b~ satisfies

~bz O(ub~) 8(vb~) ~ t + - ~ S + ~y -~V~bz. (18)

The solution of (18) is in general difficult. However, relevant results may be obtained by a flux integral and the steady state solution. The velocity components u and v (Equation (6)) have 'hexagonat periodicity', and we suppose that the initial (t= 0) magnetic field does also. Then the magnetic field will remain periodic as time goes on, and, by symmetry, no magnetic flux will cross cell boundaries - that is

~dt f f bz(x, y, t) dx dy = O, (19)

cell

a result which can also be obtained directly by the use of Equation (18). This flux integral makes possible a connection between the diffusion-limited steady state and the initial conditions. From Equation (18) (with 8/8t=0) and Equations (6) and (7), we may obtain the equation governing the steady-state field, in the form

8 /" 8~b \ 9 ( 8 ~ ) (20) R-1V2b~= ~x~b~x)+~yy bz~y ,


where R = (4rt~)-1 QLx/3 is the magnetic Reynolds number. It can be shown (after some calculation) that the solution of Equation (20) which satisfies the periodicity condition (n.Vb= = 0 on the boundary) is unique, and is given by

bz(x, y) = Ce Re. (21)

The constant C is determined by the flux integral; thus

C $5 ere dx dy = (3 L 2 ~f3) Bo, (22) cel l

where Bo is the mean initial field (3LZ~f3 is the area of the hexagon). The evaluation of C is greatly simplified by the circumstance that R is large.

We have, typically, Q~�89 kin/see, L ~ 16000 kin, and q ~ 2 x 108 cm2/sec, hence R ~ 5.5 x 104. The integral in Equation (22) may be evaluated by a two-dimensional steepest descent procedure; the main contributions come from the six vertices, since ~ is largest and V~ vanishes at a vertex. The final result for the steady-state field is

b=(x, y) ~ Bo(�89 ) e -R(�89 (23) R--+ oo

Since ~O -.~.2 -~ 3 with ~b = ~ only at a vertex, the field is exponentially small, even on the boundary, except in the immediate vicinity of a vertex. This is to be contrasted with the case of a perfect conductor for which the field grows large at all boundary points. This difference in behavior may be understood by considering the processes ~n the vicinity of an edge midpoint (D in Figure 1). Along the line between the origin and point D, the flow is converging toward D. Along DC, however, the flow is diverging from D. There is a net convergence, though, since the inflow is greater than the outflow - that is, 0(D)<0. Thus for a perfect fluid, the field builds up indefinitely. The effect of the finite conductivity is to allow the field lines to slip back through the converging flow from D toward the cell center. The flow along DC diverges from D, so that the finite conductivity has little effect on the transport of field away from D along DC. Thus the finite conductivity interferes with the accumulation of field by the inflow but not with the dispersal of field by the outflow. The consequence is that almost all of the flux eventually arrives at one of the vertices, where the flow is converging from all directions.

The behavior of the field at the vertex is of some interest. By expanding the exponent in Equation (23) about a vertex, we obtain for b~ the approximation

~ - \ ~ j _], (24)

where r is the distance from the vertex. Thus the region of large field is confined to a circular region with radius of the order of L/x/R.

Another feature of interest is the time scale for diffusive effects to be of importance. A simple estimate of this time scale is obtained by finding the time tD at which the perfect conductor solution at a vertex (Equation (12)) reaches the steady-state dif-


fusion limited value given by Equation (24) with r = 0 . The result is to=(z /3)

In (zR~/3/2) ~ 29.2 hr. This is somewhat longer than the observed cell lifetime of 20 hr, but in view of the uncertainties in the model, we conclude that diffusive effects could be of importance in determining the magnetic fields associated with supergranulation.

The magnitude of the field at the vertex predicted by Equation (24) is far too large to be relevant. However, one may conjecture that dynamical effects could create

'dead-water ' regions of finite width between the cells, f rom which the flow is excluded. Then the diffusive effects considered here could occur in thin layers separating the flow regions from the quiet regions. In any case, the converging nature of the super-

granulation flow makes it likely that some effects of finite conductivity will be ap- preciable in relatively short times, in spite of the large magnetic Reynolds numbers.

Acknowledgments

This work was supported in part by the National Science Foundation under NSF

Grant GP6491. One of the authors (H.K.J.) was supported by a NASA traineeship.

References

BUMBA, V. and HOWARD, R. : 1965a, Astrophys. J. 14L 1492. BUMBA, V. and HOWARD, R. : 1965b, Astrophys. J. 142, 796. Cr~ANDRASEKHAR, S. : 1961, Hydrodynamic and Hydromagnetic Stability. Oxford University Press,

London, p. 47-51. CHRtSTOVHERSON, D.G. : 1940, Quart. J. Math. 11, 63. CLARK, A., JR.: 1966, Phys. Fluids 9, 485. LEIGHTON, R.B., NoYzS, R.W., and SIMON, G.W. : 1962, Astrophys. J. 135, 474. MICHARD, R. : 1965, Stellar and Solar Magnetic Fields (ed. by R. Ltist). North-Holland Publ. Co.,

Amsterdam, p. 229. OSTERBROCK, D.E. : 1961, Astrophys. J. 134, 347. PARKER, E.N. : 1963, Astrophys. J. 138, 552. PIKEL'NER, S.B. : 1963, Soviet Astron. - A.J. 6, 757. SHE~LEY, N.R., JR. : 1966, Astrophys. J. 144, 723. SHEELEV, N.R., JR.: 1967, Solar Phys. 1, 171. SIMON, G.W. and LEIGHTON, R.B. : 1963, Astron. J. 68, 291. SItaON, G.W. and LEIGHTON, R.B. : 1964, Astrophys. J. 140, 1120. ZmIN, H. : 1966, The Solar Atmosphere. Blaisdell, p. 234.

Magnetic-field accumulation in supergranules

Documents

Transcript of Magnetic-field accumulation in supergranules