# Solar Magnetic Field Reversals and the Role of Dynamo Families

M.L. DeRosa11affiliation: Lockheed Martin Solar and Astrophysics Laboratory,
3251 Hanover St. B/252, Palo Alto, CA 94304, USA , A.S. Brun22affiliation:
Laboratoire AIM Paris-Saclay, CEA/Irfu Université Paris-Diderot CNRS/INSU,
91191 Gif-sur-Yvette, France , and J.T. Hoeksema33affiliation: W. W. Hansen
Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA

###### Abstract

The variable magnetic field of the solar photosphere exhibits periodic
reversals as a result of dynamo activity occurring within the solar interior.
We decompose the surface field as observed by both the Wilcox Solar
Observatory and the Michelson Doppler Imager into its harmonic constituents,
and present the time evolution of the mode coefficients for the past three
sunspot cycles. The interplay between the various modes is then interpreted
from the perspective of general dynamo theory, where the coupling between the
primary and secondary families of modes is found to correlate with large-scale
polarity reversals for many examples of cyclic dynamos. Mean-field dynamos
based on the solar parameter regime are then used to explore how such
couplings may result in the various long-term trends in the surface magnetic
field observed to occur in the solar case.

Sun: interior — Sun: photosphere — Sun: magnetic fields — Sun: activity

##  1 Introduction

The Sun is a dynamic star that possesses quasi-regular cycles of magnetic
activity having a mean period of about 22 yr. This period varies from cycle to
cycle, and over the past several centuries has ranged from 18 to 25 years
(Weiss 1990; Beer et al. 1998; Usoskin et al. 2007), as for example
illustrated by the unusual but not unprecedented length of the most recently
completed sunspot cycle 23. During each sunspot cycle (comprising half of a
magnetic cycle), the Sun emerges sunspot groups and active regions onto the
photosphere, with such features possessing characteristic latitudes, polarity,
and tilt angles. As with the period, the numbers and emergence frequencies of
active regions is observed to vary from cycle to cycle.

At activity minima when few active regions are present, the surface magnetic
field is characterized by the presence of two polar caps, i.e., largely
unipolar patches of magnetic flux dispersed across both polar regions with the
northern and southern caps possessing opposite polarities. Reversals of this
large-scale dipole represented by the polar-cap flux occur during each sunspot
cycle, allowing the subsequent sunspot cycle to begin in the opposite
configuration. After two sunspot cycles, and thus after undergoing two
polarity reversals, the photospheric field will have returned to its starting
configuration so as to complete a full activity cycle.

In response to the photospheric flux associated with various features, such as
active regions and their decay products, the coronal magnetic field possesses
structures having a broad spectrum of sizes. These structures are both evident
in observations of coronal loops, as found in narrow-band extreme ultraviolet
or soft X-ray imagery, and reproduced in models of the coronal magnetic field
(e.g., Schrijver & DeRosa 2003). In both venues, the coronal magnetic field
geometry is seen to contain a rich and complex geometry. Dynamical events
originating from the corona, such as eruptive flares and coronal mass
ejections, are likely powered by energy released by a reconfiguration of the
coronal magnetic field, which in turn is responding to changes and evolution
of photospheric fields.

Precise measurements of the time-history of photospheric magnetic field and
the ability to determine the projection of this field into its constituent
multipole components are helpful in investigating the physical processes
thought to be responsible for dynamo activity (Bullard & Gellman 1954; Moffatt
1978). In cool stars similar to the Sun, the dynamo is presumed to be a
consequence of the nonlinear interactions between convection, rotation, and
large scale flows, leading to the generation and maintenance against Ohmic
diffusion of magnetic field of various temporal and spatial scales (Weiss
1987; Cattaneo 1999; Ossendrijver 2003; Brun et al. 2004; Vögler & Schüssler
2007; Charbonneau 2010; Reiners 2012). In particular, the dependence of dynamo
activity upon rotation appears to be well established (Noyes et al. 1984; Saar
& Brandenburg 1999; Pizzolato et al. 2003; Böhm-Vitense 2007; Reiners et al.
2009). However, many details of the understanding of why many cool-star
dynamos excite waves of dynamo activity having a regular period, specifically
22 yr in the case of the Sun, remain unclear.

To investigate this question, it is useful to explore the behavior and
evolution of the lowest-degree (i.e., largest-scale) multipoles, their
amplitudes and phases, and their correlations with the solar photospheric
magnetic field. Many earlier studies (e.g., Levine 1977; Hoeksema 1984;
Gokhale et al. 1992; Gokhale & Javaraiah 1992) have illustrated how power in
these modes ebb and flow as a function of the activity level. In particular,
J. Stenflo and collaborators have performed thorough spectral analyses on the
temporal evolution of the various spherical harmonic modes. Stenflo & Vogel
(1986), Stenflo & Weisenhorn (1987), and Stenflo & Güdel (1988), and more
recently Stenflo (1994) and Knaack & Stenflo (2005), base their analysis on
Mt. Wilson and Kitt Peak magnetic data spanning the past few sunspot cycles.
As one would expect, they find that most of the power is contained in temporal
modes having a period of about 22 yr, and especially in spherical harmonics
that are equatorially antisymmetric, such as the axial dipole and octupole.
However, they find signatures of the activity cycle are present in all
axisymmetric harmonics, as significant power is present at temporal
frequencies at or near integer multiples of the fundamental frequency of 1.44
nHz [equivalent to (22 yr)-1].

In the current study, we focus on the coupling between spherical harmonic
modes, and what such coupling may indicate about the operation of the interior
dynamo. In particular, reversals of the axial dipole mode may be viewed as a
result of continuous interactions between the poloidal and toroidal components
of the interior magnetic field, i.e., the so-called dynamo loop. Currently,
one type of solar dynamo model that successfully reproduces many observed
behaviors is the flux-transport Babcock-Leighton type (e.g., Choudhuri et al.
1995; Dikpati et al. 2004; Jouve & Brun 2007; Yeates et al. 2008). A key
ingredient in producing realistic activity cycles using this type of model is
found to be the amplitude and profile of the meridional flow (Jouve & Brun
2007; Karak 2010; Nandy et al. 2011; Dikpati 2011), which result in field
reversals progressing via the poleward advection across the surface of
trailing-polarity flux from emergent bipolar regions. During the rising phase
of each sunspot cycle, polar cap flux left over from the previous cycle is
canceled, after which new polar caps having the opposite magnetic polarity
form (Wang et al. 1989; Benevolenskaya 2004; Dasi-Espuig et al. 2010).

Helioseismic analyses of solar oscillations have provided measurements and
inferences of key dynamo components, such as the internal rotation profile and
the near-surface meridional circulation (Thompson et al. 2003; Basu & Antia
2010). Complementing precise observations of the solar magnetic cycle
properties, these helioseismic inversions represent additional strong
constraints on theoretical solar dynamo models. Successful solar dynamo models
strive to reproduce as many empirical features of solar magnetic activity as
possible, including not only cycle periods, but also parity, phase relation
between poloidal and toroidal components, and the phase relation between the
dipole and higher-degree harmonic modes.

Interestingly, a recent analysis of geomagnetic records has indicated that the
interplay between low-degree harmonic modes during polarity reversals is one
way to characterize both reversals of the geomagnetic dynamo (which have a
mean period of about 300,000 yr) as well as excursions, where the dipole axis
temporarily moves equatorward and thus away from its usual position of being
approximately aligned with the rotation axis, followed by a return to its
original position without having crossed the equator (see Hulot et al. 2010
for a recent review on Earth’s magnetic field). In particular, these studies
have shown that, during periods of geomagnetic reversals, the quadrupolar
component of the geomagnetic field is stronger than the dipolar component,
while during an excursion (which can be thought of as a failed reversal), the
dipole remains dominant (Amit et al. 2010; Leonhardt & Fabian 2007; Leonhardt
et al. 2009). One may thus ask: Is a similar behavior observed for the solar
magnetic field?

In an attempt to address this question, we have performed a systematic study
of the temporal evolution of the solar photospheric field by determining the
spherical harmonic coefficients for the photospheric magnetic field throughout
the past three sunspot cycles, focusing on low-degree modes and the relative
amplitude of dipolar and quadrupolar components. Following the classification
of McFadden et al. (1991), we have made the distinction between primary and
secondary families of harmonic modes, a classification scheme that takes into
account the symmetry and parity of the spherical harmonic functions (see
Gubbins & Zhang 1993 for a detailed discussion on symmetry and dynamo).

While we recognize that the solar dynamo operates in a more turbulent
parameter regime than the geodynamo, and is more regular in its reversals, the
presence of grand minima (such as the Maunder Minimum) in the historical
record indicates that the solar dynamo can switch to a more intermittent state
on longer-term, secular time scales. In fact in the late stages of the Maunder
Minimum, the solar dynamo was apparently asymmetric, with the southern
hemisphere possessing more activity than the north (Ribes & Nesme-Ribes 1993)
for several decades, a magnetic configuration that may have been achieved by
having dipolar and quadrupolar modes of similar amplitude (Tobias 1997; Gallet
& Pétrélis 2009). Additionally, recent spectopolarimetric observations of
solar-like stars now provide sufficient resolution to characterize the
magnetic field geometry in terms of its multipolar decomposition (Petit et al.
2008). Furthermore, the analysis of reduced dynamical systems developed over
the last 20 yr describing the geodynamo and solar dynamo have emphasized the
importance of the nonlinear coupling between dipolar and quadrupolar
components (Knobloch & Landsberg 1996; Weiss & Tobias 2000; Pétrélis et al.
2009).

This article is organized in the following manner. In §2, we describe the data
sets and the data analysis methods used to perform the spherical harmonic
analysis, followed in §3 with an explanation of the temporal evolution of the
various harmonic modes, the magnetic energy spectra, and the decomposition in
terms of primary and secondary families. We interpret in §4 our results from a
dynamical systems perspective and illustrate some of these concepts using
mean-field dynamo models. Concluding remarks are presented in §5.

##  2 Observations and Data Processing

We analyze time series of synoptic photospheric magnetic field maps of the
radial magnetic field $B_{r}$ derived from line-of-sight magnetogram
observations taken by both the Wilcox Solar Observatory (WSO; Scherrer et al.
1977) at Stanford University and by the Michelson Doppler Imager (MDI;
Scherrer et al. 1995) on board the space-borne Solar and Heliospheric
Observatory (SOHO). The WSO data111Available at
http://wso.stanford.edu/synopticl.html. used in this study span the past 36
years, commencing with Carrington Rotation (CR) 1642 (which began on 1976 May
27) and ending with CR 2123 (which ended on 2012 May 25). For MDI222Available
at http://soi.stanford.edu/magnetic/index6.html., we used data from much of
its mission lifetime, starting with CR 1910 (which began on 1996 Jul. 1)
through CR 2104 (which ended on 2010 Dec. 24). In both data series, one map
per Carrington rotation was used, though maps with significant amounts of
missing data were excluded. The measured line-of-sight component of the field
is assumed to be the consequence of a purely radial magnetic field when
calculating the harmonic coefficients. Additionally, for WSO, the synoptic map
data are known to be a factor of about 1.8 too low due to the saturation of
the instrument (Svalgaard et al. 1978). Lastly, the MDI data have had
corrections applied for the polar fields using the interpolation scheme
presented in Sun et al. (2011).

For each map, we perform the harmonic analysis using the Legendre-transform
software provided by the “PFSS” package available through SolarSoft. Using
this software first entails remapping the latitudinal dimension of the input
data from the sine-latitude format provided by the observatories onto a Gauss-
Legendre grid (c.f., § 25.4.29 of Abramowitz & Stegun 1972). This regridding
enables Gaussian quadrature to be used when evaluating the sums needed to
project the magnetic maps onto the spherical harmonic functions. The end
result is a time-varying set of complex coefficients $B_{\ell}^{m}(t)$ for a
series of modes spanning harmonic degrees $\ell$ = 0, 1, …, $\ell_{\max}$,
where the truncation limit $\ell_{\max}$ is equal to 60 for the WSO maps and
192 for MDI maps. The $B_{\ell}^{m}$ coefficients are proportional to the
amplitude of each spherical harmonic mode $Y_{\ell}^{m}$ for degree $\ell$ and
order $m$ possessed by the time series of synoptic maps, so that

$B_{r}(\theta,\phi,t)=\sum_{\ell=0}^{\ell_{\max}}\sum_{m=0}^{\ell}B_{\ell}^{m}(t)\,Y_{\ell}^{m}(\theta,\phi),$
(1)

where $\theta$ is the colatitude, $\phi$ is the latitude, and $t$ is time. We
note that because the coefficients $B_{\ell}^{m}$ are complex numbers, this
naturally accounts for the rotational symmetry between spherical harmonic
modes with orders $m$ and $-m$ (for a given value of $\ell$), with the
amplitudes of modes for which $m>0$ appearing in the real part of
$B_{\ell}^{m}$, and the amplitudes of the modes where $m<0$ being contained in
the imaginary part of $B_{\ell}^{m}$. Consequently, the sum over $m$ in
equation (1) starts at $m=0$ instead of at $m=-\ell$. The coefficients
$B_{\ell}^{0}$ corresponding to the axisymmetric modes (for which $m=0$) are
real for all $\ell$.

We use the convention that, for a particular spherical harmonic degree $\ell$
and order $m$,

$Y_{\ell}^{m}(\theta,\phi)=C_{\ell}^{m}P_{\ell}^{m}(\cos\theta)\,e^{im\phi},$
(2)

where the functions $P_{\ell}^{m}(\cos\theta)$ are the associated Legendre
polynomials, and where the coefficients $C_{\ell}^{m}$ are defined

$C_{\ell}^{m}=(-1)^{m}\left[\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}\right]^{\frac{1}{2}}.$
(3)

With this normalization, the spherical harmonic functions satisfy the
orthogonality relationship

$\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\sin\theta\,d\theta\,Y_{\ell}^{m*}Y_{\ell^{\prime}}^{m^{\prime}}=\delta_{\ell\ell^{\prime}}\,\delta_{mm^{\prime}}.$
(4)

When comparing our coefficients with those from other studies, it is important
to take the normalization into account. For example, the complex
$B_{\ell}^{m}$ coefficients used here are different than (albeit related to)
the real-valued $g_{\ell}^{m}$ and $h_{\ell}^{m}$ coefficients provided by the
WSO team333Tables of $g_{\ell}^{m}$ and $h_{\ell}^{m}$ are available from the
WSO webpage at http://wso.stanford.edu/Harmonic.rad/ghlist.html.. This
difference is due to their use of spherical harmonics having the Schmidt
normalization, a convention that is commonly used by the geomagnetic community
as well as by earlier studies in the solar community such as Altschuler &
Newkirk (1969). For the WSO data used here, we have verified that the values
of $B_{\ell}^{m}$ used in this study are commensurate with the $g_{\ell}^{m}$
and $h_{\ell}^{m}$ coefficients provided by the WSO team.

Because we possess perfect knowledge of $B_{r}$ neither over the entire Sun
nor at one instant in time, the monopole coefficient function $B_{0}^{0}(t)$
does not strictly vanish and instead fluctuates around zero. In practice, we
find that the magnitude of $B_{0}^{0}(t)$ is small, and thus feel justified in
not considering it further. This assumption effectively means that from each
magnetic map we are subtracting off any excess net flux, $\oint
B_{r}(\theta,\phi)\,\sin\theta\,d\theta\,d\phi$, a practice which leads to the
introduction of small errors in the resulting analysis. However, these errors
are deemed to be less important than the inaccuracies resulting from the less-
than-perfect knowledge of the radial magnetic flux on the Sun, including
effects due to evolution and temporal sampling throughout each Carrington
rotation and due to the lack of good radial field measurements of the flux in
the polar regions of the Sun.

##  3 Multipolar Expansions and Their Evolution as a Function of Cycle

###  3.1 Dipole Field (Modes with $\boldsymbol{\ell}=1$)

The solar dipolar magnetic field can be analyzed in terms of its axial and
equatorial harmonic components. As has long been known (Hoeksema 1984), the
axial dipole component, having a magnitude of $|B_{1}^{0}|$, is observed to be
largest during solar minimum when there is a significant amount of magnetic
flux located at high heliographic latitudes on the Sun. These two so-called
polar caps possess opposite polarity, and match the polarity of the trailing
flux within active regions located in the corresponding hemisphere that
emerged during the previous sunspot cycle. Long-term observations of surface-
flux evolution indicate that a net residual amount of such trailing-polarity
flux breaks off from decaying active regions and is released into the
surrounding, mixed-polarity quiet-sun network. This flux is observed to
continually evolve as flux elements merge, fragment, and move around in
response to convective motions (Schrijver et al. 1997), but the long-term
effect is that the net residual flux is slowly advected poleward by surface
meridional flows. Such poleward advection results in a net influx of trailing-
polarity flux into the higher latitudes. At the same time, an equivalent
amount of leading-polarity flux from each hemisphere cancels across the
equator, as is necessary to balance the trailing-polarity flux advected
poleward. Over the course of a sunspot cycle, this process is repeated
throughout subsequent sunspot cycles, during which flux from the trailing
polarities of active regions eventually cancels out the polar-cap flux left
over from previous cycles. Once the leftover flux has fully disappeared, the
buildup of a new polar cap having the opposite polarity occurs by the
subsequent activity minimum.

In contrast to the axial dipole component, the equatorial dipole components,
having magnitudes $|B_{1}^{-1}|$ and $|B_{1}^{1}|$, are largest during maximum
activity intervals and weakest during activity minima. Individual active
regions on the photosphere each contribute a small dipole moment that, aside
from the small axial component arising from the Joy’s Law tilt, is oriented in
the equatorial plane. Together the equatorial dipole moments from the
collection of active regions add vectorially to form the overall dipole
moment. When many active regions are on the disk, it thus follows that the
equatorial dipole mode is likely to have a higher amplitude. During periods of
quiet activity with few active regions on disk, the equatorial dipole
amplitude is minimal.

Because the WSO data span three sunspot activity cycles, a bit of historical
perspective on the evolution of the dipole can be gained, as shown in Figure
1. Figure 1(a) shows the amplitude of the axial dipole moment since mid 1976
and its evolution as a function of the activity level, represented in the
figure by the sunspot number444Sunspot numbers with slightly different
calibrations are available from various sources worldwide. In this article, we
use the indices provided by the Solar Influences Data Center at the Royal
Observatory of Belgium, whose sunspot index data are available online at
http://www.sidc.be/sunspot-index-graphics/sidc_graphics.php. (SSN). It is also
evident that, during the most recent minimum prior to Cycle 24, the magnitude
of the axial dipole component is much lower than during any of the three
previous minima (i.e., those preceding Cycles 21–23). The connection between
the axial dipole component and the flux in the northern hemisphere is
illustrated in the time-history of the mean flux density as integrated over
the northern hemisphere, shown in Figure 2. We observe a slight lag for the
northern hemisphere averaged magnetic flux with respect to the axial dipole
coefficient due to the contribution of other axisymmetric modes possessing a
different phase.

Figure 1(b) illustrates the magnitude of the equatorial dipole since mid 1976.
In step with the relatively lower number of active regions during Cycle 23
when compared with the maxima for Cycles 21 and 22, the equatorial dipole
strength is found to be lower during the most recent maximum than during the
maxima corresponding to Cycles 21 or 22.

Given the variation in sunspot cycle strengths throughout the past few
centuries, we suspect that cycle-to-cycle variations in the magnitudes of the
axial and equatorial modes are not unusual. Proxies of the historical large-
scale magnetic field, such as cosmic-ray induced variations of isotopic
abundances measured from ice-core data (Steinhilber et al. 2012), also show
such longer-term variation and thus seem to be consistent with this view.
Interestingly, the range over which the variation in the ratio of the energies
possessed by the equatorial versus the axial dipole components is about the
same for the three sunspot cycles observed by WSO, as shown in Figure 1(c).
Longer-term measurements of this ratio unfortunately are not available due to
the lack of a sequence of long-term magnetogram maps to which the harmonic
decomposition analysis outlined in Section 2 can be applied.

Figure 1: Evolution of the dipole ($\ell=1$) modes, as characterized by the
(a) axial dipole coefficient $B_{1}^{0}$, (b) equatorial dipole magnitude
$\sqrt{(B_{1}^{-1})^{2}+(B_{1}^{1})^{2}}$, and (c) the ratio of their energies
$[(B_{1}^{-1})^{2}+(B_{1}^{1})^{2}]/(B_{1}^{0})^{2}$ for the WSO (black) and
MDI polar-corrected (magenta) data sets. Panel (d) shows the monthly smoothed
sunspot number (SSN) from Solar Influences Data Center at the Royal
Observatory of Belgium. The WSO data have not been corrected for known
saturation effects that reduce the reported values by a factor of 1.8
(Svalgaard et al. 1978). Figure 2: Northern hemispheric mean flux density
(magenta) and axial dipole coefficient from WSO (black), illustrating the
connection between the axial dipole and the flux in each hemisphere. The
downward spike in the mean hemispheric flux density occurring in 2001 is
likely related to WSO sensitivity issues occurring during that time period,
and may not be real.

###  3.2 Reversals of the Dipole

The process by which old polar caps are canceled out and replaced with new,
opposite-polarity polar caps, as described in the previous section, manifests
itself as a change in sign of the axial dipole amplitude throughout the course
of a sunspot cycle. Such dipole reversals for the past three sunspot cycles
are shown in Figure 3, where the latitude and longitude of the dipole axis are
plotted with time. It is found that the dipole axis spends much of its time in
the polar regions, and for only about 12–18 months during these cycles is it
located equatorward of $\pm$45∘.

During these reversals, which occur at or near maximal activity intervals, the
energy in the dipole never completely disappears. We find that the reduction
in the energy $(B_{1}^{0})^{2}$ in the the axial dipole is partially offset by
an increase in the energy $(B_{1}^{-1})^{2}+(B_{1}^{1})^{2}$ in the equatorial
dipole. This results in a reduction of the total energy
$\sum_{m}(B_{1}^{m})^{2}$ in all dipolar modes only by about an order of
magnitude from its axial-dipole-dominated value at solar minimum, as shown in
Figure 4(a).

Figure 3 indicates that, during a reversal when the axial dipolar component is
weak, the axis of the equatorial dipolar component wanders in longitude. This
seemingly aimless wandering occurs because the longitude of the dipole axis is
primarily determined by an interplay amongst the strongest active regions on
the photosphere at the time of observation. As older active regions decay and
newer active regions emerge onto the photosphere, the equatorial-dipolar axis
responds in kind.

Figure 3: (a)–(c) Mollweide projections of the location of the dipole axis for
the past three sunspot cycles (Cycles 21–23), as determined from WSO synoptic
charts. The solid circles indicate the longitude and latitude of the dipole
axis for each Carrington rotation, with every sixth Carrington rotation also
indicated by an open circle. Grid lines (dashed) are placed every 45∘ in
latitude and longitude for reference. The Carrington longitudes of the central
meridians of each projection are chosen to best illustrate the reversals, and
differ in each of the panels. Panel (d) illustrates the latitude of the dipole
axis as a function of time. The open circles in panel (d) correspond to same
times as the open circles in panels (a)–(c).

###  3.3 Quadrupole Field (Modes with $\boldsymbol{\ell}=2$)

The evolution of the energy contained in the quadrupolar ($\ell=2$) modes
exhibit much more variation than the dipole, as shown in Figure 4. As with the
equatorial dipole components, all of the quadrupolar modes have more power
during greater activity intervals than during quieter periods, as illustrated
in the evolution of the various quadrupolar modes plotted in Figure 5.
Furthermore, when large amounts of activity occur, it is possible for the
total energy $\sum_{m}(B_{2}^{m})^{2}$ in all quadrupolar modes to be greater
than the energy $\sum_{m}(B_{1}^{m})^{2}$ in the dipolar modes at the
photosphere. The ratio between these two groups of modes is shown in Figure
4(c), from which it is evident that during each of the past three sunspot
cycles there have been periods of time when the quadrupolar energy exceeded
the dipolar energy by as much as a factor of 10. The corona, in turn, reflects
the relative strength of a strong quadrupolar configuration of photospheric
magnetic fields by creating complex sectors and possibly multiple current
sheets that extend into the heliosphere. One example of such complex field
geometry is suggested by the potential-field source-surface model of Figure 6,
where a quadrupolar configuration having an axis of symmetry lying almost in
the equatorial plane is seen to predominate.

Figure 4: Total energy in (a) dipolar modes $\sum_{m}(B_{1}^{m})^{2}$, (b)
quadrupolar modes $\sum_{m}(B_{2}^{m})^{2}$, and (c) their ratio
$\sum_{m}(B_{2}^{m})^{2}/\sum_{m}(B_{1}^{m})^{2}$ for the WSO (black) and MDI
polar-corrected (magenta) data sets. Panel (d) shows the monthly smoothed SSN.
The WSO data have not been corrected for known saturation effects that reduce
the reported values by a factor of 1.8 (Svalgaard et al. 1978). Figure 5:
Evolution of the quadrupolar ($\ell=2$) modes, as characterized by the (a)
axial quadrupole coefficient $B_{2}^{0}$, along with higher-order magnitudes
of the (b) $m=\pm 1$ modes $\sqrt{(B_{2}^{-1})^{2}+(B_{2}^{1})^{2}}$ and (c)
$m=\pm 2$ modes $\sqrt{(B_{2}^{-2})^{2}+(B_{2}^{2})^{2}}$, for the WSO (black)
and MDI polar-corrected (magenta) data sets. Panel (d) shows the monthly
smoothed SSN. The WSO data have not been corrected for known saturation
effects that reduce the reported values by a factor of 1.8 (Svalgaard et al.
1978). Figure 6: Representation of the coronal magnetic field in October 2000
for which the large-scale field is predominantly quadrupolar. This field is
the result of a potential-field source-surface extrapolation (Schatten et al.
1969) with an upper boundary of 2.5 $R_{\odot}$ at which the coronal field is
assumed purely radial. Both closed (black) and open (magenta and green,
depending on polarity) field lines are shown in the model. Also shown is the
contour of $B_{r}=0$ at $R=2.5R_{\odot}$ (thicker black line).

###  3.4 Octupole Field (Modes with $\boldsymbol{\ell}=3$)

As with the quadrupole, the octupolar modes contain more power during periods
of high activity and less power during minimum conditions, as illustrated in
Figure 7. The exception is the axial octupolar coefficient $B_{3}^{0}$,
plotted in panel (a) of Figure 7, which is nonzero during solar minima and
exhibits sign reversals during sunspot maxima in a manner similar to the axial
dipole coefficient $B_{1}^{0}$.

The behavior of the various $m=0$ modes can be understood by considering their
functional symmetry: the $Y_{\ell}^{0}$ functions are antisymmetric in
$\theta$ (i.e., antisymmetric across the equator) when the degree $\ell$ is
odd, whereas for even $\ell$ the $Y_{\ell}^{0}$ functions are symmetric in
$\theta$. The presence of polar caps during solar minimum, a highly
antisymmetric configuration, is reflected in the similar evolution of the
$B_{1}^{0}$ and $B_{3}^{0}$ coefficients, which correspond to the axial dipole
($\ell=1$, $m=0$) and octupole ($\ell=3$, $m=0$) modes. The axial quadrupole
($\ell=2$, $m=0$) mode does not share this behavior because, as a symmetric
mode, it is not sensitive to the presence of the polar caps during solar
minima.

The dependence of the $B_{\ell}^{0}$ coefficients on the degree $\ell$ is
illustrated in Figure 8, where the time-averaged energies from the MDI data
(spanning Solar Cycle 23) as a function of degree $\ell$ are plotted. Prior to
averaging, the spectra were placed in two classes: Carrington rotations for
which the SSN is relatively large (defined as when SSN>100) and rotations for
which the SSN is relatively small (defined as when SSN<50), thus capturing the
state of the Sun when it is either overtly active or overtly quiet. The figure
indicates that the even-odd behavior is more pronounced during quiet periods,
and these occur near and during solar minimum when the polar-cap field is
significant. During active periods the even-odd trend is still recognizable,
but because the polar caps are weak and the active regions are primarily
oriented east-west (i.e., in the equatorial plane and thus contributing little
power to the axisymmetric modes) the even-odd trend is less pronounced. We
will further discuss the behavior of axisymmetric modes in the context of
Babcock-Leighton dynamo models in Section 4.2.

Figure 7: Evolution of the octupolar ($\ell=3$) modes, as characterized by the
(a) axial octupole coefficient $B_{3}^{0}$, along with higher-order magnitudes
of the (b) $m=\pm 1$ modes $\sqrt{(B_{3}^{-1})^{2}+(B_{3}^{1})^{2}}$, (c)
$m=\pm 2$ modes $\sqrt{(B_{3}^{-2})^{2}+(B_{3}^{2})^{2}}$, and (d) $m=\pm 3$
modes $\sqrt{(B_{3}^{-3})^{2}+(B_{3}^{3})^{2}}$, for the WSO (black) and MDI
polar-corrected (magenta) data sets. Panel (e) shows the monthly smoothed SSN.
The WSO data have not been corrected for known saturation effects that reduce
the reported values by a factor of 1.8 (Svalgaard et al. 1978). Figure 8:
Time-averaged energies in the axisymmetric modes
$\overline{(B_{\ell}^{0})^{2}}$ as a function of $\ell$ for MDI original
(black) and polar-corrected (magenta) data sets, for more active conditions
(solid lines; defined as when SSN>100) and for quieter periods (dashed lines;
defined as when SSN<50). The interpolation scheme used to correct the MDI
polar flux is described in Sun et al. (2011). Figure 9: Time-averaged energies
in the non-axisymmetric modes $\overline{\sum_{m>0}(B_{\ell}^{m})^{2}}$ as a
function of $\ell$ for MDI original (black) and polar-corrected (magenta) data
sets, for more active conditions (solid lines; defined as when SSN>100) and
for quieter periods (dashed lines; defined as when SSN<50). The interpolation
scheme used to correct the MDI polar flux is described in Sun et al. (2011).

###  3.5 Full Spectra and Most Energetic Modes

One property of the spherical harmonic functions $Y_{\ell}^{m}(\theta,\phi)$
is that the degree $\ell$ is equal to the number of node lines (i.e., contours
in $\theta$ and $\phi$ where $Y_{\ell}^{m}=0$). In other words, the spatial
scale represented by any harmonic mode (i.e., the distance between neighboring
node lines) is determined by its spherical harmonic degree $\ell$. As a
result, the range of $\ell$ values containing the greatest amount of energy
indicates the dominant spatial scales of the magnetic field.

To this end, we have averaged the non-axisymmetric power spectra from each of
the datasets both over time and over $m$, and have displayed the result in
Figure 9. As with Figure 8, we have divided the spectra into active and quiet
classes depending on SSN. In the figure, it can be seen that the magnetic
power spectra form a broad peak with a maximum degree occurring at
$\ell_{P_{\text{max}}}\approx$25, corresponding to a size scale of about
360${}^{\circ}/\ell_{P_{\text{max}}}\approx$15∘ in heliographic coordinates.
Stated another way, this indicates that much of the magnetic energy can be
found (not surprisingly) on the spatial scales of solar active regions or
their decay products.

Energy spectra determined from WSO charts (not shown) do not show the same
broad peak at $\ell_{P_{\text{max}}}\approx$25 as found in the curves from the
MDI-derived data shown in Figure 9. This is an effect of the lower spatial
resolution of the WSO magnetograph (which has 180$\arcsec$ pixels, and is
stepped by 90$\arcsec$ in the east-west direction and 180$\arcsec$ in the
north-south direction when constructing a magnetogram) versus MDI (which has a
plate scale of 2$\arcsec$ per pixel in full-disk mode). The WSO magnetograph,
as a result, does not adequately resolve modes higher than about $\ell=15$,
creating aliasing effects even at moderate values of $\ell$ in the energy
spectra. Accordingly, as longer time series of data from newer, higher-
resolution magnetograph instrumentation are assembled, the high-$\ell$
behavior of the energy spectra (such as those shown in Figs. 8 and 9) may
change due to better observations of finer scales of magnetic field.

###  3.6 Primary and Secondary Families

The projection of the solar surface magnetic fields onto spherical harmonic
degrees allows us to delineate the main symmetries of the magnetic field. As
noted in §2, the harmonic modes can be classified as either axisymmetric
($m=0$) or non-axisymmetric ($m\neq 0$). Separately, the harmonic modes can be
either antisymmetric (odd $\ell+m$) or symmetric (even $\ell+m$) with respect
to the equator (Gubbins & Zhang 1993). Some authors refer to antisymmetric
modes as dipolar and symmetric modes as quadrupolar (presumably because the
axial dipole and quadrupole modes usually possess the most power in their
respective categories), while others synonymously assign these modes to either
the primary and secondary family (e.g., McFadden et al. 1991 when
characterizing the Earth’s magnetic field geometry), respectively. In this
article, we adopt the primary- and secondary-family nomenclature when
describing the equatorial symmetry because this avoids the confusion that may
otherwise occur when, for example, it is realized that the equatorial dipole
mode ($\ell=1,m=1$) formally belongs to the “quadrupolar” family of modes
(since $\ell+m$ is even for this mode).

One important result put forward by the geomagnetic community is that the
relative strengths of the primary and secondary families are different during
geomagnetic field reversals and excursions. During reversals, the modes
associated with the secondary family predominate over primary-family modes,
and during excursions this is not the case (Leonhardt & Fabian 2007). We now
investigate whether analogous behavior is occurring in the solar setting, by
determining which harmonic modes are most correlated with the axial dipole and
axial quadrupole.

When applied to two variables, the Spearman rank correlation index
$\rho\in[-1,1]$ indicates the degree to which two variables are monotonically
related. The index $\rho$ is positive when both variables tend to increase and
decrease at the same points in time. A rank correlation analysis is more
general than a more common Pearson correlation analysis, which specifically
measures how well two variables are linearly related, whereas the rank
correlation analysis enables a determination of whether the time evolution of
two mode amplitudes follow a similar pattern in time without regard to their
(unknown) functional dependency.

In Table 1 we list the degrees $\ell$ and orders $m$ corresponding to the
harmonic coefficients $B_{\ell}^{m}(t)$ that have the highest $\rho$ (positive
correlation) when compared with the axial dipole and axial quadrupole
coefficients $B_{1}^{0}(t)$ and $B_{2}^{0}(t)$ (which peak at different phases
of the sunspot cycle). The corresponding harmonic modes comprise the strongest
modes in the primary and secondary families, respectively. We find that, among
the mode amplitudes that are positively correlated with $B_{1}^{0}$, two out
of three belong to the primary family. Similarly, for $B_{2}^{0}$, 7 of the 10
most-correlated modes are members of the secondary family.

These correlations indicate a preference in the solar dynamo, at least as
inferred from its surface characteristics, for modes belonging to the same
family and thus having the same north-south symmetry characteristics to be
excited nearly in phase. This preference is demonstrated further in Figures 10
and 11, in which the long-term trends of the time evolution of the first
several axisymmetric mode coefficients are shown, after smoothing with a
boxcar filter having a width of 1 yr. (We focus here on the axisymmetric mode
properties because these modes are the only ones considered in most mean-field
dynamo models, as discussed further in §4.2.) In Figure 10, there is a clear
correlation amongst the first few odd-$\ell$ and amongst the first couple of
even-$\ell$ mode coefficients, a trend which is emphasized in Figure 11 in
which these same mode coefficients are overplotted. We note that the mode
groupings are not precisely in phase, as evidenced for example by the lag in
$\ell=3$ and especially the $\ell=5$ modes reversing signs with respect to the
$\ell=1$ mode. When $\ell\geq 6$ or so, these trends become much weaker
amongst the axisymmetric modes (although Table 1 indicates that this is not
necessarily true for the non-axisymmetric modes), presumably because as
smaller and smalle1r scales are considered the effects of the global
organization associated with the 11-year sunspot cycle are less important in
structuring the surface magnetic field.

Figure 11 additionally illustrates that the modes of the secondary family
attain amplitudes of about 25% of the primary mode amplitudes. Furthermore,
the primary and secondary mode families are out of phase: during reversals the
primary modes become weak at the same time as the amplitudes of the modes
associated with the secondary family become maximal, which was shown
previously in Figure 4(c). This same pattern is observed to occur during
reversals of the axial dipole field of the geodynamo. As in the geodynamo
case, we ascribe the relative amplitudes and phase relation between the
primary and secondary families observed during solar dipole reversals as a
strong indication that the interplay of the mode families play a key role in
the process by which the axial dipole reverses. Hence, any realistic model of
the solar dynamo must excite both families of modes to similar amplitude
levels, and must exhibit similar coupling between modes belonging to the
primary and secondary families.

Most Correlated modes

|  $\ell=1$, $m=0$ |  |  |  $\ell=2$, $m=0$ |
---|---|---|---|---|---
$\ell$ | $m$ | Primary | $\ell$ | $m$ | Secondary
3 | 0 | Y | 4 | 0 | Y
5 | 0 | Y | 4 | 1 | N
2 | 0 | N | 9 | 9 | Y
|  |  | 6 | 1 | N
|  |  | 7 | 0 | N
|  |  | 9 | 1 | Y
|  |  | 5 | 1 | Y
|  |  | 1 | 1 | Y
|  |  | 7 | 7 | Y
|  |  | 3 | 3 | Y

Table 1:  Ranking of the most positively correlated modes within the primary
and secondary families for which $\ell\leq 9$. The basis for comparison in
each family is the lowest-degree axisymmetric mode belonging to each family,
specifically the axial dipole ($\ell=1$, $m=0$) and axial quadrupole
($\ell=2$, $m=0$) modes for the primary and secondary families, respectively.
The most correlated mode is the next axisymmetric mode in each family. The
equatorial dipole mode ($\ell=1$, $m=1$) is more correlated with the
axisymmetric quadrupole, as expected from its symmetry properties. We note the
presence of 4 sectoral modes (for which $\ell=m$) in the list of the secondary
family. Figure 10: (a)–(h) Coefficients $B_{\ell}^{0}$ for the axisymmetric
modes of the first eight degrees $\ell$ as a function of time, as calculated
from the WSO (black) and MDI polar-corrected (magenta) synoptic maps and after
boxcar smoothing with a width of 1 yr. Panel (i) shows the monthly smoothed
SSN. The WSO data have not been corrected for known saturation effects that
reduce the reported values by a factor of 1.8 (Svalgaard et al. 1978). Figure
11: Overplotted coefficients $B_{\ell}^{0}$ from Figure 10 of the first 3 odd
($\ell=\\{1,3,5\\}$; dark red, red, light red lines, respectively) and even
($\ell=\\{2,4,6\\}$; dark blue, blue, light blue, respectively) axisymmetric
modes, as calculated from WSO synoptic maps.

##  4 Theoretical Implications for Solar Dynamo

As demonstrated in previous sections, the amplitudes of the various harmonic
modes of the solar magnetic field are continually changing. During reversals,
as the axial dipole necessarily undergoes a change in sign, other modes
predominate such that the amplitude of the solar magnetic field never vanishes
during a reversal. As a result of such reversals occurring in the middle of
each 11-yr sunspot cycle, during the rising phase of each cycle the polar
fields and the emergent poloidal fields have opposite polarity (Babcock 1961;
Benevolenskaya 2004), while in the declining phase the polarity of sunspots
and active regions are aligned with the newly formed polar-cap field.

We have already noted how the temporal modulation of the large-scale harmonic
modes comprising the primary and secondary families during polarity reversals
appears similar to that of the magnetic field of the Earth (McFadden et al.
1991; Leonhardt & Fabian 2007). We have illustrated that as the magnitude of
the primary-family mode amplitudes (primarily those of the axisymmetric
odd-$\ell$ modes $B_{\ell}^{0}$) lessen, the secondary-family mode amplitudes
(particularly those of the equatorial dipole $B_{1}^{1}$ and axial quadrupole
$B_{2}^{0}$) simultaneously increase. Once the secondary-family modes have
peaked, the primary-family modes grow as a result of the growing polar-caps.
Such interplay between primary and secondary families provides insight toward
an understanding of the processes at play in the solar dynamo that are assumed
to be responsible for the occurrence of the observed cyclic activity (Tobias
2002).

The presence of power in members of both the primary and secondary families
indicates that the solar dynamo excites modes that are both symmetric and
antisymmetric with respect to the equator. As was demonstrated by Roberts &
Stix (1972), this cannot occur unless nonlinearities exist or unless basic
ingredients of the solar dynamo (such as, for example the $\alpha$ and/or
$\omega$ effects, or the meridional flow) possess some degree of north-south
asymmetry. In light of the the parameter regime in which the solar dynamo is
thought to operate, including large fluid and magnetic Reynolds numbers
$R_{e}$ and $R_{m}$ believed to characterize the solar convection zone (both
of order $10^{12}$–$10^{15}$; Stix 2002; Ossendrijver 2003), one expects the
Sun to possess a nonlinear dynamo. Detailed observations of the magnetic field
in the solar interior where dynamo activity is thought to occur are not
available, but the observed magnetic patterns and evolution provide
circumstantial evidence of turbulent, highly nonlinear processes that lead to
complex local and nonlocal cascades of energy and magnetic helicity (Alexakis
et al. 2005, 2007; Livermore et al. 2010; Strugarek et al. 2012). Yet, the
presence of regular patterns formed by the emergent flux on the solar
photosphere, as codified by Hale’s Polarity Law, Joy’s Law for active-region
tilts, and the approximately regular cycle lengths, suggest that some ordering
is indeed happening in the solar interior.

With the aim of distilling the necessary elements of the various nonlinear
dynamos into a manageable framework, multiple authors have created idealized
models of the solar dynamo, including (for example) Weiss et al. (1984);
Feynman & Gabriel (1990); Ruzmaikin et al. (1992); Knobloch & Landsberg
(1996); Tobias (1997); Knobloch et al. (1998); Melbourne et al. (2001); Weiss
& Tobias (2000); Spiegel (2009). Similarly, for the geodynamo there are many
efforts, including Glatzmaier & Roberts (1995); Heimpel et al. (2005);
Christensen & Aubert (2006); Busse & Simitev (2008); Nishikawa & Kusano
(2008); Takahashi et al. (2008); Christensen et al. (2010). A completely
different approach has been taken by Pétrélis & Fauve (2008) and Pétrélis et
al. (2009), who have developed simplified models of the von Karman sodium
(VKS) laboratory experiment (Monchaux et al. 2007). In all of these idealized
models, the modulations resulting from the coupling between magnetic modes
from the different families, or between the magnetic field and fluid motions
(Tobias 2002) can be analyzed in terms of the equations that describe the
underlying systems. The variability of the most prominent cycle period
develops as a result of the coupling of modes introducing a second time scale
into the dynamo system, often leading to a quasi-periodic or chaotic behavior
of the magnetic field, cycle length, and/or dominant parity. One can further
understand via symmetry considerations how reversals and excursions arise
(Gubbins & Zhang 1993).

###  4.1 Reversals and Coupling between Modes

To illustrate such a dynamical system, following the work of Pétrélis & Fauve
(2008), we assume that the axial dipole and quadrupole modes are nonlinearly
coupled. We can then define a variable $A(t)=B_{1}^{0}(t)+iB_{2}^{0}(t)$,
where we have used the time-varying mode coefficients defined in Equation 1,
and write an evolution equation that satisfies the symmetry invariance found
in the induction equation, i.e., $\boldsymbol{B}\rightarrow-\boldsymbol{B}$.
It then follows that the symmetry $A\rightarrow-A$ must also be satisfied, and
that such a equation to leading order is

$\frac{dA}{dt}=\mu_{1}A+\mu_{2}\bar{A}+\nu_{1}A^{3}+\nu_{2}A^{2}\bar{A}+\nu_{3}A\bar{A}^{2}+\nu_{4}\bar{A}^{3},$
(5)

where $\mu_{i}$ and $\nu_{i}$ are complex coefficients and
$\bar{A}=B_{1}^{0}-iB_{2}^{0}$ is the complex conjugate of $A$, and the
quadratic terms have vanished due to symmetry considerations. As discussed in
Pétrélis & Fauve (2008) and Pétrélis et al. (2009), such dynamical systems are
subject to bifurcations. In particular, they demonstrate that this dynamical
system can be characterized by a saddle-node bifurcation when comparing its
properties with so-called normal form equations (Guckenheimer & Holmes 1982).
In such a bifurcated system, both stable and unstable equilibria (fixed
points) exist, as illustrated in the left panel of Figure 12. For instance, if
the solution lies at a stable point (for example, where the dipole axis is
oriented northward), fluctuations in the system may disturb the equilibrium
and push the magnetic axis away from its stable location. If such fluctuations
are not strong enough, the evolution of the dynamical system resists the
deterministic evolution of the system and the system returns to its original
configuration (in the example, resulting in an excursion of the dipole), such
as seen in the geomagnetic field (Leonhardt & Fabian 2007). If instead the
fluctuations are large enough to push the system past the unstable point, the
magnetic field then evolves toward the opposite stable fixed point allowed by
$\boldsymbol{B}\rightarrow-\boldsymbol{B}$ (in the example, resulting in a
reversal that changes the dipole axis to a southward orientation). Such
behavior is also seen in the VKS experiment, from which is observed irregular
magnetic activity combined with both excursions and reversals. Reversals
result in an asymmetric temporal profile, with the dipole evolving slowly away
from its equilibrium followed by a swift flip (cf., Fig. 3 of Pétrélis et al.
2009).

In Figure 13 we have overplotted the last three sunspot cycle reversals such
that the zero-crossings of the axial dipole coefficients $B_{1}^{0}$ for each
cycle are aligned. It has been recently shown that the 10 major geomagnetic
reversals for which detailed records exist occurring during the past 180 Myr
possess a characteristic shape upon suitable normalization (Valet & Fournier
2012). This shape can be described as comprising a precursory event lasting of
order 2500 yr, a quick reversal not exceeding 1000 yr, and a rebound event of
order 2500 yr. Pétrélis et al. (2009) show that the magnetic field in a
simplified VKS laboratory experiment exhibits differing behavior during
reversals and excursions. During reversals, the magnetic field has an
asymmetric profile that contains a slow decrease in the dipole, followed by a
rapid change of polarity and buildup of the opposite polarity, whereas
excursions are more symmetric. Additionally, after reversals the magnetic
dipole overshoots its eventual value before settling down, whereas during
excursions no such overshooting is measured (see Fig. 3 of Pétrélis et al.
2009). For the solar cases displayed in Figure 13, we find that only the
(green) curve of the reversal of cycle 22 exhibits an overshoot, whereas the
other two cycles do not. Further, the rates at which the solar dynamo
approaches and recovers from the reversal appear to be equal, leading to a
symmetric profile, in contrast with the VKS results. Therefore, the Sun seems
to reverse its magnetic field in a less systematic way than other systems that
have shown such behavior.

Analyzing such systems from a dynamical systems perspective, when changing the
control parameter (here, $R_{m}$) past the bifurcation point, the stable and
unstable points coalesce and merge and the saddle nodes disappear, as shown in
the right panel of Figure 12. This act transforms the system from one
containing fixed points to one containing limit cycles with no equilibria
(e.g., Guckenheimer & Holmes 1982), yielding an oscillatory solution that
manifests itself as cyclic magnetic activity. Typically, large fluctuations
are required in order to put the dynamical system above the saddle-node
bifurcation threshold.

In the case of the Sun, both the primary and secondary families are excited
efficiently and a strong coupling between them is exhibited. The model of
Pétrélis & Fauve (2008), in spirit very close to the studies of Knobloch &
Landsberg (1996) or Melbourne et al. (2001), may be used to guide our
interpretation of the solar data. As illustrated in Figure 11, the
axisymmetric dipole and quadrupole are out of phase, such that their coupling
may lead to global reversals of the solar poloidal field. To the best of our
knowledge, however, the solar dynamo does not exhibit excursions of its
magnetic field (unlike the geodynamo) but instead undergoes fairly regular
reversals that take about one or two years to transpire (cf., Fig. 3). The
solar dynamo is thus better approximated by a model in which a limit cycle is
present. One may presume that the difference between the geodynamo and the
solar dynamo may be a result of the large degree of turbulence present in the
solar convection zone, whereas the mantle of the Earth has a more laminar
convective flow and thus is below the bifurcation threshold where fixed points
are still present.

It may be the case that the solar dynamo is better described by a Hopf
bifurcation, in which a limit cycle arises (branches from a fixed point) as
the bifurcation parameter is changed. The dynamo instability that occurs as a
result of the interaction of magnetic fields and fluid flows (such as
$\alpha\omega$ dynamos typically used to model the Sun, as summarized in
Tobias 2002) often arises from a Hopf bifurcation. This allows the system to
pass through domains having different properties, such as the aperiodic
oscillations that characterize the grand minima and nonuniform sunspot cycle
strengths of the solar dynamo (e.g., Spiegel 2009 and references therein). The
data analysis shown here does not favor a particular type of bifurcation, but
does indicate efficient coupling between the primary and secondary families.

Yet another approach toward investigating magnetic reversals is to develop
detailed numerical simulations solving the full set of MHD equations. Such
three-dimensional numerical simulations in spherical geometry of the Earth’s
geodynamo (Glatzmaier & Roberts 1995; Li et al. 2002; Nishikawa & Kusano 2008;
Olson et al. 2011) or of the solar global dynamo (Brun et al. 2004; Browning
et al. 2006; Racine et al. 2011) have looked at the behavior of the polar
dipole vs multipolar modes. Even though such models have large numerical
resolution and thus possess a large number of modes, all have the property
that the dominant polarity of the magnetic field follows the temporal
evolution of a few low-order modes, even if in some cases the magnetic energy
spectrum peaks at higher angular degree $\ell$. These findings suggest that
the coupling between the primary and secondary family remains an important
factor in characterizing polarity reversals for these simulations, and is
thought to be linked to a symmetry-breaking of the convective flow (Nishikawa
& Kusano 2008; Olson et al. 2011). Some studies of the geomagnetic field
(e.g., Clement 2004) even advocate for a coupling between two modes of the
same primary family, such as the axial dipole and octupole. While in the solar
data these modes are well correlated, the coupling between the primary and
secondary families of modes seems more likely to be at the origin of the
reversal, as demonstrated in §3.6.

Figure 12: Schematic diagrams (adapted from Pétrélis et al. 2009) of a
magnetic dynamo system on either side of a saddle-node bifurcation, with two
distinct polarity configurations represented by $B$ and $-B$. The coordinate
axes represent states where the primary (as represented by the axial dipole
$D$) or secondary (as represented by the axial quadrupole $Q$) families are
dominant. In the left-hand diagram, stable ($\pm B_{s}$) and unstable ($\pm
B_{u}$) states present during the system’s evolution are indicated by blue
circles and red squares. Perturbations away from a stable point can either
cause the system to evolve to the opposite stable configuration (if the
perturbation is strong enough) or simply cause an excursion in which the
system returns to the same stable state. In the right-hand diagram,
corresponding to the same system at a higher magnetic Reynolds number $R_{m}$
the stable and unstable points have merged, and the system simply oscillates
between the two configurations in a limit cycle. Figure 13: The reversals, as
defined by the magnitude of the axial dipole component for WSO, for the past
sunspot Cycles 21–23. The reversal for these three cycles occurred in Oct.
1979, Nov. 1989, and Jul. 1999, respectively.

###  4.2 Mean-Field Dynamo Models and the Axisymmetric Modes

Mean-field dynamo models are found to capture the essence of the large scale
solar dynamo (Moffatt 1978; Ossendrijver 2003; Dikpati et al. 2004;
Charbonneau 2010). At present, the most favored model is the mean-field
Babcock-Leighton (BL) dynamo model (e.g., Dikpati et al. 2004), in which the
mean magnetic induction equation is solved using empirical guidance for both
the differential rotation and meridional circulation profiles, as well as for
parameterizations of the $\alpha$-effect and poloidal-field source terms. In
this section, we use the Stellar Elements (STELEM; T. Emonet & P. Charbonneau,
1998, priv. comm.) code (see Appendix A of Jouve & Brun 2007 for more details)
to solve the axisymmetric BL dynamo equations, and investigate some of the
consequences of the coupling between modes from the primary and secondary
families. In the interest of brevity, we refer interested readers to Appendix
A.1 for a listing of the governing equations associated with BL dynamos.

In many BL solar dynamo models, the parameters governing the imposed flows and
the poloidal-field source terms are chosen based on their solar counterparts.
When carefully chosen these terms favor modes from the primary family and thus
antisymmetric field configurations, since this is what the Sun apparently
favors much of the time. This is a result of the commonly used latitudinal
profiles of the key dynamo ingredients (symmetric large-scale flows and
antisymmetric alpha effect) combined with the parity in the BL mean-field
dynamo equations, leading to a situation where modes of the primary family
remain uncoupled to modes of the secondary family that allows both dynamo
families to coexist without much interaction. We consider the symmetry of the
BL equations used here in Appendix A.2 (see also Roberts & Stix 1972 and
Gubbins & Zhang 1993 for broader discussions on this topic).

To demonstrate these characteristics, we now consider a typical solution of
the standard BL mean-field dynamo as calculated by STELEM. Figure 14 presents
the time evolution of the resulting magnetic field patterns, and is thus
analogous to the standard solar butterfly diagram. Performing a Legendre
transform on the magnetic field reveals the degrees $\ell$ of the dominant
axisymmetric modes. Figure 15 illustrates that the odd $\ell$ modes from the
primary family dominate over the even ones by about five orders of magnitude
in this model. This differs significantly from what is observed on the Sun,
where the amplitude of the quadrupole is measured to be about 25% of the
dipole amplitude for most of the time, becoming dominant only during reversals
[cf., Fig. 4(c)]. The behavior of the standard BL model of Figure 15 arises
because of the symmetry characteristics of the BL dynamo equations. Because
the model was initialized with a dipolar field, no modes from the secondary
family are excited in the standard BL model shown in Figure 15 because no
coupling exists between the primary and secondary families.

If instead the calculation were initialized with a quadrupolar field
(belonging to the secondary family), we find that the system eventually
transitions to a state in which the primary-family modes predominate, as shown
in Figure 16. The growth of the primary-family modes is due to the presence of
a very weak dipole (likely of numerical origin) at the onset of the
simulations. In these models, the BL source term of Equation (A8) quenches the
growth of the magnetic field once a certain threshold is passed, and as a
result the maximum total amplitude of the magnetic field is capped. The reason
why the primary-family modes are preferred stems from the fact that the
thresholds for dynamo action (based on the parameter $C_{s}$ in Equation [A4])
are found empirically to be lower for the dipole than for the quadrupole
($C_{s}\sim 6.12$ vs. $C_{s}\sim 6.25$), meaning the dipole-like modes have a
higher growth rate than the quadrupole-like modes. In this model, only briefly
during the transition phase does the model possess a quadrupole of order 25%
of the dipole, as in the Sun.

Observations of solar photospheric fields, however, indicate that the Sun
excites both families and does not strongly favor members of one family over
the other, a situation that has apparently existed over many centuries. Even
during the Maunder minimum, evidence suggests that this interval may have been
dominated by a hemispherical dynamo with magnetic activity located primarily
in the southern hemisphere (Ribes & Nesme-Ribes 1993), which can only be
formed by a state in which primary- and secondary-family modes possess nearly
equal amplitudes (Tobias 1997; Gallet & Pétrélis 2009). Consequently, the
solutions presented in Figures 15 and 16, in which modes from only one family
are preferred, are thus not a satisfactory model of the Sun.

As advocated by Roberts & Stix (1972) and Gubbins & Zhang (1993) following
their symmetry-based study of the solar dynamo and the induction equation, and
more recently by Nishikawa & Kusano (2008) in their geodynamo simulations, a
north-south asymmetry of the flow field specified in the BL dynamo, or
alternatively an asymmetric poloidal-field source term, may allow the co-
existence of both the primary and secondary families. To investigate this
effect we have performed two additional BL dynamo calculations, one with a BL
source term and one with a meridional circulation that each generate both
symmetric and antisymmetric fields (introduced via the parameter $\epsilon$ of
Equations (A9) and (A13) of Appendix A.1), the combination of which yields
asymmetric magnetic field patterns.

We have run several dynamo cases, with the antisymmetry arising either in the
BL source term or in the meridional flow profile, and with a range of
amplitudes for the $\epsilon$ parameter from $10^{-4}$ to $10^{-1}$. All cases
were initialized with a dipolar field. We find that when $\epsilon$ is about
$10^{-3}$, the modes in secondary family grow until they reach about 35% of
the dominant dipolar mode, as illustrated in Figures 17 and 18. This result
holds true regardless of whether the antisymmetry is introduced in the BL
source term or in the meridional circulation profile, with very little
difference in the resulting mode amplitudes. As expected, using a smaller
$\epsilon$ results in solutions where the primary-family modes dominate, while
using a larger $\epsilon$ yields a state where the secondary-family modes are
comparable to the primary-family modes. Such results may indicate that Sun
need only possess a weak degree of north-south asymmetry in order to behave as
it does.

Figure 14: Latitude-time plots of $B_{\phi}$ (at the tachocline) and $B_{r}$
(at the surface) produced by a mean-field BL dynamo model that uses empirical
guidance for the solar differential rotation and meridional flow profiles, and
that is initialized with a dipolar magnetic field. The lower panel is
analogous to the standard solar butterfly diagram.

##  5 Conclusions

Cycles of magnetic activity in many astrophysical bodies, including the Sun,
Earth, and other stars, are thought to be excited by nonlinear interactions
occurring in their interiors. Yet in some cases, such as the Sun, the cycles
have approximately regular periods and in others, such as the Earth, there is
no apparent periodicity. Dynamo theory indicates such a range of behaviors is
expected, and whether the cycles are regular depends on magnetohydrodynamic
parameters that characterize the system, including fluid and magnetic Reynolds
and Rayleigh numbers. As a consequence, the large-scale appearance of the
magnetic field may provide clues toward the type of dynamo that may be
operating.

In this article, long-term measurements of the solar photospheric magnetic
field are utilized to characterize the waves of dynamo activity that exist
within the interior of the Sun. Synoptic maps from WSO (dating back to 1976)
and MDI (spanning 1995–2010) are used to determine the spherical harmonic
coefficients of the surface magnetic field for the past three sunspot cycles.
We focus on the apparent interactions between various low-order modes
throughout the past three sunspot cycles, and interpret these trends in the
context of dynamo theory.

The multipolar expansions of the solar field as deduced from WSO and MDI data
indicate that the axial and equatorial dipole modes are out of phase. During
activity minima, the dipole component of the solar field is generally aligned
with the axis of solar rotation, while the quadrupole component is much
weaker. During activity maxima, the dipole reverses its polarity with respect
to the rotation axis, and throughout the reversal process there is more energy
in quadrupolar modes than in dipole modes. During the past three cycles, these
reversals have taken place over a time interval of about 2 yr to 3 yr on
average. More indirect measures of solar activity, such as the sunspot number
and proxies of the heliospheric field, seem to indicate that such regular
activity cycles have persisted for at least hundreds of years with a period of
approximately 11 yr. The most recently completed solar cycle (Cycle 23) lasted
for about 13 yr and while unusual, is not unprecedented. We note in passing
that such modulations of the solar dynamo may be interpreted as a type of
nonlinear interaction between the turbulent alpha effect and the field and/or
flows (Tobias 2002).

The harmonic modes can also be grouped into primary and secondary families, a
distinction that depends on the north-south symmetry of the various modes. For
example, the axial dipole harmonic is antisymmetric and is a member of the
primary family. Alternatively, the equatorial dipole and axial quadrupole
modes are both symmetric with respect to the equator and thus are grouped
together in the secondary family. When the evolution of the mode coefficients
are analyzed in this way, we find that there is a trend for members of the
same family to possess the same phasing, suggesting that modes in the same
family of modes are either excited together and/or are more coupled when
compared with modes of different families. This coupling is noticeable during
reversals of the solar dipole, as less energy is present in primary-family
modes than in secondary-family modes during these intervals.

The historical record indicates that the geodynamo also undergoes reversals of
its dipole axis (with respect to the rotation axis), but these reversals occur
much more irregularly than in the solar case. Additionally, the dipole axis of
the terrestrial magnetic field occasionally makes excursions away from the
axis of rotation of the Earth, only to later return without actually
reversing. An examination of the large-scale harmonic modes of the geomagnetic
field during these intervals indicates that the energy contained in secondary-
family modes was significantly smaller during excursions than during
reversals. A strong quadrupole during geodynamo reversals is in line with the
solar behavior; there is no parallel with excursions as excursions in the
solar case have not been observed. Analogous behavior is observed to occur in
the VKS laboratory dynamo with respect to the relative strengths of the
primary and secondary families.

We also examined the coupling of the mode families using a BL mean-field
dynamo model computed using the STELEM code. Because of the symmetries in the
magnetic induction and in the assumed profiles of the large-scale flow fields
and BL source term, we find that the standard mean-field solar dynamo model
results in a state containing largely members of the primary family. This is a
result of the dipole (a primary-family mode) being more unstable to dynamo
action than the quadrupole. With a modest amount of asymmetry, implemented
here either in the meridional flow profile or in the BL source term, we find
from the models that both the primary and secondary families can coexist in
the same model and in the same proportions as in the solar dynamo. This can
lead to a small lag between the northern and southern hemispheres as is
actually observed on the Sun (Dikpati et al. 2007).

M.L.D. acknowledges support by the Lockheed Martin SDO/HMI sub-contract
25284100-26967 from Stanford University (through Stanford University prime
contract NAS5-02139). A.S.B. acknowledges financial support by the European
Research Council through grant 207430 STARS2, and by CNRS/INSU via Programme
National Soleil-Terre. A.S.B. is grateful to Alan Title and LMSAL for their
hospitality. Collection and analysis of WSO and MDI data were supported by
NASA under contracts NNX08AG47G and NNX09AI90G. The authors thank J. Aubert,
S. Fauve, A. Fournier, M. Miesch, F. Pétrélis, E. Spiegel, A. Strugarek, S.
Tobias, J. Toomre and J.-P. Zahn for useful discussions. Facilities: WSO,
SOHO/MDI

Figure 15: Time evolution of the coefficients of the lowest-order harmonic
functions of the surface magnetic field (as grouped by primary and secondary
families) from the same BL dynamo model as shown in Fig. 14. In panel (a) are
shown the evolution of the first several primary-family coefficients
$B_{\ell}^{0}$ with $\ell=$1, 3, 5, and 7 in black, blue, green, and red,
respectively. In panel (b) are shown the evolution of the secondary-family
coefficients $B_{\ell}^{0}$ with $\ell=$2, 4, 6, and 8, respectively in black,
blue, green, and red. Panels (c) and (d) show zoomed-in sections of panels (a)
and (b). Figure 16: Time evolution of the same BL model as shown Fig. 15 (and
using the same color scheme), but initialized with a quadrupolar magnetic
field. Figure 17: Time evolution of the same BL model as shown in Fig. 15 (and
using the same color scheme), but with an asymmetric BL source term as
implemented in Eq. (A9) by setting $\epsilon=10^{-3}$. Figure 18: Time
evolution of the same BL model as shown in Fig. 15 (and using the same color
scheme), but with an asymmetric meridional flow profile as implemented in Eq.
(A13) by setting $\epsilon=10^{-3}$.

##  Appendix A Mean-Field Dynamo Formalism

In this Appendix, we provide details regarding the Babcock-Leighton mean-field
dynamo models discussed in §4.2. The pertinent equations are listed in §A.1
and their symmetry properties are discussed in §A.2.

###  A.1 Mean-Field Equations

Here, we briefly list the equations governing the axisymmetric mean-field
dynamo models calculated in §4.2. A more detailed explanation can be found in,
e.g., Jouve et al. (2008). Following (Moffatt 1978), the mean-field induction
equation is

$\frac{\partial{\left<\boldsymbol{B}\right>}}{\partial
t}=\boldsymbol{\nabla}\boldsymbol{\times}\left(\left<\boldsymbol{V}\right>\boldsymbol{\times}\left<\boldsymbol{B}\right>\right)+\boldsymbol{\nabla}\boldsymbol{\times}\left<\boldsymbol{v^{\prime}}\boldsymbol{\times}\boldsymbol{b^{\prime}}\right>-\boldsymbol{\nabla}\boldsymbol{\times}\left(\eta\boldsymbol{\nabla}\boldsymbol{\times}\left<\boldsymbol{B}\right>\right),$
(A1)

where the variables $\left<\boldsymbol{B}\right>$ and
$\left<\boldsymbol{V}\right>$ refer to the mean parts of the magnetic and
velocity fields, and $\boldsymbol{v^{\prime}}$ and $\boldsymbol{b^{\prime}}$
to their respective fluctuating components. The function $\eta$ is the
magnetic diffusivity and is not necessarily a constant. The terms “mean” and
“fluctuating” refer to the fact that a separation of scales has been
performed, such that the mean quantities are computed by averaging over some
appropriate intermediate size scale and the fluctuating quantities are the
residuals.

Working in spherical coordinates $(r,\theta,\phi)$ and under the assumption of
axisymmetry, we perform a poloidal-toroidal decomposition and write the mean
magnetic field $\boldsymbol{B}$ and mean velocity field $\boldsymbol{V}$ (for
clarity the angle brackets $\langle$ and $\rangle$ are omitted going forward)
as

$\displaystyle\boldsymbol{B}(r,\theta,t)$
$\displaystyle=\boldsymbol{\nabla}\boldsymbol{\times}\left(A_{\phi}\boldsymbol{\hat{e}}_{\phi}\right)+B_{\phi}\boldsymbol{\hat{e}}_{\phi}$
(A2) $\displaystyle\boldsymbol{V}(r,\theta)$
$\displaystyle=\boldsymbol{v_{p}}+\Omega\,r\sin\theta\,\boldsymbol{\hat{e}}_{\phi},$
(A3)

where the poloidal streamfunction $A_{\phi}(r,\theta,t)$ and toroidal field
$B_{\phi}(r,\theta,t)$ are used to generate $\boldsymbol{B}$. The velocity
field is time-independent, and is prescribed by profiles for the meridional
circulation $\boldsymbol{v_{p}}(r,\theta)$ and differential rotation
$\Omega(r,\theta)$.

Rewriting the mean induction equation (A1) in terms of $A_{\phi}$ and
$B_{\phi}$, we arrive at two coupled partial differential equations for
$A_{\phi}$ and $B_{\phi}$,

$\displaystyle\frac{\partial{A_{\phi}}}{\partial t}$
$\displaystyle=\frac{\eta}{\eta_{t}}\left(\nabla^{2}-\frac{1}{\varpi^{2}}\right)A_{\phi}-R_{e}\frac{\boldsymbol{v_{p}}}{\varpi}\boldsymbol{\cdot}\boldsymbol{\nabla}\left(\varpi
A_{\phi}\right)+C_{s}S$ (A4) $\displaystyle\frac{\partial{B_{\phi}}}{\partial
t}$
$\displaystyle=\frac{\eta}{\eta_{t}}\left(\nabla^{2}-\frac{1}{\varpi^{2}}\right)B_{\phi}+\frac{1}{\varpi}\frac{\partial(\varpi
B_{\phi})}{\partial r}\frac{\partial(\eta/\eta_{t})}{\partial
r}-R_{e}\varpi\boldsymbol{v_{p}}\boldsymbol{\cdot}\boldsymbol{\nabla}\left(\frac{B_{\phi}}{\varpi}\right)-R_{e}B_{\phi}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v_{p}}+C_{\Omega}\varpi\left[\boldsymbol{\nabla}\boldsymbol{\times}(A_{\phi}\boldsymbol{\hat{e}}_{\phi})\right]\boldsymbol{\cdot}\boldsymbol{\nabla}\Omega,$
(A5)

where $\varpi=r\sin\theta$. The contribution to the transport term in the mean
induction equation (A1) that arises from the fluctuating fields, namely the
$\boldsymbol{\nabla}\boldsymbol{\times}\left<\boldsymbol{v^{\prime}}\times\boldsymbol{b^{\prime}}\right>$
term, is present in the $A_{\phi}$ equation above and in general is assumed to
take a specific form in terms of the mean magnetic field (cf., Babcock 1961;
Leighton 1969; Wang & Sheeley 1991; Dikpati & Charbonneau 1999; Jouve & Brun
2007). Here, we use a surface Babcock-Leighton (BL) term
$S(r,\theta,B_{\phi})$ for this purpose which serves to generate new poloidal
field.

Additionally, equations (A4) and (A5) have been nondimensionalized by using
$R_{\odot}$ as the characteristic length scale and $R_{\odot}^{2}/\eta_{t}$ as
the characteristic time scale, where $\eta_{t}=10^{11}$ cm2 s-1 is
representative of the turbulent magnetic diffusivity in the convective zone.
This rescaling leads to the appearance of three dimensionless control
parameters $C_{\Omega}=\Omega_{0}R_{\odot}^{2}/\eta_{t}$,
$C_{s}=s_{0}R_{\odot}/\eta_{t}$, and the Reynolds number
$R_{e}=v_{0}R_{\odot}/\eta_{t}$, where $\Omega_{0}$, $s_{0}$, and $v_{0}$ are
respectively the rotation rate and the typical amplitude of the surface source
term and of the meridional flow.

Equations (A4) and (A5) are solved with the Stellar Elements (STELEM; T.
Emonet & P. Charbonneau, 1998, priv. comm.) code (see Appendix A of Jouve &
Brun 2007 for more details) in an annular meridional plane with the colatitude
$\theta$ $\in[0,\pi]$ and the dimensionless radius $r\in[0.6,1]$, i.e., from
slightly below the tachocline ($r\approx 0.7$) up to the solar surface
$R_{\odot}$. The STELEM code has been thoroughly tested and validated via an
international mean field dynamo benchmarking process involving 8 different
codes (Jouve et al. 2008). At the latitudinal boundaries at $\theta=0$ and
$\theta=\pi$, and at the lower radial boundary at $r=0.6$, both $A_{\phi}$ and
$B_{\phi}$ vanish. At the upper radial boundary at $r=1$, the solution is
matched to an external potential field. Usual initial conditions involve
setting a confined dipolar field configuration, i.e. $A_{\phi}$ is set to
$(\sin\theta)/r^{2}$ in the convective zone and to $0$ below the tachocline.
To create the simulation shown in Figure 16, the simulation was initialized
using a quadrupolar configuration with an $A_{\phi}$ of
$(3\cos\theta\,\sin\theta)/(2r^{3})$ in the convection zone. In both cases,
the toroidal field is initialized to 0 everywhere.

The rotation profile used in the series of models discussed in this work
captures many aspects of the true solar angular velocity profile, such as
deduced from helioseismic inversions (Thompson et al. 2003). We thus assume
solid-body rotation below $r=0.66$ and a differential rotation above this
tachocline interface as given by the following rotation profile,

$\Omega(r,\theta)=\Omega_{c}+\frac{1}{2}\left[1+\text{erf}\left(\frac{2(r-r_{c})}{d_{1}}\right)\right](\Omega_{\text{eq}}+a_{2}\cos^{2}\theta+a_{4}\cos^{4}\theta-\Omega_{c}).$
(A6)

The parameters $\Omega_{\text{eq}}=1$, $\Omega_{c}=0.93944$, $r_{c}=0.7$,
$d_{1}=0.05$, $a_{2}=-0.136076$ and $a_{4}=-0.145713$. With this profile for
$\Omega$, the radial shear is maximal at the tachocline.

We assume that the diffusivity in the envelope $\eta$ is dominated by its
turbulent contribution, whereas in the stable interior $\eta_{c}\ll\eta_{t}$.
We smoothly match the two different constant values with an error function
which enables us to continuously transition from $\eta_{c}$ to $\eta_{t}$,

$\eta(r)=\eta_{c}+\frac{\left(\eta_{t}-\eta_{c}\right)}{2}\left[1+\text{erf}\left(\frac{r-r_{c}}{d}\right)\right],$
(A7)

with ${\eta_{\rm c}}=10^{9}\,\rm cm^{2}\rm s^{-1}$ and $d=0.03$.

In BL flux-transport dynamo models, the poloidal field owes its origin to the
tilt of magnetic loops emerging at the solar surface. Thus, the source has to
be confined to a thin layer just below the surface and since the process is
fundamentally non-local, the source term depends on the variation of
$B_{\phi}$ at the base of the convection zone. We use the following expression
(which is a slightly modified version of that used in Jouve & Brun 2007) in
order to better confine the activity belt to low latitudes:

$S(r,\theta,B_{\phi})=\frac{1}{2}\left[1+\text{erf}\left(\frac{r-r_{2}}{d_{2}}\right)\right]\left[1-\text{erf}\left(\frac{r-1}{d_{2}}\right)\right]\left[1+\left({\frac{B_{\phi}(r_{c},\theta,t)}{B_{0}}}\right)^{2}\right]^{-1}\cos\theta\,\sin^{3}\theta\,B_{\phi}(r_{c},\theta,t),$
(A8)

where $r_{2}=0.95$, $d_{2}=0.01$, $B_{0}=10^{5}$G. In the particular case of
an imposed asymmetry between the north and southern hemisphere (Fig. 17) we
introduce a modified source term, modulated by the parameter $\epsilon$, as
follows,

$S_{\text{asym}}(r,\theta,B_{\phi})=\frac{1}{2}\left[1+\text{erf}\left(\frac{r-r_{2}}{d_{2}}\right)\right]\left[1-\text{erf}\left(\frac{r-1}{d_{2}}\right)\right]\left[1+\left({\frac{B_{\phi}(r_{c},\theta,t)}{B_{0}}}\right)^{2}\right]^{-1}(\cos\theta+\epsilon\sin\theta)\sin^{3}\theta\,B_{\phi}(r_{c},\theta,t).$
(A9)

In BL flux-transport dynamo models, meridional circulation is used to link the
two sources of the magnetic field, namely the base of the convection zone
(where toroidal field is created via the latitudinal shear) and the solar
surface (where poloidal field is introduced via the BL source term). In the
series of models discussed in this paper, the meridional circulation is
equatorially symmetric, having one large single cell per hemisphere. Flows are
directed poleward at the surface and equatorward at depth (as in the Sun),
vanishing at the bottom boundary at $r=0.6$. The equatorward branch penetrates
slightly beneath the tachocline. To model the single cell meridional
circulation we consider a stream function with the following expression (Jouve
et al. 2008),

$\psi(r,\theta)=-\frac{2(r-r_{b})^{2}}{\pi(1-r_{b})}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)\cos\theta\sin\theta,$
(A10)

which gives, through the relation
$\boldsymbol{v_{p}}=\boldsymbol{\nabla}\boldsymbol{\times}(\psi\boldsymbol{\hat{e}_{\phi}})$,
the following components of the meridional flow,

$\displaystyle v_{r}$ $\displaystyle=-\frac{2(1-r_{b})}{\pi
r}\frac{(r-r_{b})^{2}}{(1-r_{b})^{2}}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)(3\cos^{2}\theta-1)$
(A11) $\displaystyle v_{\theta}$
$\displaystyle=\left[\frac{3r-r_{b}}{1-r_{b}}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)+\frac{r\pi}{1-r_{b}}\frac{(r-r_{b})}{(1-r_{b})}\cos\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)\right]\frac{2(1-r_{b})}{\pi
r}\frac{(r-r_{b})}{(1-r_{b})}\cos\theta\sin\theta,$ (A12)

with $r_{b}=0.6$.

To introduce asymmetry into the model, an alternative to using the asymmetric
source term of equation (A9) is to introduce an asymmetry into the meridional
flow profile. Such an asymmetric meridional flow profile can be constructed
using the following stream function,

$\psi_{\text{asym}}(r,\theta)=-\frac{2(r-r_{b})^{2}}{\pi(1-r_{b})}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)(\cos\theta+\epsilon\sin\theta)\sin\theta,$
(A13)

which leads to the following components of the meridional flow,

$\displaystyle v_{r,\text{asym}}$ $\displaystyle=-\frac{2(1-r_{b})}{\pi
r}\frac{(r-r_{b})^{2}}{(1-r_{b})^{2}}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)(3\epsilon\sin\theta\cos\theta+3\cos^{2}\theta-1)$
(A14) $\displaystyle v_{\theta,\text{asym}}$
$\displaystyle=\left[\frac{3r-r_{b}}{1-r_{b}}\sin\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)+\frac{r\pi}{1-r_{b}}\frac{(r-r_{b})}{(1-r_{b})}\cos\left(\frac{\pi(r-r_{b})}{1-r_{b}}\right)\right]\frac{2(1-r_{b})}{\pi
r}\frac{(r-r_{b})}{(1-r_{b})}(\cos\theta+\epsilon\sin\theta)\sin\theta,$ (A15)

again with $r_{b}=0.6$.

###  A.2 Symmetry Considerations

Following Gubbins & Zhang (1993) it is straightforward to assess symmetry
properties of various mathematical operators and equations. We adopt the
superscripts A and S to indicate whether scalars or vectors are antisymmetric
or symmetric across the equator, respectively. For example, products between a
scalar and a vector of the form $a\boldsymbol{F}=\boldsymbol{G}$, where $a$
and $\boldsymbol{F}$ are of like symmetry, yield a symmetric result
($a^{S}\boldsymbol{F}^{S}\rightarrow\boldsymbol{G}^{S}$ and
$a^{A}\boldsymbol{F}^{A}\rightarrow\boldsymbol{G}^{S}$), whereas products
between quantities of differing symmetries are antisymmetric
($a^{A}\boldsymbol{F}^{S}\rightarrow\boldsymbol{G}^{A}$ and
$a^{S}\boldsymbol{F}^{A}\rightarrow\boldsymbol{G}^{A}$). For the vector cross
product $\boldsymbol{F}\boldsymbol{\times}\boldsymbol{G}=\boldsymbol{H}$, when
the two vectors $\boldsymbol{F}$ and $\boldsymbol{G}$ have the same symmetry
properties the result will be antisymmetric
($\boldsymbol{F}^{S}\boldsymbol{\times}\boldsymbol{G}^{S}\rightarrow\boldsymbol{H}^{A}$
and
$\boldsymbol{F}^{A}\boldsymbol{\times}\boldsymbol{G}^{A}\rightarrow\boldsymbol{H}^{A}$),
while the cross product between two vectors having opposing symmetries will
yield a symmetric result
($\boldsymbol{F}^{A}\boldsymbol{\times}\boldsymbol{G}^{S}\rightarrow\boldsymbol{H}^{S}$).
Additionally, the curl operator reverses symmetry
($\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{G}^{A}\rightarrow\boldsymbol{H}^{S}$
and
$\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{G}^{S}\rightarrow\boldsymbol{H}^{A}$),
while the Laplacian operator preserves symmetry
($\nabla^{2}\boldsymbol{F}^{S}\rightarrow\boldsymbol{H}^{S}$ and
$\nabla^{2}\boldsymbol{F}^{A}\rightarrow\boldsymbol{H}^{A}$).

With these properties established, the analysis of the symmetry properties of
the magnetic induction equation,

$\frac{\partial\boldsymbol{B}}{\partial
t}=\boldsymbol{\nabla}\boldsymbol{\times}(\boldsymbol{V}\boldsymbol{\times}\boldsymbol{B})+\eta\nabla^{2}\boldsymbol{B},$
(A16)

follows in a straightforward manner. For cases possessing a symmetric velocity
field $\boldsymbol{V}^{S}$ with respect to the equator, both terms on the
right-hand side of equation (A16) preserve the symmetry of $\boldsymbol{B}$.
Thus, a dynamo having only a symmetric field $\boldsymbol{B}^{S}$ will remain
symmetric over time, since both the transport term and the diffusion term of
equation (A16) generate symmetric field only. Likewise, a dynamo possessing an
antisymmetric field $\boldsymbol{B}^{A}$ will preserve its antisymmetry over
time. Because equation (A16) is linear in $\boldsymbol{B}$, it follows that a
magnetic field possessing mixed symmetry in the midst of a symmetric velocity
field can be considered to be operating two independent, noninteracting
dynamos: one that is symmetric and one that is antisymmetric.

However, in cases with an antisymmetric velocity field $\boldsymbol{V}^{A}$,
the transport term on the right-hand side of equation (A16) provides a
mechanism by which the symmetric and antisymmetric modes of $\boldsymbol{B}$
can couple. This coupling arises because an initially symmetric field
$\boldsymbol{B}^{S}$ will first generate an antisymmetric field that in turn
leads to the presence of both antisymmetric and symmetric fields, according to
the right-hand side of equation (A16). Analogously, initializing with a purely
antisymmetric field $\boldsymbol{B}^{A}$ will generate fields of mixed
symmetry over time.

This analysis procedure can further be applied to the mean-field induction
equation. For example an analysis of equation (A1) with an $\alpha$-$\omega$
dynamo (i.e., where
$\left<\boldsymbol{v}^{\prime}\boldsymbol{\times}\boldsymbol{b}^{\prime}\right>$
is set to $\alpha\left<\boldsymbol{B}\right>$),

$\frac{\partial{\left<\boldsymbol{B}\right>}}{\partial
t}=\boldsymbol{\nabla}\boldsymbol{\times}\left(\left<\boldsymbol{V}\right>\boldsymbol{\times}\left<\boldsymbol{B}\right>+\alpha\left<\boldsymbol{B}\right>\right)-\eta\nabla^{2}\left(\left<\boldsymbol{B}\right>\right)$
(A17)

indicates that, for an assumed symmetric mean velocity field
$\left<\boldsymbol{V}\right>^{S}$ and an antisymmetric alpha effect
$\alpha^{A}$ (which is the natural outcome of helical turbulence in a rotating
fluid), such a mean-field dynamo will preserve the symmetry (or antisymmetry)
of the initial fields. Hence, as pointed out by Roberts & Stix (1972) and
McFadden et al. (1991), a symmetric mean velocity field and an antisymmetric
alpha effect do not couple magnetic field modes belonging to different
families. Alternatively, if instead an antisymmetric mean flow
$\left<\boldsymbol{V}\right>^{A}$ or a symmetric alpha effect $\alpha^{S}$ are
considered, this now enables a coupling between symmetric and antisymmetric
mean fields.

In a similar vein, the BL equations (A4) and (A5), which are determined by
performing the poloidal-toroidal decomposition on equation (A17), can also be
analyzed for symmetry. It is important to note that, by equation (A2), the
poloidal streamfunction $\left<A_{\phi}\right>$ has a symmetry opposite to
that of the mean magnetic field $\left<\boldsymbol{B}\right>$ it generates
(and thus also to the corresponding toroidal field $\left<B_{\phi}\right>$).
We established above that the diffusion term in the mean-field equation (A17)
preserves the symmetry of $\left<\boldsymbol{B}\right>$, and so it follows
that the corresponding diffusion terms in the BL dynamo equations (A4) and
(A5) will serve to preserve the symmetries $A_{\phi}$ and $B_{\phi}$.
Likewise, because the large-scale transport term
$\boldsymbol{\nabla}\boldsymbol{\times}\left(\left<\boldsymbol{V}\right>\boldsymbol{\times}\left<\boldsymbol{B}\right>\right)$
term in equation (A17) preserves the symmetry of $\left<\boldsymbol{B}\right>$
whenever $\left<\boldsymbol{V}\right>$ is symmetric, it follows that the
analogous terms in the equations (A4) and (A5) also preserve the symmetry of
the system as long as the imposed velocity field is symmetric. For the BL
dynamo considered here, an antisymmetric poloidal velocity streamfunction
$\psi^{A}$, as in equation (A10), yields a symmetric meridional flow profile
$\boldsymbol{v_{p}}^{S}$, since
$\boldsymbol{v_{p}}=\boldsymbol{\nabla}\boldsymbol{\times}(\psi\boldsymbol{\hat{e}_{\phi}})$,
which in turn gives a symmetric mean velocity
$\left<\boldsymbol{V}\right>^{S}$ from equation (A3). Therefore, the imposed
velocity field as defined by equations (A3) and (A10) will preserve the
symmetries of $A_{\phi}$ and $B_{\phi}$. Lastly, the source term $S$ as
defined by equation (A8) also preserves the symmetry of $A_{\phi}$, since it
is comprised of a series of symmetric coefficients multiplied by
$\cos\theta\,B_{\phi}$. Therefore, an antisymmetric toroidal field implies a
symmetric source term that in turn serves to preserve the symmetry of
$A_{\phi}$ (and thus $\left<\boldsymbol{B}\right>$), and the same is true when
the toroidal field is symmetric.

For these reasons, the dynamo whose characteristics are illustrated in Figure
15, which was initialized with a dipolar field (which is antisymmetric),
preserves its antisymmetry with time since all of the terms in equations (A4)
and (A5) preserve the initial symmetry. Indeed, the amplitude of the
secondary-family modes (which are symmetric) remain low in this model, as
shown in Figures 15(b) and (d). In the dynamos whose characteristics are
displayed in Figures 17 or 18, this effect is responsible for the growth of
symmetric mean fields, even though both models were initialized with the same
antisymmetric mean magnetic field.

It is therefore a direct outcome of symmetry considerations that in standard
mean-field dynamo models either one or the other families of magnetic fields
is excited. In the experiments discussed earlier in §4.2, we controlled the
degree to which the symmetries were mixed via the parameter $\epsilon$ in
equation (A9) and in equations (A14) and (A15), which led to the dynamos
illustrated in Figures 17 or 18, respectively. In both cases, $\epsilon$ was
chosen to yield a dynamo where the end state contained secondary-family
amplitudes of about 25%, as is observed on the Sun; other choices of
$\epsilon$ will lead to end states with different ratios.

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