Vol.0 (20xx) No.0, 000–000

11institutetext: Institute of Solar-Terrestrial Physics SB RAS, Lermontov Str.
126A, 664033, Irkutsk, Russia<EMAIL_ADDRESS>
Received 2022 May 13; accepted 2022 September 30

#  The Dependence of Stellar Activity Cycles on Effective Temperature

Leonid Kitchatinov

###### Abstract

This paper proposes the idea that the observed dependence of stellar activity
cycles on rotation rate can be a manifestation of a stronger dependence on the
effective temperature. Observational evidence is recalled and theoretical
arguments are given for the presence of cyclic activity in the case of
sufficiently slow rotation only. Slow rotation means proximity to the observed
upper bound on the rotation period of solar-type stars. This maximum rotation
period depends on temperature and shortens for hotter stars. The maximum
rotation period is interpreted as the minimum rotation rate for operation of a
large-scale dynamo. A combined model for differential rotation and the dynamo
is applied to stars of different mass rotating with a rate slightly above the
threshold rate for the dynamo. Computations show shorter dynamo cycles for
hotter stars. As the hotter stars rotate faster, the computed cycles are also
shorter for faster rotation. The observed smaller upper bound for rotation
period of hotter stars can be explained by the larger threshold amplitude of
the $\alpha$-effect for onset of their dynamos: a larger $\alpha$ demands
faster rotation. The amplitude of the (cycling) magnetic energy in the
computations is proportional to the difference between the rotation period and
its upper bound for the dynamo. Stars with moderately different rotation rates
can differ significantly in super-criticality of their dynamos and therefore
in their magnetic activity, as observed.

###### keywords:

dynamo — stars: activity — stars: magnetic field — stars: solar-type — stars:
rotation

##  1 Introduction

Large-scale magnetic fields and large-scale flows in late-type stars are
believed to be caused by global rotation. Accordingly, many observational and
theoretical studies are focused on the dependence of stellar magnetic activity
and/or differential rotation on the rotation rate. This paper suggests that
magnetic activity cycles and differential rotation are more sensitive to
another stellar parameter of the effective temperature.

The temperature dependence can manifest itself as seeming dependence on
rotation rate because of the following. Solar-type stars are spinning-down
with age due to the angular momentum loss for magnetically coupled wind (Kraft
1967). Proportionality constant in the Skumanich (1972) law $P_{\rm
rot}\propto t^{1/2}$ for stellar spindown is a decreasing function of
temperature (Barnes 2007). Among stars of approximately the same age $t$,
cooler stars have longer rotation period $P_{\rm rot}$. Relatively fast
rotation in a sample of (solar-type) stars is usually represented by F-stars
while cooler K-stars are on the slow rotation side of the sample (see fig. 3
in Donahue et al. 1996, as a characteristic example). A rotation rate
dependence inferred from the sample can therefore include an implicit
dependence on temperature.

In the case of differential rotation, this statement is more or less evident
by now. Observations (Donati & Collier Cameron 1997; Barnes et al. 2005;
Balona & Abedigamba 2016) and theoretical modelling (Kitchatinov & Rüdiger
1999; Kitchatinov & Olemskoy 2012) both suggest that former detections of an
increasing trend in dependence of the differential rotation on rotation rate
can result from combining a strong increase of the differential rotation with
temperature with its moderate dependence on rotation rate.

This paper considers the possibility that stellar activity cycles can also be
more dependent on temperature than on the rate of rotation. Observations of
stellar cycles were mainly focused on the dependence on rotation rate. A
tendency of shorter cycles for faster rotation has been found (Noyes et al.
1984b) though with a considerable scatter and possible discontinuities in this
trend (Brandenburg et al. 1998; Saar & Brandenburg 1999; Böhm-Vitense 2007).
Several attempts at reproducing the trend with dynamo models were undertaken
with mixed success (Jouve et al. 2010; Karak et al. 2014; Hazra et al. 2019;
Pipin 2021). There is an extensive literature on direct numerical simulations
of global stellar convection including simulations of the activity cycles (see
Strugarek et al. 2017; Warnecke 2018; Strugarek et al. 2018; Brun et al. 2022,
and references therein). The decreasing trend in cycle duration with rotation
rate remains however unexplained.

Observations of stellar rotation revealed the upper bound on rotation period
of solar-type stars (Rengarajan 1984; van Saders et al. 2016). Magnetic
activity is low near this maximum rotation period (Metcalfe et al. 2016). The
maximum period can therefore be interpreted as a minimum rotation rate for
large-scale stellar dynamos (Cameron & Schüssler 2017; Kitchatinov &
Nepomnyashchikh 2017b; Metcalfe & Egeland 2019). We keep to this
interpretation and apply a joint mean-field model for differential rotation
and dynamo to stars of different mass rotating with a period close to its
observationally detected upper bound. The computations show an increase in
differential rotation and shortening of the dynamo-cycle for stars of higher
mass and effective temperature. As the maximum rotation period is shorter for
hotter stars (van Saders et al. 2019), the temperature trends also imply
shorter cycles and larger differential rotation for faster rotation, though
the rotation rate is not the main physical parameter governing the trends.

We also recall observational evidence and discuss theoretical arguments for
the presence of activity cycles in sufficiently slow rotating stars only. This
is done in the following Section 2. Section 3 explains our method and the
dynamo model. Section 4 presents and discusses the results. Section 5
summarises the results and concludes.

##  2 Should cyclic activity be expected for rapid rotators?

The project of long-term monitoring of chromospheric activity at the Mount
Wilson Observatory revealed activity cycles similar to the 11-year solar cycle
on many sun-like stars (Wilson 1978; Baliunas et al. 1995). Summarising the
results of the project, Baliunas et al. (1995) noted that cycles on young
rapid rotators are rare but slow rotators as old as the Sun have cycles
(except for the cases of flat and low activity like the solar Maunder
minimum). Recently, Boro Saikia et al. (2018) compiled a chromospheric
activity catalog of Mount Wilson data and more recent data on solar-type
stars. Their table 4 gives stars with cyclic activity.

Figure 1 shows positions of the stars with activity cycles from the catalog by
Boro Saikia et al. (2018) on the plane of the rotation period $P_{\rm rot}$
and the $B-V$ colour. Only main-sequence stars are included in the plot111Some
inaccuracies were corrected in table 4 by Boro Saikia et al. (2018) when
producing Fig. 1: The color $B-V=0.594$ given in the table for HD 160346 is
too small for the K3 star. It was corrected to the value of 0.971 observed
(Høg et al. 2000). The HD 81809 misclassified in the table as a main-sequence
star is a binary system whose active component is subgiant (Egeland 2018).
Subgiants are not included in Fig. 1..

Figure 1: Positions of the main-sequence stars showing activity cycles on the
$P_{\rm rot}-(B-V)$ plane. Red symbols show stars with well-defined solar-like
activity cycles. Green triangles show stars with multiple cycles and black
squares - the stars with uncertain “probable” cycles. The dotted and dashed
lines show the maximum rotation period $P_{\rm max}$ for stellar spindown
according to Rengarajan (1984) and van Saders et al. (2019) respectively.

Figure 1 shows also the observed upper bound $P_{\rm max}$ on the rotation
period. The dotted line in Fig. 1 is the linear approximation for the $P_{\rm
max}-(B-V)$ relation by Rengarajan (1984). Rengarajan found this approximation
for $B-V<1$. Based on a vast statistics of recent data on stellar rotation,
van Saders et al. (2019) found that the $P_{\rm max}-(B-V)$ relation is well
approximated by the constant value

${\rm Ro}=2.08$ (1)

of the Rossby number ${\rm Ro}=P_{\rm rot}/\tau_{\rm c}$; the convective
turnover time $\tau_{\rm c}$ is a function of $B-V$ colour (see eq. (4) in
Noyes et al. 1984a). The $P_{\rm max}$ after Eq. (1) is shown by the dashed
line in Fig. 1. Proximity to this line can quantify the meaning of ‘slow
rotation’ for the main-sequence dwarfs.

Following Boro Saikia et al. (2018), the quality of the activity cycles in
Fig. 1 is encoded by colour: red symbols show well defined solar-like cycles,
green triangles show stars with multiple cycles, and black squares - stars
with “probable” cycles. Almost all stars with well-defined cycles are slow
rotators with $P_{\rm rot}$ close to $P_{\rm max}$. The only exception is HD
152391. Olspert et al. (2018) found doubly periodic activity for this star
with their harmonic regression model. This may explain the position of HD
152391 in the region of green symbols in Fig. 1.

Also on physical grounds, it seems plausible to expect cyclic activity for
slow rotators only.

Hydromagnetic dynamos of any kind can be understood as the instability of
conducting fluids to magnetic disturbances: only if whatever small but finite
seed field is present, can a dynamo-instability amplify and support the field.
Similar to all other instabilities, dynamo-instability has dimensionless
governing parameters and onset when the parameters exceed a certain threshold
value. The empirical Eq. (1) can be seen as the normalised rotation rate for
the onset of large-scale stellar dynamos. The red symbols of Fig. 1 close to
the $P_{\rm max}$-line represent slightly supercritical cyclic dynamos.

Various instabilities behave similarly in dependence on their governing
parameters. For slightly supercritical parameters, new steady (change of
stability) or oscillatory (overstability) states are normally realised
(Chandrasekhar 1961). In a highly supercritical case, instabilities change -
as a rule - to a turbulent regime possibly with an intermediate stage of
multi-periodic dynamics (see, e.g., Landau & Lifshitz 1987). If large-scale
dynamo instability is not an exception to this rule, some kind of dynamo
turbulence with non-cyclic activity should be expected for the highly
supercritical regime of rapid rotation.

The expectation is hard to test with computations. This would require fully
nonlinear and highly supercritical dynamo models lacking at the moment. The
strongly nonlinear regime of dynamos in rapid rotators is in particular
indicated by their observed large torsional oscillations (Collier Cameron &
Donati 2002). Changes to turbulence usually proceed via break of symmetry of
slightly supercritical regimes. An adequate dynamo model has to be nonlinear
and non-axisymmetric.

Lacking such a dynamo model, computations of this paper are restricted to
slightly supercritical cyclic dynamos of stars with rotation period close to
$P_{\rm max}$ of Fig. 1 corresponding to the constant Rossby number of Eq.
(1).

##  3 Model and method

###  3.1 Combined model of differential rotation and dynamo

We apply a joint model of dynamo and differential rotation by Kitchatinov &
Nepomnyashchikh (2017a, b) to stars of different mass $0.7\leq M/M_{\odot}\leq
1.2$ and rotation period close to $P_{\rm max}$ of Eq. (1). Differential
rotation and meridional flow for dynamo computations are supplied by an
axisymmetric hydrodynamical mean-field model. The model differs from that of
Kitchatinov & Olemskoy (2011) only in a modification of the mixing length
$\ell$: the mixing length of its standard definition
$\ell_{0}=\alpha_{MLT}H_{\rm p}$ ($H_{\rm p}$ is the pressure scale height) is
now reduced near the inner boundary $r_{\rm i}$ of the convection zone so that
it can exceed the distance to the boundary only slightly:

$\ell=\ell_{\mathrm{min}}+\frac{1}{2}\left(\ell_{0}-\ell_{\mathrm{min}}\right)\left[1+\mathrm{erf}\left(\frac{r/r_{\mathrm{i}}-x_{\ell}}{d}\right)\right].$
(2)

In this equation, $\ell_{\rm min}=0.01R$ equals one percent of the stellar
radius, $\mathrm{erf}$ is the error function, and other parameters will be
specified later. Our differential rotation model differs from other mean-field
formulations in that it does not prescribe the eddy transport coefficients but
computes them. The eddy viscosity in particular is defined by the equation

$\nu_{{}_{\mathrm{T}}}=-\frac{\tau\ell g}{15c_{\rm p}}\frac{\partial
S}{\partial r},$ (3)

where $g$ is gravity, $\tau$ is the (position dependent) convective turnover
time, $c_{\rm p}$ is the specific heat capacity at constant pressure. The
specific entropy $S$ in Eq. (3) is a dependent variable of the model. The
entropy is controlled by the (nonlinear) heat transport equation that is one
of three equations of the model (the other two being the equations for the
meridional flow and angular velocity). Recently Jermyn et al. (2018) discussed
the performance of this closure method in the convective turbulence theory.

We avoid repeating other details of the differential rotation model all of
which can be found elsewhere (Kitchatinov & Olemskoy 2011, 2012).

Our dynamo model is a particular version of the flux-transport models
pioneered by Choudhuri et al. (1995) and Durney (1995). The models’ name
reflects the importance of magnetic field advection by the meridional flow.
The flux-transport models with the $\alpha$-effect of Babcock-Leighton (BL)
type agree closely with solar observations (Jiang et al. 2013; Charbonneau
2020).

Our dynamo model is formulated for a spherical layer of a stellar convection
zone. The standard spherical coordinate system ($r,\theta,\phi$) with the
rotation axis as the polar axis is used. The formulation assumes axial
symmetry of the mean magnetic field

${\mbox{\boldmath$B$}}=\hat{\mbox{\boldmath$\phi$}}B+{\mbox{\boldmath$\nabla$}}\times\left(\hat{\mbox{\boldmath$\phi$}}\frac{A}{r\sin\theta}\right)$
(4)

and flow

${\mbox{\boldmath$V$}}=\hat{\mbox{\boldmath$\phi$}}\,r\sin\theta\,\Omega+\rho^{-1}{\mbox{\boldmath$\nabla$}}\times\left(\hat{\mbox{\boldmath$\phi$}}\frac{\psi}{r\sin\theta}\right).$
(5)

In these equations, $B$ is the toroidal magnetic field, $A$ is the poloidal
field potential, $\Omega$ is the angular velocity, $\psi$ is the stream
function for the meridional flow, $\hat{\mbox{\boldmath$\phi$}}$ is the
azimuthal unit vector, and $\rho$ is density.

Two joint dynamo equations for the poloidal and toroidal magnetic fields read

$\displaystyle\frac{\partial A}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{\rho
r^{2}\,\sin\theta}\left(\frac{\partial\psi}{\partial r}\frac{\partial
A}{\partial\theta}-\frac{\partial\psi}{\partial\theta}\frac{\partial
A}{\partial r}\right)+r\sin\theta\,{\cal E}_{\phi}\ ,$ (6)
$\displaystyle\frac{\partial B}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{\rho r^{2}}\frac{\partial\psi}{\partial
r}\frac{\partial}{\partial\theta}\left(\frac{B}{\sin\theta}\right)-\frac{1}{r\sin\theta}\frac{\partial\psi}{\partial\theta}\frac{\partial}{\partial
r}\left(\frac{B}{\rho r}\right)$ (7) $\displaystyle+$
$\displaystyle\frac{1}{r}\left(\frac{\partial\Omega}{\partial r}\frac{\partial
A}{\partial\theta}-\frac{\partial\Omega}{\partial\theta}\frac{\partial
A}{\partial r}+\frac{\partial(r{\cal E}_{\theta})}{\partial
r}-\frac{\partial{\cal E}_{r}}{\partial\theta}\right)\ ,$

where ${\mbox{\boldmath$\cal
E$}}=\langle{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}\rangle$ is the
mean electromotive force (EMF, Krause & Rädler 1980), which results from a
correlated action of fluctuating velocities $u$ and magnetic fields $b$ and
includes all the dynamo-relevant effects of convective turbulence.

Expression for the EMF can be rather complicated (Pipin 2008). Some
simplification can be achieved by splitting the EMF in three parts,

${\mbox{\boldmath$\cal E$}}={\mbox{\boldmath$\cal
E$}}^{\alpha}+{\mbox{\boldmath$\cal E$}}^{\rm diff}+{\mbox{\boldmath$\cal
E$}}^{\rm dia},$ (8)

responsible for the $\alpha$-effect, eddy diffusion, and diamagnetic pumping
respectively.

Toroidal field generation by the $\alpha$-effect is neglected in the
$\alpha\Omega$-dynamo. Nonlocal $\alpha$-effect of BL type is prescribed in
the poloidal field equation (6)

${\cal E}^{\alpha}_{\phi}=\frac{\alpha\,B(r_{\rm i},\theta)}{1+(B(r_{\rm
i},\theta)/B_{0})^{2}}\cos\theta\sin^{n_{\alpha}}\theta\,\phi_{\alpha}(r/r_{\rm
e}),$ (9)

where the function

$\phi_{\alpha}(r/r_{\rm e})=\frac{1}{2}\left[1+{\rm erf}\left((r/r_{\rm
e}+2.5h_{\alpha}-1)/h_{\alpha}\right)\right],$ (10)

with $h_{\alpha}=0.02$ peaks near the external boundary $r_{\rm e}=0.97R$.
This boundary is placed shortly below the stellar surface to exclude the near-
surface layer of steep stratification that is difficult to include in the
differential rotation model. The $\alpha$-effect of Eq. (9) describes
generation of the poloidal field near the surface from the bottom toroidal
field. The large value of $n_{\alpha}=7$ used in the model implies that the
magnetic flux-tubes whose rise to the surface produces the $\alpha$-effect are
formed at low latitudes (Kitchatinov 2020). The value $B_{0}=10^{4}$ G of the
$\alpha$-effect quenching parameter in Eq. (9) gives reasonable results for
the Sun (Kitchatinov & Nepomnyashchikh 2017a). This value was used for the
star of one solar mass. For other masses, the parameter was re-scaled in
proportion to the square root of density at the inner boundary,
$B_{0}=10^{4}\sqrt{\rho_{\rm i}(M)/\rho{\rm{}_{i}}(1M_{\odot})}$ G, to reflect
the scaling of flux-tube rise velocity with the Alfven velocity (D’Silva &
Choudhuri 1993).

Diffusive part of the EMF of our model reads

${\mbox{\boldmath$\cal E$}}^{\rm
diff}=-\eta{\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$B$}}-\eta_{\|}{\hat{\mbox{\boldmath$\Omega$}}}\times\left({\hat{\mbox{\boldmath$\Omega$}}}\cdot{\mbox{\boldmath$\nabla$}}\right){\mbox{\boldmath$B$}},$
(11)

where ${\hat{\mbox{\boldmath$\Omega$}}}={\mbox{\boldmath$\Omega$}}/\Omega$ is
the unit vector along the rotation axis. Magnetic diffusivity of Eq. (11) is
anisotropic: the diffusivity $\eta$ for a direction normal to the rotation
axis is smaller than the diffusivity $\eta+\eta_{\|}$ along this axis. The
anisotropy is caused by rotation,

$\eta=\eta_{{}_{\mathrm{T}}}\phi(\Omega^{*}),\ \
\eta_{\|}=\eta_{{}_{\mathrm{T}}}\phi_{\|}(\Omega^{*}).$ (12)

The functions $\phi(\Omega^{*})$ and $\phi_{\|}(\Omega^{*})$ of the Coriolis
number

$\Omega^{*}=2\tau\Omega$ (13)

are given in Kitchatinov et al. (1994). The rotationally induced anisotropy is
important for the differential rotation model. Only with account for
anisotropy of the eddy heat transport, can the helioseismological rotation law
be reproduced (Rüdiger et al. 2013). We include the diffusion anisotropy in
the dynamo model for consistency and for its better performance (Pipin et al.
2012).

The diamagnetic pumping results from inhomogeneity of the turbulence intensity
(Krause & Rädler 1980). Our dynamo model employs the anisotropic pumping
effect for rotating fluids as it was derived by Kitchatinov & Nepomnyashchikh
(2016),

${\mbox{\boldmath$\cal E$}}^{\rm
dia}=-({\mbox{\boldmath$\nabla$}}{\tilde{\eta}})\times{\mbox{\boldmath$B$}}+({\mbox{\boldmath$\nabla$}}\eta_{\|})\times{\hat{\mbox{\boldmath$\Omega$}}}({\hat{\mbox{\boldmath$\Omega$}}}\cdot{\mbox{\boldmath$B$}})\,,$
(14)

with another diffusivity coefficient

${\tilde{\eta}}=\eta_{{}_{\mathrm{T}}}\phi_{1}(\Omega^{*}).$ (15)

Allowance for the pumping effect generally improves performance of the solar
dynamo models (Guerrero & de Gouveia Dal Pino 2008; Karak & Cameron 2016;
Zhang & Jiang 2022).

Magnetic eddy diffusivity $\eta_{{}_{\mathrm{T}}}$ can be estimated using the
computed eddy viscosity of Eq. (3),
$\eta_{{}_{\mathrm{T}}}=\nu_{{}_{\mathrm{T}}}/{\rm Pm}$, where Pm is the
magnetic Prandtl number. The problem however is that our differential rotation
model uses local mixing-length approximation and does not include the low
diffusivity layer of overshoot convection that is important for the dynamo. We
reduce the diffusivity near the inner boundary to model the layer:

$\eta_{{}_{\mathrm{T}}}=\frac{1}{\mathrm{Pm}}\left[\nu_{\mathrm{i}}+\frac{1}{2}(\nu_{{}_{\mathrm{T}}}-\nu_{\mathrm{i}})\left(1+\mathrm{erf}\left(\frac{r/r_{\mathrm{i}}-x_{\eta}}{d}\right)\right)\right],$
(16)

where $\nu_{\rm i}=10^{-4}\nu_{\rm max}$ ($\nu_{\rm max}$ is the maximum value
of $\nu_{{}_{\mathrm{T}}}$ within the convection zone, Pm = 3 in all
computations of this paper. Parameters in the Eqs. (2) and (16) for the star
of $1M_{\odot}$ are taken to be $x_{\ell}=1.01,\ d=0.025,\ x_{\eta}=1.1$.

Figure 2: Depth profiles of the diffusion coefficients $\eta$ (full line),
$\eta_{\|}$ (dotted) and $\tilde{\eta}$ (dashed) for a star of $1M_{\odot}$
rotating with a period of 23.5 days.

Figure 2 shows depth profiles of the diffusivity coefficients of Eqs. (11) and
(15) for the $1M_{\odot}$ star. The large (but realistic, Cameron & Schüssler
2016) diffusivity of this Figure can result in too short dynamo-cycles. Cycle
periods of about 10 yr in our model are due to the downward diamagnetic
pumping that transports the magnetic field into the near-bottom layer of low
diffusion.

The $\eta_{\|}$ coefficient decreases towards the surface in Fig. 2. This is
because the depth-dependent Coriolis number of Eq. (13) decreases towards the
surface leading to a smaller rotation-induces diffusion anisotropy.

The parameter values $x_{\ell}=1.01,\ d=0.025,\ x_{\eta}=1.1$ were formerly
applied to stars of various mass (Kitchatinov & Nepomnyashchikh 2017b). It has
been realized since then that the near bottom layer of small diffusion
occupies a larger part of the convection zone in stars of larger mass in this
case (a half of the thin convection zone of the $1.2M_{\odot}$ star). To avoid
such a non-physical prescription, we re-scale the parameters so that all the
characteristic scales constitute the same fractions of the convection zone
thickness in stars of different mass:

$d=0.0745\left(\frac{r_{\rm e}}{r_{\rm i}}-1\right),\ \
x_{\ell}=0.03\frac{r_{\rm e}}{r_{\rm i}}+0.97,\ \ x_{\eta}=0.3\frac{r_{\rm
e}}{r_{\rm i}}+0.7.$ (17)

With this prescription, Fig. 2 looks almost the same for stars of all
considered mass except for the varying minimum value of $r/R$ in the plot.
Some difference in the results of this paper with Kitchatinov &
Nepomnyashchikh (2017b) is explained by the different prescription for the
parameters of the Eq.(17) and in the definition of $P_{\rm max}$ by Eq. (1)
which was not yet aware of in 2017.

The dynamo model solves numerically the initial value problem for dynamo
equations (7) with a perfect conductor boundary condition imposed on the
bottom and vertical field condition on the top. The initial condition
prescribes the zero toroidal field and the potential

$A_{0}(r,\theta)=\frac{B_{N}(r-r_{\rm i})(2r_{\rm e}-r_{\rm i}-r)}{4(1-r_{\rm
i}/r_{\rm e})^{2}}[1-p+(1+p)\cos\theta]\sin^{2}\theta$ (18)

for the poloidal field, where $B_{N}$ is the field strength on the northern
pole and $-1\leq p\leq 1$ is the parity index ($p=1$ means a quadrupolar
equator-symmetric initial field and it is $p=-1$ for a dipolar antisymmetric
field). Starting from the initial condition of mixed parity, the dynamo code
was run for one thousand years simulated time. This preliminary run suffices
for the dynamo to converge to a periodic oscillation, for which the cycle
period and other results of Sect. 4 were obtained. If the initial condition
(18) had a certain parity ($p=\pm 1$), the numerical solution cannot depart
from this parity. The runs with so-prescribed parity helped to compute the
threshold amplitude $\alpha_{\rm c}$ of the $\alpha$-effect of Eq. (9) for the
onset of dynamo-instability for dipolar ($\alpha_{\rm c}^{\rm d}$) and
quadrupolar ($\alpha_{\rm c}^{\rm q}$) fields.

###  3.2 Estimating rotation period and structure of stars

Computations of differential rotation and dynamo require the structure and
rotation rate of a star to be specified.

We use the gyrochronology relation by Barnes (2007)

$P_{\mathrm{rot}}=at^{n}\left(B-V\ -\ 0.4\right)^{b}\ \mathrm{d},$ (19)

relating the age $t$ (in Myr) and $B-V$ colour of a star to the rotation
period. The parameter values of $a=0.77,\ n=0.512,\ b=0.6$ within their
uncertainty range reproduce closely the case with the Sun.

Figure 3: Stellar age of the maximum rotation period $P_{\rm max}$ as the
function of $B-V$ colour estimated with Eq. (20).

Gyrochronology is believed to overestimate the rotation period after the
rotation slows down to the minimum rate of Eq. (1) (van Saders et al. 2016,
2019). Stellar structure varies slowly at the age when it happens. We
therefore assume that the relation (19) applies up to the age when the maximum
rotation period of Eq. (1) is attained. This age can be roughly estimated from
the reversed Eq. (19):

$t=\left[\frac{2.08\ \tau_{\rm c}}{a(B-V\,-\,0.4)^{b}}\right]^{1/n}.$ (20)

Figure 3 shows the age of Eq. (20) as function of $B-V$ colour. This age of
the large-scale dynamo termination does not decrease monotonously with
increasing temperature.

The EZ model by Paxton (2004) was used to define the evolutionary sequence of
structure models for a star of given mass and metallicity $Z=0.02$. The
colour-temperature relation and the interpolation code by VandenBerg & Clem
(2003) was used to estimate the $B-V$ color corresponding to the structure
models. The rotation period of Eq.(19) was then compared with the $P_{\rm
max}$ of Eq. (1). The structure model with the closest values of these two
rotation periods is assumed to correspond to the star that arrived on the
dashed line of Fig. 1. The Sun is almost on this line. The solar dynamo was
estimated to be about 10% supercritical in the sense of the amplitude $\alpha$
of the $\alpha$-effect of Eq. (9) (Kitchatinov & Nepomnyashchikh 2017b). The
differential rotation of the stars of different mass which ‘arrived on the
dashed line’ of Fig. 1 was computed and then used in the simulations of their
10% supercritical dynamos as explained at the end of Sect. 3.1.

The computations cover the mass range from $0.7M_{\odot}$ to $1.2M_{\odot}$
with increment of $0.05M_{\odot}$.

##  4 Results and discussion

All computations show solar-type differential rotation with faster equatorial
rotation. Figure 4 shows the surface equator-to-pole difference in rotation
rate in dependence on the effective temperature. Hotter stars have larger
differential rotation. As the hotter stars rotate faster, this Figure also
means an increase in the differential rotation with rotation rate. Figure 4 is
a slow rotation counterpart of the observational fig. 2 by Barnes et al.
(2005) and theoretical fig. 10 by Kitchatinov & Olemskoy (2011).

Figure 4: Full line, left scale: difference in rotation rates between equator
and pole as the function of temperature. Dashed line, right scale: the
relative value $\Delta\Omega/\Omega$ of the differential rotation. The scale
on the top shows the rotation period corresponding to the dashed line of Fig.
1.

Figure 4 also shows the dimensionless ratio $\Delta\Omega/\Omega$, which
varies little with $T_{\rm eff}$. As the computations of this Figure were done
for constant Ro = 2.08, the small variation in $\Delta\Omega/\Omega$ means
that $\Delta\Omega\tau_{\rm c}$ does also vary little with $T_{\rm eff}$. The
increase in $\Delta\Omega$ with $T_{\rm eff}$ proceeds in inverse proportion
to decreasing convective turnover time. The scaling with $\tau_{\rm c}^{-1}$
can explain the strong increase in differential rotation with temperature
observed by Barnes et al. (2005).

Figure 5: Periods of the computed dynamo-cycles as the function of
temperature. Similar to Fig. 4, the scale on the top shows the corresponding
rotation period.

Figure 5 shows a similar plot for the period of computed dynamo cycles. These
are the periods of energy oscillation (half-periods of the sign-changing
magnetic cycles). Dynamo computations predict shorter cycles for hotter stars.
For fixed Rossby number of Eq. (1), this temperature trend also implies a
shorter cycle for faster rotation. Similar to the differential rotation, the
observed decrease in activity cycle duration with rotation rate can be at
least partly explained by its temperature dependence. In difference with the
differential rotation, we did not find a normalization for $P_{\rm cyc}$,
which varies little with $T_{\rm eff}$.

Figure 6: Differential rotation of the $0.8M_{\odot}$ star. (a) Rotation rate
isolines in the NW quadrant of the meridional cross-section of the convection
zone. (b) Latitudinal profile of the surface rotation rate. Figure 7:
Meridional flow in the $0.8M_{\odot}$ star. (a) Stream lines of the flow. (b)
Variation of the meridional velocity with radius at the 45∘ latitude. Positive
velocity means equator-ward flow. Figure 8: Time-latitude diagrams of the
dynamo model for $0.8M_{\odot}$ star. Surface radial field and the bottom
toroidal field are shown in the top and bottom panels respectively.

The difference with the differential rotation also is that the theoretical
dependence of cycle period on rotation rate for fixed temperature is uncertain
and may not be weak. Non-kinematic dynamo models are required to study this
dependence. Consideration of Sect. 2 suggests that activity of rapidly
rotating young stars may not be cyclic. Katsova et al. (2015) estimated that
the Sun formed its activity cycle at the age of 1 to 2 Gyr. This means about
two times faster rotation compared to its present rate. Red symbols in Fig. 1
are not very close to the dashed line of $P_{\rm max}$.

We consider next the differential rotation and dynamo for two cases of stellar
mass smaller and larger compared to the Sun. The consideration shows that
equatorial symmetry of the dominant dynamo mode does also depend on
temperature.

Figure 9: Differential rotation of $1.2M_{\odot}$ star. (a) Rotation rate
isolines. (b) Latitudinal profile of the surface rotation rate. Figure 10:
Meridional flow in the $1.2M_{\odot}$ star. (a) Stream lines of the flow. (b)
Depth profile of the meridional velocity at the 45∘ latitude. Positive
velocity means equator-ward flow. Figure 11: Time-latitude diagrams of the
dynamo model for $1.2M_{\odot}$ star. Surface radial field and the bottom
toroidal field are shown in the top and bottom panels respectively.

Figures 6 to 8 show the differential rotation, meridional flow and magnetic
time-latitude diagram computed for the $0.8M_{\odot}$ star. Similar to the
Sun, dipolar dynamo mode dominates in this case. The dynamo arrived at dipolar
parity from a mixed-parity initial state of Eq. (18).

Figure 12: Marginal amplitude $\alpha_{\rm c}$ of the $\alpha$-effect for
generation of dipolar (full line) and quadrupolar (dashed) magnetic fields as
the function of temperature.

Figures 9 to 11 show the differential rotation, meridional flow and field
diagram computed for the $1.2M_{\odot}$ star. The differential rotation
pattern of Fig. 9 is similar to Fig. 6 in spite of about five times faster
rotation of the $1.2M_{\odot}$ star. This result contrasts with the dependence
of differential rotation on rotation rate produced by our model for fixed
stellar mass. The differential rotation changes towards a cylinder-shaped
pattern with increasing rotation in a star of given mass. The change results
in a weaker meridional flow for faster rotation (see figs. 4 and 5 in
Kitchatinov & Olemskoy 2012). The weakening of meridional flow is the probable
reason for longer activity cycles in faster rotating stars (of a given mass)
found by Jouve et al. (2010) and Karak et al. (2014) with the flux-transport
dynamo models. A change towards cylinder-shaped rotation does not happen
between Figs. 6 and 9 of this paper because these Figures correspond to
computations for different stellar mass but the same Rossby number of Eq. (1).
The effect of rotation on large-scale flow is measured by the dimensionless
Coriolis number $\Omega^{*}=4\pi{\rm Ro}^{-1}$ of Eq. (13). The characteristic
Coriolis number has the same value of $\Omega^{*}\simeq 6$ in all computations
of this paper. This value is below the range of $10<\Omega^{*}<20$ where the
change to cylinder-shaped rotation occurs (Kitchatinov & Olemskoy 2012).
Deviation from cylinder-shaped rotation is caused by a slight increase of the
mean temperature with latitude in the convection zone. The differential
temperature results in our model from the rotationally induced anisotropy of
the eddy heat transport (Rüdiger et al. 2005). The anisotropy is controlled by
the Coriolis number of Eq. (13). Computations for different stellar mass and
rotation rate but the same characteristic value of $\Omega^{*}$ give similarly
shaped differential rotation.

Computations for the $1.2M_{\odot}$ star show dynamo convergence to the mixed
parity solution. The field diagram of Fig. 11 shows mixed equatorial symmetry.
Convergence to a certain symmetry requires a certain link between the northern
and southern hemispheres. The hemispheric link weakens with decreasing
thickness of the convection zone in stars of larger mass.

Another manifestation of the weaker hemispheric link in the thin convection
zones is the almost equal threshold amplitudes $\alpha_{\rm c}^{\rm d}$ and
$\alpha_{\rm c}^{\rm q}$ for generation of dipolar and quadrupolar fields in
Fig. 12. The $\alpha_{\rm c}$ of this Figure increases with temperature. A
larger $\alpha_{\rm c}$ needs faster rotation. This can explain the shorter
$P_{\rm max}$ for more massive stars (smaller $B-V$) in Fig. 1. The spindown
is caused by the large-scale magnetic fields increasing the co-rotation radius
for the stellar wind plasma. The spindown stops when rotation slows to
relatively large rate corresponding to the relatively large $\alpha_{\rm c}$
in more massive stars. Further increase in $\alpha_{\rm c}$ for
$M>1.2M_{\odot}$ may eventually mean that even $P_{\rm rot}\sim 1\,{\rm d}$ of
ZAMS stars does not suffice for a dynamo. Based on observations, Durney &
Latour (1978) concluded that stars of spectral type earlier than F6 do not
support large-scale dynamos.

The $\alpha$-effect in our computations is 10% supercritical,
$\alpha=1.1\,\alpha_{\rm c}$. Computations with other slightly supercritical
$\alpha$ show that the amplitude $B$ of magnetic cycles is proportional to the
square root of the super-criticality, $B\propto(\alpha-\alpha_{\rm c})^{1/2}$.
This relation holds not only for the dynamo-instability, but is a general rule
for any weakly nonlinear instability (cf. eq. (26.10) in Landau & Lifshitz
1987). The relation can be reformulated in terms of the rotation rate,

$B\propto\left(P_{\rm max}-P_{\rm rot}\right)^{1/2},$ (21)

and possibly explain why the Sun is observed to be less active than other
stars of comparable effective temperature and rotation rate (Reinhold et al.
2020; Zhang et al. 2020). According to Eq. (21), what matters for magnetic
activity is not the value of the rotation rate but the amount of excess by the
rate its marginal value for dynamo. The derivative of Eq. (21) on $P_{\rm
rot}$ is infinite at $P_{\rm rot}=P_{\rm max}$. Stars with close rotation
rates can differ considerably in super-criticality of their dynamos and
therefore in the level of magnetic activity.

The ratio of the cycle period to the time of advection by the meridional flow,
$P_{\rm cyc}V_{\rm bot}/r_{\rm i}$, varies little to remain between the values
of 3 and 4 in our computations; $V_{\rm bot}$ is the near-bottom maximum value
of the meridional flow velocity (see Figs 7b and 10b). This means that the
computations belong to the flux-transport dynamo regime. Faster meridional
flow in hotter stars explain their shorter cycles.

A preliminary run of one thousand years starting from the mixed-parity initial
state of Eq. (18) did not converge to a certain equatorial symmetry for
$1.2M_{\odot}$ star though $\alpha_{\rm c}^{\rm q}$ is slightly smaller than
$\alpha_{\rm c}^{\rm d}$ in this case (Fig. 11). We did not extend the run
further for the following reason. Our computations do not include fluctuations
in dynamo parameters, which are most probably present in stars. The equator-
asymmetric fluctuations couple the dipolar and quadrupolar dynamo modes so
that the amplitudes of these modes vary irregularly on a time scale comparable
to the cycle period (Schüssler & Cameron 2018; Kitchatinov & Khlystova 2021).
The amplitudes are expected to be comparable for almost equal super-
criticality of the two modes. This will result in irregularly varying north-
south asymmetry in activity of a star. Observational detection of an activity
cycle may be difficult in this case if the inclination angle of the rotation
axis is not close to $\pi/2$. This may be the reason for no detections of
high-quality cycles for stars hotter than the Sun (Fig. 1).

##  5 Conclusions

Young rapidly rotating stars are not probable to show sun-like activity
cycles. Otherwise, the large-scale stellar dynamos would be an exception among
other hydromagnetic instabilities showing turbulence rather than cyclic
overstability in a highly supercritical regime.

The upper bound on the rotation period of main-sequence dwarfs showing solar-
type magnetic activity (Rengarajan 1984; Metcalfe et al. 2016; van Saders et
al. 2016) can be interpreted as the minimum rotation rate for a large-scale
stellar dynamos. This rotation rate increases with the effective temperature.
Computations with a joint model for differential rotation and dynamo show
magnetic cycles for slightly supercritical dynamos in stars of different mass.
Hotter stars have shorter cycles in the computations and the hotter stars
rotate faster. The observed decrease in cycle duration for faster rotation
based on combined statistics of stars of different spectral types can,
therefore, be at least partly explained by the cycle dependence on
temperature.

The computations also show larger marginal values of the $\alpha$-effect for
dynamo operation in hotter stars. A larger $\alpha$ demands faster rotation.
This may be the reason for the smaller upper bound on the rotation period for
hotter stars. The amplitude of magnetic energy in the dynamo model is
proportional to the difference between rotation rate and the marginal rate for
dynamo (cf. Eq. 21). Stars with similar rotation rates can therefore differ
substantially in level of their activity as observed (Reinhold et al. 2020): a
small difference in rotation rates does not necessarily mean an equally small
difference in the super-criticality.

Large variations in differential rotation and cycle period computed for the
constant Rossby number of Eq. (1) indicate that this number may not be the
universal scaling parameter for stellar rotation and dynamos. The Rossby
number measures intensity of interaction between convection and rotation. It
can be doubted that the BL-mechanism of the solar-type dynamos is fully
controlled by this interaction.

The dynamo computations predict a change in equatorial symmetry of global
magnetic fields with temperature. The stars of solar and smaller mass show
antisymmetric fields about the equator in the computations. A change to mixed-
parity asymmetric fields is predicted for more massive stars. The mixed-parity
dynamos can impede observational detection of the activity cycles.

The well-defined activity cycle of subgiant HD 81809 is very interesting and
challenging to dynamo theory (see the footnote 1). This star exceeds the Sun
in mass $M=(1.70\pm 0.64)M_{\odot}$ (see table 2 in Egeland 2018). A main-
sequence A-star is its probable progenitor. A-stars do not have (sufficiently
thick) external convection zones and do not show activity cycles. The
convection zone formed when HD 81809 evolved from the main-sequence is
probably responsible for its cyclic activity. This example can be informative
on the role of convective envelopes for stellar dynamos. The author is
thankful to an anonymous referee for pertinent and constructive comments and
to Maria Katsova for a useful discussion. This work was financially supported
by the Ministry of Science and High Education of the Russian Federation.

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