11institutetext: Department of Astrophysics-AIM, University of Paris-Saclay
and University of Paris, CEA, CNRS, Gif-sur-Yvette Cedex 91191, France
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institut für
Sonnensystemforschung, 37077 Göttingen, Germany

# How well does surface magnetism represent deep Sun-like star dynamo action?

A. J. Finley 11 A. S. Brun 11 A. Strugarek 11 R. Cameron 22

(Received March 30, 2023; accepted - -, -)

###### Abstract

Context. For Sun-like stars, the generation of toroidal magnetic field from
poloidal magnetic field is an essential piece of the dynamo mechanism powering
their magnetism. Previous authors have estimated the net toroidal flux
generated in each hemisphere of the Sun by exploiting its conservative nature.
This only requires observations of the photospheric magnetic field and surface
differential rotation.

Aims. We explore this approach using a 3D magnetohydrodynamic dynamo
simulation of a cool star, for which the magnetic field and its generation is
exactly known throughout the entire star.

Methods. Changes to the net toroidal flux in each hemisphere were evaluated
using a closed line integral bounding the cross-sectional area of each
hemisphere, following the application of Stokes-theorem to the induction
equation; the individual line segments corresponded to the stellar surface,
base, equator, and rotation axis. The influence of the large-scale flows, the
fluctuating flows, and magnetic diffusion to each of the line segments was
evaluated, along with their depth-dependence.

Results. In the simulation, changes to the net toroidal flux via the surface
line segment typically dominate the total line integral surrounding each
hemisphere, with smaller contributions from the equator and rotation axis. The
surface line integral is governed primarily by the large-scale flows, and the
diffusive current; the later acting like a flux emergence term due to the use
of an impenetrable upper boundary in the simulation. The bulk of the toroidal
flux is generated deep inside the convection zone, with the surface
observables capturing this due to the conservative nature of the net flux.

Conclusions. Surface magnetism and rotation can be used to produce an estimate
of the net toroidal flux generated in each hemisphere, allowing us to
constrain the reservoir of magnetic flux for the next magnetic cycle. However,
this methodology cannot identify the physical origin, nor the location, of the
toroidal flux generation. In addition, not all dynamo mechanisms depend on the
net toroidal field produced in each hemisphere, meaning this method may not be
able to characterise every magnetic cycle.

###### Key Words.:

Stellar Physics – Stellar Magnetism

##  1 Introduction

During the solar cycle (see review of Hathaway, 2015, and references therein),
the classical solar dynamo scenario pictures an originally poloidal magnetic
field that is acted upon by the Sun’s internal differential rotation
transforming the field into toroidal wreaths at the base of the convection
zone (see review by Charbonneau, 2020). Once this toroidal field becomes
sufficiently strong, magnetic flux buoyantly rises towards the surface
(Browning & Priest, 1984; Birch et al., 2016), twisting in response to the
Coriolis force (D’silva & Choudhuri, 1993; Fan, 2008), and convective motions
(Jouve et al., 2012), producing Joy’s law (Schunker et al., 2020). The net
effect of which regenerates the poloidal field at the surface, but with the
opposite polarity to the original poloidal field (Mordvinov & Kitchatinov,
2019; Noraz et al., 2022). Although, the exact mechanism for regenerating the
poloidal field is still debated, and could also depend on deep-seated cyclonic
convection (Parker, 1955; Mason et al., 2002), or magnetic buoyancy
instabilities (Vasil & Brummell, 2008). An overview of these mechanisms can be
found in Brun & Browning (2017).

The evolution of the Sun’s photospheric magnetic field has been continuously
studied during the last four solar cycles (since the 1970s), from ground-based
(Scherrer et al., 1977), and later space-based (Scherrer et al., 1995, 2012),
observatories. Flux emergence begins at mid-latitudes and progresses down
towards the equator during the approximately 11 year solar activity cycle (van
Driel-Gesztelyi & Green, 2015). The emerged magnetic field locally suppresses
magnetoconvection at the surface, creating dark sunspots, and bright facular
regions. The magnetic cycles of other Sun-like stars have begun to be mapped
using magnetic activity proxies (Baliunas & Vaughan, 1985; Egeland et al.,
2016), Zeeman-Doppler imaging (Saikia et al., 2016, 2018), and the occultation
of starspots by transiting exoplanets (e.g. Morris et al., 2017). Although,
not all cycles are observed to be solar-like (see Jeffers et al., 2022)

The regeneration of toroidal magnetic field from poloidal magnetic field
remains poorly understood, despite the growing number of magnetic cycle
observations. Cameron & Schüssler (2015) proposed a method for evaluating the
net toroidal flux in each hemisphere of the Sun that only requires
observations of the photospheric magnetic field and the Sun’s differential
rotation pattern. The method assumes that the cancellation of toroidal flux
between hemispheres can be treated as a simple loss term. However, this has
not been explored in a scenario where the net toroidal flux in each hemisphere
was actually known. In addition, the net toroidal flux may not scale with the
total magnetic energy, due to cancellation effects, and thus fail to capture
the underlying magnetic cycle.

If the net toroidal flux is important to the magnetic cycle of a star, this
method has enormous value in estimating the reservoir of toroidal flux
available for the next magnetic cycle. For the solar dynamo, some factors
point to the net toroidal field being relevant including: the amount of net
toroidal field estimated using the methodology of Cameron & Schüssler (2015)
being almost the same as is lost through flux emergence (Cameron & Schüssler,
2020); changes in the amount of toroidal flux produced is reflected in the
amount lost through emergence; and at times of maxima, flux emergence occurs
with a systematic preference for the same orientation (at all latitudes in
each hemisphere) in accordance with Hale’s law (Cameron et al., 2018).
However, this may not be the case for all dynamo mechanisms.

In this study, we apply the methodology from Cameron & Schüssler (2015) to a
cyclic dynamo simulation where the net toroidal flux in each hemisphere is
known, and the fundamental assumptions of the method can be tested. Section 2
describes the methodology of Cameron & Schüssler (2015), and the
magnetohydrodynamic simulation used in this work, published in Brun et al.
(2022). Section 3 details the application of this method to the simulation,
and observations of the Sun. The accuracy of this method in reproducing the
deep internal stellar dynamo from surface measurements alone, its capability
to diagnose the dynamo process, and the potential caveats, are summarised in
Section 4.

##  2 Methodology

Figure 1: Schematic depiction of the Stokes-theorem contour, overlaid on a
snapshot of the ASH simulation. The dark contour $C_{H}$ shows the path of
integration around the area $A_{H}$ in the northern hemisphere, broken into
four segments; surface, equator, base, and axis of rotation. The left
hemisphere shows the structure in the radial field at the outer boundary of
the simulation. The right hemisphere shows a slice-through the azimuthally
averaged toroidal magnetic field in the simulation.

###  2.1 Evaluating the net toroidal flux generation

The method proposed by Cameron & Schüssler (2015) is based on the application
of Stokes-theorem to the induction equation. First the induction equation is
written in terms of the azimuthaly averaged large-scale magnetic field and
velocity, $\bf{B}$ and $\bf{V}$, plus the turbulent magnetic field and
velocity components, $\bf{b}$ and $\bf{v}$, such that,

$\frac{\partial\bf{B}}{\partial
t}=\nabla\times\bigg{(}\bf{V}\times\bf{B}+\langle\bf{v}\times\bf{b}\rangle-\eta\bf{J}\bigg{)},$
(1)

where $\eta$ is the magnetic diffusivity,
$\bf{J}=$$1/\mu_{0}$$\nabla\times\bf{B}$ is the current density, and
$\langle\bf{v}\times\bf{b}\rangle$ represents the azimuthaly averaged
correlation between the turbulent magnetic field and velocity components.

Contours that enclose the northern and southern hemispheres are selected,
denoted by $C_{H}$, each composed of four line integrals following the
surface, base, equator, and axis of rotation (see dark contour in Figure 1).
By applying Stokes-theorem to the induction equation over the area inside the
contour $A_{H}$, the time-evolution of the net toroidal flux in each
hemisphere can be recovered using,

$\displaystyle\frac{d\Phi_{H}}{dt}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\int_{A_{H}}B_{\phi}dA_{H},$ (2) $\displaystyle=$
$\displaystyle\oint_{C_{H}}\bigg{(}\bf{V}\times\bf{B}+\langle\bf{v}\times\bf{b}\rangle-\eta\bf{J}\bigg{)}\cdot
dl.$

Each line segment contributes to the evolution of the net toroidal flux via
the projection of the large-scale flows, the fluctuating flows, and magnetic
diffusion onto the direction of the line segment. In spherical coordinates
($r,\theta,\phi$), for the northern hemisphere $A_{N}$, this is explicitly
written,

$\displaystyle\frac{d\Phi_{N}}{dt}$ $\displaystyle=$
$\displaystyle\int_{\text{surface}}^{0\text{ to
}\pi/2}\bigg{(}V_{r}B_{\phi}-V_{\phi}B_{r}+\langle
v_{r}b_{\phi}\rangle-\langle v_{\phi}b_{r}\rangle-\eta
J_{\theta}\bigg{)}R_{*}d\theta$ (3) $\displaystyle-$
$\displaystyle\int_{\text{equator}}^{R_{*}\text{ to
}R_{0}}\bigg{(}V_{\theta}B_{\phi}-V_{\phi}B_{\theta}+\langle
v_{\theta}b_{\phi}\rangle-\langle v_{\phi}b_{\theta}\rangle-\eta
J_{r}\bigg{)}dr$ $\displaystyle-$
$\displaystyle\int_{\text{base}}^{\pi/2\text{ to
}0}\bigg{(}V_{r}B_{\phi}-V_{\phi}B_{r}+\langle v_{r}b_{\phi}\rangle-\langle
v_{\phi}b_{r}\rangle-\eta J_{\theta}\bigg{)}R_{0}d\theta$ $\displaystyle+$
$\displaystyle\int_{\text{axis}}^{R_{0}\text{ to
}R_{*}}\bigg{(}V_{\theta}B_{\phi}-V_{\phi}B_{\theta}+\langle
v_{\theta}b_{\phi}\rangle-\langle v_{\phi}b_{\theta}\rangle-\eta
J_{r}\bigg{)}dr,$

where, $R_{*}$ and $R_{0}$ represent the radial distance to the surface and
the base respectively. In this study, these line segments are evaluated
individually, and represent the flux of toroidal field through each of these
boundaries. Each line segment containing contributions from the large-scale
flows, the fluctuating flows, and magnetic diffusion.

Figure 2: Overview of the ASH simulation used in this work. Snapshots of the
simulation at five intervals are displayed in the top row, showing the radial
field at the surface in the left hemisphere and the azimuthally averaged
toroidal flux in the right hemisphere. The second panel shows the time-
evolution of the azimuthally averaged radial magnetic field at the surface.
The third row contains snapshots, similar to the top row, displaying the
surface rotation rate in the left hemisphere and the azimuthally averaged
rotation rate (minus the time-averaged profile) in the right hemisphere. The
fourth panel shows the time-evolution of the azimuthally averaged surface
rotation rate, with the fifth panel containing the residual rotation rate
after the temporal average is subtracted (highlighting deviations coincident
with the reversing of the radial magnetic field). Figure 3: Breakdown of the
net toroidal flux generation terms in the northern and southern hemispheres,
left and right respectively. The top panel displays the contribution of each
line integral (surface, base, equator, and axis) towards the total net
toroidal flux generation in each hemisphere. The panels below, labelled by
their corresponding line integral, further breakdown the contributions to each
line integral in terms of the large-scale flows, the fluctuating flows, and
magnetic diffusion.

###  2.2 Dynamo simulation

In this study, a global convective dynamo simulation performed using the ASH
code (Brun et al., 2004) was analysed. This simulation exhibits cyclic
reversals of its large-scale magnetic field, with a solar-like differential
rotation pattern. The star has a mass of 1.1$M_{\odot}$ and mean rotation rate
three times higher than that of the Sun. The simulation domain is composed of
a stable inner radiative zone, spanning 0.5 to 0.75 stellar radii, with an
outer convective zone reaching up to 0.97 stellar radii. The simulation domain
has a grid resolution of 769 radial ($r$) cells, 512 latitudinal ($\theta$)
cells, and 1024 longitudinal ($\phi$) cells. Impenetrable and stress-free
boundary conditions are maintained at the surface and base of the domain. The
magnetic field is perfectly conducting at the base and matches a potential
field at the surface. Convection is driven by a fixed entropy gradient at both
boundaries. The ASH simulation (M11R3m) is described fully in Brun et al.
(2022) and references therein.

The resulting cyclic reversals of the poloidal magnetic field have a
periodicity of around five years. A temporal interval of 32 years, was
selected for our analysis, spanning several magnetic cycles. Equation 3 was
computed at a weekly cadence throughout this period, i.e. on $\sim 1600$ ASH
simulation snapshots. Figure 2 contains snapshots of the simulation during
this period, along with time-series of the surface magnetic field, and
rotation rate, showing two full magnetic cycles (i.e. four polar field
reversals). The polar field reversals in each hemisphere become desynchronised
around $t=33$ years, and recover during the next few magnetic cycles. This
indicates a strong contribution of the quadrupolar mode during this time (see
discussion in DeRosa et al., 2012). This ASH simulation possesses very
interesting dynamo properties that resemble that of a cyclic solar-like star.
Though, it is by no means considered as being a perfect solar dynamo model.
Here, we aim to characterise how representative surface observables are of the
underlying cyclic dynamo, whose properties are similar to the Sun but not an
exact representation of the Sun. At the base of the convection zone, the long-
term cyclic magnetic field evolution hosts both equator-ward and polar
branches (see Appendix A). This magnetic field is processed as it emerges
through the convection zone to reach the surface. This produces a mostly
dipole field that reverses on relatively long timescales. The increased mass
and radius of the star creates some differences with respect to the solar
dynamo. Here, the convection zone is shallower, the turnover timescale is
shorter (around 24 days), and the rotation rate is three times larger, leading
to a stellar Rossby number of 0.5 (slightly smaller than the Sun). The faster
rotation also drives a differential rotation pattern with flows more
concentrated at the equator. Accordingly, the azimuthally averaged surface
magnetic field is structured into three latitudinal bands in each hemisphere.
The central band in each hemisphere, having polarity opposite to the polar
field, sits within an inflection of the surface differential rotation profile,
where the latitudinal shear changes direction. These features have developed
self-consistently in the simulation. When the surface radial fields reverse,
the new magnetic field polarity begins to strengthen in the equator-ward band.
The strengthening of the surface field then progresses towards the poles.
Despite the opposing field polarity of the central band, the locally reversed
latitudinal shear maintains a consistent sign of toroidal flux generation in
each hemisphere.

As the simulation’s internal differential rotation shears the existing
poloidal field into toroidal field, oppositely directed Maxwell stresses
develops that resist the continued winding of the field. This can
significantly quench the large-scale rotation with respect to the temporal
average profile. Figure 2 contains the azimuthally averaged surface rotation
of the simulation in the fourth row, as well as the residual rotation rate
after the temporal average is subtracted in the bottom row. Time periods of
decreased rotation (coloured blue in the bottom panel) begin in the central
bands and migrate over two to three years towards the poles. This kind of
behaviour is somewhat similar to the torsional oscillation observed in the Sun
(see review of Howe, 2009). During the period of desynchronisation in the
northern hemisphere from $t=$ 33 to 45 years, the growth of toroidal field was
insufficient to influence the internal differential rotation until $t=$ 41
years. The strength of the quenching at the surface is observed to vary from
reversal to reversal. For example, in the southern hemisphere, the feedback of
the reversals at $t=$ 43, 47, and 60 years was significantly smaller than the
reversals at $t=$ 37 and 56 years (10nHz versus 30-40nHz decreases). Once the
surface radial fields have reversed, the flow accelerates producing an
enhancement with respect to the temporal average (coloured red in the bottom
panel).

##  3 Analysis of the toroidal flux budget

###  3.1 Breakdown of each line segment

The contribution from each individual line segment (surface, base, equator,
and axis) surrounding the northern and southern hemispheres are displayed in
Figure 3. Values in Figure 3 (and later 4) have been smoothed, using boxcar-
averaging with a six-month interval, in order to more easily make visual
comparisons. The top panels show the net toroidal flux generation in each
hemisphere, with the contributions from each of the four line segments, as a
function of time. The surface line segment dominates the integral in each
hemisphere, with smaller contributions from the equator and axis. The base
line segment is located at the bottom of the ASH simulation domain, in the
stable layer, and is therefore effectively zero. The lower four panels of
Figure 3 show the net toroidal flux generation provided by each individual
line segment, which is then further divided into the contributions from the
large-scale flows, the fluctuating flows, and magnetic diffusion (see equation
3). For the surface line segment, the large-scale flows drive the overall sum,
with magnetic diffusion having a smaller contribution that is typically in
opposition. This is a result of diffusion acting as a means for the toroidal
flux (built up by the large-scale flows) to emerge through the surface. As the
boundary conditions of the ASH simulation require the radial velocity at the
surface to tend to zero, this term acts like a replacement for flux emergence
by advection. In other words, the flux of magnetic energy (or Poynting flux)
is channelled through the other non-zeroed components (see equations 4 to 7 in
Finley et al., 2022). Hence, this is a loss term with respect to the net
generation below. These observations hold for both the northern and the
southern hemispheres.

The net toroidal flux generated in the northern hemisphere is plotted with a
solid grey line in the top left panel of Figure 4. This was evaluated by
performing a temporal derivative on the net hemispherical toroidal flux using
the weekly simulation outputs, and smoothed to more easily make visual
comparisons. This is compared with the resulting net toroidal flux generation
calculated from the complete line integral with a dashed black line, and just
the surface line segment with a dotted red line. The same information is
available for the southern hemisphere in Appendix B Figure 8. The complete
line integral matches the evolution of the net toroidal flux, with some small
differences owing to the weekly cadence of the snapshots used in its
calculation. The polarity of the northern polar magnetic field ($30^{\circ}$
around the pole) is displayed in red (positive) and blue (negative) along the
bottom of the panel. The surface line integral appears to match best when the
net toroidal flux generation is largest, for example during the reversals of
the polar fields. However, the surface line integral struggles to explain the
net toroidal flux generation when the value is small and comparable with
either the flux transported through the equator between hemispheres, or
sources on the rotation axis.

Figure 4: Comparison of the net toroidal flux generation in the northern
hemisphere of the ASH simulation. The upper left panel shown the time-
evolution of the net toroidal flux generation with the true value in solid
grey, the full line integral in dashed black, and the surface line integral
only in dotted red. The true value and line integral agree, with small
differences due to the data sampling frequency (here data is taken at a weekly
cadence rather than at the much smaller time-step cadence). Invervals labelled
with vertical lines from one to eight, correspond to the smaller panels below.
These panels show, for each interval, the depth dependence of the surface line
integral, and its constituents (the large-scale flows, the fluctuating flows,
and magnetic diffusion). The series of panels on the right contain a more
continuous sampling of this parameter space, showing the time-evolving depth
dependence of the surface line integral and its contributions. Above $\sim
0.75$ stellar radii, the line integral remains roughly constant with depth.
Figure 8 shows the same information but for the southern hemisphere.

###  3.2 Varying the depth of the surface line segment

The toroidal flux in the ASH simulation is generated near the base of the
convection zone (Brun et al., 2022), far from the dynamics at the surface.
This is also shown by the small panels in Figure 4, each one representing the
calculation of the surface line segment at various depths in the northern
hemisphere (in a sense, scanning through the simulation). The contributions
from the large-scale flows, the fluctuating flows, and magnetic diffusion, are
shown with coloured dashed, dash-dot, and dotted lines respectively. The total
is then shown with a solid black line. Provided the other line segments
closing the area below each surface line segment are small, this value
represents the net toroidal flux generation in the ever diminishing area under
the surface line segment. If the net toroidal flux generation inside the area
remains the same, the value of the surface line segment should be constant. In
general, this is the observed behaviour in the outer $10-20\%$ of the
simulation, far away from the principal source of net toroidal flux. As the
line segment moves to depths that contain imbalanced toroidal flux, due to the
local generation of the opposing polarity, the value of the line segment
begins to diverge. The largest values are found near the base of the
convective zone. Notably, in the large-scale flows responsible for shearing
the poloidal field into toroidal field, but also this is accompanied by a rise
in magnetic diffusion that opposes this mechanism.

The rightmost panels of Figure 4 contain a more extensive breakdown of the
depth-dependence of the surface line segment. The top panel showing the total
surface integral, with the lower three panels breaking this down into the
large-scale flows, the fluctuating flows, and magnetic diffusion. In the top
panel, the line integral remains visibly constant down to around 0.85 stellar
radii. This depth varies little in time, showing the dynamo action responsible
for the cyclicality of the simulation is deeply-seated. Below 0.75 stellar
radii, the integral passes into the radiative zone and so diminishes towards
zero. The net toroidal flux generation, therefore lies in the lower half of
the convection zone. The constituents of the line integral, have more
structure with depth than their total. The large-scale flows, the principal
driver of the net toroidal flux evolution, peak around 0.78 stellar radii.
Magnetic diffusivity peaks slightly closer to the base of the convective zone,
directly opposing the shearing from the large-scale flows. Further away from
the base of the convection zone, the contribution from magnetic diffusion
falls off, replaced by the turbulent flows which conserve the net toroidal
flux generation up towards the surface. However, as the radial velocities are
damped progressively towards zero in the outermost $\sim 5\%$ of the
simulation, the turbulent flow contribution decreases near the surface with
magnetic diffusion picking up this contribution.

###  3.3 Comparison with the Sun

Figure 5: The method of Cameron & Schüssler (2015) applied to magnetograms
from WSO, SOHO/MDI, and SDO/HMI plus the surface rotation rate from Snodgrass
(1983). Top panels show the rotation profile and azimuthally averaged magnetic
field. These are combined to produce the middle panel, a measure of the net
toroidal field generated by the large-scale flows, from the surface
observables. The lower panel shows the net toroidal flux generation in each
hemisphere from this process. Figure 6: The method of Cameron & Schüssler
(2015) applied to the ASH simulation. Top panels show the rotation profile and
azimuthally averaged magnetic field. These are combined to produce the middle
panel, a measure of the net toroidal field generated by the large-scale flows,
from the surface observables. The lower panel shows the net toroidal flux
generation in each hemisphere from this process, in comparison to the true
value from the simulation.

We now compare our analysis of the ASH simulation, to the same approach using
observations from the Sun (as in Cameron & Schüssler, 2015). Assuming that the
dominant contribution to equation (3) is the large-scale flows at the surface,
then

$\frac{d\Phi_{H}}{dt}=\int_{0}^{\pi/2}\bigg{(}V_{r}B_{\phi}-V_{\phi}B_{r}\bigg{)}R_{*}d\theta.$
(4)

If the primary contribution to this term is the large-scale differential
rotation on the poloidal magnetic field $V_{\phi}B_{r}$. This can then be
simplified to,

$\frac{d\Phi_{H}}{dt}=\int_{0^{\circ}}^{90^{\circ}}V_{\phi}B_{r}R_{*}d\theta=\int_{0}^{1}\bigg{(}\Omega(\theta)-\Omega_{eq}\bigg{)}B_{r}R_{*}^{2}d\cos\theta,$
(5)

in a rotating frame with the equator $\Omega_{eq}$, for the northern
hemisphere (see Cameron & Schüssler, 2015). Here, magnetograms are taken from
the Wilcox Solar Observatory111Data accessed Jan. 2023:
http://wso.stanford.edu/synopticl.html (WSO, Scherrer et al., 1977) and both
the Michelson Doppler Imager (MDI, Scherrer et al., 1995), onboard the Solar
and Heliospheric Observatory (SOHO), and the Helioseismic and Magnetic Imager
(HMI, Scherrer et al., 2012), onboard the Solar Dynamics Observatory (SDO).
Both SOHO/MDI and SDO/HMI data products222Data accessed May 2023:
http://hmi.stanford.edu/data/synoptic.html include a polar field correction
(Sun et al., 2011; Sun, 2018). The radial magnetic field, shown in Figure 5,
is then multiplied by the solar surface differential rotation profile, taken
form Snodgrass (1983),

$\Omega(\theta)=\Omega_{eq}+\alpha_{2}\cos^{2}\theta+\alpha_{4}\cos^{4}\theta,$
(6)

where $\Omega_{eq}=472.6$ nHz is the equatorial rotation rate, and the values
of $\alpha_{2}=-73.9$ nHz and $\alpha_{4}=-52.1$ nHz. This forms the surface
line segment integrand, shown in the second panel of Figure 5, which is
integrated in each hemisphere to estimate the net toroidal flux generation in
the lower panel of Figure 5. For the Sun, the integrand is dominated by the
action of the differential rotation on the polar fields, which leads to the
net toroidal flux generation in each hemisphere smoothly oscillating. This
also implies that the peak net toroidal flux generation in each hemisphere
occurs when the polar fields are largest.

In comparison, the ASH simulation is more complicated to interpret. Figure 6
repeats this analysis for the simulation, only considering the surface
differential rotation acting on the radial magnetic field. Compared to the
Sun, the simulated differential rotation profile features a steeper profile
around the lower latitude, and a flattening towards the poles. In addition,
the modelled dynamo supports a latitudinally banded radial magnetic field,
where opposite neighbouring polarities in each hemisphere add complexity to
the interpretation. Nevertheless, the intergrand (second panel of Figure 6)
shares similarities with that of the Sun (in Figure 5). The toroidal flux
generation in each hemisphere peaking when the banded radial magnetic field
strengthens and becomes the same polarity in the central and pole-ward bands.
Then diminishes when these bands have mixed polarities.

Figure 3 shows that the surface line segment of the simulation also includes a
significant contribution from the magnetic diffusivity (essentially flux
emergence). So equation (5) will not perform as well as the complete surface
line segment, shown in contrast to the true value from the simulation in
Figure 4. The loss of toroidal magnetic flux through the emergence of bipolar
magnetic regions on the Sun is discussed in Cameron & Schüssler (2020) (see
also Jeffers et al., 2022). In Figure 6, the true net toroidal flux production
in the simulation is plotted with dashed lines for comparison with the
estimate using only the differential rotation and radial magnetic field
information. There are clearly significant departures from the true value,
however these arise due to the strengthening of magnetic diffusivity.
Therefore, the surface diagnostic captures the building of toroidal flux by
the large-scale flows, up to its peak. After this, this method fails as the
magnetic diffusivity increases, offsetting some of the toroidal flux being
generated. The magnetic diffusivity of the simulation is indeed much larger
than expected for the Sun, but in reality the emergence of bipolar magnetic
regions may play a similar role. This should be accounted for in future
estimations of the net toroidal flux production from surface observables
alone.

##  4 Conclusion

The methodology presented in Cameron & Schüssler (2015) allows us to estimate
the net toroidal flux generation inside the Sun, using only the observed
surface magnetic field and differential rotation pattern. This is a useful
measure of the reservoir of toroidal flux that could emerge during the next
magnetic cycle. However, this method has yet to be tested in a scenario where
the net toroidal flux is actually known. In this study, we use a 3D
magnetohydrodynamic dynamo simulation, where the net toroidal flux generation
is known throughout the entire star, to determine the applicability and
limitations of this methodology. This simulation has slightly different
stellar parameters to the Sun, but hosts a cyclic dynamo and solar-like
differential rotation. Following the method of Cameron & Schüssler (2015), the
application of Stokes-theorem to the induction equation, allows us to evaluate
the net toroidal flux generation from a closed line integral bounding the
cross-sectional area of each hemisphere. The line segments of this integral
correspond to the stellar surface, the base of the simulation domain, the
equator, and the rotation axis. In order for this method to be useful, the
surface segment should dominate the integral, as this is the only component
that can be observed for the Sun and other stars.

Based on the application of this methodology to the dynamo simulation, we find
that the surface segment is generally capable of describing the net toroidal
flux generation. This is most reliable during the initial conversion of
poloidal field into toroidal field during minima of activity, where the large-
scale flows dominate the generation of toroidal flux. Although, to ensure the
accuracy of the surface segment, the contributions of the turbulent flows and
magnetic diffusion should also be included. Future studies of the Sun’s
toroidal flux generation should consider the contributions from the turbulent
flows and magnetic diffusion (see Cameron & Schüssler, 2020; Jeffers et al.,
2022; Cameron & Schüssler, 2023). In the dynamo simulation, magnetic diffusion
is large near the surface, acting as a pseudo flux emergence due to the
impenetrable upper boundary. This component plays a significant role after the
large-scale flows have acted to generate the bulk of the net toroidal flux.
During phases of flux emergence, or maxima of activity, the toroidal flux
generation is weak and so the equatorial and rotation axis segments can be
comparable to the total integral. This introduces ambiguity in the net
toroidal flux evolution predicted by surface observables alone during these
periods.

The success of this method, however, does not imply that the surface
differential rotation acting on the poloidal magnetic field is sufficient to
generate the toroidal flux for the next magnetic cycle. It is clear that the
toroidal flux is mostly generated, and confined, near to the base of the
convective zone. Instead, the magnetic field lines are non-local and follow a
conservation law. For this method to be applicable in real stars, the exchange
of toroidal flux between hemispheres must be small or near zero in order for
the unsigned toroidal flux to be a representative measure of the magnetic
field being regenerated by the internal dynamo. Thus, if the toroidal flux is
produced closer to the equator, with an increased possibility of flux
transferring from one hemisphere to another, this method may begin to
breakdown as the reservoir of toroidal energy in one hemisphere is not tightly
linked to the surface flux evolution anymore.

Equally, there are some dynamo mechanisms where the net toroidal flux in each
hemisphere does not correlate with the magnetic activity cycle, as the net
toroidal flux does not measure directly the total magnetic energy available to
power the next cycle. One such example is the classic alpha-omega dynamo wave
with a relatively high wave number. This dynamo produces several bands of
toroidal flux of each polarity in each hemisphere. With appropriate boundary
conditions there would be zero net flux in each hemisphere, with the bands
cancelling, though substantial magnetic energy would be produced. Therefore,
this method may not be applicable to every observed magnetic cycles.

###### Acknowledgements.

This research has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 810218 WHOLESUN), in addition to funding by the Centre
National d’Etudes Spatiales (CNES) Solar Orbiter, and the Institut National
des Sciences de l’Univers (INSU) via the Programme National Soleil-Terre
(PNST). Data manipulation was performed using the numpy (Harris et al., 2020),
and scipy (Virtanen et al., 2020) python packages. Figures in this work are
produced using the python package matplotlib (Hunter, 2007).

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##  Appendix A Toroidal field evolution at the base of the convection zone

Figure 7: Time-evolution of the azimuthally averaged toroidal magnetic field
at the base of the convection zone.

The ASH simulation investigated in this work (M11R3m) possesses some
interesting dynamo properties that resemble a cyclic solar-like star (Brun et
al., 2022). This is best observed at the base of the convection zone, where
the toroidal field is organised. Figure 7 displays the azimuthally averaged
toroidal field at a depth of 0.76 stellar radii. The equator-ward and polar
branches are clearly visible, with the toroidal field in each hemisphere
reversing after around five years.

##  Appendix B Analysis of southern hemisphere

Figure 8: Same as Figure 4, but now for the southern hemisphere.

In Section 3.2, the Cameron & Schüssler (2015) method for estimating the net
toroidal flux generation, was compared against the true value in the ASH
simulation for the northern hemisphere integral using equation (3). Here the
same analysis is performed on the southern hemisphere, where the integral is
formed,

$\displaystyle\frac{d\Phi_{S}}{dt}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\int_{A_{S}}B_{\phi}dA_{S},$ (7) $\displaystyle=$
$\displaystyle\oint_{C_{S}}\bigg{(}\bf{V}\times\bf{B}+\langle\bf{v}\times\bf{b}\rangle-\eta\bf{J}\bigg{)}\cdot
dl.$

and broken down into four line segments as,

$\displaystyle\frac{d\Phi_{S}}{dt}$ $\displaystyle=$
$\displaystyle\int_{\text{surface}}^{0\text{ to
}\pi/2}\bigg{(}V_{r}B_{\phi}-V_{\phi}B_{r}+\langle
v_{r}b_{\phi}\rangle-\langle v_{\phi}b_{r}\rangle-\eta
J_{\theta}\bigg{)}R_{*}d\theta$ (8) $\displaystyle-$
$\displaystyle\int_{\text{equator}}^{R_{*}\text{ to
}R_{0}}\bigg{(}V_{\theta}B_{\phi}-V_{\phi}B_{\theta}+\langle
v_{\theta}b_{\phi}\rangle-\langle v_{\phi}b_{\theta}\rangle-\eta
J_{r}\bigg{)}dr$ $\displaystyle-$
$\displaystyle\int_{\text{base}}^{\pi/2\text{ to
}0}\bigg{(}V_{r}B_{\phi}-V_{\phi}B_{r}+\langle v_{r}b_{\phi}\rangle-\langle
v_{\phi}b_{r}\rangle-\eta J_{\theta}\bigg{)}R_{0}d\theta$ $\displaystyle+$
$\displaystyle\int_{\text{axis}}^{R_{0}\text{ to
}R_{*}}\bigg{(}V_{\theta}B_{\phi}-V_{\phi}B_{\theta}+\langle
v_{\theta}b_{\phi}\rangle-\langle v_{\phi}b_{\theta}\rangle-\eta
J_{r}\bigg{)}dr.$

The net toroidal flux generated in the southern hemisphere is plotted with a
solid grey line in the top left panel of Figure 8, in comparison to the true
value shown with a black dashed line, and the surface integral only with a
dotted red line. The smaller panels in the lower right of Figure 8 display
radial scans through the simulation of the surface line integral. This is
expanded in the right panels, showing the depth dependence of the net toroidal
flux generation estimated from the surface line integral (split into the
contributions from large-scale flows, fluctuating flows, and magnetic
diffusion).