The Astrophysical Journal, 580:1188–1196, 2002 December 1
Copyright is not claimed for this article. Printed in U.S.A.
MERIDIONAL FLOW AND THE SOLAR CYCLE VARIATION OF THE SUN’S OPEN MAGNETIC FLUX
Y.-M. Wang, N. R. Sheeley, Jr., and J. Lean
E. O. Hulburt Center for Space Research, Code 7670, Naval Research Laboratory, Washington, DC 20375-5352;
ywang@yucca.nrl.navy.mil, sheeley@spruce.nrl.navy.mil, lean@demeter.nrl.navy.mil
Received 2002 June 14; accepted 2002 July 31
ABSTRACT
We simulate the evolution of the Sun’s large-scale magnetic field during solar cycle 21, including the effect
of surface transport processes and active region emergence. As an important new constraint on the model, we
have scaled our source fluxes upward to be consistent with the average measured strength of the interplanetary magnetic field (IMF). By adopting a poleward bulk flow of amplitude 20–25 m s1 together with a
supergranular diffusion rate of 500 km2 s1, we are then able to match the observed variation of the Sun’s
polar fields and open magnetic flux. The high meridional flow speeds, peaking at low latitudes, prevent the
buildup of an overly strong axisymmetric dipole component at sunspot minimum, while accounting for the
giant poleward surges of flux and accompanying polar field fluctuations observed near sunspot maximum.
The present simulations also reproduce the large peak in the equatorial dipole and IMF strength recorded in
1982.
Subject headings: interplanetary medium — solar-terrestrial relations — Sun: activity — Sun: corona —
Sun: magnetic fields — Sun: photosphere
Earth. In that study, a potential-field source-surface (PFSS)
extrapolation was used to determine the total open flux,
which was then assumed to be uniformly distributed at
1 AU, in accordance with Ulysses magnetometer results (see
Balogh et al. 1995). In contrast, when a constant scaling factor of 1.8 was applied to the magnetograph data, the
extrapolated fields were too low by a factor of 2 and
showed a strong modulation anticorrelated with the solar
activity cycle, contrary to observations.
Recently, Mackay, Priest, & Lockwood (2002) have
employed a flux transport model with source doublets to
derive the variation of the total open flux over the solar
cycle. They found that the open flux again peaked strongly
at sunspot minimum, instead of just after sunspot maximum
as observed in recent cycles, and suggested that the currentfree coronal extrapolation procedure was inadequate. A
similar result is obtained with the transport parameters and
doublet sources of WNS89 (see x 3).
In this paper we show that the flux transport model can
be reconciled with the measured variation of the interplanetary field, provided that the source fluxes used in WNS89 are
increased by a factor of 3. We find that this correction is
consistent with both the WSO and NSO/Kitt Peak magnetograph data, and we discuss its implications for the surface
meridional flow rate.
1. INTRODUCTION
In an earlier paper (Wang, Nash, & Sheeley 1989a, hereafter WNS89), we used the magnetic flux transport code of
Sheeley, DeVore, & Boris (1985) to simulate the evolution
of the Sun’s polar fields during sunspot cycle 21. Active
regions that emerged during 1976–1986 were represented by
magnetic doublets, whose flux was transported over the
photospheric grid by means of differential rotation, supergranular diffusion, and meridional flow. For quantitative
comparisons with the model calculations, we employed
magnetograph data from the Wilcox Solar Observatory
(WSO); the measured photospheric fluxes were multiplied
by a constant factor of 1.8 to correct for the saturation of
the Fe i 5250 line profile, following the analysis of Svalgaard, Duvall, & Scherrer (1978). We then found that the
solar cycle variation of the WSO polar fields could be fitted
by assuming a diffusion rate of 600 km2 s1 and poleward
flow speeds of 10–20 m s1, with the higher velocities occurring near sunspot maximum.
A major uncertainty in our earlier simulations involved
the calibrations of both the WSO fields and the doublet
strengths. The latter were estimated visually from photographic prints of National Solar Observatory (NSO/Kitt
Peak) magnetograms taken in the Fe i 8688 line (see Sheeley et al. 1985; Wang & Sheeley 1989); the resulting photospheric fluxes were in approximate agreement with the
WSO measurements scaled upward by a factor of 1.8. However, more recent studies suggest that both our doublet
strengths and the low-latitude WSO fluxes were underestimated. By comparing magnetograph measurements made
at the Mount Wilson Observatory (MWO) in Fe i 5250
and Fe i 5233, which saturates only at field strengths as
high as 4000 G, Ulrich (1992) derived a correction factor
for the Fe i 5250 fluxes that varied from 4.5 at disk center
to 2 at the solar limb. By applying the latitude-dependent
Ulrich correction to MWO and WSO photospheric field
data during 1970–1993, Wang & Sheeley (1995) were able to
reproduce both the average magnitude and the long-term
variation of the radial interplanetary field component at
2. METHOD
Let r denote heliocentric radius, L heliographic latitude,
Carrington longitude, and t time. The radially oriented
photospheric field Br ðR ; L; ; tÞ is assumed to obey the
transport equation
@Br
@Br
¼ !ðLÞ
þ r2? Br
@t
@
1
@
½vðLÞBr cos L þ SðL; ; tÞ ;
R cos L @L
ð1Þ
where r2? represents the L and components of the Lapla1188
MERIDIONAL FLOW AND THE SUN’S OPEN FLUX
cian, !ðLÞ ¼ 13:38 2:30 sin2 L 1:62 sin4 L deg day1 is
the synodic rotation rate of the photospheric plasma
(Snodgrass 1983), is the diffusion coefficient associated
with the nonstationary supergranular convection (Leighton
1964), vðLÞ is the meridional flow velocity, and SðL; ; tÞ is
a source term describing the emergence of new bipolar magnetic regions (BMRs). Estimates of typically range
between 200 and 700 km2 s1 (see Simon, Title, & Weiss
1995 and references therein), while Doppler measurements
and magnetic-feature tracking indicate poleward flows of
amplitude 10 to 30 m s1 (Duvall 1979; Komm,
Howard, & Harvey 1993; Hathaway et al. 1996; Haber et al.
2002).
For a given initial field distribution Br ðR ; L; ; 0Þ and
choice of and vðLÞ, we solve equation (1) over the interval
1976–1986, employing the numerical code described by
Sheeley et al. (1985). The computations are done on a mesh
of dimensions 128 cells in longitude by 64 cells in latitude,
representing the whole solar surface. During the course of
each simulation, 3000 BMRs are deposited onto the photospheric grid according to their observed locations and times
of emergence. The properties of these doublet sources are as
described in Wang & Sheeley (1989), except for an important modification to be discussed in x 3.
We extrapolate the simulated photospheric field into the
corona by assuming that µ B ¼ 0 holds out to a ‘‘ source
surface ’’ at r ¼ Rss ¼ 2:5 R, where BL ¼ B ¼ 0 (Schatten, Wilcox, & Ness 1969). The total open flux is then given
by
Z
open ðtÞ ¼ R2ss jBr ðRss ; L; ; tÞj d
;
ð2Þ
D
where the integral is over all solid angle . On the assumption that the open flux is uniformly distributed in L and at
1 AU (r ¼ rE ), the radial field strength BE at Earth is related
to open by
BE ðtÞ ¼
open
:
4r2E
ð3Þ
Because of the rapid falloff of higher order magnetic multipoles between r ¼ R and r ¼ Rss , open and BE depend
principally on the dipole (l ¼ 1) and quadrupole (l ¼ 2)
components of the photospheric field (see Wang, Lean, &
Sheeley 2000a).
We also compute the total photospheric flux,
Z
ð4Þ
tot ðtÞ ¼ R2 jBr ðR ; L; ; tÞj d
;
the average fields in the north (L > þ60 ) and south
(L < 60 ) polar caps,
R
Br ðR ; L; ; tÞ d
L>þ60
R
Bn ðtÞ ¼
;
ð5Þ
d
L>þ60
R
Br ðR ; L; ; tÞ d
L<60
R
Bs ðtÞ ¼
;
ð6Þ
d
L<60
and the axial, equatorial, and total dipole strengths at
the solar surface: Dax ðtÞ, Deq ðtÞ ¼ ðH12 þ H22 Þ1=2 , and
Dtot ðtÞ ¼ ðD2ax þ D2eq Þ1=2 , where
Z
3
Br ðR ; L; ; tÞ sin L d
;
Dax ðtÞ ¼
ð7Þ
4
Z
3
Br ðR ; L; ; tÞ cos L cos d
;
H1 ðtÞ ¼
4
Z
3
Br ðR ; L; ; tÞ cos L sin d
:
H2 ðtÞ ¼
4
1189
ð8Þ
ð9Þ
(Dax and Deq correspond, respectively, to the l ¼ 1, m ¼ 0
and l ¼ 1, jmj ¼ 1 spherical harmonic components of the
photospheric field.)
3. SOURCE FLUX CORRECTION
As noted in x 1, WNS89 adjusted the flux transport
parameters to match the evolution of the WSO photospheric fields, scaled upward by a factor of f ¼ 1:8 in
accordance with the analysis of Svalgaard et al. (1978).
Ulrich (1992) subsequently found that the saturation correction for magnetograph measurements in the Fe i 5250
line (used at both MWO and WSO) was even larger and
decreased strongly from disk center to limb; his result can
be approximated by the latitude-dependent function
fU ðLÞ ¼ 4:5 2:5 sin2 L
ð10Þ
(see Wang & Sheeley 1995).
Figure 1 illustrates the evolution of the total photospheric flux and the equatorial and axial dipole strengths
during 1976–1986, as obtained with our earlier model (thick
solid curves). Here, as in WNS89 and Wang, Nash, &
Sheeley (1989b), we have set ¼ 600 km2 s1 and
jvðLÞj ¼ ð10 m s1 Þ cos L sin0:01 jLj; the initial photospheric
field was taken to be of the form ð11 GÞ sin8 L, and the
doublet sources were similar to those described in Wang &
Sheeley (1989). For comparison, we have also plotted the
WSO total photospheric flux and dipole components, corrected for line profile saturation using both the constant factor of 1.8 (thin solid curves) and the Ulrich function fU ðLÞ
(dotted curves); in both cases, the measured line-of-sight
fields were deprojected by dividing by cos L (see Wang &
Sheeley 1992). As expected, the ‘‘ old model ’’ roughly
matches the WSO data multiplied by 1.8; when compared to
the Ulrich-corrected measurements, however, the simulated
values of tot and Deq are generally lower by a factor of 2–
3, reflecting the difference between the strengths of the calculated and observed low-latitude photospheric fields.
Figure 2 compares the variation of BE determined from
the old model and from the WSO data corrected using the
two different scaling factors. Also plotted are the in situ
measurements of the radial interplanetary magnetic field
(IMF) strength (connected asterisks), extracted from the online National Space Science Data Center (NSSDC) OMNI
database. The open fluxes derived from the simulation and
from the WSO data scaled by a factor of 1.8 both show a
pronounced minimum near sunspot maximum, contrary to
the observed IMF variation, and the calculated values of BE
are too low by factors of 2–4 during 1978–1984. In contrast,
when the Ulrich correction is applied, the extrapolated
WSO fields are in much better agreement with the IMF
measurements, reproducing in particular the high peak in
1982.
From Figure 2 we conclude that the model of WNS89
greatly underestimates the radial IMF strength near sunspot
maximum. During this period, most of the open flux originates in and around active regions, rather than from the
polar coronal holes, as at sunspot minimum (see Wang et al.
WANG, SHEELEY, & LEAN
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Fig. 2.—Solar cycle 21 simulation of the near-Earth radial IMF strength
BE ¼ open =4r2E (nT), based on the doublet sources and transport parameters of WNS89 (thick solid curve). Connected asterisks show the corresponding in situ spacecraft measurements (NSSDC OMNI data). Also
plotted are the values of BE derived from the WSO photospheric data, corrected for saturation by multiplying by 1.8 (thin solid curve) and by the
Ulrich function fU ðLÞ (dotted curve).
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Fig. 1.—Solar cycle 21 simulation based on the doublet sources and
transport parameters of WNS89 (thick solid curves). Also plotted are the
corresponding WSO measurements, corrected for line profile saturation by
multiplying by 1.8 (thin solid curves) and by the Ulrich function fU ðLÞ (dotted curves). (a) Evolution of Btot tot =4R2 (G), the total photospheric
flux expressed as a surface-averaged field strength. (b) Evolution of the
Sun’s equatorial dipole strength Deq (G). (c) Evolution of the Sun’s axial
dipole component Dax (G).
2000a). The effect of the latitude-dependent Ulrich correction (eq. [10]) is to increase the relative strength of the active
region fields, so that the total open flux near sunspot maximum becomes comparable to that near sunspot minimum,
in agreement with observations. In the case of the flux transport simulations, we might expect to produce an equivalent
result by increasing the strength of the source fluxes.
Figure 3 shows the long-term variation of tot derived
from NSO/Kitt Peak maps of the photospheric field for
Carrington rotations (CRs) 1645–1984. For comparison,
we also plot the simulated photospheric fluxes during cycle
21; as remarked previously, these simulations employed
magnetic doublet sources whose strengths were estimated
visually from photographic prints of daily NSO magnetograms. It is evident that during 1978–1986, the model under-
TOTAL PHOTOSPHERIC FLUX (G)
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Fig. 3.—Variation of Btot tot =4R2 (G) during 1976–2001, as
derived from NSO/Kitt Peak synoptic maps of the photospheric field (dotted curve). For comparison, the thick solid curve shows Btot ðtÞ as calculated
using the doublet sources and transport parameters of WNS89.
No. 2, 2002
MERIDIONAL FLOW AND THE SUN’S OPEN FLUX
estimates the fluxes in the digitized NSO maps by a factor of
the order of 2. However, Figure 3 suggests that the NSO
fluxes for cycle 21 are themselves too low and should be
increased by 30% to make them comparable to those of
cycle 22, given that the sunspot-number amplitudes of these
two cycles were nearly equal. It thus appears that the magnetic doublet fluxes should be scaled upward by a factor of
close to 3.
A correction factor for the source fluxes can also be estimated by comparing the simulated and observed values of
Deq ðtÞ. The variation of the equatorial dipole component is
closely associated with low-latitude activity (see Wang et al.
2000a); since it characterizes the large-scale magnetic field,
its amplitude is relatively insensitive to the spatial resolution
of the magnetograph measurements, unlike the total photospheric flux. From Figure 1b we see that the values of Deq
derived from the Ulrich-corrected WSO data are 3 times
higher than the simulated values during the period from
7
1191
mid-1978 to 1986. In the subsequent simulations of this
study, we accordingly multiply the strengths of all doublet
sources deposited between 1978 July and 1986 January by a
factor of 3. The total flux of each polarity emerging at the
solar surface between 1976 August and 1986 April then
increases from 1:1 1025 to 2:9 1025 Mx.
4. NEED FOR HIGHER POLEWARD FLOW SPEEDS
Let us now consider the effect of increasing the source
fluxes during 1978–1986 by a factor of 3, while retaining the
transport rates used in the previous simulation and in
WNS89. Figure 4 shows the resulting behavior of Deq ðtÞ,
Dax ðtÞ, Bn ðtÞ, Bs ðtÞ, and BE ðtÞ, along with the corresponding
quantities derived from the Ulrich-corrected WSO data.
Although the model approximately reproduces the
observed variation of the equatorial dipole strength, the
40
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YEAR
Fig. 4.—Solar cycle 21 simulation in which the transport parameters are the same as in WNS89 but the source fluxes have been multiplied by a factor of 3
(thick solid curves). Dotted curves show the corresponding quantities derived from the Ulrich-corrected WSO measurements. (a) Evolution of the equatorial
dipole strength Deq (G). (b) Evolution of the axial dipole component Dax (G). (c) Evolution of Bn and Bs (G), the average fields poleward of L 60 . (d ) Evolution of the near-Earth radial IMF strength BE (nT).
WANG, SHEELEY, & LEAN
The adopted flow profile is such that the poleward velocity
attains a maximum value of 23 m s1 at jLj ¼ 6=4, falls to 12
m s1 at jLj ¼ 45 , and vanishes at the equator and at the
poles. (The relatively steep falloff toward high latitudes is
needed to prevent the final flux distribution from becoming
too sharply peaked at the poles.) The initial photospheric
field distribution in 1976 August is assumed to have the
form ð12 GÞ sin8 L. Figure 5 displays the calculated and
observed variation of Deq ðtÞ, Dax ðtÞ, Bn ðtÞ, and Bs ðtÞ. As
expected, the effect of increasing the poleward flow speeds is
to reduce the final values of the axial dipole and polar field
strengths, bringing the model into much better agreement
with observations (compare Figs. 4 and 5). Moreover, the
simulation now roughly reproduces the large fluctuations
undergone by the WSO polar fields during 1980–1983. The
nature of these fluctuations can be seen more clearly from
the latitude-time plots in Figure 6, constructed by averaging
successive maps of the simulated and observed photospheric
field over . During times of enhanced activity, flux streams
poleward from the sunspot latitudes in the form of discrete
surges; the polar field oscillations are associated with large
surges of alternating (first trailing, then leading) polarity.
(The WSO polar fields also show a spurious annual modulation caused by the Sun’s 7 axial tilt.) In WNS89, the surges
observed during 1980–1982 were modeled by increasing the
source fluxes and doubling the flow speed from 10 to 20
m s1 around sunspot maximum. In the present study, we
Vol. 580
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predicted values of the axial dipole, polar field, and IMF
strengths are far too large at the end of the cycle.
To understand the discrepancy between the simulated
and observed polar field evolution, it is necessary to first
recall how the polar fields are formed in the flux transport
model (see Wang & Sheeley 1991 and references therein).
Because of the action of Coriolis forces on the rising, toroidally oriented flux, active regions generally emerge with
their leading, westward poles shifted slightly equatorward
of their trailing poles (‘‘ Joy’s law ’’); the corresponding
north-south dipole moments are opposite to that of the
Sun’s large-scale field at the start of the cycle. The establishment of the new-cycle polar fields requires that net quantities of trailing-polarity flux be transported from the sunspot
latitudes to the polar regions. In the absence of supergranular diffusion, the meridional surface flow would carry equal
amounts of leading- and trailing-polarity flux to the poles,
and there would be no net change in the polar fields. In reality, the leader flux (being located at systematically lower latitudes) tends to diffuse across the equator and be canceled
by its opposite-hemisphere counterpart, leaving a surplus of
trailing-polarity flux in each hemisphere, which is gradually
concentrated at the poles by the surface flow.
In the simulation of Figure 4, the final polar fields are too
strong because we have increased the source fluxes without
changing the transport rates; as a result, too much trailingpolarity flux now reaches the poles. To reduce the net quantity of trailing-polarity flux in each hemisphere, it is necessary to prevent leading-polarity flux from diffusing into the
opposite hemisphere. This effect can be achieved by decreasing and/or by increasing the low-latitude flow speeds.
In our next simulation, the source fluxes are again taken
to be 3 times stronger than in WNS89, but we now set
¼ 500 km2 s1 and
jvðLÞj ¼ 25 m s1 cos2 L sin0:025 jLj :
ð11Þ
(b)
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Fig. 5.—Solar cycle 21 simulation (‘‘ revised model ’’) in which the source
fluxes of WNS89 have been multiplied by 3, the flow amplitude has been
increased from 10 to 25 m s1, and the diffusion coefficient has been
decreased from 600 to 500 km2 s1 (thick solid curves). Dotted curves show
the corresponding Ulrich-corrected WSO measurements. (a) Plot of Deq ðtÞ
(G). (b) Plot of Dax ðtÞ (G). (c) Plot of Bn ðtÞ and Bs ðtÞ (G).
have assumed a time-independent meridional flow for simplicity, although the fit to the observed polar field variation
could be improved further by relaxing this condition.
In Figure 7 we plot the solar cycle evolution of Dtot and
BE predicted by our revised model, along with the corresponding observed quantities. The variation of the radial
IMF strength (or total open flux) resembles that of the total
dipole strength, since the source surface field is generally
dominated by its dipole (l ¼ 1) component. However, BE ðtÞ
shows less overall modulation than Dtot ðtÞ because it also
contains a significant contribution from the magnetic quadrupole (l ¼ 2) component around sunspot maximum (see
Wang et al. 2000a). The simulation now reproduces the
large peak in the IMF strength observed in 1982, which in
turn was associated with a strengthening of the equatorial
dipole component (see Fig. 5a).
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Fig. 6.—Latitude-time plots of the photospheric field, constructed by
averaging successive Carrington maps over longitude. Gray scale ranges
from hBr i < 5 G (black) to hBr i > þ5 G (white). (a) Simulation (revised
transport model). (b) Ulrich-corrected WSO data. In comparing the two
plots, it should be kept in mind that the spatial resolution of the WSO photospheric field maps (30 pixels in sin L) is much coarser than that of the simulations (64 pixels in L). The strong curvature (deceleration) of the
observed poleward-directed streams is more clearly seen in the corresponding high-resolution NSO plot displayed in Fig. 1 of WNS89.
Figure 8 shows how the open flux is distributed at the
solar surface as a function of latitude and time. Here the
footpoint locations of open field lines were determined by
applying a PFSS extrapolation to the simulated and
observed photospheric field maps, and the footpoint fields
were then averaged over longitude (with their signs included
to indicate the dominant polarity at each latitude). The open
flux is seen to be concentrated at high latitudes (inside the
polar coronal holes) near sunspot minimum and at low latitudes (in and around active regions) at sunspot maximum.
During the rising and declining phases of the cycle, both
active regions and polar holes contribute significantly to the
total open flux.
Although we have adopted a flow amplitude of 25 m s1
and a diffusion rate of 500 km2 s1, it should be noted that
the observed IMF variation and polar field evolution can
also be fitted by simultaneously decreasing or increasing
both jvj and . Thus, if our flow velocities at low latitudes
are decreased by 20%, would have to be decreased from
500 to 400 km2 s1 in order for the rate at which leadingpolarity flux leaks across the equator to remain unchanged.
However, lowering also prolongs the lifetimes of active
regions and raises the total photospheric flux to levels substantially higher than indicated by the NSO/Kitt Peak
measurements. Conversely, steady flow speeds significantly
faster than 25 m s1 would be difficult to reconcile with
present helioseismic and magnetic tracer observations.
Measurements made with the Michelson Doppler Imager
on the Solar and Heliospheric Observatory during 1996–
2001 (Haber et al. 2002) indicate surface flows that reach
maximum poleward speeds of the order of 20 m s1 at lati-
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Fig. 7.—Evolution of (a) the total dipole strength Dtot (G) and (b) the
near-Earth radial IMF strength BE (nT), as predicted by the revised cycle
21 model (thick solid curves). Dotted curves indicate the corresponding
Ulrich-corrected WSO quantities, while connected asterisks denote the in
situ IMF measurements.
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( a ) SIMULATION
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MERIDIONAL FLOW AND THE SUN’S OPEN FLUX
SINE LATITUDE
No. 2, 2002
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Fig. 8.—Latitude-time plots of the open flux, constructed by averaging
the photospheric field within open regions over longitude. Gray scale
ranges from hBopen i < 1 G (black) to hBopen i > þ1 G (white); closed
field areas are neutral gray. (a) Simulation (revised transport model). (b)
Ulrich-corrected WSO data.
WANG, SHEELEY, & LEAN
5. ORIGIN OF THE 1982 IMF PEAK
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DIPOLE STRENGTHS AND TOTAL PHOTOSPHERIC FLUX (G)
During solar cycles 21 and 22, the near-Earth IMF
strength was observed to peak in 1982 and 1991, respectively, just as sunspot activity was beginning to decline.
From Figure 7 we recall that the variation of BE tends to follow that of the Sun’s total dipole strength, which in turn differs markedly from that of the total photospheric flux or the
sunspot number. To illustrate this difference, Figure 9 compares the evolution of tot, Dtot, and Deq during 1976–1986,
as derived with the revised transport model. The total dipole
moment Dtot ¼ ðD2ax þ D2eq Þ1=2 is dominated by its axial
component Dax near sunspot minimum and by its equatorial
component Deq near sunspot maximum; since jDax ðtÞj and
Deq ðtÞ have comparable amplitudes of the order of 4 G,
Dtot ðtÞ and thus BE ðtÞ undergo less long-term modulation
than tot ðtÞ. We also note that although Deq ðtÞ and tot ðtÞ
are both closely correlated with sunspot activity, the heights
of their peaks do not show a simple one-to-one correspondence, since the equatorial dipole strength depends not just
on the total flux present, but also on the longitudinal polarity organization of the photospheric field.
Let us now examine the distribution of the large-scale
solar field around the time of the 1982 peak in the open flux.
The Carrington-format maps in Figure 10 illustrate the
simulated photospheric field, open field regions, and source
surface field during CR 1725 (1982 August). A large BMR
has emerged in the northern hemisphere around longitude
290 ; it is in phase with and reinforces the background
polarity sector structure. Highly sheared open regions with
very strong footpoint fields are located to the east and west
of the BMR. The polar holes are still in the process of formation, and the source surface field has a tilted dipole
structure.
It is instructive to compare Figure 10 with Figure 11,
which displays the corresponding Carrington-format maps
for CR 1695 (1980 May). The Sun is more active at this time
than during CR 1725, and the total photospheric flux is
30% larger. However, the continual emergence of new
BMRs limits the areal sizes of the unipolar regions and their
embedded coronal holes. The polarities are so ‘‘ well
mixed ’’ that the equatorial dipole component is only onehalf as strong as during 1982 August, and open is one-half
( a ) PHOTOSPHERIC FIELD
tudes ranging from 10 to 25 ; at high latitudes, the
direction of the flow is sometimes even reversed. If we
replace the exponent 0.025 in equation (11) by 0.15, so that
our velocity profile peaks at a latitude of 15 instead of 6 ,
we find that the diffusion rate would have to be reduced to
300 km2 s1 in order to offset the leakage of flux into the
opposite hemisphere and match the observed dipole and
polar field evolution. Again, however, the resulting values
of tot would be too large around sunspot maximum.
Although the reason for the discrepancy between the latitudinal profile adopted here and that inferred from helioseismology remains unclear to us, Haber et al. (2002) note that
their measurements are accurate only at very shallow depths
(d0.02 R), and that even within this narrow layer the
velocities are highly variable; it is thus possible that these
depths are not the same as those that determine the motion
of large-scale magnetic fields.
Vol. 580
( b ) HOLES
1194
30
10
0
-30
-60
5
-90
0
90
180
270
360
LONGITUDE (DEGREES)
0
76
77
78
79
80
81
82
YEAR
83
84
85
86
87
Fig. 9.—Solar cycle variation of Dtot (thick solid curve), Deq (thin solid
curve), and Btot tot =4R2 (dotted curve), as derived with the revised
transport model. All quantities are in gauss.
Fig. 10.—Carrington-format maps for CR 1725 (1982 August), as simulated using the revised transport model. (a) Photospheric field. Gray scale
ranges between Br < 20 G (black) and Br > þ20 G (white). (b) Open field
regions (coronal holes). Gray scale ranges between Bopen < 10 G (black)
and Bopen > þ10 G (white). (c) Source surface field. Gray-scale levels are
Bss < 0:4 G (black), 0:4 G < Bss < 0 G (dark gray), 0 G < Bss <
þ0:4 G (light gray), and Bss > þ0:4 G (white).
90
60
30
0
-30
LATITUDE (DEGREES)
-60
-90
4. When a simulation is performed with the same transport parameters as in WNS89 (jvj 10 m s1, ¼ 600 km2
s1) but with the source fluxes multiplied by 3, the final values of the axial dipole, polar field, and radial IMF strengths
are 4 times higher than observed during the 1986 sunspot
minimum.
5. Approximate agreement between the calculated and
observed evolution of the photospheric and interplanetary
fields during 1976–1986 can be obtained by increasing the
flow amplitude from 10 to 25 m s1 and decreasing the
diffusion rate from 600 to 500 km2 s1.
In this study, we have emphasized the role of meridional
flow in determining the evolution of the Sun’s large-scale
field. Somewhat paradoxically, the effect of a fast poleward
flow at low latitudes is to greatly weaken—not strengthen—
the polar and heliospheric fields at the end of the cycle. Even
in the absence of flow, supergranular diffusion and differential rotation would still combine to annihilate the nonaxisymmetric field component, including the equatorial dipole;
however, since it is unaffected by rotational shearing, the
axisymmetric dipole component would build up continually
as more and more BMRs emerge. A strong poleward flow
counteracts the growth of jDax j toward sunspot minimum
by preventing the leading-polarity flux from diffusing freely
across the equator, so that it cancels a larger fraction of the
trailing-polarity flux in its own hemisphere. By raising the
maximum flow speed from 10 to 23 m s1 and decreasing from 600 to 500 km2 s1, we were able to reduce the final values of jDax j, jBn j, jBs j, and BE by as much as a factor of 4.
(The evolution of the axial dipole strength when is fixed at
600 km2 s1 and jvj is varied from 0 to 20 m s1 is illustrated
in Fig. 5 of Wang & Sheeley [1991]; the asymptotic value of
jDax j begins to decrease only for poleward speeds in excess
of 5 m s1.)
As was also noted in WNS89, the combined effect of
enhanced sunspot activity and fast poleward flow is to generate powerful surges of alternating polarity and accompanying polar field oscillations, similar to those observed
during 1980–1983. (That such surges were not unique to
solar cycle 21 is suggested by Fig. 1 of Sheeley [1991], in
which polar faculae numbers are seen to undergo large fluctuations during the post-maximum phases of earlier cycles.)
Although we have assumed a time-independent flow in the
present study, it is possible that these giant surges represent
periods of increased flow speeds, perhaps physically linked
to the enhanced rate of flux emergence; if so, their effect is to
remove most of the extra flux by sweeping both polarities to
the poles. Indeed, flux-transport dynamo models suggest
that the latitudinal progression of sunspot activity and the
solar cycle period itself are determined by the global meridional circulation, which is directed equatorward at the bottom of the convection zone (see Wang, Sheeley, & Nash
1991; Dikpati & Charbonneau 1999).
As indicated by Figure 8, there are two distinct contributions to the open flux over the solar cycle: a high-latitude
component that dominates near sunspot minimum and corresponds to the long-lived polar coronal holes and a lowlatitude component that dominates near sunspot maximum
and is rooted in and around active regions. The highlatitude flux is closely related to the Sun’s axial dipole
moment and represents the large-scale, axisymmetric remnant of the active region fields (the flux that has managed to
survive the journey to the polar regions). On the other hand,
90
60
30
0
-30
-60
-90
90
( c ) SS FIELD
60
30
0
-30
-60
-90
0
90
180
270
1195
( a ) PHOTOSPHERIC FIELD
MERIDIONAL FLOW AND THE SUN’S OPEN FLUX
( b ) HOLES
No. 2, 2002
360
LONGITUDE (DEGREES)
Fig. 11.—Simulated Carrington maps for CR 1695 (1980 May). For key,
see Fig. 10.
as large. The dipole and quadrupole components now contribute equally to the source surface field.
6. DISCUSSION
We now summarize the steps leading to our revised flux
transport model:
1. Ulysses observations (Balogh et al. 1995; Smith et al.
2001) have shown that the Sun’s open flux is distributed uniformly in latitude at heliocentric distances of the order of 1
AU. The near-Earth radial IMF strength is thus given by
BE ’ open =4r2E , where the total open flux open can be
estimated from a PFSS extrapolation of the photospheric
field (Wang & Sheeley 1995).
2. When this result is applied to the simulated photospheric fields of WNS89, the derived values of BE are 2–4
times smaller than observed during 1978–1984, and their
variation is strongly anticorrelated with the sunspot cycle,
contrary to in situ IMF measurements (see also Mackay et
al. 2002).
3. The discrepancy between the predicted and observed
IMF strengths suggests that the magnetic doublets
employed in the WNS89 simulations were too weak. In the
present study, we have therefore multiplied the source fluxes
during 1978–1986 by a factor of 3, which brings the total
photospheric fluxes into closer agreement with NSO/Kitt
Peak measurements and makes the equatorial dipole
strengths consistent with the Ulrich-corrected WSO data.
1196
WANG, SHEELEY, & LEAN
the low-latitude open flux is mainly associated with the
Sun’s equatorial dipole and quadrupole moments. Since this
flux is predominantly nonaxisymmetric, it is subject to rotational shearing and is annihilated on a timescale of 1 yr
(see Wang, Sheeley, & Lean 2000b); it must therefore be
continually replenished by new activity. The underestimate
of the source fluxes in our earlier model led to an underestimate of the equatorial dipole strength and thus of the lowlatitude open flux.
We thank J. T. Hoeksema for making available the WSO
photospheric field maps used in this study. The NSO/Kitt
Peak data were produced cooperatively by the NSF/
NOAO, NASA Goddard Space Flight Center, and NOAA
Space Environment Center. This work was supported by
NASA and by the Office of Naval Research/Naval
Research Laboratory under the ‘‘ Solar Magnetism and the
Earth’s Environment ’’ Accelerated Research Initiative.
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