2023

# Observationally guided models for the solar dynamo and the role of the
surface field

Robert H. Cameron  Manfred Schüssler  Max Planck Institute for Solar System
Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany

###### Abstract

Theoretical models for the solar dynamo range from simple low-dimensional “toy
models” to complex 3D-MHD simulations. Here we mainly discuss appproaches that
are motivated and guided by solar (and stellar) observations. We give a brief
overview of the evolution of solar dynamo models since 1950s, focussing upon
the development of the Babcock-Leighton approach between its introduction in
the 1960s and its revival in the 1990s after being long overshadowed by mean-
field turbulent dynamo theory. We summarize observations and simple
theoretical deliberations that demonstrate the crucial role of the surface
fields in the dynamo process and and give quantitative analyses of the
generation and loss of toroidal flux in the convection zone as well as of the
production of poloidal field resulting from flux emergence at the surface.
Furthermore, we discuss possible nonlinearities in the dynamo process
suggested by observational results and present models for the long-term
variability of solar activity motivated by observations of magnetically active
stars and the inherent randomness of the dynamo process.

###### keywords:

solar activity, solar cycle, dynamo

##  1 Introduction

Studies of solar and stellar dynamos face a problem of utter complexity, i.e.,
the interaction of turbulent convection with rotation and magnetic field in a
highly stratified medium, covering wide ranges of spatial and temporal scales.
Attempts to directly attack this problem with numerical 3D-MHD simulations
have made significant progress in recent years (e.g., Charbonneau, 2020;
Browning and etal, 2023), but are still severely limited in spatial
resolution, so that a faithful representation of the solar cycle has not been
achieved so far. All other approaches to the dynamo problem resort to
simplifications in order to obtain a tractable task. They largely rely on the
surprising amount of regularity shown by the solar cycle, e.g., Hale’s
polarity rules and the systematic tilt of sunspot groups (Joy’s law), the
butterfly diagram, and the regular reversals of the global dipole field
(Hathaway, 2015). These regularities suggest simplified concepts, such as the
generation of azimuthal (toroidal) magnetic flux through winding of meridional
(poloidal) magnetic field lines by differential rotation (Cowling, 1953) or
the formation of bipolar sunspot groups by the emergence of buoyantly rising
magnetic flux tubes (Parker, 1955b).

A variety of models addressing various aspects of the dynamo problem has been
developed. These range from rather ad-hoc “toy models” over sophisticated two-
scale approaches, pioneered by the model of Parker (1955a) and by mean-field
theory of turbulent MHD flows (see review by Brandenburg et al, 2023) to 2D/3D
flux-transport dynamo models including various physical processes considered
to be relevant for the dynamo (see reviews by Charbonneau, 2020; Hazra et al,
2023).

The simplest “model” in the literature is probably due to Barnes et al (1980).
These authors considered a digital narrow-band filtering of Gaussian white
noise, corresponding to a randomly disturbed periodic signal that could be
illustrated by a “pendulumn pelted with peas” (Yule, 1927). The results of
their short computer program (15 lines!) exhibit a very similar kind of
variability as shown by the sunspot record, including the occurence of
extended periods of low activity (grand minima). The important lesson from
this result for models of the solar dynamo is that even a striking similarity
of their output with records of solar activity alone does not necessarily
imply the validity of a model.

In this paper, we focus upon models of the solar dynamo as well as for the
cyclic generation and removal of magnetic flux that are guided by solar (and
also stellar) observations. The motivation for such approaches comes from the
fact that the very complex interactions in the convection zone can lead to
comparably simple results, such as the stable differential rotation in
latitude, the meridional circulation, the polarity rules of sunspot groups, or
the butterfly diagram of magnetic flux emergence. This might perhaps be
compared to the flow of water in a watermill: although the detailed turbulent
flow pattern is utterly complex, the general result is simple: water flows
down the potential well and faithfully drives the wheel. The paradigm of
observationally motivated models is the scenario of Babcock (1961), which was
later extended and put in a mathematical form by Leighton (1969). These
seminal papers paved the way for what are now generally called Babcock-
Leighton-type models of the solar dynamo.

The plan of this paper is as follows. In Sec. 2 we describe the principles of
the Babock-Leighton scenario and review the evolution of dynamo models until
the end of the 1980s, when mounting empirical evidence led to the renaissance
of Babcock-Leighton dynamo models after an extended period of near oblivion.
The generation and removal of toroidal and poloidal magnetic flux in the Sun
and the crucial role of the observable surface field are discussed in Sec. 3.
Nonlinearity of the dynamo process, predictability, and models for the long-
term variability of solar activity are considered in Sec. 4. Sec. 5 gives a
brief outlook.

##  2 The Babcock-Leighton approach and the evolution of models for the solar
dynamo

The seminal paper of Babcock (1961) lays out a purely observationally based
scenario for the 11/22-year solar cycle. In a first step, the poloidal
magnetic flux connected to the global dipole field is wound up by latitudinal
differential rotation in the convection zone. This generates oppositely
directed subsurface belts of toroidal field in both hemispheres. Subsequent
instability, buoyant rise, and emergence of toroidal flux at the surface
produces bipolar magnetic regions (BMRs) and sunspot groups in accordance with
Hale’s polarity rules. These regions are observed to be systematically tilted
in the sense that the magnetic polarity leading in the direction of rotation
is nearer to the equator than the following polarity. This tilt, together with
the dispersal of the BMRs in the course of time, leads to preferred transport
of leading-polarity flux over the equator, so that a surplus of the respective
following-polarity flux builds up in both hemispheres. Poleward transport of
this flux by convection and meridional circulation leads to the reversal of
the polar fields and the buildup of an oppositely directed dipole field, a
process already suggested by Babcock and Babcock (1955). This entails the
generation and emergence of a reversed toroidal field followed by another
reversal of the dipole field, thus giving rise to the 22-year magnetic cycle.

Babcock (1961) already noted that the systematic tilt of the sunspot groups
could be “… the result of Coriolis forces, which induce a vorticity in the
whole of the fluid associated with a BMR when it rises to the surface.” The
Coriolis effect is also invoked in the concept of “cyclonic convection” of
Parker (1955a) and in the mean-field scheme based on turbulence theory
pioneered by Steenbeck and Krause (1966) and Steenbeck et al (1966). Possibly
since both these approaches focus on the collective effect of a small-scale
process (in the sense of a two-scale approach), neither Parker nor Steenbeck
and coworkers apparently realized that the systematic tilt provides direct
observational support for the action of the Coriolis effect as well a
quantitative measure.

Leighton (1964) added a key component to Babcock’s scenario in the form of a
random-walk model for the quasi-diffusive transport of magnetic flux on the
solar surface by supergranular convection. Subsequently, he put the Babcock
model in the mathematical form of a one-dimensional dynamo equation (Leighton,
1969). It is not obvious why this Babcock-Leighton (BL) model fell into near
oblivion for about two decades thereafter. In order to provide some
understanding for this development, a brief sketch of the evolution of solar
dynamo studies from the early 1970s onward is given in what follows.

Perhaps since the mathematically involved mean-field theory was considered to
provide a strong theoretical footing, models for the solar dynamo until the
1990s mostly relied on a mean-field turbulent “$\alpha$-effect” based on
correlations between small-scale fields operating within the convection zone.
To explain the equatorward-directed migration of the activity belts shown by
the butterfly diagram, these models generally required an inwardly increasing
rotation rate, $d\Omega/dr<0$ (Köhler, 1973; Yoshimura, 1975). The effect of
the observed strong latitudinal differential rotation was thought to be of
secondary importance. Consequently, both key processes for the turbulent
dynamo, radial differential rotation and $\alpha$-effect, were assumed to
operate in the solar interior, thus inaccessable to observation at that time.
Therefore, the corresponding model parameters were largely unconstrained. The
observable evolution of the surface field was seen as a mere epiphenomenon of
a dynamo process operating deeply hidden in the convection zone and served
solely as an observational constraint for the models.

The foundations of this approach were undermined when helioseismology made it
possible to measure the differential rotation in the convection zone. It
turned out that (except for a shallow near-surface shear layer) the increase
of the rotation rate with depth required by these models is absent in the bulk
of the convection zone (see review by Howe, 2009). On the other hand, the
rotation rate at the bottom of the convection zone showed a steep radial
gradient across the “tachocline” (Brown et al, 1989), such that $d\Omega/dr<0$
in solar latitudes $\geq 45\,$deg and $d\Omega/dr>0$ in lower latitudes. Some
years before, Galloway and Weiss (1981) already had suggested that the solar
dynamo should work in a stably stratified layer of overshooting convection
below the bottom of the convection zone in order to avoid the rapid buoyant
loss of toroidal magnetic flux inferred by Parker (1975). These results led to
the concept of turbulent dynamo action within the overshoot layer/tachocline
(e.g., Rüdiger and Brandenburg, 1995) or near its interface with the
convection zone proper (Parker, 1993), where a properly chosen combination of
the signs of $\alpha$-effect and radial gradient of the rotation rate could
provide the correct conditions for equatorward propagating dynamo waves and
activity belts (e.g., Tobias, 1996; Charbonneau and MacGregor, 1997). This
approach was complemented by studies of equilibrium, stability, and dynamics
of thin magnetic flux tubes starting with Spruit and van Ballegooijen (1982),
Choudhuri and Gilman (1987), and Moreno-Insertis et al (1992). Such studies
(see reviews by Fan, 2021; Isik and etal, 2023) indicated that the toroidal
field in the overshoot region must be amplified to about $10^{5}\,$G before
becoming unstable (Ferriz-Mas and Schüssler, 1993, 1995) and rising to the
surface to form bipolar magnetic regions within the latitude range shown by
the butterfly diagram (Schüssler et al, 1994). Twisting of the rising flux
tubes due to the Coriolis force then leads to tilt angles consistent with
observation (Fan et al, 1994; Caligari et al, 1995).

Although the combination of tachocline/overshoot layer dynamos and the
dynamics of thin flux tubes offered a comprehensive picture from the
generation of magnetic flux up to its emergence at the surface, these models
relied on a number of untested assumptions and simplifications, so that their
predictive power remained rather limited. In the course of time, a number of
theoretical considerations and observational results cast severe doubt upon
the validity of this aproach:

  * •

The total energy energy of a toroidal field of $10^{5}\,$Gauss at the bottom
of the convection zone would be comparable to the energy in the differential
rotation of the tachocline (Rempel, 2006). However, a corresponding strong
variation of the differential rotation in the lower convection zone and
tachocline in the course of the 11-year activity cycle is not observed (Basu
and Antia, 2019). Moreover, the interface between radiative core and
convection zone cannot support much shear stress (Spruit, 2011). Consequently,
the radial differential rotation in the tachocline (mainly reflecting the
transition from latitudinally differential rotation in the convection zone to
nearly rigid rotation below) cannot generate a sizeable amount of toroidal
magnetic flux unless being maintained by a very powerful downward transport of
angular momentum.

  * •

The latitudinal differential rotation is sufficient to create the total
toroidal flux covered by the emergence of bipolar magnetic regions and sunspot
groups. At the same time, the contribution of the observed radial differential
rotation in the convection zone is only a few percent of that of the
latitudinal differential rotation (Cameron and Schüssler, 2015). Moreover, the
radial shear in the tachocline is strong in high latitudes and weak in low
latitudes, thus unfavorable for toroidal flux generation in low latitudes.

  * •

Helioseismology indicates that the overshoot layer is much more strongly
subadiabatic than assumed in the models for the storage and stability of
toroidal flux (Christensen-Dalsgaard et al, 2011). On the other hand, part of
the lower convection zone could be subadiabatically stratified (e.g., Spruit,
1997; Hotta, 2017).

  * •

Observations of magnetically active stars show that partly and fully
convective stars follow the same activity-rotation law (Wright and Drake,
2016; Reiners et al, 2022), thus questioning the relevance of convective
overshoot and the existence of a tachocline for the dynamo process.
Furthermore, even very cool, fully convective dwarfs (beyond spectral type M7)
can exhibit activity cycles (Route, 2016).

  * •

3D-MHD simulations demonstrate the formation of super-equipartion magnetic
flux concentrations and buoyantly rising flux loops within a simulated
convection zone without overshoot and tachocline (Nelson et al, 2014; Fan and
Fang, 2014; Chen et al, 2017).

  * •

If emerged magnetic structures were “anchored” near to the bottom of the
convection zone, they should show slower rotation than that of the surface
plasma below about $\pm 30\,$deg latitude according to the helioseismically
determined rotation profile. However, magnetic structures are observed rotate
faster than the plasma at the surface (e.g., Howard, 1996).

A new twist came when observations of a systematic poleward-directed
meridional flow at the surface pioneered by Duvall (1979) and Howard (1979)
became established during the 1980s (see review by Hanasoge, 2022). This led
to the suggestion that the associated deep return flow within the convection
zone could transport toroidal flux equatorward. Models of flux-transport
dynamos (FTDs) that rely upon this concept can provide a butterfly diagram
consistent with observations regardless of the sign of the radial gradient of
the rotation rate (Wang and Sheeley, 1991; Durney, 1995; Choudhuri et al,
1995). A recent review of FTD models has been provided by Hazra et al (2023).

The confirmation of the systematic poleward surface flow also sparked the
development of simulation models for the transport of magnetic flux at the
solar surface on solar-cycle time scales, pioneered by the NRL group (DeVore
et al, 1984; Wang et al, 1989b). Such Surface Flux Transport (SFT) models (see
reviews by Mackay and Yeates, 2012; Yeates et al, 2023) assume passive
transport of vertically (radially) orientated magnetic flux by surface flows.
Comparison of SFT simulations with observed synoptic magnetograms shows that
the evolution of the surface flux can be faithfully described by flux
emergence in systematically tilted bipolar magnetic regions followed by
passive flux transport by differential rotation, meridional flow, and
supergranular flows (the latter mostly being treated as a diffusion-like
random walk, cf. Leighton, 1964), and flux cancellation. In particular, the
models confirm the buildup of polar fields due to the preferred transport of
following-polarity flux toward the poles (Wang et al, 1989a).

The success of the SFT simulations led to the re-appraisal of the BL approach
for the (re)generation of the poloidal field in the course of the dynamo
process (Giovanelli, 1985; Wang and Sheeley, 1991), typically in connection
with equatorward flux transport by meridional flow in the convection zone as
Flux Transport Dynamo (FTD) models (Wang et al, 1991; Charbonneau, 2020; Hazra
et al, 2023). In the course of time, further lines of evidence for the
validity of the BL approach and the crucial role of the surface flux for the
solar dynamo became apparent:

  * •

SFT models reproduce the evolution of the surface field, particularly the
poleward drift of the following-polarity surface field leading to the buildup
of the polar fields and the axial dipole (e.g., Wang et al, 2002; Baumann et
al, 2004; Jiang et al, 2014b; Upton and Hathaway, 2014; Whitbread et al,
2017).

  * •

The polar field strength around cycle minimum is strongly correlated with the
amplitude of the subsequent activity maximum (Legrand and Simon, 1981; Wang
and Sheeley, 2009; Kitchatinov and Olemskoy, 2011b; Hathaway and Upton, 2016),
thus providing the most faithful predictor of cycle strength (Schatten et al,
1978; Petrovay, 2020; Kumar et al, 2021; Bhowmik et al, 2023). In addition to
confirming this correlation, Muñoz-Jaramillo et al (2013) showed that the
“memory” of the system does not extend beyond one cycle.

  * •

Cameron and Schüssler (2015) showed that the above correlation in fact
reflects a causation: the net hemispheric toroidal flux generated during a
half cycle results from the action of the latitudinal differential rotation on
the poloidal flux connected to the polar fields (see Sec. 3.1 below).

  * •

The observed azimuthal surface field (a proxy for flux emergence) evolves in
accordance with an updated BL model (Cameron et al, 2018).

The timescale for the rise of flux loops formed by the toroidal field and the
subsequent flux emergence is generally considered to be short compared to the
cycle time scale. Therefore, BL dynamo models typically do not explicitely
include these processes but rather incorporate a source term near the surface
that is related to the deep-seated toroidal field in the convection zone (for
a discussion of various approaches, see Choudhuri and Hazra, 2016). A few
examples of two-dimensional axisymmetric dynamo models with different
prescriptions for the BL source term are the studies of Durney (1997), Dikpati
and Charbonneau (1999), Nandy and Choudhuri (2001), Chatterjee et al (2004),
and Muñoz-Jaramillo et al (2010). More complete overviews have been provided
by Charbonneau (2020) and Hazra et al (2023).

FTD/BL models mostly rely on radial differential rotation in the tachocline
and often also assume penetration of the meridional flow into the stably
stratified interior in order to “store” the toroidal magnetic flux at the
bottom of the convection zone (e.g., Nandy and Choudhuri, 2002). In contrast,
the 2D model of Zhang and Jiang (2022) exhibits no such penetration, but
consistently includes the helioseismically determined differential rotation in
the convection zone (including the near-surface shear layer), a one-cell
meridional circulation, radial pumping keeping the surface field vertical and
inhibiting diffusive loss of the toroidal field, and a BL source term. The
generation of toroidal flux turns out to be strongly dominated by the
latitudinal differential rotation in the bulk of the convection zone (see also
Guerrero and de Gouveia Dal Pino, 2007; Muñoz-Jaramillo et al, 2009). The
latitudinal propagation of the toroidal flux belts in the model of Zhang and
Jiang (2022) is provided by a combination of flux transport by the equatorward
meridional return flow and the latitude dependence of the latitudinal
rotational shear generating toroidal magnetic flux, the latter as already
envisaged by Babcock (1961). Test cases with removal of the radial shear in
the tachocline or putting the numerical boundary above the tachocline does not
significantly change the results of the model, thus strongly suggesting that
the tachocline shear is indeed largely irrelevant for the dynamo (cf. Spruit,
2011; Brandenburg, 2005; Cameron and Schüssler, 2015).

Figure 1: Sketch of the updated Babcock-Leighton model of Cameron and
Schüssler (2017a). Figure 2: Results from the updated Babcock-Leighton model
of Cameron and Schüssler (2017a). Left panels: Radial surface field (upper
panel) and depth-integrated toroidal flux (lower panel) as a function of
latitude and time for a solution consistent with the properties of the solar
cycle. Right panels: Properties of model solutions as functions of the
amplitude of the effective speed of the equatorward return flow and of the
magnetic diffusivity in the convection zone. Colour shadings indicate the
dynamo period (upper panel) and the phase difference between the maximum of
the polar field and the maximum rate of flux emergence (lower panel). The
regions between the coloured lines indicate “solar-like” models: period in the
range 21–23 yr and phase difference between 80 and 100 degrees. These results
indicate a rather tight constraint for the speed of the return flow of about
2–3 m s-1.

Typically, FT/BL dynamo models are 2D (axisymmetric) or even 3D and time-
dependent. Although they exhibit solar-like solutions for properly chosen
values of the parameters (e.g., turbulent diffusivity, geometry and speed of
the meridional flow, dynamo excitation, etc.), computational limitations do
not permit a complete coverage of the parameter space. In order to
systematically investigate the parameter ranges for solar-like solutions,
Cameron and Schüssler (2017a) updated the 1D, two-layer model of Leighton
(1969). Model quantities are the azimuthally averaged radial field at the
surface and the radially integrated toroidal flux, both as a function of
latitude. Poleward meridional flow at the surface, an equatorward return flow
somewhere in the convection zone, latitudinal differential rotation as well as
radial differential rotation in the near-surface shear layer are included.
Furthermore, downward convective pumping of the near-surface horizontal field
and turbulent diffusion of the radial surface field and the toroidal field are
considered. Using the radially integrated magnetic flux has the advantage that
the model neither depends on the radial distribution of the toroidal field in
the convection zone nor on the depth location of the meridional return flow.
The solutions of this linear model depend on four parameters which represent
the driving of the dynamo (by emergence of tilted bipolar regions), the
effective speed of the meridional return flow, the turbulent diffusivity
affecting the toroidal field in the convection zone, and the effective radial
shear below the near-surface shear layer. All other parameters (such as
meridional flow and diffusivity at the surface) are taken from observations.
The simplicity of the model permits the exploration of the full four-
dimensional parameter space. Relevant ranges of parameters are identified by
requiring that the model results meet observational constraints: positive
growth rate, dipole parity, 22-year magnetic period, flux emergence
concentrated in low latitudes, and a 90 deg phase difference between the
maximum of flux emergence and maximum strength of the polar field. It turns
out that these requirements strongly constrain the parameters, yielding about
$2\,$m s-1 for the speed of the return flow, a turbulent diffusivity of about
80 km2s-1, toroidal flux generation dominated by latitudinal differential
rotation, and weak dynamo excitation not far above the threshold. A sketch of
the model is shown in Fig. 1 and an example of the results in Fig. 2.

The model results of Leighton (1969) indicate that the solar dynamo most
probably is a flux-transport dynamo operating near marginal excitation. The
model also clearly favors diffusion-dominated dynamo action (Yeates et al,
2008). The inferred speed of the equatorward flow transporting the toroidal
flux is consistent with the latitudinal drift rate of the activity belts. The
relevant values of the magnetic diffusivity affecting the toroidal flux are
also consistent with the observed evolution of the solar activity belts
(Cameron and Schüssler, 2016). Dominance of latitudinal differential rotation
entails a natural explanation for the concentration of flux emergence in low
latitudes: the latitudinal rotational shear peaks at mid latitudes and the
deep meridional return flow transports the generated toroidal flux
equatorward. Consequently, it is not necessary to assume a threshold value of
the toroidal field for the initiation of flux emergence, e.g., in the sense of
a buoyancy instability.

Further development of BL dynamo models was provided through a closer
connection between the near-surface evolution and the interior evolution of
the magnetic field. Cameron et al (2012) found that downward convective
pumping (leading to predominantly radial field near the surface) is required
to bring a 1D SFT model into accordance with the results of a 2D axisymmetric
FTD model (see also Karak and Cameron, 2016). Bhowmik and Nandy (2018) used
the surface distribution of magnetic flux during activity minima resulting
from a data-driven SFT simulation as input for a FTD model. A fully coupled
2D$\times$2D model was developed by Lemerle et al (2015) and Lemerle and
Charbonneau (2017). These authors connected an observationally calibrated SFT
model in latitude and longitude with an axisymmetric FTD model operating in
the meridional (radius-latitude) plane. The FTD model provides the source for
the SFT model by stochastic flux emergence in tilted bipolar magnetic regions.
In turn, the SFT model feeds back upon the FTD model in the form of a BL-like
source term near the upper boundary of the dynamo model, resulting in a self-
consistent coupling of the two models. Fig. 3 illustrates that properly
calibrated results from this model are consistent with the characteristic
features of the corresponding solar observations. A similar 2D$\times$2D
approach was presented by Miesch and Dikpati (2014).

Figure 3: Example result obtained by Nagy et al (2017, part of their Fig. 1)
using the 2D$\times$2D dynamo model developed by Lemerle and Charbonneau
(2017). Shown are time-latitude diagrams of the radial field at the surface
(upper panel) and of the number of flux emergences (butterly diagram, lower
panel).

The BL source in such models is typically provided by “deposition” of tilted
bipolar regions in the SFT part of the model, non-locally depending on the
strength of the deep-seated toroidal field. This ad-hoc procedure ignores the
details of the formation and rise of flux loops through the convection zone,
in particular the development of the tilt angle. It also implicitely assumes a
dynamical disconnection of the emerged bipolar region from its subsurface
roots. Bekki and Cameron (2022) considered the post-emergence evolution of
deposited bipolar regions in the framework of a nonlinear 3D simulation in a
spherical shell, including the Lorentz force as well as solar-like
differential rotation and meridional flow driven by a mean-field prescription.
These authors found that the evolution depends sensitively on the initial
shape and depth of the injected bipolar regions. When initialized with zero
tilt, the Lorentz force in combination with the Coriolis force tends to
produce negative tilt angles. The simulated BMRs also develop an systematic
asymmetry between the strength of leading and following spots, which is
consistent with observations.

Yeates and Muñoz-Jaramillo (2013) suggested a more consistent treatment of
flux emergence by assuming helical upflows that transport flux loops through
the convection zone, thus capturing aspects of buoyancy and advection by
convective flows as well as the connection between the surface and interior
field. A similar treatment was proposed by Pipin (2022). The idea of this
approach is consistent with the results of detailed observations of the
properties of emerging flux, which favor passive flux transport by convective
upflows (Birch et al, 2016). On the other hand, comparison with well-
calibrated SFT models (Whitbread et al, 2019) shows that after emergence the
surface flux needs to be dynamically disconnected from its deep toroidal roots
as suggested by Schüssler and Rempel (2005). Moreover, Schunker et al (2019)
and Schunker et al (2020) showed that the tilt of bipolar magnetic regions
develops only after emergence, possibly related to the extended flows
converging towards the bipolar magnetic regions (Martin-Belda and Cameron,
2016; Gottschling et al, 2021). In view of these results it seems fair to say
that the formation of rising flux loops, their transfer through the convection
zone, the actual emergence process, and the subsequent early evolution of the
emerged magnetic flux are still poorly understood (see also reviews by Fan,
2021; Isik and etal, 2023), so that these processes cause a significant amount
of uncertainty in all FTD/BL models presented so far.

Among the first 3D studies of FTD/BL models for the solar dynamo are the
models of Hazra et al (2017) and Karak and Miesch (2017). While these authors
used the deposition approach to represent flux emergence, Kumar et al (2019)
implemented the more consistent recipe of Yeates and Muñoz-Jaramillo (2013) in
a kinematic 3D dynamo code. Another step towards a full 3D treatment was taken
by Hazra and Miesch (2018), who introduced a 3D, stationary, convection-like
flow field in the upper part of the convection zone. This serves to replace
the diffusion term in standard SFT models by a more realistic explicit
transport process. The assumed flow pattern was based upon an observational
power spectrum of the surface flows and a downward extrapolation under the
assumption of zero divergence of the mass flux and an imposed radial profile.
The kinematic model ran into problems owing to unlimited small-scale dynamo
action driven by the imposed stationary flow. Further results obtained with
kinematic 3D models of BL dynamos have been reviewed by Hazra (2021). Bekki
and Cameron (2022) used their 3D model to perform nonlinear BL dynamo
simulations with the near-surface deposition of properly tilted bipolar
regions. The Lorentz force provides nonlinear saturation of the dynamo
amplitude. This feedback drives non-axisymmetric flows as well as solar-like
zonal flows superposed upon the differential rotation and also modifies the
meridional circulation. Other nonlinear effects (see also Sect. 4.1 that could
lead to dynamo saturation and determine the cycle amplitude have been studied
by Jiang (2020), Jiao et al (2021), and Talafha et al (2022).

##  3 Generation and removal of magnetic flux

###  3.1 Toroidal flux

Toroidal magnetic field (azimuthal field in the case of an average over
longitude) is considered to be generated in the Sun through the action of
differential rotation on a poloidal magnetic field. Part of the generated
toroidal flux emerges when radial motions carry loops of magnetic flux through
the solar surface. Flux emergence leads to the formation of two surface areas
with oppositely-directed radial field components. These bipolar magnetic
regions (BMRs) have been studied extensively (e.g. Hale et al, 1919; Harvey et
al, 1975; Martin and Harvey, 1979; Wilson et al, 1988; Harvey, 1993;
McClintock and Norton, 2016; Schunker et al, 2016). A map showing the
latitudes at which larger BMRs with sunspots emerge in time (known as the
butterfly diagram) is shown in the upper panel of Fig. 4.

BMRs show various systematic properties. One example is Hale’s law (Hale et
al, 1919), which states that the leading polarity (with respect to the
direction of rotation) of BMRs in each hemisphere has the same sign in most
(up to 95%) of the cases during an activity cycle. The leading polarity is
opposite in the two hemispheres and switches from cycle to cycle. These
properties imply a systematic East-West component of the horizontal field
during emergence, which flips sign between each hemisphere and from cycle to
cycle. This can be illustrated in a time-latitude map of the longitudinally
averaged azimuthal component of the surface field covering four activity
cycles, which is shown in the middle panel of Fig. 4 (cf. Cameron et al, 2018;
Liu and Scherrer, 2022). The latitudes where sunspot groups emerge correspond
to locations where the surface azimuthal field is strong. Weaker azimuthal
field corresponds to flux emergence of smaller BMRs (ephemeral regions, Martin
and Harvey, 1979).

Figure 4: Observationally based time-latitude diagrams of sunspots (top panel,
data from Royal Greenwich Observatory/NOAA:
http://solarcyclescience.com/AR_Database), longitudinally averaged azimuthal
surface field (middle panel, data from Wilcox Solar Observatory, WSO), and
longitudinally averaged radial field (lower panel, data from WSO).

Another systematic property of BMRs is Joy’s law (Hale et al, 1919). It refers
to the tendency for the leading part of a BMR or sunspot group to be closer to
the equator. Because of Hale’s law, the leading parts mostly have the same
polarity, so that there is a systematic tendency in both hemispheres that more
radial magnetic flux of one polarity appears closer to the equator. The radial
field of the leading parts in the other hemisphere has the opposite polarity,
and the polarities switch from cycle to cycle. These properties are
illustrated in the time-latitude diagram for the longitudinally averaged
radial surface field shown in the bottom panel of Fig. 4. The locations of
flux emergence can be clearly seen, with the leading polarity dominating on
the equatorward side of the “butterfly wings”. Plumes of magnetic field with
widths of a few months connect the butterfly wings to the polar regions, where
the field switches polarity with the period of the solar cycle.

An important feature of the time-latiude diagram of the toroidal surface field
(middle panel of Fig. 4) is that during times of activity maximum (when there
are many sunspots) the toroidal field at every latitude in each hemisphere is
of the same sign. This suggests to consider the evolution of the net
subsurface toroidal flux in each hemisphere. For this we start from the
induction equation,

$\frac{\partial{\bf B}}{\partial t}=\nabla\times\left({\bf U}\times{\bf
B}-\eta\nabla\times{\bf B}\right),$ (1)

where ${\bf B}$ is the magnetic field, ${\bf U}$ the flow velocity, and $\eta$
the (molecular) magnetic diffusivity. Following Cameron and Schüssler (2015),
Cameron and Schüssler (2020), and Jeffers et al (2022), we now consider the
area A that is enclosed by the thick outline in the left panel of Fig. 5 and
apply Stokes’ theorem to the induction equation, viz.

$\frac{\partial{\iint_{A}{\bf B}\cdot\mathrm{d}{\bf A}}}{\partial
t}=\oint_{\delta A}\left({\bf U}\times{\bf B}-\eta\nabla\times{\bf
B}\right)\cdot\mathrm{d}{\bf l}\,,$ (2)

where $\delta A$ indicates the boundary of $A$ and ${\mathrm{d}\bf l}$ is the
corresponding line element. We now take the azimuthal average of Eq. (2) and
neglect the diffusive term in the contour integral since the magnetic Reynolds
number is very large. This leads to

$\frac{\partial{\iint_{A}{\langle{\bf B}\rangle}\cdot\mathrm{d}{\bf
A}}}{\partial t}=\oint_{\delta A}\left(\langle{\bf U}\rangle\times\langle{\bf
B}\rangle+\langle{\bf u}\times{\bf b}\rangle\right)\cdot\mathrm{d}{\bf l}\,,$
(3)

where $\langle\dots\rangle$ indicates the azimuthal average. This equation
represents the temporal change of net toroidal flux in the Northern hemisphere
resulting from the inductive action of the flow field on the magnetic field.
The quantities ${\bf b}={\bf B}-\langle{\bf B}\rangle$ and ${\bf u}={\bf
U}-\langle{\bf U}\rangle$ denote, respectively, the fluctuating components of
magnetic field and flow velocity with respect to the corresponding azimuthal
averages.

Figure 5: Left panel: Area and contour used for calculating the net
axisymmetric toroidal flux in a hemisphere. The bottom part of the contour is
placed at a radius $R_{\mathrm{interior}}$, where the penetration of the 22
year cyclic component of the magnetic field is negligible. Right panel: Solar
differential rotation along segments of the contour in the left panel. Shown
are the radial dependence of $\Omega$ on the equatorial plane (red curve,
upper scale) and the (co)latitudinal dependence at the surface (blue curve,
lower scale), both relative to the equatorial surface rotation rate. The data
for this plot was obtained from Larson and Schou (2018).

Since the Sun’s radial differential rotation in the equatorial plane is small
compared to its latitudinal differential rotation (see right panel of Fig. 5)
it is convenient to use a frame of reference rotating with the surface
equatorial rate. In this frame, we have

$\displaystyle\oint_{\delta A}\left(\langle{\bf U}\rangle\times\langle{\bf
B}\rangle\right)\cdot\mathrm{d}{\bf l}$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi/2}\left(\Omega(R_{\odot},\theta)-\Omega(R_{\odot},\frac{\pi}{2})\right)\sin\theta\langle{B_{r}}\rangle
R_{\odot}^{2}\mathrm{d}\theta$ (4)
$\displaystyle-\int_{R_{\mathrm{interior}}}^{R_{\odot}}\left(\Omega(r,\frac{\pi}{2})-\Omega(R_{\odot},\frac{\pi}{2})\right)\langle{B_{\theta}}\rangle
r\mathrm{d}r\,,$

where $\theta$ is the colatitude. Only the surface part of the contour
integral (first term on right-hand side) and the part in the equatorial plane
(second term) contribute to the contour integral. The bottom part in the
interior vanishes since B is zero there and the part along the rotation axis
vanishes since we consider azimuthal averages of all quantities. The
solenoidality of the magnetic field implies

$\int_{0}^{\pi/2}\langle{B_{r}}\rangle
R_{\odot}^{2}\sin\theta\mathrm{d}\theta=-\int_{R_{\mathrm{interior}}}^{R_{\odot}}\langle{B_{\theta}}\rangle
r\mathrm{d}r,$ (5)

which means that the net magnetic flux through the equatorial plane beneath
the surface is equal in magnitude to the net flux through the solar surface in
each hemisphere.

Owing to the weak radial differential rotation in the equatorial plan), the
net toroidal flux generation is strongly dominated by the surface part of the
contour integral, i.e., the first term on the right-hand side of Eq. (4). This
can be directly evaluated from synoptic observations of the radial field on
the surface and the Sun’s differential rotation, with a net hemispheric
toroidal flux of $5$–$8\times 10^{23}$ Mx per cycle being thus generated. The
integrand for the surface part in the contour integral (Eq. 4) is strongly
dominated by the polar caps (see bottom left panel of Fig. 6), so that the
flux associated with the polar dipole field represents the relevant poloidal
source for the generation of net toroidal flux. Other poloidal flux that is
contained in the convection zone and does not cross the surface leads to equal
amounts of East-West and West-East orientated toroidal flux in a hemisphere
and thus does not contribute to the net toroidal flux required by Hale’s
polarity laws.

The term in Eq. 3 involving the fluctuating components, $\oint_{\delta
A}\langle{\bf u}\times{\bf b}\rangle\cdot\mathrm{d}{\bf l}$, is dominated by
(turbulent) diffusive fluxes across the axis and the equatorial plane. Some of
the other terms, such as those owing to the turbulent electromotoric force
($\alpha$-effect), are expected to vanish due to symmetries – along the axis
owing to the conservation of helicity and at the equator owing to the expected
vanishing of the kinetic helicity, which in its simplest form scales like
$\cos\theta$.

Another contribution from the fluctuating terms is the change of the net
toroidal flux by flux emergence and submergence through the surface, which is
described by

$\left(\frac{\mathrm{d}\Phi}{\mathrm{d}t}\right)_{\rm
em,subm}=\int^{0}_{\pi/2}\langle
u_{r}b_{\phi}\rangle|_{R_{\odot}}R_{\odot}\mathrm{d}\theta\,.$ (6)

Flux loss through the surface was part of the original Babcock-Leighton model
(Leighton, 1969), but later fell out of favour since it was thought that the
emerged flux is mostly retracted back through the surface (e.g., Wallenhorst
and Topka, 1982). This process can take place in flux cancellation events when
loops of emerged flux become sufficiently narrow, so that the tension force
dominates. Some retraction of flux is possibly required to account for the
amount of flux brought to the surface by repeated emergences in so-called
“nests of activity” (e.g., Gaizauskas et al, 1983).

Parker (1984) argued that a net loss of toroidal flux from the Sun would
require an organized sequence of reconnection events between narrowly-spaced
bipolar regions, thus effectively shedding the mass from the toroidal field
lines and allowing them to freely escape with the solar wind. However,
observations show that the bipolar regions generally are much too widely
spaced for this process to operate on the Sun, so that Parker concluded that
in dynamo models the solar photosphere should be approximated by an
impenetrable boundary.

Parker’s argument, however, ignores that what is relevant for the dynamo
models is the effect of the emergence on the mean (azimuthally averaged)
toroidal field. Cameron and Schüssler (2020) showed that the amount of the
azimuthally averaged toroidal field lost due to the emergence of a bipolar
region is time-independent, irrespective of the further evolution of its
magnetic flux. While cancellation/retraction of most of the field may occur,
at the same time the remaining surface flux spreads out in longitude
(transported by near-surface flows) to eventually fully encircle the Sun. In
this way, toroidal field detached from the solar interior is formed which can
be carrried away by the solar wind and the same amount of flux is lost from
the mean toroidal field in the interior. A quantitative analysis of the net
azimuthally averaged toroidal flux lost during a cycle arrives at $3.3\times
10^{23}$ Mx/hemisphere/cycle (Cameron and Schüssler, 2020). A similar estimate
of $5\times 10^{23}$ was obtained using in situ measurements of the magnetic
field and flows in the solar wind (Bieber and Rust, 1995). The observations
thus indicate that, integrated over a cycle, the amount of the toroidal
magnetic flux generated by the axisymmetric flows and fields ($\oint_{\delta
A}\left(\langle{\bf U}\rangle\times\langle{\bf
B}\rangle\right)\cdot\mathrm{d}{\bf l}$) is similar to the amount lost through
flux emergence ($\oint_{\delta A}\langle{\bf u}\times{\bf
b}\rangle\cdot\mathrm{d}{\bf l}$).

Figure 6: Generation and loss of net unsigned toroidal flux in the Northern
hemisphere, covering four activity cycles on the basis of synoptic
magnetograms from the Wilcox Solar Observatory. The time-latitude diagrams on
the left side show the observational input for evaluating the dominating
surface part of the contour integral (Eqs. 4 and 6). Toroidal flux is
generated proportional to the product of the azimuthally averaged surface
radial field and the surface rotation rate relative to the equatorial rate
(lower left panel). Toroidal flux is lost through flux emergence at a rate
proportional to the product of $\langle B_{\phi}\rangle$ at the surface (upper
left panel) and the radial emergence velocity estimated as
$v_{\mathrm{em}}=200\,$m s-1 (Centeno, 2012). The panel to the right shows a
phase plot of the yearly integrated values of the generated and lost toroidal
flux indicated by red dots. The connecting line segments with arrows
illustrate the change from one year to the next.

The toroidal flux budget shown in Figure 6 shows the approximate balance
between the amount of toroidal magnetic flux lost throughout a cycle due to
flux emergence and the amount produced by the winding up of poloidal flux
threading the solar surface, both being determined by surface observations of
the magnetic field ($B_{\phi}$ and $B_{r}$). There is a phase shift between
the production rate and loss of about 90∘, which is consistent with the loss
through flux emergence being proportional to the total subsurface toroidal
flux. The phase diagram is based on Eqs. (4) and (6), which confirms the
concept that the toroidal flux is mainly generated by the action of
latitudinal differential rotation on the poloidal flux threading the solar
surface in the polar dipole field.

###  3.2 Poloidal flux

The previous section has shown that the application of Stokes’ theorem to the
induction equation over a properly defined meridional area illuminates the
essence of the generation and loss of net toroidal flux during solar activity
cycles. A similar procedure can be used to determine the evolution of the net
radial flux threading a hemisphere of the Sun (cf. Durrant et al, 2004). To
this end, we integrate the radial component of the azimuthally averaged
induction equation over the photospheric surface area, $A$, of the Northern
hemisphere, NH, and apply Stokes’ theorem, which yields

$\frac{\partial\Phi_{\mathrm{pol,NH}}}{\partial t}=\frac{\partial}{\partial
t}\oint_{A}B_{r}dA=2\pi R_{\odot}\left(\langle u_{r}b_{\theta}\rangle-\langle
u_{\theta}b_{r}\rangle\right)\,,$ (7)

where the contour integral is taken over the boundary of $A$, i.e., the
equator. The two terms on the r.h.s. correspond to observable quantities. The
term $\langle u_{r}b_{\theta}\rangle$ quantifies the effect of flux emerging
across the equator. This term is important when large and highly tilted active
regions emerge with polarities on either side of the equator (e.g., Cameron et
al, 2013). The term $\langle u_{\theta}b_{r}\rangle$ corresponds to convective
motions carrying radial flux across the equator and can be approximated by a
“turbulent” diffusion term, viz.

$\langle u_{\theta}b_{r}\rangle=\eta_{\mathrm{T}}R_{\odot}\frac{\partial
B_{r}}{\partial\theta}\,.$ (8)

There is a systematic component to ${\partial B_{r}}/{\partial\theta}$
corresponding to the systematic tilt angle of bipolar regions (see the time-
latitude diagram of the radial magnetic field in Fig. 4). A random component
is introduced by large, highly tilted active regions which emerge near to the
equator (Cameron et al, 2013), also called “rogue active regions” (Nagy et al,
2017). Such regions introduce significant randomness into the net flux in each
hemisphere and the axial dipole moment (Cameron et al, 2014). This randomness
thus carries over into the production of the toroidal flux emerging in the
subsequent cycle. Jiang et al (2015) and Whitbread et al (2018) showed that
the low amplitude of solar cycle 24 can be understood in terms of rogue active
regions that emerged during cycle 23.

Figure 7: Systematic part of the generation rate and amount of net poloidal
flux in the course of four activity cycles. The left panels illustrate the
observational input: the azimuthally averaged radial surface field (upper left
panel) determines the total net poloidal surface flux (cf. Eq. 9) while the
azimuthally averaged azimuthal surface field (lower left panel) represents
flux emergence in tilted bipolar regions and thus determines the rate of
generation of poloidal flux via flux transported over the equator (cf. Eq.
15). The right panel shows a phase plot of both quantities, with each red dot
representing the the average value for one year. The line segments with arrows
represent the change from one year to the next.

Figure 7 illustrates the budget of poloidal surface flux for four activity
cycles on the basis of synoptic magnetograms from the Wilcox Solar
Observatory. The poloidal flux is determined by integration of the
longitudinally averaged radial field over the solar surface, viz.

$\Phi_{\mathrm{pol}}=\pi
R_{\odot}^{2}\int_{0}^{\pi}\langle{B_{r}}\rangle\mathrm{sign}(\pi/2-\theta)\sin\theta\mathrm{d}\theta\,,$
(9)

where the results for the two hemispheres are averaged.

The generation term for the poloidal flux results from summing the
contributions of all bipolar regions to the amount of magnetic flux which is
carried across the equator. The contribution of a single bipolar region ($i$)
depends on the magnetic flux of each polarity, $\Phi_{i}$, the angular
separation of the two polarities in latitude, $\delta_{\lambda,i}$, the
latitude of emergence, $\lambda_{i}$, and a parameter characterizing the
surface evolution of the magnetic flux near the equator, $\lambda_{R}$ (cf.
Petrovay et al, 2020). This parameter depends on the latitudinal derivative of
the surface meridional flow speed, $V$, at the equator ($\lambda=0$), and the
turbulent diffusivity at the surface, $\eta_{\mathrm{T}}$, viz.

$\lambda_{R}=\sqrt{\frac{\eta_{\mathrm{T}}}{R_{\odot}{(\mathrm{d}V}/{\mathrm{d}\lambda})_{\lambda=0}}}\,.$
(10)

Petrovay et al (2020) estimated the time-integrated amount of magnetic flux
crossing the equator (and thus the contribution of the bipolar region to the
buildup of the poloidal field) by the leading term of a Taylor expansion as

$\Phi_{{\mathrm{pol}},i}=\frac{\Phi_{i}\delta_{\lambda,i}}{\sqrt{2\pi}\lambda_{R}}\;{\mathrm{e}}^{-\lambda^{2}_{i}/(2\lambda_{R}^{2})}\,.$
(11)

During its emergence, the flux tube forming the bipolar region contributes to
the longitudinally averaged toroidal surface field, $\langle B_{\phi}\rangle$,
by the amount $\langle B_{\phi,i}\rangle$ according to

$\Phi_{i}\frac{\delta_{\phi,i}}{2\pi}=\langle B_{\phi,i}\rangle
v_{\mathrm{em}}\,\Delta t\,R_{\odot}\,\Delta\lambda\,,$ (12)

where $\delta_{\phi,i}$ is the longitudinal angular extent of the bipolar
region, $\Delta t$ is the time over which the emergence takes place,
$\Delta\lambda$ is the latitudinal width of the leading polarity of the
emerging region, and $v_{\mathrm{em}}$ is the mean emergence speed (assumed to
be the same for all emergences). The latitudinal and longitudinal extents are
related by
$\delta_{\lambda,i}=\tan(\alpha_{i})\cos(\lambda_{i})\delta_{\phi,i}$, where
$\alpha_{i}$ is tilt angle. We thus obtain

$\Phi_{i}\delta_{\lambda,i}=\langle B_{\phi,i}\rangle\,v_{\mathrm{em}}\,\Delta
t\,R_{\odot}\,\Delta\lambda\,2\pi\tan(\alpha_{i})\cos(\lambda_{i})\,.$ (13)

Inserting this result into Eq. (11), we obtain

$\Phi_{{\mathrm{pol}},i}=\frac{\langle
B_{\phi,i}\rangle\,v_{\mathrm{em},i}\,\Delta
t\,R_{\odot}\,\Delta\lambda\,\sqrt{2\pi}\,\tan(\alpha_{i})\,\cos(\lambda_{i})}{\lambda_{R}}\,\mathrm{e}^{-\lambda^{2}_{i}/(2\lambda_{R}^{2})}$
(14)

For the the tilt angle we consider Joy’s law without scatter, so that
$\alpha_{i}=\alpha(\lambda_{i})$. We further assume that the emergence
velocity is the same for all bipolar regions with a value of
$v_{\mathrm{em}}=200\,$m s-1 (Centeno, 2012). Adding together the
contributions of the individual bipolar regions to obtain the rate of
generation of poloidal flux then amounts to the latitude integral

$\frac{{\mathrm{d}}\Phi_{\mathrm{pol}}}{\mathrm{d}t}=\frac{\sqrt{2\pi}\,v_{\mathrm{em}}}{\lambda_{R}}\int_{-\pi/2}^{\pi/2}\langle
B_{\phi}\rangle
R_{\odot}\tan[\alpha(\lambda)]\cos(\lambda)\,{\mathrm{e}}^{-\lambda^{2}/(2\lambda_{R}^{2})}{\mathrm{d}}\lambda$
(15)

We choose the parameters as in Jiang et al (2014a) with
$\eta_{\mathrm{T}}=250$ km2s-1, surface meridional flow
$V(\lambda)=11\sin(180\lambda/75)\,$m s-1, and Joy’s law for a cycle of
intermediate strength in the form $\alpha(\lambda)=0.7\times
1.3\sqrt{|\lambda|}\,\mathrm{sign}(\lambda)$.

The poloidal flux budget resulting from using the observed longitudinally
averaged radial and azimuthal surface fields in Eqs. (9) and (15) is
illustrated by the phase diagram on the right side of Figure 7. It
demonstrates that poloidal flux and poloidal flux generation as determined
through Eqs. (9) and (15) are consistent with each other. This confirms the
basic concept that the poloidal field is being generated by the joint
contributions of tilted bipolar regions. Compared to the toroidal flux budget
(Figure 6), the phase diagram is somewhat less smooth. This is a consequence
of the significant random component affecting the generation of poloidal flux.
This component is only included in the poloidal flux but not in the rate of
poloidal flux generation, where we have assumed Joy’s law without scatter.

###  3.3 Flux transport

The net toroidal and poloidal flux budgets discussed in the previous sections
show that the observable magnetic flux at the surface plays a crucial role in
the solar dynamo, highlighting the role of differential rotation, flux
emergence, and Joy’s law. This section briefly discusses additional processes
which modifiy the spatial distribution of the fluxes. In terms of the poloidal
flux, the budget shown in Fig. 7 demonstrates that flux carried across the
equator is essential. Furthermore, the toroidal flux budget is dominated by
the polar fields, which means that the poleward transport of flux is an
important ingredient of the dynamo process: surface flux which crosses the
equator is transported to the poles by a combination of the large-scale
meridional flow and small-scale flows associated with convection. These
processes are captured by the surface flux transport model (Jiang et al,
2014b; Yeates et al, 2023).

The butterfly diagram of flux emergence demonstrates that there is a similar
requirement for the equatorward transport of the subsurface toroidal flux from
latitudes of about $55^{\circ}$, where the toroidal flux is most efficiently
produced (cf. Spruit, 2011), towards the equator. Low-latitude flux emergence
in accordance with Joy’s law then leads to the cross-equatorial transport of
poloidal field. The nature of this transport of toroidal flux is not directly
constrained, so that dynamo waves, turbulent pumping, and meridional
circulation are all viable candidates. Gizon et al (2020) showed that the
speed of the deep equatorward meridional flow inferred from helioseismology is
consistent with the observed migration of the activity belts if the toroidal
flux is distributed over the lower half of the convection zone. This lends
credibility to the concept of flux transport dynamos (recently reviewed by
Hazra et al, 2023).

##  4 Nonlinearity, predictability, and long-term variability

###  4.1 Nonlinear effects

An excited hydromagnetic dynamo leads to exponential growth of an initially
weak magnetic field until further amplification becomes limited by the action
of the Lorentz force, which introduces a nonlinearity into the system. The
nonlinearity can affect large-scale flows (e.g, differential rotation and
meridional flow), modify turbulence effects (e.g., the mean-field
$\alpha$-term, turbulent diffusivity, and turbulent pumping), or change the
properties of flux emergence. A nonlinearity often invoked in mean-field
models is “$\alpha$-quenching” (Steenbeck and Krause, 1969; Stix, 1972), a
parameterization of the decreasing efficiency of turbulence to produce
poloidal field from the toroidal field as the field amplitude grows. In the
case of a system with a high magnetic Reynolds number, $R_{m}$, such as the
Sun, this nonlinearity is potentially catastrophic since the ratio of the
small-scale field and the mean field scales as $\sqrt{R_{\mathrm{m}}}$
(Cattaneo and Vainshtein, 1991). This causes saturation of the dynamo at field
amplitudes many orders of magnitude smaller than observed on the Sun. However,
catastrophic quenching may be alleviated by removal of magnetic helicity from
the system (Kleeorin et al, 2000; Hubbard and Brandenburg, 2012).

In the Babcock-Leighton framework, the poloidal field generation results from
the systematic tilt angle of bipolar magnetic regions. While there is
observational evidence that the average tilt angle decreases with increasing
cycle strength (Dasi-Espuig et al, 2010; McClintock and Norton, 2013; Jiao et
al, 2021), catastrophic quenching is not expected in this case (Kitchatinov
and Olemskoy, 2011a). Sufficiently strong magnetic field may also reduces the
turbulent magnetic diffusivity (e.g., Kleeorin and Rogachevskii, 2007;
Guerrero et al, 2009), which also affects the Babcock-Leighton dynamo.

Magnetically induced changes of the large-scale differential rotation (zonal
flows, cf. Labonte and Howard, 1982) and of the meridional circulation
(systematic inflows towards active regions, cf. Gizon et al, 2001) have been
observed (see also Hathaway et al, 2022). The zonal flows are probably too
weak to have a substantial effect on the dynamo mechanism. The inflows towards
active regions explain part of the cyclic variation of the meridional flow
(Cameron and Schüssler, 2012). Their effects have been studied in the
framework of surface flux transport and Babcock-Leighton models (Martin-Belda
and Cameron, 2017; Nagy et al, 2020). It was found that, in principle, the
resulting changes to the meridional flow can provide nonlinear saturation of
the dynamo.

Another nonlinearity which has been studied in the Babcock-Leighton framework
is ’latitudinal quenching’ (Jiang, 2020; Talafha et al, 2022). This effect is
related to the observation that the average latitudes of sunspots are located
more poleward in strong cycles as compared to weak cycles (Waldmeier, 1955;
Solanki et al, 2008; Hathaway, 2015). Since bipolar regions at higher
latitudes contribute less to the flux crossing the equator and thus to the
buildup of the poloidal field (Jiang et al, 2014a), this effect provides an
amplitude-limiting nonlinearity.

A closer look at the properties of flux emergence depending on cycle strength
yields further insight into the nature of latitude quenching. In Fig. 8, we
used 13-month smoothed sunspot sunspot numbers (Version 2) between solar
cycles 12 and 24 from the SILSO data base
(https://www.sidc.be/silso/datafiles) and 12-month averages of observed times,
latitudes, and areas of sunspots given in the Royal Greenwich Observatory and
USAF/NOAA data bases (downloaded from
http://solarcyclescience.com/activeregions.html) to plot the sunspot number,
the central latitude, and the full width at half maximum (FWHM) of the sunspot
zones (“butterfly wings”), respectively, as functions of time since the start
of a cycle. Depending on their peak sunspot number, four cycles each were put
into groups of strong, medium, and weak cycles. Central latitude and FWHM of
the sunspot zones were determined from Gaussian fits of the sunspot data with
unsigned latitudes, thus merging both hemispheres.

Since the solar cycle is not perfectly periodic, comparing the evolution of
different cycles requires a definition of a reference time for each cycle. One
possibility is to fit the shape of the time evolution of the sunspot number to
a given functional relationship. Using such a procedure, Hathaway (2011) found
that the central latitude of the sunspot belts propagates equatorward in the
same way for all cycles (see also Waldmeier, 1939). We thus take the reference
time as the instant at which the central latitude is at $19^{\circ}$ and
define the start of the cycle to be 4 years before that instant. These
particular choices have no significant impact on the analysis and the results.

Figure 8: Properties of the sunspot zones as function of time from cycle
start, based upon the historical sunspot record. Shown are the sunspot number
(top panel), central latitude (middle panel), and full width at half maximum
(lower panel) of the sunspot zones. The quantities are averaged over weak
cycles ($n=12,14,24,16$, blue lines), intermediate cycles ($n=15,20,17,23$,
green lines), and strong cycles ($n=18,21,22,19$, red lines). Shading
indicates the range covered by the $\pm 1$ stderr, calculated using the
scatter between the four cycles in each group.

The figure confirms earlier results of Waldmeier (1955): (i) the activity of
stronger cycles rises faster and peaks earlier than that of weaker cycles
(often referred to as “Waldmeier effect”) while the declining phase is
independent of cycle strength, and (ii) the time profile of the propagation of
the sunspot zones towards the equator is independent of cycle strength (see
also Hathaway, 2011). Furthermore, the full wdth at half maximum of the
sunspot zones in the declining phase is also independent of cycle strength
(Cameron and Schüssler, 2016), while stronger cycles show broader sunspot
zones (see also Mandal et al, 2017; Biswas et al, 2022). These properties
reveal three aspects of latitude quenching, namely

  1. 1.

The Waldmeier effect has the consequence that flux emergence around cycle
maxima on average occurs in higher latitudes for stronger cycles than for
weaker cycles, thus being less effective for the buildup of the polar field.
This corresponds to a negative feedback.

  2. 2.

The broader wings of the sunspot zones during the maximum phases of stronger
cycles have the opposite effect since more flux emerges in lower latitudes.

  3. 3.

The fact that all three properties, sunspot number as well as central latitude
and width of the sunspot zones, behave independently of cycle strength in the
declining phase of the cycles means that, during this most critical phase for
the buildup of the poloidal field, the amount of magnetic flux transfer
transferred across the equator is independent of cycle strength, thus
corresponding to negative feedback.

These three aspects reveal the action of an underlying nonlinearity connected
to flux emergence. It is plausible that the cycle strength reflects the amount
of toroidal magnetic flux in the convection zone which is available for flux
emergence. In strong cycles, latitudinal differential rotation, which is
steepest in mid latitudes, acts upon a stronger poloidal field and thus
produces more and stronger toroidal magnetic flux. A nonlinearity matching the
cycle properties discussed above could be that flux emergence occurs when a
critical field strength of the order of the equipartition field strength is
exceeded. While the latitude drift of the sunspot zones is independent of
cycle strength (possibly being determined by a deep meridional flow towards
the equator, which is largely unaffected by the magnetic field), the critical
field strength is reached earlier in strong cycles, meaning that more flux
emerges at higher latitudes (point 1 of the list above). At the same time, the
critical field strength is exceeded in a broader range of latitudes (point 2).
Consequently, stronger cycles have lost a bigger part of their available
toroidal flux earlier in the cycle compared to weaker cycles, so that in the
later phases flux emergence and width of the sunspot zones become independent
of cycle strength. Biswas et al (2022) confirmed this conjecture using a
Babcock-Leighton flux-transport dynamo model. They suggested that, in the
declining phase of all cycles, flux emergence compensates the increase of the
toroidal field strength due to the pileup of flux near the stagnation point of
the equatorward meridional flow. Consequently, the mean field strength remains
near to the critical field strength for flux emergence and all cycles decline
in the same way.

All mechanisms discussed in this section are plausible explanations for the
nonlinearity limiting the amplitude of the solar dynamo. Distinguishing which
combination of them acts on the Sun is an open observational challenge.

###  4.2 Long-term variability

The sunspot cycle shows variability on a wide range of timescales (see reviews
by Usoskin, 2017; Biswas et al, 2023). Panel A of Fig. 9 shows the historical
sunspot record from telescopic observations since the beginning of the 17th
century. It reveals significant cycle-to-cycle fluctuations of cycle strength
together with longer-term variability. Particularly conspicuous is the period
of very low sunspot activity between 1645 to 1715 and the period of high
average sunspot activity between about 1940 and 2006\. Other such “grand
minima” and “grand maxima” are found in reconstructions of solar activity
during the past millenia (albeit at a coarser time resolution) on the basis of
various records of cosmogenic isotopes (e.g., Solanki et al, 2004; Usoskin et
al, 2016). An example of such a reconstruction is shown in panel B of Fig. 9.

Figure 9: Observed and simulated records of solar activity. The historical
sunspot record (sunspot group number, panel A) and the level of solar activity
reconstructed from cosmogenic isotopes samples (Usoskin et al, 2016) are shown
in panels A and B, respectively. Panels C and D in the middle show random
realizations of a normal-form model (near a Hopf bifurcation) model with
noise, covering the same lengths of time as panels A and B. Panels E and F at
the bottom shows realizations from a simple Babcock-Leighton dynamo model.
Figure taken from Cameron and Schüssler (2017b).

Observational studies of the rotational evolution (gyrochronology) of solar-
type stars (e.g., Metcalfe et al, 2016; van Saders et al, 2016; David et al,
2022; Metcalfe et al, 2022, 2023) indicate that magnetic braking by a stellar
wind is significantly reduced for stars near or beyond the solar age, which
suggests a decline of their large-scale magnetic fields (for a different view,
see Kotorashvili et al, 2023). This indicates that, at its current rotation
rate, the Sun could be approaching a transition point where its gobal dynamo
switches off, so that at present the excitation of the solar dynamo is only
weakly supercritical (or even subcritical, as suggested by Tripathi et al,
2021).

The observationally well-studied rotation-activity relation for magnetically
active stars (e.g., Brun and Browning, 2017) suggests the rotation rate as the
relevant control parameter for dynamo excitation. For most dynamo models, the
transition from decaying field to excited oscillatory dynamo action
corresponds to a supercritical Hopf bifurcation (e.g., Tobias et al, 1995),
when a fixed point (equilibrium) spawns a limit cycle (oscillatory solution).
The behaviour of a a weakly excited nonlinear system near a Hopf bifurcation
is fully described by a generic normal-form model that is independent of the
specific properties of the system, including also the nature of its
nonlinearity (e.g., Guckenheimer and Holmes, 1983).

Cameron and Schüssler (2017b) applied this concept to the solar dynamo and
showed that the observed power spectrum of the solar cycle (from the sunspot
record and from the reconstruction based on cosmogenic isotopes) is consistent
with a noisy normal-form model whose parameters are completely determined by
observations. The noise in the system results from the large scatter in the
observed tilt angles of active regions, which is possibly associated with the
interaction of magnetic flux and convective motions (Longcope and Fisher,
1996). As discussed in Sections 3.1 and 3.2, the tilt angle scatter leads to
randomness in the amount of flux transported across the equator, and hence to
randomness in the amount of toroidal field generated for the subsequent cycle.

Panels C and D of Fig. 9 show an example realization of the normal-form model
covering 10,000 years. Panel B of Fig. 10 gives the corresponding temporal
power spectrum, which is consistent with the observed spectrum shown in panel
A of Fig. 10. Moreover, the model also exhibits extended periods of low
activity (grand minima) whose statistical properties in terms of the
distributions of their lengths and the waiting times between grand minima is
consistent with the reconstructed record of solar activity by Usoskin et al
(2016).

Likewise, the updated 1D Babcock-Leighton dynamo model of Cameron and
Schüssler (2017a) with slightly supercritical excitation and including noise
in the source term for the poloidal field also yields time series that are
statistically similar to the empirical series (see panels E and F of Fig. 9).
The corresponding power spectra in panels C and D of Fig. 10) show a good
match to the observed spectra (panel A of Fig. 10). A similar approach was
taken by Kitchatinov and Nepomnyashchikh (2017) and Kitchatinov et al (2018),
who studied a more comprehensive dynamo model.

Figure 10: Power specta of empirical solar data and models. (A) Sunspot group
number (green), and reconstruction from cosmogenic isotopes (blue), together
with the expectation value from 10.000 realizations of the normal form model
(black curve) and the corresponding 25% to 75% quartiles (grey shading). (B)
One realization of the normal form model. (C) Same as (A), but compared to the
expectation value and quartiles from realizations of an updated Babcock-
Leighton model (black curve and pink shading). (D) One relization of the
updated Babcock-Leighton model. Figure taken from Cameron and Schüssler
(2017b).

Cameron and Schüssler (2019) carried out a detailed analysis of power pectra
obtained with the generic noisy normal-form model (incorporating only the
11/22 year base period) in order to evaluate the statistical significance of
periodicities inferred from the record of reconstructed solar activity on the
basis of cosmogenic isotopes. They showed that power spectra from realizations
covering 10,000 years of simulated time (matching the length of the
reconstructed solar data) typically exhibit spectral peaks at various periods
that are qualitatively similar to those found in the solar data. Such peaks,
which result from the stochastic noise in the dynamo excitation, can reach
significance levels of $3\sigma$. These results cast doubt on the proposition
that seemingly significant periodicities such as the 9̃0-year Gleissberg and
the 2̃10-year de Vries “cycles” are intrinsic to the solar dynamo and not just
statistical fluctuations. In fact, the sharpness of the corresponding peaks in
the power spectrum indicates a random origin since spectral peaks representing
intrinsic dynamo periodicities tend to be broadened owing to the damping
inherent to the dynamo process.

Apart from stochastic forcing as discussed above, long-term variability in the
dynamo process can also arise from magnetic feedback on the flow and from time
delays in the dynamo process. A comprehensive review of models for the long-
term variability has been provided by Karak (2023).

###  4.3 Notes on predictability

Based on sufficient understanding of the global solar dynamo, the observed
state of the solar magnetic field can be used to predict its future evolution
(for comprehensive reviews, see Petrovay, 2020; Bhowmik et al, 2023). In the
Babcock-Leighton model, the cycle-to-cycle variability is directly related to
the amount of magnetic flux that gets across the equator. At the end of a
cycle, all this flux ends up in the polar regions and the amplitude of the
polar field is strongly correlated with the strength of the subsequent cycle
(e.g., Kumar et al, 2021). The physical basis for this correlation is
explained in Sec. 3.1. Predicting the strength of a cycle thus becomes a
matter of predicting the polar field strength at the end of a cycle, which is
almost equivalent to predicting how much flux is transported across the
equator (see Sec. 3.2). This can be achieved by performing surface-flux-
transport simulations using the actual observations of latitude, magnetic
flux, and tilt angle of each emerging active region. Such simulations rather
accurately reproduce the amount of flux in each hemisphere and the resulting
axial dipole moment as a predictor for the subsequent cycle (Jiang et al,
2015; Yeates et al, 2023). A prediction for the next cycle during an ongoing
cycle (i.e., before all the active regions of the cycle have emerged) can be
made by including all active regions which have been observed and performing
Monte-Carlo simulations of the effect of the active regions which have not yet
emerged, using their average properties (Cameron et al, 2016). The result of
this approach is the prediction of the polar field at the end of the cycle
with error estimates. These can then be converted into predictions for the
strength of the subsequent cycle.

In any case, all predictions are limited by the inherent randomness of the
dynamo process (Jiang et al, 2018; Kitchatinov et al, 2018). For instance,
individual “rogue active regions” can have a strong effect and may potentially
even shut down the the large-scale dynamo and initiate a grand minimum episode
(Nagy et al, 2017).

##  5 Outlook

Considering the enormous range of scales and the complexity of physical
processes governing the interaction of turbulent convection, differential
rotation, and magnetic field in the solar convection zone, it is rather
surprising that a comparatively simple approach such as the Babcock-Leighton
scenario seems to provide such a successful description of the solar dynamo
process. An important factor here is that key ingredients of the model, such
as the properties of bipolar magnetic regions (latitude range, tilt, flux
distribution, etc.) and surface flows, can be obtained by observations. To a
large extent, such observations are not available for other stars, so that
basic input is missing for the application of the model. Furthermore, the
underlying processes leading to flux emergence, i.e., the roles of convective
flows, magnetic buoyancy, and instabilities deep in the convection zone, are
largely not understood. In the absence of direct observational evidence, 3D
MHD simulations seem to be the only possibility here. Considering the
impressive progress that simulations have seen during the last decade, there
is hope that the complexity of these processes will be better understood in
the not-too-far future, providing an even better basis for simplified models
such as the Babcock-Leighton approach. Still, we need to exercise caution and
be aware of the severe limitations of all our efforts to understand the solar
dynamo as lucidly expounded by Parker (2009) and Spruit (2011, 2012).

Acknowledgments Sunspot data were obtained from the World Data Center SILSO,
Royal Observatory of Belgium, Brussels. The data for the $aa$-index were
obtained from https://isgi.unistra.fr/indices_aa.php

## Ethics Declarations

Competing interests

The authors declare they have no conflicts of interest.

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