On a Babcock-Leighton Solar Dynamo Model with a Deep-seated

THE ASTROPHYSICAL JOURNAL, 486 :1065È1077, 1997 September 101997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ON A BABCOCK-LEIGHTON SOLAR DYNAMO MODEL WITH A DEEP-SEATEDGENERATING LAYER FOR THE TOROIDAL MAGNETIC FIELD. IV.

BERNARD R. DURNEY

University of Arizona, Physics Department, Tucson, AZ 85721 ; durney=physics.arizona.eduReceived 1996 November 25 ; accepted 1997 April 18

ABSTRACTThe study is continued of a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of

toroidal magnetic Ðeld are the source for the poloidal Ðeld) with a thin, deep seated layer (GL ), for thegeneration of the toroidal Ðeld, The partial di†erential equations satisÐed by and by the vectorBÕ. BÕpotential for the poloidal Ðeld are integrated in time with the help of a second order time- and space-centered Ðnite di†erent scheme. Axial symmetry is assumed; the gradient of the angular velocity in theGL is such that within this layer a transition to uniform rotation takes place ; the meridional motion,transporting the poloidal Ðeld to the GL , is poleward and about 3 m s~1 at the surface ; the radial di†u-sivity equals 5] 109 cm2 s~1, and the horizontal di†usivity is adjusted to achieve marginal stabil-g

rghity. The initial conditions are : a negligible poloidal Ðeld, and a maximum value of in the GL equaloBÕ o

to where is a prescribed Ðeld.1.5] Bcr, BcrFor every time step the maximum value of in the GL is computed. If this value exceeds thenoBÕ o Bcr,there is eruption of a Ñux tube (at the latitude corresponding to this maximum) that rises radially to thesurface. Only one eruption is allowed per time step (*t) and in the GL is unchanged as a consequenceBÕof the eruption. The ensemble of eruptions is the source for the poloidal Ðeld, i.e., no use is made of amean Ðeld equation relating the poloidal with the toroidal Ðeld. For a given value of *t, and since theproblem is linear, the solutions scale with Therefore, the equations need to be solved for one valueBcr.of only.BcrSince only one eruption is allowed per time step, the dependence of the solutions on *t needs to bestudied. Let be an arbitrary numerical factor (\3 for example) and compare the solutions of theF

tequations for *t) and *t/3). It is clear that there will be 3 times as many eruptions in the(Bcr, (Bcr,second case (with the shorter time step) than in the Ðrst case. However, if the erupted Ñux in case oneis multiplied by 3, then the solutions for this case become nearly identical to those of case two (*t isshorter than any typical time of the system, and the di†erence due to the unequal time steps isnegligible). Therefore, varying the time step is equivalent to keeping *t Ðxed while multiplying theerupted Ñux by an appropriate factor. In the numerical calculations *t was set equal to 105 s. The factor

can then be interpreted as the number of eruptions per 105 s. The integration of the equations showsFtthat there is a transition in the nature of the solutions for For the eruptions occurF

tB 2.5. F

t\ 2.5,

only at high latitudes, whereas for the eruptions occur for h greater than B n/4, where h is theFt[ 2.5,

polar angle. Furthermore, for the toroidal Ðeld, in the GL can become considerably largerFt\ 2.5, oBÕ o,

than while this ceases to be the case forBcr, Ft[ 2.5.

The factor is an arbitrary parameter in the model and an appeal to observations is necessary. WeFtset G. In the model, the magnetic Ñux of erupting magnetic tubes, is then about 3 ] 1021 G, ofBcr \ 103

the order of the solar values. For this value of and for the value of at which the transitionBcr Ft(B2.5)

takes place, the total erupted Ñux in 10 years is about 0.85 ] 1025 Mx in remarkable agreement with thetotal erupted Ñux during a solar cycle. Concerning the dynamo models studied here, a major drawbackencountered in previous papers has been the eruptions at high latitudes, which entail unrealistically largevalues for the radial magnetic Ðeld at the poles. The results of this paper provide a major step forwardin the resolution of this difficulty.Subject headings : MHD È Sun: interior È Sun: magnetic Ðelds

1. INTRODUCTION

Convincing arguments suggest that the toroidal large-scale magnetic Ðeld, responsible for the surface solarBÕ,activity, is generated in a thin layer (hereafter the GL ) at thebase of the solar convection zone, SCZ (see & WeissSpiegel

& van Ballegooijen & Rosner1980 ; Spruit 1982 ; SchmittSchu� ssler, & Ferriz-Mas1983 ; Moreno-Insertis, 1992 ;

Schu� ssler 1993).The generation of the toroidal Ðeld by a shear in the

angular velocity, acting on a poloidal Ðeld, is straightfor-ward. The regeneration of the poloidal Ðeld from the toroi-dal Ðeld, an essential step in the dynamo action (Parker

& Krause is on the other hand the1955 ; Steenbeck 1966),

outstanding problem in dynamo theory. Since loops oftoroidal Ðeld have to merge to form the large-scale poloidalÐeld, the precise location of the regenerating layer (RGL )must depend on the prevailing dissipation. Because thesolar surface is an appealing place for the RGL , it wasargued in De Young, & Roxburgh thatDurney, (1993)dynamo models of the Babcock-Leighton type (Babcock

should be modiÐed to incorporate1961 ; Leighton 1969)current ideas about the location of the GL . In Babcock-Leighton models, the poloidal Ðeld is regenerated by theeruption of the toroidal Ðeld in the form of large bipolaractive regions, their breakup, and di†usion. Nash,Sheeley,& Wang and Sheeley, & Nash have(1987) Wang, (1991),

1065

1066 DURNEY Vol. 486

added a poleward meridional circulation at the surface, tothese models.

It is important to keep in mind, however, that it is farfrom certain that the main source for the surfaceÏs globalpoloidal magnetic Ðeld is the breakup of large active regions(see It is certainly the most conspicuous oneStenÑo 1992).but this does not ensure that it is the most important.StenÑo argues that the large-scale Ðeld results from a con-tinuous supply of small-scale Ñux originating below thesolar surface. This small-scale Ñux has tiny but nonzerolarge-scale correlations (see Harvey which,1992, 1993),when averaged, could be an important source for the globalÐeld patterns seen at the surface.

In previous papers (Durney hereafter1995, Paper I,hereafter hereafter we1996a, Paper II, 1997, Paper III)

studied a dynamo model of the Babcock-Leighton typewith a thin, deep seated generating layer, GL , for the toroi-dal Ðeld, For every time step the maximum value ofBÕ. oBÕ oin the GL is computed. If in this layer, and for a certainvalue of the polar angle, h, exceeds a prescribed Ðeld,oBÕ o

then there is eruption of a Ñux tube at this latitude. ThisBcr,Ñux tube, which was assumed to rise radially, generates atthe surface, a bipolar magnetic region (BMR) with Ñuxes '

fand for the following and preceding spot, respectively.'pFor the purposes of the numerical calculations the BMR

was replaced by its equivalent axisymmetrical magnetic ringdoublet. Only one eruption was allowed per time step. Theensemble of eruptions acted as the source for the poloidalÐeld, i.e., no mean Ðeld equation relating the poloidal with thetoroidal Ðeld was assumed. In LeightonÏs model (see hiseq. [4] or below), the time derivative of theeq. [19]radial magnetic Ðeld, is expressed as a function ofLB

r/Lt,

L(sin where c is the angle formed by the magneticcBÕ)/Lh,axis of the BMR with the east-west line (the ““ tilt angle ÏÏ).The di†use Ðeld is assumed to be directly related to theB

rtoroidal Ðeld, and in the dynamo equations (turbulent) dif-fusion appears then as an arbitrary parameter. The equa-tions of mean Ðeld electrodynamics are equations for thelarge-scale Ðelds, and therefore no inconsistency arises ifdi†usion is small or even vanishes. The approach followedin this paper di†ers essentially since no ““mean Ðeld equa-tion,ÏÏ as LeightonÏs equation (4), is used. Di†usion, far frombeing an arbitrary parameter, uninvolved in the generationof generates now the di†use Ðeld from the eruptingB

r, B

rtoroidal Ðeld. It is true that the assumption of axial sym-metry implies an average over longitudes. The reason forthis assumption is clearly a practical one. If computers werefaster, it would be straightforward to add the /-dimensionto the dynamo equations. It cannot be overemphasized,however, that the neglect of LeightonÏs equation (4), consti-tutes already a signiÐcant step away from mean Ðeld elec-trodynamics. It is clearly a fortunate feature ofBabcock-Leighton dynamo models that such a step is notfraught with unsurmountable difficulties as it would be thecase, for example, in conventional a-u dynamos (see Stix

& Brandenburg for solar dynamos1976 ; Ru� diger 1995 ;with variable a and u e†ects see & ThomasRoald 1997).

The poloidal Ðeld, generated in the surface layers, wastransported to the GL in the lower solar convection mainlyby meridional motions (for studies on the e†ect of merid-ional motions on the dynamo action see & StixRoberts

Ballegooijen and Schu� ssler, &1972, van 1982, Choudhuri,Dikpati 1996).

In Papers and the existence of oscillatory solutionsII III,

of the dynamo equations with was established.'p\ ['

fHere, we continue the study of Papers and with aI, II, IIIsimple radial dependence of the source for the poloidalmagnetic Ðeld. The strength of the source is determined bythe magnetic Ñux of the erupting magnetic tube, whereas thelatitudinal dependence is determined by the separation ofthe following and preceding poles and by the latitudinaldimensions of both poles. Furthermore, the radial depen-dence of the source must be speciÐed. Most of the numericalcalculations of this paper were performed with a source thatvanished outside the interval with(R

a[ 5 *r, R

a] 5 *r),

cm, and was constant inside. Here *r \Ra\ 6.7 ] 1010

is the mesh size in the radial direction.(R0[ Rc)/100 R0\

6.9] 1010 cm and cm are the upper andRc\ 5 ] 1010

lower boundaries of the solar convection zone (SCZ).Therefore, the source was constant for values of r such that6.605] 1010 cm\ r \ 6.795] 1010 cm, and vanished forlarger and smaller rÏs. For a usual, surface eruption of a Ñuxtube, the appropriate choice would have been R

a\ 6.805

] 1010 cm, leading to a nonvanishing source extendingfrom the surface down to 6.71 ] 1010 cm. However, asstated above it is far from certain that the main contributionto the generation of the solar large-scale di†use magneticÐeld is the breakup of large active regions.

Clearly, the main justiÐcation of the somewhat““ abstract ÏÏ radial dependence for the source used in thispaper is its mathematical simplicity : it allows for a clearvisualization of the input that generates the poloidal Ðeld. Itis certainly not a ““ friendly ÏÏ source in the sense that it wasnot chosen to facilitate reasonable solutions of the model.On the contrary, it was chosen to severely test the model.To estimate the sensitivity of the values of (the radialB

r(R0)magnetic Ðeld at the surface), on the source depth, we used

in this paper a slightly submerged source with Ra\ 6.7

] 1010 cm, somewhat smaller than the value of Ra(\6.805

] 1010 cm) corresponding to the ““ surface-source ÏÏmentioned above. The di†erences are otherwise trivial.

Physical arguments in the favor of the source used in thispaper do exist. It has been suggested by Fisher, &Fan,McClymont that after emergence has taken place,(1994)and as a consequence of relaxation to hydrostatic equi-librium, the tubeÏs Ðeld strength at some intermediate depthbecomes sufficiently small to cause the surface portions ofthe Ñux tube to become disconnected from the base of theSCZ. If this is the case then the source should vanish belowa certain depth. The fact that the source vanishes for r [R

acan be interpreted as an incomplete eruption of the Ñuxtube. As shown by Caligari, & Schu� sslerMoreno-Insertis,

Ñux tubes with magnetic Ðelds smaller than a few(1995),times 104 G, can undergo, as they rise, a catastrophicexpansion and weakening of the magnetic Ðeld at the apexof the Ñux tube. The rising Ñux tube ““ explodes ÏÏ and it isthen conceivable that the source for the poloidal Ðeldreaches its maximum below the surface.

2. DESCRIPTION OF THE MODEL

The equation to be solved is the induction equation :

LBLt

\ $ Â MU Â B [ g$ Â BN] $ ÂA0, 0,

LSLtB

, (1)

where g is the di†usivity, S is the source term due to Ñuxeruption, and U is the meridional motion. Let A \ (0, 0, AÕ)be the vector potential that deÐnes the poloidal Ðeld, and BÕ

No. 2, 1997 BABCOCK-LEIGHTON SOLAR DYNAMO MODEL. IV. 1067

be the toroidal Ðeld. The equations satisÐed by andaÕ bÕ,deÐned below (symmetry across the equator is assumed),are written down in Appendix A :

AÕ \ aÕ sin h/r , BÕ\ bÕ sin h cos h/r , S \ s sin h/r .

(2)

Notice that in equations and of the(A1) (A2) Appendix A,di†usivities in the radial and latitudinal directions (g

r, gh)are assumed to be di†erent. The equations of Appendix A

were solved with the help of a second-order time- andspace-centered di†erence scheme. The mesh consists of101 ] 101 points in the r- and h-direction. Therefore, *r \

cm and *h\ n/200.(R0[ Rc)/100 \ 1.9] 108

The toroidal Ðeld is ampliÐed in a layer of thicknessh(\10 ] *r \ 1.9] 109 cm) at the bottom of the SCZ. Thelower boundary of this layer (the GL ), is We identify theR

c.

GL with the layer in which the latitudinal di†erential rota-tion (generated in the SCZ by the interaction of rotationwith convection) decreases with r, and ) becomes uniform.

The integration of the equations proceeds as follows, themaximum of in the GL , say is calculatedoBÕ o Bmax(rer, her),at each time step. If where is a speci-Bmax(rer, her) [Bcr, BcrÐed Ðeld, then there is eruption of a Ñux tube at time andterat colatitude that rises radially to the surface. Theh \ hertime of rise is neglected, only one eruption is allowed per timestep, and the magnetic Ðeld in the GL is unchanged as aconsequence of an eruption. The eruption of this Ñux tube atthe surface generates at a bipolar magnetic regionh \ her,(BMR) with a Ñux for the following pole and'(her) ['(her)for the preceding pole. We replace these poles by theirequivalent magnetic rings doublet and assume that the rateof Ñux eruption is constant with time and inversely pro-portional to q, the eruption time. Each eruption lasts for atime q ; at all the Ñux has erupted and the erup-t \ ter] qtion that started at does not contribute to the sourceterterm for T he total source term at time t is the sumt [ ter ] q.over the contributions from all the ““ active ÏÏ eruptions (i.e.,such that No mean Ðeld equation relatingteri ¹ t \ teri ] q).directly the poloidal to the toroidal Ðeld, as L eightonÏs Equa-tion (4), is assumed.

2.1. L atitudinal and Radial Dependence of the SourceLet S \ (0, 0, S) be the vector potential for Bs, the mag-

netic Ðeld of the source. The equation for is of the formaÕ(see with S \ s sin h/r.Appendix A) LaÕ/Lt \ É É É ] Ls/LtWe proceed now to calculate Ls/Lt. The magnetic Ðeld Bs isthe magnetic Ðeld of the erupting BMR that has beenreplaced by a magnetic ring doublet. Let " be the angularthickness of each ring and s their angular separation (see

If c is the distance between the following and preced-Fig. 1).ing poles (each of latitudinal dimension D), and c is the ““ tilt

FIG. 1.ÈParameters associated with the following and precedingannuli

angle,ÏÏ then and We assume thats \ c sin c/R0 "\D/R0.the Ðeld is radial at For a usual surface-eruption,r \Ra.

however, we consider here cases withRa\R0 ; R

a\ R0corresponding to Ñux tubes erupting only partially. The

radial Ðeld of the source is given by

Brs \ 1

r sin hLLh

(S sin h) \ 1r2 sin h

LLh

(s sin2 h) . (3)

Therefore,

s \ r2sin2 h

P0

hsin hB

rs dh , r \ R

a. (4)

For the time dependence of we take the simple lawBrs

Brs \ Ber(h, her, R

a)(t [ ter)/q , ter\ t \ ter] q , (5)

where is the radial magnetic Ðeld of theBer(h, her, Ra)

bipolar magnetic region (in our case of the preceding andfollowing annuli) generated by the eruption of a magnetictube of Ñux at colatitude is the time the erup-'(her) her ; tertion started, and q is a measure of the rate of eruption. Thesolutions are insensitive to the precise value of q. Theexplicit expression for isBer(h, her, R

a)

Ber(h, her, Ra) \ '(her)

2nRa2"

]CHhAher[

s ] "2

, her[s [ "

2BN

sinAher[

s2B

[HhAher]

s [ "2

, her]s ] "

2BN

sinAher]

s2BD

, (6)

where is the function deÐned below:HhHh(h0, h1) \ 1 (h0¹ h ¹ h1) ,

Hh(h0, h1) \ 0 (h \ h0, h [ h1) . (7)

Therefore, is unity for and zeroHh(h0, h1) h0¹ h ¹ h1otherwise. The surface of an annulus of angular thickness ",centered at is sin If the Ñux ofher[ s/2, 2nR

a2 (her [ s/2)".

such an annulus is then its magnetic Ðeld is clearly'(her),given by the Ðrst term of the right-hand side in equation (6).The magnetic Ðeld of the annulus centered at isher ] s/2,such that the total erupted Ñux exactly vanishes. The Ñux ofthe northern annulus, is the Ñux of the tube that'(her),erupted from the GL . Since this Ñux cannot be exceeded,

must vanish forLBrs/Lt t [ ter ] q.We discuss now the radial dependence of the source.

Equations and are an expression for the source(4), (5), (6)term at The radial dependence of Ls/Lt is as yetr \ R

a.

undetermined. In this paper we adopted the following radialdependence for Ls(r)/Lt,

Ls(r)Lt

\ Ls(Ra)

LtH

r(R

a[ 5 *r, R

a] 5 *r) , (8)

where is a function of r deÐned asHr(r0, r1) Hh(h0, h1)below Therefore, since *r B 2 ] 108 cm, theequation (6).

source s, is constant for cmRa[ 109 cm ¹ r ¹R

a] 109

and vanishes for other values of r. Such an expression forthe vector potential generates latitudinal magnetic Ðelds(associated with the closure of the magnetic lines of forces)at cm. The discontinuities at cmr \R

a^ 109 r \ R

a^ 109

are numerically inconsequential, i.e., the contribution ofthese latitudinal Ðelds to the generation of is negligible,BÕ


as can be checked numerically by decreasing the radialextension of the source.

To summarize, for the source term due to the eruption atwe used with given by equationster, equation (8) s(R

a, t) (4),

and The total source term at time t is the sum over the(5), (6).contributions from all the ““ active ÏÏ eruptions (i.e., such thatteri ¹ t \ teri ] q).

Most of the numerical calculation of this paper were per-formed with cm. Therefore, the sourceR

a\ 6.7] 1010

di†ers from zero in the interval (6.605] 1010 cm,6.795] 1010 cm), and vanishes for other values of r.

2.2. Shear in the Angular VelocityAn appealing location for the GL is the layer in which the

latitudinal di†erential rotation, generated in the SCZ by theinteraction of rotation with convection, decreases with rand ) becomes uniform. The thickness of this layer neednot be larger than a few 103 km (see De Young, &Durney,Passot eq. [10]). Assume that in this layer, the angular1990,momentum, A(r)dr, of a thin shell of radius r and thicknessdr is independent of r. Therefore, with decreasing rÏs theequator rotates slower and the poles faster in such a waythat the shells (r, dr) have the same angular momentum. Thethin shellÏs angular momentum is given by

A(r)dr \ 8n3

or4)o

Cu

o(r)[ u2(r)

5Ddr , (9)

where for ) we used the following expansion :

)\ )o[u

o(r)] u2(r)P2(cos h)] . (10)

Here is the second-order Legendre polynomial,P2(cos h)rad s~1, and)

o\ 2.57 ] 10~6 u

o(R0) \ 1, u2(R0) \[0.185 at the surface. Since the thickness of this layer is

much smaller than the density scale height, A(r)dr will beindependent of r if

u2(r) \ u2c(r [ Rc)/h , u

o@ (r) \ u2@ (r)/5 , (11)

where h is the thickness of the generating layer. For weu2have used a linear function of r that vanishes at the lowerboundary of the SCZ, whereas at the upper boundary of theGL , which is negative. It follows from equa-u2(r) \ u2c,tions and that ) increases inward for is(10) (11) h \ 63¡.4,constant with r for and decreases inward for h [h \ 63¡.4

in agreement with helioseismic observations (see63¡.4,It should be noticed that the value ofKosovichev 1996). u

odoes not enter into the equations for the magnetic Ðeld.Outside the GL , the angular velocity was assumed to beuniform, i.e., Again it can be readilyu

o@ \ u2@ \ u2\ 0.

shown numerically that the contribution to the generationof of the discontinuity in ) at is negligible.BÕ r \ R

c] h

2.3. Meridional MotionsFor appearing in the induction equationU \ (U

r, Uh, 0)

and equations and we used the following expres-(A1) (A2)sions :

Ur\ t2

or2 P2(cos h) ] t4or2 P4(cos h) , (12a)

Uh \ t2@6ro

LP2Lh

] t4@20ro

LP4Lh

, (12b)

that satisfy the continuity equation. The choice for thestream functions, and was based on the followingt2 t4,considerations. An integration over r and h of the steadystate azimuthal momentum equation leads to (see Durney

1993, 1996b)

t2B15or

4)ou

o

Po

nSu

ruÕT sin2 h dh , (13)

where

SuruÕT B Sv

rvÕT [ lr sin h

L)Lr

. (14)

Here is the velocity correlation for the turbulentSvrvÕTconvective velocities for solid-body rotation, and l is a turb-

ulent viscosity coefficient. As the following intuitive argu-ments suggest, is an expression for theequation (13)conservation of angular momentum in the steady state.Unless ) decreases strongly with r, the velocity correlation

is negative since, for example, for sinking particlesSuruÕT conservation of angular momentum requires that(v

r\ 0),

In consequence, the term feeds angularvÕ[ 0. SuruÕTmomentum to the inner regions of the SCZ. Let beA(R

c, r)

the angular momentum of a shell with inner radius andRcouter radius r. From the time-dependent azimuthal momen-

tum equation we readily obtain

LALt

\ 2nr3P0

nQ

rÕ sin2 h dh , QrÕ \ [oU

rUÕ[ oSu

ruÕT .

(15)

Notice that if the meridional velocities vanish, and sinceis negative, LA/Lt is positive. Therefore, in the steadySu

ruÕTstate, a meridional Ñow must be present to remove the

angular momentum transferred to the inner regions byIt is reasonable to expect that the strength of thisSu

ruÕTÑow should be proportional to some average of overSu

ruÕTa sphere, and that the Ñow should rise at the equator and

sink at the poles in order to remove angular momentumfrom the inner regions. A Ñow having these propertiesagrees with the observations. for is readilyEquation (13) t2obtained from if the time derivative vanishes.equation (15)For we used the simple expression below (for the behav-t2ior of the integral in with r, see, e.g., Fig. 1 ofeq. [13]Durney 1996b) :

t2(r) \ t2F o(r [ Rc)(R0[ r)/o

cR

c2 , (16a)

where is the density at the lower boundary of the SCZ.ocThe existence of a two-cell Ñow rising at the equator and

poles, and sinking at mid-latitudes is also suggested by thetheory (the mean azimuthal momentum equation cannot bereadily satisÐed with only a one-cell Ñow; see the papersmentioned above). For this Ñow we used

t4(r) \ t4F o(r [ Rc)(R0[ r)/o

cR

c2 . (16b)

The predicted magnitude for this Ñow, namely,(see Durney is, however, at4(B[4t2/3) 1993, 1996b),

result far less robust than because Ñows withequation (13)higher order Legendre polynomials than four wereneglected. Nevertheless, it should be mentioned that thecombined Ñow (due to and is polewardt2 t4B [4t2/3)for h [ B25¡ and equatorward for smaller values of h, inagreement with the & Dailey observ-Snodgrass (1996)ations. Snodgrass & Dailey have shown that the time-average of the motions of small magnetic features over aperiod of 26 yr is poleward for 80¡[ h [ 30¡, markedlyequatorward for h [ 80¡ and weakly equatorward forh \ 30¡. The Ñow toward the equator observed bySnodgrass & Dailey for h [ 80¡ is probably magnetically


driven. The a e†ect is small at these latitudes but this Ñowcould provide a coupling between both hemispheres. Evenif, e.g., the Ñow could be important for thet4\ [0.5t2, P4dynamo action because it decreases the values of at theUhpoles. Latitudinal velocities with this property were used by

Nash, & Sheeley et al. andWang, (1989), Wang (1991),& Sheeley in their numerical simulations.Wang (1994)

A word of caution should, however, be inserted here, themeasure of the meridional velocities (see Snodgrass 1984 ;

et al. Ceppatelli, &Ulrich 1988 ; Howard 1991 ; Cavallini,Righini Latushko Howard, &1992 ; 1993, 1994 ; Komm,Harvey & Dailey is1993 ; Snodgrass 1996 ; Hathaway 1996),a notoriously difficult problem (see the discussions in

[GONG observations] and &Hathaway 1996 SnodgrassDailey [motions of small magnetic features in the solar1996photosphere]). The activity cycle appears to strongly inÑu-ence the meridional motions and existing GONG obser-vations do not suggest an equatorward Ñow at highlatitudes.

3. NUMERICAL CALCULATIONS

As stated earlier only one eruption is allowed per timestep. Consider two integrations of the equations with timesteps *t and where is an arbitrary numerical*t/F

t, F

tfactor. The solutions are not the same mainly because theerupted Ñux per unit time is not the same. If is larger thanF

tunity, and since only one eruption is allowed per time step,there will be more eruptions if the time step is than if it*t/F

tis *t. However, since *t is shorter than any typical time ofthe system, the solutions of the equations for and(Bcr, *t)

will be very nearly identical provided that in the(Bcr, *t/Ft)

Ðrst case we multiply the erupted Ñux by Therefore, theFt.

integration of the equations with *t \ 105 s, gives practi-cally identical results than the integration of the equationswith *t \ 2 ] 104 s provided that in the Ðrst case we multi-ply the erupted Ñux by The di†erences due to theF

t\ 5.

unequal time step are negligible. The model has in conse-quence one arbitrary factor, and an appeal to obser-F

t,

vations is necessary. The numerical calculations wereperformed with *t \ 105 s and the values for were, 0.5, 1,F

t2.5, 5, and 15.

3.1. Input to the Numerical CalculationsThe equations of were solved with the help ofAppendix A

a second-order time- and space-centered di†erence scheme.The value of was set equal to 5 ] 109 cm2 s~1 and wasg

rghadjusted to achieve marginal stability. It is necessary to

specify the density (that appears in the Ðrst term of equationand in the expressions for the stream functions), the(A2)

meridional motions, the shear in the angular velocity, thesource term for the poloidal Ðeld, the initial conditions, andthe boundary conditions. We discuss these points in turn.

The SCZ was modeled by a polytrope (see Durney 1991,eq. [18]), with the density at the lower boundary of theR

c,

SCZ, equal to g cm~3. The upperoc\o(R

c) \ 0.226

boundary was chosen equal to cm,R0\ 6.9 ] 1010resulting in a value for the density at the surface equal to

g cm~3.oo\ o(R0)\ 1.26] 10~3For the meridional motions, equations (12a), (12b), (16a),

and were used with g s~1. If(16b) t2F\ [2.5] 1024 t4\0, it follows from that the latitudinal Ñow atequation (12b)the surface is given by sin h cos h/Uh(R0)\[3t2@ (R0)(6ro) B [6 sin h cos h m s~1. Stream functions for this Ñowcan be found in Figure 4 of Paper II.

For the shear in the angular velocity we used equationwith This value of at the top of the(11) u2c \ [0.15. u2(r)GL is somewhat smaller, in absolute value, than the surface

value of namely [0.185. Outside the GL , the angularu2(r),velocity was uniform. The thickness of the GL ish \ 1.9] 109 cm.

We discuss now the parameters deÐning the source termfor the poloidal Ðeld. As stated earlier, the maximum of oBÕ oin the GL , say, is calculated at each time step. IfBmax(rer, her),a speciÐed Ðeld, then there is eruption ofBmax(rer, her) [ Bcr,a Ñux tube at The latitudinal extension of the Ñuxh \ her.tube that erupts from the GL was chosen equal to 2R

c*h B

1.57] 109 cm, whereas the radial dimension was h, thethickness of the GL . The tubeÏs magnetic Ñux, was'(her),accurately calculated by an integration of over the crossBÕsection of the tube. This is necessary because (*r)2/g

rB 0.2

yr, i.e., the radial di†usion is large enough to smooth out BÕin the GL , but not, to make it uniform. The Ñux, is the'(her)input from the GL needed to evaluate the source term, s.The tube rises radially to the surface and at gener-r \R

aates a magnetic ring doublet (the axisymmetric equivalentof a BMR). The angular thickness of each ring, ", was setequal to 2*h, and the angular separation of both rings forh \ 0 was chosen equal to 4*h(\s). For values of sh D 0,is given by JoyÏs law. Let c be the distance between the twopoles of a BMR. Then, s \ (c/R

a) sin cB 0.5(c/R

a) cos h,

where c is the ““ tilt angle.ÏÏ In this paper we used

s(her) \ (c/Ra) sin cerB 0.5(c/R

a) cos her\ 4 *h cos her .

(17)

For cos becomes smaller than 0.5 andher [ 1.047, heri.e., the following and preceding annulis(her) \ 2 *h,overlap. When overlapping occurred, the Ñux was'(her),decreased by an amount proportional to the overlapping.Near the equator there can be eruptions, but their contribu-tion to the source term is negligible. A simple linear depen-dence on time was chosen for the eruption of Ñux, namely,

for the eruption of the following ring, and'(her)(t [ ter)/q for the preceding one. Here is the time['(her)(t [ ter)/q terthe eruption started, and q(\106 s) is the eruption time. Atall the Ñux has erupted and the source term fort \ ter] q,

this eruption vanishes for The total source termt [ ter] q.at time t is the sum over the contributions from all the““ active ÏÏ eruptions (i.e., such that Theteri \ t \ teri ] q).radial dependence of the source has been speciÐed inequation (8).

The choice of the initial conditions was as follows : Theequations were Ðrst integrated with a seed bipole and notoroidal Ðeld at the initial time. The resultant toroidal Ðeldat 50 yr, multiplied by an appropriate factor, was the initialtoroidal Ðeld for all the runs. The multiplicative factor waschosen so that the maximum value of in the GL wasoBÕ oequal to The initial poloidal Ðeld was set to a very1.5Bcr.small value. In all the runs the poloidal Ðeld was thereforeentirely generated by eruptions. If the equations are solvedwith these initial conditions, and since they are linear thesolutions scale with and need to be solved only for oneBcrvalue of If the toroidal Ðeld is uniform, the magnetic ÑuxBcr.of the erupting tube, is Mx,'(her), 2hR

c*hBcr Bn1018Bcrwhere is in gauss. The solar value of is estimated toBcr Bcrlie between 5] 104 Mx and 105 Mx (see & Choud-DÏSilva

huri Fisher, & McClymont et al.1993 ; Fan, 1994 ; Schu� sslerMoreno-Insertis, & Schu� ssler &1994 ; Caligari, 1995 ; Fan


Fisher For these values of the Ñux of a BMR1996). Bcr,would be orders of magnitude larger than the solar value.The reason is of course the relatively coarse resolution in hthat must be used in order to solve the equations within areasonable time. Until a higher resolution can be used, it isphysically far more meaningful to match the erupted mag-netic Ñuxes instead of We choose therefore G,Bcr. Bcr\ 103resulting in Mx, the Ñux of an active'(her) B 3 ] 1021region. can be thought of as a normalizing factor sinceBcrthe solutions scale with once a value has been assignedBcr ;to we can evaluate the erupted Ñux in say, 10 yr, denotedBcrby yr), and study the behavior of the solution for the'er(10solar values of yr).'er(10

We proceed now to discuss the boundary conditions. Atit was assumed that the derivatives with respect to rr \R0of and vanish at The poloidal Ðeld is thereforeaÕ bÕ r \ R0.radial at the surface, and the toroidal Ðeld essentially

satisÐes the boundary condition suggested by Parker (1984)and Not enough is known about theChoudhuri (1981).physics of the GL to justify the boundary conditions at

on physical grounds. The results presented here werer \Rcobtained with the same boundary conditions at thanr \R

cthose imposed at the surface. Clearly, these are ““ friendly ÏÏboundary conditions. The toroidal Ðeld is generated in thethin GL by the stretching of the poloidal Ðeld and theassumption that the r-derivatives of and vanish ataÕ bÕimposes there a minimum of constraints on ther \R

c,

magnitudes of the radial and toroidal Ðeld. It is important,however, to study the sensitivity of the solutions to theseboundary conditions. With this in mind the equations werealso solved by assuming that at the r-derivative ofr \R

c,

vanishes and that The boundary condition forbÕ aÕ\ 0. aÕimposes a severe condition on the radial Ðeld, since itimplies that which plays a dominant role in the gener-B

r,

ation of the toroidal Ðeld, vanishes at It should ber \Rc.

mentioned that these boundary conditions were used byin the solution of his dynamo model, andYoshimura (1975)

that they can be justiÐed physically if the conductivity of theregion below is very large. Runs were also made withr \R

cand atbÕ\ 0, LaÕ/Lr \ 0 r \Rc.

3.2. Results of the Numerical CalculationsWe proceed now to discuss the results of the numerical

calculations. For a given and for values of the inputFt,

parameters speciÐed above, a trial value for was chosenghand the equations were integrated for 100 yr. This resultedin growing or decaying solutions. If the solution decayed(grew) then was decreased (increased) until a value forgh ghwas reached yielding solutions with a small, growing ordecaying tendency in 100 yr. The procedure is time consum-ing since needs to be accurately determined, in particularghfor the larger values of F

t.

In Figures and we plot the toroidal magnetic2, 3, 4, 5, 6Ðeld, taken from the lower GL , versus time for di†erentvalues of and in Figures and we plot theF

t, 7, 8, 9, 10, 11

values of the latitudes at which the eruptions take placeversus time.

The numbers and letters in Figures were deÐned as2È6follows : in Figures and the values of were divided2 3 oBÕ oas speciÐed in below, whereas in Figuresequation (18a) 4, 5,and we divided these values in accordance with6 equation

(in both cases and Numbers were(18b) C1[ 0 C2[ 1).assigned to as deÐned in (18c), and letters toBÕ[ 0 BÕ\ 0as deÐned in (18d). A blank was therefore assigned to

in the lower GL vs. time forFIG. 2.ÈBÕ Ft\ 0.5 : gh \ 2.415] 1011

cm2 s~1, yr) B 2.5] 1024 Mx.C1\ 0.2, 'er(10

FIG. 3.ÈSame as in for cm2 s~1,Fig. 2 Ft\ 1 : gh \ 5.1] 1011 C1\

0.2, yr) B 4 ] 1024 Mx.'er(10


FIG. 4.ÈSame as in for cm2 s~1,Fig. 2 Ft\ 2.5 : gh\ 1.255] 1012

yr) B 0.85] 1024 Mx.C1\ 0.2, 'er(10

if it lies between or andBÕ[ 0 (B2, B3), (B4, B5), (B6, B7),correspondingly for The values of and wereBÕ\ 0. C1 C2chosen so as to convey best the general behavior of BÕ ;always proved to be a good choice. If, e.g.,C2\ 1.5 BÕ[ 0,then the value of is bracketed by andBcr Bcr(1 [ C1)For small values of was a suitableBcr(1 ] C1). F

t, C1\ 0.2

choice. With increasing values of departs less and lessFt, BÕfrom in the region of eruption, and in consequence it isBcrconvenient to decrease which equals 0.1 inC1, Figure 6

in this case) :(Ft\ 15

B1\ Bcr(1[C1)C23

, B2\ Bcr(1[C1)C22

, B3\Bcr(1[C1)C2

,

B4\ Bcr(1[C1) , B5\ Bcr(1]C1) ,

B6\ Bcr(1]C1)C2 , B7\ Bcr(1]C1)C22 , (Figs. 2 and 3) ,

(18a)

B1\ Bcr(1[C1)C24

, B2\ Bcr(1[C1)C23

, B3\Bcr(1[C1)C22

,

B4\ Bcr(1[C1)C2

, B5\ Bcr(1[C1) , B6\ Bcr(1]C1) ,

B7\ Bcr(1]C1)C2 , (Figs. 4, 5, and 6) , (18b)

blank B1 1 B2 blank B3 2 B4 blank B5 3 B6 blank B7[(18c)

blank[ B1 a [ B2 blank [ B3 b [ B4 blank

[ B5 c[ B6[ B7\ . (18d)

FIG. 5.ÈSame as in for cm2 s~1,Fig. 2 Ft\ 5 : gh \ 1.87] 1012 C1\

0.15, yr) B 1.5] 1025 Mx.'er(10

FIG. 6.ÈSame as in for cm2 s~1,Fig. 2 Ft\ 15 : gh \ 2.7925] 1012

yr) B 4.8] 1025 Mx.C1\ 0.1, 'er(10


FIG. 7.ÈLatitudes at which the eruptions of Ñux tubes take place in theGL as a function of time for The magnetic Ðeld alternatesF

t\ 0.5.

polarity.

FIG. 8.ÈSame as in forFig. 7 Ft\ 1

FIG. 9.ÈSame as in forFig. 7 Ft\ 2.5


In Figures and it is clear that for all2 3, BN~1 \B

N/C2values of and that the value of is bracketed byN D 5, Bcr B4and If for example, the blank between theB5. BÕ[ 0,

numbers 2 and 3 in these Ðgures indicates that the magneticÐeld lies between and It is0.8] Bcr 1.2 ] Bcr (C1\ 0.2).important to notice that a number 3 in Figures and2 3,means that is larger than and smaller thanBÕ 1.2 ] BcrIn the polar regions the toroidal Ðeld can there-1.8] Bcr.fore signiÐcantly exceed Bcr.In Figures and it is apparent that if4, 5, 6, B

N~1\ BN/C2and that is bracketed by and representedN D 6, Bcr B5 B6,by the number 3 in Figures and In contrast to the4, 5, 6.

cases plotted in Figures and the toroidal Ðeld does not2 3,exceed now (values that would be representedBcr(1] C1)by a blank inside the central regions of Figures and4, 5, 6).This explains the need for dividing the range of values for

di†erently in Figures and than in Figures andBÕ 2 3 3, 4, 6.T he reader should keep in mind that in Figures and the2 3number 3 or the letter c means that is signiÐcantly largeroBÕ othan whereas in Figures and it means thatBcr, 4, 5, 6 oBÕ oBBcr.In Figures the values of *t, and2È6, g

r, C2, R

a, t2F, t4F,are, cm2 s~1, cm,g

r\ 5 ] 109 C2\ 1.5, R

a\ 6.7] 1010



*t \ 105 s, g s~1, and In thet2F\ [2.5] 1024 t4\ 0.caption of these Ðgures we give the values of andF

t, C1, ghresulting in marginal stability, as well as the erupted Ñux in

10 yr, namely yr). It is instructive to estimate the'er(10value of yr) for di†erent values of assuming that'er(10 F

tthe toroidal Ðeld in the erupting Ñux tube is uniform andequal to G, and that there is one eruptionBcr\ 103per time step, *t(\105 s). Then, yr) B'er(10

This value is somewhatn1018Bcr Ft(10 yr/*t) B 1025F

tMx.

larger than those shown in Figures Of course there is2È6.never an eruption per time step specially at large values of

Another factor reducing yr) is the fact that isFt. 'er(10 BÕnot necessarily larger than at all points within the ÑuxBcrtube.It should be stressed that Figures are valid for arbi-2È6

trary values of If we integrate the equations with theBcr.initial conditions speciÐed above, and use equationsto plot we obtain the same graph, regardless(18a)È(18d) BÕ,of the value of Bcr.Figures show that at there is a transition in2È11 F

tB 2.5

the nature of the solutions. For the eruptionsFt\ B2.5

take place predominantly in the polar regions, and here,in the GL can become considerably larger thanoBÕ(max) o

whereas for the eruptions occur mainly inBcr, Ft[ B2.5

the southern (northern) quadrant of the northern (southern)hemisphere with The reason for this result,BÕ(max)B Bcr.which at Ðrst sight could appear paradoxical, is the follow-ing : for marginal stability and for a weakly driven dynamo(i.e., small), must be small also. As increases, so doesF

tgh F

tand which is transported from the surface to thegh, Br,

polar regions of the lower SCZ by meridional motions,decreases in the GL . As a consequence, the toroidal Ðeld,generated now by di†erential rotation from a weaker poloi-dal Ðeld, is unable to become larger than at values of hBcrclose to the poles. As mentioned above, in the solutions witheruptions at high latitudes, can signiÐcantly exceed inBÕ Bcrthe region of eruption. This result is not entirely satisfactorysince it could be validly argued that signiÐcantly exceedsBÕbecause we are not allowing for enough eruptions perBcrtime step. An appealing suggestion is that should beF

tincreased until the maximum value of remains close tooBÕ oin the region of eruption. Remarkably, this is the valueBcrfor at which the transition from polar- to low-latitudeF

teruptions takes place. In other words, the transition value ofand therefore also the necessity of eruptions for h [ Bn/4F

tare determined, to some extent, by the logical consistency ofthe model. Furthermore, for the transition value of theF

t,

value of yr), namely, B1025 Mx is in remarkable'er(10agreement with the solar value.

The existence of such a transition, an entirely unexpectedresult, clearly constitutes a major step forward in theresolution of a serious difficulty encountered in previouscalculations : for the type of meridional Ñows consideredhere, the eruptions always started at high latitudes, in dis-agreement with the observations. Furthermore, the trans-port of this Ðeld to the poles resulted in exceedingly largevalues of there.B

rWe discuss now the e†ect of the boundary conditions(B.C.) on the results of the numerical calculations. For thecases plotted in Figures and bracketing the transition3 5,value of the equations were also solved with B.C.F

t,

expressing that at and vanish at Asr \ Rc, aÕ LbÕ/Lr r \ R

c.

discussed above, requires that the radial Ðeld, whichaÕ\ 0plays a mayor role in the generation of the toroidal Ðeld,

vanishes at As expected, the value for leading tor \Rc. ghmarginal stability is smaller with the new boundary condi-

tions than with the old ones. However, the di†erence is notlarge : for and the di†erence isF

t\ 1, ghnew B 0.93 ] ghold,negligible for The general behavior of the solutionsF

t\ 5.

is unchanged. Since the toroidal magnetic Ðeld is in allprobability generated below the SCZ proper, in a regionwhere the e†ects of magnetic buoyancy do not dominate,there is no reason to expect that will vanish atBÕ r \R

c(see also the papers by Parker & Choudhuri quoted above).Nevertheless, the equations were also solved with the fol-lowing boundary conditions, and atLaÕ/Lr \ 0 bÕ\ 0 r \

We are therefore requiring that the toroidal magneticRc.

Ðeld, generated in a thin layer at the base of the SCZ, shouldvanish at the lower boundary of this layer. Not surprisingly,with these new B.C., the solutions do indeed change signiÐ-cantly : for and the values of for margin-F

t\ 0.5 F

t\ 15 ghal stability are now, respectively, 8.9 ] 1010 cm2 s~1 and

1.8408853] 1012 cm2 s~1. For the eruptions occurFt\ 1

exclusively in the polar regions for latitudes larger thanwhereas for the eruptions occur forh

e\ 1.4, F

t\ 15 1.15[

very much as in but shifted towardhe[ 0.94 Figure 11,

higher latitudes.In the numerical calculations discussed above, the value

of is 6.7] 1010 cm (hereafter case I) and the sourceRadi†ers from zero in the interval (6.605] 1010 cm,

6.795] 1010 cm). If, on the other hand, Ra\ 6.805] 1010

cm (case II), the source extends from r \ 6.71] 1010 cm tothe surface. The main di†erences between cases I and II arethe values of at the surface. The maximum value ofB

rnamely occurs at the poles. For case I,oBr(R0) o, B

rm(R0),is about 370, 220, 140, 220, and 400 G for theB

rm(R0)values of corresponding to Figures and ForF

t2, 3, 4, 5, 6.

case II, G for and B103 G forBrm(R0) B 2.5 ] 103 F

t\ 1,

The eruptions at high latitudes for areFt\ 15. F

t\ 1

responsible for the large value of in case II. It should alsoBrm

be mentioned that the source in case I is slightly more effi-cient than in case II as the values of leading to marginalghstability show: case I) \ 2.7925] 1012 cm2 s~1,gh(Ft

\ 15,whereas case II) \ 2.393] 1012 cm2 s~1.gh(Ft

\ 15,

4. DISCUSSION AND CONCLUSIONS

The aim of this paper has been to examine whetherBabcock-Leighton models are viable contenders to providean explanation for the generation of the solar magneticÐeld. The results of the numerical calculations presentedhere suggest that this is indeed the case. This conjecture isstrengthened by the work of & MakarovaMakarov (1996).These authors have shown that for a given cycle, thenumber of polar faculae is positively correlated with thecoming sunspot area (see their Fig. 5). Such result is to beexpected in the framework of the dynamo model studiedhere. Furthermore, the butterÑy diagram of the magneticÑux (see is also suggestive. This Ðgure clearly showsFig. 12),the transport of Ñux toward the poles.

It would of course be premature to be carried away withenthusiasm. Clearly, it will be up to future generations topronounce the Ðnal word on this difficult subject. Sinceloops of toroidal Ðeld have to merge to form the large-scalepoloidal Ðeld, the precise location of the regenerating layer(RGL ) must depend on the prevailing dissipation. InBabcock-Leighton models the a e†ect operates at thesurface. Other serious alternatives do, however, exist : the ae†ect could be the outcome of instabilities arising in strong


FIG. 12.ÈButterÑy diagram of magnetic Ñux. Synoptic NSO/KP data (Courtesy Karen Harvey).

magnetic Ðelds (see SchmittParker 1971 ; 1985, 1987 ;Schmitt, & Schu� ssler or it could beFerriz-Mas, 1994)

located at the interface between the radiative and convec-tive regions of the Sun & Char-(Parker 1993, MacGregorbonneau and & MacGregor In1997, Charbonneau 1997).this case the di†erence in di†usivity between the two regionsallows for the buildup of strong toroidal magnetic Ðeldsbelow the interface needed to explain the surface propertiesof the BMRs.

Another point deserving discussion is the following (seeit has been observed that magnetic structures inLow 1996) :

the same hemisphere generally have the same magnetic heli-city, implying a progressive accumulation of the magnetichelicity in the solar corona. However, every 11 years theentire atmospheric magnetic Ðeld is rebuilt from new cycleÐelds generated in the SCZ, i.e., the accumulation of mag-netic helicity cannot proceed beyond 11 years. It has beensuggested by Low (see that the coronal massLow 1996)ejections (CME) are the means by which the accumulatedmagnetic helicity (and Ñux) are taken out of the corona. TheCME are therefore an essential part of the cycle. Thispicture clearly harmonizes well with the dynamo modelstudied here.

We discuss now the problems facing the model studiedhere. The value of at the poles is too large, especially ifB

rthe source reaches the surface. The reason why Wang et al.and & Sheeley are not faced with(1989, 1991) Wang (1994),

this difficulty in their studies of latitudinal Ñux transport,are twofold : these authors use larger values for the di†usi-vity and a latitudinal dependence for of the form sinp hUhcosq h with values for p and q resulting in considerablysmaller velocities near the poles than at low latitudes. Con-

cerning the di†usivities we have here very little choice : Thevalue of is already rather small and if we increase theg

rvalue of the dynamo dies. It should, however, be kept ingh,mind that constant turbulent di†usivities are notoriouslysuspect concepts, particularly in MHD since the Ðeld reactsback on the Ñuid motions. In complete analogy with thetoroidal Ðeld that can only be generated in regions of weakbuoyancy, it appears likely that the poloidal Ðeld can onlybe generated in ““ regions of high di†usivity.ÏÏ If we increasethe value of in our model, we are increasing not only atgh ghthe surface where it plays an essential role in the generationof the poloidal Ðeld, but also, e.g., in the GL , where in alllikelihood cannot be large. It should also be stressed thatghif there are no eruptions in the polar regions (which is thecase for the values of at the poles, while stillF

t[B2.5), B

rsomewhat large, are considerably reduced. The results ofthis paper suggest that the large values of at the poles areB

rnot the unsurmountable difficulty suggested by previouswork.

Concerning the meridional motions, we have includedhere only the main component of the meridional Ñow,namely the poleward motion given by equation (16a).Studies should, however, be made with int4F D 0 equation

If, for example, with 0\ b \ 1, the Ñow is still(16b). t4F\b,poleward but the velocities at the poles are decreased (seeFig. 4 of Paper II).

An important point that needs to be clariÐed is the roleplayed by the small-scale Ñux in the generation of the solarlarge-scale di†use magnetic Ðeld (see There isStenÑo 1992).no doubt, for example, that the ephemeral bipolar magneticregions (EBMR) form part of the solar cycle (see Harvey

It is of course essential to understand the nature of1993).


this small-scale activity. It is tempting to speculate that theEBMR are the result of Moreno-Insertis, Caligari, &Schu� sslerÏs ““ explosion ÏÏ of Ñux tubes, and that they reveal asubsurface source for the poloidal Ðeld. A knowledge of theaverage Ñuxes, and ““ tilt angles,ÏÏ !, of the EBMR in'",annuli of latitudinal dimension " would allow to estimatethe contribution of the EBMR to the global magnetic Ðeldof the Sun. In it is shown thatAppendix B

Br(h) \ 1

2nR03" sin hL(C sin !'")

Lh, (19)

where C is the average distance between both poles ofthe EBMR. It should be noticed that if ", C, !, and '"refer to bipolar magnetic regions, is essentiallyequation (19)LeightonÏs mean Ðeld equation relating the radial with the

toroidal magnetic Ðeld. As shown in in theAppendix B,derivation of it is necessary to assume thatequation (19)

which it is not satisÐed for the case ofC sin !? R0",BMR at low latitudes (see In fact it is apparenteq. [17]).that cannot be right at low latitudes. Since, sinequation (19)!, is approximately proportional to cos h, it follows from

that for h B n/2, is proportional toequation (19) Br

BÕ,clearly a wrong functional dependence. Since equation (19)has not been used in this paper, this difficulty has beenbypassed.

I am grateful to two anonymous referees for valuablecomments, and to the sta† of the National Solar Observ-atory, and the National Science Foundation for theirsupport.

APPENDIX A

EQUATIONS FOR THE FIELDS

LaÕLt

\ [UrLaÕLr

[UhrALaÕ

Lh] 2aÕ cot h

B] g

rL2aÕLr2 ] gh

G 1r2

L2aÕLh2 ] 3

r2LaÕLh

cot h [ 2r2 aÕ

H] Ls

Lt, (A1)

LbÕLt

\o@o

UrbÕ] )0(u0@ ] u2@ P2)

ALaÕLh

tan h ] 2aÕB

] 3)0u2LaÕLr

sin2 h [ Ur

ALbÕLr

[ 2bÕrB

[ UhrALbÕ

Lh[ bÕ tan h

B

] grL2bÕLr2 ] gh

G 1r2

L2bÕLh2 ] 1

r2 (3 cot h [ 2 tan h)LbÕLh

[ 6r2 bÕ

H. (A2)

LaÕLh

\ LbÕLh

\ 0 h \ 0 ,n2

. (A3)

APPENDIX B

EBMRÏs CONTRIBUTION TO THE GLOBAL MAGNETIC FIELD OF THE SUN

Consider a vast number of ephemeral bipolar magnetic regions (EBMR) with m, c, j, and / denoting, respectively, theangular distance separating the individual poles ; the angle formed by their magnetic axis with the east-west line (the ““ tiltangle ÏÏ) ; the angular latitudinal extension of the individual magnetic poles ; and the Ñux of the following pole. We assume thatm sin c? j and, for the moment, that all quantities are independent h. The more important case where c is a function of h willbe consider later on. We divide the interval h \ 0, h \ n/2 in steps of j : 0, j, 2j, 3j, . . . Mj \ n/2. The distribution of theEBMR can depend on latitude and we denote by N(h) the number of EBMR centered at each of the above values of h. Theangular latitudinal separation of the following and preceding poles of these EBMR is clearly

*h\ c sin c/R0\ m sin c , (B1)

where c is the distance between both poles. To take advantage of the axial symmetry of the problem we replace the EBMR byring doublets of width j centered at h ^ *h/2. Then,

2nR02 j sin (h [ *h/2)Bf(h [ *h/2) \ N(h)/\ '(h) , (B2)

2nR02 j sin (h ] *h/2)Bp(h ] *h/2) \ [N(h)/\ ['(h) , (B3)

where and are the radial Ðeld of the following and preceding annulus, respectively, and is the solar radius. The ÐrstBf

Bp

R0relation in follows from the above deÐnitions, and the second one deÐnes '(h). expresses that theequation (B2) Equation (B3)total erupted Ñux is zero. The contribution of the preceding poles of the EBMR centered at h [ *h allows us to write

2nR02 j sin (h [ *h/2)Bp(h [ *h/2) \ [N(h [ *h)/\ ['(h [ *h) . (B4)

Identically, from the following poles centered at h ] *h it follows that :

2nR02 j sin (h ] *h/2)Bf(h ] *h/2) \ N(h ] *h)/\ '(h ] *h) . (B5)


The sums of equations and and those of equations and give(B2) (B4) (B3) (B5)

2nR02 j sinAh [ *h

2BB

r

Ah [ *h

2B

\ *hL'(h)Lh

, (B6)

2nR02 j sinAh ] *h

2BB

r

Ah ] *h

2B

\ *hL'(h)Lh

, (B7)

where is the global radial magnetic Ðeld. Taking the sum of equations and and neglecting terms ofBr\B

f] B

p(B6) (B7)

second order in *h we obtain

2nR02 j sin (h)Br(h) \ *h

L'(h)Lh

. (B8)

Therefore,

Br(h)\ m sin c

2nR02 j sin hL'(h)Lh

. (B9)

The integration of or givesequation (B8) equation (B9)

Phi

hf2nR02 sin (h)B

r(h)dh ] m sin c

j['(h

i)[ '(h

f)]\ 0 . (B10)

The interpretation of this equation is straightforward. The integral in is the total Ñux in Consider theequation (B10) (hi, h

f).

If the EBMR are centered at h such that then only the preceding poles contribute tohi-boundary. h

i[ *h/2 \h\ h

i] *h/2,

the Ñux in The number of EBMR centered in this interval is N(h)*h/j. The Ñux of the following poles that has not been(hi, h

f).

included in the integral of is thereforeequation (B10)

'mi B

m sin cj

'(hi) . (B11)

Similarly for the The Ñux of the preceding poles not included in the integral of ishf-boundary : equation (B10)

'mf B [m sin c

j'(h

f) . (B12)

If we add to the Ñux in we must obtain zero since the Ñux from the preceding and following poles of the'mi ] '

mf (h

i, h

f)

EBMR cancels. We recover therefore equation (B10).Assume now that m sin c is not constant with h. It is clear from the above discussion that the correct generalization of

isequation (B9)

Br(h) \ 1

2nR02 j sin hL[m sin c'(h)]

Lh(B13)

The total Ñux in the annulus is now(hi, h

f)

Phi

hf2nR02 sin (h)B

r(h)dh \

Am sin cj

'Bhf

[Am sin c

j'Bhi

. (B14)

In practice what can be measured, as a function of h, is the Ñux, in an annulus "? j. Then, and'", '\ j'"/" equationbecomes(B13)

Br(h) \ 1

2nR02" sin hL($ sin !'")

Lh, (B15)

where $ and ! are the averages m and c in ". The derivation of is as rigorous as it ever going to get ; however, itequation (B15)is clearly not a total disaster.

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