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

V I S C O U S B O U N D A R Y L A Y E R *

E. N. PARKER

Department of Physics, Department of Astronomy and Astrophysics, University of Chicago, Chicago, Ill. 60637, U.S.A.

(Received 19 February; in revised form 19 May, 1981)

Abstract. Galloway, Proctor, and Weiss have shown by numerical experiment that the magnetic field extending across a convective cell in a highly conducting viscous fluid may be concentrated into sheets with energy density B 2/8 ~r larger than the kinetic energy density �89 2 of the convection by a factor (v/'q)1/2 or more. This paper employs conventional boundary layer theory for high Reynolds number to provide a simple analytical example illustrating this remarkable effect of field concentration.

1. Introduction

Galloway, Proctor, and Weiss (1977, 1978; Peckover and Weiss, 1978) have shown

with numerical experiments how the closed convective circulation of a viscous fluid

with small resistivity ~ breaks up a broad magnetic field into isolated filaments and

then proceeds to compress those filaments to field strengths beyond equipartition with the convective motions when v >> r/. The extraordinary compression of the

magnetic field arises f rom the fact that the large kinematic viscosity v couples the

energy and m o m e n t u m of the thick viscous boundary layer into the thin resistive

boundary layer occupied by the field. Hence at a plane boundary, where the ratio of the boundary layer thicknesses is (1,/77)1/2, the characteristic stress density B 2/4 r of

the field may increase to a value (v /~) 1/2 times the Reynolds stress density p U 2 for

the fluid, so that

B ~ / 8 ~ ~ ~oU2(u /n ) ~/~ (1)

in order of magnitude. It is a remarkable and wholly unanticipated result. It merits

the closest scrutiny and understanding of anyone interested in the physics of magnetic fields and convecting fluids. This paper presents an elementary illustration

of Weiss 's viscous compression mechanism, taking advantage of the simplification at

large Reynolds numbers.

Now the breakup of the magnetic field of the Sun into many separate flux tubes is generally attributed to the quasi-steady convective cells (supergranules) (Parker,

1963, 1979; Weiss, 1966). The extreme compression of the individual tubes, to 1-2 kG, is sometimes passed off as a part of the same process, based on Weiss's viscous enhancement mechanism (1). In view of the importance of understanding the causes of the intense fibril state of the solar magnetic field, with its implications for

* This work was supported by the National Aeronautics and Space Administration under grant NGL 14-001-001.

Solar Physics 77 (1982) 3-11. 0038-0938/82/0771-0003501.35. Copyright �9 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

4 E.N. PARKER

the origins of solar activity, a closer look at the application to the Sun seems warranted.

First of all consider the circumstances under which u >> ~. In fully ionized hydrogen at a temperature T the kinematic viscosity is (Chapman, 1954)

~ 1.2 x 1 0 - 1 6 Ts/Z/o c m 2 S - 1 ,

while the resistive diffusion coefficient is (Cowling, 1953)

'r/~ 0.5 X 1013 T -3/2 .

Hence

v/~ = 2 x 10 -29 T 4 / p ,

so that ~,/r/>> 1 in high temperatures and low densities. A few hundred km below the visible surface of the Sun, where T = 104 K and p = 4 x 10 -7 g cm -3, it follows that u / r / ~ 5 x l 0 -7. At the base of the convective zone, where T ~ 2 x l 0 6 K and p ~ 0 . 2 g c m -3, it follows that ~ , / r / ~ 2 x 1 0 -3. At the center of the Sun, where T ~ 1.5 x 107 K and p ~ 102 g cm -3, it follows that u / r / ~ 10 -2. Evidently nowhere

within the Sun is u as large as r/. On the other hand, in the solar corona, where T ~- 2 x 106 K and p ~ 2 x 10 -16 g cm -3, it follows that v/r/~- 2 x 1012. Looking

elsewhere, we note that in the interstellar H n region, where T =-104K and p = 2 x 10-24gcm -3 we have u/r/~101~. It would appear, then, that the viscous

compression of field beyond equipartition, described by (1) when v > r/, may occur in the solar corona or in interstellar space but not within the Sun. Consider, then, the basic physics of Weiss's mechanism so that we may better understand the broad picture of field compression.

2. Field Concentration at the Boundary

The remarkable effect demonstrated by Weiss et al. is that the viscous flow may push the field back, compressing it to values far in excess of the equipartition value. The purpose of this paper is to provide a simple analytical illustration to complement the extensive numerical results already available in the literature. The numerical results are limited by the grid size to modest Reynolds numbers and magnetic Reynolds numbers (see also the analytic solution for small Reynolds number in Galloway et al. (1978)). On the other hand, when these numbers are very large, it is a simple matter to treat the problem analytically using the standard boundary layer approximation (Goldstein, 1938). For large Reynolds number the boundary layer is thin, so that the curvature of the boundary plays no role and can be omitted from the calculation. If the fluid encounters the field for a distance L along the boundary, the Reynolds number R is LU/v, and the magnetic Reynolds number Rm is LU/r/. We suppose t h a t Rm >> R >) 1,

Following standard boundary layer theory for large Reynolds number, note that the principal velocity and magnetic field are the tangential components along the

COMPRESSION OF MAGNETIC FIELD IN A VISCOUS B O U N D A R Y LAYER 5

boundary, while the principal gradients in the velocity and magnetic fields are perpendicular to the boundary of the flow. Denote distance measured downstream along the boundary of the flow by s, starting with s = 0 at the point where the field first penetrates through the boundary into the fluid. Denote by y the local normal distance measured from the boundary into the fluid. Denote the fluid velocity along the boundary by V(s, y) with the free stream value U at distances y large compared to the thickness of the viscous boundary layer. To keep the problem as simple as possible, suppose that the fluid experiences a free boundary except where it encounters the Lorentz force (V x B)x B/47r of the field in the resistive boundary layer. Thus the viscous boundary layer begins to form only at s = 0, and beyond, where the flow of the fluid is impeded at the wall by the Lorentz forces. Figure 1 is a sketch of the field and flow with the characteristic thickness of the resistive and viscous boundary layers indicated by the dashed lines. The field lines are not sketched in the thin resistive boundary layer for lack of space. They are shown only as

_

(4 s/U) 2 ~ y -

Fig. 1. A sketch of the setup for the analytical calculation of the compression of field in a viscous boundary layer. The solid line s represents the 'boundary ' of the flow, beyond which the flow has but little effect. The dashed lines represent the viscous and resistive boundary layers. The heavy arrows represent

the fluid mot ion and the short arrows represent the field Bo extending into the moving fluid.

6 E . N . P A R K E R

they cross the boundary. The fluid motion is indicated by the four heavy arrows. The characteristic thickness of the viscous boundary layer at a distance s downstream is (4vs/U) 1/2, of course. Similarly, the characteristic thickness of the resistive bound- ary is (4rls/a U)1/2 where a U denotes the tangential velocity of the fluid within the resistive boundary layer. Working in the limit that v >> 4, the viscosity prevents the tangential velocity from varying much across the very thin resistive layer, so that a U is essentially the velocity of slip along the boundary.

The hydromagnetic equation for steady field and flow is

Vx ( v x B - n V x B ) = 0 . (2)

In the two dimensions s and y write

Bs = +OA/Oy, By = -OA/Os. (3)

The tangential component of v is a U. The normal component is taken to be negligible within the very thin boundary layer, so that (2) reduces to

O (aU OA 02A'~ 0s- '07] =~

upon neglecting 02lOs 2 compared to 02/Oy e. Integrating once yields

OA 02A aU Os = rl Oy 2 ' (4)

where the integration constant has been set equal to zero because A = constant in the field-free region outside the resistive boundary layer. Write h ---- ~ / a U. The solution

o0

[ / k \ 1/2 "1 A(s, y)=-B--~~ { i - f dkf(k) c o s [ k s - ( ~ ) yJ exp [ - ( k ) l / e y ] }

q 0

(5)

is as convenient a form as any, so that A vanishes at y = +0o far from the boundary. Suppose that there is a field extending across the boundary with

By = Bo sin qs (6)

on y = 0 in the interval 0 < s < 2rr/q, and By = 0 elsewhere, as sketched in Figure 1. Then the vector potential on the boundary is

A(y, s) = -(Bo/q)(1 - cos qs) (7)

in 0 < s < 2rr/q and zero elsewhere. It follows that

f ( k ) = l [sin (2~r/q)(k +q) k+q

2k sin 27rk/q ~(k -q) (k +q)

sin (27r/q)(kk_q - q)]

(8)

C O M P R E S S I O N OF M A G N E T I C F I E L D IN A V I S C O U S B O U N D A R Y L A Y E R 7

The function f(k) oscillates about zero with the principal maximum (for k > 0) at

k = q. As a rough approximation, for the present exploratory purposes, replace (8) by 6(k- q), recognizing only the peak in f(k) at k = q. The result is the rather crude representation

( [" l a \ ] r ( q \112 11 A(s, y )~-B~ tl-cos[qs-t-~A)a/2yJexp[-~-~) yj.~ (9) q

for the field. Note that this is the same as would follow if (6) and (7) applied over the entire interval -oo < s < +oo instead of 0 < s < 2rr/q. The approximation throws away the higher harmonics involved in cutting off the field beyond s = 2rr/q. The lines of force A -- constant are plotted in Figure 2. The representation (9) is sufficient

to establish the basic relation between the general magnitude of the field and the fluid velocity U.

It follows from (3) and (9) that

B0 2/2 ~'] exp [ q 1/2

-t77) Y J

The distortion

(11)

of the field by the force - ( V • 2 1 5 exerted on it by the

streaming fluid is obvious from Figure 2.

30 l I

,., , 2 2O

0

j 1.0

0.9

27r qs

Fig. 2. A plot of the magnetic lines of force A - cons tan t , with the numerical values of q A / B o indicated on each curve.

8 E .N. PARKER

With Rm >> 1 it follows that Aq << 1. Hence, the tangential component of the Lorentz force is

(VxB)xB/41r~-(Boq/4~rA) sin2 qs- y x

x e x p �9

The total drag on the fluid per cm 2 of boundary is cO

D(s )= - f dy (VxB)xB/4c r

0

B~ 1 1 (2qs + - 8~-(2qA)~/2 [ - ( 2 - ~ sin 4 ) ] " (12)

The fluid is driven along the resistive boundary layer in opposition to D(s) by the shear in the fluid. Hence, for v/'q >> 1, the boundary condition

pv Ors - -~-D(s) (13) Oy

applies at the outer 'surface' of the resistive boundary layer, which can be approxi- mated by applying (13) at y = 0.

3. The Viscous Boundary Layer

The hydrodynamic equation for steady flow is

(v" V)v = -Vp/p + uV2V.

The standard boundary layer approximation for large Reynolds numbers neglects the normal component vy and replaces (v �9 V)v with (U �9 V)v, where U is the free stream velocity. The pressure p in the free stream outside the thin boundary layer is essentially unaffected, so that Op/Os-~ 0 in the free stream. Hence, Op/Os must be small in the boundary layer alongside the free stream. Neglecting 02/Os 2 compared to 02/0y 2 in V 2, we obtain the familiar equation

3V 32t~ U - - ~ v - - (14)

3S 0y 2

for the tangential velocity v(s, y) in the viscous boundary layer. As a solution to this parabolic equation let

v(s,y)= U - C exp - ~ s - y erfc (4A~)1/2 +

+ F sin [2qs- (q/A)l/2y] exp [-(q/A)l/Zy],

COMPRESSION OF MAGNETIC FIELD IN A VISCOUS BOUNDARY LAYER 9

where A is the characteristic length u~ U, and C and F are constants, to be evaluated from the boundary condition (12) and (13). It follows that

i/2 y2 ) V 2 Y + aI/2 .A(RRm)I/2 x

(4As) 1/2 U2

xsin[2qs-(q/A)l /2y]exp[-(q) l /2y]] , (15)

where R and Rm are the Reynolds numbers

R =- U/qu, R,. =- U/q•. (16)

The fluid velocity at the resistive boundary layer is, accordingly

v(s, 0)= U~l-V~(ceRRm)a/z[(~)l/2-1sin2qs]}[ U2 . (17)

This is the velocity aU that drives the resistive boundary layer. The magnetic field is compressed into the boundary layer in the manner described by (10) and (1 i) (see Figure 2). The tangential field IBsl has a maximum at qs = 5~r/4, y = 0, given by

Bmax = Bo/(qA ) i/2 = Bo(o~Rm )I/2 (18)

In terms of Bma x the fluid velocity at the boundary can be written

I)(s,O)=U{ 1 B2ax/S7-i-( R ]I/2[(qs]'/2_�88 ( 1 9 )

�89 2 \ - ~ 1 L\2~r/

Define the effective velocity o~U, employed in Section 2 in the calculation of the resistive boundary layer, to be equal to v(s, O) at some position s= si (to be specified). Then (19) can be written

O(sOS = o~2/2- a3/2, (20)

where

( qs ) l/2_ �88 O(s)~ ~ sin 2qs

and

B m.x/8, ( n lj2

We wish to determine the maximum field strength that can be produced by compression of the field Bo extending into the flow. If we suppose for the moment that B0 is very small, then B . . . . given by (18), may be well below the equipartition

10 E . N . PARKER

1 / 21:~ . value. The maximum value of o~ is one, for which Bmax = R,~ "-'o, Bmax goes to zero with B0. The fact is that the field Bo simply cannot be compressed into a layer thinner

than the resistive boundary, of thickness (4TI/qU) 1/2. The question, then, is how

strong must Bo be in order that the field interact strongly with the flow and achieve

maximum compression. If we imagine that initially the weak magnetic field B0 extended far into the fluid,

and was subsequently pushed back toward the boundary by the flow, so that it passed

into the compressed state as t ime progressed, then we would expect the flow velocity

aU along the boundary to decline from an initial value a = 1 to a < 1 as the field was

compressed. If the field became so strong that the Lorentz forces (the drag D in

Equation (12)) caused a to approach zero, then obviously the compression of the field would diminish and a would increase again. On the other hand, if a became

large, say a - 1, then the compression by the flow would increase the field, causing a

to diminishl It is evident, therefore, that there should be a stable situation with a depressed to the value allowing the largest possible field.

It is readily shown from (20) that O(Sl)S is a maximum for a = �89 The maximum value is

O(sOSmax = (2/3) 3/2.

In this case

B2ax 1 uz/Rm~ 1/2 2 8 , ~ - ~ o ~ - ~ } (3)3/20(s~) �9

It seems not unreasonable to suppose that s~ should be chosen to lie somewhere in

the vicinity of the maximum field B . . . . at s = 5~r/4q, for evaluating the reduction of

the fluid velocity V(Sl, 0) by the field. In that case O(sO = (5/8)1/2-~ ~ 0.556, and

n max - - ~ 0 . 7 �89 2. (21) 8~r

It follows f rom (18) that

Bo ~ 3 Bo~ 3 e max

87r R~ 87r

2 - - ( R m R ) I / 2 ~ p U 2 . (22)

If Bo is smaller than (22), the field is not compressed to the value of Bma x given in (21), because the field B0 can be compressed only into the resistive boundary layer thickness and not more. If B0 is larger, then the field pushes farther out into the fluid, and its outer surface is covered by the resistive boundary layer treated here. The 'boundary ' y = 0, it will be recalled, is simply the inner surface of the resistive boundary layer, beyond which the fluid motion does not penetrate effectively. The numerical experiments of Weiss et al. indicate that the problem becomes more

COMPRESSION OF MAGNETIC FIELD IN A VISCOUS BOUNDARY LAYER 11

complicated when B0 exceeds the value given by (22), often breaking up into more than one convective cell. The present results are not applicable in that case.

The main point is that the peak magnetic energy density may exceed the free stream kinetic energy density by a factor of the order of (R,,/R) I/2= (v/~) I/2. The precise value of the numerical coefficient given by the present estimate for sI = 5 rr/4q is not to be taken seriously. The coefficient is of the order of unity rather than 10 -1 or 10 +1 and that is about all that can be said.

In hot tenuous gases v/r/may be a large factor. The present rather crude analytical calculation in the limit of large Reynolds numbers (i.e. thin boundary layer) is presented to illustrate the physical principles that lead to the extraordinary dynami- cal compression of the field discovered by Weiss and collaborators. The need for numerical experiments to determine the precise degree of concentration of the field in each special case, particularly for modest Reynolds numbers, should be obvious from the difficulties of the idealized analytical solution presented here.

References

Chapman, S.: 1954, Astrophys. J. 120, 151. Cowling, T. G.: 1953, in G. P. Kuiper (ed.), The Sun, University of Chicago Press, Chicago, Chapter 8. Galloway, D. J., Proctor, M. R. E., and Weiss, N. O.: 1977, Nature 266, 686. Galloway, D. J., Proctor, M. R. E., and Weiss, N. O.: 1978, Jr- FluidMech. 87, 243. Goldstein, S.: 1938, Modern Developments in Fluid Dynamics, Clarendon Press, Oxford. Parker, E. N.: 1963, Astrophys. J. 138, 552. Parker, E. N.: 1979, Cosmical Magnetic Fields, Clarendon Press, Oxford, Chapter 16. Peckover, R. S. and Weiss, N. O.: 1978, Monthly Notices Roy. Astron. Soc. 182, 189. Weiss, N. O.: 1966, Proc. Roy. Soc. London A293, 310.