The Mechanism involved in the Reversals of the Sun's Polar Magnetic Fields

THE MECHANISM INVOLVED IN THE REVERSALS OF THE SUN’SPOLAR MAGNETIC FIELDS

C. J. DURRANT, J. P. R. TURNER and P. R. WILSONSchool of Mathematics & Statistics, University of Sydney, NSW, Australia

(Received 18 April 2004; accepted 11 May 2004)

Abstract. Models of the polarity reversals of the Sun’s polar magnetic fields based on the surfacetransport of flux are discussed and are tested using observations of the polar fields during Cycle 23obtained by the National Solar Observatory at Kitt Peak. We have extended earlier measurementsof the net radial flux polewards of ±60◦ and confirm that, despite fluctuations of ∼ 20%, there is asteady decline in the old polarity polar flux which begins shortly after sunspot minimum (although notat the same time in each hemisphere), crosses the zero level near sunspot maximum, and increases,with reversed polarity during the remainder of the cycle. We have also measured the net transport ofthe radial field by both meridional flow and diffusion across several latitude zones at various phasesof the Cycle. We can confirm that there was a net transport of leader flux across the solar equatorduring Cycle 23 and have used statistical tests to show that it began during the rising phase of thiscycle rather than after sunspot maximum. This may explain the early decrease of the mean polarflux after sunspot minimum. We also found an outward flow of net flux across latitudes ±60◦ whichis consistent with the onset of the decline of the old polarity flux. Thus the polar polarity reversalsduring Cycle 23 are not inconsistent with the surface flux-transport models but the large empiricalvalues required for the magnetic diffusivity require further investigation.

1. Introduction

The polarity reversals of the solar polar magnetic fields were first observed byHorace Babcock during Cycle 19 and it is now well established that the polarityof these fields reverses during each cycle shortly after sunspot maximum (but notsimultaneously in each hemisphere). Because the flux emerging at high latitudes isgenerally in the form of small randomly oriented magnetic bipoles, most models forthe polarity reversals (e.g., Babcock, 1959; Leighton, 1964) assume that they aredue to the advection and diffusive decay of the flux emerging in the activity beltsbelow latitude 50◦. Babcock’s model had postulated a meridional flow to carry theflux to the poles, whereas Leighton had invoked only magnetic diffusion drivenby the field gradients. However, the postulated meridional flow was confirmedby observations in 1974 (see Howard, 1974; Duvall, 1979) and has since beenrecognized as the dominant poleward transport mechanism at mid latitudes (seeCameron and Hopkins, 1998; Snodgrass and Dailey, 1996).

In these models, the decaying active region flux is assumed to be advected pass-ively towards the poles by meridional flows, wrapped azimuthally by differentialrotation and smoothed by diffusion. However, since the positive and negative flux

Solar Physics 222: 345–362, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

346 C. J. DURRANT, J. P. R. TURNER AND P. R. WILSON

which emerges in active regions must be in balance, the poleward transport ofthis flux, with or without internal diffusion, would produce zero net change in thepolarity of the polar fields over a cycle.

A mechanism by which the advection process may lead to a reversal of thispolarity was suggested by Babcock (1959) and described in more detail by Giovan-elli (1985). In the latter part of the cycle, when the active regions emerge at lowlatitudes, they have argued that some flux from these regions may diffuse across theequator before being swept polewards. Because of the tilt of the spot group axes,more leader than follower flux would be involved in this process, and the effect ofthis diffusive loss from one hemisphere is doubled by the diffusive gain from theother hemisphere. This would provide an excess of follower polarity flux in eachhemisphere so that, as a result of poleward advection and diffusion, followed byflux cancellation, the polarity of the polar fields may be reversed by the end of thecycle.

Simulations of the global synoptic fields have been carried out using the syn-optic flux transport equation (McCloughan and Durrant, 2002). These are basedon the global synoptic fields observed by the National Solar Observatory, KittPeak (NSO/KP) for two consecutive Carrington rotations (CRs), the simulatedfields below latitudes ±48◦ being updated by the observed field values after eachsimulated CR while those above are retained in the ongoing simulation. Thus thesimulated polar fields are modified by the poleward advection and diffusion of theactive region flux emerging at lower latitudes, and Durrant and Wilson (2003) haveshown that, under certain conditions, it is possible to simulate the polarity reversalsof the polar fields.

However, it is not clear that the processes modeled by the synoptic transportequation correspond to the actual solar processes, in particular, there is evidence(Durrant, Kress, and Wilson, 2001; Durrant, Turner, and Wilson, 2002) that theemergence of flux at higher latitudes contributes to the configuration of the large-scale polar fields and thus to the polarity reversals. Further, the simulations requirevalues for the diffusivity parameter which are larger than those indicated by ob-servation (see below). Schrijver, De Rosa, and Title (2002) have argued that thepolar field reversals require an additional decay process, but the physical natureof this process is unclear. Wang, Lean, and Sheeley (2002) confirmed this resultbut found that, by allowing the meridional flow rates to vary from cycle to cyclestable polarity oscillations can be maintained. However, it is unclear why the flowspeeds should change in this way. Thus the polar field reversals merit some furtherinvestigation.

POLAR FIELD REVERSALS 347

2. The Advection-Diffusion Model

2.1. FLUX CANCELLATION

Giovanelli (1985) has estimated that it might take the follower flux of an activeregion emerging near latitude 30◦ some 12 months to reach the pole, followed bythe leader flux ∼17 days later.

Since the total flux in the polar region at sunspot minimum (∼ 8 × 1021 Mx)(N.B. 1 maxwell = 10−8 weber) is of the same order as that in either the leaderor the follower footpoints of a moderate to large active region, it follows that,unless there is some significant attenuation or annihilation of this flux during itsadvection to the pole, the effect of its arrival would be considerable. If no fluxcancellation occurs in the polar regions, flux of both polarities should accumulatethere. Alternatively, if there is no attenuation during the advection process andcomplete cancellation of opposite polarities near the poles, there should be a verysignificant reduction, if not reversal of the old polarity flux, within 12 months ofthe emergence of the first significant active region after sunspot minimum.

2.2. FLUCTUATIONS

Whatever these effects, the subsequent arrival of the leader polarity flux some 17days later should cancel them out and the later arrival of flux from other decayingactive regions should give rise to significant fluctuations in the polar flux.

Durrant and Wilson (2003) have measured the net flux above (a) 60◦ (b) 70◦ and(c) 80◦ in each hemisphere using the NSO/KP data obtained during the rising phaseof Cycle 23 and these are shown in Figure 6 of that paper. Here we have extendedthese observations back to CR 1910 (June 1996, i.e., at sunspot minimum) andforward through CR 2006 (August 2003), and the net fluxes above 60◦N and below60◦S are shown in Figure 1.

These do indeed exhibit fluctuations, but only of order 20% about an initialmean value of order ∼ 14 Mx which, in the north, extended up to CR 1930(December 1997). After this, despite continuing fluctuations, the mean strength ofthe old polarity field declined steadily, crossing the zero level near CR 1960 (March2000) and increased (with opposite polarity) through CR 2006. In the southernhemisphere the decline in the mean flux strength began earlier, near CR 1914, butthe zero level crossing occurred later, near CR 1970 (December 2000). The overallbehavior in the two hemispheres was qualitatively similar, however.

2.3. ADDITIONAL EVIDENCE

Observations of the polar fields during Cycle 22 (Snodgrass, Kress, and Wilson,2000) also showed the polarity reversals and that, after each reversal, the newpolarity flux continued to increase at the poles until the next sunspot minimum.


Figure 1. The net flux density poleward of latitudes 60◦ in each hemisphere (in gauss) is plottedagainst Carrington rotation from CR 1910 at the beginning of Cycle 23 until almost the end of thecycle. Periods of unfavorable tilt of the solar axes are shaded.

Again, Wang, Nash, and Sheeley (1989) have plotted the vertical-dipole compon-ent of the photospheric field during Cycle 21 and found a similar pattern. Thedipole component declined steadily during the growth phase of the cycle, changedsign in 1980 (near sunspot maximum) and continued to increase in strength withnegative polarity through the declining phase until the next sunspot minimum in1986. Again, they found only minor fluctuation in the strength of net flux duringthis period.

Since no significant build-up of flux of either polarity is observed in the polarregions during the rising phase of the cycle, we infer that flux cancellation mustoccur within the polar region. Further, since the observed fluctuations are only oforder 20%, there must be a significant attenuation of the decaying active regionfields during their advection to the polar regions

2.4. DIFFUSION AND ATTENUATION

An obvious candidate for the attenuation mechanism is magnetic diffusion but,since flux does not accumulate at the poles from cycle to cycle, this must result,


Figure 2. The monthly mean sunspot numbers during Cycle 23.

not just in the mixing of the opposite polarity fluxes, but in the annihilation of asignificant fraction of the flux of both polarities. These processes are modeled in thetransport equation by the diffusivity term. However, it has been found (e.g. Kressand Wilson, 2000) that in order to simulate the polar field reversals it is necessaryto use values for the diffusivity of order ∼ 600 ± 100 km2 s−1 (e.g. Wang, Nash,and Sheeley, 1989; Kress and Wilson, 2000; Snodgrass, Kress, and Wilson, 2000).These values are significantly larger than those inferred from direct observationsof the granule and supergranule velocity fields (e.g., 150–250 km2 s−1, Smithson,1973; Mosher, 1977; Schrijver and Martin, 1980) and it remains to explain thisdiscrepancy between the observed and the ‘empirical’ values of the diffusivity.

2.5. FLUX-CANCELLATION BETWEEN THE HEMISPHERES

It was noted above that, in order to change the polar polarity by the end of thesunspot cycle, the Babcock–Leighton model, as refined by Giovanelli (1985), re-quires that some leader flux must diffuse across the equator during the cycle andcancel with leader flux in the opposite hemisphere, leaving an excess of followerflux in each hemisphere. However the trans-equatorial transport of leader flux has


not yet been confirmed by direct observation, and one of the aims of this paper isto assess the observational evidence that a significant quantity of leader flux doesindeed cross the equator during the cycle.

2.6. THE INITIAL DECLINE OF THE OLD POLAR POLARITY

While it is generally assumed that this trans-equatorial cancellation occurs in thelatter part of the cycle when the active regions emerge at lower latitudes, the ob-servations shown in Figure 1 indicate that the mean level of the old polarity fluxbegins to decline shortly after sunspot minimum. In order to explain this earlydecline, one of us (CJD) has noted that, during the rising phase of the cycle, therate at which active regions emerge increases with time. Although the leader andfollower flux must be in balance, he argues that, because of the equatorward tiltof the magnetic axes of these regions, the net follower flux within a given latitudeband in the activity zone should increase during this process. Thus the polewardadvection of the fields in these bands would deliver an excess of follower flux tothe polar regions during this phase. On the other hand, when activity is declining,the reverse would be true.

2.7. THE RATE OF GROWTH OF ACTIVITY

The rate of growth of activity must follow the sunspot number curve which, forCycle 23, is shown in Figure 2. This rate is obviously strongest between June,1997 (CR 1923) and March, 1998 (CR 1934), but continues through March, 2000(CR 1960), and this is consistent with the decline of the old polar polarity. AfterMarch 2000, however, the sunspot number declines and thus after (say) CR 1963,the rates of emergence of leader flux should be in approximate balance with, andmay well exceed, that of the follower flux within a given latitude zone.

According to the above argument, this should lead to a restoration of the oldpolar polarity, but it is clear from Figure 1 that this did not occur in Cycle 23. Indeedin most cycles, the mean value of the old polarity flux or the dipole component hasdeclined steadily, not only during the rising phase of the cycle but also duringthe maximum and post-maximum phases. This can be explained only if the trans-equatorial cancellation of the leader flux of active regions becomes effective muchearlier in the cycle, before the rate of emergence of active regions begins to decline.Thus if the trans-equatorial transport of flux can be confirmed from observations,it will be of some importance to determine the time of its onset during the cycle.

2.8. MULTIPLE REVERSALS?

Using Hα filaments as proxies for the neutral lines of the surface magnetic fields,Makarov and Sivaraman (1989) have reported that, in some cycles, e.g., Cycles 16,19, and 20 in the north and 12 and 14 in the south, a second reversal, restoringthe old polar polarity, was observed. However, this was soon followed by a third


reversal so that, by the next sunspot minimum, the polar polarities are the reverseof those which applied at the previous minimum.

Waldmeier (1973) also deduced the possibility of a multiple polarity reversalin the northern hemisphere during Cycle 20 from a study of the distribution ofsolar prominences. This was supported by Howard (1974) using magnetic datafrom the Mount Wilson Observatory although, since magnetograph observationsare unreliable at high latitudes, the polar fields are generally determined from theextrapolation and interpolation of fields observed at latitude 74◦ and below.

The cause of these multiple reversals is not clear, indeed since reliable mag-netograph measurements were not available before Cycle 22, their reality mustremain in some doubt. However, if the steady decline of the the old polar polarityrequires the earlier onset of the trans-equatorial cancellation of leader flux acrossthe equator, then the later onset of this cancellation in some cycles may account forthe temporary restoration of the old polar polarity.

2.9. FOCUS OF THE STUDY

In this paper, we have first extended the measurements of the net flux above latitude60◦ back to the beginning and through the maximum to the declining phase ofCycle 23, and this is shown in Figure 1 above. We then estimate directly from theobserved synoptic field distributions and surface velocities the rate of transport ofthe net radial flux across any given latitude by advection and diffusion at differentphases of the cycle. In particular, we are concerned with the transport of flux acrossthe equator.

3. The Surface Transport of Flux

The net flux crossing a surface S enclosed by a curve C is

� =∫S

B · dS,

and the time rate of change of this is

∂�

∂t= ∂

∂t

∫S

B · dS =∫S

∂B∂t

· dS.

But from the flux transport equation,

∂B∂t

= curl (v × B) + η �2 B, (1)

where v is the plasma velocity and η is the magnetic diffusivity, since

�2B = �(� · B) − curl (curl B)


and � · B = 0, we have

∂B∂t

= curl((v × B) − η curl B

), (2)

However, by Stokes theorem,∫S

curl F · dS =∮C

F · dl,

where F is any vector. Thus

∂�

∂t=

∫S

curl((v × B − η curl B)

)· dS =

∮C

(v × B − η curl B) · dl. (3)

3.1. FLUX TRANSPORT ACROSS CIRCLES OF LATITUDE

If the surface S is that part of the northern solar surface bounded by a circle ofco-latitude θ , then dl = R sin θdφφ̂ and the surface velocity v may be written

v = vmf (θ)θ̂ + vdr(θ)φ̂.

Here vmf (θ) is meridional velocity at co-latitude θ , vdr(θ) is the rotational velocityrelative to latitude 16◦ and θ̂ and φ̂ are unit vectors in the θ and φ directions.

Thus the rate of change of radial flux poleward of co-latitude θ is given by

∂�(θ)

∂t=

2π∫0

(−vmf Br + η

R

(∂Br

∂θ− ∂(rBθ)

∂r

))R sin θdφ, (4)

where the derivatives are evaluated at r = R.Since most equatorial surface fields form parts of closed loops, the components

of Bθ away from and towards the equator must be in approximate balance, thecontributions of the last term in Equation (4) should average to zero and may beignored. Thus Equation (4) reduces to

∂�(θ)

∂t=

2π∫0

(−vmf Br + η

R

∂Br

∂θ

)R sin θdφ. (5)

We note that, as θ → 0, π , ∂�(θ)/∂t → 0, which is consistent with the as-sumption of the flux-transport model that flux changes across a particular surfaceS must be due to flux-transport across the closed curve C bounding the surface.Thus we define Q(θ) = ∂�(θ)/∂t to be the ‘fluxflow’ across co-latitude θ at anyinstant, i.e., the rate at which flux is advected and diffused across this co-latitude.


3.2. FLUX TRANSPORT AS A FUNCTION OF LONGITUDE

It follows from Equation (5) that the rate of flux transport across an arc-length ofone degree at Carrington longitude φ of a circle of co-latitude θ , q(θ, φ) is givenby

q(θ, φ) =(

−vmf (θ)Br(θ, φ) + η

R

∂Br(θ, φ)

∂θ

)× πR sin θ

180. (6)

4. Flux Transport Inferred from the Observed Flux Distributions andMeridional Flows

4.1. AS A FUNCTION OF CO-LATITUDE AND LONGITUDE

A model for the observed steady meridional flow vmf (m s−1) has been derivedfrom azimuthally averaged Mount Wilson synoptic data by Cameron and Hopkins(1998), viz.

vmf (θ) = −2.85v0 cos θ sin2.5 θ, (7)

where v0 = 10. Snodgrass and Dailey (1996) have used a cross-correlation analysisof the meridional motions of the small magnetic features to estimate the meridionalflow patterns and find a similar result but with some suggestion of equatorwardflows below latitudes ±10◦. For these and other models, we note that the azi-muthally averaged meridional plasma flow across the equator is zero so that thetrans-equatorial flux as determined from Equation (5) is due to diffusion arisingfrom the meridional field gradients near the equator, and these may be inferredfrom the synoptic plots given by the NSO/KP Observatory.

Synoptic plots do not, of course, provide an instantaneous picture of the solarsurface fields at all longitudes but they do show the instantaneous field distributionalong a longitudinal strip at Carrington longitude φ when that strip is near centralmeridian of the disk. Thus estimates of the meridional gradients (i.e., with respectto the co-latitude θ) are valid for these strips.

We have examined the synoptic plots for all Carrington rotations from CR 1916to 2006 and in Figures 3(a) and 4(a) we show the plots for two of these, CRs 1916and 1969, in mercator projection. CR 1916 (December 1996) occurred shortly aftersunspot minimum and it is obvious from the polarities of the leader and followerregions that the equatorial active regions belong to the old cycle, while the re-gion seen at CL 280◦, sine-latitude 0.5, is an early new-cycle region. By contrast,CR 1969 (November 2000) occurred well into the rising phase of the cycle (justbefore sunspot maximum) when there were many new cycle regions emerging bothnear the equator and at higher latitudes.

In order to estimate these gradients it is first necessary to smooth the raw datausing a moving average square window of 7 elements on a regular grid. The


Figure 3. (a) The NSO plot for CR 1916. (b) The corresponding smoothed plot. The zero-levelcontour for this plot is shown and is also superimposed on (a). (c) The flux transported by advectionand diffusion as a function of latitude and longitude. The flux flows at each sine-latitude are indicatedat the bottom right.


smoothed plots for CRs 1916 and 1969 are shown in Figures 3(b) and 4(b). Thususing the model for vmf given by Equation (7) with v0 = 11, together with theempirical value of η (600 km2 s−1), the observed values of Br and by determin-ing ∂Br/∂θ numerically from the smoothed observed flux distributions, we mayevaluate q(θ, φ) from Equation (6). For CRs 1916 and 1969, this is plotted againstCarrington Longitude and at several sine latitudes below the corresponding fluxdistributions in Figures 3(c) and 4(c).

It should be noted that a positive flux flow may indicate the flow of posit-ive flux in the positive θ direction (i.e., away from the Sun’s northern polar re-gion). However, it may equally represent the flow of negative flux in the oppositedirection.

Because there are so few regions of activity present in CR 1916, it is possible toestimate the equatorial flux gradients from the observed distributions and it is clearthat they are qualitatively consistent with those calculated from Equations (6) and(7). At sine-latitude −0.1, Carrington longitude 170◦, for example, the fluxes arelarge but the gradients probably small and the flux transport would be dominatedby the meridional flows. Again this is consistent with the calculated flux transportq(θ, φ).

Despite the increase in activity in CR 1969, we may also qualitively confirmsome of the features of the flux transported across circles of latitude for this rota-tion. For example, between Carrington longitudes 250◦ and 265◦, the neutral linebounding a strong negative region in the northern hemisphere lies approximatelyalong the equator, so that the flux gradient would be significantly positive in thepositive θ direction. This corresponds to the transport of negative flux in the pos-itive θ direction, which is qualitatively consistent with the flux flow curves. Againat sine-latitude −0.4, where the meridional flow is obviously relevant, there arenegative peaks in the flux flow curves at Carrington longitudes 52◦ and 68◦ whichare consistent with the smoothed observed distributions.

4.2. THE NET FLUX FLOW ACROSS A GIVEN LATITUDE

By integrating these values of q(θ, φ) over 2π with respect to φ, we may obtain anestimate of the typical net flux flows, Qn(θ) across co-latitude θ during Carringtonrotation Cn, and these are shown to the right of the curves for q(θ, φ) in Figures 3and 4.

In Figure 5, we plot the flux flow across the equator against Carrington rotationnoting that, since the meridional flow at the equator, as given by Equation (7), iszero, this depends only on the meridional gradients of the radial fields. In Figure 6we plot the estimated net flux flows across latitudes ±60◦ using the correspondinggradients and the meridional flows given by Equations (6) and (7).

Of course, it is to be expected that there would be large local variations in thecalculated flux flows due to the variable emergence of active regions, and these


Figure 4. (a) The NSO plot for CR 1969. (b) The corresponding smoothed plot. Again the zero-levelcontour for the smoothed plot is shown in both (a) and (b). (c) The advected flux as a function ofsine-latitude and Carrington longitude.


Figure 5. Trans-equatorial flux for each CR.

are evident in Figures 5 and 6. However, if the mechanims described above arerelevant, it should be possible to identify some general patterns in these results.

5. Patterns of Flux Flows: Tests for Significance

5.1. FLUX CROSSING THE EQUATOR

A fundamental assumption of the Babcock–Leighton model is that a significantamount of leader flux crosses the equator from each hemisphere, where it can-cels with the local leader flux thus producing an excess of follower flux in thathemisphere.

From the plot of the trans-equatorial fluxflow shown in Figure 5, it would seemthat, despite the inevitable scatter, there is indeed a net transport of positive fluxinto the southern hemisphere during the cycle. In order to investigate this result inmore detail, we now use the ‘Student’s t-test’ to examine the hypotheses:

(i) H0, that zero net flux crosses the equator during the cycle (i.e., that Qn =∑N1 Qn/N = 0, N being the number of Carrington rotations in the sample),and(ii) H1, that Qn �= 0.For this test,

tN−1 = Qn

σ

√N,


where N is the number of samples and the variance

σ =√∑N

1 (Qn − Qn)2

(N − 1).

We consider first the complete data set from CRs 1915–2006, i.e., N = 92, Qn =3.91, σ = 7.56, so we have

tN−1 = 3.91

7.56

√92 = 4.91.

The t-tables list values of a threshold, t , for various combinations of N − 1 andquote the probability p for which tN−1 will exceed t . Obviously not all possibilitiescan be covered so the t-table is restricted to the ranges 1 ≤ N−1 ≤ 120 and 0.25 ≥t ≥ 0.001. For example, the tables show that the probability that tN−1 > t = 3.59(which is the table limit) is 0.001, and hence the probability that tN−1 > 4.91 willbe less than that.

Of course, negative values of Qn must be considered but, due to the symmetryof the distribution, the probability that |tN−1| > t is double the single-sided value.Thus the probability that |tN−1| > 4.91 is less than 0.002, and the result that tN−1

is actually 4.91 is strong evidence against H0.Both Babcock and Giovanelli envisaged that the trans-equatorial transport of

flux might be significant in the declining phase of the cycle when the active regionsemerge closer to the equator, and a visual inspection of Figure 5 indicates that itis certainly not uniform throughout the cycle. Indeed, from CR 1916 (near sun-spot minimum) through CR 1946, the flux flow appears to be small and randomlyscattered about zero but for the remainder of the cycle, despite considerable scatter,the flux flow appears to be significant and positive.

Accordingly, we consider two subsets of the data, (i) from CR 1916 throughCR 1946, and (ii) from CR 1947 through CR 2006, and in Figure 5, we also showthe least-squares best fit lines for this partition of the data. We again consider thesignificance of each of these data sets for the hypotheses, H0 and H1:

(i) from CRs 1915–1946, N = 32, Qn = 0.244, σ = 3.054, so tN−1 = 0.452.

From the t-tables we find that for N = 30 the probability that tN−1 > 0.683 (thetable limit) is 0.25 and the probability that |tN−1| > 0.683 would be 0.5.

It follows that the result tN−1 = 0.452 is entirely consistent with H0. Thus thereis no direct evidence of net flux crossing the equator during this phase of the cycle.

(ii) From CRs 1947–2006, N = 60, Qn = 5.80, σ = 8.45, so we have tN−1 =5.32. The t-tables for N = 50 give

P(|t| > 3.261) = 0.002,

and thus the result that tN−1 = 5.32 is very strong evidence against H0 during thisphase of the cycle. Indeed, between CR 1943 and 1962 the flux flow is uniformlypositive and increasing and, since CR 1962 is close to sunspot maximum, this


implies that the trans-equatorial transport of flux begins during the rising phaseof the cycle. Allowing approximately 12 months for this flux to reach the polarregions, this unexpectedly early onset of trans-equatorial flux transport may wellexplain the steady decay of the old polarity polar flux after the increase of activitylevels off near sunspot maximum.

5.2. FLUX CROSSING LATITUDES ±60◦

In Figure 1 we have shown the net flux poleward of latitude ±60◦ and we wish tosee if this is consistent with the net flux crossing latitudes ±60◦, which is shownin Figure 6. While it is clear that there is a net flux out of the polar regions duringthis part of the cycle, there appear to be three different phases. In the first phase,extending to CR 1934 in the north and CR 1931 in the south, the flux flow appearsto be randomly distributed about a mean value which may be zero. In the secondphase the old polarity flux transported away from the poles across latitudes ±60◦(or the new polarity flux transported into the polar regions) at first increases andthen declines towards zero.

For θ = +60◦ from CR 1915–1934, N = 20, Qn(π3 ) = 0.157, σ = 1.402, so

tN−1 = 0.56.Again, using the t-tables for N = 20, H0 requires that

P(|tN−1| > 0.687) = 0.5,

and thus the result that tN−1 = 0.56 is entirely consistent with H0.For θ = −60◦ from CR 1915–1931, N = 17, Qn(

−π3 ) = 0.599, σ = 0.910,

so tN−1 = 2.72.

However, according to the t-tables for N=17,

P(|t| > 2.567) = 2P(t > 2.567) = 0.02,

and thus the result that tN−1 = 2.72 is unlikely on the basis of H0. It follows that,during this phase of the cycle, there is some direct evidence of net flux crossinglatitude −60◦, unlike latitude 60◦. Thus the observation of the earlier onset of thedecline of the old polarity polar field in the southern hemisphere is consistent withthe observed flux flows across latitudes −60◦.

6. Discussion

Qualitatively, these results appear to confirm the physical processes which havebeen postulated in order to account for the details of the polarity reversals. Inparticular, they show that the diffusive transport of flux across the equator beganduring the rising phase of Cycle 23 at just the appropriate time to account for thecontinued decline of the old polarity polar flux.


Figure 6. (i) The flux crossing latitude −60◦. (ii) The flux crossing latitude + 60◦.


They also show that the outflow of old polarity flux across latitudes ±60◦ (orthe inflow of new polarity flux) is consistent with the decline of the old polaritypolar field.

We have no information about the time of the onset of trans-equatorial fluxtransport in earlier cycles but a somewhat later onset may have resulted in thetemporary increase of the old polarity polar flux after the rate of growth of activitydeclines, and thus account for the multiple reversals which, it is claimed, were ob-served during these cycles. However, this conjecture cannot be tested until synopticdata for a cycle exhibiting a triple reversal becomes available.

The discrepancy between the observed and empirical values of the diffusivityhas not, however, received adequate explanation and this will be considered in alater paper.

Acknowledgements

It is a pleasure to thank Peter Fox, Herschel Snodgrass and Pat McIntosh for valu-able discussions, and the latter for permission to reproduce his results for Cycle 23in Figure 2.

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The Mechanism involved in the Reversals of the Sun's Polar Magnetic Fields